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abstract: 'In this paper, we investigate a continuous family of notions of independence which interpolates between the classical and free ones for non-commutative random variables. These notions are related to the liberation process introduced by D. Voiculescu. To each notion of independence correspond new convolutions of probability measures, for which we establish formulae and of which we compute simple examples. We prove that there exists no reasonable analogue of classical and free cumulants associated to these notions of independence.'
address:
- 'Florent Benaych-Georges, LPMA, UPMC Univ Paris 6, Case courier 188, 4, Place Jussieu, 75252 Paris Cedex 05, France, and CMAP, ´ Ecole Polytechnique, route de Saclay, 91128 Palaiseau Cedex, France'
- 'Thierry Lévy, CNRS and École Normale Supérieure, DMA, 45, rue d’Ulm, F-75005 Paris'
author:
- 'Florent Benaych-Georges and Thierry Lévy'
title: A continuous semigroup of notions of independence between the classical and the free one
---
Introduction {#introduction .unnumbered}
============
Let $\mu$ and $\nu$ be two Borel probability measures on the real line $\R$. The classical convolution of $\mu$ and $\nu$ is the probability measure on $\R$, denoted by $\mu *\nu$, which is the distribution of the sum of two classical independent random variables with respective distributions $\mu$ and $\nu$. Let us describe $\mu*\nu$ in an alternative way. To each $n\times n$ matrix with eigenvalues $\lambda_{1},\ldots,\lambda_{n}$, we associate its spectral measure, which is the probability measure $\frac{1}{n}\sum_{i=1}^{n}\delta_{\lambda_{i}}$. Let $(A_{n})_{n\geq 1}$ and $(B_{n})_{n\geq 1}$ be two sequences of diagonal real matrices, with $A_{n}$ and $B_{n}$ of size $n$ for all $n\geq 1$, such that the spectral measure of $A_{n}$ (resp. of $B_{n}$) converges, as $n$ tends to infinity, to $\mu$ (resp. to $\nu$). For each $n\geq 1$, let $S_{n}$ be a random matrix chosen uniformly among the $n!$ permutation matrices of size $n$. Then the spectral measure of $A_{n}+S_{n}B_{n}S_{n}^{-1}$ converges, as $n$ tends to infinity, to $\mu*\nu$.
If we replace, for each $n\geq 1$, the matrix $S_{n}$ by a random matrix $U_{n}$ chosen in the unitary group $U(n)$ according to the Haar measure, then the spectral measure of $A_{n}+U_{n}B_{n}U_{n}^{-1}$ converges, as $n$ tends to infinity, to the free convolution of $\mu$ and $\nu$, a probability measure on $\R$ denoted by $\mu \boxplus \nu$.
This way of describing classical and free convolutions suggests a natural way to interpolate between them. Indeed, consider, for all $n\geq 1$, a properly scaled Brownian motion $(U_{n,t})_{t\geq 0}$ issued from the identity matrix on the unitary group $U(n)$. Given $t\in [0,+\infty)$, one may consider the spectral measure of $A_{n}+U_{n,t}S_{n} B_{n} S_{n}^{-1}U_{n,t}^{-1}$, and ask for the limit of this distribution as $n$ tends to infinity. For $t=0$, the matrix $U_{n,0}$ is the identity matrix and we find the classical convolution of $\mu$ and $\nu$. For $t=+\infty$, that is, when $U_{n,t}$ is replaced by its limit in distribution as $t$ tends to infinity, which is a uniformly distributed unitary matrix, we recover the free convolution of $\mu$ and $\nu$. For any other $t\in (0,+\infty)$, it turns out that one finds a probability measure which depends only on $\mu$, $\nu$ and $t$ and which we denote by $\mu *_t\nu$. Note that this definition of $*_{t}$ can be considered as a particular case of the so-called [*liberation process*]{} introduced by Voiculescu [@v99].
Consider for example the case where $\mu=\nu=\frac{1}{2}(\delta_1+\delta_{-1})$. Then $\mu*\nu=\frac{1}{4}\delta_{-2}+\frac{1}{2} \delta_0 + \frac{1}{4} \delta_2$ and it is well known that $\mu\boxplus \nu={\mathbbm 1}_{[-2,2]}(x) \frac{dx}{\pi \sqrt{4-x^2}}$, a dilation of the arcsine law [@NS Example 12.8]. One may wonder which probability measures interpolate between $\mu*\nu$ and $\mu\boxplus \nu$. We will prove that $$\forall t\geq 0\; , \; \; \frac{\delta_1+\delta_{-1}}{2} *_{t} \frac{\delta_1+\delta_{-1}}{2} = {\mathbbm 1}_{[-2,2]}(x) \frac{\rho_{4t}(e^{4i \arccos \frac{x}{2}})}{\pi \sqrt{4-x^2}} \; dx.$$ Here, for all $t> 0$ and $\theta\in \R$, $\rho_{t}(e^{i\theta})$ is the density at $e^{i\theta}$, with respect to the uniform probability measure on the unit circle, of the distribution of the free unitary Brownian motion at time $t$. This distribution is also the limit, as $n$ tends to infinity, of the spectral measure of $U_{n,t}$. There is no simple formula for this distribution, which apparently has to be taken as a fundamental function in any problem involving the large $n$ asymptotics of the Brownian motion on the unitary group $U(n)$. However the moments of this distribution are known since P. Biane first computed them [@b97]. It follows for instance from the previous expression that $\mu*_t \nu$ has a density with respect to the Lebesgue measure for all $t>0$ and that its support, which one can compute for all $t\geq 0$, is the whole interval $[-2,2]$ if and only if $t\geq 1$.\
The family of operations $*_{t}$ is really just a by-product of a more fundamental construction, which is that of a continuous family of independence (or dependence) structures between non-commutative random variables which interpolates between classical independence and freeness. Indeed, we will define, for all $t\in [0,+\infty]$, a notion of independence between two subalgebras of a non-commutative probability space, which we call [*$t$-freeness*]{} and which, for $t=0$ (resp. $t=+\infty$), coincides with classical independence (resp. freeness). Once this structure is defined, it is straightforward to define additive or multiplicative convolution of $t$-free self-adjoint or unitary elements, thus giving rise to several operations on probability measures: additive or multiplicative convolution of probability measures with compact support on $\R$, denoted by $*_{t}$ and $\odot_{t}$ ; multiplicative convolution of probability measures on the unit circle, also denoted by $\odot_{t}$.
The idea of seeking a continuous way of passing from classical to free independence is presumably as old as the theory of free probability itself, but the research of such a continuum has been broken off by a paper of Roland Speicher in 1997 [@s97], where he has shown that no other notion of independence than the classical and the free ones can be the base of a [*reasonable*]{} probability theory. Indeed $t$-freeness does not satisfy all the axioms enforced by R. Speicher because it is not an [*associative*]{} notion of independence. This axiom of associativity states, roughly, that if $X,Y, Z$ are three random variables such that $X$ is independent of $Y$ and $Z$ is independent of $\{X,Y\}$, then $X$ must be independent of $\{Y,Z\}$. Instead of this, what is true with $t$- freeness is that for all $s,t\geq 0$, if $X,Y, Z$ are three random variables such that $X$ is $t$-free with $Y$ and $Y$ is $s$-free with $Z$, then under certain additional hypotheses, $X$ will be $(s+t)$-free with $Z$. This is of course related to the semi-group property of the Brownian motion.\
There are several ways to characterize and deal with independence and freeness. The first one, which we have already mentioned, is to relate them with matrix models. The second one is to describe them by means of computation rules: the expectation factorizes with respect to independent subfamilies of random variables, whereas the expectation of a product of free elements can be computed using the fact that if $x_1,\ldots, x_n$ are centered and successively free, then their product is centered. The third way to describe independence and freeness is to identify integral transforms which linearize them (namely the logarithm of Fourier transform or the $R$-transform). This amounts to describing classical and free cumulants. The last way, a bit more abstract, is to consider tensor or free products: a family of random variables is independent (resp. free) if and only if it can be realized on a tensor product (resp. free product) of probability spaces.
In the present paper, we look for the analogues of all these approaches for the notion of $t$-freeness. We begin, in Section \[10.9.08.1\], by giving the definition of a $t$-free product and presenting the corresponding random matrix model. Then, in Section \[10.9.08.2\], we state the computation rules, which are best understood as a family of differential equations. Finally, in Section \[10.9.08.3\], we prove that no notion of cumulants of order greater than $6$ can be associated to the notion of $t$-freeness. More precisely, we show that there does not exist a universally defined $7$-linear form on any non-commutative probability space with the property that this form vanishes whenever it is evaluated on arguments which can be split into two non-empty subfamilies which are $t$-free, unless $t=0$ or $t=+\infty$. This can be summarized in the following diagram.\
[c|c|c|c|c|]{} & Matrix model & Computation rules & Cumulants & Algebraic structures & $A+SBS^{-1}$ & Factorization & Class. cumulants & Tensor product & $A+U_tS B S^{-1}U_t^{-1}$ & Differential system & [**Do not exist**]{} & $t$-free product & $A+UBU^{-1}$ & $\varphi(x_1\ldots x_n)=0$ & Free cumulants & Free product
Preliminaries
=============
In this section, we review the notions of non-commutative probability which are relevant to the definition of $t$-freeness.
Probability space, distribution
-------------------------------
Non-commutative probability is based on the following generalization of the notion of probability space.
A [*non-commutative probability space*]{} is a pair $({\mathcal}{A},\vfi)$, where:
- $\A$ is an algebra over $\C$ with a unit element denoted by $1$, endowed with an operation of adjunction $x\mapsto x^*$ which is $\C$-antilinear, involutive and satisfies $(xy)^*=y^*x^*$ for all $x,y\in \A$,
- $\vfi\,:\, \A\to \C$ is a linear form on $\A$, satisfying $\vfi(1)=1$, $\vfi(xy)=\vfi(yx)$, $\vfi(x^*)=\overline{\vfi(x)}$ and $\vfi(xx^*)\geq 0$ for all $x,y\in \A$.
The linear form $\vfi$ is often called the [*expectation*]{} of the non-commutative space.
Two fundamental examples are the algebra $L^{\infty-}(\Omega,\Sigma,\mathbb P)$ of complex-valued random variables with moments of all orders on a classical probability space, endowed with the complex conjugation and the expectation (we will say that this non-commutative probability space is [*inherited*]{} from $(\Omega,\mathcal A,\mathbb P)$); and the algebra $\M_{n}(\C)$ endowed with the matricial adjunction and the normalized trace.
\[def ncd\] Let $(\A,\varphi)$ be a non-commutative probability space. The [*non-commutative distribution*]{} of a family $(a_1, \ldots, a_n)$ of elements of $\A$ with respect to $\varphi$ is the linear map defined on the space of polynomials in the non-commutative variables $X_1, X_1^*, \ldots, X_n, X_n^*$ which maps any such polynomial $P$ to $\vfi(P(a_1,a_1^*,\ldots, a_n,a_n^*))$.
The link between the classical notion of distribution and the non-commutative one is the following. Consider a self-adjoint element $a$ in a non-commutative space $(\A, \vfi)$, that is, an element such that $a=a^*$. Since $\vfi(xx^*)\geq 0$ for all $x\in \A$, the distribution of $a$ is a linear form on $\C[X]$ which is non-negative on the polynomials which are non-negative on the real line. Hence, it can be represented as the integration with respect to a measure on the real line. This probability measure is unique if and only if it is determined by its moments, which is in particular the case when it has compact support, or equivalently when there exists a constant $M$ [such that ]{}for all $n\geq 0$, one has $\vfi(a^{2n})\leq M^{2n}$.
Similarly, the distribution of a unitary element $u$, that is, an element such that $uu^{*}=u^{*}u=1$, is the integration with respect to a probability measure on the unit circle of $\C$. Since the circle is compact, there is no issue of uniqueness in this case.
Independence, freeness and random matrices
------------------------------------------
### Definitions and basic properties
We shall recall the definitions of the two notions of independence in a non-commutative probability space between which our main purpose is to interpolate. The first one is a straightforward translation of the classical notion of independence in the non-commutative setting, which coincides with the original notion in the case of a non-commutative space inherited from a classical one. The second one is the notion of freeness, as defined by Voiculescu [@vdn91], which is called freeness.
In this paper, by a subalgebra of the algebra of a non-commutative space, we shall always mean a subalgebra which contains $1$ and which is stable under the operation $x\mapsto x^*$.
\[23.06.08.17h04\] Let $({\mathcal}{M},\vfi)$ be a non-commutative space. The kernel of $\vfi$ will be called the set of [*centered elements*]{}. Consider a family $(\A_i)_{i\in I}$ of subalgebras of $\M$.
- The family $(\A_i)_{i\in I}$ is said to be [*independent*]{} if
- for all $i\neq j\in I$, $\A_i$ and $\A_j$ commute,
- for all $n\geq 1$, $i_1, \ldots, i_n\in I$ pairwise distinct, for all family $(a_1, \ldots, a_n)\in \A_{i_1}\times \cdots\times \A_{i_n}$ of centered elements, the product $a_1\cdots a_n$ is also centered.
- The family $(\A_i)_{i\in I}$ is said to be [*free*]{} if for all $n\geq 1$, $i_1, \ldots, i_n\in I$ [such that ]{}$i_1\neq i_2, i_2\neq i_3, \ldots, i_{n-1}\neq i_n$, for all family $(a_1, \ldots, a_n)\in \A_{i_1}\times \cdots\times \A_{i_n}$ of centered elements, the product $a_1\cdots a_n$ is also centered.
On a classical probability space $(\Omega,\Sigma,\mathbb P)$, a family $(\Sigma_{i})_{i\in I}$ of sub-$\sigma$-fields of $\Sigma$ is independent with respect to $\mathbb P$ if and only if the subalgebras $\left(L^{\infty-}(\Omega,\Sigma_{i},\mathbb P)\right)_{i\in I}$ of $\left(L^{\infty-}(\Omega,\Sigma,\mathbb P),\mathbb E\right)$ are independent in the sense of the definition above.
In the classical setting again, a family of random variables is independent if and only if its joint distribution is the tensor product of the individual ones. In the following definition and proposition, we translate this statement into our vocabulary, and give its analogue for freeness. These definitions prepare those which we will give later for $t$-freeness.
\[23.06.08.17h09\]
Let $({\mathcal}{A}_1,\vfi_1)$ and $({\mathcal}{A}_2,\vfi_2)$ be two non-commutative spaces.
- Their [*tensor product*]{}, denoted by $({\mathcal}{A}_1,\vfi_1)\otimes({\mathcal}{A}_2,\vfi_2)$, is the non-commutative space with algebra the tensor product of unital algebras ${\mathcal}{A}_1\otimes{\mathcal}{A}_2$, on which the adjoint operation and the expectation are defined by $$\forall (x_1,x_2)\in \A_1\times \A_2, (x_1\otimes x_2)^*=x_1^*\otimes x_2^*,\quad\vfi(x_1\otimes x_2)=\vfi_1(x_1)\vfi_2(x_2).$$
- Their [*free product*]{}, denoted by $({\mathcal}{A}_1,\vfi_1)*({\mathcal}{A}_2,\vfi_2)$, is the non-commutative space with algebra the free product of unital algebras ${\mathcal}{A}_1*{\mathcal}{A}_2$, with adjoint operation and expectation defined uniquely by the fact that for all $n\geq 1$, for all $i_1\neq \cdots\neq i_n\in \{1,2\}$, for all $(x_1, \ldots, x_n)\in \A_{i_1}\times\cdots\times\A_{i_n}$, $$(x_1\cdots x_n)^*=x_n^*\cdots x_1^*$$ and $x_1\cdots x_n$ is centered whenever all $x_i$’s are.
This definition can easily be extended to products of finite or infinite families of non-commutative spaces, but we have restricted ourselves to what is needed in this article. We can now explain the link between these products and the notions of independence and freeness.
\[23.06.08.18h29\] Let $({\mathcal}{M},\vfi)$ be a non-commutative space. Let $\A_1,\A_2$ be subalgebras of $\A$. Then the family $(\A_1,\A_2)$ is
- independent if and only if $\A_1$ commutes with $\A_2$ and the unique algebra morphism defined from $\A_1\otimes \A_2$ to ${\mathcal}{M}$ which, for all $(a_1,a_2)\in \A_1\times \A_2$, maps $a_1\otimes 1$ to $a_1$ and $1\otimes a_2$ to $a_2$, preserves the expectation from $(\A_1,\vfi_{|\A_1})\otimes (\A_2,\vfi_{|\A_2})$ to $({\mathcal}{M},\vfi)$,
- free if and only if the unique algebra morphism defined from the free product of unital algebras $\A_1* \A_2$ to ${\mathcal}{M}$ which, restricted to $\A_1\cup\A_2$ is the canonical injection, preserves the expectation from $(\A_1,\vfi_{|\A_1})* (\A_2,\vfi_{|\A_2})$ to $({\mathcal}{M},\vfi)$.
Let us finally recall the definition of the free analogue of the classical convolution, which is meaningful thanks to the last proposition.
\[23.06.08.20h09\][Let $\mu$ and $\nu$ be two probability measures on $\R$. The distribution of the sum of two free self-adjoint elements with respective distributions $\mu$ and $\nu$ depends only on $\mu$ and $\nu$ and will be called the [free additive convolution]{} of $\mu$ and $\nu$, and be denoted by $\mu{\boxplus}\nu$. ]{}
### Asymptotic behavior of random matrices
In this section, we recall matrix models for the classical and free convolution. The main notion of convergence which is involved is the following.
\[23.06.08.17h10\][ Let $p$ be a positive integer and let, for each $n\geq 1$, $(M(1,n), \ldots, M(p,n))$ be a family of $n\times n$ random matrices. This family is said to [*converge in non-commutative distribution*]{} if its non-commutative distribution converges in to a non random one, that is, if the normalized trace of any word in the $M(i,n)$’s and the $M(i,n)^*$’s converges in to a constant.]{}
\[24.06.08.1\] Let us fix $p, q\geq 1$. For each $n\geq 1$, let ${\mathcal}{F}_n=(A(1,n), \ldots, A(p,n),B(1,n),\ldots, B(q,n))$ be a family of $n\times n$ random matrices and assume that the sequence $(\mathcal F_n)_{n\geq 1}$ converges in non-commutative distribution. Assume also that for all $r\geq 1$, the entries of these random matrices are uniformly bounded in $L^r$.
- Assume that these matrices are diagonal and consider, for each $n$, the matrix $S_n$ of a uniformly distributed random permutation of $\{1,\ldots, n\}$ independent of the family ${\mathcal}{F}_n$. Then the family \[26.11.08.11h40\](A(1,n), …, A(p,n),S\_nB(1,n)S\_n\^[-1]{},…, S\_nB(q,n)S\_n\^[-1]{})converges in distribution to the distribution of a commutative family $(a_1,\ldots, a_p, b_1, \ldots, b_q)$ of elements of a non-commutative space [such that ]{}the algebras generated by $\{a_1, \ldots, a_p\}$ and $\{b_1, \ldots, b_q\}$ are independent.
- Consider, for each $n$, the matrix $U_n$ of a uniformly distributed random unitary $n$ by $n$ matrix independent of the family ${\mathcal}{F}_n$. Then the family $$(A(1,n), \ldots, A(p,n),U_nB(1,n)U_n^{-1},\ldots, U_nB(q,n)U_n^{-1})$$ converges in distribution to the distribution of a family $(a_1,\ldots, a_p, b_1, \ldots, b_q)$ of elements of a non-commutative space [such that ]{}the algebras generated by $\{a_1, \ldots, a_p\}$ and $\{b_1, \ldots, b_q\}$ are free.
[The hypothesis of uniform boundedness of the entries of the matrices in each $L^r$ could be sharply weakened for the first part of the theorem if, instead of asking for the convergence of the non-commutative distribution of the family , one would ask for the weak convergence of the empirical joint spectral measure. This would amount to choosing, as set of test functions, the set of bounded continuous functions of $p+q$ variables instead of the set of polynomials in $p+q$ variables (see [@bgchapon], where this is precisely proved).]{}
The first part of this theorem is much simpler than the second but seems to be also less well-known. It is in any case harder to locate a proof in the literature, so that we offer one. We shall need the following lemma. We denote by $\Vert\cdot \Vert_{2}$ the usual Hermitian norm on $\C^{n}$.
Let, for each $n\geq 1$, $x(n)=(x_{n,1},\ldots, x_{n,n})$ and $y(n)=(y_{n,1},\ldots, y_{n,n})$ be two complex random vectors defined on the same space [such that ]{}the random variables $$\overline{x(n)}=\frac{x_{n,1}+\cdots + x_{n,n}}{n},\quad\overline{y(n)}=\frac{y_{n,1}+\cdots+ y_{n,n}}{n}$$ converge in to constant limits $x,y$ as $n$ tends to infinity. Suppose moreover that the sequences ${\frac{1}}{n}\Vert x(n)\Vert_2^2$ and ${\frac{1}}{n}\Vert y(n)\Vert_2^2$ are bounded in $L^2$. Consider, for all $n$, a uniformly distributed random permutation $\sigma_n$ of $\{1,\ldots, n\}$, independent of $(x(n), y(n))$, and define $y_{\sigma_n}(n):=(y_{n,\sigma_n(1)},\ldots,y_{n,\sigma_n(n)})$. Then the scalar product $${\frac{1}}{n}{\langle}x(n), y_{\sigma_n}(n){\rangle}=\frac{x_{n,1}y_{n,\sigma_n(1)}+\cdots+ x_{n,n}y_{n,\sigma_n(n)}}{n}$$ converges in to $xy$ as $n$ tends to infinity.
First of all, note that one can suppose that for all $n$, $ \overline{x(n)}=\overline{y(n)}=0$ almost surely. Indeed, if the result is proved under this additional hypothesis, then since for all $n$, one has $${\frac{1}}{n}{\langle}x(n), y_{\sigma_n}(n){\rangle}={\frac{1}}{n}{\langle}x(n)-\overline{x(n)}\cdot1_n, y_{\sigma_n}(n)-\overline{y(n)}\cdot1_n{\rangle}+\overline{x(n)}\cdot\overline{y(n)},\quad\textrm{(with $1_n=(1,\ldots, 1)$)},$$ the result holds for general $x(n), y(n)$. So we henceforth assume that for all $n$, $ \overline{x(n)}=\overline{y(n)}=0$. The equality $\overline{y(n)}=0$ implies, for all $n$ and all $i,j=1,\ldots, n$, that $$\E[y_{n, \sigma_n(i)}y_{n, \sigma_n(j)}\,|\,x(n),y(n)]=\begin{cases}{\frac{1}}{n}\Vert y(n)\Vert_2^2&\textrm{if $i=j$,}\\ -{\frac{1}}{n(n-1)}\Vert y(n)\Vert_2^2&\textrm{if $i\neq j$.}
\end{cases}$$ Then, using the fact that $ \overline{x(n)}=0$, we have &=&\
&=& O([n]{}),which completes the proof.
\
[[**Proof of Theorem \[24.06.08.1\].** ]{}]{}The second point is a well-known result of Voiculescu (see [@vdn91]). To prove the first one, we shall prove that the normalized trace any word in the random matrices $A(1,n), \ldots, A(p,n),S_nB(1,n)S_n^{-1},\ldots, S_nB(q,n)S_n^{-1}$ converges to a constant which is the product in two terms: the limiting normalized trace of the $A(i,n)$’s and the $A(i,n)^*$’s which appear in the word on one hand and the limiting normalized trace of the $B(j,n)$’s and the $B(j,n)^*$’s which appear in the word on the other hand. Since the $A(i,n)$’s, the $A(i,n)^*$’s, the $S_nB(j,n)S_n^{-1}$’s and the $S_nB(j,n)^*S_n^{-1}$’s commute, are uniformly bounded and their non-commutative distribution converges, this amounts to proving that if $M(n), N(n)$ are two diagonal random matrices with entries uniformly bounded in $L^r$ for all $r\geq 1$, whose normalized traces converge in to constants $m,n$, then for $S_n$ the matrix of a uniform random permutation of $\{1, \ldots, n\}$ independent of $(M(n), N(n))$, the normalized trace of $M(n)S_nN(n)S_n^{-1}$ converges to $mn$. This follows directly from the previous lemma and the proof is complete. [ $\square$]{}
Let $\mu,\nu$ be two measures on the real line. Let, for each $n\geq 1$, $M_n,N_n$ be $n$ by $n$ diagonal random matrices with empirical spectral measures converging weakly in to $\mu$ and $\nu$respectively. For each $n\geq 1$, let $S_n$ (resp. $U_n$) be a uniformly distributed $n$ by $n$ permutation (resp. unitary) random matrix independent of $(M_n,N_n)$. Then
- the empirical spectral measure of $M_n+S_nN_nS_n^{-1}$ converges weakly in to the classical convolution $\mu *\nu$ of $\mu$ and $\nu$,
- the empirical spectral measure of $M_n+U_nN_nU_n^{-1}$ converges weakly in to the free convolution $\mu {\boxplus}\nu$ of $\mu$ and $\nu$.
In the case where $\mu,\nu$ have compact supports and the entries of the diagonal matrices $M_n,N_n$ are uniformly bounded, it is a direct consequence of the previous theorem. The general case can easily be deduced using functional calculus, like in the proof of Theorem 3.13 of [@bg07].
Unitary Brownian motion, free unitary Brownian motion
-----------------------------------------------------
In this paragraph, we give a brief survey of the definition and the main convergence result for the Brownian motion on the unitary group.
Let $n\geq1$ be an integer. Let $\Herm_{n}$ denote the $n^2$-dimensional real linear subspace of $\M_{n}(\C)$ which consists of Hermitian matrices. On $\M_{n}(\C)$, we denote by $\Tr$ the usual trace and by $\tr=\frac{1}{n}\Tr$ the normalized trace. Let us endow $\Herm_{n}$ with the scalar product $\langle\cdot,\cdot\rangle$ defined by $$\forall A,B \in \Herm_{n}\; , \;\; \langle A,B\rangle =n\Tr(A^*B)=n\Tr(AB).$$ There is a linear Brownian motion canonically attached to the Euclidean space $(\Herm_{n},\langle\cdot,\cdot\rangle)$. It is the unique Gaussian process $H$ indexed by $\R_{+}$ with values in $\Herm_{n}$ such that for all $s,t\in\R_{+}$ and all $A,B\in \Herm_n$, one has $$\E[\langle H_s,A\rangle \langle H_t,B\rangle]=\min(s,t) \langle A,B\rangle.$$
Let us consider the following stochastic differential equation: $$U_{0}=I_{n} \; , \;\; {\mathrm{d}}U_{t}=i ({\mathrm{d}}H_{t})U_{t}-\frac{1}{2} U_{t} {\mathrm{d}}t,$$ where $(U_{t})_{t\geq 0}$ is a stochastic process with values in $\M_{n}(\C)$. This linear equation admits a strong solution. The process $(U^*_{t})_{t\geq 0}$, where $U^*_{t}$ denotes the adjoint of $U_{t}$, satisfies the stochastic differential equation $$U^*_{0}=I_{n} \; , \;\; {\mathrm{d}}U^*_{t}=-iU^*_{t} {\mathrm{d}}H_{t}-\frac{1}{2} U^*_{t} {\mathrm{d}}t.$$ An application of Itô’s formula to the process $U_{t}U_{t}^*$ shows that, for all $t\geq 0$, $U_{t}U^*_{t}=I_{n}$. This proves that the process $(U_{t})_{t\geq 0}$ takes its values in the unitary group $U(n)$.
[The process $(U_{t})_{t\geq 0}$ is called the [*unitary Brownian motion of dimension $n$*]{}.]{}
As $n$ tends to infinity, the unitary Brownian motion has a limit in distribution which we now describe. For all $t\geq 0$, the numbers $$e^{-\frac{kt}{2}} \sum_{j=0}^{k-1} \frac{(-t)^j}{j!} \binom{k}{j+1} k^{j-1} \; , \; \; k\geq 0,$$ are the moments of a unique probability measure on the set $\U=\{z\in\C:|z|=1\}$ invariant by the complex conjugation. We denote this probability measure by $\nu_{t}$. The following definition was given by P. Biane in [@b97].
\[20.06.08.1\]
Let $(\A,\tau)$ be a non-commutative probability space. We say that a collection $(u_{t})_{t\geq 0}$ of unitary elements of $\A$ is a [*free unitary Brownian motion*]{} if the following conditions hold.
- For all $s,t\geq 0$ such that $s\leq t$, the distribution of $u_{t}u_{s}^*$ is the probability measure $\nu_{t-s}$.
- For all positive integer $m$, for all $0\leq t_{1}\leq t_{2}\leq \ldots\leq t_{m}$, the elements $u_{t_{1}}u_{0}^*,$ $u_{t_{2}}u_{t_{1}}^*,\ldots,u_{t_{m}}u_{t_{m-1}}^*$ are free.
In the same paper, P. Biane has proved the following convergence result.
\[23.06.08.1\] For each $n\geq 1$, let $(U_{n,t})_{t\geq 0}$ be a Brownian motion on the unitary group $U(n)$. As $n$ tends to infinity, the collection of random matrices $(U_{n,t})_{t\geq 0}$ converges in non-commutative distribution to a free unitary Brownian motion.
A continuum of notions of independence {#10.9.08.1}
======================================
In this section, we shall define a family indexed by a real number $t\in [0,+\infty]$ of relations between two subalgebras of a non-commutative space which passes from the classical independence (which is the case $t=0$) to freeness (which is the “limit" when $t$ tends to infinity). We start with the definition of the $t$-free product of two non-commutative spaces. In a few words, it is the space obtained by conjugating one of them, in their tensor product, by a free unitary Brownian motion at time $t$, free with the tensor product.
Fix $t\in [0,+\infty]$ and let $(\A,\vfi_\A)$ and $(\B,\vfi_\B)$ be two non-commutative spaces. Let $({\mathcal}{U}^{(t)}, \vfi_{{\mathcal}{U}^{(t)}})$ be the non-commutative space generated by a single unitary element $u_t$ whose distribution is that of a free unitary Brownian motion at time $t$ (with the convention that a free unitary Brownian motion at time $+\infty$ is a Haar unitary element, i.e. a unitary element whose distribution is the uniform law on the unit circle of $\C$).
[The [*$t$-free product*]{} of $(\A,\vfi_\A)$ and $(\B,\vfi_\B)$, defined up to an isomorphism of non-commutative spaces, is the non-commutative space $({\mathcal}{C},\vfi_{|{\mathcal}{C}})$, where ${\mathcal}{C}$ is the subalgebra generated by $\A$ and $u_t\B u_t^*$ in $$({\mathcal}{X},\vfi):=[(\A,\vfi_\A)\otimes(\B,\vfi_\B)]*({\mathcal}{U}^{(t)}, \vfi_{{\mathcal}{U}^{(t)}}).$$]{}
A few simple observations are in order.
\[20.06.08.14h\][Both $(\A,\vfi_\A)$ and $(\B,\vfi_\B)$ can be identified with subalgebras of the algebra of their $t$-free product (namely with $(\A, \vfi_{|\A})$ and $(u_t\B u_t^*, \vfi_{|u_t\B u_t^*})$). More specifically, if one defines $$\A_{st}:=\{a\in \A{\, ;\, }\vfi_\A(a)=0, \vfi_\A(aa^*)=1\}, \; \B_{st}:=\{b\in \B{\, ;\, }\vfi_\B(b)=0, \vfi_\B(bb^*)=1\},$$ then any element in the algebra of the $t$-free product $(\A,\vfi_\A)$ and $(\B,\vfi_\B)$ can be uniquely written as a constant term plus a linear combination of words in the elements of $\A_{st}\cup u_t\B_{st}u_t^*$ where no two consecutive letters both belong to $\A_{st} $ or to $u_t\B_{st}u_t^*$.]{}
[As a consequence, since $u_t$ is unitary and $(u_t, u_t^*)$ has the same non-commutative distribution as $(u_t^*, u_t)$, the $t$-free product of $(\A,\vfi_\A)$ and $(\B,\vfi_\B)$ is clearly isomorphic, as a non-commutative space, to the $t$-free product of $(\B,\vfi_\B)$ and $(\A,\vfi_\A)$.]{}
\[20.06.08.16h\][Another consequence of Remark \[20.06.08.14h\] is that as a unital algebra, the algebra of the $t$-free product of $(\A,\vfi_\A)$ and $(\B,\vfi_\B)$ is isomorphic to the free product of the unital algebras $\A/\tilde{\A}$ and $\B/\tilde{\B}$, where $\tilde{\A}$ (resp. $\tilde{\B}$) is the bilateral ideal of the elements $x$ of $\A$ (resp. of $\B$) [such that ]{}$\vfi_\A(xx^*)=0$ (resp. $\vfi_\B(xx^*)=0$). Thus if $\A$ and $\B$ are subalgebras of the algebra of a non-commutative space $({\mathcal}{M}, \vfi)$, there is a canonical algebra morphism from the algebra of the $t$-free product of $(\A,\vfi_{|\A})$ and $(\B,\vfi_{|\B})$ to ${\mathcal}{M}$ whose restriction to $\A\cup \B$ preserves the expectation.]{}
Now, we can give the definition of $t$-freeness. A real $t\in [0,+\infty]$ is still fixed.
Let $({\mathcal}{M}, \tau)$ be a non-commutative space.
- Two subalgebras $\A, \B$ of ${\mathcal}{M}$ are said to be [*$t$-free*]{} if the canonical algebra morphism from the algebra of the $t$-free product of $(\A,\vfi_\A)$ and $(\B,\vfi_\B)$ to ${\mathcal}{M}$ mentioned in Remark \[20.06.08.16h\] preserves the expectation.
- Two subsets $X,Y$ of ${\mathcal}{M}$ are said to be [*$t$-free*]{} if the subalgebras they generate are $t$-free.
[Note that for $t=0$, $t$-freeness is simply the independence, whereas it follows from [@haag2] that in the case where $t=+\infty$, it is the freeness.]{}
The following proposition is obvious from the definition of $t$-freeness.
\[20.06.08.16h45\] Let $({\mathcal}{M}, \tau)$ be a non-commutative space. Let $\{a_1,\ldots, a_n\}$ and $\{b_1, \ldots, b_m\}$ be two $t$-free subsets of ${\mathcal}{M}$. Then the joint non-commutative distribution of the family $(a_1,\ldots, a_n,b_1, \ldots, b_m)$ depends only on $t$ and on the distributions of the families $(a_1,\ldots, a_n)$ and $(b_1, \ldots, b_m)$.
Let us fix $t\in [0,+\infty)$. Let $\mu,\nu$ be compactly supported measures on the real line (resp. on $[0,+\infty)$, on the unit circle). Let $a$, $b$ are $t$-free self-adjoint elements (resp. positive elements, unitary elements) with distributions $\mu$, $\nu$. Then the distribution of $a+b$ (resp. of $\sqrt{b}a\sqrt{b}$, of $ab$) is a compactly supported measure on the real line which depends only on $t$, $\mu$ and $\nu$, and which will be denoted by $\mu*_t\nu$ (resp. $\mu\odot_t\nu$).
Let us treat the case of the sum of two self-adjoint elements. The other cases can be treated analogously. From Proposition \[20.06.08.16h45\], it follows that the moments of $a+b$ depend only on $\mu$ and $\nu$. To see that these are the moments of a compactly supported measure on the real line, introduce $M>0$ [such that ]{}the supports of $\mu$ and $\nu$ are both contained in $[-M,M]$. Then for all $n\geq 1$, by Hölder inequalities in a non-commutative space [@Nelson], $\vfi((a+b)^{2n})\leq 2^{2n}M^{2n}$. By the remark made after Definition \[def ncd\], the result follows.
For each $n\geq 1$, let $M_n$ and $N_n$ be diagonal random matrices whose non-commutative distributions have limits. Let also, for each $n$, $S_n$ be the matrix of a uniform random permutation of $\{1,\ldots, n\}$ and $U_{n,t}$ be a random $n\times n$ unitary matrix distributed according to the law of a Brownian motion on the unitary group at time $t$. Suppose that for each $n$, the sets of random variables $\{M_n,N_n\}, \{S_n\}, \{U_{n,t}\}$ are independent. Then as $n$ tends to infinity, the non-commutative distribution of $$(M_n, \;U_{n,t}S_nN_nS_n^*U_{n,t}^*)$$ converges in to that of a pair $(a,b)$ of self-adjoint elements of a non-commutative space which are $t$-free.
By Theorem \[24.06.08.1\], the non-commutative distribution of $(M_n, S_nN_nS_n^*)$ converges to the one of a pair $(x,y)$ of independent elements. Moreover, since for all $n$, the law of $U_n$ is invariant by conjugation, by Theorems \[24.06.08.1\] and \[23.06.08.1\], the family of sets $$(\{M_n, S_nN_nS_n^*\}, \{U_{n,t}\})$$ is asymptotically free and the limit distribution of $U_{n,t}$ is that of a free unitary Brownian motion at time $t$. By definition of $t$-freeness, this concludes the proof.
In the next result, the convergences in probability of random measures towards non-random limits are understood with respect to the weak topology on the space of probability measures on the real line.
For each $n$, let $M_n,N_n$ be random $n\times n$ diagonal matrices, one of them having a distribution which is invariant under the action of the symmetric group by conjugation. Suppose that the spectral law of $M_n$ (reps. $N_n$) converges in probability to some compactly supported probability measure $\mu$ (resp. $\nu$) on the real line. Then the spectral law of $M_n+U_{n,t}N_nU_{n,t}^*$ converges in probability to the measure $\mu*_t\nu$.
Computation rules for $t$-freeness {#10.9.08.2}
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Multivariate free Itô calculus
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### Technical preliminaries
In this section, we shall extend some results of [@bs98] to the multivariate case. Let us first recall basics of free stochastic calculus. For more involved definitions, the reader should refer to sections 1-2 of [@bs98]. Let $({\mathcal}{M},\tau)$ be a faithful[^1] non-commutative space endowed with a filtration $({\mathcal}{M}_t)_{t\geq 0}$ and an $({\mathcal}{M}_t)_{t\geq 0}$-free additive Brownian motion $(X_t)_{t\geq 0}$. Let ${\mathcal}{M}^{op}$ be the opposite algebra of ${\mathcal}{M}$ (it is the same vector space, but it is endowed with the product $a\times_{op} b=ba$). We shall denote by $\sharp$ the left actions of the algebra ${\mathcal}{M}\otimes {\mathcal}{M}^{op}$ on ${\mathcal}{M}$ and ${\mathcal}{M}\otimes {\mathcal}{M}$ defined by $(a\otimes b)\sharp u=aub$ and $(a\otimes b)\sharp (u\otimes v)=au\otimes vb$. The algebras ${\mathcal}{M}$ and ${\mathcal}{M}\otimes {\mathcal}{M}^{op}$ are endowed with the inner products defined by ${\langle}a,b{\rangle}=\tau(ab^*)$ and ${\langle}a\otimes b, c\otimes d{\rangle}= \tau(ac^*)\tau(bd^*)$. The Riemann integral of functions defined on a closed interval with left and right limits at any point with values in the Hilbert space[^2] $L^2({\mathcal}{M}, \tau)$ is a well known notion. Now, we shall recall the definition of the stochastic integral. A simple adapted biprocess is a piecewise constant map $U$ from $[0,+\infty)$ to ${\mathcal}{M}\otimes {\mathcal}{M}^{op}$ vanishing for $t$ large enough [such that ]{}$U_t\in {\mathcal}{M}_t\otimes {\mathcal}{M}_t$ for all $t$. The set of simple biprocesses is endowed with the inner product $${\langle}U,V{\rangle}=\int_0^\infty {\langle}U_t,V_t{\rangle}{\mathrm{d}}t.$$ We shall denote by $\B_2^a$ the closure of the set of simple adapted biprocesses with respect to this inner product. Let $U$ be a simple adapted biprocess. Then there exists times $0=t_0\leq t_1\leq \cdots \leq t_m$ [such that ]{}$L$ (resp. $U$) is constant on each $[t_{i}, t_{i+1})$ and vanishes on $[t_m, +\infty)$. Then we define $$\int_0^\infty U_t{\mathrm{d}}X_t=\sum_{i=0}^{m-1}U_{t_i}\sharp(X_{t_{i+1}}-X_{t_i}).$$ It can be proved (Corollary 3.1.2 of [@bs98]) that the map $U\mapsto \int_0^\infty U_t{\mathrm{d}}X_t$ can be extended isometrically from $\B_2^a$ to $L^2({\mathcal}{M}, \tau)$.
### Free Itô processes
We shall call a [*free Itô process*]{} any process \[9.9.08.1\]A\_t=A\_0+\_0\^tL\_ss+\_0\^t U\_sX\_s, where $A_0\in {\mathcal}{M}_0$, $L$ is an adapted process with left and right limit at any point and $U\in \B_2^a$. In this case, we shall denote \[9.9.08.2\]A\_t=L\_tt+U\_tX\_t. The part $U_t\sharp {\mathrm{d}}X_t$ of this expression is called the [*martingale part*]{} of $A$. Note that the process $A$ is determined by $A_0$ and ${\mathrm{d}}A_t$. We shall use the following lemma, which follows from Proposition 2.2.2 of [@bs98] and from the linearity of $\tau$.
\[martpart=zero\]Let $A_t$ be as in . Then $\tau(A_t)=\tau(A_0)+\int_0^t\tau(L_s){\mathrm{d}}s$.
### Multivariate free Itô calculus
Consider $n$ elements $a_1,\ldots, a_n\in {\mathcal}{M}$ for some $n\geq 2$. Consider also two elements $u=\sum_k x_k\otimes y_k, v=\sum_lz_l\otimes t_l$ of ${\mathcal}{M}\otimes {\mathcal}{M}^{op}$. For all $1\leq i<j\leq n$, we define an element of $\M$ by setting $$\begin{aligned}
{\langle}{\langle}a_1,\ldots, a_{i-1},u,a_{i+1}, \ldots, a_{j-1},v, a_{j+1}, \ldots, a_n{\rangle}{\rangle}_{i,j}=&\\
& \hskip -3cm \sum_{k,l}a_1\cdots a_{i-1}x_k\tau(y_ka_{i+1}\cdots a_{j-1}z_l)t_l a_{j+1} \cdots a_n.\end{aligned}$$
The following theorem follows from Theorem 4.1.12 and the remark following in [@bs98].
\[9.9.08.3\]Let $A_t= A_0+\int_0^tL_s{\mathrm{d}}s+\int_0^t U_s{\mathrm{d}}X_s$ and $B_t= B_0+\int_0^tK_s{\mathrm{d}}s+\int_0^t V_s{\mathrm{d}}X_s$ be two Itô processes with respect to the same free Brownian motion $(X_t)$. Then $AB$ is a free Itô process and with the notations of , $${\mathrm{d}}(AB)_t=A_t{\mathrm{d}}B_t+({\mathrm{d}}A_t)B_t+{\langle}{\langle}U_t,V_t{\rangle}{\rangle}_{1,2} {\mathrm{d}}t.$$
In order to prove computation rules for $t$-freeness, we shall need the following theorem.
\[Ito.multidim\]Let $A_1,\ldots, A_n$ be free Itô processes with respect to the same Brownian motion. For all $k$, denote $A_{k,t}=A_{k,0}+\int_0^tL_{k,s}{\mathrm{d}}s+\int_0^t U_{k,s}{\mathrm{d}}X_s$. Then $A_1\cdots A_n$ is a free Itô process and $$\begin{aligned}
{\mathrm{d}}(A_1\cdots A_n)_t=&\sum_{k=1}^n A_{1,t}\cdots A_{k-1,t}({\mathrm{d}}A_{k,t})A_{k+1,t}\cdots A_{n,t}\\
& +\sum_{1\leq k<l\leq n}{\langle}{\langle}A_{1,t},\ldots, A_{k-1,t},U_{k,t},A_{k+1,t},\ldots, A_{l-1,t},U_{l,t}, A_{l+1,t},\ldots, A_{n,t}{\rangle}{\rangle}_{k,l} {\mathrm{d}}t.\end{aligned}$$
Let us prove this theorem by induction on $n$. For $n=1$, it is obvious. Let us suppose the result to hold at rank $n$. Then the martingale part of $A_1\cdots A_n$ is $$\sum_{k=1}^n A_{1,t}\cdots A_{k-1,t}(U_{k,t}\sharp {\mathrm{d}}X_{k,t})A_{k+1,t}\cdots A_{n,t}.$$Thus by Theorem \[9.9.08.3\], $A_1\cdots A_{n+1}$ is a free Itô process and $$\begin{aligned}
{\mathrm{d}}(A_1\cdots A_{n+1})_t=& (A_1\cdots A_{n})_t{\mathrm{d}}A_{n+1,t}+({\mathrm{d}}(A_1\cdots A_n)_t)A_{n+1,t}\\
& +\sum_{k=1}^n {\langle}{\langle}A_{1,t},\ldots, A_{k-1,t},U_{k,t},A_{k+1,t},\ldots, A_{n,t},U_{n+1,t}{\rangle}{\rangle}_{k,n+1} {\mathrm{d}}t\\
=& \sum_{k=1}^{n+1} A_{1,t}\cdots A_{k-1,t}({\mathrm{d}}A_{k,t})A_{k+1,t}\cdots A_{n,t}\\
&\hskip -2cm +\sum_{1\leq k<l\leq n}{\langle}{\langle}A_{1,t},\ldots, A_{k-1,t},U_{k,t},A_{k+1,t},\ldots, A_{l-1,t},U_{l,t}, A_{l+1,t},\ldots, A_{n,t}, A_{n+1,t}{\rangle}{\rangle}_{k,l} {\mathrm{d}}t\\
&\hskip -2cm + \sum_{k=1}^n {\langle}{\langle}A_{1,t},\ldots, A_{k-1,t},U_{k,t},A_{k+1,t},\ldots, A_{n,t},U_{n+1,t}{\rangle}{\rangle}_{k,n+1} {\mathrm{d}}t,\\\end{aligned}$$ which concludes the proof.
Computation rules for $t$-freeness {#computation-rules-for-t-freeness}
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### Main result
In order to do computations with elements which are $t$-free, we have to find out a formula for the expectation of a product of elements of the type \[gouffran.11.9.08.15h38\]x\_1u\_ty\_1u\_t\^\*x\_2u\_ty\_2u\_t\^\*x\_[n]{}u\_ry\_nu\_t\^\*,for $\{x_1,\ldots,x_n\}$ independent with $\{y_1,\ldots, y_n\}$ and $\{x_1,y_1,\ldots, x_n,y_n\}$ free with $u_t$, free unitary Brownian motion. Actually, for the result which follows, the independence of the $x_i $’s and the $y_i$’s will not be useful, thus we consider a non-commutative space $({\mathcal}{M},\tau)$, an integer $n\geq 1$, $a_1,\ldots, a_{2n}\in \M$ and a free unitary Brownian motion $(u_t)$ which is free with $\{a_1,\ldots, a_{2n}\}$. In order to have some more concise formulae, it will be useful to multiply the product of by $e^{nt}$. So we define $$f_{2n}(a_1,\ldots,a_{2n},t)=e^{nt}\tau(a_1u_ta_2u_t^*\cdots a_{2n-1}u_ta_{2n}u_t^*).$$ We shall use the convention $f_0(a,t)=\tau(a)$ for all $a\in\M$.
Since $f_{2n}(a_1,\ldots,a_{2n} ,0)=\tau(a_1\cdots a_{2n})$, the following theorem allows us to deduce all functions $f_{2n}(a_1,\ldots,a_{2n},t)$ (thus the expectation of any product of the type of ) from the joint distribution of the $a_i$’s.
\[suazo.11.9.08\] For all $n\geq1$ and all $a_{1}, \ldots, a_{2n}\in \M$ free with the process $(u_t)$, the following differential relations are satisfied: $$\begin{aligned}
{\frac}{\partial}{\partial t}f_{2n}(a_1,\ldots,a_{2n},t)=&-\!\!\!\!\!\!\!\!\sum_{\substack{1\leq k<l\leq 2n\\ k=l \; {\rm mod}\; 2}} \!\!\!\!\!\!\!\!
f_{2n-(l-k)}(a_1,\ldots, a_k,a_{l+1},\ldots, a_{2n},t)f_{l-k}(a_{k+1},\ldots, a_l,t)\\
&\hskip -3cm +e^t\!\!\!\!\!\!\sum_{\substack{1\leq k<l\leq 2n\\ k\neq l \; {\rm mod}\; 2}} \!\!\!\!\!\!\!\! f_{2n-(l-k)-1}(a_1,\ldots, a_{k-1},a_ka_{l+1},a_{l+2},\ldots, a_{2n} ,t)f_{l-k-1}(a_la_{k+1},a_{k+2},\ldots, a_{l-1} ,t).\\\end{aligned}$$
Let us introduce the process $(v_t)$ defined by $v_t=e^{t/2}u_t$ for all $t$. As explained in the beginning of section 2.3 of [@b97], this process can be realized as an Itô process, with the formula $$v_t=1+i\int_0^t {\mathrm{d}}X_s v_s.$$ Thus one can realize the family of non-commutative random variables $a_1,\ldots, a_{2n}, (v_t)_{t\geq 0}$ in a faithful non-commutative space $(\M,\tau)$ endowed with a filtration $(\M_t)_{t\geq 0}$ and an additive free Brownian motion $(X_t)_{t\geq 0}$ [such that ]{}$a_1,\ldots,a_{2n}\in \M_0$ and for all $t$, $v_t=1+i\int_0^t {\mathrm{d}}X_s v_s$ and $v_t^*=1-i\int_0^t v_s^*{\mathrm{d}}X_s $. By definition of $f_{2n}(a_1,\ldots, a_{2n},t)$, one has $$f_{2n}(a_1,\ldots, a_{2n},t)=\tau(a_1v_ta_2v_t^*\cdots a_{2n-1}v_ta_{2n}v_t^*).$$Note that since all $a_i$’s belong to $\M_0$, the processes $A_{1}:=(a_1v_t), A_{2}:=(a_2v_t^*), \ldots,A_{2n-1}:= (a_{2n-1}v_t), A_{2n}:= (a_{2n}v_t^*)$ are all free Itô processes: if one defines $U_{k,t}=a_k\otimes iv_t$ for $k$ odd and $U_{k,t}=-ia_kv_t^*\otimes 1 $ for $k$ even, then for all $k$, ${\mathrm{d}}A_{k,t}=U_{k,t}\sharp {\mathrm{d}}X_t$. Thus by theorem \[Ito.multidim\], $A_1\cdots A_{2n}$ is an Itô process [such that ]{}for all $t$, $$\begin{aligned}
(A_1\cdots A_{2n})_t=&(A_1\cdots A_{2n})_0+\int_0^t \sum_{k=1}^{2n} A_{1,s}\cdots A_{k-1,s}(U_{k,s}\sharp {\mathrm{d}}X_s)A_{k+1,s}\cdots A_{2n,s} \\
&\hskip -1cm +\int_0^t \sum_{1\leq k<l\leq 2n} {\langle}{\langle}A_{1,s},\ldots, A_{k-1,s},U_{k,s},A_{k+1,s},\ldots, A_{l-1,s},U_{l,s}, A_{l+1,s},\ldots, A_{2n,s}{\rangle}{\rangle}_{k,l} {\mathrm{d}}s.\\\end{aligned}$$
Hence by lemma \[martpart=zero\], for all $t$, $$\begin{aligned}
\label{cold.war.kids.11.09.08}
{\frac}{\partial}{\partial t}f_{2n}(a_1,\ldots,a_{2n},t)=& \\
&\hskip -2cm \!\!\!\!\sum_{1\leq k<l\leq 2n}\!\!\!\! \tau({\langle}{\langle}A_{1,t},\ldots, A_{k-1,t},U_{k,t},A_{k+1,t},\ldots, A_{l-1,t},U_{l,t}, A_{l+1,t},\ldots, A_{2n,t}{\rangle}{\rangle}_{k,l}). \nonumber\end{aligned}$$ Now, fix $1\leq k<l\leq 2n$ and discuss according to the parity of $k$ and $l$.
$\bullet$ If $k=l\mod 2$. Suppose for example that $k,l$ are both odd (the other case can be treated in the same way). Then $U_{k,t}=a_k\otimes iv_t$ and $U_{l,t}=a_l\otimes iv_t$, which implies that $$\begin{aligned}
\tau({\langle}{\langle}A_{1,t},\ldots, A_{k-1,t},U_{k,t},A_{k+1,t},\ldots, A_{l-1,t},U_{l,t}, A_{l+1,t},\ldots, A_{2n,t}{\rangle}{\rangle}_{k,l})= &\\
&\hskip -8cm i\tau(a_1v_ta_2v_t^*\cdots a_{k-1}v_t^*a_kv_ta_{l+1}v_t\cdots a_{2n}v_t^*)i\tau(v_ta_{k+1}v_t^*\cdots a_{l-1}v_t^*a_l).\end{aligned}$$ Note that since $\tau$ is tracial and the joint distribution of $a_1,\ldots, a_{2n}, (v_t)_{t\geq 0}$ is the same as the one of $a_1,\ldots, a_{2n}, (v_t^*)_{t\geq 0}$, we have $\tau(v_ta_{k+1}v_t^*\cdots a_{l-1}v_t^*a_l)=\tau(a_{k+1}v_t\cdots a_{l-1}v_ta_lv_t^*)$. Hence $$\begin{aligned}
\label{cocorosie.11.9.08}
\tau({\langle}{\langle}A_{1,t},\ldots, A_{k-1,t},U_{k,t},A_{k+1,t},\ldots, A_{l-1,t},U_{l,t}, A_{l+1,t},\ldots, A_{2n,t}{\rangle}{\rangle}_{k,l})=&\\
&\hskip -8cm -f_{2n-(l-k)}(a_1,\ldots, a_k,a_{l+1},\ldots, a_{2n},t)f_{l-k}(a_{k+1},\ldots, a_l,t).\nonumber\end{aligned}$$
$\bullet$ If $k\neq l\mod 2$. Suppose for example $k$ to be odd and $l$ to be even (the other case can be treated in the same way). Then $U_{k,t}=a_k\otimes iv_t$ and $U_{l,t}=-a_liv_t^*\otimes 1$, which implies that $$\begin{aligned}
\tau({\langle}{\langle}A_{1,t},\ldots, A_{k-1,t},U_{k,t},A_{k+1,t},\ldots, A_{l-1,t},U_{l,t}, A_{l+1,t},\ldots, A_{2n,t}{\rangle}{\rangle}_{k,l})=&\\
&\hskip -9cm \tau(a_1v_ta_2v_t^*\cdots a_{k-1}v_t^*a_ka_{l+1}v_t\cdots a_{2n}v_t^*)(-i^2)\tau(v_ta_{k+1}v_t^*\cdots a_{l-1}v_ta_lv_t^*).\end{aligned}$$ Note that since $v_t^*v_t=e^t$, $\tau$ is tracial and the joint distribution of $a_1,\ldots, a_{2n}, (v_t)_{t\geq 0}$ is the same as the one of $a_1,\ldots, a_{2n}, (v_t^*)_{t\geq 0}$, we have $\tau(v_ta_{k+1}v_t^*\cdots a_{l-1}v_ta_lv_t^*)=e^t\tau(a_la_{k+1}v_t\cdots a_{l-1}v_t^*)$. Hence $$\begin{aligned}
\label{cocorosie.bis.11.9.08}
\tau({\langle}{\langle}A_{1,t},\ldots, A_{k-1,t},U_{k,t},A_{k+1,t},\ldots, A_{l-1,t},U_{l,t}, A_{l+1,t},\ldots, A_{2n,t}{\rangle}{\rangle}_{k,l})=&\\
&\hskip -11cm e^tf_{2n-(l-k)-1}(a_1,\ldots, a_{k-1},a_ka_{l+1},a_{l+2},\ldots, a_{2n} ,t)f_{l-k-1}(a_la_{k+1},a_{k+2},\ldots, a_{l-1} ,t).\nonumber\end{aligned}$$
Equations , and together conclude the proof.
\
The following proposition, which we shall use later, is a consequence of the previous theorem.
\[gulf.shores.13.09.08\] In a non-commutative space $({\mathcal}{M},\tau)$, consider two independent normal elements $a,b$ with symmetric compactly supported laws. Let $(u_t)$ be a free unitary Brownian motion which is free with $\{a,b\}$. Then the function $$G(t,z)=\sum_{n\geq 1}\tau((au_tbu_t^*)^{2n})e^{2nt}z^n$$ is the only solution, in a neighborhood of $(0,0)$ in $[0,+\infty) \times \C$, to the nonlinear, first order partial differential equation [G]{}[t]{}+4zG [G]{}[z]{}&=&0\[allaround.13.09.08\]\
G(0,z)&=& \_[n1]{}(a\^[2n]{})(b\^[2n]{})z\^n.\[allaround.13.09.08.1\]
Let us define, for all $n\geq 1$, $g_{n}(t)=\tau((au_tbu_t^*)^{n})e^{nt}$. For $n=0$, we set $g_0(t)=0$. Let us fix $n\geq 1$. In order to apply the previous theorem, let us define, for $i=1,\ldots, 2n$, $a_i=a$ if $i$ is odd and $a_i=b$ if $i$ is even. By the previous theorem, for all $n\geq 1$, we have $$\begin{aligned}
\label{13.09.08.1}
{\frac}{\partial}{\partial t}g_{n}(t)=&-\!\!\!\!\!\!\!\!\sum_{\substack{1\leq k<l\leq 2n\\ k=l \; {\rm mod}\; 2}} \!\!\!\!\!\! g_{n-(l-k)/2}(t)g_{(l-k)/2}(t)\\
&\hskip -1.5cm +e^t\!\!\!\!\!\!\sum_{\substack{1\leq k<l\leq 2n\\ k\neq l \; {\rm mod}\; 2}} \!\!\!\!\!\! f_{2n-(l-k)-1}(a_1,\ldots, a_{k-1},a_ka_{l+1},a_{l+2},\ldots, a_{2n} ,t)f_{l-k-1}(a_la_{k+1},a_{k+2},\ldots, a_{l-1} ,t).\nonumber\end{aligned}$$
Now, note that since for any ${\varepsilon}, {\varepsilon}'=\pm 1$, the joint distribution of $(a,b, u_t)$ is the same as the one of $({\varepsilon}a,{\varepsilon}' b,u_t)$, $g_p(t)=0$ when $p$ is odd. Thus in the first sum of only pairs $(k,l)$ [such that ]{}$k=l\mod 4$ have a non null contribution. For the same reason, all terms in the second sum are null. Indeed, for any $1\leq k<l\leq 2n$ [such that ]{}$k\neq l \mod 2$, the set $\{k+1, k+2, \ldots, l\}$, whose cardinality is odd, has either an odd number of odd elements or an odd number of even elements. To sum up, for all $n\geq 1$, we have $$\begin{aligned}
\frac{\partial}{\partial t}g_{2n}(t) &= - \sum_{\substack{1\leq k<l\leq 4n \\ k=l \; {\rm mod} \; 4}} g_{2n-(l-k)/2}(t)g_{(l-k)/2}(t)\\
&= - 4 \sum_{i=1}^{n-1} (n-i) g_{2(n-i)}(t) g_{2i}(t)\\
&= - 2n \sum_{i=1}^{n-1} g_{2(n-i)}(t) g_{2i}(t).\end{aligned}$$
Thus since $g_0(t)=0$ and $G(t,z)=\sum_{n\geq 1}g_{2n}(t)z^n=\sum_{n\geq 0}g_{2n}(t)z^n$, the last computation implies $${\frac}{\partial G}{\partial t}=-2z{\frac}{\partial G^2}{\partial z},$$ which proves . The formula is obvious.
To prove the uniqueness, let $H(t,z)=\sum_{n\geq 0}h_n(t)z^n$ be another solution of and . By , for all $n\geq 0$, we have $h_n(0)=g_{2n}(0)$ and by , for all $n\geq 0$, we have ${\frac}{\partial}{\partial t}h_{n}(t)=
-2n\sum_{m=0}^{n}h_{n-m}(t)h_{m}(t)$, which implies that $h_0=0$ and that by induction on $n$, $h_n=g_{2n}$.
### Examples
Let us give examples of applications of the computation rules that we have just established. The third example below is a rather big formula, but we shall need it when we study the problem of existence of $t$-free cumulants. So, let $\A$ and $\B$ be two independent subalgebras of a non-commutative space $(\M,\tau)$ and $(u_t)$ be a free unitary Brownian motion free from $\A\cup \B$.
1\) For $a\in \A, b\in \B$, for all $t\geq 0$, we have \[eddie.vedder.11.9.08\](au\_tbu\_t\^\*)=(a)(b).(In fact, it even follows from theorem \[suazo.11.9.08\] that without the assumption that $a$ and $b$ are independent, for all $t$, we have $\tau(au_tbu_t^*)=e^{-t}\tau(ab)+(1-e^{-t})\tau(a)\tau(b)$).
2\) For $a,a'\in \A, b,b'\in \B$, for all $t\geq 0$, we have \[eddie.vedder.11.9.08.bis\](au\_tbu\_t\^\* a’u\_tb’u\_t\^\*)=$$(\tau(a)\tau(a')\tau(bb')+\tau(aa')\tau(b)\tau(b')-\tau(a)\tau(a')\tau(b)\tau(b'))(1-e^{-2t})+\tau(aa')\tau(bb')e^{-2t}.$$
3\) For $a,a',a''\in \A$ and $b,b',b''\in \B$, we have $$\begin{aligned}
\label{eddie.vedder.11.9.08.third}
&&\hskip 1cm \tau(au_tbu_t^*a'u_tb'u_t^*a''u_tb''u_t^*)\\ &=&\tau(a)\tau(a')\tau(a'')\tau(b)\tau(b')\tau(b'')(2-6e^{-2t}+4e^{-3t})\nonumber\\
&&-(1-3e^{-2t}+2e^{-3t})\tau(a)\tau(a')\tau(a'')\tau(b)\tau(b'b'')\nonumber\\
&&-(1-3e^{-2t}+2e^{-3t})\tau(a)\tau(a')\tau(a'')\tau(bb')\tau(b'')\nonumber\\
&&-(1-3e^{-2t}+2e^{-3t})\tau(a)\tau(a')\tau(a'')\tau(bb'')\tau(b')\nonumber\\
&&-(1-3e^{-2t}+2e^{-3t})
\tau(aa')\tau(a'')\tau(b)\tau(b')\tau(b'')\nonumber\\
&&-(1-3e^{-2t}+2e^{-3t})\tau(aa'')\tau(a')\tau(b)\tau(b')\tau(b'')\nonumber\\
&&-(1-3e^{-2t}+2e^{-3t})\tau(a)\tau(a'a'')\tau(b)\tau(b')\tau(b'')\nonumber\\
&&+(1-3e^{-2t}+2e^{-3t})[\tau(a)\tau(a')\tau(a'')\tau(bb'b'')+\tau(aa'a'')\tau(b)\tau(b')\tau(b'')]\nonumber\\
&&-(e^{-2t}-e^{-3t}) [\tau(aa')\tau(a'')\tau(bb')\tau(b'')+\tau(aa')\tau(a'')\tau(bb'')\tau(b')+\tau(aa'')\tau(a')\tau(bb'')\tau(b')\nonumber\\ &&+\tau(aa'')\tau(a')\tau(b)\tau(b'b'')+\tau(a)\tau(a'a'')\tau(b'')\tau(bb')+\tau(a)\tau(a'a'')\tau(b)\tau(b'b'')]\nonumber\\
&&+(1-2e^{-2t}+e^{-3t}) [\tau(aa')\tau(a'')\tau(b'b'')\tau(b)+\tau(aa'')\tau(a')\tau(bb')\tau(b'')+\tau(a)\tau(a'a'')\tau(bb'')\tau(b')]\nonumber\\
&&+(e^{-2t}-e^{-3t})\tau(bb'b'')[\tau(a)\tau(a'a'')+\tau(aa')\tau(a'')+\tau(aa'')\tau(a')]\nonumber\\
&&+(e^{-2t}-e^{-3t})\tau(aa'a'')[\tau(b)\tau(b'b'')+\tau(bb')\tau(b'')+\tau(bb'')\tau(b')]\nonumber\\
&&+e^{-3t}\tau(aa'a'')\tau(bb'b'')\nonumber\end{aligned}$$
It can be verified that the last formula actually corresponds to the formula of $\E(aba'b'a''b'')$ with $\{a,a',a''\}$ and $\{b,b',b''\}$ independent when $t=0$, and to the formula of $\tau(aba'b'a''b'')$ with $\{a,a',a''\}$ and $\{b,b',b''\}$ free when $t$ tends to infinity.
Multiplicative and additive $t$-free convolutions of two symmetric Bernoulli laws
---------------------------------------------------------------------------------
### Multiplicative case
Here, we shall compute the multiplicative $t$-free convolution of ${\frac}{\delta_{-1}+\delta_1}{2}$ (considered as a law on the unit circle) with itself.
For all $t\geq 0$, ${\frac}{\delta_{-1}+\delta_1}{2}\odot_t{\frac}{\delta_{-1}+\delta_1}{2}$ is the only law on the unit circle which is invariant under the symmetries with respect to the real and imaginary axes and whose push-forward by the map $z\mapsto z^2$ has the law of $u_{4t}$, a free unitary Brownian motion taken at time $4t$.
[The moments of $u_{4t}$ have been computed by P. Biane at Lemma 1 of [@b97]: for all $n\geq 1$, $$\label{moments mbul}
\tau(u_{4t}^n)={\frac}{e^{-2nt}}{n}\sum_{k=0}^{n-1}{\frac}{(-4nt)^k}{k!}\binom{n}{k+1}.$$]{}
In a non-commutative space $({\mathcal}{M},\tau)$, consider two independent normal elements $a,b$ with law ${\frac}{\delta_{-1}+\delta_1}{2}$. Let $(u_t)$ be a free unitary Brownian motion which is free with $\{a,b\}$. Then ${\frac}{\delta_{-1}+\delta_1}{2}\odot_t{\frac}{\delta_{-1}+\delta_1}{2}$ is the distribution of the unitary element $au_tbu_t^*$. Since the joint distribution of $(a,b,u_t)$ is the same as the one of $(-a,b ,u_t)$, ${\frac}{\delta_{-1}+\delta_1}{2}\odot_t{\frac}{\delta_{-1}+\delta_1}{2}$ is invariant under the transformation $z\mapsto -z$. Moreover, $(au_tbu_t^*)^*=u_tbu_t^*a$ has the same distribution as $au_tbu_t^*$ (because $\tau$ is tracial and $u_t$ has the same law as $u_t^*$), hence ${\frac}{\delta_{-1}+\delta_1}{2}\odot_t{\frac}{\delta_{-1}+\delta_1}{2}$ is invariant under the transformation $z\mapsto \bar{z}$. This proves that ${\frac}{\delta_{-1}+\delta_1}{2}\odot_t{\frac}{\delta_{-1}+\delta_1}{2}$ is invariant under the symmetries with respect to the real and imaginary axes.
Since any distribution on the unit circle is determined by its moments, to prove that the push-forward of ${\frac}{\delta_{-1}+\delta_1}{2}\odot_t{\frac}{\delta_{-1}+\delta_1}{2}$ by the map $z\mapsto z^2$ is the law of $u_{4t}$, it suffices to prove that for all $n\geq 1$, $$\tau((au_tbu_t^*)^{2n})=\tau(u_{4t}^{n}),$$ i.e. to prove that the functions $$F_1(t,z)=\sum_{n\geq 1} \tau((au_tbu_t^*)^{2n})e^{2nt}z^n\quad\textrm{ and }\quad F_2(t,z)=\sum_{n\geq 1} \tau(u_{4t}^{n})e^{2nt}z^n$$ are equal. It follows from Proposition \[gulf.shores.13.09.08\] that $F_1$ is the only solution, in a neighborhood of $(0,0)$ in $[0,+\infty) \times \C$, to equation satisfying $F_1(0,z)={\frac}{z}{1-z}$. But it follows from Lemma 1 of [@b97] that $F_2$ is also a solution of with the same initial conditions. By uniqueness, it closes the proof.
For all $t\in [0,1]$, let us define $\beta(t)=2\sqrt{t(1-t)} +\arccos(1-2t)$. Then $\beta(t)$ is an increasing function of $t$ which goes from $0$ to $\pi$ when $t$ goes from $0$ to $1$. P. Biane has proved in [@b97b Prop. 10] that the distribution of $u_{4t}$ is absolutely continuous with respect to the Lebesgue measure on the unit circle, that its support is the full unit circle for $t\geq 1$, and the set $\{e^{i\theta} : |\theta|\leq \beta(t)\}$ for all $t\in [0,1]$. Moreover, the density of this distribution with respect to the uniform probability measure on the unit circle, which we denote by $\rho_{4t}$, is positive and analytic on the interior of its support for all $t\geq 0$, except at $-1$ for $t=1$.
\[26.11.08.11h26\][Since there is no simple formula for the density of $u_{t}$, it may be worth explaining how we got the picture above. The expression of the moments of the distribution of $u_{t}$ given by (\[moments mbul\]) is numerically ineffective, because it is an alternated sum of very large numbers. It only allows one to compute the first few dozens of moments of the distribution. Nevertheless, this expression of the moments allows one to prove that, for all $t\geq 0$, the function $\kappa_{t}$ defined on the interior of the complex unit disk by the formula $$\kappa_{t}(z)=\int_{0}^{2\pi} \frac{e^{i\theta}+z}{e^{i\theta}-z} \rho_{t}(e^{i\theta})\; \frac{d\theta}{2\pi}=1+2 \sum_{k=1}^{+\infty} \tau(u_{t}^{k})z^{k},$$ satisfies the equation $$\label{calc dens}
\frac{\kappa_{t}(z)-1}{\kappa_{t}(z)+1}e^{\frac{t}{2}\kappa_{t}(z)}=z.$$ This fact can be established using the Lagrange inversion formula (see [@b97]), see also [@b97b 4.2.2]. Now, on one hand, a computer seems to be able to solve this equation more reliably than it computes the moments of the distribution. On the other hand, $\kappa_{t}$ is the holomorphic function in the unit disk whose real part is the harmonic extension of the density of the distribution of $u_{t}$. Thus, we evaluated $\rho_{t}(e^{i\theta})$ by taking the real part of a numerical solution of (\[calc dens\]) with $z=e^{i\theta}$. ]{}
From the facts exposed above Remark \[26.11.08.11h26\], one deduces easily the next result.
\[support mult\] For all $t>0$, the measure ${\frac}{\delta_{-1}+\delta_1}{2}\odot_t{\frac}{\delta_{-1}+\delta_1}{2}$ has a density with respect to the uniform probability measure on the unit circle, which we shall denote by $\sigma_t$ and which is given by the formula $\sigma_t(z)=\rho_{4t}(z^2)$ for all $z$ in the unit circle. In particular, the support of this measure is the full unit circle for $t\geq 1$ and the set $\{e^{i\theta} : |\theta|\leq \frac{1}{2} \beta(t) \mbox{ or } |\pi- \theta|\leq \frac{1}{2} \beta(t)\}$ for $t\in [0,1]$. The density $\sigma_t$ is positive and analytic on the interior of its support for all $t\geq 0$, except at $\pm i$ for $t=1$.
### Additive case
Here, we shall compute the additive $t$-free convolution of ${\frac}{\delta_{-1}+\delta_1}{2}$ (considered as a law on the real line) with itself.
For all $t\geq 0$, ${\frac}{\delta_{-1}+\delta_1}{2}*_t{\frac}{\delta_{-1}+\delta_1}{2}$ is the only symmetric law on the real line whose push-forward by the map $x\mapsto x^2$ has the law of $2+v+v^*$, with $v$ unitary element distributed according to ${\frac}{\delta_{-1}+\delta_1}{2}\odot_t{\frac}{\delta_{-1}+\delta_1}{2}$.
In a non-commutative space $({\mathcal}{M},\tau)$, consider two independents normal elements $a,b$ with law ${\frac}{\delta_{-1}+\delta_1}{2}$. Let $(u_t)$ be a free unitary Brownian motion which is free with $\{a,b\}$. Then ${\frac}{\delta_{-1}+\delta_1}{2}*_t{\frac}{\delta_{-1}+\delta_1}{2}$ is the distribution of $a+u_tbu_t^*$. Since the joint distribution of $(a,b,u_t)$ is the same as the one of $(-a,-b ,u_t)$, ${\frac}{\delta_{-1}+\delta_1}{2}*_t{\frac}{\delta_{-1}+\delta_1}{2}$ is symmetric. Note that since $a^2$ and $b^2$ have $\delta_1$ for distribution, one can suppose that $a^2=b^2=1$. In this case, $$(a+u_tbu_t^*)^2=2+a u_tbu_t^*+ u_tbu_t^*a=2+a u_tbu_t^*+(a u_tbu_t^*)^*,$$and the result is obvious by definition of $\odot_t$.
From the last result and Proposition \[support mult\], we deduce the following.
For all $t>0$, the measure ${\frac}{\delta_{-1}+\delta_1}{2}*_t {\frac}{\delta_{-1}+\delta_1}{2}$ has a density with respect to the Lebesgue measure on $[-2,2]$, which we shall denote by $\eta_t$ and which is given by the formula $$\forall x\in [-2,2] \; , \;\; \eta_t(x)= \rho_{4t}(e^{4i\arccos \frac{x}{2}}) \frac{1}{\pi\sqrt{4-x^2}}.$$ The support of this measure is the interval $[-2,2]$ for $t\geq 1$, and the set $$\left[-2,-2 \cos \frac{\beta(t)}{4}\right] \cup \left[-2 \sin \frac{\beta(t)}{4},2 \sin \frac{\beta(t)}{4}\right] \cup \left[2 \cos \frac{\beta(t)}{4}, 2\right]$$ for $t\in [0,1]$. The density $\eta_t$ is positive and analytic on the interior of its support for all $t\geq 0$, except at $\pm \sqrt{2}$ for $t=1$.
The lack of cumulants {#10.9.08.3}
=====================
In this section, we investigate the existence of an analogue of classical and free cumulants in the context of $t$-freeness. Informally, the problem is to find multilinear forms defined on any non-commutative probability space which vanish when evaluated on a family of elements which can be split into two non-empty subfamilies which are $t$-free.
More precisely, given a non-commutative probability space $({\mathcal}{M},\varphi)$, we would like to know if there exists a family $(k_{n})_{n\geq 2}$ of multilinear forms on ${\mathcal}{M}$, with $k_n$ an $n$-linear form for all $n\geq 2$, such that, for all $n\geq 2$, all $n_{1},n_{2}>0$ such that $n_{1}+n_{2}=n$, all $m_{1},\ldots,m_{n}$ in ${\mathcal}{M}$ such that $\{m_{1},\ldots,m_{n_{1}}\}$ and $\{m_{n_{1}+1},\ldots,m_{n_{1}+n_{2}}\}$ are $t$-free, and finally for all $\sigma \in \Sym_{n}$, one has $k_{n}(m_{\sigma(1)},\ldots,m_{\sigma(n)})=0$.
Our main result is negative: there does not exist in general such a family of multilinear forms, at least in a large class which we describe now.
Let $({\mathcal}{M},\varphi)$ be a non-commutative probability space. Let $n\geq 1$ be an integer. Let $\sigma$ be an element of $\Sym_{n}$. We define the $n$-linear form $\varphi_{\sigma}$ on ${\mathcal}{M}$ as follows: $$\forall m_{1},\ldots,m_{n} \in {\mathcal}{M} \; , \;\; \varphi_{\sigma}(m_{1},\ldots,m_{n})=\prod_{\substack{c {\rm \; cycle \; of\; }\sigma\\c=(i_{1}\ldots i_{r})}} \varphi(m_{i_{1}}\ldots m_{i_{r}}).$$
Using only the algebra structure of ${\mathcal}{M}$ and the linear form $\varphi$, a linear combination of the forms $\{\varphi_\sigma:\sigma\in \Sym_n\}$ seems to be the most general $n$-linear form that one can construct on ${\mathcal}{M}$. We seek cumulants within this wide class of $n$-linear forms. Our definition does not require that the vanishing of cumulants characterize $t$-freeness. We only insist that mixed cumulants of $t$-free variable vanish.
Let $n\geq 2$ be an integer. Let $t\geq 0$ be a real number. A [*$t$-free cumulant of order $n$*]{} is a collection $(c(\sigma))_{\sigma\in \Sym_{n}}$ of complex numbers such that $\displaystyle \sum_{\sigma \; n{\rm -cycle}} c(\sigma)\neq 0$ and the following property holds for every non-commutative probability space $({\mathcal}{M},\varphi)$ : for any pair $(\A,{\mathcal}{B})$ of sub-algebras of ${\mathcal}{M}$ which are $t$-free with respect to $\varphi$, for any family $(m_{1},\ldots,m_{n})$ of elements of ${\mathcal}{A}\cup {\mathcal}{B}$, which do not all belong to ${\mathcal}{A}$, and not all to ${\mathcal}{B}$, we have $$\label{cum 1}
\sum_{\sigma\in \Sym_{n}} c(\sigma) \varphi_{\sigma}(m_{1},\ldots,m_{n})=0.$$
Let us emphasize that what we call [*cumulant*]{} is not a specific multilinear form, but rather a collection of coefficients which allows one to define a multilinear form on any non-commutative probability space.
If $(c(\sigma))_{\sigma\in \Sym_{n}}$ is a $t$-free cumulant of order $n$ and $m_{1},\ldots,m_{n}$ are elements of a non-commutative probability space $({\mathcal}{M},\varphi)$, at least one of which is equal to $1$, then $$\label{cum 2}
\sum_{\sigma\in \Sym_{n}} c(\sigma) \varphi_{\sigma}(m_{1},\ldots,m_{n})=0.$$ Indeed, the subalgebra $\C.1$ of ${\mathcal}{M}$ is $t$-free with any subalgebra of ${\mathcal}{M}$.
We extend the previous definition by including the free case $t=+\infty$. We will mainly consider collections $(c(\sigma))_{\sigma\in \Sym_{n}}$ with the property that $c(\rho\sigma\rho^{-1})=c(\sigma)$ for all $\sigma,\rho \in \Sym_{n}$. We call such collections [*conjugation-invariant*]{}. They are in fact indexed by conjugacy classes of $\Sym_{n}$, that is, integer partitions of $n$. Thus, we write use as well the notation $(c_{\lambda})_{\lambda \vdash n}$ for a conjugation-invariant collection.
Our main results are the following.
\[existence\] For all $t\in [0,+\infty]$ and all $n\in \{2,3,4,5,6\}$, there exists, up to scaling, a unique conjugation-invariant $t$-free cumulant of order $n$.
\[non existence\] There exists a $t$-free cumulant of order $7$ if and only if $t=0$ or $t=+\infty$.
Let us start by proving that we lose nothing by focusing on conjugation-invariant $t$-free cumulants.
\[cum ci\] If for some $t$ and some $n$ there exists a $t$-free cumulant of order $n$, then there exists such a cumulant $(c(\sigma))_{\sigma\in \Sym_{n}}$ such that moreover $c(\sigma)=c(\rho \sigma \rho^{-1})$ for all $\sigma,\rho\in \Sym_{n}$.
The point is that the order of the arguments is arbitrary in (\[cum 1\]). Hence, if (\[cum 1\]) is satisfied, then for all $\rho\in \Sym_{n}$, $$\begin{aligned}
0=\sum_{\sigma\in \Sym_{n}} c(\sigma) \varphi_{\sigma}(m_{\rho^{-1}(1)},\ldots,m_{\rho^{-1}(n)})&=\sum_{\sigma\in \Sym_{n}} c(\sigma)\varphi_{\rho^{-1}\sigma\rho}(m_{1},\ldots,m_{n})\\
&=\sum_{\sigma\in \Sym_{n}} c(\rho\sigma\rho^{-1})\varphi_{\sigma}(m_{1},\ldots,m_{n}).\end{aligned}$$ Hence, if $(c(\sigma))_{\sigma\in \Sym_{n}}$ is a $t$-free cumulant, then so is $(c(\rho\sigma \rho^{-1}))_{\sigma\in \Sym_{n}}$. By averaging over $\rho$, we get a conjugation-invariant cumulant.
Observe that the assumption made in the definition of a cumulant that the sum of $c(\sigma)$ when $\sigma$ spans the $n$-cycles is nonzero implies that $c_n\neq 0$ for any conjugation-invariant cumulant.
Let us introduce some notation. Given a permutation $\sigma$ of $\{1,\ldots,n\}$, we denote by $\{\{\sigma\}\}$ the partition of $\{1,\ldots,n\}$ whose blocks are the sets underlying the cycles of $\sigma$. Let ${\mathcal}{P}(n)$ denote the set of partitions of the set $\{1,\ldots,n\}$. Let $({\mathcal}{A},\varphi)$ be a commutative non-commutative probability space. For each partition $\pi\in {\mathcal}{P}(n)$, we define an $n$-linear form $\varphi_{\pi}$ on ${\mathcal}{A}$ by setting $\varphi_{\pi}=\varphi_{\sigma}$, where $\sigma$ is any permutation of $\{1,\ldots,n\}$ such that $\{\{\sigma\}\}=\pi$. Since ${\mathcal}{A}$ is commutative, this definition does not depend on the choice of $\sigma$. Finally, when $\sigma$ is a permutation of $\{1,\ldots,n\}$, we say that $i,j\in \{1,\ldots,n\}$ are [*consecutive*]{} in a cycle of $\sigma$ if $\sigma(i)=j$ or $\sigma(j)=i$. We will use repeatedly the following fact, which is a consequence of Proposition \[20.06.08.16h45\] and Proposition 1 of [@b98].
\[c sigma pi pi\] Choose two integers $k,l>0$ and set $n=k+l$.\
1. There exists universal coefficients $(C(\sigma,\pi,\pi'))_{\sigma\in \Sym_n,\pi\in {\mathcal}{P}(k),\pi'\in {\mathcal}{P}(l)}$ such that the following property holds:
Let ${\mathcal}{A}$ and ${\mathcal}{B}$ be two commutative sub-algebras of some non-commutative probability space $({\mathcal}{M},\varphi)$ which are $t$-free with respect to $\varphi$. Consider $\sigma\in \Sym_{n}$. For all $a_{1},\ldots,a_{k}\in {\mathcal}{A}$ and all $b_{1},\ldots,b_{l}\in {\mathcal}{B}$, $$\label{expand}
\varphi_{\sigma}(a_{1},\ldots,a_{k},b_{1},\ldots,b_{l})=\sum_{\pi\in {\mathcal}{P}(k),\pi'\in {\mathcal}{P}(l)} C(\sigma,\pi,\pi') \varphi_{\pi}(a_{1},\ldots,a_{k})\varphi_{\pi'}(b_{1},\ldots,b_{l}).$$ 2. The coefficient $C(\sigma,\pi,\pi')$ can be non-zero only if every block of $\pi$ is contained in a block of $\{\{\sigma\}\}$.\
3. If two elements $i$ and $j$ of $\{1,\ldots,k\}$ are consecutive in a cycle of $\sigma$, then $C(\sigma,\pi,\pi')$ can be non-zero only if $i$ and $j$ are in the same block of $\pi$.\
4. The parts 2. and 3. of this lemma are also valid for $\pi'$ (modulo a translation of $k$ of the indices, since $\pi'$ is a partition of $\{1,\ldots, l\}$ and not of $\{k+1,\dots, k+l\}$).
With the notation of the lemma above, we associate to every collection $(c(\sigma))_{\sigma\in \Sym_{n}}$ the following family of coefficients: $$\label{def D}
\forall \pi\in {\mathcal}{P}(k), \pi'\in {\mathcal}{P}(l), \;\; D_{c}(\pi,\pi')=\sum_{\sigma\in \Sym_{n}} c(\sigma)C(\sigma,\pi,\pi'),$$ which will play an important role in the proofs of Theorems \[non existence\] and \[existence\].\
(Theorem \[non existence\]) Let us choose $t>0$ a positive real. We prove by contradiction that there exists no $t$-free cumulant of order $7$. So, let us assume that there exists one and let $(c(\sigma))_{\sigma\in \Sym_7}$ be one of them, which we choose to be conjugation-invariant thanks to Lemma \[cum ci\]. Thus, we denote it also by $(c_{\lambda})_{\lambda\vdash 7}$. Since $c_7\neq 0$, we may and will assume that $c_{7}=1$. Then, we proceed as follows.
Let us consider a non-commutative probability space $({\mathcal}{M},\varphi)$ and two commutative sub-algebras ${\mathcal}{A}$ and ${\mathcal}{B}$ of ${\mathcal}{M}$ which are $t$-free with respect to $\varphi$. Let us choose $a_{1},a_{2},a_3\in {\mathcal}{A}$ and $b_{1},b_2,b_3,b_4,b_5 \in {\mathcal}{B}$, which we assume to be all centered. Set $k_{7}=\sum_{\sigma\in \Sym_{7}} c(\sigma)\varphi_{\sigma}$. By using the $t$-freeness of $\A$ and ${\mathcal}{B}$, we will express $k_7(a_1,a_2,b_1,b_2,b_3,b_4,b_5)$ and $k_7(a_1,a_2,a_3,b_1,b_2,b_3,b_4)$ in terms of the coefficients $(c_\lambda)_{\lambda\vdash 7}$, the joint moments of $a_1,a_2,a_3$, and the joint moments of $b_1,b_2,b_3,b_4,b_5$. By the assumption that $k_7$ is a $t$-free cumulant, the two expressions that we get must vanish. Since the joint distributions of the $a$’s and of the $b$’s are both arbitrary among those of families of centered elements, every coefficient of a given product of moments of the $a$’s and $b$’s must vanish. This gives us linear relations on the coefficients $(c_\lambda)_{\lambda\vdash 7}$, which will turn out to be incompatible.\
Let us start with $k_{7}(a_{1},a_{2},b_{1},b_2,b_3,b_4,b_{5})$. By Lemma \[c sigma pi pi\], this quantity can be written as $$\begin{aligned}
\label{dev k7 52}
&\sum_{\sigma\in \Sym_7,\pi\in{\mathcal}{P}(2),\pi'\in{\mathcal}{P}(5)} c(\sigma) C(\sigma,\pi,\pi') \varphi_\pi(a_1,a_2) \varphi_{\pi'}(b_1,b_2,b_3,b_4,b_5) \nonumber \\
&\hskip 4cm =\sum_{\pi\in{\mathcal}{P}(2),\pi'\in{\mathcal}{P}(5)} D_{c}(\pi,\pi')\varphi_\pi(a_1,a_2) \varphi_{\pi'}(b_1,b_2,b_3,b_4,b_5).\end{aligned}$$ We are thus interested in computing, for each pair $(\pi,\pi')$, the coefficient $D_{c}(\pi,\pi')$. It turns out to be convenient to think of $b_1,\ldots,b_5$ as occupying the slots $3$ to $7$ rather than $1$ to $5$ and to see $\pi'$ as a partition of the set $\{3,\ldots,7\}$ accordingly. We hope that no ambiguity will result from this change in our conventions.
Since we have chosen to consider elements which are centered, the sum (\[dev k7 52\]) can be restricted to pairs of partitions without singletons. This leaves us with the following pairs $(\pi,\pi')$: $(\{\{1,2\}\},\{\{3,4,5,6,7\}\})$, $(\{\{1,2\}\},\{\{3,4\},\{5,6,7\}\})$ and those which are deduced from the latter by permuting $3,4,5,6,7$.\
Let us compute $D_{c}(\{\{1,2\}\},\{\{3,4,5,6,7\}\})$. By the second assertion of Lemma \[c sigma pi pi\], the permutations $\sigma$ which contribute to this term must have $1,2$ on one hand, and $3,4,5,6,7$ on the other hand, in the same cycle. This can occur if $\sigma$ is either a $7$-cycle or the product of a $2$-cycle and a $5$-cycle.
Let us first compute the contribution of $7$-cycles. The coefficient $C(\sigma,\{1,2\},\{3,4,5,6,7\})$ is not the same for all $7$-cycles $\sigma$. We must distinguish between those in which $1$ and $2$ are consecutive and those in which they are not. There are $2!5!$ $7$-cycles in which $1$ and $2$ are consecutive. For each such cycle $\sigma$, $C(\sigma,\{1,2\},\{3,4,5,6,7\})=1$, thanks to (\[eddie.vedder.11.9.08\]). In a cycle where $1$ and $2$ are not consecutive, there may be one, two, three or four elements of $\{3,4,5,6,7\}$ between $1$ and $2$. In each case, there are $5!$ cycles, each contributing a factor $e^{-2t}$, thanks to (\[eddie.vedder.11.9.08.bis\]).
Let us now compute the contribution of products of a transposition and a $5$-cycle. There are $1!4!$ permutations with two cycles, one which contains $1,2$ and the other $3,4,5,6,7$. Each such permutation contributes a factor $c_{52}$.
Altogether, we have found that $$\label{d25 1}
D_{c}(\{\{1,2\}\},\{\{3,4,5,6,7\}\})=24(c_{52}+10(1+2e^{-2t})).$$
Let us now compute $D_{c}(\{\{1,2\}\},\{\{3,4\},\{5,6,7\}\})$. By the second assertion of Lemma \[c sigma pi pi\], there are five possibilities for the partition $\{\{\sigma\}\}$ underlying a permutation $\sigma$ which contributes to this coefficient. We study them one after the other.
${\scriptstyle \bullet} \{\{\sigma\}\}=\{\{1,2,3,4,5,6,7\}\}$. Since, by the third assertion of Lemma \[c sigma pi pi\], any two elements of $\{3,4,5,6,7\}$ which are consecutive in $\sigma$ must be in the same block of $\pi'=\{\{3,4\},\{5,6,7\}\}$, no element of $\{3,4\}$ can be consecutive to an element of $\{5,6,7\}$ in $\sigma$. Since there are only two $a$’s, the only possibility is that $3$ and $4$ on one hand, and $5$, $6$, and $7$ on the other hand, are consecutive in $\sigma$ and separated by $1$ and $2$. There are $2!2!3!$ $7$-cycles with this property. Each of them contributes to the sum with a factor $1-e^{-2t}$, according to (\[eddie.vedder.11.9.08.bis\]).
${\scriptstyle \bullet} \{\{\sigma\}\}=\{\{1,2\},\{3,4,5,6,7\}\}$. By the third assertion of Lemma \[c sigma pi pi\], these permutations do not contribute.
${\scriptstyle \bullet} \{\{\sigma\}\}=\{\{1,2,3,4\},\{5,6,7\}\}$. There are two possible structures for the $4$-cycle of $\sigma$ in this case. Either the $a$’s and the $b$’s are consecutive, or they are intertwined. In the first situation, there are $2!2!2!$ permutations, each of which contributes $c_{43}$, thanks to (\[eddie.vedder.11.9.08\]). In the second situation, there are $2!2!$ permutations, because of a higher symmetry, each of which contributes $e^{-2t}c_{43}$, thanks to (\[eddie.vedder.11.9.08.bis\]).
${\scriptstyle \bullet} \{\{\sigma\}\}=\{\{1,2,5,6,7\},\{3,4\}\}$. Again, there are two possible structures for the $5$-cycle of $\sigma$, depending on whether the $a$’s are consecutive or not. There are $2!3!$ permutations where they are, and each contributes $c_{52}$. There are also $2!3!$ permutations where they are not, each contributing $e^{-2t} c_{52}$.
${\scriptstyle \bullet} \{\{\sigma\}\}=\{\{1,2\},\{3,4\},\{5,6,7\}\}$. This is the simplest situation. There are $2$ permutations with this cycle structure and each contributes $c_{322}$.
Finally, $$\label{d52 2}
D_{c}(\{\{1,2\}\},\{\{3,4\},\{5,6,7\}\})=2\left(c_{322}+2(2+e^{-2t}) c_{43} + 6(1+e^{-2t})c_{52} + 12(1-e^{-2t})\right).$$
Let us perform the same kind of computations on $$k_7(a_1,a_2,a_3,b_1,b_2,b_3,b_4)=\sum_{\pi\in{\mathcal}{P}(\{1,2,3\}), \pi'\in {\mathcal}{P}(\{4,5,6,7\})} D_{c}(\pi,\pi') \varphi_{\pi}(a_1,a_2,a_3) \varphi_{\pi'}(b_1,b_2,b_3,b_4).$$ Since our variables are centered, the only pairs of partitions which occur in the sum are $(\{\{1,2,3\}\},\{\{4,5,6,7\}\})$, $(\{\{1,2,3\}\},\{\{4,5\},\{6,7\}\})$ and those which are deduced from the latter by permuting $4,5,6,7$.\
Let us compute $D_{c}(\{\{1,2,3\}\},\{\{4,5,6,7\}\})$. We shall again use formulae , and several times, without citing them every time. The permutations which contribute to this coefficient are $7$-cycles and products of a $3$-cycle and a $4$-cycle. As before, all $7$-cycles do not contribute in the same way. If the $a$’s are consecutive, which is the case for $3!4!$ $7$-cycles, the contribution is simply $1$. If two $a$’s are consecutive and the third is on its own, the $7$-cycle contributes $e^{-2t}$. In this case, there can be one, two or three $b$’s between the isolated $a$ and the pair of consecutive $a$’s, in the cyclic order. In each case, there are $3!4!$ possible $7$-cycles. Finally, the three $a$’s can be isolated. This happens in $3!4!$ $7$-cycles, and each contributes $e^{-3t}$, thanks to (\[eddie.vedder.11.9.08.third\]). So far, we have a contribution of $144(1+3e^{-2t}+e^{-3t})$. The contribution of products of a $3$-cycle and a $4$-cycle is much simpler to compute: it is $2!3!c_{43}$. We find $$\label{d34 1}
D_{c}(\{\{1,2,3\}\},\{\{4,5,6,7\}\})=12(c_{43}+12(1+3e^{-2t}+e^{-3t})).$$
Let us finally compute $D_{c}(\{\{1,2,3\}\},\{\{4,5\},\{6,7\}\})$. Again, by Lemma \[c sigma pi pi\], there are five possibilities for the partition $\{\{\sigma\}\}$, which we examine one after the other.
${\scriptstyle \bullet} \{\{\sigma\}\}=\{\{1,2,3,4,5,6,7\}\}$. No element of $\{4,5\}$ can be consecutive with an element of $\{6,7\}$ in $\sigma$. Still, there are several possibilities. Let us first consider the $7$-cycles where $4,5$ on one hand and $6,7$ on the other hand are consecutive. These two groups must be separated by $a$’s. There are $2!2!2!3!$ such $7$-cycles, and each contributes for $1-e^{-2t}$, according to (\[eddie.vedder.11.9.08.bis\]). Since there are only three $a$’s, one at least of the two pairs $\{4,5\}$ and $\{6,7\}$ must be consecutive. However, it can happen that one is not. This happens in $2!2!2!3!$ $7$-cycles, and according to (\[eddie.vedder.11.9.08.third\]), each contributes for $e^{-2t}-e^{-3t}$.
${\scriptstyle \bullet} \{\{\sigma\}\}=\{\{1,2,3\},\{4,5,6,7\}\}$. These permutations do not contribute.
${\scriptstyle \bullet} \{\{\sigma\}\}=\{\{1,2,3,4,5\},\{6,7\}\}$. As usual by now, there are two possibilities in the $5$-cycle of $\sigma$. Either the two $b$’s are consecutive, which happens in $2!3!$ cases with the contribution $c_{52}$, or they are not. This happens in $2!3!$ cases, and each case contributes for $e^{-2t}c_{52}$.
${\scriptstyle \bullet} \{\{\sigma\}\}=\{\{1,2,3,6,7\},\{4,5\}\}$. By symmetry, this contribution is equal to the one above.
${\scriptstyle \bullet} \{\{\sigma\}\}=\{\{1,2,3\},\{4,5\},\{6,7\}\}$. There are $2$ permutations, each contributing for $c_{322}$.
Finally, $$\label{d34 2}
D_{c}(\{\{1,2,3\}\},\{\{4,5\},\{6,7\}\})=2(c_{322}+12(1+e^{-2t})c_{52} + 24 (1-e^{-3t})).$$
Let us summarize our results. We have proved that, if there exists a $t$-free cumulant of order $7$, whose associated $7$-linear form is denoted by $k_7$, then for all centered $a_1,a_2,a_3\in \A$ and $b_1,\ldots,b_5 \in {\mathcal}{B}$, the following equalities hold: $$\begin{aligned}
k_7(a_1,a_2,b_1,b_2,b_3,b_4,b_5) &=24(c_{52}+10(1+2e^{-2t})) \varphi(a_{1}a_{2})\varphi(b_{1}b_{2}b_{3}b_{4} b_{5}) \\
& \hskip -3cm + 2\left(c_{322}+2(2+e^{-2t}) c_{43} + 6(1+e^{-2t})c_{52} + 12(1-e^{-2t})\right) \varphi(a_{1}a_{2})\varphi(b_{1}b_{2})\varphi(b_{3}b_{4}b_{5})\\
& \hskip -3cm + 2\left(c_{322}+2(2+e^{-2t}) c_{43} + 6(1+e^{-2t})c_{52} + 12(1-e^{-2t})\right) \varphi(a_{1}a_{2})\varphi(b_{1}b_{3})\varphi(b_{2}b_{4}b_{5})\\
& \hskip -3cm + \ldots,\end{aligned}$$ where all partitions of $\{b_1,b_2,b_3,b_4,b_5\}$ into a pair and a triple appear, and
$$\begin{aligned}
k_7(a_1,a_2,a_3,b_1,b_2,b_3,b_4) &=12(c_{43}+12(1+3e^{-2t}+e^{-3t})) \varphi(a_{1}a_{2}a_3)\varphi(b_{1}b_{2}b_{3}b_{4}) \\
&\hskip -2cm + 2(c_{322}+12(1+e^{-2t})c_{52} + 24 (1-e^{-3t})) \varphi(a_{1}a_{2}a_3)\varphi(b_{1}b_{2})\varphi(b_{3}b_{4})\\
&\hskip -2cm + 2(c_{322}+12(1+e^{-2t})c_{52} + 24 (1-e^{-3t})) \varphi(a_{1}a_{2}a_3)\varphi(b_{1}b_{3})\varphi(b_{2}b_{4})\\
&\hskip -2cm + 2(c_{322}+12(1+e^{-2t})c_{52} + 24 (1-e^{-3t})) \varphi(a_{1}a_{2}a_3)\varphi(b_{1}b_{4})\varphi(b_{2}b_{3}).\end{aligned}$$
Since $k_7$ is a $t$-free cumulant, these two expressions must be zero for all choices of $a$’s and $b$’s. Since the joint distributions of the $a$’s and of the $b$’s are both arbitrary among those of families of centered elements, this implies that the coefficients which appear in these equalities in front of the various products of moments of $a$’s and $b$’s must vanish. This implies the following relations: $$\begin{aligned}
c_{52}&=-10(1+2e^{-2t}),\\
c_{43}&=-12(1+3e^{-2t}+e^{-3t}),\\
c_{322}&=-2(2+e^{-2t}) c_{43} - 6(1+e^{-2t})c_{52} - 12(1-e^{-2t}),\\
c_{322}&=-12(1+e^{-2t})c_{52} - 24 (1-e^{-3t}).\end{aligned}$$
It does not take a long computation to see that the two expressions of $c_{322}$ are different, since the first involves $e^{-5t}$, whereas the second does not. We leave it to the reader to check that the difference between the two values of $c_{322}$ that we have obtained is equal to $24 e^{-3t} (1-e^{-t})^2$. This quantity vanishes only for $t=0$ or $t=+\infty$.
In order to prove that $t$-free cumulants of order at most $6$ exist, we are going to construct them. We prove first a lemma which settles the problem of the coefficients $c_\lambda$ for the partitions $\lambda$ whose smallest part is $1$.
Let us introduce some notation. Let $\mu=(\mu_1\geq \ldots \geq \mu_r)$ be a partition of some integer $n$. We denote by $\ell(\mu)$ the number of non-zero parts of $\mu$ and we write $\mu \vdash \!\! \vdash n$ if $\mu_{\ell(\mu)}\geq 2$, that is, if $\mu$ has no part equal to $1$. Let $i\geq 1$ an integer. We denote by $\mu+\delta_i$ the partition of $n+1$ whose parts are $\mu_1,\ldots,\mu_{i-1},\mu_i+1,\mu_{i+1},\ldots,\mu_r$, rearranged in non-increasing order. If $i>\ell(\mu)$, then $\mu+\delta_i$ is simply the partition $\mu$ to which a part equal to $1$ has been appended.
\[cond part avec 1\] \[determine\] Let $n\geq 2$ be an integer. Choose $t\in [0,+\infty]$. A collection $(c_\lambda)_{\lambda\vdash n}$ is a $t$-free cumulant if and only if the following two conditions hold:\
1. The relation (\[cum 1\]) is satisfied for all $m_{1},\ldots,m_{n}$ which are centered.\
2. For all $\mu\vdash n-1$, $$\label{part avec 1}
c_{\mu+\delta_{\ell(\mu)+1}}=-\sum_{i=1}^{\ell(\mu)} \mu_i c_{\mu+\delta_i}.$$
Moreover, a collection of complex numbers $(c_\lambda)_{\lambda \vdash\!\!\vdash n}$ which satisfies (\[cum 1\]) for all $m_1,\ldots,m_n$ which are centered can be extended in a unique way into a $t$-free cumulant of order $n$.
When $\sigma$ is a permutation of $\{1,\ldots,n\}$, let us denote by $[\sigma]$ the partition of the integer $n$ given by the lengths of the cycles of $\sigma$.
Let $c$ be a $t$-free cumulant of order $n$. In order to check that (\[part avec 1\]) is satisfied, let us choose $m_1,\ldots,m_{n-1}$ in some probability space $({\mathcal}{M},\varphi)$ and write the fact that $k_n(m_1,\ldots,m_{n-1},1)=0$. We find $$\label{mu 1}
\sum_{\lambda\vdash n}c_\lambda \sum_{\substack{\sigma \in \Sym_n\\ [\sigma]=\lambda}} \varphi_\sigma(m_1,\ldots,m_{n-1},1)=0.$$ Let $r_n:\Sym_n\to \Sym_{n-1}$ denote the following function: for all $\sigma\in \Sym_n$, $r_n(\sigma)$ is the permutation of $\{1,\ldots,n-1\}$ obtained by removing $n$ from the cycle of $\sigma$ which contains it. For each $\sigma$, we have the equality $\varphi_\sigma(m_1,\ldots,m_{n-1},1)=\varphi_{r_n(\sigma)}(m_1,\ldots,m_{n-1})$. Now a permutation $\tau\in \Sym_{n-1}$ has exactly $n$ preimages by $r_n$. Moreover, if $[\tau]=\mu=(\mu_1\geq \ldots \geq \mu_{\ell(\mu)}>0) \vdash n-1$, then all preimages of $\tau$ belong to one of the conjugacy classes $\mu+\delta_i$ for $i=1,\ldots,\ell(\mu)+1$. Finally, $r_n^{-1}(\tau)$ contains exactly one element of $\mu+\delta_{\ell(\mu)+1}$ and $\mu_i$ elements of $\mu+\delta_i$ for $i=1,\ldots,\ell(\mu)$. We can thus rewrite (\[mu 1\]) as follows: $$\label{mu 2}
\sum_{\mu\vdash n-1} \left(\sum_{i=1}^{\ell(\mu)} \mu_i c_{\mu+\delta_i} + c_{\mu+\delta_{\ell(\mu)+1}} \right) \sum_{\substack{\tau \in \Sym_{n-1}\\ [\tau]=\mu}} \varphi_\tau(m_1,\ldots,m_{n-1})=0.$$ Since the distribution of $(m_1,\ldots,m_{n-1})$ is arbitrary, all the coefficients between the brackets must vanish. It follows that (\[part avec 1\]) is satisfied.
Conversely, let $(c(\sigma))_{\sigma\in \Sym_{n}}$ be a collection which satisfies (\[cum 1\]) for centered elements and (\[part avec 1\]). Then, by the computation that we have just done, this collection satisfies (\[cum 2\]) and hence, by multilinearity, (\[cum 1\]) for arbitrary elements.
Let us prove the last assertion. For any $\lambda\vdash n$ with at least one part equal to $1$, the relation (\[part avec 1\]) expresses the value of $c_\lambda$ in terms of $c_{\lambda'}$ for partitions $\lambda'$ of $n$ which have strictly less parts equal to $1$ than $\lambda$. The collection $(c_\lambda)_{\lambda\vdash n}$ is thus completely and uniquely determined by $(c_\lambda)_{\lambda \vdash\!\!\vdash n}$. The fact that the resulting collection is a $t$-free cumulant is granted by the first part of the proposition.
The last result simplifies greatly the search for $t$-free cumulants, since it allows one to restrict to centered elements and partitions in parts at least equal to $2$. We apply it to find cumulants of order less than $6$.
(Theorem \[existence\]) Let us prove that there exist $t$-free cumulants up to order $6$. We proceed by first establishing enough conditions that their coefficients must satisfy, in order to determine these coefficients. Then, we check that we actually have a $t$-free cumulant.
We will always normalize our cumulants by the condition $c_n=1$.
${\scriptstyle\bullet} \; n=2.$ By Proposition \[determine\], the condition $c_2=1$ suffices to determine the whole cumulant, and $c_{11}=-1$. The relation (\[eddie.vedder.11.9.08\]) implies that we have indeed got a $t$-free cumulant.
${\scriptstyle\bullet} \; n=3.$ Again, the condition $c_3=1$ determines completely the cumulant. Using (\[part avec 1\]), we find $c_{21}=-2$ and $c_{111}=4$. The relation (\[eddie.vedder.11.9.08\]) implies again that the collection thus obtained is a $t$-free cumulant. Indeed, by , the product of any three centered elements, one being $t$-free with the two others, is centered. Hence, our collection satisfies (\[cum 1\]) on centered elements.
${\scriptstyle\bullet} \; n=4.$ This is the first case where the relation $c_4=1$ does not suffice determine the cumulant. Indeed, we must compute $c_{22}$. For this, let us choose in some probability space elements $a_1,a_2,\ldots$ and $b_1,b_2,\ldots$, such that $\{a_{1},a_{2},\ldots\}$ and $\{b_{1},b_{2},\ldots\}$ are $t$-free. We will use this notation again in this proof without redefining it. Let us assume that a $t$-free cumulant $c$ of order $4$ is given and let us compute $D_{c}(\{\{1,2\}\},\{\{3,4\}\})$ (see (\[def D\])). Again, we shall use formulae and several times, without citing them every time. There are $4$-cycles which contribute to this coefficient. In $2!2!$ of them, $1$ and $2$ are consecutive and they contribute for $1$ each. In $2!$ others, $1$ and $2$ are not consecutive and each such cycle contributes for $e^{-2t}$. There is also one product of two $2$-cycles, which contributes for $c_{22}$. Finally, $D_{c}(\{\{1,2\}\},\{\{3,4\}\})=c_{22}+2(2+e^{-2t})$. The nullity of this coefficient implies $c_{22}=-2(2+e^{-2t})$. Using (\[part avec 1\]), we determine the remaining coefficients, and find $$\begin{aligned}
c_4=1, \; c_{31}=-3,\; c_{22}=-2(2+e^{-2t}), \; c_{211}=2(5+e^{-2t}), c_{1111}=-6(5+e^{-2t}).\end{aligned}$$ Now let us check that the collection thus defined satisfies (\[cum 1\]) for centered elements. Set $k_{4}=\sum_{\sigma} c(\sigma)\varphi_{\sigma}$. If we expand $k_{4}(a_{1},b_{1},b_{2},b_{3})$ according to (\[expand\]), then all terms involve $\varphi(a_{1})$ and vanish. Now $k_{4}(a_{1},a_{2},b_{1},b_{2})$ also vanishes, because this is how we have chosen the value of $c_{22}$. Finally, we do have a $t$-free cumulant of order $4$.
${\scriptstyle\bullet} \; n=5.$ Let $c$ be a $t$-free cumulant of order $5$. Let us compute $c_{32}$ by writing the nullity of $D_{c}(\{\{1,2,3\}\},\{\{4,5\}\})$ and using formulae and . There are $3!2!$ $5$-cycles in which $4$ and $5$ are consecutive, and they contribute for $1$ each. There are also $3!2!$ $5$-cycles in which they are not consecutive, and each contributes for $e^{-2t}$. There are finally $2!$ products of a $3$-cycle and a $2$-cycle, which contribute for $c_{32}$ each. Hence, we must have $c_{32}=-6(1+e^{-2t})$. Using as usual (\[part avec 1\]), we find that the other values of $c$ must be $$\begin{aligned}
&c_5=1, \; c_{41}=-4, \; c_{32}=-6(1+e^{-2t}), \; c_{311}=6(3+e^{-2t}), c_{221}=12(1+e^{-2t}), \\
& c_{2111}=-12(5+2e^{-2t}), \; c_{11111}=48(5+2e^{-2t}).\end{aligned}$$ The fact that $k_{5}\sum_{\sigma} c(\sigma)\varphi_{\sigma}$ is a cumulant is checked just as in the case $n=4$: the identity $k_{5}(a_{1},b_{1},b_{2},b_{3},b_{4})=0$ is granted by (\[expand\]) and $k_{5}(a_{1},a_{2},b_{1},b_{2},b_{3})=0$ by the choice of $c_{32}$.
${\scriptstyle\bullet} \; n=6.$ Let $c$ be a $t$-free cumulant of order $5$. The value of $c_{42}$, deduced as usual from the nullity of $D_{c}(\{\{1,2,3,4\}\},\{\{5,6\}\})$, is $c_{42}=-4(2+3e^{-2t})$. Similarly, $D_{c}(\{\{1,2,3\}\},\{\{4,5,6\}\})=0$ gives us $c_{33}=-3(3 + 6 e^{-2t} + e^{-3t})$. Finally, $D_{c}(\{\{1,2\}\},\{\{3,4\},\{5,6\}\})=0$ implies $c_{222}=8(7+17e^{-2t}+6e^{-4t})$. The other coefficients follow as usual from (\[part avec 1\]) and we find $$\begin{aligned}
&c_{6}=1, \; c_{51}=-5, \; c_{42}=-4(2+3e^{-2t}), \; c_{411}=4(7+3e^{-2t}), \; c_{33}=-3(3+6e^{-2t}+e^{-3t})\\
&c_{321}=6(7+e^{-3t}+12e^{-2t}), \; c_{3111}=-12(14+15e^{-2t}+e^{-3t}), \;c_{222}=8(7+17e^{-2t}+6e^{-4t})\\
&c_{2211}=-8(28+53e^{-2t}+3e^{-3t}+6e^{-4t}), \; c_{21111}=48(21+34e^{-2t}+2e^{-3t}+3e^{-4t})\\
&c_{111111}=-240(21+34e^{-2t}+2e^{-3t}+3e^{-4t}). \end{aligned}$$ Let us set $k_{6}=\sum_{\sigma} c(\sigma) \varphi_{\sigma}$. The nullity of $k_{6}(a_{1},b_{1},\ldots,b_{5})$ follows as usual from (\[expand\]). That of $k_{6}(a_{1},a_{2},b_{1},b_{2},b_{3},b_{4})$ results from the choices of $c_{42}$ and $c_{222}$. Finally, $k_{6}(a_{1},a_{2},a_{3},b_{1},b_{2},b_{3})=0$ is granted by the choice of $c_{33}$.
Nowhere there has been any freedom in the definition of the cumulants. This shows the uniqueness of conjugation-invariant $t$-free cumulants of order at most than $6$.
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[^1]: A non-commutative space $({\mathcal}{M},\tau)$ is said to be [*faithful*]{} if for all $x$ in ${\mathcal}{M}\setminus\{0\}$, $\tau(xx^*)>0$. Any non-commutative space can be quotiented by a bilateral ideal into a faithful space.
[^2]: The Hilbert space $L^2({\mathcal}{M}, \tau)$ is the completion of ${\mathcal}{M}$ with respect to the inner product ${\langle}x,y{\rangle}=\tau(xy^*)$.
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abstract: 'We analyze the one parameter family of tronquée solutions of the Painlevé equation ¶1 in the pole-free sectors together with the region of the first array of poles. We find a convergent expansion for these solutions, containing one free parameter multiplying exponentially small corrections to the Borel summed power series. We link the position of the poles in the first array to the free parameter, and find the asymptotic expansion of the pole positions in this first array (in inverse powers of the independent variable). We show that the tritronquées are given by the condition that the parameter be zero. We show how this analysis in conjunction with the asymptotic study of the pole sector of the tritronquée in [@inprep] leads to a closed form expression for the Stokes multiplier directly from the Painlevé property, not relying on isomonodromic or related type of results.'
address:
- |
Mathematics Department\
The Ohio State University\
Columbus, OH 43210
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Mathematics Department\
The Ohio State University\
Columbus, OH 43210
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Mathematics Department\
The University of Chicago, IL 60637
author:
- 'O. Costin, R.D. Costin and M. Huang'
title: 'Tronquée solutions of the Painlevé equation ¶1'
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$ $ -2cm
Introduction
============
The Painlevé equation ¶1 and its tronqueé solutions {#sec1}
----------------------------------------------------
The Painlevé equation ¶1 $$\label{p1}
y''= 6y^2 + z$$ has a five-fold symmetry: if $y(z)$ solves , then so does $\rho^2y(\rho z)$ if $\rho^5=1$.
Relatedly, there are five special directions of (see also Note\[SpecDir\]) which border the sectors $$\label{eq:sectorsz}
S_k=\left\{z\in\CC\, \Big|\, \frac{2k-1}{5}\pi
<\arg z<\frac{2 k+1}{5}\pi \right\},\ k\in\ZZ_5$$ Generic solutions have poles accumulating at $\infty$ in all $S_k$. Any solution has poles in at least one $S_k$ [@KitaevKapaev]. For any [*two adjacent*]{} sectors $S_k$ there is a one-parameter family of solutions, called [*[tronquée]{}*]{} solutions, with the behavior $y=\pm i\frac{\sqrt{z}}{\sqrt{6}}(1+o(1))$ as $z\to\infty$ in these sectors, and moreover, they are asymptotic to a divergent power series $$\label{tro_ser}
y(z)\sim \pm i\frac{\sqrt{z}}{\sqrt{6}}\left(1+\sum_{n=1}^\infty\frac{a_n}{z^{5n/2}}\right)\equiv\tilde{y}_0(x)$$ whose coefficients $a_n$ are uniquely determined; the free parameter indexing the tronquée solutions is not visible in their asymptotic behavior .
In any set of [*four*]{} sectors $S_k$ there is exactly one solution with behavior , see [@Kapaev-2004], [@Masoero]; these particular tronquée solutions, which are maximally regular solutions (and have poles in only one sector), are called [*tritronquées*]{}. The terms “tronquée”, “bitronquée”, and “tritronquée” together with the corresponding solutions to P1 were first introduced in the pioneering work by Boutroux [@Boutroux]. For an overview of the asymptotic behavior of solutions of ¶1 equation, and a wealth of references, see [@Clarkson].
The tronquée solutions we study satisfy $$\label{newsectrinz}
y(z)=i{\frac{\sqrt{z}}{\sqrt{6}}}\,\left(1+o(1)\right)\ \text{as $|z|\to \infty$ with }\arg z\in
\left[-\frac{3\pi}{5},\frac{\pi}{5}\right]$$ and are analytic for large $z$ in this sector. The other families of tronqueés are obtained from it by the five-fold symmetry.
Main results
============
Overview
--------
In Theorem\[asir2\] we establish a one-parameter convergent representation of tronqueé solutions valid in the pole free sector, using results in [@imrn] and [@duke]. The natural parameter is the constant multiplying the exponential “beyond all orders” of . We show that if the constant is zero, then the solution is a tritronquée solution and describe the Stokes phenomenon for these solutions in Theorem\[TritCis0\]. We find an asymptotic expansion uniformly valid throughout the sector in as well as in a region containing the first array of poles, (Theorem \[FAPIn\]) and determine the position of the poles in the first array to $O(x^{-4})$ in Proposition \[RegP\]. These results are obtained using transseries representations of solutions; an overview of this topic is found in the Appendix, §\[IntroTran\]-\[historic\].
One of the most significant developments in a century of study of Painlevé equations is the isomonodromic approach (and related ones), originating in [@Fuchs], [@Garnier], [@Jimbo-Miwa-Ueno], [@Flaschka] and further developed [@Clarkson], [@Deift], [@Deift1], [@joshi], [@Fokas]. A question is whether a complete asymptotic description and explicit connection formulae are a direct consequence of the Painlevé property, or more structure is needed, e.g., the existence of an isomonodromic representation; this paper together with [@inprep] are a step toward a positive answer to this question. In §\[find\] we show that the transseries representation together with a new method to describe solutions in singular regions [@inprep] allows for a closed form calculation of the Stokes multiplier by direct asymptotic methods using the meromorphicity of solutions (not relying on isomonodromic-type methods).
A convergent one-parameter representation of tronqueé solutions
---------------------------------------------------------------
We normalize as described in [@invent]: the following refinement of the Boutroux substitution $$\label{chvarx}
z={24}^{-1}{30^{4/5}}x^{4/5}e^{-\pi i /5};\ y(z)=i\sqrt{z/6}(1-\tfrac{4}{25}x^{-2}+h(x))$$ where the branch of the square root is the usual one, which is positive for $z>0$, brings to the Boutroux-like form $$\label{eq:eqp}
h''+\frac{h'}{x}-h-\frac{h^2}{2}-\frac{392}{625}\,\frac{1}{ x^4}=0$$ [ This normalization is associated with an interesting maximal regularity property, see Proposition \[RegP\] below.]{}
A sector $S_k$ in $z$ corresponds, after the normalization , to a quadrant in $x$, and the sector $-\pi<{\rm{arg}}\ z\leq \pi$ corresponds to the sector $-\pi<{\rm{arg}}\ x\leq 3\pi/2$. The fact that the solutions of ¶1 are meromorphic implies that the solutions of (which have a branch point at $x=0$) return to the same values after analytic continuation around $0$ by an angle of $\tfrac{5\pi}{2}$.
Relatedly, is invariant under the transformations $h(x)\mapsto h(xe^{\pm i\pi})$ and under the conjugation symmetry $h(x)\mapsto\overline{h(\overline{x})}$.
The tronqueé solutions correspond to solutions $h$ satisfying $$\label{asyht}
h(x)=o(1)\ \text{as $x\to \infty$ with }\arg x\in \left[-\frac{\pi}{2},\frac{\pi}{2}\right]$$
Convergent expansions are obtained using general formal solutions described in Proposition\[Formh\] (see §\[IntroTran\]-\[Sbeta\] for a brief introduction to transseries solutions and their correspondence to actual solutions).
\[Formh\][*Formal small solutions of the normalized ¶1.*]{}
\(i) There is a unique power series solution of which is $o(1)$ as $x\to\infty$, and it has the form \_0(x)= \_[k=4;k]{}\^c\_k x\^[-k]{}, c\_4=[-]{}
\(ii) Transseries solutions of have the form (x)=\_[0]{}(x)+\_[k1]{}C\^ke\^[-kx]{}\_[k]{}(x) x(-,) where $\tilde{h}_{k}(x)=x^{-k/2}\tilde{t}_{k}(x)$, with $\tilde{t}_{k}(x)$ a nonnegative integer power series in $x^{-1}$, and (x)=\_[0]{}(x)+\_[k1]{}C\^ke\^[kx]{}\_[k]{}(x) x(-,)i where $\tilde{\tilde{h}}_{k}(x)=x^{-k/2}\,e^{\mp ik\pi/2}\,\tilde{t}_{k}(-x)$.
The proof of Proposition\[Formh\] is found in §\[PfFormh\].
We note that $\tilde{h}_n(x)$ are linked, via , to the $n$-instanton of ¶1 [@Its_G_K].
[**[Notations.]{}**]{} In the following the Laplace transform of $Y(p)$ in the direction $e^{i\phi}$ is defined as \_Y(x)=\_[e\^[i]{}\_+]{} [[e]{}]{}\^[-px]{} Y(p) dp where, by convention, $\phi=-\arg x$. The convolution is defined as $(f*g)(p)=\int_0^p f(s)g(p-s)ds$. The Borel transform of a series is, as usual, (x\^[-]{}\_[n=0]{}\^c\_nx\^[-n]{})=p\^\_[n=0]{}\^c\_[n]{} >0 (For an introduction to the Borel transform and Borel summation see §\[LBsum\].)
Next theorem describes the tronqué solutions in coordinates .
\[asir2\] I. Assume $h$ solves and satisfies $$\label{assph}
h(x)=o(1)\ \text{as }
x\to\infty\text{ with } \arg(x)\in (-\tfrac{\pi}2,\tfrac{\pi}{2})$$ Then
\(i) We have $h\sim \tilde{h}_0$ as $x\to +\infty$ and the asymptotic expansion is differentiable.
Also, there exist constants $C_\pm$ so that $$\begin{array}{l}
h(x)\sim C_+x^{-1/2}e^{-x}\ \text{as}\ x\to +i\infty, \\ \\
h(x)\sim C_-x^{-1/2}e^{-x}\ \text{as}\ x\to -i\infty
\end{array}$$
\(ii) Let $$\label{defHk}
H_k=\mathcal{B}\tilde{h}_k$$ be the Borel transforms of the series in the transseries solution .
Then $h(x)$ has the Borel summed transseries representations: $$\label{eq:transh}
h(x)=\left\{
\begin{array}{l}
\mathcal{L}_\phi H_0\,(x)+ \sum_{k=1}^{\infty} C_+^k e^{-kx}\,\mathcal{L}_\phi H_k\,(x)\ \ \ \text{for }
- \phi= \arg x\in(0,\tfrac{\pi}{2}]\\ \\
\mathcal{L}_\phi H_0\,(x)+\sum_{k=1}^{\infty} C_-^k e^{-kx}\,\mathcal{L}_\phi H_k\,(x)\ \ \ \text{for }
-\phi= \arg\, x\in[-\tfrac{\pi}{2},0)
\end{array}\right.$$ where $\mathcal{L}_\phi H_k$ are analytic for large $x$ with $\arg x\in (-\pi/2, 3\pi/2)$ and the series converge for $|x|$ large enough with $0<|\arg x|\Le\tfrac{\pi}{2}$; $\mathcal{L}_\phi H_k=O(x^{-k/2})$ in these regions.
II\. Similar Borel summed transseries representations hold for small solutions in the sectors $\arg(x)\in (-\tfrac{\pi}2,\tfrac{\pi}{2})\pm i\pi$, with $e^{-kx}$ replaced by $e^{kx}$, all $H_k(\cdot)$ with $k\Ge 1$ replaced by $-H_k(\cdot e^{\pm i\pi})$ and with the constants $C$ changing only at the Stokes lines $\arg x=\pm \pi$.
The proof of Theorem\[asir2\] is found in §\[PfTasir2\].
Description of the Stokes phenomenon {#S2.2}
------------------------------------
The proposition below links the Stokes constant $C_+-C_-$ in to the leading behavior of $H_0=\mathcal{B}\tilde{h}_0$ at $p=1$.
\[H0\_behave\]
\(i) Near $p=1$ $H_0(p)$ has the form ${H}_0(p)=S\,(1-p)^{-1/2}{A}(p)+{B}(p)$ with ${A}(p),\, {B}(p)$ analytic at $p=1$, ${A}(p)=1+O(p-1)$ and $S$ is a constant.
\(ii) Denote $$\label{defhpm}
\displaystyle{\begin{array}{l} h^+(x)=\mathcal{L}_{\phi}H_0\,(x)\ \text{for }-\phi=\arg(x)\in(0,\tfrac{\pi}{2})\\ \\
h^-(x)=\mathcal{L}_{\phi}H_0\,(x)\ \text{for }-\phi=\arg(x)\in(-\tfrac{\pi}{2},0)
\end{array} }$$
Then $h^+(x)$ can be analytically continued for large $x$ with $\arg x\in[-\tfrac{\pi}{2},\pi]$, $h^-(x)$ can be analytically continued for large $x$ with $\arg x\in[-\pi,\tfrac{\pi}{2}]$ and $$\label{jumph01}
h^+(x)-h^-(x)= -\mu e^{-x}x^{-1/2}(1+o(1))\ {\rm{as\ }}x\to+\infty\ \ \text{where }\ \ \mu=2iS\sqrt{\pi}$$ with $S$ defined in (i).
\(iii) The constants $C_\pm$ in satisfy $C_+-C_-=-\mu$ with $\mu$ as in .
The proof of [Proposition]{}\[H0\_behave\] is found in §\[PfP3\].
[The [*Stokes constant*]{} $\mu$ was calculated in closed form, $ \mu=i\sqrt{\frac{6}{5 \pi }}$ first by Kapaev using the method of isomonodromic deformations [@Kapaev] (some corrections were made in [@KitaevKapaev]). The [*existence*]{} a constant $\mu$ (independent of $C_\pm$) such that $C_--C_+=\mu$ is known for a wide class of differential equations, see formula following (1.15) in [@imrn], and also [@duke], . An [*[explicit expression]{}*]{} for $\mu$ is expected only in special cases such as integrable equations. The explicit value also follows, without isomonodromic-type methods from the asymptotic analysis in this paper together with [@inprep].]{}
Arrays of poles near regular sectors of tronquée solutions {#firstpoles}
----------------------------------------------------------
Theorem\[FAPIn\] describes the asymptotic behavior of tronquée solutions for large $x$ in the pole free sectors together with the region of the first array of poles: and give a uniform expansion of the tronquées in the sector of analyticity up to and including the first array of poles near $i\RR^+$. Based on this, Proposition\[RegP\] gives the position of these poles, and the way it depends on the value of the constant beyond all orders $C_+$.
\[FAPIn\]
\(i) [@invent] Let $h(x)$ be a solution as in Theorem\[asir2\]I with [$C_+\ne 0$.]{} Denote $$\label{eq:eqdefxi}
\xi=C_+x^{-1/2}e^{-x}$$ Then the leading behavior $h(x)$ for large $|x|$ with ${\rm{arg}}\, x$ close to $\tfrac{\pi}{2}$ is $$\label{eq:eq51}
h\sim F_0(\xi)+\frac{F_{1}(\xi)}{x}+\frac{F_{2}(\xi)}{x^2}+\cdots\ \ (|x|\to \infty,\ x\in\mathcal{D}_{x})$$ where $$\label{regZx}
{ \mathcal{D}_{x}=\left\{x\in\CC\,|\, |x|>R,\,\arg x\in (-\tfrac{\pi}{2}+\delta,\tfrac{\pi}{2}+\delta),\,|\xi(x)-12|>\epsilon,\, |\xi(x)|<\epsilon^{-1}\right\} }$$ for any $\delta,\epsilon>0$ small enough and $R=R(\epsilon,\delta)$ large enough, and where $$\label{eq:f11}
F_0(\xi)= \tfrac{144\, \xi}{(\xi-12)^2},\, F_{1}(\xi)=\tfrac{
\frac{1}{60}\xi^4-3\xi^3-210\xi^2
-216\xi}{(\xi-12)^3},\ldots,\, F_{n}(\xi)=\tfrac{P_n(\xi)}{(\xi-12)^{n+2}}$$ with $P_n$ polynomials of degree $2n+2$ for $n\Ge 1$.
\(ii) An expansion similar to holds for $g=h(1+\frac13h)^{-1}$ in the region $\mathcal{D}'_x$ given by with $12$ replaced by $-12$: $$\label{eq:devh1}
g\sim G_0(\xi)+\frac{1}{x}G_1(\xi)+\frac{1}{x^2}G_2(\xi)+\cdots$$ with $$G_0(\xi)=\frac{144\xi}{(\xi+12)^2},\ \ G_1(\xi)=-\frac{1}{60}\frac{\xi (\xi-12)(\xi^3-180\xi^2-12600\xi-12960)}{(\xi+12)^4}$$ At any point in the region $|\xi(x)|<\epsilon^{-1}$ one, or both of the expansions , holds.
\(iii) The symmetry $h\to\overline{h({\overline{x}}})$ implies that a similar representation exists in the fourth quadrant with $C_-$ instead of $C_+$.
The proof is found in §\[PfRem8\]. The counterpart of expansion in the original variables of P$_I$ is given in [@invent].
\[RegP\] (i) Any solution as in Theorem\[asir2\] with $C_+\ne 0$ in is analytic for large $x$ in the region $|\xi(x)|<12$ and has a first array of poles near the curve $\xi(x)=12$, located at $$\label{eq:pospoles}
x_n=2n\pi i+L-\frac{\frac{109}{120}+\frac12 L}{2 n\pi i}+\frac{\frac{4699}{2400}+\frac{139}{120}L+\frac14L^4}{(2 n\pi i)^2}\\ - \frac{\frac{41402111}{6480000}-\frac{899}{200}L-\frac{77}{60}L^2-\frac16 L^3}{(2 n\pi i)^3}+O(n^{-4})$$ for $n\to\infty$, where $ L=\displaystyle \ln\(\frac{C^+}{12(2 n\pi i)^{1/2}}\)$.
\(ii) The following maximal regularity property holds. If $\mathcal{A}$ is an analytic function tangent to the identity, $\mathcal{A}(0)=0,\mathcal{A}'(0)=1$, then $\mathcal{A}(h(x))$ is singular for some large $x$ in the region $|\xi(x)|<R$ if $R>12$.
The proof of Proposition\[RegP\] is found in §\[PfPr10\].
[*Note:*]{} rotating $x$ further into the second quadrant, $h$ develops successive arrays of poles separated by distances $O(\ln x)$ of each other as long as $\arg(x)=\tfrac{\pi}{2}+o(1)$ [@invent].
Tritronqué solutions
--------------------
The following theorem characterizes tritronqué solutions as the tronquées with a zero constant beyond all orders and establishes a representation as convergent series in all four pole free sectors.
\[TritCis0\] (i) The tritronqueé solution $y_t$ of with $$\label{sectorinz}
y_t(z)=i\sqrt{\frac{z}{6}}\,\left(1+o(1)\right)\ \text{as $|z|\to \infty$ with }\arg z\in
\left(-\frac{3\pi}{5},\pi\right)$$ is the unique solution analytic for large $x$ in some sector in the first quadrant, having $C_+=0$ in its representation .
\(ii) Define $\hsig=\mathcal{L}_\phi H_0$ for $-\phi=\arg x\in(\pi,\tfrac{3\pi}{2})$ and let $h^+(x), h^-(x)$ be as in .
Then $h^+$ and $\hsig$ can be analytically continued to $\arg x=\pi$ and $$\label{jumphpi}
h^+(x)-\hsig(x)= \mu e^{-|x|}|x|^{-1/2}(1+o(1))\ {\rm{for\ }}x\to-\infty$$ with $\mu$ as in .
\(iii) We have $$\label{ht_trans}
h_t(x)=\left\{
\begin{array}{l}h^+(x)
\ \ \ \text{for }
\arg x=- \phi\in(0,\pi)\\ \\
h^-(x)+\sum_{k=1}^{\infty} (-\mu)^k e^{-kx}\,\mathcal{L}_\phi H_k\,(x)\ \ \ \text{for }
\arg x=- \phi\in[-\tfrac{\pi}{2},0)\\ \\
\hsig(x)+\sum_{k=1}^{\infty} (i\mu)^k e^{kx}\,\mathcal{L}_\phi H_k\,(x)\ \ \ \text{for }
\arg x=- \phi\in(\pi,\tfrac{3\pi}{2})
\end{array}\right.$$
The proof of Theorem\[TritCis0\] and the choice of branches are given in §\[PfTritCis0\].
\(i) The uniqueness of $h_t$ and the symmetry of the equation imply $h_t(x)=\overline{h_t(-\overline{x})}$, and the last expansion in follows from the middle one, symmetry and the choice of branches.
\(ii) Also by symmetry, we see that $C_-=0$ corresponds to tritronquée solutions which are pole free in the sector $\arg z\in
(-\frac{7\pi}{5},\frac{\pi}{5})$.
[$y_t$ is now known to be pole free in the whole sector, up to the origin ([*[Dubrovin’s conjecture]{}*]{} holds); more precisely, $y_t$ is analytic in a ball near the origin and in the closed sector $\{z:||z|>0, \arg z \in [-3\pi/5,\pi]\}$, [@DubrovinC]]{}.
Calculating the constant beyond all orders from the values of the tronquée
--------------------------------------------------------------------------
The tronquée solutions are distinguished by the constant $C_\pm$ beyond all orders. We recall the following result which allows for the calculation of $C_{\pm}$ using the actual solution.
[@OC-MDK] The constant $C_\pm$ in satisfies $$\label{eq:eqCpm}
C_\pm=\lim_{\begin{subarray}{c}x\rightarrow\infty\\\arg(x)=-\phi\end{subarray}}\, e^x\, x^{1/2}\left(h(x)-\sum_{{k\le |x|}}\frac{c_{k}}{x^k}\right)$$ (see ) for $x$ in the corresponding quadrant.
In the direction $\arg x=0$ the limit is $\tfrac{1}{2}C_++\tfrac{1}{2}C_-$.
Application: finding the Stokes multiplier {#find}
==========================================
Let $y_t$ be the tritronquée of Theorem\[TritCis0\]. The continuation of $y_t(z)$ for $\arg z$ from $-\tfrac{3\pi}{5}$ up to $-\pi$, through the pole sector, is given with all technical details in [@inprep]. We describe below the method used, the intuition and formal calculations behind [@inprep].
After normalization, $y_t(z)$ corresponds to $h_t(x)$ analytic for large $x$ in the sector $\arg x\in(-\tfrac{\pi}{2},\tfrac{3\pi}{2})$ and having poles in the sector $$\label{def-Sigma}
\Sigma=\{x\, |\, -\pi<{\rm{arg}}\, x<-\pi/2\}$$ The first array of poles near the edges of this sector is described in §\[firstpoles\].
We show that in the pole region $h_t$ can be described by constants of motion which are valid starting with $\arg x$ close to $-\tfrac{\pi}{2}$ (the first array of poles) up to $\arg x$ close to $-\pi$, close to another “first” array of poles, where $h_t$ can be again [*matched to a transseries*]{}; the asymptotic expansions of the constants of motion that we obtain depend explicitly on the constant beyond all orders $\mu$. The transseries representation of $h_t$ also depends on $\mu$ in a way visible in the leading order asymptotics when $\arg x=-\pi$ or $3\pi/2$, cf. Theorem \[asir2\] (we note that both arguments of $x$ correspond to $z\in\RR^-$).
The connection problem
----------------------
The solution $y_t$ is meromorphic; this was known since Painlevé, and proving meromorphicity does not require a Riemann-Hilbert reformulation, see e.g. [@OCRDCP1; @Hinkkanen; @Gromak] for direct proofs and references. Starting with a large $z\in\RR^+$ we analytically continue $y_t$ (i) anticlockwise on an arccircle until $\arg z=\pi$ and (ii) clockwise on an arccircle until $\arg z=-\pi$. The continuation (ii) traverses the pole sector, $\arg z\in (-\pi,-3\pi/5)$. Because of the above-mentioned meromorphicity, we must have $$\label{eq:mer1}
y_t(|z|e^{i\pi})=y_t(|z|e^{-i\pi})$$ In variables this corresponds to the following. We start with large $x$ with $\arg x=\tfrac{\pi}{4}$ and (i’) analytically continue $h_t(x)$ anticlockwise, until $\arg
x=\tfrac{3\pi}{2}$, and also (ii’) clockwise, until $\arg x=-\pi$. The single-valuedness equation and eq. imply $$\label{hpm}
h_t(|x|e^{3\pi i/2}) = -h_t(|x|e^{-\pi i})-2+\frac{8}{25|x|^2}$$ Eq. implies a nontrivial equation for $\mu$, below, explained in §\[polreg\], and which determines $\mu$ uniquely. The fact that $\mu$ is uniquely determined relates to the fact that there is only one solution, the tritronquée, with algebraic behavior in the region , cf. [@invent], Proposition 15. Relatedly, the last expansion in is not used in the calculation.
Asymptotics of the tronqueés in the pole region $\arg x\in (-\pi,-\pi/2)$: heuristics {#polreg}
-------------------------------------------------------------------------------------
Our calculations rely on the construction of two functionally independent adiabatic invariants. In a nutshell, an adiabatic invariant is a conserved quantity given as an asymptotic expansion with controlled errors (the expansion here is in powers of $1/x$.) Adiabatic invariants are useful when dealing with small perturbations of integrable systems (usually in the stronger sense of explicit integrability). In our problem, for large $x$, ¶1 is treated as a small perturbation of the elliptic equation $f''-f-f^2/2=0$. Relying on adiabatic invariants is reminiscent of KAM techniques: indeed, conserved quantities are the complex-analogs of action-angle variables. To our knowledge the adiabatic invariants (48) and (50), refined in [@inprep] are new. Their construction is facilitated by the Boutroux “cycle” technique [@Boutroux].
Note that for large $x$ is close to the autonomous Hamiltonian system $$\label{eq:eqell}
h''-h-h^2/2=0\ \ \text{with Hamiltonian $s/2$}$$ where $$\label{def_s}
s={h'}^2-h^2-h^3/3$$ The solutions of are elliptic functions, doubly periodic in $\CC$. For we expect solutions to be asymptotically periodic, and $s$ to be a slow varying quantity; this is certainly the case in the region where holds. It is then natural to take $h=:u$ as an independent [*angle-like*]{} variable and treat $s$ and $x$ as dependent variables. With $w=u'$ we first rewrite equation (\[eq:eqp\]) as a system $$\begin{aligned}
\label{eqqq1}u' &=w\\
\label{eqqq2}w' &=u+\frac{u^2}{2}-\frac{w}{x}+\frac{392}{625}\,\frac{1}{ x^4}\end{aligned}$$ and then, with $$\label{def_R}
R(u,s)=\sqrt{u^3/3+u^2+s}$$ we transform , into a system for $s(u)$ and $x(u)$: $$\begin{aligned}
\label{syst2}
&\frac{ds}{du}=-\frac{2w}{x}+\frac{784}{625}\,\frac{1}{x^4}=-\frac{2R(u,s)}{x}+\frac{784}{625}\,\frac{1}{x^4}\\
\label{syst3}
&\frac{dx}{du}=\frac{1}{w}=\frac{1}{R(u,s)}
\end{aligned}$$
Let $h$ be a tronquée solution of having poles for large $x$ in the sector $\Sigma$ in . It turns out that there are closed curves $\mathcal{C}$, similar to the classical [*cycles*]{} [@Kitaev], such that $R(u,s(u))$ does not vanish on $\mathcal{C}$ and $x(u)$ traverses $\Sigma$ from edge to edge as $u$ travels along $\mathcal{C}$ a number $N$ times.
Written in integral form, and become $$\begin{aligned}
\label{eq:eqds2n}
& s(u)=s_n-2\int_{u_n}^u \(\frac{R(v,s(v))}{x(v)}-\frac{392}{625}\frac{1}{x(v)^4}\)dv \\
& x(u)=x_n+\int_{u_n}^u \frac{1}{R(v,s(v))}dv \label{eq:eqdx2n}\end{aligned}$$ where the integrals are along the path $\mathcal{C}$ and we write $u_n$ to denote that $u$ has traveled $n$ times along $\mathcal{C}$, and $s_n=s(u_n)$ and $x_n=x(u_n)$.
### Initial conditions and iteration {#IC1}
We take initial values $x_0,\, s_0$ so that $|x_0|$ is large and close to the first array of poles (see §\[firstpoles\]) i.e. $\arg(x_0)=-\pi/2(1+o(1))$ for large $|x_0|$, and $s_0$ sufficiently small; $x_0,s_0$ are arbitrary otherwise.
\[IC2\] The parameters $x_0, s_0$ are free constants in an open set in $\CC^2.$ By Theorem \[FAPIn\] (iii), and we see that for the tronquées $x_0, s_0$ depend on the constant beyond all orders $C_-$.
Starting with $u_0\in\mathcal{C}$ the following hold[^1]. For some $N=N_{m}(x_0)$ of the order $|x_0|$, $x_N$ is close to the last array of poles (i.e. $\arg(x_N)=-\pi(1+o(1))$), and $|x_n|$ is of the order $|x_0|$ for all $n\leq N$. Also, two roots of $R(u_n,s_n),n=0,1..,N$ are in the interior of $\mathcal{C}$ and a third one is in its exterior.
To establish these facts, an important ingredient in [@inprep] is the study of the Poincaré map for , , namely the study of $(s_{n+1},x_{n+1})$ as a function of $(x_n,s_n)$. The Poincaré map is used to eliminate the fast evolution. The asymptotic expansions of $s(u)$ and $x(u)$ when $u$ is between $u_n$ and $u_{n+1}$ are straightforward local expansions of and . We denote $$\label{eq:eqJL}
\JN (s)=\oint _\mathcal{C} R(v,s)\,dv;\ \ \LN(s)=\oint_\mathcal{C} \,\frac{dv}{R(v,s)}$$ It is easily checked that $$\label{eq:difeq00}
\JN\,''+\frac{1}{4}\rho(s)\JN=0;\ \ \text{where}\ \ \rho(s)=\frac {5}{3s \left( 3\,s+4 \right) }$$ and, since $\JN\,'=\LN/2$ we get $$\label{eq:difeq01}
\LN\, ''-\frac{\rho'(s)}{\rho(s)}\LN\, '+\frac{1}{4}\rho(s)\LN=0$$
The points $s=0$ and $s=-4/3$ are regular singular points of (and of ) and correspond to the values of $s$ for which the polynomial $u^3/3+u^2+s$ has repeated roots. Simple asymptotic analysis of and shows that the Poincaré map satisfies $$\label{sna0}
s_{n+1}=s_n-\frac{2J_n}{x_n}\left(1+{o}(1)\right)\ \ \ \ \ \ \ {\rm{with\ }}J_n=\JN(s_n)$$ $$\label{xna0}
x_{n+1}=x_n+L_n\left(1+{o}(1)\right)\ \ \ \ \ \ \ {\rm{with\ }}L_n=\LN(s_n)$$ Here, and in the following [*heuristic*]{} outline, $o(1)$ stands for terms which are small for large $x_n$ and large $n$. The rigorous justification of these estimates is the subject of [@inprep].
### Solving and ; asymptotically conserved quantities
We see from that $s_{n+1}-s_n\ll s_n$ and $x_{n+1}-x_n\ll x_n$. It is natural to take a “continuum limit” and approximate $s_{n+1}-s_n\text{ by } ds/dn$ and $x_{n+1}-x_n\text { by }dx/dn$. We get $$\label{eqapprx1}
\frac{ds}{dx}=\frac{ds/dn}{dx/dn}=\frac{-2\JN(s)}{x\LN(s)}\left(1+o(1)\right)=-\frac{\JN(s)}{x\JN\, '(s)}\left(1+o(1)\right)$$ which implies, by separation of variables and integration, $$\label{eq:eqapprx}
\mathcal{Q}(x,s):= x\JN(s)= x_0\JN(s_0)\, \left(1+o(1)\right)=\mathcal{Q}(x_0,s_0)\, \left(1+o(1)\right)$$ That is, $\mathcal{Q}$ is asymptotically constant. A second (nonautonomous) one is obtained using and as follows. We write $$\label{eq:2ndc1}
\frac{1}{\JN\, ^2(s)}\, \frac{ds}{dn}=-\frac{2}{x_0\JN(s_0)}\left(1+o(1)\right)$$ Let $\hat{J}$ be an independent solution of with $\hat{J}(0)=0$ and denote $$\label{defK}
\mathcal{K}(s):=\kappa_0\int_{0}^s\frac{ds}{\JN(s)^2}
=\frac{\hat{J}(s)}{\JN(s)}$$ where $\kappa_0$ is the Wronskian of $\JN$ and $\hat{J}$. Integrating both sides of from $0$ to $n$ we get $$\label{eq:2ndc}
\mathcal{K}(s)-\mathcal{K}(s_0) =-\frac{2n}{\kappa_0\, x_0\JN(s_0)}\,\left(1+o(1)\right)\Rightarrow \mathcal{K}(s)+\frac{2n}{\kappa_0\, x_0\JN(s_0)}=\mathcal{K}(s_0)+o(1)$$ (for $n=O(x_0)$). $\mathcal{K}$ is a Schwarzian triangle function. We thus obtain two functionally independent asymptotically conserved quantities , from which we can retrieve the asymptotics of the tronquée solutions in the pole sector. The rigorous proof is the subject of [@inprep], where three asymptotic orders of the expansion are obtained. Near the antistokes lines the expansion takes a slightly different form.
[ From Note \[IC2\] it is seen that in the pole region, the classical asymptotic expansion contains two free constants (which for the tronquées depend on $C_-$). The fact that the constants are classically visible makes possible to calculate their [*change*]{} as the pole region is traversed, and thus to [*calculate explicitly $\mu$*]{} from the requirement that $h_t$ has a transseries representation at both edges of one pole sector. This condition leads to the equation ([@inprep]): ]{}
$$\begin{gathered}
\label{stok2}
-\frac{4 \sqrt{3}-24 i}{5 \pi}=\frac{24 i N}{5 \pi}+\frac{1}{5 \pi ^2}\bigg[12 \ln \left(\frac{(6+6 i) \left(\sqrt{3}+i\right)}{\mu}\right)+i \pi -4 \sqrt{3} \pi
-6 \ln (240 \pi )\bigg]\end{gathered}$$
A straightforward calculation shows that implies $$\label{eq:mu-value}
\mu=\sqrt{\frac{6}{5 \pi }}i$$
Proofs
======
Normal form for
----------------
We first transform to a normal form, to which we apply the general results of [@imrn], [@duke], then use this information to obtain results about the tronquée solutions of .
Simple algebra brings to the normal form
$$\label{sysP1}
\mathbf{y}'+\left({\Lambda}+\frac{1}{x}{B}\right)\mathbf{y}=\mathbf{g}(x^{-1},\mathbf{y})$$
with ${\Lambda}=\text{diag}(\lambda_1,\lambda_2),{B}=\text{diag}(\beta_1,\beta_2)$ where $$\label{noP1}
\lambda_{1}=1,\ \lambda_{2}=-1,\ \beta_{1}=\beta_{2}=\tfrac12$$ and $\mathbf{g}=(g_1,g_2)^T$ is analytic in $(x^{-1},\bfy)$ in a neighborhood of zero and $\mathbf{g}=O(x^{-4})+O(\bfy x^{-2})+O(|\mathbf{y}|^2)$ as $x^{-1},\bfy\to 0$; see §\[rewrP1\] for details.
Transseries solutions {#PfFormh}
---------------------
For an introduction to transseries see §\[IntroTran\].
[*General formal solution of the system*]{}
The transseries solutions of the system - are the following.
I. For $\arg x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ the transseries solutions are (x)=\_0(x)+\_[k=1]{}\^C\^ke\^[-kx]{}\_k(x) \_k(x)=x\^[-k/2]{} \_k(x) where $C$ is an arbitrary constant, $\tilde{\bfs}_k(x)$ is an entire power series in $x^{-1}$ and $$\tilde{\bfy}_0= \sum_{k\Ge 4}\left(\begin{array}{c}1\\(-1)^k\end{array}\right)c_k x^{-k},\ \ \ \ \tilde{\bfs}_{1}=\left(1+\frac{1}{8x}\right)\left(\begin{array}{c}1\\0\end{array}\right)+O(x^{-2})$$
II\. For $\arg x\in\left(\frac{\pi}{2},\frac{3\pi}{2}\right)$ the transseries solutions are (x)=\_0(x)+\_[k=1]{}\^C\^ne\^[kx]{}\_k(x) \_k(x)= x\^[-k/2]{}\_k(x) where $C$ is an arbitrary constant, and $\tilde{\tilde{\bfs}}_k(x)={\tilde{\bfs}}_k(-x)$, $\tilde{\tilde{\bfy}}_0(x)={\tilde{\bfy}}_0(-x)$.
[*Proof.*]{} This is an application of Theorem 2 in [@imrn].
### Proof of Proposition\[Formh\]
\(i) is obtained by a straightforward calculation. Furthermore, this follows from the fact that is a re-writing of ¶1, whose asymptotic expansions are known [@Clarkson].
The general formal solution of systems with a rank one irregular singularity has the form below; the system - has $d=2$ and the transseries must have $C_1$ or $C_2$ equal to zero. Since by below we have h=(1-)y\_1+(1+)y\_2 part (ii) follows. $\Box$
Proof of Theorem\[asir2\] {#PfTasir2}
-------------------------
After establishing Lemma\[L42\], we apply [@imrn] to the system -; we then establish properties of the Borel transform of the series $\tilde{h}_k$ in Lemma\[Lemma2\], then complete the proof of Theorem\[asir2\].
\[L42\] [(i)]{} Let $\bfy(x)$ be a solution of - so that $\bfy(x)=o(x^{-3})$ as $|x|\to\infty$ with $\arg(x)=a$ (for some $a$).
Then $\bfy(x)$ has a formal asymptotic power series in powers of $x^{-1}$ and the asymptotics is differentiable.
\(ii) For any $k\in\ZZ$ there is a unique solution of -
such that $$\label{eq:eqasyh}
\bfy=o(x^{-2}\ln(x)^2)$$ as $|x|\to\infty$ with $\arg(x)=(2k+1)\pi i/2$, $k\in \ZZ$; these are tritronqueées.
\(i) Take $a\in[-\tfrac{\pi}{2},\tfrac{\pi}{2}]$ (the proof for other $a$ is similar). Let $\tfrac{1}{x_0}$ and $\epsilon$ be small enough, where $\arg x_0=a$. We write in the integral form $$\begin{aligned}
\label{eq:systy1}
y_1(x)={A_1}x^{-\frac12} e^{-x}+x^{-\frac12}e^{-x}\int_{x_0}^xs^{\frac12}e^s g_1(s^{-1},\bfy(s))ds\nonumber \\ y_2(x)={A_2}x^{-\frac12}e^{x}+x^{-\frac12}e^{x}\int_{\infty e^{i\phi}}^{x}s^{\frac12}e^{-s} g_2(s^{-1},\bfy(s))ds
\end{aligned}$$ Since $\bfy=o(x^{-3})$ and from the properties of $\bf g$ we see that the second integral is convergent and (again since $\bfy=o(x^{-3})$) we must have ${A}_2=0$. It is straightforward to show that is contractive in the ball of radius $\epsilon$ in the norm $\|\bfy\|=\sup_{|x|>|x_0|}\|x^3 \bfy(|x|e^{ia})\|$, if $\epsilon$ and $1/|x_0|$ are small. It then follows that $\bfy ={\bf a}_4x^{-4}+o(x^{-4})$, where ${\bf a}_4=\frac{392}{625}(1,1)$; feeding back this estimate into it follows that $\bfy$ is of the form $\bfy ={\bf a}_4x^{-4}+{\bf a}_5x^{-5}+o(x^{-5})$, etc. Differentiability of the asymptotics follows from the integral form .
\(ii) We can w.l.o.g. take $k=0$. The proof is similar to that of (i), with $O(x^{-2}\ln^2 x)$ replacing $o(x^{-3})$ and the lower limit of integration $x_0$ replaced by $\infty e^{i\pi/2}$.
The scale $x^{-2}\ln^2 x$ is chosen for technical reasons, since [@inprep] finds the asymptotics in the pole region up to $O(x^{-2}\ln^2 x)$ errors.
Next theorem establishes that generalized Borel summation of the transseries , and of , produces actual solutions, , of the system (see also §\[LBsum\]).
\[Note43\] Consider the system -.
I. Assume $\arg x\in {[}-\tfrac{\pi}{2},\tfrac{\pi}{2}]$. There exist functions $\mathbf{Y}_k,\, (k\Ge 0)$ such that the following hold.
\(i) ${\bf Y}_k$ are the Borel transforms of $\tilde{\bfy}_k$ and for any $\phi\in [-\tfrac{\pi}{2},0)\cup(0,\tfrac{\pi}{2}{]}$ we have $$\label{eq:y_k}
\mathcal{L}_\phi{\bf Y}_k \sim\tilde{\bfy}_k(x)\ \ \ \ \text{for }x\to\infty,\ \ x\in e^{-i\phi}\RR^+$$
\(ii) Let $\phi{\in {[}-\tfrac{\pi}{2},\tfrac{\pi}{2}{]}}$. If $\mathbf{y}(x)$ solves (\[sysP1\]) with $\mathbf{y}=o(x^{-3})\ \ \text{for }x\to\infty,\ x\in e^{-i\phi}{\RR^+}$ then $\mathbf{y}$ has a [*[unique expansion as a Borel summed transseries]{}*]{}: for some constant $C$ $$\label{BoSumTrans}
{\mathbf{y}}(x;C)=\mathcal{L}_\phi{\bf Y}_0\, (x)\, +\sum_{k=1}^\infty C^ke^{-kx}\,\mathcal{L}_\phi{\bf Y}_k\, (x)\ \ \ \text{for }\ x\in e^{-i\phi}\RR^+,\ |x|\ \text{large}$$ If $\phi=\pm \tfrac{\pi}{2}$ then $C=0$.
\(iii) The constant $C$ in depends on the direction $\phi$, $C=C(\arg x)$, and is piecewise constant; it can only change at the Stokes direction [[arg]{}]{}$\, x=0$.
\(iv) $\mathbf{Y}_k$ have the following regularity properties:
(a) \[itema\] $\mathbf{Y}_0(p)=p^3\mathbf{A}_0(p)$, and, for $k\Ge 1$, $\mathbf{Y}_k(p)=p^{k/2-1}\mathbf{A}_k(p)$, with $\mathbf A_k(p)$ analytic on the universal covering of $\CC\setminus\{\pm1,\pm2,\ldots\}$; and $\mathbf A_1(0)$ normalized to equal $\mathbf{e}_1$. Also \_0(p)=(1-p)\^[-1/2]{}S\_Y(p)+(p) with $\mathbf{A}(p), \mathbf{B}(p)$ analytic at $p=1$, $\mathbf{A}(p)=\mathbf{e}_1+O(p-1)$, and $S_Y$ is a constant;
(b) all $\mathbf{Y}_k(|p|e^{i\phi})$ are left and right continuous in $\phi$ at $\phi=0$ and $\phi=\pi$ and are in $L^1_{loc}{(\RR_+)}$.
(c) For any $\delta>0$ there is a large enough $b$ so that $\|\mathbf{Y}_k\|_b<\delta^k$ for $k=0,1\ldots$ where $\|f\|_b:=\int_0^\infty e^{-bt}|f(te^{i\phi})|dt$. As a consequence, each $\mathbf{Y}_k$ is Laplace transformable along any direction of argument $\phi\in (0,\pi)\cup(\pi,2\pi)$.
II\. Analogously, for $x$ with $\arg x\in {[}\tfrac{\pi}{2},\tfrac{3\pi}{2}{]}$ (or $\arg x\in {[}-\tfrac{\pi}{2},-\tfrac{3\pi}{2}{]}$) similar Borel summed transseries representations exist for solutions $\mathbf{y}(x)$ of (\[sysP1\]) with $\mathbf{y}(x)=o(x^{-3})\ \ \text{for }x\to\infty,\ x\in e^{-i\phi}{\RR^+}$. The constant $C$ can only change at the Stokes directions [[arg]{}]{}$\, x=\pm\pi$.
Part [*[I.]{}*]{} follows from general results in [@imrn], [@duke]; part of the theorem is also found in [@invent]. More precisely, using Lemma \[L42\], Proposition\[Formh\] and it follows that $\tilde{\bfy}_0$ is unique, so that [Theorem]{}\[Note43\] (i)-(ii) follows from Theorem [2]{} (ii) in [@imrn], while [Theorem]{}\[Note43\] (iii) (a),(b) follows from Proposition 1 in [@imrn] and (c) from Proposition20 in [@imrn]. The fact that $\mathbf{Y}_0(p)=O(p^3)$ as $p\to 0$ follows from $\tilde{\bfy}_0=O(x^{-4})$ (by , ) and Proposition 1ii) in [@imrn].
We note that along Stokes directions, when arg$\, x$ is $0$ or $\pm\pi$, the Laplace transform does not exist as a usual integral, and must be considered in a generalized sense [@duke].
Part [*[II.]{}*]{} follows due to the symmetry of under $h(x)\mapsto h(xe^{
\pm i\pi})$. $\Box$
### Properties of $H_k$
\[Lemma2\] Let $H_k$ be as in . Then $$\label{asyHk}
\mathcal{L}_\phi H_k\,(x)\sim \tilde{h}_{k}(x); \ \ \ \tilde{h}_{k}(x)=O(x^{-k/2})\ \ \text{for }x\to\infty,\,\arg x\in
(-\tfrac{\pi}{2},\tfrac{\pi}{2})$$ and the analytic continuation of $H_k$ have the following regularity properties.
\(a) The function $H_0(p)$ satisfies the inverse Laplace transformed equation : $$\label{eq:Bplane}
(p^2-1)H=\frac{196}{1875}\, p^3+p*H+\frac{1}{2}H*H$$ and $H_0$ is the unique solution of which is analytic at $p=0$.
\(b) $H_0$ is an odd function.
\(c) ${H}_0(p)=p^3{A_0}(p)$ with $A_0(p)$ analytic on the universal covering of $\CC\setminus\{\pm1,\pm2,\ldots\}$;
\(d) $H_0$ satisfies the statement of Proposition\[H0\_behave\](i), and therefore also ${H}_0(p)=-S\,(1+p)^{-1/2}{A}(-p)-{B}(-p)$ for the same $S,A,B$ ; also, for $k\geq 1$, $H_k(p)=p^{k/2-1}A_k(p)+B_k(p)$ with $A_k,B_k$ analytic at $p=0$ and $A_1(0)=1$.
\(e) $H_0(|p|e^{i\phi})$ is left and right continuous in $\phi$ at $\phi=0$ and $\phi=\pi$, and is in $L^1_{loc}{(\RR_+)}$.
\(f) For any $\delta>0$ there is a large enough $b$ so that $\|H_k\|_b<\delta^k$ for $k=0,1\ldots$ where $\|f\|_b:=\int_0^\infty e^{-bt}|f(te^{i\phi})|dt$. As a consequence, each $H_k$ is Laplace transformable along any direction of argument $\phi\in (0,\pi)\cup(\pi,2\pi)$.
[*[Proof.]{}*]{}
Solutions $h(x)$ are obtained from solutions of via . Using the known asymptotic forms of solutions of , see [@Clarkson], it follows that implies that $h(x)=O(x^{-4})$ in the same sector, hence $\bfy$ given by is $o(x^{-3})$.
Note that we have, in view of , \_k=(1-)\_[k;1]{}+(1+)\_[k;2]{} (where $\tilde{\bfy}_{k}=(\tilde{y}_{k;1},\tilde{y}_{k;2})^T$). We have (see ) \_H\_k (x)=\_\_k(x)=(1-)y\_[k;1]{}(x)+(1+)y\_[k;2]{}(x)=h\_k(x) Using Theorem \[Note43\] this implies .
Relation follows from the fact that $H_0=\mathcal{B}\tilde{h}_0$, hence it satisfies the inverse Laplace transformed equation .
\(b) Equation has a unique solution analytic at $p=0$ [@imrn] and it can be easily checked that if $H(p)$ is a solution, then so is $-H(-p)$.
To prove the other properties, note that we have, using , and the fact that $\mathcal{B}(x^{-1}\tilde{\mathbf{Y}}_k)=1*\mathcal{B}(\tilde{\mathbf{Y}}_k)$, $$\begin{gathered}
\label{e42}
H_k=\mathcal{B}\tilde{h}_k=\frac{1}{2}\left(Y_{k;1}-\frac{1}{4}\,1*Y_{k;1}\right)+\frac{1}{2}\left(Y_{k;2}+\frac{1}{4}\,1*Y_{k;2}\right)\\
=\frac{1}{2}\left(Y_{k;1}+Y_{k;2}\right)+\frac{1}{8}\int_0^p \left(-Y_{k;1}(q)+Y_{k;2}(q)\right)\, dq\end{gathered}$$ and the properties follow from [Theorem]{}\[Note43\]. $\Box$
### Proof of Theorem\[asir2\] {#proof-of-theoremasir2}
I. follows from Lemma\[Lemma2\], Theorem\[Note43\] and -.
II\. follows due to the symmetry of under $h(x)\mapsto h(xe^{
\pm i\pi})$.
Proof of Proposition\[H0\_behave\] {#PfP3}
----------------------------------
These statements are true in a general setting, see [@imrn]; see also §\[Sbeta\] for an overview in the general case and more details. The main ideas are as follows.
\(i) was proved in Lemma\[Lemma2\].
\(ii) By rotating the angle $\phi$ into $\phi\in[-\pi, -\frac{\pi}{2}]$, and using the estimates of Lemma\[Lemma2\](c),(d) it is clear that $h^+(x)$ can be analytically continued for $x$ in the second quadrant. Continuation of $h^+(x)$ for $x$ in the fourth quadrant is done by deformation of the path of integration $\arg p=\phi<0$ to a direction with $\arg p>0$ plus an infinite sum of paths coming from $\infty$ in the first quadrant, encircling only one point $k\in\ZZ_+$ counterclockwise, and going back to $\infty$. The series obtained converges due to Lemma\[Lemma2\](d) and we obtain a Borel summed transseries for this continuation of $h^+(x)$. The analytic continuation of $h^-(x)$ is similar.
Formula is proved below, but it is more general, see §\[Sbeta\] in the Appendix.
We have $h^+(x)-h^-(x)=\int_\ell e^{-xp}H_0(p)\, dp$ where $\ell$ is a path coming from $+\infty$ above $[1,+\infty)$, going counterclockwise around $p=1$ and returning to $+\infty$ below $[1,+\infty)$. Then, choosing the usual branch of the radical (with $(1-p)^{1/2}>0$ for $p<1$) we have $$\begin{gathered}
h^+(x)-h^-(x)\sim S\int_\ell e^{-xp}(1-p)^{-1/2}\, dp\\
=S\int_{+\infty}^1i(p-1)^{-1/2}e^{-xp}\, dp+S\int_1^{+\infty}(-i)(p-1)^{-1/2}e^{-xp}\, dp\\
=-2i\sqrt{\pi}Se^{-x}x^{-1/2}
\end{gathered}$$
\(iii) Similar arguments for $\mathcal{L}_{\phi}H_1\,(x)$ show that the analytic continuation produces only terms of order $e^{-2x}$ or smaller, see Theorem\[Note43\](iv)(a), ,. $\Box$
Specification of branches and proof of Theorem\[TritCis0\] {#PfTritCis0}
----------------------------------------------------------
### Branches {#brachoise}
We recall that $H_0(p)$ is analytic at $p=0$, see Lemma\[Lemma2\](c). Each directional Laplace transform of $H_0$ uses the analytic continuation of this germ of analytic function at $p=0$ along the direction of the transform.
$H_k(p)$ with $k\geq 1$ may have a square root branch point at $p=0$, see Lemma\[Lemma2\](d). We use the analytic continuation of the usual branch of the square root, with $\arg p=0$ for $p>0$; for instance in the integral in the third expansion in , the functions are continued through clockwise rotation, starting with $\arg p=0$.
### Proof of Theorem\[TritCis0\]
\(i) By Theorem\[asir2\] a tritronquée has a series representation of the form for some $C_+$. This $C_+$ must be zero, otherwise $h_t$ has poles beyond $\arg x=\tfrac{\pi}{2}$, by Proposition \[RegP\], hence it does not correspond to the tritronquée .
\(ii) Note that for $x$ in the second quadrant ($\arg x=\pi-\epsilon$), $h^+=\mathcal{L}_\phi H_0$ with $\phi=-\pi+\epsilon$ and that after clockwise rotation in the $p$-plane we have $(1+p)^{1/2}=i |1+p|^{1/2}$ for $p<-1$.
We have $h^+(x)-\hsig(x)=\int_{\tilde{\ell}} e^{-xp}H_0(p)\, dp$ where $\tilde{\ell}$ is a path coming from $-\infty$ above $(-\infty,-1]$, going clockwise around $p=-1$ and returning to $-\infty$ below $(-\infty,-1]$. Then $$h^+(x)-\hsig(x)\sim
2i\int_{-1}^{-\infty} |p+1|^{-1/2} e^{-xp}(-S)\, dp=2iS\sqrt{\pi}e^x|x|^{-1/2}$$ which gives (noting that analytic continuation in $x$ is counterclockwise).
\(iii) Formula for $|\arg x|\Le\tfrac{\pi}{2}$ follows by , since $C_+=0$ by (i) and by Proposition\[H0\_behave\](iii).
Then, for $\arg x\in [\tfrac{\pi}{2},\pi)$, in $h^+=\mathcal{L}_\phi H_0$ can be analytically continued by rotating of $\phi$, since $H_0$ is analytic and Laplace transformable by Lemma\[Lemma2\](c),(f).
To obtain the series for $\arg x\in (\pi, \tfrac{3\pi}{2})$ we use (ii) and the proof is analogous to the proof of Proposition\[H0\_behave\] (ii), only the branches are different; alternatively, this follows from the symmetry $h(x)\to\overline{h(-\overline{x})}$.
Proof of Theorem\[FAPIn\] {#PfRem8}
-------------------------
\(i) is proved in [@invent]. We include the main steps in the calculation of $F_n$ in §\[CalcHn\].
\(ii) It is straightforward to check that the transformation $g=h(1+\frac13h)^{-1}$ leads to a system of equations satisfying the same assumptions as , and the construction of the expansion mirrors the one for . Expectedly, the expansion in coincides with the formal asymptotic expansion of $h(1+\frac13h)^{-1}$ in powers of $1/x$ using .
Proof of Proposition \[RegP\] {#PfPr10}
-----------------------------
The fact that this is the first array of poles is guaranteed by which shows that for $|\xi|<a<12$, $h$ is bounded.
[(i) Let $g$ be as in Theorem(ii). We first note that a pole of $h$ is a regular point of $g$, one in which $g$ assumes the value $3$. Formula is simply obtained from the implicit function theorem and the expansion of $g$ in Theorem [@invent] (ii), by solving the implicit equation $g(x)-3=0$, writing $\ln\xi\sim \ln 12+2 n\pi i...$ and retaining three orders in $n^{-1}$ in the calculation.]{}
[(ii) The roots $r_{1,2}$ of the quadratic $H_0(r)=y$ satisfy $r_1 r_2=12$. Thus $H_0$ maps the disk $\{z:|z|\le 12\}$ onto the Riemann sphere $\CC\cup\{\infty\}$. If $\mathcal{A}(0)=0,\mathcal{A}'(0)=1$ and $\mathcal{A}$ is analytic, it follows that $\mathcal{A}(h(x))$ is singular in $\{z:|z|\le 12\}$. The rest is immediate.]{}
Appendices {#Appendix}
==========
Sections §\[IntroTran\]-\[Sbeta\] contain an outline of some results found in [@imrn],[@duke] and illustration of these results on some simple examples. Section §\[historic\] contains a brief overview on the development of the subject.
Representation of solutions as transseries {#IntroTran}
------------------------------------------
Very few differential equations can be explicitly solved, and even when this is possible, their expression may be too complicated for easily extracting useful information about solutions; by contrast, formal solutions can often be obtained algorithmically as asymptotic expansions, from which properties of solutions such as rate of decay/increase, or approximations, can be easily read. Sometimes, free parameters are “hidden” beyond all orders of a classical asymptotic series; in such cases transseries are instrumental in uncovering these parameters.
It is well known that equations written in terms of analytic functions have convergent power series solutions at any regular point of the equation; convergent expansions for solutions also exist at regular singularities (generically). But at irregular singularities the asymptotic expansions of solutions are often divergent.
To illustrate, consider the simple equation y’+y= The point $x=\infty$ is an irregular singular point of the equation (indeed, the substitution $x=1/z$ maps $\infty$ to $0$ and brings to the form $-z^2\tfrac{dy}{dz}+y=z^2$ for which $z=0$ is an irregular singularity).
It is easy to see that there exists a unique asymptotic power series solution for $x\to\infty$, \_0(x)=\_[n=2]{}\^ and it is divergent. On the other hand the general solution of is y(x;C)=y\_0(x)+Ce\^[-x]{}, y\_0(x)=e\^[-x]{}\_[x\_1]{}\^x ds\~\_0(x) x+and $C$ is a free parameter.
This simple example illustrates main phenomena at irregular singularities: the power series solutions are divergent, there is loss of information (in there is a one parameter family of solutions asymptotic to the same power series), and asymptoticity holds only in sectors ($y_0(x)\sim \tilde{y}_0$ only for $x\to\infty$ with $|\arg x|<\tfrac{\pi}{2}$).
In view of , it is natural to consider that the [*[complete formal solution]{}*]{} of is (x)=\_0(x)+Ce\^[-x]{}, x+The formal expression , which satisfies , is not an asymptotic series in the sense of Poincaré if $|\arg x|<\tfrac{\pi}{2}$, as the term $Ce^{-x}$ is much smaller than all the powers of $x$ in $\tilde{y}_0(x)$: it is a [*[term beyond all orders]{}*]{} of the main series. The formal solution is the simplest example of a [*[transseries]{}*]{}.
Consider next a nonlinear example, namely plus a nonlinear term: y’+y=+y\^4 Again, equation has a unique power series solution $\tilde{y}_0(x)=\frac{1}{x^2}+\frac{2}{x^3}+\frac{6}{x^4}+\ldots$ which can be shown to be divergent too. To find possible further terms in a formal expansion we search for a perturbation: substituting $y=\tilde{y}_0+\delta$ in and using the fact that $\tilde{y}_0$ is already a formal solution we get $\delta'+\delta\sim 2\tilde{y}_0\delta$, giving $\delta\sim Ce^{-x}\tilde{y}_1(x)$ where $\tilde{y}_1(x)$ is a (divergent) power series. Since $\tilde{y}_0+Ce^{-x}\tilde{y}_1(x)$ does not solve the equation, further corrections are required, yielding a complete formal solution of in the form =(x;C)= \_0(x) +C[[e]{}]{}\^[-x]{}\_1(x) + C\^2[[e]{}]{}\^[-2x]{}\_2(x)+…where $\tilde{y}_k(x)$ are divergent power series and $C$ is an arbitrary parameter. The formal solution is a transseries for $x\to\infty$ along directions in the complex plane for which the terms can be well ordered decreasingly, namely for $x\in e^{ia}\RR_+$ with $ |a|<\tfrac{\pi}{2}$.
Scalar equations more general than , of the form y’+(-)y=g(x\^[-1]{},y) g=O(x\^[-2]{})+O(y\^2) x, y0 have formal series solution of the form =(x;C)=\_0(x)+\_ [k=1]{}\^C\^k[[e]{}]{}\^[-kx]{}\_k(x), \_k(x)=x\^[k]{}\_k(x) (with $\tilde{s}_k(x)$ an integer power series in $x^{-1}$) which is a transseries for $x\to\infty$ along any direction for which $|\arg (\lambda x)|<\tfrac{\pi}{2}$.
Systems have transseries solutions which are similar: generic equations can be brought to the normal form (in [@imrn] this is eq. (1.1))
[l]{}\
\
\
\
Under appropriate nonresonance conditions[^2] systems have formal solutions =(x;)=\_(x)+\_ [\^d]{} \^\^[-x]{} \_(x) \_(x)=x\^\_(x) which is a transseries for $x\to\infty$ along any direction along which the terms can be well ordered, meaning that all the exponentials are decaying, therefore along any direction in the sector
Correspondence between transseries and actual solutions: generalized Borel summation {#LBsum}
------------------------------------------------------------------------------------
Consider the linear equation . Since its formal solution is factorially divergent and $\mathcal{L}(p^{n-1})=(n-1)!x^{-n}$, heuristically, it is natural to attempt to write $\tilde{y}_0$ as a Laplace transform: this is central to Borel summation.
Recall that the Borel transform is defined as the formal inverse Laplace transform, $\mathcal{B}(x^{-\alpha})=\tfrac{p^{\alpha-1}}{\Gamma(\alpha)}$ for $\alpha>0$ (where $\mathcal{L}$ is the Laplace transform), and, more generally, we have .
Taking the inverse Laplace transform, equation becomes $(1-p)Y(p)=p$ therefore y=\_Y Y(p)= and $\mathcal{L}_\phi$ is the Laplace transform along a direction of argument $\phi$, see .
Note that we cannot take the Laplace transform along $\RR_+$ in (except in the sense of distributions [@duke]), but we can integrate on any half-lines above, or below $\RR_+$, obtaining $$y_0^+(x)=\mathcal{L}_\phi Y(x)\ \ \ \ \text{for }-\phi=\arg x\in(0,\frac{\pi}{2})$$ and $$y_0^-(x)=\mathcal{L}_\phi Y(x)\ \ \ \ \text{for }-\phi=\arg x\in(-\frac{\pi}{2},0)$$ The values of $y_0^\pm(x)$, do not depend on the value of $\phi$ in its specific quadrant; they can be both analytically continued in the right half-plane (and beyond) $ y_0^+(x)\ne y_0^-(x)$ and in fact the difference $ \tfrac{1}{2\pi i}(y_0^+(x)- y_0^-(x))= e^{-x}$ recovers the exponentially small term in .
These facts generalize to nonlinear equations. Consider the example ; taking the inverse Laplace transform one obtains the convolution equation (1-p)Y(p)=p+Y\^[\*4]{}(p) which has a unique solution $Y=Y_0(p)$ analytic at $p=0$. In fact, $Y_0(p)$ is analytic for $|p|<1$, and it is singular at $p=1$. Due to the convolution term in ,the singularity at $p=1$ gives rise to an equally spaced array of singularities in the Borel plane at $p=2,3,4,\ldots$. $Y_0(p)$ is analytic along any direction $p=|p|e^{i\phi}$ with $0<|\phi|<\tfrac{\pi}{2}$, is Laplace transformable, and $y_0=\mathcal{L}_\phi Y_0=\mathcal{L}_\phi\mathcal{B}\tilde{y}_0$ is an actual solution of for $x$ large with $\arg x=-\phi$: $y_0(x)$ is the Borel summation, along the direction of $x$, of $\tilde{y}_0(x)$.
The Borel sum $y_0^+=\mathcal{L}_\phi\mathcal{B}\tilde{y}_0$ of $\tilde{y_0}$ is the same for all $-\phi=\arg x\in (0,\tfrac{\pi}{2})$ and $y_0^-=\mathcal{L}_\phi\mathcal{B}\tilde{y}_0$ is the same for all $-\phi=\arg x\in (-\tfrac{\pi}{2},0)$, both $y_0^\pm$ can be analytically continued in the right half-plane, and $y_0^+-y_0^-$ is exponentially small.
The other series $\tilde{y}_k$ in are Borel summed similarly (using convolutions equations which are found for $Y_k=\mathcal{B}\tilde{y}_k$ in [@imrn],[@duke]), yielding functions $y_k=\mathcal{L}_\phi Y_k=\mathcal{L}_\phi\mathcal{B}\tilde{y}_k$ analytic for large $x$; the series $$y_0(x)+\sum_{k=1}^\infty C^ke^{-kx}y_k(x)$$ converges for $x$ large with $\arg x=-\phi\in(-\tfrac{\pi}{2},0)\cup(0,\tfrac{\pi}{2})$ to a solution of [^3] [@imrn],[@duke].
The general one-dimensional case is similar; $Y_0(p)$ will have equally spaced arrays of singularities along $\arg p=\arg \lambda$. Along any other direction $p=|p|e^{i\phi}$ with $0<|\phi-\arg\lambda|<\tfrac{\pi}{2}$ the continuation of $Y_0(p)$ is analytic (generally on a Riemann surface) and Laplace transformable, and $\mathcal{L}_\phi Y_0=\mathcal{L}_\phi\mathcal{B}\tilde{y}_0$ is an actual solution of for $x$ large with $\arg x=-\phi$.
[**[Remark.]{}**]{} [*To Borel sum the series in for $k\Ge 1$ we may consider $y_k=\mathcal{L}_\phi\mathcal{B}\tilde{y}_k$ (if $\alpha<0$), or we can choose any $m$ (large enough so that $\alpha-m<0$), find solutions as Borel summed transseries in the form y\_0(x)+\_[k=1]{}\^C\^ke\^[-kx]{}x\^[mk]{}\_(x\^[-mk]{}\_k) The final result does not depend on $m$, since x\^N\_(x\^[-N]{}x\^[-n]{})=\_(x\^[-n]{})* ]{}
Finally, [the series]{} to actual solutions for $x\in S_{an}$ where $$\displaystyle{ S_{an}=\{x\, \big|\, -\frac{\pi}{2}+\epsilon<\arg(x)<\frac{\pi}{2}-\epsilon,\ |x|>R\} }$$
Generic nonlinear equations have their transseries solutions summed similarly along directions $d$ in $\CC$. Furthermore, there exists a one-to-one correspondence between solutions s.t. $\bfy(x)\to 0$ ($ x\in d,\,x\to\infty$) and (generalized) Borel sums of $\tilde{\bfy}(x;C)$ transseries solutions along $d$. These solutions $\bfy(x;C)$ are analytic in a sector containing $d$ for $|x|$ large. These results are stated and proved in [@imrn], [@duke].
Theorem\[Note43\] is an application of these results for the system (\[sysP1\]) associated to the Painlevé equation ¶1.
The Stokes phenomenon {#Sbeta}
---------------------
The directions $\pm i\overline{\lam}_j\RR_+$ are called [*[antistokes lines]{}*]{}; along these directions, some exponential $e^{-\lambda_jx}$ in is purely oscillatory. Antistokes directions border the sectors where transseries exist, . Directions with $\lambda_jx\in\RR_+$ (for some $j$) are called [*[Stokes lines]{}*]{}; along these, some exponential $e^{-\lambda_jx}$ has fastest decay. At Stokes directions the constants beyond all order in the one-to-one association between small solutions and transseries may change: this is the [*[Stokes phenomenon]{}*]{}.
To illustrate this consider . As noted above, solutions can be written using $Y_0$, the Borel sum of $\tilde{y}_0$ as $$\label{eq:try}
y(x)=\left\{
\begin{array}{l}
\mathcal{L}_\phi Y_0\,(x)+ C_+ e^{-x}\ \ \ \text{for }
- \phi= \arg x\in(0,\tfrac{\pi}{2})\\
\mathcal{L}_\phi Y_0\,(x)+ C_- e^{-x}\ \ \ \text{for }
- \phi= \arg x\in(-\tfrac{\pi}{2},0)
\end{array}\right.$$
The value of the jump in the constant beyond all orders, $C_+ -C_-$, is called the [*[Stokes constant]{}*]{}.
More generally, a fixed solution of can be written as Borel summed transseries for some fixed $C$ for all $\phi$ with $\arg\phi\in\arg\lambda+(0,\tfrac{\pi}{2})$, and with a different $C$ for all $\phi$ with $\arg\phi\in\arg\lambda+(-\tfrac{\pi}{2},0)$.
For general equations the situation is similar: the vector parameter $\mathbf{C}$ in a transseries associated via Borel summation along a direction to a true solution does not change when this direction varies between two consecutive Stokes or antistokes lines, but it generally changes across a Stokes line.
Consider systems , with $\lambda_1=1$, $|\lambda_j|\Ge 1$ and $\beta:=\beta_1<1$ (which can be arranged by a suitable substitution) and solutions obtained by Borel summation of the transseries solution along directions slightly above and below the Stokes line $\arg x=0$: $$\label{GenTrans}
\bfy(x)=\left\{
\begin{array}{l}
\mathcal{L}_\phi \mathbf{Y}_\mathbf{0}\,(x)+ \sum_ {\mathbf{k}\in\NN^d\setminus\mathbf{0}} \mathbf{C_+}^\mathbf{k}{\rm{e}}^{-\boldsymbol\lambda \cdot \mathbf{k}x} \mathcal{L}_\phi\mathbf{Y}_\mathbf{k}(x) \ \ \text{for }
- \phi= \arg x\in(0,a_2)\\
\mathcal{L}_\phi \mathbf{Y}_\mathbf{0}\,(x)+ \sum_ {\mathbf{k}\in\NN^d\setminus\mathbf{0}} \mathbf{C_-}^\mathbf{k}{\rm{e}}^{-\boldsymbol\lambda \cdot \mathbf{k}x} \mathcal{L}_\phi\mathbf{Y}_\mathbf{k}(x) \ \ \ \text{for }
- \phi= \arg x\in(a_1,0)
\end{array}\right.$$ where $\mathbf{Y}_\mathbf{k}=\mathcal{B}_\phi\tilde{\bfy}_\mathbf{k}$ (is the analytic continuation of the Borel transform of $\tilde{\bfy}_\mathbf{k}$ along the direction of argument $\phi$), and the sector $a_1<\arg x<a_2$ does not contain another Stokes or antistokes line besides $\arg x=0$.
The first component $C_1$ of the constant beyond all orders in changes when $\arg x$ crosses the Stokes line $\arg x=0$, corresponding to $\lambda_1=1$ [@duke].
Changes in the constant beyond all orders occur upon analytic continuation across\* a Stokes line; the leading order change, which is exponentially small, is due to the continuation of $\mathcal{L}_\phi\mathbf{Y}_\bfk$. The continuations of $\mathcal{L}_\phi\mathbf{Y}_\bfk$ generally add further, but these are of order $e^{-x}$ or smaller, and for $|\bfk|\Ge 1$, the $\mathcal{L}_\phi\mathbf{Y}_\bfk$ already multiplies an exponential, so this change does not affect the coefficient of $e^{-x}$. The fact that the changes in all $\mathbf{C}^\bfk$ with $|\bfk|\Ge 1$ match to give an overall jump equivalent to $\mathbf C_+\to \mathbf C_-$ is due to the so-called [*[resurgence]{}*]{}, which links the singularities of all $\mathbf{Y}_\bfk$ in a precise manner.)
### The Stokes multiplier
A calculation analogous to the one in the proof of Proposition\[H0\_behave\] gives the change in $C_1$, and the argument is as follows. To analytically continue $\mathcal{L}_{0^-}\mathbf{Y}_0(x)$ past $\arg x=0$ we write $\mathcal{L}_{0^-}\mathbf{Y}_0=\mathcal{L}_{0^+}\mathbf{Y}_0+\delta$ where $\delta=(\mathcal{L}_{0^-}-\mathcal{L}_{0^+})\mathbf{Y}_0$. Since $\mathbf{Y}_0(p)$ is analytic for $|p|<1$ (by [@imrn] Proposition 1), the path of integration in $\delta$ can be deformed to the path from $\infty$ to $1$ below $[1,+\infty)$, going around $1$ and then going to $+\infty$ above $[1,+\infty)$.
Using the fact that $\mathbf{Y}_0(p)=S_{\beta}(1-p)^{\beta-1} (\mathbf{e}_1+o(1))$ (by [@imrn] Proposition 1), and that $(1-p)^{\beta-1}=e^{\mp i\pi(\beta-1)}|1-p|^{\beta-1}$ for $|p|>1, \arg(p)=\pm 0$ we obtain, using Watson’s Lemma, $$\delta= -2i S_{\beta} \sin (\pi \beta) \int_1^\infty |1-p|^{\beta-1}e^{-px}dp (\mathbf{e}_1+o(1))=-2iS_{\beta} \sin (\pi \beta)\Gamma(\beta) e^{-x}x^{-\beta} (\mathbf{e}_1+o(1))$$ so that the jump in the constant $C_1$ across the Stokes line $\arg x=0$ is[^4] $$\label{eq:sjump}
C_{1;+}-C_{1;-}=-S=-\mu\ \text{with}\ S=2iS_{\beta} \sin (\pi \beta)\Gamma(\beta),\ \ \beta=\beta_1;\ \ \ \Re(\beta)\in (0,1)$$
For general equations, the values of Stokes constants are transcendental.
\[SpecDir\] The five special directions of ¶1 are Stokes or antistokes lines of its normalized form .
Further references. {#historic}
-------------------
Double expansions of solutions of linear equations as power series multiplying exponentials have been studied starting with Fabry [@Fabry] (1885), and then Cope [@Cope] (1936). Iwano (1957-’59) analyzed solutions of nonlinear systems as a convergent series of functions analytic in sectors, multiplying exponentials [@Iwano]. The subject has been developed and expanded substantially after the fundamental work of Ecalle (1981), with results in multisummability of power series of linear ODEs [@BBRS], nonlinear ones, [@Bra], transseries of nonlinear ODEs [@imrn],[@duke], similar results for discrete equations [@Bra_dis],[@Bra_dis2] and for PDEs [@OC-ST-PDE]; singularity formation near antistokes lines was studied in a general setting in [@invent].
Rewriting as a normalized system {#rewrP1}
--------------------------------
We write as usual, $$\label{eq:eqsys1}
\begin{pmatrix}
h\\ h'
\end{pmatrix}'=
\begin{pmatrix}
0\\\tfrac{392}{625}x^{-4}
\end{pmatrix}+
\begin{pmatrix}
0 & 1\\1 &0
\end{pmatrix}
\begin{pmatrix}
h\\ h'
\end{pmatrix}+
\begin{pmatrix}
0 & 0 \\ 0 & -\tfrac{1}{x}
\end{pmatrix}\begin{pmatrix}
h\\ h'
\end{pmatrix}+
\begin{pmatrix}
0\\\tfrac12 h^2
\end{pmatrix}$$ The transformation $$\label{eq:trsf3}
\begin{pmatrix}
h\\ h'
\end{pmatrix}=
\frac{1}{2}\begin{pmatrix}
1-\tfrac{1}{4x} & 1+\tfrac{1}{4x}\\ -1-\tfrac{1}{4x} & 1-\tfrac{1}{4x}
\end{pmatrix}
\begin{pmatrix}
y_1\\y_2
\end{pmatrix}$$ brings (\[eq:eqsys1\]) to ,, which is in the normal form .
More precisely, $$\begin{gathered}
\nonumber
g_1(x,\bfy)=-{\frac {1568}{625}}\,{\frac {4\,x+1}{ \left( 16\,{x}^{2}+1 \right) {x
}^{3}}}-\frac{1}{16}\,{\frac { \left( 4\,x-1 \right) \left( 4\,x+1 \right) ^{
2}{y_1} { y_2} }{ \left( 16\,{x
}^{2}+1 \right) x}}\\
-\frac{1}{32}\,{\frac { \left( 4\,x+1 \right) \left( 4\,x-
1 \right) ^{2} { y_1}\, ^{2}}{
\left( 16\,{x}^{2}+1 \right) x}}-\frac{1}{32}\,{\frac { \left( 4\,x+1
\right) ^{3} { y_2}\, ^{2}}{ \left(
16\,{x}^{2}+1 \right) x}}\\
-{\frac { \left( 2\,x-1 \right) { y_1}
}{ \left( 16\,{x}^{2}+1 \right) x}}+\frac{1}{2}\,{\frac {
\left( 8\,x-1 \right) { y_2} }{ \left( 16\,{x}^{2}
+1 \right) x}}\end{gathered}$$ $$\begin{gathered}
\nonumber
g_2(x,\bfy)={\frac {1568}{625}}\,{\frac {4\,x-1}{ \left( 16\,{x}^{2}+1 \right) {x}
^{3}}}+\frac{1}{16}\,{\frac { \left( 4\,x+1 \right) \left( 4\,x-1 \right) ^{2
}{ y_1} { y_2} }{ \left( 16\,{x}
^{2}+1 \right) x}}\\
+\frac{1}{32}\,{\frac { \left( 4\,x-1 \right) ^{3} {
y_1} \, ^{2}}{ \left( 16\,{x}^{2}+1 \right)
x}}+\frac{1}{32}\,{\frac { \left( 4\,x-1 \right) \left( 4\,x+1 \right) ^{2}
{ y_2} \,^{2}}{ \left( 16\,{x}^{2}+1
\right) x}}\\
-\frac{1}{2}\,{\frac { \left( 8\,x+1 \right) { y_1} }{ \left( 16\,{x}^{2}+1 \right) x}}+{\frac { \left( 2\,x+1
\right) { y_2} }{ \left( 16\,{x}^{2}+1 \right) x}}\end{gathered}$$
Calculation of the functions $F_n(\xi)$ {#CalcHn}
---------------------------------------
Substituting the two scale expansion in we obtain an asymptotic series, for $1\ll x\ll \xi$ and $F_0(\xi)\ll x$, in integer powers of $x^{-1}$, with coefficients functions of $\xi$; the first term is $${\xi}^{2}\, {\frac
{d^{2}}{d{\xi}^{2}}}F_{{0}} \left( \xi \right) + \xi\,
{\frac {d}{d\xi}}F_{{0}} \left( \xi \right)-\frac{1}{2}\, F_{{0}}(\xi) ^{2} -F_{{0
}} \left( \xi \right) =O \left( {x}^{-1} \right)$$ and we look for $F_0$ analytic at $\xi=0$ and $F_0(0)=0,\, F_0'(0)=1$.
Substituting $F_0(\xi)=G_0(s)$, $s=\ln\xi$ we get $G_0''-\tfrac{1}{2}G_0^2-G_0=0$, an equation having, as expected, Weierstrass elliptic functions as general solution, a one parameter family of rational solutions, as well as two constant solutions: multiplying the equation by $2G_0'$ we obtain $G_0'^2=\tfrac{1}{3}G_0^3+G_0^2+Const.$ whose solution contains a term $s=\ln\xi$ unless $Const.=0$, in which case we obtain $F_0(\xi)= 12\xi/[c(1-\xi/c)^{2}]$ (degenerate elliptic) and $F_0'(0)=1$ implies the formula in .
The coefficient of $x^{-1}$ gives the equation for $F_1(\xi)$: $$\xi^2F_1''+\xi F_1'-(1+F_0)F_1=-\xi^2 F_0''$$ which shows that the only possible singularities for $F_1$ are at $\xi=0$ and $\xi=12$. Similarly, the differential equation for all $F_n$ are linear, with coefficients depending on $F_0,\ldots,F_{n-1}$, and by induction, the only possible singularities for $F_1$ are at $\xi=0$ and $\xi=12$.
To determine $F_1$ we need two constants; one is determined from the condition that $F_1$ be analytic at $0$ (thus the coefficient multiplying $\ln\xi$ must vanish), and the other constant is determined at the next step, when solving for $F_2$ (from the condition that $F_2$ does not contain $\ln\xi$ terms). This pattern continues for all $F_n$, and is typical for generic equations.
An additional potential obstruction to $F_n$ rational occurs at $n=6$: $F_6$ also contains, in principle a term $\ln(\xi-12)$ multiplied by a constant; this term vanishes precisely when the coefficient of $x^{-4}$ in equals $-\tfrac{392}{625}$: any other value of this coefficient produces an equation with movable branch points, hence not having the Painlevé property! This is the special feature of integrability of P$_I$.
For practical calculation of $F_n$, for $n\Ge 3$ it is better to substitute $F_n(\xi)={\frac {\xi\, \left( \xi+12 \right) }{ \left( \xi-12 \right) ^{3}}}V_n(\xi)$; the functions $V_n(\xi)$ can be calculated recursively using only two successive integrations of rational functions.
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[^1]: Later on we will choose $u_0=-4\in\mathcal{C}$.
[^2]: It is required that $\lambda_1,\ldots,\lambda_d$ be linearly independent over $\ZZ$ (otherwise its expression has a slightly more general form) and it suffices that the Stokes lines be distinct.
[^3]: For $\arg x=0$ solutions are obtained using weighted averages of Laplace transforms along paths going toward $\infty$ avoiding the singularities in prescribed ways, independent of the equation.
[^4]: We note that in [@imrn], the constants $C(0\pm)$ correspond here to $C_\mp$, that in the formula below (1.15), the factor $\Gamma(\beta)$ is missing, in (1.19) $S_\beta$ should be $S$, and in (1.2) $\beta=\hat{B}_{1,1}$ with $\Re \beta\in (0,1]$.
|
---
abstract: 'We present ALMA observations and multiwavelength spectral energy distribution (SED) analysis in a [*WISE*]{}-selected, hyperluminous dust-obscured quasar [W0533$-$3401]{} at $z=2.9$. We derive its physical properties of each component, such as molecular gas, stars, dust and the central supermassive black hole (SMBH). Both the dust continuum at 3 mm and the [CO$(3-2)$]{} line are detected. The derived molecular gas mass $M_{\rm gas}=8.4\times10^{10}\ M_\odot$ and its fraction $f_{\rm gas}=0.7$ suggest that [W0533$-$3401]{} is gas-rich. The star formation rate (SFR) has been estimated to be $\sim3000-7000\ M_\odot$ yr$^{-1}$ by using different methods. The high values of SFR and specific SFR suggest that [W0533$-$3401]{} is a maximum-starburst. The corresponding gas depletion timescales are very short ($t_{\rm depl}\sim12-28$ Myr). The [CO$(3-2)$]{} emission line is marginally resolved and has a velocity gradient, which is possibly due to a rotating gas disk, gas outflow or merger. Finally, we infer the black hole mass growth rate of [W0533$-$3401]{} (${\dot{M}}_{\rm BH}$ = 49 $M_\odot$ yr$^{-1}$), which suggests a rapid growth of the central SMBH. The observed black hole to stellar mass ratio $M_{\rm BH}/M_\star$ of [W0533$-$3401]{}, which is dependent on the adopted Eddington ratio, is over one order of magnitude higher than the local value, and is evolving towards the evolutionary trend of unobscured quasars. Our results are consistent with the scenario that [W0533$-$3401]{}, with both a gas-rich maximum-starburst and a rapid black hole growth, is experiencing a short transition phase towards an unobscured quasar.'
author:
- Lulu Fan
- 'Kirsten K. Knudsen'
- Yunkun Han
- 'Qing-hua Tan'
title: ' ALMA REVEALS A GAS-RICH, MAXIMUM-STARBURST IN THE HYPERLUMINOUS, DUST-OBSCURED QUASAR [W0533$-$3401]{} AT $z\sim2.9$'
---
Introduction {#sec:intro}
============
Super-massive black holes (SMBH) have been discovered in the centers of local elliptical galaxies, and the stellar bulge mass in galaxies is found to be correlated with the mass of their central SMBHs [e.g., @Magorrian1998; @Ferrarese2005], implying that galaxies and their central SMBHs could co-evolve [@Kormendy2013]. In the popular framework of massive galaxy formation and co-evolution with the central SMBH [e.g. @Hopkins2008], galaxy gas-rich mergers trigger intense starbursts and also provide the fuel for central SMBH accretion. An evolutionary sequence is predicted that the evolution of massive galaxies will experience several phases: starburst, dust-obscured quasar, unobscured quasar and finally a passively evolved galaxy [e.g., @Sanders1988; @Granato2004; @Alexander2012]. Dust-obscured quasars have been believed to represent a brief transition phase linking intense starbursts and unobscured quasars, and will be good candidates for studying the interplay between host galaxies and their central SMBHs[e.g., @Hickox2018].
Among several selection techniques commonly used to identify and characterize obscured quasar [see a recent review by @Hickox2018], the mid-IR color-color diagnostics [@Padovani2017] are efficient and effective for identifying heavily dust-obscured, powerful quasars. Recently, a new population of luminous, dust-obscured galaxies has been discovered based on a so-called [W1W2]{}-dropout color-selected method [@Eisenhardt2012; @Wu2012] which uses only four mid-IR wavebands of [*Wide-field Infrared Survey Explorer*]{} [[*WISE*]{}; @Wright2010] all-sky survey. Follow-up studies, including UV/optical spectral analysis [@Wu2012; @Wu2018], IR spectral energy distribution (SED) analysis [@Tsai2015; @Fan2016b; @Fan2018b], X-ray observations [@Stern2014; @Piconcelli2015; @Assef2016; @Ricci2017; @Vito2018; @Zappacosta2018] and high-resolution radio imaging [@Frey2016], suggest that these [*WISE*]{}-selected galaxies are mainly powered by accreting SMBHs and substantially dust-obscured quasars.
Multiwavelength observations have been carried out to investigate the physical properties of each component in these dust-obscured quasars, such as stars, dust, gas, the central SMBH, galaxy morphology and the environment they reside in [e.g., @Wu2014; @Assef2015; @Diaz-santos2016; @Diaz-santos2018; @Jones2015; @Jones2017; @Fan2016a; @Fan2017a; @Fan2018a; @Tsai2018]. All results are generally consistent with the merger-driven SMBH-host co-evolution scenario.
In this paper, we report the results of ALMA observations and a thorough UV-to-millimeter SED analysis of a [*WISE*]{}-selected dust-obscured quasar [W0533$-$3401]{} at $z\sim2.9$, which is among the most luminous obscured quasars with the total IR luminosity greater than $10^{14}L_\odot$ [@Tsai2015]. Our previous study based on IR SED decomposition suggests that [W0533$-$3401]{} has simultaneously rapid growth of SMBH and intense starburst ($>3000 M_\odot$ yr$^{-1}$) [@Fan2016b]. With the present ALMA observations and multiwavelength SED analysis, we can further derive the properties of each component in host galaxy and explore the potential relation between the assembly of host galaxy and the central SMBH accretion. In Section \[sec:alma\], we present our ALMA observations and data analysis on [W0533$-$3401]{}. In Section \[sec:multised\], we compile the multiwavelength data and introduce our SED modeling method. We show our main results and discussion in Section \[sec:res\]. In Section \[sec:sum\], we summarize our conclusions. Throughout this work we assume a standard, flat ${\rm \Lambda}$CDM cosmology [see @Komatsu2011], with $H_0 = 70$ km s$^{-1}$, $\Omega_M = 0.3$, and $\Omega_\Lambda = 0.7$.
ALMA observations and data analysis {#sec:alma}
===================================
Observations of [W0533$-$3401]{} were obtained with ALMA using the band-3 receiver as a part of project 2017.1.00441.S. The observations were carried out on 2017-Dec-20 using 45 antennas in a configuration with baseline length ranging from 15m to 2460m. The on-source integration time was 517s. The sources J0538$-$4405 and J0522$-$3627 were used for bandpass and flux calibration, and for gain calibration, respectively. The uncertainty on the absolute flux calibration was estimated to be about 5%. The precipitable water vapour (PWV) was measured to be 4.2-4.5mm and the weather conditions were stable during this relatively short period. The receiver settings were used as follows: the lower sideband has two spectral windows tuned to 87.418 and 89.219GHz with spectral line mode using 960 spectral channels each, where the tuning was selected to target [CO$(3-2)$]{} using the optical redshift $z_{\rm opt}=2.904$ [@Tsai2015], and the upper sideband has two spectral windows tuned to 99.421 and 101.221GHz in continuum mode with 128 channels each.
The data were processed using CASA (Common Astronomy Software Application[^1]; @Mcmullin07). We checked the data calibration from observatory delivered pipeline processing. We found that the calibration was sufficient and made no further adjustments. The calibrated visibilities were re-imaged using task [tclean]{}. For natural weighting, the angular resolution of the observations (clean beam size) is $0.61''\times0.53''$, at a position angle (P.A.) = $-78.4$degrees, and the rms is 0.43mJy/beam in 53kms$^{-1}$ channels. A summary of target properties and ALMA measurements can be found in Table \[tab:obssum\].
[lc]{}\
Name & [W0533$-$3401]{}\
R.A.$_{\it WISE}$ (J2000) & 05:33:58.44\
Dec$_{\it WISE}$ (J2000) & $-34$:01:34.5\
$z_{\rm opt}$ & 2.904\
Date of ALMA observations & 2017-Dec-20\
Number of antennas & 45\
R.A.$_{\rm CO(3-2)}$ (J2000) & 05:33:58.42\
Dec$_{\rm CO(3-2)}$ (J2000) & $-34$:01:34.5\
$z_{\rm CO(3-2)}$ & $2.9026\pm0.0003$\
Size$_{\rm CO(3-2)}$ (arcsec$^2$) & (0.73 $\pm$ 0.14)$\times$(0.37 $\pm$ 0.14)\
P.A.$_{\rm CO(3-2)}$ (deg) & 126 $\pm$ 18\
FWHM (km s$^{-1}$) & 566 $\pm$ 44\
$I_{\rm CO(3-2)}$ (Jy km s$^{-1}$) & 2.01 $\pm$ 0.13\
$L'_{\rm CO(3-2)}$ ($10^{10}$ K km s$^{-1}$ pc$^2$) & $8.4\pm0.5$\
$S_{\rm 3mm}$ \[mJy\] & 0.140$\pm$0.033\
Multiwavelength data and SED modeling {#sec:multised}
=====================================
UV-to-millimeter SED {#subsec:sed}
--------------------
We construct the multiwavelength SED of [W0533$-$3401]{} by compiling the optical to millimeter broadband photometry from various catalogs available in the literature (see Table \[tab:photometry\]). Optical/near-infrared photometry in five broad bands, $grizY$, are retrieved from the first public data release of the Dark Energy Survey [DES DR1, @desdr1] [^2]. Near-infrared $J$ band photometry has been obtained by SOAR/OSIRIS [@Assef2015]. The [[*WISE *]{}]{}W3 and W4 photometry of [W0533$-$3401]{} are from the ALLWISE Data Release [@Cutri2013]. W3 and W4 flux densities and uncertainties have been converted from catalog Vega magnitude by using zero points of 29.04 and 8.284 Jy, respectively [@Wright2010]. While for [[*WISE *]{}]{}W1 and W2 photometry, we do the aperture photometry based on the unblurred coadded [[*WISE *]{}]{}images[^3] [unWISE, @Lang2014; @Meisner2017]. The photometry errors have been estimated based on the inverse variance images. We also collect the FIR photometry of [W0533$-$3401]{} obtained with [[*Herschel *]{}]{}[@Pilbratt2010] PACS [@Poglitsch2010] at 70 and 160 $\mu$m and SPIRE [@Griffin2010] at 250, 350 and 500 $\mu$m in our previous work [@Fan2016b]. Our ALMA observations show a marginal detection ($\sim4.2\sigma$) of 3mm dust continuum. The measured observed-frame continuum flux density is $0.140\pm0.033$ mJy (see Figure \[fig:dust\]).
Multiwavelength SED analysis {#subsec:sed}
----------------------------
[lccc]{} CTIO/DECam $g$ & 0.36 & 833333 & 0.0052 $\pm$ 0.0002\
CTIO/DECam $r$ & 0.54 & 555556 & 0.0075 $\pm$ 0.0003\
CTIO/DECam $i$ & 0.64 & 468750 & 0.0099 $\pm$ 0.0005\
CTIO/DECam $z$ & 0.77 & 389610 & 0.0135 $\pm$ 0.0011\
CTIO/DECam $Y$ & 0.90 & 333333 & 0.0119 $\pm$ 0.0033\
SOAR/OSIRIS $J$ & 1.05 & 285714 & 0.0128 $\pm$ 0.0013\
[*WISE*]{}/W1 & 3.4 & 88174.2 & 0.0351 $\pm$ 0.0016\
[*WISE*]{}/W2 & 4.6 & 65172.3 & 0.0726 $\pm$ 0.0036\
[*WISE*]{}/W3 & 12 & 24982.7 & 3.0 $\pm$ 0.1\
[*WISE*]{}/W4 & 22 & 13626.9 & 11.9 $\pm$ 1.1\
[*Herschel*]{}/PACS & 70 & 4143.65 & 39.3 $\pm$ 5.9\
[*Herschel*]{}/PACS & 160 & 1805.05 & 97.4 $\pm$ 14.0\
[*Herschel*]{}/SPIRE & 250 & 1199.17 & 107.5 $\pm$ 4.8\
[*Herschel*]{}/SPIRE & 350 & 856.55 & 76.3 $\pm$ 7.3\
[*Herschel*]{}/SPIRE & 500 & 599.585 & 48.9 $\pm$ 4.5\
ALMA & 3000 & 100.00 & $0.140\pm0.033$\
For the multiwavelength SED analysis of [W0533$-$3401]{}, we use a forthcoming version (Han et al., in prep) of the Bayesian SED modeling and interpreting code [^4][@Han2012a; @HanY2014a; @HanY2019a]. In the new version of , the stellar emission, dust attenuation, and dust emission can be consistently connected by assuming an energy balance, a technique similar to that employed in [@dacunha2008] and [@Noll2009a; @BoquienM2019a]. The stellar emission is modeled by using the [@bc03] SSP with a [@chabrier2003] initial mass function (IMF), an exponentially declining star formation history (SFH) and the [@Calzetti2000a] dust attenuation law. The energy of stellar emission absorbed by dust is assumed to be totally re-emitted at IR band, which is modeled by a graybody. The graybody model is defined as: $S_{\lambda}\propto(1-e^{-(\frac{\lambda_0}{\lambda})^{\beta}}) B_\lambda(T_{dust})$, where $B_\lambda$ is the Planck blackbody spectrum and $\lambda_0$ = 125$\,\mu$m. Dust temperature $T_{\rm dust}$ and the emissivity index $\beta$ are two free parameters. Finally, as in [@Fan2016b], the AGN torus emission is modeled independently with the extensive database [^5] of 1,247,400 SEDs from the torus model [@Nenkova2008a; @Nenkova2008b]. A k-dimensional tree based nearest-neighbor searching technique has been employed to allow us to evaluate the torus model at any point in its 6-D parameter space. We remind that the model includes not only the torus dust emission, but also a part of the AGN accretion disk emission that is scattered into our line of sight or not absorbed by torus dust. Thus the model can provide a consistent description of AGN UV-to-millimeter SED. In total, the three-component model, including stars, AGN and cold dust emissions, has 12 free parameters. The priors for them are summarized in Table \[tab:priors\].
In Figure \[fig:sedfitting\], we show the best-fit three-component SED model (black solid line) with to the observed UV-to-millimeter SED of [W0533$-$3401]{} (red points). With a newly-developed function in the new version of , we can provide the confidence regions (CR) for our best-fit SED model. We plot the 68% and 95% CR with color-filled regions (cyan and purple, respectively). Absorbed stellar emission, AGN emission and cold dust emission have been shown in green dotted line, blue dashed line and gray dot-dashed line, respectively. The derived properties have been shown in Table \[tab:obsprop\] and the results will be discussed in the next section.
[lllll]{}\
${\rm log}(age/{\rm yr})$ & Uniform, $age<age_{\rm U}(z)$ & 5 & 10.3 & $6.7^{+0.1}_{-0.1}$\
${\rm log}(\tau/{\rm yr})$ & Uniform & 6 & 12 & $9.1^{+1.8}_{-1.9}$\
${\rm log}(Z/{\rm Z_{\odot}})$ & Uniform & -2.3 & 0.7 & $0.28^{+0.18}_{-0.38}$\
$A_{\rm V}/{\rm mag}$ & Uniform & 0 & 4 & $1.81^{+0.08}_{-0.08}$\
\
$T_{\rm dust}/{\rm K}$ & Uniform & 10 & 100 & $78.1^{+5.9}_{-5.2}$\
$\beta$ & Uniform & 1 & 3 & $1.84^{+0.14}_{-0.13}$\
\
$N_0$ & Uniform & 1 & 15 & $5.9^{+1.3}_{-1.0}$\
$Y$ & Uniform & 5 & 100 & $48.2^{+32.6}_{-28.2}$\
$i$ & Uniform & 0 & 90 & $37.8^{+20.0}_{-23.8}$\
$q$ & Uniform & 0 & 3 & $2.2^{+0.4}_{-0.4}$\
$\sigma$ & Uniform & 15 & 70 & $62.3^{+8.5}_{-5.1}$\
$\tau_{\rm V}$ & Uniform & 10 & 300 & $22.2^{+5.4}_{-5.1}$\
Results and discussion {#sec:res}
======================
CO(3-2) line emission {#subsec:gas}
---------------------
{width="49.00000%"} {width="49.00000%"}
In Figures \[fig:coimg01\] and \[fig:cospec\], we show the resulting line detection, both as spectrum and moment-maps. The [CO$(3-2)$]{} emission line peaks at a frequency corresponding to a redshift of $z_{\rm CO(3-2)}=2.9026\pm0.0003$. The detected [CO$(3-2)$]{} line is spatially extended and shows a velocity gradient. We fit the velocity-integrated visibilities with an elliptical Gaussian function. The derived source size deconvolved from beam is (0.73 $\pm$ 0.14)$\times$(0.37 $\pm$ 0.14) arcsec$^2$ with P.A. = $126\pm18$. We measure the line flux $I_{\rm CO(3-2)}=2.01\pm0.13$ Jy km s$^{-1}$. The rms is determined in the inner 10$''$ of the moment map excluding the central part that contains the source itself. A single Gaussian fit to the continuum-subtracted [CO$(3-2)$]{} spectrum gives [CO$(3-2)$]{} FWHM = $566\pm44$ km s$^{-1}$. From the [CO$(3-2)$]{} line flux, we can derive the CO line luminosity $L'_{\rm CO(3-2)}=(8.4\pm0.5)\times10^{10}$ K km s$^{-1}$ pc$^2$, by using Equation 3 in @solomon2005. The continuum is marginally detected, as shown in Figure \[fig:dust\]. The extend of the continuum emission is smaller than that of the [CO$(3-2)$]{} line, which is likely due to a combination of the optical depth effects and the signal-to-noise ratio of continuum detection. The results are summarized in Table \[tab:obssum\].
We can infer the molecular gas mass directly from the measured CO line luminosity. We adopt the excitation ratio between the [CO$(3-2)$]{} and CO$(1-0)$ line of $r_{32/10} = 0.8$ as suggested by [@Banerji2017]. The adopted ratio is intermediate between the typical values for submillimeter galaxies (SMGs, $r_{32/10} = 0.66$) and optical quasars ($r_{32/10} = 0.97$) from [@Carilli2013], and consistent with the expectation that obscured quasars represent the transition phase from SMGs to unobscured quasars. The calculated CO$(1-0)$ line luminosity is $L'_{\rm CO(1-0)}=(10.5\pm0.6)\times10^{10}$ K km s$^{-1}$ pc$^2$. We also adopt the CO-to-H$_2$ conversion factor, $\alpha_{\rm CO}=0.8~M_\odot$(K km s$^{-1}$ pc$^2$)$^{-1}$, which is suggested to be appropriate for starbursts and quasar hosts [@Carilli2013]. Under these considerations, we obtain a molecular gas mass of $M_{\rm H_2}=(8.4\pm0.5)\times10^{10}\,M_\odot$.
We show the [CO$(3-2)$]{} velocity map in the right panel of Figure \[fig:coimg01\] and the Position versus velocity (PV) diagram extracted along the major axis at P.A. = 330$^{\circ}$ in Figure \[fig:pv\]. The [CO$(3-2)$]{} emission line is marginally resolved. Both figures show the possible presence of a velocity gradient. Such a velocity gradient, which has also been observed in many other high-redshift quasars [e.g., @Leung2017; @Brusa2018; @Feruglio2018; @Talia2018], could be possibly the signature of a rotating disk of molecular gas, although the explanation is not unique. Especially, tentatively emission is seen at $\sim3\sigma$ level extending to the west side of the source (see the left panel of Figure \[fig:coimg01\]). This suggests that the gas-rich late-stage major merger and gas outflow are also possible for originating such a velocity gradient [e.g., @Springel2005; @Hopkins2009; @Ueda2014; @Hung2015]. However, due to limited depth and resolution, we cannot make a solid conclusion. Deeper observation would be required to distinguish them.
Assuming that the gas is distributed in a rotating disk, we can investigate the kinematic properties of the molecular gas traced by the [CO$(3-2)$]{} line, using [$^{3D}$BAROLO]{}, a tool for fitting 3D tilted-ring models to emission-line data cubes [@DiTeodoro2015]. We assume a disc model with three rings and a ring width of 0.2$''$. We fix P.A.$ = 330^{\circ}$ and adopt an inclination of 59$^{\circ}\pm 14^{\circ}$, which is inferred from the observed ratio of minor to major axis. The derived rotation velocity ($V_{\rm rot}$) and the intrinsic velocity dispersion $\sigma$ are about $\sim240$ km s$^{-1}$ and 60 km s$^{-1}$, respectively. The ratio of rotation velocity to velocity dispersion $V_{\rm rot}/\sigma$ for [W0533$-$3401]{} is about 4, which is close to the typical value $\sim7$ for the molecular gas in $z>1$ star-forming galaxies [@Tacconi2013]. The large ratio $V_{\rm rot}/\sigma$ indicates that the molecular gas is turbulent, which is possibly due to a thick, dynamically hot disk in [W0533$-$3401]{}. A similar finding has been reported by [@Tadaki2018] that they found a gravitationally unstable, rotating gas disk in an extreme starburst galaxy at $z\sim4$. The dynamical mass within the CO-emitting region can be estimated by applying the relation in [@Wang2013]: $M_{\rm dyn}/M_\odot=1.16\times10^5\times(0.75\times{\rm FWHM_{\rm CO}})^2\times D/{\rm sin}i$, where $D$ is the disk diameter in kpc from the [CO$(3-2)$]{} measurement and $i$ is the inclination angle. This gives $M_{\rm dyn}=1.6\times10^{11}M_\odot$.
Dust properties {#subsec:dust}
---------------
Cold dust emission heated by stars has been modeled with a graybody model in our multiwavelength SED analysis (see Section \[subsec:sed\] and Figure \[fig:sedfitting\]). The best-fit model gives the estimations of dust properties: IR luminosity $L_{\rm GB}$, dust temperature $T_{\rm dust}$ and the emissivity index $\beta$, which are listed in Table \[tab:obsprop\]. The derived values are well consistent with those in our previous work based on only IR SED decomposition [@Fan2016b]. Dust mass can also be calculated by using the following equation: $$\label{equ:mdust}
M_{\rm dust}=\frac{D^2_{\rm L}}{(1+z)}\times\frac{S_{\nu_{\rm obs}}}{\kappa_{\nu_{\rm rest}} B(\nu_{\rm rest},T_{\rm dust})}$$ where $D_{\rm L}$ is the luminosity distance, $S_{\nu_{\rm obs}}$ is the flux density at observed frequency $\nu_{\rm obs}$, $\kappa_{\nu_{\rm rest}}=\kappa_0(\nu/\nu_0)^\beta$ is the dust mass absorption coefficient at the rest frequency of the observed band, and $B(\nu_{\rm rest},T_{\rm dust})$ is the Planck function at temperature $T_{\rm dust}$. Adopting the best-fit values $T_{\rm dust}=78.1$ K, $\beta=$1.84 and $\kappa_{\rm 1THz}=20$ cm$^2$ g$^{-1}$ which is the same as in [@Wu2014] and [@Fan2016b], we derive the dust mass $M_{\rm dust}=8.9\pm0.5\times10^7M_\odot$. The small uncertainty of dust mass estimation only takes the uncertainties of the derived $T_{\rm dust}$ and $\beta$ values into account. We note that the largest uncertainty can arise from the adopted $\kappa_{\nu_{\rm rest}}$ value, which can vary by over one order of magnitude at a certain frequency/wavelength. For instance, $\kappa_{850\mu m}$ can vary from $\sim$0.4 to $\sim$11 cm$^2$ g$^{-1}$ in the literature [e.g., @james2002; @dunne2003; @draine2003; @Siebenmorgen2014].
We derive the gas-to-dust mass ratio of [W0533$-$3401]{}, $\delta_{\rm GDR} = 944\pm77$, based on the estimations of $M_{\rm H_2}$ and $M_{\rm dust}$. The derived $\delta_{\rm GDR}$ value of [W0533$-$3401]{} is thus about one order of magnitude higher than the typical value $\sim50-150$ derived for the Milky Way [@Jenkins2004], the local star-forming galaxies (SFGs) and ultraluminous IR galaxies (ULIRGs) at solar metallicity [@Draine2007; @Remy2014] and high-redshift SMGs [@Magnelli2012; @Miettinen2017]. The high $\delta_{\rm GDR}$ value in [W0533$-$3401]{} may be due to several possible reasons: the uncertainty of dust mass estimation, the low efficiency of dust formation and/or the high efficiency of dust destruction. The dust mass derived by using a graybody model is about half of that derived by [@Draine2007b] model [@Magdis2011]. This will result in an overestimation of $\delta_{\rm GDR}$ by a factor of two. The $\delta_{\rm GDR}$ value is reported to increase with the decreasing metallicity and the increasing redshift [e.g., @Remy2014; @Miettinen2017]. It is possible that [W0533$-$3401]{} has a low, sub-solar metallicity. It is also possible that dust destruction in [W0533$-$3401]{} may be efficient due to the strong radiation field from massive young stars and AGN, and the supernova shock waves which are expected to be frequent in this maximum-starburst galaxy [@Jones2004].
Stellar mass and star formation rate {#subsec:stars}
------------------------------------
[lclc]{} $M_\star$ & (3.5 $\pm$ 0.9) $\times10^{10}$ & $M_\odot$ & (1)\
SFR & 6985 $\pm$ 3006 & $M_\odot$ yr$^{-1}$ & (2)\
${\rm sSFR}$ & 200 $\pm$ 100 & Gyr$^{-1}$ & (3)\
$L_{\rm AGN}$ & (7.0 $\pm$ 0.6)$\times10^{13}$ & $L_\odot$ & (4)\
$L_{\rm GB}$ & (3.5 $\pm$ 0.4)$\times10^{13}$ & $L_\odot$ & (5)\
$L_\star^{\rm unabs}$ & (3.6 $\pm$ 0.4)$\times10^{13}$ & $L_\odot$ & (6)\
$L_\star^{\rm abs}$ & (9.7 $\pm$ 1.0)$\times10^{11}$ & $L_\odot$ & (7)\
$M_{\rm H_2}$ & ($8.4\pm0.5$)$\times10^{10}$ & $M_\odot$ & (8)\
$M_{\rm dyn}$ & $1.6\times10^{11}$ & $M_\odot$ & (9)\
$M_{\rm BH}$ & $2.2\times10^9$ & $M_\odot$ & (10)\
$M_{\rm dust}$ & (8.9 $\pm$ 1.6) $\times 10^{7}$ & $M_\odot$ & (11)\
T$_{\rm dust}$ & 78.1 $\pm$ 5.6 & K & (12)\
$\beta$ & 1.84 $\pm$ 0.14 & & (13)\
${\rm A_V}$ & 1.81 $\pm$ 0.08 & mag & (14)\
${\rm t_{depl}}$ & $\sim12-28$ & Myr & (15)\
Based on our best-fit SED model presented in Section \[subsec:sed\], we derive the stellar mass and star formation rate (SFR) of [W0533$-$3401]{} and list them in Table \[tab:obsprop\]. The stellar mass $M_\star$ is derived by adopting an exponentially declining SFH, which is represented by a young stellar population. We check if there is an old stellar population which is omitted by the present SED fitting procedure. We add an SSP with an age of 1Gyr to our multiwavelength SED model. The result is that the contribution of the old SSP to the derived stellar mass is not larger than 20%, though the constraint from the SED fitting is loose. Given $M_\star=3.5\times10^{10}M_\odot$ and $M_{\rm H_2}=8.4\times10^{10}M_\odot$, we can calculate the molecular gas fraction $f_{\rm gas}=M_{\rm H_2}/(M_{\rm H_2}+M_\star)=0.71$, which indicates that [W0533$-$3401]{} is gas-rich. The derived SFR is $\sim7000\ M_\odot$yr$^{-1}$. The specific SFR or sSFR = 200 Gyr$^{-1}$ of [W0533$-$3401]{} is over one order of magnitude higher than the star-forming main-sequence (MS) at $z\sim3$ [@Speagle2014], suggesting that it is a maximum-starburst. The uncertainty of SFR is large ($\sim3000\ M_\odot$yr$^{-1}$) due to a wide range of possible SFH. By using CO line luminosity, we can estimate SFR with an individual way. Firstly, we convert the measured $L'_{\rm CO(3-2)}$ to $L'_{\rm CO(5-4)}$ by taking $L'_{\rm CO(5-4)}/L'_{\rm CO(3-2)}=0.7$ which is intermediate between the typical values for SMGs and quasars [@Carilli2013]. Then we estimate FIR luminosity using high-J CO versus FIR luminosity relation presented in [@Liu2015]. Using the relation SFR ($M_\odot$yr$^{-1}$)=$4.5\times10^{-44}L_{\rm FIR}$ (erg s$^{-1}$), we calculate SFR$\sim3000\ M_\odot$yr$^{-1}$ [@Kennicutt1998], which is generally consistent with the best-fit SED result. From the derived SFR and molecular gas mass, we can estimate the gas depletion timescale $t_{\rm depl}=M_{\rm gas}/$SFR. We infer $t_{\rm depl}\sim 12-28$ Myr using SFRs based on the best-fit SED result and the CO line luminosity, respectively. The gas depletion timescale of [W0533$-$3401]{} is similar to other obscured quasars [e.g., @Aravena2008; @Brusa2018], but is much shorter than MS galaxies and SMGs [@Bothwell2013; @Sargent2014], indicating that [W0533$-$3401]{} as an obscured quasar is consuming its residual gas more rapidly.
In Figure \[fig:sfe\], we report the star formation efficiency (SFE), traced by the ratio of IR to CO$(1-0)$ line luminosities, as a function of IR luminosity for [W0533$-$3401]{} and the compiled samples of SMGs, unobscured and obscured quasars at $z>1$ in [@Perna2018]. For all three samples, SFE shows a positive correlation with IR luminosity. [W0533$-$3401]{} shows a high SFE=$L_{\rm IR}/L'_{\rm CO}=464\pm56\ L_\odot/$(K km s$^{-1}$ pc$^2$), which is well above the best-fit relation and 1$\sigma$ scatter for massive MS galaxies [@Sargent2014]. Similar as [W0533$-$3401]{}, both unobscured and obscured quasars at $z>1$ show a higher SFE at a given IR luminosity than MS galaxies and SMGs. Higher SFE in unobscured and obscured quasars relative to MS galaxies and SMGs is possibly due to the presence of starburst activity and/or the depletion of cold gas by AGN feedback in quasar host galaxies. The former enhances $L_{\rm IR}$ and the latter reduces $L'_{\rm CO}$. It is possible that $L_{\rm IR}$ can be overestimated in quasars, as AGN-heated dust on kpc scales can contribute significantly to IR luminosity [@Duras2017; @Symeonidis2017].
Rapid growth of both the stellar component and the central SMBH
---------------------------------------------------------------
We derive the AGN bolometric luminosity $L_{\rm AGN}$ of [W0533$-$3401]{} by integrating the AGN component of the best-fit UV-to-millimeter SED (see Section \[subsec:sed\]). Although we do not have a direct measurement of the black hole mass in [W0533$-$3401]{}, we can make a rough estimate from the derived $L_{\rm AGN}$. By assuming Eddington ratios of 0.3, 1.0 and 3.0, the corresponding black hole masses are $7.3\times10^9$, $2.2\times10^9$ and $7.3\times10^8\ M_\odot$, respectively. [@Wu2018] reported the black hole masses and Eddington ratios of five luminous obscured quasars at $z\sim2$, which are taken from the same parent sample of [W0533$-$3401]{}, based on broad H$_\alpha$ lines. They found that the average black hole mass is about $10^9\ M_\odot$ and the derived Eddington ratios are close to unity. [@Tsai2018] also reported the measurement of $M_{\rm BH}$ and $L_{\rm Edd}$ for an extremely luminous, obscured quasar W2246$-$0526 at $z=4.6$, which is selected with the same criterion as [W0533$-$3401]{}. They found that the central SMBH in it is growing rapidly by accreting at a super-Eddington ratio ($\lambda_{\rm Edd}=2.8$). Adopting the Eddington ratio $\lambda_{\rm Edd}=1.0$ for [W0533$-$3401]{} will be a reasonable approximation.
From SED-based stellar mass $M_\star$ and the estimated SMBH mass, we can infer the black hole to stellar mass ratio of [W0533$-$3401]{}. In Figure \[fig:mbh2mstar\], we plot the observed $M_{\rm BH}/M_\star$ as a function of redshift for [W0533$-$3401]{} and several other samples in the literature. If assuming $\lambda_{\rm Edd}=1.0$, the black hole to stellar mass ratio of [W0533$-$3401]{} ($M_{\rm BH}/M_\star=0.063$) is close to that of an X-ray selected unobscured quasar, CID-947, which has the highest $M_{\rm BH}/M_\star$ (1/8) at $z\sim3.3$ known so far. A low Eddington ratio in [W0533$-$3401]{}, for instance $\lambda_{\rm Edd}<0.3$, seems not likely, otherwise $M_{\rm BH}/M_\star$ of [W0533$-$3401]{} will be higher than 0.2. Even if taking $\lambda_{\rm Edd}=1.0$, the inferred $M_{\rm BH}/M_\star$ of [W0533$-$3401]{} is not only over one order of magnitude higher than the typical values $\sim0.0002-0.0005$ in the local Universe [@Kormendy2013], but also $\sim5$ times higher than the expected value by the evolutionary trend of $M_{\rm BH}/M_\star$ [@McLure2006; @Peng2006; @Merloni2010; @Targett2012; @Bongiorno2014; @Matsuoka2018]. We derive the black hole mass growth rate ${\dot{M}}_{\rm BH}$ of [W0533$-$3401]{} from the bolometric luminosity by using the relation $L_{\rm AGN}=(\eta {\dot{M}}_{\rm BH} c^2)/({1-\eta})$ and adopting $\frac{\eta}{1-\eta}=0.1$. We infer ${\dot{M}}_{\rm BH}$ = 49 $M_\odot$ yr$^{-1}$, which suggests a rapid growth of the central SMBH. We then calculate the ratio between the black hole mass growth rate and star formation rate and obtain ${\dot{M}}_{\rm BH}$/SFR = 0.007, which is close to the local $M_{\rm BH}/M_\star$ values. The result indicates that [W0533$-$3401]{} is evolving towards the evolutionary trend of $M_{\rm BH}/M_\star$ (the dashed line in Figure \[fig:mbh2mstar\]).
Summary and Conclusions {#sec:sum}
=======================
We present ALMA observations of cold dust and molecular gas and multiwavelength SED analysis in a [*WISE*]{}-selected, hyperluminous dust-obscured quasar [W0533$-$3401]{} at $z=2.9$. We derive the physical properties of each component, such as molecular gas, stars, dust and the central SMBH. We summarize our main results as follows.
1. Our ALMA band-3 observations detect both the dust continuum and the [CO$(3-2)$]{} line. We derive molecular gas mass, $M_{\rm gas}=8.4\times10^{10}\ M_\odot$ and its fraction $f_{\rm gas}=0.7$ suggest that [W0533$-$3401]{} is gas-rich. Based on the FWHM of the [CO$(3-2)$]{} line, we estimate the dynamical mass $M_{\rm dyn}=1.6\times10^{10}\ M_\odot$, which is generally consistent with the sum of the molecular gas mass and stellar mass. The velocity map of the [CO$(3-2)$]{} emission line showing a velocity gradient is possibly due to a rotating gas disk. However, other possibilities, such as mergers and outflows, cannot be ruled out. Under the assumption of rotation disk, we can roughly estimate the rotation velocity and velocity dispersion. The ratio $V_{\rm rot}/\sigma$ is large, indicating that the gas disk is possibly unstable, which may be partly responsible for the observed high star formation efficiency in [W0533$-$3401]{}.
2. By assuming a graybody model, we derive the cold dust temperature $T_{\rm dust}=78.1\pm5.6$ K and dust mass $M_{\rm dust}=(8.9\pm1.6)\times10^7\ M_\odot$. The gas-to-dust ratio of [W0533$-$3401]{}, $\sigma_{\rm GDR}=944\pm77$, is about one order of magnitude higher than the typical values for the Milky Way, the local SFGs/ULIRGs and high-redshift SMGs.
3. Based on the UV-to-millimeter SED modeling, we derive the stellar mass and SFR of [W0533$-$3401]{}. The stellar mass of [W0533$-$3401]{} is $(3.5\pm0.9)\times10^{10}\ M_\odot$. The star formation rate (SFR) is estimated to be $\sim3000-7000\ M_\odot$ yr$^{-1}$ by using different methods. The high values of SFR and specific SFR suggest [W0533$-$3401]{} is a maximum-starburst. The corresponding gas depletion timescales are very short ($t_{\rm depl}\sim12-28$ Myr).
4. Finally, we infer the black hole mass growth rate of [W0533$-$3401]{} (${\dot{M}}_{\rm BH}$ = 49 $M_\odot$ yr$^{-1}$), which suggests a rapid growth of the central SMBH. The observed black hole to stellar mass ratio $M_{\rm BH}/M_\star$ of [W0533$-$3401]{}, which is dependent on the adopted Eddington ratio, is over one order of magnitude higher than the local value if assuming a reasonable Eddington ratio $\lambda_{\rm Edd}=1.0$, and is evolving towards the evolutionary trend of unobscured quasars.
5. Our results suggest that [W0533$-$3401]{} has both a gas-rich maximum-starburst and a rapid black hole growth within it. All results are consistent with the scenario that [W0533$-$3401]{} is experiencing a short transition phase towards an unobscured quasar.
We thank the anonymous referee for constructive comments and suggestions. We thank Dr. Hu Zou (NAOC) for his help on optical photometry. This work is supported by National Key R&D Program of China (No. 2017YFA0402703). We thank the staff of the Nordic ALMA Regional Center (ARC) node for their support and helpful discussions. The Nordic ARC node is based at Onsala Space Observatory and funded through Swedish Research Council grant No 2017-00648. LF acknowledges the support from the National Natural Science Foundation of China (NSFC, Grant Nos. 11822303, 11773020 and 11433005) and Shandong Provincial Natural Science Foundation, China (ZR2017QA001, JQ201801). KK acknowledges support from the Knut and Alice Wallenberg Foundation and the Swedish Research Council. YH acknowledges the support from NSFC (Grant No. 11773063) and Natural Science Foundation of Yunnan Province (Grant No. 2017FB007). Q.-H.T acknowledges the support from the NSFC (Grant No. 11803090).
This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2017.1.00441.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.
This paper makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.
This paper used public archival data from the Dark Energy Survey (DES). Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Funda[ç]{}[ã]{}o Carlos Chagas Filho de Amparo [à]{} Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cient[í]{}fico e Tecnol[ó]{}gico and the Minist[é]{}rio da Ci[ê]{}ncia, Tecnologia e Inova[ç]{}[ã]{}o, the Deutsche Forschungsgemeinschaft, and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energ[é]{}ticas, Medioambientales y Tecnol[ó]{}gicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgen[ö]{}ssische Technische Hochschule (ETH) Z[ü]{}rich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ci[è]{}ncies de l’Espai (IEEC/CSIC), the Institut de F[í]{}sica d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universit[ä]{}t M[ü]{}nchen and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, The Ohio State University, the OzDES Membership Consortium, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University. Based in part on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.
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[^1]: <https://casa.nrao.edu/>
[^2]: <https://des.ncsa.illinois.edu/releases/dr1/>
[^3]: <https://unwise.me/>
[^4]: <https://bitbucket.org/hanyk/bayesed/>
[^5]: [http://www.pa.uky.edu/clumpy/models/ clumpy\_models\_201410\_tvavg.hdf5/](http://www.pa.uky.edu/clumpy/models/ clumpy_models_201410_tvavg.hdf5/)
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abstract: 'Christos Athanasiadis [@Christos] studies an effective technique to show that Gorenstein sequences coming from compressed polytopes are unimodal. In the present paper we will use such the technique to find a rich class of Gorenstein toric rings with unimodal $h$-vectors arising from finite graphs.'
author:
- Hidefumi Ohsugi and Takayuki Hibi
title: Special simplices and Gorenstein toric rings
---
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Introduction {#introduction .unnumbered}
============
Let $\PP \subset \RR^N$ be an integral convex polytope, i.e., a convex polytope each of whose vertices has integer coordinates. Let $K[\xb, \xb^{-1}, t] =
K[x_1, x_1^{-1}, \ldots, x_N, x_N^{-1}, t]$ denote the Laurent polynomial ring in $(N + 1)$ variables over a field $K$. The [*toric ring*]{} of $\PP$ is the subalgebra $K[\PP]$ of $K[\xb, \xb^{-1}, t]$ generated by those Laurent polynomials $\xb^{\ab}t = x_1^{a_1} \cdots x_N^{a_N} t$ such that $\ab = (a_1, \ldots, a_N)$ is a vertex of $\PP$. We will regard $K[\PP]$ as a homogeneous algebra [@BruHer p. 147] by setting each $\deg \xb^{\ab}t = 1$ and write $F(K[\PP], \lambda)$ for its Hilbert series. One has $F(K[\PP], \lambda) =
(h_0 + h_1 \lambda + \cdots + h_s \lambda^s) / (1 - \lambda)^{d+1}$, where each $h_i \in \ZZ$ with $h_s \neq 0$ and where $d$ is the dimension of $\PP$. The sequence $(h_0, h_1, \ldots, h_s)$ is said to be the $h$-[*vector*]{} of $K[\PP]$. If the toric ring $K[\PP]$ is normal, then $K[\PP]$ is Cohen–Macaulay. If $K[\PP]$ is Cohen–Macaulay, then the $h$-vector of $K[\PP]$ is nonnegative, i.e., each $h_i \geq 0$. Moreover, if $K[\PP]$ is Gorenstein, then the $h$-vector of $K[\PP]$ is symmetric, i.e., $h_i = h_{s-i}$ for all $i$.
An outstanding conjecture (which is still open) is that the $h$-vector of a Gorenstein toric ring is unimodal, i.e., $h_0 \leq h_1 \leq \cdots \leq h_{[s/2]}$. One of the established techniques to show that the $h$-vector $(h_0, h_1, \ldots, h_s)$ of a Gorenstein toric ring $K[\PP]$ is unimodal is to find a simplicial convex polytope [@Stanley] of dimension $s - 1$ whose $h$-vector coincides with $(h_0, h_1, \ldots, h_s)$. On the other hand, however, given a Gorenstein toric ring $K[\PP]$, it seems difficult to find such a simplicial convex polytope.
Christos Athanasiadis [@Christos] introduces the concept of a special simplex of a convex polytope. Let $\PP \subset \RR^N$ be a convex polytope. A $(q-1)$-simplex $\Sigma$ each of whose vertices is a vertex of $\PP$ is said to be a [*special simplex*]{} in $\PP$ if each facet of $\PP$ contains exactly $q - 1$ of the vertices of $\Sigma$. Recall that an integral convex polytope $\PP \subset \RR^N$ is [*compressed*]{} [@Sta p. 337] (and [@OhHicompressed]) if all “pulling triangulations” of $\PP$ are unimodular. The toric ring $K[\PP]$ of a compressed polytope $\PP$ is normal. It turns out [@Christos Theorem 3.5] that if $\PP$ is compressed and if there is a special simplex in $\PP$, then the $h$-vector of $K[\PP]$ is equal to the $h$-vector of a simplicial convex polytope.
In the present paper we will use [@Christos Theorem 3.5] to study the $h$-vector of the toric ring of the edge polytope of a finite graph satisfying the odd cycle condition as well as that of the stable polytope of a perfect graph.
Two polytopes arising from finite graphs
========================================
Let $G$ be a finite graph on the vertex set $[n] = \{ 1, 2, \ldots, n \}$ having no loops and no multiple edges, and $E(G)$ the edge set of $G$. We associate each subset $W \subset [n]$ with the $(0, 1)$-vector $\rho(W) = \sum_{j \in W} \eb_j \in \RR^n$. Here $\eb_j$ is the $j$-th unit coordinate vector in $\RR^n$. Thus in particular $\rho(\emptyset)$ is the origin of $\RR^n$. A subset $W \subset [n]$ is called [*stable*]{} (resp. a [*clique*]{}) if $\{i, j \} \not\in E(G)$ (resp. $\{i, j \} \in E(G)$) for all $i, j \in W$ with $i \neq j$. Note that the empty set as well as each single-element susbset of $[n]$ is both stable and a clique. Let $S(G)$ denote the set of stable sets of $G$.
We now introduce two convex polytopes arising from a finite graph $G$ on $[n]$. First, the [*edge polytope*]{} [@OhHinormal] of $G$ is the $(0, 1)$-polytope $\PP_G \subset \RR^n$ which is the convex hull of $\{ \rho(e) \, : \, e \in E(G) \}$. Second, the [*stable polytope*]{} [@Chvatal] of $G$ is the $(0, 1)$-polytope $\QQ_G \subset \RR^n$ which is the convex hull of $\{ \rho(W) \, : \, W \in S(G) \}$.
\[posetpolytope\] [*Let $P$ be a finite poset on $[n]$ and $\com(P)$ its comparability graph. Thus $\com(P)$ is the finite graph on $[n]$ such that $\{ i , j \}$ with $i \neq j$ is an edge of $\com(P)$ if and only if $i$ and $j$ are comparable in $P$. Then the stable polytope of $\com(P)$ coincides with the chain polytope [@twoposetpolytopes] of $P$.* ]{}
The problem when the toric ring $K[\PP_G]$ is normal and the problem when the edge polytope $\PP_G$ possesses a unimodular covering [@OhHinormal p. 420] are studied in [@OhHinormal] (and [@SVV]).
\[normal\] Given a finite connected graph $G$, the following conditions are equivalent:
1. The toric ring $K[\PP_G]$ is normal;
2. The edge polytope $\PP_G$ possesses a unimodular covering;
3. $G$ satisfies the odd cycle condition, i.e., if each of $C$ and $C'$ is an odd cycle (a cycle of odd length) of $G$ and if $C$ and $C'$ possess no common vertex, then there exists an edge $\{ i, j \}$ of $G$ such that $i$ is a vertex of $C$ and $j$ is a vertex of $C'$.
Thus in particular the edge polytope of a finite connected bipartite graph possesses a unimodular covering and its toric ring is normal.
A [*chromatic number*]{} of a finite graph $G$ on $[n]$ is the smallest integer $\ell > 0$ for which there is a map $\varphi : [n] \to [\ell]$ with the property that $\varphi(i) \neq \varphi(j)$ if $\{i, j \} \in E(G)$. A finite graph $G$ is called [*perfect*]{} if, for all induced subgraphs $H$ of $G$ including $G$ itself, the chromatic number of $H$ is equal to the maximal cardinality of cliques contained in $H$. The comparability graph of a finite partially ordered set is perfect ([@Berge]).
The facets of the edge polytope $\PP_G$ of a finite connected graph $G$ is completely determined ([@OhHinormal Theorem 1.7]). On the other hand, the facets of the stable polytope $\QQ_G$ is completely determined when $G$ is a perfect graph ([@Chvatal Theorem 3.1]).
Gorenstein toric rings
======================
When $K[\PP_G]$ (resp. $K[\QQ_G]$) is normal, it follows easily that $K[\PP_G]$ (resp. $K[\QQ_G]$) coincides with the [*Ehrhart ring*]{} [@Hibi p. 97] of $\PP_G$ (resp. $\QQ_G$). On the other hand, since the equations of the facets of $\PP_G$ (resp. $\QQ_G$) are known, when $K[\PP_G]$ (resp. $K[\QQ_G]$) is normal, by using the criterion [@DeNegri--Hibi Corollary (1.2)] one can determines the finite graphs $G$ for which the toric ring $K[\PP_G]$ (resp. $K[\QQ_G]$) is Gorenstein.
Let $G$ be a finite connected graph on $[n]$. Given a subset $V \neq \emptyset$ of $[n]$, write $G_V$ for the induced subgraph of $G$ on $V$. We say that $G$ is [*$2$-connected*]{} if $G$ together with $G_{[n] \setminus \{ i \}}$ for all $i \in [n]$ is connected. If $i \in [n]$, then $N(G;i)$ stands for the set of vertices $j$ with $\{ i, j \} \in E(G)$. If $T \subset [n]$, then $N(G;T) = \Union_{i \in T} N(G;i)$. The [*bipartite graph induced by*]{} a stable set $T \neq \emptyset$ of $G$ is the bipartite graph on the vertex set $T \Union N(G;T)$ consisting of those edges $\{ i, j \}$ of $G$ with $i \in T$ and $j \in N(G;T)$.
Recall that a [*matching*]{} of $G$ is a set of edges $\{ e_1, \ldots, e_m \}$ such that $e_i \Sect e_j = \emptyset$ for all $i \neq j$. A matching $\{ e_1, \ldots, e_m \}$ of $G$ is called [*perfect*]{} if $\Union_{i=1}^{m} e_i = [n]$. In particular $n$ is even and $m = n / 2$ if $G$ possesses a perfect matching $\{ e_1, \ldots, e_m \}$. It follows that $G$ possesses a perfect matching if and only if the monomial $x_1 x_2 \cdots x_n$ belongs to the toric ring $K[\PP_G]$.
\[Gorenstein\] [*(a)*]{} Let $G$ be a finite connected graph on $[n]$ satisfying the odd cycle condition and suppose that every connected component of $G_{[n] \setminus \{ i \}}$ possesses at least one odd cycle for all $i \in [n]$. Then the toric ring $K[\PP_G]$ of the edge polytope $\PP_G$ of $G$ is Gorenstein if and only if (i) $G$ possesses a perfect matching, (ii) one has $|N(G;T)| = |T| + 1$ for each stable set $T$ of $G$ such that the bipartite graph induced by $T$ is connected with $T \Union N(G;T) \neq [n]$ and that every connected component of $G_{[n] \setminus (T \Union N(G;T))}$ has at least one odd cycle and (iii) one has $|T| = n/2 - 1$ for each stable set $T$ of $G$ such that the bipartite graph induced by $T$ is connected with $T \Union N(G;T) = [n]$.
[*(a’)*]{} Let $G$ be a bipartite graph on $[n] = V_1 \Union V_2$ and suppose that $G$ is $2$-connected. Then the toric ring $K[\PP_G]$ of the edge polytope $\PP_G$ of $G$ is Gorenstein if and only if (i) $G$ possesses a perfect matching and (ii) one has $|N(G;T)| = |T| + 1$ for every subset $T \subset V_1$ such that $G_{T \Union N(G;T)}$ is connected and that $G_{[n] \setminus (T \Union N(G;T))}$ is a connected graph with at least one edge.
[*(b)*]{} The toric ring $K[\QQ_G]$ of a stable polytope $\QQ_G$ of a perfect graph $G$ is Gorenstein if and only if all maximal cliques have the same cardinality.
\(a) The edge polytope $\PP_G \subset \RR^n$ lies on the hyperplane $\HH$ defined by the equation $z_1 + \cdots + z_n = 2$. Let $\pi : \RR^{n-1} \to \HH$ denote the affine map defined by setting $\pi(z_1, \ldots, z_{n-1})
= (z_1, \ldots, z_{n-1}, 2 - (z_1 + \cdots + z_{n-1}))$. Then $\pi$ is an affine isomorphism with $\pi(\ZZ^{n-1}) = \HH \Sect \ZZ^n$. Hence $\pi^{-1}(\PP_G) \subset \RR^{n-1}$ is an integral convex polytope with $\dim \pi^{-1}(\PP_G) = n - 1$ and the toric ring $K[\pi^{-1}(\PP_G)]$ is isomorphic to $K[\PP_G]$ as homogeneous algebras over $K$.
Let $\delta$ denote the smallest integer for which the interior of $\delta (\pi^{-1}(\PP_G))$ contains at least one integer point $(a_1, \ldots, a_{n-1})$. Since every connected component of $G_{[n] \setminus \{ i \}}$ possesses at least one odd cycle, it follows from [@OhHinormal Theorem 1.7 (a)] that the hyperplane defined by the equation $z_i = 0$ is a supporting hyperplane which defines a facet of $\PP_G$ for each $1 \leq i \leq n$. Thus the hyperplane defined by the equation $z_i = 0$ is a supporting hyperplane which defines a facet of $\pi^{-1}(\PP_G)$ for each $1 \leq i < n$. Thus by using [@DeNegri--Hibi Corollary (1.2)] one has each $a_i = 1$ if $K[\pi^{-1}(\PP_G)]$ is Gorenstein. If $\eb_1 + \cdots + \eb_{n-1}$ belongs to the interior of $\delta (\pi^{-1}(\PP_G))$, then $\eb_1 + \cdots + \eb_{n-1} + q\eb_n$ belongs to the interior of $\delta \PP_G$ for some integer $q > 0$. Since $K[\PP_G]$ coincides with the Ehrhart ring of $\PP_G$, it follows that there are edges $e_1, \ldots, e_m$ of $G$ with $\eb_1 + \cdots + \eb_{n-1} + q\eb_n
= \rho(e_1) + \cdots \rho(e_m)$. Hence $q = 1$ and $\eb_1 + \cdots + \eb_n
= \rho(e_1) + \cdots \rho(e_m)$. Thus $G$ possesses a perfect matching with $\delta = n / 2$ if $K[\pi^{-1}(\PP_G)]$ is Gorenstein.
For a while, suppose that $G$ possesses a perfect matching with $\delta = n / 2$. Let $\PP^{\flat} \subset \RR^{n-1}$ denote the integral convex polytope $\delta (\pi^{-1}(\PP_G)) - (\eb_1 + \cdots + \eb_{n-1})$. Then $\PP^{\flat}$ is of standard type, i.e., $\dim \PP = n - 1$ and the origin of $\RR^{n-1}$ belongs to the interior of $\PP$. Then [@DeNegri--Hibi Corollary (1.2)] guarantees that the toric ring $K[\pi^{-1}(\PP_G)]$ is Gorenstein if and only if the dual polytope [@DeNegri--Hibi p. 631] of $\PP^{\flat}$ is integral.
Now, by using [@OhHinormal Theorem 1.7 (a)] again, it turns out that the equations of the supporting hyperplanes which defines the facets of $\PP^{\flat}$ are the followings:
- $z_i = - 1$ for each $1 \leq i < n$;
- $\sum_{i \in [n] \setminus (T \Union N(G;T))} z_i
+ 2 \sum_{j \in N(G;T)} z_j = |T| - |N(G;T)|$ if $n \in T$;
- $2 \sum_{i \in T} z_i
+ \sum_{j \in [n] \setminus (T \Union N(G;T))} z_j =
|N(G;T)| - |T|$ if $n \in N(G;T)$;
- $\sum_{i \in T} z_i - \sum_{j \in N(G;T)} z_j =
|N(G;T)| - |T|$ if $n \not\in T \Union N(G;T)$,
where $T \neq \emptyset$ is a stable set of $G$ for which the bipartite graph induced by $T$ is connected and for which either $T \Union N(G;T) = [n]$ or every connected component of the induced subgraph $G_{[n] \setminus (T \Union N(G;T))}$ has at least one odd cycle.
Hence the dual polytope of $\PP^{\flat}$ is integral if and only if ($\alpha$) one has $|N(G;T)| = |T| + 1$ for each nonempty stable set $T$ of $G$ such that the bipartite graph induced by $T$ is connected with $T \Union N(G;T) \neq [n]$ and that every connected component of $G_{[n] \setminus (T \Union N(G;T))}$ has at least one odd cycle and ($\beta$) one has $|T| = n/2 - 1$ for each stable set $T$ of $G$ such that the bipartite graph induced by $T$ is connected with $T \Union N(G;T) = [n]$.
Consequently, when the toric ring $K[\PP_G]$ is Gorenstein, the conditions (i), (ii) and (iii) are satisfied. Conversely, suppose that the conditions (i), (ii) and (iii) are satisfied. Since the hyperplane defined by the equation $z_i = 0$ is a supporting hyperplane which defines a facet of $\PP_G$ for each $1 \leq i \leq n$, if $\gamma \PP_G$, where $\gamma > 0$, contains at least one integer point $(a_1, \ldots, a_n)$, then each $a_i > 0$ and $\gamma \geq [(n + 1)/2]$. It follows from (i), (ii) and (iii) that $n$ is even and $\eb_1 + \cdots + \eb_n$ belongs to the interior of $(n / 2) \PP_G$. Thus the smallest number $\delta > 0$ for which $\delta \PP_G$ contains at least one integer point is $\delta = n / 2$ and $\eb_1 + \cdots + \eb_n$ belongs to the interior of $\delta \PP_G$. Our discussion done already in the preceding paragraph guarantees that the toric ring $K[\PP_G]$ is Gorenstein, as desired.
(a’) In imitation of the preceding proof of (a) by using [@OhHinormal Theorem 1.7 (b)] instead of [@OhHinormal Theorem 1.7 (a)], one can easily give a proof of (a’).
\(b) The facets of the stable polytope $\QQ_G$ is completely determined when $G$ is a perfect graph ([@Chvatal Theorem 3.1]). In fact, when $G$ is perfect, the equations of the supporting hyperplanes which defines the facets of $\QQ_G$ are either $z_i = 0$ for $1 \leq i \leq n$ or $\sum_{W \subset [n]} z_i = 1$, where $W$ is a maximal cliques of $G$. Let $\delta$ denote the smallest integer $\delta > 0$ for which the interior of $\QQ_G$ contains at least one integer point. Then $\delta - 1$ coincides with the maximal cardinality of cliques of $G$ and $\eb_1 + \cdots + \eb_n$ belongs to the interior of $\delta \QQ_G$. It follows that the dual polytope of the integral polytope $\delta \QQ_G - (\eb_1 + \cdots + \eb_n)
\subset \RR^n$ of standard type is integral if and only if all maximal cliques of $G$ have the cardinality $\delta - 1$. Hence [@DeNegri--Hibi Corollary (1.2)] guarantees that the toric ring $K[\QQ_G]$ is Gorenstein if and only if all maximal cliques have the same cardinality.
\[graph\] [*The toric ring of the edge polytope of each of the finite connected graphs $G_1$ and $G_2$ drawn below is normal and Gorenstein.* ]{}
0.1in
(41.60,20.15)(7.00,-29.15)
(16.0000,-30.0000)[(0,0)[$G_1$]{}]{}(40.0000,-30.0000)[(0,0)[$G_2$]{}]{}
Unimodal Gorenstein sequences
=============================
Let $\PP \subset \RR^N$ be a convex polytope. Recall that a $(q-1)$-simplex $\Sigma$ each of whose vertices is a vertex of $\PP$ is said to be a [*special simplex*]{} [@Christos] in $\PP$ if each facet of $\PP$ contains exactly $q - 1$ of the vertices of $\Sigma$.
\[specialsimplex\] [*(a)*]{} Let $G$ be a finite connected graph as in Theorem \[Gorenstein\] (a) or (a’) and suppose that the toric ring $K[\PP_G]$ of the edge polytope $\PP_G$ of $G$ is Gorenstein. Then there is a special simplex in $\PP_G$.
[*(b)*]{} Let $G$ be a perfect graph and suppose that the toric ring $K[\QQ_G]$ of the stable polytope $\QQ_G$ of $G$ is Gorenstein. Then there is a special simplex in $\QQ_G$.
\(a) Let $[n]$ be the vertex set of $G$. Since $G$ possesses a perfect matching, it follows that $n = 2m$ is even and there exist $m$ edges $e_1, \ldots, e_m$ of $G$ with $\rho(e_1) + \ldots + \rho(e_m)
= \eb_1 + \cdots + \eb_n$. Let $\Sigma$ denote the $(m - 1)$-simplex whose vertices are $\rho(e_1), \ldots, \rho(e_m)$. We claim that $\Sigma$ is special in $\PP_G$.
Theorem \[Gorenstein\] together with [@OhHinormal Theorem 1.7] give the complete information about the equations of the supporting hyperplanes which define the facets of the edge polytope $\PP_G$. Let $\HH_i$ denote the hyperplane defined by the equation $z_i = 0$. Then $\rho(e_j) \in \HH_i$ if and only if $i \not\in e_j$. Let $\HH_T$ denote the hyperplane defined by the equation $\sum_{i \in T} z_i = \sum_{j \in N(G;T)} z_j$, where $T$ is a stable set of $G$, which is the supporting hyperplane of a facet of $\PP_G$. Let $C$ denote the set of those $1 \leq i \leq m$ with $\rho(e_i) \bigcap T \neq \emptyset$ and $D$ the set of those $1 \leq i \leq m$ with $\rho(e_i) \Sect N(G;T) \neq \emptyset$. In either the case of $|N(G;T)| = |T| + 1$ or the case of $|N(G;T)| -1 = |T| + 1 = m$, one has $C \subset D$ with $|C| = |D| - 1$. Let $i_0 \in D \setminus C$. Then $\rho(e_j) \in \HH_T$ if and only if $j \neq i_0$. Thus $\Sigma$ is special in $\PP_G$ as desired.
\(b) Let $G$ be a perfect graph on $[n]$ and suppose that all maximal cliques have the cardinality $q$. Since $G$ is perfect, the chromatic number of $G$ is equal to $q$. Thus there is a map $\varphi : [n] \to [q]$ with the property that $\varphi(i) \neq \varphi(j)$ if $\{i, j \} \in E(G)$. Let $W'_\ell$ denote the stable set $\{ i \in [n] \, : \, \varphi(i) = \ell \}$ for each $1 \leq \ell \leq q$. We assume that $\QQ_G$ is not a simplex. Thus one of the stable sets $W'_1, \ldots, W'_q$ contains at least two vertices. Let, say, $W'_1$ contain at least two vertices and fix $i_0 \in W'_1$. Let $W_0 = \{ i_0 \}, W_1 = W'_1 \setminus \{ i_0 \}$ and $W_\ell = W'_\ell$ for $2 \leq \ell \leq q$. Each of $W_0, W_1, \ldots, W_q$ is a stable set of $G$ and $\sum_{\ell = 0}^{q} \rho(W_\ell)
= \eb_1 + \cdots + \eb_n$. Let $\Sigma$ denote the $q$-simplex with $q + 1$ vertices $\rho(W_0), \rho(W_1), \ldots, \rho(W_q)$. We claim that $\Sigma$ is special in $\QQ_G$.
Recall that the equation of the supporting hyperplanes which defines the facets of the stable polytope of $G$ are either (i) $x_i = 0$ for $1 \leq i \leq n$ or (ii) $\sum_{i \in W} x_i = 1$, where $W$ is a maximal clique of $G$. If $F_i$ is the facet defined by $x_i = 0$, then $\rho(W_\ell) \in F_i$ if and only if $i \not\in W_\ell$. Since $[n]$ is the disjoint union $W_0 \Union W_1 \Union \cdots \Union W_q$, it follows that $F_i$ contains exactly $q$ of the vertices of $\Sigma$. Let $F'_W$ denote the facet defined by $\sum_{i \in W} x_i = 1$, where $W$ is a maximal clique of $G$. Since each of the subsets $W \cap (W_0 \Union W_1),
W \cap W_2, \ldots, W \cap W_q$ of $[n]$ consists of one element, it follows that each of the vertices $\rho(W_2), \ldots, \rho(W_q)$ belongs to $F'_W$ and that $\rho(W_0) \in F'_W$ (resp. $\rho(W_1) \in F'_W$) if and only if $i_0 \in W$ (resp. $i_0 \not\in W$). Hence $F'_W$ contains exactly $q$ of the vertices of $\Sigma$.
By using Example \[posetpolytope\] together with [@twoposetpolytopes Theorem 3.2], it turns out that the above Theorem \[specialsimplex\] (b) generalize Reiner–Welker [@RW Corollary 3.8].
Now, by virtue of [@Christos Theorem 3.5], one has a rich class of unimodal Gorenstein sequences [@StanleyGreenBook p. 66]. It is known [@OhHicompressed Example 1.3 (c)] the stable polytope of a perfect graph is compressed.
The edge polytope of a finite connected graph is unimodular, i.e., all of its triangulations are unimodular, if and only if any two odd cycles of $G$ possess at least one common vertex. In particular the edge polytope of a finite connected bipartite graph is unimodular. The edge polytope of $G_1$ of Example \[graph\] is compressed ([@OhHimultipartite]) but not unimodular, and that of $G_2$ is unimodular. A combinatorial characterization of finite graphs $G$ for which the edge polytope $\PP_G$ is compressed is given in [@Ohsugi Theorem 4.1].
[*(a)*]{} Let $G$ be a finite connected graph as in Theorem \[Gorenstein\] (a) and suppose that the edge polytope $\PP_G$ is compressed and that the toric ring $K[\PP_G]$ is Gorenstein. Then the $h$-vector of $K[\PP_G]$ is unimodal.
[*(a’)*]{} Let $G$ be a finite $2$-connected bipartite graph and suppose that the toric ring $K[\PP_G]$ is Gorenstein. Then the $h$-vector of $K[\PP_G]$ is unimodal.
[*(b)*]{} Let $G$ be a perfect graph and suppose that the toric ring $K[\QQ_G]$ of the stable polytope $\QQ_G$ of $G$ is Gorenstein. Then the $h$-vector of $K[\QQ_G]$ is unimodal.
We conclude the present paper with
\[final\] [*Let $n \geq 3$ and $G$ the finite connected graph on $[2n]$ drawn below. Let $n$ be odd. (If $n$ is even, then $K[\PP_G]$ is not Gorenstein by Theorem \[Gorenstein\] (a’).) By virtue of [@Ohsugi Theorem 4.1] it turns out that the edge polytope $\PP_G$ of $G$ is compressed. By using Theorem \[Gorenstein\] (a) it follows that the toric ring $K[\PP_G]$ is (normal and) Gorenstein. Moreover, we can compute the $h$-vector explicitly. Since the graph $G$ satisfies the condition in [@OhHiquadratic Theorem 1.2], the “toric ideal" $I_G$ of $G$ is generated by quadratic binomials which correspond to even cycles of $G$ of length $4$. There exists a reverse lexicographic order such that the initial monomials of the quadratic binomials are relatively prime. Since the set of quadratic binomials is a Gröbner basis of $I_G$ with respect to $<_{rev}$, it follows that the initial ideal is generated by $n$ monomials which are squarefree, quadratic and relatively prime. Thus the $h$-vector of $K[\PP_G]$ is $\left(1, n, {n \choose 2}, \cdots, {n \choose n - 2}, n, 1\right)$.* ]{}
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(36.0000,-12.0000)[(0,0)[$n+1$]{}]{}(32.1000,-14.3000)[(0,0)[$n+2$]{}]{}(29.5000,-17.8000)[(0,0)[$n+3$]{}]{}(40.0000,-14.0000)[(0,0)[$2 n$]{}]{}
[99]{}
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Hidefumi Ohsugi Takayuki Hibi\
Department of Mathematics Department of Pure and Applied Mathematics\
Faculty of Science Graduate School of Information Science and Technology\
Rikkyo University Osaka University\
Toshima, Tokyo 171–8501, Japan Toyonaka, Osaka 560–0043, Japan\
E-mail:ohsugi@@rkmath.rikkyo.ac.jp E-mail:hibi@@math.sci.osaka-u.ac.jp
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abstract: 'Matter with an equation of state $p=-\rho/3$ may arise in certain scalar field theories, and the energy density of this matter decreases as $a^{-2}$ with the scale factor $a$ of the Universe. In this case, the Universe could be closed but still have a nonrelativistic-matter density $\Omega_0<1$. Furthermore, the cosmic microwave background could come from a causally-connected region at the other side of the Universe. This model is currently viable and might be tested by a host of forthcoming observations.'
address: 'Department of Physics, Columbia University, New York, New York 10027'
author:
- 'Marc Kamionkowski[^1]and Nicolaos Toumbas'
date: January 1996
title: 'A Low-Density Closed Universe'
---
Of the three possibilities, a closed Universe receives far less attention in the current literature than an open or a flat Universe. Observations that find a matter density less than critical suggest an open Universe. Theoretical arguments, such as the Dicke coincidence and inflation, favor a flat Universe. However, there are heuristic arguments for a closed Universe that involve, for example, consistency of quantum field theories on a compact space or the idea that it is easier to create a finite Universe with zero energy, charge, and angular momentum. Even so, given the observations, it requires some [*chutzpah*]{} to suggest that the matter density is greater than critical. For these reasons, models that are closed by virtue of a cosmological constant ($\Lambda$) have been recently considered [@scottwhite]. In this paper, we consider a variation: a low-density closed Universe, which at low redshifts is entirely indistinguishable from a standard open Friedmann-Robertson-Walker (FRW) Universe with the same non-relativistic matter density.
If some form of matter with an equation of state $p=-\rho/3$ exists, then its energy density decreases with the scale factor $a$ of the Universe as $a^{-2}$ and thus mimics a negative-curvature term in the Friedmann equation. In this case, the Universe could be closed and still have a nonrelativistic-matter density $\Omega_0<1$.
In fact, the energy density contributed by a scalar field with a uniform gradient-energy density would scale as $a^{-2}$. However, such a scalar-field configuration would collapse within a Hubble time unless it was somehow stabilized. Davis [@davisone] pointed out that if there was a manifold of degenerate vacua with nontrivial mappings into the three-sphere \[which could be accomplished if there was a global symmetry $G$ broken to a subgroup $H$ with $\pi_3(G/H) \neq 1$\], then a texture—a topological defect with uniform gradient-energy density—would be stabilized provided that it was wound around a closed Universe [@davisone]. Non-intersecting strings would also provide an energy density that scales as $a^{-2}$.
Moreover, if the energy density contributed by the texture is chosen properly, the observed cosmic microwave background (CMB) comes from a causally-connected patch at the antipode of the closed Universe [@daviscmb].[^2] Although the homogeneity problem is still not addressed, we find it illustrative and interesting that one can still construct a viable model, which looks remarkably like an open Universe at low redshifts, even though the largest-scale structure differs dramatically.
The Friedmann equation for a closed Universe with nonrelativistic matter and some other form of matter (perhaps a stable texture) with an equation of state $p=-\rho/3$ is $$\begin{aligned}
H^2 &\equiv& \left({\dot a \over a}\right)^2 = {8\pi G \over 3}
\rho_m + {\gamma - 1 \over a^2} \nonumber \\
&=&H_0^2 [ \Omega_0 (1+z)^3 +(1-\Omega_0)(1+z)^2]
\equiv H_0^2 [E(z)]^2,
\label{friedmann}\end{aligned}$$ where $H=\dot a/a$ is the Hubble parameter (and the dot denotes derivative with respect to time), $z=(a_0/a)-1$ is the redshift, $G$ is Newton’s gravitational constant, $\rho_m$ is the density of nonrelativistic matter, and $\gamma$ is a parameter that quantifies the contribution of the energy density of the texture. The second line defines the function $E(z)$. This is exactly the same as the Friedmann equation for an open Universe with the same $\Omega_0$, so this closed Universe has the same expansion dynamics. At the current epoch (denoted by the subscript “0”), $$\Omega_0=1 + {1-\gamma \over a^2 H^2} = 1-\Omega_t +
{1\over a_0^2 H_0^2},
\label{Omegaequation}$$ where $\Omega_t=\gamma(a_0 H_0)^{-2}$ is the contribution of the texture to closure density today. So, $\Omega_0<1$ if $\gamma>1$ even though the Universe is closed, and we require that $\Omega_t + \Omega_0>1$.
If the metric of a closed Universe is written as $$ds^2= dt^2 - a^2(t) \left[ d\chi^2 + \sin^2 \chi (
d\theta^2 + \sin^2 \theta d\phi^2) \right],$$ then the polar-coordinate distance between a source at a redshift $z_1$ and another source along the same line of sight at a redshift $z_2$ (for $\Omega_0<1$) is $$\chi_2 -\chi_1 = \sqrt{\Omega_0 + \Omega_t -1}
\int_{z_1}^{z_2} \, {dz \over E(z)}.
\label{chiequation}$$ If $\Omega_t$ is chosen such that the polar-coordinate distance of the CMB surface of last scatter is $\chi_{LS}\simeq \pi$, then the CMB we observe comes from a causally-connected patch at the antipode of the Universe. Since this Universe expands forever, we could also choose $\chi_{LS}\simeq2\pi$, in which case the CMB photons have traveled precisely once around the Universe. This introduces the intriguing possibility that when we observe the CMB we are looking at the [*local*]{} (rather than some distant) region of the Universe as it was at a redshift $z\simeq1100$. In fact, for $\chi_{LS}\simeq n\pi$ with $n=1,2,3,...$, CMB photons have traveled $n/2$ times around the Universe, and the CMB comes from a causally-connected patch on the other side of the Universe (for $n$ odd) or from the local neighborhood (for $n$ even). From Eq. \[chiequation\], the condition on $\Omega_t$ for $\chi_{LS}=n\pi$ is $$\Omega_t=\left[ {n\pi \sqrt{1-\Omega_0} \over {\rm arcsinh}
( 2 \sqrt{1-\Omega_0}/\Omega_0)} \right]^2 +1 - \Omega_0.
\label{conditioneqn}$$ For $n=1$ ($n=2$), $\Omega_t$ increases from 1.6 to 2.5 (4 to 10) for $\Omega_0$ between 0.1 and 1.
Is this a realistic possibility? For $n\geq2$, it requires a radius of curvature for the Universe that is probably too small to be consistent with observations. The $n=1$ case is still consistent with our current knowledge of the Universe. However, forthcoming observations may be used to distinguish it from a standard open Universe, as we now explain.[^3]
Since the expansion dynamics is the same as for an open FRW Universe, quantities that depend only on the expansion, such as the deceleration parameter, the age of the Universe, or the distribution of quasar absorption-line redshifts, do not probe $\Omega_t$. Furthermore, the growth of density perturbations is the same as in a standard open Universe, so dynamical measurements of $\Omega_0$ (e.g., from peculiar-velocity flows) will also be insensitive to $\Omega_t$. Effects due to geometry arise only at ${\cal
O}(z^3)$ since $\sin\chi$ and $\sinh\chi$ differ only at ${\cal
O}(\chi^3)$; therefore, this Universe will differ from an open Universe only at $z\gtrsim1$.
Ergo, we now turn to cosmological tests that probe the geometry of the Universe. Underlying these is the angular-diameter distance between a source at a redshift $z_2$ and a redshift $z_1<z_2$, $$d_A(z_1,z_2)= { \sin(\chi_2-\chi_1) \over (1+z_2) H_0
\sqrt{\Omega_0+\Omega_t-1}}.
\label{angulardiametereqn}$$ The angular size of an object of proper length $l$ at a redshift $z$ is $\theta\simeq l/d_A(0,z)$. Consider first the case where $\Omega_t$ is fixed by $n=2$. Then the antipode $\chi=\pi$ of the Universe must be at some redshift $z_a<1100$. One finds that $z_a\lesssim5$ for $\Omega_0\gtrsim0.3$, and therefore, the angular sizes of the highest-redshift quasars must be very large. Additional arguments against an antipode at $z\lesssim5$ for a closed $\Lambda$ Universe have been given Refs. [@gott]. These arguments probably apply to the model considered here, although we have not done a complete analysis. Therefore, a closed Universe with $n\geq2$ is highly unlikely and we pursue it no further.
In Fig. \[angleplot\], we plot the angular size as a function of redshift fixing $\Omega_t$ so that the CMB comes from the antipode \[i.e., Eq. \[conditioneqn\] with $n=1$\]. We also plot the results for a FRW Universe. The Figure shows that the angular sizes in a flat matter-dominated Universe can be roughly similar to those in a low-density closed Universe. Therefore, an analysis of the angular sizes of some compact radio sources, which shows consistency with a flat Universe [@radios], may also be consistent with a low-density closed Universe. Proper-motion distances of superluminal jets in radio sources at large redshift may provide essentially the same probe as do flux–redshift relations. The common caveat is that evolutionary effects must be understood if these are to provide reliable cosmological tests. It has been proposed that these effects may conceivably be understood well enough to discriminate between open and flat $\Lambda$ models [@kraussschramm]. Fig. \[angleplot\] illustrates, however, that the difference between the angular sizes for the FRW Universe and the closed model for the same value of $\Omega_0$ is quite a bit more dramatic than the difference between open FRW and flat $\Lambda$ models (c.f., Fig. 13.5 in Ref. [@peebles]). Therefore, if the angle-redshift relation can distinguish open and flat $\Lambda$ models, then the distinction between these and the closed model will be even clearer.
Another classical cosmological test is the number-redshift relation. In the low-density closed Universe, the differential number of galaxies per steradian per unit redshift is, $${dN_{\rm gal} \over dz d\Omega} = {n_0 \sin^2[\chi(z)]
\over H_0^3 (\Omega_0 +\Omega_t-1)E(z)},$$ where $n_0$ is the local number density of galaxies, and the number per comoving volume is assumed to remain constant. In Fig. \[numbersfig\], we plot the number-redshift relation for the low-density closed Universe with $\Omega_t$ chosen so that the CMB comes from the antipode and for standard open and flat FRW models. The Figure shows that an application of this test, which finds values of $\Omega_0$ near unity in a FRW Universe [@lohspillar], can also be consistent with a low-density closed Universe. However, galactic evolutionary effects are realistically quite significant, so this remains a controversial test.
A test for $\Lambda$ discussed by Alcock and Paczyński [@alcock] may also be an especially effective probe of $\Omega_t$. The redshift thickness $\delta z$ and angular size $\delta \theta$ of a roughly spherical structure that grows with the expansion of the Universe will have a ratio $${1 \over z}{\delta z \over \delta \theta}=
{E(z)\sin[\chi(z)] \over z \sqrt{\Omega_0+\Omega_t-1}}.$$ As shown in Fig. \[dzdthetafig\], this function is significantly lower in a low-density closed Universe than it is in an open Universe (and in a $\Lambda$ Universe; c.f., Fig. 13.9 in Ref. [@peebles]). Furthermore, it depends only very weakly on the value of $\Omega_0$ and therefore provides an $\Omega_0$-independent determination of the geometry. A precise measurement may be feasible with forthcoming quasar surveys [@phillips].
We have also checked the probability for gravitational lensing of sources at high redshift. This test provides perhaps the strongest constraint on $\Lambda$ models [@gravlenses], and makes it unlikely that the CMB comes from the antipode of a Universe that is closed with the addition of a cosmological constant [@scottwhite]. The probability for lensing of a source at redshift $z_s$ for $\Omega_0<1$ and $\Omega_t+\Omega_0>1$ relative to the fiducial case of a standard flat Universe is $$\begin{aligned}
P_{\rm lens} & = & {15\over4} \left[ 1- {1\over
(1+z_s)^{1/2} } \right]^{-3} \nonumber \\
& & \times \int_{0}^{z_s}\, {(1+z)^2 \over E(z)} \left[
{d_A(0,z)d_A(z,z_s) \over d_A(0,z_s)} \right]^2\,dz.\end{aligned}$$ The current observational constraint is roughly $P_{\rm lens}\lesssim5$. If $\Omega_t$ is chosen so that the CMB comes from the antipode, then $P_{\rm
lens}<2.5$ for $0<\Omega_0<1$. Hence the model is consistent with current data and is likely to remain so.
Finally, if ours is actually a low-density closed Universe, it will probably have a dramatic signature in the anisotropy spectrum of the CMB, especially if the CMB comes from the antipode of the Universe. Although the detailed shape of the anisotropy spectrum depends on a specific model for structure formation, it quite generically has structure (known as “Doppler peaks”) on angular scales smaller than that subtended by the horizon at the surface of last scatter. The angle subtended by the horizon at last scatter depends on the cosmological model; in a standard FRW Universe, it is $\theta_{LS} \simeq \Omega^{1/2}\,1^\circ$. Therefore, measurement of the location of the first Doppler peak provides a determination of the geometry of the Universe [@kss], and with forthcoming all-sky CMB maps with sub-degree angular resolution, this measurement may be quite precise [@jkks].
The angular scale subtended by the horizon in a low-density closed Universe may be approximated by $$\theta_{LS} \simeq 2^\circ\, {\sqrt{\Omega_0+\Omega_t-1}
\over \Omega_0^{1/2} \sin\chi_{LS}},$$ when $\theta_{LS}$ evaluates to small angles; otherwise, $\theta_{LS}={\cal O}(\pi)$. Here, $$\chi_{LS}=\sqrt{\Omega_0 +\Omega_t-1 \over 1-\Omega_0} {\rm
arcsinh} \left( {2 \sqrt{1-\Omega_0} \over \Omega_0} \right)$$ is the polar-coordinate distance traversed by the CMB photons since last scatter. As expected, this is always larger than $\theta_{LS}$ for a flat or open FRW Universe. Moreover, if $\chi_{LS}\simeq \pi$, the Doppler-peak structure of the CMB is shifted to the largest angular scales, and the suppression of CMB anisotropies due to Silk damping is also shifted to larger angular scales. The precise shift depends on exactly how close the last-scattering surface is to the antipode.[^4] It is almost certain that these signatures will be distinguishable in forthcoming CMB maps if they are indeed there.
Although there is no horizon problem in this model, at earlier or later epochs, the CMB is not generally at the antipode. Furthermore, the homogeneity of the Universe is not necessarily explained even if the CMB comes from a causally-connected region. Even so, it is worth noting that one can construct a viable model, which is indistinguishable from an open Universe at redshifts $z\lesssim1$, with a closed geometry. Furthermore, the model will be tested by forthcoming observations of the Universe at large redshifts, especially through angular sizes, $\delta z/\delta\theta$, and the CMB.
We have focussed in our numerical work on the case where $\Omega_t$ is such that the CMB comes precisely from the antipode. However, one could explore other values of $\Omega_t$, perhaps within the context of flat inflationary models.
Finally, what about the homogeneous matter with an energy density which scales as $a^{-2}$? If this is due to a topologically stabilized scalar-field configuration, as discussed above, then the symmetry-breaking scale must be of order the Planck scale if $\Omega_t$ is of order unity. Furthermore, the global symmetry must be [*exact*]{}. This model would therefore have significant implications for Planck-scale physics if verified [@global].
We thank D. Helfand and D. Spergel for useful comments. This work was supported in part by the D.O.E. under contract DEFG02-92-ER 40699 and by NASA under contract NAG5-3091.
[99]{}
[^1]: kamion@phys.columbia.edu
[^2]: This could similarly be accomplished with $\Lambda\neq0$, but these models are likely ruled out by lensing statistics [@scottwhite].
[^3]: For an excellent review of many classical cosmological tests, see Ref. [@peebles].
[^4]: For example, the anisotropy spectrum might resemble those shown in Fig. 6 of Ref. [@scottwhite] for the analogous case with a cosmological constant for a flat scale-invariant spectrum of density perturbations. However, the overall tilt of the spectrum depends on the model of primordial perturbations and could therefore be considerably different.
|
---
abstract: 'This paper reviews the current status of experimental results on radiative kaon decays. Several experiments at BNL, CERN and FNAL have recently or will soon complete data collection; as a result, there are several new results.'
author:
- |
S. Kettell\
[*Brookhaven National Laboratory*]{}\
title: ' Experimental Results on Radiative Kaon Decays[^1] '
---
Introduction
============
Radiative kaon decays provide a testing ground for Chiral Perturbation Theory (ChPT). ChPT provides a framework for calculating the decay rates for several modes, either directly or relative to other measured modes. The radiative modes are important for determining long distance contributions to other decays of interest: the two-photon contribution to $K^\circ_L \! \rightarrow \! \mu^+ \mu^-$, and the CP-conserving and indirect CP-violating contributions to $K^\circ_L \!
\rightarrow \! \pi^\circ e^+ e^-$ and $K^\circ_L \! \rightarrow \!
\pi^\circ \mu^+ \mu^-$. They are also important as backgrounds to other modes (e.g. the $K^\circ_L \! \rightarrow \! e^+
e^-\gamma\gamma$ background to $K^\circ_L \! \rightarrow \! \pi^\circ
e^+ e^-$).
A number of recent results have been reported in the literature, as well as in several recent conferences[@ichep98; @dpf99; @moriond99; @panic99; @epshep99; @kaon99].
Radiative K$_{\pi2}$ Decays
===========================
The radiative K$_{\pi2}$ decays: $K^+ \! \rightarrow \! \pi^+
\pi^\circ \gamma$, $K^\circ_L \! \rightarrow \! \pi^+ \pi^- \gamma$ and $K^\circ_{\rm S} \! \rightarrow \! \pi^+ \pi^- \gamma$ have two contributions. One is inner bremsstrahlung (IB) radiation from one of the charged particles. The second is direct emission (DE) from the vertex. The branching ratio of the IB contribution scales with the underlying K$_{\pi2}$ decay rate. Whereas, the rate for direct emission is expected to be roughly comparable for all three modes.
A new result[@klppg] for $K^\circ_L \! \rightarrow \! \pi^+ \pi^-
\gamma$ from KTeV is shown in Fig. \[fig\_klppg\]. The energy of the photon is shown, along with the contributions from IB and DE.
The DE component is modified by a “$\rho$-propagator” that serves to soften the DE spectrum. The branching ratio for the direct emission component (see eq.\[eqn\_klppg\]) is $${\rm BR(K^\circ_L \! \rightarrow \! \pi^+ \pi^- \gamma;DE)=
(3.70\pm0.10)\times 10^{-5} (E_{\gamma}^{*}>20 MeV) }
\label{eqn_klppg}$$ The ratio of direct emission to DE+IB is (see eq.\[eqn\_klppg\_de\]) $${\rm DE/(DE+IB) = 0.685\pm 0.009\pm0.017 }
\label{eqn_klppg_de}$$ This result is based on $\sim$5% of the total KTeV data for this mode.
There are new results from E787[@k+ppg] in the charged decay mode ($K^+ \! \rightarrow \! \pi^+ \pi^\circ \gamma$) as well. This result is striking, in that the branching ratio is a factor of 4 lower than the previous value. The data is traditionally expressed in terms of the variable W, which is defined as: $$\begin{aligned}
{\rm W^{2}} & \equiv & {\rm (p\cdot q)/{m_{K^+}^{2}} \times
(p_{+}\cdot q)/{m_{\pi^+}^{2}}} \\ \nonumber
& = & {\rm E_{\gamma}^2\times (E_{\pi^+} - P_{\pi^+}\times
\cos{\bf\theta_{\pi^+\;\gamma}})/({m_{K^+}^2}\times{m_{\pi^+}^{2}})}
\label{eqn_w}\end{aligned}$$ The new result from E787, shown in Figure \[fig\_kppg\],
has about 8 times higher statistics than the old one. The branching ratio for the direct emission component, from a fit to IB and DE (see eq.\[eqn\_kppg\]) is $${\rm BR(K^+ \! \rightarrow \! \pi^+ \pi^\circ \gamma;DE)=
(4.72\pm0.77)\times 10^{-6} \, (55<\!T_{\pi^+}\!<90 MeV) }
\label{eqn_kppg}$$ This represents half of the E787 data that is currently on tape. The interference term is small, $(-0.4\pm1.6)$% and the direct emission is $(1.85\pm0.30)$%. The decay rate, corrected to full phase space[^2], is now measured to be similar to that for $K_L$: $\Gamma(K^+ \! \rightarrow \! \pi^+ \pi^\circ \gamma;DE)=808\pm132
s^{-1}$ vs. $\Gamma(K^\circ_L \! \rightarrow \! \pi^+ \pi^-
\gamma;DE)=617\pm18 s^{-1}$.
KTeV also has new results on $K^\circ_L \! \rightarrow \! \pi^+ \pi^-
e^+ e^- $, where the photon has internally converted to two electrons. In addition to measuring the branching ratio[@klppee], a T-odd observable in the angular distribution of the plane of the $\pi$-pair vs. the plane of the electron pair is observed[@klppee2]. This data represents one quarter of the final KTeV sample.
A summary of the current experimental status of radiative K$_{\pi2}$ decays is shown in Table \[tab\_kzppg\].
0.1 in
Decay Mode Citation
-------------------------------------------------------- ----------------------------------- --------------------
$K^\circ_L \! \rightarrow \! \pi^+\pi^-\gamma(DE)$ $(3.70\pm0.10)\times10^{-5}$ KTeV-99[@klppg]
$K^+ \! \rightarrow \! \pi^+\pi^\circ\gamma(DE)$ $(4.72\pm0.77)\times10^{-6}$ E787-99[@k+ppg]
$K^\circ_L \! \rightarrow \! \pi^+\pi^-e^+e^-$ $(3.63\pm.11\pm.14)\times10^{-7}$ KTeV-99[@klppee]
$K^\circ_L \! \rightarrow \! \pi^\circ\pi^\circ\gamma$ $< 5.6\times10^{-6}$ NA31-94[@klp0p0g]
$K^\circ_S \! \rightarrow \! \pi^+\pi^-\gamma$ $(1.78\pm0.05)\times10^{-3}$ E731-93[@ksppg]
$K^\circ_S \! \rightarrow \! \pi^+\pi^-\gamma(DE)$ $<0.06\times10^{-3}$ CERN-76[@ksppg_de]
: *Summary of Radiative K$_{\pi2}$ results.*
\[tab\_kzppg\]
$K \! \rightarrow \! \pi \gamma \gamma$ Decays
==============================================
The decay $K^\circ_L \! \rightarrow \! \pi^\circ \gamma\gamma$ is very interesting, since to ${\cal O}(p^4)$ of ChPT the decay rate and spectral shape are completely determined, without any free parameters[@chpt_klpgg]. The prediction of the spectral shape is a striking success of ChPT; however, the decay rate is a factor of 3 too small. To match the experimental number a model dependent contribution from ${\cal O}(p^6)$ is needed, which is usually parameterized with a constant $a_V$[@chpt_av]. The CP-conserving contribution to $K^\circ_L \! \rightarrow \! \pi^\circ e^+ e^-$ depends on the value of $a_V$. Based on half of the total data sample, KTeV has recently measured $a_V = -0.72\pm0.05\pm0.06$[@klpgg], implying a contribution of $1-2\times10^{-12}$.
The charged mode $K^+ \! \rightarrow \! \pi^+ \gamma \gamma$ is more complicated, requiring an unknown parameter, $\hat{c}$, even at ${\cal
O}(p^4)$. Both the decay rate and spectral shape are predicted with this single parameter. E787 has measured $\hat{c} =
1.8\pm0.6$[@kpgg].
The experimental measurements of $K \! \rightarrow \! \pi \gamma
\gamma$ are summarized in Table \[tab\_kzpgg\].
0.1 in
Decay Mode Citation
------------------------------------------------------ ------------------------------------- ------------------
$K^\circ_L \! \rightarrow \! \pi^\circ\gamma\gamma$ $(1.68\pm0.07\pm0.08)\times10^{-6}$ KTeV-99[@klpgg]
$K^+ \! \rightarrow \! \pi^+\gamma\gamma$ $(6.0\pm1.5\pm0.7)\times10^{-7}$ E787-97[@kpgg]
$K^\circ_S \! \rightarrow \! \pi^\circ\gamma\gamma$ no limit (NA48-02?)
$K^\circ_L \! \rightarrow \! \pi^\circ e^+e^-$ $< 5.6\times10^{-10}$ KTeV-99[@klpee]
$K^\circ_L \! \rightarrow \! \pi^\circ\mu^+\mu^-$ $< 3.4\times10^{-10}$ KTeV-99[@klpmm]
$K^+ \! \rightarrow \! \pi^+e^+e^-$ $(2.94\pm0.05\pm0.13)\times10^{-7}$ E865-99[@kpee]
$K^+ \! \rightarrow \! \pi^+\mu^+\mu^-$ $(9.22\pm0.60\pm0.49)\times10^{-8}$ E865-99[@kpmm]
$K^\circ_S \! \rightarrow \! \pi^\circ e^+e^-$ $< 1.1\times10^{-6}$ NA31-93[@kspee]
$K^\circ_L \! \rightarrow \! \pi^\circ e^+e^-\gamma$ $(2.20\pm0.48\pm0.11)\times10^{-8}$ KTeV-99[@klpeeg]
$K^+ \! \rightarrow \! \pi^+e^+e^-\gamma$ $\sim$30 events E865-99[@kpeeg]
: *Summary of $K \! \rightarrow \! \pi \gamma \gamma$ results.*
\[tab\_kzpgg\]
The KTeV measurement of $K^\circ_L \! \rightarrow \! \pi^\circ e^+ e^-
\gamma $ should improve by $\times$3; the measurements of $K^\circ_L
\! \rightarrow \! \pi^\circ \ell^+ \ell^-$ are background limited, and will improve by $\sqrt{3}$.
K$^0$ to Two Real or Off-shell Photons
======================================
The decay $K^\circ_{\rm S} \! \rightarrow \! \gamma \gamma$ is predicted in ${\cal O}(p^4)$ of ChPT, without any free parameters, to occur with BR($K^\circ_{\rm S} \! \rightarrow \! \gamma \gamma$) = $2.0\times10^{-6}$[@chpt_klpgg]. This is in good agreement with the experimental value[@ksgg] (see Table \[tab\_kgg\]), although the experimental errors need to be reduced.
The decay $K^\circ_L \! \rightarrow \! \gamma\gamma$ is of interest for its importance in interpreting the measurement of $K^\circ_L \!
\rightarrow \! \mu^+ \mu^-$. The decay $K^\circ_L \! \rightarrow \!
\mu^+ \mu^-$ is sensitive to internal top quark loops, that would allow a determination of the fundamental SM parameter $\rho$. The decay is, however, dominated by the decay $K^\circ_L \! \rightarrow \!
\gamma\gamma$ with the photons converting to a $\mu^\pm$ pair. For this reason a precise measure of $K^\circ_L \! \rightarrow \!
\gamma\gamma$ is needed. With the improved precision on $K^\circ_L \!
\rightarrow \! \mu^+ \mu^-$ from E871, the uncertainties on $K^\circ_L
\! \rightarrow \! \gamma\gamma$ and $\frac{K^\circ_{\rm L} \!
\rightarrow \! \pi^\circ \pi^\circ} {K^\circ_{\rm S} \! \rightarrow \!
\pi^+ \pi^- }$ are now contributing significantly[@klgg; @klmm; @pdg] to the uncertainty on the ratio $$\begin{aligned}
\frac{\Gamma(K^\circ_L \! \rightarrow \! \mu^+ \mu^-)}
{\Gamma(K^\circ_L \! \rightarrow \! \gamma\gamma)} & = &
\left[ \begin{array}{c}B(K^\circ_L \! \rightarrow \! \mu^+ \mu^-) \\
\hline B(K^\circ_L \! \rightarrow \! \pi^+ \pi^-)\end{array} \right] \times \\
\nonumber
& & \hspace{-1.5cm} \left[ \left| \begin{array}{c}\eta_{+-}\\ \hline
\eta_{\circ\circ} \end{array} \right|
\begin{array}{c}B(K^\circ_S \! \rightarrow \! \pi^+ \pi^-)
\\ \hline B(K^\circ_{\rm S} \! \rightarrow \! \pi^\circ \pi^\circ) \end{array}
\right] \times
\left[ \begin{array}{c}B(K^\circ_{\rm L} \! \rightarrow \! \pi^\circ
\pi^\circ)
\\ \hline B(K^\circ_L \! \rightarrow \! \gamma\gamma)
\end{array} \right] \\ \nonumber
& & [1.55\%] [(0.23\%)(1.28\%)][1.42\%] \\ \nonumber
& = & (1.213\pm0.030)\times10^{-5} \nonumber
\label{eqn_kmm}\end{aligned}$$ KLOE should be able to contribute to improving both of these measurements. Finally there is a long distance dispersive contribution, from two off-shell photons, for which additional input from ChPT and measurements of the decays $K^\circ_L \! \rightarrow \!
e^+ e^-\gamma$, $K^\circ_L \! \rightarrow \! \mu^+ \mu^- e^+ e^-$ and $K^\circ_L \! \rightarrow \! e^+ e^- e^+ e^-$ are needed[@chpt_klmm].
Results of kaon decays to two real or off-shell photons are summarized in Table \[tab\_kgg\].
0.1 in
Decay Mode Citation
-------------------------------------------------------- ------------------------------------------------------- ------------------
$K^\circ_S \! \rightarrow \! \gamma\gamma$ $(2.4\pm0.9)\times10^{-6}$ NA31-95[@ksgg]
$K^\circ_L \! \rightarrow \! \gamma\gamma$ $(5.92\pm0.15)\times10^{-4}$ NA31-87[@klgg]
$K^\circ_L \! \rightarrow \! \mu^+\mu^-$ $(7.24\pm0.17)\times10^{-9}$ E871-99[@klmm]
$K^\circ_L \! \rightarrow \! e^+e^-$ $8.7^{+5.7}_{-4.1}\times10^{-12}$ E871-98[@klee]
$K^\circ_L\rightarrow e^+e^-\gamma$ $\!(1.06\pm\!.02\pm\!.02\pm\!.04)\!\times\!10^{-5}\!$ NA48-99[@kleeg]
$K^\circ_L \! \rightarrow \! \mu^+\mu^-\gamma$ $(3.23\pm0.23\pm0.19)\times10^{-7}$ E799-95[@klmmg]
$K^\circ_L \! \rightarrow \! e^+e^-e^+e^-$ $(4.14\pm0.27\pm0.31)\times10^{-8}$ KTeV-98[@kleeee]
$K^\circ_L \! \rightarrow \! e^+e^-\mu^+\mu^-$ $\sim$40 events KTeV-99[@klmmee]
$K^\circ_L \! \rightarrow \! \! \mu^+\mu^-\mu^+\mu^-$ no limit
$K^\circ_S \! \rightarrow \! \mu^+\mu^-$ $< 3.2\times10^{-7}$ CERN-73[@ksmm]
$K^\circ_S \! \rightarrow \! e^+e^-$ $< 1.4\times10^{-7}$ CPLEAR[@ksee]
$K^\circ_L \! \rightarrow \! e^+e^-\gamma\gamma$ $(6.31\pm0.14\pm0.42)\times10^{-7}$ KTeV-99[@kleegg]
$K^\circ_L \! \rightarrow \! \mu^+\mu^-\gamma\gamma$ $(1.42^{+1.02}_{-0.81}\pm0.14)\times10^{-9}$ KTeV-99[@klmmgg]
: *Summary of results of decays to two photons.*
\[tab\_kgg\]
The KTeV measurements of $K^\circ_L \! \rightarrow \! e^+ e^- e^+ e^-$ and $K^\circ_L \! \rightarrow \! \mu^+ \mu^- e^+ e^-$ should improve by $\times$4 and the modes $K^\circ_L \! \rightarrow \! e^+
e^-\gamma\gamma$ and $K^\circ_L \! \rightarrow \! \mu^+ \mu^-
\gamma\gamma$ should improve by $\times$3 with the final KTeV data set. The $K^\circ_L \! \rightarrow \! e^+ e^-\gamma$ should improve by $\times$20 and $K^\circ_L \! \rightarrow \! \mu^+ \mu^- \gamma$ should improve by $\times$30. The $K_S$ modes may be improved by NA48 in a special run, after $\epsilon'/\epsilon$. The $K^\circ_L \!
\rightarrow \! \gamma\gamma$ and $K^\circ_S \! \rightarrow \!
\gamma\gamma$ as well as several other modes will be improved by KLOE. There is no improvement in the foreseeable future for $K^\circ_L \!
\rightarrow \! e^+ e^-$ or $K^\circ_L \! \rightarrow \! \mu^+ \mu^-$.
Radiative K$_{\ell2}$ Decays
============================
The form factors in the decays $K^+ \! \rightarrow \! \ell^+\nu_\ell
\gamma$, A and V, and, R, in the decays $K^+ \! \rightarrow \!
\ell^+\nu_\ell \ell'^+\ell'^-$, are predicted by ChPT. Recent measurements should allow precise experimental determinations of all three parameters. The most recent determination of $|F_V + F_A| =
0.165\pm0.007\pm0.011$ from the E787 measurement of the direct emission component of $K^+ \! \rightarrow \! \mu^+\nu\gamma$, usually called Structure Dependent (SD$^+$) radiation, is consistent with the previous determination of $|F_V + F_A| = 0.148\pm0.010$ from $K^+ \!
\rightarrow \! e^+\nu\gamma$. A limit of $-0.25 < F_V - F_A < 0.07$ is derived from the $K^+ \! \rightarrow \! \mu^+\nu\gamma(SD^+)$. An improved measure of $F_V - F_A$ along with a measure of $R$ should be available soon from E865.
A summary of the recent radiative K$_{\ell2}$ results is presented in Table \[tab\_kl2g\].
0.1 in
Decay Mode Citation
--------------------------------------------- ----------------------------------- -------------------
$K^+ \! \rightarrow \! \mu^+\nu\gamma$ $(5.50\pm0.28)\times10^{-3}$ KEK-85[@kmng]
$K^+ \! \rightarrow \! \mu^+\nu\gamma(DE)$ $(1.33\pm.12\pm.18)\times10^{-5}$ E787-97[@kmng_de]
$K^+ \! \rightarrow \! e^+\nu\gamma(DE)$ $(1.52\pm0.23)\times10^{-5}$ CERN-79[@keng]
$K^+ \! \rightarrow \! \mu^+\nu\mu^+\mu^-$ $< 4.1\times10^{-7}$ E787-89[@kmnmm]
$K^+ \! \rightarrow \! e^+\nu\mu^+\mu^-$ $< 5.0\times10^{-7}$ E787-98[@kenmm]
$K^+ \! \rightarrow \! \mu^+\nu e^+e^-$ $\sim$1500 events E865-99[@kmnee]
$K^+ \! \rightarrow \! e^+\nu e^+e^-$ $\sim$400 events E865-99[@kmnee]
: *Summary of Radiative K$_{\ell2}$ results.*
\[tab\_kl2g\]
Other Radiative Kaon Decays
===========================
The experimental sensitivity for the other radiative kaon decays K$_{\pi3\gamma}$, K$_{\ell3\gamma}$ and K$_{\pi4\gamma}$ are such as to only be sensitive to IB contributions. All of these measurements are consistent with theoretical predictions. A summary of the results is given in Table \[tab\_kpppg\].
0.1 in
Decay Mode Citation
------------------------------------------------------------ --------------------------------------- -------------------- --
$K^+ \! \rightarrow \! \pi^+\pi^+\pi^-\gamma$ $(1.04\pm0.31)\times10^{-4}$ ITEP-89[@kpppg]
$K^- \! \rightarrow \! \pi^-\pi^\circ\pi^\circ\gamma$ $(7.5^{+5.5}_{-3.0})\times10^{-6}$ IHEP-95[@kpp0p0g]
$K^+ \! \rightarrow \! \pi^\circ\mu^+\nu_\mu\gamma$ $< 6.1 \times10^{-5}$ ZGS-73[@kpmng]
$K^\circ_L \! \rightarrow \! \pi\mu\nu_\mu\gamma$ $(5.7^{+0.6}_{-0.7})\times10^{-4}$ NA48-98[@klpmng]
$K^+ \! \rightarrow \! \pi^\circ e^+\nu_e\gamma$ $(2.62\pm0.20)\times10^{-4}$ ITEP-91[@kpeng]
$K^\circ_L \! \rightarrow \! \pi e\nu_e\gamma$ $(3.62^{+0.26}_{-0.21})\times10^{-4}$ NA31-96[@klpeng]
$K^+ \! \rightarrow \! \pi^\circ\pi^\circ e^+\nu_e\gamma$ $< 5\times10^{-6}$ ITEP-92[@kp0p0eng]
$K^+ \! \rightarrow \! \pi^+\pi^- e^+\nu_e\gamma$ no limit (E865-97?)
: *Summary of Radiative 3- and 4-body decays.*
\[tab\_kpppg\]
A couple of modes should be seen for the first time in existing data, $K^+ \! \rightarrow \! \pi^\circ\mu^+\nu_\mu\gamma$ (E787) and $K^+
\! \rightarrow \! \pi^+\pi^- e^+\nu_e\gamma$ (E865). Improvements in other modes may be possible, particularly at IHEP.
Conclusions
===========
Several new results are expected from KTeV and NA48, as well as a few more from E787,E865 and E871. With the turn on of DA$\Phi$NE and KLOE, which is well equipped for the radiative modes, we can expect another round of new measurements. Finally, the next generation of rare kaon experiments, designed to fully constrain the CKM unitarity triangle, by measuring the ‘Golden modes’ $K^+ \! \rightarrow \! \pi^+ \nu
\overline{\nu}$ and $K^\circ_L \! \rightarrow \! \pi^\circ \nu
\overline{\nu}$ , are under construction (E391a, E949) or being designed (KOPIO, CKM, KAMI). These experiments will provide even more precise measurements of several radiative modes.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank several people from various experiments for providing data and discussions for this talk, including: Hong Ma, Mike Zeller, Bob Tschirhart, John Belz, Lutz Koepke, Bill Molzon, Leonid Landsberg, Takeshi Komatsubara, Takashi Nakano and Laurie Littenberg. This work was supported under U.S. Department of Energy contract \#DE-AC02-98CH10886.
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[^1]: To be published in the Proceedings of the 3${rd}$ Workshop on PHYSICS AND DETECTORS FOR DA$\Phi$NE, Frascati, Italy, Nov. 16–19, 1999 (Frascati Physics Series Vol. XVI, 2000)
[^2]: This correction assumes that the form factor has no energy dependence.
|
---
abstract: 'POLICAN is a near-infrared imaging linear polarimeter developed for the Cananea Near-infrared Camera (CANICA) at the $2.1\,\mathrm{m}$ telescope of the Guillermo Haro Astrophysical Observatory (OAGH) located in Cananea, Sonora, México. POLICAN is mounted ahead of CANICA and consist of a rotating super-achromatic ($1-2.7\,\mathrm{\mu m}$) half-wave plate (HWP) as the modulator and a fixed wire-grid polarizer as the analyzer. CANICA has a 1024 $\times$ 1024 HgCdTe detector with a plate scale of $0.32\,\mathrm{arcsec/pixel}$ and provides a field of view of $5.5 \times 5.5\,\mathrm{arcmin^2}$. The polarimetric observations are carried out by modulating the incoming light through different steps of half-wave plate angles $(0^{\circ}, 22$.$5^{\circ}, 45^{\circ}, 67$.$5^{\circ})$, to establish linear Stokes parameters ($I, Q$, and $U$). Image reduction consists of dark subtraction, polarimetric flat fielding, and sky subtraction. The astrometry and photometric calibrations are performed using the publicly available data from the Two Micron All Sky Survey. Polarimetric calibration includes observations of globular clusters and polarization standards available in the literature. Analysis of multiple observations of globular clusters yielded an instrumental polarization of 0.51%. Uncertainties in polarization range from 0.1% to 10% from the brightest $7\,\mathrm{mag}$ to faintest $16\,\mathrm{mag}$ stars. The polarimetric accuracy achieved is better than 0.5% and the position angle errors less than $5^{\circ}$ for stars brighter than $13\,\mathrm{mag}$ in $H$-band. POLICAN is mainly being used to study the scattered polarization and magnetic fields in and around star-forming regions of the interstellar medium.'
author:
- 'Devaraj R.'
- 'A. Luna'
- 'L. Carrasco'
- 'M. A. Vázquez-Rodríguez'
- 'Y. D. Mayya'
- 'J. G. Tánori'
- 'E. O. Serrano Bernal'
title: 'POLICAN: A near-infrared imaging polarimeter at the $2.1\,\mathrm{m}$ OAGH telescope'
---
Introduction
============
Imaging polarimeters offer great opportunity to study various astrophysical topics ranging from galactic regions to extragalactic sources such as active galactic nuclei (AGNs). Polarimeters built to function in optical wavelength are plenty [e.g. IAGPOL, IMPOL, Dipol-2, RoboPol; @mag96; @ram98; @pir14; @king14], but they are limited to provide only partial insight into some of the science cases. On the other hand, near-infrared (NIR) polarimetry offers a unique window to observe new regions, revealing different physical phenomena. One of the main subjects of interest for polarimetric studies with $2\,\mathrm{m}$ class telescopes is the interstellar dust and cool galactic star-forming regions. The linearly polarized light from the stars, caused by dichroic extinction [@hall49; @hiltner49] from dust grains, which are aligned to local magnetic fields [@davis51; @lazarain07], is very useful in understanding the interplay of interstellar matter and magnetic fields. Combining theory and observations, we can begin to understand dust properties and magnetic fields from dense cores to star-forming regions in and around molecular cloud complexes [e.g. @jones89; @nishi09; @chapman11]. Existing NIR polarimeters like SIRPOL [@kandori06] and Mimir [@clemens07] have been used to conduct various observations and surveys like GPIPS [@clemens12a] to study the magnetic fields in the galactic medium. Additionally, NIR polarimetry can aid in investigating the circumstellar structures of young stellar objects (YSOs) whose radiation is scattered by dust and is observed as infrared reflection nebulae [@tamura06; @hashimoto08]. Polarization data from spiral galaxies can be used to study galaxian magnetic field properties and the field’s orientation to the disk [e.g. @jones97; @clemens13; @mont14].
With all of these diverse astrophysical topics to exploit, and considering that only few NIR polarimeters are available, we built a new instrument called POLICAN[^1] to function as an imaging linear polarimeter. POLICAN operates at the $2.1\,\mathrm{m}$ telescope of the Guillermo Haro Astrophysical Observatory (OAGH) in Cananea, Sonora, México. It is attached to the Cananea Near-infrared Camera (CANICA) [@car17] to operate as one of the primary backend instruments at the telescope. The instrument development was completed in the year 2012 with the support of funding from the Mexican science agency CONACyT.
POLICAN consists of basic polarizing elements: a rotating super-achromatic ($1-2.7\,\mathrm{\mu m}$) half-wave plate (HWP) as the modulator and a fixed wire-grid polarizer as the analyzer. These are housed in an external assembly placed between the telescope and CANICA as shown in Figure \[fig1\].
To meet the scientific requirements and to obtain good quality polarization data, it is important to make POLICAN function well and to understand its characteristics and operational behavior. Obtaining accurate polarimetric data requires optimization of observation methods and a robust data processing and analysis tool kit. Further, a detailed calibration of the instrument is necessary to calculate the true polarization. This led to development of various strategies and methods for operation and calibration of POLICAN. Additionally, software pipelines were developed for handling the large amounts of data to be processed into science-quality results. The core of POLICAN capabilities depends on the CANICA characteristics, which have been characterized and evaluated in @dev17; hereafter Paper I. POLICAN incorporates a mechanical design similar to SIRPOL and most of its calibration scheme are derived from the Mimir team’s approach [@clemens12b]. Preliminary descriptions about POLICAN are reported in @dev15 [@dev17a]. In the following sections, we present the instrument overview, polarimeter operation, observational properties, data processing methods, calibration, and observational results of POLICAN.
Instrument overview
===================
The reflector telescope at the OAGH observatory is a Ritchey-Chrétien configuration on an equatorial mount with a primary mirror of $2.1\,\mathrm{m}$ diameter and a secondary hyperbolic mirror of $50\,\mathrm{cm}$, yielding a focal length of $25\,\mathrm{m}$. The CANICA optics re-image the f/12 beam of the telescope to f/6 onto the detector. This optical setup provides observed images with an average point spread function (PSF) of $1.{\arcsec}5$ FWHM. CANICA operates primarily in the three broadbands $J (1.24\,\mathrm{\mu m})$, $H (1.63\,\mathrm{\mu m})$ and $K^{'} (2.12\,\mathrm{\mu m})$ including other multiple narrow bands. The camera unit is made up of a cryostat assembly including collimator, filter wheels, focusing system and a detector. The detector is a HgCdTe HAWAII array of 1024 $\times$ 1024 pixels with a plate scale of $0.32\,\mathrm{arcsec/pixel}$ and provides a field of view (FOV) of $5.5 \times 5.5\,\mathrm{arcmin^2}$. CANICA is operated using a correlated double sampling (CDS) readout method, and hence the raw data delivered after POLICAN observations are the CDS images. Details of CANICA construction and design are presented in @car17. A complete description of CANICA characteristics and performance are presented in Paper I.
Polarimeter
-----------
The CANICA has a light entrance window that is off-centered with respect to the cryostat so as to align it with the optical axis designed to accommodate filter wheels. To adjust for the displaced entrance window, an adapter couples the telescope and the CANICA. As a result, it was observed that the polarimeter POLICAN can be implemented in the space next to the adapter putting the polarizing elements externally to the CANICA at room temperature (see Figure \[fig1\]).
POLICAN’s polarizing elements consists of a retarder (half-wave plate) and an analyzer (wire-grid polarizer). The retarder is a super-achromatic ($1-2.7\,\mathrm{\mu m}$) HWP of diameter $50\,\mathrm{mm}$, which is made by cementing pairs of MgF$_{2}$ and Quartz plates. It has very low path difference of $\pm 0.04\%$ and the change in orientation of the optical axis is negligible $\pm 0.2^{\circ}$ across the entire spectral range. The retarder is manufactured by Bernhard Halle Nachfl, Germany. The analyzer is a holographic high extinction ratio (HER) polarizer of diameter $71\,\mathrm{mm}$ deposited on a CaF$_{2}$ substrate, with a large spectral range from $1-10\,\mathrm{\mu m}$. It has a grid spacing of 4000 lines/mm with a transmission efficiency of 84% at $2.5\,\mathrm{\mu m}$. The polarizer is manufactured by Specac company, UK.
The polarizing elements are housed inside an aluminum mechanical assembly with a detachable system. The detachable system can be manually removed from the beam along a translating stage, so as to switch the observations between normal photometry and imaging polarimetry. The change in back focal length during each mode is corrected by positioning the secondary mirror accordingly. The top and bottom ends of the mechanical assembly are provisioned with circular flanges for attaching to the telescope and to CANICA. The flexure in the mechanical assembly due to the weight of CANICA at different declinations is found to be negligible. The adapter for coupling POLICAN with the telescope is a rotating system that can be used to orient the instrument. The entire setup with the adapter is aligned to a setting of 328$^{\circ}$ to orient the observations along north-up and east-right direction on the detector. Figure \[fig2\] shows a zoomed view of the mechanical assembly with the stepper motor, HWP, and the polarizer. Details of the mechanical assembly design and construction are presented in @vazrod12.
Operation and Control {#opcntrl}
=====================
Linear polarimetric observations can be achieved from a combination of rotating modulators and analyzers whereby the orthogonal components of polarization are produced. Dual-beam polarimeters [e.g. PLANETPOL, DBIP, MMTPOL; @hough06; @masiero07; @pack12] that have a Glan-Thompson prism or a Wollaston prism as analyzers can produce two orthogonal polarizing components simultaneously for a single position of the modulator, thereby operating faster and reducing sky-dependent noise. However, they are restricted to observing well separated or isolated sources across a fairly narrower field. POLICAN, on the other hand, is designed to operate in a single-beam mode which has the advantage of observing both point and extended sources with a medium FOV. However, it must compromise on time cadence and deal with the effects of sky-dependent noise. Such a design of POLICAN requires a minimum of four modulation angles to obtain the polarizing components needed to establish the linear Stokes parameters [@shur62]. The change in the polarization state of the light as it passes through different optical components can be described by a Mueller matrix formalism [@clarke10]. From analysis of Mueller matrices, we find that the input linear Stokes parameters $I$, $Q$, and $U$ can be established from the output lights intensity, by modulating the incoming light with four HWP angles of the first quadrant$\mbox{---}0^{\circ}, 22$.$5^{\circ}, 45^{\circ}, 67$.$5^{\circ}$. Appendix \[mlrmat\] describes the detail derivation of the modulation scheme with the use of Mueller matrices. Other sets of HWP angles from different quadrants can be used similarly to obtain the Stokes parameters. The results from each quadrant can then be combined to reduce HWP dependent errors. However, with POLICAN setup, we chose only the angles of the first quadrant to limit the large integration time during observations.
The control for HWP rotation for modulation is handled by a stepper motor that is integrated into the polarimeter mechanical assembly. The stepper motor connects to the HWP with a gear-to-gear transmission system and the rotation delivered is followed in precise step angle of 0.3$^{\circ}$. The reference home position of the HWP (i.e. the zero-phase axis) is identified by a Hall effect sensor. Connection to the motor controller from the main observation computer is through a RS232 serial communication line. The stepper motor unit is provided by Parker Motion Control Systems, USA, and the motor controller program is a software in the Visual BASIC. To simultaneously achieve control of CANICA and POLICAN, the stepper motor control and the CANICA control are integrated. This allows for scripted observations to be acquired in sequence for each HWP angle at each dithered position.
Observation goals {#obgoals}
=================
The chief scientific goal of POLICAN is to study magnetic fields in the star-forming regions of the nearby (distances $\sim$few kpc) interstellar medium (ISM). Hence, it was important to define various observational properties to carry out polarimetric studies meeting the scientific requirements.
Area coverage
-------------
Most of the polarimeters available in the NIR can either do wide-field, large-scale surveys (e.g. Mimir, SIRPOL), spanning a few tens of parsecs, or use adaptive optics to study narrow field regions [e.g. ZIMPOL, GPI; @rol10; @perrin15], across scales of $100-1000\,\mathrm{AU}$. With POLICAN we aim to bridge this gap to obtain polarization information at intermediate scales, between few-parsec to sub-parsec scales in the ISM. On the spatial range, we can achieve this by targeting areas of size from $\sim0.2\,\mathrm{arcmin^2}$ to $200\,\mathrm{arcmin^2}$.
Waveband
--------
At NIR bands, the magnetic field information in the ISM obtained from starlight polarization is revealed due to dichroic extinction. While $K$-band offers the best window to probe regions of high extinction ($A_{V}$), the thermal emission from the sky is large. On the other hand, at $J$-band the sky emission is low, but the $A_{V}$ values probed are limited. The $H$-band offers the best compromise, where we can sufficiently probe regions of moderate $A_{V}$ with low thermal emission from the sky. Hence, current observations with POLICAN are concentrated in $H$-band, which are presented in this article. However, calibration and study of sources in $J$ and $K$-band is being carried out simultaneously.
Sensitivity and sampling goals
------------------------------
The starlight polarization information from the reddened stars in the ISM is usually weak, of the order of $1\%-5\%$ [@mat70]. To accurately perform starlight polarimetry for studying magnetic field properties requires measurements with polarization uncertainties below 1% [@clemens12a]. Further, to have adequate stellar density for tracing magnetic field at sub-parsec scales means the angular sampling must be greater than $10-20$ reliable sources per square arcmin. From the Two Micron All Sky Survey (2MASS) [@skrut06] data, to reach the above angular sampling we need to observe magnitude depths of $13-14\,\mathrm{mag}$ in the galactic plane between latitudes $b\pm1^{\circ}$. Hence, to achieve a polarization uncertainty of around 1% for stars as faint as $13\,\mathrm{mag}$ (e.g. Figure \[fig13\]) requires per image exposure times ranging from $30\,\mathrm{s}$ to $40\,\mathrm{s}$ in $H$-band with POLICAN.
Signal-to-noise ratio goals
---------------------------
The accuracy of polarimetric measurements also depends on the source signal-to-noise ratio (S/N). 2MASS data reached $15\,\mathrm{mag}$ at a S/N of 10 in $H$-band. We aim to obtain a $\mathrm{S/N}>10$ for $15\,\mathrm{mag}$ sources in order to reach the best polarimetric accuracy. This can be achieved by combining more number of images acquired per field for each HWP. Hence, we considered typically a minimum of 15 images are required for each HWP angle, leading to a total of 60 images for a given observing field. This leads to a total integration time of around 30 to $40\,\mathrm{mins}$ for a given field.
Essentially, these values and estimates form the basis for starlight polarimetry with POLICAN. However, observations of extragalactic and other sources can be customized to have different integration times as desired.
Observation scheme {#obscheme}
==================
Ground-based images obtained in the NIR are contaminated by the atmospheric sky emission (OH line and thermal continuum emission), sky transmission noise, and thermal emission from the telescope and optics. Successfully isolating these effects during data processing is essential to consider in an observing scheme. A standard practice for observing in the NIR is to obtain multiple images using telescope dither. This facilitates estimating the aforementioned “sky” contributions. Typically, a minimum of five dithered images is sufficient to establish the sky image. However, as noted in the previous section, we want to obtain more images per field to boost the S/N of the combined image. Hence, with POLICAN, we implemented a sequence to acquire 15 dithered images for each HWP angle for a given observing field.
The Stokes parameters are calculated by the difference in flux between two orthogonal polarization measurements (see Section \[polanal\]). The flux difference is more accurate when the two orthogonal polarizations are obtained in sequence, as the change in sky transmission is minimum [@clemens12a]. Based on this, we designed the observing scheme such that the HWP images corresponding to each Stokes parameter are acquired consecutively for each dithered position (i.e. in the order $0^{\circ}$ and $45^{\circ}$ for Stokes $Q$, $22$.$5^{\circ}$ and $67$.$5^{\circ}$ for Stokes $U$). This sequence is followed for all the 15 dither positions leading to a total of 60 images per observed field.
The dithering strategy varies depending on the field of interest and size of the source.\
1) For studying magnetic field properties through starlight polarimetry, the sequence of observations consists of 15 dithers distributed in a non-repetitive random pattern within a diameter of $30\,\mathrm{arcsec}$ around the targeted center. The dither size of $30\,\mathrm{arcsec}$ makes sure that the extended emission from bright stars do not overlap in each image.\
2) For studying scattered polarization from extended sources, the sequence of observation consists of a dithering pattern of 8 source images and 7 off-field images. The source and off-field images are obtained alternatively. The off-field images are taken by dithering completely outside the source field in one or more different cardinal directions (typically along the north-south direction).
To obtain the dark contribution, a set of 10 dark images are acquired at the end of the observing night for all values of exposure time used. The 10 darks for each exposure are then averaged to obtain mean dark images.
The automatic image acquisition sequence for observations is passed to the telescope control system by a JavaScript from the main computer operated by the observer. The script includes functions for telescope dither, camera exposure time, and HWP rotation. The user can modify each parameter as desired. Once the script is run, user intervention is not needed. A log of the POLICAN observing runs conducted to date is saved both in hard copy and in digital format. On a typical night, POLICAN produces around 600 images, which approximately sum up to $2.5\,\mathrm{GB}$ of memory.
Flat fielding strategy {#flat}
----------------------
There are both temporal and spatial variations in the detector that affect the quality of the images. Temporal variations from noise and sky amplify with spatial variations introduced by non-uniform illumination and pixel-to-pixel variations in the detector. Other effects include dust on the HWP, which rotates during modulation and is not canceled in the Stokes combination scheme. To correct for all of the above effects, a suitable flat-fielding strategy is required. Because polarimetric observations vary with each HWP angle, the flats should be obtained at the same HWP angles.
Based on Mimir calibration [@clemens12b] methods, we implemented a similar technique for obtaining these “polarimetric flats.” The flats consisted of sequence of images ($\sim30-$empirical estimation) acquired with dome lights ON and OFF for each HWP angle. Exposure times were set so as to fill half of the full-well-depth in the ON images. The lights ON and OFF images are differenced to eliminate the effects of dark counts and thermal backgrounds. The differenced flats are then averaged to obtain a master flat with high S/N. This is repeated for each HWP angle to obtain four master flats. The master flats are normalized using the mean value obtained from a region of 40 $\times$ 40 pixels in the central zone \[460:500, 540:580\] of the detector. Figure \[fig3\] shows different characteristics of a normalized master flat. The center panel shows the 3D surface plot of the normalized master flat where the illumination profile is seen along with large-scale non-uniform pixel-to-pixel variations. The last panel shows row and column cuts obtained for a particular region of pixels.
Data Processing {#dataproc}
===============
The images obtained with POLICAN contain a number of instrumental effects in addition to the NIR atmospheric contamination, leading to a challenge to obtain high-quality, linear polarimetric data. These include 1) the non-linear response of the detector array pixels; 2) the dark current that is pixel, time, and temperature dependent; 3) the bad pixels, about 0.2% spread across the detector array; 4) the non-uniform illumination profile and the pixel-to-pixel variations; 5) detector crosstalk effects, seen as residual negative charges in different quadrants; 6) the atmospheric sky emission and transmission that varies with time and position; and 7) the thermal emission from telescope and optics that is FOV and time-dependent.
To extract useful polarimetric information from POLICAN observations, all the above problems need to be canceled or mitigated. Hence, a robust data processing method was developed to obtain accurate values of degree of polarization and position angles for each observing field while also managing all necessary corrections. The procedure involves two distinct stages of data processing. The first stage consists of the basic steps of reduction and image correction leading to science-quality images. The second stage utilizes the multiple measurements and analyses from the reduced images to yield degree of polarization, position angles, and their corresponding uncertainties.
Basic data processing {#BDR}
---------------------
Stage 1: The first stage of data processing that will lead to science-quality images uses a custom pipeline called *POLREDUCE*, developed in the IRAF[^2] environment. Image display and interactive functions are carried out with the help of SAO DS9 software. The individual dithered images obtained for a given set of observation are associated based on their HWP angle and filter band. The corresponding mean dark images and polarimetric flats are grouped in the same set. The reduction process is carried out separately for each HWP angle leading to four science-quality images. The steps in reduction are as follows:
1\) The first step in reduction involves non-linearity correction. The correction coefficients are estimated by characterizing the illumination response of each pixel using dome flats as described in Paper I. Using the correction coefficients, the individual images are corrected for each pixel to establish linearity corrected images.\
2) Next, the images are subtracted with the mean dark image to remove the dark count contribution.\
3) The non-uniform illumination, pixel-to-pixel variations, and any effects introduced by the HWP are corrected by flat fielding using polarimetric flats (see Section \[flat\]).\
4) The individual dithered images after Step 3 are stacked and median combined using the *imcomb* task to obtain the sky image. The images are then subtracted with the sky image to obtain the reduced “clean” images. The reduced images have an offset in their zero level introduced by the median sky.\
A modal filter is applied to each image with a $3\sigma$ lower and upper threshold. The modal value gives the zero level offset, which is then subtracted from the reduced images, resulting in uniform background nearly to zero. The images are next corrected for detector crosstalk effects as described in Paper I.\
Step 4 is slightly modified if the reduction is carried out for extended sources. Because dithering is performed as alternating source and off-field, only the off-field images are median combined to obtain the sky image. The rest of the procedure remains the same.\
5) Because the images are shifted because of dithering, they have to be aligned before combining. The aligning procedure is carried out by selecting the centroids ($Xcen$ and $Ycen$) of a common star in all of the images. Using the centroids, the shift in each image, both in $X$ and $Y$ directions, is computed, keeping the first image as reference. From the computed image shifts, the dithered images are aligned using the *imalign* task. The aligned images are next average combined with a minmax rejection to produce the final reduced image.\
Image aligning and combination remains the same for extended sources with only the source images used in the process.\
6) The final corrected image is then transformed to the equatorial system with north-up and east-left direction. Next, the image is cropped to the central 4$\times$4 arcmin$^2$, which removes major optical aberrations (as shown in Paper I).\
7) The last step in the pipeline involves finding all of the point sources using the *daofind* task, with a detection threshold of $5\sigma$. The source positions as centroids $Xcen$ and $Ycen$ are stored in a file. Additionally, all the important parameters involved in the reduction process are saved and can be used in future for rapid re-processing.\
Steps 1 to 7 are performed for all four sets of images corresponding to each HWP angle, obtaining the final reduced images and source positions. The reference image used in aligning all four sets of images is the same, and hence the final four reduced images are matched in the image coordinate system. The reduced images and results form the basis for polarimetric analysis, which is explained in Section \[polanal\].
### Astrometry corrections
Observations made with POLICAN have astrometric information updated to the image headers based on telescope and dithering data. These values were found to differ by a few arcsec to a few arcmin when compared with astrometric-based coordinates. The images were also found to have rotation offsets of a few degrees and to possess slight geometric distortions. Hence, scientific analysis of the reduced images required accurate astrometric corrections. Existing astrometry software such as [astrometry.net](astrometry.net) [@lang10] failed to obtain correct solutions, due large offsets in POLICAN image headers, leading to the necessity of implementing a customized program. This is carried out in two steps, both relying on the astrometric information provided by the publicly available 2MASS data.
The first step involves making coarse corrections to center the images such that the astrometric errors are withing a few arcsec. A reference star is chosen in the image, and its corresponding 2MASS coordinates are obtained. The image header is then updated for the central reference value (CRVAL) with the 2MASS coordinates of the reference star. This transforms the image field with astrometric information close to the true coordinates. Users satisfied with this information have the choice to skip the second step and move directly to polarimetric analysis. Most of the time this is the case for observations of extended sources. However, starlight polarimetry requires further corrections.
The second step involves obtaining solutions to rectify the image rotation and geometric distortions. This is performed with the help of tasks in IRAF *imcoords* package. As the images at this stage have a roughly acceptable astrometry, the 2MASS catalog for all the point sources in the field are obtained within a given search radius. Next, the $Xcen$ and $Ycen$ for a minimum of six point sources in the image are associated with their 2MASS coordinates. These are then used to obtain the plate solutions in the equatorial system. The astrometry correction for one HWP image is sufficient to correct the other 3 HWP images, as they are aligned with respect to each other. The astrometry information of the first image is directly copied to the other three, making them equal both in image and celestial coordinates.
Polarimetric Analysis {#polanal}
---------------------
Stage 2: The second stage of polarimetric data processing is carried out using a pipeline developed in Interactive Data Language (IDL). The pipeline is called *FLX2POL*, and it combines functions and procedures that are found in the IDL Astronomy Library [@land93] and in the Coyote Graphics Library[^3]. The main steps in polarimetric analysis are to accurately measure the flux of sources; establish the Stokes parameters; and compute polarizations, position angles, and their corresponding uncertainties.
The output file from Stage 1 consists of source positions for all point sources. The first step is to match source positions among the four HWP images and obtain a common identification and location list. The source-matching algorithm selects the centroids $Xcen$ and $Ycen$ for all sources in the four HWP images and runs a cross-correlation search within a given radius, typically 2 pixels ($\sim0.{\arcsec}6$). Once the sources are matched, they are sorted in sequence and given a common identification number.
Polarimetric analysis of point source involves measuring the total flux in each of the four HWP images, which is mainly achieved by synthetic aperture photometry. In POLICAN pipeline, synthetic aperture photometry is performed using the *aper* routine adapted from the DAOPHOT [@stet87] package. The pipeline includes two runs of photometry that allow aperture selection for obtaining accurate flux values. The first run of photometry involves calculating the S/N of the source with an optimum aperture radius. In the second run, the final flux of the source is measured with an aperture whose value is chosen depending on the S/N of the source. The necessity of aperture selection based on S/N is required, as the PSF of the sources vary both in time and across the FOV as shown in Paper I.
[cc]{} $<$ 10 & 7\
10 - 50 & 8\
50 - 100 & 9\
100 - 500 & 10\
500 - 1000 & 11\
1000 - 5000 & 12\
5000 - 10000 & 13\
$>$ 10000 & 14\
Based on magnitude growth analysis [@howell89; @stet90], we find that an aperture radius of 10 pixels is optimum for photometry. Using this radius, we perform first run of photometry and obtain the S/N of all the sources. Next, a new aperture is chosen between a range of 7 to 14 pixels (empirical estimation) for each source depending on its S/N (see Table \[tbl-1\]). Using the new aperture, a second run of photometry is performed on each source, obtaining the final flux and flux error values which are used to establish the Stokes parameters and their uncertainties. The final flux values in each of the HWP image can be denoted as $I_{0}, I_{22.5}, I_{45}, I_{67.5}$.
The Stokes $I$, $Q$, and $U$ in the instrumental reference system are now computed as follows: $$\label{eqn1}
I = (I_{0}+I_{45}+I_{22.5}+I_{67.5})/2$$ $$Q = (I_{0}-I_{45})/I$$ $$U = (I_{22.5}-I_{67.5})/I$$
These are then scaled by polarization efficiency $\eta$ and rotated by the HWP zero-phase offset angle, $\theta$ (see Section \[pcalstnd\]) to obtain the equatorial Stokes values. The polarization efficiency $\eta$ was obtained from SIRPOL measurements [@kandori06], as both instruments use the same polarizing elements from the same manufacturers. The $\eta$ values are $J=0.955$, $H=0.963$, and $K=0.985$. The calculation is as follows: $$Q_{eq} = (Q cos(2\theta) - U sin(2\theta))/\eta \label{eqn4}$$ $$U_{eq} = (U cos(2\theta) + Q sin(2\theta))/\eta \label{eqn5}$$
Next, the Stokes values are corrected for instrumental polarization, derived from globular cluster observations (see Section \[pcalgc\]): $$Q_c = Q_{eq} - Q_{inst} \label{eqn6}$$ $$U_c = U_{eq} - U_{inst} \label{eqn7}$$
Finally, the corrected Stokes values are combined to form the equatorial degree of polarization, $P_{eq}$, and the position angles, $P.A$, measured from the north-up to east-left direction:
$$P_{eq} = 100\sqrt{Q_{c}^{2}+U_{c}^{2}} \label{eqn8}$$
$$P.A = \frac{1}{2}\tan^{-1}\left(\frac{U_{c}}{Q_{c}}\right). \label{eqn9}$$
The $P.A$ values have an ambiguity in the calculation because the arctangent function produces values between $-\pi/2$ to $\pi/2$ radians. They are corrected to represent them within the range of 0 to 180$^{\circ}$. The polarization uncertainty $\sigma_{P}$ is computed from the corresponding Stokes errors which is described in Appendix \[polerr\]. The calculated polarization values have a positive bias because of the quadrature combination of the Stokes parameters. The Ricean correction prescription of @wardle74 which works well for polarization S/N values greater than 2 (i.e. $P/\sigma_{P} \ge 2)$ is used for de-biasing the polarization values as follows:
$$P = \sqrt{P_{eq}^{2}- \sigma_{P}^{2}} \label{eqn10}$$
Photometric measurements for each source is next obtained from the deep co-added intensity image calculated from the four HWP images as in equation \[eqn1\]. Based on the previously obtained image source list, aperture photometry is performed on all the sources to obtain their magnitude values. A broad zeropoint correction on instrumental magnitudes is applied by using the average zeropoint value. The source coordinates are converted from pixel values to celestial coordinates using the *xyad* routine, which uses the astrometric information available in the image header.
The values of polarimetric analysis together with astrometric and photometric measurements are combined to form a catalog of results. The catalog contains the source information as follows:\
$ID$, $RA$, $DEC$, $Xcen$, $Ycen$, $P\%$, $Perr$, $P.A$, $P.Aerr$, $Mag$, $MagErr$, $Q\%$, $U\%$, $Qerr$, $Uerr$
Polarization visualization are carried out by producing map of vectors representing $P$ and $P.A$ for each source using a separate customized program (e.g Figure \[fig14\]).
Polarimetric analysis for extended sources remain similar to the above description. Because the four HWP images are aligned with each other, the surface brightness in each pixel is used for analysis instead of flux of a point source. The Stokes parameters and polarization values are obtained in order up to equation \[eqn10\] for each pixel. The results are then stored into an image array. The visualization and map making procedure is carried out by binning the polarization values in each pixel, with a certain threshold as required by user. For each bin, a vector is plotted to represent the $P$ and $P.A$ value (e.g Figure \[fig8\]).
Polarimetric Calibration
========================
Design factors and optical setup introduce instrumental polarization that need to be carefully removed to faithfully recover the true polarization. As the polarizing elements in POLICAN are located ahead of CANICA, the only contribution for instrumental polarization should be the telescope mirrors. POLICAN is designed for single-beam linear polarimetric observations with just two polarizing components: the HWP and polarizer. There exists no room for other optical elements such as prisms in the setup for producing artificial polarization for calibrations. Hence, all the calibration needs to be done through observations of astronomical objects. For POLICAN, we used two standard steps of polarimetric calibration as described for the Mimir instrument [@clemens12b]: 1) removal of instrumental polarization across the FOV from observations of globular clusters and 2) converting instrumental polarization position angle to equatorial position angle from observations of polarimetric standards. In the following sections, we describe the methods and results obtained for polarimetric calibration.
Globular Clusters {#pcalgc}
-----------------
Globular clusters are known to have stars with low polarization levels. Clusters with large angular extent are useful to calculate the instrumental polarization across the entire FOV. Mimir calibration observations of multiple globular clusters (M2, M3, M5, M12, M13 $\mathrm{etc.}$) showed that has polarization values below $0.1\%$ with low color excess $E(B-V)$, representing the best cluster with distributed stars. Hence, we concentrated on observing only to determine the instrumental polarization for POLICAN. M5 observations were conducted in various runs over a period of three years from 2013 to 2016. In total 37 sets of clear sky observations were obtained, which was used for analysis. The images were acquired using dithering methodology for 15 positions as described for extended sources in Section \[obscheme\]. In all observing runs, the image acquisition order remained same, with exposure time set to $20\,\mathrm{s}$. The M5 fields were distributed to fall on the detector center as well as at different quadrants of the detector. This allowed to have distributed polarization values in all the pixels, making it feasible to map the instrumental polarization across the entire FOV.
The images from each observation were reduced as described in Section \[BDR\]. All the stars in each field were selected and analyzed up to calculations of Stokes values in equatorial system (i.e. up to equation \[eqn5\]), as described in Section \[polanal\]. By the time of analysis, the HWP zero-phase offset angle $\theta$ was determined and hence we converted the Stokes values into equatorial system. Next, the polarimetric and photometric values of the stars were obtained up to equation \[eqn10\], omitting equation \[eqn6\] and \[eqn7\]. This produced a catalog of results for thousands of stars in each observing field. The results were filtered to obtain Stokes values only for bright stars with low polarimetric uncertainties. (i.e. for stars with $\mathrm{mag}<13$ and $\sigma_{P}<1.0$). This avoided the faint stars with larger errors and forced the measurements to true instrumental polarization. Further, the central crowded field of stars were excluded, because aperture photometry fails to obtain accurate measurements in crowded regions. An alternative to aperture photometry is PSF photometry, but this could not be applied, as CANICA images have varying PSF across the FOV as described in Paper I.
Next, for each observation, the mean Stokes $\langle Q\rangle$ and $\langle U\rangle$ were calculated from the individual Stokes values of the selected sample of stars. These were then combined to form the grand mean Stokes value $\langle Q_{inst}\rangle$ and $\langle U_{inst}\rangle$. Figure \[fig4\] shows the distribution of mean Stokes $\langle Q\rangle$ and $\langle U\rangle$ in plus symbol for the 37 observations of M5. The grand mean Stokes value is shown at the center with a circle and plus symbol. The estimated values of grand mean instrumental Stokes values are $\langle Q_{inst}\rangle=-0.53\%\pm0.23$, $\langle U_{inst}\rangle=0.14\%\pm0.2$.
To obtain the final value of instrumental polarization across the FOV, we combined results of all the selected sample of stars in the 37 observations. A total of 10,700 stars were obtained in the combined result. The combined Stokes $Q$ and $U$ values for all the stars were examined by a histogram and a Gaussian[^4] was fitted to the distribution as shown in Figure \[fig5\]. The peak of the histogram matched with the peak of the Gaussian fit, which represented the final instrumental Stokes value of POLICAN. The values were estimated to be $Q_{inst}=-0.50\%$ and $U_{inst}=0.12\%$. These values remained consistent within $0.02\%$ levels when compared with the grand mean Stokes value. The uncertainties in the instrumental Stokes $\sigma_{Qinst}$ and $\sigma_{Uinst}$ were taken as the standard deviation of the fit, which gave $\sigma_{Qinst}=0.42\%$ and $\sigma_{Uinst}=0.36\%$.
The full-field instrumental Stokes values were next calculated from the same sample of 10,700 stars. The Stokes values for all the stars were sorted by their positions and gridded into an image array with resolution of 1 pixel. The image arrays were next smoothed to 10 pixels and converted into a contour map to represent the Stokes value for a FOV of 4$\times$4 arcmin$^2$. Figure \[fig6\] shows the map of instrumental Stokes $Q_{inst}$ and $U_{inst}$, along with the $P_{inst}$ value computed from the former Stokes values. The variations in instrumental Stokes, both $Q$ and $U$, across the FOV are around $\pm0.2\%$. This variation is below the Stokes uncertainties, indicating FOV dependence of instrumental polarization is minimum. In summary, the final instrumental Stokes $Q_{inst}$ and $U_{inst}$ are calculated to be $-0.50\%$ and $0.12\%$, and the mean instrumental polarization is $0.51\%$.
Polarimetric Standards {#pcalstnd}
----------------------
POLICAN’s mechanical assembly and the polarizing elements are fixed along their axis and mounted stationarily to the telescope. It is not definitive that the HWP zero-phase angle is aligned to the equatorial north. This results in calculations of polarization position angle to be based in the instrumental coordinates. Hence, it is important to determine the HWP zero-phase offset, to correct the position angle to standard equatorial system. Observation of polarimetric standards are the best way to determine the offset angle. Mimir’s calibration observations of polarimetric standards provided a large sample of stars for the study. They were mainly derived from @whittet92, who studied wavelength dependence of polarization. The $H$-band filter central wavelength and bandwidths remained the same for POLICAN and Mimir, which avoided any corrections for wavelength dependence.
A large number of calibration observations during each run were directed towards two bright standard stars that are available during most times of the year. These were and in the fields of Orion and Cygnus. Here, we present results obtained for HD38563C which was observed for a total of 19 nights during a period of 6 months.
The observation scheme for HD38563C was similar to point sources as described in Section \[obscheme\] with exposure time set to $5\,\mathrm{s}$. The HD38563C field was targeted to fall on the detector center to avoid any effects introduced by optical aberrations. The images were reduced and analyzed up to equation \[eqn6\] and \[eqn7\] as described in Section \[polanal\], to obtain the corrected Stokes values $Q_c$ and $U_c$. The only change in the analysis was that the HWP offset angle was set to zero. The instrumental polarization calculated from globular clusters remained in the instrumental system for this analysis.
The individual Stokes values, $Q_c$ and $U_c$ from each of the 19 observations were averaged to obtain the mean Stokes value. Next, the mean Stokes value was used to compute the Ricean corrected polarization and position angle as described in equation \[eqn8\] to \[eqn10\]. The computed position angle was then compared with the Mimir’s published value to find the difference. The difference gave the HWP zero-phase offset angle for POLICAN, which was determined to be $139^{\circ}$.
Figure \[fig7\] shows results of HD38563C corrected for position angle for all the 19 observations. Results from each observing night are represented by a black-filled circle with their corresponding error bars. The final computed mean polarization and position angle are expressed on the top left corner of each panel. Overall observations showed good agreement with published values. Over the course of the last few years, there were times when the motor sensor failed to locate the HWP home position. This was corrected immediately and the sensor was brought back to the original setting to keep the home position constant, in turn the offset angle for POLICAN remained consistent.
Observations
============
POLICAN saw its first light at the end of 2012. Over the past few years, the majority of the observations were concentrated towards polarimetric calibration and pilot studies. These observations were spread evenly in each semester for an average telescope time of around 7 to 8 weeks in a year. Early science proposals were targeted for well known sources to provide comparison of POLICAN results with the literature data. In parallel, a few new regions of interest were also observed to compliment the current studies with priority. These included galactic molecular clouds, H[ii]{} regions, planetary nebulas, and post-AGB stars.
The starlight polarimetry towards the center of the galactic plane is usually around 3% polarization due to the presence of large columns of dust. These regions provide a good starting point for pilot studies. The GPIPS survey [@clemens12a] spanning from $18^\circ\le l\le 56^\circ$ and $-1^\circ\le b\le 1^\circ$, contains an excellent catalog of polarimetric data for comparing the POLICAN observations.
To evaluate the performance of POLICAN, we observed two regions that represented good examples of scattered polarization and starlight polarimetry. They were Sharpless H[ii]{} region and the GPIPS field number 182. In the following sections, we present the observation details for each of them.
Scattered polarization
----------------------
is an emission nebula estimated to be at a distance of $600\,\mathrm{pc}$ in the Cygnus constellation. At the center of the nebula is a young massive star (type O8) of approximately 15 solar masses, which emits jets of gas forming a bipolar structure. We carried out $H$-band polarimetric observation towards the central 4$\times$4 arcmin$^2$ region surrounding the massive star. The observation scheme was similar to extended sources as described in Section \[obscheme\] with exposure time set to $20\,\mathrm{s}$. Image reduction and analysis followed the steps described in Section \[BDR\] and \[polanal\]. The final polarimetric results were saved into an image array.
Analysis of the results showed that the region surrounding the central star had high polarization levels to $\sim50\%$. The distribution of position angles appear to be in a centro-symmetric pattern, as had been previously shown by @saito09. This indicates that the nebulosity is illuminated by the central star which causes the strongly polarized light via dust scattering. Further, the distribution of bipolar structure shows the circumstellar matter associated to the massive star, similar to a disk/envelope system. In Figure \[fig8\] we show the results of SIRPOL [@saito09] and POLICAN observations of Sh 2-106. The right panel displays POLICAN polarization vectors plotted for surface brightness above 3$\sigma$, binned for every $3\times3$ pixels. Comparing the results in Figure \[fig8\], we see that the angular and spatial resolution achieved with POLICAN is better than SIRPOL. This allows us to resolve dense regions around massive stars to have a clear distinction of nebulosities and circumstellar matter. Such observations with POLICAN will help to obtain scattered polarization at smaller scales making it vital for star-forming studies.
Starlight polarization
----------------------
The GPIPS field 182 (hereafter GP182) is centered around the galactic coordinates of $l=20.456$ and $b=-0.645$ with a size of $10 \times 10\,\mathrm{arcmin^2}$. @clemens12c showed that GP182 field contains high stellar density with significant polarization detections. Further, the stars in the field have high polarization S/N ($P_{S/N}=P/\sigma_{P}$) and their galactic position angles are oriented along the galactic plane. Hence, GP182 field is an ideal region for evaluating POLICAN’s performance. We chose a total region covering $20 \times 12\,\mathrm{arcmin^2}$ for mapping GP182 and its surrounding areas. The useful FOV with POLICAN is $4 \times 4\,\mathrm{arcmin^2}$, therefore the observations need to span multiple pointings to cover the entire region. By equally placing $4 \times 4\,\mathrm{arcmin^2}$ fields distributed over the entire region, we obtained a total of 15 pointings for POLICAN. Figure \[fig9\] shows the background 2MASS image overlaid with 10 arcmin field of GP182 in black color. The 15 POLICAN $4\,\mathrm{arcmin}$ fields are shown in blue color and are marked from R1 to R15, based on their observing orders. The fields are not overlapped with each other, as the full FOV of POLICAN is $5.5 \times 5.5\,\mathrm{arcmin^2}$ and during image reduction they are cropped to $4\,\mathrm{arcmin}$ fields.
GP182 observations with POLICAN were conducted for four nights during 2017 April. Each field was observed with 15 dither positions totaling 60 images for all the HWP angles. The exposure time was fixed to $20\,\mathrm{s}$ with a dither diameter of $30\,\mathrm{arcsec}$. During each night three to four fields were observed and started at the same universal time to keep the airmass and time-dependent variations minimum. The total clock time taken to complete the 15 fields was $7.5\,\mathrm{hours}$. The basic reduction included linearity correction, dark subtraction, polarimetric flat-fielding, sky subtraction, and image combination. Thereduced images were astrometry corrected and processed for polarimetric analysis through aperture photometry. The detailed steps in reduction and analysis followed the description in Section \[BDR\] and \[polanal\]. Once the astrometric, photometric, and polarimetric results were obtained for all the 15 fields, they were combined into a catalog representing POLICAN data for the entire region. The results and comparison to GPIPS data are discussed in the Section \[res\] and \[per\]. The mapping strategy by having multiple pointings for large regions forms the basis for polarimetric observations of molecular clouds, filaments and H[ii]{} in the ISM. Given the large clock time for such observations, the regions are limited to sizes within $20 \times 20\,\mathrm{arcmin^2}$.
Results {#res}
=======
Stellar properties
------------------
The combined starlight polarimetric catalog towards GP182 region for all the 15 POLICAN fields resulted in a total stellar count of 13,635 stars. These were obtained by selecting sources above $5\sigma$ in the deep co-added intensity image. Out of this entire stellar population, 9556 stars had definite polarization detections. They formed the essential sample of stars for all future analysis. Analyzing 2MASS survey data for the same region, we find the number of stellar count obtained is 4453 stars. This showed that POLICAN observations had twice the number of detections to 2MASS. Similarly, analyzing GPIPS data for the same region, we obtain a total of 7230 stars with definite polarization detections. This indicated the number of polarization detections with POLICAN is much higher for the chosen integration time. Figure \[fig10\] shows the stellar count histogram against magnitude for POLICAN, GPIPS and 2MASS data. Also plotted is the cumulative distribution function for each data. It is seen that POLICAN observations reached depth of many orders better than 2MASS. The majority of stars were in the magnitude range from $13\,\mathrm{mag}$ to $16\,\mathrm{mag}$, with 50% probability of detection for $14\,\mathrm{mag}$ stars. The stellar density achieved with POLICAN in this region is about $30-40$ stars per square arcmin, meeting the sampling goals as described in Section \[obgoals\].
Photometric properties
----------------------
The photometric results obtained from the deep co-added intensity image for the 9556 polarization detections were analyzed for their magnitude properties. Because the photometric values had only broad corrections on the magnitudes, as described in Section \[polanal\], a post correction was implemented to obtain accurate magnitude values. This was carried out by zeropoint corrections using the 4453 2MASS matched stars, as described in Paper I. The resultant photometry showed that the stars in GP182 region spanned magnitude ranges from $7\,\mathrm{mag}$ to $18\,\mathrm{mag}$. Errors in photometric magnitudes were below 1% up to $13\,\mathrm{mag}$ stars and 10% up to $15.5\,\mathrm{mag}$ stars. Figure \[fig11\] shows a plot of S/N against magnitude with the magnitude error in the right axis. The S/N values were obtained from the flux and flux error values of the stars. It is seen that the S/N achieved is greater than 10 for stars up to $15.5\,\mathrm{mag}$. This matched the desired signal-to-noise ratio goals described in Section \[obgoals\].
The 4453 2MASS matched stars were compared with their magnitudes for estimating photometric accuracy. The difference in POLICAN and 2MASS magnitudes are plotted against their magnitude along with their corresponding magnitude errors in Figure \[fig12\]. The dispersion in magnitude differences were better than $0.05\,\mathrm{mag}$ up to $11\,\mathrm{mag}$ stars. For stars up to $13\,\mathrm{mag}$, the dispersion was around $0.1\,\mathrm{mag}$. For fainter stars, the dispersion increased to around $0.3\,\mathrm{mag}$, due to larger photometric uncertainties.
Polarimetric properties
-----------------------
The polarimetric data available for all the 9556 stars provided a full range of polarization properties. Uncertainties in polarization ($\sigma_{P}$) values are useful to calculate the polarization S/N ($P_{S/N}$). Examining the polarization uncertainty against the magnitude showed POLICAN observations had polarization uncertainties of 1% up to $13\,\mathrm{mag}$ and 2% up to $14\,\mathrm{mag}$. This matched our polarization sensitivity goals as described in Section \[obgoals\]. Figure \[fig13\] shows the log plot of polarization uncertainty against POLICAN magnitude for stars from 7 to $18\,\mathrm{mag}$.
[cc]{} UF = 0 & $\sigma_{P}<1$
&
mag$<$13\
&
&
$P_{S/N}>2.5$\
UF = 1 & $\sigma_{P}<1$
&
mag$<$13\
UF = 2 & $\sigma_{P}<2$
&
mag$<$14\
UF = 3 & $\sigma_{P}>2$
&
mag$>$14\
The reliability of polarization data can be determined from combination of $\sigma_{P}$, $P_{S/N}$ and magnitude. @clemens12c classified the stellar polarizations based on their reliability into usage flags (UF) to allow easy identifications. As magnetic field studies using POLICAN’s starlight polarimetry have well-established observing goals and scheme (Section \[obgoals\] and \[obscheme\]), it will be useful to classify POLICAN data. Based on the observed polarization properties, we formed a new set of usage flags for POLICAN data as follows: UF = 1 represented the high-quality polarization values having $\sigma_{P}$ within 1% and magnitude $<$ 13.0. Further, the stars with UF = 1 having $P_{S/N}>2.5$ were categorized under UF = 0 category. UF = 0 represents the highest-quality of polarization values that can directly trace magnetic field directions with lowest dispersion in position angles. UF = 2 sample is categorized for stars with $\sigma_{P}$ within 2% and magnitude $<$ 14.0. They represent the moderately resolved magnetic field directions. The UF = 2 sample can mostly be used to produce a mean magnetic field for a region with higher stellar density. The rest of stars belong to UF = 3 category. They can be averaged with UF = 1 and 2 stars to provide a very low resolution map of magnetic field. The average polarization values from UF = 3 can also predict the mean polarization in large regions of the local ISM. Table \[tbl-2\] lists the complete classification of stars based on their usage flags for POLICAN. Figure \[fig13\] shows the usage flag classification on the plot of polarization uncertainty against magnitude.
Performance {#per}
===========
After establishing POLICAN’s GP182 stellar photometric and polarimetric properties, the results could be compared with the GPIPS data. The polarimetric data from GPIPS survey was derived from *GPIPS data release 3.1*. This included the most recent version with the best compilation of up to date data. Individual stars from POLICAN and GPIPS data were matched to obtain common detections. A total of 1298 stars had common detection for UF = 1 category, with also common detection in the 2MASS data. Out of these, there were 817 stars that matched UF = 0 category. Figure \[fig14\] shows the background 2MASS image of GP182 region with polarization vectors for UF = 0 stars. The polarization values of POLICAN are overlaid in blue vectors, with the GPIPS values overlaid in red vectors. Visually, the polarization vectors align with each other in their length and position angle.
[lccc]{} Plate scale & 0.32 arcsec/pixel & On the detector\
Field of view & $4\times4$ arcmin$^{2}$ & Cropped from $5.5\times5.5$ arcmin$^{2}$\
Photometric accuracy & $<0.1\,\mathrm{mag}$ & For stars brighter than $13\,\mathrm{mag}$\
Polarimetric accuracy & $<$ 0.5% & For stars with UF = 1\
Position angle accuracy & $<5^{\circ}$ & For stars with UF = 1\
Instrumental polarization & 0.51% & $Q_{inst}=-0.50\%$ and $U_{inst}=0.12\%$\
Polarization uncertainties & 0.1% to 10% & From $7\,\mathrm{mag}$ to $16\,\mathrm{mag}$\
HWP zero-phase offset angle & $139^{\circ}$ & Correction angle\
The values of UF = 1 subset form the core information to determine POLICAN’s polarimetric accuracy and performance. Since UF = 1 are considered reliable and high-quality data, they need to equal the GPIPS data consistently. Comparison of both the data sets were carried out for all the 1298 matched stars. The differences between POLICAN and GPIPS stars were established by subtracting their individual polarization values. The difference in polarization percentage were below or around $0.5\%$ for stars brighter than $11\,\mathrm{mag}$. For fainter stars the difference reached up to $\sim1.5\%$. The position angle difference were below or around $5^{\circ}$ for stars brighter than $11\,\mathrm{mag}$. For fainter stars, the difference in position angle reached up to $\sim15^{\circ}$, with some exceeding it. Figure \[fig15\] shows the plot of POLICAN and GPIPS polarization and position angle differences against 2MASS $H$-band magnitude.
A histogram examination of the polarization differences gives the accuracy of POLICAN. In Figure \[fig16\] we show the histogram distribution for both polarization and position angle differences. A Gaussian is fitted for each distribution to determine the peak and standard deviation. The peak in the polarization and position angle differences are close to zero, indicating there is no offset in the calculated values. This means that the polarimetric efficiency, HWP zero-phase offset angle, instrumental polarization and polarization de-biasing, are well established for POLICAN. Because the data set represented high-quality UF = 1 stars, the standard deviation of the Gaussian fit should give the polarimetric accuracy of POLICAN. From the fit, we see that the polarization accuracy is found to better than 0.5% and the position angle accuracy is below $5^{\circ}$. The values of accuracy are minimum and within the expected levels of uncertainties with POLICAN data.
The established accuracies with POLICAN allow to obtain precise calculation of magnetic field strengths using @chan53 method. Overall, POLICAN’s UF = 1 subset showed good agreement with archival data, adequately meeting the polarimetric goals for magnetic field studies.
Summary
=======
We have described the important aspects in the operation, data processing, calibration, and performance of the newly developed polarimeter: POLICAN, at the $2.1\,\mathrm{m}$ OAGH telescope in México. POLICAN consists of a HWP and a polarizer that are housed in a mechanical assembly attached to CANICA and the telescope. The setup with plate scale of $0.32\,\mathrm{arcsec/pixel}$ and useful FOV of $4 \times 4\,\mathrm{arcmin^2}$ enables deep high-resolution medium field linear polarimetric imaging. The observation schemes are optimized to study polarization properties of both point sources and extended sources in the interstellar medium. POLICAN’s large data sets of raw images are handled by the robust image reduction and analysis techniques implemented into custom pipelines in IRAF and IDL environment.
Polarimetric calibrations were carried out from observations of globular clusters and polarimetric standards. The analysis of 10,700 stars from 37 observations of globular cluster M5, determined the instrumental polarization to be 0.51%. Observations of polarimetric standard HD38563C, determined the HWP zero-phase offset angle to be 139$^\circ$. Pilot studies were carried out for both extended and point source regions to obtain POLICAN’s observational results and performance. Scattered polarization was compared with SIRPOL data for Sh 2-106 object. Starlight polarimetry was compared with GPIPS data for GP182 field. Mapping a GP182 region of $20 \times 12\,\mathrm{arcmin^2}$ produced 9556 polarization detections reaching sensitivity many orders better than 2MASS survey. The polarimetric data were classified with usage flags to deem their reliability. A total of 1298 stars with reliable polarization under UF = 1 category were compared with the GPIPS data. POLICAN achieved polarization accuracy better than 0.5% and position angle errors below $5^\circ$ up to $13\,\mathrm{mag}$ stars in $H$-band. The entire performance of POLICAN is summarized in Table \[tbl-3\]. Based on background starlight polarimetry, POLICAN data can be used to trace the plane-of-sky magnetic field directions in the interstellar medium. Various observations on star-forming regions are being conducted towards the galactic plane to study their magnetic field properties. POLICAN features all characteristics of a sensitive NIR polarimeter capable of delivering reliable polarization data in the coming years.
We would like to thank the anonymous referee for useful comments on improving the article. We thank all the OAGH staff for their help in development and observations with the instrument. We are deeply indebted to the the valuable comments and feedback provided by Dan Clemens, Boston University, on improving the data processing techniques and calibration methods. We thank Eswaraiah Chakali, NTHU Taiwan, for helpful discussion on polarimetric analysis. This work has been carried out at Instituto Nacional de Astrofísica, Óptica y Electrónica, México, with support from CONACyT under the project CB-2012-01 182841. D.R. with CVU 555629 acknowledges CONACyT for the grant 370405. SAOImage DS9 software is developed with the funding from the Chandra X-ray Science Center (CXC), the High Energy Astrophysics Science Archive Center (HEASARC) and JWST Mission office at Space Telescope Science Institute. This work makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research used data from the Boston University (BU) Galactic Plane Infrared Polarization Survey (GPIPS), funded in part by NSF grants AST 06-07500, 09-07790, and 14-12269. GPIPS used the Mimir instrument, jointly developed at BU and Lowell Observatory and supported by NASA, NSF, and the W.M. Keck Foundation.
.
Stokes parameters and Mueller matrices {#mlrmat}
======================================
The Stokes parameters define the polarization state of a non-coherent electromagnetic radiation. Originally described by G. G. Stokes in his classic paper @stokes52, the Stokes parameters were re-introduced to modern astronomy by @chan47, who denoted the polarization states by $I$, $Q$, $U$ and $V$. The Stokes parameters can be combined to form a vector $S$ as $$S = \begin{pmatrix}
I \\ Q \\ U \\ V
\end{pmatrix}$$ where $I$ is the total intensity of the radiation; $Q$ is the intensity difference between horizontal and vertical linearly polarized components; $U$ is the intensity difference between linearly polarized components oriented at $\pm45^{\circ}$; and $V$ is the intensity of the circularly polarized radiation.
When electromagnetic radiation interacts with matter, it is likely to change its polarization state. The change in polarization state can be algebraically represented by matrix transformations of the input Stokes vector and the final measured Stokes vector. @mueller48 described the matrix calculus for different states of polarization, each represented by its $4\times 4$ matrix transformation equation. The Mueller matrix formalism is always carried out by matrix multiplication in a particular order, which is from the final measured Stokes vector to the input vector.
In the POLICAN setup, we use two polarizing components, which are the HWP and a polarizer. The matrix formalism for POLICAN with an input Stokes vector $S_{in}$ and the final measured Stokes vector $S_{out}$ can be represented as $$S_{out} = M_{pol} * M_{HWP} * S_{in}$$ where $M_{pol}$ is the Mueller matrix of the linear polarizer with fast axis oriented at $0^{\circ}$, and $M_{HWP}$ is the Mueller matrix of the HWP with arbitrary fast axis orientation $\theta$. These are represented by their respective Mueller matrices [@shur62] as
$$M_{pol} = \frac{1}{2}\begin{pmatrix}
1 & -1 & 0 & 0 \\
-1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \end{pmatrix}
\quad\mathrm{and}\quad
M_{HWP} = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & cos(2\theta)^2-sin(2\theta)^2 & 2sin(2\theta)cos(2\theta) & 0 \\
0 & 2sin(2\theta)cos(2\theta) & cos(2\theta)^2-sin(2\theta)^2 & 0 \\
0 & 0 & 0 & 0 \end{pmatrix}$$
Substituting the above matrices in equation A2, we can write the final Stokes parameter $S_{out}$ as $$S_{out} = \frac{1}{2}\begin{pmatrix}
I_{in} + Q_{in}(cos(2\theta)^2-sin(2\theta)^2) - U_{in}2sin(2\theta)cos(2\theta) \\
-I_{in} - Q_{in}(cos(2\theta)^2-sin(2\theta)^2) + U_{in}2sin(2\theta)cos(2\theta) \\
0 \\
0 \end{pmatrix}$$
The final measured intensity $I_{out}$ of the Stokes vector $S_{out}$, depends on the HWP fast axis $\theta$. As noted in Section \[opcntrl\], four HWP modulation angles are necessary for estimating the input linear Stokes parameters: $Q_{in}$ and $U_{in}$. Given that $\theta$ can have number of values between $0$ and 360$^{\circ}$, we can chose values such that $sine$ and $cosine$ functions in equation A4 can cancel out to remain $Q_{in}$ and $U_{in}$. The first four angles of $\theta$ that fulfill the conditions are $$\begin{matrix}
at & \theta = 0^{\circ} \\
& \theta = 22.5^{\circ} \\
& \theta = 45^{\circ} \\
& \theta = 67.5^{\circ}
\end{matrix} \quad\quad\quad
\begin{matrix}
sin(2\theta) = 0 \quad\mathrm{and}\quad cos(2\theta) = 1 \\
sin(2\theta) = 1/\sqrt{2} \quad\mathrm{and}\quad cos(2\theta) = 1/\sqrt{2} \\
sin(2\theta) = 1 \quad\mathrm{and}\quad cos(2\theta) = 0 \\
sin(2\theta) = 1/\sqrt{2} \quad\mathrm{and}\quad cos(2\theta) = -1/\sqrt{2}
\end{matrix}$$
Substituting these values in equation A4 and combining the Stokes vector $S_{out}$ as $I_{out}$, we get the output intensity for each HWP angle as $$\begin{matrix}
I_{0} = \frac{1}{2}[\pm I_{in}\mp Q_{in}] \\
I_{22.5} = \frac{1}{2}[\pm I_{in}\mp U_{in}] \\
I_{45} = \frac{1}{2}[\pm I_{in}\pm Q_{in}] \\
I_{67.5} = \frac{1}{2}[\pm I_{in}\pm U_{in}] \\
\end{matrix}$$
Now, we can re-arrange equation A6 to establish the input Stokes parameters as
$$\begin{matrix}
I_{in} = (I_{0} + I_{45} +I_{22.5} + I_{67.5})/2 \\
Q_{in} = I_{0} - I_{45} \\
U_{in} = I_{22.5} - I_{67.5}
\end{matrix}$$
Polarimetric Error analysis {#polerr}
===========================
Polarimetric analysis of point sources (mainly stars) are obtained by measuring the fluxes (integrated counts) on the stars in images corresponding to each of the orthogonal polarized components of linear Stokes parameters. The flux measurement in POLICAN is performed through synthetic aperture photometry on brightness profiles of the stars in the observed images based on the use of DAOPHOT package in IDL (see Section \[polanal\]).
The *phot/aper* function of the DAOPHOT package applied to the image with stars, measures the flux of a source in values of analog-to-digital units (ADU) as follows [@stet87]: $$I_{s} = I_{tot} - (area*skymod)$$ where $I_{s}$ is the total flux measured of the source within an aperture, $I_{tot}$ is the total flux measured within an aperture, $area$ is the total area of pixels in the aperture, and $skymod$ is the sky/background modal value per pixel measured from all the pixel values within a sky annulus (In IDL this is obtained by *mmm*).
The error in flux measurement $\sigma_{s}$ in ADU is given as follows: $$\sigma_{s} = \sqrt{(\frac{I_{s}}{gN_{i}}) + (area*skyvar) + (\frac{area^2*skyvar}{nsky})}$$ where $skyvar$ is the variance in sky measurement per pixel for the final image, $g$ is the gain in electrons/ADU, $N_{i}$ is the number of images used for constructing the final image, and $nsky$ is the number of pixels used in the sky annulus during photometry.
Based on the equation of Stokes parameters as described in Section \[polanal\] and Appendix \[mlrmat\], the error in Stokes parameters can be given by standard error propagation as follows:
$$\sigma_{I} = \frac{\sqrt{\sigma_{0}^2 + \sigma_{22.5}^2 + \sigma_{45}^2 + \sigma_{67.5}^2}}{2}$$
$$\sigma_{Q} = \sqrt{\frac{\sigma_{0}^2 + \sigma_{45}^2}{I^2} + (\frac{Q}{I}\sigma_{I})^2}$$
$$\sigma_{U} = \sqrt{\frac{\sigma_{22.5}^2 + \sigma_{67.5}^2}{I^2} + (\frac{U}{I}\sigma_{I})^2}$$
where $\sigma_{0}, \sigma_{22.5}, \sigma_{45}, \sigma_{67.5}$ are flux errors for the fluxes $I_{0}, I_{22.5}, I_{45}, I_{67.5}$ measured in each HWP angle.
The Stokes parameters are next scaled by polarization efficiency $\eta$ and rotated by the HWP zero-phase offset angle, $\theta$ as shown in equation \[eqn5\]. The corresponding Stokes errors in equatorial system are $$\sigma_{Qeq} = \sqrt{(\frac{cos2\theta}{\eta}\sigma_{Q})^2 + (\frac{sin2\theta}{\eta}\sigma_{U})^2}$$ $$\sigma_{Ueq} = \sqrt{(\frac{cos2\theta}{\eta}\sigma_{U})^2 + (\frac{sin2\theta}{\eta}\sigma_{Q})^2}$$
Next, the Stokes values are corrected for instrumental polarization as in equation \[eqn7\] and are represented with their errors as $$\sigma_{Qc} = \sqrt{\sigma_{Qeq}^2 + \sigma_{Qinst}^2}$$ $$\sigma_{Uc} = \sqrt{\sigma_{Ueq}^2 + \sigma_{Uinst}^2}$$ where $\sigma_{Qinst}$ and $\sigma_{Uinst}$ are error in instrumental Stokes values calculated from globular cluster observations (see Section \[pcalgc\]).
The final Stokes values are combined to give the equatorial degree of polarization $P_{eq}$ and the position angle $P.A$ as in equation \[eqn8\] and \[eqn9\]. The error in polarization is calculated as $$\sigma_{P} = \frac{100^2}{P_{eq}}\sqrt{(Q_{c}^2*\sigma_{Qc}^2) + (U_{c}^2*\sigma_{Uc}^2)}$$
After obtaining the de-biased polarization value $P$ as in equation \[eqn10\], the $P.A.$ uncertainty ($\sigma_{P.A}$) is computed as $$\sigma_{P.A} = 28.65(\frac{\sigma_{P}}{P})$$
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[^1]: POLICAN: arimetro nfrarojo para ICA
[^2]: Image Reduction and Analysis Facility (IRAF) is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. <http://iraf.noao.edu/>
[^3]: <http://www.idlcoyote.com/>
[^4]: In each plot of Gaussian fit, \# indicates the total number of measurements in the distribution, $\mu$ indicates the peak value of the Gaussian fit, and $\sigma$ indicates the standard deviation of the Gaussian fit.
|
---
abstract: 'We quantize a scalar field at finite temperature $T$ in the background of a classical black hole, adopting ’t Hooft’s “brick wall” model with generic mixed boundary conditions at the brick wall boundary. We first focus on the exactly solvable case of two dimensional space-time. As expected, the energy density is integrable in the limit of vanishing brick wall thickness only for $T=T_H$ - the Hawking temperature. Consistently with the most general stress energy tensor allowed in this background, the energy density shows a surface contribution localized on the horizon. We point out that the usual divergences occurring in the entropy of the thermal atmosphere are due to the assumption that the third law of thermodynamics holds for the quantum field in the black hole background. Such divergences can be avoided if we abandon this assumption. The entropy density also has a surface term localized on the horizon, which is open to various interpretations. The extension of these results to higher space-time dimensions is briefly discussed.'
---
\[section\] \[theo\][Corollary]{} \[theo\][Proposition]{} \[theo\][Definition]{} \[theo\][Conjecture]{} \[theo\][Lemma]{}
[**Aspects of Finite Temperature Quantum Field Theory in a Black Hole Background**]{}\
[Giuseppe Milanesi $^{a,}$[^1] and Mihail Mintchev $^{b,}$[^2]]{}\
[*$^a$ Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2-4, 34014 Trieste, Italy and INFN, Sezione di Trieste*]{}\
[*$^b$ INFN and Dipartimento di Fisica, Universitá di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy*]{}
Introduction and outline
========================
Since the original proposal by Bekenstein [@Bekenstein:1973ur] that a black hole carries an entropy proportional to its area and later discovery of the Hawking radiation [@Hawking:1974sw], the microscopic origin of the thermodynamical behavior of black holes has been one of the main puzzles of theoretical physics and possibly one of the key problems for the understanding of quantum gravity. The literature is really vast and here we will give an account of just the references strictly relevant to our context.
A very simple but nonetheless instructive model to address the problem of black hole entropy is the so called “brick wall” by ’t Hooft [@'tHooft:1984re]. ’t Hooft considers a quantum field in the background of a classical black hole and using the WKB approximation he derives the thermal entropy of the field outside the horizon of the black hole. In performing the computation, two spatial cutoffs are employed: a large distance one, needed to avoid large volume divergences in the asymptotically flat region, and a short distance one, the “brick wall”, localized just outside the horizon and suppressing the divergences due to the growing number of modes close to the horizon. On the boundaries of the space slice arising in this way, Dirichlet boundary conditions are imposed. As noted in [@Mukohyama:1998rf; @Mukohyama:1998ng], the finite temperature state used in this quantization is a thermal excitation of the Boulware vacuum [@Boulware:1974dm]. The entropy obtained in this model is divergent in the limit of vanishing brick wall thickness. These divergences were later recognized as quantum corrections to the Bekenstein-Hawking formula which can be absorbed into renormalization of the one loop effective gravitational lagrangian [@Susskind:1994sm; @Demers:1995dq; @Fursaev:1994ea; @Cognola:1995km; @deAlwis:1995cr; @winstanley_01] (see also [@Brustein:2005vx] for a recent perspective). Within this context the “brick wall” takes the role of a useful mathematical tool to regularize the theory. An interesting different interpretation has been recently proposed in [@Barbon:2003aq; @Solodukhin:2005qy].
The introduction of the brick wall cutoff as a more physical device can be considered consistent with recent proposals arising in string theory in which the nature of space-time “inside” the horizon has deep quantum mechanical structure (see [@Mathur:2005zp]).
In this paper we revisit the brick wall model. We first consider in detail the case of 1+1 dimensional Schwarzschild black hole, because it can be solved exactly without resorting to the WKB approximation. We then argue that our results can be extended to higher dimensions. In order to clarify the role of the boundary conditions, we adopt a generic mixed (Robin) boundary condition, which is the most general linear and local one. The cyclic state used in the quantization is a Kubo-Martin-Schwinger (KMS) state[^3], with respect to the Schwarzschild time. It accounts for the thermal excitations over the Boulware vacuum. The Boulware vacuum polarization can be exactly calculated in 1+1 dimensions, via the energy-momentum conservation and the trace anomaly [@Mukohyama:1998ng]. At this point one can derive the energy inside the “shell” between the brick wall and any fixed point outside the horizon. In the limit of vanishing brick wall thickness, the energy of the shell is divergent unless we choose the KMS state temperature $T$ to coincide with the Hawking temperature $T_H$. In other words, the brick wall can be removed only provided that $T=T_H$. This somewhat expected result does not depend on the boundary conditions being in this sense universal [@Hartle:1976tp; @Jacobson:1994fp].
In all the above considerations one has to keep in mind that the expectation value of the energy-momentum tensor is a distribution rather than a function. We observe in this respect that when considering the horizon as part of the space-time in exam, the above expectation value admits in general a non-vanishing Dirac delta contribution, localized on the horizon. To our knowledge this term has not been previously taken into account and indeed appears in our computation in the limit of vanishing brick wall thickness. We will show that it will also lead to a corresponding surface term in the entropy density.
The entropy of a thermodynamical system is determined by its energy up to an arbitrary constant. In most of the cases this constant is fixed via the third principle of thermodynamics (Nernst theorem), which requires that the entropy vanishes at zero temperature. In our case, the computation of the entropy from the energy of a shell outside the horizon is however a subtle matter. The derivation of the entropy for generic $T$ must be performed before removing the brick wall, since the removal is possible only for $T=T_H$. Any attempt to determine at this stage the arbitrary constant by the third principle, leads to divergences when later removing the brick wall. These divergences are of the same kind encountered in usual computation in the brick wall model. We can get rid of them if we do not ask the quantum fields on such a background to satisfy the third principle. We note here that the issue on the third principle is a priori non related to the observations regarding extremal black holes and Nernst theorem (see for example [@Wald:1997qp; @Belgiorno:2002pm]). Note also that abandoning the third principle turns out to be a different kind of regularization of the theory with respect to the one previously considered in the literature [@Demers:1995dq; @Fursaev:1994ea; @Cognola:1995km; @deAlwis:1995cr; @winstanley_01; @Brustein:2005vx] and involving infinite renormalization of the coupling constants.
Our calculations yield also a surface term for the entropy density localized on the horizon. In our setting this term, together with the corresponding surface term in the energy density, is understood as a boundary effect: taking the brick wall boundary as a pure mathematical tool, it can be considered as a (potentially finite) contribution to the one loop renormalization of Newton constant; on the other hand, taking the brick wall and the boundary condition as somewhat more physical, we can interpret these terms as an indication of the presence of physical degrees of freedom localized on the horizon. We leave some more detailed discussion on this for the last section of the paper.
The paper is organized as follows. The model is introduced in the next section. In Sect. 3 we show that the total energy of a shell outside the horizon is finite even in the limit of vanishing brick wall thickness, provided that the temperature of the thermalized field equals $T_H$. We also show the appearance of a surface term in the energy density. In Sect. 4 we derive the entropy and discuss the issue related to the third principle of thermodynamics. Sect. 5 is devoted to a comparison with ’t Hooft results in the WKB approximation. In Sect. 6 we look for a possible generalization to more realistic models in higher dimensions. Finally, Sect. 7 contains a further discussion and our conclusions.
The model {#statement}
=========
We consider a free real scalar field in a generic $n+1$ dimensional space-time $\{{\mathcal{M}},\, g\}$ with a time-like boundary ${\partial}{\mathcal{M}}$. The action $$\label{azioneSULbd} S=\int_{\mathcal{M}}{\mathrm{d}}^{n+1}x\,\sqrt{|g|}\left(\frac{1}{2}\, g^{\mu\nu}{\partial}_\mu
\varphi {\partial}_\nu \varphi - \frac{1}{2} m^2 \varphi^2\right) -
\int_{{\partial}{\mathcal{M}}} {\mathrm{d}}^n x\,\sqrt{|g_{\mathrm{ind}}|} \frac{\eta}{2}
\varphi^2 \, ,$$ where $|g_{\mathrm{ind}}|$ is the determinant of the induced (lorentzian) metric on ${\partial}{\mathcal{M}}$, implies both the Klein Gordon equation $$\label{KG} (g^{\mu\nu}\nabla_\mu\nabla_\nu + m^2) {\varphi}=0$$ and the Robin boundary condition $$\label{condbordo} (g^{\mu\nu}N_\mu{\partial}_\nu{\varphi}-\eta{\varphi})|_{{\partial}{\mathcal{M}}}
= 0\, ,$$ where $N^\mu$ is the unit vector normal to ${\partial}{\mathcal{M}}$. Using the boundary condition , one can also express the action as $$\label{azioneCONbdINTERNA} S=\int_{\mathcal{M}}{\mathrm{d}}^{n+1}x\,\sqrt{|g|}\left[\frac{1}{2}\,g^{\mu\nu} {\partial}_\mu
\varphi {\partial}_\nu \varphi - \frac{1}{2} m^2
\varphi^2-\frac{1}{2}g^{\mu\nu}\nabla_\mu({\varphi}\,{\partial}_\nu
{\varphi})\right]\, .$$ The stationarity condition for and the boundary condition are equivalent to the stationarity condition for the action . We note that the second expression for the action is suitable also for the Dirichlet boundary condition. From one derives the energy-momentum tensor $$\label{tmnINbd}
\theta_{\mu\nu}(x)=\frac{2}{\sqrt{|g|}}\frac{\delta
S}{\delta\left[ g^{\mu\nu}(x)\right]} =
-{\varphi}\nabla_\mu\nabla_\nu{\varphi}+ \frac{1}{2}g_{\mu\nu}
\left(g^{\rho\sigma}{\varphi}\nabla_\rho\nabla_\sigma{\varphi}+m^2{\varphi}^2\right)
\, .$$
Let us focus now on a massless ($m=0$) scalar field in the background of a $1+1$ dimensional classical black hole, with metric $$\label{1+1 metric} {\mathrm{d}}s^2=f(r){\mathrm{d}}t^2 -\frac{1}{f(r)}{\mathrm{d}}r^2 \, .$$ We require asymptotic flatness $$\lim_{r\to \infty} f(r) = 1$$ and assume that $f$ has one and only one zero in $r=r_0$ (the horizon) with positive *surface gravity* $$\kappa_0 \equiv \frac{1}{2}f'(r_0) > 0 \, . \label{surfgrav}$$ We are thus considering a non-extremal black hole.
Following [@'tHooft:1984re], we insert a brick wall at $r=\rho
> r_0$ and study the region $r\geq\rho$, which represents a static space-time with time-like boundary. We consider there the dynamics with respect to the time-translation defined by the flux of $\frac{\partial}{{\partial}t}$, the time-like Killing vector of the metric. This is the Schwarzschild time for our model.
The classical equation of motion for the scalar field is $$\frac 1 {f(r)}{\partial}^2_t{\varphi}-{\partial}_r\left [f(r){\partial}_r{\varphi}\right ] =0\, .$$ The Robin boundary condition takes the form $$\label{condbordo1+1}
\left.\left(\sqrt{f(r)}\,{\partial}_r{\varphi}-\eta{\varphi}\right)\right|_{r=\rho}=0
\, ,$$ where $\eta $ in general can depend on $\rho$: $$\eta=\eta(\rho) \, .$$ In the coordinates $(t,y)$, with the “tortoise coordinate" $y$ defined by $$y=y(r)\, , \qquad\frac{{\mathrm{d}}y}{{\mathrm{d}}r}=\frac{1}{f(r)} \, ,$$ the metric is conformally flat. Notice that $f(r_0)=0$ and $f'(r_0)=2\kappa_0 > 0$ imply that for $r{\rightarrow}r_0$, to leading order $$y(r) \approx \frac{1}{2\kappa_0}\ln 2\kappa_0(r-r_0)\, , \qquad
f(r) \approx {\mathrm{e}}^{2\kappa_0 y(r)}\, .$$ In the coordinates $(t,y)$ the Klein-Gordon equation assumes its flat space form $$\label{eqm}
({\partial}_t^2-{\partial}_y^2){\varphi}=0 \, ,$$ and one is left with the problem of a scalar field on the half-line $y>Y\equiv y(\rho)$ with the boundary condition $$\label{condbordoY}
\left.\left({\partial}_y{\varphi}-{\eta_\rho}{\varphi}\right)\right|_{y=Y}=0\, , \qquad
{\eta_\rho}=\eta\sqrt{f(\rho)}\, .$$ In order to avoid imaginary energies we ask $$\eta\geq0\, .$$ The quantization of , is easily performed. The initial conditions are fixed by the canonical equal-time commutation relations $$\label{ccr} [{\varphi}(t,y_1)\, ,\, {\varphi}(t,y_2)]= 0 \, , \qquad
[{\partial}_t{\varphi}(t,y_1)\, ,\, {\varphi}(t,y_2)]=-i\delta(y_1-y_2) \, .$$ We introduce a class of quasi-free states $G_{\beta,\rho}$. Their two point functions satisfy the Kubo-Martin-Schwinger condition. They are Gibbs states at temperature $T=\beta^{-1}$ for our model with brick wall position $\rho$. The relative expectation values are denoted by $\langle{\varphi}(t_1,y_1)\cdots
{\varphi}(t_n,y_n)\rangle_{\beta,\rho}$. The basic one is the two-point function [@Liguori:1996xr; @Mintchev:2004jy] $$\begin{aligned}
\label{twopoint}
\langle{\varphi}(t_1,y_1){\varphi}(t_2,y_2)\rangle_{\beta,\rho} =
\qquad \qquad \qquad \qquad \qquad \nonumber \\
\int_{\mathbb R}\frac{{\mathrm{d}}p}{4\pi} \frac{|p|_\ell^{-1}}
{{\mathrm{e}}^{\beta |p|}-1}
\left[{\mathrm{e}}^{\beta |p|}{\mathrm{e}}^{-{\mathrm{i}}|p|(t_1-t_2)}+{\mathrm{e}}^{{\mathrm{i}}|p|(t_1-t_2)}\right]
\left[{\mathrm{e}}^{-{\mathrm{i}}p (y_1-y_2)}+B(p,\rho){\mathrm{e}}^{{\mathrm{i}}p(y_1+y_2-2Y)}\right]\, ,\end{aligned}$$ where $B(p,\rho)$ is the reflection factor from the boundary $$B(p,\rho)= \frac{p-{\mathrm{i}}{\eta_\rho}}{p+{\mathrm{i}}{\eta_\rho}} \label{rfactor}$$ and the distribution $|p|_\ell^{-1}$ is defined by $$|p|_\ell^{-1} \equiv \frac{{\mathrm{d}}}{{\mathrm{d}}p}\, \varepsilon(p) \ln (|p|
\ell )\, , \label{distribution}$$ $\varepsilon$ being the sign function. The derivative in is understood in the sense of distributions. The scale parameter $\ell$ has a well-known infrared origin [@Grignani:1988fx]. Note that $$\label{p ell property}
p\, |p|_\ell^{-1} = \varepsilon (p)\, ,$$ which implies, as we shall see later on, that $\ell$ is irrelevant in the calculation of the energy density.
The states $G_{\infty,\rho}$ can be considered as the analogous of the Boulware vacuum. Indeed they appear as usual vacuum to an observer at rest in the $r$ coordinate in the asymptotically flat region. They are annihilated by every destruction operator associated to a normal mode with respect to the Schwarzschild time.
Energy
======
In this Section we discuss in detail the derivation of the energy density for the states introduced in Section 2. At the Hawking temperature, the divergences occurring in the thermal energy of a shell outside the horizon are perfectly balanced by the Boulware vacuum polarization. From general considerations on the structure of the stress energy tensor, we cannot exclude a contribution to the energy density in the form of a surface term localized on the horizon. The introduction and following removal of a brick wall as a regularization confirms the presence of a finite term of this form. In general, the Boulware vacuum polarization could give a further surface contribution which cannot be determined in this setting. In principle a complete determination of such a term can be performed by some sort of measurement or deduced by a better understanding of this semiclassical picture in the context of a quantum theory of both matter and gravity.
Wald’s axioms and definition
----------------------------
Given one of the Gibbs states described in the last section, we want to calculate the expectation value of the thermal excitations of the energy-momentum tensor over the Boulware-like vacuum. Wald showed [@Wald:1977up; @Wald:1978pj; @wald] that a correctly renormalized energy-momentum tensor $T^\mu_{\phantom\mu\nu}$ obeying certain assumptions is essentially unique. Wald’s requirements are
1. *Conservation*. Given any state $\alpha$ $$\nabla_\mu \langle T^\mu_{\phantom\mu\nu}\rangle_\alpha = 0\, .\label{e-m
tensor conservation}$$
2. *Consistency*. Given any regular enough couple of states $\alpha_1,\alpha_2$ we have that $\langle T^\mu_{\phantom\mu\nu}\rangle_{\alpha_1}-\langle
T^\mu_{\phantom\mu\nu}\rangle_{\alpha_1}$ is defined by the usual point-splitting procedure.
3. *Causality holds* in the form of a locality requirement.
4. *Normalization*. In Minkowski space-time, being $\Omega$ the usual Fock vacuum, $\langle T^\mu_{\phantom\mu\nu}\rangle_\Omega=0$.
In a generic $n+1$ dimensional space-time, from the contraction of a Killing vector $K$ with $T^\mu_{\phantom{\mu}\nu}$ we can construct a form $J_K$ whose expectation values satisfy the following relations $$\begin{gathered}
\langle J_K\rangle=\sqrt{|g|}\epsilon_{\mu_1\cdots\mu_n\nu}\langle
T^\nu_{\phantom\nu \rho}\rangle K^\rho{\mathrm{d}}x^{\mu_1}\wedge\cdots\wedge{\mathrm{d}}x^{\mu_n}\, ,\qquad\\
{\mathrm{d}}\langle J_K\rangle=0\, .\end{gathered}$$ In our model we consider the Killing vector $K={\partial}_t$. For any state $\alpha$, integrating the second relation above, we can define, the *energy* inside a “shell” (actually a segment) $(r_1,r_2)$ $$\label{first energy in a shell}
E(r_1,r_2) = \int_{r_1}^{r_2} \langle T^t_{\phantom t
t}(t,r)\rangle_\alpha{\mathrm{d}}r\, .$$ We define $$\begin{gathered}
\label{point splitting definition}
\langle\theta^\mu_{\phantom{\mu}\nu}(x)\rangle_{\beta,\rho}\equiv
\qquad \qquad \qquad \qquad \qquad \qquad \\\notag \lim_{x'\to x}
\left(-\nabla^\mu\nabla_\nu+ \frac{1}{2}g^\mu_{\phantom{\mu}\nu}
g^{\rho\sigma}\nabla_\rho\nabla_\sigma \right) \left
[\langle{\varphi}(x'){\varphi}(x)\rangle_{\beta,\rho}-
\langle{\varphi}(x'){\varphi}(x)\rangle_{\infty,\rho}\right ]\, .\end{gathered}$$ Then, the second of Wald’s requirements implies $$\label{second Walds for us}
\langle\theta^\mu_{\phantom{\mu}\nu}\rangle_{\beta,\rho}=\langle
T^\mu_{\phantom{\mu}\nu} \rangle_{\beta,\rho}-
\langle T^\mu_{\phantom{\mu}\nu}
\rangle_{\infty,\rho}$$ and thus the energy inside a shell for the Gibbs state at temperature $\beta$ is given by $$\label{energy inside a shell definition}
E_{\beta,\rho}(r_1,r_2)=\int_{r_1}^{r_2}\left[\langle\theta^t_{\phantom
t t}(r)\rangle_{\beta,\rho}+\langle T^t_{\phantom t
t}(r)\rangle_{\infty,\rho}\right]
{\mathrm{d}}r\, .$$ In view of the point-splitting procedure , the expression for the two point function and the property , an integration by parts and a change of variable give $$E_{\beta,\rho}(r_1,r_2)=\int_{y(r_1)}^{y(r_2)}{\varepsilon}_{\beta,\rho}(y)
{\mathrm{d}}y+\int_{r_1}^{r_2}\langle T^t_{\phantom t
t}(r)\rangle_{\infty,\rho}{\mathrm{d}}r\,,$$ where $$\label{epsilon beta rho}
{\varepsilon}_{\beta,\rho}(y)= \int_{\mathbb R}\frac{{\mathrm{d}}p}{2\pi}\frac{|p|}
{{\mathrm{e}}^{\beta|p|}-1}\left [1+ B(p,\rho) {\mathrm{e}}^{2{\mathrm{i}}p(y-Y)}\right ]\, ,$$ which can also be expressed in a manifestly real form as $${\varepsilon}_{\beta,\rho}(y) = \int_0^\infty\!\frac{{\mathrm{d}}p}{\pi}\frac{p}{{\mathrm{e}}^{\beta p}-1} \left\{
1+\frac{p^2-{\eta_\rho}^2}{p^2+{\eta_\rho}^2}\cos [2p(y-Y)]+
\frac{2p{\eta_\rho}}{p^2+{\eta_\rho}^2}\sin [2p(y-Y)]\right \}\, .$$ We can single out the usual Stefan-Boltzmann contribution to the thermal energy and a specific contribution due to the boundary $$\begin{gathered}
{\varepsilon}_{\beta,\rho}(y) = \frac\pi{6\beta^2} +
h_{\beta,\rho}(y-Y)\, ,\\
h_{\beta,\rho}(\xi)\equiv
\frac{1}{\beta^2}\mathcal{F}\left[\frac{|p|}{({\mathrm{e}}^{|p|}-1)}
\frac{(p-{\mathrm{i}}{\eta_\rho}\beta )}{(p+{\mathrm{i}}{\eta_\rho}\beta )}\right]
\left(\frac{2\xi}{\beta}\right) \, ,\end{gathered}$$ $\mathcal{F}$ being the Fourier transform $$\mathcal{F}[g(p)](x)=\int_{-\infty}^{+\infty} \frac{{\mathrm{d}}p}{2\pi}g(p)\,{\mathrm{e}}^{{\mathrm{i}}p x}\, .$$ We note that since the function $\frac{|p|}{({\mathrm{e}}^{|p|}-1)}\frac{(p+{\mathrm{i}}{\eta_\rho}\beta )}{(p-{\mathrm{i}}{\eta_\rho}\beta
)}$ is continuous and $L^1$, its Fourier transform is continuous, $L^1$ and infinitesimal at infinity.
Vacuum polarization and boundary contribution
---------------------------------------------
In order to get a complete expression for the energy in we still have to determine $\langle T^\mu_{\phantom\mu\nu}\rangle_{\infty,\rho}$, that is the Boulware vacuum polarization. In our 1+1 dimensional model the expectation value $\langle
T^\mu_{\phantom\mu\nu}\rangle$ for any state that does not imply transport, i. e. $\langle T^t_{\phantom t r}\rangle=0$, is almost completely determined [@Mukohyama:1998ng] by its conservation law and the trace anomaly $$\langle T^\mu_{\phantom\mu\mu}\rangle_{\beta,\rho}=\frac1{24\pi}R \, ,
\label{e-m tensor trace anomaly}$$ where $R$ is the scalar curvature and in our case $R=f''$. The integration gives $$\label{1+1 integra trr}
f(r)\langle T_{\phantom r r}^r(r)\rangle=\frac1{24\pi}\kappa^2(r)-C\, ,$$ where $C$ is an integration constant and $\kappa(r)=\frac{1}{2}f'(r)$. Since this is an equation involving distributions, the general solution is given by $$\langle T_{\phantom r
r}^r(r)\rangle=\frac1{f(r)}\left[\frac1{24\pi}\kappa^2(r
)-C\right]-U\,\delta(r-r_0)\, ,$$ where $U$ is an arbitrary constant with the dimensions of an energy. We can call the $U \delta$ term a “boundary” term. Different values of the constants $C$ and $U$ identify different states. By means of the trace anomaly one thus derives $$\label{1+1 ttt rin con costanti arbitrarie}
\langle T^t_{\phantom t
t}(r)\rangle=\frac1{f(r)}\left[C-\frac1{24\pi}\kappa^2(
r)\right]+\frac1{24\pi}f''(r)+U\delta(r-r_0)\, .$$ As we noted in the previous section, for any $\rho$, the Boulware like state appears as vacuum to an observer in the asymptotically flat region; since $\kappa(r){\rightarrow}0$ when $r{\rightarrow}\infty$, it is identified by the choice $C=0$. Moreover, since we are dealing with the $r\geq\rho>r_0$ region, the “boundary” term is irrelevant. We can thus write $$\langle T^t_{\phantom t t}(r)\rangle_{\infty,\rho}=
-\frac{1}{24\pi}\frac{\kappa^2(r)}{f(r)}+\frac{1}{24\pi}f''(r)\, .$$ Note that it is actually $\rho$ independent.
We now write the energy in a segment or “shell” $[r_1,r_2]$ as $$\begin{gathered}
E_{\beta,\rho}(r_1,r_2)= \int_{r_1}^{r_2}\langle T^t_{\phantom t
t}(r)\rangle_{\beta,\rho} {\mathrm{d}}r=E^{(b)}_{\beta,\rho}(r_1,r_2)+E^{HH}_{\beta,\rho}(r_1,r_2)\, ,\\
E^{(b)}_{\beta,\rho}(r_1,r_2)
=\int_{y(r_1)}^{y(r_2)}h_{\beta,\rho}(y-Y)\, ,\\
E^{HH}_{\beta,r_0}(r_1,r_2)=\int_{r_1}^{r_2}\!\left [\frac\pi{6\beta^2}\frac
1{f(r)}+ \langle T^t_{\phantom t t}(r)\rangle_{\infty,\rho}\right ]\, .\end{gathered}$$ We will show in the following that the first term can be regarded as a pure boundary contribution while the second term, with the appropriate choice for $\beta$, can be regarded as the contribution from the “Hartle-Hawking” state [@Hartle:1976tp; @Jacobson:1994fp].
Removing the brick wall
-----------------------
Now we study the limit $\rho{\rightarrow}r_0$, that is the limit of vanishing brick wall thickness. We first consider the case of fixed $r_1$ and $r_2$ satisfying $r_2>r_1>\rho$. As $\rho{\rightarrow}r_0$, $Y{\rightarrow}- \infty$ and so, as noted at the end of Section 3.1, $h_{\beta,\rho}(y-Y){\rightarrow}0$ pointwise inside all of the segment, and we get $$\label{Er1r2}
\begin{split}
&E^{(b)}_{\beta,r_0}(r_1,r_2)=0\, ,\\
&E^{HH}_{\beta,r_0}(r_1,r_2)=\int_{r_1}^{r_2}\!\left [\frac\pi{6\beta^2}\frac
1{f(r)}+ \langle T^t_{\phantom t t}(r)\rangle_{\infty,r_0}\right ]
{\mathrm{d}}r.
\end{split}$$
In the case of the segment $r_1=\rho,r_2=\sigma$ with $\sigma$ independent on $\rho$, we have in the limit $$\begin{gathered}
\label{energy boundary term definition}
E^{(b)}_{\beta,\rho}(\rho,\sigma)=\int_Y^{y(\sigma)}h_{\beta,\rho}(y-Y){\mathrm{d}}y=\\=\int_0^{y(\sigma)-Y}\!
h_{\beta,\rho}(\xi){\mathrm{d}}\xi {\rightarrow}\int_0^\infty\!
h_{\beta,r_0}(\xi){\mathrm{d}}\xi=E^{(b)}_\beta\, ,\end{gathered}$$ where $E^{(b)}_\beta$ depends only on $\beta$ and $\eta_{r_0}=\lim_{\rho{\rightarrow}r_0}\eta_\rho$: $$\begin{gathered}
E^{(b)}_{\beta}=\frac{1}{\beta^2}\int_0^\infty\!\mathcal{F}\left[\frac
{{\lvertp\rvert}}{{\mathrm{e}}^{|p|}-1}\frac{p-{\mathrm{i}}\eta_{r_0}\beta}{p+{\mathrm{i}}\eta_{r_0}\beta}
\right]\left(\frac{2\xi}{\beta}\right){\mathrm{d}}\xi=
\\
=\frac1\beta\int_0^\infty\!\mathcal{F}\left[\frac{|p|}{{\mathrm{e}}^{|p|}-1}
\frac{p-{\mathrm{i}}\eta_{r_0}\beta}{p+{\mathrm{i}}\eta_{r_0}\beta}\right]\!(2\xi){\mathrm{d}}\xi.\end{gathered}$$ From equations , we can deduce that the function $h_{\beta,\rho}$ determines a pure boundary term in the stress energy tensor localized on the horizon.
We consider now $$\begin{gathered}
E^{HH}_{\beta,\rho}
(\rho,\sigma)=\int_\rho^\sigma\!\left
[\frac\pi{6\beta^2}\frac1{f(r)}+\langle
T^t_{\phantom t t}(r)\rangle_{\infty, \rho}\right ]{\mathrm{d}}r=
\\
\int_\rho^\sigma\!\left [\frac\pi{6\beta^2}\frac1{f(r)}-\frac{1}{24\pi}
\frac{\kappa^2(r)}{f(r)}+\frac{1}{2 4\pi}f''(r)\right ]{\mathrm{d}}r\, .\end{gathered}$$ This term may diverge when $\sigma{\rightarrow}\infty$ or $\rho{\rightarrow}r_0$. The first one is a well understood volume divergence and it is not interesting to us. We thus keep $\sigma$ constant and finite. Then, $$\begin{gathered}
\label{energia HH sviluppata}
E^{HH}_{\beta,\rho}
(\rho,\sigma)=\left(\frac\pi{6\beta^2}-\frac{\kappa_0^2}{24\pi}\right)
X(\rho,\sigma)+
\frac{\kappa_0^2}{24\pi}\Delta(\rho,\sigma)+\frac{1}{12\pi}
\left [\kappa(\sigma)-\kappa(\rho)\right ]\, ,\end{gathered}$$ where $$\begin{gathered}
X(r_1,r_2)\equiv \int_{r_1}^{r_2}\!\frac 1 {f(r)} {\mathrm{d}}r=y(r_2)-y(r_1)\, , \\
\int_{r_1}^{r_2}\!\frac 1 {24\pi} \frac {\kappa(r)^2} {f(r)} {\mathrm{d}}r
= \frac {\kappa_0^2}{24\pi} \left [X(r_1,r_2)-\Delta (r_1,r_2)\right ]\end{gathered}$$ In the limit $\rho\approx r_0$, $$\begin{gathered}
X(\rho,\sigma)\approx\frac 1 {2\kappa_0}\ln\frac a {\rho-r_0}\, ,\\
\Delta (\sigma,\rho)\quad \text{is finite}\, ,\end{gathered}$$ where $a$ depends on $\sigma$. The quantity in is thus divergent in the limit $\rho{\rightarrow}r_0$, unless we put $$\beta=\beta_H=\frac1{T_H}=\frac{2\pi}{\kappa_0}.$$ In this case the divergences due to thermal excitations are perfectly balanced by the Boulware vacuum polarization and we get a finite energy for every $\sigma$. This can be considered as a definition of the Hawking temperature $T_H$ for our model.
At $T=T_H$ it is thus possible to remove the brick wall and we are left with the expression: $$\begin{gathered}
\langle T_t^t\rangle_{\beta_H,
r_0}=\frac1{f(r)}\left[\frac\pi{6\beta_H^2}-\frac{\kappa^2(r)}{24\pi}\right]+
\frac{f''(r)}{24\pi}+E^{(b)}_{\beta_H}\delta(r-r_0)=\\
=\langle T^t_t\rangle_{HH}+E^{(b)}_{\beta_H}\delta(r-r_0)\, ,\end{gathered}$$ where $\langle T^\mu_{\phantom\mu \nu}\rangle_{HH}$ is known as the expectation value of $ T^\mu_{\phantom\mu \nu}$ in the Hartle Hawking state[@Hartle:1976tp; @Jacobson:1994fp; @sew80; @Fredenhagen:1989kr; @wald]. Indeed, by definition, the expectation value of the energy-momentum tensor in the Hartle Hawking state is regular on both the past and the future horizon of the black hole; this corresponds to the choice $$C=\frac\pi6\left(\frac{\kappa_0}{2\pi}\right)^2 =
\frac\pi{6\beta_H^2}$$ in equation . To an observer at rest in the asymptotically flat region, the Hartle Hawking state appears as a thermal bath at the Hawking temperature $T_H$.
Entropy
=======
In this Section we describe the derivation of the entropy of a shell of our space-time from the previously calculated energy density. Even at the Hawking temperature, the entropy in a shell attached to the horizon is divergent. The divergence can be resolved allowing that the system of the quantum field in this background does not satisfy Nernst theorem. The price for this is the presence of an arbitrary constant in the entropy of any shell. One can consider this freedom as a consequence of a breakdown of the semiclassical picture or as an intrinsic feature of the QFT in this background (analogous to renormalization effects), or as a combination of both.
The entropy density admits a surface term localized on the horizon. It indicates the presence of physical degrees of freedom there, which can give rise to a (potentially finite) renormalization of Newton constant.
Entropy from energy
-------------------
We will now perform the calculation of the entropy inside a shell. Since $$E=\frac{{\partial}(\beta F)}{{\partial}\beta}\, ,\qquad F=E-S/\beta\, ,$$ we have $$\label{energia entropia generiche F E}
S(\beta)=\beta
E(\beta)-\bar{\beta}F(\bar{\beta})-\int_{\bar{\beta}}^\beta\!E(b){\mathrm{d}}b\, .$$ When the relations $$\label{1+1 ipotesi terzo principio}
E(\beta){\rightarrow}E_0\,, \quad \beta [E(\beta)-E_0]{\rightarrow}0
\qquad\text{for}\quad \beta {\rightarrow}\infty$$ are satisfied, the arbitrary constant $\bar{\beta}F(\bar{\beta})$ can be determined via the third principle of thermodynamics in the form $$S(\beta){\rightarrow}0\qquad\text{for}\quad \beta{\rightarrow}\infty \, ,$$ that gives $$F(\bar{\beta})=-\frac{1}{\bar{\beta}}\int_{\bar{\beta}}^\infty\!(E(b)-
E_0){\mathrm{d}}b+E_0\, .$$ It is straightforward to show that the conditions in are necessary and sufficient for the system to verify the third principle.
Removing the brick wall with and without third principle
--------------------------------------------------------
Since the removal of the brick wall is not possible “off shell”, i.e. for $\beta\neq\beta_H$, we are forced to perform the integration in at non zero brick wall thickness and then to perform the limit $\rho{\rightarrow}r_0$.
Let’s consider the energy in a shell $(r_1,r_2)$. The analysis of the previous section gives $$\begin{gathered}
E_{\beta,\rho}(r_1,r_2)=\int_0^{X(r_1,r_2)}\!h_{\beta,\rho}(\xi){\mathrm{d}}\xi+
\left(\frac\pi{6\beta^2}-\frac{\kappa_0^2}{24\pi}\right)X(r_1,r_2)+\\+
\frac{\kappa_0
^2}{24\pi}\Delta(r_1,r_2)+
\frac{1}{24\pi}\left[2\kappa(r_2)-2\kappa(r_1)\right]\, .\end{gathered}$$ We recall that $$h_{\beta,\rho}(\xi) =
\frac{1}{\beta^2}\mathcal{F}\left[\frac{|p|}{({\mathrm{e}}^{|p|}-1)}
\frac{(p-{\mathrm{i}}{\eta_\rho}\beta )}{(p+{\mathrm{i}}{\eta_\rho}\beta )}\right]
\left(\frac{2\xi}{\beta}\right)\, .$$ Using equation and putting we have $$\begin{gathered}
S_{\beta,\rho}(r_1,r_2)=\frac{\pi}{6\beta}X(r_1,r_2)+
\beta\int_0^{X(r_1,r_2)}\!h_{\beta,\rho}(\xi){\mathrm{d}}\xi+\\+ \beta
_H\frac{1}{24\pi}\left[-\kappa_0^2
X(r_1,r_2)+\kappa_0^2\Delta(r_1,r_2)+2\kappa(r_2)-2\kappa(r_1)\right]+\\
-\beta_H F_H-X(r_1,r_2)\int_{\beta_H}^\beta\!\frac{\pi}{6b^2}{\mathrm{d}}b-\int_{\beta_H}^\beta \!\int_0^{X(r_1,r_2)}\!h_{b,\rho}(\xi){\mathrm{d}}b
{\mathrm{d}}\xi.\end{gathered}$$ The third principle is satisfied if $$\begin{gathered}
\beta_H
F_H=-X(r_1,r_2)\int_{\beta_H}^\infty\!\frac{\pi}{6b^2}{\mathrm{d}}b
-\int_{\beta_H}^\infty\!\int_0^{X(r_1,r_2)}\!h_{b,\rho}(\xi){\mathrm{d}}\xi
{\mathrm{d}}b \,+\\
+\beta_H\frac{1}{24\pi}\left[-\kappa_0^2
X(r_1,r_2)+\kappa_0^2\Delta(r_1,r_2)+2\kappa(r_2)-2\kappa(r_1)\right]\, .\end{gathered}$$ Hence, going “on shell”, we have: $$\begin{gathered}
\label{Sbhrs}
S_{\beta_H,\rho}(r_1,r_2)=\\
=\frac{\pi}{3\beta_H}X(r_1,r_2)+
\beta_H\int_0^{X(r_1,r_2)}\!h_{\beta_H,\rho}(\xi){\mathrm{d}}\xi+
\int_{\beta_H}^\infty\!\int_0^{X(r_1,r_2)}\!h_{b,\rho}(\xi){\mathrm{d}}\xi{\mathrm{d}}b\, .\end{gathered}$$ We note here that the function $$\int_{\beta_H}^\infty\!h_{b,\rho}(\xi){\mathrm{d}}b=-\frac{1}{\beta_H}\mathcal{F}\left[\ln
(1-{\mathrm{e}}^{-|p|})\frac{p-{\mathrm{i}}{\eta_\rho}\beta_H}{p+{\mathrm{i}}{\eta_\rho}\beta_H}\right]\left(\frac
{2\xi}{\beta_H}\right)$$ is not $L^1$ but it is infinitesimal at infinity, since it is the Fourier transform of a non continuous $L^1$ function. One can analyze in analogy of what we did in the previous section for the energy.
In the limit $\rho{\rightarrow}r_0$ with fixed $r_1,r_2$ satisfying $r_2>r_1>\rho$, the second and third term in expression vanish and we are left with an usual volume term: $$S_{\beta_H,\rho}(r_1,r_2)=\frac{\pi}{3\beta_H}X(r_1,r_2)\, .$$
When $r_1=\rho$ and $r_2=\sigma$, with $\sigma$ constant and finite, we have in the limit $\rho{\rightarrow}r_0$ $$X(\rho,\sigma) {\rightarrow}\infty\quad,\quad \Delta(\rho,\sigma) \,\text{is
finite}\, .$$ The first term in is linearly divergent in $X$, and the last one can also be divergent. The function $\int_{\beta_H}^\infty\!h_{b,\rho}(\xi){\mathrm{d}}b$ is not $L^1$ but is infinitesimal at infinity: the possible divergence from the last term in can not be linear in $X$[^4].\
Its origin can be traced back to the $\beta$ dependence of the so called “surface” term in the energy $$E^{(b)}_{\beta,\rho}=\int_0^X\!h_{\beta,\rho}(\xi){\mathrm{d}}\xi=\int_0^X\!
\frac1{\beta^2}\mathcal{F}\left[\frac{|p|}{{\mathrm{e}}^{|p|}-1}\frac{p-{\mathrm{i}}\eta
_\rho\beta}{p+{\mathrm{i}}\eta_\rho\beta}\right]\left(\frac{2\xi}{\beta}\right)
{\mathrm{d}}\xi\, .$$ As we showed in the preceding section, in the limit $\rho{\rightarrow}r_0$ we have $$E^{(b)}_\beta=\frac1\beta\int_0^\infty\!\mathcal{F}\left[\frac{|p|}{{\mathrm{e}}^{|p|}-1}\frac{p-{\mathrm{i}}\eta_{r_0}\beta}{p+{\mathrm{i}}\eta_{r_0}\beta}\right]\!(2\xi){\mathrm{d}}\xi\, .$$ When $\beta{\rightarrow}\infty$, or more precisely $\beta\gg\eta_{r_0}^{-1}$, we have, up to leading order: $$E^{(b)}_\beta=-\frac1\beta\int_0^\infty\!\mathcal{F}\left[\frac{|p|}
{{\mathrm{e}}^{|p|}-1}\right]\!(2\xi){\mathrm{d}}\xi\propto\frac1\beta$$ and the $\beta$ dependence is incompatible with the third principle. Such a behavior can be qualitatively understood looking at the description of the system in the $(t,y)$ coordinates: when $\rho{\rightarrow}r_0$ we have $Y{\rightarrow}-\infty$ and thus the specific boundary condition cannot appear independently in the leading order expansion of thermodynamical quantities. Due to the conformal symmetry of our two dimensional model, the only dimensional parameter relevant for the expansion is $\beta$ and thus $\beta^{-1}$ is the only possible leading term.
The different behavior of the two divergences reflects their different physical origin. The one linear in $X$ is basically due to the thermal excitations of the model close to the horizon and we expect it to be present also in more realistic models; the other one can be regarded as a specific feature of our oversimplified model.
We have thus shown that the entropy of a shell contains two distinct terms: $$\begin{gathered}
S_{\beta_H,\rho}(r_1,r_2)=S_{\beta_H,\rho}^{HH}(r_1,r_2)+S_{\beta_H,\rho}^{(b)}(r_1,r_2)\\
S_{\beta_H,\rho}^{HH}(r_1,r_2)\equiv\frac{\pi}{3\beta_H}X(r_1,r_2)\\
S_{\beta_H,\rho}^{(b)}(r_1,r_2)\equiv\beta_H\int_0^{X(r_1,r_2)}\!h_{\beta_H,\rho}(\xi){\mathrm{d}}\xi+
\int_{\beta_H}^\infty\!\int_0^{X(r_1,r_2)}\!h_{b,\rho}(\xi){\mathrm{d}}\xi{\mathrm{d}}b\, .\end{gathered}$$ The first one is a volume term, which is also obtained in the usual treatment of the brick wall model (see Section 5). The second one is a surface term localized on the horizon which is inherited from the surface term in the energy density.
Both these terms are divergent in the limit $\rho{\rightarrow}r_0$ but both divergences can be avoided if we do not use the third principle in order to fix the arbitrariness in the determination of the entropy. With the substitution $$\beta_H F_H{\rightarrow}\beta_H F_H+\frac{\pi}{3\beta_H}\frac{1}{2\kappa_0}X
+\int_{\beta_H}^\infty\!\int_0^X\!h_{b,\rho}(\xi){\mathrm{d}}\xi{\mathrm{d}}b-S^\Omega_{\beta_H}(\rho,\sigma)$$ we have $$\label{SbetaHfinale}
S_{\beta_H,r_0}(r_0,\sigma)=\beta_H\int_0^\infty\!
\frac{1}{\beta_H^2}\mathcal{F}\left[\frac{|p|}{({\mathrm{e}}^{|p|}-1)}
\frac{(p-{\mathrm{i}}\eta_{r_0}\beta_H )}{(p+{\mathrm{i}}\eta_{r_0}\beta_H )}\right]
\left(\frac{2\xi}{\beta_H}\right){\mathrm{d}}\xi+S^\Omega_{\beta_H}(r_0,\sigma)\, ,$$ where $S^\Omega_{\beta_H}(\rho,\sigma)$ is an arbitrary continuous function, which parameterize the inability of determining the exact value of the entropy without any external information input such as, for example, a derivation from a more fundamental theory which correctly identifies the degrees of freedom of the system. This picture recalls the standard situation one has in renormalization theory, where we pay the finiteness of the theory with an indetermination.
As we already noted, we expect that the divergences appearing in the surface term of the entropy are a specific feature of our extremely simplified model. Thus we expect that a surface term is generically present and potentially finite, showing an accumulation of degrees of freedom on the horizon which is different from the one usually considered as a consequence of the growing number of modes close to the horizon. This can be interpreted as a first order quantum correction to the Bekenstein-Hawking formula due to the interaction of the matter field with the gravitational field. Also this term can be considered as affected by some sort of indetermination since we cannot exclude a priori the presence of a surface contribution also in the arbitrary function $S^\Omega_{\beta_H}(r_0,\sigma)$. In general, the boundary contribution is non-vanishing whenever $$S_{\beta_H,r_0}(r_0,r_0)=\beta_H\int_0^\infty\!h_{\beta_H,\rho}(\xi)\d
d\xi+S^\Omega_{\beta_H}(r_0,r_0)\neq0\, .$$
The expression for the entropy in a shell far outside the horizon can be deduced from $$\label{Sr1r2Omega}
S_{\beta_H,r_0}(r_1,r_2)=S^\Omega_{\beta_H}(r_0,r_2)-S^\Omega_{\beta_H
}(r_0,r_1)\, .$$ If we assume that the origin of the awkward behavior of the entropy is in some sort of interaction with the black hole and its horizon, it seems reasonable to ask for the expression to become equal to the expected one in the asymptotically flat region, that is $$S^\Omega_{\beta_H}(r_0,r_2)-S^\Omega_{\beta_H}(r_0,r_1){\rightarrow}\frac\pi{3
\beta}(r_2-r_1)\qquad
\text{for}\quad r_1,r_2{\rightarrow}\infty \, .$$
The WKB approximation {#wkb}
=====================
It is instructive to reexamine the model described in the previuos sections within the WKB approximation. For this purpose we consider the shell $(\rho,\sigma)$ and define the wave number $k(r)$ $$k(r)=\frac E{f(r)}\, .$$ The density of states in the WKB approximation is given by $$\begin{gathered}
n(E)=\frac{{\partial}N(E)}{{\partial}E}\, ,\\
\pi N(E)=\int_\rho^\sigma\!{\mathrm{d}}r \frac E{f(r)}\, .\end{gathered}$$ We note that for Neumann or Dirichlet boundary conditions in 1+1 dimensions the WKB approximation gives the right eigenvalues of the energy. The free energy $F$ reads $$F_{\rm WKB}=\int\!n(E) \ln(1-{\mathrm{e}}^{-\beta E}){\mathrm{d}}E=\int\!\frac{N(E)}{{\mathrm{e}}^{\beta
E}-1}=\frac\pi{6\beta^2}\int_\rho^\sigma\frac{{\mathrm{d}}r}{f(r)}\, .$$ Recalling that $S=\beta^2{\partial}_\beta F$ and neglecting subleading terms in the limit of $\rho\approx r_0$, one obtains $$S_{\rm WKB}=\frac{\pi}{3\beta_H}X(\rho,\sigma)=
S_{\beta_H,\rho}^{HH}(\rho,\sigma)\, .$$ Again neglecting subleading terms and comparing with one finds $$S_{\beta_H,\rho}(\rho,\sigma)=S_{\rm
WKB}+S^{(b)}_{\beta_H,\rho}(\rho,\sigma)\, ,$$ showing that the the boundary term in the entropy is lost in the WKB approximation. In both cases however the entropy is divergent in the limit $\rho{\rightarrow}r_0$ and the considerations at the end of the previous section apply.
Higher dimensions {#highdim}
=================
In the higher dimensional and massive extension of our model we have to deal with two main issues:
- resolving the spectral problem for the field, i.e. the determination of the its normal modes;
- determining the Boulware vacuum polarization.
These problems are not conceptual but technical. The following steps towards their solution can be made in the spherically symmetric case.
The metric for an $n+1$ dimensional spherical black hole is given by $${\mathrm{d}}s^2=U(r){\mathrm{d}}t^2-\frac1{U(r)}{\mathrm{d}}r^2+r^2\Omega^{(n-1)}_{ij}{\mathrm{d}}\theta^i {\mathrm{d}}\theta^j \, ,$$ where $U$ is assumed to have one and only one zero in $r_0$ and . We insert a brick wall and consider a scalar field only in the exterior of the sphere of radius $\rho=r_0$, with a boundary condition $$\left. \left(\sqrt{U}{\partial}_r{\varphi}=\frac h r_0
{\varphi}\right)\right|_{r=\rho}.$$ Using the “tortoise” coordinate $r^*$ defined by $$\frac{{\mathrm{d}}r^*}{{\partial}r}=\frac1U$$ the thermal energy in a spherical shell is given by $$E_{\beta,\rho}=\int_{r_1}^{r_2}\!\langle\theta^t_{\phantom t
t}(r)\rangle_{\beta,\rho}\Sigma_{n-1}
r^{n-1}{\mathrm{d}}r=\int_{r^*_1}^{r^*_2}\! {\varepsilon}(r^*)_{\beta,\rho}\Sigma_{n-1}
(r(r^*))^{n-1}{\mathrm{d}}r^*\, ,$$ where $\Sigma_{n-1}r^{n-1}$ is the area of the $(n-1)$ dimensional sphere of radius $r$, $\Sigma_{n-1}=\frac{2\pi^{n/2}}{\Gamma\left(\frac n2\right)}$. At this point one can separate the angular dependence and pass to the reduced radial wave functions $f=r^{\frac{n-1}2}\psi$, where $\psi$ is the original radial wave function. Then one is left with the study of the operators $$D_l=-{\partial}_{r^*}^2+\frac{n-1}{2r}U\left({\partial}_rU\right)+
\frac{(n-1)(n-3)}{4}\frac{U^2}{r^2}+U\frac{l(l+n-2)}{r^2}+
U m^2 \, ,$$ where $r=r(r^*)$, $l(l+n-2)$ is the eigenvalue of the angular part of the flat Laplace operator in $n$ dimension (the squared total angular momentum) and $m$ is the mass of the field. Now, one has to investigate the spectral problem for $D_l$ with the boundary condition $$\left.\left(\frac{1}{\sqrt{U}}{\partial}_{r^*}f=\frac{h+\frac{n-1}{2}}{r_0}f
\right)\right|_{r^*=\rho^*}\, ,$$ where $\rho^*=r^*(\rho)$. Being $f_{\lambda l}(r^*)$ the complete set of orthonormal (improper) eigenfunctions, i. e. $$\begin{gathered}
D_l f_{\lambda l}=\lambda^2 f_{\lambda _l}\, ,\\
\int\!\sigma_{\lambda l}f_{\lambda l}(r_1^*)f_{\lambda
l}(r_2^*){\mathrm{d}}\lambda=\delta(r_1^*-r_2^*)\, ,\end{gathered}$$ we get $${\varepsilon}(r^*)_{\beta,\rho}=\sum_{l=0}^\infty\frac{d_l
}{\Sigma_n}\frac1{r^{n-1}}\int\!\sigma_{\lambda
l}\frac{\lambda}{{\mathrm{e}}^{\beta \lambda}-1}f_{\lambda
l}^2(r^*){\mathrm{d}}\lambda \, ,$$ where $d_l= \frac{(2l+n-2)(l+n-3)!}{l!(n-2)!}$ is the multiplicity of the eigenvalue $l(l+n-2)$ of the squared total angular momentum.
We expect that $\langle
T^\mu_{\phantom\mu\nu}\rangle_{\beta,\rho}$ has a structure analogous to the one arising in the two dimensional case: a volume term with singular behaviour for all values of $\beta$ except $\beta=\beta_H$ and a boundary term, which appears when considering a shell attached to the boundary. When performing an analysis similar to the one of Section 4 we expect analogous diverging contribution from the bulk and a potentially finite (see discussion in Section 4) horizon contribution. Again, the leading divergence could be traced back to the third principle. These issues need a further investigation.
Outlook and conclusions
=======================
In this work we analysed the behavior of a quantum field at finite temperature $T$ in the backgroud of a classical black hole. We applied the brick wall approach of ’t Hooft, introducing in addition a parameter $\eta$, which fixes the boundary conditions for the field on the brick wall and which can be interpreted as a parametrization of the interaction of the field with gravity close and behind the horizon. Focusing mainly on a 1+1 dimensional black hole space-time, we computed both the energy and the entropy densities. The energy density contains an $\eta$-dependent boundary term, which is localized on the horizon and respects the conservation and the trace anomaly of the energy-momentum tensor. Taking into account the Boulware vacuum polarization, we have shown that the energy density remains finite in the limit of vanishing brick wall thickness, provided that $T$ equals the Hawking temperature $T_H=\frac{\kappa_0}{2\pi}$. We recall in this respect that in the original brick wall model the determination of the brick wall thickness is made by requiring that the full classical black hole entropy is due to the thermal atmosphere of all the fundamental fields at the Hawking temperature[^5], which enters the model as an external input.
The entropy density shows analogous regular behavior for vanishing brick wall thickness, provided we release the third principle of thermodynamics in deriving the thermal entropy from the thermal energy. We note here that there are several examples of incomplete models that do not verify Nernst theorem. Likely the most known and simple one is the perfect Boltzmann gas, where the entropy diverges when $T\to 0$. In this case the problem is solved observing that for sufficiently low temperatures the particle interactions and quantum statistical effects cannot be neglected and the model is no longer valid. This means that the physical degrees of freedom of the system are not correctly identified by the model - a situation which looks similar to ours.
The entropy we obtained has two main features. First, it is also endowed with a boundary term localized on the horizon and determined by the boundary condition. Second, the entropy in the bulk is determined up to a function, since a new input, substituting the third principle, is needed for its complete determination. We believe that this input should come from a better understanding of the interplay between quantum mechanics and general relativity in systems containing horizons and black holes.
For example, in the context of string theory, a recently quite popular proposal [@Mathur:2005zp] suggests that the structure of space-time inside the horizon is deeply quantum mechanical. In this picture a model in which the semiclassical picture is confined outside the horizon can be consistent. The specific boundary condition can thus mimic some sort of interaction with the physics inside the horizon. However, it could appear non reasonable to perform the limit of vanishing brick wall thickness since we cannot expect the transition between quantum and semiclassical behaviour to happen sharply at the horizon. Also in this picture some of our results are relevant: in particular the term that gives rise to the “would be” boundary contribution, still remain as a an indication of a peculiar accumulation of degrees of freedom in the near horizon region.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank S. Liberati and V. Moretti for encouragement and useful discussions. This research is supported by the Italian MIUR under the program, “Teoria dei Campi, Superstringhe e Gravità”.
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[^1]: milanesi@sissa.it
[^2]: mintchev@df.unipi.it
[^3]: See e.g. [@haag; @brattellirobinsonII]
[^4]: In principle, we can not be sure of the actual appearance of such a divergence since the integral could be convergent in improper way.
[^5]: For specific issues concerning the two dimensional case we refer to [@Mann:1990fk].
|
---
author:
- 'J. J. Cuenca-García'
- 'G. Martínez-Pinedo'
- 'K. Langanke'
- 'F. Nowacki'
- 'I. N. Borzov'
date: 'Received: / Revised version: '
title: 'Shell-model half-lives for r-process $N=82$ nuclei'
---
Introduction
============
The astrophysical r-process produces about half of the heavy elements in the Universe by a sequence of fast neutron-capture reactions interrupted by photodissociations and followed by $\beta$ decays, running through extremely neutronrich nuclei far off the valley of stability [@Burbidge.Burbidge.ea:1957; @Cameron:1957]. The $\beta$ decays of the r-process waiting points, associated with nuclei with magic neutron numbers $N=50, 82$ and 126 play a crucial role for the r-process dynamics and elemental abundance distributions [@Cowan.Thielemann.Truran:1991]. Despite their importance only a few half-lives of waiting points with magic neutron numbers $N=50$ and 82 are known experimentally [@Pfeiffer.Kratz.ea:2001; @Hosmer.Schatz.ea:2005], while no experimental data exist yet for the $N=126$ waiting points. This paper is concerned with the $N=82$ waiting points, and fortunately here the half-lives of $^{131}$In, $^{130}$Cd and $^{129}$Ag have been measured and serve as stringent constraints for models which have to be used to predict the unknown half-lives of the other waiting points and r-process nuclei.
Beta-decays are notoriously difficult to model as they are determined by the weak low-energy tails of the Gamow-Teller strength distribution, mediated by the operator $\bm{\sigma \tau_-}$. There have been several previous estimates for the half-lives of the $N=82$ waiting points based on the Quasiparticle Random Phase Approximation on top of semi-empirical global models [@Moeller.Nix.Kratz:1997; @Borzov.Goriely:2000], the energy-density functional (DF3) method [@Borzov:2006] or the Hartree-Fock-Bogoliubov (HFB) method [@Engel.Bender.ea:1999]. A comparison to the data show that the predicted half-lives are in the right order of magnitude, but are often somewhat too long. This might imply that the models underestimate the correlations among nucleons which pull down the Gamow-Teller (GT) strength to low excitation energies. It is well known that the interacting shell model is the method of choice to describe the Gamow-Teller distribution in nuclei [@Brown.Wildenthal:1988; @Caurier.Langanke.ea:1999; @Langanke.Martinez-Pinedo:2000] and in fact, the best agreement with the measured half-lives has been achieved within a shell model approach [@Martinez-Pinedo.Langanke:1999].
This shell model calculation as well as the other theoretical approaches have been challenged recently by the experimental determination of the excitation energy of the first $1^+$ state in $^{130}$In, which carries most of the GT$_-$ strength of the $^{130}$Cd decay [@Dillmann.Kratz.ea:2003]. Here, gamma rays observed in the beta decay of $^{130}$Cd [@Dillmann.Kratz.ea:2003] yield an excitation energy of $E_x=2.16$ MeV, in contrast to the shell model predictions which place this state at noticeably lower energy, $E_x= 1.4$ MeV [@Dillmann.Kratz.ea:2003] and 1.5 MeV [@Martinez-Pinedo.Langanke:1999]. As the halflife has a strong energy dependence which approximately scales like $(Q_\beta-E_x)^5$, where $Q_\beta=M_i-M_f$ is the difference of the masses of the parent and daughter nuclei, respectively, the misplacement of the $1^+$ excitation energy has been fortuitiously cancelled in the shell model calculation [@Martinez-Pinedo.Langanke:1999] by the use of too small a $Q_\beta$ value which was taken from the Duflo-Zuker mass model yielding $7.56$ MeV [@Duflo.Zuker:1995], while the experimental $Q_\beta$ value is 8.34 MeV. Thus the shell model calculation of [@Martinez-Pinedo.Langanke:1999] despite its successful desription of the halflives of the $N=82$ waiting point nuclei, has to be improved to account also for the recent experimental structure data concerning the respective decays. It is the aim of this manuscript to present such an improved study.
Formalism
=========
We have performed large-scale shell model calculations using the code ANTOINE [@Antoine]. We have improved the previous shell model study of the half-lives of the $N=82$ r-process nuclei [@Martinez-Pinedo.Langanke:1999] in several ways. At first, we used a larger model space which includes the $0g_{7/2},1d_{3/2,5/2},2s_{1/2},0h_{11/2}$ orbitals outside the $N=40$ core for neutrons, thus assuming a closed $N=82$ shell configuration in the parent nucleus. For protons our model space was spanned by the $0g_{9/2,7/2},1d_{3/2,5/2},2s_{1/2}$ orbitals. Thus our model space avoids spurious center-of-mass excitations by omitting the $h_{11/2}$ orbit for protons and the $g_{9/2}$ orbit for neutrons. Secondly, we have performed the calculations of the parent ground states and the GT strength distributions in the daughter nucleus for the $N=82$ parents with charge numbers $Z=43-49$ including all possible correlations within the defined model space, a clear improvement with respect to the calculations in ref. [@Martinez-Pinedo.Langanke:1999]. But most importantly we have modified the residual interaction adopted in our studies to reproduce relevant experimental nuclear structure information. These modifications were guided by the observation that for the nuclei of interest here, the halflives are dominated by Gamow-Teller transitions to states at low excitation energy in the daughter nuclei and that these transitions are mainly determined by a single transition matrix element in which a $g_{7/2}$ neutron is changed into a $g_{9/2}$ proton. This implies that the transition matrix elements are quite insensitive to the relative position of the $g_{7/2}$ neutron orbit but its position determines the excitation energy of the daughter states and the $Q_\beta$ value for the evaluation of the half-lives. Starting from the interaction used in the previous shell-model calculations [@Martinez-Pinedo.Langanke:1999] we have implemented several monopole modifications aiming to reproduce the known $Q_\beta$ values of $^{131}$In [@Fogelberg.Gausemel.ea:2004] and $^{130}$Cd [@Dillmann.Kratz.ea:2003], the position of the $h_{11/2}$ neutron orbit [@Fogelberg.Gausemel.ea:2004] and the experimental excitation energy of the first $1^+$ state in $^{130}$In (for details see ref. [@gniady.others:2007]). This has resulted in two effective interactions: one that allows proton excitations from the $p_{1/2}$ orbit (that means uses a $^{88}$Sr core) and another one where these excitations are supressed (uses a $^{90}$Zr core). Fig. \[fig:in130\] compares our calculated low-energy spectrum of $^{130}$In with the data and find good agreement for the excitation energies of the first $3^+$ state. Importantly our improved shell model calculations also reproduces the unexpectedly high excitation energy of the first $1^+$ state, which is of key importance for the calculation of the $^{130}$Cd halflife. However, the low energy $5^+$ state and the states with possible assigments $0^-$ and $1^-$ are missed by the calculation that uses a $^{90}$Zr core. These are reproduced by the interactions that includes the $p_{1/2}$ orbital which is energetically relatively close to the $g_{9/2}$ orbital, but carries the opposite parity. Once, the $p_{1/2}$ orbital is included in the model space we are forced to truncate the number of protons excited across the $g_{9/2}$ shell gap. The calculations shown at the right of figure \[fig:in130\] where performed allowing for 6 protons to be excited from the $p_{1/2}$ and $g_{9/2}$ orbitals. Shell model calculations performed within the same model space and using the same interaction also reproduce the recently measured spectrum of $^{130}$Cd [@Jungclaus.Caceres.ea:2007]. It is particularly noteworthy that this includes the excitation energy of the first excited $2^+$ state which is calculated at $E_x=1.325$ MeV in close agreement with the experimental value of 1.346 MeV, hence not confirming the tentative suggestion that this state would reside at $E_x=0.957$ MeV which had been interpreted as an onset of shell quenching already in $^{130}$Cd [@Kautzsch.Walters.ea:2000].
![Comparison of low-energy shell model spectrum for $^{130}$In with the data. The shell model spectrum of the left is calculated for the model space assuming a $^{90}$Zr core, the one on the right for the model space with a $^{88}$Sr core. Details of the calculations are given in the text.\[fig:in130\]](in130spec.eps){width="\linewidth"}
As stated above, our calculation of the halflives are based on the valence space on top of the $^{90}$Zr as this allows for untruncated calculations and the changes in the computed values of the half-lives are negligible if we adopt the model space with the $^{88}$Sr core (and the appropriate interaction).
Results
=======
![Comparison of shell model $Q_\beta$ values for the $N=82$ isotones to the Audi-Wapstra systematics [@Audi.Wapstra.Thibault:2003] and to predictions of other models: FRDM [@Moeller.Nix.Kratz:1997], ETFSI-Q [@Pearson.Nayak.Goriely:1996], DF3 [@Borzov:2006] and Duflo-Zuker [@Duflo.Zuker:1995]. \[fig:Qvalue\]](fig2.eps){width="\linewidth"}
As discussed above, $\beta$ half-lives depend sensitively on the $Q_\beta$ value. Unfortunately this important quantity is experimentally not known for most of the relevant $N=82$ r-process nuclei and hence has to be estimated based on theoretical models or by systematic extrapolations from data for neighboring nuclei. Our $Q_\beta$ values were calculated from the energies of the isobaric analog states with a systematic correction for the Coulomb displacement energies [@Antony.Pape.Britz:1997]. In Fig. \[fig:Qvalue\] we compare the shell model $Q_\beta$-values with experiment, the Audi-Wapstra systematics [@Audi.Wapstra.Thibault:2003] and other theoretical calculations. As stated above, our shell model calculation is tuned by proper monopole modifications to reproduce the experimental $Q_\beta$ value for $^{130}$Cd. However, we find also a good agreement to the other $Q_\beta$ values as given in the compilation of Audi-Wapstra [@Audi.Wapstra.Thibault:2003]. Thus as the shell model $Q_\beta$ values are close to the experimental values and agree with those of most other models within the theoretical uncertainties, we will in the following adopt the shell model $Q_\beta$ values in our calculation of the half-lives. However, the exception are the predictions of the quenched Extended Thomas-Fermi with Strutinski Integral (ETFSI-Q) model [@Pearson.Nayak.Goriely:1996] which predicts $Q_\beta$-values which are noticeably larger than those of the other models, most noticeably for the most neutron deficient $N=82$ isotones.
The final ingredients of our half-life calculations are the GT$_-$ strength functions. We have calculated them within our shell model approach using the Lanczos method with 60 iterations for each possible final $J$-value. The calculated GT$_-$ strength functions for the $N=82$ nuclei from $^{124}$Mo ($Z=42$) to $^{131}$In ($Z=49$) are shown in Figs. \[fig:Gtstrength\_Cd\]-\[fig:Gtstrength\_Mo\]. For even-even nuclei the GT transitions lead to $J_f^\pi=1^+$ states in the daughters, while for the odd-$A$ nuclei the final states can have $J_f
= J_i-1, J_i, J_i+1$, where $J_i$ is the angular momentum of the parent ground state. For the calculation of the half-lives, the choice of the appropriate Gamow-Teller quenching factor is an important issue. It is wellknown that shell model calculations reproduce the GT strength distributions (total strength and fragmentation) very well within complete 0$\hbar\omega$ calculations (i.e. model spaces which include a complete major oscillator shell), if the GT operator is quenched by a constant factor. This factor has been determined for $sd$ shell [@Brown.Wildenthal:1985; @Brown.Wildenthal:1988], where it is 0.77, and the $pf$ shell, where it is 0.74 [@Martinez-Pinedo.Poves.ea:1996b]. Unfortunately the appropriate constant for nuclei in the mass $A=130$ region has not yet been determined. Thus we adopt a quenching factor of 0.71 adjusted to reproduce the experimental half-life of $^{130}$Cd ($t_{1/2}= 162\pm
7$ ms).
Here one word about first-forbidden transitions is in order. While a study by Möller and collaborators, based on the FRDM/QRPA for GT transitions and the statistical gross theory for forbidden theories, indicates sizable contributions of first-forbidden transitions to the half-lives, the only consistent microscopic treatment of GT and first-forbidden transitions for the $N=82$ half-lives by Borzov, based on the energy density-functional method, implies that forbidden transitions accelerate the halflives only slightly by about $10\%$ or less. Based on the later result our halflives calculated purely from GT transitions are meaningful, as a small, but roughly constant contribution of first-forbidden transition can be absorbed into a modification of the quenching factor which would be $0.73$ (very close to the standard quenching factor), if first-forbidden transitions yield a $10\%$ contribution.
Our shell model half-life for $^{131}$In (260 ms) agrees also with the measured values ($280\pm30$ s). However, we overestimate the one for $^{129}$Ag slightly (70 ms to be compared with $46^{+5}_{-9}$ s) [@Kratz.Others:1998; @Pfeiffer.Kratz.ea:2001]. The present shell-model half-lives are longer for nuclei with $Z\leq 47$ than the ones computed in reference [@Martinez-Pinedo.Langanke:1999] (see table \[tab:lives\]). There are two reasons for the change in half-lives. First, for nuclei with $Z=42$–44 the $Q_\beta$-values obtained in the present shell-model calculations are around 1 MeV smaller than the Duflo-Zuker values used in ref. [@Martinez-Pinedo.Langanke:1999] (see fig. \[fig:Qvalue\]). Second, for all the nuclei the low lying Gamow-Teller strength is shifted upwards around 0.5 MeV in excitation energy. This shift results in longer half-lives even if the $Q_\beta$-value were unmodified as it is the case in $^{129}$Ag. This shift is due to the monopole modifications introduced in the interaction used for the present calculations to increase the excitation energy of the $1^+$ state in $^{130}$In from 1.6 MeV [@Martinez-Pinedo.Langanke:1999] to the experimental value of 2.12 MeV [@Dillmann.Kratz.ea:2003].
![Comparison of half-lives of the $N=82$ isotones as calculated in the FRDM [@Moeller.Nix.Kratz:1997], HFB [@Engel.Bender.ea:1999], and the present shell model approaches with data [@Pfeiffer.Kratz.ea:2001; @Dillmann.Kratz.ea:2003].\[fig:halflives\]](fig1.eps){width="\linewidth"}
------------ ---------------- ------- ------
Nucleus
Expt.
$^{131}$In $280 \pm 30$ 260 177
$^{130}$Cd $162 \pm 7$ 162 146
$^{129}$Ag $46^{+5}_{-9}$ 70 35.1
$^{128}$Pd 46 27.3
$^{127}$Rh 27.65 11.8
$^{126}$Ru 19.76 9.6
$^{125}$Tc 9.44 4.3
$^{124}$Mo 6.13 3.5
------------ ---------------- ------- ------
: Comparison of the present shell model half-lives and the ones of reference [@Martinez-Pinedo.Langanke:1999] with experiment . All half-lives are in ms.[]{data-label="tab:lives"}
![GT$_-$ strength distribution for $^{131}$In. \[fig:Gtstrength\_In\]](In131.eps){width="\linewidth"}
![GT$_-$ strength distribution for $^{130}$Cd. \[fig:Gtstrength\_Cd\]](Cd130.eps){width="\linewidth"}
![GT$_-$ strength distribution for $^{129}$Ag. \[fig:Gtstrength\_Ag\]](Ag129.eps){width="\linewidth"}
![GT$_-$ strength distribution for $^{128}$Pd. \[fig:Gtstrength\_Pd\]](Pd128.eps){width="\linewidth"}
![GT$_-$ strength distribution for $^{127}$Rh. \[fig:Gtstrength\_Rh\]](Rh127.eps){width="\linewidth"}
![GT$_-$ strength distribution for $^{126}$Ru. \[fig:Gtstrength\_Ru\]](Ru126.eps){width="\linewidth"}
![GT$_-$ strength distribution for $^{125}$Tc. \[fig:Gtstrength\_Tc\]](Tc125.eps){width="\linewidth"}
![GT$_-$ strength distribution for $^{124}$Mo. \[fig:Gtstrength\_Mo\]](Mo124.eps){width="\linewidth"}
In Fig. \[fig:halflives\] and Table \[tab:lives\] our results are compared with those of other theoretical models and the data. The current half-lives are quite similar to those obtained within the HFB model of [@Engel.Bender.ea:1999] and are also in good agreement with the DF3-QRPA approach [@Borzov:2003]. The latter is reassuring as the DF3-QRPA model also reproduces the half-lives of other spherical nuclei in the vicinity of $N=82$ quite well [@Borzov.Others:2007]. Our half-lives show a mild odd-even effect, with the half-lives of the even-even waiting point nuclei slighly enlarged with respect to the neighboring odd-A nuclei. This is caused by a partial cancellation of the odd-even staggering in the $Q_\beta$ values (Fig. \[fig:Qvalue\]) by the larger excitation energies of the lowest GT states in the odd-$A$ daughter nuclei (Figs. \[fig:Gtstrength\_Cd\]-\[fig:Gtstrength\_Mo\]).
As shown in Fig. \[fig:Qvalue\] the various models predict $Q_\beta$ values within a variation of about 1 MeV, which, due to the strong energy dependence of the phase space, translates into the largest uncertainties of the $\beta$ half-lives. To estimate this uncertainty we have recalculated the half-lives by replacing the shell model $Q_\beta$ values by those of the different models and find half-lives which agree with the present ones within a factor of two or better, except for the larger ETFSI-Q $Q_\beta$ values which result in significantly smaller half-lives for the most proton-deficient nuclei. It is also worth noting that by replacing our shell model $Q_\beta$ value for $^{129}$Ag by the systematic Audi-Wapstra value, we find agreement with the experimental half-life within the uncertainties of the systematic $Q_\beta$ value.
![Comparison of the shell model neutron separation energies for the $N=81$ isotones to the data [@Audi.Wapstra.Thibault:2003] and predictions of other models: FRDM [@Moeller.Nix.Kratz:1997], Duflo-Zuker [@Duflo.Zuker:1995] and ETFSI-Q [@Pearson.Nayak.Goriely:1996]. \[fig:Sn\]](fig3.eps){width="\linewidth"}
![Comparison of the shell model two-neutron separation energies for the $N=82$ nuclei to the data [@Audi.Wapstra.Thibault:2003] and predictions of other models: FRDM [@Moeller.Nix.Kratz:1997], Duflo-Zuker [@Duflo.Zuker:1995] and ETFSI-Q [@Pearson.Nayak.Goriely:1996]. \[fig:S2n\]](fig4.eps){width="\linewidth"}
As the neutron separation energies in the daughter nuclei are quite small, some of the GT$_-$ strengths resides actually at energies above the neutron emission threshold and $\beta$ decays to these states are followed by neutron emission. The probability for $\beta$-delayed neutron emission depends sensitively on the neutron separation energies $S_n$ which are not known experimentally for most of the nuclei of interest here and have to be estimated theoretically. In Fig. \[fig:Sn\] we compare our shell model neutron separation energies for the $N=81$ daughter nuclei with those of various models. Importantly for $^{131}$Sn and $^{130}$In the neutron separation energies are known experimentally and our present results agree quite nicely with the data. The neutron separation energies predicted in the FRDM model [@Moeller.Nix.Kratz:1997] are larger than all other model predictions and also exceed the experimental values for $^{131}$Sn and $^{130}$In. Our shell model results predict a quite similar slope of the $S_n$ values as found in the Duflo-Zuker mass model [@Duflo.Zuker:1995] yielding quite sizable neutron separation energies even in the proton-deficient nuclei $^{125}$Ru and $^{124}$Tc. This is in difference to the ETFSI-Q values [@Pearson.Nayak.Goriely:1996] which predict a pronounced weakening of the $S_n$ values with increasing neutron excess. A quite similar behavior is found if one compares the 2-neutron separation energies of the $N=82$ r-process nuclei (Fig. \[fig:S2n\]). Again, our shell model results agree with the available data ($^{131}$In and $^{130}$Cd) and show a significantly slower decrease of the $S_{2n}$ values than predicted by the ETFSI model, while the FRDM and Duflo-Zuker models yield 2-neutron separation energies which agree reasonably well with the shell model ones.
![One neutron emission probabilities.\[fig:P1n\]](Pn.eps){width="\linewidth"}
Using our shell model GT$_-$ strength functions and $S_n$ values we have calculated the probability $P_n$ that the $\beta$ decay is accompanied by the emission of (at least) one neutron, defined as the relative probability of the $\beta$-decay rate above the neutron emission threshold $S_n$. The results are shown in Fig. \[fig:P1n\], indicating that within our shell model study most of the $\beta$ decays go to states below the neutron threshold for the even-even parent nuclei, yielding probabilities for $\beta$-delayed neutron emission of 40% or less. The shell model predicts quite a strong odd-even staggering in the $P_n$ values, indicating $P_n$ values of $80\%$ or larger for $^{127}$Rh and $^{125}$Tc. The FRDM model [@Moeller.Nix.Kratz:1997] also predicts an odd-even dependence in the neutron emission probabilities, however, this effect is somewhat smaller than for the shell model values. For $^{130}$Cd, the $P_n$ value is known experimentally [@Dillmann.Kratz.ea:2003] and it agrees with the FRDM and shell model predictions.
Conclusion
==========
We have recalculated shell model half-lives and neutron emission probabilities for the $N=82$ waiting point nuclei in the r-process, improving a previous shell model study by enlargement of the model space and by modification of the residual interaction which reproduces recent spectroscopic findings for nuclei in this regime of the nuclear chart. In particular our modified calculation reproduces the unexpectedly high excitation energy of the first $1^+$ state in $^{130}$In [@Dillmann.Kratz.ea:2003]. This, as a good description of the $Q_\beta$ value, is crucial as the GT transition to this low-lying state dominates the $^{130}$Cd half-life. We find good agreement with the experimentally known half-lives for the $Z=48,49$ nuclei and overestimate the one of $^{129}$Ag slightly.
As in most other models [@Moeller.Nix.Kratz:1997; @Engel.Bender.ea:1999; @Borzov.Goriely:2000] the shell model half-lives have been computed solely on the assumption of an allowed Gamow-Teller transition. The importance of forbidden transitions for the half-lives of the $N=82$ waiting point nuclei is somewhat controversial. Using the gross theory for the forbidden transitions, Möller [@Moeller.Pfeiffer.Kratz:2003] finds non-negligible contributions, while Borzov, adopting the QRPA approach within the Fermi-liquid theory, concludes that forbidden transitions play only a minor role [@Borzov:2003]. In view of these differences, shell model calculations of the forbidden $\beta$ decays of the $N=82$ r-process waiting point nuclei are desirable.
I. N. Borzov is supported by a grant from the German DFG under contract number 436 RUS 113/907/0-1.
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|
---
abstract: 'In this paper we present and discuss the great difference in OH absorption spectra against PSR B1849+00 and SNR G33.6+0.1 along the same line-of-sight. This finding is important as it clearly demonstrates that statistics of absorbing molecular gas depends on the size of the background source.'
author:
- 'S.'
- 'J. M.'
- 'J. M.'
- 'A.'
- 'K.'
- 'A.'
- 'S. B.'
title: 'PSR B1849+00 probes the tiny-scale molecular gas?'
---
Introduction
============
Studies of the absorption of signals from background continuum sources by the intervening medium have been a very powerful way of probing the properties of the interstellar medium (ISM). Pulsars are particularly suitable as background sources because of their pulsed radiation which allows us to investigate, in both emission and absorption, [*almost exactly the same line-of-sight*]{} (Weisberg et al. 1995; Koribalski et al. 1995). Another great advantage pulsars have is that their continuum emission subtends over an extremely small solid angle, allowing us to probe needle-thin samples of the ISM.
Motivated by the pulsars’ unique capabilities for studying the ISM, we measured the absorption spectra of several pulsars at the wavelength of the hydroxyl radical (OH), $\lambda = 18$ cm, using the Arecibo telescope. We detected OH absorption against one of our sources – PSR B1849+00. The line-of-sight toward B1849+00 is particularly interesting as it passes right through the Galactic plane ($b=0^{\circ}$) and is very close (8 arcmin south) to a nearby supernova remnant (SNR) G33.6+0.1.
Observations and Data Processing
================================
OH observations were undertaken with the Arecibo (305 m) radio telescope[^1]. The FWHM of the Arecibo telescope beam is approximately $2'.6 \times 3'.0$ at 1.6 GHz. The Caltech Baseband Recorder was used as a fast-sampling backend, simultaneously covering both OH mainlines (at 1665 and 1667 MHz). Two types of spectra of astrophysical interest were formed during the off-line data processing stage: the pulsar absorption spectrum, which depicts the pulsar signal alone (as absorbed by any intervening OH); and the “pulsar–off” spectrum, which registers all emission and absorption lying in the telescope beam during the time that the pulsar signal is not present. The final spectra have velocity resolution of 0.9 . More information about our observations and data processing can be found in Stanimirovic et al. (2003).
Pulsar and SNR OH Absorption Spectra
====================================
![[*Top two panels:*]{} Pulsar absorption spectra toward B1849+00 produced against the pulsar continuum emission alone at 1665 and 1667 MHz. [*Bottom two panels:*]{} Pulsar-off spectra toward B1849+00 produced against the continuum emission from G33.6+0.1 at 1665 and 1667 MHz. In addition to the absorption system at 102 , an absorption system at 10 (‘B’) and an emission feature at 70 (‘C’) are seen.](sstanimi1_fig1.ps){width="3.7in"}
Pulsar OH absorption spectra at 1665 and 1667 MHz are shown in Fig. 1 (top two panels). At both frequencies, narrow absorption lines were detected at a velocity of about 102 . The [*only*]{} previous OH absorption detected against a pulsar at 1667 MHz, to our knowledge, was by Slysh (1972). Spectra in Fig. 1 depict the pulsar signal [*[alone]{}*]{} as being absorbed by intervening OH. The absorption system shown in Fig. 1, which we label as ‘A’, has higher optical depth than what is typically found towards extragalactic sources (Dickey et al. 1981; Colgan et al. 1989). The ratio of the equivalent widths for the 1665 and 1667 MHz lines is however very close to 5:9 which is expected for thermalized level populations.
The pulsar-off spectra are presented in the same figure (bottom two panels). As the SNR G33.6+0.1 is partially covered by the Arecibo beam absorption features in these spectra are effectively produced against the continuum emissiom from G33.6+0.1. The absorption features at 102 differ greatly, in both peak intensity and linewidth, from corresponding features seen in the pulsar absorption spectra at the same velocity. In particular, feature ‘A’ in the 1667 MHz line is almost 15 times wider and 30 times shallower that its corresponding feature in the PSR absorption spectrum.
Possible Geometrical Explanations
=================================
{width="3.3in"}
The pulsar absorption spectra are very deep and narrow, tracing dense molecular gas with $N_{\rm H} \sim 10^{23}$ cm$^{-2}$, if the OH excitation temperature $T_{\rm ex}=10$ K is assumed. However, the most striking observational result is that the pulsar absorption and pulsar-off spectra appear to trace very different absorption features along [*the same*]{} line-of-sight and with the same central velocity of 102 . This result has to account for two additional constraints. First, a large molecular cloud was observed in $^{12}$CO(1-0) in the direction toward the SNR and the PSR by Green & Dewdney (1992). Second, it was suggested that the two objects are interacting with each other (Green 1989; Green & Dewdney 1992). Below we investigate two different geometrical scenarios that can explain the large difference in OH optical depths found against the PSR and the SNR.
\(1) [*An additional molecular cloud could be located in front of the PSR yet behind the SNR.*]{} This would be the simplest explanation whereby OH absorption features in the PSR and the pulsar-off spectra originate from two physically unrelated molecular clouds (see Fig. 2, case 1). The sharp and deep absorption lines, seen in the PSR absorption spectra, are produced by an additional molecular cloud located in front of the PSR yet behind the SNR. On the other hand, broad OH absorption lines in the pulsar-off spectrum are most likely due to the the interaction between the SNR with the molecular cloud.
This additional molecular cloud located behind the SNR could be of any size. However, the large-scale $^{12}$CO(1-0) distribution presented in Green & Dewdney (1992) does not show any obvious features that could be associated with this secondary cloud. In addition, we compared the hydrogen column densities derived from OH and $^{12}$CO(1-0) in the PSR direction and found a good agreement. This suggests that, most likely, all OH seen in absorption and CO seen in emission coexist in the same region making the existence of an additional molecular cloud along the line of sight unlikely.
\(2) [*All molecular gas is in front of the SNR and the PSR.*]{} An alternative possibility is that the OH absorption features seen in the PSR absorption and pulsar-off spectra originate from [*the same*]{} general molecular cloud located in front of both the PSR and the SNR. This could happen in the case where the PSR absorption is produced by a small clump (‘cloudlet’), while the shallower, broader absorption features against the SNR are caused by an ensemble of ‘cloudlets’ of varying properties (Fig. 2, case 2). More interestingly, the small ‘cloudlet’ could represent a typical building block for the molecular cloud.
The solid angle subtended by the ‘cloudlet’ intercepts solid angles of both PSR and SNR continuum emission regions, however the pulsar-off spectrum does not appear to have a significant contribution from the ‘cloudlet’. This suggests that the ‘cloudlet’ covers a very small fraction of the SNR and can be used to place an upper limit on its size. By assuming that the molecular cloud is at the distance of 7 kpc, we estimate that the ‘cloudlet’ radius must be $< 1$ pc while its hydrogen volume density is $n>10^{5}$ cm$^{-3}$.
Discussion
==========
As discussed in the previous section PSR absorption spectra reveal existence of fine spatial structure in the absorbing OH gas on scales $<1$ pc. Also, all molecular gas seen in absorption is most likely located in front of both the SNR and the PSR. This is a clear demonstration that a pencil-sharp OH absorption sample against the PSR differs [*dramatically*]{} from a large-angle absorption sample against the SNR. The example of B1849+00 and G33.6+0.1 shows that measured optical depths in OH depend heavily on the size of the background source. This OH result is very different from HI absorption findings (Dickey et al. 1979; Dickey et al. 1981; Payne al. 1982) where absorption statistics was compared for a wide range of angular size sources and no significant difference was found. This led to the conclusion that the ‘cloudlet’ model of the interstellar HI, whereby HI clouds are composed of a large number of randomly distributed smaller clumps (or ‘cloudlets’), is not prominent. However, the difference at HI and OH is not totally unexpected: the solid-angle effect is expected to be more pronounced for molecular gas where clumpiness is known to be significant.
We have investigated whether the PSR OH optical depth profiles could be building blocks for the molecular cloud by modeling the SNR optical depth profiles with an ensemble of PSR profiles (see Stanimirovic et al. 2003). It was shown that the ‘cloudlet’ model is not appropriate. A more complex structure of the molecular cloud is required to explain the observed OH line profiles. In order to constrain better molecular cloud geometry detailed OH observations of the whole SNR are crucial. Another open question is whether the line-of-sight toward B1849+00 and G33.6+0.1 is unique or similar examples exist elsewhere in the ISM. We would like to encourage further OH observations towards pulsars to constrain how common this phenomenon may be.
Conclusions
===========
We presented here the second ever detection of the OH absorption against a pulsar. Absorption lines were detected against PSR B1849+00 in both OH mainlines. In addition we detected OH absorption against a nearby SNR, G33.6+0.1. The two sets of absorption profiles differ greatly but most likely trace the same molecular cloud located in front of both the SNR and the PSR. This surprising result indirectly points to the existence of small scale ($<1$ pc) structure in the absorbing OH gas. Also, it shows that angular size of background sources can influence greatly optical depth measurements in OH. This is opposite to what was found for the HI absorbing gas.
This work was supported in part by NSF grants AST-0097417 and AST-9981308.
[00]{}
, S. W. J., E. E. [Salpeter]{}, and Y. [Terzian]{}: 1989, ‘[Arecibo emission-absorption observations of Galactic OH]{}’. , 231–242.
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, B., S. [Johnston]{}, J. M. [Weisberg]{}, and W. [Wilson]{}: 1995, ‘[H I line measurements of eight southern pulsars]{}’. , 756–764.
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, S., J. [Weisberg]{}, J. M. [Dickey]{}, A. [de la Fuente]{}, K. [Devine]{}, A. [Hedden]{}, and S. B. [Anderson]{}: 2003, ‘[Detection of OH absorption against PSR B1849+00]{}’. . in press.
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[^1]: The Arecibo Observatory is part of the National Astronomy and Ionosphere Center, operated by Cornell University under a cooperative agreement with the National Science Foundation.
|
---
abstract: 'We derive the equations of motion for the planar somersault, which consist of two additive terms. The first is the dynamic phase that is proportional to the angular momentum, and the second is the geometric phase that is independent of angular momentum and depends solely on the details of the shape change. Next, we import digitised footage of an elite athlete performing 3.5 forward somersaults off the 3m springboard, and use the data to validate our model. We show that reversing and reordering certain sections of the digitised dive can maximise the geometric phase without affecting the dynamic phase, thereby increasing the overall rotation achieved. Finally, we propose a theoretical planar somersault consisting of four shape changing states, where the optimisation lies in finding the shape change strategy that maximises the overall rotation of the dive. This is achieved by balancing the rotational contributions from the dynamic and geometric phases, in which we show the geometric phase plays a small but important role in the optimisation process.'
address: '${}^1$School of Mathematics and Statistics, The University of Sydney$\phantom{*}\;\;\;{}^2$Australian Institute of Health Innovation, Macquarie University'
author:
- 'William Tong$^{1,2}$ and Holger R. Dullin$^1$'
bibliography:
- 'PlanarSomersault.bib'
title: Using the Geometric Phase to Optimise Planar Somersaults
---
Introduction
============
The somersault is an acrobatic manoeuvre that is essential in Olympic sports such as diving, trampolining, gymnastics and aerial skiing. Today, athletes seek to better understand the scientific theory behind somersaults in the hope of gaining an edge in competition. Within diving alone there is extensive literature ranging from books aimed at athletes and coaches (e.g. [@batterman; @Fairbanks; @obrien]), to journal articles targeting the scientific community. Yeadon has provided great insight into the biomechanics behind the twisting somersault, which include a series of classical papers [@yeadon93a; @yeadon93b; @yeadon93c; @yeadon93d]. In this paper we focus on optimising somersaults without twist, which we will refer to as the planar somersault. There are several studies focusing on the planar somersault, e.g. [@Blajer2003; @King2004471; @MurthyK93; @schuler2011optimal], but here we take a different approach that utilises the geometric phase in order to maximise somersault rotation. The geometric phase is also important in the twisting somersault, see [@TwistSom], and has been used to generate a new dive in [@513XD]. The main focus for the twisting somersault is the generation of twist, while in the planar case the particulars of the shape change can instead be used to generate additional somersault. Our initial formulas are a special case of those derived in [@TwistSom], but since the resulting differential equation for overall rotation is only a single first order equation, a much more thorough analysis is possible.
The splitting into the dynamic and geometric phases is a well known modern concept in geometric mechanics, see e.g. [@MarsdenRatiu13; @Holm08; @Holm09]. To our knowledge this is the first application of geometric phase to diving. When modelling the rotation of a non-rigid (and hence a shape changing) body the contributions to the overall rotation can be split into two terms. The more familiar term is the dynamic phase, which is proportional to the angular momentum of the body and thus expresses the obvious fact that the body rotates faster when it has larger angular momentum. The less familiar term is [*not*]{} proportional to the angular momentum, and is only present when a shape change occurs. This term is called the geometric phase, and it gives a contribution to the overall rotation of the body even when the angular momentum vanishes. This is the reason a cat can change orientation by changing its body shape even when it has no angular momentum. In diving angular momentum is non-zero, but the geometric phase is still present, and both together determine the overall rotation.
The structure of the paper is as follows. In section \[sec:model\] we take the mathematical model of an athlete introduced in [@WTongthesis], and simplify it to analyse planar somersaults. In section \[sec:eqofmotion\] we then present the generalised equations of motion for coupled rigid bodies in space, and perform planar reduction to reduce the 3-dimensional vector equations into the 2-dimensional scalar variants. Next in section \[sec:realworld\] we analyse a real world dive and demonstrate how the geometric phase can be used to improve overall rotation obtained by the athlete. Finally, in section \[sec:theoreticalplanar\] we propose a new theoretical planar somersault using realistic assumptions to find optimal shape change trajectories that maximise overall rotation. In these instances the dynamic and geometric phases are accessed for different values of angular momentum, and the role of the geometric phase in optimising overall rotation is demonstrated.
A preliminary version of this study was presented at the 1st Symposium for Researchers in Diving at Leipzig, Germany. The conference proceedings can be found in [@divingsym]. However, here the model presented in section 2 has been tweaked, the analysis of the real world dive in section 4 has been extended to show how the geometric phase can be utilised to increase the amount of somersault produced, and we present a new optimisation procedure in section 5 that maximises the theoretical planar somersault using the geometric phase.
Model {#sec:model}
=====
We begin with the 10-body model proposed in [@WTongthesis], which is a slight modification of Frohlich’s [@Frohlich] 12-body model that uses simple geometric solids connected at joints to represent the athlete. The components of the 10-body model consist of the torso, head, $2\times$ upper arm, $2\times$ forearm with hand attached, $2\times$ thigh and $2\times$ lower leg with foot attached. Although more sophisticated models exist (like those proposed by Jensen [@Jensen76] and Hatze [@Hatze]) that use the elliptical zone method to estimate segment parameters, this sophistication only affects the tensor of inertia $I_i$, centre of mass ${\boldsymbol{C}}_i$ and joint location ${\boldsymbol{E}}_i^j$ for each body $B_i$ connected to body $B_j$. As the dynamics are driven only by $I_i$, ${\boldsymbol{C}}_i$ and ${\boldsymbol{E}}_i^j$, we can use the 10-body model since it provides similar estimates for these quantities.
In the case of the planar somersault we enforce the shape change to be strictly about the somersault axis and require that both left and right limbs move together, so that the normalised angular velocity vector is constant. By combining the corresponding left and right limb segments we obtain the 6-body planar model shown in Figure \[fig:schematic\], whose parameters are listed in Table \[tab:model\].
body $B_i$ mass $m_i$ moi $I_i$ joint position ${\boldsymbol{E}}_i^j$
-------------------------- -------------- ------------- ---------------------------------------
$B_1=$ torso $m_1=32.400$ $I_1=1.059$ ${\boldsymbol{E}}_1^2=(0,0.25)^t$
${\boldsymbol{E}}_1^4=(0.08,-0.3)^t$
${\boldsymbol{E}}_1^6=(0,0.3)^t$
$B_2=$ upper arms $m_2=4.712$ $I_2=0.038$ ${\boldsymbol{E}}_2^1=(0,0.2)^t$
${\boldsymbol{E}}_2^3=(0,-0.15)^t$
$B_3=$ forearms & hands $m_3=4.608$ $I_3=0.055$ ${\boldsymbol{E}}_3^2=(0,0.183)^t$
$B_4=$ thighs $m_4=17.300$ $I_4=0.294$ ${\boldsymbol{E}}_4^1=(0.08,0.215)^t$
${\boldsymbol{E}}_4^5=(0,-0.215)^t$
$B_5=$ lower legs & feet $m_5=11.044$ $I_5=0.310$ ${\boldsymbol{E}}_5^4=(0.08,0.289)^t$
$B_6=$ head $m_6=5.575$ $I_6=0.027$ ${\boldsymbol{E}}_6^1=(0,-0.11)^t$
: The parameters of the 6-body planar model obtained by reducing the 10-body model proposed in [@WTongthesis]. We abbreviate moment of inertia as moi in the table above.[]{data-label="tab:model"}
Equations of motion {#sec:eqofmotion}
===================
The equations of motion for a rigid body in 3-dimensional space is $${\boldsymbol{\dot{L}}}={\boldsymbol{L}}\times {\boldsymbol{\Omega}},\label{eq:eom}$$ where ${\boldsymbol{L}}$ is the angular momentum and ${\boldsymbol{\Omega}}$ is the angular velocity. Now if the rigid body is replaced by a system of coupled rigid bodies, then holds if $${\boldsymbol{\Omega}} = I^{-1}({\boldsymbol{L}}-{\boldsymbol{A}}).\label{eq:omega}$$ Here, $I$ is the overall tensor of inertia and ${\boldsymbol{A}}$ is the total momentum shift generated by the shape change. Thus in the absence of shape change ${\boldsymbol{A}}={\boldsymbol{0}}$, which gives the classical result ${\boldsymbol{\Omega}} = I^{-1}{\boldsymbol{L}}$. The proof of is provided in Theorem 1 in [@TwistSom], along with $I$ and ${\boldsymbol{A}}$, which are $$\begin{aligned}
I&=\sum_{i=1}^6\Big(R_{\alpha_i} I_i R_{\alpha_i}^t+m_i\left[|{\boldsymbol{C}}_i|^2\mathbb{1}-{\boldsymbol{C}}_i{\boldsymbol{C}}_i^t\right]\Big)\label{eq:I}\\
{\boldsymbol{A}}&=\sum_{i=1}^6\Big(m_i {\boldsymbol{C}}_i\times {\boldsymbol{\dot{C}}}_i +R_{\alpha_i} I_i{\boldsymbol{\Omega}}_{\alpha_i}\Big).\label{eq:A}\end{aligned}$$ In and for each body $B_i$ we have: the mass $m_i$, the tensor of inertia $I_i$, the centre of mass ${\boldsymbol{C}}_i$, the relative orientation $R_{\alpha_i}$ to the reference body (chosen to be $B_1$), and the relative angular velocity ${\boldsymbol{\Omega}}_{\alpha_i}$, such that the angular velocity tensor is $\hat{\Omega}_{\alpha_i} = R_{\alpha_i}^t \dot{R}_{\alpha_i}$. It is clear that when there is no shape change $\dot{{\boldsymbol{C}}}_i$ and ${\boldsymbol{\Omega}}_{\alpha_i}$ both vanish, and hence ${\boldsymbol{A}}$ vanishes. Manipulating by using the definition of ${\boldsymbol{C}}_i$ found in [@WTongthesis] and the vector triple product formula, we can factorise out the relative angular velocity ${\boldsymbol{\Omega}}_{\alpha_i}$ to obtain $${\boldsymbol{A}}=\sum_{i=1}^6\Big(R_{\alpha_i}\Big[\sum_{j=1}^6 m_j [R_{\alpha_i}^t {\boldsymbol{C}}_j \cdot {\boldsymbol{\tilde{D}}}_i^j \mathbb{1} - {\boldsymbol{\tilde{D}}}_i^j {\boldsymbol{C}}_j^t R_{\alpha_i}] + I_i \Big]{\boldsymbol{\Omega}}_{\alpha_i}\Big),\label{eq:A2}$$ where ${\boldsymbol{\tilde{D}}}_i^j$ is a linear combination of ${\boldsymbol{E}}_i^j$’s defined in Appendix \[app:D\].
For planar somersaults the angular momentum and angular velocity vector are only non-zero about the somersault axis, so we write ${\boldsymbol{L}}=(0,L,0)^t$ and ${\boldsymbol{\Omega}}=(0,\dot{\theta},0)^t$ to be consistent with [@WTongthesis]. Substituting ${\boldsymbol{L}}$ and ${\boldsymbol{\Omega}}$ in shows that ${\boldsymbol{\dot{L}}}={\boldsymbol{0}}$, meaning the angular momentum is constant. As the athlete is symmetric about the median plane, the centre of mass for each $B_i$ takes the form ${\boldsymbol{C}}_i = (C_{i_x},0,C_{i_z})^t$ and the relative angular velocity ${\boldsymbol{\Omega}}_{\alpha_i}=(0,\dot{\alpha}_i,0)^t$. Substituting these results in and , we can then simplify to obtain the 2-dimensional analogue of $I$ and ${\boldsymbol{A}}$ giving $$\begin{aligned}
I^{(2D)} &= \sum_{i=1}^6 I_{i_y}+m_i(C_{i_x}^2+C_{i_z}^2)\\
A^{(2D)} &= \sum_{i=1}^6\Big( \sum_{j=1}^6 m_j\big[{\boldsymbol{C}}_j\cdot {\boldsymbol{\tilde{D}}}_i^j \cos{\alpha_i} + P{\boldsymbol{C}}_j\cdot {\boldsymbol{\tilde{D}}}_i^j \sin{\alpha_i}\big]+I_{i_y}\Big)\dot{\alpha}_i,\end{aligned}$$ where $I_{i_y}$ is the (2,2) entry of $I_i$ and $P = \text{antidiag}(-1,1,1)$ is an anti-diagonal matrix used to permute the components of ${\boldsymbol{C}}_j$. The 2-dimensional analogue of with the arguments ${\boldsymbol{\alpha}}=(\alpha_2,\dots,\alpha_6)^t$ (note $\alpha_1=0$ because $B_1$ is the reference segment) explicitly written and $(2D)$ superscripts suppressed is then $$\dot{\theta} = I^{-1}({\boldsymbol{\alpha}})L+{\boldsymbol{F}}({\boldsymbol{\alpha}})\cdot{\boldsymbol{\dot{\alpha}}},\label{eq:theta}$$ where we write ${\boldsymbol{F}}({\boldsymbol{\alpha}})\cdot{\boldsymbol{\dot{\alpha}}} = -I^{-1}({\boldsymbol{\alpha}})A({\boldsymbol{\alpha}},{\boldsymbol{\dot{\alpha}}})$ to match the differential equation found in [@pentagon]. In that paper $L=0$, and the study focused on maximising the geometric phase. However, here $L\neq 0$ and the differential equation is composed of two parts: the dynamic phase $I^{-1}({\boldsymbol{\alpha}})L$, which is proportional to $L$, and the geometric phase ${\boldsymbol{F}}({\boldsymbol{\alpha}})\cdot{\boldsymbol{\dot{\alpha}}}$, which is independent of $L$. Solving with the initial condition $\theta(0)=\theta_0$ gives the orientation of the athlete as a function of time.
The sequence of shape changes an athlete goes through while performing a dive can be represented by a curve on shape space, which closes into a loop provided the athlete’s take-off and final shape is the same. The dynamic and geometric phases are both dependent on the path of the loop, however the dynamic phase also depends on the velocity with which the loop is traversed, while the geometric phase is independent of the velocity. Traversing the same loop with different velocities therefore contributes different amounts to the dynamic phase, while the contribution to the geometric phase is unchanged. For planar somersaults, we generally expect the dominating term to be the dynamic phase as it is proportional to $L$ (which is large), and the geometric phase to play a lesser role.
The main idea behind our optimisation is that we assume $L$ is already as large as possible and cannot be increased further. Also, we assume that the athlete is holding tuck or pike as tight as possible in the middle of the dive, so again no further improvement is possible. Both increasing $L$ and decreasing $I$ change the dynamic phase, which increases overall rotation and thus the number of somersaults. We will show that the contribution from the second term in , the geometric phase, can be used in principle to increase the overall rotation.
A real world dive {#sec:realworld}
=================
Footage of a professional male athlete performing a 107B dive (forward 3.5 somersaults in pike) off the 3m springboard was captured at the New South Wales Institute of Sport (NSWIS) using a 120 FPS camera. SkillSpector [@skillspector] was used to manually digitise the footage, which commenced from the moment of take-off and ended once the athlete’s hand first made contact with the water upon entry. The total airborne time spanned 1.55 seconds, creating a total of 187 frames, and the digitisation of the dive is shown in Figure \[fig:digitise\]. For convenience we shall refer to the initial frame as the zeroth frame, and write ${\boldsymbol{\alpha}}[j]=\{\alpha_2[j],\dots\}$ to denote the collection of shape angles of the $j$th frame, where $0 \leq j \leq 186$.
![Illustration of the digitised dive for frames 0, 15, 30, 45, 60, 75, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180 and 186, from left to right. To avoid clutter, each illustrated frame has been shifted right by a small constant amount to provide better visualisation of the dive sequence.[]{data-label="fig:digitise"}](planarsketch.pdf){width="14cm"}
In each frame we locate the joint positions of the ankle, knee, hip, shoulder, elbow, wrist, and ear (which serves as a decent approximation for the centre of mass of the head). To reduce digitisation errors a discrete Fourier cosine transform is applied to the data, so that by keeping the first fifteen Fourier coefficients the data is smoothed when inverting the transformation. The spatial orientation $\theta_i$ is the angle between an orientation vector constructed from appropriate joint positions (e.g. hip and shoulder positions for the torso) and the reference vector given by the anatomical neutral position vector when standing upright. From $\theta_i$ the relative orientation $\alpha_i$ can be obtained by using $\alpha_i = \theta_i - \theta_\mathit{obs}$ for $i\in\{2,\dots,6\}$, where $\theta_\mathit{obs}=\theta_1$ is the observed spatial orientation of the torso. The spatial orientation $\theta_\mathit{obs}$ and relative orientations $\alpha_2,\dots,\alpha_6$ for the digitised dive are shown in Figure \[fig:angles\].
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To validate the 6-body planar model we compute $\theta$ using and compare the theoretical result to the observed $\theta_\mathit{obs}$. Small variations between the two curves are expected as the segment parameters are taken from Table \[tab:model\] and not from the particular athlete used in the data. Comparison is made by using the initial orientation $$\theta_0=\theta_\mathit{obs}[0]=0.6478\label{eq:theta0}$$ and collection of shape angles ${\boldsymbol{\alpha}}[\mathit{fr}]$, where square brackets denote frame $\mathit{fr}$ and round brackets indicate continuous time $t$. A cubic interpolation is used to obtain ${\boldsymbol{\alpha}}(t)$, which is then substituted in to obtain $\theta$. The angular momentum constant $L$ is found by a least square fit that reveals $L=122.756$, and the difference $\Delta\theta=\theta-\theta_\mathit{obs}$ is shown in Figure \[fig:deltatheta\]. From the moment of take-off until the diver hits the water we see that the discrepancy remains small, with the maximum difference being $0.331$ radians at time $t=0.650$. When the dive is completed the difference is only $0.045$ radians, which gives us confidence in using the planar model.
![The difference between $\theta$ computed with and the observed $\theta_\mathit{obs}$ from the data.[]{data-label="fig:deltatheta"}](shape5.pdf){width="9cm"}
![An impossible shape configuration of the athlete with $\theta = 0.648$ and ${\boldsymbol{\alpha}}=\{1.5,-2.5,2,-2,1\}$.[]{data-label="fig:weird"}](weirdshape.pdf){width="5cm"}
Looking at the graphs in Figure \[fig:angles\] it appears the upper arms and forearms move together relative to the torso, and similarly the thighs and lower legs. Therefore it is reasonable to assume that the elbows and knees remain straight throughout the dive, in particular the knees during pike somersaults. The head can be included in the torso to remove another degree of freedom. Using these assumptions thus further reduces the segment count to three so that the shape space is some subset of a 2-dimensional torus. We say subset as there are shapes deemed impossible or unrealistic for diving, e.g. the shape shown in Figure \[fig:weird\]. This motivates us to constrain the shape space to $\{\alpha_2,\alpha_4\}\in[-\pi,0]^2$ for the set of all possible shapes obtainable by the athlete. An important point to note is that the general theory is not dependent on the segment count reduction, and that these assumptions merely make it easier to understand the main principles behind the theory. Using $L=120$ for comparison purposes, we found the overall difference in rotation obtained to be less than $6.53^\circ$, which is small considering the athlete completes 3.5 somersaults for the dive.
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The athlete’s 2-dimensional shape change trajectory is shown in Figure \[fig:shapespace\], where the loop $C$ is constructed by letting ${\boldsymbol{\alpha}}(t)$ run from zero to the airborne time $T_\mathit{air}$. As take-off and entry shapes are not identical, the overall rotation obtained is gauge dependent, see [@LittlejohnReinsch97] for details. To eliminate this ambiguity we add one additional frame where ${\boldsymbol{\alpha}}[187] = {\boldsymbol{\alpha}}[0]$, such that ${\boldsymbol{\alpha}}(t)$ closes and the overall rotation is well-defined. As we now have a total of 188 frames captured at 120 FPS, the additional frame adds $1/120$th of a second to the airborne time, thus making $T_\mathit{air} = 187/120$. To find the change in orientation of the dive we solve with giving $$\theta(T_\mathit{air})= \theta_\mathit{dyn} + \theta_\mathit{geo} + \theta_0,$$ where the dynamic phase contribution is $$\theta_\mathit{dyn} = L \int_0^{T_\mathit{air}} I^{-1}\big({\boldsymbol{\alpha}}(t)\big)\,\mathrm{d}t= 0.1705L\label{eq:dyn}$$ and the geometric phase contribution is $$\theta_\mathit{geo} = \int_0^{T_\mathit{air}}{\boldsymbol{F}}\big({\boldsymbol{\alpha}}(t)\big)\cdot{\boldsymbol{\dot{\alpha}}}(t)\,\mathrm{d}t= 0.0721.\label{eq:geo}$$ As the angular momentum $L\approx 120$ is large, this confirms our hypothesis that the dynamic phase is the dominant term and the geometric phase only plays a minor role. For this particular dive the contribution is less than one degree of rotation, and hence is negligible. However, we will show that this contribution can be increased to something tangible by changing the approach into and out of pike in the next section. Instead of directly integrating the second term in the differential equation , the geometric phase can be obtained from certain areas in shape space. If the loop in shape space is closed the integral can be rewritten in terms of a certain function over the enclosed area. The reformulation is obtained by applying Green’s theorem, which gives $$\int_{C} F({\boldsymbol{\alpha}})\cdot\dot{{\boldsymbol{\alpha}}}\,dt = \iint_A B({\boldsymbol{\alpha}})\,d\tilde{A},\label{eq:green}$$ where $A$ is the region enclosed by $C$ and $B({\boldsymbol{\alpha}})$ is interpreted as the magnetic field with constant contours shown in Figure \[fig:shapespace\]. The advantage of the area formulation is that it is easy to visualise the function $B(\alpha)$ in shape space, and wherever this function is large the potential contribution to the geometric phase is large as well. By writing out ${\boldsymbol{F}}({\boldsymbol{\alpha}})=F_1({\boldsymbol{\alpha}}){\boldsymbol{i}}+F_2({\boldsymbol{\alpha}}){\boldsymbol{j}}$ explicitly, we have $$\begin{aligned}
F({\boldsymbol{\alpha}})\cdot\dot{{\boldsymbol{\alpha}}} &= F_1({\boldsymbol{\alpha}})\,\frac{d\alpha_2}{dt}+F_2({\boldsymbol{\alpha}})\,\frac{d\alpha_4}{dt} &\text{ and }&&
B({\boldsymbol{\alpha}}) &= \frac{\partial F_2({\boldsymbol{\alpha}})}{\partial \alpha_2}-\frac{\partial F_1({\boldsymbol{\alpha}})}{\partial \alpha_4}\nonumber\end{aligned}$$ appearing in . As $C$ is self-intersecting, the geometric phase can be computed by partitioning the loop into 10 pieces labelled $C_1,\dots, C_{10}$ (as shown in Figure \[fig:shapespace\]) and then taking the appropriate pieces that make up the sub-loops. The total geometric phase is therefore the sum of the individual geometric phase contributions from each sub-loop. Specifically, the total geometric phase is composed of $$\begin{aligned}
(C_2,C_4,C_6,C_8) &= 0.0953 & C_3 &= -0.0149 & C_5 &= -0.0012\nonumber\\
(C_1,C_9,C_{10}) &= -0.0023 & C_7 &= -0.0048, &\nonumber\end{aligned}$$ where summing the contributions from these five sub-loops yields $\theta_\mathit{geo}$ found in . We observe that most contributions towards the geometric phase are negative, meaning improvement can be made by reversing the direction of travel along the loop. As $B < 0$ throughout the dive, loops orientated clockwise will provide a positive contribution to the geometric phase, while loops oriented counterclockwise will provide a negative contribution.
We will now show that the geometric phase can be increased without changing the dynamic phase. Originally the pieces were ordered from $C_1$ to $C_{10}$, but after orientating the sub-loops the order becomes $-C_{10} \rightarrow -C_9 \rightarrow C_2\rightarrow -C_3\rightarrow C_4\rightarrow -C_5\rightarrow C_6\rightarrow -C_7\rightarrow C_8\rightarrow -C_1$, where a negative means we traverse along the piece in the opposite direction. The before and after shape trajectories are shown in Figure \[fig:loops\], where the improved geometric phase is $$\tilde{\theta}_\mathit{geo} = 0.1186.$$ While the effect from the geometric phase is small, it is still an improvement of $64\%$ compared to the original geometric phase . Clearly these adjustments are not practical in an actual dive. However, we want to emphasise that we can increase the overall rotation achieved simply by reordering and reversing certain parts of the loop $C$. This additional rotation is obtained by maximising the geometric phase while leaving the dynamic phase unchanged.
Optimising Planar Somersaults {#sec:theoreticalplanar}
=============================
![The moment of inertia $I$ plotted frame by frame.[]{data-label="fig:moi"}](moi.pdf){width="14cm"}
The following observations can be made by analysing Figure \[fig:digitise\] and Figure \[fig:moi\]:
1. The athlete takes off with a large moment of inertia.
2. The athlete transitions quickly into pike position.
3. The athlete maintains pike position during which the moment of inertia is small.
4. The athlete completes the dive with (roughly) the same shape as during take-off.
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Figure \[fig:shapevelocity\] illustrates the relative velocities of the athlete’s arms and legs, where the black boxes indicate transitions into and out of pike position that take at least a quarter second to complete. The interval between the two transition stages reveals small oscillations in the velocities, which are the result of the athlete making micro-adjustments to maintain pike position under heavy rotational forces.
Let ${\boldsymbol{\alpha}}_\mathit{max}\in[-\pi,0]^2$ correspond to the shape with maximum moment of inertia $I_\mathit{max}$, and let ${\boldsymbol{\alpha}}_\mathit{min}\in[-\pi,0]^2$ correspond to the shape with minimum moment of inertia $I_\mathit{min}$. We then find $$\begin{aligned}
{\boldsymbol{\alpha}}_\mathit{max}&=(-\pi, -0.3608) & I_\mathit{max} &= 21.0647\nonumber\\
{\boldsymbol{\alpha}}_\mathit{min}&=(-0.3867,-\pi) & I_\mathit{min} &= 5.2888\nonumber\end{aligned}$$ and illustrate these shape configurations in Figure \[fig:Iextreme\]. The figure reveals hip flexion in the shape corresponding to ${\boldsymbol{\alpha}}_\mathit{max}$, which is due to the positioning of the hip joint in the model. This resembles reality as athletes exhibit some hip flexion during take-off and entry into the water, as seen in Figure \[fig:digitise\].
![The black curve illustrates the quickest way to move into and out of pike position so that the transition time is a quarter second. Constant $I({\boldsymbol{\alpha}})$ contours have been plotted, which show that at ${\boldsymbol{\alpha}}_\mathit{max}$ and ${\boldsymbol{\alpha}}_\mathit{min}$ the moments of inertia are $I_\mathit{max}$ and ${\boldsymbol{\alpha}}_\mathit{min}$, respectively.[]{data-label="fig:fastest"}](fastest.png){width="12cm"}
We now propose a theoretical dive by using the structure observed in the digitised dive as a guideline. In the idealised dive the athlete takes off with shape ${\boldsymbol{\alpha}}_\mathit{max}$ and immediately transitions into pike position specified by shape ${\boldsymbol{\alpha}}_\mathit{min}$. The athlete maintains pike without oscillations, then reverts back into the original shape ${\boldsymbol{\alpha}}_\mathit{max}$ to complete the dive. It appears obvious that this process will yield the maximal amount of somersault. However, as we will show, this is only true when the transition from ${\boldsymbol{\alpha}}_\mathit{max}$ to ${\boldsymbol{\alpha}}_\mathit{min}$ is instantaneous, which is of course unrealistic. When a maximum speed of shape change is imposed we will show that the amount of somersault can be increased slightly by using a different manoeuvre. The explanation behind this surprising observation is the geometric phase. We cap $|\dot{\alpha}_2| = 11.0194$ (arms) and $|\dot{\alpha}_4| = 11.1230$ (legs) when the limbs move at maximum speed, so that the transition from ${\boldsymbol{\alpha}}_\mathit{max}$ to ${\boldsymbol{\alpha}}_\mathit{min}$ (and vice versa) takes precisely a quarter second. Moving the limbs at maximum speed into and out of pike position maximises the time spent in pike, but there is no contribution to the amount of somersault from the geometric phase. We know the geometric phase is zero because there is no area enclosed by the loop $C$, as illustrated in Figure \[fig:fastest\]. The loop $C$ appears to be a line (and hence has no enclosed area) because the motions into pike and back into the layout position are happening in exactly the same way. The essential idea is to break this symmetry by making the shape change into pike and the shape change back into the layout position different.
![Here $\{s_\mathit{in}, s_\mathit{out}\}=\{0.9818, 0.3158\}$, so the transition from ${\boldsymbol{\alpha}}_f= (-0.3867, -3.0910)$ to ${\boldsymbol{\alpha}}_\mathit{min}$ takes $0.0046$ seconds, and from ${\boldsymbol{\alpha}}_\mathit{min}$ to ${\boldsymbol{\alpha}}_b=(-2.2717, -\pi)$ takes $0.1711$ seconds. The loop is shown with constant $B({\boldsymbol{\alpha}})$ contours, so the enclosed region gives an idea of the expected geometric phase contribution.[]{data-label="fig:slowest"}](slowest.png){width="12cm"}
This idea is illustrated in Figure \[fig:slowest\], which has a slower leg movement when moving into pike, so that after a quarter second the arms are in place while the legs are not. This is indicated by the shape ${\boldsymbol{\alpha}}_f$, and the black vertical curve shows the additional leg movement required to reach ${\boldsymbol{\alpha}}_\mathit{min}$. When leaving pike position the arms move first to reach shape ${\boldsymbol{\alpha}}_b$, before both pairs of limbs move concurrently for a quarter second to complete the dive with shape ${\boldsymbol{\alpha}}_\mathit{max}$. In this part of the shape change the legs move at maximal speed. While this results in rotational contribution from the geometric phase, the contribution from the dynamic phase is less due to the reduced time spent in pike position.
As the geometric phase given by involves integrating $B({\boldsymbol{\alpha}})$ over the region enclosed by the loop $C$, having the absolute maximum of $B({\boldsymbol{\alpha}})$ and its neighbouring large absolute values contained in this region will provide a more efficient (in terms of geometric phase per arc length) contribution towards the geometric phase. Maximising the overall rotation obtained therefore involves finding the balance between the dynamic and geometric phase contributions. We will now perform optimisation to determine the speed at which the arms and legs should move to achieve this.
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![Loop with $\{s_\mathit{in}, s_\mathit{out}\}=\{1, 0.8593\}$ shown with constant $B({\boldsymbol{\alpha}})$ contours, and the point ${\boldsymbol{\alpha}}_b=(-0.7743, -\pi)$.[]{data-label="fig:optimal"}](optimal.png){width="12cm"}
We define $s_\mathit{in}\in[0,1]$ to be the fraction of maximum speed for the leg movement into pike position, and $s_\mathit{out}\in[0,1]$ to be the fraction of maximum speed for the arm movement out of pike. These reduced speeds (for either arms or legs) will only be used when moving from ${\boldsymbol{\alpha}}_\mathit{max}$ to ${\boldsymbol{\alpha}}_f$ and ${\boldsymbol{\alpha}}_b$ to ${\boldsymbol{\alpha}}_\mathit{max}$, as the transitions from ${\boldsymbol{\alpha}}_f$ to ${\boldsymbol{\alpha}}_\mathit{min}$ and ${\boldsymbol{\alpha}}_\mathit{min}$ to ${\boldsymbol{\alpha}}_b$ will involve the appropriate limb moving at maximum speed to minimise the extra time spent in shape change. With this construction, the extra time required to move into and out of pike is $\tau_E(s_\mathit{in})$ and $\tau_E(s_\mathit{out})$, where $$\tau_E(s) = (1-s)/4.\label{eq:extT}$$ For the dive illustrated in Figure \[fig:fastest\] we have $\{s_\mathit{in}, s_\mathit{out}\}=\{1,1\}$, thus both $\tau_E(s_\mathit{in}) = \tau_E(s_\mathit{out}) =0$. However, in general when $\tau_E(s_\mathit{in}) \neq 0$ and $\tau_E(s_\mathit{out}) \neq 0$ the dive is composed of four pieces: the transitions from ${\boldsymbol{\alpha}}_\mathit{max} \rightarrow {\boldsymbol{\alpha}}_f$, ${\boldsymbol{\alpha}}_f \rightarrow {\boldsymbol{\alpha}}_\mathit{min}$, ${\boldsymbol{\alpha}}_\mathit{min} \rightarrow {\boldsymbol{\alpha}}_b$ and ${\boldsymbol{\alpha}}_b \rightarrow {\boldsymbol{\alpha}}_\mathit{max}$, which we denote as ${\boldsymbol{\alpha}}_i(t)$ for $i\in\{1,2,3,4\}$, respectively. Let the transition time for each ${\boldsymbol{\alpha}}_i(t)$ be from zero to $\tau_i$, then $\tau_1=\tau_4=1/4$, $\tau_2=\tau_E(s_\mathit{in})$, $\tau_3=\tau_E(s_\mathit{out})$, giving the cumulative shape change time $$T_\Sigma = 1/2 + \tau_E(s_\mathit{in}) + \tau_E(s_\mathit{out}).$$ Combined with $T_\mathit{pike}$, which is the duration the athlete maintains pike position, we obtain the total airborne time $$T_\mathit{air} = T_\Sigma + T_\mathit{pike}.\label{eq:Tair}$$ Defining the shape change velocities in vector form to be $${\boldsymbol{v}}(s_\mathit{in},s_\mathit{out}) = 4\operatorname{diag}{(s_\mathit{out}, s_\mathit{in})}({\boldsymbol{\alpha}}_\mathit{min}-{\boldsymbol{\alpha}}_\mathit{max}),$$ the shape changing transitions can be written as $$\begin{aligned}
{\boldsymbol{\alpha}}_1(t) &= {\boldsymbol{\alpha}}_\mathit{max}+t{\boldsymbol{v}}(s_\mathit{in}, 1) & {\boldsymbol{\alpha}}_2(t) &= {\boldsymbol{\alpha}}_f+t{\boldsymbol{v}}(1,0) \nonumber\\
{\boldsymbol{\alpha}}_3(t) &= {\boldsymbol{\alpha}}_\mathit{min}-t{\boldsymbol{v}}(0,1) & {\boldsymbol{\alpha}}_4(t) &= {\boldsymbol{\alpha}}_b-t{\boldsymbol{v}}(1, s_\mathit{out}),\nonumber\end{aligned}$$ where ${\boldsymbol{\alpha}}_f = {\boldsymbol{\alpha}}_1(\tau_1)$ and ${\boldsymbol{\alpha}}_b = {\boldsymbol{\alpha}}_3(\tau_3)$. Phases $2$ and $3$ disappear when $s_{in} = s_{out} = 1$ because the corresponding time spent in these phases is $\tau_E(1) = 0$. Solving for the overall rotation obtained via we get $$\theta(s_\mathit{in},s_\mathit{out})=\theta_\mathit{dyn}(s_\mathit{in},s_\mathit{out})+\theta_\mathit{geo}(s_\mathit{in},s_\mathit{out}),\label{eq:maxtheta}$$ where the components are $$\begin{aligned}
\theta_\mathit{dyn}(s_\mathit{in},s_\mathit{out}) &= L\left[\sum_{i=1}^4 \int_0^{\tau_i}I^{-1}\big({\boldsymbol{\alpha}}_i(t;s_\mathit{in},s_\mathit{out})\big)\,\mathrm{d}t + I^{-1}({\boldsymbol{\alpha}}_\mathit{min})T_\mathit{pike}\right]\nonumber\\
\theta_\mathit{geo}(s_\mathit{in},s_\mathit{out}) &= \sum_{i=1}^4 \int_0^{\tau_i}{\boldsymbol{F}}\big({\boldsymbol{\alpha}}_i(t;s_\mathit{in},s_\mathit{out})\big)\cdot{\boldsymbol{\dot{\alpha}}}_i(t;s_\mathit{in},s_\mathit{out})\,\mathrm{d}t.\nonumber\end{aligned}$$ The above components can be further simplified by substituting in to eliminate $T_\mathit{pike}$, and combining the shape change segments ${\boldsymbol{\alpha}}(t) = \bigcup_{i=1}^4 {\boldsymbol{\alpha}}_i(t)$ so that $t$ runs from zero to $T_\mathit{air}$ for the complete dive. This then gives $$\begin{aligned}
\theta_\mathit{dyn} &= L\left[\int_0^{T_\mathit{air}}I^{-1}({\boldsymbol{\alpha}})\,\mathrm{d}t -I^{-1}({\boldsymbol{\alpha}}_\mathit{min})T_\Sigma + I^{-1}({\boldsymbol{\alpha}}_\mathit{min})T_\mathit{air}\right]\label{eq:thetadyn}\\
\theta_\mathit{geo} &= \int_0^{T_\mathit{air}} F({\boldsymbol{\alpha}})\cdot\dot{{\boldsymbol{\alpha}}}\,dt = \iint_A B({\boldsymbol{\alpha}})\,d\tilde{A},\end{aligned}$$ where the arguments $t, s_\mathit{in}$ and $s_\mathit{out}$ have been suppressed to avoid clutter. We see in that the constant $T_\mathit{air}$ provides an overall increase in $\theta_\mathit{dyn}$, but otherwise plays no role in the optimisation strategy. To obtain maximal rotation there is competition between having a large $B$ and a small $I$ for as long as possible. The angular momentum $L$ is an important parameter: decreasing $L$ reduces the contribution from $\theta_\mathit{dyn}$ while $\theta_\mathit{geo}$ is unaffected, thus the geometric phase contribution has a greater impact on $\theta$ when $L$ is small. The essential competition in the optimisation in this model comes from the extra time $T_\Sigma$ taken to make the loop larger around large $B$ (which increases the geometric phase) and $T_\Sigma$ taken away from the time spent in pike position (which decreases the dynamic phase). Our main result is that for a fixed total time $T_\mathit{air}$ and a moderate $L$ the total amount of somersault can be improved by using a shape change that has a different motion into pike than out of pike, and hence generates a geometric phase. The detailed results are as follows.
Figure \[fig:s\] shows the optimal $\{s_\mathit{in}, s_\mathit{out}\}$ that maximises $\theta$ for different values of $L$. When $L=0$ this implies $\theta_\mathit{dyn}=0$, meaning the rotation is purely governed by $\theta_\mathit{geo}$. So by choosing $\{s_\mathit{in}, s_\mathit{out}\} = \{0,0\}$, the geometric phase is maximised and so too is the overall rotation. The case $L=0$ corresponds to the problem of the falling cat reorienting itself: rotation can be generated without angular momentum via a shape change. As $L$ increases the dynamic phase becomes proportionally larger, making it more important to enter pike position earlier, and hence $\{s_\mathit{in}, s_\mathit{out}\}\rightarrow\{1,1\}$ as $L$ gets larger. The limiting point $\{s_\mathit{in}, s_\mathit{out}\} = \{1,1\}$ occurs when $L=193.65$, where beyond this point maximum overall rotation is achieved by transitioning to pike position as fast as possible, as demonstrated in Figure \[fig:fastest\]. For comparison, Figure \[fig:slowest\] shows the optimal shape change trajectory when $L=30$.
When optimising $\theta$ in for a typical planar somersault with $L=120$, we see in Figure \[subfig:s120\] that $\{s_\mathit{in}, s_\mathit{out}\}=\{1, 0.8593\}$ yields the maximal rotation, whose shape change trajectory is illustrated in Figure \[fig:optimal\]. When compared to the dive with $\{s_\mathit{in}, s_\mathit{out}\}=\{1, 1\}$, the gain in overall rotation is $0.0189$ radians or $1.0836^\circ$. We see that in the optimal dive the limbs move at maximum speed into pike position, but when leaving the pike the arms move slower, thus providing a geometric phase contribution that exceeds the dynamic phase contribution lost by $1.0836^\circ$. This behaviour is due to the take-off and pike shapes being ${\boldsymbol{\alpha}}_\mathit{max}$ and ${\boldsymbol{\alpha}}_\mathit{min}$, but had we chosen different shapes then the observed result may differ. The dynamic phase benefits from being at (or close to) the minimum moment of inertia $I_\mathit{min}$ located at ${\boldsymbol{\alpha}}_\mathit{min}$, whereas the geometric phase favours enclosing a region with large magnitudes of the magnetic field $B$, where the absolute maximum is $0.3058$ occurring at $(-1.3148, -3.0043)$.
Repeating the same computation with $L=30$ yields the optimal shape change trajectory shown in Figure \[fig:slowest\], and comparing this with the $\{s_\mathit{in}, s_\mathit{out}\}=\{1, 1\}$ dive yields an additional $0.2327$ radians or $13.3354^\circ$ in rotation, which is more significant than the $1.0836^\circ$ found for $L=120$. Although the addition of $1^\circ$ is irrelevant in practice, the model used is extremely simple and these results should only be considered as a proof of principle. We hope that by using more realistic models with asymmetric movements, the additional rotation obtained via geometric phase optimisation will yield a small but important contribution that transforms a failed dive into a successful one. The principles here are not limited to planar somersaults, but can also be utilised in other fields such as robotics and space aeronautics, where the benefit of the geometric phase will be large whenever the angular momentum is small.
Acknowledgement {#acknowledgement .unnumbered}
===============
This research was supported in part by the Australian Research Council through the Linkage Grant LP100200245 “Bodies in Space” in collaboration with the New South Wales Institute of Sport.
Definition of ${\boldsymbol{D}}_i^j$, ${\boldsymbol{\bar{D}}}_i^j$ and ${\boldsymbol{\tilde{D}}}_i^j$ {#app:D}
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Consider a rooted tree where each $B_i$ is treated as a node with $B_1$ being the root (top most node). Let $B_{p(j)}$ denote the node who is the parent of $B_j$ in the tree, and $B_{c(j,i)}$ be the child of $B_j$, who is either $B_i$ or is the node with the direct line to $B_i$, i.e. an ancestor of $B_i$. An ancestor of $B_i$ is any node reachable by repeated proceedings from child to parent, e.g. the root of the tree $B_1$ is an ancestor to every other node. We can now give the general definition of the constant vectors ${\boldsymbol{D}}_j^i$ using the idea of trees. We have ${\boldsymbol{D}}_j^i={\boldsymbol{0}}$ for $1\leq i\leq n$ and $1\leq j\leq n$, unless either $j=i$ and $j \neq 1$, or if $B_j$ is an ancestor of $B_i$ in the tree. When this occurs we have $${\boldsymbol{D}}_j^i=\begin{cases}
-{\boldsymbol{E}}_j^{p(j)} & \mbox{if } j=i \mbox{ and } j \neq 1\\
{\boldsymbol{E}}_{j=\mathit{ref}}^{c(j,i)} & \mbox{if } j\neq i \mbox{ and } j = 1\\
-{\boldsymbol{E}}_j^{p(j)}+{\boldsymbol{E}}_j^{c(j,i)} & \mbox{if } j\neq i \mbox{ and } j \neq 1.
\end{cases}$$ Next, we have $$\begin{aligned}
{\boldsymbol{\bar{D}}}_j&=\frac{1}{M}\sum_{i=1}^{n}m_i {\boldsymbol{D}}_j^i\end{aligned}$$ where $M$ is the total mass, and this can be interpreted as the weighted mean of the ${\boldsymbol{D}}_j^i$’s. Finally, $${\boldsymbol{\tilde{D}}}_i^j = {\boldsymbol{D}}_i^j-{\boldsymbol{\bar{D}}}_i$$ is the difference between ${\boldsymbol{D}}_i^j$ and the weighted mean ${\boldsymbol{\bar{D}}}_j$.
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abstract: 'Magnetic resonance image (MRI) reconstruction is a severely ill-posed linear inverse task demanding time and resource intensive computations that can substantially trade off [*accuracy*]{} for [*speed*]{} in real-time imaging. In addition, state-of-the-art compressed sensing (CS) analytics are not cognizant of the image [*diagnostic quality*]{}. To cope with these challenges we put forth a novel CS framework that permeates benefits from generative adversarial networks (GAN) to train a (low-dimensional) manifold of diagnostic-quality MR images from historical patients. Leveraging a mixture of least-squares (LS) GANs and pixel-wise $\ell_1$ cost, a deep residual network with skip connections is trained as the generator that learns to remove the [*aliasing*]{} artifacts by projecting onto the manifold. LSGAN learns the texture details, while $\ell_1$ controls the high-frequency noise. A multilayer convolutional neural network is then jointly trained based on diagnostic quality images to discriminate the projection quality. The test phase performs feed-forward propagation over the generator network that demands a very low computational overhead. Extensive evaluations are performed on a large contrast-enhanced MR dataset of pediatric patients. In particular, images rated based on expert radiologists corroborate that GANCS retrieves high contrast images with detailed texture relative to conventional CS, and pixel-wise schemes. In addition, it offers reconstruction under a few milliseconds, two orders of magnitude faster than state-of-the-art CS-MRI schemes.'
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title: 'Deep Generative Adversarial Networks for Compressed Sensing (GANCS) Automates MRI'
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Introduction
============
Owing to its superb soft tissue contrast, magnetic resonance imaging (MRI) nowadays serves as the major imaging modality in clinical practice. Real-time MRI visualization is of paramount importance for diagnostic and therapeutic guidance for instance in next generation platforms for MR-guided, minimally invasive neurosurgery [@clearpoint]. However, the scan is quite slow, taking several minutes to acquire clinically acceptable images. This becomes more pronounced for high-resolution and volumetric images. As a result, the acquisition typically undergoes significant undersampling leading reconstruction to a seriously ill-posed linear inverse problem. To render it well-posed, the conventional compressed-sensing (CS) incorporates the prior image information by means of sparsity regularization in a proper transform domain such as Wavelet (WV), or, Total Variation (TV); see e.g., [@pualy_mri20017]. This however demands running iterative optimization algorithms that are time and resource intensive. This in turn hinders [*real-time*]{} MRI visualization and analysis.
Recently, a few attempts have been carried out to [*automate*]{} medical image reconstruction by leveraging historical patient data; see e.g., [@Majumdar'15; @lowdose_ct2017]. They train a network that maps the aliased image to the gold-standard one using convolutional neural networks (CNN) with residuals for computed tomography (CT) [@lowdose_ct2017], denoising auto-encoders for MRI [@Majumdar'15]. Albeit, speed up, they suffer from blurry and aliasing artifacts. This is mainly due to adopting a pixel-wise $\ell_1$/$\ell_2$ cost that is oblivious of high-frequency texture details, which is crucial for drawing diagnostic decisions. See also the recent DeepADMM scheme in [@deepADMM2016] for CS MRI that improves the quality, but it is as slow as the conventional CS. Generative adversarial networks (GANs) have been lately proved very successful in [@gan-goodfellow2014; @dcgan2016] modeling a low-dimensional distribution (manifold) of natural images that are perceptually appealing [@Zhu; @et; @al'16]. In particular, for image super-resolution tasks GANs achieve state-of-the-art perceptual quality under $4\times$ upscaling factor for natural images e.g., from ImageNet [@leding; @et; @al'16; @Sonderby; @et; @al'14]. GANs has also been deployed for image inpaitning [@inpainting-yeh-2016], style transfer [@johnson2016], and visual manipulation [@Zhu; @et; @al'16].
Despite the success of GANs for [*local*]{} image restoration such as super-resolution and inpainting, to date, they have not been studied for removing [*aliasing*]{} artifacts in biomedical image reconstruction tasks. This is indeed a more difficult image restoration tasks. In essence, aliasing artifacts (e.g., in MRI) emanate from data undersampling in a different domain (e.g., Fourier, projections) which [*globally*]{} impact image pixels. Inspired by the high texture quality offered by GANs, and the high contrast of MR images, we employ GANs to learn a low-dimensional manifold of diagnostic-quality MR images. To this end, we train a tandem network of a generator (G) and a discriminator (D), where the generator aims to generate the ground-truth images from the complex-valued aliased ones using a deep residual network (ResNet) with skip connections, with refinement to ensure it is consistent with measurement (data consistency). The aliased input image is simply obtained via inverse Fourier Transform (FT) of undersampled data. D network then scores the G output, using a multilayer convolutional neural network (CNN) that scores one if the image is of diagnostic quality, and, zero if it contains artifacts. For training we adopt a mixture of LSGAN [@lsgan2017] and $\ell_1$ pixel-wise criterion to retrieve high-frequency texture while controlling the noise. We performed evaluations on a large cohort of pediatric patients with contrast-enhanced abdominal images. The retrieved images are rated by expert radiologists for diagnostic quality. Our observations indicate that GANCS results have almost similar quality to the gold-standard fully-sampled images, and are superior in terms of diagnostic quality relative to the existing alternatives including conventional CS (e.g., TV and WV), $\ell_2$-, and $\ell_1$-based criteria. Moreover, the reconstruction only takes around $10-20$ msec, that is two orders of magnitude faster than state-of-the-art conventional CS toolboxes.
Last but not least, the advocated GANCS scheme tailors inverse imaging tasks appearing in a wide range of applications with budgeted acquisition and reconstruction speed. All in all, relative to the past work this paper’s main contributions are summarized as follows:
- Propose GANCS as a data-driven regularization scheme for solving ill-posed linear inverse problems that appear in imaging tasks dealing with (global) aliasing artifacts
- First work to apply GAN as a automated (non-iterative) technique for aliasing artifact suppression in MRI with state-of-the-art image diagnostic quality and reconstruction speed
- Proposed and evaluated a novel network architecture to achieve better trade-offs between data-consistency (affine projection) and manifold learning
- Extensive evaluations on a large contrast-enhanced MRI dataset of pediatric patients, with the reconstructed images rated by expert radiologists
The rest of this paper is organized as follows. Section 2 states the problem. Manifold learning using LSGANs is proposed in Section 3. Section 4 also reports the data evaluations, while the conclusions are drawn in Section 5.
Problem Statement {#sec:problem_statement}
=================
Consider an ill-posed linear system $\by=\bPhi \bx + \bv$ with $\bPhi \in \mathbb{C}^{M \times N}$ where $M \ll N$, and $\bv$ captures the noise and unmodeled dynamics. Suppose the unknown and complex-valued image $\bx$ lies in a [*low-dimensional*]{} manifold, say $\cM$. No information is known about the manifold besides the training samples $\cX:=\{\bx_k\}_{k=1}^K$ drawn from it, and the corresponding (possibly) noisy observations $\mathcal{Y}:=\{\by_k\}_{k=1}^K$. Given a new observation $\by$, the goal is to recover $\bx$. For instance, in the MRI context motivated for this paper $\bPhi$ refers to the partial 2D FT that results in undersampled $k$-space data $\by$. To retrieve the image, in the first step we learn the manifold $\cM$. Subsequently, the second step projects the aliased image, obtained via e.g., pseudo inverse $\bPhi^{\dagger}\by$ onto $\cM$ to discard the artifacts. For the sake of generality, the ensuing is presented for a generic linear map $\bPhi$.
Manifold Learning via Generative Adversarial Networks {#sec:gans}
=====================================================
The inverse imaging solution is to find solutions of the intersection between two subspaces defined by acquisition model and image manifold. In order to effectively learn the image manifold from the available (limited number of) training samples we first need to address the following important questions:
- How to ensure the trained manifold contains plausible images?
- How to ensure the points on the manifold are data consistent, namely $\by \approx \bPhi\bx, ~\forall \bx \in \cM$?
To address the first question we adopt GANs, that have recently proven very successful in estimating prior distribution for images. GANs provide sharp images that are visually plausible [@gan-goodfellow2014]. In contrast, variational autoencoders [@leding; @et; @al'16], a important class of generative models, use pixel-wise MSE costs that results in high pick signal-to-noise ratios but often produce overly-smooth images that have poor perceptual quality. Standard GAN consists of a tandem network of G and D networks. Consider the undersampled image $\tilde{\bx}:=\bPhi^{\dagger}\by$ as the input to the G network. The G network then projects $\tilde{\bx}$ onto the low-dimensional manifold $\cM$ containing the high-quality images $\cX$. Let $\hat{\bx}$ denote the output of G, it then passes through the discriminator network D, that outputs one if $\hat{\bx} \in \cX$, and zero otherwise.
The output of G, namely $\check{\bx}$, however may not be consistent with the data. To tackle this issue, we add another layer after G that projects onto the feasible set of $\by=\bPhi\bx$ to arrive at $\hat{\bx}=\bPhi^{\dagger} \by + (\bI-\bPhi^{\dagger}\bPhi) \check{\bx}$. Alternatively, we can add a soft LS penalty when training the G network, as will be seen later in (P1). To further ensure that $\hat{\bx}$ lies in the intersection of the manifold $\cM$ and the space of data consistent images we can use a mutlilayer network that alternates between residual units and data consistency projection as depicted in Fig. \[fig:fig\_net\] (b). We have observed that using only a couple of residual units may improve the performance of G in discarding the aliasing artifacts. The overall network architecture is depicted in Fig. \[fig:fig\_net\] (a), where $\mathbf{P}_{\cN}:=(\bI-\bPhi^{\dagger}\bPhi)$ signifies projection onto the nullspace of $\bPhi$.
![(a) GANCS structure for manifold learning, where the dashed module is projection on the feasible set. (b) The multilayer residual blocks (RB) for data consistency. []{data-label="fig:fig_net"}](network_structure_may19.png)
Training the network in Fig. \[fig:fig\_net\] amounts to playing a game with conflicting objectives between the adversary G and the discriminator D. D network aims to score one the real images drawn from the data distribution $p_{x}$, and score zero the rest. G network also aims to map the input images $\tilde{\bx}=\bPhi^{\dagger}\by$ with the distribution $p_{\tilde{x}}=p_{x}(\bPhi^{\dagger}\bPhi \bx)$ to the fake images $\hat{\bx}$ that fool the D network. Various strategies have been devised to reach the equilibrium. They mostly differ in terms of the cost function adopted for the G and D networks [@gan-goodfellow2014], [@lsgan2017]. The standard GAN uses a sigmoid cross-entropy loss that leads to vanishing gradients which renders the training unstable, and as a result it suffers from sever degrees of mode collapse. In addition, for the generated images classified as the real with high confidence (i.e., large decision variable), no cost is incurred. Hence, the standard GAN tends to pull samples away from the decision boundary, that introduces non-realistic images [@lsgan2017]. LSGN instead pulls the generated samples towards the decision boundary by using a LS cost.
One issue with GAN however is that it introduces high frequency noise all over the image. $\ell_1$ criterion has proven well in discarding the noise from natural images as it does appropriately penalize the low-intensity noise [@lossfunction_zhao2017]. Accordingly, to reveal fine texture details while discarding noise, we are motivated to adopt a mixture of LSGAN and $\ell_1$ costs to train the generator. The overall procedure aims to jointly minimize the discriminator cost $$\begin{aligned}
{\rm(P1.1)} \quad \quad \min_{\bTheta_d}~ \mathbb{E}_{\bx} \Big[\Big(1-\cD(\bx;\bTheta_d)\Big)^2\Big] + \mathbb{E}_{\by} \Big[\Big(\cD(\cG(\bPhi^{\dagger} \by; \bTheta_g);\bTheta_d)\Big)^2\Big] \nonumber\end{aligned}$$ and the generator cost $$\begin{aligned}
{\rm(P1.2)} \quad \quad \min_{\bTheta_g}~ \mathbb{E}_{\by}\Big[\Big\|\by - \bPhi \cG(\bPhi^{\dagger} \by; \bTheta_g) \Big\|^2 \Big] & + \eta \mathbb{E}_{\bx,\by}\Big[\Big\|\bx - \cG(\bPhi^{\dagger} \by; \bTheta_g) \Big\|_1 \Big] \nonumber \\ & + \lambda \mathbb{E}_{\by} \Big[\Big(1-\cD \big(\cG(\bPhi^{\dagger} \by; \bTheta_g);\bTheta_d\big)\Big)^2\Big] \nonumber\end{aligned}$$ The first LS fitting term in (P1.2) is a soft penalty to ensure the input to D network is data consistent. Parameters $\lambda$ and $\eta$ also control the balance between manifold projection, noise suppression and data consistency.
Looking carefully into (P1.2) the generator reconstructs image $\cG(\bPhi^{\dagger} \by; \bTheta_g)$ from the data $\by$ using an expected regularized-LS estimator, where the regularization is learned form training data via LSGAN and $\ell_1$-net. Different from the conventional CS formulation which also optimize the reconstruction with $\ell_1$-regularized LS estimation, the entire optimization only happens in training and the generator learned can be directly applied to new samples to achieve fast reconstruction.
As argued in [@lsgan2017], it can be shown that LSGAN game yields minimizing the Pearson-$\chi^2$ divergence. For (P1) following the same arguments as of the standard GANS in [@gan-goodfellow2014] and [@lsgan2017] it can be readily shown that even in the presence of LS data consistency and $\ell_1$ penalty, the distribution modeled by G network, say $p_g$, coincides with the true data distribution. This is formally stated next.
**Lemma 1.** [*For the noise-free scenario ($\bv=\mathbf{0}$), suppose D and G have infinite capacity. Then, for a given generator network G, i) the optimal discriminator D is $\cD^{*}(\bTheta_d;\check{\bx})=p_x(\check{\bx})/(p_x(\check{\bx})+p_g(\check{\bx}))$; and ii) $p_g=p_x$ achieves the equilibrium for the game (P1).* ]{}
[*Proof.*]{} The first part is similar to the one in [@lsgan2017] with the same cost for D. The second part also readily follows as the LS data consistency and $\ell_1$ penalty are non-negative, and become zero when $p_g=p_x$. Thus, according to Pearson-$\chi^2$ divergence still bounds (P1.2) objective from below, and is achievable when $p_g=p_x$. $\blacksquare$
Stochastic alternating minimization {#subsec:alt_min}
-----------------------------------
To train the G and D networks, a mini-batch stochastic alternating minimization scheme is adopted. At $k$-th iteration with the mini-batch training data $\{(\bx_{\ell},\by_{\ell})\}_{\ell=1}^L$, assuming that G is fixed, we first update the discriminator $\bTheta_d$ by taking a single descent step with momentum along the gradient of D cost, say $f_d$. Similarly, given the updated $\bTheta_d$, the G network is updated by taking a gradient descent step with momentum along the gradient of G cost, say $f_g$. The resulting iterations are listed under Algorithm \[tab:alg\_gan\_training\], where the gradients $\nabla_{\bTheta_g}\cG(\bTheta_g;\tilde{\bx}_{\ell})$, $\nabla_{\bTheta_d}\cD(\hat{\bx}_{\ell};\bTheta_d)$, and $\nabla_{\bTheta_g}\cD(\cG(\bTheta_g;\tilde{\bx}_{\ell});\bTheta_d)$ are readily obtained via backpropagation over D and G networks. Also, $G_n$ refers to the $n$-th output pixel of G network, and $[.]_n$ picks the $n$-th pixel.
\[tab:alg\_gan\_training\]
Experiments {#sec:eval}
===========
Effectiveness of the novel GANCS scheme is assessed in this section via tests for MRI reconstruction. A single-coil MR acquisition model is considered where for $n$-th patient the acquired $k$-space data abides to $y_{i,j}^{(n)} = [\cF(\bX_n)]_{i,j} + v_{i,j}^{(n)},~~(i,j) \in \Omega$. Here, $\cF$ is the 2D FT, and the set $\Omega$ indexes the sampled Fourier coefficients. As it is conventionally performed with CS MRI, we select $\Omega$ based on a variable density sampling with radial view ordering \cite{} that tends to pick low frequency components from the center of $k$-space (see sampling mask in Fig. 4 (left) of the supplementary document). Throughout the test we assume $\Omega$ collects only $20\%$ of the Fourier coefficients, and we choose $\lambda=0.1$.
[**Dataset**.]{} High contrast abdominal image volumes are acquired for $350$ pediatric patients after gadolinium-based contrast enhancement. Each 3D volume includes contains $151$ axial slices of size $200 \times 100$. Axial slices used as input images for training a neural network. $300$ patients ($45,300$ images) are considered for training, and $50$ patients ($7,550$ images) for test. All in vivo scans were acquired at the Stanford’s Lucile Packard Children’s Hospital on a 3T MRI scanner (GE MR750) with voxel resolution $1.07 \times 1.12 \times 2.4$ mm.
Under this setting, the ensuing parts address the following questions:
Q1. How does the perceptual cost learned by GANCS improve the image quality compared with the pixel-wise $\ell_2$ and $\ell_1$ costs?
Q2. How much speed up and quality improvement one can achieve using GANCS relative to conventional CS?
Q3. What MR image features derive the network to learn the manifold and remove the aliasing artifacts?
Q4. How many samples/patients are needed to achieve a reasonable image quality?
Training and network architecture {#subsec:training}
---------------------------------
The input and output are complex-valued images of the same size and each include two channels for real and imaginary components. The input image $\tilde{\bx}$ is simply generated using inverse 2D FT of the sampled $k$-space, which is severely contaminated by artifacts. Input channels are then convolved with different kernels and added up in the next layer. Note, all network kernels are assumed real-valued. Inspired by super-resolution ideas in [@johnson2016; @leding; @et; @al'16], and the network architecture in [@srez] we adopt a deep residual network for the generator with $8$ residual blocks. Each block consists of two convolutional layers with small $3 \times 3$ kernels and $64$ feature maps that are followed by batch normalization and ReLU activation. It then follows by three convolutional layers with map size $1 \times 1$, where the first two layers undergo ReLU activation, while the last layer has sigmoid activation to return the output. G network learns the projection onto the manifold while ensuring the data consistency at the same time, where the manifold dimension is controlled by the number of residual blocks and feature maps and the settings of discriminator D network.
To satisfy data consistency term, previous work in the context of image super-resolution [@Sonderby; @et; @al'14] used (hard) affine projection after the G network. However, the affine projection drifts $\hat{\bx}$ away from the manifold landscape. As argued in Section 3, we instead use a multilayer succession of affine projection and convolutional residual units that project back $\hat{\bx}$ onto the manifold. We can repeat this procedure a few times to ensure $\hat{\bx}$ lies close to the intersection. This amounts to a soft yet flexible data consistency penalty.
The D network starts from the output of the G network with two channels. It is composed of $8$ convolutional layers. In all the layers except the last one, the convolution is followed by batch normalization, and subsequently ReLU activation. No pooling is used. For the first four layers, number of feature maps is doubled from $8$ to $64$, while at the same time convolution with stride $2$ is used to reduce the image resolution. Kernel size $3 \times 3$ is adopted for the first 5 layers, while the last two layers use kernel size $1 \times 1$. In the last layer, the convolution output is averaged out to form the decision variable for binary classification. No soft-max is used.
Adam optimizer is used with the momentum parameter $\beta=0.9$, mini-batch size $L_b=8$, and initial learning rate $\mu=10^{-5}$ that is halved every $5,000$ iterations. Training is performed with TensorFlow interface on a NVIDIA Titan X Pascal GPU, 12GB RAM. We allow $20$ epochs that takes around $6$ hours for training. The implementation is available online at [@github-gancs-2017].
As a figure of merit for image quality assessment we adopt SNR (dB), and SSIM that is defined on a cropped window of size $50 \times 50$ from the center of axial slices. In addition, we asked Radiologists Opinion Score (ROS) regarding the diagnostic quality of images. ROS ranges from $1$ (worse) to $5$ (excellent) based on the overall images quality in terms of sharpness/blurriness, and appearance of residual artifacts.
Observations and discussion {#subsec:obs}
---------------------------
Retrieved images by various methods are depicted in Fig. \[fig:fig\_recon\_gancs\_vs\_cs\_mse\_4fold\] with $5$-fold undersampling of $k$-space. For a random test patient, representative slices from axial, and coronal orientations, respectively, are shown from top to bottom. Columns from left to right also show, respectively, the images reconstructed by zero-filling (ZF), CS-WV, CS-TV, $\ell_2$-net, $\ell_1$-net, GAN, GANCS with $\lambda=\eta=10$, and the gold-standard (GS). Note, we propose $\ell_1$-net and $\ell_2$-net using the same network structure and training as in Section \[subsec:training\], with only changing the G net cost function in (P1). CS reconstruction is performed using the Berkeley Advanced Reconstruction Toolbox (BART) [@bart2016], where the tunning parameters are optimized for the best performance. GANCS, $\ell_1$-net and $\ell_2$-net are trained with ZF images that apparently contain aliasing artifacts.
Quantitative metrics including the SNR (dB), SSIM, and the reconstruction time (sec) are also reported in Table I. These metrics are averaged out over all axial slices for test patients. As apparent from the magnified regions, GANCS returns the most detailed images with high contrast and texture details that can reveal the small structures. $\ell_2$-net images are seen somehow over-smoothed as the $\ell_2$ cost encourages finding pixel-wise averages of plausible solutions. Also, $\ell_1$-net performs better than $\ell_2$-net, which was also already reported in a different setting [@lossfunction_zhao2017], but still not as sharp as GANCS which leverages both $\ell_1$-net and GAN. GAN results with $\eta=0$ also introduces sharp images but noise is still present all over the image. CS-based results are also depicted as the benchmark MR reconstruction scheme nowadays, where evidently introduce blurring artifacts.
CS-based scheme achieve higher SNR and SSIM, but they miss the high frequency textures as evidenced by Fig. \[fig:fig\_recon\_gancs\_vs\_cs\_mse\_4fold\]. In addition, they demands iterative algorithms for solving non-smooth optimization programs that takes a few seconds for reconstruction using the optimized BART toolbox [@bart2016]. In contrast, the elapsed time for GANCS is only about $10$ msec, which allows reconstructing $100$ frames per second, and thus a suitable choice for real-time MRI visualization tasks. Regarding the convergence, we empirically observe faster and more stable training by imposing more weight on the data consistency which restricts the search space for the network weights.
To assess the perceptual quality of resulting images we also asked the opinion of expert radiologists. We normalize the scores so as the gold-standard images are rated excellent (i.e., ROS=$5$). Statistical ROS is evaluated for the image quality, residual artifacts, and image sharpness. It is shown in the bar plot of Fig. \[fig:fig\_barplot\_ros\], which confirms GANCS almost perceptually pleasing as the gold-standard scan. This demonstrates the superior diagnostic quality of GANCS images relative to the other alternatives.
For the sake of completeness, the evolution of different (empirical) costs associated with the generator cost in (P1.2) over batches are also depicted in Fig. \[fig:fig\_loss\]. It is observed that the data consistency cost and GAN loss tend to improve alternatively to find the distribution at the intersection of manifold and dats consistency space.
![Representative coronal (1st row) and axial (3rd row) images for a test patient retrieved by ZF (1st), CS-WV (2nd), $\ell_2$-net (3th), $\ell_1$-net (4th), GAN (5th), GANCS (6th), and gold-standard (7th). []{data-label="fig:fig_recon_gancs_vs_cs_mse_4fold"}](fig_comp_gancs_vs_cs_zoomed_7schemes.png)
[c]{}\
\[tab:table\_comp\_quantitative\]
[|c|c|c|c|c|c|c|c|]{} Scheme & ZF & CS-WV & CS-TV & $\ell_2$-net & $\ell_1$-net & GAN & GANCS\
SNR & $15.28$ & $20.74$ & $21.33$ & $18.96$ & $18.64$ & $16.6$ & $20.48$\
SSIM & $0.72$ & $0.88$ & $0.87$ & $0.81$ & $0.79$ & $0.78$ & $0.87$\
Recon. time & $5\hspace{-1mm} \times \hspace{-1mm} 10^{-4}$ & $5.27$ & $1.51$ & $0.02$ & $0.02$ & $0.02$ & $0.02$\
![Representaitve $k$-space axial image retrieved by ZF (1st column), CS-WV (2nd), CS-TV (3rd), and GANCS (4th), and gold-standard (5th). []{data-label="fig:fig_recon_kspace_gancs_vs_cs_mse_4fold"}](fig_recon_kspace_gancs_vs_cs_mse_4fold.png)
![Evolution of different costs contributing in the overall training cost of G network. []{data-label="fig:fig_loss"}](fig_loss_batch.png)
**Manifold landscape.** We visualize what the discriminator learns by showing the feature maps in different layers as heat-maps superimposed on the original images. Since there are several feature maps per layer, we computed the Principle Component maps for each layer and visualize the first $8$ dominant ones. Fig. indicates that after learning from tens of thousands of generated MRI images by the G network and their gold standards including different organs, is able to detect anatomically valuable features. It is observed that the first layers reveal the edges, while the last layers closer to the classification output reveal more regions of interests that include both anatomy and texture details. This observation is consistent with the way expert radiologist inspect the images based on their diagnosis quality.
![Heat-map of discriminator feature maps at four layers for four different images. Each 4 row from top to bottom represent the results from one MR image. The first row shows the MR image and the Principle Components of the network features from the first layer. The second row shows an overlay view of the MR image and the heat-map. The third row shows the MR slice image with the Principle Components of the network features from the last layer of discriminator; while the fourth row shows the overlay view of the MR image and the heat-maps.[]{data-label="fig:fig_heatmap"}](heatmap_vertical-min.png)
**Performance with different number of patients** We also experimented on the number of patients needed for training and achieving good reconstruction quality in the test phase. It is generally valuable for the clinicians how much training data is needed as in the medical applications, patient data is not easily accessible due to privacy concerns. Fig. \[fig:fig\_performance\] plots the normalized RMSE on a test set versus the percentage of patients used for training (normalized by the maximum patient number $350$). Note, the variance differences for different training may be due to the training with fewer samples has better convergence, since we are using the same epoch numbers for all the training cases. More detailed study is the subject of our ongoing research.
![Performance changes with different size of dataset used for training (output about 45,300 images) []{data-label="fig:fig_performance"}](Performance_vs_Datasize.png)
Conclusions and Future Work {#sec:conclusion}
===========================
This paper caters a novel CS framework that leverages the historical data for faster and more diagnosis-valuable image reconstruction from highly undersampled observations. A low-dimensional manifold is learned where the images are not only sharp and high contrast, but also consistent with both the real MRI data and the acquisition model. To this end, a neural network based on LSGANs is trained that consists of a generator network to map a readily obtainable undersmapled image to the gold-standard one. Experiments based on a large cohort of abdominal MR data, and the evaluations performed by expert radiologists confirm that the GANCS retrieves images with better diagnostic quality in a real-time manner (about $10$ msec, more than $100$ times faster than state-of-the-art CS MRI toolbox). This achieves a significant speed-up and diagnostic accuracy relative to standard CS MRI. Last but not least, the scope of the novel GANCS goes beyond the MR reconstruction, and tailors other image restoration tasks dealing with aliasing artifacts. There are still important question to address such as using 3D spatial correlations for improved quality imaging, robustifying against patients with abnormalities, and variations in the acquisition model for instance as a result of different sampling strategies.
[10]{}
\[online\] https://github.com/gongenhao/GANCS.html
\[online\] http://mridata.org/fullysampled/knees.html
\[online\] https://github.com/david-gpu/srez.html
\[online\] http://www.mriinterventions.com/clearpoint/clearpoint-overview.html
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[^1]: The authors are with the Stanford University, Departments of Electrical Engineering$^{1}$, Radiology$^{2}$, Radiation Oncology$^{3}$, and Computer Science$^{4}$.
|
---
abstract: 'The identification of influential nodes in complex network can be very challenging. If the network has a community structure, centrality measures may fail to identify the complete set of influential nodes, as the hubs and other central nodes of the network may lie inside only one community. Here we define a bipartite clustering coefficient that, by taking differently structured clusters into account, can find important nodes across communities.'
address: 'School of Mathematical and Geospatial Sciences, RMIT University, Melbourne 3001, Australia'
author:
- 'J. Liebig'
- 'A. Rao'
bibliography:
- 'bibliography.bib'
title: Identifying Influential Nodes in Bipartite Networks Using the Clustering Coefficient
---
Bipartite networks ,Clustering coefficient ,Influential nodes
Introduction {#sec:intro}
============
Locating important nodes in a network is often crucial as this could aid in terminating the spread of diseases or alternatively assist the spread of knowledge and information [@che12]. A number of centrality measures are currently used to identify important nodes, but as @kit10 point out, these measures may not reveal the truly important nodes. This is especially the case if the network has a community structure, where centrality measures may only reveal important nodes from one of the communities [@zha13]. This paper uses a very different approach, defining new clustering coefficients and using these to find influential nodes across communities.
Here we focus on bipartite networks. Many real world systems are best modelled as such. Examples are collaboration networks [@new01], roosting spatial networks [@for09] and board interlocks [@pie13]. Networks are bipartite if their nodes can be partitioned into two disjoint sets (primary and secondary) such that the edges only connect nodes from different sets [@asr98]. In a collaboration network, for instance, the authors are only connected to papers and form the primary set. The papers form the secondary set.
In order to analyse bipartite networks it is very common to study a one-mode mapping of the original bipartite network. This approach is called one-mode projection [@was94] or ‘conversion’ [@bor11]. The bipartite network is projected onto a one-mode network by dropping one of the two node sets and connecting two nodes in the one-mode network if they share a neighbour in the bipartite network. This popular approach is necessitated by the fact that many network measures cannot be directly applied to bipartite networks [@lat08]. However, the projection onto a one-mode network leads to loss of information [@con12; @vog10; @zho07] and more recently many measures have been specifically redefined to suit the analysis of bipartite networks [@bor12].
In this paper we work directly on the bipartite network, defining a clustering coefficient for the analysis. We show that clusters in bipartite networks may have different structures and that ignoring these leads to inaccurate results. We then use our measure to identify nodes that drive the clustering behaviour of the network and show that these are indeed influential nodes.
The rest of the paper is organised as follows: Section \[sec:preMethods\] discusses previous definitions of the bipartite clustering coefficient and identifies their limitations. Section \[sec:formation\] shows that differently structured bipartite clusters have distinct origins that depend on the way in which the network develops over time. In Section \[sec:improvedClustering\] we give a bipartite clustering coefficient that can be used to identify important nodes. The proposed method that finds important nodes is outlined in Section \[sec:importantNodes\]. Sections \[sec:women\] and \[sec:Noordin\] discuss results that are obtained from applying our clustering coefficient to real world data sets. The results give insight into how clusters are structured and reveal the nodes that drive the formation of clusters and their particular structure.
The Bipartite Clustering Coefficient {#sec:preMethods}
====================================
The clustering coefficient is one particular measure that, as originally defined, cannot be applied to bipartite networks. In a one-mode network, the clustering coefficient measures the concentration of triangles. It is an important measure in the analysis of networks, since it gives insight into how well the neighbourhood of a node is connected. As bipartite networks are triangle free, this measure cannot be directly applied to these networks. Several definitions of the bipartite clustering coefficient exist [@lin05; @ops13; @rob04; @zha08]. They are, however, inconsistent and hence require further investigation.
Most existing bipartite clustering coefficients measure the concentration of 4-cycles instead of triangles ([@lin05; @rob04; @zha08]). A triangle and a 4-cycle are the smallest possible cycles in a one-mode and bipartite network respectively.
In a bipartite network a 4-cycle shows that two primary nodes are connected twice via two secondary nodes. However, since the one-mode clustering coefficient measures closure between three nodes, @ops13 chooses to define the clustering coefficient for bipartite networks in terms of paths of length 4 and cycles of length 6:
$$\label{eqn:biClust}C^* = \frac{\textrm{closed 4-paths}}{\textrm{4-paths}} = \frac{\tau^*_{\Delta}}{\tau^*},$$
where $\tau^*$ is the number of 4-paths and $\tau^*_{\Delta}$ is the number of these 4-paths that are closed. A closed 4-path is equivalent to a cycle of length 6. We give an explanation in Section \[sec:formation\].
The local clustering coefficient of a node $v_i$ is given as follows:
$$\label{eqn:localBiClust}C^*(i) = \frac{\tau^*_{i,\Delta}}{\tau^*_i},$$
where $\tau^*_i$ is the number of 4-paths that are centred at node $v_i$ and $\tau^*_{i,\Delta}$ is the number of these 4-paths that are closed.
A path is defined as a sequence of unique nodes and edges. In other words, when traversing a path in a network, no node is revisited. The length of a path is equal to the number of its edges. A cycle is a path that starts and ends at the same node.
Since the bipartite clustering coefficient should measure closure between three nodes of the same type (as it does in one-mode networks) the idea of triadic closure is the obvious direction to follow. Figure \[im:clustering\] shows the two subgraphs that may be considered as a closed connection between three primary nodes. Primary nodes are represented by circles and secondary nodes are represented by squares. However, star subgraphs are the reason for the count of triangles in a projected one-mode network to be higher than expected [@ops13]. Hence Eq. , does not consider this structure as a closed connection between three primary nodes, only counting cycles of length 6.
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The structure of bipartite clusters {#sec:formation}
===================================
This section looks at how clusters in bipartite networks are structured.
A bipartite 6-cycle may be formed by connecting a secondary node to the two end nodes of a 4-path (see Fig. \[im:closed4path\]). Hence the terminology in Eq. and Eq. .
There is an important difference in the cluster formation in bipartite networks as opposed to the formation in one-mode networks. In a one-mode network, additional edges between the nodes of a triangle and between the nodes of a 2-path cannot exist without introducing multiple edges. However, in a bipartite network, the nodes of a 6-cycle as well as the nodes of a 4-path may be connected to each other by additional edges (see Fig. \[im:innerConnections\]). Additional edges in the bipartite subgraphs give rise to different structures with distinct meanings that depend on the network in question.
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It is possible to form different 6-cycles between the same six nodes by traversing different edges. The number of different 6-cycles between the same set of nodes depends on the number of edges that connect these six nodes to each other. A bipartite 6-cycle can have at most three additional edges (see Fig. \[im:innerConnections6cycle\]). We call the structures shown in Fig. \[im:noIC6cycle\], \[im:oneIC6cycle\], \[im:twoIC6cycle\] and \[im:threeIC6cycle\] an unconnected 6-cycle, a sparsely connected 6-cycle, a highly connected 6-cycle and a completely connected 6-cycle respectively.
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By traversing the different edges in the structures shown in Fig. \[im:innerConnections6cycle\], one can confirm that an unconnected 6-cycle contributes one to the overall count of 6-cycles. A sparsely connected 6-cycle contains a single 6-cycle and hence also contributes one to the overall count of 6-cycles. A highly connected 6-cycle contributes two to the overall count of 6-cycles and finally, a completely connected 6-cycle contributes six to the overall count of 6-cycles. As we look at undirected networks the direction of the cycle does not matter. It is also irrelevant at which of the six nodes the cycle starts.
This clearly shows that counting the number of 6-cycles and not distinguishing the structures shown in Fig. \[im:innerConnections6cycle\] leads to an over count of 6-cycles. Our clustering coefficient treats each of the different types of 6-cycles separately. Distinguishing between the different structures gives insight into how interconnected any set of three nodes of the same type can be.
A new improved bipartite clustering coefficient {#sec:improvedClustering}
===============================================
We now give a bipartite clustering coefficient that distinguishes between the 6-cycles identified in Section \[sec:formation\]. We develop four bipartite clustering coefficients, one for each type of 6-cycle, that provide new information about the network not revealed by previously defined clustering coefficients.
In order to calculate the clustering coefficient, we need to determine all possibilities by which the different types of 6-cycles may be formed. We wish to analyse a snapshot of a network at a particular point in time, however, this needs awareness of how the network was formed over time. But first we show why it is necessary to distinguish between different types of bipartite networks.
In some bipartite networks a primary node can only connect to a particular secondary node at a particular point in time. We call these networks time dependent networks. In a network that models the attendance of people at events, where each event takes place at a specific point in time, ties cannot be formed within an existing 6-cycle. For instance, assume there exists a 6-cycle that is part of a network of people and events (see Fig. \[im:exampleType1\]). Events 1, 2 and 3 take place at times $t_1, t_2$ and $t_3$ respectively. Figure \[im:exampleType1\] clearly shows that connections within 6-cycles cannot be formed at a later point in time, as all three events have passed.
In networks where nodes are allowed to connect to secondary nodes at any point in time, it is possible to form connections within an existing 6-cycle. A network of online forums and users, where the forums form the secondary node set, is an example of such a network. Each forum is accessible over a period of time and hence it is possible for connections to be formed within an existing 6-cycle. It is obvious that the origins of the 6-cycles shown in Fig. \[im:innerConnections6cycle\] are different in the two types of networks. Here, we only consider time dependent networks, as defined in the paragraph above.
Any 6-cycle in a time dependent bipartite network is formed by connecting a secondary node to the two primary end nodes of a 4-path. Figure \[im:formation6cycles\] shows all the possibilities by which the distinct 6-cycles may be formed. We call the 4-path in Fig. \[im:4-pathNoIC\] an unconnected 4-path, the 4-path in Fig. \[im:4-pathOneIC\] a connected 4-path and the 4-path in Fig. \[im:4-pathTwoIC\] a completely connected 4-path.
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Using the origins of 6-cycles, we define equations - to measure four different clustering coefficients $cc_{(k)}$ in a time dependent bipartite network.
The unconnected clustering coefficient:
$$\label{eq:cc0}
cc_{(0)} = \frac{\lambda_{(0)}^*}{\lambda_{(0)}},$$
where $\lambda^*_{(0)}$ is the number of closed 4-paths that form an unconnected 6-cycle and $\lambda_{(0)}$ is the total number of unconnected 4-paths. The unconnected clustering coefficient $cc_{(0)}$ measures the proportion of unconnected 4-paths that are closed and form an unconnected 6-cycle.
The sparsely connected clustering coefficient:
$$\label{eq:cc1}
cc_{(1)} = \frac{\lambda_{(1)}^* }{\lambda_{(0)} + \lambda_{(1)}},$$
where $\lambda^*_{(1)}$ is the number of closed 4-paths that form a sparsely connected 6-cycle and $\lambda_{(1)}$ is the total number of connected 4-paths. The sparsely connected clustering coefficient $cc_{(1)}$ measures the proportion of 4-paths that are closed and form a sparsely connected 6-cycle.
The highly connected clustering coefficient:
$$\label{eq:cc2}
cc_{(2)} = \frac{\lambda_{(2)}^* }{\lambda_{(1)} + \lambda_{(2)}},$$
where $\lambda^*_{(2)}$ is the number of closed 4-paths that form a highly connected 6-cycle and $\lambda_{(2)}$ is the total number of completely connected 4-paths. The highly connected clustering coefficient $cc_{(2)}$ measures the proportion of 4-paths that are closed and form a highly connected 6-cycle.
The completely connected clustering coefficient:
$$\label{eq:cc3}
cc_{(3)} = \frac{\lambda_{(3)}^*}{\lambda_{(2)}},$$
where $\lambda^*_{(3)}$ is the number of closed 4-paths that form a completely connected 6-cycle. The clustering coefficient $cc_{(3)}$ measures the proportion of 4-paths that are closed and form a completely connected 6-cycle.
The local clustering coefficients $cc_{(i,k)}$ of a node $v_i$ can be measured in a similar manner. For example, the local clustering coefficient $cc_{(i,0)}$ of the node $v_i$ is measured by dividing the number of closed 4-paths that are centred at $v_i$ and form an unconnected 6-cycle by the number of all unconnected 4-paths that are centred at $v_i$.
Identifying Important Nodes {#sec:importantNodes}
===========================
In order to find the driving nodes of a network, we calculate a score that indicates the extent to which a node is driving the clustering behaviour of the complete network, by first comparing the global clustering coefficients to the clustering coefficients of random networks and then comparing the local clustering coefficients of each node to the global clustering coefficients.
There are two cases:
1. $cc_{(k)} < CI_{(k)}$ or
2. $cc_{(k)} \geq CI_{(k)}$,
where $CI_{(k)}$ is the mid point of the confidence interval calculated for $cc_{(k)}$ in an ensemble of random networks. The random networks need to have the same size, density and degree distribution as the original network. At least 100 random networks have to be generated in order to to achieve a small enough confidence interval. To be able to compare the scores of different nodes, we first calculate a global driving score, denoted $ds_{(global)}$, for the network by measuring how far each of the global clustering coefficients $cc_{(k)}$ for the given network is from $CI_{(k)}$ and then averaging over all four scores, see Eq. . Note that $ds_{(global)}$ lies between 0 and 1. The greater the difference between the global clustering coefficients of the given network $cc_{(k)}$ and the respective $CI_{(k)}$, the higher the global driving score.
$$\label{gds}
ds_{(global)} = \frac{1}{4}\sum_{k=0}^{3} g(k),$$
where $$\label{gk}
g(k) = \left\{
\begin{array}{ll}
\frac{|CI_{(k)} - cc_{(k)}|}{CI_{(k)}} & \textrm{if}\quad cc_{(k)} < CI_{(k)}, \\
\frac{|CI_{(k)} - cc_{(k)}|}{1 - CI_{(k)}} & \textrm{if}\quad cc_{(k)} \geq CI_{(k)}.
\end{array}
\right.$$
A node with a local clustering coefficient $cc_{(i,k)}$ that is close to $CI_{(k)}$, behaves as expected and hence does not contribute to a clustering behaviour that is different to a random network. If the global clustering coefficient $cc_{(k)}$ of the given network is smaller than $CI_{(k)}$, i.e. $cc_{(k)} < CI_{(k)}$, then either
1. $cc_{(i,k)} < CI_{(k)}$ or
2. $cc_{(i,k)} \geq CI_{(k)}$.
If the local clustering coefficient $cc_{(i,k)}$ of node $i$ also lies below $CI_{(k)}$, then node $i$ contributes to the global clustering behaviour of the whole network and we assign a score between 0 and 1 to node $i$, depending on the difference between the local clustering coefficient and $CI_{(k)}$. If on the other hand, $cc_{(i,k)}$ lies above $CI_{(k)}$ then node $i$ drives against the clustering behaviour and we assign node $i$ a score between 0 and -1. Similarly, when the global clustering coefficient $cc_{(k)}$ lies above the mid point of the confidence interval, there are two cases.
The driving score, $ds_{(i)}$, of node $i$ is thus given by the following equation:
$$\label{ds}
ds_{(i)} = \frac{1}{4}\sum_{k=0}^{3} f(k),$$
where
$$\label{fk}
f(k) = \left\{
\begin{array}{ll}
\frac{|CI_{(k)} - cc_{(i,k)}|}{CI_{(k)}} & \textrm{if}\quad cc_{(k)} < CI_{(k)} > cc_{(i,k)}, \\
-\frac{|CI_{(k)} - cc_{(i,k)}|}{1 - CI_{(k)}} & \textrm{if}\quad cc_{(k)} < CI_{(k)} \leq cc_{(i,k)}, \\
\frac{|CI_{(k)} - cc_{(i,k)}|}{1 - CI_{(k)}} & \textrm{if}\quad cc_{(k)} \geq CI_{(k)} \leq cc_{(i,k)},\\
-\frac{|CI_{(k)} - cc_{(i,k)}|}{CI_{(k)}} & \textrm{if}\quad cc_{(k)} \geq CI_{(k)} > cc_{(i,k)}.
\end{array}
\right.$$
The four different structures, shown in Fig. \[im:innerConnections6cycle\], are considered to be equally important, hence the factor of $\frac{1}{4}$ in Eq. . The driving score depends solely on how the network under investigation compares to random networks.
In order to achieve a high driving score, a node does not necessarily have to have high clustering coefficients (see Fig. \[im:ds\]). For instance, if the global clustering coefficients are low, a node that also has low clustering coefficients, receives a high driving score.
In the following sections we apply the four clustering coefficients (Eq. - Eq. ) to different networks from distinct areas and analyse the obtained results. We further identify the important nodes in the different networks.
The Southern Women Network {#sec:women}
==========================
We first calculate the clustering coefficients of a popular, often analysed, data set that was collected by @dav41. This so called southern women network consists of 18 women and 14 events. An edge between a woman and an event only exists if the woman attended the event. This bipartite network is clearly a time dependent network as the events take place at a certain time only and cannot be attended afterwards.
Table \[resultsWomen\] shows the clustering coefficients of the southern women network and the coefficients of 100 randomly generated bipartite networks and their 95% confidence intervals.
Southern Women network random networks
------------ ------------------------ --------------------
$cc_{(0)}$ 0.4446 \[0.6261, 0.6478\]
$cc_{(1)}$ 0.6532 \[0.5483, 0.5658\]
$cc_{(2)}$ 0.5984 \[0.3972, 0.4237\]
$cc_{(3)}$ 0.5604 \[0.3018, 0.3457\]
: The four clustering coefficients of the southern women network and the average clustering coefficients of 100 randomly generated networks with their 95% confidence intervals. []{data-label="resultsWomen"}
We found that none of the clustering coefficients lie within the 95% confidence intervals and hence, none of the values are as expected in a random network. The coefficient $cc_{(0)}$ lies below the lower bound of the confidence interval whereas $cc_{(1)}, cc_{(2)}$ and $cc_{(3)}$ lie above the interval. In the average random network $cc_{(0)}$ has the highest value ($cc_{(0)} = 0.6226$), as opposed to the southern women network, where $cc_{(0)}$ takes the lowest value ($cc_{(0)}=0.4446$). Hence, a greater proportion of 4-paths are closed to form an unconnected 6-cycle in random networks than in the southern women network. The remaining three clustering coefficients lie above the 95% confidence interval, showing that the proportion of closed 4-paths with additional edges is much higher than expected. Intuitively, as the southern women network is a social network, one would assume that any of the 18 women would rather attend an event with friends than by herself. Our results confirm the assumption that three women tend to cluster if they are already connected to each other by at least one event.
woman $i$ $cc_{(i,0)}$ $cc_{(i,1)}$ $cc_{(i,2)}$ $cc_{(i,3)}$ $ds_{(i)}$
----------------- --------------------- --------------------- --------------------- --------------------- ------------
[**Evelyn**]{} 0.3957 $\downarrow$ 0.6986 $\uparrow$ 0.6732 $\uparrow$ 0.6545 $\uparrow$ 0.4083
[**Laura**]{} 0.4468 $\downarrow$ 0.6610 $\uparrow$ 0.7218 $\uparrow$ 0.7364 $\uparrow$ 0.4179
[**Theresa**]{} 0.0619 $\downarrow$ 0.7228 $\uparrow$ 0.7951 $\uparrow$ 0.6667 $\uparrow$ 0.6092
[**Brenda**]{} 0.3455 $\downarrow$ 0.656 $\uparrow$ 0.7241 $\uparrow$ 0.7565 $\uparrow$ 0.4633
Charlotte 1 $\uparrow$ 0.84 $\uparrow$ 0.6093 $\uparrow$ 0.6 $\uparrow$ 0.0962
Frances 0.6667 $\uparrow$ 0.684 $\uparrow$ 0.5164 $\uparrow$ 0.7742 $\uparrow$ 0.2626
[**Eleanor**]{} 0.5094 $\downarrow$ 0.662 $\uparrow$ 0.6302 $\uparrow$ 0.6234 $\uparrow$ 0.3133
Pearl 0.4074 $\downarrow$ 0.6931$\uparrow$ 0.4278 $=$ 0.0652 $\downarrow$ -0.0254
[**Ruth**]{} 0.2869 $\downarrow$ 0.697 $\uparrow$ 0.6254 $\uparrow$ 0.3704 $=$ 0.3248
Verne 0.3778 $\downarrow$ 0.613$\uparrow$ 0.6188 $\uparrow$ 0.3429 $=$ 0.2253
Myrna 0.6735 $\uparrow$ 0.5221 $\downarrow$ 0.504 $\uparrow$ 0.4615 $\uparrow$ 0.04978
Katherine 0.7260 $\uparrow$ 0.569 $\uparrow$ 0.5572 $\uparrow$ 0.5254 $\uparrow$ 0.0822
[**Sylvia**]{} 0.3395 $\downarrow$ 0.6694 $\uparrow$ 0.653 $\uparrow$ 0.5444 $\uparrow$ 0.3646
Nora 0.7185 $\uparrow$ 0.7555 $\uparrow$ 0.4021 $\downarrow$ 0.5238 $\uparrow$ 0.1247
Helen 0.7143 $\uparrow$ 0.6273 $\uparrow$ 0.4703 $\uparrow$ 0.375 $=$ 0.0308
Dorothy 0.4667 $\downarrow$ 0.4557 $\downarrow$ 0.163 $\downarrow$ 0 $\downarrow$ -0.3793
Olivia 1 $\uparrow$ 0.3103 $\downarrow$ 0 $\downarrow$ 0 $\downarrow$ -0.8607
Flora 1 $\uparrow$ 0.3103 $\downarrow$ 0 $\downarrow$ 0 $\downarrow$ -0.8607
: The local clustering coefficients of the 18 women and their driving scores. The 7 women who were identified to drive the clustering behaviour of the whole network are printed in bold.[]{data-label="localWomen"}
The global driving score of the southern women networks is $ds_{(global)} = 0.297$. The local clustering coefficients of the southern women network, together with the respective driving scores are displayed in Table \[localWomen\]. The arrows next to the entries in the table, indicate if the local clustering coefficient is higher or lower than the respective global clustering coefficient.
The driving scores of the women, reveal that Evelyn, Laura, Theresa, Brenda, Eleanor, Ruth and Sylvia drive the clustering behaviour of the whole network. A woman drives the clustering behaviour if her driving score lies above the global driving score of the network. All nodes with a negative score drive against the overall clustering behaviour.
We repeat the analysis for the secondary node set that represents the 14 events. Table \[resultsWomenPassive\] shows the clustering coefficients of the southern women network with respect to the events.
Southern Women network random networks
------------ ------------------------ --------------------
$cc_{(0)}$ 0.3578 \[0.7164, 0.7412\]
$cc_{(1)}$ 0.597 \[0.6272, 0.65\]
$cc_{(2)}$ 0.8556 \[0.4871, 0.5209\]
$cc_{(3)}$ 0.7903 \[0.422, 0.4757\]
: The four clustering coefficients of the southern women network with respect to the passive node set of events and the average clustering coefficients of 100 randomly generated networks with their 95% confidence interval.[]{data-label="resultsWomenPassive"}
Again, none of the four clustering coefficients lie within the 95% confidence interval of the randomly generated networks. The coefficients $cc_{(0)}$ and $cc_{(1)}$ lie below the lower bound of the respective confidence interval whereas the $cc_{(2)}$ and $cc_{(3)}$ lie above the interval.
Calculation of the driving scores of the 14 events, displayed in Table \[localWomenPassive\], shows that events 3, 5, 6 and 8 drive the clustering behaviour of the network.
event $i$ $cc_{(i,0)}$ $cc_{(i,1)}$ $cc_{(i,2)}$ $cc_{(i,3)}$ $ds_{(i)}$
----------- --------------------- --------------------- --------------------- --------------------- ------------
1 1 $\uparrow$ 0.9556 $\uparrow$ 0.7714 $\downarrow$ 0.6 $\downarrow$ -0.2659
2 0.8 $\uparrow$ 0.9574 $\uparrow$ 0.8571 $\uparrow$ 0.5143 $\downarrow$ -0.0785
[**3**]{} 0.3043 $\downarrow$ 0.7113 $\uparrow$ 0.9727 $\uparrow$ 0.8824 $\uparrow$ 0.5281
4 0.9 $\uparrow$ 0.9529 $\uparrow$ 0.8803 $\uparrow$ 0.6427 $\uparrow$ -0.0976
[**5**]{} 0.2545 $\downarrow$ 0.7952 $\uparrow$ 0.9895 $\uparrow$ 0.9029 $\uparrow$ 0.505
[**6**]{} 0.3421 $\downarrow$ 0.5482 $\downarrow$ 0.8913 $\uparrow$ 0.8791 $\uparrow$ 0.5584
7 0.3195 $\downarrow$ 0.6965 $\uparrow$ 0.8165 $\uparrow$ 0.7051 $\uparrow$ 0.374
[**8**]{} 0.38 $\downarrow$ 0.5918 $\downarrow$ 0.9429 $\uparrow$ 0.8672 $\uparrow$ 0.5489
9 0.3062 $\downarrow$ 0.6823 $\uparrow$ 0.7968 $\uparrow$ 0.6923 $\uparrow$ 0.3727
10 0.48 $\downarrow$ 0.7023$\uparrow$ 0.7891 $\uparrow$ 0.8049 $\uparrow$ 0.3465
11 1 $\uparrow$ 0.7949 $\uparrow$ 0.1 $\downarrow$ 0 $\downarrow$ -0.8085
12 0.3889 $\downarrow$ 0.7348 $\uparrow$ 0.8187 $\uparrow$ 0.875 $\uparrow$ 0.4019
13 1 $\uparrow$ 0.6098 $\downarrow$ 0.5323 $\uparrow$ 0.6923 $\uparrow$ -0.114
14 1 $\uparrow$ 0.6098 $\downarrow$ 0.5323 $\uparrow$ 0.6923 $\uparrow$ -0.114
: The local clustering coefficients of the 14 events and their driving scores. The events that were identified to drive the clustering behaviour of the whole network are printed in bold. The global driving score with respect to the events equals 0.4756.[]{data-label="localWomenPassive"}
Discussion {#subsec:discussion}
----------
The first analysis of the southern women dataset was carried out by Davis, Gardner and Gardner [@dav41] in the form of interviews, with the aim to categorise the 18 women into groups. They found two different groups that were further divided into core, primary and secondary members. Figure \[core\] shows the southern women network with the two groups identified in [@dav41]. Our analysis found that all the core women of the first group are influential as well as one core woman of the second group. Interestingly, our results show that Eleanor and Ruth should also be considered as important. Both attended only four events, however, these events were also attended by members from both groups. This observation indicates that Eleanor and Ruth are an important connection between the two groups. Davis, Gardner and Gardner also found that Ruth had some affiliation with both groups. Clearly our clustering coefficient identifies important nodes across the communities that were identified by Davis, Gardner and Gardner. Our analysis shows that the importance of a woman does not depend on her degree. For instance, if Ruth, who has a low degree, is removed from the network, information would spread less easily between the two groups.
![This figure shows the southern women network. The two groups and their core, primary and secondary members that were identified by Davis, Gardner and Gardner [@dav41] are labelled. The darker shaded nodes are driving the clustering behaviour of the network, as identified by our analysis. The size of the nodes corresponds to their degrees.[]{data-label="core"}](swn){width="80.00000%"}
Dorothy, Olivia, Flora and Pearl received negative driving scores. This result is consistent with the results presented in [@bon78], [@dor79] and [@eve93] which found that Dorothy, Olivia, Flora and Pearl were not associated with any of the groups.
The events that our analysis identified seem to be important not only because they have a high degree, but also because they were attended by women from both groups and hence act as a connection between the two communities.
The Noordin Top Terrorist Network {#sec:Noordin}
=================================
We now investigate a subset of the Noordin Top Terrorist network [@eve12]. This particular subset models the attendance of 26 members of the terrorist network at 20 different meetings. The subset contains a total of 64 connections between members and meetings. Table \[resultsNoordin\] shows the clustering coefficients of the members of the terrorist ring.
Noordin Top Terrorist Network random network
------------ ------------------------------- --------------------
$cc_{(0)}$ 0.0303 \[0.1768, 0.1973\]
$cc_{(1)}$ 0.1108 \[0.0542, 0.0676\]
$cc_{(2)}$ 0.2 \[0.0199, 0.0376\]
$cc_{(3)}$ 0 \[0, 0.0148\]
: The four clustering coefficients of the terrorist network and the average clustering coefficients of 100 randomly generated networks with their 95% confidence interval with respect to the members of the terrorist ring. []{data-label="resultsNoordin"}
The clustering coefficient $cc_{(0)}$ of the members lies below the confidence interval, whereas $cc_{(1)}$ and $cc _{(2)}$ lie above the interval. The clustering coefficient $cc_{(3)}$, however, lies within the 95% confidence interval.
In the terrorist network, the proportion of 4-paths that are closed and form a highly connected 6-cycle is much higher than in a random network ($cc_{(2)} = 0.2$). As in the southern women network, it seems that three members of the terrorist ring would cluster if they were already connected through at least one previous meeting. The results from the southern women network can be explained by the underlying friendship network. In case of the terrorist network, it is rather unlikely that the members decided which meetings to attend, based on friendships to other members. Examination of the local clustering coefficients reveal the important members (see Table \[localNoordin\]).
member $i$ $cc_{(i,0)}$ $cc_{(i,1)}$ $cc_{(i,2)}$ $cc_{(i,3)}$ $ds_{(i)}$
------------------------------ --------------------- ------------------- ------------------- -------------- ------------
Abdullah Sunata n/a n/a n/a n/a n/a
Abu Dujanah n/a 0 $\downarrow$ 0.1667 $\uparrow$ 0$=$ 0.0473
Abu Fida 0.1667 $\downarrow$ 0.1333 $\uparrow$ 0 $\downarrow$ n/a -0.2713
Adung n/a n/a n/a n/a n/a
[**Ahmad Rofiq Ridho**]{} 0.0408 $\downarrow$ 0.1818 $\uparrow$ 0.2414 $\uparrow$ 0$=$ 0.5324
Akram n/a n/a n/a n/a n/a
Asep Jaja n/a n/a n/a n/a n/a
[**Azhari Husin**]{} 0 $\downarrow$ 0.0842 $\uparrow$ 0.2857 $\uparrow$ 0 $=$ 0.5723
Cholily n/a n/a n/a n/a n/a
Heri Sigu Samboja n/a n/a n/a n/a n/a
Imam Bukhori n/a n/a n/a n/a n/a
Ismail n/a n/a n/a n/a n/a
Iwan Dharmawan 0.1429 $\downarrow$ 0.2609 $\uparrow$ 0 $\downarrow$ n/a -0.1836
Jabir n/a n/a n/a n/a n/a
Joko Triharmanto n/a n/a n/a n/a n/a
Misno n/a n/a n/a n/a n/a
Mohamed Saifuddin 0 $\downarrow$ 0 $\downarrow$ n/a n/a 0
[**Noordin Mohammed Top**]{} 0.0141 $\downarrow$ 0.124 $\uparrow$ 0.2079 $\uparrow$ 0$=$ 0.5442
Purnama Putra 0.1429 $\downarrow$ 0.3333 $\uparrow$ 0.3333 $\uparrow$ 0$=$ 0.46
Qotadah n/a 0 $\downarrow$ 0.1667 $\uparrow$ 0$=$ 0.0473
Saptono n/a n/a n/a n/a n/a
Son Hadi 0.1667 $\downarrow$ 0 $\downarrow$ n/a n/a -0.4454
Suramto n/a n/a n/a n/a n/a
Ubeid n/a n/a n/a n/a n/a
Urwah 0.2 $=$ 0.3333 $\uparrow$ 0 $\downarrow$ n/a -0.2419
Usman bin Sef 0 $\downarrow$ 0 $\downarrow$ n/a n/a 0
: This table shows the local clustering coefficients of the 26 members of the Noordin terrorist network. The four members that were identified to drive the clustering behaviour of the whole network are printed in bold. The global driving score with respect to the members equals 0.2692. []{data-label="localNoordin"}
The driving scores clearly show the influential nodes who are driving the clustering behaviour of the network are Ahmad Rofiq Ridho, Azhari Husin and Noordin Mohammed Top. Noordin Mohammed Top and Azhari Husin worked together to plan the terrorist attacks, with Noordin Mohammed Top financing the attacks and Azhari Husin being in charge of building the bombs [@bbc14]. Ahmad Rofiq Ridho was acting as a communicator between the members [@eve12]. Purnama Putra also received a very high driving score and was taking the role of a communicator similar to that Ahmad Rofiq Ridho.
The driving scores of the secondary node set revealed that meetings 16 and 18 are driving the clustering behaviour. Unfortunately, we do not have enough information about the meetings or the terrorist ring to explain our results.
![This figure shows the Noordin Top terrorist network. The darker shaded nodes are driving the clustering behaviour of the network, as identified by our analysis. The size of the nodes corresponds to their degrees.[]{data-label="im:noordin"}](ntn){width="80.00000%"}
Figure \[im:noordin\] shows the terrorist network. The influential nodes have a darker shading. Even though Noordin Top is one of the most influential members in the network and also has the highest degree, in general the importance of a member does not depend on its degree.
Conclusion and Future Work {#sec:conclusion}
==========================
Many real world networks are bipartite. However, not every network measure can be directly applied to this type of network [@lat08]. In order to analyse bipartite networks, one can either project the network to a one-mode network or redefine those networks measure that are not suitable for the analysis of bipartite networks.
A measure that has received much recent interest is the clustering coefficient. However, the existing methods do not consider the different structures that a bipartite cluster may have.
This paper showed that it is important to distinguish between different types of 6-cycles that are identified by the number of additional edges that connect nodes within the cycle. Ignoring additional edges results in an over-count of 6-cycles. We showed that the formation of the different types of 6-cycles depends on the network. For instance, in a network where primary nodes may connect to secondary nodes at any point in time, a sparsely connected 6-cycle can originate from an unconnected 6-cycle. This is not possible in a time dependent network where any 6-cycle can only originate from a 4-path. We defined four clustering coefficients that correspond to the different types of 6-cycles.
Applying the four clustering coefficients to real world networks gives valuable insight into how clusters are structured and how they form. The driving scores of the individual nodes in a network revealed those nodes that are driving the clustering behaviour and have influence on the network structure. Previous analyses supports the results we obtained in Section \[sec:women\].
Here, we tested our approach on relatively small networks. The next step would be to test the performance of our method on large-scale bipartite networks.
Considering time stamps of a network, if they are available, could give further insight into the network of interest and make a more dynamic analysis of the network possible. The clustering coefficients that were introduced in this paper can only be applied to time dependent bipartite networks. Further work needs to be done on other types of networks in which 6-cycles are formed differently. We will also focus future work on finding whole communities in bipartite networks by applying the clustering coefficients proposed in this paper.
|
---
abstract: |
Nowadays, we are surrounded by a large number of complex phenomena such as virus epidemic, rumor spreading, social norms formation, emergence of new technologies, rise of new economic trends and disruption of traditional businesses. To deal with such phenomena, social scientists often apply reductionism approach where they reduce such phenomena to some lower-lever variables and model the relationships among them through a scheme of equations (e.g. Partial differential equations and ordinary differential equations). This reductionism approach which is often called equation based modeling (EBM) has some fundamental weaknesses in dealing with real –world complex systems, for example in modeling how a housing bubble arises from a housing market, the whole market is reduced into some factors (i.e. economic agents) with unbounded rationality and often perfect information, and the model built from the relationships among such factors is used to explain the housing bubble while adaptability and the evolutionary nature of all engaged economic agents along with network effects go unaddressed. In tackling deficiencies of reductionism approach, in the past two decades, the Complex Adaptive System (CAS) framework has been found very influential. In contrast to reductionism approach, under this framework, the socio-economic phenomena such as housing bubbles are studied in an organic manner where the economic agents are supposed to be both boundedly rational and adaptive. According to CAS framework, the socio-economic aggregates such as housing bubbles emerge out of the ways agents of a socio-economic system interact and decide. As the most powerful methodology of CAS modeling, Agent-based modeling (ABM) has gained a growing application among academicians and practitioners. ABMs show how simple behavioral rules of agents and local interactions among them at micro-scale can generate surprisingly complex patterns at macro-scale. Despite a growing number of ABM publications, those researchers unfamiliar with this methodology have to study a number of works to understand (1) the why and what of ABMs and (2) the ways they are rigorously developed. Therefore, the major focus of this paper is to help social sciences researchers get a big picture of ABMs and know how to develop them both systematically and rigorously.\
Keywords:\
Complexity, Reductionism, Equation-based modeling (EBM), Complex adaptive system (CAS), Agent-based Modeling (ABM).
author:
- |
Hossein Sabzian\
[^1]\
- |
Mohammad Ali Shafia\
[^2]\
- |
Ali Maleki\
[^3]
- |
Seyeed Mostapha Seyeed Hashemi\
[^4]
- |
Ali Baghaei\
[^5]
- |
Hossein Gharib\
[^6]
bibliography:
- 'refs.bib'
title: '**Theories and Practice of Agent based Modeling: Some practical Implications for Economic Planners** '
---
Introduction {#section.Intro}
============
A large number of social phenomena such as cultural changes, cooperation formation, innovation, norm formation, technology diffusion, and even evolution of states happen not just due to separate choices by constituent individuals but mainly because of dynamic interactions among them over time. As a matter of fact, such phenomena have a nature entirely different from their constituents. Modeling the formation of these collective phenomena has been a great target for mainstream socio-economic modeling approach but it has not captured it sufficiently. This mainstream modeling approach which often called equation-based modeling (EBM) has been frequently used in different disciplines of social sciences. However, EBMs lack a needed functionality in explaining how the interactions among micro-components of a system can lead to an interestingly different macro-behavior for that system. In fact, they perform very poorly in modeling the emergent properties of real-life systems, namely how a whole arises from the interactions among its simpler and lower-level parts so that it exhibits properties that its simpler and lower-level parts can never exhibit. For tackling such a limitation, the agent based models (ABMs) have been developed. An ABM is a kind of computational model which explores systems of multiple interacting agents which are spatially situated and evolve over the time. ABMs are highly effective in explaining how complex patterns emerge from micro-level rules during a period of time. In contrast to EBMs that are based on deductive reasoning, ABMs properly work not only as an inductive reasoning technique where a conclusion is formed from a series of observations but also as a pure form of abductive reasoning where the best explanation for the phenomena under study is inferred via simulation. ABMs have become a major modeling trend in a large number of domains ranging from spread of epidemics[@Situngkir2004] and the threat of bio-warfare[@Caplat2008] to formation of norms[@Axelrod1986], supply chain optimization[@VanDykeParunak1998; @Jetly2014; @Swaminathan1998] and collaboration in project teams[@Son2010]. In contrast to EBMs which majorly focus on relationship among macro-variables of a system in top-down manner, ABMs try to model how local and predictable interactions among micro-components of a system can generate a complex system-level behavior[@Macy2002]. ABM methodology is rooted in complexity theory and network science. In terms of complexity theory, ABMs are developed to explain how simple rules generate complex emergence (i.e. a process model) and in terms of network science ABMs are used to analyze the pattern that arise from agents’ interactions over the time (i.e. a pattern model)[@wilensky2015introduction].In this paper, we want to explore ABMs systematically and show their great potentiality for modeling a large number of real world problems () that contemporary methods cannot model properly.\
The rest of this paper is organized as follows: the deals with why and what of ABMs.The unique characteristics of ABMs in comparison to EBMs are discussed in . is concerned with main uses of ABMs.ABM building blocks are discussed in . ABM development process is unraveled in . Some critical considerations are offered in .Two economics-related applications of ABMs are presented in and a conclusion is provided .
Why and What of ABM {#section.whyandwhat}
===================
We are living in complex world which itself includes an unlimited number of complexities ranging from highly micro-level complexities such as interacting atoms to highly macro-level ones such as nations. With an eye to socio-economic organizations like banks, insurance companies, hospitals and automobile producers, it becomes clear that all of these organizations are in turn a type of complex system so that each of them owns a distinguished whole (or ensemble) beyond its constituent parts (or components). Complex systems should be considered different from complicated systems. Actually, a complex system includes multiple interacting components forming a whole irreducible to its parts, therefore, it doesn’t lend itself to divide-and-conquer logic while a complicated system is composed of multiple related components forming a while reducible to its part and can be understood by divide-and-conquer logic. When a complex system is studied, the uncertainty of its outcomes never decreases to zero but as soon as a complicated system is analyzed and understood, the certainty of its outcomes increases to a large degree[@Snyder2013].
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One example for illustrating the difference between a complex system and a complicated one is Figure\[Engine\] and Figure\[Team\]. A car engine is assembled by several number of parts. When it is well understood by a team of experts, it can be decomposed and integrated over and over again without losing any of its expected functionalities.A car engine is assembled by several number of parts. When it is well understood by a team of experts, it can be decomposed and integrated over and over again without losing any of its expected functionalities. In contrast, a team including a number of interacting persons can show a surprisingly unexpected performance even the experts disarrange it from its initial conditions and rearrange it completely the same as its prior initial conditions.[^7]Complexity theory (CT) is an interdisciplinary filed of studying complex systems ranging from biophysical complex systems such as molecules and organs to socio-economic complex systems such as small firms and multi-national corporations. According to CT, complex systems which absorb information from surrounding environment and accumulate knowledge that can help action are usually called complex adaptive systems (CASs). A CAS represents the notion of a system where “The whole is more than the parts”. Actually, these are systems where multiple and perhaps very simple parts interact in a nonlinear and non-trivial manner to give rise to global often unpredictable behaviors observable and discoverable at a higher level of abstraction[@Holland2002]. A number of fundamental characteristics of CASs are listed in Table \[table1\].\
In the domain of CASs modeling methodologies[^8] , ABMs as a micro-scale computational models[^9] have shown a much better performance than equation-based models (EBMs) such as analytical models and statistical modeling methods[@EliotR.Smith2007; @niazi2017towards; @Sun2005; @VanDykeParunak1998; @wilensky2015introduction]. Developed from the fields of complexity, cybernetics, cellular automata and computer science, ABMs have gained lots of popularity in the 1990s and shows a growing migration not only from equation based models (EBMs) such as econometric models, analytical models and statistical modeling techniques but also from the more classical simulation approaches such as the discrete-event simulation[@Heath2009; @Siebers2008; @Siebers2010]. ABMs have a wide range of application domains ranging from biological systems[@Caplat2008; @Situngkir2004] to engineered ones [@Olfati-Saber2004]. The primary reason widely held by ABM practitioners is its very high strength in modeling complex adaptive systems (CAS) in comparison with other modeling methods.
[| p[.25]{} | p[.75]{} |]{}
\
Characteristics & Descriptions\
Multiplicity and heterogeneity of constituent components & It is composed of a number of components usually called” agent” [@wilensky2015introduction; @Holland2002].These agents can be very heterogeneous.\
Non-linear interactions &Its agents interact with each other in a non-linear (non-additive) way [@wilensky2015introduction; @Holland2002].\
Learnability and adaptability & Its agents can adapt or learn[@Holland2002] so agents can experience and accumulate knowledge.\
Non-ergodicity &It is non-ergodic[@Kauffman2000; @Moss2008].Therefore, it is highly sensitive to initial conditions.\
Self-organization & It self-organizes and its control is intensely distributed among its agents[@Chan2001; @wilensky2015introduction].\
Emergence & It exhibits emergence [@wilensky2015introduction; @Holland2002; @Chan2001]. It means, from the interactions of individual agents arises a global pattern or an aggregate behavior which is characteristically novel and irreducible to behavior(s) of agent(s).\
Co-evolution &Its agents can co-evolve and change the system’s behavior gradually[@Kauffman2000].\
Far from equilibrium &It shows “far from equilibrium” phenomenon[@nicolis1989exploring]. Isolated systems have a high tendency towards equilibrium and this will cause them to die. The “far from equilibrium” phenomenon shows how systems that are forced to explore possibilities space will create different structures and novel patterns of relationships[@Chan2001; @wilensky2015introduction].\
Time asymmetry and irreversibility & It is time asymmetric and irreversible. One characteristic of a CAS is time asymmetry. Asymmetry in time occurs when a system passes a bifurcation point, a pivotal or decisional point where an option is taken over another or others, leading to time irreversibility. Irreversibility means that the system cannot be run backwards— rewound or reversed—so as to reach its exact initial conditions. Systems which, when run in reverse, do not necessarily or typically return to their original state are said to be asymmetric in time[@Prigogine1997], and asymmetry in time is important in testing for a complex adaptive system. If system-time is symmetric in both directions, then it is reversible, and it is not a CAS but a deterministic system. Complex adaptive systems are asymmetric in time, irreversible and nondeterministic. So, in a CAS one can neither predict nor “retrodict,” even with infinite information on initial conditions, because the system “chooses” its forward path. Its “choice” is indeterminate, a function of statistical probability rather than certainty[@Rogers2005].\
Distributed control & The behavior of a CAS is not controlled by a centralized mechanism, rather, it is completely distributed among its constituent parts. The interactions of these constituent parts cause a CAS to exhibit a coherent macro-level behavior[@Chan2001].\
In thermodynamics, systems that does not have any exchange of energy and matter with their surrounding environment are called “isolated systems”. Such systems have a tendency to evolve towards equilibrium. But, our surrounding is enriched by phenomena arising from conflations far from equilibrium. Some examples can be turbulences, fractals and even life itself [@Jaeger2010].\
The philosophy of agent-based modeling comes directly from the idea that a CAS can be effectively modeled and explained by creating agents and environment, characterizing behavioral rules of agents, and specifying interactions among them [@wilensky2015introduction]. Modeling a CAS needs a specific type of methodology. EBMs such as statistical modeling techniques or PDEs lack a needed functionality for this purpose because they just decompose a system into its main parts and model the relationship among them (a top-down approach) while neglecting the fact that the system itself is entity beyond its constituent parts and it needs to be analyzed as an emergence of its constituent parts (a bottom-up approach).
Unique characteristics of ABMs in comparison with EBMs {#section.unique}
======================================================
EBM and ABM have stemmed from two distinct epistemological frameworks. The former is grounded on reductionism approaches such as neoclassical economic theories (NET) where the issues such as unbounded rationality, perfect information, deductive reasoning and low-rate heterogeneity are discussed while the latter is built upon complexity theory (CT)where the issues such as bounded rationality, information asymmetry, network interaction, emergence and inductive reasoning are taken into consideration [@Al-suwailem2008]. This has made ABMs specifically advantaged in modeling CASs. Some of these advantages can be summarized as Table\[table2\]
Advantage Description
----------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Bounded rationality The environment in which agents interact is highly complex and unbounded rationality is not a viable assumption for it [@Al-suwailem2008; @wilensky2015introduction], agents have limited possibilities not only for receiving information but also for its processing. AB modelers contend that socio-economic systems have an inherently non-stationary nature, due to continuous novelty (e.g., new patterns of aggregated behavior) endogenously introduced by the agents themselves[@Windrum2007]. Therefore, it is extremely difficult for agents to learn and adapt in such a turbulent and endogenously changing environments. On this basis, ABM researchers argue that assumption of unbounded rationality is an unsuitable for modelling real world systems and agents should not only have bounded rationality but also adapt their expectations in different periods of time.
Exhibition of emergence Since ABMs can model how micro-dynamics result in a high-level macro-dynamic they can be used as the best method for exhibiting emergent properties. On this basis, ABM does not require knowledge of the aggregate phenomena, in fact, researchers do not need to know what global pattern results from the individual behavior. When modeling an outcome variable with EBM, you need to have a good understanding of the aggregate behavior and then test out your hypothesis against the aggregate output [@wilensky2015introduction].
Bottom-up perspective A macro-system is an outcome of the way its sub-systems interact so the properties of macro-dynamics can only be properly understood as the outcome of micro-dynamics involving basic entities/ agents[@Tesfatsion2002]. This contrasts with the top-down nature of traditional neoclassical models, where the bottom level typically comprises a representative individual and is constrained by strong consistency requirements associated with equilibrium and unbounded rationality[@EliotR.Smith2007; @Macy2002]. Conversely, AB models describe strongly heterogeneous agents living in complex systems that evolve through time [@Kirman1997; @Kirman2005]. Therefore, aggregate properties are interpreted as emerging out of repeated interactions among simple entities rather than from the consistency requirements of rationality and equilibrium imposed by the modeler [@Dosi1994].
Heterogeneity and discrete nature An ABM can nicely model a heterogeneous population, whereas equational models typically must make assumptions of homogeneity. In many models, most notably in social science models, heterogeneity plays a key role. Furthermore, when you model individuals, the interactions and results are typically discrete and not continuous. Continuous models do not always map well onto real-world situations [@wilensky2015introduction].
Networked interactions: Interactions among economic agents in AB models are direct and inherently non-linear [@Fagiolo1998; @Silverberg1988] . Agents interact directly because current decisions directly depend, through adaptive expectations, on the past choices made by other agents in the population (i.e. a widespread presence of externalities). These may contain structures, such as subgroups of agents or local networks. In such structures, members of the population are in some sense closer to certain individuals in the socio-economic space than others. These interaction structures may themselves endogenously change over time, since agents can strategically decide with whom to interact according to the expected payoffs. When combined with heterogeneity and bounded rationality, it is likely that aggregation processes are non-trivial and, sometimes, generate the emergence of structurally new objects [@Lane1993a; @Lane1993].
Comprehensiveness:e It exhibits emergence [@wilensky2015introduction; @Holland2002; @Chan2001]. Results generated by ABMs are more detailed than those generated by EBMs. ABMs can provide both individual and aggregate level detail at the same time. Since ABMs operate by modeling each individual and their decisions, it is possible to examine the history and life of any one individual in the model, or aggregate individuals and observe the overall results. This “bottom-up” approach of ABMs is often in contrast with the “top-down” approach of many EBMs, which tell you only how the aggregate system is behaving and do not tell you anything about individuals. Many EBMs assume that one aspect of the model directly influences, or causes, another aspect of the model, while ABMs allow indirect causation via emergence to have a larger effect on the model outcomes [@wilensky2015introduction].
Randomness and indeterminacy One important feature of agent-based modeling, and of computational modeling in general, is that it is easy to incorporate randomness into your models. Many equation-based models and other modeling forms require that each decision in the model be made deterministically. In agent-based models this is not the case; instead, the decisions can be made based on a probability [@Siebers2008; @wilensky2015introduction].
: Major advantages of ABM over EBM[]{data-label="table2"}
With regard to Table \[table2\], it makes sense that social structures such as teams, organizations, governments and nations or even galaxial systems are few examples of CASs each of which can exhibits a number of emergent properties. For instance, organizations are a type of CAS out of which phenomena such as cooperation, aggregation of core competencies or even the ways employees interactively reinforce or weaken organizational routines emerge[@Wall2016]. In a wider economic system, macro-level phenomena such as inflation, stagflation, stock markets dynamics and economic inequality are aggregates (complex problems) emerging out of the economic systems. In recent years, the literature about complexity economics has been developed in so many areas including evolutionary models inspired by Nelson and Winter [@Nelson1982]and Hodgson [@Hodgson1998], Brock and Durlauf’s study of social interaction[@Brock2001] , study of firm size by Axtell[@Axtell2001], Alan Kirman and his colleagues models of financial markets[@Kirman2005] and the agent-based simulation of general equilibrium[@Gintis2006a; @Gintis2006].[^10]\
However, regarding complex nature of real world, it goes clear that EBMs (such as constrained optimization models used in econometrics) cannot capture the behavior of complex adaptive systems. This is an essential departure from the presumptions existing in conventional economic theories. Such systems should be analyzed ‘in’ time and this limits the way that mathematics can be used. Standard economic theory includes the applying of an ahistorical body of logical clauses to display attitudes perceived in the historical domain. In opposition, complex adaptive system theory copes directly with the fundamental principles that rule the behavior of systems in history. Therefore, it can be said that thinking about the economy and its sub-components as complex adaptive systems can allow us to evade such scientific impasses .In economic thought, Schumpeter’s contributions toward the process of “creative destruction” conform to complex adaptive systems theory[@Foster2001]. However, the core idea of Agent-Based Modeling is rooted in the fact that a CAS can be productively modeled with agents, an environment, and the rules of interactions among them. An agent is an autonomous entity with particular properties, behaviors, and even goals. The environment is a landscape over which agents have interactions and can be spatial, network-based, or mixture of them. The interactions can be non-linear and quite complex. Agents can have interaction with other agents or with the environment and they can not only change their interaction rules but also can change the strategies used to decide what behavior to do at a particular time [@wilensky2015introduction]. So, ABMs can be considered as a revolutionary methodology for modeling and simulating systems (i.e. real-world CASs) that are tremendously difficult and often impossible to be studied by EBMs[@Bankes2002].
Main uses of ABMs {#section.main}
=================
ABMs can be used in description and explanation. Like all models, an AMB is a simplification of a real world system which doesn’t entail all of its aspects so it is distinguishable from real world system and can help its understanding. The exploratory nature of ABM indicates that they can be used to pinpoint the essential mechanisms underlying the phenomena under study. a subject matter expert can use an AMB as a proof that his or her hypothesized mechanisms sufficiently account for the aggregate behavior under study [@wilensky2015introduction]. Explanation is strongly believed to be a major function of ABMs because it helps understand how simple rules generate complex structures. ABMs’ explanatory power is highly generative, especially in social sciences due to the fact that it explains which macro-structures such as epidemic dynamics or social evolutions emerge in population of heterogeneous agents that interact locally and in non-trivial way under a set of tenable behavioral rules [@Epstein2008].\
ABMs facilitate the experimentation process [@Leal2017]. They can be run repeatedly to discern variations in their dynamics and in their outputs [@wilensky2015introduction]. Some models show a very little variations during several runs. Some have a path-dependency nature [@Brown2005] and some exhibit tremendous variations from run to run. Through experimentation, system modelers get informed of how input parameters affect model’s outputs. Therefore, they can make various scenarios for achieving the targeted behavior.\
ABMs are sometimes used for prediction purposes. Subject matter experts frequently use models to get a picture about possible future states. Like every model the quality of ABMs’ prediction relies on the accuracy of its input parameters and since society is a complex system with an unspecified degree of uncertainty and very high sensitivity to small-scale events, no prediction can be deemed as absolutely right [@Moss2008; @wilensky2015introduction]. Prediction differs from description where the modeler describes the past or present states of the system, for example when a modeler describes what changes first occurred in the system. Moreover, prediction is also district from explanation, for example Plate tectonics definitely explains earthquakes, but does not help us to predict the time and place of their occurrence or evolution is commonly accepted as explaining speciation, but it is impossible to predict next year’s flu strain [@Epstein2008].Nonetheless, when subject matter experts claim to have used ABMs for purpose of prediction, they actually use ABMs either for description or explanation [@wilensky2015introduction].\
ABMs has a high functionality for education and analysis [@Blikstein2009; @Wilensky2006; @Sengupta2009]. Educators can develop models for people that they have never seen before. For example, educators can model some examples of mutualism between individuals of different species when both individuals benefit[^11]Moreover, models can simulate a system that may not be readily available from real-world observations, therefor they can be very thought-provoking and enable learners to go beyond their observations and conduct experiments just like scientists.\
When a subject matter expert is going to gain a deeper understanding of a phenomena about which there is not enough theory, thought experiment can be very useful. Though experiment is another suitable area for ABMs. This type of experiment is done to achieve its purpose without benefit of execution [@Sorensen1998].Thought experiment is conducted when the real-world experiments are neither affordable nor possible to execute [@Rangoni2014]. It has a wide application in social and natural sciences. Through this method, researchers can get aware of the logical consequences of their hypotheses. For example, what will happen if a half of a company’s staff suddenly leave it? ABMs can be very useful in thought experiments especially when people want to deal with complex systems such organization and society. Such systems are far from a real-word laboratory where it is possible to control some variable (as control group) and measure the effect of test on other variables (as treatment group). As a matter of fact, in such systems, there are numerous causal factors that are mainly interdependent over which we have on or a very limited control[@Savona2005]. So real-world experiments can rarely be executed in such systems. This has led researchers of social fields to utilize the potential of though experiment in simulating the consequences of their hypothesized mechanism.\
ABM Building Blocks {#section.Buildingblocks}
===================
ABMs include three building blocks of (1) agents, (2) environment and (3) interactions [@Epstein1997; @niazi2017towards; @wilensky2015introduction]. As the first building block of ABMs, agents are the basic computational units of agent based models. They are defined by two main aspects of (1) properties and (2) behaviors (or actions). Agent’s properties are internal or external states that can be changed by its behaviors (actions). Suppose you want to model an economic system including individual human agents. Some properties for these agents can be status of employment, income level, number of bank account and age or even if necessary blood type! Actions of such agents can be searching for a job, opening a bank account, taking a loan and so on. As it is sensible, actions affect properties, for example, when a person opens a new bank account, the number of his or her bank accounts increases. Or when a person finds a job, his or her status of employment is changed and subsequently his or her income level is positively influenced. As the first building block of any ABM, agents are in three specific types of mobile agents, stationary agents and connecting agents. Mobile agents have the capability of movement for example a human is a type of a mobile agent. Stationary agents are those static agents that have no moving capability. For example, an organization or in wider sense, an environment are types of stationary agents[^12]. Connecting agents are those agents that connect agents together. One clear example of this can be “links” among agents (\[bblocks\]). Additionally, in modeling agents, two major factors have to be taken into consideration, the first factor is about the granularity (grain-size) of agents. For example, when you want to model an economic system, you chose to model the individual actors or prefer to model institutions. The second important factor deals with the cognitive level of agents. In fact, how much is the capability of agents to observe (and sense) the surrounding world and make decisions?\
According to cognitive level of agents, they can be classified into four types of (1) reflective or myopic agents, (2) utility-based agents, (3) goal-based agents, (4) adaptive agents [@wilensky2015introduction]. The reflective agents are very simple if-then agents so that if they face situation A, they immediately do action B. Utility-based agents are very similar to reflective ones but there is a utility function that they do want to maximize it under all conditions. Goal-based agents are more advanced form of utility-based function so that they have a goal that dictates their actions. The most advanced form of agents are adaptive agents. They have enough cognitive capabilities to change their actions in similar conditions based on prior experience. Namely, if they do action A in situation B and lose some payoffs, when they face situation B again, they don’t do action A according to their prior experiences.[^13]\
As the second building blocks of ABMs, the environment is composed of all conditions surrounding the agents as they interact within the model. In other words, the environment is where an artificial social life unfolds [@Epstein1997]. Environments can come into three different major forms of (1) spatial environment, (2) networked environment and (3) mixed environment. The spatial environment is often a discrete environment including several discrete points[^14] . The most common form of spatial environment is lattice structure which can be two or three dimensional (as Figure \[bblocks\]). In spatial environments, when agent A (here the purple agent) reaches a border on the far right side of the environment (i.e., the world) and wants to go farther right, boundary conditions of the environment come to play. The topology of an environment deals with such boundary conditions. For a spatial lattice structure such as Figure \[bblocks\], there can be three types of topologies. The first type is a toroidal topology where agent A reappears in the far left side of the lattice. The second type is bounded topology where agent A cannot move farther right and finally, the third type is infinite plane topology where agent A can keep going right for ever [@wilensky2015introduction]. In real world situations, such as socio-economic settings, agents have more networked interactions than spatial (geographical) interactions. In two different stock markets, a rumor spreads through the individual agents of a network. Therefor an environment can be in a network form where the mobile agents are “nodes” and the connections among them are “links”. There are several types of networks that three of them are widely used which are “random networks” [@Erdos1959], “watts-strogatz small-world” [@Watts1998] and “ scale-free networks”[@Albert2002].[^15] All these networks have been visualized in Figure \[Random\], Figure \[SW\] and Figure \[PA\] respectively.\
Using network structures as an ABM environment provides lots of opportunities to synthesize social network theory (SNT) with ABM. As a matter of fact, ABMs are developed to explain how simple rules generate complex emergence (i.e. a process model) and in terms of network science ABMs are used to analyze the pattern that arise from agents’ interactions over the time (i.e. a pattern model)[@wilensky2015introduction]. When spatial environment and networked environment come together, they form a mixed environment as visualized in Figure\[bblocks\].\
As the third building block of ABMs, interactions refer to rules of behaviors for both agents and the environment[@Epstein1997]. Actually, these rules enable agents to interact with both themselves and others. There are five basic classes of interactions: agent-self, environment-self, agent-agent, environment-agent, and environment-environment. In agent-self interactions, an agent checks its internal states and decides according to them. Environment-self interactions are when areas of the environment alter or change themselves. For instance, they can change their internal state variables as a result of some calculations. Agent-agent Interactions are usually the most important type of action within ABMs. Agent-Environment Interactions happen when the agent manipulates or examines an area of the world in which it exists, or when the environment in some way observes or alter the agent’s internal states. Environment-Environment Interactions between different areas of the environment are probably the least commonly used interaction type in ABMs [@wilensky2015introduction].
Development of ABMs {#section.development}
===================
Generally, an ABM can be developed through three sequential and often iterative phases[^16]:(1) designing phase, (2) programming phase and finally, (3) examination and analysis phase[@wilensky2015introduction]. In designing phase, the initial skeleton of ABMs is constructed. This initial skeleton is actually the textual model of ABM through which all behavioral rules and properties of agents and environment along with the way they interact with each other are verbally documented. As the second phase, programming phase deals with how to translate the ABM’s textual model to a computational model by agent based programming languages and toolkits. The examination and analysis as the third phase are conducted for getting insights concerning (1) model verification, (2) model validation, (3) model replication and reproducibility and (4) model’s output analysis. In the following, these three phases would be discussed.
Designing Phase {#section.Development.Designing}
---------------
Through designing phase, all behavioral rules and properties of agents and environment along with the way they interact with each other are documented by natural language of subject matter experts (SMEs) in this phase. This documentation actually serves as a textual model.[^17]8 basic stages have to be well addressed as detailed in Table\[table3\]:
[| p[.25]{} | p[.75]{} |]{}
\
Stage & Description\
1- Underlying questions of model & What questions does the model want to answer? Which aspects of the real system under study are going to be described in the model?\
2- Types of agents and granularity & What types of agents are going to be created in the model?\
3- Granularity of agents & What is the granularity of the agents? Are agents coarse-grained or fine-grained or both of them?\
4- Properties of agents & What properties do agents hold?\
5- Behavioral rules & What are the behavioral rules of agents? How different are these behavioral rules?\
6- Environment structure & What are the external forces affecting agents? Is environment spatial or networked or both of them?\
7- Input parameters & What are input variables to model? What is the type of input variables? (Boolean, string, continuous,..)\
8- Outputs and measures & What measures and outputs are going to be collected from the model?\
9- scheduling & What is the sequence and time-step of model’s events?\
While designing an ABM, three main factors associated with ABMs’ modeling approaches should be taken into consideration. Actually, In ABM literature, modeling approaches can be divided based on a number of aspects. Three of the most important aspects include 1-goal of modeling, 2-development method and 3-elaboration strategy. In terms of modeling goal, ABMs can be grouped into two major categories of phenomena-based modelling and exploratory modelling[@wilensky2015introduction]. In phenomena-based modeling researchers begin with a known target phenomenon. Typically, that phenomenon has a characteristic pattern, known as a reference pattern. Examples of reference patterns might be common housing segregation patterns in cities, diffusion of a specific ICT technology, spiral-shaped galaxies in space or oscillating population levels in interacting species[@wilensky2015introduction]. These reference patterns are those statistical regularities that econometricians suppose as stylized facts for example, the way price affects supply or demand. The goal of phenomena-based modeling is to create a model that will somehow capture the reference pattern. In ABM, this translates to finding a set of agents, and rules for those agents that will generate the known reference pattern. Once you have generated the reference pattern you have a candidate explanatory mechanism for that pattern and may also vary the model parameters to see if other patterns emerge, and perhaps try to find those patterns in data sets or by conducting experiments. Phenomena-based modeling can also be used with other forms of modeling, such as equation-based modeling. In equation-based modeling, this would mean writing down equations that will give rise to the reference pattern. Evidently all empirical validations perform well in case of phenomena-based modelling where there is a reference pattern against which the accuracy of model’s results is measured.\
The second core modeling form is exploratory modeling. This form is perhaps less common in equational contexts than it is in ABM literature. In exploratory modeling with ABM, a researcher can create a set of agents, define their behavior, and explore the patterns that emerge. One might explore them solely as abstract forms, much like cellular automata developed by Conway[@Conway1976] but to count as a modeling practice, we must note similarities between the behavior of our model and some phenomena in the world just as patterns generated by cellular automata like oscillators and spaceships [@Wolfram1983]. Then our ABM should be refined in the direction of perceived similarities with these phenomena and converge toward an explanatory model of some phenomenon.\
Phenomena-based modelling stems from the notion that there is an objective and real but unobservable data generating mechanism and that the purpose of any model is to represent elements of that mechanism in ways that generate some of the same data. But, in the case of exploratory based modeling the purpose of the models is the representation of perceptions of policy analysts and other stakeholders in the relevant social processes[@Moss2008]. The phenomenon based modeling is a class of agent based models that has much in common with mainstream economic models. They incorporate utility functions; they employ numerical representations of phenomena and attributes naturally described in qualitative terms by the individuals being represented and by other stakeholders. The exploratory modeling is a class of models emerging from a process that is embedded in the social process of policy and strategy formation. Such models are typically described in linguistic terms (i.e. mental models) used by stakeholders rather than numerical variables intelligible and meaningful only to modelers. Such models are developed to facilitate stakeholder participation in the model design and validation process. They are intended precisely to represent the perceptions of stakeholders in order to bring clarity to scenarios built to explore the possibilities - the opportunities and threats - of an uncertain future. The major function of exploratory modeling is to enable the subject matter experts (SMEs) to see what outcomes their mental model(s) can produce when implemented in a real-world system. Therefore, they can have lots of advantages for those who deal with understanding complex systems.\
In terms of elaboration strategy, ABMs can be grouped into KISS, KIDS and TAPAS strategies. KISS strategy that stands for “Keep It Simple, Stupid”. This notion is rooted in the Occam’s razor principle stating “while being faced with a number of competing hypotheses for a problem, one should select a hypothesis that has a fewer assumption” This principle advocates the law of parsimony and agent-based modelers that use this principle “start from simple models and gradually sophisticate it to answer their question”. KIDS strategy which stands for “Keep It Descriptive, Simple” is in opposite direction of KISS strategy. Advocates of KIDS strategy start from descriptive models and gradually simplify them to answer their questions[@Windrum2007; @Elsenbroich2014].[^18] TAPAS strategy that stands for (Take A Previous model and Add Something). In this strategy, modelers take an existing model and successively modify it through adding new features or relating initial assumption[@Frenken2006].\
In terms of development, ABMs can be grouped into two major categories of theory-based modelling and evidence-based modelling[@Moss2008]. ABMs can be developed via a theory. Actually a theory that specifies the behavioral rules of agents or the statistical regularities that the model is designed to explain them. Since theory-based ABMs are built upon prior empirical studies and aimed at stimulating real-world aggregates such as technology diffusion, disease spread and inflation formation, they are basically used for phenomena-based modeling where the modelling of a real-world pattern is the purpose of the modeler. Besides theories, ABMS can also be developed based on evidence. The evidence-based modeling is used when researchers have a mental model concerning behavioral rules of agents of a system and they are interested in understanding the collective behavior of that system when its agents interact with each other. As it can be inferred, this approach is the foundation of exploratory modeling approach where a model is developed for representing the emergent properties of researcher’s mental models.\
Evidence-based ABMs can be developed either as participatory modeling or individual thought experiments. One of major forms of participatory modeling is “companion modeling”(ComMod) which is an iterative participatory approach where multidisciplinary researchers and stakeholders work together continuously throughout a three-stage cycle *field work $->$ modelling $->$ simulation $->$ field work* again[@Barreteau2003].\
ComMod follows two basic objectives. First, it increases the understanding of complex systems through its three-stage cycle. Second, it supports collective decision-making Processes in Complex Situations. In this case, the approach facilitates collective these kinds of processes by making more explicit the various points of view and subjective criteria, to which the different stakeholders refer implicitly or even unconsciously. Indeed, as demonstrated in past research[@Funtowicz1994]when facing a complex situation, the decision-making process is evolving, iterative, and continuous. It means that this process produces always imperfect “decision acts”, but following each iteration they are less imperfect and more shared. Principally, the main principle of the ComMod approach is to develop simulation models integrating various stakeholders’ points of view and to use them within the context of platforms for collective learning. This is a modeling approach in which stakeholders participate fully in the construction of models to improve their relevance and increase their use for the collective assessment of scenarios. The general objective of ComMod is to facilitate dialogue, shared learning, and collective decision-making through interdisciplinary and “implicated” action-oriented research to strengthen the adaptive management capacity of local communities[@Barreteau2003]. As a software engineering method Virtual Overlay Multi Agent System (VOMAS) has been used to improve ComMod methodology[@niazi2012cognitive].VOMAS is used for facilitating last two stages of ComMod, namely modeling and simulation. This method has been very successful in building verified and validated agent based models[@NiaziMuazA;HussainAmir;Kolberg2017].In a nutshell, the textual model (i.e. the conceptual model) of an ABM is the output of designing phase.
Programming Phase {#section.Development.Programming}
-----------------
When the textual model of an ABM is designed, it should be simulated through an agent based programming languages or simulation toolkits.[^19] Such simulation toolkits are a type of simulation software specifically for translating the textual model of an ABM into a computational model. A simulation is an understandable manifestation of a model, coded and visualized by a computer program which provides insights regarding the system under study. A simulation model basically refers to the computing algorithms or mathematical expression that entail the performance and total behavior of a system in the real world scenarios[@Abar2017]. In the early 1990s, general purpose programming languages (GPPLs) were basically used for simulation. SMALLTALK, C++ and Java were the most common GPPLs in ABM community of practice. Using GPPLs for agent based simulation have some visible disadvantages. For example, modelers have to implement basic functions and plotting from scratch and they should be very familiar with the programming language[@Gilbert2002]. Such problems have resulted in development of agent-based simulation toolkits which help modelers a lot to simulate the complex system under study. As presented in Table\[Toolkits and Languages\], the majority of toolkits support the primary GPPLs including Java, C++, C and Logo variant.
[|c|c|c|c|c|]{}
\
& &\
& Java &C++ & C & Logo\
ADK & \* & & &\
AgentBuilder & \* &\* &\* &\
AnyLogic & \*& & &\
Ascape &\* & & &\
DeX & & \* & &\
Echo & & &\* &\
iGen & \*& \* &\* &\
LSD & & \* & &\
MadKit &\* & \*& \*&\
MAML & & & \*&\
Mason &\* & & &\
Netlogo & & & &\*\
RepastS &\* & & &\
Starlogo & & & &\*\
StarlogoT & & & &\*\
Swarm &\* & & &\
As it can be seen in above table, some of platforms are supported by multiple languages such as AgentBuilder, iGen and Madkits that are supported by Java, C++ and C. During this paper, Nelogo 6.0.1 has been used to simulate all ABMs. As one of the most frequently used agent based modeling and simulation toolkit, Netlogo was developed by Uri Wilensky in 1999. Since its development, it has been regularly updated in sequence of versions and a number of extensions.[^20] When model designers and model programmers are different persons, before offering their conceptual models to programmers, model authors had better put it in a unified modeling language (UML) format. Because, it helps programmers a lot to accurately discern what model authors want[@Bersini2012].
Examination Phase {#section.Development.Examination}
-----------------
When ABMs are designed and programmed, they should be examined. Examination phase is highly critical for showing (1) is ABM right designed? (2) Is a right ABM designed? (3) Is ABM replicable by other researchers? And (4) how should outputs be analyzed? Each of these questions will be addressed in the following:
### Verification {#section.Development.Examination.Verification}
In verification process the main purpose is to determine whether the designed model corresponds to programmed model. As it is sensible, through verification process the researchers try to understand the gap between the designed model and its implementation (computational model) and fill it through correction and code debugging. As a matter of facts, verification process comes to play when the model author (i.e., subject matter expert or designer) and model implementer (i.e., simulation specialist or programmer) are different persons which is very common among academic researchers cooperating in a team[@NiaziMuazA;HussainAmir;Kolberg2017; @niazi2012cognitive; @niazi2017towards; @Rand2011; @wilensky2015introduction]. In such a situation, the model author should iteratively discuss the model with the model programmer so that the model programmer understands and programs what exactly the model author wants. This process is called “iterative modelling”[@wilensky2015introduction]which can help model verification process when model author and model programmer are different persons (See the backward curve with the red arrow in Figure\[ABMDevelopmentProcess\]). One very noteworthy point for facilitating the verification process is that the SMEs frequently check textual model developed in designing phase and find its inconsistencies with the computational model developed in programming.
### Validation {#section.Development.Examination.Validation}
Validation of computational models has always been a major concern for Simulation specialists[@Carley1996; @Garcia2007; @Rand2011; @Windrum2007]. Validation is defined as when the simulated model produces the results that are in a satisfactory range of accuracy matching up with the real-world data[@Windrum2007]. Model validation is the process of determining whether the implemented model corresponds to, and explains some phenomenon in the real world[@wilensky2015introduction]. When a model is implemented, its validation becomes so essential. A valid model assures the researchers of the model’s rightness[@PullumLauraL;Cui2012]. Therefore, the model’s results are supposed to be useful out of the model and can be confidently used for policy making. A number of studies have been conducted about types of validation[@PullumLauraL;Cui2012; @Windrum2007], levels through which validation can be done[@Rand2011; @wilensky2015introduction]and various methodologies for conducting validation[@Moss2008; @NiaziMuazA;HussainAmir;Kolberg2017; @Windrum2007]. In a big picture, all validation methods can be classified as qualitative and quantitative methods. qualitative methods like face validity are very subjective and expert-based while quantitative ones such as empirical validation approaches are objective and based on real-world data[@PullumLauraL;Cui2012] (Pullum, Laura L; Cui, 2012). Face validation is the process of showing that the mechanisms and properties of the model look like mechanisms and properties of the real world. It is mainly conducted by subject matter experts (SMEs). Empirical validation makes sure that the model generates data that can be demonstrated to correspond to similar patterns of data in the real world. Validation approaches of ABMs essentially depend on modelling approaches[@Moss2008]. When ABMs are used to simulate a real-life statistical regularity as is the case of phenomena-based modeling, empirical validation approaches are used. But when ABMs are used to simulate the mental models of systems stakeholders and show what will emerge from them as is the case of exploratory modeling, qualitative validation methods such as face validity are used. Since ABMs have different micro-macro behavior levels (e.g., an agent’s individual behavior vs model’s behavioral aggregate),they should be validated according to two axes[@Rand2011]as shown in Table\[Validation Axes\].
[|c|c|c|c|]{}
& Macro& Macro face Validation& Macro empirical Validation\
&Micro&Micro face Validation& Micro empirical Validation\
& & Face& Eimpirical\
\
In the macro –face validation, the SMEs want to know how much the aggregated patterns produced by the implemented model correspond “on face” to the real-world aggregated patterns? In the micro-face validation, the SMEs plan to discern how much properties and behavior rules of agents correspond “on face” to reality? In the macro-empirical validation, the SMEs want to know how much the outputs or targeted aggregates produced by the implemented model correspond to real-word data? In the micro-empirical validation, the SMEs want to discern how much input parameters, properties and behavior rules of agents correspond to real-word data? In all empirical validations, most often the model should be calibrated both at micro and macro levels. The calibration is a process through which the suitable parameters and initial conditions are found so that the implemented model generates a pattern similar to real-world pattern[@Carley1996; @Rand2011; @Windrum2007; @wilensky2015introduction].
### Replication {#section.Development.Examination.Replication}
If a model is going to be acceptable among a scientific community, it must be replicable. Replication of a model is actually the re-implementation of its conceptual model according to the previous results produced by its implemented model. When a computational model is replicated, it not only shows its verification is trustable but also reexamines its reevaluation and facilitates a common language among modelers[@wilensky2015introduction]. One effective way for facilitating the replication of a model is publication of its programming codes by its authors in open access platforms such as for Netlogo programmers andfor all programmers. Another effective way is a full publication of (1) hardware specifications of a computer used for simulation and (2) the programming language and toolkits used for simulation. However, one very noticeable point is the fact that when a model is successfully replicated it should be able to produce outputs A successful replication is one in which the replicators are able to establish that the replicated model creates outputs sufficiently similar to the outputs of the original. This does not necessarily mean that the two models have to generate the exact same results[@wilensky2015introduction].[^21]
### Output Analysis {#section.Development.Examination.Analysis}
ABMs generate a great deal of data both at micro-level and macro-level. Because of such a data comprehensiveness, a number of analyses can be conducted on such models. As the eighth stage of table 3, the output analysis is related to measures that the SMEs want to analyze the effect of model input parameters on them. One very noteworthy point in analysis of ABMs’ outputs is the stochastisity of such computational models. Actually, like CASs, ABMs have stochastic nature. Even with the same set of parameters (parameter combinations), They produce different results in different runs. This requires researchers to do such simulations in multiple runs and analyze statistical distributions of targeted results (behavior under study) through inferential statistics methods[@axtell2000agents; @wilensky2015introduction]. To have a good analysis, in addition to descriptive statistics measures, the researchers can use inferential statistics methods such as various statistical tests in analyzing the statistical distribution of ABMs’ outputs. Moreover, if a network environment is used, several number of social network theory (SNT) measures can be very useful for extracting specific insights from ABMs’ outputs. In the nutshell, the big picture of ABM development process is visualized in Figure\[ABMDevelopmentProcess\].
Two Critical Considerations {#section.considerations}
===========================
Complex adaptive systems (CASs) are real world systems which have a number of characteristics (As discussed in Table\[table1\]. In contrast, Agent based models (ABMs) are a type of computational methodology believed to be very promising in modeling CASs. Every developed ABM only shows one or some aspects of a CAS not all of its aspects. Therefore, what ABM practitioners produce via simulating a CAS (e.g., a society) is a simplified and artificial picture of that CAS as it is visualized in Figure\[realsimulated\]. Two significant factors that have to be taken in to consideration by every ABM practitioner would be discussed in the following.
Emergence {#section.considerations.Emergence}
---------
CASs exhibit emergent properties. A property of a CAS that emerges out of non-linear (non-trivial) interactions among its constituent components so that it is beyond and irreducible to them is called “emergence”. In comparison to other simulation techniques such as discrete event simulation (DES), system dynamics (SD) or even game theory, one of the greatest advantages of ABM is its outstanding prowess in showing the emergent properties of CASs. Our world exhibits several observable CASs, in biological sciences, molecules emerge out of interacting atoms, organelles emerge out of interacting molecules, cells emerge out of interacting organelles. Organs emerge of interacting cells and finally body emerges out of interacting organs. Such examples of emergent properties of our body indicates that it is a biological CAS full of interesting emergent properties. In a sociological perspective, there are also several examples of CAS, as an instance, the society can be interpreted as an emergent property of interacting humans or even norms can be studied as emergent property of social system. The emergence is very difficult to forecast and completely depends on the observation[@sun2006cognition]. In facing the challenge of emergence, there can be two kinds of human thinking way[@wilensky2015introduction].
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As shown in \[Integrative\], integrative thinking refers to when the SMEs have a CAS in mind (e.g., an organization into which a number of people with different religious backgrounds and specialties work) but they don’t know its targeted emergent property (e.g., the pattern of cooperation or formation of hierarchy). In contrast, when SMEs have an observable emergent property but don’t know its CAS (i.e., from which CAS, that property emerged) they are facing differential thinking as visualized in \[differential\].\
Essentially, integrative thinking comes to play when the exploratory modeling is assumed while differential thinking takes effect when the phenomena-based modeling is adopted. However, whereas the core of integrative thinking is to discern what properties will emerge out of the CAS under study, differential thinking is used to grasp what CASs can lead to the emergent property under study.
Non-Ergodicity of ABMs and the necessity of sensitivity analysis {#section.Development.Non-ergodicity}
----------------------------------------------------------------
Like CASs, ABMs are non-ergodic. Non-ergodic systems are highly sensitive to initial conditions. If their initial conditions are a little changed, their output behaviors will be drastically influenced. So, it is necessary to do sensitivity analysis in ABMs after they are programmed and evaluate how their final behaviors (measures) are sensitive to change of their input parameters[@Al-suwailem2008; @wilensky2015introduction]. A good sensitivity analysis can help SMEs (1) to get aware of how emergent properties are produced in ABMs, (2) to get able to examine the robustness of emergent properties and (3) to get able to quantify the changeability of ABM’s outputs after input parameters are changed. A number of methods have been proposed for sensitivity analysis of ABMs, three important if which are (1) one-factor-at-a-time (OFAT), model-free output decomposition of variance and (3) variance decomposition of model based output[@TenBroeke2016].
Two Simulations {#section.simulations}
===============
To illustrate the implementation of ABM, two economic systems have been simulated using an exploratory agent-based modeling approach[@Sabzian2018]. Exploratory ABM has a wide potentiality for thought experiment[@Rangoni2014]. As discussed above, this approach is grounded on evidence based modelling where a SME (or a team of them) likes to make sense of what will possibly emerge out of their mental models before being executed in real world. Outcomes produced by an exploratory ABM can find a number of empirical supports in real world data. Therefore, it can also play a vital role in theory development[@Macy2002].Through this implementation, an exploratory ABM has been used to show (1) how economic inequality emerges within an economic system (2) how charity and allocation strategies of charity entities can help reduce this emergent economic inequality.\
The firs system (system I) is mainly inspired by the work of Wilensky and Rand (2015). However, to make this model more suitable for this study, some new features have been added to its original version including (1) the possibility of changing the number of agents in the interface view, (2) the possibility of choosing money amount for agents in the interface view, (3) the possibility of simulation when agents start with equal or unequal amount of money in initial condition (see \[equal amount\] for equal amount and see \[unequal amount\] for unequal amount) and (4) the possibility of showing the amount of money of each agent by its color in the world view so that the richer agents get a darker color and go north wise while the poorer agents get a lighter color and go south wise (Figure \[unequal amount\]).\
The second system (system II) is an extension of system I which includes a number of new features such as (1) the ability of human agents to give charity, (2) the establishment of charity agents, (3) the strategies that charity agents (entities) can take to allocate the charity among the needy agents and so on . In order to make the replication of this model easier, the agent-based simulation of system II can be directly searched and downloaded from[ ]{} that is a useful platform for communicating and discussing agent-based models written in Netlogo 6.0.1. The conceptual models (textual models) of system I and system II along with their simulation and analysis would be discussed in the following:
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System I {#section.simulations.sysI}
--------
This system is built upon five assumptions including (1) there is a society in which N number of persons live, (2) there is a global clock that shows the time by each tick and it is completely discrete, (3) each person has M amount of money that can be equal or unequal in the initial conditions, (4) there is a money gap between the five low deciles (bottom 50% and the tenth decile (top 10%). It can be positive, zero or negative. When it is positive, it shows the money of five low deciles is more than that of tenth decile. In case of being zero, it shows the money of five low deciles is equal to that of tenth decile and if it is negative, it indicates the money of five low deciles is less than that of tenth decile, (5) there is a critical threshold which shows the criticality of economic situation when the value of money gap becomes lower than it. According to this system, if all people (N = 500) of the society have an equal amount of money (M =100) in the initial conditions and donate a unit of money to each other randomly and by each time tick as long as each person’s money is more than zero, what would be the answers to following questions: 1- How will the probability distribution of money be in tick of 100? 2- How will the probability distribution of money be in tick of 1000? 3- How will the probability distribution of money be in tick of 9000? Because of the stochastic nature of agent-based models, the probability distribution of money in system I has been simulated in 10 different runs . In run 1, According to \[100 ticks\], After 100 ticks, the probability distribution of money is similar to a normal distribution as M N (100, 110.612). This distribution shows that money of each person is very inclined to the average money of society. Therefore, all deciles of society have a somewhat similar amount of money. In addition, as shown in \[Distance in 100 ticks\],there is a great distance between the money volume of all five lower deciles (bottom 50% with 22885 units of money) and that of tenth decile (top 10% with 5933 units of money) meaning that bottom 50% has 16952 units of money more than top 10%.\
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As demonstrated in r\[1000 ticks\], the probability distribution of money has become flatter when the system I is in tick of 1000. This normal distribution has a mean of 100 and variance of 972.661 implying that the majority of the society have a money inclined to 100 units of money and just a few of them own a very high amount of money. However, regarding \[Distance in 1000 ticks\] it can be concluded, the total amount of money of bottom 50% has become mitigated to 18835 while the top 10% has accumulated a remarkable amount of money 7783 and money distance has decreased by a large degree to 11052.
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According to \[9000 ticks\], when system I comes to tick of 9000, it shows a Boltzman-Gibbs distribution implying that just a very few of persons have accumulated a great amount of money and a large number of them have just gained a little amount of money. In this case the bottom 50% has an amount of money of 9134 while top 10% has an amount equal to 13052 that is 3918 units more than that of bottom 50%. The money volume of top 10% has exceeded that of bottom 50% in tick of 5800 since then the gap has increased (see \[Distance in 9000 ticks\]). Statistical summary of all 10 runs for ticks of 100, 1000 and 9000 is described in Table\[System up to tick of 100\], Table\[System up to tick of 1000\] and Table\[System up to tick of 9000\] respectively.
\# of runs Mean Variance Total money of top 10% Total money of bottom 50% diff Tick of critical stage
------------ ------ ---------- ------------------------ --------------------------- ------- ------------------------
RUN 1 100 110.621 5933 22885 16952 -
RUN 2 100 104.561 5877 22959 17082 -
RUN 3 100 102.997 5908 22988 17080 -
RUN 4 100 92.641 5824 23114 17290 -
RUN 5 100 100.917 5867 22974 17107 -
RUN 6 100 97.002 5850 23025 17175 -
RUN 7 100 89.254 5840 23132 17292 -
RUN 8 100 90.509 5884 23149 17265 -
RUN 9 100 100.044 5913 23028 17115 -
RUN 10 100 106.052 5963 22987 17024 -
: System up to tick of 100[]{data-label="System up to tick of 100"}
\# of runs Mean Variance Total money of top 10% Total money of bottom 50% diff Tick of critical stage
------------ ------ ---------- ------------------------ --------------------------- ------- ------------------------
RUN 1 100 972.661 7783 18835 11052 -
RUN 2 100 950.793 7578 18871 11293 -
RUN 3 100 1031.490 7926 18555 10629 -
RUN 4 100 1092.669 7763 18279 10516 -
RUN 5 100 1054.322 7837 18425 10588 -
RUN 6 100 986.492 7645 18778 11133 -
RUN 7 100 1001.531 7975 18765 10790 -
RUN 8 100 1259.535 8191 17949 9758 -
RUN 9 100 958.436 7712 18822 11110 -
RUN 10 100 1063.314 7840 e18480 0640 -
: System up to tick of 1000[]{data-label="System up to tick of 1000"}
[|p[.1]{} | p[.1]{} | p[.1]{} |p[.1]{} |p[.1]{}|p[.1]{}| p[.1]{}|]{}
\
\# of runs & Mean &Variance &Total money of top 10% &Total money of bottom 50% &diff &Tick of critical stage\
RUN 1 &100&6256.793&13052&9134&-3918&5800\
RUN 2 &100&5359.6312&12692&10807&-1885&6476\
RUN 3 &100&5042.136&12207&10878&-1329&6310\
RUN 4 &100&4959.899&11890&10662&-1228&6919\
RUN 5 &100&5443.066&12562&10225&-2337&5801\
RUN 6&100&4869.382&11871&10737&-1134&6941\
RUN 7 &100&5597.174&12459&9723&-2736&5520\
RUN 8 &100&5326.837&12560&10373&-2187&5752\
RUN 9 &100&5533.523&12648&10046&-2602&2725\
RUN 10 &100&5469.022&12428&10011&-2417&5668\
As a fundamental law of equilibrium statistical mechanics, Boltzman-Gibbs law states that any conserved quantity in a big system should follow an exponential probability distribution. According to Boltzman-Gibbs law, in a closed economic system, the total amount of money is conserved because it is not manufactured, consumed or destroyed. $$\label{1}
p(m) = {{Ce} ^ \frac{-m}{t}}$$ Here m stands for money, C is a normalizing constant and T is an effective temperature equal to the average amount of money per agent. Some studies have shown that the money distribution follows a Boltzman-Gibbs law when it is conserved and exchanged in a closed system. According to this property of money distribution a very few of persons will accumulate a great amount of money while a large number of them just gain a little amount of money[@Dragulescu2000; @Ferrero2004; @Yakovenko2009]. As it can be seen in table 3, in all ten runs, the top 10% has outweighed the bottom 50% in terms of accumulated money and consequently economic situation entered a critical stage in all runs. Thus, in order to prevent this economic system from becoming more critical, system I has been extended into system II by adding some more mechanisms to it.
System II {#section.simulations.sysII}
---------
M number of charity organizations are added to system I. These organizations come to scene when the economic situation becomes critical (i.e., the money of 10% of the society becomes equal to or more than that of its 50%). The mission of these organizations is to help the low five deciles not to deteriorate more in the depth of poverty. These beneficiaries work based on the charity amount that benefactors give to them in order to distribute it among five lower deciles. In terms of charity -giving and distribution, these charities can take one of three allocation strategies of A, B or C. Strategy A is applied when just the richest person of the society gives a unit of his or her money to the charity organization and it allocates that money to the poorest person of the society. When c% of members of decile 10 give a unit of money to the charity organization and it allocates that of amount of money among d% of members of five lower deciles, the charity organization has used strategy B. The charity entities use strategy C when k% of members of decile 10, p% of members of decile 9 and v% of members of decile 8 give money (everybody one unit of money) to them and they distribute that amount of money among x% of members of decile 1, y% of members of decile 2 and z% of members of decile 3 respectively. This model can help answering the following questions: 1- How will strategy A affect the economic system when it enters a critical stage? 2- How will strategy B affect the economic system when it enters a critical stage? 3- How will strategy C affect the economic system when it enters a critical stage? According to table 3, for run 1, the system has entered the critical stage in tick of 5800. The charity organization has used strategy A to help system exit this stage but it has not got out of the critical stage in the next ticks (up to 9000). It means that as long as the charity entity uses strategy A to help system get out of the critical stages, it fails to exit and returns to critical stage by each tick. The return period or recurrence interval of the critical stage is a key indicator for measuring the sustainability of allocation strategies. The visualization of how charity organization uses strategy A in tick of 5800 is shown in Figure\[Strategy A in tick 5800\].
According to Figure\[Strategy A in tick 5800\], the red line indicates charity from the richest person of the society (benefactor) to charity and the green line indicates the charity distributed by charity to the poorest person of the society. As it can be noticed, the spatial position and color of rich persons are different from those of the poorer ones. Such differences are because of a mechanism embedded into this model which forces the rich get darker color and move north wise while making the poor get brighter and move south wise. As shown in figures 20 to 25, for run 1, strategy A, strategy B (with parameters of c=100 and d= 20) and Strategy C (with parameters of k = 100, p = 60, v= 40, x= 100, y=60 and z= 40) have been applied for handling the economic critical stages up to 9000 ticks.
[0.7]{}
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As demonstrated in \[Distribution and Strategy A for 9000 ticks\],\[Distribution and Strategy B for 9000 ticks\] and \[Distribution and Strategy C for 9000 ticks\], by each of allocation strategies, charity organization have tried to help the economic system exit the critical stage while entering it in every time. In this case when the total money of top 10% becomes equal to or exceeds that of bottom 50%, the total money of top 10% will be forced back to a value less than that of bottom 50% depending on how many of three higher deciles participate in giving charity and also which allocation strategies the charity organizations use to distribute the charity among poorer deciles. According to the Boltzman-Gibbs law after one of allocation strategies is implemented the money (as a conserved quantity here) will tend to become accumulated into hands of a very few of people. Thus the systems enter the critical stage again. The number of times that economic system returns to the critical stage after a typical strategy is used for resource redistribution in it, is a key factor for measuring how sustainable that strategy is.
[0.7]{}
[0.7]{}
[0.7]{}
As it can be seen from \[Distance and Strategy A for 9000 ticks\],\[Distance and Strategy B for 9000 ticks\],\[Distance and Strategy C for 9000 ticks\], when a typical strategy is used, the society will have a specific form of money distribution in the final tick (here 9000). Therefore, the less variance the distribution has, the better resource allocation strategy has been used. Because it has more reduced the economic inequality among people (in terms of money distribution). Thus, the variance of money distribution in the society when it enters the final tick is another key factor for measuring the efficiency of the resource allocation strategy. The number of return periods of critical stages and variance of the money distribution in all runs have been presented for all of strategies in Table\[Application of strategy A up to 9000 ticks\], Table\[Application of strategy B up to 9000 ticks\] and Table\[Application of strategy C up to 9000 ticks\].\
\# of runs Mean Variance Total money of top 10% Total money of bottom 50% diff
------------ ------ ---------- ------------------------ --------------------------- ------
RUN 1 2015 11328 11219 4465.974 -109
RUN 2 1517 10792 10888 4398.300 96
RUN 3 2287 11121 10784 4596.4128 -337
RUN 4 1386 10815 10650 4516.769 -165
RUN 5 2150 11001 0838 4489.615 -163
RUN 6 1656 11208 11314 4341.547 106
RUN 7 2638 10745 10716 4479.010 -29
RUN 8 1590 11383 11298 4419.070 -85
RUN 9 2368 10725 10364 4624.541 -361
RUN 10 2059 10997 10727 4459.478 -270
: Application of strategy A up to 9000 ticks[]{data-label="Application of strategy A up to 9000 ticks"}
\# of runs Mean Variance Total money of top 10% Total money of bottom 50% diff
------------ ------ ---------- ------------------------ --------------------------- ------
RUN 1 42 11032 11182 4399.110 150
RUN 2 47 11353 11459 4415.555 106
RUN 3 48 11475 11592 4443.134 117
RUN 4 18 11357 11474 4364.448 117
RUN 5 34 11256 11316 4367.595 60
RUN 6 26 11594 11673 4378.737 79
RUN 7 37 11243 11351 4377.206 108
RUN 8 37 11213 11272 4426.685 59
RUN 9 39 11351 11420 4435.002 69
RUN 10 22 11252 11292 4379.559 40
: Application of strategy B up to 9000 ticks[]{data-label="Application of strategy B up to 9000 ticks"}
\# of runs Mean Variance Total money of top 10% Total money of bottom 50% diff
------------ ------ ---------- ------------------------ --------------------------- ---------
RUN 1 31 11224 11520.3 4302.669 296.33
RUN 2 19 11681 11760 4388.751 79
RUN 3 28 11464 11544.3 4409.166 80.33
RUN 4 14 11305 11565.3 4252.689 260.33
RUN 5 30 11371 11519.7 4315.761 148.66
RUN 6 22 11667 11850 4394.670 182.99
RUN 7 24 11525 12082.7 4166.427 562.666
RUN 8 27 11579 11699 4363.750 119.99
RUN 9 26 11286 11379.7 4326.632 93.66
RUN 10 27 11635 11707.7 4311.794 72.66
: Application of strategy C up to 9000 ticks[]{data-label="Application of strategy C up to 9000 ticks"}
According to Table\[Application of strategy A up to 9000 ticks\], Table\[Application of strategy B up to 9000 ticks\] and Table\[Application of strategy C up to 9000 ticks\], when strategy A is used, the average number of return periods is 1966.6 (roughly 1967) times Meaning that in all 10 runs, the economic system has returned to critical stage with the average of 1967 times. By applying strategy B, the system has shown a remarkably small average number of return periods which is 35. This number shows the second strategy is far more sustainable than strategy A in helping the system exit the critical stage for long time intervals. Strategy C has shown the number of 24.8 (roughly 25) for average of return periods in all runs.The number of return periods for each strategy has been visualized in Figure\[sustainability\]: The sustainability of strategies. The vertical axis of this figure is the number of return periods that is made logarithmic (for better representation) and the horizontal axis stands for number of runs of simulation. As it can be seen, the strategy A has had the lowest level of sustainability while the strategy C has the highest one. Except run 10, strategy C has shown the lowest number of return periods for all other runs. Though this number doesn’t significantly differ from that of strategy B, it shows more sustainability.\
Another information that has be inferred from Table\[Application of strategy A up to 9000 ticks\], Table\[Application of strategy B up to 9000 ticks\] and Table\[Application of strategy C up to 9000 ticks\], refers to how efficient each of allocation strategies has been in reducing the overall variance among people in terms of money distribution. This indicator shows how much agents differ from each other in terms of money volume. When strategy A is applied, the average overall variance of money distribution is 4624.5 meaning that in all 10 runs of simulation, the average overall variance of distribution of money in economic system has been 4479.07. By executing strategy B, the average overall variance has decreased to 4398.70. While the charity paid in strategy B is 50 times more than that of strategy A (when systems enter critical stage), the average overall variance has not decreased remarkably (only 80.37 units). Strategy C has shown a better performance in reducing the variance of money distribution almost in all runs (except run 6). This strategy has had the overall average variance of 4323.2 for all 10 runs showing the fact that it is the most efficient strategy. The variance values of money distribution for each strategy have been visualized in Figure\[Variance\]. The vertical axis of this figure is values of variance (for better representation) and the horizontal axis stands for number of runs of simulation. As it can be seen, the strategy C has had the lowest values of variances in all runs except run 6. Thus, this strategy is regarded to have the highest efficiency.
Remarks on Simulations {#section.simulations.Remarks}
-----------------------
These simulations have served two purposes. The first purpose was the explanation of how economic inequality emerges in an economic system. Results indicate that in a closed economic system when agents exchange the money, it tends to be accumulated in the hands of a very few number of agents over time. Therefore, the distribution of money in the society will follow a power law distribution. these results have found empirical support from the field of equilibrium statistical mechanics[@Dragulescu2000; @Yakovenko2009]. As a fundamental law of this field, Boltzman-Gibbs law states that in a closed economic system, the total amount of money is conserved because it is not manufactured, consumed or destroyed so any conserved quantity in a big system should follow an exponential probability distribution. The second purpose was to simulate how charity and allocation strategies of charity entities can help reduce the economic inequality emerged in a system. The results showed that charity is highly effective in reducing the gap among economic deciles. So the more people pay charity the more they can decrease the gap. A part from charity, the results imply that the way charity entities allocate the money among lower economic deciles (i.e., bottom 50%) is of paramount importance. In strategy A when system enters a critical stage, just one unit of money (as charity) is transited from the richest agent to poorest one. In strategy B, that amount of money becomes fifty times lager in each transition but the overall average variance (in comparison to average number of return periods) has not had a remarkable decrease. The money is made two times larger in strategy C but it shows a great decrease in overall average variance in contrast to that of strategy B. The main reason for this is not only because of money volume increase but also largely because of the ways money resources are taken from higher deciles and allocated to lower ones.
Conclusion {#section.conclusion}
===========
Agent based modeling (ABM) has a high potentiality for modeling systems that are very hard or often impossible to capture by traditional modeling techniques such as PDEs, ODEs and even statistical modeling methods. In addition, ABM has shown a performance far better than a number of simulation methods such as discrete-event simulation (DES) and system dynamics (SD). A number of works have been published on this subject most often each of which has particularly dealt with one aspect of ABMs. For examples some works have only discussed the difference between ABM and EBM[@Sun2005; @VanDykeParunak1998]. Some have only dealt with what of ABMs[@axtell2000agents; @Chattoe-Brown2013; @Epstein1999; @Epstein1997; @Heath2010a; @Macy2002]and a number of works have just been conducted on issues of ABM verification and validation[@Law2008; @NiaziMuazA;HussainAmir;Kolberg2017; @PullumLauraL;Cui2012; @Stone1974; @Windrum2007; @Xiang2005], and replication and output analysis[@Axtell1996; @Lee2015; @wilensky2015introduction]. The major focus of this paper has been to help social sciences researchers (particularly economic planners) not only understand ABMs and gain a clear-cut big picture about them but also learn a step-by-step framework for developing them both systematically and rigorously.\
Like any scientific work, this work has had some limitations. The first limitation is the simplicity of the simulated economic systems. So extending the model to more elaborated levels can be a good subject for future studies. For example, human agents can be made more intelligent. They can have memory capacity, networked interactions, abilities for production, consumption and destruction, educational level, tendency for entrepreneurship and many other psychophysiological features. Moreover, new agent types can be added, for example, banks, factories, venture capital funds and so on. Such an extension can yield very interesting results. For instance, when factories hire more of their staff among those educated agents that belong to lower economic deciles, venture capital funds define some priorities for poor agents possessing high entrepreneurial tendency and a professional interaction is set among agents, some behavioral patterns will surprisingly emerge both at micro-scale (agent-level) and macro-scale (system-level).The second limitation is that just distribution of money has been the subject of this study and distribution of wealth or income has not been simulated. So, extending the system in order to show distribution of wealth and income will be a great contribution that can be made via future studies. The third limitation refers to the fact that in this paper, just two thought experiment examples have been provided and the validation process has not been done to full scale because of the space limitation (though it can be inferred via face validation). Therefore, applying the proposed framework for conducting real-world examples with empirical validations can be pursued in future studies. As the fourth limitation, the Netlogo toolkit has not been analyzed in details whereas it has some interestingly applied features for sensitivity analysis (i.e. BehaviorSpace), participatory simulation (i.e. Hubnet) and parameter regimes selection (i.e. Behaviorsearch). So having a specific study on Netlogo features and applications will be very useful for those social scientists who are eager to learn an easy but powerful ABM toolkit.
[^1]: Iran University of Science and Technology.
[^2]: Iran University of Science and Technology.
[^3]: Sharif University of Technology.
[^4]: Tadbir Economic Development Group.
[^5]: Tadbir Economic Development Group.
[^6]: Tadbir Consulting Group.
[^7]: It can be inferred that what Merton calls “unintended consequences” can just be observed in complex systems such as society[@Merton1936]
[^8]: CAS modeling methodologies have been comprehensively discussed in [@Niazi2011]
[^9]: Microscale models form a broad class of computational models that simulate fine-scale details, in contrast with macroscale models, which amalgamate details into select categories. Microscale and macroscale models can be used together to understand different aspects of the same problem [@Gustafsson2007; @Gustafsson2010]
[^10]: For a comprehensive overview of computational methods in complexity economics, look at [@Amman1996; @Tesfatsion2002]
[^11]: An interesting example can be the mutualism between a goby and a shrimp. The shrimp digs a burrow in the sand and cleans it up where both species can live. Since the shrimp is almost blind, it has a high vulnerability to predators outside the burrow. When the shrimp is under dangerous conditions the goby goes over to warn the shrimp by touching it with its tail. This causes both the shrimp and goby quickly back into the burrow [@Helfman2009]
[^12]: According to Netlogo programming language, an environment is built of several static agents called “patch” and all agents are on the environment. ABM programming languages and tools will be discussed in
[^13]: For a more comprehensive study of agent cognition, look at [@Russell2016]
[^14]: It can also be continuous, see [@wilensky2015introduction]
[^15]: For a more comprehensive study of networks, look at [@Newman2010; @Wasserman1994]
[^16]: There is not still a completely agreed-upon and standard protocol for ABM development among all practitioners[@Windrum2007]. One good protocol for ABM development is in[@VanDam2012]
[^17]: A textual model is the conceptual model of an ABM which is documented in natural language.
[^18]: Technically, KISS and KIDS can be supposed as two opposite ends of spectrum. In an effort for developing a unified strategy, Rand and Wilensky (2007) developed full spectrum modeling strategy. In this strategy models are developed in a progressive way ( either from simple to descriptive or descriptive to simple) and the phenomenon under study is modelled at multiple levels of details[@Rand2007]
[^19]: Sometime an ABM can be directly developed in the programming phase through participatory simulation platforms. This type of simulation is useful for simulating systems that there is not enough data about them therefore designing an initial conceptual model for them is almost impossible. According to this approach, an agent-based model is directly developed through direct participation of stakeholders of the problems in distributed platforms such as client-server network. This simulation is very useful in research and education[@Colella2000; @Frey2013].
[^20]: Readers can refer to Netlogo home page in order to get more information of this agent based modeling toolkit.
[^21]: For a more comprehensive study of ABMs’ replication , look at[@Axtell1996].
|
---
abstract: 'The graph partition problem is the problem of partitioning the vertex set of a graph into a fixed number of sets of given sizes such that the sum of weights of edges joining different sets is optimized. In this paper we simplify a known matrix-lifting semidefinite programming relaxation of the graph partition problem for several classes of graphs and also show how to aggregate additional triangle and independent set constraints for graphs with symmetry. We present an eigenvalue bound for the graph partition problem of a strongly regular graph, extending a similar result for the equipartition problem. We also derive a linear programming bound of the graph partition problem for certain Johnson and Kneser graphs. Using what we call the Laplacian algebra of a graph, we derive an eigenvalue bound for the graph partition problem that is the first known closed form bound that is applicable to any graph, thereby extending a well-known result in spectral graph theory. Finally, we strengthen a known semidefinite programming relaxation of a specific quadratic assignment problem and the above-mentioned matrix-lifting semidefinite programming relaxation by adding two constraints that correspond to assigning two vertices of the graph to different parts of the partition. This strengthening performs well on highly symmetric graphs when other relaxations provide weak or trivial bounds.'
author:
- 'Edwin R. van Dam [^1]'
- '[Renata Sotirov]{}[^2]'
title: 'Semidefinite programming and eigenvalue bounds for the graph partition problem[^3]'
---
Keywords: graph partition problem, semidefinite programming, eigenvalues, strongly regular graph, symmetry
Introduction {#sec:intro}
============
The graph partition problem (GPP) is the problem of partitioning the vertex set of a graph into a fixed number, say $k$, of sets of given sizes such that the sum of weights of edges joining different sets is optimized. Here we also refer to the described GPP problem as the $k$-partition problem. The GPP is a NP-hard combinatorial optimization problem, see [@GarJoSt:76]. It has many applications such as VLSI design [@Len:90], parallel computing [@BisHenKa:00; @HeKo:00; @Simon91], network partitioning [@FidMatt:82; @Sanch89], and floor planing [@DaKuh87]. For recent advances in graph partitioning, we refer to [@BMSSS:13].
There are several approaches for deriving bounds for the GPP. Here we are interested in eigenvalue and semidefinite programming (SDP) bounds. Donath and Hoffman [@DoHo:73] derived an eigenvalue-based bound for the GPP that was further improved by Rendl and Wolkowicz [@ReWo:95]. Falkner, Rendl, and Wolkowicz [@FeReWo:92] derived a closed from bound for the minimum $k$-partition problem when $k=2,3$ by using the bound from [@ReWo:95]. Their bound for $k=2$ coincides with a well-established result in spectral graph theory; see e.g., Juvan and Mohar [@JuMo:92]. Alizadeh [@Aliz95] and Karish and Rendl [@KarRend:98] showed that the Donath-Hoffman bound from [@DoHo:73] and the Rendl-Wolkowicz bound from [@ReWo:95], respectively, can be reformulated as semidefinite programs. Other SDP relaxations of the GPP were derived in [@KarRend:98; @WoZh:99; @KaReCl:00; @dKPaDoSo:10; @Sot11]. For a comparison of all these relaxations, see [@Sot10; @Sot11].
Armbruster, Helmberg, Fügenschuh, and Martin [@ArmHeFueMa:11] evaluated the strength of a branch-and-cut framework for linear and semidefinite relaxations of the minimum graph bisection problem on large and sparse instances. Their results show that in the majority of the cases the semidefinite approach outperforms the linear one. This is very encouraging since SDP relaxations are widely believed to be of use only for instances that are small and dense. The aim of this paper is to further investigate eigenvalue and SDP bounds for the GPP.\
[**Main results and outline**]{}
Symmetry in graphs is typically ascribed to symmetry coming from the automorphism group of the graph. However, symmetry may also be interpreted (and exploited) more broadly as what we call combinatorial symmetry. In Section \[sec:SyminGr\] we explain both types of symmetry. In particular, in Section \[sec:perm\] we explain symmetry coming from groups, while in Section \[sec:comb\] we describe combinatorial symmetry and related coherent configurations. The associated coherent algebra is an algebra that can be exploited in many combinatorial optimization problems, in particular the GPP. In the case that a graph has no (or little) symmetry, one can still exploit the algebraic properties of its Laplacian matrix. For this purpose, we introduce the Laplacian algebra and list its properties in Section \[Ex:LaplAlg\].
In Section \[sect:EffRelax\] we simplify the matrix-lifting SDP relaxation from [@Sot11] for different classes of graphs and also show how to aggregate triangle and independent set constraints when possible, see Section \[sec:aggregateConstr\]. This approach enables us, for example, to solve the SDP relaxation from [@Sot11] with an additional $3{\genfrac(){0pt}{}{n}{3}}$ triangle constraints in less than a second (!) for highly symmetric graphs with $n=100$ vertices. In Section \[sec:SRG\] we present an eigenvalue bound for the GPP of a strongly regular graph (SRG). This result is an extension of the result by De Klerk et al. where an eigenvalue bound is derived for the graph equipartition problem for a SRG. In Section \[sec:SRG\] we also show that for all SRGs except for the pentagon, the bound from [@Sot11] does not improve by adding triangle inequalities. In Section \[sec:JohnKne\] we derive a linear program that is equivalent to the SDP relaxation from [@Sot11] when the graph under consideration is a Johnson or Kneser graph on triples.
In Section \[sec:AnyGaph\] we derive an eigenvalue bound for the GPP for any, not necessarily highly symmetric, graph. This is the first known closed form bound for the minimum $k$-partition when $k>3$ and for the maximum $k$-partition when $k> 2$ that is applicable to any graph. Our result is a generalization of a well-known result in spectral graph theory for the $2$-partition problem to any $k$-partition problem.
In Section \[sec:impVec\], we derive a new SDP relaxation for the GPP that is suitable for graphs with symmetry. The new relaxation is a strengthened SDP relaxation of a specific quadratic assignment problem (QAP) by Zhao, Karisch, Rendl, and Wolkowicz [@ZhKaReWo:98] by adding two constraints that correspond to assigning two vertices of the graph to different parts of the partition. The new bound performs well on highly symmetric graphs when other SDP relaxations provide weak or trivial bounds. This is probably due to the fact that fixing breaks (some) symmetry in the graph under consideration. Finally, in Section \[sec:impMat\] we show how to strengthen the matrix-lifting relaxation from [@Sot11] by adding a constraint that corresponds to assigning two vertices of the graph to different parts of the partition. The new matrix-lifting SDP relaxation is not dominated by the relaxation from [@ZhKaReWo:98], or vice versa.
The numerical results in Section \[sec:NumRez\] present the high potential of the new bounds. None of the presented SDP bounds strictly dominates any of the other bounds for all tested instances. The results indicate that breaking symmetry strengthens the bounds from [@Sot11; @ZhKaReWo:98] when the triangle and/or independent set constraints do not (or only slightly) improve the bound from [@Sot11]. For the cases that the triangle and/or independent set constraints significantly improve the bound from [@Sot11], the fixing approach does not seem to be very effective.
Symmetry and matrix algebras {#sec:SyminGr}
============================
A matrix $*$-algebra is a set of matrices that is closed under addition, scalar multiplication, matrix multiplication, and taking conjugate transposes. In [@GatPa:04; @EdeKl:10; @deKlDoPa:11] and others, it was proven that one can restrict optimization of an SDP problem to feasible points in a matrix $*$-algebra that contains the data matrices of that problem. In particular, the following theorem is proven.
[*[@deKlDoPa:11]*]{} \[thmAlg\] Let $\mathcal A$ denote a matrix $*$-algebra that contains the data matrices of an SDP problem as well as the identity matrix. If the SDP problem has an optimal solution, then it has an optimal solution in $\mathcal A$.
When the matrix $*$-algebra has small dimension, then one can exploit a basis of the algebra to reduce the size of the SDP considerably, see e.g, [@EdeKl:10; @dKPaDoSo:10; @KOP]. In the recent papers [@EdeKl:10; @dKPaDoSo:10; @deKlSot:10; @deKlSotNaTr:10], the authors considered matrix $*$-algebras that consist of matrices that commute with a given set of permutation matrices that correspond to automorphisms. Those $*$-algebras have a basis of $0$-$1$ matrices that can be efficiently computed. However, there exist also such $*$-algebras that are not coming from permutation groups, but from the ‘combinatorial symmetry’, as we shall see below. We also introduce the Laplacian algebra in order to obtain an eigenvalue bound that is suitable for any graph.
Every matrix $*$-algebra ${\mathcal{A}}$ has a canonical block-diagonal structure. This is a consequence of the theorem by Wedderburn [@Wedderburn] that states that there is a $*$-isomorphism $$\varphi:{\mathcal{A}}\longrightarrow \oplus_{i=1}^p \mathbb{C}^{n_i\times n_i},$$ i.e., a bijective linear map that preserves multiplication and conjugate transposition. One can exploit such a $*$-isomorphism in order to further reduce the size of an SDP.
Symmetry from automorphisms {#sec:perm}
----------------------------
An [*automorphism*]{} of a graph $G=(V,E)$ is a bijection $\pi:V \rightarrow V$ that preserves edges, that is, such that $\{\pi(x),\pi(y)\} \in E$ if and only if $\{x,y\}\in E$. The set of all automorphisms of $G$ forms a group under composition; this is called the [*automorphism group*]{} of $G$. The [*orbits*]{} of the action of the automorphism group acting on $V$ partition the vertex set $V$; two vertices are in the same orbit if and only if there is an automorphism mapping one to the other. The graph $G$ is [*vertex-transitive*]{} if its automorphism group acts transitively on vertices, that is, if for every two vertices, there is an automorphism that maps one to the other (and so there is just one orbit of vertices). Similarly, $G$ is [*edge-transitive*]{} if its automorphism group acts transitively on edges. Here, we identify the automorphism group of the graph with the automorphism group of its adjacency matrix. Therefore, if $G$ has adjacency matrix $A$ we will also refer to the automorphism group of the graph as $\operatorname{aut}(A) :=\{P
\in \Pi_n: AP=PA\}$, where $\Pi_n$ is the set of permutation matrices of size $n$.
Assume that $\mathcal G$ is a subgroup of the automorphism group of $A$. Then the centralizer ring (or commutant) of ${\mathcal G}$, i.e., ${\mathcal A}_{\mathcal G}=\{ X\in {\mathbb{R}}^{n\times n}: XP=PX, ~\forall P \in {\mathcal G} \}$ is a matrix $*$-algebra that contains $A$. One may obtain a basis for ${\mathcal A}_{\mathcal G}$ from the orbitals (i.e., the orbits of the action of $\mathcal G$ on ordered pairs of vertices) of the group $\mathcal G$. This basis, say $\{A_1$,…, $A_r\}$ forms a so-called coherent configuration.
\[def:coherent config\] A set of zero-one $n\times n$ matrices $ \{A_1,\ldots, A_r\}$ is called a *coherent configuration* of rank $r$ if it satisfies the following properties:
(i) $\sum_{i \in \mathcal{I}} A_i = I$ for some index set $\mathcal{I} \subset \{1,\ldots,r\}$ and $\sum_{i=1}^r A_i = J$,
(ii) $A_i^\mathrm{T} \in \{A_1,\ldots, A_r\}$ for $i=1,\ldots, r$,
(iii) There exist $p^h_{ij}$, such that $A_iA_j =\sum_{h=1}^r p^h_{ij}A_h$ for $i,j\in\{1,\ldots,r\}$.
As usual, the matrices $I$ and $J$ here denote the identity matrix and all-ones matrix, respectively. We call $\mathcal{A}:=\operatorname{span}\{A_1,\dots,A_r\}$ the associated [*coherent algebra*]{}, and this is clearly a matrix $*$-algebra. Note that in the case that $A_1,\dots,A_r$ are derived as orbitals of the group $\mathcal G$, it follows indeed that $\mathcal{A}={\mathcal A}_{\mathcal G}$. If the coherent configuration is commutative, that is, $A_iA_j=A_jA_i$ for all $i,j=1,\dots,r$, then we call it a (commutative) [*association scheme*]{}. In this case, $\mathcal{I}$ contains only one index, and it is common to call this index $0$ (so $A_0=I$), and $d:=r-1$ the number of classes of the association scheme.
In the case of an association scheme, all matrices can be diagonalized simultaneously, and the corresponding $*$-algebra has a canonical diagonal structure $\oplus_{j=0}^d \mathbb{C}$. The $*$-isomorphism $\varphi$ is then given by $\varphi(A_i)=\oplus_{j=0}^d P_{ji}$, where $P_{ji}$ is the eigenvalue of $A_i$ on the $j$-th eigenspace. The matrix $P=(P_{ji})$ of eigenvalues is called the [*eigenmatrix*]{} or [*character table*]{} of the association scheme.
Centralizer rings are typical examples of coherent algebras, but not the only ones. In general, the centralizer ring of the automorphism group of $A$ is not the smallest coherent algebra containing $A$, even though this is the case for well-known graphs such as the Johnson and Kneser graphs that we will encounter later in this paper. We could say that, in general, the smallest coherent configuration captures more symmetry than that coming from automorphisms of the graph. In this case, we say that there is more combinatorial symmetry.
Combinatorial symmetry {#sec:comb}
----------------------
Let us look at coherent configurations and the combinatorial symmetry that they capture in more detail. One should think of the (non-diagonal) matrices $A_i$ of a coherent configuration as the adjacency matrices of (possibly directed) graphs on $n$ vertices. The diagonal matrices represent the different ‘kinds’ of vertices (so there are $|\mathcal{I}|$ kinds of vertices; these generalize the orbits of vertices under the action of the automorphism group). The non-diagonal matrices $A_i$ represent the different ‘kinds’ of edges and non-edges.
In order to identify the ‘combinatorial symmetry’ in a graph, one has to find a coherent configuration (preferably of smallest rank) such that the adjacency matrix of the graph is in the corresponding coherent algebra $\mathcal{A}$. As mentioned before, not every coherent configuration comes from the orbitals of a permutation group. Most strongly regular graphs — a small example being the Shrikhande graph — indeed give rise to such examples. A (simple, undirected, and loopless) $\kappa$-regular graph $G=(V,E)$ on $n$ vertices is called [*strongly regular*]{} with parameters $(n, \kappa, \lambda, \mu)$ whenever it is not complete or edgeless and every two distinct vertices have $\lambda$ or $\mu$ common neighbors, depending on whether the two vertices are adjacent or not, respectively. If $A$ is the adjacency matrix of $G$, then this definition implies that $A^2=\kappa I+\lambda A+\mu(J-I-A)$, which implies furthermore that $\{I,A,J-I-A\}$ is an association scheme. The combinatorial symmetry thus tells us that there is one kind of vertex, one kind of edge, and one kind of non-edge. For the Shrikhande graph, a strongly regular graph with parameters $(16,6,2,2)$ (defined by $V=\mathbb{Z}_4^2$, where two vertices are adjacent if their difference is $\pm(1,0),\pm(0,1),$ or $\pm(1,1)$) however, the automorphism group indicates that there are two kinds of non-edges (depending on whether the two common neighbors of a non-edge are adjacent or not), and in total there are four (not three) orbitals. Doob graphs are direct products of $K_4$s and Shrikhande graphs, thus generalizing the Shrikhande graph to association schemes with more classes. In many optimization problems the combinatorial symmetry, captured by the concept of a coherent configuration or association scheme, can be exploited, see Section \[sec:aggregateConstr\] and e.g., . In Section \[ces:CombSymNum\], we will mention some numerical results for graphs that have more combinatorial symmetry than symmetry coming from automorphisms.
The Laplacian algebra {#Ex:LaplAlg}
---------------------
Let $A$ be an adjacency matrix of a connected graph $G$ and $L:=\operatorname{Diag}(Au_n)-A$ the Laplacian matrix of the graph. We introduce the matrix $*$-algebra consisting of all polynomials in $L$, and call this algebra the Laplacian algebra $\mathcal L$. This algebra has a convenient basis of idempotent matrices that are formed from an orthonormal basis of eigenvectors corresponding to the eigenvalues of $L$. In particular, if the distinct eigenvalues of $L$ are denoted by $0=\lambda_0 < \lambda_1 < \cdots < \lambda_d$, then we let $F_i=U_iU_i^{\mathrm{T}}$, where $U_i$ is a matrix having as columns an orthonormal basis of the eigenspace of $\lambda_i$, for $i=0, \ldots,d$. Then $\{F_0, \dots, F_d\}$ is a basis of $\mathcal L$ that satisfies the following properties:
- $F_0= \frac{1}{n} J$, $\sum\limits_{i=0}^dF_i=I$, $\sum\limits_{i=0}^d \lambda_iF_i~=L$
- $F_iF_j = \delta_{ij} F_i$, $\forall i,j$
- $F_i=F_i^*$, $\forall i$.
Note that $\operatorname{tr}F_i=f_i$, the multiplicity of eigenvalue $\lambda_i$ of $L$, for all $i$. Clearly, the operator $P$, where $$P(Y)=\sum_{i=0}^d \frac{\operatorname{tr}YF_i}{f_i} F_i$$ is the orthogonal projection onto $\mathcal L$.
We note that the Laplacian algebra of a strongly regular graph is the same as the corresponding coherent algebra $\operatorname{span}\{I,A,J-I-A\}$ (and a similar identity holds for graphs in association schemes).
The graph partition problem {#sec:gpp}
===========================
The minimum (resp. maximum) graph partition problem may be formulated as follows. Let $G=(V,E)$ be an undirected graph with vertex set $V$, where $|V|=n$ and edge set $E$, and $k\geq 2$ be a given integer. The goal is to find a partition of the vertex set into $k$ (disjoint) subsets $S_1,\ldots, S_k$ of specified sizes $m_1\geq \ldots \geq m_k $, where $\sum_{j=1}^k m_j =n$, such that the sum of weights of edges joining different sets $S_j$ is minimized (resp. maximized). The case when $k=2$ is known as the [*graph bisection problem*]{} (GBP). If all $m_j$ ($j=1,\ldots,k$) are equal, then we refer to the associated problem as the [*graph equipartition problem*]{} (GEP).
We denote by $A$ the adjacency matrix of $G$. For a given partition of the graph into $k$ subsets, let $X=(x_{ij})$ be the $n\times k$ matrix defined by $$x_{ij} = \left \{
\begin{array}{ll}
1 & \mbox{if $i\in S_j$ } \\
0 & \mbox{otherwise}.
\end{array} \right .$$ Note that the $j$th column of $X$ is the characteristic vector of $S_j$. The sum of weights of edges joining different sets, i.e., the cut of the partition, is equal to $\frac{1}{2} \operatorname{tr}A(J_n -XX^{\mathrm{T}})$. Thus, the minimum GPP problem can be formulated as $$\label{GPP}
\begin{array}{ll}
\min & \frac{1}{2} \operatorname{tr}A(J_n -XX^{\mathrm{T}}) \\[1ex]
{\rm s.t.} & Xu_k=u_n\\[1ex]
& X^{\mathrm{T}}u_n=m\\[1ex]
& x_{ij}\in \{0,1\},~~\forall i,j,
\end{array}$$ where $m=(m_1,\ldots, m_k)^\mathrm{T}$, and $u_k$ and $u_n$ denote all-ones vectors of sizes $k$ and $n$, respectively. It is easy to show that if $X$ is feasible for (\[GPP\]), then $$\label{obj1}
\frac{1}{2}\operatorname{tr}A(J_n -XX^{\mathrm{T}})=\frac{1}{2} \operatorname{tr}LXX^{\mathrm{T}},$$ where $L$ is the Laplacian matrix of the graph. We will use this alternative expression for the objective in Section \[sec:AnyGaph\].
A simplified and improved SDP relaxation for the GPP {#sect:EffRelax}
====================================================
In [@Sot11], the second author derived a matrix lifting SDP relaxation for the GPP. Extensive numerical results in [@Sot11] show that the matrix lifting SDP relaxation for the GPP provides competitive bounds and is solved significantly faster than any other known SDP bound for the GPP. The goal of this section is to further simplify the mentioned relaxation for highly symmetric graphs. Further, we show here how to aggregate, when possible, certain types of (additional) inequalities to obtain stronger bounds.
The matrix lifting relaxation in [@Sot11] is obtained after linearizing the objective function $\operatorname{tr}A(J_n -XX^{\mathrm{T}})$ by replacing $XX^{\mathrm{T}}$ with a new variable $Y$, and approximating the set $${\rm conv} \left \{ XX^{\mathrm{T}}: X\in {\mathbb{R}}^{n\times k}, ~Xu_k=u_n, ~X^{\mathrm{T}}u_n=m, ~x_{ij}\in \{0,1\} \right \}.$$ The following SDP relaxation for the GPP is thus obtained. $$({\rm GPP}_{\rm m})\quad
\begin{array}{rl}
\min & \frac{1}{2} \operatorname{tr}A(J_n -Y) \\[1ex]
{\rm s.t.} & \operatorname{diag}(Y)=u_n \\& \operatorname{tr}JY =\sum\limits_{i=1}^k m_i^2 \\& kY - J_n\succeq 0, ~~Y\geq 0.
\end{array}$$ We observe the following simplification of the relaxation ${\rm GPP}_{\rm m}$ for the bisection problem.
For the case of the bisection problem the nonnegativity constraint on the matrix variable in ${\rm GPP}_{\rm m}$ is redundant.
[*Proof*]{}. Let $Y$ be feasible for ${\rm GPP}_{\rm m}$ with $k=2$. We define $Z:= 2Y - J_n$. Now from $\operatorname{diag}(Y)=u_n$ it follows that $\operatorname{diag}(Z)=u_n$. Because $Z \succeq 0$, it follows that $-1\leq z_{ij} \leq 1$, which implies indeed that $y_{ij} \geq 0$.\
In order to strengthen ${\rm GPP}_{\rm m}$, one can add the triangle constraints $$\label{triangle}
y_{ab} + y_{ac}\leq 1 + y_{bc}, \quad \forall (a,b,c).$$ For a given triple $(a,b,c)$ of (distinct) vertices, the constraint ensures that if $a$ and $b$ belong to the same set of the partition and so do $a$ and $c$, then also $b$ and $c$ do so. There are $3{\genfrac(){0pt}{}{n}{3}}$ inequalities of type . For future reference, we refer to ${\rm GPP}_{\rm m\triangle}$ as the SDP relaxation that is obtained from ${\rm GPP}_{\rm m}$ by adding the triangle constraints.
One can also add to ${\rm GPP}_{\rm m}$ and/or ${\rm GPP}_{\rm m\triangle}$ the independent set constraints $$\label{indepSet}
\sum\limits_{a<b, ~a,b\in W} y_{ab}\geq 1, ~\mbox{for all}~ W ~\mbox{with}~ |W|=k+1.$$ These constraints ensure that the graph with adjacency matrix $Y$ has no independent set ($W$) of size $k+1$. There are ${\genfrac(){0pt}{}{n}{k+1}}$ inequalities of type . For future reference, we refer to ${\rm GPP}_{\rm m-ind}$ as the SDP relaxation that is obtained from ${\rm GPP}_{\rm m}$ by adding the independent set constraints.
Constraints and are also used by Karish and Rendl [@KarRend:98] to strengthen the SDP relaxation for the graph equipartition problem, and by the second author [@Sot11] to strengthen the SDP relaxation for the (general) graph partition problem. By adding constraints and/or to ${\rm GPP}_{\rm m}$, one obtains — in general — stronger relaxations that are computationally more demanding than ${\rm GPP}_{\rm m}$. In the following sections we will show how to efficiently compute, for graphs with symmetry, all above derived relaxations.
Symmetry and aggregating triangle and independent set constraints {#sec:aggregateConstr}
-----------------------------------------------------------------
It is well known how to exploit the symmetry in problems such as ${\rm GPP}_{\rm m}$ by using coherent configurations (or association schemes). Aggregating triangle inequalities was suggested by Goemans and Rendl [@GoeRend:99] in the context of the maximum cut problem for graphs in association schemes. Surprisingly, the suggestion by Goemans and Rendl was not followed so far in the literature, as far as we know. Here we will extend the approach successfully to coherent configurations. Moreover, we will aggregate the independent set inequalities for the case $k=2$.
Let us now consider graphs with symmetry, and assume that the data matrices of ${\rm GPP}_{\rm m}$ belong to the coherent algebra of a coherent configuration $\{ A_1,\ldots, A_r\}$. We will first show how this allows us to efficiently solve ${\rm GPP}_{\rm m}$, and subsequently how to aggregate additional triangle and/or independent set constraints.
Because of our assumption, we may consider $Y=\sum_{j=1}^r y_jA_j$ (see Theorem \[thmAlg\]) and the SDP relaxation ${\rm GPP}_{\rm m}$ reduces to $$\label{RSred}
\begin{array}{rl}
\min & \frac{1}{2} \operatorname{tr}AJ_n - \frac{1}{2}\sum\limits_{j=1}^r y_j \operatorname{tr}AA_j \\[1.5ex]
{\rm s.t.} & \sum\limits_{j\in {\mathcal I}} y_j \operatorname{diag}(A_j) =u_n \\[1ex]
& \sum\limits_{j=1}^r y_j \operatorname{tr}JA_j = \sum\limits_{i=1}^k m_i^2 \\[2.5ex]
& k\sum\limits_{j=1}^r y_jA_j - J_n\succeq 0, ~~y_j\geq 0,~~ j=1,\ldots,r,
\end{array}$$ where $\mathcal I$ is the subset of $\{1,\ldots, r\}$ that contains elements of the coherent configuration with nonzero diagonal (as in Definition \[def:coherent config\]). Note that solves significantly faster than ${\rm GPP}_{\rm m}$ when $r \ll n^2/2$. Also, the linear matrix inequality in can be block-diagonalized. In the following sections we will show that the SDP relaxation can be further simplified for some special types of graphs.
Next, we will reduce ${\rm GPP}_{\rm m\triangle}$ by adding aggregated triangle inequalities to . Because we cannot express a single triangle inequality in terms of the new variables in , we consider all inequalities, at once, of the same ‘type’, as follows. For a given triple of distinct vertices $(a,b,c)$ consider the triangle inequality $y_{ab} + y_{ac}\leq 1 + y_{bc}$. If $(A_i)_{ab}=1$, $(A_h)_{ac}=1$, and $(A_j)_{bc}=1$, then we say that this triangle inequality is of type $(i,j,h)$ ($i,j,h \in \{1,\ldots, r\} \setminus {\mathcal I}$; note that the indices $i,j,h$ are not necessary distinct). From Definition \[def:coherent config\] (iii) it follows that if $(A_i)_{ab}=1$, then the number of (directed) triangles containing the (directed) edge $(a,b)$ and for which $(A_h)_{ac}=1$ and $(A_{j'})_{cb}$ is equal to $p^i_{hj'}$ (here $j'$ is the index for which $A_{j'}=A_j^{\mathrm T}$). Now, if $Y$ is feasible for , then by summing all triangle inequalities of a given type $(i,j,h)$, the aggregated triangle inequality becomes $$\label{CohTriangAgg}
p^i_{hj'} \operatorname{tr}A_iY + p^h_{ij} \operatorname{tr}A_hY \leq p^i_{hj'} \operatorname{tr}A_iJ + p^j_{i'h} \operatorname{tr}A_{j}Y.$$ After exploiting the fact that $Y=\sum_{j=1}^r y_jA_j$, the aggregated inequality reduces to a linear inequality that can be added to the relaxation . The number of aggregated triangle inequalities is bounded by $r^3$ which may be significantly smaller than $3{\genfrac(){0pt}{}{n}{3}}$. So the SDP relaxation ${\rm GPP}_{\rm m\triangle}$ can be efficiently computed for small $r$.
For the bisection problem (i.e., $k=2$), the independent set constraints can be aggregated in a similar way as the triangle inequalities, and we obtain that $$\label{CohIndependAgg}
p^i_{hj'} \operatorname{tr}A_iY + p^h_{ij} \operatorname{tr}A_hY + p^j_{i'h} \operatorname{tr}A_{j}Y \geq p^i_{hj'} \operatorname{tr}A_iJ.$$ It is not clear how to aggregate the independent set constraints for $k\geq 3$. Note that in the case that the considered coherent configuration is an association scheme, all matrices are symmetric which simplifies the above aggregation processes.
Strongly regular graphs {#sec:SRG}
-----------------------
In this section we derive a closed form expression for the optimal objective value of the SDP relaxation ${\rm GPP}_{\rm m}$, for a strongly regular graph. A similar approach was used in to derive an eigenvalue bound for the equipartition problem from the SDP relaxation presented by Karish and Rendl [@KarRend:98]. Furthermore, we show that the triangle inequalities are redundant in ${\rm GPP}_{\rm m}$ for connected SRGs, except for the pentagon.
Let $A$ be the adjacency matrix of a strongly regular graph $G$ with parameters $(n, \kappa, \lambda, \mu)$, see Section \[sec:comb\]. Using the matrix equation $A^2=\kappa I+\lambda A+\mu(J-I-A)$, we can determine the eigenvalues of the matrix $A$ from the parameters of $G$, see e.g., [@BrHa]. Since $G$ is regular with valency $\kappa$, it follows that $\kappa$ is an eigenvalue of $A$ with eigenvector $u_n$. The matrix $A$ has exactly two distinct eigenvalues associated with eigenvectors orthogonal to $u_n$. These two eigenvalues are known as [*restricted eigenvalues*]{} and are usually denoted by $r \geq 0$ and $s<0$. The character table of the corresponding association scheme is $$\label{charTabSRG}
P= \left (
\begin{array}{ccc}
1 & \kappa & n-1-\kappa \\[1ex]
1 & r & -1-r\\[1ex]
1 & s & -1-s
\end{array}
\right ).$$ From Theorem \[thmAlg\] it follows that there exists an optimal solution $Y$ to ${\rm GPP}_{\rm m}$ in the coherent algebra spanned by $\{I, A, J-A-I \}$. Because of the constraints $\operatorname{diag}(Y)=u_n$ and $Y \geq 0$, there exist $y_1, y_2 \geq 0$ such that $$\label{strY}
Y= I + y_1A + y_2(J-A-I),$$ which we shall use to get an even simpler form than . The constraint $\operatorname{tr}JY = \sum_{i=1}^k m_i^2$ reduces to $$\label{eqtr}
n + n\kappa y_1 + (n^2-n\kappa -n)y_2 = \sum\limits_{i=1}^k m_i^2.$$ Since the matrices $\{I, A, J-A-I \}$ may be simultaneously diagonalized, the constraint $kY - J_n\succeq 0$ becomes a system of linear inequalities in the variables $y_1$ and $y_2$. In particular, after exploiting , the constraint $kY - J_n\succeq 0$ reduces to the three constraints $$\begin{aligned}
k+k \kappa y_1+ k(n-\kappa-1)y_2 -n &\geq 0, \label{lineq1} \\[1ex]
1+r y_1 - (r+1)y_2 &\geq 0, \label{lineq2} \\[1ex]
1+s y_1-(s+1)y_2 &\geq 0. \label{lineq3}\end{aligned}$$ Because $\sum\limits_{i=1}^k m_i^2 \geq n^2/k$ by Cauchy’s inequality, is actually implied by , so we may remove this first constraint. It remains only to rewrite the objective function, i.e., $$\frac{1}{2} \operatorname{tr}{A(J_n -Y)}= \frac{\kappa n(1-y_1)}{2}.$$ To summarize, the SDP bound ${\rm GPP}_{\rm m}$ can be obtained by solving the following linear programming (LP) problem $$\label{LPsrg}
\begin{array}{ll}
\min & \frac{1}{2}\kappa n(1-y_1) \\[1ex]
{\rm s.t.} & \kappa y_1 + (n-\kappa -1)y_2 = \frac{1}{n}\sum\limits_{i=1}^k m_i^2 -1 \\[1ex]
& 1+r y_1 - (r+1)y_2 \geq 0 \\[1ex] & 1+s y_1-(s+1)y_2 \geq 0 \\[1ex] & y_1 \geq0, ~y_2 \geq 0.
\end{array}$$ It is straightforward to derive a closed form expression for the optimal objective value of that is given in the following theorem.
\[strgClosedForm\] Let $G=(V,E)$ be a strongly regular graph with parameters $(n,\kappa,\lambda,\mu)$ and restricted eigenvalues $r \geq
0$ and $s<0$. Let $k$ and $m_i$ ($i=1,\ldots,k$) be positive integers such that $\sum_{j=1}^k m_j=n$. Then the SDP bound ${\rm GPP}_{\rm m}$ for the minimum GPP of $G$ is given by $$\max \left \{\frac{\kappa-r}{n} \sum_{i<j} m_i m_j, ~\frac{1}{2} \left ( n(\kappa+1) - \sum_i m_i^2 \right) \right \}.$$ Similarly, the SDP bound ${\rm GPP}_{\rm m}$ for the maximum GPP is given by $$\min \left \{ \frac{\kappa -s}{n} \sum_{i<j} m_i m_j, ~\frac{1}{2}\kappa n \right \}.$$
We remark that in general, the point where this minimum ${\rm GPP}_{\rm m}$ is attained is the intersection point of the first constraint and the boundary of the third constraint, with objective value $\frac{\kappa-r}{n} \sum_{i<j} m_i m_j$. However, if $\frac{1}{n}\sum\limits_{i=1}^k m_i^2 -1 \leq -\kappa/s$, then the minimum is attained at the $y_1$-axis, with objective value $\frac{1}{2} ( n(\kappa+1) - \sum_i m_i^2)$. For the case of the GEP, the results of Theorem \[strgClosedForm\] coincide with the results from [@deKlSotNaTr:10] and [@dKPaDoSo:10]. To see that, one should use the equation $n(\kappa +
rs)=(\kappa -s)(\kappa -r)$ (which follows from taking row sums of the equation $(A-rI)(A-sI)=(\kappa+rs) J$) and other standard equations for the parameters of strongly regular graphs (see e.g., [@BrHa]), and the fact that ${\rm GPP}_{\rm m}$ is equivalent to the SDP relaxation for the GEP problem by Karish and Rendl [@KarRend:98] (see [@Sot11]).
Next, we consider ${\rm GPP}_{\rm m\triangle}$ for SRGs. From and , with $A_1:=A$ and $A_2:=J-A-I$, it follows that for given $i,j,h \in \{1, 2\}$ the aggregated triangle inequality reduces to $$(\operatorname{tr}BA)y_1 + (\operatorname{tr}B(J-A-I))y_2 \leq b,$$ where $B=p^i_{hj}A_i + p^h_{ij} A_h- p^j_{ih} A_{j}$ and $b=p^i_{hj} \operatorname{tr}A_i J$. After simplifying and removing equivalent inequalities, at most the inequalities $$\begin{aligned}
y_1\leq 1, \quad y_2\leq 1,\label{tr1}\\
1+y_1 - 2y_2 \geq 0,\label{tr2}\\
1-2y_1 + y_2 \geq 0, \label{tr3}\end{aligned}$$ remain, and when some of the intersection numbers $p^h_{ij}$ vanish, even fewer remain (we omit details for the sake of readability). It is not hard to see that the constraints are always redundant to the constraints of (for example by drawing the feasible region), and that (cf. ) and (cf. ) are redundant except for $r<1$ and $s>-2$, respectively, which occurs only for the pentagon, disconnected SRGS, and complete multipartite graphs. However, for the disconnected SRGs and complete multipartite graphs, the ‘nonredundant’ constraints and (respectively) don’t occur precisely because of the vanishing of the relevant intersection numbers. In other words, adding triangle inequalities to ${\rm GPP}_{\rm m}$ for strongly regular graphs does [*not*]{} improve the bound, except possibly for the pentagon. On the other hand, if we consider the pentagon and add the triangle inequalities to ${\rm GPP}_{\rm m}$ with $m=(2,3)^{\mathrm T}$, then the bound improves and is tight.
For the bisection problem the aggregated independent set constraints are of the form , again with $A_1:=A$ and $A_2:=J-A-I$. Our numerical tests show that for many strongly regular graphs, the independent set constraints do not improve ${\rm GPP}_{\rm m}$, but there are also graphs for which ${\rm GPP}_{\rm m-ind}$ dominates ${\rm GPP}_{\rm m}$, see Section \[sec:bisec\] and \[sec:Aggregate\].
Johnson and Kneser graphs {#sec:JohnKne}
-------------------------
In this section we show that for the Johnson and Kneser graphs (on triples), the SDP bound ${\rm GPP}_{\rm m}$ can be obtained by solving a linear programming problem. We also present aggregated triangle and independent set inequalities that one may add to ${\rm GPP}_{\rm m}$. The Johnson graphs were also studied by Karloff [@HK:99] in the context of the max cut problem, in order to show that it is impossible to add valid linear inequalities to improve the performance ratio for the celebrated Goemans-Williamson approximation algorithm. Our results show that ${\rm GPP}_{\rm m}$ improves after adding the independent set constraints.
The Johnson and Kneser graphs are defined as follows. Let $\Omega$ be a fixed set of size $v$ and let $d$ be an integer such that $1\leq d\leq v/2$. The vertices of the Johnson scheme $J(v,d)$ are the subsets of $\Omega$ with size $d$. The adjacency matrices of the association scheme are defined by the size of the intersection of these subsets, in particular $(A_i)_{\omega,\omega'}=1$ if the subsets $\omega$ and $\omega'$ intersect in $d-i$ elements, for $i=0,\dots,d$. We remark that $A_1$ represents a so-called distance-regular graph $G$ — the [*Johnson graph*]{} — and $A_i$ represents being at distance $i$ in $G$. The [*Kneser graph*]{} $K(v,d)$ is the graph with adjacency matrix $A_d$, that is, two subsets are adjacent whenever they are disjoint. The Kneser graph $K(5,2)$ is the well-known Petersen graph.
For the case $d=2$, the Johnson graph is strongly regular and also known as a triangular graph. Consequently the bound ${\rm GPP}_{\rm m}$ of $J(v,2)$ has a closed form expression (apply Theorem \[strgClosedForm\] with $\kappa=2(v-2)$, $r=v-4$, and $s=-2$). Similarly, the Kneser graph $K(v,2)$ is strongly regular and the closed form expression for the GPP follows from Theorem \[strgClosedForm\] with $\kappa={\genfrac(){0pt}{}{v}{2}}-1-2(v-2)$, $r=1$, and $s=3-v$.
Here we focus on the next interesting group of Johnson and Kneser graphs, i.e., those on triples ($d=3$), but we also note that the restriction to the case $d=3$ is not essential. The eigenvalues (character table) of the Johnson scheme can be expressed in terms of Eberlein polynomials; see Delsarte’s thesis [@Delsarte73 Thm. 4.6]. For $d=3$, the character table is $$\label{charTabJoh}
P= \left (
\begin{array}{cccl}
1 & \theta_0 & \varphi(\theta_0) & {\genfrac(){0pt}{}{v}{3}}-1 -\theta_0-\varphi(\theta_0) \\[1ex]
1 & \theta_1 & \varphi(\theta_1) & -1 -\theta_1-\varphi(\theta_1)\\[1ex]
1 & \theta_2 & \varphi(\theta_2) & -1 -\theta_2-\varphi(\theta_2) \\[1ex]
1 & \theta_3 & \varphi(\theta_3) & -1 -\theta_3-\varphi(\theta_3)
\end{array}
\right ),$$ where $\varphi(\theta) = \frac{1}{4} \left (\theta^2-(v-2)\theta -3(v-3) \right) $, $\theta_0 = 3(v-3)$, $\theta_1 = 2v-9$, $\theta_2=v-7$, and $\theta_3 = -3$.
Let $A_1$ denote the adjacency matrix of $J(v,3)$, and $A_3$ the adjacency matrix of $K(v,3)$. We first simplify ${\rm GPP}_{\rm m}$ for the case that the graph under consideration is the Johnson graph $J(v,3)$. From Theorem \[thmAlg\], it follows that there exists an optimal solution $Y$ to ${\rm GPP}_{\rm m}$ which belongs to the coherent algebra spanned by $\{I,
A_1, A_2, A_3 \}$. Thus, there exist $y_1,y_2, y_3 \geq 0$ such that $$\label{strYJohn}
Y= I + y_1A_1 + y_2A_2 + y_3A_3.$$ Now, similar to the case of strongly regular graphs (see also [@dKPaDoSo:10; @deKlSotNaTr:10]), we can rewrite the objective function and constraints from ${\rm GPP}_{\rm m}$ by using . The derived LP for the minimum GPP is $$\label{LPJohn}
\begin{array}{ll}
\min & \frac{3}{2}{\genfrac(){0pt}{}{v}{3}}(v-3) (1-y_1) \\[1ex]
{\rm s.t.} & A_{eq}y = b_{eq} \\[1ex]
& A_{neq} y \geq b_{neq} \\[1ex]
& y\geq 0, ~y\in {\mathbb{R}}^3,
\end{array}$$ where $$\label{Aeq}
A_{eq} := \left ( 3(v-3), ~3{\genfrac(){0pt}{}{v-3}{2}}, ~{\genfrac(){0pt}{}{v-3}{3}} \right ),$$
$$\label{Aineq}
A_{neq}
:= \left (
\begin{array}{ccc}
2v-9 & \frac{1}{2} (v^2-13v +36) & \frac{1}{2} (-v^2+9v-20) \\[1.5ex]
v-7 & -2v+11 & v-5 \\[1.5ex]
-3 & 3 & -1
\end{array} \right ),$$
$$\label{beq}
b_{eq}:=\frac{1}{n} \sum\limits_{i=1}^k m_i^2 -1, \quad
b_{neq} := -(1, 1, 1)^{\mathrm{T}},$$
and $n={\genfrac(){0pt}{}{v}{3}}$ is the number of vertices of $J(v,3)$. To derive (\[LPJohn\]) we exploited the fact that the matrices $\{I, A_1, A_2, A_3 \}$ may be simultaneously diagonalized and we used the character table . Note that the computation time for solving is negligible and does not increase with the order of the Johnson graph.
\[JohnsonLP\] Let $J(v,3)$ be the Johnson graph, with $n$ vertices, and let $k$ and $m_i$ ($i=1,\ldots,k$) be positive integers such that $\sum_{i=1}^k m_i=n$. Then the SDP bound ${\rm GPP}_{\rm m}$ for the minimum GPP of $J(v,3)$ is equal to the optimal value of the linear programming problem .
Similarly, we simplify ${\rm GPP}_{\rm m}$ for the Kneser graph $K(v,3)$. Clearly, the only difference is the objective function which corresponds to the partition of the Kneser graph. The resulting LP relaxation is $$\label{LPKnes}
\begin{array}{ll}
\min & \frac{1}{2}{\genfrac(){0pt}{}{v}{3}}{\genfrac(){0pt}{}{v-3}{3}} (1-y_3) \\[1ex]
{\rm s.t.} & A_{eq}y = b_{eq} \\[1ex]
& A_{neq} y \geq b_{neq} \\[1ex]
& y\geq 0, ~y\in {\mathbb{R}}^3,
\end{array}$$ where $A_{eq}$-$b_{neq}$ are as in -. This leads to the following result.
\[KneserLP\] Let $K(v,3)$ be the Kneser graph on $n$ vertices, and let $k$ and $m_i$ ($i=1,\ldots,k$) be positive integers such that $\sum_{i=1}^k m_i=n$. Then the SDP bound ${\rm GPP}_{\rm m}$ for the minimum GPP of $K(v,3)$ is equal to the optimal value of the linear programming problem .
We can add to and the aggregated triangle inequalities . For given $i,j,h \in \{1, 2, 3\}$, these reduce to $$(\operatorname{tr}BA_1)y_1 + (\operatorname{tr}BA_2)y_2 + (\operatorname{tr}BA_3)y_3 \leq b,$$ where $B=p^i_{hj}A_i + p^h_{ij} A_h- p^j_{ih} A_{j}$ and $b=p^i_{hj} \operatorname{tr}A_i J$. After taking into consideration all possible choices of $i,j,h \in \{1, 2, 3\}$, there remain only seven (aggregated) triangle inequalities when $v=6$ and eleven when $v>6$. Numerical results indicate that these additional inequalities do not improve the solution obtained by solving and .
Since we know how to aggregate the independent set constraints when $k=2$, we tested the effect on the bound ${\rm GPP}_{\rm m}$ of adding these constraints. The numerical results show that the bound may improve, e.g., for the bisection of $J(7,3)$ with $m=(17,18)^{\mathrm T}$, it improves from $62$ to $64$.
A new eigenvalue bound for the GPP {#sec:AnyGaph}
==================================
In this section we present a closed form expression for the optimal value of a relaxation for the GPP for any graph and any $k\geq 2$. To the best of our knowledge, in the literature there are such general closed form bounds for the GPP only for $k=2$ (see e.g., Juvan and Mohar [@JuMo:92] and Falkner, Rendl, and Wolkowicz [@FeReWo:92]) and $k=3$ (see [@FeReWo:92]). Recently, Pong at al. [@PoSuWaWolkowicz] and Rendl et al. [@RendlLissPia] derived eigenvalue bounds for the vertex separator problem, which is closely related to the GPP.
In order to derive an eigenvalue bound for the GPP, we relax several constraints in ${\rm GPP}_{\rm m}$. In particular, we relax $\operatorname{diag}(Y)=u_n$ to $\operatorname{tr}Y=n$ and remove nonnegativity constraints. Moreover, we use to rewrite the objective in terms of the Laplacian matrix $L$, which leads to the relaxation $$\label{RSrelax}
\begin{array}{rl}
\min & \frac{1}{2} \operatorname{tr}LY \\[1ex]
{\rm s.t.} & \operatorname{tr}Y=n \\& \operatorname{tr}JY=\sum\limits_{i=1}^k m_i^2 \\[1.5ex]
& kY - J_n\succeq 0.
\end{array}$$ Recall from Section \[Ex:LaplAlg\] that we denote the distinct Laplacian eigenvalues of the graph by $0=
\lambda_0 < \lambda_1 < \ldots < \lambda_d$, and their corresponding multiplicities $f_i$, for $i=0,\ldots,d$, and let $\mathcal L=\operatorname{span}\{ F_0,\ldots,F_d\}$ be the Laplacian algebra of the graph. By Theorem \[thmAlg\], there exists an optimal solution $Y$ to in $\mathcal L$, and therefore we may assume that $Y=\sum_{i=0}^d y_iF_i,$ where $y_i\in {\mathbb{R}}$ ($i=0,\ldots,d$) (as before, these are the new variables). We will exploit this to rewrite . The objective is $$\operatorname{tr}LY= \operatorname{tr}(\sum_{i=0}^d \lambda_i F_i )( \sum_{j=0}^d y_j F_j ) =\sum_{i=0}^d \lambda_if_iy_i.$$ The constraint $ \operatorname{tr}Y=n $ reduces to $ \sum_{i=0}^d f_i y_i = n, $ while the constraint $\operatorname{tr}JY=\sum_{i=1}^k
m_i^2 $ reduces to $y_0= (\sum_{i=1}^k m_i^2)/n$. It remains only to reformulate the semidefinite constraint: $$kY - J =k\sum_{i=0}^d y_iF_i - J=
\frac{k\sum_{i=1}^k m_i^2 -n^2 }{n^2}J + \sum_{i=1}^d y_i F_i \succeq 0.$$ From this, it follows that $y_i\geq 0$. To conclude, the SDP relaxation reduces to $$\label{LPGraph}
\begin{array}{rl}
\min & \frac12 \sum\limits_{i=1}^d \lambda_if_iy_i \\[2ex]
{\rm s.t.} & \sum\limits_{i=1}^d f_i y_i = \frac{2}{n} \sum\limits_{i<j}m_im_j \\[2ex]
& y_i \geq 0, \quad i=1,\ldots,d.
\end{array}$$
\[thm:AnyGraph\] Let $G$ be a graph on $n$ vertices, and let $k$ and $m_i$ $(i=1,\ldots,k)$ be positive integers such that $\sum_{i=1}^k m_i=n$. Then the SDP lower bound for the minimum GPP of $G$ is equal to $$\frac{\lambda_1}{n} \sum\limits_{i<j}m_im_j,$$ and the SDP upper bound for the maximum GPP of $G$ that is obtained by replacing $\min$ by $\max$ in is $$\frac{\lambda_d}{n} \sum\limits_{i<j}m_im_j.$$
[*Proof*]{}. This follows from .\
Our eigenvalue bound for the bisection problem (the case $k=2$) coincides with a well-known result in spectral graph theory, see [@JuMo:92; @MohPolj:93]. Therefore, Theorem \[thm:AnyGraph\] may be seen as a generalization of this result for the $2$-partition problem to any $k$-partition problem. Falkner, Rendl, and Wolkowicz [@FeReWo:92] derived a closed form bound for the minimum $3$-partition problem of the form $$\label{FRW}
\frac{1}{2} \theta_1 \mu_1 + \frac{1}{2} \theta_2 \mu_2,$$ where $\mu_{1,2}= (m_1m_2+m_1m_3+m_2m_3 \pm \sqrt{ m_1^2m_2^2 + m_1^2m_3^2 + m_2^2m_3^2 -nm_1m_2m_3 })/n$, and $\theta_1$ and $\theta_2$ are the two smallest nonzero (not necessarily distinct) Laplacian eigenvalues. It is clear that this lower bound coincides with ours when $\theta_1=\theta_2$ ($=\lambda_1$). To the best of our knowledge there are no other closed form bounds for the minimum $k$-partition problem when $k>3$, or for the maximum $k$-partition problem when $k> 2$ that is applicable to any graph. Although the bounds from Theorem \[LPGraph\] are, in general, dominated by the bounds obtained from ${\rm GPP}_{\rm m}$, they may be useful in the theoretical analysis of the GPP and related problems. Still, our numerical results show that for many problems the new eigenvalue bound is equal to ${\rm GPP}_{\rm m}$, see Section \[sec:NumRez\]. We finally remark that for strongly regular graphs, the eigenvalue bounds also follow from Theorem \[strgClosedForm\] because $\lambda_1=\kappa-r$ and $\lambda_d=\kappa-s$ (indeed, they are even the same unless $\sum_i m_i^2 <
\frac{r+1}{n-\kappa+r}n^2$ or $\sum_i m_i^2 < \frac{-s}{\kappa-s}n^2$, respectively). This is related to the fact that the Laplacian algebra and the used coherent algebra are the same for strongly regular graphs.
Improved relaxations for the GPP {#sect:EQP}
================================
An improved relaxation from the quadratic assignment problem {#sec:impVec}
------------------------------------------------------------
In this section, we derive a new SDP relaxation for the GPP that is obtained by strengthening the SDP relaxation of the more general quadratic assignment problem by Zhao, Karisch, Rendl, and Wolkowicz [@ZhKaReWo:98], by adding two constraints that correspond to assigning two vertices of the graph to different parts of the partition. A similar approach was used by the authors [@vDamSo:12] to derive the best known bounds for the bandwidth problem of graphs with symmetry. For the equipartition problem, the new relaxation dominates the relaxation for the GEP by De Klerk et al. [@dKPaDoSo:10], that is obtained by fixing one vertex of the graph. Our new bound is however not restricted to the equipartition problem, and it is also suitable for graphs with symmetry.
The GPP is a special case of the quadratic assignment problem $$\min_{X\in \Pi_n} \frac{1}{2} \operatorname{tr}AXBX^{\mathrm T},$$ where $A$ and $B$ are given symmetric $n \times n$ matrices, and $\Pi_n$ is the set of $n\times n$ permutation matrices. For the graph partition problem, $A$ is the adjacency matrix of the relevant graph $G$ with $n$ vertices, and $B$ is the adjacency matrix of the complete multipartite graph $K_{m_1,\ldots,m_k}$ with $k$ classes of sizes $m_1$,…,$m_k$ (with $m_1+\ldots +m_k=n$). For example, for the $k$-equipartition problem with $n=km$, $$\label{defBEQ}
B:=(J_k-I_k)\otimes J_m,$$ (where $\otimes$ is the Kronecker product). In the general case, $B$ has the same block structure, but the sizes $m_1$,…,$m_k$ of the blocks vary. In particular, for the bisection problem with $m=(m_1,m_2)^{\mathrm T}$, $$\label{defBBis}
B:=\left (
\begin{array}{cc}
0_{m_1\times m_1} & J_{m_1\times m_2}\\
J_{m_2\times m_1} & 0_{m_2\times m_2}\\
\end{array} \right ).$$ Now it follows that the following ‘vector-lifting’ SDP relaxation of this particular QAP (see Zhao et al. [@ZhKaReWo:98] and Povh and Rendl [@PoRe:09]) is also a relaxation for the GPP: $$(\mbox{GPP}_{\rm QAP})~~~~
\begin{array}{rcl}
\min && \frac{1}{2} \operatorname{tr}(B\otimes A)Y\\[1ex]
{\rm s.t.} && \operatorname{tr}(I_n\otimes E_{jj})Y=1, ~~\operatorname{tr}(E_{jj}\otimes I_n)Y=1, \quad j=1,\ldots,n\\[1ex]
&& \operatorname{tr}(I_n\otimes(J_n-I_n)+(J_n-I_n)\otimes I_n)Y=0\\[1ex]
&& \operatorname{tr}JY=n^2 \\[1ex]
&& Y\geq 0, ~~Y \succeq 0,
\end{array}$$ where (here and below) $E_{ij}=e_i e_j^{\mathrm{T}}$. In [@Sot10] (see also [@Sot11]) it is proven that for the equipartition problem, the relaxations $\mbox{GPP}_{\rm QAP}$ and $\mbox{GPP}_{\rm m}$ are equivalent, and in [@Sot11] that the first dominates the second for the bisection problem. De Klerk et al. [@dKPaDoSo:10] strengthened $\mbox{GPP}_{\rm QAP}$ for the GEP by adding a constraint that corresponds to assigning an arbitrary vertex of the complete multipartite graph to a vertex in the graph.
Here, we extend the approach from [@dKPaDoSo:10] (see also [@vDamSo:12]) and assign (several times) a pair of vertices of $G$ to an edge in $K_{m_1,\ldots,m_k}$. By symmetry, we have to do this for one pair of vertices in each orbital (recall from Section \[sec:SyminGr\] that the orbitals actually represent the ‘different’ kinds of pairs of vertices; (ordered) edges, and (ordered) nonedges in the graph $G$). Let us assume that there are $t$ such orbitals ${\mathcal O}_h$ ($h=1,2\ldots,t$) of edges and nonedges, and note that for highly symmetric graphs, $t$ is relatively small. We formally state the above idea in the following theorem.
\[thmFixnew\] Let $G$ be an undirected graph on $n$ vertices with adjacency matrix $A$, and let $\mathcal{O}_h$ $(h=1,2,\dots,t)$ be the orbitals of edges and nonedges coming from the automorphism group of $G$. Let $(s_1,s_2)$ be an arbitrary edge in $K_{m_1,\ldots,m_k}$ while $(r_{h1},r_{h2})$ is an arbitrary pair of vertices in $\mathcal{O}_h$ $(h=1,2,\dots,t)$. Let $\Pi_n(h)$ be the set of matrices $X \in \Pi_n$ such that $X_{r_{h1},s_1}=1$ and $X_{r_{h2},s_2}=1$ $(h=1,2,\dots,t)$. Then $$\min_{X \in \Pi_n} \operatorname{tr}X^{\mathrm{T}}AXB =
\min_{h=1,\dots,t} \min_{X \in \Pi_{n}(h)} \operatorname{tr}X^{\mathrm{T}}AXB.$$
Similar to the proof of Theorem 10 in [@vDamSo:12].\
Clearly, exploiting this requires solving several SDP (sub)problems. However, if we assign to an edge $(s_1,s_2)$ in $K_{m_1,\ldots,m_k}$ a pair of vertices $(r_{h1},r_{h2})$ from $\mathcal{O}_h$ $(h=1,2,\dots,t)$, then we can add to $\mbox{GPP}_{\rm QAP}$ the constraints $$\operatorname{tr}(E_{s_i,s_i} \otimes E_{r_{hi},r_{hi}})Y=1, \quad i=1,2.$$ Thus, we obtain several SDP problems of the form $$\label{ZWGEPh}
\begin{array}{rcl}
\mu^h := \min && \frac{1}{2} \operatorname{tr}(B\otimes A)Y\\[1ex]
{\rm s.t.} && \operatorname{tr}(I_n\otimes E_{jj})Y=1, ~~\operatorname{tr}(E_{jj}\otimes I_n)Y=1, \quad j=1,\ldots,n\\[1ex]
&& \operatorname{tr}(E_{s_i,s_i} \otimes E_{r_{hi},r_{hi}})Y=1, \quad i=1,2 \\[1ex]
&& \operatorname{tr}(I_n\otimes(J_n-I_n)+(J_n-I_n)\otimes I_n)Y=0\\[1ex]
&& \operatorname{tr}JY=n^2 \\[1ex]
&& Y\geq 0, ~~Y \succeq 0,
\end{array}$$ where $h=1,\ldots,t$, and the new lower bound for the GPP is \[relaxFix\] $${\rm GPP}_{\rm fix}=\min\limits_{h=1,\ldots,t}\mu^h.$$ We remark that $\mu^h$ is a relaxation that depends on the particular edge $(s_1,s_2)$ (but not on the particular pair $(r_{h1},r_{h2}) \in \mathcal{O}_h$). However, for the GEP and GBP (but not in general!), the lower bound ${\rm GPP}_{\rm fix}$ is independent of the edge $(s_1,s_2)$. This is due to the fact that $K_{m,\ldots,m}$ and $K_{m_1,m_2}$ are edge-transitive.
The following proposition follows directly from .
\[cor:newbounddominates\] Let $(s_1,s_2)$ be an arbitrary edge in $K_{m_1,\ldots,m_k}$ and $(r_{h1},r_{h2})$ be an arbitrary pair of vertices in $\mathcal{O}_h$ $(h=1,2,\dots,t)$. Then the SDP bound ${\rm GPP}_{\rm fix}$ dominates ${\rm GPP}_{\rm QAP}$.
Similarly, the following corollary for the GEP follows.
\[cor:newbounddominates1\] Consider the equipartition problem. Let $(s_1,s_2)$ be an arbitrary edge in $K_{m,\ldots,m}$, and $(r_{h1},r_{h2})$ be an arbitrary pair of vertices in $\mathcal{O}_h$ $(h=1,2,\dots,t)$. Then the SDP relaxation ${\rm GPP}_{\rm fix}$ dominates the SDP relaxation from [@dKPaDoSo:10 Eq. 10].
It is, in general, hard to solve (and thus ${\rm GPP}_{\rm fix}$) for $n\geq 16$, see e.g., [@ReSo]. Therefore we need to further exploit the symmetry of $K_{m_1,\ldots,m_k}$ (in particular, consider pointwise stabilizers) and the graphs under consideration, see also [@vDamSo:12]. We do this by applying the general theory of symmetry reduction to the SDP subproblems in a mechanical way, as described in, e.g., [@vDamSo:12; @deKlSot:10; @dKPaDoSo:10; @dKSo:12]. After symmetry reduction of the largest linear matrix inequality contains matrices of size $3n$ (resp. $2n$) for the GEP (resp. GBP).
Our numerical results show that ${\rm GPP}_{\rm fix}$ can be a significantly stronger bound than ${\rm GPP}_{\rm m}$ for highly symmetric graphs (i.e., for which $t$ is very small) and for cases that the bound obtained by solving ${\rm GPP}_{\rm m}$ cannot be improved by adding triangle and independent set constraints. This could be a consequence of the fact that (some) symmetry in the graph has been broken.
An improved matrix-lifting relaxation {#sec:impMat}
-------------------------------------
Clearly, we can exploit the idea of fixing a pair of vertices in a graph also in the context of the matrix lifting relaxation ${\rm GPP}_{\rm m}$. Assume again that for the given graph $G$ there are $t$ orbitals ${\mathcal O}_h$ ($h=1,2\ldots,t$) of edges and nonedges. Now, in order to assign two (arbitrary) vertices $(r_{h1},r_{h2})\in {\mathcal O}_h$ of the graph $G$ to two different subsets, we add to ${\rm GPP}_{\rm m}$ the constraint $$\operatorname{tr}( E_{r_{h1},r_{h2}}+E_{r_{h2},r_{h1}})Y=0.$$ Therefore, computing this new lower bound reduces to solving $t$ subproblems of the form $$\label{RSfixPom}
\begin{array}{rl}
\nu^*_h :=\min & \frac{1}{2} \operatorname{tr}A(J_n -Y) \\[1ex]
{\rm s.t.} & \operatorname{diag}(Y)=u_n \\[1ex]
& \operatorname{tr}JY=\sum\limits_{i=1}^k m_i^2 \\[2ex]
& \operatorname{tr}( E_{r_{h1},r_{h2}}+E_{r_{h2},r_{h1}})Y=0 \\[2ex]
& kY - J_n\succeq 0, ~~Y\geq 0,
\end{array}$$ ($h=1,\ldots,t$). Consequently, the new matrix-lifting lower bound is a minimum over $t$ SDP bounds, i.e., $$\label{RSfix}
\min\limits_{h=1,\ldots, t} \nu^*_h.$$ The following result follows immediately.
The SDP bound dominates ${\rm GPP}_{\rm m}$.
In order to solve (and thus ) we further exploit symmetry in the graphs under consideration in a similar way as described in Section \[sec:aggregateConstr\].
Our numerical results suggest that the new SDP bound is dominated by $\mbox{GPP}_{\rm fix}$, and also that is not dominated by $\mbox{GPP}_{\rm QAP}$, or vice versa.
Numerical results {#sec:NumRez}
=================
In this section we present numerical results for the graph partition problem. In particular, we compare bounds from all the presented relaxations and several relaxations from the literature. All relaxations were solved with SeDuMi [@sedumi] using the Yalmip interface [@YALMIP] on an Intel Xeon X5680, $3.33$ GHz dual-core processor with 32 GB memory. To compute orbitals, we used GAP [@gap].
Why symmetry?
-------------
We first show the importance of exploiting symmetry in graphs, when applicable, in order to compute SDP bounds. In Table \[tab1\] we consider the planar unweighted [grid graphs]{}, where $|V|= \sharp$ rows $\times$ $\sharp$ columns. They are generated by the rudy graph generator [@Rinaldi]. Table \[tab1\] presents computational times, in seconds, required to solve ${\rm GPP}_{\rm m}$ with and without exploiting symmetry (see also Table $3$ in [@Sot11 online supplement]). The table reads as follows. In the first two columns, the sizes of the graphs and the sizes of the partitions are specified. The third column lists computational times required to solve ${\rm GPP}_{\rm m}$ without exploiting symmetry. The fourth column provides the rank of the associated matrix $*$-algebra that is obtained as the centralizer ring of the automorphism group of the graph, and the last column contains computational times required to solve ${\rm GPP}_{\rm m}$ after exploiting symmetry. Note that even though the graphs are not highly symmetric, the reduction in computational times after exploiting symmetry is significant.
[ccccc]{} $|V|$ & $m^{\mathrm T}$ & no symmetry & $r_{\rm aut}$ & symmetry\
$9 \times 9$ & $(35, 30, 16)$ & 198.35 & 861 & 1.70\
$10 \times 10$ & $(50, 25, 25)$ & 799.21 & 1275 & 3.41\
Combinatorial symmetry vs. group symmetry {#ces:CombSymNum}
-----------------------------------------
In this section we list numerical results for several graphs that have (substantially) more combinatorial symmetry than symmetry coming from the automorphism group. We provide the eigenvalue bound of Theorem \[thm:AnyGraph\] and ${\rm GPP}_{\rm m}$ for the GEP of those graphs.
Table \[tab:other\] reads as follows. In the first three columns, we list the graphs, the number of vertices, and the number of parts $k$ of the equipartition, respectively. [Chang3]{} is one of the strongly regular graphs introduced by Chang [@Chang] (see also [@BrChang]). For a description of the [Doob]{} graph, see Section \[sec:comb\]. Graphs [A64v30]{} and [A64vEnd]{} are strongly regular graphs with parameters $(64,18,2,6)$ obtained by Haemers and Spence [@HaeSpe:01], where 30 (resp. End) means that it is the 30th (resp. last) graph in the list, see also <http://www.maths.gla.ac.uk/~es/SRGs/64-18-2-6>. The [design]{} graph is the bipartite incidence graph of a symmetric $2$-$(45,12,3)$-design, see <http://www.maths.gla.ac.uk/~es/polar/45-12-3.36>. In the fourth column of Table \[tab:other\] we give the eigenvalue lower bound from Theorem \[thm:AnyGraph\], while in the fifth column, we list the SDP bound ${\rm GPP}_{\rm m}$. All presented bounds are rounded up to the closest integer. In the last four columns we list the rank of the coherent configuration corresponding to the graph’s combinatorial symmetry, computational times required to solve ${\rm GPP}_{\rm
m}$ after exploiting its combinatorial symmetry, the rank of the coherent configuration coming from the automorphism group, and the corresponding computational times, respectively. If a graph is strongly regular we do not report the computational time since for such graphs we use the closed form expression from Theorem \[strgClosedForm\].
Note that [A64vEnd]{} does not have any symmetry coming from automorphisms, but it has lots of combinatorial symmetry. We remark that the listed graphs are [*not*]{} isolated cases, but only a sample that shows that combinatorial symmetry may differ significantly from group symmetry.
[ccccccccr]{} $G$ & $n$ & $k$ & eig & ${\rm GPP}_{\rm m}$ & $r_{\rm comb}$ & time & $r_{\rm aut}$ & time\
[Chang3]{} & 28 & 7 & 96 & 126 & 3 & – & 14 & 0.23\
[A64v30]{} & 64 & $8$ & 448 & 448 & 3 & – & 90 & 0.61\
[Doob]{} & 64 & $8$ & 112 & 160 & 4 & 0.34 & 8 & 0.41\
[A64vEnd]{} & 64 & $4$ & 384 & 384 & 3 & – & – & 14.33\
[design]{} & 90 & $9$ & 360 & 360 & 4 & 0.40 & 2074 & 4.56\
The graph equipartition problem
-------------------------------
In this section we compare different relaxations for the equipartition problem. We first present results for the [Higman-Sims]{} graph [@HigSim:68], see Table \[tab2\]. The [Higman-Sims]{} graph is a strongly regular graph with parameters $(100,22,0,6)$. The max and min $k$-equipartition problem for this graph was studied in [@dKPaDoSo:10; @deKlSotNaTr:10].
Table \[tab2\] reads as follows. The first column specifies whether we are solving a minimization or maximization problem, while the second column shows the number of parts $k$ of the equipartition. The third column provides the new eigenvalue bound, see Theorem \[thm:AnyGraph\]. The fourth column lists $\mbox{GPP}_{\rm m}$ which is known to be equivalent to $\mbox{GPP}_{\rm QAP}$ for the case of the equipartition (for a proof, see [@Sot10; @Sot11]). The fifth column provides bounds obtained by solving the relaxation from [@dKPaDoSo:10] (that is, the improved $\mbox{GPP}_{\rm QAP}$ by adding a constraint that corresponds to fixing a vertex in the graph). In the sixth column, we list the bounds obtained by solving GPP$_{\rm fix}$, see page , while the seventh column contains the bounds obtained by solving the SDP relaxation from [@deKlSotNaTr:10]. The latter relaxation is the ‘level two reformulation-linearization technique-type relaxation for the QAP with an additional linear matrix inequality constraint’, which is known to be at least as strong as $\mbox{GPP}_{\rm QAP}$.
In Table \[tab2\], the bounds improve along with increasing complexity of the relaxations; the strongest bound is from [@deKlSotNaTr:10]. Note however that the bound from [@deKlSotNaTr:10] is appropriate only for vertex-transitive graphs, while our bounds do [*not*]{} have such a restriction. The last column provides bounds obtained from heuristics that are taken from Table $2$ and $3$ in [@deKlSotNaTr:10].
We remark that the SDP bound provides the same bounds as $\mbox{GPP}_{\rm m}$ for all problems in Table \[tab2\].\
[lccccccc]{} $\max$ & $k$ & eig & $\mbox{GPP}_{\rm m}$ & [@dKPaDoSo:10] & GPP$_{\rm fix}$ & [@deKlSotNaTr:10] & lower bound\
& 4 & 1125 & 1100 & 1097 & 1094 & 1048 & 1006\
& 5 & 1200 & 1100 & 1100 & 1100& 1100& 1068\
$\min $ & &&&&& & upper bound\
& 20 & 950 & 950 & 951 & 951 & 975 & 980\
& 25 & 960 & 960 & 963 & 964 & 1000 & 1000\
In Table \[tab:maxKpart1\] we present results for the maximum equipartition problem for several Johnson and Kneser graphs (see Section \[sec:JohnKne\]). The table reads as follows. In the first column we list the graphs, in the second column the number of vertices, and in the third column the number of parts $k$ of the equipartition. In the fourth and fifth column we list the new eigenvalue bound and ${\rm GPP}_{\rm m}$, respectively. In the last two columns of Table \[tab:maxKpart1\] we list GPP$_{\rm fix}$ and the corresponding required computational times.
In Table \[tab:maxKpart1\] we do not report computational times for the new eigenvalue bound and ${\rm GPP}_{\rm m}$ since they are negligible, see Sections \[sec:SRG\], \[sec:JohnKne\], and \[sec:AnyGaph\]. Also, since adding triangle inequalities to ${\rm GPP}_{\rm m}$ for the problems in Table \[tab:maxKpart1\] do not improve ${\rm GPP}_{\rm m}$, we did not make a separate column for ${\rm GPP}_{\rm m\triangle}$. All presented bounds are rounded down to the closest integer.
The results show that the new bound GPP$_{\rm fix}$ can be significantly stronger than ${\rm GPP}_{\rm m}$, in particular for problems when the eigenvalue bound and ${\rm GPP}_{\rm m}$ provide the same bound. The results also show that the eigenvalue bound performs well for most of the instances.
[ccccccr]{} $G$ &$n$ & $k$ & eig & ${\rm GPP}_{\rm m}$ & GPP$_{\rm fix}$ & time\
$K(8,2)$ & 28 & 4 & 210 & 210 & 204 & 7.66\
$K(9,2)$ & 36 & 3 & 324 & 324 & 317 & 14.43\
$K(9,2)$ & 36 & 12 & 444 & 378 & 378 & 5.32\
$K(12,2)$ & 66 & 6 & 1485 & 1485 & 1473 & 31.79\
$J(8,3)$ & 56 & 4 & 378 & 378 & 377 & 148.54\
$K(9,3)$ & 84 & 3 & 840 & 840 & 828 & 551.99\
$K(15,2)$ & 105 & 5 & 3780 & 3780 & 3772 & 106.97\
$K(10,3)$ & 120 & 3 & 2000 & 2000 & 1979 & 1097.10\
The graph bisection problem {#sec:bisec}
---------------------------
In this section we present numerical results for the graph bisection problem. All graphs in Table \[tab:GBP\] are strongly regular. The table reads as follows. In the first column we list the graphs. The [Johnson]{} graphs are defined in Section \[sec:JohnKne\], whereas the [Hoffman-Singleton]{} ([HS]{}) graph, the [Gewirtz]{} graph, and the $M_{22}$ graph are the unique strongly regular graphs with parameters $(50,7,0,1)$, $(56,10,0,2)$, and $(77,16,0,4)$, respectively. In the second column of Table \[tab:GBP\] we list the number of vertices in the corresponding graph, while the third column contains the sizes of the subsets. We choose these sizes arbitrarily. In the remaining columns we provide the lower bounds $\mbox{GPP}_{\rm m}$, , $\mbox{GPP}_{\rm QAP}$, $\mbox{GPP}_{\rm m-ind}$, and GPP$_{\rm fix}$, respectively. All bounds are rounded up to the closest integer. In Table \[tab:GBPtime\] we provide computational times required to solve the problems from Table \[tab:GBP\] (the times to compute GPP$_{\rm fix}$ and are sums of computational times of all subproblems needed to obtain the bounds).
From Table \[tab:GBP\] it follows that is not dominated by $\mbox{GPP}_{\rm QAP}$, or vice versa. Similarly, we may conclude that $\mbox{GPP}_{\rm QAP}$ and GPP$_{\rm fix}$ are not dominated by $\mbox{GPP}_{\rm m-ind}$, or vice versa. One more interesting observation is that for all tested instances, the new eigenvalue bound is equal to the bound obtained by solving $\mbox{GPP}_{\rm m}$.
[cccccccc]{} $G$ & $n$ & $m^\mathrm{T}$ & $\mbox{GPP}_{\rm m}$ & & $\mbox{GPP}_{\rm QAP}$ & $\mbox{GPP}_{\rm m-ind}$ & GPP$_{\rm fix}$\
$J(6,2)$& 15 & (8,7) & 23 & 23 & 23 & 26 & 24\
$J(7,2)$& 21 & (12,9) & 36 & 37 & 36 & 38 & 38\
$J(9,2)$& 36 & (26,10) & 65 & 66 & 65 & 65 & 67\
[HS]{} & 50 & (46,4) & 19 & 19 & 19 & 19 & 21\
[Gewirtz]{} & 56 & (53,3) & 23 &23 & 24 & 23 & 26\
$J(12,2)$& 66 & (33,33) & 198 & 199& 198 & 198 & 199\
$M_{22}$ & 77 & (74,3)& 41 & 41 & 42 & 41 & 44\
$J(15,2)$& 105 & (85,20) & 243 & 243 & 243 & 243 & 246\
[ccccccc]{} $G$ & $n$ & & $\mbox{GPP}_{\rm QAP}$ & $\mbox{GPP}_{\rm m-ind}$ & GPP$_{\rm fix}$\
$J(6,2)$& 15 & 0.42 & 0.20 & 0.17 & 2.30\
$J(7,2)$& 21 & 0.66 & 0.23 & 0.18 & 2.70\
$J(9,2)$& 36 & 1.06 & 0.48 & 0.23 & 5.21\
HS & 50 & 0.68 & 0.62 & 0.30 & 6.58\
Gewirtz & 56 & 1.67 & 1.66 & 0.34 & 15.87\
$J(12,2)$& 66 & 1.02 & 0.95 & 0.34 & 8.87\
$M_{22}$ & 77 & 1.55 & 3.15 & 0.30 & 19.12\
$J(15,2)$& 105& 2.27 & 3.41 & 0.54 & 29.19\
Aggregated triangle and independent set constraints {#sec:Aggregate}
---------------------------------------------------
In Section \[sec:aggregateConstr\] we showed how to aggregate triangle and independent set constraints for the case that the data matrices of $\mbox{GPP}_{\rm m}$ belong to a coherent algebra, and that this is efficient when the rank of this algebra is small. In this section we provide numerical results for graphs whose adjacency matrices indeed belong to a coherent algebra of small rank. In particular, besides the Johnson graph $J(7,2)$, we consider the distance-regular [Pappus]{}, [Desargues]{}, [Foster]{}, and [Biggs-Smith]{} graphs (see [@bcn]), as well as the [Dyck]{} graph (the graph on the triangles of the Shrikhande graph, where two triangles are adjacent if they share an edge). In the third column of Table \[tab:Aggregate\], we list the rank of the smallest coherent configuration containing the corresponding adjacency matrix (i.e., coming from the combinatorial symmetry). In all cases, this coherent configuration is the same as the one coming from the automorphism group. In columns five to eight, we list bounds obtained by solving $\mbox{GPP}_{\rm m}$, $\mbox{GPP}_{\rm m\triangle}$, $\mbox{GPP}_{\rm m-ind}$, and $\mbox{GPP}_{\rm
m}$ with (aggregated) triangle and independent set inequalities, respectively.
The numerical results in [@Sot11] show that to solve $\mbox{GPP}_{\rm m\triangle-ind}$ for a graph without symmetry and $n=100$ takes more than 3 hours. However, each bound presented in Table \[tab:Aggregate\] is computed in [*less*]{} than a second (!).
The results here also show that, for most of the cases, adding triangle inequalities to $\mbox{GPP}_{\rm m}$ increases the bound [*more*]{} than adding the independent set inequalities to $\mbox{GPP}_{\rm m}$. Note that $J(7,2)$ is a strongly regular graph for which $\mbox{GPP}_{\rm m}$ improves after adding all independent set constraints.\
[cccccccc]{} $G$ & $n$ & $r_{\rm aut}$ & $m^\mathrm{T}$ & $\mbox{GPP}_{\rm m}$ & $\mbox{GPP}_{\rm m\triangle}$ & $\mbox{GPP}_{\rm m-ind}$ & $\mbox{GPP}_{\rm m\triangle-ind}$\
& 18 & 5 & $(10,8)$ & 6 & 7 & 7 & 7\
[Desargues]{} & 20 & 6 & $(15,5)$ & 4 & 5 & 4 & 5\
$J(7,2)$ & 21 & 3 & $(11,10)$ & 37 & 37 & 40 & 40\
[Dyck]{} & 32 & 10 & $(16,16)$ & 7 & 8 & 7 & 8\
[Foster]{} & 90 & 9 & $(45, 45)$ & 13 & 18 & 14 & 19\
[Biggs-Smith]{} & 102 & 8 & $(70,32)$ & 10 & 15 & 10 & 15\
In Table \[tab:Aggregate22\], we also list the eigenvalue bound, , $\mbox{GPP}_{\rm QAP}$, and GPP$_{\rm fix}$ for the same problems as in Table \[tab:Aggregate\]. Due to memory restrictions, we couldn’t compute GPP$_{\rm fix}$ for the [Foster]{} and [Biggs-Smith]{} graph. For these graphs, we computed without exploiting their symmetry.\
The computational results show that in all cases the eigenvalue bound coincides with $\mbox{GPP}_{\rm m}$. The results also show that $\mbox{GPP}_{\rm QAP}$ equals $\mbox{GPP}_{\rm m}$ in all cases except for the [Desargues]{} graph, and that for the listed graphs fixing edges does not improve $\mbox{GPP}_{\rm m}$ and/or $\mbox{GPP}_{\rm QAP}$ while adding triangle and/or independent set constraints does.
[cccccc]{} $G$ & $m^\mathrm{T}$ & eig & & $\mbox{GPP}_{\rm QAP}$ & GPP$_{\rm fix}$\
& $(10,8)$ & 6 & 6 & 6 & 6\
[Desargues]{} & $(15,5)$ & 4 & 4 & 5 & 6\
$J(7,2)$ & $(11,10)$ & 37 & 38 & 37 & 38\
[Dyck]{} & $(16,16)$ & 7 & 7 & 7 & 7\
[Foster]{} & $(45, 45)$& 13 & 13 & 13 & –\
[Biggs-Smith]{}& $(70,32)$ & 10 & 10 & 10 & –\
Conclusion
==========
In this paper, we presented several new bounds for the graph partition problem and also showed how to simplify existing ones, when possible.
In particular, in Theorem \[thm:AnyGraph\] we derived an eigenvalue bound for the GPP that is applicable for any graph, extending a well-known result in spectral graph theory. Further, we simplified the relaxation ${\rm GPP}_{\rm m}$ for different classes of graphs such as strongly regular graphs and certain Johnson and Kneser graphs. We showed how to strengthen ${\rm GPP}_{\rm m}$ by aggregating triangle and independent set constraints when possible, which leads to huge improvements in computational abilities and times. The numerical results show that, in general, adding triangle inequalities to ${\rm GPP}_{\rm m}$ strengthened the bound more than adding the independent set constraints.
Finally, we showed how to strengthen the matrix and vector lifting relaxation for the GPP, i.e., ${\rm GPP}_{\rm m}$ and ${\rm GPP}_{\rm QAP}$, respectively, by adding constraints that correspond to assigning two vertices of the graph to different parts of the partition. Such an approach performs very well on highly symmetric graphs when other SDP and eigenvalue-based relaxations provide trivial or weak bounds.
[**Acknowledgments**]{}\
The authors would like to thank two anonymous referees for suggestions that led to an improvement of this paper.
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[^1]: Department of Econometrics and OR, Tilburg University, The Netherlands. [edwin.vandam@uvt.nl]{}
[^2]: Department of Econometrics and OR, Tilburg University, The Netherlands. [r.sotirov@uvt.nl]{}
[^3]: This version is published in Mathematical Programming, DOI:[10.1007/s10107-014-0817-6](http://dx.doi.org/10.1007/s10107-014-0817-6)
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title: First measurement of the Sivers asymmetry for gluons from SIDIS data
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abstract.tex
Keywords: deep inelastic scattering, gluon, Sivers, TMD, PDF.
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---
abstract: '[We study the possibility of extended inflation in the effective theory of gravity from strings compactified to four dimensions and find that it strongly depends on the mechanism of supersymmetry breaking. We consider a general class of string–inspired models which are good candidates for successful extended inflation. In particular, the $\omega$–problem of ordinary extended inflation is automatically solved by the production of only very small bubbles until the end of inflation. We find that the inflaton field could belong either to the untwisted or to the twisted massless sectors of the string spectrum, depending on the supersymmetry breaking superpotential.]{}'
author:
- |
[**J. García–Bellido**]{}[^1] and [**M. Quirós**]{}[^2]\
Instituto de Estructura de la Materia\
Serrano, 123 E–28006 Madrid. Spain
title: '[**Extended Inflation from Strings**]{} [^3]'
---
-16cm
It is nowadays commonly believed by most cosmologists that the inflationary paradigm may solve most of the problems of the standard cosmological model. However, it is not yet clear how the inflationary scenario can be successfully implemented. In fact, the first proposed inflationary model (known as ’old’ inflation) [@G], based on a first order phase transition, could not provide a satisfactory explanation of how to get out from the inflationary phase without disturbing the good properties of the standard cosmological model [@G; @GW]. The first models proposed to solve this ’graceful exit’ problem, known as ’new’ inflation [@NI], with a second order phase transition, were plagued with severe fine–tunings. Soon after, a different solution without phase transition, known as ’chaotic’ inflation [@CI], was proposed. (Chaotic inflation has been recently shown to considerably soften the fine–tuning problems of new inflation [@CEI].)
Recently, La and Steinhardt [@EI] proposed an inflationary model, known as ’extended’ inflation, based again on a first order phase transition, where the graceful exit problem was solved by using a Jordan–Brans–Dicke [@JBD] theory of gravity with a scalar field, instead of General Relativity [@WEI]. It was soon realized that the anisotropy at decoupling produced by large bubbles [@EW] made extended inflation incompatible with the post–Newtonian bounds [@SCT] of General Relativity. This desease could be cured either by using a more general scalar–tensor theory of gravity [@MQ] or by means of a cosmological constant with a runaway dependence on the scalar field [@HKW].
Most particle physicists believe that the theory of gravity at low energies (General Relativity, scalar–tensor theories or whatever) is an effective approximation of some fundamental theory of quantum gravity at scales beyond the Planck mass ($M_P$). The only reliable candidates for such a fundamental theory are superstrings [@GSW]. They are known to describe gravity as a low energy effective theory. It is therefore of interest to know whether or not strings could lead to any kind of cosmological inflation in the low energy effective theory.
The effective theory of superstrings exhibits three properties that makes it a good candidate for some kind of extended inflation. First of all, the scalar fields of the gravitational multiplet (the dilaton and the moduli) are always coupled to the curvature scalar of the four–dimensional metric, in the same way as the scalar field in scalar–tensor theories of gravity. Second, the dilaton and moduli are also coupled to the matter sector giving rise to a scalar field dependent potential. Finally, the existence of flat directions in the potential follows from very general principles [@GSW]. In the presence of supersymmetry breaking terms and a positive vacuum energy, flat directions become runaway fields, and so are good candidates for solving the anisotropy/post–Newtonian bounds conundrum of extended inflation.
In a previous paper [@FT27] we studied the conformal properties and cosmological solutions in the low energy effective theory of gravity from closed strings compactified to four dimensions, for different supersymmetric and non–supersymmetric string scenarios, during the radiation and matter dominated eras. In this paper we study the possibility of extended inflation in string scenarios with spontaneously broken supersymmetry. This problem has been recently studied under some assumptions (in particular, constant values for the moduli) with negative results [@LIN]. However, we will show that the possibility of extended inflation strongly relies on the mechanism of supersymmetry breaking and find the conditions under which it could happen. We will argue that a general necessary condition is the existence of a positive ’metastable’ minimum with some runaway direction along the scalar fields. This runaway direction should become flat at the true minimum in order to solve the cosmological constant problem. In this case, the same symmetry principle (if any) that could help solving the cosmological constant problem, could also help extended inflation. A non–constant value of the moduli along the runaway direction will help overcoming the problems found in Ref.[@LIN].
At string tree level, and keeping only linear terms in the string tension $\alpha'$, the effective Lagrangian for the dilaton $(\phi)$, the modulus $(\sigma)$ [^4] and the matter fields $(C_n)$, can be written in the Einstein frame as [^5] [@D4; @DKL] $$\label{L4W}
%
{\cal L}_{eff}= \sqrt{-\tilde{g}} \left[
\tilde{R} - \frac{1}{2}(\partial_\mu\phi)^2
- 6(\partial_\mu\sigma)^2 - \sum^3_{n=1} \frac{\alpha_n}{2^n}
e^{-n(\sigma+\frac{1}{2}\phi)}\mid D_{\mu}C_n\mid^2
- V(\phi, \sigma, C_n, C_n^*) \right]$$ where $C_1$ are untwisted matter fields, $C_2$ twisted moduli (blowing up modes) and $C_3$ twisted matter fields, and $\alpha_n$ are some positive constants ($\alpha_1=3$, $\alpha_2=\alpha_3=1$).
For the purpose of this paper, in order to establish the necessary conditions for extended inflation, it will be enough to expand the potential $V$ in (\[L4W\]) as $$\label{Vm}
V(\phi, \sigma, C_n, C_n^*) = V_o(\phi,\sigma) +
\sum^3_{n=1} V_n(\phi,\sigma) \mid C_n\mid^2 + \ ...$$ In fact, the Lagrangian (\[L4W\]) and the potential (\[Vm\]) can be put in the standard supergravity form [@CFG] by means of the Kähler potential [@D4; @DKL] $$\label{Kp}
K = - \ln(S+S^*) - 3\ln(T+T^*) +
\sum^3_{n=1} \alpha_n(T+T^*)^{-n} \mid C_n\mid^2 +\ ...$$ and the superpotential $$\label{Wp}
W(S,T,C_n) = W_o(S,T) + \sum^3_{n=1} W_n(S,T)\ C_n^3 +\ ...$$ where $$\begin{array}{l}
\label{ST}
Re S = e^{3\sigma-\frac{1}{2}\phi} \vspace{2mm}\\
Re T = e^{\sigma+\frac{1}{2}\phi}
\end{array}$$ It is important to stress that a superpotential $W_o$ different from zero, necessary for supersymmetry breaking, and a non–constant superpotential $W_n$ could be generated by string non–perturbative effects. Also note that we are consistently studying the Lagrangian along the (strong CP–conserving) real directions [^6] ($Im S=Im T=0$).
The properties of the potential (\[Vm\]), and in particular its ability to produce extended inflation, will depend in general on the form of the superpotential (\[Wp\]). We will first give some (by no means sufficient) conditions on the potential (\[Vm\]), and their implications on the superpotential $W_o$, in order to produce extended inflation:
a\) We assume that the potential $V_o$ has a minimum along some field direction, $e.g.$ $$\label{sb}
\sigma = - b \phi + d$$ (with $b$ and $d$ some real parameters) and a runaway direction along the orthogonal field[^7]. This condition can be fulfilled depending on the functional form of the superpotential. For instance, if $W_o=W_o(X)$ with $X=S^\alpha T^{3\beta}$ ($\alpha$ and $\beta$ real), then [^8] $$\label{b}
b=\frac{3\beta-\alpha}{6(\alpha+\beta)}$$ and the potential $V$ takes the form [@FIL] $$\begin{array}{l}
\label{Von}
V_o=\frac{1}{16} e^{-6\sigma-\phi} \ v_o(\sigma+b\phi)
\vspace{2mm}\\
V_n=\frac{\alpha_n}{16\cdot 2^n} e^{-(6+n)\sigma-(1-\frac{n}{2})\phi}
\ v_n(\sigma+b\phi)
\end{array}$$ where $$\begin{array}{l}
\label{von}
v_o(\sigma+b\phi)=f^2_\alpha + 3f^2_\beta - 3f^2_o
\vspace{2mm}\\
v_n(\sigma+b\phi)=f^2_\alpha + (3-n)f^2_\beta - 2f^2_o
\end{array}$$ with $$\label{fl}
f_\lambda (\sigma+b\phi)\equiv W_o-2\lambda X\frac{\partial W_o}
{\partial X} \ .$$ The minimization of $v_o$ in (\[Von\]) should provide the condition (\[sb\]).
Notice that condition (\[sb\]) is not essential for extended inflation. It is just a simplifying assumption where one direction in the $(\sigma,\phi)$ configuration space is fixed to its vacuum expectation value and so the remaining theory of gravity has only one scalar field. However, more general situations suitable for extended inflation are thinkable. For instance, the case where both $\sigma$ and $\phi$ are runaway directions (no field is fixed to its vacuum expectation value) can be easily realized in many models. In particular, in the simple case where $W_o=$ constant. (A constant superpotential can be triggered by the vacuum expectation value of some field.) In this case, $\ v_o\ =\ \mid
W_o\mid^2\ $ and $\ v_n\ =\ (2-n)\\
\mid W_o\mid^2$.
b\) There should be a positive cosmological constant at the minimum (\[sb\]), $i.e.$ $$\label{cc}
v_o(d)>0 \ .$$ In particular, this implies that supersymmetry is broken at the minimum (\[sb\]) in such a way that $$\label{cm}
f^2_\alpha(d) + 3f^2_\beta(d) > 3f^2_o(d) \ .$$ In the case $W_o=$ constant, condition (\[cc\]) is automatically satisfied.
c\) The last condition is that the minimum (\[sb\]) is required to be stable along the inflaton field direction $C_n$, $i.e.$ $$\label{cd}
v_n(d)>0$$ or $$\label{cn}
f^2_\alpha(d) + (3-n)f^2_\beta(d) > 2f^2_o(d)$$ where $n$ is the sector to which the inflaton belongs. In this way, the inflaton potential can trigger a first order phase transition from the false vacuum at $C_n=0$ to the true physical vacuum at $C_n\neq0$, which we assume to correspond to a zero cosmological constant [^9].
In the simple case of $W_o=$ constant, condition (\[cd\]) is always satisfied for $n=1$ (untwisted matter sector) but never satisfied for $n=3$ (twisted matter sector). For $n=2$ (blowing–up modes) $v_2\equiv 0$ and so the stability along the inflaton direction $C_2$ would rely upon higher order derivatives of the potential and therefore upon the superpotential $W_2$.
In what follows we will assume that conditions (\[sb\]), (\[cc\]) and (\[cd\]) hold and therefore will write the Lagrangian (\[L4W\]) for $\phi$ and the inflaton field $C_n$ as $$\label{L4}
{\cal L}_{eff}= \sqrt{-\tilde{g}} \left[
\tilde{R} - (6b^2+\frac{1}{2})(\partial_\mu\phi)^2
- e^{-n(\frac{1}{2}-b)\phi}\mid D_\mu C_n\mid^2
-\ e^{-(1-6b)\phi}\rho_o + ...\right]$$ where $\rho_o$ is a constant energy density, we have used Eq.(\[sb\]) and absorbed all constant coefficients in the definition of $C_n$. Notice that the energy density $\rho(\phi)$ in (\[L4\]) is proportional to $m^2_{3/2}$, the scale of supersymmetry breaking (the gravitino mass), as expected, $$\label{m3/2}
m^2_{3/2}\propto e^{-(1-6b)\phi} \mid W_o \mid^2 \ .$$ The mass of the observable fields at the true vacuum depends on the global structure of the potential $V$ in (\[L4W\]), which is very poorly known in most cases. In fact, it depends on the total structure of the Kähler potential (\[Kp\]) and the superpotential (\[Wp\]), which could in turn depend on non–perturbative effects at high energy scales (string effects) and/or at low energy scales (QCD condensates, ...). We will assume for the masses a simple dependence $$\label{m2}
\tilde{m}^2 \sim e^{-a\phi}\ m^2_o$$ where $m_o$ is a constant mass and $a$ is a real coefficient parametrizing our ignorance on the details of supersymmetry breaking in string theory and the low energy non–perturbative effects. The case of constant masses ($a=0$), considered in the analysis of Ref.[@LIN], is particularly interesting and will be commented later on.
Under a conformal redefinition [@DGG; @KKO; @LIN; @FT27] of the metric $$\label{DG}
\tilde{g}_{\mu \nu} = e^{c\phi}\ g_{\mu \nu}$$ $$\label{RR}
\tilde{R}=e^{-c\phi}\left[ R - c(D-1) D^{2}\phi-\frac{1}{4}
c^2(D-1)(D-2) g^{\mu\nu}\partial_{\mu}\phi \partial_{\nu}\phi
\right] \ ,$$ the masses transform as $$\label{mp}
m^2=e^{c\phi}\ \tilde{m}^2 \ .$$ It is therefore convenient to make the conformal redefinition of $g_{\mu\nu}$ (\[DG\], \[RR\]) with parameter $c=a$ such that the mass of the observable fields, see Eqs.(\[m2\], \[mp\]), become $\phi$–independent [@FT27]. Then (\[L4\]) can be written as [^10] $$\label{L4P}
{\cal L}= \sqrt{-g}\left[\Phi R -
\frac{\omega}{\Phi}(\partial_\mu\Phi)^2
- \frac{1}{2}\Phi^{1-\beta'}(\partial_\mu C_n)^2
- \Phi^{2(1-\beta)}\rho_o \right]$$ where $$\label{F}
\Phi= e^{a\phi}$$ and the parameters $\omega$, $\beta$ and $\beta'$ in (\[L4P\]) are defined as functions of $a$ and $b$ as $$\label{OM}
2\omega+3=\frac{1+12b^2}{a^2}$$ $$\label{beta}
\beta=\frac{1-6b}{2a}$$ $$\label{beta'}
\beta'= n\left(\frac{1-2b}{2a}\right) \ .$$ Written in terms of a Robertson–Walker metric, $\Phi(t)$ is a dimensionless scalar related to the variation of the Plank mass $$\label{Phi}
\Phi(t) = \frac{M^2_P(t)}{M^2_P}$$ where $M_P^2$ stands for $M_P^2(t_o)\equiv 1/G_N$ ($t_o$ is the present age of the universe), and we assume $\Phi(t_e)\simeq 1$ at the end of inflation. We can also define the scales $M$ and $m_P$ through $$\label{M}
\rho(0) = \Phi(0)^{2(1-\beta)} \rho_o \equiv M^4$$ $$\label{Ph0}
\Phi(0) \equiv \frac{m^2_P}{M^2_P} \ .$$ The equations of motion then read $$\label{SPF}
\begin{array}{c}
{\displaystyle
\left(\frac{\dot{a}}{a}
\right)^2+\frac{k}{a^2}=
\frac{\rho_o}{6}\Phi^{1-2\beta} + \frac{\omega}{6}
\left(\frac{\dot{\Phi}}{\Phi} \right)^2-\frac{\dot{a}}{a}
\frac{\dot{\Phi}}{\Phi} }\vspace{2mm}\\
{\displaystyle
\ddot{\Phi} + 3\frac{\dot{a}}{a}\dot{\Phi}
=\frac{2\beta}{2\omega+3} \rho_o \Phi^{2(1-\beta)} }
\end{array}$$ with solutions for $k=0$ [@HKW] $$\label{SS}
\begin{array}{l}
a(t)=a(0) (1+Bt)^p, \hspace{2cm} {\displaystyle p=
\frac{2\omega + 3 -2\beta}{2\beta(2\beta-1)} } \vspace{2mm} \\
\Phi(t)=\Phi(0) (1+Bt)^q, \hspace{2cm}
{\displaystyle q=\frac{2}{2\beta-1} }
\end{array}$$ where $$\label{B}
B^2 = \frac{2\beta^2(2\beta-1)^2\rho_o \Phi(0)^{1-2\beta}}
{(2\omega+3)(6\omega+9-4\beta)} =
\frac{2\beta^2(2\beta-1)^2\ M_P^2}{(2\omega+3)(6\omega+9-4\beta)}
\left(\frac{M}{M_P}\right)^4\left(\frac{M_P}{m_P}\right)^2 \ .
\vspace{2mm}$$ We now raise the question of the sufficient conditions for extended inflation and whether or not a ’gracefull exit’ can be achieved.
First of all, we require that quantum gravity effects be negligible. In other words, that the kinetic energy due to de Sitter fluctuations (maximal at beginning of inflation [@ST; @EI]) be less than $\rho$, see Eqs.(\[M\]–\[SPF\]), $i.e.$ ${\displaystyle H^4(0) \approx M^8
\left(\frac{M_P}{m_P}\right)^4 < \rho(0) }$. This gives the constraint $$\label{mP}
m_P > M \ .$$ We are assuming that the universe at $T_c$ goes through a first order phase transition in which the high-temperature phase remains metastable down to $T=0$ [@GW], where bubble nucleation is dominated by quantum mechanical tunneling [@COL]. Bubbles are assumed to be formed with zero radius at $t_B$ and then expand at the speed of light. A bubble radius at a later time $t>t_B$ is given by $$\label{rad}
r(t,t_B)=\int^t_{t_B}\frac{dt'}{a(t')} \ .$$ We now define the asymptotic radius of a bubble nucleated at $t$ as $$\label{ras}
r_{as}(t)=\int^\infty_t \frac{dt'}{a(t')}
= \frac{p}{p-1}\ \frac{1}{a(t) H(t)}$$ where $H(t)$ is the Hubble expansion factor $$\label{H}
H(t)=pB \left(\frac{\Phi(0)}{\Phi(t)}\right)^{\beta-1/2} =
pB \left(\frac{a(0)}{a(t)}\right)^{1/p} \ .$$ The end of inflation is determined by the competition between the bubble nucleation rate and the cosmic expansion rate. The dimensionless parameter which controls the percolation of the phase transition can be computed as [@GW] $$\label{eps}
\epsilon(t)=\int^t_{t_B} dt' \lambda(t') a^3(t') \frac{4\pi}{3}
r^3(t,t') \simeq \frac{\lambda(t)}{H^4(t)} \hspace{2cm} (p\gg 1)$$ where $\lambda(t)$ is the bubble nucleation rate per unit volume. In our model, $\lambda(t)$ is time dependent since the energy density which drives inflation is itself time dependent through $\Phi(t)$, see Eq.(\[L4P\]). Holman [*et al.*]{} [@HKVW] compute this dependence to be $$\label{lam}
\lambda(t)=\lambda_o \Phi(t)^{2(1-\beta')}
e^{-B_o\left[\Phi(t)^{2(\beta-\beta')}-1\right]}$$ where [^11] $\lambda_o=Ae^{-B_o}$. $B_o$ is the Euclidean bounce action [@COL; @GW], which depends on the inflaton potential and can acquire very big values $O(10^2)$, while the prefactor $A$ is of order one and has dimensions of $T_c^4$, where $T_c\sim M$ is the mass scale of the phase transition.
The epsilon parameter can then be written as $$\begin{array}{rl}
\label{eps0}
\epsilon(t)&=\epsilon_o \ \Phi(0)^{2(1-2\beta)}
\Phi(t)^{2(2\beta-\beta')}
e^{-B_o\left[\Phi(t)^{2(\beta-\beta')}-1\right]} \vspace{2mm}\\
&=\epsilon(t_e) \Phi(t)^{2(2\beta-\beta')}
e^{-B_o\left[\Phi(t)^{2(\beta-\beta')}-1\right]}
\end{array}$$ where ${\displaystyle \epsilon_o\equiv\frac{\lambda_o}{H^4(0)} }$ is the usual parameter considered in the literature.
A measure of the progress of the transition is the fraction of space which remains in the high temperature phase (’false vacuum’), $p(t)=e^{-\epsilon(t)}$. We need a very small epsilon parameter at the beginning of inflation which increases very quickly to above a critical value, thus allowing for percolation of the low temperature phase (’true vacuum’). Therefore we require $$\label{conE}
\epsilon(t_e)=\epsilon_o\ \Phi(0)^{2(1-2\beta)} \ > \epsilon_{cr}$$ where ${\displaystyle 1.1\times 10^{-6}<\epsilon_{cr}<\frac{3}{4\pi} }$ was computed in Ref.[@GW] for solving the ’gracefull exit’ problem. Thus $$\label{e0}
\epsilon_o > \left(\frac{M}{M_P}\right)^{4(2\beta-1)} \epsilon_{cr}$$ which gives ample room for very small values of $\epsilon_o$, provided that $2\beta>1$ (which is anyhow necessary for an increasing $\Phi(t)$). We must be sure, however, that $\epsilon(t)$ is increasing, that is $$\label{deps}
\frac{\dot{\epsilon}(t)}{\epsilon(t)} = 2(\beta'-\beta)
\ \frac{\dot{\Phi}(t)}{\Phi(t)} \left[B_o\Phi(t)^{-2(\beta'-\beta)}
-\frac{\beta'-2\beta}{\beta'-\beta}\right] > 0
\vspace{1mm}$$ which is satisfied for $$\label{cde}
\beta' > \beta$$ and $$\label{cds}
B_o\Phi(t)^{-2(\beta'-\beta)} >
\frac{\beta'-2\beta}{\beta'-\beta} \ .$$ This condition is very easily satisfied as we will see.
We are now ready to analyze our string model for inflation, see Eq.(\[L4P\]). The peculiarity of this model is the fact that $\omega$, $\beta$ and $\beta'$ are not independent but determined by the string effective action, see Eqs.(\[OM\]–\[beta’\]). This dependence corresponds to the conformal redefinition of the metric tensor for which observable matter have constant masses, as discussed above. We will now impose further constraints on our model.
A necessary condition for inflation is that ${\displaystyle \frac{\ddot{a}}{a}>0 }$, or $p>1$, which becomes $$\label{b1/2}
0 < b < \frac{1}{2} \ .$$ We must also impose that $\Phi(t)$ increases, which gives the condition $$\label{a6b}
a < 1-6b \ .$$ The condition that $\epsilon(t)$ increases, see Eqs.(\[cde\], \[cds\]), then becomes $$\label{a0}
a \geq 0$$ $$\label{B0}
B_o > 1$$ which are both sufficient conditions for all values of $n$, see Eq.(\[beta’\]).
Assuming $N$ orders of magnitude increase in the scale factor, $$\label{10N}
10^N=\frac{a(t_e)}{a(0)}=\left(\frac{\Phi(t_e)}{\Phi(0)}\right)^{p/q}
< \left(\frac{M_P}{M}\right)^{p/q}$$ imposes the constraint $$\label{abM}
a < \left(\frac{1+12b^2}{1-6b}\right)\
\frac{\log \left(\frac{M_P}{M}\right)}
{N+\log\left(\frac{M_P}{M}\right)} \ .$$ Furthermore, in order to solve the horizon problem we need sufficient inflation such that [^12] ${\displaystyle d_{H_o}<d_{H(0)}\frac{a_o}{a(0)} }$. However, since $H(t)\sim t^{-1}\sim T^2$ and $a T\sim$ constant during the post–inflationary period, and assuming ’good reheating’ for recovering all the latent energy density of the phase transition ($T_e\equiv T(t_e)\sim T_c\sim M$), we obtain the condition $$\label{N}
N > \frac{p}{p-1} \log \left(\frac{M}{T_o}\right) \ .$$ Therefore, the required number of orders of magnitude of inflation depends crucially on the energy scale of the phase transition $M$.
Inflation must occur after the production of monopoles or any other topological defects whose densities might affect cosmology. For the same reason, the universe must also reheat before baryogenesis. These conditions place the constraint $10^2$ GeV $<M<$ $10^{18}$ GeV [@EI].
However, solving the horizon and monopole problems is not enough. We must be sure that the phase transition ends and that all the bubbles percolate without disturbing too much the isotropy and homogeneity of the cosmic background radiation. Therefore, we expect that the volume fraction contained in bubbles with radius greater than a given comoving radius $r=r(t_e,t)$ at the end of inflation be less than $10^{-n}$ at a temperature $T$: $$\label{V}
{\cal V}_>(r,t_e) = 1-p(t) \simeq \epsilon(t) = \epsilon(t_e)
\ \left(\frac{T}{M}\right)^\delta \ e^{-B_o \left[
\left(\frac{M}{T}\right)^{\delta'}-1\right]} \ < \ 10^{-n}$$ where we have used $$\label{del}
\Phi(t)^{2(2\beta-\beta')}=\left(\frac{r_o}{r}\right)^\delta
\simeq \left(\frac{T}{M}\right)^\delta
\vspace{2mm}$$ where $r_o\equiv r_{as}(t_e)$ is the asymptotic radius of a bubble nucleated at the end of inflation, $\ {\displaystyle \delta\equiv\frac{8\beta(2\beta-\beta')}
{2\omega+3-4\beta^2} }\ $ and $\ {\displaystyle \delta'\equiv\frac{8\beta(\beta'-\beta)}
{2\omega+3-4\beta^2} > 0}\ $. In particular, for the cosmic background radiation, we require that [@EW] $n\simeq 5$ at $T\simeq 1$ eV in (\[V\]). This condition is trivially satisfied thanks to the exponential, using ${\displaystyle \frac{M}{T}>10^{11} }$ and condition (\[B0\]). In this way, the extended inflation problem of anisotropy at decoupling produced by large bubbles is successfully solved in this kind of models [^13].
We still have to be sure of reestablishing a common Robertson–Walker frame in all the bubble clusters that will coalesce to form our universe. There must be some way to remember the original (pre–bubble–nucleation) coordinates; such a record can be found in the evolution of $a(t)$ or $\Phi(t)$. Since constant $H(t)$ corresponds in General Relativity to a de Sitter universe with no distinguished frame, we must require sufficient variations of this quantity, $e.g.$ $m$ orders of magnitude in $H(t)$ [@EW; @HKW]. In particular, we expect that homogeneity and isotropy must hold by the time of nucleosynthesis ($T_{NS}\simeq 1$ MeV, $m\simeq 1$), thus $$\label{Hm}
\frac{H(t)}{H(t_e)}=\left(\frac{r+r_o}{r_o}\right)^{\frac{1}{p-1}}
\simeq\left(\frac{M}{T}\right)^{\frac{1}{p-1}}\ > \ 10^m$$ corresponding to $$\label{ppo}
p < 1 + \log\left(\frac{M}{T_{NS}}\right) \equiv p_o$$ which is an explicit bound on the power of the scale factor and gives an extra condition on our parameters $$\label{apo}
a < \frac{p_o}{p_o-1} (1-6b) - \frac{1}{p_o-1}
\left(\frac{1+12b^2}{1-6b}\right) \ .$$ Furthermore, quantum fluctations of the scalar field $\Phi$ create a spectrum of adiabatic fluctuations, which can be computed for power–law solutions in the Einstein frame [@LM] $$\label{rpl}
\frac{\delta\tilde{\rho}}{\tilde{\rho}}\simeq
\frac{\tilde{H}^2}{\pi\dot{\phi}}\simeq
\frac{\tilde{p}^{3/2}}{\pi}\cdot\frac{1}{\tilde{t}}$$ and must be bounded in the conformal frame ($\rho=\Phi^2\tilde{\rho}$) to be compatible with the observed density perturbations [@KKO] $$\label{drho}
\frac{\delta\rho}{\rho}\simeq\left(\frac{M}{M_P}\right)^2 \tilde{p}
\ \ k^{-\frac{1}{\tilde{p}-1}} < 10^{-4} \hspace{2cm} (\tilde{p}\gg 1)$$ where $k$ is the dimensionless physical scale of reentering perturbations and $\tilde{p}$ is the power of the scale factor in the Einstein frame ${\displaystyle \left(
\tilde{a}(\tilde{t})\sim \tilde{t}^{\ \tilde{p}},\ \ \tilde{p}=
\frac{2\omega+3}{4\beta^2}\right) }$. This imposes a very mild constraint on $M$ $$\label{MMp}
M < \left(\frac{1-6b}{\sqrt{1+12b^2}}\right)\ 10^{-2} M_P < 10^{-2} M_P \ .$$ Finally, the most stringent bounds will come from the post–Newtonian experiments of time delay [@VIK; @SCT] and the nucleosynthesis bound [@FT26] on the $\omega$ parameter $$\label{pN}
2\omega + 3 > 500 \hspace{2cm} (2\sigma)$$ which gives a very strong constraint on our parameters $$\label{aom}
a < \sqrt{\frac{1+12b^2}{500}} \ .$$ It is interesting to notice that the anisotropy of the cosmic background radiation, which was the main problem for extended inflation, does not impose any significant bound on our model, see Eq.(\[V\]). The most stringent bound comes from the post–Newtonian experiments and nucleosynthesis bound, see Eq.(\[aom\]), which constraint the parameter $a$. On the other hand, the strongest constraint on $b$ comes from the isotropy and homogeneity at nucleosynthesis, see Eqs.(\[ppo\], \[apo\]).
Most of the previous bounds depend on the energy scale $M$ of the phase transition. We have studied those bounds for two typical values of $M$. For a phase transition driven by phenomenological supersymmetry breaking ($m_{3/2}\simeq 1$ TeV) we have $M=(m_{3/2} M_P)^{1/2}\simeq 10^{11}$ GeV, while for the usual grand unified theory we take $M=10^{16}$ GeV. The inflationary scenario is characterized by two parameters, the power $p$ of the scale factor and the number $N$ of orders of magnitude increase during inflation. Both parameters depend on the energy scale of the phase transition, see Eqs.(\[N\], \[ppo\]). For $M=10^{11}$ GeV we have $p<16$ and $N>26$, while for $M=10^{16}$ GeV, $p<21$ and $N>31$ to solve the horizon problem without disturbing the isotropy and homogeneity at nucleosunthesis. The actual value of $N$ depends on the parameters of the theory. Using the bounds (\[apo\]), (\[MMp\]) and (\[aom\]) we obtain $N>45$, which widely solves the flatness problem.
In Fig.1 we show the region in parameter space $(a, b)$ allowed by all the inflationary conditions, for $M=10^{11}$ GeV and $M=10^{16}$ GeV (dashed and dotted curves respectively). The allowed region is the one below the curves. The condition associated with neglecting the quantum gravity effects (\[mP\], \[abM\]) strongly depends on the energy scale of the phase transition, as expected, and as we can see from the lines labelled QG. Other conditions depend slightly on $M$, like those associated with reestablishing the isotropic and homogeneous Robertson–Walker frame (\[Hm\], \[apo\]), and labelled RW in Fig.1 . Finally, there are those conditions which do not depend at all on the energy scale of the phase transition, like the post–Newtonian bounds (\[pN\], \[aom\]) and the condition (\[a6b\]) that $\Phi$ increases from $m_P$ to $M_P$, labelled pN–NS and $\Phi$ respectively. However, as we can see from Fig.1, the final allowed region in parameter space does not depend much on the scale $M$ since it is bounded by the post–Newtonian and nucleosynthesis bound and the isotropy and homogeneity condition at nucleosynthesis.
As we can see from Fig.1, the case of constant observable masses ($a=0$) is consistent with all inflationary and post–Newtonian bounds. This can be easily obtained by taking the limit $a\rightarrow0$ in our explicit solution (\[SS\]), which corresponds to $$\label{sol}
\begin{array}{l}
a(t) \sim t^{\ \frac{1+12b^2}{(1-6b)^2}} \\
\Phi(t) \sim 1
\end{array}$$ On the other hand the direction $b=0$, see Eq.(\[sb\]), is incompatible with the necessary condition for inflation, Eq.(\[b1/2\]), and corresponds to the case of constant moduli. We thus agree with the negative results found in Ref.[@LIN].
In conclusion, we have studied in this paper the general conditions under which the effective theory of gravity from strings compactified to four dimensions could lead to extended inflation. We have found that a necessary condition is the existence of runaway directions in the space of fields coupled to the curvature scalar (dilaton and moduli fields). However, the existence of runaway directions is a usual feature of the effective theory of strings (through classical invariance arguments [@GSW]). It is satisfied for many supersymmetry breaking potentials. In the simplest case of supersymmetry breaking, a constant $W_o$ superpotential in (\[Wp\]), all moduli and the dilaton are runaway fields with a positive potential ($\sim\mid W_o\mid^2$) and extended inflation may follow. However, to simplify the study of the equations of motion, we have assumed just one runaway direction and parametrized it by a real parameter $b$. This is just a simplifying hypothesis since extended inflation might occur under much more general circumstances.
A second necessary condition for extended inflation is the existence of a metastable minimum along some (matter) inflaton field. This condition is necessary to enforce a first order phase transition. It also depends on the particular structure of the supersymmetry breaking superpotential, but this condition (see Eq.(\[cd\])) is easily satisfied in many models. For instance, in the simple case $W_o=$ constant, it holds when the inflaton belongs to the untwisted sector ($n=1$), and does not hold if it belongs to the twisted sector ($n=3$). The case of the inflaton as a blowing–up mode ($n=2$) would require the precise knowledge of the total superpotential.
Third, we assume a simple behaviour of the mass of observable fields on the runaway direction, and parametrize this behaviour with a real parameter $a$. (The case of constant masses corresponds to $a=0$.) We make a conformal redefinition of the metric in order to go into the ’physical’ frame, where the masses of the observable fields are constant. Of course, if the functional dependence of masses were more complicated, we would have needed a more general conformal transformation and the theory would look different, in particular it would have a non–constant $\omega$ parameter. However, we should stress here that a conformal redefinition is not essential since Physics cannot depend on it. In other words, we could redefine the physical scale factor by taking its ratio with respect to the Compton wavelength [@HKWW] which is then manifestly independent of the conformal transformation [@LIN; @FT27].
Finally, we have imposed all the conditions for successful extended inflation on the solution of our model and found an allowed region in parameter space ($a,b$). Our results are summarized in Fig.1. The direction $b=0$ (the region of constant moduli) is excluded from the allowed region, while $a=0$ (the case of constant masses) is inside the permitted region and therefore consistent with all experimental bounds. Notice that our model successfully solves the $\omega$–problem of extended inflation (namely, that the condition of isotropy of the cosmic background radiation at decoupling is in conflict with the post–Newtonian bounds on $\omega$) by producing very small bubbles until the end of inflation when the epsilon parameter increases exponentially up to the critical value.
Acknowledgements {#acknowledgements .unnumbered}
================
One of us (J.G.–B.) would like to thank M. Cvetič, B. Ovrut and P. Steinhardt for valuable discussions, and the Theoretical Physics Department of Pennsylvania University for the hospitality given to him. The other (M.Q.) thanks F. Quevedo for discussions and the T–8 Division of Los Alamos National Laboratory for its warm hospitality.
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Figure Captions {#figure-captions .unnumbered}
===============
Fig.1
: Plot of the region in parameter space $(a, b)$ allowed by the inflationary conditions in the text. The solid lines represent those bounds which do not depend on the scale $M$ of the phase transition. The dashed curve correspond to the bounds associated with the scale $M=10^{11}$ GeV and the dotted curve to those related to $M=10^{16}$ GeV. The allowed region is the one below the curves. The border $b=0$ is excluded from it. We have labelled the curves as follows: QG corresponds to the condition associated to neglecting quantum gravity effects, $\Phi$ corresponds to the condition for an increasing scalar field, RW corresponds to the bound on isotropy and homogeneity at the time of nucleosynthesis and pN–NS corresponds to the bounds from post–Newtonian experiments and the nucleosynthesis bound on $\omega$.
[^1]: Supported by FPI Grant. e–mail: bellido@iem.csic.es
[^2]: e–mail: imtma27@cc.csic.es and quiros@cernvm.
[^3]: Work partially supported by CICYT under contract AEN90–0139.
[^4]: We take the diagonal direction in the space of moduli fields.
[^5]: We will work hereafter, unless explicit mention, in units in which $M_P\equiv 1$.
[^6]: Notice that a minimum along a different direction would just amount to a field redefinition and so the general results of this paper should remain valid.
[^7]: Otherwise $\phi$ and $\sigma$ would be fixed to their vacuum expectation values and no extended inflation could be present. Since we are concerned in this paper with extended inflation from strings, we will not consider the latter case. On the other hand, the possibility of new inflation from strings was studied some years ago and shown to require a huge amount of fine–tuning [@ENQ]. Although these negative results are not general enough to exclude other kinds of inflation based on General Relativity ($e.g.$ chaotic) which could arise from string theories, they make us search for inflation in more general theories of gravity.
[^8]: The case $\alpha=0$, $\beta=1/3$, giving $b=1/2$, has recently been considered [@FIL] and shown to be consistent with target space modular invariance. However, we will consider a more general case since non–perturbative effects could break modular invariance [@GK].
[^9]: Of course this would impose extra conditions on the total superpotential $W$, which we will not study here.
[^10]: Recall that under a conformal redefinition of the Robertson–Walker metric, the scale factor and the time variable transform as $\tilde{a}(\tilde{t})=\Phi(t)^{1/2} a(t)$ and $d\tilde{t}=\Phi(t)^{1/2} dt$ respectively.
[^11]: In ordinary JBD theories and GR, this rate is essentially time independent and given by $\lambda_o$.
[^12]: We use here the notation $H_o\equiv H(t_o)$, $a_o\equiv a(t_o)$ and $T_o\equiv T(t_o)$.
[^13]: In fact, this solution was proposed on general grounds in Ref.[@LIN].
|
---
abstract: 'We study the destabilization mechanism in a unidirectional ring of identical oscillators, perturbed by the introduction of a long-range connection. It is known that for a homogeneous, unidirectional ring of identical Stuart-Landau oscillators the trivial equilibrium undergoes a sequence of Hopf bifurcations eventually leading to the coexistence of multiple stable periodic states resembling the Eckhaus scenario. We show that this destabilization scenario persists under small non-local perturbations. In this case, the Eckhaus line is modulated according to certain resonance conditions. In the case when the shortcut is strong, we show that the coexisting periodic solutions split up into two groups. The first group consists of orbits which are unstable for all parameter values, while the other one shows the classical Eckhaus behavior.'
author:
- 'Jan Philipp Pade$^{1}$, Leonhard Lücken$^{1}$ and Serhiy Yanchuk$^{1}$'
bibliography:
- 'DISS.bib'
title: The dynamical impact of a shortcut in unidirectionally coupled rings of oscillators
---
$^{1}$Humboldt-University of Berlin, Institute of Mathematics, Unter den Linden 6, 10099 Berlin, Germany
Introduction
============
The theory of dynamics on networks is a rapidly growing field of research. Motivated by applications from neuroscience, biological systems, and social networks, scientists from various areas have investigated the connection between network structure and dynamics in the last few decades. Since then, the importance of special topologies like scale-free or small world networks was discovered and dynamics on these structures were studied extensively [@Boccaletti2006].
In this article, we study a special class of network topologies, namely unidirectional rings which are perturbed by the insertion of a single additional non-local link [\[]{}Fig. \[fig:Coupling-scheme\][\]]{}. One reason for the interest in ring structures is that they emerge in many natural systems [@Koseska2010; @Restrepo2004a; @Takamatsu2001]. Also, rings can be seen as motifs of larger and more complex networks [@Milo2002]. A lot of research has been done for bidirectional rings in the last few years [@Abrams2004; @Daido1997; @Restrepo2004; @Waller1984; @Zou2009]. In contrary, less is known about dynamics in unidirectional rings [@horikawa2012b; @Horikawa2012a; @Perlikowski2010a; @Yanchuk2008a; @Popovych2011; @Perlikowski2010], although these structures play an important role in various applications [@Bressloff1997; @collins1994; @strelkowa2011; @Takamatsu2001; @vishwanathan2011; @VanderSande2008]. As a simple, paradigmatic model, we consider unidirectional rings of $N$ identical Stuart-Landau oscillators with an additional shortcut from node $\ell$ to node $N$, to which we refer as a perturbation of the homogeneous system. The dynamics on the perturbed ring are described by the following equations: $$\begin{aligned}
\dot{z}_{j}\left(t\right) & = & \left(\mu-\left|z_{j}\left(t\right)\right|^{2}\right)z_{j}+z_{j+1}\left(t\right),\quad j=1,...,N-1,\nonumber \\
\dot{z}_{N}\left(t\right) & = & \left(\mu-\left|z_{N}\left(t\right)\right|^{2}\right)z_{N}+z_{1}\left(t\right)+sz_{\ell}\left(t\right),\label{eq:SL-system}\end{aligned}$$ where $\mu=\alpha+i\beta$, $\beta>0$, $s>0$ is the shortcut strength, and $z_{j}\left(t\right)\in\mathbb{C}$. For an unperturbed ring ($s=0$), the destabilization mechanism is described in [@Yanchuk2008a]. In particular, when the parameter $\alpha=\Re\left(\mu\right)$ in (\[eq:SL-system\]) is increased, the zero equilibrium $$z_{1}=...=z_{N}=0\label{eq:zero-equilibrium}$$ loses stability and undergoes a sequence of Hopf bifurcations. The first half of the emerging branches of periodic solutions stabilizes at appropriate values of $\alpha$ (see chapter \[sub:BifurcationsOfEquilibrium\]). Remarkably, this phenomenon resembles the Eckhaus scenario in spatially extended diffusive systems [@Eckhaus1965; @Tuckerman1990]. The aim of this article is to investigate the transformation of this scenario under non-local perturbations which destroy the rotational symmetry of the system. We investigate two different asymptotic cases of small and large perturbation size $s$. For small $s$, the Eckhaus scenario persists qualitatively with a modulated Eckhaus line, while for large $s$, there is a qualitative difference to the unperturbed case that reflects the new network topology, i.e. the existence of a new loop.
The article is structured as follows. In section \[sec:stability-of-equilibrium\] we discuss the stability of the zero solution (\[eq:zero-equilibrium\]), its spectrum and its bifurcations. In section \[sec:Emergent-periodic-orbits\] we find asymptotic expressions for the emerging periodic solutions in the case of small perturbation size $s$. We also reduce the case of asymptotically large $s$ to the analysis of an inhomogeneous ring. Section \[sec:Stability-of-the-psols\] deals with the stability analysis of the periodic solutions and the results are compared to numerical simulations. Finally, we discuss our findings and give an outlook on possible extensions and applications of the results in section \[sec:Discussion-1\].
![*\[fig:Coupling-scheme\]*Coupling scheme of a unidirectional ring with a shortcut.](crosslinks_fig1){width="30.00000%"}
Stability of the zero solution\[sec:stability-of-equilibrium\]
==============================================================
To study the stability of system (\[eq:SL-system\]) it is convenient to identify each variable $z_{j}\left(t\right)\in\mathbb{C}$ with a two-dimensional real variable $\boldsymbol{z}_{j}\left(t\right)=\left(z_{j,1}\left(t\right),z_{j,2}\left(t\right)\right)^{\top}=\left(\Re\left(z_{j}\left(t\right)\right),\Im\left(z_{j}\left(t\right)\right)\right)^{\top}\in\mathbb{R}^{2}$. Then, system (\[eq:SL-system\]) is equivalent to the real system $$\begin{aligned}
\dot{\boldsymbol{z}}_{j}\left(t\right) & = & \left(M_{\mu}-\left(z_{j,1}^{2}+z_{j,2}^{2}\right)\right)\boldsymbol{z}_{j}+\boldsymbol{z}_{j+1},\nonumber \\
\dot{\boldsymbol{z}}_{N}\left(t\right) & = & \left(M_{\mu}-\left(z_{N,1}^{2}+z_{N,2}^{2}\right)\right)\boldsymbol{z}_{N}+\boldsymbol{z}_{1}+s\boldsymbol{z}_{\ell},\label{eq:SL-system-real}\end{aligned}$$ where $M_{\mu}=\left[\begin{matrix}\alpha & -\beta\\
\beta & \alpha
\end{matrix}\right]$ is the representation of the multiplication with $\mu$ in $\mathbb{R}^{2\times2}$.
Spectrum of the equilibrium
---------------------------
We linearize system (\[eq:SL-system-real\]) in $\boldsymbol{z}_{1}=..=\boldsymbol{z}_{N}\equiv0$ and obtain the variational equation $$\begin{aligned}
\frac{d}{dt}\delta\boldsymbol{z}_{j}\left(t\right) & =M_{\mu}\delta\boldsymbol{z}_{j}+\delta\boldsymbol{z}_{j+1}\left(t\right),\ j=1,...,N-1,\\
\frac{d}{dt}\delta\boldsymbol{z}_{N}\left(t\right) & =M_{\mu}\delta\boldsymbol{z}_{N}+\delta\boldsymbol{z}_{1}\left(t\right)+s\delta\boldsymbol{z}_{\ell}\left(t\right),\end{aligned}$$ which can be written as $$\frac{d}{dt}\delta Z\left(t\right)=\left[\mathrm{Id}_{N}\otimes M_{\mu}+G_{s}\otimes\mathrm{Id}_{2}\right]\delta Z\left(t\right),\label{eq:variational-eq}$$ where $\delta Z=\left(\delta\boldsymbol{z}_{1},...,\delta\boldsymbol{z}_{N}\right)^{\top}$, $\mathrm{Id}_{N}\in\mathbb{R}^{N\times N}$ is the $N$-dimensional identity matrix, $$G_{s}=\left[\begin{array}{cccc}
0 & 1 & & 0\\
\vdots & \ddots & \ddots\\
0 & & \ddots & 1\\
1 & 0 & s & 0
\end{array}\right]\label{eq:coupling-scheme-Gs}$$ is the coupling matrix of the perturbed ring, $A\otimes B$ denotes the tensor product of the two matrices $A$ and $B$. Equation (\[eq:variational-eq\]) is a simple example of a system, which is treatable by a master stability function (MSF) approach [@Pecora1998]. In our case the MSF $M:\mathbb{C}\longrightarrow\mathbb{R}$ simply reads $M\left(\lambda\right)=\alpha+\Re\left(\lambda\right)$, where $\lambda$ is an eigenvalue of the coupling matrix $G_{s}$. Indeed, the spectrum of (\[eq:variational-eq\]) is $$\sigma\left(\mathrm{Id}_{N}\otimes M_{\mu}+G_{s}\otimes\mathrm{Id}_{2}\right)=\sigma\left(M_{\mu}\right)+\sigma\left(G_{s}\right)=\left\{ \mu,\bar{\mu}\right\} +\sigma\left(G_{s}\right).\label{eq:spectrum-equilibrium}$$ Taking the real part, we obtain $M\left(\lambda\right)=\alpha+\Re\left(\lambda\right)$. The spectrum of the coupling matrix $G_{s}$ equals the set of solutions of the characteristic equation $$\chi_{G_{s}}\left(\lambda\right)=\lambda^{N}-s\lambda^{\ell-1}-1=0.\label{eq:CharPoly}$$ For $s=0$, $\sigma\left(G_{0}\right)$ consists of the $N$ roots of unity $$\gamma_{N,k}=e^{i\frac{2\pi k}{N}},\ k=0,...,N-1.$$ For small $s\ne0$, the roots $\lambda_{k}$, $k=1,...,N$, of eq. (\[eq:CharPoly\]) are given (asymptotically) as $$\lambda_{k}\left(s\right)=\gamma_{N,k}+\frac{s}{N}\gamma_{N,k}^{\ell}+{\cal O}\left(s^{2}\right),\label{eq:asymp-evs-G}$$ as one can readily compute by applying the implicit function theorem to (\[eq:CharPoly\]) with base points $(s_{0},\lambda_{0})=(0,\gamma_{N,k})$. Hence, the spectrum of $G_{s}$ is a weak modulation of the spectrum of the circulant matrix $G_{0}$ [\[]{}see Fig. \[fig:Spectra\](a), (b)[\]]{}. For large $s$, $\sigma\left(G_{s}\right)$ can be computed in a similar manner. In Appendix \[sec:Adjacency-spectrum-appendix\] we show that in this case the spectrum of $G_{s}$ splits into two parts: there are $\ell-1$ roots $$\lambda_{1,k}\left(s\right)\approx s^{-\nicefrac{1}{\ell-1}}\gamma_{\ell-1,k},\ k=0,...,\ell-2,\label{eq:ev-inner-circle}$$ located close to an inner circle of amplitude $\sim s^{-\nicefrac{1}{\ell-1}}$ and $N-\ell+1$ roots $$\lambda_{2,k}\left(s\right)\approx s^{\nicefrac{1}{N-\ell+1}}\gamma_{N-\ell+1,k},\ k=0,...,N-\ell,\label{eq:ev-outer-circle}$$ which are close to an outer circle of amplitude $\sim s^{\nicefrac{1}{N-\ell+1}}$ [\[]{}see Fig. \[fig:Spectra\][\]]{}.
![\[fig:Spectra\]Spectra of the coupling matrix $G_{s}$ (\[eq:coupling-scheme-Gs\]) for $N=20$ oscillators, a shortcut at node $\ell=6$ and for different coupling strengths $s$: a) $s=0.1$, b) $s=0.6$, c) $s=1$, d) $s=5$.](crosslinks_fig2_v2){width="100.00000%"}
Bifurcations of the equilibrium\[sub:BifurcationsOfEquilibrium\]
----------------------------------------------------------------
From the formula (\[eq:spectrum-equilibrium\]) for the spectrum of the equilibrium $\boldsymbol{z}_{1}=..=\boldsymbol{z}_{N}\equiv0$, it follows that it is asymptotically stable iff $$\Re\left(\lambda\right)<-\alpha,\ \text{for all }\lambda\in\sigma\left(G_{s}\right).\label{eq:stability-equilibrium}$$ When the parameter $\alpha$ is increased starting from a value which satisfies (\[eq:stability-equilibrium\]), a bifurcation takes place whenever for some $\lambda\in\sigma\left(G_{s}\right)$: $$\alpha=-\Re\left(\lambda\right).\label{eq:bifurcation-points}$$ Note that there is always one purely real eigenvalue $\lambda_{1}\left(s\right)=1+\frac{s}{N}+{\cal O}\left(s^{2}\right)$ of $G_{s}$ which has maximal real part among all eigenvalues of $G_{s}$. Indeed, for any solution $\lambda=\varrho e^{i\varphi}$, $\varrho\ge0$, of (\[eq:CharPoly\]), we have $$1=\left|\varrho^{N}e^{iN\varphi}-s\varrho^{\ell-1}e^{i\left(\ell-1\right)\varphi}\right|\ge\varrho^{N}-s\varrho^{\ell-1}=\chi_{G_{s}}\left(\varrho\right)+1.$$ Hence, $\chi_{G_{s}}\left(\varrho\right)\le0$, and $\lim_{x\to\infty}\chi_{G_{s}}\left(x\right)=+\infty$ implies that there exists a real solution $\varrho_{0}\ge\varrho$ such that $\chi_{G_{s}}\left(\varrho_{0}\right)=0$. Therefore, for small $s\ge0$ the equilibrium switches stability at $$\alpha_{1}\left(s\right)\approx-\left(1+\frac{s}{N}\right).$$ Since we assume $\beta\ne0$, this bifurcation is a Hopf bifurcation and the emerging periodic orbit has frequency $\beta$ at onset. As for $s=0$, the bifurcation is supercritical for small $s>0$, because the cubic term of the corresponding normal form depends continuously on $s$. Therefore, a branch of stable periodic solutions emerges and exists for $\alpha>\alpha_{1}\left(s\right)$. A further increase of $\alpha$ leads to a sequence of Hopf bifurcations which give rise to $N-1$ branches of periodic solutions. As in the case $s=0$, all bifurcations are supercritical if $s>0$ is small (by continuity). The same is true if $s>0$ is large, as we show in Appendix \[sec:Supercriticality-of-HB\]. The latter $N-1$ periodic solutions are initially unstable, inheriting instability from the steady state. The initial frequency $\omega$ of the emerging periodic solution equals the imaginary part of the crossing eigenvalue, that is $\omega=\beta+\Im\left(\lambda\right)$ for the corresponding $\lambda\in G_{s}$. For the unperturbed ring ($s=0$) the first $\left\lfloor (N-1)/2\right\rfloor $ ($\left\lfloor x\right\rfloor :=\max\left\{ n\in\mathbb{N}:\,n\le x\right\} $) branches stabilize when they cross the Eckhaus stabilization line [@Yanchuk2008a] $$\frac{1}{N}\vert Z\vert^{2}=\frac{3\alpha}{4}+\sqrt{\left(\frac{\alpha}{4}\right)^{2}+\frac{1}{2}},\label{eq:eckhaus-line-seq0}$$ where $\left|Z\right|^{2}$ is the (constant) amplitude of a periodic solution $Z\left(t\right)=(z_{1}\left(t\right),...,z_{N}\left(t\right))^{T}$. This observation is in striking analogy with the well known Eckhaus destabilization in diffusive systems [@Eckhaus1965; @Tuckerman1990]. Remarkably, it is also found in this unidirectional system which does not extend to a spatially extended system in a natural manner. In section \[sec:Stability-of-the-psols\] we investigate how the added shortcut changes this scenario.
Emergent periodic orbits\[sec:Emergent-periodic-orbits\]
========================================================
Let $Z\left(t\right)$ be a periodic solution of (\[eq:SL-system\]), which emerges from a Hopf bifurcation at $$\alpha\left(s\right)=-\Re\left(\lambda\left(s\right)\right),\label{eq:alpha-k-of-s}$$ and which is associated to the eigenvalue $\lambda\left(s\right)$ of the coupling matrix $G_{s}$, see (\[eq:asymp-evs-G\])–(\[eq:ev-outer-circle\]). Because of the $S^{1}$-symmetry of the system, we employ the ansatz $$Z\left(t\right)=\sqrt{\varepsilon}e^{i\omega\left(\varepsilon,s\right)t}V\left(\varepsilon,s\right),\label{eq:PerAnsatz}$$ where $$\varepsilon:=\alpha-\alpha\left(s\right)\ge0\label{eq:eps-k-of-s}$$ is the parameter distance from the bifurcation point, $$V\left(\varepsilon,s\right)=\left(v_{1}\left(\varepsilon,s\right),...,v_{N}\left(\varepsilon,s\right)\right)^{T}\in\mathbb{C}^{N},\label{eq:profile-psols}$$ is the profile vector and the frequency $\omega\left(\varepsilon,s\right)$ of $Z\left(t\right)$ is $$\omega\left(\varepsilon,s\right)=\beta+\Im\left(\lambda\left(s\right)\right)+{\cal O}\left(\varepsilon\right).\label{eq:freq-psols}$$ The emerging periodic orbit is $\varepsilon$-close to the complex plane spanned by the eigenvector $b\left(s\right)$ of $G_{s}$ corresponding to $\lambda\left(s\right)$ and tangential at the bifurcation point itself [@Kuznetsov1995]. This means, $V\left(0,s\right)=b\left(s\right)$ with $$b\left(s\right)=a\left(s\right)\cdot\left(1,\lambda\left(s\right),\lambda^{2}\left(s\right),...,\lambda^{N-1}\left(s\right)\right)^{T},\label{eq:eigenvector_Gs}$$ where one may assume $a\left(s\right)\in\mathbb{R}$ due to the $S^{1}$-symmetry of the system. Figure \[fig:Profiles\] shows several examples of the profile shapes for different $s$ and $\lambda$. For $\left|\lambda\right|>1$ the emerging solutions become stronger localized at the $N$-th node $z_{N}$ with increasing $s$ since it scales as $z_{N}\sim\lambda^{N-1}$. For $\left|\lambda\right|<1$, the localization takes place at $z_{1}\left(t\right)$ for the same reason.
![\[fig:Profiles\]Moduli of the components of the initial profiles $V\left(0,s\right)=b\left(s\right)$ of emerging periodic solutions [\[]{}see (\[eq:profile-psols\]) and (\[eq:eigenvector\_Gs\])[\]]{} for different wave numbers $k$ (i.e., different eigenvalues $\lambda_{k}\left(s\right)$ of $G_{s}$) and coupling strengths $s$ as indicated in the figure. For all panels: $N=100$ and $\ell=26$.](crosslinks_fig3_v2){width="100.00000%"}
Case I: small perturbation\[sub:orbits-small-s\]
------------------------------------------------
In this section we consider $s$ to be small. Our aim is to determine a formal asymptotic expansion for the frequencies (\[eq:freq-psols\]) and the profiles (\[eq:profile-psols\]) of the periodic solutions and to derive evaluable approximate conditions for their stability. In particular, we are interested in the deformation of the Eckhaus stabilization line (\[eq:eckhaus-line-seq0\]). In order to obtain asymptotic expressions for the profiles in case that $s,\varepsilon>0$, we linearize the vector field in $\varepsilon=s=0$. For a periodic solution (\[eq:PerAnsatz\]), we introduce rescaled, rotating coordinates $u_{j}\left(t\right)\in\mathbb{C}$, $j=1,...,N$ according to $$z_{j}=\sqrt{\varepsilon}e^{i\omega t}v_{j}u_{j},\label{eq:scaled-rotating-coords}$$ with $\omega=\omega\left(\varepsilon,s\right)$ and $v_{j}=v_{j}\left(\varepsilon,s\right)$. Then (\[eq:SL-system\]) becomes $$\begin{aligned}
\dot{u}_{j} & =\left(\alpha+i\left(\beta-\omega\right)-\varepsilon\left|v_{j}u_{j}\right|^{2}\right)u_{j}+\frac{v_{j+1}}{v_{j}}u_{j+1},\label{eq:RotatingEquation1}\\
\dot{u}_{N} & =\left(\alpha+i\left(\beta-\omega\right)-\varepsilon\left|v_{N}u_{N}\right|^{2}\right)u_{N}+\frac{v_{1}}{v_{N}}u_{1}+s\frac{v_{\ell}}{v_{N}}u_{\ell},\label{eq:RotatingEquation2}\end{aligned}$$ In rotating coordinates, the equilibrium solution $$u_{j}\equiv1,\ j=1,...,N,\label{eq:u-eq-1-sln}$$ corresponds to the periodic solution (\[eq:PerAnsatz\]) of (\[eq:SL-system\]) and the stability of (\[eq:PerAnsatz\]) and (\[eq:u-eq-1-sln\]) is the same. In Appendix \[sec:Expansion-of-profiles\] we show that for each eigenvalue $\lambda_{k}\left(s\right)=\gamma_{N,k}+\frac{s}{N}\gamma_{N,k}^{\ell}+{\cal O}\left(s^{2}\right)$ of $G_{s}$, the corresponding branch of periodic solutions has frequencies $$\omega_{k}\left(\varepsilon,s\right)=\beta+\Im\left(\gamma_{N,k}\right)+\frac{s}{N}\Im\left(\gamma_{N,k}^{\ell}\right)+{\cal O}\left(\left(\left|\varepsilon\right|+\left|s\right|\right)^{2}\right),\label{eq:omega-ebene1}$$ and profiles $$v_{j}\left(\varepsilon,s\right)=\gamma_{N,k}^{j-1}\left(1+s\frac{j-1}{N}\gamma_{N,k}^{\ell-1}\right)+{\cal O}\left(\left(\left|\varepsilon\right|+\left|s\right|\right)^{2}\right).\label{eq:vjEbene1}$$
Case II: large perturbation\[sub:Orbits-large-s\]
-------------------------------------------------
In this section we consider $s$ to be large. To treat (\[eq:SL-system\]) as a weakly perturbed system we perform a change of variables $$y_{j}\left(t\right)=\varsigma^{j}z_{j}\left(\varsigma^{2N}t\right),\label{eq:variable-transfo}$$ with a small parameter $\varsigma=s^{-\nicefrac{1}{N-\ell+1}}$, which is the inverse radius of the outer spectral circle of eigenvalues (\[eq:ev-outer-circle\]). This transformation normalizes the emerging profiles (\[eq:eigenvector\_Gs\]) corresponding to the eigenvalues (\[eq:ev-outer-circle\]). The transformed variables (\[eq:variable-transfo\]) satisfy $$\begin{aligned}
\dot{y}_{j}\left(t\right) & = & \left(\varsigma^{2N}\mu-\varsigma^{2\left(N-j\right)}\left|y_{j}\left(t\right)\right|^{2}\right)y_{j}\left(t\right)+\varsigma^{2N-1}y_{j}\left(t\right),\ j=1,...,N-1,\nonumber \\
\dot{y}_{N}\left(t\right) & = & \left(\varsigma^{2N}\mu-\left|y_{N}\left(t\right)\right|^{2}\right)y_{N}\left(t\right)+\varsigma^{3N-1}y_{1}+\varsigma^{2N-1}y_{\ell}\left(t\right).\label{eq:transformed-sys-full-large-s}\end{aligned}$$ Since $\varsigma^{3N-1}y_{1}=\varsigma^{N}\left(\varsigma^{2N-1}y_{1}\right)$ and $\varsigma^{N}$ is small, we consider system (\[eq:transformed-sys-full-large-s\]) as a small perturbation of $$\begin{aligned}
\dot{y}_{j}\left(t\right) & = & \left(\varsigma^{2N}\mu-\varsigma^{2\left(N-j\right)}\left|y_{j}\left(t\right)\right|^{2}\right)y_{j}\left(t\right)+\varsigma^{2N-1}y_{j}\left(t\right),\ j=1,...,N-1,\nonumber \\
\dot{y}_{N}\left(t\right) & = & \left(\varsigma^{2N}\mu-\left|y_{N}\left(t\right)\right|^{2}\right)y_{N}\left(t\right)+\varsigma^{2N-1}y_{\ell}\left(t\right).\label{eq:transformed-sys-large-s-1}\end{aligned}$$ Although we cannot show that results for the persistence of hyperbolic invariant manifolds [@fenichel1971] apply and assure that (\[eq:transformed-sys-large-s-1\]) possesses the same hyperbolic invariant manifolds as does (\[eq:transformed-sys-full-large-s\]), the truncated system (\[eq:transformed-sys-large-s-1\]) is a natural approximation to (\[eq:transformed-sys-full-large-s\]). Note that in (\[eq:transformed-sys-large-s-1\]) the components $y_{1},\,...,\,y_{\ell-1}$ do not couple back to the rest of the system since the weak link from $y_{1}$ to $y_{N}$ was taken out. Therefore, the dynamics of the subsystem $y_{\ell},\,...,\,y_{N}$ is independent and, apart from the zero solution, acts as a periodic force on the attached subsystem $y_{1},\,...,\,y_{\ell-1}$. In original variables (\[eq:transformed-sys-large-s-1\]) reads $$\begin{aligned}
\dot{z}_{j}\left(t\right) & = & \left(\mu-\left|z_{j}\left(t\right)\right|^{2}\right)z_{j}+z_{j+1}\left(t\right),\ j=1,...,N-1,\nonumber \\
\dot{z}_{N}\left(t\right) & = & \left(\mu-\left|z_{N}\left(t\right)\right|^{2}\right)z_{N}+sz_{\ell}\left(t\right).\label{eq:SL-system-large-s-all-nodes}\end{aligned}$$ The linearization of (\[eq:SL-system-large-s-all-nodes\]) at the zero solution has eigenvalues $$\mu+\nu,\ \,\text{and}\,\ \bar{\mu}+\nu$$ where $\nu$ is a root of the characteristic equation $$\left(\nu^{N-\ell+1}-s\right)\nu^{\ell-1}=0$$ of the reduced coupling matrix $H_{s}$, which is obtained by erasing the link from $z_{1}$ to $z_{N}$ from $G_{s}$. It has eigenvalues $$\nu=0\ \,\text{and}\ \,\nu=s^{\nicefrac{1}{N-\ell+1}}\gamma_{N-\ell+1,k},\ k=1,...,N-\ell+1.\label{eq:eigenvalue-H-s}$$ Periodic orbits which correspond to the algebraically $\left(\ell-1\right)$-fold eigenvalue $\nu=0$ are localized on the nodes $z_{1},\,...,\,z_{\ell-1}$.
Stability of the periodic orbits\[sec:Stability-of-the-psols\]
==============================================================
Case I: small perturbation\[sub:stab-small-s\]
----------------------------------------------
The variational equation for system (\[eq:RotatingEquation1\])–(\[eq:RotatingEquation2\]) at the stationary solution (\[eq:u-eq-1-sln\]) is
$$\begin{aligned}
\dot{u}_{j} & = & \left(\alpha+\varepsilon+i\left(\beta-\omega\right)-2\varepsilon\left|v_{j}\right|^{2}\right)u_{j}+\frac{v_{j+1}}{v_{j}}u_{j+1},\\
\dot{u}_{N} & = & \left(\alpha+\varepsilon+i\left(\beta-\omega\right)-2\varepsilon\left|v_{N}\right|^{2}\right)u_{N}+\frac{v_{1}}{v_{N}}u_{1}+s\frac{v_{\ell}}{v_{N}}u_{\ell}.\end{aligned}$$
We transform the system into $\mathbb{R}^{2N}$ ($\boldsymbol{x}_{j}=(\Re(u_{j}),\Im(u_{j}))^{T}$) and insert the expansions (\[eq:omega-ebene1\]) and (\[eq:vjEbene1\]) to obtain
$$\begin{aligned}
\dot{\boldsymbol{x}}_{j} & = & -\left[M_{\tilde{\lambda}}+2\varepsilon\boldsymbol{\delta}_{11}\right]\boldsymbol{x}_{j}+M_{\tilde{\lambda}}\boldsymbol{x}_{j+1}+{\cal O}\left(\left(\left|\varepsilon\right|+\left|s\right|\right)^{2}\right)\nonumber \\
\dot{\boldsymbol{x}}_{N} & = & -\left[M_{\tilde{\lambda}}+2\varepsilon\boldsymbol{\delta}_{11}\right]\dot{\boldsymbol{x}}_{N}+M_{\tilde{\lambda}}\dot{\boldsymbol{x}}_{1}+sM_{\lambda_{0}^{\ell}}\left[\dot{\boldsymbol{x}}_{\ell}-\dot{\boldsymbol{x}}_{1}\right]+{\cal O}\left(\left(\left|\varepsilon\right|+\left|s\right|\right)^{2}\right)\label{eq:FloquetGl}\end{aligned}$$
with $\lambda_{0}=\gamma_{N,k}$, $\tilde{\lambda}\left(s\right)=\lambda_{0}+\frac{s}{N}\lambda_{0}^{\ell}$, the matrix representation $M:\mathbb{C}\to\mathbb{R}^{2\times2}$ [\[]{}as in (\[eq:SL-system-real\])[\]]{} and $\boldsymbol{\delta}_{mn}=\left(\delta_{jm}\delta_{kn}\right)_{j,k}$ is the matrix which has the entry $1$ at position $\left(j,k\right)$ and zeros everywhere else. We drop the higher order terms in (\[eq:FloquetGl\]) and write the system in the form $$\dot{\boldsymbol{X}}=A\left(\varepsilon,s\right)\boldsymbol{X}=\left[-\mathrm{Id}_{N}\otimes\left(M_{\tilde{\lambda}}+2\varepsilon\boldsymbol{\delta}_{11}\right)+G_{0}\otimes M_{\tilde{\lambda}}+\left(\boldsymbol{\delta}_{N1}-\boldsymbol{\delta}_{N\ell}\right)\otimes sM_{\lambda_{0}^{\ell}}\right]\boldsymbol{X}.\label{eq:FloquetGlcompact}$$ Clearly, an MSF approach as in section \[sec:stability-of-equilibrium\] is not feasible anymore. However, we have reduced the dynamical problem to an algebraic one. The eigenvalues of system (\[eq:FloquetGlcompact\]) can be computed by standard numerical procedures to determine approximately the stability of the periodic solutions in the vicinity of a bifurcation point. The eigenvalues of (\[eq:FloquetGlcompact\]) approximate the eigenvalues of the exact system (\[eq:RotatingEquation1\])–(\[eq:RotatingEquation2\]) at the steady state (\[eq:u-eq-1-sln\]) up to first order in $\varepsilon$ and $s$. Anyway, this first order approximation leads to good predictions even for moderate values of the parameters, in particular for $\varepsilon$ [\[]{}see Fig. \[fig:Eckhaus-freq-vs-a\][\]]{}.
![\[fig:Eckhaus-freq-vs-a\]Bifurcation diagrams for different strengths $s$ of the shortcut in a ring of $N=100$ oscillators with a shortcut from node $\ell=26$ to node $N=100$; $\beta=2.5$. Dashed lines indicate unstable periodic orbits and solid lines stable periodic orbits. In black the stabilization line for $s=0$ is shown. The frequency of the solutions is plotted against the bifurcation parameter $\alpha$. The perturbation strength $s$ increases from left to right: a), d) $s=0.05$; b), e) $s=0.1$; c), f) $s=0.2$. The upper panels a), b), c) show the approximated diagram obtained from (\[eq:FloquetGlcompact\]). The lower panels d), e), f) show results of numerical bifurcation analysis of the full system for comparison. These calculations were carried out with the program AUTO [@Doedel2006].](crosslinks_fig4){width="100.00000%"}
### Resonances {#resonances .unnumbered}
An important observation for the perturbed system is that in comparison to the unperturbed case $s=0$, for small $s>0$ the point of stabilization of a periodic solution may be altered or not. It remains nearly the same if the corresponding eigenvalue $\lambda_{k}=\gamma_{N,k}+{\cal O}\left(s\right)$ fulfills $$\arg\left(\gamma_{N,k}\right)\approx\arg\left(\gamma_{N,k}^{\ell}\right),\label{eq:phase-matching-cond}$$ or equivalently, $\frac{k\left(l-1\right)}{N}\in\mathbb{Z}$. This corresponds to a situation where both components of the input $z_{1}\left(t\right)+sz_{\ell}\left(t\right)$ to node $z_{N}\left(t\right)$ possess approximately the same phase. At the same time it is a condition for maximizing the modulus $\left|\lambda_{k}\right|\approx\left|\gamma_{N,k}+\frac{s}{N}\gamma_{N,k}^{\ell}\right|$ and for $\gamma_{N,k}$ to span an eigenmode of both, the unperturbed cycle of length $N$ and of a cycle which has the same length $N-\ell-1$ as the newly created cycle $\ell\to N\to\left(N-1\right)\to...\to\ell$, i.e. to be an $N$-th and an $\left(N-\ell+1\right)$-th root of unity. On the contrary, the antiphase condition $$\arg\left(\gamma_{N,k}\right)\approx\arg\left(\gamma_{N,k}^{\ell}\right)+\pi.\label{eq:antiphase-cond}$$ causes the point of stabilization to grow rapidly with increasing $s$, i.e., it destabilizes the corresponding periodic orbit. The more precisely the equality (\[eq:antiphase-cond\]) holds, the more pronounced is the destabilizing effect of the additional link [\[]{}see fig. \[fig:Eckhaus-freq-vs-a\][\]]{}.
Case II: Large $s$\[sub:stab-large-s\]
--------------------------------------
For large $s$ both systems, the original (\[eq:SL-system\]) and the truncated (\[eq:SL-system-large-s-all-nodes\]), admit two types of periodic solutions emerging in bifurcations corresponding to eigenvalues of scale $\left|\lambda\right|\approx0$ or $\left|\lambda\right|\approx s^{\nicefrac{1}{N-\ell+1}}$, respectively [\[]{}see eq. (\[eq:ev-inner-circle\]), (\[eq:ev-outer-circle\]), and (\[eq:eigenvalue-H-s\])[\]]{}. In case of system (\[eq:SL-system\]) all these periodic solutions emerge in form of rotating eigenvectors (\[eq:PerAnsatz\])–(\[eq:eigenvector\_Gs\]) of the coupling matrix $G_{s}$ which leads to a locally pronounced activity in $z_{1}\left(t\right)$ for a corresponding eigenvalue $\left|\lambda\right|\approx0$, and in $z_{N}\left(t\right)$ for $\left|\lambda\right|\approx s^{\nicefrac{1}{N-\ell+1}}$ [\[]{}see fig. \[fig:Profiles\][\]]{}. The same picture applies for bifurcations of (\[eq:SL-system-large-s-all-nodes\]) corresponding to the simple eigenvalues ($\left|\nu\right|\approx s^{\nicefrac{1}{N-\ell+1}}$) of $H_{s}$, while the $\ell-1$-fold bifurcation at $\alpha=0$ corresponding to $\nu=0$ simultaneously creates several solutions which are completely localized in the attached subsystem $\left(z_{1},...,z_{\ell-1}\right)$ where only the oscillators $z_{1},\,...,\,z_{k}$, $k=1,...,\ell-1$, are active and all others silent. For each of these solutions the first active element $z_{k}\left(t\right)=\sqrt{\alpha}e^{i\beta t}$ is located on the limit cycle of an isolated Stuart-Landau oscillator and all other $z_{j}$, $j<k$, lock either in phase or antiphase to their input signal $z_{j+1}$. However, these solutions can never stabilize, since the zero solution of the subsystem $z_{\ell},\,...,\,z_{N}$ is unstable after the first bifurcation corresponding to $\left|\nu\right|\approx s^{\nicefrac{1}{n}}$. Therefore, it suffices to study the inhomogeneous ring $$\begin{aligned}
\dot{z}_{j}\left(t\right) & = & \left(\mu-\left|z_{j}\left(t\right)\right|^{2}\right)z_{j}+z_{j+1}\left(t\right),\ j=1,...,n-1,\nonumber \\
\dot{z}_{n}\left(t\right) & = & \left(\mu-\left|z_{n}\left(t\right)\right|^{2}\right)z_{n}+sz_{1}\left(t\right),\ n=N-\ell+1,\label{eq:SL-system-large-s}\end{aligned}$$ to understand the possibly stable dynamics of (\[eq:SL-system-large-s-all-nodes\]). To approximate the stability of the emerging periodic solutions one can proceed as for the case of small $s$: write the system in scaled rotating coordinates (\[eq:scaled-rotating-coords\]), linearize around the equilibrium solution (\[eq:u-eq-1-sln\]), expand the variational equations in powers of $\varepsilon$ and truncate terms of order higher than ${\cal O}\left(\varepsilon\right)$.We obtain the following approximate variational equation [\[]{}see Appendix \[sec:Expansion-of-profiles-inhom\][\]]{}: $$\begin{aligned}
\dot{\boldsymbol{x}}_{j} & = & -\left[s^{\frac{1}{n}}M_{\gamma_{n,k}}+\varepsilon\left(\left|v_{j}^{0}\left(s\right)\right|^{2}\left(\begin{array}{cc}
3 & 0\\
0 & 1
\end{array}\right)-\left(\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right)\right)\right]\boldsymbol{x}_{j}\nonumber \\
& & +\left[s^{\frac{1}{n}}M_{\gamma_{n,k}}+\varepsilon\left(\left|v_{j}^{0}\left(s\right)\right|^{2}-1\right)\left(\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right)\right]\boldsymbol{x}_{j+1},\label{eq:appr-vareq-large-s-1}\end{aligned}$$ where $\left|v_{j}^{0}\left(s\right)\right|^{2}=ns^{\frac{2(j-1)}{n}}\frac{s^{\frac{2}{n}}-1}{s^{2}-1}$. Although the loss of symmetry prevents us from applying an MSF approach, (\[eq:appr-vareq-large-s-1\]) enables us to approximate the Floquet exponents by solving numerically the characteristic equation of (\[eq:appr-vareq-large-s-1\]).
![\[fig:bifdiag-freq-vs-a-intermediate-s\]Bifurcation diagram for intermediate strength $s=5$ of the shortcut. Fixed parameters are $N=100$, $\ell=26$, $\beta=2.5$. Branches of unstable periodic orbits are indicated by dashed lines and stable branches by solid lines. Along the black line (\[eq:eckhaus-line-seq0\]) the stabilization takes place for $s=0$. The frequency of the solutions is plotted against the bifurcation parameter $\alpha$. The calculations were carried out with the program AUTO [@Doedel2006].](crosslinks_fig5){width="95.00000%"}
Discussion\[sec:Discussion-1\]
==============================
We have investigated how the introduction of a shortcut alters the dynamics in a unidirectional ring of Stuart-Landau oscillators. In absence of a shortcut, the system exhibits a bifurcation scenario similar to the Eckhaus instability observed in dissipative media. For small shortcut strengths $s$ we have found that the Eckhaus stabilization line is modulated in the following manner: The destabilizing impact on periodic solutions is stronger for non-resonant modes than for resonant ones. The latter correspond to wavenumbers that are compatible with the lengths of both cycles which exist in the perturbed system, i.e. for the cycle of length $N-\ell+1$ which contains the shortcut, as well as for the full cycle of length $N$. In contrary to the non-resonant solutions, the stabilization of the resonant periodic solutions occurs for similar parameter values as in the case without shortcut. As a result, one can control the destabilization of a specific set of wavenumbers via the link position $\ell$ and its strength $s$. In the case of a large shortcut strength $s$ we have provided an argument that the cycle of length $N-\ell+1$ dominates the dynamics and stable solutions can be treated as solutions of a single unidirectional inhomogeneous ring which has coupling strength $s$ at only one link.
Further investigations will be dedicated to how small perturbations may be used to select solutions with a specific wavenumber by adding a corresponding set of resonant links. More generally, our findings may help to understand how perturbations with a more complicated structure, consisting of several shortcuts, can influence the dynamics of a unidirectional ring. Moreover, our observations might even help to locate an unkown shortcut when one is only allowed to vary a bifurcation parameter and observe the dynamics, since the modulated Eckhaus line can be used to identify the shortcut. A strong shortcut can be used in arbitrary networks in order to localize the activity on the cycles in which they are contained and amplify the activity in particular on their targets.
Another point which was not investigated here but deserves a closer study is the development of profiles far from the bifurcation point. The simplest, most important phenomenon is that, for increasing $\alpha$, the exponential tails of the profiles develop into plateaus, where the profile amplitude is locally constant as a function of the component index. By taking the limit $N\to\infty$ one can argue that solutions may consist of several, sharply separated plateaus where each plateau can possess a different wavenumber. This complies with numerical observations, although all observed solutions with more than one plateau lie on the unstable branches which correspond to the inner spectral circle for large $s$. For sufficiently large values of $s$, those branches begin to curl with increasing $\alpha$ and seem to be unstable for arbitrary large values of $\alpha$ [\[]{}see Fig. \[fig:bifdiag-freq-vs-a-intermediate-s\][\]]{}.
Adjacency spectrum for large $s$\[sec:Adjacency-spectrum-appendix\]
===================================================================
To apply the implicit function theorem and continue roots of the characteristic equation (\[eq:CharPoly\]) for the case of large $s$, we define $$F\left(\lambda,\tau,\vartheta\right)=\tau\lambda^{N}-\lambda^{\ell-1}-\vartheta.\label{eq:large-s-function}$$ Then $F\left(\lambda,\frac{1}{s},\frac{1}{s}\right)=0$ is equivalent to (\[eq:CharPoly\]). We now apply the implicit function theorem twice to find the two distinct families (\[eq:ev-inner-circle\]) and (\[eq:ev-outer-circle\]) of small and large solutions of (\[eq:CharPoly\]). Let us first compute $$\begin{aligned}
\partial_{\lambda}F\left(\lambda,\tau,\vartheta\right) & = & \tau N\lambda^{N-1}-\left(\ell-1\right)\lambda^{\ell-2},\nonumber \\
\partial_{\tau}F\left(\lambda,\tau,\vartheta\right) & = & \lambda^{N},\qquad\partial_{\vartheta}F\left(\lambda,\tau,\vartheta\right)=-1.\label{eq:large-s-function-derivatives}\end{aligned}$$ Consider the equation $$F\left(\lambda,0,\vartheta\right)=-\lambda^{\ell-1}-\vartheta=0.\label{eq:large-s-small-ev1}$$ It possesses $\ell-1$ solutions $$\lambda_{1,k}\left(0,\vartheta\right)=\vartheta^{\frac{1}{\ell-1}}e^{i\frac{\pi}{\ell-1}}\gamma_{\ell-1,k},$$ $k=0,...,\ell-2$. Assuming (we show that below) that for $\left(\tau,\vartheta\right)\ne0$ one can extend these solutions to smooth functions $\left(\tau,\vartheta\right)\mapsto\lambda_{1,k}\left(\tau,\vartheta\right)$ which solve $F\left(\lambda_{1,k}\left(\tau,\vartheta\right),\tau,\vartheta\right)=0$. These can be expanded in $\tau=0$ as $$\begin{aligned}
\lambda_{1,k}\left(\tau,\vartheta\right) & = & \lambda_{1,k}\left(0,\vartheta\right)-\partial_{\lambda}F\left(\lambda_{1,k}\left(0,\vartheta\right),0,\vartheta\right)^{-1}\partial_{\tau}F\left(\lambda_{1,k}\left(0,\vartheta\right),0,\vartheta\right)\tau+{\cal O}\left(\tau^{2}\right)\\
& = & \lambda_{1,k}\left(0,\vartheta\right)+\left(\left(\ell-1\right)\lambda_{1,k}^{\ell-2}\left(0,\vartheta\right)\right)^{-1}\lambda_{1,k}^{N}\left(0,\vartheta\right)\tau+{\cal O}\left(\tau^{2}\right)\\
& = & \lambda_{1,k}\left(0,\vartheta\right)+\frac{\lambda_{1,k}^{N-\ell+2}\left(0,\vartheta\right)}{\ell-1}\tau+{\cal O}\left(\tau^{2}\right)\end{aligned}$$ and, for $\vartheta=\tau$, $$\lambda_{1,k}\left(\tau,\tau\right)=\lambda_{1,k}\left(0,\tau\right)+\frac{\lambda_{1,k}^{N-\ell+2}\left(0,\tau\right)}{\ell-1}\tau+{\cal O}\left(\tau^{2}\right).\label{eq:large-s-small-ev-asymp}$$ This gives a family of $\ell-1$ solutions of (\[eq:CharPoly\]) situated near a small circle of radius $\sim\left(1/s\right)^{\frac{1}{\ell-1}}$. For $\tau\ne0$ , the equation $$F\left(\lambda,\tau,0\right)=\tau\lambda^{N}-\lambda^{\ell-1}=0\label{eq:large-s-large-ev1}$$ has an $\left(\ell-1\right)$-fold root at $\lambda=0$ which corresponds to the solutions of (\[eq:large-s-small-ev1\]) and $N-\ell+1$ roots $$\lambda_{2,k}\left(\tau,0\right)=\left|\tau\right|^{-\frac{1}{N-\ell+1}}\gamma_{N-\ell+1,k}\label{eq:Large-s-Large-Eigenvalues}$$ $k=0,...,N-\ell$. As before, we obtain an asymptotic representation
$$\begin{aligned}
\lambda_{2,k}\left(\tau,\tau\right) & = & \lambda_{2,k}\left(\tau,0\right)-\partial_{\lambda}F\left(\lambda_{2,k}\left(\tau,0\right),\tau,0\right)^{-1}\partial_{\vartheta}F\left(\lambda_{2,k}\left(\tau,0\right),\tau,0\right)\tau+{\cal O}\left(\tau^{2}\right)\nonumber \\
& = & \lambda_{2,k}\left(\tau,0\right)+\chi_{G_{s}}^{\prime}\left(\lambda_{2,k}\left(\tau,0\right)\right)^{-1}\tau+{\cal O}\left(\tau^{2}\right)\label{eq:large-s-largel-ev-asymp}\end{aligned}$$
This is the family of solutions which lie near a larger circle of radius $\sim s^{\frac{1}{N-\ell+1}}$.
Now let us show that the interval of existence of the implicit functions $\tau\mapsto\lambda_{1,k}\left(\tau,\vartheta_{0}\right)$ and $\vartheta\mapsto\lambda_{2,k}\left(\tau_{0},\vartheta\right)$, respectively, indeed contain the points $\tau=\vartheta_{0}$ and $\vartheta=\tau_{0}$, respectively. Let us denote the implicit function in question simply $\lambda\left(\tau,\vartheta\right)$. If the implicit function theorem fails to provide an extension of $\lambda\left(\tau,\vartheta\right)$ in some point $\left(\tau_{\ast},\vartheta_{\ast}\right)>0$, we must have $$\partial_{\lambda}F\left(\lambda_{\ast},\tau_{\ast},\vartheta_{\ast}\right)=\tau_{\ast}N\lambda_{\ast}^{N-1}-\left(\ell-1\right)\lambda_{\ast}^{\ell-2}=0\label{eq:singular-point-ift}$$ where $\lambda_{\ast}=\lambda\left(\tau_{\ast},\vartheta_{\ast}\right)$. Since $F\left(0,\tau,\vartheta\right)=0$ is equivalent to $\vartheta=0$, we may assume $\lambda_{\ast}\ne0$. Thus, (\[eq:singular-point-ift\]) is equivalent to $$\lambda_{\ast}^{N-\ell+1}=\frac{\ell-1}{N\tau_{\ast}}.\label{eq:ift-ex-cond1}$$ Furthermore, from $(\ref{eq:large-s-function})=0$ we obtain $\tau_{\ast}=\lambda^{\ell-1-N}+\vartheta_{\ast}\lambda^{-N}$ which we insert into (\[eq:singular-point-ift\]) to obtain $$\lambda_{\ast}^{\ell-1}=-\frac{N}{N-\ell+1}\vartheta_{\ast}.\label{eq:ift-ex-cond2}$$ From (\[eq:ift-ex-cond1\]) and (\[eq:ift-ex-cond2\]), we obtain $$\tau_{\ast}=\Gamma\left(\vartheta_{\ast}\right):=\frac{\ell-1}{N}\left(\frac{N-\ell+1}{N}\right)^{\frac{N-\ell+1}{\ell-1}}\vartheta_{\ast}^{-\frac{N-\ell+1}{\ell-1}}.$$ Since $\Gamma:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ is monotonic, we have $$\partial_{\lambda}F\left(\lambda\left(\tau,\vartheta_{\ast}\right),\tau,\vartheta_{\ast}\right)\ne0$$ for all $\tau<\tau_{\ast}=\Gamma\left(\vartheta_{\ast}\right)$. Since $\Gamma\left(\tau\right)\to\infty$, for $\tau\searrow0$, there exists $\tau_{0}>0$ such that $\tau<\Gamma\left(\tau\right)$ and $\lambda\left(\tau,\tau\right)$ is defined uniquely, for $0<\tau<\tau_{0}$.
Supercriticality of the Hopf bifurcations\[sec:Supercriticality-of-HB\]
=======================================================================
To show that the bifurcations at $\alpha=-\Re\left(\lambda\right)$ of system (\[eq:SL-system-real\]) are supercritical for sufficiently large $s\ge0$, we use the projection method for center manifolds [@Kuznetsov1995]. In the following, we write the vector field of (\[eq:SL-system-real\]) as $$f_{\alpha}\left(\boldsymbol{z}\right)=A\boldsymbol{z}+\frac{1}{6}C\left(\boldsymbol{z},\boldsymbol{z},\boldsymbol{z}\right),$$ where $A=\mathrm{Id}_{N}\otimes M_{\mu}+G_{s}\otimes\mathrm{Id}_{2}$ is the linearization of $f_{\alpha}$ at $\boldsymbol{z}=0$ and the trilinear function $C$ contains all cubic terms. Furthermore, let $\boldsymbol{v}\in\mathbb{R}^{2N}$ be an eigenvector of $A$ corresponding to the eigenvalue $\mu+\lambda$, $\lambda\in\sigma(G_{s})$. (The case of an eigenvalue $\bar{\mu}+\lambda$ can be treated analogously.) Let $w\in\mathbb{R}^{2N}$ be the normalized adjoint eigenvector corresponding to $\boldsymbol{v}$, i.e. $A^{T}\boldsymbol{w}=(\bar{\mu}+\bar{\lambda})\boldsymbol{w}$ and $\left\langle \boldsymbol{w},\boldsymbol{v}\right\rangle =1$. We have $$\begin{aligned}
\boldsymbol{v} & = & \left(1,\lambda,...,\lambda^{N-1}\right)^{T}\otimes\left(i,1\right)^{T},\label{eq:ev}\\
\boldsymbol{w} & = & \frac{1}{\bar{\kappa}}\left(\overline{\lambda}{}^{\ell-1},...,\overline{\lambda},\overline{\lambda}^{N},...,\overline{\lambda}^{\ell}\right)^{T}\otimes\left(i,1\right)^{T},\label{eq:adjoint-ev}\end{aligned}$$ with $\kappa=2\lambda^{\ell-1}(\left(\ell-1\right)+\left(N-\ell+1\right)\lambda^{N})$. A Hopf bifurcation at $\alpha$ is supercritical for a negative and subcritical for a positive first Lyapunov coefficient $$l_{1}\left(0\right)=\frac{1}{2\omega_{0}^{2}}\Re\left(\left\langle \boldsymbol{w},C\left(\boldsymbol{v},\boldsymbol{v},\overline{\boldsymbol{v}}\right)\right\rangle \right),\label{eq:first-lyap-coeff}$$ where $\omega_{0}=\beta+\Im\left(\lambda\right)\ne0$. Writing $C=(C_{1,1},C_{1,2},\dots,C_{N,1},C_{N,2})$ we obtain $$C_{j,1}\left(\boldsymbol{v},\boldsymbol{v},\overline{\boldsymbol{v}}\right)=-6v_{j,1}\left|v_{j,1}\right|^{2}-2\left(2v_{j,1}\left|v_{j,2}\right|^{2}+v_{j,2}^{2}\overline{v}_{j,1}\right)=iC_{j,2}\left(\boldsymbol{v},\boldsymbol{v},\overline{\boldsymbol{v}}\right),\label{eq:components-cubic-form-C}$$ for $1\le j\le N$. Using this, we get $$\left\langle \boldsymbol{w},C\left(\boldsymbol{v},\boldsymbol{v},\overline{\boldsymbol{v}}\right)\right\rangle =-8\left(\frac{\sum_{j=1}^{\ell-1}\left|\lambda\right|^{2\left(j-1\right)}+\lambda^{N}\sum_{j=\ell}^{N}\left|\lambda\right|^{2\left(j-1\right)}}{\left(\ell-1\right)+\left(N-\ell+1\right)\lambda^{N}}\right)\label{eq:LyapCoeffKompakt}$$\
For large $s>0$ we distinguish two cases.For the case $\left|\lambda\right|\sim s^{\nicefrac{1}{N-\ell+1}}$, we find that $$\Re\left(\left\langle \boldsymbol{w},C\left(\boldsymbol{v},\boldsymbol{v},\overline{\boldsymbol{v}}\right)\right\rangle \right)\to-\infty,\ \text{as }s\to\infty,$$ and for the case $\left|\lambda\right|\sim s^{-\nicefrac{1}{\ell-1}}$, $$\Re\left(\left\langle \boldsymbol{w},C\left(\boldsymbol{v},\boldsymbol{v},\overline{\boldsymbol{v}}\right)\right\rangle \right)\nearrow0,\ \text{as }s\to\infty.$$ This means that for sufficiently large $s$, we have $l_{1}\left(0\right)<0$ for all eigenvalues. Thus, all bifurcations are supercritical.
Supercriticality for an inhomogeneous ring
==========================================
Consider the case $\ell=1$ and $s$ arbitrary, i.e. a ring with inhomogeneous coupling strengths. Here, using (\[eq:LyapCoeffKompakt\]) and the characteristic equation, $\lambda^{N}=1+s$, we get $$\begin{aligned}
\left\langle \boldsymbol{w},C\left(\boldsymbol{v},\boldsymbol{v},\overline{\boldsymbol{v}}\right)\right\rangle & = & -\frac{8\left(1-\left(1+s\right)^{2}\right)}{N\left(1-\left|1+s\right|^{2/N}\right)}.\label{eq:lyap-coeff-inhom-ring}\end{aligned}$$ Thus, in the case of the inhomogeneous ring the bifurcation is supercritical for arbitrary $s$.
Expansion of the solution profiles for small perturbations\[sec:Expansion-of-profiles\]
=======================================================================================
The linearization of (\[eq:RotatingEquation1\])–(\[eq:RotatingEquation2\]) at (\[eq:u-eq-1-sln\]) is $$\begin{aligned}
0 & = & \left(\alpha+i\left(\beta-\omega\right)-\varepsilon\left|v_{j}\right|^{2}\right)+\frac{v_{j+1}}{v_{j}},\label{eq:V-and-omega-full-eqn}\\
0 & = & \left(\alpha+i\left(\beta-\omega\right)-\varepsilon\left|v_{N}\right|^{2}\right)+\frac{v_{1}}{v_{N}}+s\frac{v_{\ell}}{v_{N}},\nonumber \end{aligned}$$ From these equations we will now find expressions for the first terms of the Taylor expansions of the unknown functions $$\omega_{k}\left(\varepsilon,s\right)=\omega_{00}+\varepsilon\omega_{10}+s\omega_{11}+{\cal O}\left(\left(\left|\varepsilon\right|+\left|s\right|\right)^{2}\right)$$ and $$v_{j}\left(\varepsilon,s\right)=v_{j}^{00}+\varepsilon v_{j}^{10}+sv_{j}^{01}+{\cal O}\left(\left(\left|\varepsilon\right|+\left|s\right|\right)^{2}\right),\ j=1,...,N.$$
Terms at order ${\cal O}\left(1\right)$ {#terms-at-order-cal-oleft1right .unnumbered}
---------------------------------------
Considering the terms in $s=\varepsilon=0$ in (\[eq:V-and-omega-full-eqn\]) yields the circular equations (let $v_{N+1}^{00}:=v_{1}^{00}$) $$\begin{aligned}
0 & = & \alpha^{0}+i(\beta-\omega_{00})+\frac{v_{j+1}^{00}}{v_{j}^{00}},\label{eq:circ-eq-O1}\end{aligned}$$ with the shorthand $\alpha^{0}=\alpha_{k}\left(0\right)=-\cos\left(2\pi k/N\right)$. This leads to $$\left(-\alpha^{0}-i\left(\beta-\omega_{00}\right)\right)^{N}=1$$ which contains no new information, since it only determines $\alpha^{0}=-\Re\left(\lambda_{0}\right)$ and $\omega_{00}=\beta+\Im\left(\lambda_{0}\right)$ with an $N$-th root of unity $$\lambda_{0}=\lambda_{k}\left(0\right)=e^{i\frac{2\pi k}{N}}.$$ For the profile, (\[eq:circ-eq-O1\]) yields $$v_{j+1}^{00}=\lambda_{0}^{j}r_{0},\label{eq:profile-O1}$$ with a hitherto unknown scale factor $r_{0}:=v_{1}^{00}$. Without loss of generality, one may choose $v_{1}\left(\varepsilon,s\right)\in\mathbb{R}_{\ge0}$, because of the phase shift invariance of (\[eq:PerAnsatz\]). In particular, we then have $r_{0}\in\mathbb{R}_{+}$.
Terms at order ${\cal O}\left(\varepsilon\right)$ {#terms-at-order-cal-oleftvarepsilonright .unnumbered}
-------------------------------------------------
At first order in $\varepsilon$ we obtain another set of circular equations (let again $v_{N+1}^{10}=v_{1}^{10}$) $$\begin{aligned}
0 & = & 1-i\omega_{10}-r_{0}^{2}+\frac{\lambda_{0}}{r_{0}}\left(\frac{v_{j+1}^{10}}{\lambda_{0}^{j}}-\frac{v_{j}^{10}}{\lambda_{0}^{j-1}}\right)\end{aligned}$$ which leads us to the following recursion $$\frac{v_{j+1}^{10}}{\lambda_{0}^{j}}=\left(r_{0}^{2}-\left(1-i\omega_{10}\right)\right)\frac{r_{0}}{\lambda_{0}}+\frac{v_{j}^{10}}{\lambda_{0}^{j-1}}\label{eq:Rekursion1}$$ Defining $a_{j}=\frac{v_{j}^{10}}{\lambda_{0}^{j-1}}$ and $A=\left(r_{0}^{2}-(1-i\omega_{10})\right)\frac{r_{0}}{\lambda_{0}}$, equation (\[eq:Rekursion1\]) can be written as $a_{j+1}=A+a_{j}$ with the solution $a_{j+1}=jA+a_{1}$. Because of the circularity $a_{N+1}=a_{1}$ we then have $a_{1}=NA+a_{1}$ which determines $A=0$. Therefore, $r_{0}^{2}=(1-i\omega_{10})$ and finally $$\omega_{10}=0\ \text{and}\ r_{0}=1\label{eq:omega10-and-r0}$$ That means at first order the frequency of the oscillations does not depend on $\varepsilon$. For the profiles $v_{j}^{10}$ it follows (as at order ${\cal O}\left(1\right)$) $$v_{j+1}^{10}=\lambda_{0}^{j}r_{1\text{0}},\label{eq:profile-O-eps}$$ with $r_{10}:=v_{1}^{10}\in\mathbb{R}$.
Terms at $O\left(s\right)$ {#terms-at-oleftsright .unnumbered}
--------------------------
Due to the fact that $s$ perturbs the network symmetry, the equations at order ${\cal O}\left(s\right)$ are more complex than those at order ${\cal O}\left(\varepsilon\right)$. Again, the linear terms in $s$ of (\[eq:V-and-omega-full-eqn\]) give a recursive formula $$\begin{aligned}
\frac{v_{j+1}^{01}}{\lambda_{0}^{j}} & = & \frac{1}{\lambda_{0}}\left(i\omega_{01}-\alpha^{1}\right)+\frac{v_{j}^{01}}{\lambda_{0}^{j-1}},\nonumber \\
v_{1}^{01} & = & \frac{1}{\lambda_{0}}\left(i\omega_{01}-\alpha^{1}\right)-\lambda_{0}^{\ell-1}+\frac{v_{N}^{01}}{\lambda_{0}^{N-1}},\label{eq:v01Relation}\end{aligned}$$ where $\alpha^{1}=-\frac{d}{ds}\Re\left(\lambda_{k}\left(s\right)\right)\mid_{s=0}$. For $j<N$ we have $\frac{v_{j+1}^{01}}{\lambda_{0}^{j}}=\frac{j}{\lambda_{0}}\left(i\omega_{01}-\alpha^{1}\right)+v_{1}^{01}$. Inserting the resulting expression for $v_{N}^{01}$ in the second equation gives $v_{1}^{01}=\frac{N}{\lambda_{0}}\left(i\omega_{01}-\alpha^{1}\right)-\lambda_{0}^{\ell-1}+v_{1}^{01}$. Therefore, $N\left(i\omega_{01}-\alpha^{1}\right)=\lambda_{0}^{\ell}$ or equivalently $$\omega_{01}=\frac{1}{N}\Im\left(\lambda_{0}^{\ell}\right)\ \text{and}\ \alpha^{1}=-\frac{1}{N}\Re\left(\lambda_{0}^{\ell}\right).\label{eq:omega-and-alpha-O-s}$$ The perturbations $v_{j+1}^{01}$ of the profile are then determined as $$\begin{aligned}
v_{j+1}^{01} & = & \left(\frac{j}{N}\lambda_{0}^{\ell-1}+r_{01}\right)\lambda_{0}^{j}.\label{eq:profile-O-s}\end{aligned}$$ up to a scaling $r_{01}:=v_{1}^{01}\in\mathbb{R}$ as above.
Higher order terms {#higher-order-terms .unnumbered}
------------------
To determine the amplitudes $r_{10}$ and $r_{01}$ of the perturbations $v_{j}^{10}$ and $v_{j}^{01}$, $j=1,...,N$, we need to calculate the second order terms ${\cal O}\left(\varepsilon^{2}\right)$, ${\cal O}\left(s^{2}\right)$, and ${\cal O}\left(\varepsilon\cdot s\right)$ of (\[eq:V-and-omega-full-eqn\]) due to the nonlinear term. We omit this here and just give the resulting values $$r_{10}=0\ \text{and}\ r_{01}=0.\label{eq:r10-and-r01}$$ The vanishing $r_{10}$ means that at first order the profiles do not depend on $\varepsilon$.
Expansion of the solution profiles for the inhomogeneous ring \[sec:Expansion-of-profiles-inhom\]
==================================================================================================
A periodic solution of (\[eq:SL-system-large-s\]) corresponds to a fixed point in co-rotating coordinates (\[eq:scaled-rotating-coords\]). The stability of this fixed point is governed by its variational equations $$\begin{aligned}
\dot{\boldsymbol{x}}_{j} & = & \left(M_{\alpha+i\left(\beta-\omega\right)}-\varepsilon\left(s\right)\left|v_{j}\right|^{2}\left[\begin{matrix}3 & 0\\
0 & 1
\end{matrix}\right]\right)\dot{\boldsymbol{x}}_{j}+M_{\frac{v_{j+1}}{v_{j}}}\dot{\boldsymbol{x}}_{j+1},\label{eq:vareqn-psol-large-s-11}\\
\dot{\boldsymbol{x}}_{n} & = & \left(M_{\alpha+i\left(\beta-\omega\right)}-\varepsilon\left(s\right)\left|v_{n}\right|^{2}\left[\begin{matrix}3 & 0\\
0 & 1
\end{matrix}\right]\right)\dot{\boldsymbol{x}}_{n}+sM_{\frac{v_{1}}{v_{n}}}\dot{\boldsymbol{x}}_{1},\label{eq:vareqn-psol-large-s-12}\end{aligned}$$ Higher order terms of $\omega=\omega\left(\varepsilon,s\right)$ and $v_{j}=v_{j}\left(\varepsilon,s\right)$ can be determined by the equations $$\begin{aligned}
0 & = & \left(\alpha+i\left(\beta-\omega\right)\right)-\varepsilon\left|v_{j}\right|^{2}+\frac{v_{j+1}}{v_{j}},\label{eq:V-and-omega-inh-1}\\
0 & = & \left(\alpha+i\left(\beta-\omega\right)\right)-\varepsilon\left|v_{n}\right|^{2}+s\frac{v_{1}}{v_{n}},\label{eq:V-and-omega-inh-2}\end{aligned}$$ which are obtained from inserting the solution ansatz (\[eq:PerAnsatz\]) into (\[eq:SL-system-large-s\]). Solving for real and imaginary parts yields the conditions $$\begin{aligned}
\omega-\beta & = & \Im\left(\frac{v_{j+1}}{v_{j}}\right)=s\Im\left(\frac{v_{1}}{v_{n}}\right),\label{eq:cond-frequency}\\
\alpha & = & \varepsilon\left|v_{j}\right|^{2}-\Re\left(\frac{v_{j+1}}{v_{j}}\right)=\varepsilon\left|v_{n}\right|^{2}-s\Re\left(\frac{v_{1}}{v_{n}}\right).\label{eq:cond-profile}\end{aligned}$$ Note that we expand the unknown functions only in $\varepsilon$, keeping $s$ arbitrary. Using (\[eq:cond-frequency\]) and (\[eq:cond-profile\]) the variational equations (\[eq:vareqn-psol-large-s-11\])–(\[eq:vareqn-psol-large-s-12\]) write $$\begin{aligned}
\dot{\boldsymbol{x}}_{j} & = & -\left[M_{\frac{v_{j+1}}{v_{j}}}+2\varepsilon\mid v_{j}\mid^{2}\left(\begin{array}{cc}
1 & 0\\
0 & 0
\end{array}\right)\right]\boldsymbol{x}_{j}+M_{\frac{v_{j+1}}{v_{j}}}\boldsymbol{x}_{j+1},\label{eq:vareqn-psol-large-s-21}\\
\dot{\boldsymbol{x}}_{n} & = & -\left[sM_{\frac{v_{1}}{v_{n}}}+2\varepsilon\mid v_{n}\mid^{2}\left(\begin{array}{cc}
1 & 0\\
0 & 0
\end{array}\right)\right]\boldsymbol{x}_{n}+sM_{\frac{v_{1}}{v_{n}}}\boldsymbol{x}_{1}.\label{eq:vareqn-psol-large-s-22}\end{aligned}$$ In the rest of this section, we find expressions for $\frac{v_{j+1}}{v_{j}}\left(0,s\right)$ and $\frac{\partial}{\partial\varepsilon}\left[\frac{v_{j+1}}{v_{j}}\left(\varepsilon,s\right)\right]_{|\varepsilon=0}$ which can be inserted into (\[eq:vareqn-psol-large-s-21\])–(\[eq:vareqn-psol-large-s-22\]) to obtain (\[eq:appr-vareq-large-s-1\]). Let $$\omega\left(\varepsilon,s\right)=\omega_{0}\left(s\right)+\varepsilon\omega_{1}\left(s\right)+{\cal O}\left(\varepsilon^{2}\right)$$ and $$v_{j}\left(\varepsilon,s\right)=v_{j}^{0}\left(s\right)+\varepsilon v_{j}^{1}\left(s\right)+{\cal O}\left(\varepsilon^{2}\right),\ j=1,...,N.$$ For $\varepsilon=0$, equations (\[eq:V-and-omega-inh-1\])–(\[eq:V-and-omega-inh-2\]) yield $$\begin{aligned}
v_{j+1}^{0} & = & \left(i\left(\omega_{0}-\beta\right)-\alpha_{0}\right)v_{j}^{0}=...=\left(i(\omega_{0}-\beta)-\alpha_{0}\right)^{j}v_{1}^{0},\\
v_{1}^{0} & = & \frac{1}{s}\left(i\left(\omega_{0}-\beta\right)-\alpha_{0}\right)v_{n}^{0},\end{aligned}$$ where $\alpha_{0}=-\Re\left(\lambda_{k}\left(s\right)\right)=-s^{\nicefrac{1}{n}}\cos\left(2\pi k/n\right)$ is the critical value at which the periodic solution emerges, $\omega_{0}=\beta+\Im\left(\lambda_{k}\right)$ is the initial frequency, and the initial profile is given by $v_{j}^{0}=\lambda_{k}^{j-1}v_{1}^{0}$.
For terms of order ${\cal O}\left(\varepsilon\right)$ one obtains $$\omega_{1}=0,\ \left|v_{1}^{0}\right|^{2}=n\frac{s^{\frac{2}{n}}-1}{s^{2}-1},\ \text{and }v_{j}^{1}=v_{j}^{0}\sum_{l=0}^{j-1}\left(\vert v_{j-l}^{0}\vert^{2}-1\right)+\lambda_{k}^{j}v_{1}^{1}$$ To approximate (\[eq:vareqn-psol-large-s-21\])–(\[eq:vareqn-psol-large-s-22\]), we calculate the first order terms of the quotients $v_{j+1}/v_{j}$, which gives at ${\cal O}(\varepsilon^{0})$ $$\frac{v_{j+1}^{0}}{v_{j}^{0}}=s^{\nicefrac{1}{n}}\gamma_{n,k},$$ and at ${\cal O}(\varepsilon)$ $$\begin{aligned}
\left(\frac{v_{j+1}}{v_{j}}\right)^{1} & = & \left|v_{j}^{0}\right|^{2}-1.\end{aligned}$$
|
---
abstract: |
We show that the problem of constructing *tree-structured descriptions of data layouts* that are optimal with respect to space or other criteria, from given sequences of displacements, can be solved in *polynomial time*. The problem is relevant for efficient compiler and library support for communication of non-contiguous data, where tree-structured descriptions with low-degree nodes and small index arrays are beneficial for the communication soft- and hardware. An important example is the Message-Passing Interface (MPI) which has a mechanism for describing arbitrary data layouts as trees using a set of increasingly general constructors. Our algorithm shows that the so-called MPI *datatype reconstruction problem by trees* with the full set of MPI constructors can be solved optimally in polynomial time, refuting previous conjectures that the problem is NP-hard. Our algorithm can handle further, natural constructors, currently not found in MPI.
Our algorithm is based on dynamic programming, and requires the solution of a series of shortest path problems on an incrementally built, directed, acyclic graph. The algorithm runs in $O(n^4)$ time steps and requires $O(n^2)$ space for input displacement sequences of length $n$.
author:
- |
Robert Ganian\
Algorithms and Complexity Group\
Vienna University of Technology\
Austria\
`rganian@gmail.com`
- |
Martin Kalany\
Parallel Computing Group\
Vienna University of Technology\
Austria\
`kalany@par.tuwien.ac.at`
- |
Stefan Szeider\
Algorithms and Complexity Group\
Vienna University of Technology\
Austria\
`stefan@szeider.net`
- |
Jesper Larsson Träff\
Parallel Computing Group\
Vienna University of Technology\
Austria\
`traff@par.tuwien.ac.at`
bibliography:
- 'traff.bib'
- 'parallel.bib'
title: 'Polynomial-time Construction of Optimal Tree-structured Communication Data Layout Descriptions[^1]'
---
Introduction {#sec:introduction}
============
It is a common situation for instance in parallel, numerical libraries that substructures of large, static data structures have to be communicated among processors [@ChoiDongarraOstrouchovPetitetWalkerWhaley96; @PoulsonMarkerHammondRomerovandeGeijn13], e.g., row- or column vectors or sub-matrices of multi-dimensional matrices, or irregular substructures corresponding to the non-zeros or other special elements of larger structures. This requires efficient access to the typically non-contiguously stored substructure elements in some predefined order, either for the application which “(un)packs” the elements (from) to some structured communication buffer, or for the communication soft- or hardware to handle the non-consecutive communication in a way that is transparent to the application. For the latter approach, concise and efficient descriptions of such substructures are needed. For instance, lists of element addresses or displacements are neither concise (space proportional to the number of elements is required) nor efficient (processing time is at least doubled, since also the list has to be traversed). For substructures with some regularities, much better representations are obviously possible. Often, tree representations are used with leaves describing base-types and interior constructor nodes how subtrees are repeated. For example, complex data types in C-like languages can be built recursively using a small number of constructors (like arrays and `struct`s) from given primitive types (`int`s, `char`s, `double`s, etc.), and the resulting type trees describe to the compiler how data are laid out in memory. The same kind of mechanism could be used to describe substructures of such data types (but is not a part of C). The Message-Passing Interface (MPI) [@MPI-3.0] is an important example of a parallel communication interface, indeed often used to implement parallel numerical libraries [@ChoiDongarraOstrouchovPetitetWalkerWhaley96; @PoulsonMarkerHammondRomerovandeGeijn13], which provides a generic, explicit mechanism for describing non-consecutive application data to allow the library implementation to perform non-consecutive communication in an efficient way, possibly by directly exploiting hardware features for, e.g., strided, non-consecutive communication. Given such a tree-structured description of an application data layout, it is a natural question to ask whether this description is optimal under some given cost model reflecting the cost of storing or processing the description. Likewise, given a trivial description of a data layout in the form of a long list of addresses (or offsets, or displacements), it is natural to ask for an algorithm for constructing an efficient, that is, cost-optimal representation as a tree with some given set of constructors. In the MPI community, the former problem is referred to as *type normalization*, and the latter as *type reconstruction* [@Traff11:typeguide]. Both problems are eventually important for the implementation of very high-quality MPI libraries. The problems would be similarly important in other parallel interfaces or languages supporting communication of arbitrarily structured, non-consecutive data. Ideally, a compiler would be able to perform the normalization (optimization) of data layout descriptions given more or less explicitly by the application programmer in the code with the constructs available in the parallel language [@SchneiderKjolstadHoefler13].
In this paper, we investigate primarily the type reconstruction problem for a given set of constructors, that is, the problem of finding the most concise tree representation of a given substructure specified by an explicit list of displacements. As the set of constructors, we use a convenient abstraction of the type constructors found in MPI [@MPI-3.0 Chapter 4]. This is both a natural and powerful set that includes constructors for the case where a single substructure is repeated in a regular or irregular pattern as well as the case where different substructures are concatenated with given displacements. Our main result is to show that an optimally concise tree representation can be found in polynomial time for the whole set of constructors, and thus as a corollary that both type reconstruction and type normalization for the whole set of MPI derived data type constructors can be solved in polynomial time. This is an interesting result since the computational hardness of the problem was not known before. Indeed, the problem was believed not to be in $P$ by parts of the MPI community. Specifically, we give an algorithm that finds an optimal type tree description for a sequence of displacements of length $n$ in $O(n^4)$ operations. The algorithm is based on a non-trivial use of dynamic programming requiring the solution of a single-source shortest path problem for each new subproblem solution. Using standard dynamic programming techniques, the space requirement is $O(n^2)$.
MPI libraries typically employ simple forms of type normalization to derived data types set up by the application programmer (this is folklore, but see [@KjolstadHoeflerSnir11; @KjolstadHoeflerSnir12; @RossMillerGropp03] for explicit descriptions). In recent papers [@Traff14:normalization; @Traff15:mpilinear], the problem was more systematically analyzed, and it was shown that when restricted to certain homogeneous constructors (those having a single child) the reconstruction and normalization problems can be solved quite efficiently in low, polynomial time. It was explicitly conjectured that the problems with the full set of MPI derived data type constructors would be NP-hard [@Traff15:mpilinear; @Traff11:typeguide]. We stress that when it is allowed to fold the constructed trees into even more concise, directed acyclic graphs (DAGs), the optimality of our construction is no longer guaranteed. We discuss this problem at the end of the paper.
The notion of an optimal tree-like representation of a data layout is of course relative to the way the tree will be used and processed by the parallel programming language or library implementation. Processing typically includes the ability to pack and unpack parts of the layout independently using hardware support for blocked, strided memory access and similar features of the communication subsystem. We do not deal with the problem of efficient datatype-tree processing here, but abstract storage and processing costs with a simple, parameterized cost model, which must be adapted to the concrete situation. The literature on optimization of the processing of tree representations of data layouts in MPI is large; some pointers are given in [@Traff14:normalization].
The paper is structured as follows. We define the set of considered constructors and precisely formulate the type reconstruction problem in Section \[sec:problem\]. Our main result is given in Section \[sec:treereconstruc\], which describes our dynamic programming algorithm, proves correctness and establishes the complexity bound. In Section \[sec:generalizations\] we discuss how our approach can be extended to include other convenient and in specific situations more concise constructors, and how the problem changes when trees can be folded into DAGs. Concluding remarks, including a discussion of relevant future work in this area are given in Section \[sec:conclusion\].
The type reconstruction problem {#sec:problem}
===============================
A *data layout* is an ordered sequence of relative (integer) displacements, each indexing a certain base data type (integer, char, floating point number) relative to some base address. Since the semantics of base-types will not be important for the following, we abstract the problem to consider from here onward *displacement sequences* which we write as $D={\langle d_0,d_1,\ldots,d_{n-1} \rangle}$ with the displacements $D[i]=d_i$ being indexed from $0$ to $n-1$. We point out that the complexity of the problems that we investigate does not change by considering full *type maps* consisting of sequences of displacements with their associated basetype (and number of bytes occupied), as would have to be done in a concrete implementation of our algorithms for real libraries, although of course the structure of the reconstructed types may look different. A *segment* of an $n$-element displacement sequence from index $i$ to index $j$ is denoted by $D[i,j]={\langle d_i,d_{i+1},\ldots,d_{j} \rangle}$, $0\leq i\leq
j<n$. A *prefix* of length $c$ is the segment $D[0,c-1]$. The displacements of the sequence are arbitrary (non-negative, negative) integers, and the same displacement can appear more than once (although this will normally not be the case, and is often disallowed, e.g., for some uses of derived data types in MPI). Thinking of displacements as (Byte) addresses, it is clear that any application data layout can be described by a displacement sequence. The ordering constraint (displacement sequence, *not* displacement set) implies that data are accessed in a specific order. This is often important for data layouts used in communication operations.
Displacement sequences typically contain regularities and some form of structure, since they can be thought of as arising from a specific application, and this can be exploited to obtain more concise descriptions. We do this by type trees, where interior *constructor nodes* describe some ordered catenation of the layout(s) described by the child(ren) node(s). It is natural to ask for an efficient, polynomial time algorithm for computing the most concise and efficient representation for a given set of constructors and cost model.
We consider the following set of constructors that subsume constructors found in C-like programming languages, as well as the derived data type constructors found in MPI:
\[def:baseconstructors\] A *basic tree* may be constructed from the following four *basic constructors*:
1. A *leaf* ${\mathsf{con}}(c)$ with *count* $c$ describes a sequence of $c$ adjacent relative displacements $0,1,2,\ldots,c-1$.
2. A *(homogeneous) vector* ${\mathsf{vec}}(c,d,C)$ with *count* $c$ and *stride* $d$ describes the catenation of $c$ sequences $C$ at relative displacements $0, d, 2d, \ldots, (c-1)d$.
3. A *(homogeneous) index* ${\mathsf{idx}}(c,{\langle i_0,i_1,\ldots,i_{c-1} \rangle},C)$ with *count* $c$ and *indices* ${\langle i_0,i_1,\ldots,i_{c-1} \rangle}$ describes the catenation of $c$ sequences $C$ at relative displacements $i_0,i_1,\ldots,i_{c-1}$.
4. A *heterogeneous index*, or *struct*, ${\mathsf{strc}}(c,\allowbreak {\langle i_0,i_1,\ldots,i_{c-1} \rangle},\allowbreak {\langle C_0,C_1,\ldots,C_{c-1} \rangle})$, with *count* $c$ and *indices* ${\langle i_0,i_1,\ldots,i_{c-1} \rangle}$ describes the catenation of $c$ sequences $C_0,C_1,\ldots,C_{c-1}$ at relative displacements $i_0,i_1,\ldots,i_{c-1}$.
For example, the displacement sequence ${\langle 3,5,7,9,11 \rangle}$ can be described by ${\mathsf{idx}}(1,\allowbreak {\langle 3 \rangle},{\mathsf{vec}}(5, \allowbreak 2,
\allowbreak {\mathsf{con}}(1)))$. A more involved example is shown in Figure \[fig:typetreeExample\]. Note that any displacement sequence $D$ of length $n$ can trivially be represented as ${\mathsf{idx}}(n, D,
{\mathsf{con}}(1))$.
\[.${\mathsf{strc}}(2,{\langle 0,60 \rangle})$ \[.${\mathsf{con}}(5)$ \] \[.${\mathsf{vec}}(5,-10)$ \[.${\mathsf{idx}}(3,{\langle 0,-4,7 \rangle})$ \[.${\mathsf{con}}(1)$ \] \] \] \]
We refer to vertices of type trees as *nodes*, where each node is one of the constructors.
It can easily be shown that each of the MPI derived data type constructors (for contiguous, vector, index, and structured subtrees) [@MPI-3.0 Chapter 4] is expressible by the basic constructors of Definition \[def:baseconstructors\], and that the mapping is almost one-to-one. For instance, the `MPI_Type_vector` constructor denotes a layout consisting of a strided sequence of blocks, each being a strided sequence of some type $B$. This is expressed as ${\mathsf{vec}}(c,s,{\mathsf{vec}}(b,e,B))$ where $c$ is the number of blocks, $s$ their stride, $b$ the number of elements in each block, and $e$ the stride used within each block. We treat base types as sequences of bytes which can be expressed by leaf nodes, e.g., a 32-bit entity like `int` would be expressed by ${\mathsf{con}}(4)$. The ${\mathsf{idx}}$ constructor makes it possible to express the repetition of the same layout $B$ each at some arbitrary displacement; for this only the sequence of start indices (and the size of this sequence) needs to be represented. The most expressive, arbitrary *branching constructor* ${\mathsf{strc}}$ can express the catenation of a sequence of possibly different, smaller layouts each starting at an arbitrary displacement. This is the only constructor node with arity greater than one. In contrast to the similar MPI constructor `MPI_Type_create_struct`, which also takes a repetition count (blocklength) for each substructure, the ${\mathsf{strc}}$ constructor saves this extra sequence. If a substructure is indeed a repetition of some even smaller substructure, this information is part of the substructure and not of the ${\mathsf{strc}}$ node itself. The basic constructors increase in generality and storage cost: an ${\mathsf{idx}}$ node is a ${\mathsf{strc}}$ node where all substructures are similar, and therefore does not need to store a sequence of subtypes; a ${\mathsf{vec}}$ node is an ${\mathsf{idx}}$ node with regularly strided displacements, which can be computed from a single scalar instead of storing an explicit index sequence. As the example in Figure \[fig:typetreeExample\] shows, the ${\mathsf{strc}}$ constructor makes unbounded compression possible over the ${\mathsf{idx}}$ constructor.
To make it possible to express further common patterns without redundancy, we also consider a few auxiliary constructors. The patterns that these constructors capture can all be expressed by two-level nestings of basic constructors, but possibly at a higher cost. For practical purposes and depending on the application usage patters that are intended to be supported, it might therefore make sense to have a richer set of constructors. For instance, MPI has both an `MPI_Type_create_indexed_block` (which is captured by the ${\mathsf{idx}}$ basic constructor node) and an `MPI_Type_indexed` constructor which stores also a repetition count for each index. In cases where all substructures are repeated the same number of times, this is strictly redundant, and there are therefore use cases for both constructors. We include the auxiliary constructors to argue informally that our algorithm can handle a large set of reasonable constructors.
\[def:auxconstructors\] An *extended tree* may contain also the following two *auxiliary constructors*:
1. A *strided bucket*, ${\mathsf{vecbuc}}(c,\allowbreak d,\allowbreak e,\allowbreak
{\langle b_0,b_1,\ldots,b_{c-1} \rangle},\allowbreak C)$ with *count* $c$ and *strides* $d,e$ describes the catenation of $c$ sequences at relative displacements $0, d, 2d,\ldots (c-1)d$. The $i$-th sequence is the catenation of $b_i$ sequences $C$ at relative displacements $0,e,2e,\ldots (b_i-1)e$.
2. An *indexed bucket*, ${\mathsf{idxbuc}}(c,\allowbreak e, \allowbreak {\langle i_0,i_1,\ldots i_{c-1} \rangle}, \allowbreak {\langle b_0,b_1,\ldots b_{c-1} \rangle},C)$, with *count* $c$ and *substride* $e$ describes the catenation of $c$ sequences at relative indices $i_0,i_1,\ldots i_{c-1}$. The $i$-th sequence is the catenation of $b_i$ sequences $C$ at relative displacements $0,e,2e,\ldots (b_i-1)e$.
As can be seen from the discussion above, the indexed bucket constructor corresponds to the `MPI_Type_indexed` constructor. There is no MPI counterpart of the other, arguably natural constructor. We discuss these constructors in more detail in Section \[sec:auxiliary\].
Each basic or extended tree represents one displacement sequence, obtained by an ordered traversal of the nodes of the type tree. This process is called *flattening* and is captured by the algorithm in Listing \[alg:flattening\] for the basic constructors; the auxiliary constructors can be handled similarly. The converse is not true: a displacement sequence will almost always have several possible type tree representations.
We make no claim that Listing \[alg:flattening\] depicts a particularly good way of implementing flattening [@Traff99:flattening]. Note that the size of the displacement sequence described by a type tree $T$ could be much larger than the number of nodes in $T$. Within this paper, we assume that all numbers can be represented by a constant number of bits; otherwise, our main result still holds, but the upper bound on space requirements increases by a logarithmic factor.
By the *conciseness* of a type tree we mean the space taken by the representation. This is constant for vector and leaf nodes and proportional to the size of the index and type sequences for the other constructors. Processing costs are related to conciseness: the concise vector constructor that describes a strided repetition of a sub-pattern can often be handled by strided memory-copy or strided communication operations, whereas constructors with sequences of displacements or types need at least a traversal of the corresponding sequences and typically entails a more irregular and expensive access to memory. We will therefore first focus on a simple cost model for optimizing conciseness.
The *cost* of a type node shall be proportional to the number of words that must be stored to process the node. This includes the node type (${\mathsf{con}}, {\mathsf{vec}}, {\mathsf{idx}}, {\mathsf{strc}}$), count, displacement or pointer to index or type array, pointer to child node(s), and a lookup cost for the elements in lists of indices or types: $$\begin{aligned}
{\mathrm{cost}}({\mathsf{con}}(c)) & = &K_{{\mathsf{con}}} \\
{\mathrm{cost}}({\mathsf{vec}}(c,d,C)) & = & K_{{\mathsf{vec}}} \\
{\mathrm{cost}}({\mathsf{idx}}(c,{\langle \ldots \rangle},C)) & = & K_{{\mathsf{idx}}}+cK_{{\mathrm{lookup}}}\\
{\mathrm{cost}}({\mathsf{strc}}(c,{\langle \ldots \rangle},{\langle \ldots \rangle})) & = & K_{{\mathsf{strc}}}+2cK_{{\mathrm{lookup}}}\\\end{aligned}$$ The constants can be adjusted to reflect other overheads related to representing and processing a node. We define the *cost* of a type tree $T$ to be the *additive cost* of its nodes $T_i$: ${\mathrm{cost}}(T) = \sum_i {\mathrm{cost}}(T_i)$.
For the examples given in this paper, we take $K_{{\mathsf{con}}} = K_{{\mathsf{vec}}} =
K_{{\mathsf{idx}}} = K_{{\mathsf{strc}}}$, and $K_{{\mathrm{lookup}}}=1$. For instance, with a C-style structure as shown in Listing \[lst:typenode\] to represent any of the type constructors, all constructors indeed have the same constant in the cost (which we could take as 6 units). We remark that our algorithm is not dependent on the specific choice of the cost function, and that our results also hold for other reasonable cost functions where the cost of a node is a function of the node itself and the costs of its children.
We can now formally define the problem that we will solve in the next section. Recall that a type tree $T$ *represents* a displacement sequence $D$ if $\texttt{Flatten}(T,0) = D$.
> <span style="font-variant:small-caps;">Basic Type Reconstruction Problem</span>\
> *Instance*: A displacement sequence $D$ of length $n$.\
> *Task*: Find a least-cost (or optimal) basic tree $T$ representing $D$; that is, ${\mathrm{cost}}(T)\leq{\mathrm{cost}}(T')$ for any basic tree $T'$ representing $D$.
Basic tree reconstruction in polynomial time {#sec:treereconstruc}
============================================
We now present our main result, namely that the <span style="font-variant:small-caps;">Basic Type Reconstruction Problem</span> can be solved in polynomial time. subsequently show that extending the set of the auxiliary constructors of Definition \[def:auxconstructors\].
\[thm:polytree\] For any input displacement sequence $D$ of length $n$, the <span style="font-variant:small-caps;">Basic Type Reconstruction Problem</span> can be solved in $O(n^4)$ time and $O(n^2)$ space.
<span style="font-variant:small-caps;">Proof outline:</span> We first give a characterization of the structure of optimal basic trees (Lemma \[lemma:niceTypeTree\]) which allows for a simple and elegant procedure to solve the special case of displacement sequences in *normal form* (Definition \[def:normalForm\]).
The fundamental observation for the proof is that any (non-trivial) displacement sequence can be described by either a catenation of the same kind of shorter displacement sequences (and thus by either a vector or an index constructor) or by a catenation of different, but shorter displacement sequences (and thus by a struct constructor). In both cases, for an optimal description, the description of the shorter sequences must likewise be optimal, and the principle of optimality applies. This intuition is formalized in Lemma \[lemma:repetition\] and Lemma \[lemma:strc\]. Lemma \[lemma:optimalForNF\] proves the claim for the special case of displacement sequences in normal form, with a detailed procedure given in Listing \[alg:typetree\].
Finally, Lemma \[lemma:optimalForAny\] shows how to construct an optimal basic tree for any displacement sequence out of an optimal basic tree representation of its normal form.
A *repetition* in a displacement sequence $D$ of length $n$ is a prefix $C = D[0,q-1]$ of length $q$ s.t. $q$ is a divisor of $n$ and for all $i$,$j$, $1\leq i <n/q$, $0\leq j<q$ we have that $D[j] - D[0]
= D[iq+j] - D[iq]$. A *strided repetition* of length $q$ additionally fulfills $D[(i+1)q] - D[iq] = D[q] - D[0]$ for all $i$, $0\leq i < n/q - 1$, where $d = D[q] - D[0]$ is the *stride* of the repetition.
The intention of the functions and (see Listing \[lst:subroutines\]) is to find (strided) repetitions $C$ of a displacement sequence $D$ that can be exploited to represent $D$ via an ${\mathsf{idx}}$ or ${\mathsf{vec}}$ constructor with subsequence $C$. It is easy to see that and as outlined both take linear time.
As mentioned above, any displacement sequence $D$ can be described by either a catenation of the same kind of shorter displacement sequences or by a catenation of different, but shorter displacement sequences. Additionally, a representation via a ${\mathsf{con}}$ node is possible if $D$ is a trivial displacement sequence ${\langle 0,1,\dots,\allowbreak n-1 \rangle}$. In terms of type trees, this means that an optimal basic tree $T$ for a displacement sequence $D$ is either
1. $T = {\mathsf{con}}(n)$, a single ${\mathsf{con}}$ node with count $n$; or
2. $T = {\mathsf{vec}}(c, d, S)$, where the prefix $D[0,q-1]$ of length $q =
n/c$ is a strided repetition in $D$ with stride $d$ and $S$ is an optimal basic tree for the prefix ${\langle D[0],\dots, D[q-1] \rangle}$; or
3. $T = {\mathsf{idx}}(c, {\langle i_0,\dots,i_{c-1} \rangle},S)$, where the prefix $D[0,q-1]$ of length $q = n/c$ is a repetition in $D$, $S$ is an optimal basic tree for the sequence ${\langle D[0]-i_0,\dots,
D[q-1]-i_0 \rangle}$ and the indices $i_0,\dots,i_{c-1}$ are such that $\FnFlatten(T,0) = D$; or
4. $T = {\mathsf{strc}}(c, {\langle i_0,\dots,i_{c-1} \rangle}, {\langle S_0,\dots,
S_{c-1} \rangle})$, where the $S_j$ for $0\leq j < c$ are optimal basic trees for some sequences $C_j$ which together with the indices $i_0,\dots,i_{c-1}$ are such that $\FnFlatten(T,0) = D$.
While the first case can be handled with a single scan of $D$, the others are more involved. In the following, we give a more detailed characterization of (optimal) basic trees to tackle the problem.
We call an index node ${\mathsf{idx}}(c,\allowbreak {\langle i_0,\dots \rangle},\allowbreak
C)$ or a struct node ${\mathsf{strc}}(c, \allowbreak
{\langle i_0,\dots \rangle},\allowbreak {\langle \dots \rangle})$ with $i_0 \neq 0$ a *shifted node*; $s= i_0$ is called the node’s *shift*.
Note that adding some value $s$ to all indices of an ${\mathsf{idx}}$ or ${\mathsf{strc}}$ node $N$ shifts the sequence represented by the basic tree rooted at $N$ by $s$.
\[def:nicetree\] A *nice basic tree* contains at most one shifted node, which is the first ${\mathsf{idx}}$ or ${\mathsf{strc}}$ node on every root to leaf path.
\[lemma:niceTypeTree\] For any basic tree $T$ representing a displacement sequence $D$, a nice basic tree representation $\tilde{T}$ of $D$ of equal cost exists.
A node is *bad* if it is a shifted node and it is not the first ${\mathsf{idx}}$ or ${\mathsf{strc}}$ node on every root to leaf path. Let $D$ be a fixed displacement sequence and let $T$ be a basic tree representing $D$ with a minimum number of bad nodes. We will show that $T$ is, in fact, nice.
Assume that a bad index node (the proof is analogous for a bad struct node) $N_I = {\mathsf{idx}}(c, \allowbreak {\langle i_0,\dots,\allowbreak
i_{c-1} \rangle},\dots)$ is present in the $k$-th subtree of a struct node $N_S = {\mathsf{strc}}(c',\allowbreak {\langle {i'_0},\dots,\allowbreak i'_k,\dots,
\allowbreak i'_{c'-1} \rangle},{\langle \dots \rangle})$ s.t. there is no other shifted node on the path from $N_I$ to $N_S$. We can change $N_I$ to a non-shifted index node by subtracting its shift $s=i_0$ from all indices $i_j$, for $0\leq j < c$ and adding $s$ to the $k$-th index $i'_k$ of $N_S$, i.e., $\tilde{N_I} = {\mathsf{idx}}(c,\allowbreak
{\langle 0,i_1-s,\dots,\allowbreak i_{c-1}-s \rangle},\dots)$ and $\tilde{N_S} =
{\mathsf{strc}}(c',\allowbreak {\langle {i'_0},\dots,\allowbreak i'_k+s,\dots ,
i'_{c'-1} \rangle},{\langle \dots \rangle})$. Notice that the basic tree obtained in this way still represents the same displacement sequence $D$ but contains one less bad node, and hence the existence of such a node $N_I$ would contradict our choice of $T$.
Hence there is no ${\mathsf{strc}}$ node on the path from a bad node $N_I$ to the root node $R$. If this path contains an index node $N'_I \neq
N_I$, proceed analogously to the previous case: $\tilde{N_I} =
{\mathsf{idx}}(c,\allowbreak {\langle 0,i_1-s,\dots,\allowbreak i_{c-1}-s \rangle},\dots)$ and $\tilde{N'_I} = {\mathsf{idx}}(c',\allowbreak {\langle i'_0+s,\dots,\allowbreak
i'_{c'-1}+s \rangle},\dots)$. Again, the obtained basic tree also represents $D$ but contains one less bad node, contradicting our original choice of $T$. Consequently, $T$ does not contain any bad nodes and thus must be a nice basic tree.
\[corollary:atMostOneIndexWithCount1\] Any optimal basic tree $T$ contains at most one index node with count 1, i.e., at most one node of the form $N =
{\mathsf{idx}}(1,{\langle i_0 \rangle},\dots)$. Additionally, there is no other ${\mathsf{idx}}$ or ${\mathsf{strc}}$ node on the path from $N$ to the root.
Assume that $T$ contains two index nodes with count 1. Since $T$ is a tree, there is an index node $N$ with count 1 s.t. the path from $N$ to the root node of $T$ contains another ${\mathsf{idx}}$ or ${\mathsf{strc}}$ node. In a cost-equivalent nice basic tree representation $\tilde{T}$ (obtained by applying the procedure from the proof of Lemma \[lemma:niceTypeTree\]), the corresponding index node is $\tilde{N} = {\mathsf{idx}}(1,{\langle 0 \rangle},T')$. Note that the type tree rooted at $\tilde{N}$ represents exactly the same displacement sequence as its subtype $T'$. Thus a representation $T'$ of less cost exists, which contradicts the assumption that $T$ is optimal.
The following proposition, although not directly required for the analysis, provides some additional insight into the structure of optimal basic trees.
\[proposition:heightOfOptimalTypeTree\] The height of an optimal basic tree is $O(\log n)$.
It is easy to see that an optimal basic tree does not contain two consecutive ${\mathsf{strc}}$ nodes, as they can always be merged into one while reducing the cost. For any basic tree $T$ that represents a sequence of length $n$, a basic tree ${\mathsf{idx}}(c,{\langle \dots \rangle},T)$ or ${\mathsf{vec}}(c,\dots,T)$ with $c\geq 2$ represents a sequence of length at least $2n$. Let $P$ be a maximum-length path from a leaf to the root of an arbitrary optimal basic tree. Since any optimal basic tree contains at most one ${\mathsf{idx}}$ node with count $c=1$ (Corollary \[corollary:atMostOneIndexWithCount1\]) and no ${\mathsf{vec}}$ node with $c=1$, the length of the represented sequence at least doubles with at least every other node on $P$.
\[def:normalForm\] The *normal form* $\hat{D}$ of a displacement sequence $D$ of length $n$ is defined as $\hat{D}[i] = D[i] - D[0]$, for all $i$, $0\leq i < n$.
In other words, the normal form $\hat{D}$ of a displacement sequence $D$ is obtained by shifting $D$ so that its first element is $0$.
\[corollary:structureOfNFSolutions\] An optimal basic tree $T$ for a displacement sequence $\hat{D}$ in normal form does not contain any shifted nodes or any ${\mathsf{idx}}$, ${\mathsf{vec}}$ or ${\mathsf{strc}}$ node with count 1.
It follows directly from Lemma \[lemma:niceTypeTree\] and Corollary \[corollary:atMostOneIndexWithCount1\] that there exists an optimal basic tree $T$ for $\hat{D}$ which does not contain any shifted nodes. Note that a non-shifted ${\mathsf{idx}}$, ${\mathsf{vec}}$ or ${\mathsf{strc}}$ node with count 1 does not change the represented sequence. Thus, removing such nodes from a basic tree reduces the cost while not changing the represented displacement sequence. It follows that no such node can be part of an optimal basic tree.
Observe that since there are no shifted nodes in an optimal basic tree $T$ for $\hat{D}$, any subtree of $T$ represents a segment of $\hat{D}$ in normal form. In the following, we will use $T_{i,j}$ to denote an optimal basic tree representation for the normalized segment $\hat{D}[i,j]$ of $\hat{D}$.
For convenience, we define the function $\FnMinCost(S, T)$ which, given two basic trees $S$ and $T$, returns the one with least cost (if either is `null`, the other is returned). Note that the cost of a basic tree can trivially be computed by a simple traversal. However, when constructing basic trees from the bottom up (as we will do in this section), we keep for each node the cost of the subtree rooted at that node. This allows for the cost of a basic tree to be queried in constant time and thus for a constant-time implementation of $\FnMinCost$.
\[lemma:repetition\] Let $\hat{D}$ be any displacement sequence of length $n$ in normal form and assume that optimal basic tree representations for all normal form prefixes of length less than or equal to $\lfloor n/2\rfloor$ are known. A representation $T_{r}$, where the root node of $T_{r}$ is either an ${\mathsf{idx}}$ or a ${\mathsf{vec}}$ node and $T_r$ is of least cost w.r.t. all possible representations of that form, can be computed in $O(n\sqrt{n})$ time.
Listing \[alg:checkForRepetitions\] enumerates all possible representations of the desired form and chooses the one with least cost among them. Note that for the divisor $q=1$, the trivial representation ${\mathsf{idx}}(n, \hat{D}, {\mathsf{con}}(1))$ (which exists for any displacement sequence $\hat{D}$), is generated and thus a valid representation for $\hat{D}$ is guaranteed to be found. For the same reasons as given in Corollary \[corollary:structureOfNFSolutions\], ${\mathsf{idx}}$ nodes with count 1 cannot be part of a least-cost representation of the desired form and thus need not be considered.
The number of divisors of $n$ is upper-bounded by $2\lfloor\sqrt{n}\rfloor$ and, by assumption, optimal representations for all prefixes of $\hat{D}$ of length less than or equal to $\lfloor
n/2 \rfloor$ are known, i.e., $T_{0,j}$ is known for all $j$, $O \leq
j \leq n/2$. This implies the claimed runtime bound.
\[lemma:strc\] Let $\hat{D}$ be any displacement sequence of length $n$ in normal form and assume that optimal basic tree representations are known for all normal form segments of length strictly less than $n$. A representation $T_{s}$, where the root node of $T_{s}$ is a ${\mathsf{strc}}$ node and $T_s$ is of least cost w.r.t. to all possible representations of that form, can be computed in $O(n^2)$ time.
Construct a weighted, directed acyclic graph $G=(V,E,w)$ with $V =
\{v_0,\dots,v_{n}\}$, $E = \{(v_i,v_j) \mid 0 \leq i < j \leq n,\; j -
i < n\}$ and the weight function $w$ which is defined for all edges $(v_i,v_j)$ in $E$ as $w(v_i,v_j) = 2K_{{\mathrm{lookup}}} + {\mathrm{cost}}(T_{i,j-1})$. The intended meaning of this construction is as follows. A node $v_i$ corresponds to the $i$-th element of $\hat{D}$ ($v_n$ is a special vertex that corresponds to the hypothetical first element after the end of $\hat{D}$) and an edge $(v_i,v_j)$ with $i<j$ corresponds to the segment $\hat{D}[i,j-1]$ in normal form. The weight of an edge $(v_i,v_j)$ is equal to the cost of the optimal representation $T_{i,j-1}$ of the segment $\hat{D}[i,j-1]$ (which exists by the assumption) plus a cost of $2K_{{\mathrm{lookup}}}$ for including this representation as a subtype in a ${\mathsf{strc}}$ node. The edge $(v_0, v_n)$, which is not part of the constructed graph, can be thought of as corresponding to the type tree $T_{0,n-1}$, i.e., the optimal type tree representation of $\hat{D}$ we want to compute.
Let $P = {\langle v_{0}, u_1,\dots, u_k, v_{n} \rangle}$ be a shortest path in $G$ from $v_0$ to $v_n$ with $u_i \in V$ for $1\leq i\leq k$. Then the basic tree ${\mathsf{strc}}(k+1,
{\langle \hat{D}[0],\hat{D}[u_1],\dots,\hat{D}[u_k] \rangle},
{\langle T_{0,u_1-1},T_{u_1,u_2-1},\dots T_{u_{k},n-1} \rangle})$ is a valid representation of $\hat{D}$. Note that by construction, for any valid representation of $\hat{D}$ of the desired form, a corresponding path from $v_0$ to $v_n$ exists in $G$ and thus a shortest path represents the desired solution of least cost. Given $P$, this representation can be constructed in linear time, since optimal representations for all required segments are known by the assumption. The resulting graph has $n\choose 2$ edges and the runtime is dominated by the cost of $O(n^2)$ time for finding a shortest path in a DAG.
We can now give the complete dynamic programming algorithm for constructing optimal basic trees for displacement sequences in normal form, which proves Lemma \[lemma:optimalForNF\]. Due to Lemma \[lemma:niceTypeTree\], it suffices to construct an optimal nice basic tree which according to Corollary \[corollary:structureOfNFSolutions\] cannot contain any shifted nodes nor any ${\mathsf{idx}}$, ${\mathsf{vec}}$ or ${\mathsf{strc}}$ nodes with count 1. The algorithm is shown in Listing \[alg:typetree\].
\[lemma:optimalForNF\] For any input displacement sequence $\hat{D}$ of length $n$ in normal form, the <span style="font-variant:small-caps;">Basic Type Reconstruction Problem</span> can be solved in $O(n^4)$ time and $O(n^2)$ space.
The input to the algorithm is an $n$-element displacement sequence $\hat{D}$ in normal form. The algorithm computes an optimal basic tree $T[i,j]$ for each normalized segment $\hat{D}[i,j]$, $0\leq i\leq
j<n$, which is stored with edge $(i,j+1)$ in the constructed graph $G$. Note that the solution for the whole input sequence $\hat{D}$ can be read off of the edge $(v_0,v_{n})$.
The algorithm starts with a preprocessing step to find all segments whose normal form is representable with a single ${\mathsf{con}}$ node. Note that the normal form of any segment of length 1 can trivially be represented as ${\mathsf{con}}(1)$ and since no other valid representations exist for this particular kind of displacement sequence, this representation is optimal. A straight forward implementation of this preprocessing step as in Listing \[alg:typetree\] is clearly feasible in time $O(n^2)$.
The algorithm computes optimal basic tree representations for all normalized segments of $\hat{D}$, via a bottom up dynamic programming approach. The dynamic programming table to be filled in is implicit in the graph $G$, where each segment $\hat{D}[i,j]$ is associated with an edge $(v_i, v_{j+1}$). Note that after the preprocessing step, solutions for all segments of length 1 are known. By incrementally computing optimal representations for all segments of length $2,\dots,n$, it is ensured that Lemmas \[lemma:repetition\] and \[lemma:strc\] can be applied to compute an optimal representation for each segment as follows. A basic tree $T_r$, whose root node is either an ${\mathsf{idx}}$ or a ${\mathsf{vec}}$ node, and a basic tree $T_s$, whose root node is a ${\mathsf{strc}}$ node, are computed. Both are of least cost w.r.t. all basic tree representations of the desired form. The optimal basic tree for a normalized segment $\hat{D}[i,i+l-1]$ is necessarily one of $T_r$, $T_s$ or a representation via a ${\mathsf{con}}$ node (if such a representation is possible), which was already computed in the preprocessing step.
To compute $T_r$, a small, technical extension of procedure $\FnRepetition$ (Listing \[alg:checkForRepetitions\]) for finding representations via ${\mathsf{idx}}$ or ${\mathsf{vec}}$ nodes is necessary. The procedure requires access to optimal representations of the prefixes of the argument displacement sequence $D$. However, in the general case, $D$ is a segment of $\hat{D}$, that is, $D = \hat{D}[i,j]$, and its prefixes therefore start with $\hat{D}[i]$. To account for this (and avoid copying $\hat{D}[i,j]$), we pass an additional argument $o$ representing the offset of the segment within the input displacement sequence $\hat{D}$ (i.e., for a segment $\hat{D}[i,j]$, we have $o=i$), and in lines 10 and 12 replace the argument $T_{0, q-i}$ with $T_{o, o+q-i}$.
To compute $T_s$ in Listing \[alg:typetree\], contrary to Lemma \[lemma:strc\], we do not construct a new graph for each segment when computing its representation $T_s$. Instead a single dynamic, incrementally built graph $G$ suffices to solve the problem for all segments of $\hat{D}$. By construction, when computing the desired representation of a segment $\hat{D}[i,i+l-1]$, $G$ contains edges representing optimal representations for all segments of length less than $l$ (and possibly some edges representing solutions of length $l$). A shortest path from node $v_i$ to $v_{i+l}$ in $G$ therefore leads to the same representation as the one constructed by Lemma \[lemma:strc\].
To find such a shortest path, for each segment $\hat{D}[i,i+l-1]$ of length $l$, one single-source shortest path (SSSP) problem on a weighted DAG with $l+1$ nodes and $O(l^2)$ edges has to be solved. Since $G$ is a topologically sorted DAG by construction, SSSP is solvable in $O(|V| + |E|)$ time, where $|V|$ denotes the number of vertices and $|E|$ denotes the number of edges in $G$ [@CormenLeisersonRivestStein09]. To compute the desired representations for all segments of length $l$, a shortest path has to be computed for each of the $n+1-l$ node pairs $(v_i,v_{i+l})$, for $0\leq i \leq n+1-l$. The total runtime is thus upper bounded by $\sum_{l=1}^{n+1} l^2 (n+1-l)$, which is $O(n^4)$.
The algorithm constructs a graph with $O(n^2)$ edges, where a basic tree $T_{i,j}$, representing the solution for the normalized segment $\hat{D}[i,j]$, is associated with each edge $(v_i, v_j)$. Note that for each edge $(v_i, v_j)$ it suffices to store the root node of the associated basic tree $T_{i,j}$ plus pointers to its child nodes, which are already stored with the respective edges. To meet the desired space bound, only a constant amount of space may be used by each edge and associated basic tree. This is trivially true for ${\mathsf{con}}$ nodes (apart from one word indicating the node’s kind and the cost of the type tree rooted at the node, only the count $c$ needs to be stored) as well as ${\mathsf{vec}}$ nodes (two integer values and one pointer to the child node are required in addition to the node’s kind and the cost of the type tree rooted at this node). However, ${\mathsf{idx}}$ and ${\mathsf{strc}}$ nodes may require $\Omega(n)$ space in the worst case (e.g., if ${\mathsf{idx}}(n, \hat{D}, {\mathsf{con}}(1))$ is the optimal representation of $\hat{D}$). We employ a standard trick often used in dynamic programming algorithms and store for each node only the information required to reconstruct the full solution once the algorithm in Listing \[alg:typetree\] has terminated. If for an ${\mathsf{idx}}$ node the count $c$ is known, the full ${\mathsf{idx}}$ node is easily derived as ${\mathsf{idx}}(c,
{\langle \hat{D}[0], \hat{D}[q],\dots, \hat{D}[(c-1)q] \rangle}, T_{0,q-1})$ with $q = n/c$. The parameters of a ${\mathsf{strc}}$ node associated with an edge $(v_i, v_j)$ can be reconstructed by again computing the shortest path from node $v_i$ to $v_j$ and mapping it to a ${\mathsf{strc}}$ node as done in Lemma \[lemma:strc\]. Note that this reconstruction step does not change the asymptotic runtime bound and that the required space for each node is $O(1)$, from which the claimed upper bound of $O(n^2)$ space follows directly.
The following Corollary \[corollary:idxWithCount1IsRoot\] and Lemma \[lemma:optimalForAny\] show how the algorithm of Lemma \[lemma:optimalForNF\] can be applied to general displacement sequences.
\[corollary:idxWithCount1IsRoot\] For any optimal basic tree with an index node $N$ with count 1, i.e., a node $N = {\mathsf{idx}}(1,{\langle i_0 \rangle},\dots)$, a representation $T'$ of equal cost s.t. $N$ is the root node of $T'$, exists.
Due to Corollary \[corollary:atMostOneIndexWithCount1\], there is no ${\mathsf{idx}}$ or ${\mathsf{strc}}$ node on the path from $N$ to the root and thus $N$ shifts the whole sequence by $i_0$. This shift can be represented equivalently by removing $N$ from the basic tree and adding a new root node to represent the shift, i.e., by letting $T' = {\mathsf{idx}}(1,
{\langle i_0 \rangle}, T \setminus N)$.
\[lemma:optimalForAny\] Given optimal basic trees $\hat{T}_{i,j}$ for all normalized segments $\hat{D}[i,j]$ of a displacement sequence $D$, an optimal basic tree $T$ representing $D$ can be computed in $O(n^2)$ time and $O(n)$ space.
By Lemma \[lemma:niceTypeTree\], for any optimal basic tree $T$ a cost-equivalent nice basic tree $\tilde{T}$ representing the same displacement sequence $D$ exists and it therefore suffices to find an optimal nice basic tree representation $\tilde{T}$ for $D$. By assumption, an optimal nice basic tree representation $\hat{T} =
\hat{T}_{0,n-1}$ for the normalized sequence $\hat{D}$ exists. To construct $\tilde{T}$, find the first node $N$ on any root to leaf path in $\hat{T}$ that is either an ${\mathsf{idx}}$ or a ${\mathsf{strc}}$ node and add the displacement sequence’s shift $s = D[0]$ to the indices of this node, i.e., if $N = {\mathsf{idx}}(c, {\langle i_0,\dots,i_{c-1} \rangle}, \hat{T'})$ in $\hat{T}$, set $\tilde{N} = {\mathsf{idx}}(c, {\langle i_0 + s,\dots,i_{c-1} + s \rangle},
\hat{T'})$ in $\tilde{T}$ and analogously for the case of $N$ being a ${\mathsf{strc}}$ node. Note that $\tilde{T}$ has the same cost as $\hat{T}$ and thus is an optimal basic tree representation for $D$.
If such a node does not exist, it follows from Lemma \[lemma:niceTypeTree\] and Corollary \[corollary:idxWithCount1IsRoot\] that the optimal solution is either
- $\tilde{T} = {\mathsf{idx}}(c,{\langle \dots \rangle}, \hat{T}_{0,n/c-1})$, for some divisor $c$ of $n$, or
- $\tilde{T} = {\mathsf{strc}}(c, {\langle \dots \rangle},{\langle \hat{T}_0,\dots, \hat{T}_{c-1} \rangle})$, for some $c$, $1< c<n$.
Note that for ${\mathsf{idx}}$ nodes, both the trivial representation ${\mathsf{idx}}(n,
D, {\mathsf{con}}(1))$ as well as the representation ${\mathsf{idx}}(1, {\langle D[0] \rangle},
\tilde{T}$ which only adds a shifted node to $\tilde{T}$ need to be checked. Since solutions for all normalized segments are already known, this construction is feasible in $O(n^2)$ time and $O(n)$ space.
The <span style="font-variant:small-caps;">Basic Type Reconstruction Problem</span> for a displacement sequence $D$ of length $n$ can be solved by computing an optimal basic tree representation for the normalized displacement sequence $\hat{D}$ (Lemma \[lemma:optimalForNF\]) and the post-processing step given in Lemma \[lemma:optimalForAny\]. The claimed space and time bounds follow directly from the given Lemmas.
Computing more concise representations {#sec:generalizations}
======================================
In this section we discuss possibly more space efficient tree representations by allowing a richer set of constructors, exemplified by the auxiliary constructors introduced in Definition \[def:auxconstructors\]. We then explain why computing representations by DAGs is an apparently harder problem. Finally, we discuss the applicability of our algorithms to the type normalization problem.
Handling the auxiliary constructors {#sec:auxiliary}
-----------------------------------
The auxiliary constructors of Definition \[def:auxconstructors\] can be handled by slight extensions to our algorithm in a way that polynomial-time type reconstruction is still possible. Basically, only the part that checks for vector or index patterns shown in Listing \[alg:checkForRepetitions\] needs to be extended. Assume that a repeated prefix $C$ of length $q$ has been found in the given displacement sequence $D$, and that $D'$ is the displacement sequence consisting of every $q$th element of $D$, i.e., $D'=[D[0],D[q],D[2q],\ldots]$.
The *strided bucket*, ${\mathsf{vecbuc}}(c, d, e,
{\langle b_0,b_1,\ldots,b_{c-1} \rangle}, C)$ constructor can concisely describe application data layouts consisting of buckets each with some maximum number of elements (the stride $d$) where each bucket contains some (possibly different) number of elements $b_i$ with bucket stride $e$. This description is likely to be less costly than describing such a layout by a ${\mathsf{strc}}$ constructor with each subtype describing one bucket. To incorporate the strided bucket it simply has to be checked in Listing \[alg:checkForRepetitions\] whether $D'$ follows the strided bucket pattern, and this can easily be done in linear time. There are two cases to consider. If the first bucket has more than one element, take as bucket stride $e=D'[1]-D'[0]$ and scan the index list for repetitions at stride $e$. The first violation at some position $i$ forces the maximum bucket size to be $d=D'[i]-D'[0]$. Now continue to scan till the end of $D'$, checking that the $e,d$ strided pattern repeats and counting the number of elements $b_i$ in each bucket of $e$-strided displacements. Otherwise, the first bucket has only one element. Take instead as maximum bucket size $d=D'[1]-D'[0]$, and scan for repetitions with stride $d$. The first violation at some position $i$ forces the bucket stride to be $e=D'[i]-D'[i-1]$. As in the other case, the bucket sizes $b_i$ are counted by scanning $D'$ till the end. If an index $i$ is found where $D'[i]-D'[i-1]\neq e$ and $D'[i]-D'[j]\neq d$ where $j$ is the start of the current bucket in $D'$, then $D'$ is not a displacement sequence of a strided bucket layout.
The strided bucket constructor is in a sense the opposite of the index constructor. Instead of an index sequence it takes a sequence of bucket sizes, and has (roughly) the same cost. Interestingly, there is no such constructor in the MPI standard.
The *indexed bucket*, ${\mathsf{idxbuc}}(c,\allowbreak e,\allowbreak
{\langle i_0,i_1,\ldots i_{c-1} \rangle},\allowbreak {\langle b_0,b_1,\ldots
b_{c-1} \rangle},\allowbreak C)$, on the other hand corresponds closely to the `MPI_Type_indexed` constructor. For each index, a repetition count $b_i$ gives the number of repeats of $C$ in the bucket starting at that index; all repetitions use the same stride $e$ (the constructor could trivially be extended to the case where each index has its own stride). For each possible bucket stride, the number of buckets that this stride will give rise to has to be counted. The stride $e$ leading to a smallest number of buckets is a candidate for the representation of $D'$ and $C$ as an ${\mathsf{idxbuc}}$ node. We observe that each $i$ with $D'[i+1]-D'[i]=e$ joins two $e$-strided segments $D'[j,i]$ and $D'[i+1,k]$ into one bucket starting at index $j$. Therefore, the stride that occurs most often in the stride sequence $S[i]=D'[i+1]-D'[i]$, $0\leq i < n -1$, will lead to the smallest number of buckets. To count the number of occurrences of each stride, we either sort $S$ or count by hashing during the scan of $D'$. Let $e$ be a stride with the most occurrences. A final scan of $D'$ suffices to compute the start indices and sizes of the buckets with stride $e$.
Type reconstruction into DAGs {#sec:dagreconstruc}
-----------------------------
A type tree describing some given displacement sequence may have multiple instances of the same subtree. Our algorithm in particular constructs nice type trees (Definition \[def:nicetree\]) in which all displacement sequences in index and struct nodes except perhaps one start at index $0$, and it can well happen that the same index or struct node occurs many times. A more concise representation results if such trees are folded into directed acyclic graphs with only one node for each substructure.
Type DAGs represent displacement sequences by the same flattening procedure as shown in Listing \[alg:flattening\] for trees. Each path from the root node in the type DAG to a leaf is traversed in order to generate the corresponding displacement sequence. Thus the processing cost of a type DAG would arguably be similar to the processing costs of a tree. By a similar traversal of a DAG an equivalent tree can be constructed, simply by making a new copy each time a node is visited.
The space required for the DAG can be much smaller than the space required for the corresponding tree. One can therefore define also for DAGs our cost model for optimizing conciseness as the additive cost of the nodes in the DAG; and *not* as the sum of the costs of all paths traversed. The type reconstruction problem into DAGs is now to find the least-cost DAG representing the given displacement sequence.
One crucial difficulty which arises when dealing with such type DAGs is that the best representation for a subsequence no longer needs to be locally optimal, since costs savings can be achieved by reusing other nodes of the DAG. This is illustrated in Figure \[fig:DAGExample\].
\[.${\mathsf{strc}}(3,{\langle 0,110,130 \rangle})$ \[.${\mathsf{strc}}(2,{\langle 0,5 \rangle})$ \[.${\mathsf{con}}(3)$ \] \[.${\mathsf{idx}}(20,X)$ \[.${\mathsf{con}}(3)$ \] \] \] \[.${\mathsf{con}}(1)$ \] \[.${\mathsf{idx}}(20,X)$ \[.${\mathsf{con}}(3)$ \] \] \]
In particular, this implies that the type tree constructed by unfolding a cost-optimal DAG is not necessarily a cost-optimal tree, and conversely, that the DAG obtained by folding a given, cost-optimal type tree is not necessarily a cost-optimal DAG. This constitutes a fundamental problem for our general approach for handling type trees, and new ideas are needed to solve the type reconstruction problem into DAGs.
The type normalization problem
------------------------------
The type normalization problem subsumes the type reconstruction problem that we have considered so far. Type normalization asks to improve the cost of an already given tree description of the data layout. Since any data layout can be represented as a single ${\mathsf{idx}}$ node with the whole displacement sequence as index sequence, type normalization includes type reconstruction as a special case. Type normalization is the problem that compiler or library implementors are typically faced with: application data structures described as trees are given by the programmer as part of the code, and an internal, optimal representation is to be constructed by the programming system.
The trivial solution is to flatten the given type tree and apply the type reconstruction algorithm on the resulting displacement sequence. Since the size of the resulting displacement sequence is not bounded by the size or conciseness of the tree, this is highly undesirable. We would like a procedure where the complexity can be bounded by the conciseness of the type trees, specifically the total size of the index sequences in the tree.
As shown in [@Traff14:normalization], if the set of basic constructors is restricted to exclude the ${\mathsf{strc}}$ constructor, it is possible to perform type normalization by only rechecking optimality of the ${\mathsf{idx}}$ nodes. In this case, type normalization can be done in time proportional to the conciseness of the given tree. When the ${\mathsf{strc}}$ constructor is allowed, arbitrarily more concise representations can be possible as shown in Figure \[fig:typetreeExample\]. Optimality of a subtree that does not use the ${\mathsf{strc}}$ constructor does therefore not imply optimality when ${\mathsf{strc}}$ is allowed. It is therefore necessary to flatten the whole tree and apply the tree reconstruction algorithm on the resulting displacement sequence.
Conclusion {#sec:conclusion}
==========
The main result of this paper is that the type reconstruction problem into trees is actually solvable in polynomial time. However, an $O(n^4)$ algorithm is not useful for larger values of $n$ as might be the case in parallel applications where $n$ could be proportional to the number of processors which in itself could be in the range of tens to hundreds of thousands. We note that our bottom-up dynamic programming algorithm performs a considerable amount of almost redundant checking for (strided) repetitions in displacement sequence segments. An asymptotically more efficient algorithm, perhaps based on a top-down approach, is likely to exist. Whether an exact, practically efficient algorithm for the full problem is possible, we do not know at the point of writing.
Restricting the power of the constructors can permit more efficient algorithms. As shown in [@Traff14:normalization], if only ${\mathsf{con}},{\mathsf{vec}}$ and ${\mathsf{idx}}$ nodes are allowed, then the type reconstruction problem for a displacement sequence of length $n$ can be solved in $O(n\sqrt{n})$ time. However, the resulting restricted trees can and often will be much more costly, as shown in Figure \[fig:typetreeExample\]. The high complexity of our algorithm is caused by the unbounded branching constructor ${\mathsf{strc}}$ node. A slightly better, $O(n^3)$ time algorithm would result from allowing only bounded branching, for instance a binary struct constructor that catenates only two subtrees. For such a constructor, the shortest path computation of Lemma \[lemma:strc\] could be done in linear time. In some contexts, bounded branching might be sufficiently expressive.
An alternative approach would be to look for low-complexity approximation algorithms with provable approximation guarantees. Or, even weaker, for heuristics that perhaps work well for the intended application cases. This reflects the state in current MPI libraries.
As discussed, type trees can be represented more concisely as directed acyclic graphs (DAGs). To the best of our knowledge, it is still open whether a cost-optimal DAG representation for an arbitrary displacement sequence can likewise be constructed in polynomial time.
A related problem to consider is the following. Given two displacement sequences of the same length, construct a least-cost tree (or DAG) representing a mapping between the two sequences. Such a tree (DAG) has uses when copying between different data layouts; this arises, e.g., in matrix transposition. In the MPI context this operation has been called *transpacking* [@Traff08:iotypes; @RossLathamGroppLuskThakur09]. Our dynamic programming algorithm may extend to this case as well.
Our work was specifically inspired by the derived data type mechanism of MPI. We believe that this idea is applicable in a much wider context of (parallel) programming interfaces and languages, and that the type normalization and reconstruction problems as defined here, as well as the associated processing of data layouts represented by trees, have relevance extending beyond the motivating context.
[^1]: This work was co-funded by the European Commission through the EPiGRAM project (grant agreement no. 610598).
|
---
abstract: |
Consider a network in which $n$ distributed nodes are connected to a single server. Each node continuously observes a data stream consisting of one value per discrete time step. The server has to continuously monitor a given parameter defined over all information available at the distributed nodes. That is, in any time step $t$, it has to compute an output based on all values currently observed across all streams. To do so, nodes can send messages to the server and the server can broadcast messages to the nodes. The objective is the minimisation of communication while allowing the server to compute the desired output.
We consider monitoring problems related to the domain $D_t$ defined to be the set of values observed by at least one node at time $t$. We provide randomised algorithms for monitoring $D_t$, (approximations of) the size $|D_t|$ and the frequencies of all members of $D_t$. Besides worst-case bounds, we also obtain improved results when inputs are parameterised according to the similarity of observations between consecutive time steps. This parameterisation allows to exclude inputs with rapid and heavy changes, which usually lead to the worst-case bounds but might be rather artificial in certain scenarios.
author:
- Pascal Bemmann
- Felix Biermeier
- Jan Bürmann
- Arne Kemper
- Till Knollmann
- Steffen Knorr
- Nils Kothe
- Alexander Mäcker
- Manuel Malatyali
- Friedhelm Meyer auf der Heide
- Sören Riechers
- Johannes Schaefer
- |
Jannik Sundermeier\
\[0.4em\] Heinz Nixdorf Institute & Computer Science Department\
Paderborn University, Germany\
{pbemmann, felixbm, jbuerman, kempera, tillk, stknorr,\
nkothe, amaecker, malatya, fmadh, soerenri, jschaef, janniksu}\
@mail.uni-paderborn.de
bibliography:
- 'bibliography.bib'
title: '[Monitoring of Domain-Related Problems in Distributed Data Streams]{}[^1]'
---
Introduction
============
Consider a system consisting of a huge amount of nodes such as a distributed sensor network. Each node continuously observes its environment and measures information such as temperature, pollution or similar parameters. Given such a system, we are interested in aggregating information and continuously monitoring properties describing the current status of the system at a central server. To keep the server’s information up to date, the server and the nodes can communicate with each other. In sensor networks, however, the amount of such communication is particularly crucial, as communication translates to energy consumption, which determines the overall lifetime of the network due to limited battery capacities. Therefore, algorithms aim at minimizing the communication required for monitoring the respective parameter at the server.
One very basic parameter is the domain of the system defined to be the values currently observed across all nodes. We consider different notions related to the domain and propose algorithms for monitoring the domain itself, (approximations of) its size and (approximations of) the frequencies of values comprising the domain, respectively. Each of these parameters can provide useful information, e.g. the information about the (approximated) frequency of each value allows to approximate very precisely the histogram of the observed values, and this allows to determine (approximations of) several functions of the input, e.g. heavy hitters, quantiles, top-$k$, frequency moments or threshold problems.
Model and Problems {#DR:section:goal}
------------------
We consider the continuous distributed monitoring setting, introduced by Cormode, Muthukrishnan, and Yi in [@cormodeModel], in which there are $n$ distributed nodes, each uniquely identified by an identifier (ID) from the set $ {\left\{ 1,\dots,n \right\}}$, connected to a single server. Each node observes a stream of values over time and at any discrete time step $t$ node $i$ observes one value $v_i^t \in {\left\{ 1,\dots,\Delta \right\}}$. The server is asked to, at any point $t$ in time, compute an output $f(t)$ which depends on the values $v_i^{t'}$ (for $t' \leq t$, and $i = 1, \ldots, n$) observed across all distributed streams up to the current time step $t$. The exact definition of $f(\cdot)$ depends on the concrete problems under consideration, which are defined in the section below. For the solution of these problems, we are usually interested in approximation algorithms. An $\varepsilon$-approximation of $f(t)$ is an output $\tilde f(t)$ of the server such that $(1-\varepsilon)f(t) \leq \tilde f(t) \leq (1+\varepsilon)f(t)$. We call an algorithm that, for each time step, provides an $\varepsilon$-approximation with probability at least $1-\delta$, an $(\varepsilon, \delta)$-approximation algorithm. To be able to compute the output, the nodes and the server can communicate with each other by exchanging single cast messages or by broadcast messages sent by the server and received by all nodes. Both types of communication are instantaneous and have unit cost per message. That is, sending a single message to one specific node incurs cost of one and so does one broadcast message. Each message has a size of $O(\log \Delta + \log n + \log\log\frac{1}{\delta})$ bits and will usually, besides a constant number of control bits, consist of a value from $\{1, \ldots, \Delta\}$, a node ID and an identifier to distinguish between messages of different instances of an algorithm applied in parallel (as done when using standard probability amplification techniques). Having a broadcast channel is an extension to [@cormodeModel], which was originally proposed in [@cormodeBroadcast] and afterwards applied in [@ipdps1; @ipdps2]. For ease of presentation, we assume that not only the server can send broadcast messages, but also the nodes. This changes the communication cost only by a factor of at most two, as a broadcast by a node can always be implemented by a single cast message followed by a broadcast of the server. Between any two time steps we allow a communication protocol to take place, which may use polylogarithmic ${\mathcal{O}}(\log^c n)$ rounds, for some constant $c$. The optimisation goal is the minimisation of the communication cost, given by the number of exchanged messages, required to monitor the considered problem.
### Monitoring of Domain-Related Functions.
In this paper, we consider the monitoring of different problems related to the *domain* of the network. The domain at time $t$ is defined as $D_t \coloneqq \{ v \in \{1, \ldots, \Delta \} \mid \exists i \text{ with } v_i^t = v \}$, the set of values observed by at least one node at time $t$. We study the following three problems related to the domain:
- **Domain Monitoring.** At any point in time, the server needs to know the domain of the system as well as a *representative* node for each value of the domain. Formally, monitor $D_t = \{v_1, \ldots, v_{|D_t|}\} \subseteq \{1,\ldots,\Delta\}$, at any point $t$ in time. Also, maintain a sequence $R_t = (j_1, \ldots, j_\Delta)$ of nodes such that for all observed values $v \in D_t$ a representative $i$ is determined with $j_v = i$ and $v_i^t = v$. For each value $v \notin D_t$ which is not observed, no representative is given and $j_v = \text{nil}$.
- **Frequency Monitoring.** For each $v \in D_t$ monitor the frequency $|N_t^v|$ of nodes in $N_t^v \coloneqq \{i \in \{1,\ldots, n\} \mid v^t_i = v\}$ that observed $v$ at $t$, i.e. the number of nodes currently observing $v$.
- **Count Distinct Monitoring.** Monitor $|D_t|$, i.e. the number of distinct values observed at time t.
We provide an exact algorithm for the Domain Monitoring Problem and $(\varepsilon, \delta)$-approximations for the Frequency and Count Distinct Monitoring Problem.
Our Contribution
----------------
For the Domain Monitoring Problem, an algorithm which uses $\Theta(\sum_{t \in T}|D_t|)$ messages on expectation for $T$ time steps is given in \[se:domain\]. This is asymptotically optimal in the worst-case in which $D_t \cap D_{t+1} = \emptyset$ holds for all $t \in T$. We also provide an algorithm and an analysis based on the minimum possible number $R^*$ of changes of representatives for a given input. It exploits situations where $D_t \cap D_{t+1} \neq \emptyset$ and uses ${\mathcal{O}}(\log n \cdot R^*)$ messages on expectation.
For an [($\varepsilon$,$\delta$)-approximation]{}of the Frequency Monitoring Problem for $T$ time steps, we first provide an algorithm using $\Theta(\sum_{t \in T}|D_t|\frac{1}{\varepsilon^2} \log \frac{|D_t|}{\delta})$ messages on expectation in \[sec:frequencies\]. We then improve this bound for instances in which observations between consecutive steps have a certain similarity. That is, for inputs fulfilling the property that for all $v\in\{1,\ldots,\Delta\}$ and some $\sigma \leq 1/2$, the number of nodes observing $v$ does not change by a factor larger than $\sigma$ between consecutive time steps, we provide an algorithm that uses an expected amount of ${\mathcal{O}}(|D_1| (\max(\delta, \sigma)T+1) \frac{1}{\varepsilon^2} \log \frac{|D_1|}{\delta})$ messages. In \[sec:countDistinct\], we provide an algorithm using $\Theta(T \cdot \frac{1}{\varepsilon^2} \log \frac{1}{\delta})$ messages on expectation for the Count Distinct Monitoring Problem for $T$ time steps. For instances which exhibit a certain similarity an algorithm is presented which monitors the problem using $\Theta \left( \left( 1+T\cdot \max\{2\sigma, \delta \} \right) \frac{\log (n) \cdot R^*}{|D_t| \cdot \varepsilon^2} \log \frac{1}{\delta} \right)$ messages on expectation.
Related Work {#DR:section:relatedWork}
------------
The basis of the model considered in this paper is the *continuous monitoring model* as introduced by Cormode, Muthukrishnan and Yi in [@cormodeModel]. In this model, there is a set of $n$ distributed nodes each observing a stream given by a multiset of items in each time step. The nodes can communicate with a central server, which in turn has the task to continuously, at any time $t$, compute a function $f$ defined over all data observed across all streams up to time $t$. The goal is to design protocols aiming at the minimisation of the number of bits communicated between the nodes and the server. In [@cormodeModel], the monitoring of several functions is studied in their (approximate) threshold variants, in which the server has to output $1$ if $f\geq \tau$ and $0$ if $f \leq (1-\varepsilon)\tau$, for given $\tau$ and $\varepsilon$. Precisely, algorithms for the frequency moments $F_p= \sum_i m_i^p$ where $m_i$ denotes the frequency of item $i$ for $p=0,1,2$ are given. $F_1$ represents the simple sum of all items received so far and $F_0$ the number of distinct items received so far. Since the introduction of the model, monitoring of several functions has been studied such as the monitoring of frequencies and ranks by Huang, Yi and Zhang in [@huang]. The frequency of an item $i$ is defined to be the number of occurrences of $i$ across all streams up to the current time. The rank of an item $i$ is the number of items smaller than $i$ observed in the streams. Frequency moments for any $p>2$ are considered by Woodruff and Zhang in [@woodruff]. A variant of the Count Distinct Monitoring Problem is considered by Gibbons and Tirthapura in [@gibbons]. The authors study a model in which each of two nodes receives a stream of items and at the end of the streams a server is asked to compute $F_0$ based on both streams. A main technical ingredient is the use of so called public coins, which, once initialized at the nodes, provide a way to let different nodes observe identical outcomes of random experiments without further communication. We will adopt this technique in \[sec:countDistinct\]. Note that the previously mentioned problems are all defined over the items *received so far*, which is in contrast to the definition of monitoring problems which we are going to consider and which are all defined only based on the *current time step*. This fact has the implication that in our problems the monitored functions are no longer monotone, which makes its monitoring more complicated.
Concerning monitoring problems in which the function tracked by the server only depends on the current time step, there is also some previous work to mention. In [@lam], Lam, Liu and Ting study a setting in which the server needs to know, at any time, the order type of the values currently observed. That is, the server needs to know which node observes the largest value, second largerst value and so on at time $t$. In [@yi], Yi and Zhang consider a system only consisting of one node connected to the server. The node continuously observes a $d$-dimensional vector of integers from $\{1, \ldots, \Delta\}$. The goal is to keep the server informed about this vector up to some additive error per component. In [@davis], Davis, Edmonds and Impagliazzo consider the following resource allocation problem: $n$ nodes observe streams of required shares of a given resource. The server has to assign, to each node, in each time step, a share of the resource that is as least as large as the required share. The objective is then given by the minimization of communication necessary for adapting the assignment of the resource over time.
The Domain Monitoring Problem {#se:domain}
=============================
We start by presenting an algorithm to solve the Domain Monitoring Problem for a single time step. We analyse the communication cost using standard worst-case analysis and show tight bounds. By applying the algorithm for each time step, we then obtain tight bounds for monitoring the domain for any $T$ time steps. The basic idea of the protocol as given in \[alg:p1\] is quite simple: Applied at a time $t$ with a value $v \in \{1,\ldots, \Delta\}$, the server gets informed whether $v \in D_t$ holds or not. To do so, each node $i$ with $v_i^t=v$ essentially draws a value from a geometric distribution and then those nodes having drawn the largest such value send broadcast messages. By this, one can show that on expectation only a constant number of messages is sent.
Furthermore, if applied with $v = nil$, the server can decide whether $v' \in D_t$ for all $v' \in \{1,\ldots,\Delta\}$ at once with $\Theta(|D_t|)$ messages on expectation. To this end, for *each* $v' \in \{1,\ldots,\Delta\}$ independently, the nodes $i$ with $v_i^t=v'$ drawing the largest value from the geometric distribution send broadcast messages. In the presentation of \[alg:p1\], we assume that $v^t_i = v$ is always true if $v = nil$. Also, in order to apply it to a subset of nodes, we assume that each node maintains a value $status_i \in \{0,1\}$ and only nodes $i$ take part in the protocol for which $status_i = status$ holds.
1. Each node $i$ for which $status_i = status$ and $\left(v \neq nil\Rightarrow v^t_i = v\right)$ hold, draws a value $\hat h_i$ from a geometric distribution with success probability $p \coloneqq 1/2$.
2. Let $h_i = \min\{\log n, \hat h_i \}$.
3. Node $i$ broadcasts its value in round $\log n -h_i$ unless a node $i'$ with $v^t_i = v^t_{i'}$ has broadcasted before.
We have the following lemma, which bounds the expected communication cost of \[alg:p1\] and has already appeared in a similar way in [@ipdps2] (Lemma III.1).
\[lemma:protone\_message\_complexity\] Applied for a fixed time $t$, [<span style="font-variant:small-caps;">ConstantResponse</span>]{}($v, 1$) uses $\Theta(1)$ messages on expectation if $v \neq nil$ and $\Theta(|D_t|)$ otherwise.
First consider the case where $v \neq nil$. Regarding the expected communication of [<span style="font-variant:small-caps;">ConstantResponse</span>]{}($v, 1$) we introduce some notation. Let $ X_{i} $ be a $ {\left\{ 0,1 \right\}} $-random variable indicating whether the node $i \in {N_t^v}$ sends a message to the server, and $X\coloneqq \sum X_i$. According to the algorithm a sensor $ i $ sends a message if and only if its height $ h_{i} $ matches the round specified for that height and no other sensor $ i' $ has sent its value beforehand. We obtain $$\begin{aligned}
{\ensuremath{\Pr \left[ X_{i} = 1 \right]}}
&= {\ensuremath{\Pr \left[ \exists r \in {\left\{ 1,\dots,\log n \right\}}: h_{i} = r \land \forall i' \in {N_t^v}\setminus{\left\{ i \right\}} : h_{i'} \leq r \right]}}\\
&\leq \sum_{r=1}^{\log n} \frac{1}{2^r}\left(1-\frac{1}{2^r}\right)^{n^v-1} .
\end{aligned}$$ We know that $ \textnormal{E}[X_{i}] = {\ensuremath{\Pr \left[ X_{i}=1 \right]}} $ and thus $$\begin{aligned}
\textnormal{E}[X]
\leq n^v \cdot \sum_{r=1}^{\log n} \frac{1}{2^r}\left(1-\frac{1}{2^r}\right)^{n^v-1}.
\end{aligned}$$ Observing that $f(r) = n^v \cdot \frac{1}{2^{r}}{\left( 1-\frac{1}{2^{r}} \right)}^{n^v-1}$ has only one extreme point and $f(r)\leq 2$ for all $r\in[0,\log(n)]$, we use the integral test for convergence to obtain $$\begin{aligned}
\textnormal{E}[X]
&\leq n^v \cdot\sum_{r=1}^{\log n}\frac{1}{2^{r}}{\left( 1-\frac{1}{2^{r}} \right)}^{n^v-1}
\leq n^v \int_{0}^{\log n}\frac{1}{2^{r}}{\left( 1-\frac{1}{2^{r}} \right)}^{n^v-1}\textnormal{dr} + 2\\
&\leq \left[\frac{1}{\ln{\left( 2 \right)}}{\left( 1-\frac{1}{2^{r}} \right)}^{n^v}\right]_{0}^{\log n}+2
\leq \frac{1}{\ln{\left( 2 \right)}}+2
<4.
\end{aligned}$$
For the case $v= nil$ we can apply the same argumentation independently for each value $v \in D_t$. This concludes the proof of the lemma.
In order to solve the domain monitoring problem for $T$ time steps, the server proceeds as follows: In each step $t$ the server calls [<span style="font-variant:small-caps;">ConstantResponse</span>]{}($nil, 1)$ to identify all values belonging to $D_t$ as well as a valid sequence $R_t$. By the previous lemma we then have an overall communication cost of $\Theta(|D_t|)$ for each time step $t$. For monitoring $T$ time steps, the cost is $\Theta(\sum_{t \in T} |D_t|)$. This is asymptotically optimal in the worst-case since on instances where $D_t \cap D_{t+1} = \emptyset$ for all $t$, any algorithm has cost $\Omega(\sum_{t \in T} |D_t|)$.
Using [<span style="font-variant:small-caps;">ConstantResponse</span>]{}($v, 1$), the Domain Monitoring Problem for $T$ time steps can be solved using $\Theta(\sum_{t \in T}|D_t|)$ messages on expectation.
A Parameterised Analysis {#a-parameterised-analysis .unnumbered}
------------------------
Despite the optimality of the result, the strategy of computing a new solution from scratch in each time step seems unwise and the analysis does not seem to capture the essence of the problem properly. It often might be the case that there are some similarities between values observed in consecutive time steps and particularly, that $D_t \cap D_{t+1} \neq \emptyset$. In this case, there might be the chance to keep a representative for several consecutive time steps, which should be exploited. Due to these observations we next define a parameter describing this behavior and provide a parameterised analysis. To this end, we consider the number of component-wise differences in the sequences of nodes $R_{t-1}$ and $R_t$ and call this difference the *number of changes of representatives* in time step $t$. Let $R^*$ denote the minimum possible number of changes of representatives (over all considered time steps $T$). The formal description of our algorithm is given in \[alg:domainMonitoring\]. Roughly speaking, the algorithm defines, for each value $v$, phases, where a phase is defined as a maximal time interval during which there exists one node observing value $v$ throughout the entire interval. Whenever a node being a representative for $v$ changes its observation, it informs the server so that a new representative can be chosen (from those observing $v$ throughout the entire phase, which is indicated by $status_i=1$). If no new representative is found this way, the server tries to find a new representative among those observing $v$ and for which $status_i=0$ and ends the current phase. Additionally, if a node observes a value $v$ at time $t$ for which $v \notin D_t$, a new representative is determined among these nodes. Note that this requires each node to store $D_t$ at any time $t$ and hence a storage of ${\mathcal{O}}(\Delta)$.
**(Node $i$)**
1. Define $status_i \coloneqq 1$.
2. If at some time $t$, $v^t_i \neq v^{t-1}_{i}$, then
1. If $v^t_i \notin D_{t-1}$, set $status_i = 0$ and apply [<span style="font-variant:small-caps;">ConstantResponse</span>]{}($v^t_i, 0$).
2. If $v^t_i \in D_{t-1}$, set $status_i = 0$. Additionally inform server in case $i \in R_{t-1}$.
3. If server starts a new phase for $v = v^t_i$, set $status_i = 1$.
**(Server)**
*\[Initialisation\]*
Call [<span style="font-variant:small-caps;">ConstantResponse</span>]{}($nil, 1)$ to define $D_0$ and for each $v \in D_0$ choose a representative uniformly at random from all nodes which have sent $v$.
*\[Maintaining $D_t$ and $R_t$ at time $t$\]*
Start with $D_t = D_{t-1}$ and $R_t = R_{t-1}$ and apply the following rules:
1. *\[Current Phase, (try to) find new representative\]*
2. If informed by representative of a value $v \in D_{t-1}$,
1. Call [<span style="font-variant:small-caps;">ConstantResponse</span>]{}($v, 1)$.
2. If node(s) respond(s), choose new representative among the responding sensors uniformly at random. \[step:changeRep\]
3. Else call [<span style="font-variant:small-caps;">ConstantResponse</span>]{}($v, 0)$. End current phase for $v$ and, if there is no response, delete $v$ from $D_t$ and the respective representative from $R_t$.
3. *\[If [<span style="font-variant:small-caps;">ConstantResponse</span>]{}($v,0$) leads to received message(s), start new phase\]*
4. Start a new phase for value $v$ if message from an application of [<span style="font-variant:small-caps;">ConstantResponse</span>]{}($v, 0)$ (by \[step:removed\] initialised by the server or initialised in Step 2.1. by a node) is received. Add or replace respective representative in $R_t$ by choosing a node uniformly at random from those responding to [<span style="font-variant:small-caps;">ConstantResponse</span>]{}($v, 0)$. \[step:restart\]
<span style="font-variant:small-caps;">DomainMonitoring</span> as described in \[alg:domainMonitoring\] solves the Domain Monitoring Problem using ${\mathcal{O}}(\log n \cdot R^*)$ messages on expectation, where $R^*$ denotes the minimum possible number of changes of representatives.
We consider each value $v \in \bigcup_t D_t$ separately. Let $N_{t_1, t_2} \coloneqq \{i \mid v_i^t = v \, \forall t_1 \leq t \leq t_2\}$ denote the set of nodes that observe the value $v$ at each point in time $t$ with $t_1 \leq t\leq t_2$. Consider a fixed phase for $v$ and let $t_1$ and $t_2$ be the points in time where the phase starts and ends, respectively. A phase only ends in \[step:removed\], hence there was no response from [<span style="font-variant:small-caps;">ConstantResponse</span>]{}$(v,1)$, which implies $N^v_{t_1,t_2} = \emptyset$. Thus, to each phase for $v$ we can associate a cost of at least one to $R^*$ and this holds for each $v \in \bigcup_t D_t$. Therefore, $R^*$ is at least the overall number of phases of all values.
Next we analyze the expected cost of \[alg:domainMonitoring\] during the considered phase for $v$. Let w.l.o.g. $N_{t_1} \coloneqq N_{t_1,t_1}= \{1, 2, \ldots, k\}$. With respect to the fixed phase, only nodes in $N_{t_1}$ can communicate and the communication is bounded by the number of changes of the representative for $v$ during the phase. Let $t'_i$ be the first time after $t_1$ at which node $i$ does not observe $v$. Let the nodes be sorted such that $i<j$ implies $t'_i \geq t'_j$. Let $a_1, \ldots, a_m$ be the nodes \[alg:domainMonitoring\] chooses as representatives in the considered phase. We want to show that $\mathbb{E}[m] = {\mathcal{O}}( \log k)$. To this end, partition the set of time steps $t'_i$ into groups $G_i$. Intuitively, $G_i$ represents the time steps in which the nodes continuously observe value $v$ since time $t_1$ and the size of the initial set of nodes that observed $v$ is halved $i$ times. Formally, $G_i$ contains all time steps $t_{\ell_{i-1}+1},\ldots,t_{\ell_i}$ (where $\ell_{-1}\coloneqq 0$ for convenience) such that $\ell_i$ is the largest integer fulfilling $|N_{t_1, t'_{\ell_i}}| \in (k/2^{i+1}, k/2^i]$.
Let $S_i$ be the number of changes of representatives in time steps belonging to $G_i$. We have $\mathbb{E}[m] = \sum_{i=0}^{\log k} \mathbb{E}[S_i]$. Consider a fixed $S_i$. Let $\mathcal{E}_j$ be the event that the $j$-th representative chosen in time steps belonging to $G_i$ is the first one with an index in $\left\{1, \ldots, \lfloor\frac{k}{2^{i+1}}\rfloor\right\}$. Observe that as soon as this happens, the respective representative will be the last one chosen in a time step belonging to group $G_i$.
Now, since the algorithm chooses a new representative uniformly at random from the index set $\left\{1, \ldots, \lfloor\frac{k}{2^{i}}\rfloor\right\}$, the probability that it chooses a representative from $\left\{1, \ldots, \lfloor\frac{k}{2^{i+1}}\rfloor\right\}$ is at least $1/2$ except for the first representative of $v$, where it might be slightly smaller due to rounding errors. $\mathcal{E}_j$ occurs only if the first $j-1$ representatives were each *not* chosen from this set, i.e. ${\ensuremath{\Pr \left[ \mathcal{E}_j \right]}}\leq \left(\frac{1}{2}\right)^{j-2}$. Hence, ${\mathbb{E}}[S_i] = \sum_{j} {\mathbb{E}}[S_i | \mathcal{E}_j] \cdot \Pr[\mathcal{E}_j]
\leq \sum_j j \cdot (\frac{1}{2} )^{j-2}
= \sum_j \frac{j}{2^{j-2}} = {\mathcal{O}}(1)$.
The Frequency Monitoring Problem {#sec:frequencies}
================================
In this section we design and analyse an algorithm for the Frequency Monitoring Problem, i.e. to output (an approximation) of the number of nodes currently observing value $v$. We start by considering a single time step and present an algorithm which solves the subproblem to output the number of nodes that observe $v$ within a constant multiplicative error bound. Afterwards, and based on this subproblem, a simple sampling algorithm is presented which solves the Frequency Monitoring Problem for a single time step up to a given (multiplicative) error bound and with demanded error probability.
While in the previous section we used the algorithm [<span style="font-variant:small-caps;">ConstantResponse</span>]{} with the goal to obtain a representative for a measured value, in this section we will use the same algorithm to estimate the number of nodes that measure a certain value $v$. Observe that the expected maximal height of the geometric experiment increases with a growing number of nodes observing $v$. We exploit this fact and use it to estimate the number of nodes with value $v$, while still expecting constant communication cost only. For a given a time step $t$ and a value $v \in D_t$, we define an algorithm [<span style="font-variant:small-caps;">ConstantFactorApproximation</span>]{} as follows: We apply [<span style="font-variant:small-caps;">ConstantResponse</span>]{}($v,1$) with $status_i=1$ for all nodes $i$. If the server receives the first response in communication round $r \leq \log n$, the algorithm outputs $\tilde n_{\text{const}}^v=2^r$ as the estimation for ${|{N_t^v}|}$.
We show that we compute a constant factor approximation with constant probability. Then we amplify this probability using multiple executions of the algorithm and taking the median (of the executions) as a final result.
\[lemma:constant\_factor\_approximation\] The algorithm [[<span style="font-variant:small-caps;">ConstantFactorApproximation</span>]{}]{}estimates the number ${|{N_t^v}|}$ of nodes observing the value $v$ at time $t$ up to a factor of $8$, i.e. $\tilde n^{v}_{const} \in [{|{N_t^v}|}/8, {|{N_t^v}|}\cdot 8]$ with constant probability.
Let $n^v$ be the number of nodes currently observing value $v$, i.e. $n^v \coloneqq {|{N_t^v}|}$. Recall that the probability for a single node to draw height $h$ is $\Pr[h_i = h]=\frac{1}{2^h}$, if $h < \log n$, and $\Pr[h_i = h] = \frac{2}{2^{h}}$, if $h = \log n$. Hence, $\Pr[h_i \geq h]=\frac{1}{2^{h-1}}$ for all $h\in\{1,\ldots,\log n\}$.
We estimate the probability of the algorithm to fail, by analysing the cases that $\tilde{n}^v_{const}$ is larger than $\log n^v+3$ or smaller than $\log n^v-3$. We start with the first case and by applying a union bound we obtain: $$\begin{aligned}
\Pr[\exists i : h_i > \log n^v + 3]
&\leq \Pr[\exists i : h_i \geq \lceil\log n^v\rceil + 3] \\
&= n^v \cdot \left(\frac{1}{2} \right)^{\lceil\log n^v\rceil + 2}
\leq\frac{1}{4}.\end{aligned}$$
For the latter case we bound the probability that each node has drawn a height strictly smaller than $\log n^v - 3$ by $$\begin{aligned}
\Pr[\forall i :h_i < \log n^v - 3]
& \leq \prod_{i} \Pr[h_i < \lceil\log n^v\rceil - 3]\\
&= \left(1- \frac{1}{2^{\lceil\log n^v\rceil - 4}} \right)^{n^v}
\leq \left(1- \frac{8}{n^v} \right)^{n^v}
\leq \frac{1}{e^8}.\end{aligned}$$
Thus, the probability that we compute an 8-approximation is bounded by $$\begin{aligned}
\Pr \left[\frac{n^v}{8} \leq 2^{h_i} \leq 8 n^v \right]
&=1- \bigl( \Pr[\exists i: h_i >\log n^v + 3] + \Pr[ \forall i:h_i < \log n^v - 3] \bigr)\\
&\geq 1- \left(\frac{1}{4} + \frac{1}{e^8} \right)
> 0.7\end{aligned}$$
We apply an amplification technique to boost the success probability to arbitrary $1-\delta'$ using $\Theta(\log \frac{1}{\delta'})$ parallel executions of the [[<span style="font-variant:small-caps;">ConstantFactorApproximation</span>]{}]{}algorithm and choose the median of the intermediate results as the final output.
\[corollary:constant\_factor\_approximation\_delta\] Applying $\Theta\left( \log \frac{1}{\delta'} \right)$ independent, parallel instances of [[<span style="font-variant:small-caps;">ConstantFactorApproximation</span>]{}]{}, we obtain a constant factor approximation of ${|{N_t^v}|}$ with success probability at least $1-\delta'$ using $\Theta \left( \log \frac{1}{\delta'} \right)$ messages on expectation.
Choose $d=\frac{45}{2}\ln \frac{1}{\delta'}$ to be the number of copies of the algorithm and return the median of the intermediate results. Let $\mathcal{I}_j$ be the indicator variable for the event that the $j$-th experiment does not result in an 8-approximation. By \[lemma:constant\_factor\_approximation\] the failure probability can be upper bounded by a constant, i.e. ${\ensuremath{\Pr \left[ \mathcal{I}_j \right]}}\leq 0.3$. Hence, using a Chernoff bound, the probability that at least half of the experiments do meet the required approximation factor of $8$ is $$\begin{aligned}
{\ensuremath{\Pr \left[ \sum_{j=1}^d \mathcal{I}_j \geq \frac{1}{2}d \right]}}
&\leq{\ensuremath{\Pr \left[ \sum_{j=1}^d \mathcal{I}_j \geq \left(1+\frac{2}{3} \right)\cdot 0.3\cdot d \right]}}\\
&\leq e^{-\left(\frac{2}{3}\right)^2\cdot\frac{1}{3}\cdot 0.3\cdot d}
=e^{-\frac{2}{45}\cdot d}
=e^{-\frac{2}{45}\cdot \frac{45}{2}\ln \frac{1}{\delta'} }
=\delta'.
\end{aligned}$$ Observe that if at least half of the intermediate results are within the demanded error bound, so is the median. Thus, the algorithm produces an $8$-approximation of ${|{N_t^v}|}$ with success-probability of at least $1-\delta'$, concluding the proof.
To obtain an $(\varepsilon, \delta)$-approximation, in \[alg:epsilon\_factor\_approximation\] we first apply the [[<span style="font-variant:small-caps;">ConstantFactorApproximation</span>]{}]{}algorithm to obtain a rough estimate of ${|{N_t^v}|}$. It is used to compute a probability $p$, which is broadcasted to the nodes, so that every node observing value $v$ sends a message with probability $p$. Since the [[<span style="font-variant:small-caps;">ConstantFactorApproximation</span>]{}]{}result $\tilde n_{\text{const}}^v$ in the denominator of $p$ is close to ${|{N_t^v}|}$, the number of messages sent on expectation is independent of ${|{N_t^v}|}$. The estimated number of nodes observing $v$ is then given by the number of responding nodes $\bar n^v$ divided by $p$, which, on expectation, results in ${|{N_t^v}|}$.
**(Node $i$)**
1. Receive $p$ from the server.
2. Send a response message with probability $p$.
**(Server)**
1. Set $\delta' \coloneqq \frac{\delta}{3}$
2. Call [<span style="font-variant:small-caps;">ConstantFactorApproximation</span>]{}($v$, $\delta'$) to obtain $\tilde n^{v}_{\text{const}}$.
3. Broadcast $p=\min \left( 1, \frac{24}{\varepsilon^2 \tilde n^{v}_{\text{const}}}\cdot \ln \frac{1}{\delta'} \right)$.
4. Receive ${\bar n^v}$ messages.
5. Compute and output estimated number of nodes in ${N_t^v}$ as ${\tilde n^v}= {\bar n^v}/ p$.
\[lemma:single\_shot\_epsilon\] The algorithm [[<span style="font-variant:small-caps;">EpsilonFactorApprox</span>]{}]{}as given in \[alg:epsilon\_factor\_approximation\] provides an [($\varepsilon$,$\delta$)-approximation]{}of ${|{N_t^v}|}$.
The algorithm obtains a constant factor approximation $\tilde n^{v}_{\text{const}}$ with probability $1-\delta'$. The expected number of messages is ${\mathbb{E}}\left[{\bar n^v}\right]=n^v \cdot p$.
We start by estimating the conditional probability that more than $(1+\varepsilon)\, n^v p$ responses are sent under the condition that $\tilde n^{v}_{\text{const}}\leq8n^v$ and $p<1$. In this case we have $$p = \frac{24}{\varepsilon^2 \tilde n^{v}_{\text{const}}}\cdot \ln \frac{1}{\delta'} \geq \frac{3}{\varepsilon^2 n^v}\cdot \ln \frac{1}{\delta'},$$ hence using a Chernoff bound it follows $$\begin{aligned}
p_1\coloneqq {\ensuremath{\Pr \left[ {\bar n^v}\geq (1+\varepsilon) n^v p \left| \tilde n^{v}_{\text{const}}\leq8n^v\wedge p<1\right. \right]}}
\leq e^{-\frac{\varepsilon^2}{3} n^v \cdot \frac{3}{\varepsilon^2 n^v}\cdot \ln \frac{1}{\delta'}}
= \delta'.\end{aligned}$$ Likewise the probability that less than $(1-\varepsilon)\, n^v p$ messages are sent under the condition that $\tilde n^{v}_{\text{const}}\leq8n^v$ and $p<1$ is $$\begin{aligned}
p_2 &\coloneqq {\ensuremath{\Pr \left[ {\bar n^v}\leq (1-\varepsilon)n^v p\left| \tilde n^{v}_{\text{const}}\leq8n^v\wedge p<1\right. \right]}}\\
&\leq e^{-\frac{\varepsilon^2}{2}n^v\cdot \frac{3}{\varepsilon^2 n^{v} }\cdot \ln \frac{1}{\delta'}}
\leq e^{-\frac{3}{2} \ln \frac{1}{\delta'}}
< \delta'
.\end{aligned}$$ Next consider the case that $\tilde n^{v}_{\text{const}} > 8n^v$ and $p <1$ holds. Using $${\ensuremath{\Pr \left[ \tilde n^{v}_{\text{const}} > 8n^v \right]}} \leq {\ensuremath{\Pr \left[ \tilde n^{v}_{\text{const}} > 8n^v \vee \tilde n^{v}_{\text{const}} < \frac{n^v}{8} \right]}} \leq \delta'$$ and $p_i\cdot {\ensuremath{\Pr \left[ \tilde n^{v}_{\text{const}} \leq 8n^v \right]}}\leq p_i$ for $i\in\{1,2\}$, $$\begin{aligned}
{\ensuremath{\Pr \left[ (1-\varepsilon)n^v p < {\bar n^v}< (1+\varepsilon) n^v p\left| p < 1\right. \right]}} \hspace{3cm} \\
\geq 1 - \left({\ensuremath{\Pr \left[ \tilde n^{v}_{\text{const}} > 8n^v \right]}} + (p_1+p_2)\right)
\geq 1-3\delta' = 1-\delta.\end{aligned}$$ For the last case $p=1$, we have $
{\ensuremath{\Pr \left[ (1-\varepsilon)n^v p < {\bar n^v}< (1+\varepsilon)n^v p\left| p \geq 1\right. \right]}} = 1,
$ by using ${\bar n^v}= n^v$. Now, ${\ensuremath{\Pr \left[ (1-\varepsilon)n^v p < {\bar n^v}< (1+\varepsilon)n^v p \right]}} \geq 1-\delta$ directly follows.
\[lemma:msg\_epsilonApprox\] Algorithm [[<span style="font-variant:small-caps;">EpsilonFactorApprox</span>]{}]{}as given in \[alg:epsilon\_factor\_approximation\] uses $\Theta(\frac{1}{\varepsilon^2}\log \frac{1}{\delta})$ messages on expectation.
Recall that each of the $n^v$ nodes sends a message with probability $p$, leading to $n^v \cdot p$ messages on expectation. First assume that the constant factor approximation was successful, i.e. $\frac{n_1}{8} \leq \tilde n^v_\text{const} \leq 8 n_1$. If $p<1$, we have $$n^v \cdot p = n^v \frac{24}{\varepsilon^2 \tilde n^{v}_{\text{const}}}\cdot \ln \frac{1}{\delta'} \leq \frac{24\cdot 8}{\varepsilon^2}\cdot \ln \frac{1}{\delta'}=\Theta\left( \frac{1}{\varepsilon^2}\log \frac{1}{\delta} \right).$$ If $p=1$, by definition $\frac{24}{\varepsilon^2 \tilde n^{v}_{\text{const}}}\cdot \ln \frac{1}{\delta'} \geq 1$, hence $\tilde n^{v}_{\text{const}} = {\mathcal{O}}\left( \frac{1}{\varepsilon^2}\cdot \log \frac{1}{\delta'} \right) $. Thus, $n^v p \leq 8 \tilde n^{v}_{\text{const}} p = {\mathcal{O}}\left( \frac{1}{\varepsilon^2}\cdot \log \frac{1}{\delta'} \right)$.
For the case that the constant factor approximation was not successful, note that ${\ensuremath{\Pr \left[ \tilde n^{v}_{\text{const}} < \frac{1}{8 \cdot 2^i} n^{v} \right]}} \leq \frac{1}{e^{2^{i+3}}}$ holds analogously to the calculation in \[lemma:constant\_factor\_approximation\]. Also, for $\tilde n^{v}_{\text{const}} \geq \frac{1}{8 \cdot 2^i} n^{v}$ and $p<1$, we have $$n^v p \leq 8 \cdot 2^i\cdot \tilde n^{v}_{\text{const}}\cdot \frac{24}{\varepsilon^2 \tilde n^{v}_{\text{const}}}\cdot \ln \frac{1}{\delta}= 2^i\cdot \Theta\left( \frac{1}{\varepsilon^2}\log \frac{1}{\delta} \right).$$ Similarly, for $p=1$, we have $n^v p \leq 8 \cdot 2^i\cdot \tilde n^{v}_{\text{const}} = 2^i\cdot \Theta\left( \frac{1}{\varepsilon^2}\log \frac{1}{\delta} \right)$ as in this case, $\tilde n^{v}_{\text{const}} = {\mathcal{O}}\left( \frac{1}{\varepsilon^2}\cdot \log \frac{1}{\delta'} \right)$. Hence, we can conclude $$\begin{aligned}
\mathbb{E}\left[{\bar n^v}\right] &\leq \Theta\left( \frac{1}{\varepsilon^2}\log \frac{1}{\delta} \right) \cdot {\ensuremath{\Pr \left[ \tilde n^v_{\text{const}}\geq \frac{1}{8}n^v \right]}} \\&\qquad + \sum_{i=0}^\infty {\ensuremath{\Pr \left[ \frac{1}{8 \cdot 2^{i+1}} n^{v} \leq \tilde n^v_{\text{const}} < \frac{1}{8 \cdot 2^i} n^{v} \right]}}\cdot 2^{i+1}\cdot \Theta\left( \frac{1}{\varepsilon^2}\log \frac{1}{\delta} \right)
\end{aligned}$$ $$\begin{aligned}
&\leq \Theta\left( \frac{1}{\varepsilon^2}\log \frac{1}{\delta} \right) \left(1 + \sum_{i=0}^\infty \frac{2^{i+1}}{e^{2^{i+3}}}\right)
\leq \Theta\left( \frac{1}{\varepsilon^2}\log \frac{1}{\delta} \right) \left(1 + \sum_{i=0}^\infty 2^{i+1-2^{i+3}}\right)\\
&\leq \Theta\left( \frac{1}{\varepsilon^2}\log \frac{1}{\delta} \right) \left(1 + \sum_{i=0}^\infty 2^{-i}\right) = \Theta\left( \frac{1}{\varepsilon^2}\log \frac{1}{\delta} \right).
\end{aligned}$$
\[theorem:epsilon\_factor\_approximation\] There exists an algorithm that provides an ($\varepsilon$,$\delta$)-approximation for the Frequency Monitoring Problem for $T$ time steps with an expected number of $\Theta\left(\sum_{t \in T}|D_t|\frac{1}{\varepsilon^2} \log \frac{|D_t|}{\delta}\right)$ messages.
In every time step $t$ we first identify $D_t$ by applying [<span style="font-variant:small-caps;">ConstantResponse</span>]{} using $\Theta \left( |D_t| \right)$ messages on expectation. On every value $v \in D_t$ we then perform algorithm [<span style="font-variant:small-caps;">EpsilonFactorApprox</span>]{}($v$,$\varepsilon$,$\frac{\delta}{|D_t|}$), resulting in an amount of $\Theta \left( |D_t| \frac{1}{\varepsilon^2} \log \frac{|D_t|}{\delta} \right)$ messages on expectation for a single time step, while achieving a probability (using a union bound) of $1-\frac{|D_0|\delta}{|D_0|}=1-\delta$ that in one time step the estimations for every $v$ are $\varepsilon$-approximations. Applied for each of the $T$ time steps, we obtain a bound as claimed.
A Parameterised Analysis {#a-parameterised-analysis-1 .unnumbered}
------------------------
Applying [[<span style="font-variant:small-caps;">EpsilonFactorApprox</span>]{}]{}in every time step is a good solution in worst case scenarios. But if we assume that the change in the set of nodes observing a value is small in comparison to the size of the set, we can do better.
We extend the [[<span style="font-variant:small-caps;">EpsilonFactorApprox</span>]{}]{}such that in settings where from one time step to another only a small fraction $\sigma$ of nodes change the value they measure, the amount of communication can be reduced, while the quality guarantees remain intact. We define $\sigma$ such that $$ \forall t: \sigma \geq \frac{| N_{t-1}^v \setminus {N_t^v}| + | {N_t^v}\setminus N_{t-1}^v|}{{|{N_t^v}|}}.$$ Note that this also implies that $D_t=D_{t-1}$ holds for all time steps $t$, i.e. the set of measured values stays the same over time.
The extension is designed so that compared to [<span style="font-variant:small-caps;">EpsilonFactorApprox</span>]{}, also in settings with many changes the solution quality and message complexity asymptotically does not increase. The idea is the following: For a fixed value $v$, in a first time step [[<span style="font-variant:small-caps;">EpsilonFactorApprox</span>]{}]{}is executed (defining a probability $p$ in \[setp\] of \[alg:epsilon\_factor\_approximation\]). In every following time step, up to $1/\delta$ consecutive time steps, nodes that start or stop measuring a value $v$ send a message to the server with the same probability $p$, while nodes that do not observe a change in their value remain silent. In every time step $t$, the server uses the accumulated messages from the first time step and all messages from nodes that started measuring $v$ in time steps $2 \dots t$, while subtracting all messages from nodes that stopped measuring $v$ in the time steps $2\dots t$. This accumulated message count is then used similarly as in [[<span style="font-variant:small-caps;">EpsilonFactorApprox</span>]{}]{}to estimate the total number of nodes observing $v$ in the current time step. The algorithm starts again if a) $1/\delta$ time steps are over, so that the probability of a good estimation remains good enough, or b) the sum of estimated nodes to start/stop measuring value $v$ is too large. The latter is done to ensure that the message probability $p$ remains fitting to the number of nodes, ensuring a small amount of communication, while guaranteeing an $(\varepsilon, \delta)$-approximation.
Let $n_t^+, n_t^-$ be the number of nodes that start measuring $v$ in time step $t$ or that stop measuring it, respectively, i.e.$n_t^+ = | {N_t^v}\setminus N_{t-1}^v|, n_t^- = | N_{t-1}^v \setminus {N_t^v}|$, and $\bar n_t^+$ and $\bar n_t^-$ the number of them that sent a message to the server in time step $t$. In the following we call nodes contributing to $n^+_t$ and $n^-_t$ *entering* and *leaving*, respectively.
**(Node $i$)**
1. If $t=1$, take part in [<span style="font-variant:small-caps;">EpsilonFactorApprox</span>]{} called in \[alg:ContinuousEpsilonApproximation:start\] by the server.
2. If $t>1$, broadcast a message with probability $p$ if $v_i^{t-1}=v \wedge v_i^t \neq v$\
or $v_i^{t-1} \neq v \wedge v_i^t = v$.
**(Server)**
1. Set $\delta' \coloneqq \delta^2$.
2. Set $t \coloneqq 1$ and run [<span style="font-variant:small-caps;">EpsilonFactorApprox</span>]{}($v$, $\varepsilon/3$, $\delta$) to obtain $\bar n_1, p$.
3. Output $\tilde n_1=\frac{\bar n_1}{p}$.
4. Repeat at the beginning of every new time step $t>1$:
1. Receive messages from nodes changing the observed value to obtain $\bar n_t^+$ and $\bar n_t^-$.
2. Break if $t \geq 1/\delta$ or $\left(\sum_{i=1}^t \bar n_i^+ + \sum_{i=1}^t \bar n_i^-\right)/p \geq \bar n_1/2$.
3. Output $\tilde n_t=\left(\bar n_1+\sum_{i=1}^t \bar n_i^+ - \sum_{i=1}^t \bar n_i^-\right)/p$.
5. Go to \[alg:ContinuousEpsilonApproximation:start\].
\[lemma:frequencies\_multiple\_step\_correctness\] For any $v \in D_1$, the algorithm [[<span style="font-variant:small-caps;">ContinuousEpsilonApprox</span>]{}]{}provides an [($\varepsilon$,$\delta$)-approximation]{}of ${|{N_t^v}|}$.
By the same arguments as in \[lemma:single\_shot\_epsilon\], we obtain an ($\varepsilon$,$\delta'$)-approximation of $n_1$. In any further time step we compute our estimate over the sum of all received messages ($\bar n_1$, arrivals and departures). If too many nodes change their measured value, we redo a complete estimation of the nodes in ${N_t^v}$.
Recall that $\tilde n_t$ is the random variable giving the estimated number of nodes by the algorithm, and $\tilde n_t^+=\frac{\bar n^+}{p}, \tilde n_t^-=\frac{\bar n^-}{p}$ are the random variables giving the estimated arrivals and departures in that time step. We look at any time step $t>1$ where the restart criteria are not met: Since $\tilde n_t=\tilde n_1 + \sum_{i=2}^t \left( \tilde n_i^+ - \tilde n_i^- \right)$ and the linearity of expectation, for any time $t \geq 1$ we can use a Chernoff bound as in \[lemma:single\_shot\_epsilon\] to show that the estimation is an $(\varepsilon,\delta')$-approximation.
Using a union bound on the fail probability of up to $1/\delta$ time steps, we get a $1 - \frac{1}{\delta} \cdot \delta'=1-\delta$ probability of having a correct estimation in any time step.
\[lemma:frequencies\_multiple\_step\_complexity\] For a fixed value $v$ and $T'=\min\{\frac{1}{2\sigma},\frac{1}{\delta}\}$, $\sigma \leq \frac{1}{2}$, time steps, [[<span style="font-variant:small-caps;">ContinuousEpsilonApprox</span>]{}]{}uses $\Theta \left(\frac{1}{\varepsilon^2} \log \frac{1}{\delta} \right)$ messages on expectation.
The message complexity depends on the initial size $|N_1^v|$ and on the number of nodes leaving and entering $N^v$ in those time steps, which is bounded by $\sigma$. If [<span style="font-variant:small-caps;">EpsilonFactorApprox</span>]{} obtained a correct probability $p$ in \[alg:ContEpsApprox:1\], i.e. $p = \Theta(\frac{1}{n_1})$, the expected number of messages (in case $p<1$) is
$$\begin{aligned}
\mathbb{E} \left[\sum_{t=1}^{T'} \bar n_t \, \middle|\, p = \Theta \left(\frac{1}{n_1} \right)\right]
&=\mathbb{E} \left[\bar n_1 + \sum_{i=2}^{T'} \bar n_i^+ + \bar n_i^- \, \middle| \, p = \Theta \left(\frac{1}{n_1} \right)\right]
\\
&=\left(n_1 + \sum_{i=2}^{T'} n_i^+ + n_i^- \right) p
\leq\left(n_1 + T' \sigma n_1\right)p
\\
&= n_1 \left(1 + T' \sigma \right) \cdot 24 \cdot \frac{1}{\varepsilon^2 \tilde n^v_{\text{const}}} \ln \frac{1}{\delta'}
\\
&= \Theta \left( \left(1+ \min\left\{\frac{1}{2\sigma},\frac{1}{\delta}\right\} \sigma \right) \cdot 1/\varepsilon^2 \log \frac{1}{\delta} \right).
\end{aligned}$$
Considering the case where [[<span style="font-variant:small-caps;">EpsilonFactorApprox</span>]{}]{}estimated wrong, the message complexity could increase greatly if the probability $p$ is too large for the actual number of nodes (i.e. an underestimation leads to high message complexity). But the probability to misestimate by some constant factor (which would increase the message complexity by that factor) decreases exponentially in this factor (as shown in \[lemma:msg\_epsilonApprox\] for [<span style="font-variant:small-caps;">EpsilonFactorApprox</span>]{}), leaving the expected number of messages to be $\Theta \left( \left(1+ \min\left\{\frac{1}{2\sigma},\frac{1}{\delta}\right\} \sigma \right) \cdot \frac{1}{\varepsilon^2}\cdot \log \frac{1}{\delta} \right)
=\Theta \left( \frac{1}{\varepsilon^2} \log \frac{1}{\delta} \right)$.
\[th:fmptheorem\] There exists an ($\varepsilon$,$\delta$)-approximation algorithm for the Frequency Monitoring Problem for $T$ consecutive time steps which uses an amount of $\Theta \left( |D_1| \left( 1+T\cdot \max\{2\sigma, \delta \} \right) \frac{1}{\varepsilon^2} \log \frac{|D_1|}{\delta} \right)$ messages on expectation, if $\sigma \leq 1/2$.
The algorithm works by first applying [<span style="font-variant:small-caps;">ConstantResponse</span>]{}($nil$,$1$) to obtain $D_1$ and then applying [<span style="font-variant:small-caps;">ContinuousEpsilonApprox</span>]{}($v$, $\varepsilon$, $\delta/|D_1|$) for every $v \in D_1$. By \[lemma:frequencies\_multiple\_step\_correctness\] we know that in every time step and for all $v\in D_1$, the frequency of $v$ is approximated up to a factor of $\varepsilon$ with probability $1-\delta/|D_1|$. We divide the $T$ time steps into intervals of size $T'=\min\{\frac{1}{2\sigma},\frac{1}{\delta}\}$ and perform [[<span style="font-variant:small-caps;">ContinuousEpsilonApprox</span>]{}]{}on each of them for every value $v \in D_1$. There are $\lceil \frac{T}{T'} \rceil \leq 1+T\cdot \max\{2\sigma,\delta\}$ such intervals. For each of those, by \[lemma:frequencies\_multiple\_step\_complexity\] we need $\Theta \left( \left(1+ \min\{\frac{1}{2\sigma},\frac{1}{\delta}\} \sigma \right) \cdot 1/\varepsilon^2 \log \frac{|D_1|}{\delta} \right)$ messages on expectation for each $v \in D_1$. This yields a complexity of $\Theta \left( |D_1| \left( 1+T\cdot \max\{2\sigma, \delta \} \right) \frac{1}{\varepsilon^2} \log \frac{|D_1|}{\delta} \right)$ due to $\min\{\frac{1}{2\sigma},\frac{1}{\delta}\} \sigma\leq \frac{1}{2\sigma} \cdot \sigma = \Theta(1)$. Using a union bound over the fail probability for every $v\in D_1$, a success probability of at least $1-\frac{|D_1|\delta}{|D_1|}=1-\delta$ follows.
By \[theorem:epsilon\_factor\_approximation\], trivially repeating the single step algorithm [[<span style="font-variant:small-caps;">EpsilonFactorApprox</span>]{}]{}needs $\Theta \left( T |D_1| \frac{1}{\varepsilon^2}\log\frac{|D_1|}{\delta}\right)$ messages on expectation for $T$ (because the number of nodes in ${N_t^v}$ for any $v \in D_1$ is at least $N_1^v/2$ in every time step of that interval). Hence, the number of messages sent when using [<span style="font-variant:small-caps;">ContinuousEpsilonApprox</span>]{} is reduced in the order of $\max\{2\sigma, \delta \}$.
The Count Distinct Monitoring Problem {#sec:countDistinct}
=====================================
In this section we present an [($\varepsilon$,$\delta$)-approximation]{}algorithm for the Count Distinct Monitoring Problem. The basic approach is similar to the one presented in the previous section for monitoring the frequency of each value. That is, we first estimate $|D_t|$ up to a (small) constant factor and then use the result to define a protocol for obtaining an $(\varepsilon,\delta)$-approximation. If we could assume that, at any fixed time $t$, each value was observed by at most one node, it would be possible to solve this problem with expected communication cost of $O(\frac{1}{\varepsilon^2} \log \frac{1}{\delta})$ (per time step $t$ and per value $v \in D_t$) using the same approach as in the previous section. Since this assumption is generally not true, we aim at simulating such behaviour that for each value in the domain only one random experiment is applied. We apply the concept of *public coins*, which allows nodes measuring the same value to observe identical outcomes of their random experiments. To this end, nodes have access to a shared random string $R$ of fully independent and unbiased bits. This can be achieved by letting all nodes use the same pseudorandom number generator with a common starting seed, adding a constant number of messages to the bounds proven below. We assume that the server sends a new seed in each phase by only loosing at most a constant factor in the amount of communication used. However, we can drop this assumption by checking whether there are nodes that changed their value such that only in rounds in which there are changes new public randomness is needed. The formal description of the algorithm for a constant factor and an $\varepsilon$-approximation are given in \[alg:publicCoin1\] and \[alg:publicCoin2\], respectively.
We consider the access of the public coin to behave as follows: Initialised with a seed, a node accesses the sequence of random bits $R$ bitwise, i.e. after reading the $j$’th bit, the node next accesses bit $j+1$. Observe the crucial fact that as long as each node accesses the exact same number of bits, each node observes the exact same random bits simultaneously. \[alg:publicCoin1\] essentially works as follows: In a first step, each node draws a number from a geometrical distribution using the public coin. By this, all nodes observing the same value $v$ obtain the same height $h_v$. In the second step we apply the strategy as in the previous section to reduce communication if lots of nodes observe the same value: Each node $i$ draws a number $g_i$ from a geometrical distribution without using the public coin. Afterwards, all nodes with the largest height $g_i$ among those with the largest height $h_v$ broadcast their height $h_v$.
**(Node $i$, observes value $v = v_i$)**
1. Draw a random number $h_v$ as follows:\
Consider the next $\Delta \cdot \log n$ random bits $b_1, \ldots, b_{\Delta \cdot \log n}$ from $R$. Let $h$ be the maximal number of bits $b_{v \cdot \log n + 1}, \ldots, b_{v \cdot \log n + 1+ h}$ that equal $0$. Define $h_v \coloneqq \min\{h,\log n\}$.
2. Let $g'_i$ be a random value drawn from a geometric distribution with success-probability $p = 1/2$ and define $g_i = \min(g'_i, \log n)$ (without accessing public coins).
3. Broadcast drawn height $h_v$ in round $r = \log^2 n - (h_v - 1) \cdot \log n - g_i$ unless a node $i'$ has broadcasted before.
**(Server)**
1. Receive a broadcast message containing height $h$ in round $r$.
2. Output $\hat{d}_t = 2 ^h$.
Note that only (at most $n$) nodes that observe value $v$ with $h_v$ = $max_{v'} \, h_{v'}$ may send a message in \[alg:publicCoin1\]. Now, all nodes observing the same value observe the same outcome of their random experiments determining $h_v$. Hence, by a similar reasoning as in \[lemma:constant\_factor\_approximation\], one execution of the algorithm uses ${\mathcal{O}}(1)$ messages on expectation.
Using the algorithm given in \[alg:publicCoin1\] and applying the same idea as in the previous section, we obtain an $(\varepsilon, \delta)$-approximation as given in \[alg:publicCoin2\]: Each node tosses a coin with a success probability depending on the constant factor approximation (for which we have a result analogous to \[corollary:constant\_factor\_approximation\_delta\]). Again, all nodes use the public coin so that all nodes observing the same value obtain the same outcome of this coin flip. Afterwards, those nodes which have observed a success apply the same strategy as in the previous section, that is, they draw a random value from a geometric distribution, and the nodes having the largest height send a broadcast.
**(Node $i$)**
1. Flip a coin with success probability $p = 2^{-q} = \frac{ c \log 1/\delta} {\varepsilon^2 \hat{d}_t}$, $q \in \mathbb{N}$ as follows:\
Consider the next $\Delta \cdot q$ random bits $b_1, \ldots b_{\Delta \cdot q}$. The experiment is successful if and only if all random bits $b_{v \cdot q + 1}, \ldots, b_{v \cdot q + q}$ equal $0$. The node deactivates (and does not take part in Steps 2. and 3.) if the experiment was not successful.
2. Draw a random value $h'_i$ from a geometric distribution and define $h_i = \min(h'_i, \log n)$ (without accessing public coins).
3. Node $i$ broadcasts its value in round $\log n - h_i$ unless a node $i'$ with $v^t_i = v^t_{i'}$ has broadcasted before.
**(Server)**
1. Let $S_t$ be the set of received values.
2. Output $\tilde{d}_t \coloneqq |S_t| / p$
Using arguments analogous to \[lemma:single\_shot\_epsilon,lemma:msg\_epsilonApprox\] and applying [[<span style="font-variant:small-caps;">EpsilonFactorApprox</span>]{}]{}for $T$ time steps, we obtain the following theorem.
There exists an $(\varepsilon, \delta)$-approximation algorithm for the Count Distinct Monitoring Problem for $T$ time steps using ${\mathcal{O}}(T \cdot \frac{1}{\varepsilon^2} \log \frac{1}{\delta})$ messages on expectation.
A Parameterised Analysis {#a-parameterised-analysis-2 .unnumbered}
------------------------
In this section we consider the problem for multiple time steps and parameterise the analysis with respect to instances in which the domain does not change arbitrarily between consecutive time steps. Recall that for monitoring the frequency from a time step $t-1$ to the current time step $t$, all nodes that left and all nodes that entered toss a coin to estimate the number of changes. However, to identify that a node observes a value which was not observed in the previous time step, the domain has to be determined exactly.
We apply the following idea instead: For each value $v \in \{1, \ldots, \Delta\}$ we flip a (public) coin. We denote the set of values with a successful coin flip as the *sample*. Afterwards, the algorithm only proceeds on the values of the sample, i.e. in cases in which a node observes a value with a successful coin flip and no node observed this value in previous time steps, this value contributes to the estimate $\tilde d_t^+$ at time $t$. Regarding the (sample) of nodes that leave the set of observed values, the [[<span style="font-variant:small-caps;">DomainMonitoring</span>]{}]{}algorithm is applied to identify which (sampled) values are not observed any longer (and thus contribute to $\tilde d_t^-$).
1. Compute $\delta'= 2\,\delta^2$
2. Broadcast a new seed value for the public coin.
3. Compute an $(\varepsilon, \delta')$-approximation $\tilde d_1$ of $|D_1|$ using \[alg:publicCoin2\]. Furthermore, obtain the success-probability $p$.
4. Repeat for each time step $t>1$:
1. Each node $i$ applies \[alg:domainMonitoring\] if the observed value $v_i$ is in the sample set. Let $\hat d_t^-$ be the number of values (in sample set) which left the domain and $\hat d_t^+$ the number of nodes that join the sample.
2. Server computes $\tilde d_t=\tilde d_1 + \sum_{i=2}^t \hat d_i^+/p - \sum_{i=2}^t \hat d_i^-/p$.
3. Break if $t=1/\delta$ or $\left(\sum_{i=2}^t \tilde d_i^+ + \sum_{i=2}^t \tilde d_i^-\right)/p$ exceeds $\tilde d_1/2$.
5. Set $t=1$ and go to \[alg:multiple\_step\_epsilon:start\].
Analogous to \[lemma:frequencies\_multiple\_step\_correctness\], we have the following lemma.
\[lemma:multiple\_step\_correctness\] [[<span style="font-variant:small-caps;">ContinuousEpsilonApprox</span>]{}]{}achieves an ($\varepsilon$,$\delta$)-approximation of $|D_t|$ in any time step $t$.
For the number of messages, we argue based on the previous section. However, in addition the [[<span style="font-variant:small-caps;">DomainMonitoring</span>]{}]{}algorithm is applied. Observe that the size of the domain changes by at most $n/2$, and consider the case that this number of nodes observed the same value $v$. The expected cost (where the expectation is taken w.r.t. whether $v$ is within the sample) is ${\mathcal{O}}( \log n \cdot R^* \cdot p) = {\mathcal{O}}\bigl(\frac{\log n \cdot R^*}{|D_t| \varepsilon^2} \log \frac{1}{\delta} \bigr)$. Similar to \[th:fmptheorem\], we then obtain the following theorem.
[[<span style="font-variant:small-caps;">ContinuousEpsilonApprox</span>]{}]{}provides and $(\varepsilon,\delta)$-approximation for the Count Distinct Monitoring Problem for $T$ time steps using an amount of $\Theta \left( \left( 1+T\cdot \max\{2\sigma, \delta \} \right) \frac{\log (n) \cdot R^*}{|D_t| \cdot \varepsilon^2} \log \frac{1}{\delta} \right)$ messages on expectation, if $\sigma \leq 1/2$.
[^1]: This work was partially supported by the German Research Foundation (DFG) within the Priority Program “Algorithms for Big Data” (SPP 1736) and by the Federal Ministry of Education and Research (BMBF) as part of the poject “Resilience by Spontaneous Volunteers Networks for Coping with Emergencies and Disaster” (RESIBES), (grant no 13N13955 to 13N13957).
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abstract: 'Our goal in this work is to present some function spaces on the complex plane ${\mathbb{C}}$, $X({\mathbb{C}})$, for which the quasiregular solutions of the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f (z)$, have first derivatives locally in $X({\mathbb{C}})$, provided that the Beltrami coefficient $\mu$ belongs to $X({\mathbb{C}})$.'
author:
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title: '[**Beltrami equation with coefficient in Sobolev and Besov spaces** ]{}'
---
[^1]
Introduction
============
A function $f : {\mathbb{C}}\longrightarrow {\mathbb{C}}$ is called $\mu$-quasiregular if it belongs to the Sobolev space $W^{1,2}_{\text{loc}}({\mathbb{C}})$ (functions with distributional first order derivatives locally in $L^2$) and satisfies the Beltrami equation $$\label{eq1}
\overline\partial f (z) = \mu(z) \partial f (z), \quad a.e. \; z \in {\mathbb{C}}\,,$$ where $\mu$, called the Beltrami coefficient of $f$, is a Lebesgue measurable function on the complex plane ${\mathbb{C}}$ satisfying $\|\mu\|_\infty < 1$. If, in addition, $f$ is a homeomorphism, then we say that f is $\mu$-quasiconformal. Quasiconformal and quasiregular mappings are a central tool in modern geometric function theory and have had strong impact in other areas.
It is well-known that quasiregular functions are locally in some Hölder class (Mori’s Theorem), and moreover they actually belong to $W^{1,p}_{\text{loc}}$ for some $p>2$. In this paper we are interested in studying how the regularity of the Beltrami coefficient affects the regularity of the solutions of . Thus, if the Beltrami coefficient $\mu$ belongs to the Hölder class $C^{l,s}$, $0<s<1$, using Schauder estimates (see for instance [@AIM chapter 15]), then $\mu$-quasiregular functions belong to $C^{l+1,s}_{\text{loc}}$. For the borderline cases $s=0$ and $s=1$, the $C^{l+1,s}$ regularity fails (e.g. [@AIM p. 390]). If $\mu\in W^{1,p}$, $2<p <\infty$, then one can read in Ahlfors’ book [@Ah p. 56] the result that quasiregular functions are locally in $W^{2,p}$. The cases $\mu\in W^{1,p}$, $p\le 2$, were studied in [@CFMOZ]; for instance, when $p=2$ one gets that the solutions are locally in $ W^{2,q}$ for every $q< 2$.
Our goal in this work is to present some function spaces $X$ for which all quasiregular solutions of have first derivatives locally in $X$, provided that the Beltrami coefficient belongs to $X$. These function spaces will enjoy the additional property of being an algebra (that is, the product of two functions in $X$ is again in $X$) and this feature will play an important role in our arguments. We deal with Triebel-Lizorkin spaces $F^{s}_{p,q}({\mathbb{C}})$ and Besov spaces $B^{s}_{p,q}({\mathbb{C}})$ with $s>0$, $1<p<\infty$, $1<q<\infty$ and $sp>2$. Let $A^{s}_{p,q}({\mathbb{C}})$ denote any of these function spaces with the indices as we have determined. In any case, the condition $sp>2$ ensures that we have bounded continuous functions and multiplication algebras (e.g. [@RS 4.6.4]). In Section \[preli\] we will give the precise definitions of these function spaces involved in the statement of the our first theorem.
\[T1\] Suppose that $\mu \in A^{s}_{p,q}({\mathbb{C}})$ is compactly supported with $\| \mu \|_{\infty}= k<1$. Then any $f\in W^{1,2}_{\text{loc}}({\mathbb{C}})$ satisfying the Beltrami equation has first derivatives locally in $A^{s}_{p,q}({\mathbb{C}})$.
When the Beltrami coefficient is compactly supported there is a unique $W^{1,2}_{\text{loc}}({\mathbb{C}})$ solution of normalized by the condition $z + O (1/z)$ near $\infty$. Moreover, it is a homeomorphism of the complex plane. It is called the principal solution of . By Stoilow’s Factorization Theorem (e.g. [@AIM section 5.5]), for any quasiregular function $f$ there exists a holomorphic function $h$ such that $f= h\circ \phi$, where $\phi$ is the associated principal solution. Therefore, we will only concentrate on principal solutions. As is well known, $\phi$ is given explicitly by the formula [@AIM p. 165] $$\phi(z) = z + \mathsf C(h)(z)\,,$$ where the operator $$\label{cauchy}
\mathsf C \, h(z) = \frac{1}{\pi}\, \int_{\mathbb C} h(z-w)\frac{1}{w}\, \mathrm{d}w$$ is the Cauchy transform of $h$. When $h\in L^p$, $1<p<\infty$, one has the identity $\bar\partial \mathsf C(h) =h$. Consequently, our theorem immediately follows from next proposition.
\[Prop1\] Suppose that $\mu$ is compactly supported with $\| \mu \|_{\infty} = k<1$ and $\phi(z) = z + \mathsf C(h)(z)$ is the principal solution of the Beltrami equation . Let $s > 0$, $1 < p <\infty$, $1 < q <\infty$ and $sp > 2$. If $\mu \in A^{s}_{p,q}({\mathbb{C}})$, then $h \in A^{s}_{p,q}({\mathbb{C}})$.
<span style="font-variant:small-caps;">Sketch of the proof.</span> The Beurling transform is the principal value convolution operator
$$Bf(z)= - \frac{1}{\pi}\, \text{p.v.} \int_{\mathbb C} f(z-w)\frac{1}{w^2}\, \mathrm{d}w\,.$$ The Fourier multiplier of $B$ is $\frac{\overline{\xi}}{\xi}$, or, in other words, $$\widehat{Bf}(\xi) = \frac{\overline{\xi}}{\xi}\,\, \hat{f}(\xi)\,.$$ Thus $B$ is an isometry on $L^2({\mathbb{C}})$ and is well-known that $B$, as any Calderón-Zygmund convolution operator, is bounded on $ A^{s}_{p,q}({\mathbb{C}})$.
Recall the relation between the Cauchy and the Beurling transforms: $\partial \mathsf C = B$. Thus, $\partial\phi = 1+ B(h)$ and $\overline{\partial}\,{\phi} =h$, and consequently the function $h$ is determined by the equation $$(I-\mu\,B)(h) = \mu\, .$$ So, we only need to invert the Beltrami operator $I-\mu\,B$ on the corresponding function space. This task is completed in Section \[sec3\].
For the critical case $sp = 2$, we consider a Riesz potential space $I_1(L^{2,1}(\mathbb C))$, the set of functions with first order derivatives in the Lorentz space $L^{2,1}(\mathbb C)$. Even though close to $L^2$ , the Lorentz space $L^{2,1}(\mathbb C)$ is strictly contained in $L^2$. This small improvement on the derivatives allows us to have continuous functions vanishing at infinity (by the way, remind that functions with first order derivatives in $L^{2}$ may not be continuous).
\[Prop2\] Suppose that $\mu\in I_1(L^{2,1}(\mathbb C))$ is compactly supported with $\| \mu \|_{\infty}= k<1$ and $\phi(z) = z + \mathsf C(h)(z)$ is the principal solution of the Beltrami equation . Then $h \in I_1(L^{2,1}(\mathbb C))$.
As we mentioned ago, Proposition \[Prop2\] does not hold when the Beltrami coefficient only has first derivatives in $L^{2}$ . However, the analogous result would remain valid if we replace $I_1(L^{2,1}(\mathbb C))$ by $I_s(L^{\frac{2}{s},1}(\mathbb C))$, $0 < s < 2$.
The main result of [@MOV] identifies a class of non-smooth Beltrami coefficients which determine bilipschitz quasiconformal mappings. In particular, one proved the following result.
Let $\Omega$ be a bounded domain of ${\mathbb{C}}$ with boundary of class $\mathcal{C}^{1,\varepsilon}$, $0< \varepsilon <1$, and let $\mu\in \mathcal{C}^{0,\varepsilon}(\Omega)$ with $\| \mu\|_{\infty}<1$. Let $\phi(z) = z + \mathsf C(h)(z)$ be the principal solution of the Beltrami equation . Then $h \in \mathcal{C}^{0,\varepsilon '}(\Omega)$ for any $\varepsilon' <\varepsilon$ and moreover $\phi$ is billipschitz .
Now, we replace the Hölder smoothness of the Beltrami coefficient by a Sobolev (or Besov) condition restricted on a domain. (See definitions in the next section).
\[T2\] Let $0<s< \varepsilon<1$ and $1<p<\infty$ such that $sp>2$ and let $\Omega$ be a bounded domain of ${\mathbb{C}}$ with boundary of class $\mathcal{C}^{1,\varepsilon}$. Suppose that $\mu$ is supported in $\overline\Omega$ with $\| \mu \|_{\infty}= k<1$ and $\phi(z) = z + \mathsf C(h)(z)$ is the principal solution of the Beltrami equation .
1. If $\mu \in W^{s,p}(\Omega)$, then $h \in W^{s,p}(\Omega)$.
2. If $\mu \in B^{s}_{p,p}(\Omega)$, then $h \in B^{s}_{p,p}(\Omega)$.
The proof runs in parallel to that of the above propositions, but now a new obstacle appears: the boundedness of the Beurling transform on $W^{s,p}(\Omega)$ (or $B^{s}_{p,p}(\Omega)$). In general, it is not clear if Calderón-Zygmund convolution operators are bounded on $W^{s,p}(\Omega)$ (or $B^{s}_{p,p}(\Omega)$). Of course, the answer depends on the operator and on the boundary of the domain. We will study this question in domains $\Omega $ of $\mathbb R^{n}$, $n\ge 2$.
In $\mathbb R^n$ we consider the kernel $K(x)= \frac{\omega(x)}{|x|^n}$, $x\ne 0$, where $\omega$ is a homogeneous function of degree $0$, with zero integral on the unit sphere and $\omega \in \mathcal{C}^{1}(S^{n-1})$. Then, the singular integral $$T f(x)= \text{p.v.} \int f(y) K(x-y)\, \mathrm{d}y\,$$ is bounded on $L^p(\mathbb R^n)$, $1<p<\infty$. (Really, the condition $\omega \in \mathcal{C}^{1}(S^{n-1})$ could be weakened but it is enough for our purpose). On the other hand, Sobolev spaces $W^{s,p}(\mathbb R^n)$ ($=F^{s}_{p,2}(\mathbb R^n)$) are described as spaces of Bessel potentials, that is, $f\in W^{s,p}$ if and only if $f=G_{s} * g$, where $G_{s}$ denotes the Bessel kernel of order $s$ and $g\in L^p$ (e.g. [@St chapter 5]). Remember that the Bessel kernel of order $s$, $G_{s}$ , is the $L^1$ function with Fourier transform $(1+ |\xi|^2)^{-\frac{s}{2}}$. Then, because $T$ is a convolution operator, one has the identity $$T(f) = T(G_{s}*g) = G_{s}*(Tg)$$ and one gets the boundedness of $T$ on $W^{s,p}$, $1<p< \infty$. But if one takes $f\in W^{s,p}(\Omega)$, $\Omega$ a domain of $\mathbb R^n$, then $$T_{\Omega} f(x) := \text{p.v.} \int_{\Omega} f(y) K(x-y)\, dy\,$$ clearly belongs to $L^p(\Omega)$. However, perhaps $T_{\Omega}f\notin W^{s,p}(\Omega)$. For instance, let $Q$ denote a rectangle in ${\mathbb{C}}$ and $\chi_Q$ denote its characteristic function. A computation shows that the Beurling transform of $\chi_{Q}$, $B \chi_{Q}$, has logarithmic singularities at the vertices of the rectangle and, therefore, its first derivatives belong to $L^p(Q)$ only if $p<2$ (e.g. [@AIM p. 147]). For positive results, we restrict our attention to operators with even kernel, that is, $K(-x) = K(x)$. In Section \[CZdo\] we will deal with Theorem \[T3\].
\[T3\] Let $\Omega$ be a bounded domain of $\mathbb R^n$ with boundary of class $\mathcal{C}^{1,\beta}$, $\beta >0$, and let $T$ be an even smooth homogeneous Calderón-Zygmund operator.
1. If $T\chi_{\Omega} \in B^{s}_{p,p}(\Omega)$, $0<s<1$, $n<sp<\infty$, then $T_{\Omega}: B^{s}_{p,p}(\Omega) \longrightarrow B^{s}_{p,p}(\Omega)$.
2. If $T\chi_{\Omega} \in W^{s,p}(\Omega)$, $0<s<1$, $n<sp<\infty$, then $T_{\Omega}: W^{s,p}(\Omega) \longrightarrow W^{s,p}(\Omega)$.
3. If $T\chi_{\Omega} \in W^{1,p}(\Omega)$, $n<p <\infty$, then $T_{\Omega}: W^{1,p}(\Omega) \longrightarrow W^{1,p}(\Omega)$.
In any case the norm operator depends on the domain $\Omega$ and the Calderón-Zygmund constant of the kernel of $T$ (see for the definition).
The result reduces the study of the boundedness of the operator $T_{\Omega}$ to the behaviour of $T_{\Omega}$ on the function $\chi_{\Omega}$. Thus, we have a necessary and sufficient condition of type $T(1)$. In the proof of Theorem 3, we follow the same method of Y. Meyer in [@Me], where he studied the continuity of generalised Calderón-Zygmund operators on Sobolev spaces $W^{s,p}(\mathbb R^n)$.
Since T is bounded on $L^p$, using complex and real interpolation, one could think that items 1 and 2 of the above theorem are a consequence of the third one. But this it not the case because the conditions on items 1 and 2 are weaker than $T\chi_{\Omega} \in W^{1,p}(\Omega)$. When $\Omega$ is a bounded domain of $\mathbb R^n$ with boundary of class $ \mathcal{C}^{1,\varepsilon}$, $0<s<\varepsilon <1$, and $n<sp<\infty$ then $T_{\Omega}$ is bounded on $W^{s,p}(\Omega)$ and $B^{s}_{p,p}(\Omega)$ (see details in Section \[CZdo\]). In particular, the assumptions on the domain $\Omega$, in the statement of Theorem \[T2\], are to ensure that the Beurling transform is bounded on the corresponding function space. Recently, V. Cruz and X. Tolsa( [@CT], [@To]) have showed that if the outward unit normal $N$ on $\partial\Omega$ belongs to the Besov space $B^{s -1/p}_{p,p}(\partial\Omega)$, then $B {\chi_{\Omega}}\in W^{s,p}(\Omega)$.\
In Section 2 we shall introduce some basic notation and set up some necessary preliminaries. The proof of Proposition \[Prop1\] and Proposition \[Prop2\] are in Section \[sec3\]. In Section \[CZdo\] we study even smooth homogeneous Calderón-Zygmund operators on domains. The proof of the Theorem \[T2\] is explained in Section 5.
As usual, the letter $C$ will denote a constant, which may be different at each occurrence and which is independent of the relevant variables under consideration.
Preliminaries {#preli}
=============
We start reviewing some basic facts concerning Triebel-Lizorkin spaces and Besov spaces. Let $\mathcal S(\mathbb R^n)$ be the usual Schwartz class of rapidly decreasing $\mathcal C^{\infty}$-functions and $\widehat g$ stands for the Fourier transform of $g$. Let $\psi\in \mathcal S(\mathbb R^n)$ with $\widehat\psi(\xi) =1$ if $|\xi| \le 1$ and $\widehat\psi(\xi) =0$ if $|\xi | \ge 3/2$. We set $\psi_{0} = \psi$ and $\widehat\psi_{j}(\xi) = \widehat\psi(2^{-j}\xi) - \widehat\psi(2^{-j+1}\xi)$, $j\in\mathbb N$. Since $\sum_{j=0}^{\infty}\widehat\psi_{j}(\xi)=1$ for all $\xi\in \mathbb R^n$, the $\widehat\psi_{j}$ form a dyadic resolution of unity. Then, for $f\in L^1_{\operatorname{loc}}(\mathbb R^n)$, $1\le p,q <\infty$, and $s>0$, one defines the norms $$\| f\|_{B^s_{p,q}} = \left( \sum_{j=0}^{\infty} \| 2^{js}\psi_{j}* f \|_{p}^q \right)^{\frac{1}{q}}$$ and $$\| f\|_{F^s_{p,q}} = \left\| \left( \sum_{j=0}^{\infty} | 2^{js}\psi_{j}* f |^q \right)^{\frac{1}{q}}
\right\|_{p}$$ The Besov space $B^s_{p,q}(\mathbb R^n)$ consists of the functions such that $\| f\|_{B^s_{p,q}} <\infty$, while the functions in the Triebel-Lizorkin space $F^s_{p,q}(\mathbb R^n)$ are those such that $\| f\|_{F^s_{p,q}} <\infty$.
The spaces $F^s_{p,2}$ , $1 < p <\infty$, are known as Sobolev spaces of fractional order or Bessel-potential spaces and we prefer denote them by $W^{s,p}$. Since $p\ge 1$ and $q \ge 1$, both $B^s_{p,q}$ and $F^s_{p,q}$ are Banach spaces. A systematic treatment of these spaces may be found in [@Tri1], [@RS] and [@Gr Chapter 6]. A remarkable fact when $sp > n$ is that $B^s_{p,q}$ and $F^s_{p,q}$ form an algebra with respect to pointwise multiplication, that is, $$\label{algebra1}
\| f \cdot g\|_{A^s_{p,q}} \le C \| f\|_{A^s_{p,q}} \| g\|_{A^s_{p,q}},$$ where $A^s_{p,q}$ denotes the corresponding Besov space or Triebel-Lizorkin space (e.g. [@RS 4.6.4]). Moreover, functions in these spaces satisfy some Hölder condition and so they are continuous functions with $$\label{algebra2}
\| f\|_{\infty}\le C \| f\|_{A^s_{p,q}}\, .$$
We say that a bounded domain $\Omega \subset {{\mathbb R}^n}$ has a boundary of class $\mathcal{C}^{1,\varepsilon}$ if ${\partial\,\Omega}$ is a $C^1$ hyper-surface whose unit normal vector satisfies a Lipschitz (Hölder) condition of order ${\varepsilon}$ as a function on the surface. To state an alternative condition, for $x = (x_1, \dots ,x_n)\in {{\mathbb R}^n}$ we use the notation $x = (x',x_n)$, where $x'= (x_1,\dots,x_{n-1})$. Then $\Omega$ has a boundary of class $\mathcal{C}^{1,\varepsilon}$ if for each point $a \in \partial\,\Omega$ one may find a ball $B(a,r)$ and a function $x_n = \varphi(x')$, of class $\mathcal{C}^{1,\varepsilon}$, such that, after a rotation if necessary, $\Omega \cap B(a,r)$ is the part of $B(a,r)$ lying below the graph of $\varphi$. Thus we get $$\label{eq6}
\Omega \cap B(a,r)= \{x \in B(a,r) : x_n <
\varphi(x_1,\dots,x_{n-1})\}\,.$$ We say that $\Omega$ is a bounded Lipschitz domain if the function $\varphi$ in is of class $\mathcal{C}^{0, 1}$.
In general, if one has a function space $X$ defined on $\mathbb R^n$ and a domain $\Omega \subset {{\mathbb R}^n}$, one defines the space $X(\Omega)$ as the restrictions of functions of $X$ from $\mathbb R^n$ to $\Omega$. In addition, the restriction space is endowed with the quasi-norm quotient. In the cases that we are considering we have an intrinsic characterization of elements of $X(\Omega)$. We will use these characterizations in the proofs of Theorems 2 and 3. Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^n$, $1<p<\infty$ and $0<s<1$. Then:
1. $f\in B_{p,p}^{s}(\Omega)$ if and only if $f\in L^{p}(\Omega)$ and $$\int_{\Omega}\int_{\Omega}\frac{|f(x)-f(y)|^{p}}{|x-y|^{n+s p}}\mathrm{d}x\mathrm{d}y<\infty.$$ (e.g. [@Tar p. 169])
2. $f\in W^{s,p}(\Omega)$ if and only if $f\in L^{p}(\Omega)$ and $$\label{sobolevs}
\int_{\Omega}\left(\int_{\Omega}\frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2 s}}\mathrm{d}x\right)^{\frac{p}{2}}\mathrm{d}y<\infty.$$ (e.g [@Str p. 1051])
3. $f\in W^{1,p}(\Omega)$ if and only if $f\in L^{p}(\Omega)$ and $$\label{sobolev1}
\lim_{\alpha \to0}\alpha\int_{\Omega}\int_{\Omega}\frac{|f(x)-f(y)|^{p}}{|x-y|^{n+p-\alpha}}\mathrm{d}x\mathrm{d}y<\infty.$$ (e.g. [@Br2 p. 703])
A smooth (of class $\mathcal C^1$) homogeneous Calderón-Zygmund operator is a principal value convolution operator of type $$\label{eq6bis}
T(f)(x)= \text{p.v.} \int f(y)\,K(x-y) \, \mathrm{d}y \,,$$ where $$K(x)= \frac{\omega(x)}{|x|^{n}}\,,\quad x \neq 0\,,$$ $\omega(x)$ being a homogeneous function of degree $0$, continuously differentiable on ${{\mathbb R}^n}\setminus\{0\}$ and with zero integral on the unit sphere. Note that one trivially has $$|K(x-y)|\leq \frac{C}{|x-y|^n}$$ and $$|K(x-y)-K(x-y')|\leq C\frac{|y-y'|}{|x-y|^{n+1}}\quad\text{whenever }|x-y|\geq2|y-y'| .$$ The Calderón-Zygmund constant of the kernel of $T$ is defined as $$\| T \|_{CZ} = \|K(x)\,|x|^n \|_\infty + \|\nabla
K(x)\,|x|^{n+1}\|_\infty\,.\label{CZ}$$ The operator $T$ is said to be even if the kernel is even, namely, if $\omega(-x)=\omega(x)\,,$ for all $ x \neq 0\,.$ The even character of $T$ gives the cancellation $T(\chi_{B})\chi_{B}=0 $ for each ball $B$, which should be understood as a local version of the global cancellation property T (1) = 0 common to all smooth homogeneous Calderón-Zygmund operators. This extra cancellation property is essential for proving Lemma \[lema:acotacion\] and so Theorem \[T3\].
It is well known that Calderón-Zygmund convolution operators are bounded on $L^p(\mathbb R^n)$ and also on $W^{s,p}(\mathbb R^n)$ (because $W^{s.p} = G_{s}*L^p$). Using the method of real interpolation, one easily gets that these operators are also bounded on $B^s_{p,q}(\mathbb R^n)$ (see also [@Gr 6.7.2] for a direct proof). The boundedness of Calderón-Zygmund convolution operators on $F^s_{p,q}(\mathbb R^n)$ was proved in [@FTW Theorem 3.7] (see [@JHL Theorem 1.2] for a nice proof). Summarizing, if $s>0$ and $1<p, q<\infty$ we have $$\label{CZdes}
\|T f\|_{A^s_{p,q}}\le C \| f\|_{A^s_{p,q}},$$ where $C$ is a constant which depends on $s,p,q,n$ and $\| T \|_{CZ}$.\
Lorentz spaces are defined on measure spaces $(Y,m)$, but we only need the case $Y=\mathbb C$ and $m$ is the Lebesgue planar measure. The classical definition of Lorentz spaces use the rearrangement function. For any measurable function $f$ we define its nonincreasing rearrangement by $$f^{*}(t) := \inf \{ s: m\{z\in\mathbb C\colon|f(z)|>s\} \le t \}.$$ For $1\leq p,q<\infty$, the Lorentz space $L^{p,q}(\mathbb C)$ is the set of functions $f$ such that $\|f\|_{L^{p,q}}<\infty $, with $$\| f\|_{L^{p,q}(\mathbb C)}:= \left\{ \begin{array}{ll}
(\int_0^{\infty} [ t^{1/p} f^{*}(t)]^q t^{-1}dt)^{1/q}, & \textrm{for $ 1\le q<\infty$} \\[1mm]
\sup_{t>0} t^{1/p} f^{*}(t), & \textrm{for $q=\infty$}
\end{array}
\right .$$ A second definition of Lorentz spaces, which is equivalent to the first one, is given by real interpolation between Lebesgue spaces: $$(L^{p_0}, L^{p_1})_{\theta, q}=L^{p,q},$$ where $1\le p_0 <p<p_1 \le\infty$, $1\le q\le\infty$, $0<\theta<1$ and $\frac{1}{p}=\frac{1-\theta}{p_0}+ \frac{\theta}{p_1}$. Lorentz spaces inherited from Lebesgue spaces the stability property of the multiplication by bounded function, that is, if $f\in L^\infty$ and $g\in L^{p,q}$ then $fg\in L^{p,q}$ and we have $$\label{eq:vi1}
\|fg\|_{L^{p,q}}\leq \|f\|_\infty \|g\|_{L^{p,q}}\, .$$ Let $1\leq p,q<\infty$ and consider $0<\alpha<2$. The Lorentz potential space, $I_\alpha(L^{p,q}(\mathbb C))$, is the set of functions $f$ such that $f=I_\alpha*g$, where $g\in L^{p,q}(\mathbb C)$ and $I_\alpha (x) = c_{\alpha}|x|^{\alpha -2}$ is the Riesz potential of order $\alpha$. The norm in this space is given by $$\|f\|_{I_\alpha(L^{p,q}(\mathbb C))}=\|g\|_{L^{p,q}}.$$ Note that when $\alpha =1$, one has $ \|f\|_{I_1(L^{p,q}(\mathbb C))}\approx \|\nabla f\|_{L^{p,q}}$.
It is well known [@St2] that functions $f$ of $I_1(L^{2,1}(\mathbb C))$ are continuous and there exists a constant $C$ such that $$\label{eq:vi2}
\|f\|_\infty\leq C\|f\|_{I_1(L^{2,1}(\mathbb C))}.$$ In general $I_\alpha(L^{\frac{2}{\alpha},1}(\mathbb C))$ are embedded in $\mathcal C_0$, the space of continuous functions vanishing at the infinity (see [@Ba]). Again, a remarkable property of these spaces $I_\alpha(L^{\frac{2}{\alpha},1}(\mathbb C))$ is that they are multiplication algebras, that is, $$\label{LorenAlg}
\| fg\|_{I_\alpha(L^{\frac{2}{\alpha},1})} \le C \| f\|_{I_\alpha(L^{\frac{2}{\alpha},1})} \| g\|_{I_\alpha(L^{\frac{2}{\alpha},1})} .$$
Finally, note that Calderón-Zygmund convolution operators are bounded on $L^{p,q}(\mathbb R^n)$ and so also on Lorentz potential space, $I_\alpha(L^{p,q}(\mathbb C))$, with constant depending on .
Invertibility of the Beltrami operator {#sec3}
======================================
As we mentioned in the Introduction, to prove Proposition \[Prop1\] (and then Theorem \[T1\]) and Proposition \[Prop2\] we only have to consider the invertibility of the Beltrami operator $I-\mu\,B$ on $A^{s}_{p,q}({\mathbb{C}})$ and on $I_{1}(L^{2,1}) ({\mathbb{C}})$. Following the idea of Iwaniec [@I1 p. 42–43] we define $$P_{m}=I+\mu B+\cdots+(\mu B)^{m}\, ,$$ so that we have $$(I-\mu B)P_{n-1}=P_{n-1}(I-\mu B)=I - (\mu B)^{n} =I-\mu^{n}B^{n}+K ,\label{eq:fredh}$$ where $K= \mu^{n}B^{n} - (\mu B)^{n} $ can be easily seen to be a finite sum of operators that contain as a factor the commutator $[\mu , B] =
\mu B -B\mu$. In Lemma \[Le2\] (and in Lemma \[Le2bis\]) we will prove that $[\mu, B ]$ is compact on $A^{s}_{p,q}({\mathbb{C}})$ (and on $I_{1}(L^{2,1}) ({\mathbb{C}})$) , so that $K$ is also compact. In Lemma \[Le1\] we will check that the operator norm of $\mu^{n}B^{n}$ on $A^{s}_{p,q}({\mathbb{C}})$ (and on $I_{1}(L^{2,1}) ({\mathbb{C}})$) is small if $n$ is large. Therefore, $I - \mu B$ is a Fredholm operator on $A^{s}_{p,q}({\mathbb{C}})$ (and on $I_{1}(L^{2,1}) ({\mathbb{C}})$). Clearly $I - t \mu B$, $0\le t\le 1$, is a continuous path from the identity to $I - \mu B$ . By the index theory of Fredholm operators on Banach spaces (e.g. [@Sch]), the index is a continuous function of the operator. Hence $I-\mu B$ has index 0. On the other hand, $I-\mu B$ is injective on $A^{s}_{p,q}({\mathbb{C}})$ (and on $I_{1}(L^{2,1}) ({\mathbb{C}})$) because by [@I1 p. 43] it is injective on $L^p({\mathbb{C}})$ for all $1<p<\infty$. That concludes that $I-\mu B$ is invertible.
\[Le1\]
(a) The operator norm of $\mu^{n}B^{n}$ on $A^{s}_{p,q}({\mathbb{C}})$ is small if $n$ is large.
(b) The operator norm of $\mu^{n}B^{n}$ on $I_{1}(L^{2,1} ({\mathbb{C}}))$ is small if $n$ is large.
Let $b_n=\dfrac{(-1)^n n}{\pi } \dfrac{\bar z^{n-1}}{z^{n+1}}$ the kernel of iterated Beurling transform $B^n$. Then, the Calderón-Zygmund constant of $B^n$ is $$\|b_{n}(z)|z|^{2}\|_{\infty}+\|\nabla
b_{n}(z)|z|^{3}\|_{\infty}\leq Cn^{2}.$$
\(a) It is an easy consequence of well-known results. Since $\| g^m \|_{A^{s}_{p,q}} \le C\|g \|^{m-1}_{\infty} \| g \|_{A^{s}_{p,q}}$ (see [@RS Teorem 5.3.2/4]), using and , we have $$\begin{aligned}
\|\mu^{n}B^{n}(f)\|_{A^{s}_{p,q}} & \leq & C\:\|\mu^{n}\|_{A^{s}_{p,q}}\|B^{n}(f)\|_{A^{s}_{p,q}} \\
& \leq & C\:\|\mu^{n}\|_{A^{s}_{p,q}} n^2 \|f\|_{A^{s}_{p,q}} \\
& \leq & C\: n^2 \|\mu \|^{n-1}_{\infty} \|\mu \|_{A^{s}_{p,q}} \|f\|_{A^{s}_{p,q}} \end{aligned}$$ and the norm becomes small if $n$ is big enough because $\| \mu \|_{\infty}= k<1$.\
(b) Using $ \|f\|_{I_1(L^{2,1})}\approx \|\nabla f\|_{L^{2,1}}$, , and the boundedness of Calderón-Zygmund convolution operators, we have $$\begin{aligned}
\|\mu^{n}B^{n}(f)\|_{I_{1}(L^{2,1})} & \leq& C\:\|\mu^{n}\|_{I_{1}(L^{2,1})}\|B^{n}(f)\|_{I_{1}(L^{2,1})} \\
& \leq& C\:\|\mu^{n}\|_{I_{1}(L^{2,1})} n^2 \|f\|_{I_{1}(L^{2,1})} \\
& \leq& C\: n^3 \|\mu \|^{n-1}_{\infty} \|\mu \|_{I_{1}(L^{2,1})} \|f\|_{I_{1}(L^{2,1})} \end{aligned}$$ and the norm becomes small if $n$ is big enough because $\| \mu \|_{\infty}= k<1$.
\[Le2\] The commutator $[\mu, B ]$ is compact on $A^{s}_{p,q}({\mathbb{C}})$.
First we have $$\begin{aligned}
\|[\mu,B]f\|_{A^{s}_{p,q}} & = & \|\mu Bf-B(\mu f)\|_{A^{s}_{p,q}}\\
& \leq & \|\mu\|_{A^{s}_{p,q}}\|Bf\|_{A^{s}_{p,q}}+C \|\mu f\|_{A^{s}_{p,q}}\\
& \leq & C \|\mu\|_{A^{s}_{p,q}}\|f\|_{A^{s}_{p,q}}
\end{aligned}$$ and so the commutator is bounded in $A^{s}_{p,q}$.
Using that the limit of compact operators is a compact operator, we can assume that $\mu \in\mathcal C_{c}^{\infty}(\mathbb{C})$, with its support contained in the disk $D(0, R)$. Now we use a trick from [@AIM p. 145]. Consider an arbitrary function $g= \mathsf C \, f$ with $f\in A^{s}_{p,q} $, where $\mathsf C\, f$ denotes the Cauchy transform of $f$ (see ). As $ \partial g= B(f)$, $\bar \partial g = f$ and $B( \bar\partial (\mu g))= \partial (\mu g)$, $$\begin{split}
\mu B(f) -B(\mu f) & = \mu \partial g - B(\mu \bar\partial g) = \mu \partial g - B( \bar\partial (\mu g)) +B(\bar\partial\mu \, g)\\
& = \mu \partial g - \partial (\mu g) + B(\bar\partial\mu \, g) = B(\bar\partial\mu \, g) - \partial\mu \, g\\
& = B(\bar\partial\mu \, \mathsf C\, f) - \partial\mu \, \mathsf C\, f
\end{split}$$ From this representation one can see that $[\mu, B ]$ is compact. Given $\varphi\in \mathcal C_{c}^{\infty}( D(0, R))$ the operator $\varphi\, \mathsf C\, f$ is a compact operator on $A^{s}_{p,q}({\mathbb{C}})$, because by the lifting property (see [@RS 2.1.4]) $\varphi\, \mathsf C\, f\in A^{s+1}_{p,q}({\mathbb{C}})$, obviously $\varphi\, \mathsf C\, f(z)=0$ if $|z|\ge R$ and the inclusion of $ A^{s+1}_{p,q}( D(0, R))$ into $ A^{s}_{p,q}(D(0, R))$ is compact (e.g. [@RS 2.4.4]).
\[Le2bis\] The commutator $[\mu, B ]$ is compact on $I_{1}(L^{2,1} ({\mathbb{C}})$.
As above we have $$\label{acot}
\begin{split}
\|[\mu,B]f\|_{I_{1}(L^{2,1})} & = \|\mu Bf-B(\mu f)\|_{I_{1}(L^{2,1})}\\
& \leq \|\mu\|_{I_{1}(L^{2,1})}\|Bf\|_{I_{1}(L^{2,1})}+C \|\mu f\|_{I_{1}(L^{2,1})}\\
& \leq C \|\mu\|_{I_{1}(L^{2,1})}\|f\|_{I_{1}(L^{2,1})}
\end{split}$$ and so the commutator is bounded in $I_{1}(L^{2,1})$. So, by density, we only need to prove the compactness of the commutator when $\mu \in \mathcal C^{\infty}_c.$
On the other hand, $$\begin{aligned}
\| [\mu,B]f\|_{I_{1}(L^{2,1})} & = & \sum_{j=1}^{2}\|\partial_{j}(\mu B(f)-B(\mu f))\|_{L^{2,1}}\\
& = & \sum_{j=1}^{2}\| [\partial_{j}\mu,B]f + [\mu,B](\partial_{j}f)\|_{L^{2,1}} \\
& \leq & \sum_{j=1}^{2}\|[\partial_{j}\mu,B]f\|_{L^{2,1}}+\|[\mu,B](\partial_{j}f)\|_{L^{2,1}}.
\end{aligned}$$
Since the commutator is a compact operator in $L^p$ when $\mu$ is smooth [@Uchi] and using real interpolation of compact operators [@CoP] we have that $[\mu,B]\colon L^{2,1}(\mathbb{C})\to L^{2,1}(\mathbb{C})$ is compact.
Therefore we only have to prove that $[a,B]\colon I_{1}(L^{2,1}(\mathbb{C}))\to L^{2,1}(\mathbb{C})$ is a compact operator when $a\in \mathcal C_{c}^{\infty}(B(0,R))$ for some $R>0.$ Given $\eta >0$ we consider a regularization of the Beurling transform
$$B^{\eta}f(z)= - \frac{1}{\pi}\, \text{p.v.} \int f(z-w)K_{\eta}(w) \mathrm{d}w \,,$$ where $K_{\eta}(z)=\dfrac{\varphi_{\eta}(z)}{z^2}$ and $0\le \varphi_{\eta}(z) \le 1$ is a radial $\mathcal C^{\infty}$ function satisfying $\varphi_{\eta}(|z|)=0$ if $|z|<\frac{\eta}{2}$ and $\varphi_{\eta}(|z|)=1$ if $|z|> \eta.$ It is easy to check that $B^{\eta}$ is a convolution Calderón-Zygmund operator with constants depending on $\eta.$
In the rest of this proof we will use the estimate without any mention. For any $f\in {I_{1}(L^{2,1})},$ the function $[a,B-B^{\eta}](f)$ has compact support. On the other hand,
$$\begin{aligned}
|[a,B-B^{\eta}](f)(z)| =\left|\frac{-1}{\pi}\int(a(z)-a(y))\left(\frac{1}{(z-y)^2}-\frac{\varphi_{\eta}(z-y)}{(z-y)^2}\right)f(y) \mathrm{d}y\right|\\[2mm]
\le C\|f\|_{\infty}\|\nabla a\|_{\infty} \int_{|z-y|<\eta}\frac{1}{|z-y|}\,\mathrm{d}y\le C\eta\|f\|_{I_1( L^{2,1})}\|\nabla a\|_{\infty}.\end{aligned}$$
Consequently the operator $[a,B^{\eta}]$ tends to $[a,B]$ when $\eta\to 0.$ To prove that $[a,B^{\eta}]\colon I_{1}(L^{2,1})\to L^{2,1}$ is compact we will use Fréchet-Kolgomorov Theorem for Lorentz spaces (e.g. [@Br1 p. 111] for $L^p$ spaces).
By , the image by $[a,B^{\eta}]$ of the unit ball of $I_{1}(L^{2,1}(\mathbb C))$ is uniformly bounded in $L^{2,1}(\mathbb C).$ To get the equicontinuity, take $f\in I_{1}(L^{2,1})$ and $|z-w|<\frac{\eta}{8}.$ Then,
$$\begin{aligned}
[a,B^{\eta}]f(z)-[a,B^{\eta}]f(w) & = & \frac{-1}{\pi}\left((a(z)-a(w)\right)\int_{\mathbb{C}}\frac{\varphi_{\eta}(z-\xi)}{(z-\xi)^2}f(\xi)\mathrm{d}\xi\\
& + & \frac{-1}{\pi} \int_{\mathbb{C}}\left(\frac{\varphi_{\eta}(z-\xi)}{(z-\xi)^2}-\frac{\varphi_{\eta}(w-\xi)}{(w-\xi)^2}\right)\left(a(w)-a(\xi)\right)f(\xi)\mathrm{d}\xi\\
& = & \theta_{1}(z,w)+\theta_{2}(z,w).\end{aligned}$$
Since $B^{\eta}$ is a convolution Calderón-Zygmund operator $$|\theta_{1}(z,w)| = \frac{1}{\pi}|\left(a(z)-a(w)\right)B^{\eta}f(z)| \le C_{\eta}|z-w|\|\nabla a\|_{\infty}\|f\|_{I_{1}(L^{2,1})}$$ and $$\begin{aligned}
|\theta_{2}(z,w)| & = & \frac{1}{\pi} \left|\int_{\mathbb{C}\setminus B(z,\frac{\eta}{4})}\left(\frac{\varphi_{\eta}(z-\xi)}{(z-\xi)^2}-
\frac{\varphi_{\eta}(w-\xi)}{(w-\xi)^2}\right)
\left(a(w)-a(\xi)\right)f(\xi)\mathrm{d}\xi\right|\\
& \leq &C |z-w|\|f\|_{\infty}\|a\|_{\infty}\left\{ \int_{|z-\xi|>\frac{\eta}{8}} \frac{1}{|z-\xi|^3} \mathrm{d}\xi + \int_{2\eta >|z-\xi|>\frac{\eta}{8}} \frac{\| \nabla \varphi_{\eta} \|_{\infty}}{|z-\xi|^2} \mathrm{d}\xi \right\} \\
&\leq & \frac{C}{\eta} |z-w| \|f\|_{I_1(L^{2,1})}.\end{aligned}$$
Therefore $$| [a,B^{\eta}]f(z)-[a,B^{\eta}]f(w) | \le C |z-w| \|f\|_{I_1(L^{2,1})},\label{equi1}$$ where the constant $C$ depends on $a$ and $\eta$.
On the other hand, if $|z|>M>2R$ $$\begin{aligned}
|[a,B^{\eta}]f(z)| & = & \left|\int_{\mathbb{C}}(a(z)-a(w))\frac{\varphi_{\eta}(z-w)}{(z-w)^2}f(w)\mathrm{d}w\right|\\
& \leq & \|f\|_{\infty}\|a\|_{\infty}\int_{|w|<R}\frac{1}{|z-w|^2}\mathrm{d}w\\
& \leq & C\|f\||_{I_1(L^{2,1})}\|a\|_{\infty}\frac{1}{|z|^{2}},\end{aligned}$$ and then $$\|[a,B^{\eta}](f)\chi_{\mathbb C\setminus B(0,M)}\|_{L^{2,1}}\le C\|f\|_{I_1{L^{2,1}}}\|a\|_{\infty}\|\frac{1}{|z|^2}
\chi_{\mathbb C\setminus B(0,M)}\|_{L^{2,1}} ,\label{equi2}$$ which tends to $0$ as $M\rightarrow 0$. Combining and , by Fréchet-Kolgomorov Theorem for Lorentz spaces, one gets that $[a,B^{\eta}]$ is a compact operator from $I_{1}(L^{2,1}(\mathbb{C}))$ to $L^{2,1}(\mathbb{C})$ as we desired.
Calderón-Zygmund operators on domains {#CZdo}
======================================
In this section we will prove Theorem \[T3\]. Let $X(\Omega)$ denote any of function spaces in the statement of Theorem \[T3\] and let $f\in X(\Omega)$. It is clear from the Calderón-Zygmund theory that $T_{\Omega}f\in L^p(\Omega)$. So, in order to study the behaviour of $T_{\Omega}$ on $X(\Omega)$, we must deal with $T_{\Omega}f(x) -T_{\Omega}f(y)$ because we have a characterization of $X(\Omega)$ using first differences. Following [@Me] we consider the next decomposition.
\[lema:diferenciasT\]
Let $\psi \in \mathcal{C}^{\infty}_{c}$ such that $\psi(u)=1$ on $|u|\leq2$ and $\psi(u)=0$ if $|u|\geq 4$. Define $\eta(u)=1-\psi(u)$. Then: $$T_{\Omega}f(y)-T_{\Omega}f(x):=\sum_{i=1}^{4}g_{i}(x,y)+f(x)(T\chi_{\Omega}(y)-T\chi_{\Omega}(x)),\label{eq:difer}$$ where $$\begin{aligned}
g_{1}(x,y) & = & \int_{\Omega}(K(y-u)-K(x-u))(f(u)-f(x))\ \eta\left(\frac{u-x}{|y-x|}\right)\mathrm{d}u,\\
g_{2}(x,y) & = & -\int_{\Omega}K(x-u)(f(u)-f(x))\ \psi\left(\frac{u-x}{|y-x|}\right)\mathrm{d}u,\\
g_{3}(x,y) & = & \int_{\Omega}K(y-u)(f(u)-f(y))\ \psi\left(\frac{u-x}{|y-x|}\right)\mathrm{d}u,\\
g_{4}(x,y) & = & (f(y)-f(x))\int_{\Omega}K(y-u)\ \psi\left(\frac{u-x}{|y-x|}\right)\mathrm{d}u.\end{aligned}$$
Note that if $\tilde\psi(w) +\tilde\eta(w)=1$ we can write $$\begin{aligned}
T_{\Omega}f(x) & = & f(x)T_{\Omega}\tilde\psi(x)+\int_{\Omega}K(x- w)(f(w)-f(x))\ \tilde\psi(w)\mathrm{d}w\\
& & +\int_{\Omega} K(x- w)f(w)\tilde\eta(w)\mathrm{d}w,
\end{aligned}$$ and then $$\begin{aligned}
T_{\Omega}f(y)-T_{\Omega}f(x)& = & \int_{\Omega}(K(y-u)-K(x-u))(f(u)-f(x))\ \tilde\eta(u)\mathrm{d}u\\
& - & \int_{\Omega}K(x-u)(f(u)-f(x))\ \tilde\psi(u)\mathrm{d}u\\
& + & \int_{\Omega}K(y-u)(f(u)-f(y))\ \tilde\psi(u)\mathrm{d}u\\
& + & (f(y)-f(x))\int_{\Omega}K(y-u)\ \tilde\psi(u)\mathrm{d}u\\
& + & f(x)(T\chi_{\Omega}(y)-T\chi_{\Omega}(x)).
\end{aligned}$$ Given $x\ne y$, take $\tilde\psi(u) = \psi\left(\frac{u-x}{|y-x|}\right)$ and $\tilde\eta(u) = \eta\left(\frac{u-x}{|y-x|}\right)$ and that is what we wished to prove.
Let $B=B(x_{0},r)$ be the ball in $\mathbb R^n$ of center $x_{0}$ and radius $r$ and $\varphi_{B}$ denotes a smooth function supported in $B$ such that $\| \varphi_{B} \|_{\infty}\le 1$ and $\| \nabla \varphi_{B} \|_{\infty}\le r^{-1}$. To deal with the term $g_{4}$ we will use the next lemma, which is an application of the Main Lemma of [@MOV].
\[lema:acotacion\] Let $\Omega$ be a bounded domain of $\mathbb R^n$ with boundary of class $\mathcal{C}^{1,\beta}$, $\beta >0$, and let $T$ be an even smooth homogeneous Calderón-Zygmund operator. Then, there exists a constant $C=C(\Omega)$ such that $\|T_{\Omega}\varphi_{B}\|_{\infty}\leq C$.
Since the $\mathcal{C}^{0,\beta}$ norm of $\varphi_{B}$ is bounded by $1+ r^{-\beta}$, by the Main Lemma of [@MOV] we have $$\begin{array}{ll}
| T_{\Omega}\varphi_{B}(x) | \le C(1+ r^{-\beta})\, , & \text{for all $x\in\mathbb C$ and} \\[2mm]
| T_{\Omega}\varphi_{B}(x) -T_{\Omega}\varphi_{B}(y) | \le C r^{-\beta} |x-y|^{\beta}, & \forall x,y \in \Omega.
\end{array}$$
Associated to the domain $\Omega$ there is a $r_{0}>0$ satisfying . Then, if $3r\ge r_{0}$ one has $ | T_{\Omega}\varphi_{B}(x) | \le C(1+ (\frac{3}{r_{0}})^{\beta})$ for all $x\in \mathbb C$. If $3r <r_{0}$ we write $$\begin{aligned}
T_{\Omega}\varphi_{B}(x) & = & \int_{\Omega}K(x-y)\varphi_{B}(y)\mathrm{d}y = \int_{\Omega\cap 3B}K(x-y)\varphi_{B}(y)\mathrm{d}y \\
& = & \int_{\Omega\cap 3B}K(x-y)(\varphi_{B}(y)-\varphi_{B}(x))\mathrm{d}y+\varphi_{B}(x)\int_{\Omega\cap 3B}K(x-y)\mathrm{d}y\\[1mm]
& = & p(x)+ q(x)\end{aligned}$$ For $p(x)$ we have $$|p(x)|\le C\int_{\Omega\cap 3B}\frac{|\varphi_{B}(x)-\varphi_{B}(y)|}{|x-y|^{n}}\mathrm{d}y\le C\|\nabla\varphi_{B}\|_{\infty}\int_{3B}\frac{\mathrm{d}y}{|x-y|^{n-1}}\leq C.$$ If $x\notin B$, $q(x)=0$, and for $x\in B$ one can prove $$\left| \int_{\Omega\cap 3B}K(x-y)\mathrm{d}y\right| \le C(\Omega),$$ proceeding as in the proof of the Main Lemma of [@MOV p. 408-410]. Observe that for $x\in B$ the function $T_{\Omega} (\chi_{3B})$ has the same behaviour that $T_{\Omega} (1)=T(\chi_{\Omega})$
Let’s continuous with the proof of Theorem \[T3\]. In the case that $f\in B^{s}_{p,p}(\Omega)$, $0<s<1$, $n<sp<\infty$, we have to prove that $$\int_{\Omega}\int_{\Omega}\frac{|T_{\Omega}f(x)-T_{\Omega}f(y)|^{p}}{|x-y|^{n+s p}}\mathrm{d}x\mathrm{d}y<\infty.\label{eq:carcoroBesov}$$ By Lemma \[lema:diferenciasT\], $$T_{\Omega}f(y)-T_{\Omega}f(x)=\sum_{i=1}^{4}g_{i}(x,y)+f(x)(T\chi_{\Omega}(y)-T\chi_{\Omega}(x))$$ and we will study each term separately. Since $f$ is bounded (because $n<sp<\infty$) and $T\chi_{\Omega}\in B^{s}_{p,p}(\Omega)$ $$\int_{\Omega}\int_{\Omega} \frac{|f(x)(T\chi_{\Omega}(y)-T\chi_{\Omega}(x))|^{p}}{|x-y|^{n+s p}}\mathrm{d}x\mathrm{d}y
\le \| f\|_{\infty} \int_{\Omega}\int_{\Omega} \frac{|T\chi_{\Omega}(y)-T\chi_{\Omega}(x)|^{p}}{|x-y|^{n+s p}}\mathrm{d}x\mathrm{d}y<\infty.$$
Fix $t$ such that $s<t<1$. Using the properties of the kernel $K$ and the Hölder’s inequality ($ \frac{1}{p} +\frac{1}{q} =1$), we have $$\begin{aligned}
|g_{1}(x,y)| & \le & C \int_{\Omega\cap\{|u-x|>2|x-y|\}}|K(x-u)-K(y-u)||f(u)-f(x)|\mathrm{d}u\\
& \le & C \int_{\Omega\cap\{|u-x|>2|x-y|\}}\frac{|x-y|}{|x-u|^{n+1}}|f(u)-f(x)|\mathrm{d}u\\
& = & C |x-y|\int_{\Omega\cap\{|u-x|>2|x-y|\}}\frac{|f(u)-f(x)|}{|x-u|^{\frac{n}{p}+t}}\frac{1}{|x-u|^{\frac{n}{q}-t+1}}\mathrm{d}u\\
& \le & C|x-y|\left(\int_{\Omega\cap\{|u-x|>2|x-y|\}}\frac{|f(u)-f(x)|^{p}}{|x-u|^{n+t p}}\mathrm{d}u\right)^{\frac{1}{p}}\\
& &\qquad\qquad
\cdot\left(\int_{\{|u-x|>2|x-y|\}}\frac{\mathrm{d}u}{|x-u|^{n-t q+q}}\right)^{\frac{1}{q}}\\
& \le & C|x-y|^{t}\left(\int_{\Omega\cap\{|u-x|>2|x-y|\}}\frac{|f(u)-f(x)|^{p}}{|x-u|^{n+t p}}\mathrm{d}u\right)^{\frac{1}{p}}.
\end{aligned}$$ Thus $$\frac{|g_{1}(x,y)|^{p}}{|x-y|^{n+s p}}\le \frac{C}{|x-y|^{n+s p-t p}}\int_{\Omega\cap\{|u-x|>2|x-y|\}}\frac{|f(u)-f(x)|^{p}}{|x-u|^{n+t p}}\mathrm{d}u,$$ and then, by the Fubini’s theorem, $$\begin{aligned}
\int_{\Omega}\int_{\Omega}\frac{|g_{1}(x,y)|^{p}}{|x-y|^{n+s p}} &\mathrm{d}x\mathrm{d}y \le \\
\le & C\int_{\Omega}\int_{\Omega}\frac{1}{|x-y|^{n+s p-t p}}\int_{\Omega\cap\{|u-x|>2|x-y|\}}\frac{|f(u)-f(x)|^{p}}{|x-u|^{n+t p}}\mathrm{d}u\mathrm{d}x\mathrm{d}y\\
= & C\int_{\Omega}\int_{\Omega}\int_{\Omega\cap\{|u-x|>2|x-y|\}}\frac{|f(u)-f(x)|^{p}}{|x-u|^{n+p-s p}}\frac{1}{|x-y|^{n+s p-t p}}\mathrm{d}y\mathrm{d}u\mathrm{d}x\\
\le & C\int_{\Omega}\int_{\Omega}\frac{|f(u)-f(x)|^{p}}{|x-u|^{n+t p}}\frac{1}{|x-u|^{s p-t p}}\mathrm{d}u\mathrm{d}x\\
= & C\int_{\Omega}\int_{\Omega}\frac{|f(u)-f(x)|^{p}}{|x-u|^{n+s p}}\mathrm{d}u\mathrm{d}x<\infty.
\end{aligned}$$
Since the terms $g_{2}$ and $g_{3}$ are symmetric, we only consider one of them. Take $t$ such that $0<t<s$. As before, using the properties of the kernel $K$ and the Hölder’s inequality ($ \frac{1}{p} +\frac{1}{q} =1$), $$\begin{aligned}
|g_{2}(x,y)| & \le & C \int_{\Omega\cap\{|x-u|<4|x-y|\}}\frac{|f(u)-f(x)|}{|x-u|^{n}}\mathrm{d}u\\
& = & C \int_{\Omega\cap\{|x-u|<4|x-y|\}}\frac{|f(u)-f(x)|}{|x-u|^{\frac{n}{p}+t}}\frac{1}{|x-u|^{\frac{n}{q}-t}}\mathrm{d}u\\
& \le & C \left(\int_{\Omega\cap\{|x-u|<4|x-y|\}}\frac{|f(u)-f(x)|^{p}}{|x-u|^{n+t p}}\mathrm{d}u\right)^{\frac{1}{p}} \\
& & \qquad\qquad \cdot\left(\int_{\{|x-u|<4|x-y|\}}\frac{\mathrm{d}u}{|x-u|^{n-t q}}\right)^{\frac{1}{q}}\\
& \le & C |x-y|^{t}\left(\int_{\Omega\cap\{|x-u|<4|x-y|\}}\frac{|f(u)-f(x)|^{p}}{|x-u|^{n+t p}}\mathrm{d}u\right)^{\frac{1}{p}}.
\end{aligned}$$ Then $$\begin{aligned}
\frac{|g_{2}(x,y)|^{p}}{|x-y|^{n+s p}} & \le & \frac{C}{|x-y|^{n+s p- t p}}\int_{\Omega\cap\{|x-u|<4|x-y|\}}\frac{|f(u)-f(x)|^{p}}{|x-u|^{n+ t p}}\mathrm{d}u\end{aligned}$$ and therefore $$\begin{aligned}
\int_{\Omega}\int_{\Omega}\frac{|g_{2}(x,y)|^{p}}{|x-y|^{n+s p}}&\mathrm{d}x\mathrm{d}y\le \\
\le & C \int_{\Omega}\int_{\Omega}\int_{\Omega\cap\{|x-u|<4|x-y|\}}\frac{|f(u)-f(x)|^{p}}{|x-y|^{n+s p-t p}|x-u|^{n+t p}}\mathrm{d}y\mathrm{d}u\mathrm{d}x\\
\le & C \int_{\Omega}\int_{\Omega}\frac{|f(u)-f(x)|^{p}}{|x-u|^{n+s p}}\mathrm{d}u
\mathrm{d}x < \infty.
\end{aligned}$$
Finally, by Lemma \[lema:acotacion\] we have $$\left| \int_{\Omega}K(y-u)\ \psi\left(\frac{u-x}{|y-x|}\right)\mathrm{d}u\right| \leq C$$ and consequently $$\int_{\Omega}\int_{\Omega}\frac{|g_{4}(x,y)|^{p}}{|x-y|^{n+s p}}\mathrm{d}u\mathrm{d}x\le C \int_{\Omega}\int_{\Omega}\frac{|f(x)-f(y)|^{p}}{|x-y|^{n+s p}}\mathrm{d}x\mathrm{d}y< \infty .$$
Combining all these inequalities we get .\
Using the characterizations and one can see that the proofs for $f\in W^{s,p}(\Omega)$ or $f\in W^{1,p}(\Omega)$ are very similar to that we just explained for $f\in B^s_{p,p}(\Omega)$.\
**Remark:** If $\Omega$ is a bounded domain of $\mathbb R^n$ with boundary of class $\mathcal{C}^{1,\varepsilon}$, $\varepsilon >0$, and $T$ is an even smooth homogeneous Calderón-Zygmund operator we have (see [@MOV Main Lemma]) $$| T(\chi_{\Omega})( x) - T(\chi_{\Omega})(y) |\le C|x-y|^{\varepsilon}, \qquad \forall x,y\in\Omega.$$ Therefore $ T(\chi_{\Omega})$ belongs to $W^{s,p}(\Omega)$ and $B_{p,p}^{s}(\Omega)$ for any $s\in (0,\varepsilon)$.
Proof of the Theorem \[T2\]
===========================
Consider the Beurling transform restricted on the domain $\Omega$ of class $\mathcal{C}^{1,\varepsilon}$ $$B_{\Omega}g(z)= -\frac{1}{\pi}\int_{\Omega} \frac{g(w)}{(z-w)^{2}}\mathrm{d}w.$$ By [@MOV Main Lemma], $| B (\chi_{\Omega})(z) - B (\chi_{\Omega})(w) |\le C |z -w|^{\varepsilon}$ for all $z,w\in \Omega$. Now, applying Theorem \[T3\] we have that $B_{\Omega}$ is bounded on the spaces $B_{p,p}^{s}(\Omega)$ and $W^{s,p}(\Omega)$. Let’s denote by $X(\Omega)$ any of these two spaces. We will show that the Beltrami operator $I-\mu B_{\Omega}$ is invertible on $X(\Omega)$. Then, taking $h= (I-\mu B_{\Omega})^{-1}(\mu)$ we get the conclusions.
As in the proof of the Proposition \[Prop1\], we claim that $I-\mu B_{\Omega}$ is a Fredholm operator on $X(\Omega)$. Define $P_{m}=I+\mu B_{\Omega}+\cdots+(\mu B_{\Omega})^{m}$ so that $$(I-\mu B_{\Omega})P_{m-1}=P_{m-1}(I-\mu B_{\Omega})=I-\mu^{m}(B_{\Omega})^{m}+R, \label{eq:estrella}$$ where $R=\mu^{m}(B_{\Omega})^{m}-(\mu B_{\Omega})^{m}$ can be easily seen to be a finite sum of operators that contain the commutator $[\mu,B_{\Omega}]$ as a factor. We will prove that $[\mu,B_{\Omega}]\colon X(\Omega)\to X(\Omega)$ is a compact operator. On the other hand, for $z\in \Omega$ $$\begin{aligned}
(I-\mu^{m}(B_{\Omega})^{m})f(z) & =(I-\mu^{m}(B^{m})_{\Omega})f (z)+\mu^{m}(z)((B^{m})_{\Omega} f(z)-(B_{\Omega})^{m}f(z))\\
& =(I-\mu^{m}(B^{m})_{\Omega})f (z)+\mu^{m}(z)K_{m} f(z),\end{aligned}$$ where $B^{m}$ is the $m$-iterated Beurling transform and $K_{m}f := (B^{m})_{\Omega} f -(B_{\Omega})^{m}f $. As in the proof of Lemma \[Le1\], if $F\in X(\Omega)$ we get $$\|\mu^{m} F\|_{X(\Omega)}\leq C\: m \|\mu\|_{\infty}^{m-1} \|\mu\|_{X(\Omega)}
\| F\|_{X(\Omega)}.
\label{d11}$$ Remind that the kernel of $B^m$ is $\dfrac{(-1)^m m \bar z^{m-1} }{\pi z^{m+1} }$ and then, by Theorem \[T3\], if $f\in X(\Omega)$ we have $$\| (B^{m})_{\Omega} f\|_{X(\Omega)} \leq C\: m^{2}\| f \|_{X(\Omega)}.
\label{d12}$$ Consequently, combining and , $$\|\mu^{m}(B^{m})_{\Omega} f\|_{X(\Omega)}\leq C\: m^{3} \|\mu\|_{\infty}^{m-1} \|\mu\|_{X(\Omega)} \| f\|_{X(\Omega)},$$ which implies that $I-\mu^{m}(B^{m})_{\Omega}$ is invertible if $m$ is large. Assume for a moment that the operators $K_{m}$ are compacts on $X(\Omega)$. Thus, $I-\mu B_{\Omega}$ is a Fredholm operator and in addition has index zero. Since $X(\Omega) \subset L^p(\Omega)$ we also have that $I-\mu B_{\Omega}$ is injective (see [@I1]) and therefore invertible on $X(\Omega)$.
The compactness of the operators $[\mu,B_{\Omega}]$ and $K_{m}$ on $X(\Omega)$ follows arguments parallels. Since $X(\Omega)$ is an algebra and the Beurling transform $B_{\Omega}$ is bounded on $X(\Omega)$ we have $$\|[\mu,B_{\Omega}]f\|_{X} = \| \mu B_{\Omega}f -B_{\Omega}(\mu f) \|_{X} \leq C\|\mu\|_{X}\|f\|_{X}.$$ Moreover, because the domain $\Omega$ is Lipschitz, there exists a sequence of functions $\mu_{j}\in C^{\infty}(\overline{\Omega})$ such that $\mu_{j}$ converges to $\mu$ in $X(\Omega)$. So, we have reduced to prove the compactness when $\mu\in C^{\infty}(\overline{\Omega})$. In this case, the kernel of the commutator $$[ \mu,B_{\Omega}]f (z) = -\frac{1}{\pi}\int_{\Omega}\frac{\mu(z)-\mu(w)}{(z-w)^{2}}f(w) \mathrm{d}w
:= \int_{\Omega} k(z,w) f(w) \mathrm{d}w$$ clearly satisfies $$\begin{aligned}
|k(z,w)| & \le & \frac{C}{|z-w|}\quad\text{for all } z,w\in\Omega,\label{eq:Pnuc1}\\
|k(z',w)-k(z,w)| & \le & C \frac{|z-z'|}{|z-w|^{2}}\quad\text{if \ensuremath{|z-w|>2|z-z'|}.}\label{eq:Pnuc2}\end{aligned}$$ Then, a simple computation gives (see [@MOV p. 419]), for $ z_1\,,\,z_2 \in \Omega$, $$\label{eq18}
|[\mu,B_{\Omega}]f(z_1)- [\mu,B_{\Omega}]f (z_2)| \le C\,|z_{1}-z_{2}| \,(1+\log
\frac{d}{|z_{1}-z_{2}|})\,\|f\|_\infty\,,$$ where $d$ denotes the diameter of $\Omega$. From one immediately gets that $[ \mu,B_{\Omega}]f$ belongs to $B_{p,p}^{\beta}(\Omega)$ and to $W^{\beta, p}(\Omega)$ for any $\beta <1$. The compact embedding $W^{\beta, p}(\Omega) \hookrightarrow W^{s, p}(\Omega)$, $s<\beta$, (and $B_{p,p}^{\beta}(\Omega) \hookrightarrow B_{p,p}^{s}(\Omega)$) gives the compactness for the commutator (e.g. [@Tri2 Proposition 7]).\
We have $K_{m}f = (B^{m})_{\Omega} f -(B_{\Omega})^{m}f $. To prove that $K_{m}$ is compact on $X(\Omega)$ we will proceed by induction. For $m\geq2$, $$\begin{aligned}
(B_{\Omega})^{m}f & = & B_{\Omega}((B_{\Omega})^{m-1}f) = B([(B_{\Omega})^{m-1}f] \chi_{\Omega})\\
& = & B([ B^{m-1}(f \chi_{\Omega}) - K_{m-1}f] \chi_{\Omega})\\
& = & B( B^{m-1}(f \chi_{\Omega}) - (B^{m-1}(f \chi_{\Omega}))\chi_{\Omega^{c}} - (K_{m-1}f )\chi_{\Omega})\\
& = & B^{m}(f \chi_{\Omega}) - B(\chi_{\Omega^{c}} B^{m-1}(f \chi_{\Omega}) ) - B_{\Omega}(K_{m-1}f)
\end{aligned}$$ It is then enough to prove that, for $m\geq1$, the operator $$Q_{m} f := B( (B^{m} (f\chi_{\Omega})) \chi_{\Omega^{c}}))$$ is compact in $X(\Omega)$. For $z\in \Omega$, we write $$\begin{aligned}
Q_{m}f (z) & = & B( (B^{m} (f \chi_{\Omega})) \chi_{\Omega^{c}} ) (z) \\ [2mm]
& = & -\frac{1}{\pi}\int_{\Omega^{c}}\frac{B^{m}(f \chi_{\Omega})(w)}{(z-w)^{2}}\mathrm{d}w\\
& = & - \frac{1}{\pi} \int_{\Omega^c} \frac{1}{(z-w)^2} \frac{(-1)^m m}{\pi}\,\int_\Omega
\frac{(\overline{w-\xi})^{m-1}}{(w-\xi)^{m+1}}\,f(\xi)\, \mathrm{d} \xi
\,\mathrm{d} w \\
& = & \int_{\Omega}K_{m}(z,\xi)f(\xi)\mathrm{d}\xi,\end{aligned}$$ where $$K_{m}(z,\xi):= \frac{(-1)^{m+1}}{\pi^{2}} \int_{\Omega^{c}}\frac{1}{(z-w)^{2}}\frac{m\overline{(w-\xi})^{m-1}}{(w-\xi)^{m+1}}\mathrm{d}w .$$ In [@MOV p. 418–419], it is proved that if $\Omega$ is a bounded domain of class $\mathcal{C}^{1,\varepsilon}$ and $f\in L^{\infty}(\Omega)$ then $$\begin{aligned}
| Q_{m}f (z)| & \le & C d^{\varepsilon} \| f\|_{\infty}\, , \quad z\in \Omega \, , \\
| Q_{m}f(z_1)- Q_{m}f (z_2)|& \le & C\,|z_{1}-z_{2}|^{\varepsilon} \,(1+\log
\frac{d}{|z_{1}-z_{2}|})\,\|f\|_{\infty}\, , \quad z_{1}, z_{2}\in \Omega \, ,\end{aligned}$$ where $d$ denotes the diameter of $\Omega$ and $C$ depends on $m$ and $\Omega$.
Consequently, if $f\in X(\Omega)$ then $Q_{m}f$ belongs to $B_{p,p}^{\beta}(\Omega)$ and to $W^{\beta, p}(\Omega)$ for any $\beta <\varepsilon$. Choose $\beta$ such that $s<\beta< \varepsilon$. Again, the compact embeddings $W^{\beta, p}(\Omega) \hookrightarrow W^{s, p}(\Omega)$ and $B_{p,p}^{\beta}(\Omega) \hookrightarrow B_{p,p}^{s}(\Omega)$ give the compactness of $Q_{m}$.
The authors were partially supported by grants 2009SGR420 (Generalitat de Catalunya) and MTM2010-15657 (Ministerio de Ciencia e Innovación, Spain).
[MOV]{}
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[l]{} Victor Cruz\
Instituto de Física y Matemáticas\
Universidad Tecnológica de la Mixteca\
69000 Huajuapan, Oaxaca, México\
[*E-mail:*]{} [victorcruz@mixteco.utm.mx]{}\
\
Joan Mateu\
Departament de Matemàtiques\
Universitat Autònoma de Barcelona\
08193 Bellaterra, Barcelona, Catalonia\
[*E-mail:*]{} [mateu@mat.uab.cat]{}\
\
Joan Orobitg\
Departament de Matemàtiques\
Universitat Autònoma de Barcelona\
08193 Bellaterra, Barcelona, Catalonia\
[*E-mail:*]{} [orobitg@mat.uab.cat]{}\
\
[^1]: [*Key words*]{}: quasiregular mappings, Beltrami equation, Sobolev spaces.
|
---
abstract: 'Asteroseismology is a promising tool to study Galactic structure and evolution because it can probe the ages of stars. Earlier attempts comparing seismic data from the [*Kepler*]{} satellite with predictions from Galaxy models found that the models predicted more low-mass stars compared to the observed distribution of masses. It was unclear if the mismatch was due to inaccuracies in the Galactic models, or the unknown aspects of the selection function of the stars. Using new data from the K2 mission, which has a well-defined selection function, we find that an old metal-poor thick disc, as used in previous Galactic models, is incompatible with the asteroseismic information. We show that spectroscopic measurements of \[Fe/H\] and \[$\alpha$/Fe\] elemental abundances from the GALAH survey indicate a mean metallicity of $\log (Z/Z_{\odot})=-0.16$ for the thick disc. Here $Z$ is the effective solar-scaled metallicity, which is a function of \[Fe/H\] and \[$\alpha$/Fe\]. With the revised disc metallicities, for the first time, the theoretically predicted distribution of seismic masses show excellent agreement with the observed distribution of masses. This provides an indirect verification of the asteroseismic mass scaling relation is good to within five percent. Using an importance-sampling framework that takes the selection function into account, we fit a population synthesis model of the Galaxy to the observed seismic and spectroscopic data. Assuming the asteroseismic scaling relations are correct, we estimate the mean age of the thick disc to be about 10 Gyr, in agreement with the traditional idea of an old $\alpha$-enhanced thick disc.'
author:
- Sanjib Sharma
- Dennis Stello
- 'Joss Bland-Hawthorn'
- 'Michael R. Hayden'
- 'Joel C. Zinn'
- Thomas Kallinger
- Marc Hon
- Martin Asplund
- Sven Buder
- 'Gayandhi M. De Silva'
- 'Valentina D’Orazi'
- Ken Freeman
- Janez Kos
- 'Geraint F. Lewis'
- Jane Lin
- Karin Lind
- Sarah Martell
- 'Jeffrey D. Simpson'
- 'Rob A. Wittenmyer'
- 'Daniel B. Zucker'
- Tomaz Zwitter
- 'Timothy R. Bedding'
- Boquan Chen
- Klemen Cotar
- James Esdaile
- Jonathan Horner
- Daniel Huber
- 'Prajwal R. Kafle'
- Shourya Khanna
- Tanda Li
- 'Yuan-Sen Ting'
- 'David M. Nataf'
- Thomas Nordlander
- Hafiz Saddon
- Gregor Traven
- Duncan Wright
- 'Rosemary F. G. Wyse'
bibliography:
- 'main.bib'
title: 'The K2-HERMES Survey: Age and Metallicity of the Thick Disc'
---
Introduction
============
In recent years, asteroseismology has emerged as a powerful tool to study Galactic structure and evolution However, previous attempts based on data from the original [*Kepler*]{} mission [@2010Sci...327..977B; @2013ApJ...765L..41S], which was designed for detecting transiting planets, have struggled to match the predictions of stellar-population-synthesis Galactic models to observations, with the models producing too many low mass stars [@2016ApJ...822...15S; @2017ApJ...835..163S]. There are three possible causes for this mismatch: (i) inaccuracies in the selection function of stars in the observational catalog, (ii) an incorrect Galactic model, and (iii) systematics in the scaling relations used to relate asteroseismic observables ($\Delta \nu, \nu_{\rm max}$) to density and surface gravity of the stars. Using data from the K2 Galactic Archaeology Program [K2GAP, @2015ApJ...809L...3S; @2017ApJ...835...83S], which is a program to observe oscillating giants with the ’second-life’ [*Kepler*]{} mission [K2, @2014PASP..126..398H] following a well-defined selection function, illuminates the first cause. In this paper we therefore, first use the K2GAP data to test if predictions of the Galactic models that are constrained independently of the asteroseismic data match the observed asteroseismic data. This provides an indirect way to test the scaling relations. Having shown that the scaling relations are fairly accurate, next, we make use of them and the asteroseismic data to fit some of the parameters in the Galactic model, and discuss the implications for our understanding of the Galaxy. Unlike [*Kepler*]{}, however, the seismic detection completeness of K2 is not 100%. This is because the time span of K2 light curves (typically 80 days) is much shorter than that of [*Kepler*]{} (typically more than a year). Hence, we carefully study the detection completeness in K2, and devise ways to take them into account when comparing observations to models.
In a Galactic model, the mass distributions of giants is sensitive to the age and the metallicity of the stellar populations in the model. While the role of age was investigated in @2016ApJ...822...15S, the possibility of an inaccurate prescription of metallicity being responsible for the mismatch between observed and predicted mass distributions has not been investigated so far. Many studies have attempted to characterize the metallicity distribution of the thin and thick discs. For the thin disc, there is a well defined radial metallicity gradient but the age-metallicity relation is almost flat [@Bensby2014; @2016MNRAS.455..987C; @2017ApJS..232....2X; @2018MNRAS.475.5487S]. For the thick disc, there is a lack of consensus regarding its properties. The Besançon model adopted a mean metallicity value of \[Fe/H\]=-0.78 based upon spectroscopic measurements by @1995AJ....109.1095G (mean thick disc \[Fe/H\] $\sim -0.6$) and photometric (U, B, V bands) measurements by . The [*Galaxia*]{} model [@2011ApJ...730....3S] also adopted the same prescription for metallicity distribution of Galactic components as the Besançon model. However, at least four separate studies have compared predictions of the Besançon Galaxy model with that of spectroscopic observations and find that away from the midplane and in regions where the thick disc dominates, the metallicity distribution of the model is inconsistent with observations and that a shift of the thick disc metallicity from -0.78 to about -0.48 is required to make the model agree with observations. Specifically, concluded the thick disc metallicity to be \[Fe/H\]$=-0.48 \pm 0.05$ by spectroscopically studying about 400 red clump stars in the direction of the North Galactic Pole at a height of $200 < z/{\rm pc} < 800$. Their spectra covered 390-680 nm at $R \sim 42000$. reached their conclusions by studying a sample of about 700 F, G, and K dwarfs at a height $1 < z/{\rm kpc} < 4$ using the GIRAFFE spectrograph (820.6-940 nm at $R\sim 6500$). They suggested overall metallicity \[M/H\] $\sim -0.48$ for the thick disc. Note, the \[M/H\] of is probably close to \[Fe/H\] but the exact relationship is not known. and used [*Galaxia*]{} to compare predictions of the Besançon model with stars from RAVE (841-879.5 nm at $R\sim 7500$) and found that for $|z|> 800$ pc, a better match to observations is obtained if \[Fe/H\] of the thick disc is set to $-0.5$. The former study makes use of dwarfs while the latter uses giants. More recently, results of @2015ApJ...808..132H using giants from the APOGEE survey (1.51-1.70 $\mu$m at $R\sim 22500$) also suggest a higher metallicity (\[Fe/H$]=-0.36$) for the thick disc, by considering stars between $1 < z/{\rm kpc} < 2$ and Galactocentric radius of $5 < R/{\rm kpc} < 7$ to be thick disc stars. The TRILEGAL Galactic model uses an effective metallicity of Z=0.008 for the thick disc, which implicitly takes the $\alpha$ enhancement into account. This translates to \[Fe/H\] $\sim -0.5$ (assuming $Z_{\odot}=0.0152$ and \[$\alpha$/Fe\]=0.24), which compared to the Besançon model is more in line with recent spectroscopic measurements but is still lower than the APOGEE measurements.
We now have a large sample of stars with very precise metallicity measurements from spectroscopic surveys like the Galactic Archaeology for HERMES [GALAH @2015MNRAS.449.2604D] , K2-HERMES [a GALAH-like survey dedicated to K2 follow-up, @2018AJ....155...84W] and Apache Point Observatory Galactic Evolution Experiment [APOGEE @2017AJ....154...94M] surveys. In this paper, we use data from the GALAH survey to determine the metallicity distribution of the stellar populations in the Besançon-based Galactic model that we later use for asteroseismic analysis. Observationally, it is difficult to measure the metallicity of the stellar populations like the thin and the thick discs that are used in Galactic models. This is because the thick and thin discs overlap considerably such that it is difficult to identify individual stars belonging to each of the discs. Hence, we adopt a forward modeling approach where we fit a Galactic model to the observed data and try to answer the following question. What is the metallicity of the thick and thin discs that best describes the spectroscopic data from GALAH? Next, we use data from the APOGEE survey to verify our best fit model. Unlike spectroscopic studies of [*Kepler*]{} seismic targets, the selection function of the K2-HERMES stars is the same as for the K2 seismic targets. We take advantage of this fact to then directly check if the metallicity distribution of the asterosesimic data in K2, whose mass distributions we wish to compare with Galactic models, is in agreement with the models.
Finally, unlike the original [*Kepler*]{} survey, which was confined to one direction of the sky, the K2 targets span along the ecliptic allowing us to test our Galactic models in various regions of the Galaxy. Of particular importance is the ability of K2 to investigate the thick disc of the Milky Way. The thick disc is one of the most intriguing components of the Galaxy and its origin is not well understood. Compared to the thin disc, it is old, alpha-enhanced, metal poor, has higher velocity dispersion, and has a larger scale height. A complication with the thin disc vs. thick disc nomenclature, is that the scale-length for the thick disc is shorter than for the thin disc [@2012ApJ...753..148B; @2017ApJS..232....2X; @2017MNRAS.471.3057M]; the thick disc truncates near the solar circle where the thin disc dominates and beyond the solar circle the thin disc flares with increasing Galactic radius [see discussion and Fig. 1 in @blandhawthorn2018]. Although numerous spectroscopic surveys have targeted thick disc stars, a characterization of the thick disc using asteroseismic data has not been carried out. It was not possible to do using the [*Kepler*]{} data because its field of view was close to the Galactic plane. To move beyond this limitation, a number of K2GAP campaigns were selected at high Galactic latitudes, which means that a significant fraction of stars in the K2GAP are expected to be thick disc stars. Here, we use the K2 data to answer the following question. What is the thick disc age that best describes the seismic masses and spectroscopic data?
The paper is structured as follows. In Section \[data\], we describe the asteroseismic and spectroscopic data used for the study. In Section \[methods\], we discuss the methods that we use. Here we describe the selection function of the sample and discuss how we take it into account when forward modeling the simulated Galactic data. In Section \[results\], we present our results where we compare model predictions with observations and also tune the metallicity and the age distributions in our model to fit the data. In Section \[conclusions\] we discuss and conclude our findings.
Data
====
Target selection
----------------
{width="80.00000%"}
{width="95.00000%"}
---------- ---------- ---------- ----------- ---------------------- ------------------ ------------------------ ----------------------------------------------- --
Campaign Proposed Observed Following N$_{\rm giants}$ N$_{\rm giants}$ N$_{\rm giants}$ Selection function
selection with $\nu_{\rm max}$ with $\Delta\nu$ $\nu_{\rm max}$ +spec.
1 9108 8630 8598 1104 583 455 $((J-K_s)>0.5)\&(7 < H < 12.927)$
6 8371 8311 8301 1951 1452 504 $((J-K_s)>0.5)\&(9 < V_{JK} < 15.0)$
4 17410 6357 4937 1839 945 702 $((J-K_s)>0.5)\&(9 < V_{JK} < 13.447)$
7 8698 4361 4085 1541 1041 930 $(J-K_s)>0.5)\&$
($(9 < V_{JK} < 14.5)\&(c \in \{6,17\})$ OR
($(14.276 < V_{JK} < 14.5)\&(c \in \{14\})))$
---------- ---------- ---------- ----------- ---------------------- ------------------ ------------------------ ----------------------------------------------- --
\[tab:tb2\]
------- -------------- -------------- ---------------------------
fFlag K2GAP K2-HERMES Description
criterion criterion
Qflag $\leq$ ’BBB’ $=$ ’AAA’ J,H,K photometric quality
Bflag $=$ ’111’ $=$ ’111’ blend flag
Cflag $=$ ’000’ $=$ ’000’ contamination flag
Xflag $=$ 0 $=$ 0
Aflag $=$ 0 $=$ 0
prox $>$ 6 arcsec $>$ 6 arcsec distance to nearest star
------- -------------- -------------- ---------------------------
: 2MASS quality selection criteria
\[tab:tb3\]
The stars in this study were observed by K2 as part of the K2GAP Guest Observer program [@2015ApJ...809L...3S; @2017ApJ...835...83S]. The stars that we use span four K2 campaigns $-$ C1, C4, C6, and C7 $-$ whose sky distributions are shown in . These K2 campaigns cover different regions of the Galaxy and sample a wide variety of Galactic stellar populations including old, young, thin disc, thick disc, inner disc, and outer disc. C1 and C6 are at high galactic latitudes and hence are likely to have more old thin disc and thick disc stars owing to the larger scale height of such stars. C4 and C7 are at lower latitudes and are likely to be dominated by young thin disc stars. C4 is towards the Galactic anti-center and samples the outer disc, whereas C7 is towards the Galactic center and samples the inner disc.
The stars follow a simple color magnitude selection based on the 2MASS photometry, which is given in . The following equation was used to convert 2MASS magnitude to an approximate $V$ band magnitude [@2018MNRAS.473.2004S]. $$V_{JK}=K + 2.0 (J-K_s+0.14)+0.382\exp[(J-K-0.2)/0.50]$$ Stars having good quality photometry from 2MASS were used; the exact criterion is shown in . Also listed are the criteria for the spectroscopic sample from the K2-HERMES survey, which for the ’Qflag’ is slightly stricter than for the K2GAP targets. Only a subset of the K2GAP stars observed by K2 were observed by the K2-HERMES survey the number of which is also listed in . The K2-HERMES survey observes K2GAP stars that have magnitudes in range $10<V_{JK}<15.0$ and that lie in 1 degree radius circular zones. There are 19 such K2-HERMES zones for each K2 campaign and their layout for campaign C7 is shown in d. In , $c$ denotes the pointing identifier of these 1 degree radius zones.
In the K2GAP survey, the proposed stars were ranked in priority by $V$ magnitude. For the dense field of C7, they were additionally restricted to only three circular zones, to make the spectroscopic follow-up more efficient. During the final K2 mission-level target selection process for each campaign, the K2GAP target list was truncated at an arbitrary point ($V$-magnitude) based on target allocation. Hence those selected stars will follow the K2GAP selection function. However, targets from other successful Guest Observer programs that overlap with lower ranked (fainter) K2GAP targets could still end up being observed. These stars would not satisfy the K2GAP selection function. It is straightforward to locate the truncation point from the lists of proposed and observed targets by plotting the fraction of proposed to observed stars as a function of row number. A sharp fall in this ratio identifies the location of the truncation point. The K2GAP-proposed stars, the K2GAP-observed stars, and the K2GAP-observed stars following the K2GAP selection function are listed in .
Asteroseismic data
------------------
Our primary asteroseimic sample comes from stars observed by K2. The K2 time-series photometry is sampled roughly every 30 minutes, and span about 80 days per campaign. This allows us to measure the seismic signal in giants brighter than [*Kepler*]{} magnitude, $Kp$, of $\sim15$ in the range $10 \lesssim \nu_{\rm max}/\mu{\rm Hz} \lesssim 270$ ($1.9 \lesssim \log g \lesssim 3.2$), with a slight detection bias against the faint high $\log g$ stars due to their higher noise levels and lower oscillation amplitudes [@2017ApJ...835...83S]. We adopt the sesimic results from @2017ApJ...835...83S (C1) and Zinn et al. (2019,in prep) (C4, C6, and C7). Throughout the paper we focus on stars with $10 < \nu_{\rm max}/\mu{\rm Hz}<270$. A detailed description of the seismic analysis is given in @2017ApJ...835...83S and reference therein. In summary, automated analysis pipelines perform the measurements of the two seismic quantities used here: the frequency of maximum oscillation power, $\nu_{\rm max}$, and the frequency separation between overtone oscillation modes, $\Delta \nu$. Here, we use the results of the two pipelines called BAM (Zinn et al. 2019 in prep) and CAN , which are both based on Bayesian MCMC schemes for accessing whether oscillations are detected in a given dataset, and to obtain statistically robust uncertainties on each measurement. Typically, only 50-70% of stars for which oscillation are detected (meaning $\nu_{\rm max}$ is determined) do the pipelines also measure a robust $\Delta \nu$ [@2017ApJ...835...83S Zinn et al. 2019 accepted].
In this paper, in addition to comparing the predictions of our new Galactic model against results from K2, we also compare against the results from the [*Kepler*]{} mission. For this we use the catalog of oscillating giants by @2013ApJ...765L..41S, in which the global seismic parameters were estimated using the @2009CoAst.160...74H pipeline (SYD). The exact selection function of oscillating giants in [*Kepler*]{} is not known. However, an approximate formula 3.731R\_<R\_[KIC]{}<, was derived by @2016ApJ...822...15S and we use this to sub-select targets from the above catalog. Here, $R_{\rm KIC}$ is the photometry-based stellar radius as given in the [*Kepler*]{} input catalog of @2011AJ....142..112B, \_[LC]{}=(1/c\_[Kepler]{}) \[equ:jenkins\] is the long cadence (LC) noise to signal ratio, and $c_{\rm Kepler}=3.46\times10^{0.4(12- Kp)+8}$ is the number of detected $e^{-1}$ per LC sample [@2010ApJ...713L.120J]. For comparing the [*Kepler*]{} results with Galactic models, the synthetic $g$ band SDSS photometry was corrected using Equation 4 from [@2016ApJ...822...15S] and then $R_{\rm KIC}$ was estimated from synthetic photometery using the procedure outlined in @2011AJ....142..112B.
Spectroscopic data
------------------
The spectroscopic data come from the K2-HERMES (for seismic K2 targets) and the GALAH surveys (non seismic targets) being conducted at the 3.9-m AAT located at Siding Spring observatory in Australia. The spectra were collected using the multi object High Efficiency and Resolution Multi-Element Spectrograph (HERMES) spectrograph [@2015JATIS...1c5002S]. The K2-HERMES survey uses the same instrument setup as the GALAH survey [@2017MNRAS.465.3203M] and the TESS-HERMES survey [@2018MNRAS.473.2004S]. The reduction is done using a custom IRAF based pipeline [@2017MNRAS.464.1259K]. The spectroscopic analysis is done using the GALAH pipeline and is described in [@2018MNRAS.478.4513B]. It uses Spectroscopy Made Easy (SME) to first build a training set by means of a model driven scheme . Next, [*The Cannon*]{} [@2015ApJ...808...16N] is used to estimate the stellar parameters and abundances by means of a data driven scheme.
Methods
=======
Galactic models
---------------
In this paper we perform two kinds of analysis, one is to compare the predictions of theory with observations and the other is to fit Galactic models to the observed data. For this, we use population synthesis based Galactic models. The models consist of four different Galactic components, the thin disc, the thick disc, the bulge, and the stellar halo. The full distribution of stars in space, age, and metallicity $Z$, is given by p(R,z,Z,|)=\_k p(k) p(R,z,Z,|,k), with $k$ denoting a Galactic component, $\theta$ the parameters governing the Galactic model, $R$ the Galactocentric cylindrical radius, $z$ the Galactic height, $\tau$ the age, and $Z$ the metallicity of the stars in the Galactic component. To sample data from a prescribed population synthesis model we use the [*Galaxia*]{}[^1] code [@2011ApJ...730....3S]. It uses a Galactic model that is initially based on the [*Besançon*]{} model by but with some crucial modifications. The density laws and the initial mass functions for the various components are given in Table 1 of @2011ApJ...730....3S and are same as in . The density normalizations for various components are given in , these differ slightly from and a discussion of the changes is given in Section 3.6 of @2011ApJ...730....3S. The thin disc spans an age range of 0 to 10 Gyr and has a star formation rate which is almost constant. The thin disc has a scale height which increases with age according to Equation 18 in @2011ApJ...730....3S. In this paper, we leave the thick disc normalization as a free parameter and solve for it using data from Gaia DR2.
Other differences between [*Galaxia*]{} and the [*Besançon*]{} model are as follows. [*Galaxia*]{} is a robust statistical sampler, it provides continuous sampling over any arbitrary volume of the Galaxy. This enables rigorous comparisons with observed stellar surveys for an arbitrary selection function. [*Galaxia*]{} has a 3D extinction scheme that is based on @1998ApJ...500..525S dust maps. We also apply a low-latitude correction to the dust maps as described in @2014ApJ...793...51S. The isochrones used to predict the stellar properties are from the Padova database using CMD 3.0 (<http://stev.oapd.inaf.it/cmd>), with PARSEC-v1.2S isochrones [@2012MNRAS.427..127B; @2014MNRAS.445.4287T; @2014MNRAS.444.2525C; @2015MNRAS.452.1068C], the NBC version of bolometric corrections [@2014MNRAS.444.2525C], and assuming Reimers mass loss with efficiency $\eta=0.2$ for RGB stars. The isochrones are computed for scaled-solar composition following the $Y=0.2485+1.78Z$ relation and their solar metal content is $Z_{\odot}=0.0152$.
Model Thick Thin
------- ----------------------------- ------- -------------------------------------
MP $\langle{\rm [M/H]}\rangle$ -0.78 \[0.01, 0.03, 0.03, 0.01,
-0.07, -0.14, -0.37\]
$\sigma_{\rm [M/H]}$ 0.33 \[0.12, 0.12, 0.10, 0.11, 0.18,
0.17, 0.2\]
Min(Age) 11 \[0, 0.15, 1, 2, 3, 5, 7\]
Max(Age) 11 \[0.15, 1, 2, 3, 5, 7, 10\]
MR $\langle{\rm [M/H]}\rangle$ -0.16 \[0.01, 0.03, 0.03, 0.01, 0, 0, 0\]
$\sigma_{\rm [M/H]}$ 0.17 same as MP
Min(Age) 9 same as MP
Max(Age) 11 same as MP
FL $\langle{\rm [M/H]}\rangle$ -0.14 0.0
$\sigma_{\rm [M/H]}$ 0.30 0.3
Min(Age) 6 same as MP
Max(Age) 13 same as MP
: Galactic models with different age and metallicity distribution functions.
\[tab:gmodels\]\
\
The details of the different Galactic models that we use in this paper are given in . The base [*Galaxia*]{} model denoted by MP (metal poor) is from @2011ApJ...730....3S, it has an old metal poor thick disc and a thin disc whose mean metallicity decreases with age as in . The model denoted by MR (metal rich) has metal rich thick and thin discs. The FL (flat) model also has a metal rich thick and thin disc, but unlike other models its thick disc spans an age range from 6 to 13 Gyr with a uniform star formation rate and no variation of metallicity with age. For each Galactic component $k$, the IMF, the formula for spatial distribution of stars, and the density normalizations are given in @2011ApJ...730....3S.
Galactic Component Normalization $\alpha_1$ $\alpha_2$
----------------------------- ---------------------------------------------- ------------ ------------
Thin ($0<{\rm Age/Gyr}<7$) 2.37 ${\rm M_{\odot} yr^{-1}}$ -1.6 -3.0
Thin ($7<{\rm Age/Gyr}<10$) 1.896 ${\rm M_{\odot} yr^{-1}}$ -1.6 -3.0
Thick $\rho_{\rm \odot,thick}$ -0.5 -0.5
Stellar Halo $10.252\times 10^{3} {\rm M_{\odot}pc^{-3}}$ -0.5 -0.5
Bulge 13.76 ${\rm stars\ pc^{-3}}$ -2.35 -2.35
: The IMFs and the density normalizations of Galactic components. The parameters $\alpha_1$ and t$\alpha_2$ are used to specify the IMF (number density of stars as a function of mass stellar mass $M$), which is of the following form, $\propto M^{\alpha_1}$ for $M/{\rm M}_{\odot}<1$ and $\propto M^{\alpha_2}$ for $M/{\rm M}_{\odot} >1$.
\[tab:gmodels1\]\
\
\
To compare predictions of Galactic models to asteroseismic data, we need to estimate the observed seismic quantities $\nu_{\rm max}$ and $\Delta \nu$ for the synthetic stars. The seismic quantities are estimated from effective temperature $T_{\rm eff}$, surface gravity $g$, and density $\rho$ using the following asteroseismic scaling relations . =()\^[-0.5]{} \[equ:scaling\_numax\]\
=f\_()\^[0.5]{} \[equ:scaling\_dnu\] Here, f\_=()()\^[-0.5]{} is the correction factor derived by @2016ApJ...822...15S by analyzing theoretical oscillation frequencies with GYRE [@2013MNRAS.435.3406T] for stellar models generated with MESA [@2011ApJS..192....3P; @2013ApJS..208....4P]. We used the code ASFGRID[^2] [@2016ApJ...822...15S] that computes the correction factor as a function of metallicity $Z$, initial mass $M$, evolutionary state $E_{\rm state}$ (pre or post helium ignition), $T_{\rm eff}$, and $g$.
Importance-sampling framework {#sec:impsamp}
-----------------------------
To constrain the parameters of a Galactic model from the observed data we developed and used an importance-sampling framework, which we now describe. Suppose we have collected some data regarding some variable $x$, such as metallicity Z or seismic mass, subject to some selection function $S$. Then suppose that we have a Galactic model parameterized by $\theta$ from which we can draw samples subject to the same selection function $S$. To constrain the model, we start with a base model parameterized by some $\theta_0$, then to change the model to one parameterized by a new $\theta$, we simply reweight the samples from the simulation parametrized by $\theta_0$ instead of drawing from a new simulation. When the model changes from $\theta_0$ to $\theta$, the new weights for a star $i$ belonging to a Galactic component $k$ are given by w\_i=p( R\_i,z\_i,Z\_i,\_i|,k)/p(R\_i,z\_i,Z\_i,\_i|\_0,k). In general, such a change can alter the number of visible stars of your synthetic Galaxy, but as long as the parameters governing the density distribution of the stars are unaltered, the changes are minimal. In this paper, we are mainly concerned with only altering the thick disc parameters like mean age and metallicity. We also alter the metallicity of the old thin disc, but this change is minor and can be ignored for the present discussion related to the number of visible stars. The base model that we use is based on the Besançon model, which was constructed by to satisfy the observed star counts in the Galaxy. When the thick disc parameters, like mean age and/or the metallicity are modified, we adopt the following procedure to address the slight change that is expected in the number of visible thick disc stars. We measure $f_{\rm SGP}$, the ratio of stars that lie between $2<|z|/{\rm kpc}<3$ out of all stars that are in a $30^{\circ}$ radius cone around the south Galactic pole ($b=-90.0^{\circ}$) and have Gaia magnitudes $0<G<14$ from Gaia DR2 . Using $f_{\rm SGP}$ estimated from Gaia DR2, we solve for the normalization factor $\rho_{\rm \odot, thick}$ and reweight the thick disc of the model such that $f_{\rm SGP}$ in the selection-function-matched mock sample matches with that of the Gaia DR2 data. Following this global normalization, the stars are further reweighted to satisfy the color magnitude selection function of the observational data to which the model is being fitted.
To fit the model to the data we need to compute the likelihood of the data given the model and this is done as follows. Let $x_q$ be the $q_{\rm th}$ percentile of the distribution of some variable $x$. For this variable, suppose we have observed samples $X_{o}$ and samples from some model $X_m$, with the model being parameterized by $\theta$ and $S$ being the selection function. The probability of the observed data given the model can then be written as p(X\_o|,S)=\_q (-), [ with ]{} \[equ:likelihood\]\
\_x\^2=(\_[x,[o]{}]{}\^2/n\_[eff,o]{}+\_[x,[m]{}]{}\^2/n\_[eff,m]{}) Here, $n_{\rm eff}$ is the effective number of stars, which for stars with different weights is given by $\left(\sum w_i\right)^2/\sum w_i^2$ according to Kish’s formula. We make use of 16, 50 and 84 percentiles to compute the likelihood of the data given the model. The $x_{o,q}$ and $x_{m,q}$ denote the $q_{\rm th}$ percentile obtained from samples $X_o$ and $X_m$ respectively. For multiple data sets, $X = \{X_{o}^{1}, ... , X_{o}^{ns}\}$ with each of them having their own selection function $S = \{S_1, ... , S_{ns}\}$, the full likelihood is p(X|,S) = \_i p(X\_[o]{}\^[i]{}|,S\_i).
In this paper, the importance-sampling framework is used for estimating the metallicity of the thick disc using spectroscopic data from the GALAH survey and to estimate the age of the thick disc from the asteroseismic data from K2. For the former (metallicity estimation), we bin up the stars lying in $5<R/{\rm kpc}<11$ and $1<|z|/{\rm kpc}<3$ using bin sizes of 0.5 kpc in $R$ and 0.33 kpc in $|z|$. We use $\log Z/Z_{\odot}$ as the observed variable $x$ and fit for the mean and the dispersion of the thick disc metallicity and the mean metallicity of the old thin disc (age greater than 3 Gyr) in the Galactic model. For this we use the MR model from . For the latter (age estimation), we bin up the stars into different K2 campaigns and 3 different giant classes. We follow @2016ApJ...822...15S by using the temperature-independent seismic mass proxy \[equ:scaling\_m1\] \_[M]{} & = & ()\^[3]{}()\^[-4]{}\
as the variable $x$ and fit for the age (mean) and metallicity (mean and dispersion) of the thick disc. For this we use the FL model from . For each selection of stars the likelihood is computed using . The $\kappa_M$ is closely related to the stellar mass $M$, which is given by \[equ:scaling\_m\] & = & \_[M]{} ()\^[1.5]{}. Given that temperatures are not always readily available for the observed stars, we use $\kappa_M$ instead of mass when comparing theoretical predictions to observations. This also removes any ambiguity in temperature scale differences between the models and the data. For simplicity we will in the following refer to $\kappa_M$ as mass.
Detection completeness
----------------------
Before we can compare the mass distributions, we have to make sure that the observed and simulated data satisfy the same selection function. In other words, we have to properly forward-model the simulated data and make it satisfy the same observational constraints that the observed data satisfies.
The duration of the K2 campaigns sets a lower limit on the detectable $\nu_{\rm max}$ of about $10 \mu$Hz below which the seismic detection efficientcy drops. The observational cadence sets an upper limit of about $\nu_{\rm max}=270 \mu$Hz [@2015ApJ...809L...3S]. The amplitude of oscillations decreases with increasing $\nu_{\rm max}$ (less luminous stars) and the photometric noise increases towards fainter stars. This makes it harder to detect oscillations for stars that have higher $\nu_{\rm max}$ and/or are faint. This bias is clearly visible as missing stars in the top right corner of (a,c,e,g), which shows the distribution of observed stars in the $(\nu_{\rm max},V_{JK})$ plane.
![Distribution of observed (left panels) stars in the $(\nu_{\rm max},V_{JK})$ plane for four K2 campaigns and the [*Kepler*]{} field. The right panels plot the ratio of observed to predicted oscillating giants in each bin. The predictions are based on simulations using [*Galaxia*]{}. The dashed line represents the equation $\nu_{\rm max}=-60(V_{JK}-17)$. The upper right region (above the dashed line) indicates where we cannot detect oscillations due to too low signal-to-noise. \[fig:vmag\_numax\]](can_c1467_vmag_lognumax.pdf){width="48.00000%"}
To model the seismic detection probability we followed the scheme presented by @2011ApJ...732...54C and @2016ApJ...830..138C. For this, we used the mass, radius, and effective temperature of each synthetic star to predict its total mean oscillation power and granulation noise in the power spectrum. The oscillation amplitude was estimated as $A=2.5\left(L/{\rm L}_{\odot}\right)^{0.9}\left(M/{\rm M}_{\odot}\right)^{-1.7}\left(T_{\rm eff}/{\rm T_{eff,\odot}}\right)^{-2}$ following @2011ApJ...737L..10S. The granulation power was estimated using the model. The apparent magnitude was used to compute the instrumental photon-limited noise in the power spectrum, which combined with granulation noise gave the total noise. For the instrumental noise we use [formula given by @2010ApJ...713L.120J]. For K2, we scaled the noise by a factor of three to take into account the higher noise in the K2 data compared to the [*Kepler*]{} data and also applied a minimum threshold of 80 ppm. The mean oscillation power and the total noise were then used to derive the probability of detecting oscillations, $p_{\rm detect}$, with less than 1% possibility of false alarm. Stars with $p_{\rm detect}> 0.9$ were assumed to be detectable.
![The probability distribution of $\nu_{\rm max}$ for observed and predicted oscillating giants. The dashed line, $\nu_{\rm max}=30.5\ \mu$Hz, shows the approximate location of the peak in the distribution of the predicted stars. The peak corresponds to the location of the red clump giants. The location of the peak is not sensitive to the choice of the Galactic model, but the distribution is sharper for the MR model. The peak for the CAN pipeline is systematically lower as compared to the BAM pipeline. For C4 the location of the peak for both the CAN and the BAM pipelines is higher as compared to predictions. \[fig:numax2\_dist\]](canbam_c1467_dist_numax2.pdf){width="48.00000%"}
The results of applying the detection probability on the [*Galaxia*]{} simulated stars are shown in (b,d,f,h) as the ratio between the number of observed to predicted stars. The $\nu_{\rm max}$ was estimated using . The figure shows that the fraction of predicted to observed stars is close to one over most of the regions where we have observed stars. However, C4, C6, and C7 show a slight tendency of having a lower than predicted number of stars towards the top right corner of each panel (higher $V_{JK}$ and higher $\nu_{\rm max}$), where the signal-to-noise ratio of the oscillations is low. This is probably because for these campaigns, the detections are based solely on automated pipelines, with no additional visual inspection as for C1 [@2017ApJ...835...83S]. The mean detection fraction is 0.72, and the cause for fewer detections is not clear. Using the deep-learning-based pipeline of @2018MNRAS.476.3233H resulted in slightly more $\nu_{\rm max}$ detections, raising the mean detection fraction to 0.78, but the fraction still remained significantly less than one.
The distribution of $\nu_{\rm max}$ and apparent magnitude
----------------------------------------------------------
We now check the distribution of apparent magnitudes and $\nu_{\rm max}$ in more detail. In , the $\nu_{\rm max}$ distributions show a peak, which corresponds to the red clump stars. For the simulated data (orange line) the peak is close to $30.5 \mu$Hz. The location of the peak varies very little across different campaigns, it is about 1 $\mu$Hz higher for the low latitude campaigns C4 and [*Kepler*]{}. For the observed data analyzed with the BAM pipeline, except for C4, the location of the peak does not show any obvious shift with respect to the predicted peak. For C4 the BAM peak is about 3 $\mu$Hz higher. For all campaigns, the location of the peak for the CAN pipeline is systematically lower by 2 $\mu$Hz compared to the BAM pipeline. This suggests that the CAN pipeline systematically underestimates $\nu_{\rm max}$ for stars around the red clump region compared to the scaling relation prediction. The peak for the [*Kepler*]{} data obtained using the SYD pipeline also did not show any shift with respect to the predicted peak. To conclude, we see systematic differences between campaigns and between pipelines, they are small but could be important for certain applications and hence should be investigated further in future.
![ Magnitude distribution of observed oscillating giants from K2 along with predictions from [*Galaxia*]{} corresponding to model MP and MR. The number of stars with $\nu_{\rm max}$ detections in the observed sample and those predicted by model MP, and MR are also listed in each panel.[]{data-label="fig:vmag_dist"}](can_c1467_dist_vmag.pdf){width="48.00000%"}
The corresponding distributions of $V_{JK}$ for K2 are shown in . Overall the observed distributions match well with the model predictions. For C6, the model predicts more stars for $V_{JK}>13.5$, the cause of which is not yet clear, but we found that this has no impact on our conclusions related to the mass distribution of stars that we present in this paper.
Classifying giants into different classes {#sec:giant_class}
-----------------------------------------
In the seismic analysis, $\nu_{\rm max}$ is easier to detect as compared to $\Delta \nu$. Hence, there are stars with a $\nu_{\rm max}$ measurement but no $\Delta \nu$ measurement and this needs to be taken into account when comparing model predictions with observations. To accomplish this, we first study the $\Delta \nu$-detection completeness of our sample and then devise ways to account for it when comparing model predictions with observations.
The probability, $p_{\Delta \nu}$, of having a $\Delta \nu$ measurement given that we have a measurement of $\nu_{\rm max}$ is shown in , as a function of $\nu_{\rm max}$. This was derived by binning the stars in $\nu_{\rm max}$ and then computing in each bin the ratio of the number of stars with a $\Delta \nu$ measurement ($N_{\Delta \nu}$) to those with a $\nu_{\rm max}$ measurement ($N_{\nu_{\rm max}}$). We see three distinct phases. The first is for $\nu_{\rm max}<25 \mu$Hz, where $p_{\Delta \nu}$ is constant but low. The second is for $25<\nu_{\rm max}<50 \mu$Hz, where $p_{\Delta \nu}$ increases with $\nu_{\rm max}$. And the third is for $\nu_{\rm max}>50 \mu$Hz, where $p_{\Delta \nu}$ is again constant and close to 1 (except for C7 where $p_{\Delta \nu}$ is lower for $\nu_{\rm max}>100 \mu$Hz). The drop in $p_{\Delta \nu}$ as $\nu_{\rm max}$ decreases from 50 to 30 $\mu$Hz, coincides with the increase in fraction of red-clump stars as predicted by a [*Galaxia*]{} simulation (see orange dots in ). This drop could be because the power spectra of red-clump stars are more complex than RGB stars and this makes the $\Delta \nu$ measurement harder to obtain. For $\nu_{\rm max}< 25 \mu$Hz, we mainly have RGB stars, but the $p_{\Delta \nu}$ is still low, and this could be due to the limited frequency resolution of the K2 data starting to affect our ability to obtain a clear $\Delta \nu$ measurement towards the low $\nu_{\rm max}$ stars.
![The ratio of the number of stars with and without $\Delta \nu$ measurements for various K2 campaigns. The black dot marks the frequency of the peak, $\nu_{\rm max,RC}$, in the $\nu_{\rm max}$ distribution of the observed stars and is due to RC stars. The ratio shows a sharp increase for stars with $\nu_{\rm max}>\nu_{\rm max,RC}$. \[fig:numax1\_ratio\_dnu\]](can_c1467_dist_numax1_ratio_dnu.pdf){width="48.00000%"}
Although all four campaigns show similar $p_{\Delta \nu}$ for high $\nu_{\rm max}$ stars, we note that for $\nu_{\rm max}<30 \mu{\rm Hz}$, $p_{\Delta \nu}$ is about a factor of two lower for C1 and C4 compared to C6 and C7. The cause for this different behavior is not clear.
We have seen in that $\Delta \nu$ detections are incomplete with a completeness that depends on stellar type (evolution stage). This suggests that we should study the different types of giants separately. Below we describe a scheme to segregate stars into three giant classes, the high luminosity RGB stars, the RC stars, and the low luminosity RGB stars. The segregation is done in the $(\kappa_M, \nu_{\rm max})$ plane. By construction the high luminosity RGB class will have some contamination from AGB stars and the RC class will have contamination from RGB stars.
![ Distribution of stars in the $(\kappa_M, \nu_{\rm max})$ plane. The blue lines split the plane into three distinct regions, the predominantly high-luminous RGB stars (left), the predominantly red-clump stars (middle) and the low-luminosity RGB stars (right). Left panels (a,c,e,g) show results from K2 based on the CAN pipeline. Right panels (b,d,f,h) show predictions from [*Galaxia*]{}. The overplotted orange points denote the red clump stars. \[fig:numax\_kappa\_m\]](can_c1467_numax_kappa_m.png){width="48.00000%"}
shows the distribution of stars in the $(\kappa_M, \nu_{\rm max})$ plane both for the observed and ${\sl Galaxia}$-simulated data. Although RC stars typically have $\nu_{\rm max} \sim 30 \mu{\rm Hz}$, shows that the high $\kappa_M$ stars can have $\nu_{\rm max}$ reaching up to $\sim 100 \mu{\rm Hz}$. This is the main reason why we decided not to isolate RGB stars solely from their $\nu_{\rm max}$. Based on simulations by ${\sl Galaxia}$ we instead fit and obtain two curves \_[max]{}\^[lower]{} & = & 6.8478 \_[M]{}\^2 -14.489 \_[M]{}+26.914\
\_[max]{}\^[upper]{} & = & 33.598 \_[M]{}\^2 -73.523 \_[M]{} +72.647 that enclose about 92% of the RC stars (blue lines). In , it can be seen that the red clump stars are nicely enclosed by the blue lines.
These curves are then used to classify stars into the three categories; a) $\nu_{\rm max}<\nu_{\rm max}^{\rm lower}$ (high-luminosity RGB stars or hRGB), b) $\nu_{\rm max}^{\rm lower}<\nu_{\rm max}<\nu_{\rm max}^{\rm upper}$) (RC stars), and c) $\nu_{\rm max}>\nu_{\rm max}^{\rm upper}$ (low-luminosity RGB stars or lRGB). Based on [*Galalxia*]{} simulations, the fraction of RGB stars in the three categories averaged across all campaigns was found to be a) 0.87, b) 0.18, and c) 0.97, suggesting that each category is dominated by the desired stellar type in that category, i.e., RGB, RC and RGB respectively.
Results
=======
Constraints from spectroscopic surveys
--------------------------------------
Large scale surveys of the Milky Way were not available at the time the [*Besançon*]{} model was constructed as implemented in [*Galaxia*]{}. The situation has changed now, with surveys like APOGEE and GALAH providing high-resolution spectra for hundreds of thousands of stars, which sample the Galaxy well beyond the solar neighborhood. Hence it is possible to characterize the the thick disc better than before.
To study the elemental composition of the thick disc we need to identify stars belonging to the thick disc. This can be done using height above the Galactic plane or rotational velocity. We choose the former approach as the overlap of the thin and thick disc is quite strong in rotational velocity. To isolate thick disc stars, we select stars with $(5<R/{\rm kpc}<7)$ and $1<|z|/{\rm kpc}<2$. This region provides the largest number of thick disc stars with the least amount of contamination from the thin disc, as can be seen in Figure 4 from @2015ApJ...808..132H.
![Distribution of GALAH giants in the (\[Fe/H\],\[$\alpha$/Fe\]) plane. Giants were selected using $\log g < 3.5$. (a) Distribution of all giants. (b) Giants restricted to $5<R/{\rm kpc}<7$ and $1<|z|/{\rm kpc}<2$. The color bar shows probability density which is normalized such that the maximum density is 1. \[fig:alpha\_feh\_dist\]](alpha_feh_dist.pdf){width="\columnwidth"}
In , we further illustrate this using data from the GALAH survey [@2018arXiv180406041B]. Three populations are visible in a; the stellar halo at \[Fe/H\]$\sim -1.75$, the thin disc at \[Fe/H\]$\sim 0$, and the thick disc at \[Fe/H\]$\sim -0.39$. After selecting stars by location, the thin disc sequence almost vanishes and the halo can be identified as a separate over-density in b.
------------ -------- ------ ------- ------ -------- ------
Source
med sdev med sdev med sdev
APOGEE -0.294 0.28 0.186 0.08 -0.160 0.24
GALAH DR2 -0.367 0.24 0.218 0.08 -0.196 0.21
GALAH DR2c -0.316 0.21 0.239 0.07 -0.131 0.18
GALAH mock -0.170 0.25
GALAH DR2c -0.162 0.17
------------ -------- ------ ------- ------ -------- ------
: Abundance of iron and alpha elements for thick disc stars. Median and standard deviation based on 16-th and and 84-th percentile values are listed. The first four rows show the abundances for stars with $5<R/{\rm kpc}<7$ and $1<|z|/{\rm kpc}<2$ and positive rotation about the Galaxy. The last row shows the result obtained by fitting a Galactic model.
\[tab:al\_feh\]\
\
and positive rotation.
For this particular spatial selection, the median and the spread of the distribution of \[Fe/H\] and \[$\alpha$/Fe\] are listed in and compared with that of APOGEE. Also given are metallicity estimates \[M/H\] constructed using the formula $$\begin{aligned}
{\rm [M/H]} & = &
\log\left(\frac{Z}{Z_{\odot}}\right) \nonumber \\
& = & {\rm [Fe/H]}
+ \log(10^{[\alpha/{\rm Fe}]} 0.694+ 0.306).\end{aligned}$$ by @2005essp.book.....S. Given an isochrone grid constructed for metallicities $Z$ using solar-scaled composition with a specified $Z_{\odot}$, the above formula provides an approximate estimate of metallicity $Z$ or \[M/H\] for a given \[Fe/H\] and \[$\alpha$/Fe\]. In , although we choose to show the median, the mean values were also very similar with the difference between the two being less than 0.01 dex (after discarding stars with \[Fe/H\] $<-1.25$, which most likely belong to the stellar halo).
![Comparison of GALAH-DR2 Cannon-based (data-driven) estimates to that of SME-based (model-driven) estimates. The plots shows systematic trends as a function of Cannon-based iron abundance \[Fe/H\]. The giants (blue) and dwarfs (orange) are shown separately. The giants shown are seismic giants from K2, and for them SME was run using $\nu_{\rm max}$ estimated from asteroseismology as a prior. The seismic giants show strong systematic trends while dwarfs have negligible systematics. The dotted line is a two degree polynomial fit to the trends for the seismic giants with $-1.5<{\rm [Fe/H]}<0.3$. \[fig:galah\_vs\_seism\]](galah_vs_seism2.pdf){width="48.00000%"}
![Comparison of GALAH DR2 stellar parameters with APOGEE stellar parameters. Results corresponding to both uncalibrated and calibrated GALAH DR2 data are shown. \[fig:galah\_vs\_apogee\]](galahdr2_vs_apogee.pdf){width="48.00000%"}
$y$ $c_2$ $c_1$ $c_0$
--------------------- ------------- ------------- -------------
$\log g$ +3.4987e-01 +7.4591e-01 +1.5727e-01
$T_{\rm eff}$ K +1.5658e+02 +2.1861e+02 -3.9895e+00
${\rm [Fe/H]}$ +1.9087e-01 +2.7875e-01 +2.4761e-02
$[\alpha/{\rm Fe}]$ -2.5775e-02 -4.1510e-02 -3.2592e-02
: Polynomial coefficients of calibration equation $y_{\rm calib}=y+c_0+c_{1}{\rm [Fe/H]}+c_{2}{\rm [Fe/H]}^2$ to correct for systematics in the Cannon based estimates against the SME based estimates. The equation was derived using giants having $\nu_{\rm max}$ estimates from asteroseismology and with $-1.5<{\rm [Fe/H]}<0.3$. The calibration is applied to giants with ${[\rm Fe/H]}>-1.5$, the giants are identified using the @2011AJ....141..108C definition.
\[tab:cannon\_sme\_poly\]
{width="100.00000%"}
In , for GALAH two different estimates are given, the first is based on the GALAH DR2 pipeline, the second named GALAH DR2c is based on a calibration correction that we derive and apply to the GALAH DR2 estimates. GALAH DR2 estimates are based on [*The Cannon*]{} method [@2015ApJ...808...16N], which was trained on results from the SME pipeline . However, as shown in , for giants we find subtle systematics in the GALAH DR2 stellar parameters compared to that of SME estimates, where $\nu_{\rm max}$ estimated from asteroseismology was used as a prior. The systematics are particularly significant for stars with \[Fe/H\]> 0.0. We use the seismic giants in the SME training set to recalibrate the GALAH results. The coefficients of the calibration equation are given in . A comparison of GALAH stellar parameters (both calibrated and uncalibrated) with those from APOGEE for common stars is shown in . The calibrated GALAH gravities and temperatures match better with APOGEE. In the range $-0.5<{\rm [Fe/H]}<0.0$, where the majority of the sample is found, the calibrated \[Fe/H\] also matches better with APOGGE. Outside this range some systematics exist. The \[$\alpha$/Fe\] shows slight offsets in zero points but no significant trend is seen.
We see from that the estimates for the mean metallicity of stars above the midplane from APOGEE, the GALAH DR2, and the GALAH DR2c agree to within 0.07 dex. The lowest values are for GALAH DR2c and the highest are for GALAH DR2. We now investigate if the metallicity of stars in $5<R/{\rm kpc}<11$ and $1<|z|/{\rm kpc}<2$ is really representative of the thick disc metallicity. We show in estimates from a mock [*Galaxia*]{} sample matched to the GALAH survey, with a thick disc having a mean \[M/H\] metallicity of -0.18, for stars lying in the same spatial selection. The estimated metallicity is higher by only 0.01 dex compared to the metallicity of the thick disc that was used in the model. This suggests that the metallicity of stars with $5<R/{\rm kpc}<11$ and $1<|z|/{\rm kpc}<2$ is indeed close to the actual metallicity of the thick disc but is probably higher by 0.01 dex. Note, GALAH and APOGEE are magnitude limited surveys, so their samples are not volume complete and this can bias the estimates of the mean metallicity.
We now measure the metallicity of the thick disc more accurately by taking the selection function into account. For this, we fitted a Galactic model to the GALAH DR2c data lying within $5<R{\rm kpc}<11$, $0.75<|z|/{\rm kpc}<3$, $12<V_{JK}<14$, and with $\texttt{field\_id}< 6546$, and having positive rotation, using our importance-sampling framework ().
{width="95.00000%"}
{width="95.00000%"}
The fitting procedure gave a mean metallicity of $-0.16$ with a spread of 0.17 for the thick disc and a mean metallicity of 0.0 for the oldest three thin disc subpopulations. While fitting the model, the spread of the thin disc metallicity with age, and the metallicity of the youngest four thin disc subpopulations was left unchanged as in the Besançon model . We also assumed that the three oldest thin disc subpopulations (age 3 to 10 Gyr) have the same mean metallicity. We make these assumptions because we do not make use of the age information in our fitting, and without ages it is difficult to constrain the age metallcity relation. In our fitting, the limited ability to constrain the age metallcity relation comes from the fact that the vertical height of a star is correlated with age. Additionally, the age of the thick disc was assumed to span from 9 to 11 Gyr. We also checked with an age span of 10 to 12 Gyr for the thick disc and found that it gives the same best fit parameters. The slight decrease in the mean metallicity of the thick disc compared to the estimate based on simply measuring the metallicity of stars within $5<R/{\rm kpc}<11$ and $1<|z|/{\rm kpc}<2$, is due to the inclusion of stars lying between $2<|z|/{\rm kpc}<3$ in the fitting process. This suggests that there is a small vertical gradient in the metallicity of the thick disk as found by others previously . Note, stars in $2<|z|/{\rm kpc}<3$ were not included in the former scheme as they could be contaminated with stars from the metal-poor stellar halo and could suffer from volume incompleteness. However, they are included in the later scheme because it takes both these effects into account.
In we show the mean metallicity as function of Galactocentric radius $R$ for different slices in height $|z|$. Results from APOGEE and GALAH DR2c are shown separately. We also plot [*Galaxia*]{} predictions from the MP and MR models. To eliminate stars belonging to the halo we restrict the analysis to stars having positive rotation about the Galaxy. The MP model clearly has a thick disc, which is too metal poor to fit the observed data for $|z|>1$ kpc. The new MR model is a very good fit to the GALAH data. It also fits the APOGEE data very well, except for the slice closest to the midplane and the slice furthest from the midplane. Compared to GALAH, stars in APOGEE are metal rich close to the midplane, but they progressively become metal poor with increasing height above the midplane. These differences could be due to systematics in abundances between the two surveys, but could also be due to the different selection function of the surveys. The effect of selection function is clearly visible in f corresponding to the top most slice ($2<|z|/{\rm kpc}<3$). Here, the MR model has a thick disc with a mean metallicity of -0.16. However, the mean metallicity of stars in this slice is below -0.2. This is because metal poor stars are more luminous and are visible furthest in a magnitude limited survey. shows that close to the midplane there is a strong radial metallicity gradient. As we move away from the midplane the gradient diminishes progressively to zero. Close to the plane, a radial gradient of -0.07 (dashed line), as used in the Besançon (MP) and MR models, is roughly consistent with the observed data.
We now study the distribution of metallicity for oscillating giants detected by K2 for which we also have HERMES spectra. In we show results separately for different campaigns (C1, C6, C4, C7, and [*Kepler*]{} (Kep)) and different seismic classes (hRGB, RC, lRGB). Predictions from [*Galaxia*]{}-MP (orange) and the new [*Galaxia*]{}-MR (green) are shown alongside the observed data (blue). The [*Galaxia*]{}-MP samples have many more metal poor stars with \[M/H\]$<-0.5$ than the observed stars. In some panels a double peaked distribution can also be seen with one of the peaks being at -0.78, corresponding to the metallicity of the thick disk in the model. The new model [*Galaxia*]{}-MR samples, which has a metal rich thick disc ($\langle {\rm [M/H]} \rangle \sim -0.16$), does not show a bimodal behavior and its distribution matches very well with observations. However, slight mismatches can be seen for low latitudes fields. For RC and lRGB stars in C4 and the [*Kepler*]{} field the [*Galaxia*]{}-MR samples are still too metal poor. For hRGB in C7 the [*Galaxia*]{}-MR samples are too metal rich.
Constraints from asteroseismology
---------------------------------
In this section we present results making use of the asteroseismic data. We first compare the observed distribution of seismic masses against the predictions of fiducial Galactic models. Next, we restrict our analysis to thick disc stars and assuming reasonable priors on the thick disc parameters, we demonstrate that the asteroseismic scaling relations are fairly accurate. Finally, assuming the scaling relations to be correct we estimate the age of the thick disc.
### Comparing observed distribution of seismic masses against predictions from Galactic models
In , we study the distribution of $\kappa_M$. The order of the panels is the same as in . For hRGB stars, the overall sample size is too small to assess the quality of how well the models match the data. Both models seem to perform equally well. However, for the large hRGB [*Kepler*]{} sample we do see that the new model provides a visibly better match. Now turning to the RC stars, we see across the board that the new MR model performs better than the old MP model, which predicts too many stars with $\kappa_M< 1$. Finally, for lRGB stars the MR model is significantly better than the MP model, which predicts too many stars with $\kappa_M< 1.25$.
Having seen the qualitative trends, we now move on to do a quantitative comparison of the observed distributions with predictions from [*Galaxia*]{}. Specifically, we want to answer the following questions: a) does the MR model match the K2 data better than the MP model does, b) and does the MR model also provide a better match to the [*Kepler*]{} data, which had issues with the selection function.
-------------- ------------------- ------------------- ------------------- ------------------- ------------------- -------------------
Campaign [*Galaxia*]{}(MP) [*Galaxia*]{}(MR) [*Galaxia*]{}(MP) [*Galaxia*]{}(MR) [*Galaxia*]{}(MP) [*Galaxia*]{}(MR)
1 $1.23 \pm 0.05$ $1.01 \pm 0.04$ $1.15 \pm 0.03$ $1.05 \pm 0.03$ $1.242 \pm 0.009$ $0.992 \pm 0.007$
6 $1.07 \pm 0.03$ $0.87 \pm 0.02$ $1.11 \pm 0.02$ $0.97 \pm 0.01$ $1.287 \pm 0.007$ $1.002 \pm 0.005$
4 $1.07 \pm 0.04$ $0.98 \pm 0.04$ $1.11 \pm 0.02$ $1.01 \pm 0.01$ $1.18 \pm 0.01$ $1.027 \pm 0.009$
7 $1.05 \pm 0.03$ $0.83 \pm 0.03$ $1.07 \pm 0.02$ $0.92 \pm 0.01$ $1.3 \pm 0.01$ $1 \pm 0.01$
[*Kepler*]{} $1.1 \pm 0.01$ $0.96 \pm 0.01$ $1.021 \pm 0.003$ $1.009 \pm 0.003$ $1.086 \pm 0.003$ $1.037 \pm 0.002$
-------------- ------------------- ------------------- ------------------- ------------------- ------------------- -------------------
\[tab:kappa\_m\]
The $\kappa_M$ distributions are in general unimodal. At the most basic level a unimodal distribution over a finite domain can be characterized by a median. We first estimate the medians and then compute the ratio of medians between the observed and predicted distributions, which we show in . Ideally, we expect the ratio to be close to one, but in previous work based on [*Kepler*]{} data, we found the median ratio to be larger than one (1.06).
The new MR model, anchored on GALAH metilicities of the thick disc, is undoubtedly better than the old MP model. For almost all giant classes and campaigns, the median ratio for the new MR model is closer to unity than for the old MP model. The only two exceptions are hRGB for C6 and C7, where the ratio is about 0.85, i.e., the model overpredicts the masses. However, these samples suffer from low number statistics. Additionally for the hRGB stars in C7, we also noticed that the MR model overpredicts the metallicity , and this will lead to overestimation of masses in the MR model.
### Testing the accuracy of the asteroseismic mass scaling relation
![The distribution of $\kappa_M$ for lRGB stars in K2 campaigns C1 and C6 that lie between $1<|z|/{\rm kpc}<3$. Shown alongside are mass distributions corresponding to simple stellar populations with a Gaussian metallicity distribution and a uniform age distribution (with a width of 2 Gyr). The mean metallicity and the mean age of each stellar population is given in the legend. \[fig:can\_c16\_kappa\_m\]](can_c16_dist_kappa_m_1z2.pdf){width="48.00000%"}
![The posterior distribution of $f_{M}$ and mean age of the thick disc $\tau$ obtained using lRGB stars in K2 campaigns C1 and C6 that lie between $1<|z|/{\rm kpc}<3$. The width $\Delta \tau$ of the age distribution was assumed to be 2 Gyr. \[fig:likelihood\_2\]](likelihood_2.pdf){width="48.00000%"}
The fact that the mass distribution of the new model MR matches the observed seismic masses so well, suggests that the asteroseismic scaling relations are fairly accurate. In the following we will explore this more quantitatively by limiting the analysis to a single Galactic component and imposing reasonable non-seismic priors on its parameters. To do this, we study the mass distribution of stars lying between $1<|z|/{\rm kpc}<3$. The Galactic model predicts that about 90% of these stars should be thick disc stars, so we can model them as a simple stellar population characterized by some age distribution and metallicity distribution. We have already shown that the metallicity distribution of this population can be represented by $\mathcal{N}(-0.16,0.17^2)$. In the following we present several pieces of evidence suggesting that the mean age of this high $|z|$ population should be between 8 to 12 Gyr. Firstly, shows that stars between $1<|z|/{\rm kpc}<2$ are enhanced in $\alpha$ element abundances and form a distinct sequence in the abundance space. Using dwarf and subgiants in the solar neighborhood, it has been shown that the stars in the $\alpha$-enhanced sequence are typically older than 10 Gyr (@Bensby2014 Figure 22 and Figure 3). Secondly, chemical evolution models predict that $\alpha$-enhanced stars must have formed within the first 1 Gyr of the star formation history of the Milky Way, or else the contribution from Type-Ia supernovae would have introduced too much iron and hence brought the value of \[$\alpha$/Fe\] down [@pagel_book_2009]. When the above fact is combined with Figure 3 from , which suggest that the oldest thin disc stars (stars not enhanced in \[$\alpha$/Fe\]) are around 8 to 10 Gyr old, we reach the conclusion that the $\alpha$-enhanced population must be older than 8 to 10 Gyr. Finally, @2017ApJ...837..162K provide one of the most precise and accurate estimates on the mean age of the thick disc using nearby white dwarfs. They estimate the mean thick disc age to be between 9.5 to 9.9 Gyr, with a random uncertainty of about 0.2 Gyr. Hence, based on these observational evidence, a reasonable prior for the mean age of the thick disc is 8 to 12 Gyr.
To test the asteroseismic mass scaling relation we select the lRGB stars in K2 campaigns C1 and C6 that lie between $1<|z|/{\rm kpc}<3$. We avoid campaigns C4 and C7 because they point into the Galactic plane and hence lack high $|z|$ stars. We restrict our test to lRGB stars because for these stars there is almost 100% probability both to detect $\nu_{\rm max}$ and to detect $\Delta \nu$ when a $\nu_{\rm max}$ has been measured. The distribution of $\kappa_M$ for the lRGB stars is shown in . The distributions of $\kappa_M$ for a stellar population with a metallicity distribution of $\mathcal{N}(-0.16,0.17^2)$ and a mean age of $\tau=10$ Gyr is also shown alongside, showing a good match to the observed distribution. However, the distribution for the stellar population with $\tau=7$ Gyr but the same metallicity distribution as before, is shifted too far to the right. Now, to quantify the accuracy of the asteroseismic mass scaling relation (Eq. \[equ:scaling\_m\]), we introduce a factor $f_{M}$, that is multiplied to $\kappa_M$ for stars in the model to get a ‘corrected’ mass, and then we investigate how close to unity this correction factor is when enforcing that the observed and model mass distributions match.
The posterior distribution of $f_{M}$ and the age, $\tau$, conditional on our data $D$ is given in . For the mean age of the high $|z|$ population we assume a flat prior in the range 8 to 12 Gyr. The analysis was done using the importance sampling framework discussed in and taking the photometric selection function into account. The figure shows that $f_{M}$ depends upon $\tau$ and varies between 0.97 and 1.05 for the adopted range of $\tau$. This would translate into a maximum deviation of the [$\nu_{\rm{max}}$]{} scaling relation (Eq. \[equ:scaling\_numax\]) of 1-2% if the [$\Delta\nu$]{} scaling relation (Eq. \[equ:scaling\_dnu\]) is true. Or alternatively, that the maximum deviation of the [$\Delta\nu$]{} scaling relation would be about 1% if the [$\nu_{\rm{max}}$]{} scaling relation is true. Now, if both the [$\Delta\nu$]{} and the [$\nu_{\rm{max}}$]{} relations are incorrect but conspire to cancel out their inaccuracy when using the mass scaling relation (Eq. \[equ:scaling\_m\]), one could in principle have a scenario where large deviations of the [$\Delta\nu$]{} and [$\nu_{\rm{max}}$]{} relations could be hidden in our mass test. However, this seems not to be the case because when testing the radius scaling relation =()()\^[-2]{}()\^[0.5]{}, which is based on different powers of [$\Delta\nu$]{} and [$\nu_{\rm{max}}$]{}, Zinn et al (submitted) finds agreement between seismic and Gaia radii at the 1% level. Hence, in combination these mass and radius scaling relation tests show strong evidence that the individual [$\Delta\nu$]{} and [$\nu_{\rm{max}}$]{} scaling relations that go into the mass and radius scaling relations are in fact astonishingly accurate.
### Constraining the age of the thick disc
Having established that the asteroseismic scaling relations are good to a high degree of accuracy, it would seem reasonable to now turn the problem around. Hence, in the following we assume the relations to be true and use the observed values of $\kappa_M$ to estimate the age and metallicity of the thick disc. We do this using the importance sampling framework discussed in . Here, we use the FL Galactic model from as the base model and reweight it to simulate samples corresponding to different values of the parameters of the model. We compute the likelihood of the observed $\kappa_M$ values given the model for different values of the mean metallicity, $\log Z/Z_{\odot}$, and mean age for the thick disc. Given the unknown selection function of the [*Kepler*]{} data, only data from the K2 campaigns were used. The results are shown in . We adopted a duration of 2 Gyr for the star formation episode of the thick disc. We also investigated shorter (1 Gyr) and longer (3 Gyr) star formation durations and found that the results were not too sensitive to the exact choice of the duration.
a shows the likelihood when considering all giants. b shows the likelihood when only lRGB giants are used. It can be seen that when we only consider the asteroseismic information, age is degenerate with metallicity. A decrease in the adopted metallicity by 0.1 dex can decrease the inferred age by about 2 Gyr. a shows that a metal poor thick disc cannot be old. For example, a thick disc with $\log Z/Z_{\odot}=-0.3$ will have an age of about 8 Gyr and would be even younger if it was more metal poor (such as the old MP model). For a star with a given mass, the decrease in age with a decrease in metallicity is expected because a low metallicity star evolves much faster along the HR diagram, compared to a high metallicity star.
b,d shows the likelihood as a function of age when we fix the metallicity to -0.16 as suggested by the spectroscopic data. Using all giants we get a mean age of 10 Gyr, if only lRGB stars are used we obtain 9.2 Gyr. Both estimates are consistent with the traditional idea of an old thick disc.
![(a,c) Likelihood of age and metallicity of the thick disc using asteroseismic information from K2 campaigns C1, C4, C6, and C7. (b-d) The likelihood of thick disc age assuming the thick disc metallicity to be $\log (Z/Z_{\odot})=-0.162$, as estimated using the GALAH survey in . In the top panels, the likelihood is computed using all oscillating giants, while in the bottom panels, only low luminosity giants () are used. \[fig:likelihood\]](likelihood_1.pdf){width="50.00000%"}
Discussion and Conclusions {#conclusions}
==========================
Asteroseismology can provide ages for giant stars and hence is a promising tool for studying Galactic structure and evolution. However, it has proven to be difficult to check the accuracy of the ages and masses estimated by asteroseismology, due to the shortage of independent estimates of mass and age. Population synthesis based Galactic models, provide an indirect way to validate the asteroseismic estimates. However, previous studies using the [*Kepler*]{} mission revealed that the models predict too many low mass stars as compared to observed mass distributions, raising doubts on the accuracy of the asteroseismic estimates, the Galactic models, and/or the selection function. In this paper, we revisit this important problem by analyzing asteroseismic data from the K2 mission, which has a well defined selection function. For the first time, we show that if the metallicity distribution in the Galactic models is updated to measurements from recent spectroscopic surveys, the distribution of asteroseismic masses is in good agreement with the model predictions. Using thick disc stars we show that the asteroseismic mass scaling relation for low luminosity red giants should be accurate to 5%. This is in agreement with findings of @2018MNRAS.476.3729B who tested the seismic relations using three eclipsing binary systems.
We identify three main factors, which, if not taken into account, can lead to discrepancies between observed asteroseismic masses and model predictions. First, in addition to age, the mass distribution giant stars in a stellar population is very sensitive to its metallicity, hence it is important to get the metallicity distribution of the various Galactic components in a model to agree with observations. Second, certain Galactic components are significantly enhanced in abundance of $\alpha$ elements and this should be taken into account, either directly by using $\alpha$ enhanced isochrones, or indirectly by increasing the effective metallicity of the solar scaled isochrones. Third, the $\Delta {\nu}$ scaling relation is not strictly valid and there exists theoretically motivated corrections, which should be applied. It was already shown in a previous study [@2016ApJ...822...15S] that the correction is such that it helps to reduce the mass discrepancy.
Using a forward modelling approach, where we take the Besançon Galactic model as a prior, we fit for the effective metallicity $Z$ (taking $\alpha$ enhancement into account) of the thin and the thick disc using the GALAH data. We find the mean $\log Z/Z_{\odot}$ of the thin disc to be $0.0$ and that of the thick disc to be -0.162 (with a dispersion of 0.17, see ). This is in good agreement with data from the APOGEE survey. This is a significant revision of thick disc from a value of \[Fe/H\]$=-0.78$ as used in the Besançon model. An increase of about 0.14 dex in $\log Z/Z_{\odot}$ is due to taking the $\alpha$ enhancement into account, but about 0.5 dex is due to revision of \[Fe/H\]. For example, if we consider stars in $5<R/{\rm kpc}<7$ and $1<|z|/{\rm kpc}<2$, which mostly come from the thick disc, both GALAH and APOGEE suggest a mean \[Fe/H\]$\sim-0.30$ for the thick disc.
Using a forward modelling approach, we also fit for the age of the thick disc using the asteroseismic data. We find the mean age to be about $9.2 - 10 \pm 0.25$ Gyr (redshift of about 1.6), which is broadly consistent with the idea of the thick disc being old and formed early on in the history of the Galaxy. What exactly do we mean by thick disc? Traditionally the thick disc was identified as the component with higher scale height in the solar annulus. Observations also suggest the thick disc to be distinct in elemental abundances from the thin disc. Two sequences $\alpha_{+}$ and $\alpha_{\rm o}$ can be seen in the (\[$\alpha$/Fe\],\[Fe/H\]) plane, with the former (having higher \[$\alpha$/Fe\]) being the thick disc and the later the thin disc. New results [@2011ApJ...735L..46B; @2012ApJ...753..148B; @2017ApJS..232....2X; @2017MNRAS.471.3057M] suggest that the scale length of the $\alpha_{+}$ sequence is shorter than that of the $\alpha_{\rm o}$ sequence. Chemical evolution models require the $\alpha_{+}$ sequence to be old. In our forward modelling we do not identify the thick disc using elemental abundances. Instead the thick disc is indirectly identified by our prior for the spatial distribution of thin and thick disc stars. In the model, stars with $|z|>1$ kpc are dominated by thick disc. The majority of the thick disc stars in our model come from the high latitude campaigns C1 and C6, and these stars have Galactocentric radius similar to that of the Sun. So our thick disc metallicity and age measurements are representative of the properties of the stellar population that roughly dominates in the region $|z|/{\rm kpc}>1$ and $6<R/{\rm kpc}<10$.
Our thick disc age estimate is consistent with previous studies that estimated the mean age independent of asteroseismology. For example, it is consistent with results by @Bensby2003 who estimate the age to be $11.2 \pm 4.3$ using F and G dwarfs. It is consistent with @2017ApJS..232....2X from LAMOST using main sequence turn-off stars and subgiants, where they show that stars with, $|z|>1$ kpc have a median age close to 10 Gyr and are $\alpha$ enhanced. It is consistent with results by @2017MNRAS.471.3057M from APOGEE using giants, where they show that $\alpha$ enhanced stars have significantly larger scale height and their mean age is close to or larger than 10 Gyr. However, the age estimates in @2017MNRAS.471.3057M are anchored on the asteroseismic age scale. Finally, our estimate ($9.2-10\pm 0.25$ Gyr) is in excellent agreement with estimates of @2017ApJ...837..162K of $9.5-9.9\pm 0.2$ Gyr using white dwarfs, an estimate that is very accurate and independent of both asteroseismology and the isochrones.
Although we find that the observed mass distributions are in good agreement with predictions by Galactic models, some small unexplained differences do remain. For lRGB, the predicted mean of the mass distributions for K2 campaign C4 and [*Kepler*]{} are higher by about 3%. We also see differences in metallicity distributions for these samples and this could potentially be responsible for the mass differences. For hRGB and red clumps, the mean predicted mass is lower than observed, for campaigns C6 and C7. This could be due to imperfections in the model, but could also be related to the fact that the detection of $\Delta \nu$ is not complete for these stars.
We present the selection function for four K2 campaigns and discuss detection biases associated with the K2 data, which should be taken into account when using the K2 data. Probability to detect $\nu_{\rm max}$ varies with both $\nu_{\rm max}$ and apparent magnitude. Low-luminosity stars have lower oscillation amplitudes and cannot be detected at fainter magnitudes. Even after we account for the effect of oscillation amplitude and apparent magnitude, comparison with Galactic models show that the overall detection rate for $\nu_{\rm max}$ is about 72%. Using a deep-learning-based pipeline improves the detection rate to 78%, which is still quite low. It is not yet clear as to why the detection rate is low. It could be that certain specific type of stars (e.g., red clumps or metal poor stars) have lower than expected oscillation amplitudes, or it could be an unknown instrumental effect, or even a problem with the Galactic model. There are also biases related to detecting $\Delta \nu$ in the K2 data. The probability to detect $\Delta \nu$ has a strong dependence on $\nu_{\rm max}$, it is less than 1 for $\nu_{\rm max}<50\ \mu$Hz, but is otherwise close to 1. Significant campaign to campaign differences are also seen, which needs further investigation. To take the detection biases into account, we propose to split up the stars into different giant classes based on their detection probabilities. Asteroseismic pipelines also show small systematic offsets in estimation of $\nu_{\rm max}$ which need further investigation.
Using the seismic sample, we find that the stellar parameters for giants in GALAH DR2, which are based on the data-driven [*The Cannon*]{} scheme, have systematic differences with respect to estimates based on the model-driven SME scheme that is anchored to seismic $\nu_{\rm max}$ values. Differences are most significant for stars with \[Fe/H\]>0. We provide analytical functions to correct for them. The reason for the systematic offsets is because the giants in the training set used by [*The Cannon*]{} were dominated by non seismic giants. In the absence of seismic $\nu_{\rm max}$, the SME gives biased results. SME with Gaia DR2 parallaxes as prior alleviates this problem, however, Gaia DR2 parallaxes were not available at the time of publication of GALAH DR2.
In near future, we will have a much larger sample of stars with asteroseismology from both the K2 and the TESS [@2015ApJ...809...77S] missions. This will allow us to fit more detailed models of our Galaxy than done here. Specifically, we can study the properties of the stellar populations as a function of age with much finer age resolution. Future, spectroscopic surveys, such as the, second phase of GALAH, 4MOST [@2016SPIE.9908E..1OD], WEAVE [@2018SPIE10702E..1BD], and SDSSV [@2017arXiv171103234K], will also produce large samples of stars with age estimates purely from spectroscopy, based on main sequence turnoff and subgiant stars or based on giants making use of the age information encoded in carbon and nitrogen abundances. Asteroseismology in this regard is going to play a crucial role by providing independent age estimates.
S.S. is funded by University of Sydney Senior Fellowship made possible by the office of the Deputy Vice Chancellor of Research, and partial funding from Bland-Hawthorn’s Laureate Fellowship from the Australian Research Council. The GALAH Survey is supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D),through project number CE170100013. D.S. is the recipient of an Australian Research Council Future Fellowship (project number FT1400147). JBH is supported by an ARC Australian Laureate Fel- lowship (FL140100278). MJH is sup- ported by an ASTRO-3D Fellowship. S.B. and K.L. acknowledge funds from the Alexander von Humboldt Foundation in the framework of the Sofja Kovalevskaja Award endowed by the Federal Ministry of Education and Research. K.L. acknowledges funds from the Swedish Research Council (Grant nr. 2015-00415\_3) and Marie Sklodowska Curie Actions (Cofund Project INCA 600398). T.Z. acknowledges financial support from the Slovenian Research Agency (research core funding No. P1-0188). DMN was supported by the Allan C. and Dorothy H. Davis Fellowship. J. Z. acknowledges support from NASA grants 80NSSC18K0391 and NNX17AJ40G.
[^1]: <http://galaxia.sourceforge.net>
[^2]: <http://www.physics.usyd.edu.au/k2gap/Asfgrid>
|
---
abstract: 'Cosmological perturbations generated quantum-mechanically (as a particular case, during inflation) possess statistical properties of squeezed quantum states. The power spectra of the perturbations are modulated and the angular distribution of the produced temperature fluctuations of the CMBR is quite specific. An exact formula is derived for the angular correlation function of the temperature fluctuations caused by squeezed gravitational waves. The predicted angular pattern can, in principle, be revealed by the COBE-type observations.'
address:
- 'McDonnell Center for the Space Sciences, Physics Department'
- 'Washington University, St.Louis MO 63130'
- and
- 'Sternberg Astronomical Institute, Moscow University'
- '119899 Moscow, V-234, Russia'
author:
- 'L. P. Grishchuk'
title: 'Cosmological Perturbations of Quantum-Mechanical Origin and Anisotropy of the Microwave Background'
---
The recent discovery by COBE \[1\] of the angular variations of CMBR makes it necessary to analyze in greater detail the observational consequences of the quantum-mechanical generation of cosmological perturbations. The underlying physical reason for the generating process is the parametric (superadiabatic) amplification of classical perturbations and the associated quantum-mechanical particle pair creation in the variable gravitational field of the homogeneous isotropic Universe. As a result of the parametric coupling between the quantized perturbations and the variable classical “pump” field, the initial vacuum state of the perturbations evolves (in the ${\rm Schr\ddot o dinger}$ picture) into a strongly squeezed vacuum state possessing very specific statistical properties. The generated fluctuations can be viewed, classically, as a stochastic collection of standing waves. The mechanism itself and its main results concerning squeezing are valid for gravitational waves and progenitors of density perturbations \[2,3\]. A particular variable gravitational field, that may be responsible for the amplification process, is provided by one or another type of the inflationary expansion. It is often stated that inflation generates “Gaussian perturbations with randomly distributed phases”. However, this is not the case: the phases of all modes of perturbations are essentially constant and fixed \[3\] which leads to standing waves, modulated spectra of the generated perturbations and a specific angular distribution of the temperature fluctuations of CMBR over the sky, as will be shown below.
In this paper we will analyze, mostly, gravitational waves. For our purposes it is sufficient to consider perturbations in a spatially-flat FLRW universe\
$ds^2=a^2( \eta )(d \eta^2 -dx^2 - dy^2 -dz^2 )$ where $a(\eta)$ is the cosmological scale factor.
The quantum-mechanical operator for the gravitational-wave field can be written in the general form $$h_{ij}(\eta , {\bf x})=C \int_{-\infty}^\infty d^3 {\bf n}
\sum_{s=1}^2 p_{ij}^s ({\bf n}) [a_{\bf n}^s (\eta ) e^{i{\bf nx}}
+a_{\bf n}^{s+} (\eta )e^{-i{\bf nx}} ]$$ where $C$ is a constant combining all the numerical coefficients, $p_{ij}^s({\bf n})$ are two $(s=1,2)$ polarization tensors and $a_{\bf n}^s(\eta)$, $a_{\bf n}^{s+}(\eta)$, are (Heisenberg) operators for each mode $\bf n$ and for each polarization state $s$.
The polarization tensors $p_{ij}^s(\bf n )$ satisfy the “transverse-traceless” conditions $p_{ij}^s n^j=0$, $p_{ij}^s\delta^{ij}=0$ and leave independent only two components of $h_{ij}$ for each ${\bf n}$-mode of the field. For a wave travelling in the direction\
${\bf n}/n=(\sin \theta\cos\varphi ,\sin\theta\sin\varphi ,\cos\theta )$ the polarization tensors are\
$p_{ij}^1({\bf n})= l_i l_j -m_i m_j$, $p_{ij}^2({\bf n})= l_i m_j +l_j m_i$, where $l_j$, $m_j$ are two unit vectors orthogonal to ${\bf n}$ and to each other: $l_j=(\sin\varphi ,-\cos\varphi ,0)$, $m_j=(\cos\theta\cos\varphi\, ,\cos\theta\sin\varphi\, ,-\sin\theta )$ for $\theta < \pi /2$ and $m_j=-(\cos\theta\cos\varphi ,\cos\theta\sin\varphi ,-\sin \theta$) for $\theta > \pi /2$.
The operators $a_{\bf n}^s(\eta )$, $a_{\bf n}^{s+}(\eta )$ are annihilation and creation operators for waves (particles) travelling in the direction ${\bf n}$. The time evolution of $a_{\bf n}^s(\eta )$, $a_{\bf n}^{s+}(\eta )$ is governed by the Heisenberg equations of motion for each mode ${\bf n}$ and for each polarization state $s$ (index $s$ is omitted here but will be restored later): $da_{\bf n}/d\eta=-i[a_{\bf n},H],$ $da_{\bf n}^+/d\eta=-i[a_{\bf n}^+,H]$. The Hamiltonian $H$ to be used in these equations has the form $H=na_{\bf n}^+ a_{\bf n} + na_{-{\bf n}}^+ a_{-{\bf n}}
+ 2\sigma (\eta ) a_{\bf n}^+ a_{-{\bf n}}^+
+ 2\sigma^{\ast} (\eta ) a_{{\bf n}} a_{-{\bf n}}$ where the coupling function $\sigma (\eta )=ia^\prime /2a$ and ${}^\prime =d/d\eta$. The solution to the Heisenberg equations of motion can be written as $$a_{\bf n}(\eta ) = u_n(\eta )a_{\bf n} (0)
+ v_n(\eta )a_{-{\bf n}}^+ (0) \, , \quad
a_{\bf n}^+(\eta ) = u_n^\ast(\eta )a_{\bf n}^+ (0)
+ v_n^\ast(\eta )a_{-{\bf n}} (0)$$ where $a_{\bf n}(0)$, $a_{\bf n}^+(0)$, are the initial values of the operators $a_{\bf n}(\eta )$, $a_{\bf n}^+(\eta )$ taken at some initial time long before the coupling became significant and the amplification process has started, and the complex functions $u_n$, $v_n$ satisfy the equations $$iu_n^\prime = nu_n + i(a^\prime /a) v^\ast_n \, , \quad
iv_n^\prime = nv_n + i(a^\prime /a)u^\ast_n$$ where $| u_n |^2-|v_n|^2=1$ and $u_n(0)=1$, $v_n(0)=0$. It follows from these equations that the function $u_n+v_n^\ast \equiv \mu_n$ obeys the equation $\mu_n^{\prime\prime}+(n^2-a^{\prime\prime}/a)\mu_n =0$ which is precisely the equation for classical complex $\mu$-amplitude \[2\] of the gravitational-wave field. Note that the solutions $u_n(\eta )$, $v_n(\eta )$ to Eq. (3) depend only on the absolute value of the vector ${\bf n}$, $n=(n_1^2+n_2^2+n_3^2)^{1/2}$, not its direction. Also, these solutions are identical for both polarizations: they obey the same equations with the same initial conditions.
The two complex functions $u_n$, $v_n$ restricted by one constraint $|u_n|^2 -|v_n|^2=1$ can be parameterized by the three real functions $r_n(\eta )$, $\phi_n(\eta )$, $\varepsilon_n(\eta )$: $$u_n=e^{i\varepsilon_n}ch\, r_n \, , \qquad
v_n=e^{-i(\varepsilon_n-2\phi_n)}sh \, r_n \, .$$ For each $n$ these functions obey the equations $$r^\prime = (a^\prime /a)\cos 2\phi \, , \quad
\phi^\prime=-n-(a^\prime /a)\sin 2\phi \, cth\, 2r \, , \quad
\varepsilon^\prime = -n-(a^\prime /a) \sin 2\phi \, th\, r$$ which can be used for an explicit calculation of $r_n$, $\phi_n$, $\varepsilon_n$ if a time-dependent scale factor $a(\eta )$ is given.
The operators $a_{\bf n}(0)$, $a_{\bf n}^+(0)$ (${\rm Schr\ddot o dinger}$ operators) satisfy the usual commutation relations $[a_{\bf n}(0),a_{\bf m}^+(0)]=\delta^3({\bf n}-{\bf m})$ and the same is true for the evolved operators: $[a_{\bf n}(\eta ),a_{\bf m}^+(\eta )]=\delta^3({\bf n}-{\bf m})$. By using Eq. (4) the (Bogoliubov) transformation (2) can be cast in the form $$a_{\bf n} (\eta ) = RSa_{\bf n} (0)S^+R^+ \, , \quad
a_{\bf n}^+ (\eta ) = RSa_{\bf n}^+ (0)S^+R^+$$ where $$S(r,\phi )=\exp \left[ r \left( e^{-2i\phi} a_{\bf n} (0)a_{-\bf n} (0)
-e^{2i\phi}a_{\bf n}^+(0)a_{-\bf n}^+(0) \right) \right]$$ is the unitary two-mode squeeze operator and $$R(\varepsilon )=\exp
\left[ -i\varepsilon \left( a_{\bf n}^+ (0)a_{\bf n}(0)
+ a_{-\bf n}^+ (0)a_{-\bf n}(0) \right) \right]$$ is the unitary rotation operator. The functions $r_n$, $\phi_n$, $\varepsilon_n$ are called squeeze parameter, squeeze angle and rotation angle. (For a description of squeezed states see, for example, \[4\].) Equations (2), (6) demonstrate explicitely the inevitable appearance of squeezing in the problems of this kind. In this paper we use the presentation based on travelling waves and two-mode squeezed states but standing waves and one-mode squeezed states are equally good \[3\].
We assume that the quantum state of the field is the vacuum state defined by the requirement $a_{\bf n}(0)|0> =0$ for each ${\bf n}$ and for both $s$. In the Heisenberg picture the state of the field does not change in time but the operators do. The values of $a_{\bf n}(\eta )$, $a_{\bf n}^+(\eta )$ determine all the statistical properties of the field at the later times. It follows from Eq. (2) that the mean values of $a_{\bf n}(\eta )$, $a_{\bf n}^+(\eta )$ are zero: $<0|a_{\bf n}(\eta )|0> = 0$, $<0|a_{\bf n}^+(\eta )|0>=0$, but the mean values of the quadratic combinations of $a_{\bf n}(\eta )$, $a_{\bf n}^+(\eta )$ (variances) are not zero:
$$\begin{aligned}
<0 |a_{\bf n}(\eta )a_{\bf m} (\eta )|0>
& = & u_n(\eta) v_m(\eta )\delta ^3 ({\bf n}+{\bf m}) \nonumber \\
<0 |a_{\bf n}^+(\eta )a_{\bf m}^+ (\eta )|0>
& = & v_n^\ast(\eta) u_m^\ast (\eta )\delta ^3 ({\bf n}+{\bf m}) \\
<0 |a_{\bf n}(\eta )a_{\bf m}^+ (\eta )|0>
& = & u_n(\eta) u_m^\ast (\eta )\delta ^3 ({\bf n}-{\bf m}) \nonumber \\
<0 |a_{\bf n}^+ (\eta )a_{\bf m} (\eta )|0>
& = & v_n^\ast(\eta) v_m (\eta )\delta ^3 ({\bf n}-{\bf m}) \nonumber\end{aligned}$$
These relationships (the first two) show explicitely that the waves (modes) with the opposite momenta are not independent. On the contrary, they are strongly correlated which is the reason for the appearance of standing waves. This fact finds its reflection in the correlation functions of the field.
To simplify the discussion of the correlation functions, we will first ignore the tensorial indices in Eq. (1) and consider a scalar field $$h(\eta ,{\bf x)} = \int_{-\infty}^\infty d^3{\bf n}
[a_{\bf n}(\eta )e^{i{\bf nx}} + a_{\bf n}^+(\eta )e^{-i{\bf nx}}] \, .$$ Physically, the field $h(\eta ,{\bf x})$ may be a scalar variable associated with the density perturbations (see Ref. \[5\] and the third paper in Ref. \[3\]). The mean value of the field $h$ is zero in every spatial point and at every moment of time. The variance of the field is not zero, it can be calculated with the help of Eq. (7): $$<0|h(\eta ,{\bf x})h(\eta ,{\bf x})|0>
= 4\pi \int_0^\infty n^2
dn(|u_n|^2 + |v_n|^2 + u_n v_n + u_n^\ast v_n^\ast ) \, .$$ In terms of the squeeze parameters the result can be written as $$<0|h(\eta ,{\bf x}) h(\eta ,{\bf x})|0>
= 4\pi \int_0^\infty n^2dn(ch2r_n+sh2r_n \cos 2\phi_n )$$ (this expression includes the vacuum energy term $4\pi\int_0^\infty n^2dn$ which should be subtracted at the end). The variance of the field does not depend on the spatial coordinate ${\bf x}$ but does depend, in general, on time. The function under the integral in Eq. (8) is usually called the power spectrum of the field: $P(n) =n^2(ch2r_n+sh2r_n\cos2\phi_n)$. The important property of squeezing is that, for a given time, the function $P(n)$ is not a smooth function of $n$ but is modulated and contains many zeros or, strictly speeking, very deep minima. To see this, one can return to Eqs. (5). For late times, that is, well after the completion of the amplification process, the function $a^\prime /a$ on the right-hand side of Eqs. (5) can be neglected. (This is equivalent to saying that one is considering waves that are well inside the Hubble radius.) At these late times, the squeeze parameter $r_n$ is not growing any more and the squeeze angle is just $\phi_n=-n\eta -\phi_{0n}$. Since $r_n \gg 1$ for the frequencies of our interest \[3\], the $P(n)$ can be written as $P(n) \approx n^2e^{2r_n}\cos^2(n\eta +\phi_{0n})$. The factor $\cos^2(n\eta+\phi_{0n})$ vanishes for a series of values of $n$; at these frequencies the function $P(n)$ goes to zero. The position of zeros, as a function of $n$, varies with time. The similar conclusions hold for the spatial auto-correlation function: $$<0|h(\eta ,{\bf x})h(\eta ,{\bf x}+{\bf l})|0>
= 4\pi \int_0^\infty n^2 {\sin nl \over nl}
(ch2r_n + sh2r_n\cos2\phi_n)dn \,.$$ The resulting expression depends on the distance between the points but not on their coordinates. The power spectrum of this correlation function is also modulated by the same factor $\cos^2(n\eta +\phi_{0n})$.
We return now to the tensor field (1). There is one combination of the components $h_{ij}$ which has a special meaning: $h(e^k)=h_{ij}e^ie^j$, where\
$e^k=(\sin\bar\theta\cos\bar\phi ,\sin\bar\theta\sin\bar\phi ,\cos\bar\theta)$ is an arbitrary unit vector. The $h(e^k)$ enters the calculation of the CMBR temperature variation seen in the direction $e^k$ (Sachs-Wolfe effect \[6\]): $${\delta T \over T}(e^k) = {1\over 2} \int_0^{w_1}
\left( {\partial h_{ij} \over \partial\eta}e^ie^j \right) dw$$ where $w=\eta_R-\eta$, $x^k=e^kw$, $w_1=\eta_R -\eta_E$ and $h_{ij}$ in this formula is $a^{-1}(\eta )$ times $h_{ij}$ introduced in Eq. (1). For a quantized $h_{ij}$-field, the $\delta T/T$ becomes an operator: $$\begin{aligned}
{\delta T \over T} (e^k)
& = & {1 \over 2} C \int_0^{w_1} dw
\int_{-\infty}^\infty d^3{\bf n} \sum_{s=1}^2 p_{ij}^s({\bf n})
e^ie^j \{ [\alpha_n^s a_{\bf n}^s (0)
+ \beta_n^s a_{-\bf n}^{s+}(0)] e^{in_ke^kw} \nonumber \\
& + & [\alpha_n^{s\ast} a_{\bf n}^{s+} (0)
+ \beta_n^{s\ast}a_{-\bf n}^s(0)] e^{-in_k x^k w} \} \nonumber\end{aligned}$$ where $\alpha_n^s(\eta) \equiv (u_n^s/a)^\prime$, $\beta_n^s(\eta) \equiv (v_n^s/a)^\prime$. The mean value of $\delta T/T$ is zero while the variance of the expected temperature fluctuations can be written as $$\begin{aligned}
<0| {\delta T \over T} (e^k){\delta T \over T} (e^k)|0>
& = & {1 \over 4} C^2 \int_0^{w_1} dw \int_0^{w_1} d \bar w
\int_{-\infty}^\infty d^3{\bf n} \cos(n_ke^k\xi) \nonumber \\
& \times & \pi^1({\bf n},e^k) f(n,w,\bar w )\end{aligned}$$ where $\xi =w-\bar w$ and $$\begin{aligned}
\pi^1 ({\bf n},e^k) & \equiv & (p_{ij}^1 ({\bf n})e^ie^j)^2
+ (p_{ij}^2 ({\bf n})e^ie^j)^2 , \nonumber \\
f(n,w,\bar w ) & \equiv & \alpha_n(w)\alpha_n^\ast (\bar w )
+ \beta_n^\ast (w)\beta_n(\bar w )
+ \alpha_n(w)\beta_n (\bar w )
+ \beta_n^\ast (w)\alpha_n^\ast (\bar w ), \nonumber \\
\alpha_n^1 & = & \alpha_n^2 \equiv \alpha_n \, , \quad
\beta_n^1 = \beta_n^2 \equiv \beta_n \, . \nonumber\end{aligned}$$
The integration over the variables $\varphi$, $\theta$ in Eq. (9) allows one to reduce this formula to $$\begin{aligned}
<0| {\delta T \over T}(e^k) {\delta T \over T} (e^k) |0>
= C^2 8\pi \int_0^{w_1} dw \int_0^{w_1} d\bar w \int_0^{\infty} n^2
W_1(n\xi )f(n,w,\bar w) dn\end{aligned}$$ where $$\begin{aligned}
W_1(n\xi )
= (\pi /2)^{1/2} (n\xi)^{-5/2} {\it J}_{5/2} (n\xi)\, . \nonumber\end{aligned}$$ The term $W_1(n\xi)$ depends on the interval between the points but not on the direction of sight. Thus, variancies seen in all directions $e^k$ are the same. They are also position independent as for $x^k = e^kw+x_0^k$ the coordinates $x_0^k$ of the observer drop out of the final result.
We will now turn to the derivation of the angular correlation function\
$<0|\delta T/T(e_1^k)\delta T/T(e_2^k)|0>$ where $e_1^k$ and $e_2^k$ are two different unit vectors. The general formula for this function can be written as $$\begin{aligned}
<0|{\delta T \over T} (e_1^k){\delta T \over T} (e_2^k)|0>
& = & {1\over 4} C^2 \int_0^{w_1} dw \int_0^{w_1} d\bar w
\int_{-\infty}^\infty d^3{\bf n} \cos(n_i\zeta^i) \nonumber \\
& \times & \pi^2 ({\bf n},e_1^k,e_2^k)f(n,w, \bar w)\end{aligned}$$ where $\zeta^i =e_1^iw-e_2^i \bar w$ and $$\begin{aligned}
\pi^2({\bf n},e_1^k,e_2^k)
\equiv (p_{ij}^1({\bf n})e_1^ie_1^j)(p_{lm}^1({\bf n})e_2^le_2^m)
+ (p_{ij}^2(n)e_1^ie_1^j)(p_{lm}^2({\bf n})e_2^le_2^m). \nonumber\end{aligned}$$ The integration over the variables $\varphi ,\theta$ in Eq. (11) reduces this formula to $$\begin{aligned}
<0|{\delta T \over T} (e_1^k){\delta T \over T} (e_2^k)|0>
= C^2 8\pi\int_0^{w_1} dw \int_0^{w_1} d\bar w \int_0^\infty n^2
W_2(n\zeta , \cos\delta ) f(n,w, \bar w ) dn \nonumber\end{aligned}$$ $$\begin{aligned}
\end{aligned}$$ where $\zeta =(w^2-2w\bar w \cos\delta +\bar w^2)^{1/2}$, $\delta$ is the angle between the two directions of observation, $\cos\delta = e_1^1e_2^1+e_1^2e_2^2+e_1^3e_2^3$, and $$\begin{aligned}
W_2(n\zeta ,\cos\delta )
& = & {1\over2}(3\cos^2\delta -1)(\pi /2)^{1/2} (n\zeta )^{-5/2}
J_{5/2}(n\zeta ) \nonumber \\
& + & \cos\delta(\cos^2\delta -1)(nw)(n\bar w)(\pi /2)^{1/2}
(n\zeta )^{-7/2} J_{7/2}(n\zeta ) \nonumber \\
& + & {1\over 8}(cos^2 \delta -1)^2(nw)^2(n\bar w )^2
(\pi /2)^{1/2} (n\zeta )^{-9/2} J_{9/2}(n\zeta ).\end{aligned}$$ Expression (12) depends only on $\cos\delta$ and, hence, the correlation function is rotationally symmetric. In the limit $\cos\delta =1$, the parameter $\zeta$ goes over into $\xi$ and Eq. (12) coincides with Eq. (10).
Expression (12) gives the angular correlation function in the general and universal form. It can be used with arbitrary functions $\alpha_n(w),\beta_n(w)$, that is, it is applicable for arbitrary (not necessarily inflationary) cosmological models generating squeezed gravitational waves. The remaining integrations in Eq. (12) assign concrete numerical values to the correlations attributed to different separation angles $\delta$, but they do not change the general angular pattern represented by the function $W_2(n\zeta ,\cos\delta )$. Consistency with the data of the COBE-type observations may lead to the determination of the functions $\alpha_n(w),\beta_n(w)$ and, eventually, to the knowledge of the expansion rate of the early universe. This will be a subject of a separate discussion. The implications of the COBE observations for inflationary models are under active analysis (see, for instance, a recent paper \[7\] and references therein). Some new results based on the correlation function (12), (13) have been derived in \[8\].
This work was supported in part by NASA grant NAGW 2902 and NSF grant 89-22140.
G. F. Smoot [*et al*]{}, submitted to Astrophys. J.; J. Mather, invited talk at GR 13, Cordoba, Argentina (to be published). L. P. Grishchuk, Zh. Eks. Teor. Fiz. [**67**]{}, 825 (1974) \[Sov. Phys. JETP [**40**]{}, 409 (1975)\]. L. P. Grishchuk and Y. V. Sidorov, Phys. Rev. D [**42**]{}, 3413 (1990);\
L. P. Grishchuk, NASA Conference Publication 3135, p.329, eds. D. Han, Y. S. Kim and W. W. Zachary, 1991; L. P. Grishchuk, in [*Proceed. 5-th\
M. Grossmann Meeting, Japan*]{} (World Scientific), to appear; L. Grishchuk, H. A. Haus and K. Bergman, Phys. Rev. D [**46**]{}, 1440, (1992). D. Stoler, Phys. Rev. D [**1**]{}, 3217 (1970); H. P. Yuen, Phys. Rev. A [**13**]{}, 2226 (1976); B. L. Schumaker, Phys. Rep. [**135**]{}, 317 (1986). V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Phys. Rep. [**215**]{}, \#5, 6 (1992). R. K. Sachs and A. M. Wolfe, Astrophys. J. [**147**]{}, 73 (1967). R. L.Davis, H. M. Hodges, G. F. Smoot, P. J. Steinhardt, and M. S. Turner, Phhys. Rev. lett. [**69**]{}, 1856 (1992). L. P. Grishchuk, Phys. Rev. D (submitted).
|
---
abstract: |
The turn-over frequency of the catalytic oxidation of CO at RuO$_2$(110) was calculated as function of temperature and partial pressures using [*ab initio*]{} statistical mechanics. The underlying energetics of the gas-phase molecules, dissociation, adsorption, surface diffusion, surface chemical reactions, and desorption were obtained by all-electron density-functional theory. The resulting CO$_2$ formation rate \[in the full ($T, p_{\rm CO},
p_{\rm O_2}$)-space\], the movies displaying the atomic motion and reactions over times scales from picoseconds to seconds, and the statistical analyses provide insights into the concerted actions ruling heterogeneous catalysis and open thermodynamic systems in general.
author:
- 'Karsten Reuter$^{1,2}$'
- 'Daan Frenkel$^{2}$'
- 'Matthias Scheffler$^{1}$'
title: |
[(Phys. Rev. Lett., accepted)]{}\
The steady-state of heterogeneous catalysis,\
studied by first-principles statistical mechanics
---
Under realistic conditions materials surfaces are in contact with a rich environment [@Stampfl-SS500-2002]. Often, the resulting surface composition and surface-actuated material function are determined by equilibrium thermodynamics. A first-principles description is then possible with the “[*ab initio*]{} atomistic thermodynamics” approach, that has been successfully applied to various systems (see e.g. Refs. [@Stampfl-SS500-2002; @Reuter-PRB-2002; @Reuter-PRL/B-2003; @Michaelides-2003; @Norskov-2003] and references therein). However, under many conditions equilibrium thermodynamics does not provide the appropriate description, and heterogeneous catalysis is a particularly interesting and important example. Here one is dealing with an open system, i.e., a supply of gases or liquids comes into contact with a solid surface, where a chemical reaction produces a new substance that is then transported away. The entire concert of the various underlying, interlinked atomistic processes is in this case determined by kinetics. Still, the temperature and partial pressures of the reactants must be set such that the system runs under [*steady-state*]{} conditions. Only then the catalyst is not getting destroyed, but is stably enhancing the rate of the desired chemical reaction. An [*ab initio*]{} description of the full steady-state situation of heterogeneous catalysis has not been achieved so far, and a microscopic understanding of the competing and concerting actions of the various atomistic processes is lacking.
The present paper describes the “[*ab initio*]{} statistical mechanics” methodology appropriate for an open thermodynamic system with a continuous conversion of chemicals $A$ and $B$ into $C$, using the oxidation of CO at a RuO$_2$ model catalyst as example. We employ density-functional theory (DFT) to obtain the energetics of all relevant processes, as there are: motion of the gas-phase molecules, dissociation, adsorption, surface diffusion, surface chemical reactions, and desorption. These calculations use the all electron, full-potential linear augmented plane wave (FP-LAPW) approach [@wien; @basisset]. The only notable approximations are the generalized gradient approximation (GGA) for the exchange-correlation functional [@PBE], and the assumption that transition-state theory (TST) [@Ruggerone-1997] is applicable. In fact, both approximations are well justified for the present study, and we will particularly address the GGA below when analyzing the results. A combination of DFT with TST, and subsequently solving the statistical mechanics problem by the kinetic Monte Carlo (kMC) approach [@Ruggerone-1997; @Landau-2000], has been employed before (see, e.g. Refs. [@Ruggerone-1997; @Ovesson-1999; @Kratzer-2002; @Neurock-2000]). In distinction to (by now standard) “empirical kMC” calculations, that use just a few [*effective parameters*]{}, which have only limited (if any) microscopic meaning, these “[*ab initio*]{} kMC” calculations include an extended set of elementary processes with full and direct physical meaning. And describing an open system with a continuous conversion of chemicals $A$ and $B$ into $C$ implies additional complexity. For the example discussed below (the catalytic oxidation of CO) we will solve the statistical mechanics problem, gaining microscopic insight into the full dynamics from picoseconds to microseconds and even to seconds. The reported results demonstrate the new quality of and the novel insights gained by such description. We will also compare the results to those obtained by “constrained thermodynamics” [@Reuter-PRL/B-2003]. Although (as expected) noticeable differences occur for certain environmental conditions, we will confirm that the much simpler “constrained thermodynamics” approach can provide important guidance on where in $(T,p)$-space high reaction rates are likely to be expected, and when results obtained for certain $(T,p)$ conditions can be extrapolated to others.
The now obtained turn-over-frequencies (TOFs) of the catalytic oxidation of CO are in unexpected and unprecedented agreement with experimental results of Peden and Goodman [@Peden-1986] and Wang [*et al.*]{} [@Wang-2002]. This is found to be a consequence of the fact that the high reaction conditions are not just ruled by a singular chemical reaction pathway. Instead, for the steady-state and high TOF conditions it is necessary that various processes play together in a most efficient manner. For the present example one crucial point is to realize an optimum disordered and dynamic “phase” at the surface. Modest errors due to the approximate treatment of the exchange-correlation functional (e.g. the GGA) are then not crucial, as long as the trends of the energetics of the various interplaying processes are described correctly. In this sense, the frequently requested [*chemical accuracy*]{} for the description of individual processes appears to be a misleading concept. At least for the present system, a careful combination of DFT and statistical mechanics is more important.
---------------- ------- ------- -------- -------------------- ---------------------
$E_b$
to br to cus with CO$^{\rm br}$ with CO$^{\rm cus}$
CO$^{\rm br}$ -1.6 0.6 1.6 - -
CO$^{\rm cus}$ -1.3 1.3 1.7 - -
O$^{\rm br}$ -2.3 0.7 2.3 1.5 1.2
O$^{\rm cus}$ -1.0 1.0 1.6 0.8 0.9
---------------- ------- ------- -------- -------------------- ---------------------
: \[table1\] DFT binding energies, $E_b$, for CO and O (with respect to (1/2)O$_2$) at br and cus sites (cf. Fig. \[snapshots\]a), diffusion energy barriers, $\Delta E^{\rm b}_{\rm diff}$, to neighboring br and cus sites, and reaction energy barriers, $\Delta E^{\rm b}_{\rm reac}$, of neighboring species at br and cus sites. All values are in eV.
The methodology will be used to study CO oxidation over RuO${}_2$(110) (the surface is sketched in Fig. \[snapshots\]a), as this system has recently received considerable attention as a highly active model catalyst (see Ref. [@Wang-2002] and references therein). In fact, this was previously called the Ru catalyst, but recent experimental and theoretical work has shown that at realistic O$_2$ pressure the Ru(0001) surface is transformed into an epitaxial RuO$_2$(110) film (see Ref. [@Reuter-CPL-2002] and references therein). It is by now also established that this RuO$_2$(110) surface actuates the catalytic reaction, and, although domain boundaries and steps are present, their influence is not significant [@Jacobi-Ertl]. The surface unit cell is rectangular and contains two adsorption sites [@Reuter-PRL/B-2003]: bridge (br) and coordinatively unsaturated (cus) sites (cf. Fig. \[snapshots\]a). Either of them can be empty or occupied by O or CO, and adsorbate diffusion can go br-to-br, br-to-cus, cus-to-cus, or cus-to-br. Since this comprises the possibility of missing O$^{\rm br}$ atoms, we note that our treatment implicitly includes the effect of O surface vacancies. A total of 26 different elementary processes are possible on this lattice, and all were carefully analyzed by DFT to obtain their pathways and energy barriers [@Reuter-PRL/B-2003; @long-version]. Table \[table1\] summarizes the adsorption energies and the diffusion and reaction energy barriers used for the kMC study, as obtained from DFT calculations with a $(1 \times 2)$ surface unit-cell. In a systematic study of CO and O (co)adsorption at various coverages we found adsorbate-adsorbate interaction to be always smaller than 150meV. Thus, lateral interactions in this system are small (compared to the other energies), and will therefore be neglected.
CO adsorption into vacant cus or bridge sites is non-dissociative, while oxygen adsorption is dissociative and requires two vacant neighboring sites, i.e. a br-br, cus-cus, or br-cus pair. The adsorption rate per free site is given by the local sticking coefficient, $\tilde S$, and the kinetic impingement: $\Gamma_{\rm ad} = {\tilde S} \, p /
\sqrt{2\pi m k_{\rm B} T}$. Here $m$ is the mass of the gas-phase molecule, and $k_{\rm B}$ is the Boltzmann constant. The rate of the time reversed process (desorption) then follows from the relation: $\Gamma_{\rm des} / \Gamma_{\rm ad} = {\rm exp} \left( (F_{b} - \mu)/
(k_{\rm B}T) \right)$, where $F_{b}$ is the free energy of the adsorbed species (approximated by $E_b$), and $\mu(T, p)$ is the chemical potential of the gas-phase molecule [@Reuter-PRB-2002]. The $\tilde{S}(T)$ are thus obtained from the calculated total energy surfaces of desorption together with detailed balance. The only uncertainty here arises from the vibrational properties of the transition state which translates into uncertainties in the desorption rate by a factor of 10 (at most 100). For the diffusion processes we use a prefactor of $10^{12}$Hz, which has an uncertainty of a factor of 10. We carefully checked that these uncertainties do not affect our below reported results and conclusions. Details of the employed new methodology and of the various test calculations will be published elsewhere [@long-version]. Detailed balance of the scenario was carefully checked by confirming that the earlier results obtained by “atomistic thermodynamics” [@Reuter-PRL/B-2003] are exactly reproduced by the present statistical mechanics treatment, if surface reactions are not allowed to occur.
Kinetic Monte Carlo runs were performed for about 1000 different $(T,p_{\rm CO},p_{\rm O_2})$ conditions covering the temperature range $300\,{\rm K} < T < 800\,{\rm K}$ and partial pressures from $10^{-10}$ to 10$^{3}$ atmospheres. Several calculations were done on a system with $(30 \times 30)$ surface sites (450 bridge plus 450 cus sites), but the vast amount of calculations was done for a 400 sites system. The results for both system sizes were identical. The kMC simulations were run until steady-state is reached (cf. Fig. \[snapshots\]b). Then the movies were recorded (cf. Fig. \[snapshots\]c) and the statistical analyses of the frequencies of the various elementary processes performed. The latter also gives the total TOFs of the CO$_2$ formation.
The results show that catalytic reaction conditions are only established when both CO and O$_2$ partial pressures exceed certain values below which the surface is in a thermodynamic equilibrium phase of RuO${}_2$(110), characteristic for low oxygen pressures and routinely observed under ultra-high vacuum (UHV) conditions: All bridge sites are occupied by oxygen atoms and all cus sites are empty. For higher pressures three surface conditions are worth mentioning, which are e.g. obtained for $T = 600$K, $p_{\rm O_2}
= 1$atm, and $p_{\rm CO} = 10^{-2}, 20,$ and $10^{3}$atm.
$(i)$ At the low $p_{\rm CO}$, all bridge and all cus sites are covered with oxygen atoms (cf. Fig. \[snapshots\]a). This is essentially the thermodynamic high $p_{\rm O_2}$ phase [@Reuter-PRB-2002; @Reuter-PRL/B-2003]. There is a noticeable desorption/adsorption dynamics, i.e., about every 40$\mu$s two of the 900 O adatoms of the $(30 \times 30)$ simulation cell desorb (mainly from cus sites) as O$_2$. The resulting vacancies are then rapidly filled again (within 1ns), most of the time with oxygen. Only rarely CO adsorbs, and even if it does, it rather desorbs again than initiating a reaction. The overall CO${}_2$ formation rate under these conditions is with $0.9\!\cdot\!10^{12}$ cm$^{-2}$s$^{-1}$ very low.
$(ii)$ For $p_{\rm CO} = 20$atm the situation corresponds to what was previously suggested to be that of high chemical activity [@Reuter-PRL/B-2003], and this suggestion is now confirmed. The time to reach steady-state is remarkably long, in particular when compared to the pico second timescale of the underlying atomistic processes (cf. Fig. \[snapshots\]b). Finally, an interesting mix of CO molecules and O atoms at bridge and cus sites is established (a typical snapshot is shown in Fig. \[snapshots\]c), with average occupation numbers $N_{\rm CO}^{\rm br} = 0.11$, $N_{\rm CO}^{\rm cus} = 0.70$, $N_{\rm O}^{\rm br} = 0.89$, and $N_{\rm O}^{\rm cus} = 0.29$. The dynamics of this surface is extremely fast. It is mainly due to CO desorption and adsorption. The rate of CO$_2$ formation is a factor of 0.0004 lower, but truly significant, namely $4.6 \cdot 10^{18}\,{\rm cm}^{-2}\,{\rm s}^{-1}$. It is dominated by the reaction CO$^{\rm cus}+ {\rm O}^{\rm cus} \longrightarrow {\rm CO}_2$. The CO$^{\rm br} + {\rm O}^{\rm cus} \longrightarrow {\rm CO}_2$ and CO$^{\rm cus}+ {\rm O}^{\rm br} \longrightarrow {\rm CO}_2$ reactions also contribute, though their rates are by a factor of 0.30 and 0.01 lower than the first one. The CO$^{\rm br} + {\rm O}^{\rm br}
\longrightarrow {\rm CO}_2$ reaction is insignificant. This result is different from what one would expect from the reaction energy barriers (cf. table \[table1\]), which would give the lead to CO$^{\rm br}+
{\rm O}^{\rm cus}$, demonstrating the importance of the proper mix of surface-site occupations.
$(iii)$ When the CO pressure is increased further, to $p_{\rm CO} =
10^3$atm, the surface becomes fully covered with CO. This result is at variance with the “constrained thermodynamics” study in which the CO$_2$ formation was forbidden. Now we find that catalysis is practically poisoned by adsorbed CO. The encountered situation is in fact close to that where the RuO$_2$ catalyst will be reduced by CO to the pure Ru metal.
Figure \[TOF\] summarizes the results by showing the various steady-state surface structures as well as a map of the TOFs in ($p_{\rm CO}$, $p_{\rm O_2}$)-space. The highest activity is found to be in very good agreement with the early experimental results by Peden and Goodman [@Peden-1986], and occurs whenever the environmental conditions lead to the dynamic coexistence “phase” described above under $(ii)$. Recently, Wang [*et al.*]{} [@Wang-2002] have also measured the TOFs for the RuO$_2$(110) surface at $T =350$K for various pressures, and in Fig. \[Jacobi\] we compare their results with ours. The agreement is again far better than what one would have expected: The theoretical and experimental TOF values at the optimum pressures are practically identical, and also the optimum pressure conditions (when the “dynamic phase” is realized) agree very well. Because the errors due to the GGA are in the range of 0.1-0.3eV, this good agreement between theory and experiment, seen in Fig. \[Jacobi\], may seem fortuitous. However, it is worth noting that the position of the optimum catalytic efficiency in $(T,p)$-space and to some extent also the value of the TOF are not determined by the energetics of a singular process alone, but by the action of many players. Apparently, the DFT-GGA calculations describe the differences between the various surface processes better than the individual absolute values. In this respect it is important to realize that a combination of different calculations (employing different approximations) or of theory and experiment could have spoiled the description. The consistent treatment of all participating processes implies that the [*optimum mix*]{} of O and CO at the surface is described well: The abundance of CO$^{\rm cus}$-O$^{\rm cus}$ nearest neighbor pairs (as well as CO$^{\rm br}$-O$^{\rm cus}$ nearest neighbor pairs) is apparently playing a role of similar importance as the energy barriers. Particularly at the optimum TOF conditions there is also an effective compensation, e.g. when too high adsorption energies result in enhanced adsorption, but also in reduced reaction barriers etc. Away from the optimum TOF conditions such compensation effects become less effective, and modest differences between the theoretical and experimental results arise. Here DFT-GGA errors are more influential, and for the experimental data we expect that contributions from surface imperfections may play a bigger role.
In summary, we computed the surface kinetics of CO oxidation catalysis at RuO$_2$(110). The TOFs are presented in $(T,p_{\rm CO},p_{\rm
O_2}$)-space, clearly identifying a narrow region of highest catalytic activity. In this region kinetics builds an adsorbate composition that is not found anywhere in the thermodynamic surface phase diagram. The statistical analysis of the surface dynamics and of the various processes reveals several surprising results. For example, the chemical reaction with the most favorable energy barrier happens a factor of 0.30 less frequently than the energetically second favorable reaction.
The results also clarify how and when a bridging of the pressure gap between UHV studies and realistic pressure conditions is possible. The considered system has in fact two important components to this issue: At first, an O$_2$-rich environment changes the material from Ru to RuO$_2$. Thus, earlier high pressure studies on the Ru catalyst were actually looking at RuO$_2$. Second, after RuO$_2$ has been formed it is important to know how to set the pressure conditions correctly, as also on RuO$_2$(110) there is a low-pressure surface phase (the orange region in Fig. \[TOF\]a), that has little in common with the catalytically active situation. The obtained agreement between the theoretical results and experimental data confirms furthermore (cf. also Ref. [@Wang-2002]) that CO$_2$ is primarily formed from [*adsorbed*]{} CO and O, and that the metal-oxide, once it was created, does not play an active role, i.e., there is no indication of significant bulk diffusion. Thus, the catalysis is explained in terms of a Langmuir-Hinschlwood mechanism. Some contributions via the Eley-Rideal mechanism (e.g. scattering of gas-phase CO at O$^{\rm cus}$) may play a minor role, but this process requires further calculations and experiments. With the advent of first-principles TOF-maps obtained from DFT calculations (cf. Fig. \[TOF\]), a more detailed data base is in general becoming available for comparison with experiments, which will eventually advance the microscopic understanding of catalysis.
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|
---
abstract: 'We present a catalog of low-mass dense cores observed with the SHARC-II instrument at 350 $\mu$m. Our observations have an effective angular resolution of 10$''''$, approximately 2.5 times higher than observations at the same wavelength obtained with the [*Herschel Space Observatory*]{}, albeit with lower sensitivity, especially to extended emission. The catalog includes 81 maps covering a total of 164 detected sources. For each detected source, we tabulate basic source properties including position, peak intensity, flux density in fixed apertures, and radius. We examine the uncertainties in the pointing model applied to all SHARC-II data and conservatively find that the model corrections are good to within $\sim$3$''''$, approximately $1/3$ of the SHARC-II beam. We examine the differences between two array scan modes and find that the instrument calibration, beam size, and beam shape are similar between the two modes. We also show that the same flux densities are measured when sources are observed in the two different modes, indicating that there are no systematic effects introduced into our catalog by utilizing two different scan patterns during the course of taking observations. We find a detection rate of 95% for protostellar cores but only 45% for starless cores, and demonstrate the existence of a SHARC-II detection bias against all but the most massive and compact starless cores. Finally, we discuss the improvements in protostellar classification enabled by these 350 [$\mu$m]{} observations.'
author:
- 'Akshaya Suresh, Michael M. Dunham, Héctor G. Arce, Neal J. Evans II, Tyler L. Bourke, Manuel Merello, & Jingwen Wu'
bibliography:
- 'Suresh\_citations.bib'
- 'source\_citations.bib'
title: 'A Catalog of Low-Mass Star-Forming Cores Observed with SHARC-II at 350 microns'
---
Introduction
============
Stars form in dense cores of dust and molecular gas [e.g., @DiFrancesco2007; @wardthompson2007:ppv]. The ultraviolet, optical, and infrared radiation from both stars forming inside cores and the interstellar radiation field (ISRF) is absorbed by the dust within these cores, heating the dust to typical temperatures of $10-20$ K [e.g., @DiFrancesco2007]. The dust then re-radiates this emission at submillimeter and millimeter wavelengths. Thus, to study the very roots of star formation it is necessary to observe dense cores at these wavelengths.
Most nearby low-mass star-forming regions have been extensively surveyed at wavelengths between 850 [$\mu$m]{} and 1.3 mm due to the large number of bolometers available at these wavelengths and the high availability of weather suitable for $\sim$1 mm observations at most telescope sites [e.g., @Motte1998; @Testi1998; @Shirley2000; @Johnstone2000; @Johnstone2001; @Motte2001a; @Motte2001b; @Young2003; @kirk2005; @1MMSurveyPers; @Stanke2006; @Young2006; @1MMSurveyOph; @1MMSurveySerp; @Kauffmann2008]. Because they are both optically thin and in the Rayleigh-Jeans limit for $10-20$ K dust, continuum observations at $\sim$1 mm are ideally suited for tracing the total dust mass and easily pick out the dense star-forming cores [e.g., @1MMSurveySerp]. However, since the peaks of $10-20$ K blackbodies occur at $250-500$ [$\mu$m]{}, these millimeter wavelength surveys do not constrain the peaks of the spectral energy distributions (SEDs). Observations at shorter submillimeter wavelengths are needed to measure total source luminosities [@dunham2008:lowlum; @dunham2013:luminosities; @dunham2014:ppvi; @enoch2009:protostars], separate the contributions from internal (from a protostar) and external (from the ISRF) heating [@dunham2006:iram04191], and accurately classify protostars into an evolutionary sequence [@andre1993:class0; @chen1995:tbol; @dunham2008:lowlum; @dunham2014:ppvi; @frimann2015:synthetic1]. Such observations are being provided by the [*Herschel*]{} Gould Belt Survey, which has obtained $70-500$ $\mu$m images of all of the nearby, star-forming clouds in the Gould Belt [e.g., @andre2010:hgbs]. While this survey is providing unprecedented coverage and sensitivity of nearby star-forming regions at submillimeter wavelengths, it is doing so at relatively low angular resolution (ranging from approximately 9$''$ at 70 [$\mu$m]{} to 36$''$ at 500 [$\mu$m]{}).
In an effort to provide a submillimeter catalog of dense cores with high spatial resolution, we present in this paper 350 $\mu$m continuum observations of low-mass protostellar cores taken with the Submillimeter High Angular Resolution Camera II (SHARC-II) at the Caltech Submillimeter Observatory (CSO) on Mauna Kea, Hawaii. SHARC-II was a 350 and 450 $\mu$m “CCD-style” bolometer array of 12 $\times$ 32 pixels giving an instantaneous field of view (FOV) of 2.59$'$ $\times$ 0.97$'$, with the pixels filling over 90% of the focal plane and separated by their projected size on the sky of 4.85$''$ [@dowell2003:sharc]. With good focus and pointing, the 350 $\mu$m beam has a full-width at half-maximum (FWHM) of 8.5$''$. In practice the effective resolution at 350 $\mu$m is 10$''$ (see Section \[sec\_datareduction\]), providing 2.5 times higher angular resolution than the [*Herschel*]{} Gould Belt survey at the same wavelength (albeit at lower sensitivity, especially to extended emission). Targets were selected to provide complementary data to several large surveys of nearby, low-mass star-forming regions, including the [*Spitzer Space Telescope*]{} c2d [@evans2003:c2d; @evans2009:c2d] and Gould Belt [@gutermuth2008:serpsouth; @dunham2015:gb] Legacy Surveys and the [*Herschel Space Observatory*]{} DIGIT [e.g., @sturm2010:digit; @vankempen2010:digit; @green2013:digit; @green2016:cdf] open time Key Project. Some of the observations presented here were originally published by @Wu2007, but we have re-analyzed and included them here using an updated version of the data reduction software and improved pointing model corrections. In general, we present observations of regions that are already well documented at other wavelengths so that this catalog will complement existing data in studies of the characteristics of protostellar regions. Though this is the primary goal of the paper, we also use our data to examine the sensitivity of SHARC-II to extended emission and to the observing mode used on the telescope, and to assess two different methods of classifying protostars.
We organize this paper as follows: First we describe the target selection and observation strategy in Sections \[sec\_targets\] and \[sec\_observations\], respectively. We then discuss the data reduction and calibration processes in Sections \[sec\_datareduction\] and \[sec\_calibration\], respectively. We discuss the source extraction procedure in Section \[sec\_extraction\], including an analysis of the effects of optimizing the data reduction pipeline for the recovery of extended emission. We present our source catalog in Section \[sec\_results\], including source positions, flux densities, and radii. A comparison of the results from two different observing modes is presented in Section \[sec\_mode\]. In Section \[sec\_extended\] we discuss the sensitivity of our observations to extended emission, and in Section \[sec\_classification\] we investigate the effects of including SHARC-II 350 [$\mu$m]{} photometry when classifying protostars. A summary of our results is presented in Section \[sec\_summary\].
Observations
============
Target Selection {#sec_targets}
----------------
Table 1 lists the targets of this survey, including the name of the core/cloud, the scan type (see below), the map center coordinates ordered by increasing Right Ascension, the distance to the target and reference for the distance determination, a representative reference for each target, the observation date, the 1$\sigma$ rms noise in units of mJy beam$^{-1}$, and the large cloud complex in which each target is located. As there is significant ambiguity in choosing a single representative reference for each target, we refer the reader to the SIMBAD database[^1] for a comprehensive list of references for each object. The noise is measured as the standard deviation of all off-source pixels, calculated using the [sky]{} procedure in the IDL Astronomy Library. Two versions of each map are produced, one with and one without extended emission preserved (see Section \[sec\_data\_reduction\_calibration\] below for details); the noise is measured in the maps without extended emission preserved.
As described in Section 1, the main purpose of this survey is to provide complementary 350 $\mu$m observations of low-mass star-forming clouds and cores observed in various [*Spitzer*]{} and [*Herschel*]{} survey programs. Thus targets were selected from the lists of regions included in those surveys, often focusing on individual studies that would benefit from these data. As a result, the data presented here do not represent an unbiased submillimeter survey of star-forming regions, but a targeted survey designed to provide a catalog of useful complementary data.
Observations {#sec_observations}
------------
Observations were conducted at the CSO in 14 observing runs spread over seven years, ranging from May 2003 through December 2010. All of the data obtained between May 2003 and November 2005 were previously published (Wu et al. 2007); here we present updated images and catalogs using a newer version of the data reduction software (see Section 3). These data were obtained using the sweep mode of SHARC-II, which moves the telescope following Lissajous curves in both the x and y dimensions. This mode, which utilizes scan rates between 5–10 arcseconds s$^{-1}$ depending on the exact size mapped, results in a map with a fully sampled central region of uniform coverage, beyond which the coverage decreases and thus the noise increases. The size of this central region depends on the exact observing parameters, but is typically $\sim$ 1$'$ $-$ 2$'$ for our observations. Beginning in December 2006, we began experimenting with using the box-scan observing mode to map larger areas. This mode, which utilizes faster scan rates (typically in the range of 20–40 arcseconds s$^{-1}$), moves the telescope in a straight line at a 45$^{\rm o}$ angle until it hits the boundary of a box, changes direction such that the angle of reflection equals the angle of incidence, and continues until the box is fully sampled. The exact size of the box depends on the observing parameters, but is typically $\sim$ 6$'$ $-$ 10$'$ for our observations. As the box-scan mode is optimized for mapping both larger areas and regions with extended emission, all data obtained during and after July 2008 were obtained exclusively in this mode. Some sources were observed in both the Lissajous and box-scan observing modes. In those cases, only the box-scan observations are listed in Table 1. The Lissajous observations for these sources will be discussed in Section 5, where we compare results from the two observing modes.
The total integration time was typically $30-120$ minutes per map, depending on weather conditions and the expected brightness of sources in each map. The noise levels of the final maps span nearly two orders of magnitude, ranging from approximately 8 to 500 mJy beam$^{-1}$ (see below). Thus we caution that these maps form a very heterogeneous dataset in terms of sensitivity. Integrations for each map were broken into individual scans, each with a duration ranging from $5-15$ minutes depending on the stability of the atmosphere and the minimum time required to complete one scan in the chosen observing mode. The zenith optical depth at 225 GHz ranged from $0.03-0.09$, with values of $\sim 0.05-0.07$ most typical. With an approximate scaling factor of 20, these correspond to 350 $\mu$m zenith optical depths of $\sim 0.6-1.8$, with values of $\sim 1-1.4$ most typical. During all of our observations except those obtained in June 2005 (see Wu et al. 2007 for more details), the Dish Surface Optimization System (DSOS)[^2] was used to correct the dish surface figure for gravitational deformations as the dish moves in elevation during observations. The pointing and focus were both checked and updated every $1-2$ hr each night, primarily with the planets Mars, Uranus, and Neptune, but occasionally with other secondary calibrators chosen from the SHARC-II website[^3]. The pointing was further corrected in reduction based on a pointing model (see Section 3).
Data Reduction and Calibration {#sec_data_reduction_calibration}
==============================
Data Reduction {#sec_datareduction}
--------------
The data were reduced using the Comprehensive Reduction Utility for SHARC-II (CRUSH) version 2.12-1, a publicly available[^4], Java-based software package that iteratively solves for both the source signal and the various correlated noise components [e.g., @Kovaks2008a; @Kovaks2008b]. For increased redundancy, we use CRUSH to add together the bolometer time streams from individual scans before obtaining the solutions, taking into account various noise elements and changing atmospheric conditions for each scan. The only exception to this general method is NGC1333, which had to be broken into five smaller pieces first due to computer memory limitations. These five pieces were then coadded together using the [coadd]{} function of the CRUSH package, after which point they were handled exactly the same way as the other sources. As described in more detail in Section \[sec\_extraction\] below, two versions of each map were produced: one with the extended flag given to CRUSH to optimize the software pipeline for the recovery of extended emission, and one without it given. The differences between and uses of the two versions of each map are discussed below.
The atmospheric opacity during each scan was determined from an online database[^5] of measurements of the zenith optical depth at 225 GHz, $\tau_{\rm 225 GHz}$, obtained in ten minute intervals. This database includes polynomial fits to $\tau_{\rm 225 GHz}$ versus time for each night, where the orders of the polynomials were treated as free parameters. The orders range from 3 to 90 over the full database, but are typically less than 20, with a mean value of 13. We visually inspected plots of $\tau_{\rm 225 GHz}$ and the resulting polynomial fits for each night to verify that the fits accurately trace the variations in $\tau_{\rm 225 GHz}$ throughout the night. CRUSH uses these polynomial fits to calculate the optical depth at the time each observation scan was taken and uses this value to correct the signal. Pointing corrections were applied to each scan with CRUSH to correct for residual telescope pointing errors. These corrections were determined using a publicly available model[^6] fit to all pointing data from that observing run. We applied this model using the option to also correct for a random drift with time by evaluating model residuals for pointing scans taken within a few hours before and after each science scan. The final maps were generated with 1.5$''$ pixels. A Gaussian smoothing function with a FWHM of 4 pixels was applied to each map to reduce pixelation artifacts, resulting in a final effective beam of approximately 10$''$.
In order to determine the residual pointing uncertainty after applying the pointing model corrections, Figure \[fig\_boxlissdist\] shows a histogram of the distances between measured peak positions for several sources that were observed twice on two different dates, after applying the pointing model corrections. The mean distance is 2.9$''$. In order to interpret this in terms of a residual rms uncertainty in the pointing model, we construct a Monte Carlo model in which a source is placed on a polar coordinate grid twice, with each angle drawn randomly from a uniform distribution and each radius drawn randomly from a Gaussian distribtuion with $\sigma = 1$. Note that this $\sigma$ has no units as we are only concerned with obtaining a dimensionless ratio that will characterize the pointing model uncertainty, as described below. The distribution of distances between the two “observations” of each source in the Monte Carlo model is shown in Figure \[fig\_montecarlo\]; this distribution has a mean of 1.2. Since the input to the simulation was a pointing model with an assumed rms of 1, and the resulting mean of the distance distribution is 20% larger, we thus infer that our observed mean distance between peak positions of 2.9$''$ implies an underlying pointing model residual rms of 2.4$''$. Given the small number of sources observed twice and uncertainties in the assumptions of the Monte Carlo model, we thus conservatively estimate that the pointing model corrections are good to within $\sim 3''$.
Once the maps were created using CRUSH, we used the [imagetool]{} function of the CRUSH package to eliminate map edges with increased noise by removing all pixels in the map with less than 25% of the maximum integration time. The average 1$\sigma$ rms for each map was calculated by using the [sky]{} routine in the IDL Astronomy Library to measure the standard deviation of all off-source pixels and then calibrating with the peak calibration factor, as defined below.
Calibration {#sec_calibration}
-----------
While CRUSH adopts an approximate calibration factor to produce maps in calibrated units of Jy beam$^{-1}$, it does not account for the fact that the instrument calibration changes both with observing conditions and randomly with time[^7]. Furthermore, the presence of beam sidelobes mean that the flux of a point source measured in apertures of increasing size will also increase, whereas ideally the flux of a point source should be independent of aperture size. We thus recalibrate all of our data as follows.
To derive calibration factors, we used observations of Mars, Neptune, and Uranus, which were observed every few hours to check and update the telescope pointing. These planet scans, which we hereafter refer to as calibration scans, were reduced with CRUSH and used to calculate the calibration factors. All calibration scans were observed using the Lissajous observing mode on the telescope, regardless of what observing mode was used for the science sources to which these calibrations were applied. As shown in Section \[sec\_mode\], the calibration factors do not depend on observing mode, validating this strategy. For each calibration scan, we measured the peak intensity and flux densities in 20$''$ and 40$''$ diameter apertures using custom IDL routines. We choose these aperture sizes to match previous (sub)millimeter continuum surveys that measure flux densities in standard apertures of 20$''$, 40$''$, 80$''$, and 120$''$ in diameter [e.g., @1MMSurveyPers; @1MMSurveyOph; @Young2006; @1MMSurveySerp; @Wu2007; @Kauffmann2008]. Here we only adopt the two smallest apertures since our calibration images are too small to use larger apertures. By comparing the measured flux densities to the known fluxes of these planets, we calculate three calibration factors: C$_{\rm peak}$, C$_{\rm 20"}$, and C$_{\rm 40"}$. C$_{\rm peak}$ is simply the factor required to obtain calibrated maps in units of Jy beam$^{-1}$. Since the maps are already calibrated with an approximate calibration factor, the values of C$_{\rm peak}$ are unitless and are generally close to unity, as seen below. C$_{\rm 20"}$ and C$_{\rm 40"}$ are “aperture calibration factors” and have units of Jy / (Jy beam$^{-1}$) $=$ beam. Multiplying the flux densities measured through aperture photometry by the aperture calibration factors for the same size aperture will give the flux density in that aperture in Jy. As mentioned above, this method corrects for the beam sidelobes such that the measured flux density of a point source is independent of aperture size [e.g., @Shirley2000; @1MMSurveyPers].
In practice, C$_{\rm peak}$, C$_{\rm 20"}$, and C$_{\rm 40"}$ are calculated as follows. For C$_{\rm peak}$, the expected peak intensity from the planet was calculated by convolving the SHARC-II beam (assumed to be Gaussian) with a disk of uniform brightness, using the known size and flux density of each planet on that observation date. The resulting values are then divided by the measured peak intensity in each calibration scan. For C$_{\rm 20"}$ and C$_{\rm 40"}$, the known total flux density of the planet on each observation date was divided by the measured flux density in 20$''$ and 40$''$ diameter apertures, respectively, using the uncalibrated maps. The flux densities were measured using standard aperture photometry with no sky subtraction since CRUSH removes the background sky emission. Table \[tab\_calibrators1\] lists our derived calibration factors for each observation night, and Table \[tab\_calibrators2\] lists the means and standard deviations of the calibration factors for each observing run. Each map is calibrated using the mean calibration factors for that run; maps consisting of scans obtained over multiple runs are calibrated using the mean values over those runs. For maps observed in runs where no planet scans were taken, the mean calibration factors over all runs were used.
Figure \[fig\_histcal\] shows histograms for each of the calibration factors. To investigate whether the derived calibration factors vary with time or depend on the properties of the calibration source, Figures \[fig\_calovertime\], \[fig\_caloverradius\], and \[fig\_caloverflux\] plot the calibration factors versus observation date, angular size of the calibration source, and total flux of the calibration source. No systematic variations are seen, and linear least squares fits to each panel in Figures \[fig\_caloverradius\] and \[fig\_caloverflux\] give better fits (lower reduced $\chi^2$ values) for zeroth order polynomial than for first order polynomials, indicating that any dependences of the calibration factors on source properties are smaller than the overall calibration uncertainties. We calculate these calibration uncertainties by dividing the standard deviations by the means, resulting in values of 18% for C$_{\rm peak}$, 21% for C$_{\rm 20"}$, and 17% for C$_{\rm 40"}$. We thus conservatively adopt an overall calibration uncertainty of 25%.
Source Extraction {#sec_extraction}
-----------------
As noted above, two versions of each map were produced, one with and one without the extended flag given to CRUSH. This flag optimizes the CRUSH pipeline for extended sources and results in maps that preserve extended emission at the cost of increased noise [see @Kovaks2008a for details]. Since our targets are star-forming regions expected to feature extended emission, we investigated the possibility of using this flag to recover more extended emission. However, we found that the added noise is dominated by sky noise that is temporally correlated in the time stream and thus spatially correlated in the final maps, rather than random, and thus especially problematic for source extraction. Many false sources are detected and extracted regardless of the detailed implementation of source extraction. Figure \[fig\_wandwcompare\] shows an example of a map reduced with and without the extended flag. Similar color figures are presented for all of the regions mapped below. All maps are displayed with a linear intensity scaling with the minimum (black) and maximum (white) intensities given in their respective figure captions. Figure \[fig\_scalebar\] displays a normalized intensity color scale bar that, combined with the minimum and maximum intensities given in the figure captions, can be used to determine the absolute intensities of each map.
Figure \[fig\_wandwcompare\] illustrates that the extended flag results in large areas of correlated noise that are extracted as sources. Thus, in order to ensure a reliable catalog of sources, we perform source extraction on the maps produced without the extended flag (such as the one shown in the right panel of Figure \[fig\_wandwcompare\]). We extracted sources from each map using the Bolocat[^8] source extraction routine [@rosolowsky2010:bolocat]. Bolocat works by identifying regions of statistically significant emission based on their significance with respect to a local estimate of the noise in the maps. These regions of high significance are then subdivided into multiple sources based on the presence of local maxima within the originally defined regions, with each pixel assigned to one of the sources using a seeded watershed algorithm, similar to the Clumpfind or Source EXtractor algorithms [@williams1994:clumpfind; @bertin1996]. Bolocat was previously used to extract sources from SHARC-II images of massive star-forming clumps by @merello2015, demonstrating the feasibility of using this source extraction routine on data from the SHARC-II instrument.
Bolocat requires three input parameters, all of which are measured in units of the map rms: $P_{\rm amp}$, the minimum required amplitude for a source to be extracted; $P_{\rm base}$, the base level of emission out to which the initial detected source is expanded; and $P_{\rm deb}$, a source deblending parameter. In practice, Bolocat first masks all regions of the map below $P_{\rm amp}$ and extracts one or more initial sources based on the number of regions of contiguous pixels remaining in the masked map. It then expands each of these initial sources to encompass all contiguous pixels down to the level $P_{\rm base}$, with the rationale being that marginally significant emission levels spatially connected to those of higher significance are likely to be real. Finally, each initial source is deblended into one or more sources by finding pairs of local maxima and identifying them as separate sources if the lower of the two is at least $P_{\rm deb}$ above the highest contour level that contains both maxima [see @rosolowsky2010:bolocat for details]. We adopted values of $P_{\rm amp} = 3$, $P_{\rm base} = 3$, and $P_{\rm deb} = 2$ based on matching “by-eye” extractions in an initial exploration of the parameter space (note that, since $P_{\rm amp} = P_{\rm base}$, we did not expand the sources beyond their initial detections). While @merello2015 also adopted $P_{\rm amp} = 3$, they adopted lower values of the other two parameters ($P_{\rm base} = 1$ and $P_{\rm deb} = 1$). We found that the adoption of such low values for $P_{\rm base}$ and $P_{\rm deb}$ resulted in extractions that did not match either our best “by-eye” results or known sources from catalogs at other wavelengths. In particular, the lower value of $P_{\rm deb}$ resulted in several objects that were clearly single being broken into multiple objects due to small noise variations. The advantage of studying known regions with copious multiwavelength data in the literature, as opposed to @merello2015, allowed us to refine the best values of the source extraction parameters.
After extracting sources and measuring their deconvolved radii using the maps reduced without the extended flag, aperture photometry was performed on the maps produced with the extended flag, at the peak position of each extracted source. We used custom IDL routines to measure the peak intensities and flux densities in 20$''$ and 40$''$ diameter apertures. In cases of overlapping apertures with nearby sources, only those up to the largest in which overlap did not occur were kept. We measured the uncertainties in the flux densities in each aperture by adding in quadrature the statistical uncertainty from the measurement itself and the overall calibration uncertainty of 25%.
This method of using both sets of maps ensures that a reliable catalog of sources is extracted while also preserving as much extended emission as possible in the measured flux densities. Figure \[fig\_histwandw\] shows the distribution of ratios of flux densities measured in the maps with the extended flag to those measured in the maps without the extended flag, for the peak intensities as well as the flux densities measured in 20$''$ and 40$''$ diameter apertures. The mean (median) ratios are 1.7 (1.5), 2.2 (1.8), and 3.4 (2.6) for the peak, 20$''$, and 40$''$ measurements. Thus, as expected, we see that the extended flag improves the flux recovery, especially in the larger apertures that are more sensitive to the amount of extended emission.
Results {#sec_results}
=======
Figures \[fig\_L1448\] – \[fig\_multiple12\] show contour maps overlaid on images for each of the 81 regions listed in Table \[tab\_properties1\]. Since the source extraction routine uses a local noise measurement but the contours are plotted using a global noise measurement, weak sources in low-noise regions may lack associated 3$\sigma$ contours. The images are of the maps reduced without the extended emission flag. Most maps are displayed in six-panel figures except for the largest and most crowded maps, which are instead displayed as one-panel figures. Maps with multiple sources have each source labeled. All of the reduced FITS files are available in calibrated units of Jy beam$^{-1}$ following the calibration procedures described above; versions with and without the extended flag can be accessed through the Data Behind the Figures (DBF) feature of the journal.
We detect a total of 164 sources in the 81 maps listed in Table \[tab\_properties1\] and presented in Figures \[fig\_L1448\] – \[fig\_multiple12\]. Table \[tab\_sources\] lists, for each detected source, the name of the source, the map in which the source is covered, the peak position of the source, the deconvolved angular source radius as determined by Bolocat [see @rosolowsky2010:bolocat for details], the peak intensity, the flux density in a 20$''$ diameter aperture, the flux density in a 40$''$ diameter aperture, and a flag noting whether each source is starless or protostellar, determined mostly from a search for an infrared point source in [*Spitzer Space Telescope*]{} images from the c2d [e.g., @evans2009:c2d] and Gould Belt [e.g., @dunham2015:gb] legacy projects. The peak intensities are given in units of Jy beam$^{-1}$ (for the SHARC-II beam, 1 Jy beam$^{-1}$ = 519.7 MJy sr$^{-1}$).
Effects of Scan Mode {#sec_mode}
====================
As noted in Section \[sec\_observations\], all observations prior to 2006 were taken in the Lissajous scan mode. However, over the course of taking observations we were made aware of a different observing mode, the box scan mode, that might yield better results. In particular, maps observed in the box scan mode have much larger areas than those observed in the Lissajous mode. Since all emission on scales larger than the map size will be treated as sky emission and removed by CRUSH, the larger map areas provided by the box scans may provide better recovery of extended emission. The use of the box scan mode also allows us to map larger areas in reasonable amounts of time. Thus, all science data obtained during and after the July 2008 observing run were obtained exclusively in the box scan mode, with observations between December 2006 and July 2008 obtained in both modes for testing purposes.
Calibration scans were taken with the Lissajous mode, even after July 2008, since one Lissajous mode scan can be completed in much less time than one box scan. To justify this choice, we obtained several calibration scans in the box-scan observing mode. The calibration factors obtained from these scans are listed in Table \[tab\_calibrators3\], using the same method as above for the Lissajous calibration scans. Averaged over all of the box calibration scans, we calculate C$_{\rm beam} = 1.63 \pm 0.15$, C$_{\rm 20} = 0.034 \pm 0.004$, and C$_{\rm 40} = 0.025 \pm 0.003$, where the uncertainties are the standard deviations. Comparison to Table \[tab\_calibrators2\] shows that the mean Lissajous and box-scan calibration factors agree to within 10%, well within the overall calibration uncertainty of 25%, indicating that any dependence on observing mode in the SHARC-II instrument calibration has a negligible impact on our results.
We also compared the SHARC-II beams resulting from the Lissajous and box scans. Figure \[fig\_boxlisscal\] compares the size and shape of the beams derived from the Lissajous and box calibration scans. The beam profiles are derived by deconvolving the measured properties (sizes and elongations) of the calibration sources with their known, intrinsic properties. For the beam FWHM, the means and standard deviations of the means are 9.3$''$ $\pm$ 0.2$''$ for the Lissajous scans and 10.0$''$ $\pm$ 0.2$''$ for the box scans. For the beam aspect ratio, the means and standard deviations of the means are 1.06 $\pm$ 0.01 for the Lissajous scans and 1.12 $\pm$ 0.01 for the box scans. Thus, observations in the box scan mode have a beam profile that is, on average, 8% larger and 6% more elongated than observations in the Lissajous mode. While the degradation of the beam profile in the box scan mode is a statistically robust finding, resulting from both the faster scan rates used by the box scan modes and astrometric errors in the SHARC-II pixel plate scale that build up over larger scan areas, the overall effects are less than 10% and have no significant impact on our results.
Finally, we also investigated whether the measured flux densities of our science targets depended on the observing mode. To do this, we observed several science sources in both observing modes. For the 23 maps observed in both modes, Tables \[tab\_properties1\] and \[tab\_sources\] only present information for the box scans; the target and source information for the Lissajous scans are given in Tables \[tab\_lissajous\] and \[tab\_lissajous2\]. We extracted sources from these extra Lissajous maps and measured flux densities using the same methods as described above. Figures \[fig\_lissmultiple1\] – \[fig\_lissmultiple4\] show contour maps overlaid on images for these extra Lissajous maps, using the maps reduced without the extended emission flag. All of the reduced FITS files used to produce these figures are available through the Data Behind the Figures (DBF) feature of the journal. These maps are given in calibrated units of Jy beam$^{-1}$ following the calibration procedures described above, and both the versions with and without the extended flag are provided. For the sources detected in both modes, Figure \[fig\_radboxliss\] plots the ratios of the flux densities calculated in maps observed in the box mode to those observed in the Lissajous mode versus the radius of each source determined from the box scan observations. The means and standard deviations of these ratios are $0.93 \pm 0.18$, $0.97 \pm 0.17$, and $1.09 \pm 0.29$ for the peak intensities, 20$''$ flux densities, and 40$''$ flux densities, respectively. In all three cases the mean ratios are within 10% of unity. Given the overall calibration uncertainty of 25% and the fact that the ratios show no dependence on source radius, we conclude that similar amounts of flux are recovered between the two observing modes on scales up to at least 40$''$.
Sensitivity to Extended Emission {#sec_extended}
================================
In addition to the detected sources listed in Table \[tab\_sources\], Table \[tab\_undetected\] lists an additional 48 cores covered by our maps but not detected in our SHARC-II observations. These are cores identified by other observations at submillimeter and millimeter wavelengths, based on SIMBAD[^9] searches of the total area covered by our maps. Combining the information from these tables, our maps cover a total of 137 protostellar cores, 130 of which are detected, and a total of 75 starless cores, 34 of which are detected. Our detection rate of 95% for protostellar cores is thus much higher than our detection rate of 45% for starless cores. These results suggest that SHARC-II is well suited for identifying and characterizing protostellar cores, but is not ideal for studying starless cores.
Of the maps observed in both the Lissajous and box scan modes, there are six starless cores. Two are detected in both modes (L1455-IRS2E and B18-4), two are detected in only the box scan mode (L1544 and L694-2), and two are undetected in both observing modes (Perseus Bolo 107 and L1521B). The lower detection rate for starless cores in the Lissajous mode (33%) versus the box scan mode (45%), coupled with the fact that two starless cores detected in the box scan mode are not detected in the Lissajous mode, suggest that the box scan mode is somewhat better suited to detecting starless cores, although we caution that the sample sizes are very small.
The low detection rate for starless cores is likely explained by the fact that, compared to protostellar cores, starless cores feature flatter density profiles and colder temperatures in their central regions [e.g., @wardthompson2007:ppv; @evans2001:starless]. Consequently starless cores exhibit 350 $\mu$m intensity profiles that are significantly shallower and less centrally condensed than those for protostellar sources, as confirmed by @Wu2007 using simple, one-dimensional radiative transfer models. The more extended nature of their emission profiles make them harder to separate from sky emission, thus they are less reliably detected. To further quantify this effect, we examined all of the starless cores covered by our maps that were identified in 1.1 mm Bolocam surveys of Perseus [@1MMSurveyPers], Ophiuchus [@1MMSurveyOph], and Serpens [@1MMSurveySerp]. Figure \[fig\_starless\] shows histograms of the core masses, sizes, and mean densities, as derived from the Bolocam observations, for both the detected and undetected populations of starless cores in our dataset. The detected starless sources span nearly the full range of masses, FWHM angular sizes, and mean densities, indicating there is no one unique property that determines the detectability of the core.
Figure \[fig\_starless2\] plots the FWHM angular size versus core mass for these same starless cores, again taking the properties from the 1.1 mm Bolocam surveys cited above. Inspection of this figure shows that, for low-mass cores ($M < \sim$2 ), only the most compact cores are detected. These cores will have the steepest intensity profiles among all cores with such masses, allowing them to be separated from sky emission and reliably detected. Only for relatively high-mass (and thus relatively bright) cores ($M > \sim$3 ) are more extended cores able to be separated from sky emission and detected by SHARC-II observations. Thus, starless cores revealed by SHARC-II surveys of star-forming regions are biased toward the most compact or highest mass cores, and even in cases where starless cores are detected, the extended nature of their intensity profiles means that the measured flux densities are likely lower limits to the true flux densities. These cautions should be kept in mind when interpreting the results from @zhang2014, who derive a prestellar core mass function based on SHARC-II observations of Ophiuchus. This core mass function for detected prestellar cores may not be representative of the full population of such cores in this cloud, and furthermore the measured masses of the detected prestellar cores likely underestimate their true masses. Since the detectability of a starless core with SHARC-II is not a simple function of the total flux density of the core, but also its emission profile, measured upper limits for undetected cores do not necessarily represent true limits to the flux densities of these cores. While we do list the 1$\sigma$ upper limits in Table \[tab\_undetected\] (taken directly from Table \[tab\_properties1\]), this caution should be kept in mind when interpreting the non-detections.
While we caution that the measured flux densities of starless cores are likely lower limits, the measured values for the protostellar cores are much more reliable. @Wu2007 used simple, one-dimensional radiative transfer models to show that protostellar cores exhibit significantly steeper intensity profiles compared to starless cores, even for protostars with luminosities as low as 0.1 . These steeper intensity profiles lead to more compact emission that is fully recovered by SHARC-II, at least on scales up to the 40$''$ considered here. This is confirmed by comparing the SHARC-II observations of IRAM 04191+1522 and L1521F, two protostars with luminosities less than 0.1 , to published radiative transfer models; in both cases the observed flux densities in 40$''$ diameter apertures agree with the models to within 2$\sigma$ [@dunham2006:iram04191; @bourke2006:l1521f].
Classification of Protostars {#sec_classification}
============================
Protostars are commonly classified into one of two classes, Class 0 and Class I, based on observational signatures that trace the underlying evolutionary state. Class 0 protostars were first defined by @andre1993:class0, who defined such objects observationally as protostars emitting a relatively large fraction (greater than 0.5%) of their total luminosity at wavelengths $\lambda \geq 350$ [$\mu$m]{}. Defining such luminosity as the submillimeter luminosity, , Class 0 objects are then protostars with $>$ 0.005. The corresponding physical Stage 0 objects are young, embedded protostars with greater than 50% of their total system mass still in the core [@andre1993:class0]. Another quantity used to classify protostars is the bolometric temperature , defined by @myers1993:tbol as the temperature of a blackbody with the same flux-weighted mean frequency as the source. By calculating for a large sample of young stars, Chen et al. (1995) showed that Class 0 objects have $<$ 70 K whereas Class I protostars have $\geq$ 70 K. Since has historically been difficult to calculate accurately due to the difficulty in obtaining high-quality submillimeter data from the ground, particularly at 350 $\mu$m, the criterion introduced by Chen et al. is often used instead for classifying protostars into their two Classes [e.g., @enoch2009:protostars; @dunham2013:luminosities; @tobin2016:vandam]. However, as several studies have shown that is less sensitive to viewing geometry and a better tracer of underlying physical stage than [@andre2000:ppiv; @young2005:evolmodels; @dunham2010:evolmodels; @frimann2015:synthetic1], protostellar classification must be revisited as additional data becomes available.
With the 350 [$\mu$m]{} photometry presented here, we can now accurately calculate [to within 20%–60%; see @dunham2008:lowlum; @enoch2009:protostars for details] both and and compare classification via the two quantities. To ensure as uniform a dataset as possible, we consider only the protostellar cores in Perseus, and construct SEDs for each source consisting of [*Spitzer Space Telescope*]{} 3.6–70 [$\mu$m]{} photometry from @evans2009:c2d and @dunham2015:gb and Bolocam 1.1 mm photometry from @1MMSurveyPers. We calculate twice, once without the SHARC-II 350 [$\mu$m]{} photometry included and once with it included, and we also calculate .
Figure \[fig\_classification\_tbol\] compares the two values of , calculated with and without the 350 [$\mu$m]{} photometry included. Leaving out the 350 [$\mu$m]{} photometry increases the value of , with the effect growing in significance with decreasing evolutionary stage (colder values of ). This result is explained by the fact that, for more deeply embedded protostars, the emission peaks at longer wavelengths, and more of this emission is lost when no submillimeter photometry is available. As demonstrated by Figure \[fig\_classification\_tbol\], Class 0 protostars with very low values of can masquerade as more evolved objects when 350 [$\mu$m]{} photometry is lacking. These results are in qualitative agreement with earlier investigations by @dunham2008:lowlum and @enoch2009:protostars.
Figure \[fig\_classification\_compare\] plots versus for these same protostars as a way of comparing the two different classification methods. Such a comparison is only possible when 350 [$\mu$m]{} data is available, since accurate calculation of requires 350 [$\mu$m]{} data. With the Class boundaries as defined by @chen1995:tbol and the Class boundaries as defined by @andre1993:class0, the two classification methods agree for only 65% of the protostars considered (35 out of 54). In particular, there are many protostars classified as Class I by ( $>$ 70 K) but Class 0 by ( $>$ 0.005). Revised Class 0/I boundaries have been proposed in the literature, including 0.01 [@andre2000:ppiv; @sadavoy2014:class0] and 0.03 [@maury2011:aquila]. Adopting these boundaries increases the agreement between classification methods to 67% and 74%, respectively.
While the primary focus of this publication is to provide a catalog of SHARC-II 350 [$\mu$m]{} observations of nearby star-forming regions, these results demonstrate the critical role played by 350 [$\mu$m]{} observations in both accurate classification of protostars and assessing the reliability of different classification methods in tracing the underlying evolutionary stages of protostars. A more complete investigation of protostellar classification, using complete far-infrared and submillimeter SEDs provided by both ground-based surveys such as this effort and surveys with the *Herschel Space Observatory*, will be presented in a future paper (M. M. Dunham et al., in preparation).
Summary {#sec_summary}
=======
In this paper we have presented a catalog of low-mass star-forming cores observed with the SHARC-II instrument at 350 $\mu$m. Our observations have an effective angular resolution of 10$''$, approximately three times higher than observations at the same wavelength obtained with the [*Herschel Space Observatory*]{}. A summary of our results is as follows:
- We present 81 maps covering a total of 164 detected sources. We tabulate basic source properties including position, peak intensity, flux density in fixed apertures, and radius.
- We examine the uncertainties in the pointing model applied to all SHARC-II data and conservatively find that the model corrections are good to within $\sim$3$''$, approximately $1/3$ of the SHARC-II beam.
- We examined the differences between the Lissajous and box scan observing modes. We find that the calibration factors, beam size, and beam shape are similar between the two modes, and we also show that the same flux densities are measured when sources are observed in the two different modes. Thus we conclude that there are no systematic effects in our catalog introduced by switching observing modes during the course of taking observations.
- We find that less than half of the starless cores observed are detected by SHARC-II (45% to be precise), and show that the detections are biased toward the most compact or highest mass starless cores. We argue that, even for the detected starless cores, the measured flux densities are likely lower limits to the intrinsic flux densities.
- For protostellar cores, our SHARC-II observations fully recover the emission, at least up to the 40$''$ scales considered here.
- We demonstrate that the inclusion of 350 [$\mu$m]{} photometry significantly improves the accuracy of calculated values of , and enables comparison between two different measures of protostellar Class, and . The latter can only be calculated when 350 [$\mu$m]{} photometry is available.
We thank the referee for helpful comments that have improved the quality of this publication. We gratefully acknowledge the assistance provided by the staff of the CSO in obtaining the observations presented here. We also acknowledge the numerous students and postdocs from the University of Texas at Austin who participated in observing runs over the years, and we thank Darren Dowell and Attila Kov[á]{}cs for technical assistance with SHARC-II and CRUSH. Finally, we express our profound gratitude to everybody who played a role in the construction, commissioning, and operation of the CSO over its three decades of operation. This work is based on data obtained with the Caltech Submillimeter Observatory (CSO), which was operated by the California Institute of Technology under cooperative agreement with the National Science Foundation (AST-0838261). This publication makes use of data products from the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. These data were provided by the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA. This research has made use of NASA’s Astrophysics Data System (ADS) Abstract Service, the IDL Astronomy Library hosted by the NASA Goddard Space Flight Center, and the SIMBAD database operated at CDS, Strasbourg, France. MMD acknowledges support from the Submillimeter Array as an SMA postdoctoral fellow, and from NASA ADAP grant NNX13AE54G. NJE acknowledges support from NSF Grant AST-1109116 to the University of Texas at Austin.
[^1]: Available at: http://simbad.u-strasbg.fr
[^2]: See http://www.cso.caltech.edu/dsos/DSOS\_MLeong.html
[^3]: See http://www.submm.caltech.edu/$\sim$sharc/analysis/calibration.htm
[^4]: See http://www.submm.caltech.edu/$\sim$sharc/crush/index.htm
[^5]: http://www.submm.caltech.edu/$\sim$sharc/analysis/taufit.htm
[^6]: See http://www.submm.caltech.edu/$\sim$sharc/analysis/pmodel
[^7]: As described in the online CRUSH documentation for SHARC-II data reduction (available at: http://www.submm.caltech.edu/$\sim$sharc/crush/instruments/sharc2/), the exact calibration factor that CRUSH applies depends on the line-of-sight optical depth at 350 $\mu$m. Other factors that affect the calibration factor but are not automatically accounted for by CRUSH include the detector temperature, optical configuration, cleanliness of the mirrors, focus quality, and DSOS status.
[^8]: Available at https://github.com/low-sky/idl-low-sky/tree/master/bolocat
[^9]: http://simbad.u-strasbg.fr
|
---
abstract: 'Let $1\rightarrow (K,K_1)\rightarrow (G,N_G(K_1))\rightarrow(Q,Q_1)\rightarrow 1$ be a short exact sequence of pairs of finitely generated groups with $K$ strongly hyperbolic relative to proper subgroup $K_1$. Assuming that for all $g\in G$ there exists $k\in K$ such that $gK_1g^{-1}=kK_1k^{-1}$, we prove that there exists a quasi-isometric section $s\colon Q \to G$. Further we prove that if $G$ is strongly hyperbolic relative to the normalizer subgroup $N_G(K_1)$ and weakly hyperbolic relative to $K_1$, then there exists a Cannon-Thurston map for the inclusion $i\colon\Gamma_K\to \Gamma_G$.'
author:
- Abhijit Pal
title: 'Relative Hyperbolic Extensions of Groups and Cannon-Thurston Maps'
---
[Stat-Math Unit, Indian Statistical Institute, 203 B.T.Road,\
Kolkata 700108, Email: abhijit\_r@isical.ac.in]{}
Introduction {#sec:1}
============
Let us consider the short exact sequence of finitely generated groups $$1\rightarrow K\rightarrow G\rightarrow Q\rightarrow 1.$$ such that $K$ is non-elementary word hyperbolic. In [@mosher], Mosher proved that if $G$ is hyperbolic, then $Q$ is hyperbolic. To prove $Q$ hyperbolic, Mosher (in [@mosher]) constructed a quasi-isometric section from $Q$ to $G$, that is, a map $s\colon Q\to G$ satisfying $$\frac{1}{k}d_Q(q,q')-\epsilon \leq d_G(s(q),s(q'))\leq kd_Q(q,q')+\epsilon,$$ for all $q,q'\in Q$, where $d_G$ and $d_Q$ are word metrics and $k\geq1$,$\epsilon\geq 0$ are constants. In [@mit0], existence of a Cannon-Thurston map for the embedding $i\colon\Gamma_K\to\Gamma_G$ was proved, where $\Gamma_K$, $\Gamma_G$ are respectively the Cayley graphs of $K$, $G$. Here in this paper, we will generalize these results to the case where the kernel is strongly hyperbolic relative to a cusp subgroup.
One of our main theorems states:\
[**[Theorem]{} \[sec thm\]**]{} Suppose we have a short exact sequence of finitely generated groups $$1\rightarrow K\rightarrow G\stackrel{p}{\rightarrow} Q\rightarrow 1,$$ with $K$ strongly hyperbolic relative to a subgroup $K_1$ and for all $g\in G$ there exists $k\in K$ such that $gK_1g^{-1}=kK_1k^{-1}$ then there exists a $(k,\epsilon)$ quasi-isometric section $s\colon Q \to G$ for some constants $k\geq1$,$\epsilon\geq 0$.
As a corollary of this theorem, under the same hypothesis, we can take the image of quasi-isometric section to be in $N_G(K_1)$.
Let $S$ be a once-punctured torus then its fundamental group $\Pi_1(S)={\mathbb F}(a,b)$ is strongly hyperbolic relative to the peripheral subgroup $H=<aba^{-1}b^{-1}>$. Let $M$ be a $3$-manifold fibering over circle with fiber $S$ such that the fundamental group $\Pi_1(M)$ is strongly hyperbolic relative to the subgroup $H\bigoplus\mathbb{Z}$. Then we have a short exact sequence of pairs of finitely generated groups:
$$1\rightarrow(\Pi_1(S),H)\rightarrow(\Pi_1(M),H\bigoplus{{{\mathbb Z}}})\rightarrow({{\mathbb Z}},{{\mathbb Z}})\rightarrow {1}.$$ Let $K=\Pi_1(S)$, $G=\Pi_1(M)$ and let $\Gamma_K$, $\Gamma_G$ be Cayley graphs of $K$, $G$ respectively. Bowditch (in [@bow; @ct1]) and Mahan Mj. (in [@Mj; @bdd]), proved the existence of a Cannon-Thurston map for the embedding $i\colon\Gamma_K\to\Gamma_G$.
Motivated by this example, we will prove the following theorem :\
**Theorem \[end thm\]** Suppose we have a short exact sequence of pairs of finitely generated groups $$1\rightarrow (K,K_1)\stackrel{i}\rightarrow (G,N_G(K_1))\stackrel{p}{\rightarrow}(Q,Q_1)\rightarrow 1$$ with $K$ strongly hyperbolic relative to the cusp subgroup $K_1$ and for all $g\in G$ there exists $k\in K$ such that $gK_1g^{-1}=kK_1k^{-1}$. If $G$ is strongly hyperbolic relative to $N_G(K_1)$ and weakly hyperbolic relative to the subgroup $K_1$, then there exists a Cannon-Thurston map for the embedding $i\colon \Gamma_K\to \Gamma_G$, where $\Gamma_K$ and $\Gamma_G$ are Cayley graphs of $K$ and $G$ respectively.
**Acknowledgments:** I would like to thank my advisor, Mahan Mj., for suggesting me this problem and for his useful comments.
Relative Hyperbolicity and Quasi-isometric Section {#rel hyp}
===================================================
For generalities of hyperbolic groups and hyperbolic metric spaces refer to [@hyp]. Gromov defined relatively hyperbolic group as follows:
(Gromov [@gro]) Let $G$ be a finitely generated group acting freely and properly discontinously by isometries on a proper and $\delta$-hyperbolic metric space $X$, such that the quotient space $X/G$ is quasi-isometric to $[0,\infty)$. Let $H$ denote the stabilizer subgroup of the endpoint on $\partial X$ of a lift of this ray to $X$. Then $G$ is said to be strongly hyperbolic relative to $H$. The subgroup $H$ is said to be a *Parabolic or Cusp* subgroup and the end point on $\partial X$ as *parabolic* end point.
Thus for a group $G$ strongly hyperbolic relative to the subgroup $H$ there is a natural bijective correspondence between parabolic end points and parabolic subgroups of $G$. Infact, a parabolic end point corresponds to a subgroup of the form $aHa^{-1}$ for some $a\in G$.
(Farb [@farb]) Let $G$ be a finitely generated group, and let $H$ be a finitely generated subgroup of $G$. Let $\Gamma _G$ be the Cayley graph of $G$. Let ${\widehat}\Gamma_G$ be a new graph obtained from $\Gamma_G$ as follows:
For each left coset $gH$ of $H$ in $G$, we add a new vertex $v(gH)$ to $\Gamma_G$, and add an edge $e(gh)$ of length $1/2$ from each element $gh$ of $gH$ to the vertex $v(gH)$. We call this new graph the *Coned-off* Cayley graph of $G$ with respect to $H$, and denote it by ${\widehat}\Gamma_G={\widehat}\Gamma_G(H)$.
We say that $G$ is **weakly hyperbolic** relative to the subgroup $H$ if the Coned-off Cayley Graph ${\widehat}\Gamma_G$ is hyperbolic.
Geodesics in the coned-off space ${\widehat}\Gamma_G$ will be called as electric geodesics. For a path $\gamma \subset \Gamma_G$, there is an induced path ${\widehat}{\gamma}$ in ${\widehat}{\Gamma_G}$ obtained by identifying $\gamma$ with ${\widehat}{\gamma}$ as sets. If ${\widehat}{\gamma}$ is an electric geodesic (resp. $P$-quasigeodesic), $\gamma$ is called a [*relative geodesic*]{} (resp.[*relative $P$-quasigeodesic*]{}). If ${\widehat}\gamma$ passes through some cone point $v(gH)$, we say that ${\widehat}\gamma$ [*penetrates*]{} the coset $gH$.
(Farb [@farb]) ${\widehat}\gamma$ is said to be an [**electric $(K, \epsilon)$-quasigeodesic in (the electric space) ${\widehat}{\Gamma_G}$ without backtracking** ]{} if ${\widehat}\gamma$ is an electric $K$-quasigeodesic in ${\widehat}{\Gamma_G}$ and ${\widehat}\gamma$ does not return to any left coset after leaving it.
(Farb [@farb])[**Bounded Coset Penetration Properties:**]{}\[bcp\] The pair $(G,H)$ is said to satisfy *bounded coset penetration property* if, for every $P\geq 1$, there is a constant $D=D(P)$ such that if $\alpha$ and $\beta$ are two electric $P$-quasigeodesics without backtracking starting and ending at same points, then the following conditions hold :
1. If $\alpha$ penetrates a coset $gH$ but $\beta$ does not penetrate $gH$, then $\alpha$ travels a $\Gamma_G$-distance of at most $D$ in $gH$.
2. If both $\alpha$ and $\beta$ penetrate a coset $gH$, then vertices in $\Gamma_G$ at which $\alpha$ and $\beta$ first enter $gH$ lie a $\Gamma_G$-distance of at most $D$ from each other; similarly for the last exit vertices.
Next we recall Farb’s definition of relatively hyperbolic group (in the strong sense) from [@farb]:
(Farb [@farb]) $G$ is said to be strongly hyperbolic relative to $H$ if $G$ is weakly hyperbolic relative to $H$ and the pair $(G,H)$ satisfies bounded coset penetration property.
(Bowditch [@bow], Szczepanski [@ski]) $G$ is strongly hyperbolic relative to the cusp subgroup $H$ in the sense of Farb if and only if $G$ is strongly hyperbolic relative to $H$ in the sense of Gromov.
(Bowditch [[@bow]]{}) [**Relative Hyperbolic Boundary:**]{} Let $G$ be a strongly hyperbolic group relative to $H$, then by Gromov’s definition $G$ acts properly discontinuously on a proper hyperbolic space $X$. The relative hyperbolic boundary of $G$ is the Gromov boundary, $\partial X$, of $X$. We denote the relative hyperbolic boundary of the pair $(G,H)$ by $\partial \Gamma(G,H)$.
Bowditch in [@bow] showed that the relative hyperbolic boundary $\partial \Gamma(G,H)$ is well defined.
Let $1\rightarrow K\rightarrow G\rightarrow Q\rightarrow 1$ be a short exact sequence of finitely generated groups with $K$ strongly hyperbolic relative to $K_1$. We say that **$G$ preserves cusps** if $gK_1g^{-1}=a_gK_1a^{-1}_g$ for some $a_g\in K$.
(Mosher [@mosher]) [**[Quasi-isometric section :]{}**]{} Let $1\rightarrow K\rightarrow G\rightarrow Q\rightarrow 1$ be a short exact sequence of finitely generated groups. A map $s\colon Q\to G$ is said to be a $(R,\epsilon)$ quasi-isometric section if $$\frac{1}{R}d_Q(q,q')-\epsilon \leq d_G(s(q),s(q'))\leq Rd_Q(q,q')+\epsilon,$$ for all $q,q'\in Q$, where $d_G$ and $d_Q$ are word metrics and $R\geq1$,$\epsilon\geq 0$ are constants.
Let $K$ be a strongly hyperbolic group relative to a cusp subgroup $K_1$. For each parabolic point $\alpha\in \partial \Gamma(K,K_1)$, there is a unique subgroup of the form $aK_1a^{-1}$. Now Hausdorff distance between two sets $aK_1$ and $aK_1a^{-1}$ is uniformly bounded by the length of the word $a$. Hence $\alpha$ corresponds to a left coset $aK_1$ of $K_1$ in $K$.
Let $1\rightarrow K\stackrel{i}\rightarrow G\stackrel{p}{\rightarrow} Q\rightarrow 1$ be a short exact sequence of finitely generated groups with $K$ non-elementary and strongly hyperbolic relative to a subgroup $K_1$.
We use the following notations for our further purpose:
- Let $\Pi$ be the set of all parabolic end points for the relatively hyperbolic group $K$ with cusp subgroup as $K_1$.
- Let $\Pi^2= \{(\alpha_1,\alpha_2): \alpha_1~\mbox{and}~\alpha_2~\mbox{are distinct elements in}~\Pi \}$.
- For $a\in K$, let $i_a\colon K\to K$ denote the inner automorphism and $L_a\colon K\to K$ the left translation.
- For $g\in G$, let $I_g\colon K\to K$ be the outer automorphism, that is, $I_g(k)=gkg^{-1}$ and $L_g\colon G\to G$ be the left translation.
$G$ preserves cusps, so for each $g\in G$ there exists $a_g\in K$ such that $a_g^{-1}g\in N_G(K_1)$. If $b\in K$, then it can be easily proved that $d_K(a_gK_1,gbg^{-1}a_gK_1)\leq d_K(K_1,bK_1) + 2l_K(a_g^{-1}g)$. Since $I_g(bK_1)=g(bK_1)g^{-1}=gbg^{-1}a_gK_1a_g^{-1}$ and Hausdorff distance between $gbg^{-1}a_gK_1$ and $gbg^{-1}a_gK_1a_g^{-1}$ is bounded, $I_g$ will induce a map $\tilde I_g\colon \Pi \to \Pi$ and $\tilde I_g$ is a bijection. Therefore, $\tilde I_g$ will induce a bijective map $\tilde I^2_g\colon \Pi^2\to \Pi^2$. For the sake of convenience of notation we will use $I_g$ for $\tilde I_g$ and $\tilde I^2_g$. Similarly, for $a\in K$, $i_a$ and $L_a$ will induce bijective maps (with same notation) from $\Pi$ to $\Pi$ and $\Pi^2$ to $\Pi^2$.
The following theorem is generalization of Mosher’s technical result Quasi-isometric section lemma to the relatively hyperbolic case.
\[sec thm\] Suppose we have a short exact sequence of finitely generated groups $$1\rightarrow K\stackrel{i}\rightarrow G\stackrel{p}{\rightarrow} Q\rightarrow 1,$$ such that $K$ is strongly hyperbolic relative to a subgroup $K_1$ and $G$ preserves cusps, then there exists a $(R,\epsilon)$ quasi-isometric section $s\colon Q \to G$ for some $R\geq1$,$\epsilon\geq 0$.
[**Proof.**]{} Let $\alpha=(\alpha_1,\alpha_2)\in \Pi^2$, then stabilizer subgroups of $\alpha_i$’s are $a_iK_1a_i^{-1}$ for some $a_i\in K$, where $i=1,2$. Let $\lambda$ be a relative geodesic in $\Gamma_K$ with starting at some point of $a_1K_1$ and ending at some point of $a_2K_1$ and ${\widehat}\lambda$ passing through cone points $v(a_iK_1)$, where $i=1,2$. Let $x$ be the exit point of $\lambda$ from the left coset $a_1K_1$. If $\mu$ is another such relative geodesic with end points same as $\lambda$ and $y$ as its exit point from $a_1K_1$, due to bounded coset penetration property 2, $d_{K}(x,y)\leq D$, where $D$ is the constant as in definition \[bcp\]. Let $B_{\alpha}$ be the set of all exit points of relative geodesics $\lambda$ with starting at some point of $a_1K_1$ and ending at some point of $a_2K_1$ and ${\widehat}\lambda$ passing through $v(a_iK_1)$’s, $i=1,2$. Then $B_{\alpha}$ is a bounded set with diameter less than or equal to $D$.
Let $C=\{\alpha\in \Pi^2\colon e_K\in B_\alpha\}$, where $e_K$ is the identity element in $K$. We fix an element $\eta = (\eta_1,\eta_2)\in \Pi ^2$. Let $\Sigma=\{g\in G : \eta \in I_g(C)\}$. $\Sigma$ will be proved to be image of a quasi-isometric section.\
We first prove that, for any $g\in G$, ${\cup}_{a\in K}I_{ga}(C)=\Pi^2$:
Let $\alpha = (\alpha_1,\alpha_2)\in \Pi^2$. Now $\alpha_i$ corresponds to left coset $a_iK_1$, where $i=1,2$. Let $\lambda$ be a relative geodesic in $\Gamma_K$ with starting at some point of $a_1K_1$ and ending at some point of $a_2K_1$ and ${\widehat}\lambda$ passing through cone points $v(a_iK_1)$, $i=1,2$, and $x_\alpha$ its exit point from $a_1K_1$. Then $x_\alpha\in B_\alpha$. Now there exists $k\in K$ such that $L_k(x_\alpha)=e_K$. Since $L_k$ is an isometry $L_k(\lambda)$ will be a relative geodesic joining points from $ka_iK_1$’s, $i=1,2$, with ${\widehat}{L_k(\lambda)}$ containing cone points $v(ka_iK_1)$, $i=1,2$, and $e_K$ being the exit point of $L_k(\lambda)$ from $ka_1K_1$. There exists $\beta_i\in \Pi$ such that $\beta_i$ corresponds to left coset $ka_iK_1$, $i=1,2$. Then $\beta=(\beta_1,\beta_2)\in \Pi^2$ and $e_K\in B_\beta$. Therefore $L_k(\alpha)=\beta\in C$. Since $L_k$ and $i_k$ are same on relative hyperbolic boundary, we have $i_k(\alpha)\in C$. Thus $\cup_{a\in K}(i_a(C))=\Pi^2$. Consequently, for any $g\in G$, $\cup_{a\in K}I_{ga}(C)=\cup_{a\in K}I_gi_a(C)=I_g(\cup_{a\in K}(i_a(C)))=I_g(\Pi^2)=\Pi^2$.
Now we prove that $p(\Sigma)=Q$:
Let $q\in Q$, then there exists $g\in G$ such that $p(g)=q$. Now ${\cup}_{a\in K}I_{ga}(C)=\Pi^2$ for any $g\in G$. Therefore for $\eta\in \Pi^2$ there exists $a\in K$ such that $\eta\in I_{ga}(C)$. Hence $ga\in \Sigma$ and $p(ga)=p(g)=q$.
Now we prove that there exists constant $R\geq 1,\epsilon \geq 0$ such that for all $g,g'\in \Sigma$ $$\frac{1}{R}d_Q(p(g),p(g'))-\epsilon \leq d_G(g,g') \leq R d_Q(p(g),p(g'))+\epsilon.$$
We can choose a finite symmetric generating set $S$ of $G$ such that $p(S)$ is also a generating set for $Q$. Obviously, $d_Q(p(g),p(g'))\leq d_G(g,g')$ for all $g,g'\in G$. To prove $d_G(g,g')\leq R d_Q(p(g),p(g'))+ \epsilon$ for all $g,g'\in \Sigma$, it suffices to prove that there exists $R\geq 1$ such that if $d_Q(p(g),p(g'))\leq 1$ for some $g,g'\in \Sigma$, then $d_G(g,g')\leq R$.
Let $d_Q(p(g),p(g'))\leq 1$ for some $g,g'\in \Sigma$. Then $g^{-1}g'=ka$ for some $k\in K$ and $a$ is either identity in $G$ or a generator of $G$. Since $g,g'\in \Sigma$, $I_g(C)\cap I_{g'}(C)\neq \Phi$. Hence $I_{ka}(C)\cap C = I_{g^{-1}g'}(C)\cap C \neq \Phi$. Now $I_{ka}=i_k(I_a)$, therefore $i_k(I_a(C))\cap C \neq \Phi$.
For each $\alpha\in\Pi^2$, we choose an element $a_\alpha\in B_\alpha$. Define a map $F\colon\Pi^2\to\Gamma_K$ by $F(\alpha)=a_\alpha$.
Since $L_k$ is an isometry, for $k\in K$, $ka_{\alpha}\in B_{k\alpha}$ and hence $$\label{hd}
d_K(a_{k\alpha},ka_\alpha)=d_K(F(k\alpha),kF(\alpha))\leq D,$$ where $k\alpha$ denotes the image of $\alpha$ under the map $L_k\colon\Pi^2\to\Pi^2$.
Let $B_D(e_K)$ be the closed $D$-neighborhood of $e_K$. Now $F(C)$ is contained in union of $B_\alpha$’s containing identity $e_K$. Therefore $F(C)$ is contained in $B_D(e_K)$. Since $G$ preserves cusps, there exists $s\in K$ such that $F(I_a(C))$ is contained in union of $B_\alpha$’s containing $s$ and hence $F(I_a(C))\subset B_D(s)$, where $B_D(s)$ is a closed $D$-neighborhood of $s$. From (\[hd\]), Hausdorff distance between two sets $F(kI_a(C))$ and $kF(I_a(C))$ is bounded by $D$. For a set $A\subset\Gamma_K$, let $N_D(A)$ denotes the closed $D$-neighborhood of $A$. Thus $$\begin{aligned}
F(kI_a(C))\subset N_D(kF(I_a(C)))=kN_D(F(I_a(C)))\subset kB_{2D}(s).\end{aligned}$$ Now $K$ acts properly discontinuously on $\Gamma_K$, therefore $$B_D(e_K)\cap kB_{2D}(s)\neq \Phi$$ for finitely many $k$’s in $K$. This implies $F(C)\cap F(kI_a(C))\neq\Phi$ for finitely many $k$’s in $K$. And hence $C\cap L_k(I_a(C))=C\cap kI_a(C)\neq\Phi$ for finitely many $k$’s in $K$. $L_k=i_k$ on relative hyperbolic boundary, so $C\cap (I_{ka}(C))\neq\Phi$ for finitely many $k$’s in $K$. Thus $g^{-1}g'= ka$ for finitely many $k$’s. Since number of generators of $G$ is finite, there exists a constant $R\geq 1$ such that $d_G(g,g')\leq R$.
Now we define $s\colon Q\to G$ as follows:
Let $q\in Q$ and let there exists $g,g'\in \Sigma$ such that $p(g)=p(g')=q$. Then by above inequality $d_G(g,g')\leq R$. We choose one element $g\in p^{-1}(q)\cap\Sigma$ for each $q\in Q$ and define $s(q)=g$. Then $s$ defines a single valued map satisfying : $$\frac{1}{R}d_Q(q,q')-\epsilon \leq d_G(s(q),s(q'))\leq Rd_Q(q,q')+\epsilon.$$ for some constants $R\geq 1$, $\epsilon \geq 0$ and for all $q,q'\in Q$.
\[pair thm\] Suppose we have a short exact sequence of pairs of finitely generated groups
$$1\rightarrow (K,K_1)\rightarrow (G,N_G(K_1))\stackrel{p}{\rightarrow}(Q,Q_1)\rightarrow 1$$ with $K$ strong relative hyperbolic with respect to the cusp subgroup $K_1$. If $G$ preserves cusps, then $Q_1=Q$ and there is a quasi-isometric section $s\colon Q\to N_G(K_1)$ satisfying
$$\frac{1}{R}d_Q(q,q')-\epsilon \leq d_{N_G(K_1)}(s(q),s(q'))\leq Rd_Q(q,q')+\epsilon$$ where $q,q'\in Q$ and $R\geq1$,$\epsilon \geq 0$ are constants. Further, if $G$ is weakly hyperbolic relative to $K_1$, then $Q$ is hyperbolic.
[**Proof.**]{}Let $q\in Q$, then there exists $g\in G$ such that $p(g)=q$. Since $G$ preserves cusps, $gK_1g^{-1}=aK_1a^{-1}$ for some $a\in K$. Therefore $a^{-1}g\in N_G(K_1)$ and $q=p(a^{-1}g)\in Q_1$. Thus $Q_1=Q$.
Let $\Pi^2_{K_1}=\{(\alpha_1,\alpha_2)\in\Pi^2:\alpha_1~\mbox{corresponds to subgroup}~K_1\}$ and $C=\{\alpha\in\Pi^2_{K_1}:e_K\in B_\alpha\}$, where $B_\alpha$ is defined as in above theorem. We fix an element $\eta\in\Pi^2_{K_1}$ and set $\Sigma=\{g\in N_G(K_1): \eta\in I_g(C)$. We choose a finite generating set $S$ of $G$ such that it contains a finite generating set of $N_G(K_1)$ and $p(S)$ is also a generating set of $Q$. Using argument same as above theorem, by replacing $G$ with $N_G(K_1)$, we get a quasi-isometric section $s\colon Q\to N_G(K_1)$ satisfying : $$\frac{1}{R}d_Q(q,q')-\epsilon \leq d_{N_G(K_1)}(s(q),s(q'))\leq Rd_Q(q,q')+\epsilon.$$ for some constants $R\geq 1$, $\epsilon \geq 0$ and for all $q,q'\in Q$.
Since $d_Q(q,q')\leq d_G(s(q),s(q'))\leq d_{N_G(K_1)}(s(q),s(q'))$, we can take the quasi-isometric section $s\colon Q \to N_G(K_1)$ such that
$\frac{1}{R}d_Q(q,q')-\epsilon \leq d_G(s(q),s(q'))\leq Rd_Q(q,q')+\epsilon$.
Now, let $\Gamma_G^{pel}$ denotes the electrocuted space obtained from $\Gamma_G$ by coning left cosets of $K_1$ in $G$. Since $G$ is weakly hyperbolic with respect to $K_1$, $\Gamma_G^{pel}$ is hyperbolic. We will prove that $Q$ is hyperbolic.
The quasi-isometric section $s\colon Q\to N_G(K_1)(\subset G)$ will induce a map ${\widehat}s\colon Q\to \Gamma_G^{pel}$. Now for all $q,q'\in Q$, $d_{G^{pel}}({\widehat}s(q),{\widehat}s(q'))\leq d_G(s(q),s(q'))\leq R~d_Q(q,q')+\epsilon$, where $d_{G^{pel}}$ is the metric on $\Gamma_G^{pel}$. Obviously, $d_Q(q,q')\leq d_{G^{pel}}({\widehat}s(q),{\widehat}s(q'))$. Hence ${\widehat}s$ is a quasi-isometric section from $Q$ to $\Gamma_G^{pel}$. Therefore $s(Q)$ is quasiconvex in $\Gamma_G^{pel}$. Since $\Gamma_G^{pel}$ is hyperbolic, $Q$ is hyperbolic.
Existence of Cannon-Thurston Maps {#c-t map}
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Let $G$ be a group strongly hyperbolic relative to a subgroup $G_1$. Then there exists a complete hyperbolic space $Z$ such that $G$ acts properly discontinuously on $Z$ by isometries and $Z/G$ is quasi-isometric to $[0,\infty)$. Let $p$ be the parabolic end point corresponding to a lift of this ray. Then the subgroup stabilising $p$ is equal to $gG_1g^{-1}$ for some $g\in G$. Let $\Pi$ be the set of all parabolic end points and for each $p\in \Pi$, let $B(p)$ be a closed $G_1$ invariant such that $B(p)\cap B(q)=\Phi$ for all distinct parabolic end points $p,q$. Let $X'=Z\setminus\bigcup_{p\in\Pi}B(p)$. Then $X'$ is quasi-isometric to $\Gamma_G$.
Let $X=\Gamma_G$ and $H_{gG_1}$ be the closed set in $\Gamma_G$ corresponding to the left coset $gG_1$ of $G_1$ in $G$. Let $\mathcal{H}_G$ $=\{H_{gG_1}:g\in G\}$. Let $X_h$ be the space obtained from $X$ by gluing $H_{gG_1}\times [0,\infty)$ to $H_{gG_1}$ for all $H_{gG_1}\in \mathcal{H}_G$ where $H_{gG_1}\times [0,\infty)$ is equipped with the path metric $d_h$ such that
1. $d_{h,t}((x,t),(y,t)) = 2^{-t}d_H(x,y)$, where $d_{h,t}$ is the induced path metric on $H_t=H\times \{t\}$.
2. $d_h((x,t),(x,s))=\vert t-s \vert$ for all $x\in H$ and for all $t,s\in [0,\infty)$.
Then $X_h$ is quasi-isometric to $Z$.
Thus if $G$ is strongly hyperbolic relative to $G_1$, there exists a complete hyperbolic metric space $X_h$ such that $G$ acts properly discontinuously on $X_h$ and $X_h$ is obtained from $X$ as above. $H_{gG_1}\in \mathcal{H_G}$ was referred to as **horosphere-like sets** by Mahan Mj. and Lawerence Reeves in [@Mj; @com] and $H_{gG_1}\times [0,\infty)$ was referred to as **hyperbolic cones or horoball-like sets** in [@pal].
\[Mj ct\] Let $\hat \lambda$ be an electric quasi-geodesic in the electric space $\hat X$ without backtracking. For any $H$ penetrated by $\hat \lambda$, let $x_H$ and $y_H$ be the first entry point and the last exit point of $\hat \lambda$. We join $x_H$ and $y_H$ by a hyperbolic geodesic segment in $H_h$ (identifying ${\widehat}{X}$ with the space obtained from $X_h$ by coning off the $H_h$’s. This results in a path $\lambda_h$ in $X_h$. The path $\lambda$ will be called an [**electro-ambient quasigeodesic**]{}.
An electro-ambient quasigeodesic is a quasigeodesic in the hyperbolic space $X_h$.
Consider the inclusion between pairs of relatively hyperbolic group $(H,H_1)\stackrel{i}\hookrightarrow (G,G_1)$. $i$ will induce a proper embedding $i\colon \Gamma_H \to \Gamma_G$. Let $X=\Gamma_G$ and $Y=\Gamma_H$. Recall that $X_h$ is the space obtained from $X$ by gluing the hyperbolic cones. Inclusion of a horosphere-like set in its hyperbolic cone is uniformly proper, therefore inclusion of $X$ in $X_h$ is uniformly proper i.e. for all $M>0$ and $x,y\in X$ there exists $N>0$ such that if $d_{X_h}(x,y)\leq M$ then $d_G(x,y)=d_X(x,y)\leq N$.
Since $G$ preserves cusps, $i$ will induce a proper embedding $i_h\colon Y_h\to X_h$.
\[cann-thu map\] A Cannon-Thurston map for $i\colon (\Gamma_H,{\mathcal{H_H}})\to (\Gamma_G,\mathcal{H_G})$ is said to exist if there exists a continuous extension $\tilde{i_h}\colon Y_h\cup \partial Y_h\to X_h\cup \partial X_h$ of $i_h\colon Y_h\to X_h$.
To prove the existence of Cannon-Thurston map for the inclusion $i\colon (K,K_1)\to (G,N_G(K_1))$, we need the notion of Partial Electrocution.
(Partial Electrocution) Let $(X, {{\mathcal H}}, {{\mathcal G}}, {{\mathcal L}})$ be an ordered quadruple such that the following holds:
1. $X$ is hyperbolic relative to a collection of subsets $H_\alpha$.
2. For each $H_\alpha$ there is a uniform large-scale retraction $g_\alpha : H_\alpha \rightarrow L_\alpha$ to some (uniformly) $\delta$-hyperbolic metric space $L_\alpha$, i.e. there exist $\delta , K, \epsilon > 0$ such that for all $H_\alpha$ there exists a $\delta$-hyperbolic $L_\alpha$ and a map $g_\alpha : H_\alpha \rightarrow L_\alpha$ with $d_{L_\alpha} (g_\alpha (x), g_\alpha (y)) \leq Kd_{H_\alpha}(x,y)
+ \epsilon $ for all $x, y \in H_\alpha$. Further, we denote the collection of such $g_\alpha$’s as ${{\mathcal G}}$.
The [**partially electrocuted space**]{} or [*partially coned off space*]{} corresponding to $(X, {{\mathcal H}}, {{\mathcal G}}, {{\mathcal L}})$ is obtained from $X$ by gluing in the (metric) mapping cylinders for the maps $g_\alpha : H_\alpha \rightarrow L_\alpha$.
(Mahan Mj. and Lawrence Reeves [@Mj; @com])\[pel-track\] Given $K, \epsilon \geq 0$, there exists $C > 0$ such that the following holds:\
Let $\gamma_{pel}$ and $\gamma$ denote respectively a $(K, \epsilon )$ partially electrocuted quasigeodesic in $(X,d_{pel})$ and a hyperbolic $(K, \epsilon )$-quasigeodesic in $(X_h,d_h)$ joining $a, b$. Then $\gamma \cap X$ lies in a (hyperbolic) $C$-neighborhood of (any representative of) $\gamma_{pel}$. Further, outside of a $C$-neighborhood of the horoballs that $\gamma$ meets, $\gamma$ and $\gamma_{pel}$ track each other.
Let $G$ be a group strongly hyperbolic relative to the subgroup $G_1$. Let $X_h$ be the complete hyperbolic space obtained from $X$. We describe a special type of quasigeodesic in $X_h$ which will be essential for our purpose:
(Mahan Mj. [@Mj; @ct]) We start with an electric quasi-geodesic $\hat \lambda$ in the electric space $\hat X$ without backtracking. For any horosphere-like set $H$ penetrated by $\hat \lambda$, let $x_H$ and $y_H$ be the respective entry and exit points to $H$. We join $x_H$ and $y_H$ by a hyperbolic geodesic segment in $H\times [0, \infty)$. This results a path in, say $\lambda$, in $X_h$. The path $\lambda$ will be called an [**electro-ambient quasigeodesic**]{}.
(Mahan Mj. [@Mj; @ct])\[e-a thm\]: An electro-ambient quasigeodesic is a quasigeodesic in the hyperbolic space $X_h$.
For the rest of paper, we will work with the following pair of short exact sequence of finitely generated groups :
$$1\rightarrow (K,K_1)\rightarrow (G,N_G(K_1))\stackrel{p}{\rightarrow}(Q,Q_1)\rightarrow 1$$
Since all groups are finitely generated, we can choose a finite generating set $S$ of $G$ such that $S$ contains finite generating set of $K,K_1,N_G(K_1)$ and $p(S)$ is also a finite generating set of $Q$. We will assume the hypothesis of Theorem \[pair thm\]. As a consequence, $Q_1=Q$ and there exists a $(R,\epsilon)$ quasi-isometric section $s\colon Q \to N_G(K_1)$ such that
$\frac{1}{R}d_Q(q,q')-\epsilon \leq d_G(s(q),s(q'))\leq Rd_Q(q,q')+\epsilon$.
Further, we assume that $G$ is strongly hyperbolic relative to the subgroup $N_G(K_1)$.
Also, using a left translation $L_k$ by an element $k\in K_1$, we can assume that $s(Q)$ contains identity element $e_K$ of $K$ and $s(Q)\subset N_G(K_1)$.
We have assumed that $G$ is weakly hyperbolic relative to the subgroup $K_1$ and hence the coned-off space $\Gamma_G^{pel}$ obtained by coning left cosets $gK_1$ of $K_1$ to a point $v(gK_1)$ is hyperbolic. As $\Gamma_Q$ is quasi-isometrically embedded in $\Gamma_G^{pel}$, $Q$ is hyperbolic. We have also assumed that $G$ is strongly hyperbolic relative to $N_G(K_1)$. Thus $\Gamma_G^{pel}$ becomes a partially electrocuted space obtained from $\Gamma_G$ by partial electrocuting the closed sets (horosphere-like sets) $H_{gN_G(K_1)}$ in $\Gamma_G$ corresponding to the left cosets $gN_G(K_1)$ to the hyperbolic space $g(s(Q))$, where $g(s(Q))$ denotes the image of $s(Q)$ under the left translation $L_g$ for $g\in G$.
Let $\lambda^b={\widehat}\lambda\setminus\mathcal{H_K}$ denotes the portions of ${\widehat}\lambda$ that does not penetrate horosphere-like sets in $\mathcal{H_K}$. The following Lemma gives a sufficient condition for the existence of Cannon-Thurston map for the inclusion $i\colon (\Gamma_K,{\mathcal{H_K}})\to (\Gamma_G,\mathcal{H_G})$. For instance see [@pal].
\[suff cond\][@pal] A Cannon-Thurston map for $i\colon (\Gamma_K,{\mathcal{H_K}})\to (\Gamma_G,\mathcal{H_G})$ exists if there exists a non-negative function $M(N)$ with $M(N)\rightarrow\infty$ as $N\rightarrow\infty$ such that the following holds:
Given $y_0\in \Gamma_K$ and an electric quasigeodesic segment ${\widehat}\lambda$ in ${\widehat}\Gamma_K$ if $\lambda^b={\widehat}\lambda\setminus\mathcal{H_K}$ lies outside an $N$-ball around $y_0\in \Gamma_K$, then for any partially electrocuted quasigeodesic $\beta_{pel}$ in $\Gamma_G^{pel}$ joining end points of ${\widehat}\lambda$, $\beta^b=\beta_{pel}\setminus\mathcal{H_G}$ lies outside an $M(N)$-ball around $i(y_0)$ in $\Gamma_G$.
Construction of Quasiconvex Sets and Retraction Map {#Qc set}
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Recall that for $g\in G$, $L_g\colon G\to G$ denotes the left translation by $g$ and $I_g\colon K\to K$ denotes the outer automorphism. Let $\phi_g= I_{g^{-1}}$ then $\phi_g(a)=g^{-1}ag$. Since $L_g$ is an isometry, $L_g$ preserves distance between left cosets of $G_1$ in $G$. Hence $L_g$ induces a isometry ${\widehat}L_g\colon \Gamma_G^{pel}\to\Gamma_G^{pel}$. The embedding $i\colon\Gamma_K\to \Gamma_G$ will induce an embedding $\hat i\colon{\widehat}\Gamma_K\to\Gamma_G^{pel}$.
Let ${\widehat}\lambda$ be an electric geodesic segment in ${\widehat}\Gamma_K$ with end points $a$ and $b$ in $\Gamma_K$. Let ${\widehat}\lambda_g$ be an electric geodesic in ${\widehat}\Gamma_K$ joining $\phi_g(a)$ and $\phi_g(b)$.
Define $$B_{{\widehat}\lambda}=\bigcup_{g\in s(Q)}{{\widehat}L_g.\hat{i}({\widehat}\lambda_g)}.$$
On ${\widehat}\Gamma_K$, define a map $\pi_{{\widehat}\lambda_g}\colon{\widehat}\Gamma_K\to {\widehat}\lambda_g$ taking $k\in {\widehat}\Gamma_K$ to one of the points on ${\widehat}\lambda_g$ closest to $k$ in the metric $d_{{\widehat}K}$.
(Mitra [@mit1])\[npp\] For $\pi_{{\widehat}\lambda_g}$ defined above, $$d_{{\widehat}K}(\pi_{{\widehat}\lambda_g}(k),\pi_{{\widehat}\lambda_g}(k'))\leq Cd_{{\widehat}K}(k,k')+C$$ for all $k,k'\in {\widehat}\Gamma_K$, where $C$ depends only on the hyperbolic constant of ${\widehat}\Gamma_K$.
Now define $\Pi_{{\widehat}\lambda}\colon \Gamma_G^{pel}\to B_{{\widehat}\lambda}$ as follows:
For $g\in s(Q)$, $\Pi_{{\widehat}\lambda}.{{\widehat}L_g}.{{\widehat}i}(k)= {\widehat}{L_g}.{\widehat}{i}.\pi_{{\widehat}\lambda_g}(k)$ for all $k\in \Gamma_{{\widehat}K}$.
For every $g'\in \Gamma_G$, there exists a unique $g\in s(Q)$ such that $g'=L_g( i(k))$ for some unique $k\in K$. Hence, $\Pi_{{\widehat}\lambda}$ is well defined on the entire space $\Gamma_G^{pel}$.
(Mitra [@mit0; @mit1])\[ret\] There exists $C_0>0$ such that $$d_{{\widehat}G}(\Pi_{{\widehat}\lambda}(g),\Pi_{{\widehat}\lambda}(g'))\leq C_0d_{{\widehat}G}(g,g')+C_0$$ for all $g,g'\in \Gamma_G^{pel}$. In particular, if $\Gamma_G^{pel}$ is hyperbolic then $B_{{\widehat}\lambda}$ is uniformly (independent of ${\widehat}\lambda$) quasiconvex.
Proof of Theorem {#main thm}
----------------
Since $i\colon \Gamma_K\to\Gamma_G$ is an embedding we identify $k\in K$ with its image $i(k)$.
Let
- ${\widehat}\mu_g={\widehat}L_g({\widehat}\lambda_g)$, where $g\in s(Q)$.
- $\mu^b_g={\widehat}\mu_g\setminus\mathcal{H_G}$.
- $B_{\lambda^b}=\bigcup_{g\in s(Q)}\mu^b_g$.
- $Y=\Gamma_K$ and $X=\Gamma_G$.
\[end lem\] There exists $A>0$ such that for all $x\in \mu^b_g\subset B_{\lambda^b}\subset B_{{\widehat}\lambda}$ if $\lambda^b$ lies outside $B_N(p)$ for a fixed reference point $p\in\Gamma_K$ then $x$ lies outside an $\frac{f(N)}{A+1}$ ball about $p$ in $\Gamma_G$, where $f(N)\rightarrow\infty$ as $N\rightarrow\infty$.
[**Proof.**]{} Let $x\in \mu^b_g$ for some $g\in s(Q)$. Let $\gamma$ be a geodesic path in $\Gamma_Q$ joining the identity element $e_Q$ of $\Gamma_Q$ and $p(x)\in\Gamma_Q$. Order the vertices on $\gamma$ so that we have a finite sequence $e_Q=q_0,q_1,...,q_n=p(x)=p(g)$ such that $d_Q(q_i,q_{i+1})=1$ and $d_Q(e_Q,p(x))=n$. Since $s$ is a quasi-isometric section, this gives a sequence $s(q_i)=g_i$ such that $d_G(g_i,g_{i+1})\leq R+\epsilon = R_1\mbox{ (say)}$. Observe that $g_n=g$ and $g_0=e_G$. Let $B_{R_1}(e_G)$ be a closed ball around $e_G$ of radius $R_1$, then $B_{R_1}(e_G)$ is finite. Now for each $g\in G$, the outer automorphism $\phi_g$ is a quasi-isometry. Thus there exists $K\geq 1$ and $\epsilon\geq 0$ such that for all $g\in B_{R_1}(e_G)$, $\phi_g$ is a $(K,\epsilon)$ quasi-isometry and $K,\epsilon$ are independent of elements of $G$. Let $s_i=g^{-1}_{i+1}g_i$, then $s_i\in B_{R_1}(e_G)$, where $i=0,...,n-1$. Hence $\phi_{s_i}$ is $(K,\epsilon)$ quasi-isometry.
Since $s(Q)\subset N_G(K_1)$, we have $s_i\in N_G(K_1)$ for all $i$. Therefore $\phi_{s_i}$ will induce a $(\hat K,\hat\epsilon)$ quasi-isometry ${\widehat}\phi_{s_i}$ from ${\widehat}\Gamma_K$ to ${\widehat}\Gamma_K$, where $\hat K,\hat \epsilon$ depends only $K$ and $\epsilon$.
Now $x\in \mu^b_{g_n}$ and $L_g$ preserves distance between left cosets for all $g\in G$, hence there exists $x_1\in \lambda^b_{g_n}$ such that $x=L_{g_n}(x_1)$.
Let $[p,q]_{g_n}\subset \lambda^b_{g_n}$ be the connected portion of $\lambda^b_{g_n}$ on which $x_1$ lies. Since ${\widehat}\phi_{s_{n-1}}$ is a quasi-isometry, ${\widehat}\phi_{s_{n-1}}([p,q]_{g_n})$ will be an electric quasigeodesic lying outside horosphere-like sets and hence it is a quasigeodesic in $Y_h$ lying at a uniformly bounded distance $\leq C_1$ from $\lambda^h_{g_{n-1}}$ in $Y_h$ (and hence in $X_h$), where $\lambda^h_{g_{n-1}}$ is electroambient representative of ${\widehat}\lambda_{g_{n-1}}$ and $Y_h, X_h$ are respectively the complete hyperbolic metric spaces corresponding to $Y,X$. Thus there exist $x_2\in \lambda^h_{g_{n-1}}$ such that $d_{X_h}(\phi_{s_{n-1}}(x_1),x_2)\leq C_1$. But $x_2$ may lie inside horoball-like set penetrated by ${\widehat}\lambda_{g_{n-1}}$. Due to bounded coset (horosphere) penetration properties there exists $y\in\lambda^b_{g_{n-1}}$ such that $d_{X_h}(x_2,y)\leq D$.
Thus $d_{X_h}(\phi_{s_{n-1}}(x_1),y)\leq C_1+D$. Since $X=\Gamma_G$ is properly embedded in $X_h$, there exists $M>0$ depending only upon $C_1,D$ such that $d_G(\phi_{s_{n-1}}(x_1),y)\leq M$.
Hence $d_G(L_{g_{n-1}}(\phi_{s_{n-1}}(x_1)),L_{g_{n-1}}(y))=d_G(\phi_{s_{n-1}}(x_1),y)\leq M$ and $L_{g_{n-1}}(y)\in \mu^b_{g_{n-1}}$.
Let $z=L_{g_{n-1}}(y)$, then $$\begin{aligned}
d_G(x,z)&\leq& d_G(x,L_{g_{n-1}}(\phi_{s_{n-1}}(x_1)))+d_G(L_{g_{n-1}}(\phi_{s_{n-1}}(x_1)),L_{g_{n-1}}(y))\\&\leq& d_G(x,xs_{n-1})+M\\&\leq& R_1+M=A (say). \end{aligned}$$ Thus, we have shown that for $x\in\mu^b_{g_n}$ there exists $z\in\mu^b_{g_{n-1}}$ such that $d_G(x,z)\leq A$. Proceeding in this way, for each $y\in\mu^b_{g_i}$ there exists $y'\in\mu^b_{g_{i-1}}$ such that $d_G(y,y')\leq A$.
Hence there exists $x'\in\lambda^b$ such that $d_G(x,x')\leq An$.
Since $\Gamma_K$ is properly embedded in $\Gamma_G$ there exists $f(N)$ such that $\lambda^b$ lies outside $f(N)$-ball about $p$ in $\Gamma_G$ and $f(N)\rightarrow\infty$ as $N\rightarrow\infty$.
Therefore $d_G(x',p)\geq f(N)$.
Thus $$\begin{aligned}
d_G(x,p)\geq f(N)-d_G(x,x')\geq f(N)-An.\end{aligned}$$
Also $d_G(x,p)\geq n$.
Therefore, $d_G(x,p)\geq \frac{f(N)}{A+1}$, that is, $x$ lies outside $\frac{f(N)}{A+1}$-ball about $p$ in $\Gamma_G$.
\[end thm\] Given a short exact sequence of pairs of finitely generated groups
$$1\rightarrow (K,K_1)\rightarrow (G,N_G(K_1))\stackrel{p}{\rightarrow}(Q,Q_1)\rightarrow 1$$ with $K$ non-elementary and strongly hyperbolic relative to the cusp subgroup $K_1$. If $G$ preserves cusps, strongly hyperbolic relative to $N_G(K_1)$ and weakly hyperbolic relative to the subgroup $K_1$, then there exists a Cannon-Thurston map for the embedding $i\colon \Gamma_K\to \Gamma_G$, where $\Gamma_K$ and $\Gamma_G$ are Cayley graphs of $K$ and $G$ respectively.
[**Proof.**]{} It suffices to prove the condition of Lemma \[suff cond\].
So for a fixed reference point $p\in \Gamma_K$, we assume that ${\widehat}\lambda$ is an electric geodesic segment in ${\widehat}\Gamma_K$ such that $\lambda^b(\subset \Gamma_K)$ lies outside an $N$-ball $B_N(p)$ around $p$. Let $\beta_{pel}$ be a quasigeodesic in the partially electrocuted space $\Gamma_G^{pel}$ joining the end points of ${\widehat}\lambda$. Recall from Theorem \[ret\] that $\Pi_{{\widehat}\lambda}$ is a retraction map from $\Gamma_G^{pel}$ onto the quasiconvex set $B_{{\widehat}\lambda}$ which satisfies the Lipschitz’s condition. Let $\beta'_{pel}=\Pi_{{\widehat}\lambda}(\beta_{pel})$, then $\beta'_{pel}$ is a quasigeodesic in $\Gamma_G^{pel}$ lying on $B_{{\widehat}\lambda}$. So $\beta'_{pel}$ lies in a $P$-neighborhood of $\beta_{pel}$ in $\Gamma_G^{pel}$. But $\beta'_{pel}$ might backtrack. $\beta'_{pel}$ can be modified to form a quasigeodesic $\gamma_{pel}$ in $\Gamma_G^{pel}$ of the same type (i.e. lying in a $P$-neighborhood of $\beta_{pel}$) without backtracking with end points remaining the same. By Lemma \[pel-track\], $\beta_{pel}$ and $\gamma_{pel}$ satisfy bounded coset (horosphere) penetration properties with the closed sets (horosphere-like sets) in $\Gamma_G$ corresponding to the left cosets of $N_G(K_1)$ in $G$. Thus if $\gamma_{pel}$ penetrates a horosphere-like set $C_{gN_G{(K_1)}}$ corresponding to the left coset $gN_G(K_1)$ of $N_G(K_1)$ in $G$ and $\beta _{pel}$ does not, then length of geodesic traversed by $\gamma^h$, where $\gamma^h$ is the electroambient quasigeodesic representative of $\gamma_{pel}$, inside $C_{gN_G(K_1)}\times [0,\infty)$ is uniformly bounded. Let ${\mathcal{C}}=\{C_{gN_G{(K_1)}}:g\in G\}$. Thus there exists $C_1\geq 0$ such that if $x\in \beta ^b_{pel} = \beta _{pel}\setminus \mathcal C$, then there exists $y\in \gamma ^b_{pel} = \gamma _{pel}\setminus \mathcal C$ such that $d_G(x,y)\leq C_1$.
Since $y\in\gamma ^b_{pel}\subset B_{\lambda^b}$, by Lemma \[end lem\], $d_G(y,p)\geq \frac{f(N)}{A+1}$.
Therefore, $d_G(x,p)\geq \frac{f(N)}{A+1}-C_1 (=M(N),\mbox{ say})$ and $M(N)\rightarrow\infty$ as $N\rightarrow\infty$. By Lemma \[suff cond\], a Cannon-Thurston map for $i\colon\Gamma_K\to\Gamma_G$ exists.
[1]{}
J. M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustia, M. Mihalik, M. Shapiro, H. Short, Sept. 1990.
B. H. Bowditch.
B.H.Bowditch.
Benson Farb.
M. Gromov. .
Mahan Mitra.
Mahan Mitra.
Mahan Mj.
Mahan Mj.
Mahan Mj. and Abhijit Pal.
Mahan Mj. and Lawrence Reeves.
Lee Mosher.
Andrzej Szczepanski
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---
abstract: 'We solve the Einstein vacuum-equations for the case of static and axisymmetric solutions in a system of coordinates different from the Weyl standard one. We prove that there exists a class of solutions with the appropriate asymptotical behaviour which can be written in a simple compact form, in terms of a function that must satisfies certain Cauchy-Newman problem. The relation between the choice of coordinates and the form of the metric functions that describe the solution is given by providing that analytic function which characterizes the metric as well as the gauge.'
author:
- |
J.L. Hernández-Pastora[^1] $ ^{1,2}$ and J. Ospino$^1$\
\
$^1$ Departamento de Matemática Aplicada.\
Universidad de Salamanca. Salamanca, España.\
\
$^2$ Instituto Universitario de Física Fundamental y Matemáticas.\
Universidad de Salamanca. Salamanca, España
date: 19 de Julio de 2010
title: NEW REPRESENTATION OF SOME STATIC AND AXISYMMETRIC VACUUM SOLUTIONS
---
PACS numbers: 02.00.00, 04.20.Cv, 04.20.-q, 04.20.Jb
Introduction
============
The general axially symmetric line element of an stationary space-time is commonly used to be written in isotropic coordinates, the so-called canonical Weyl coordinates [@kramer]. For the static case, all the solutions with a good asymptotically behaviour are given, in this system of coordinates, by the Weyl family [@weyl].
This choice of coordinates allow us to handle with only three metric functions to describe the kind of stationary solutions. The gauge imposes a determined value for the quantity $2\xi_{[\alpha}\eta_{\beta]}\xi^{\alpha}\eta^{\beta}$ involving the stationary ($\xi$) and axial ($\eta$) Killing vectors, which makes the length of both Killing vectors to depend on the same unique metric function. Nevertheless, there is no reason except for simplicity to perform the researching of stationary and axisymmetric solutions in that system of coordinates. In fact, there exist another representations of that kind of solutions, like the Erez-Rosen-Quevedo [@erq] and Gutsunayev-Manko [@gm] families, expressed in prolate spheroidal coordinates. In particular, the Weyl family of solutions is known to be specially ugly to describe the apparently more simple spherical solution. For this case, the Erez-Rosen-Quevedo representation arises as one of the most suitable ways to write the Schwarzschild solution because one releases, in contrast with its Weyl form, the use of a series to describe the corresponding metric functions. Indeed, the standard Schwarzschild coordinates are a better adapted way to describe the spherical solution because, in addition to be a simpler expression of the metric function, the system of coordinates itself is related to the existence of a symmetry of the field equation which allow us to characterize the radial coordinate [@cqg2].
There is a wide class of static and axisymmetric solutions with a very relevant physical characteristics for the exact description of the non-spherical space-time, the so-called [*Pure Multipole Moments*]{} [@mio], which adopt a very cumbersome form in standard Weyl coordinates. These are reasons for trying to look for static and axisymmetric solutions by solving the vacuum Einstein equations releasing the coordinate condition imposed by the use of Weyl canonical coordinates. As will be seen, the results obtained show the possibility of introducing from the beginning a new system of coordinates for the resolution of the static case. It is proved that the field equations can be solved in a non isotropic coordinate system, leading to solutions represented in terms of three metric functions (one more than the Weyl family). This solution adopt a very simple form than its corresponding expression in the Weyl family, and the relation between them can be obtained by a well-defined change of coordinates.
This work is organized as follows. In section 2 we make a brief review about the different descriptions of the general stationary and axisymmetric space-times. We want to put the attention on the fact that the Ernst equation for this kind of solutions still hold even for a different choice of coordinates since that equation represents an intrinsic description of the field equations.
In section 3 we solve the field equations for the static case providing explicit expressions for the two independent metric functions and the other one derived by quadratures, and we discuss about the good asymptotical behaviour of the solution obtained. In this section, a theorem is proved that allow to establish a family of solutions depending on the election of determined analytic function that solves a Cauchy-Newman problem. In particular, a series of analytic functions fulfilling the conditions of the theorem is explicitly obtained, which leads to a concrete family of solutions of the field equations. Finally we show that the family of solutions obtained are of course included at the Weyl family but they can be used to define a different representation. The gauge of coordinates to change from one representation to another is explicitly shown in terms of the above-mentioned analytic function.
A conclusion section contains some comments about the aims and results of the work as well as possible future extensions of it.
The stationary and axisymmetric vacuum field equations
======================================================
Let $\nu_4$ be a Riemmanian manifold endowed with a stationary and axisymmetric metric $g_{\alpha \beta}$. Let $\xi$ and $\eta$ the corresponding Killing vectors associated to the time and axial isometries of this space-time respectively. As is known, there exist coordinates $\{x^{\beta}\}=\{t,\rho,z,\phi\}$ ($\beta=0..3$) adapted to both Killing vectors [@papa], [@carter], and the more general stationary and axially symmetric line element can be written as follows: $$ds^2=-e^{2\sigma}\left(dt^2-\omega d\phi\right)^2+e^{2\beta
-2\sigma}\left(dz^2+d \rho^2\right
)+J^2 e^{-2\sigma}d\phi^2 \ ,\label{metric}$$ where $\sigma$, $\beta$, $\omega$ and $J$ are metric functions depending on $x^1\equiv \rho$ and $x^2\equiv z$. The Einstein’s vacuum field equations for this metric (\[metric\]) are the following:
$$\sigma ^{\prime \prime}+\ddot \sigma+\frac{\dot J}{J}\dot
\sigma+\frac{J^\prime}{J}\sigma^\prime+\frac{1}{2}\frac{e^{4\sigma}}{J^2}\left ((\omega^\prime)^2+\dot \omega ^2\right )=0\label{1}$$
$$\omega ^{\prime \prime}+\ddot \omega+4\omega^\prime\sigma^\prime+4\dot \omega \dot
\sigma-\frac{J^\prime}{J}\omega ^\prime-\frac{\dot J}{J}\dot\omega=0\label{2}$$
$$(\sigma ^\prime)^2+\dot \sigma^2+\frac{1}{4}\frac{e^{4\sigma}}{J^2}\left (\dot \omega^2+(\omega^\prime)^2\right )+\beta ^{\prime \prime}+\ddot \beta=0
\label{3}$$
$$2\dot \sigma \sigma ^\prime+\frac{\dot
J^\prime}{J}-\frac{1}{2}\frac{e^{4\sigma}}{J^2}\dot \omega \omega^\prime=\frac{J^\prime}{J}\dot \beta+\frac{\dot J}{J}\beta^\prime\label{4}$$
$$(\sigma ^\prime)^2-\dot \sigma ^2+\frac{J^{\prime\prime}}{J}+\frac{1}{4}\frac{e^{4\sigma}}{J^2}\left(\dot
\omega^2-(\omega ^\prime)^2\right)=\frac{J^\prime}{J}\beta^\prime-\frac{\dot J}{J}\dot \beta \label{5}$$
$$\ddot J+J^{\prime\prime}=0 \ ,
\label{6}$$
where $\prime$ and $\dot{}$ denote derivatives with respect to the coordinate $\rho$ and $z$ respectively. The metric functions $\sigma$, $\omega$ and $J$ can be characterized intrinsically from the Killing vectors by the following scalars: $$\begin{aligned}
\xi^{\alpha}\xi_{\alpha}&=&e^{-2\sigma} \quad , \quad \xi^{\alpha}\eta_{\alpha}=e^{2\sigma}\omega \nonumber\\
\eta^{\alpha}\eta_{\alpha}&=&e^{-2\sigma}J^2-e^{2\sigma}\omega^2 \quad , \quad 2\xi_{[\alpha}\eta_{\beta]}\xi^{\alpha}\eta^{\beta}=J^2 \ .
\label{scalars}\end{aligned}$$
An alternative intrinsic and compact form of writing the line element (\[metric\]) is the following $$ds^2=-f(dx^0-\varphi_i dx^i)^2+\hat g_{ij} dx^i dx^j ,$$ $\hat g_{ij}$ being the quotient metric on the manifold $\nu_3$, and $f\equiv e^{2\sigma}$, $g_{0i}=f\varphi_i$, $g_{0i}=0, i\neq3$, $\varphi_3 \equiv \omega$. Bel [@bel] and Geroch [@geroch] proposed independently to write the Einstein equations by using the conformal metric $\bar g_{ij}=f\hat g_{ij}$, as follows $$\begin{aligned}
& f \bar{\bigtriangleup}_2f-\bar{\bigtriangleup}_1f+\bar{\bigtriangleup}_1W=0\nonumber\\
& f \bar{\bigtriangleup}_2W-2\bar{\bigtriangleup}_1(f,W)=0 \nonumber\\
&\bar R_{ij}=\frac 12 f^{-2}\left(\bar{\nabla}_iW\bar{\nabla}_jW+\bar{\nabla}_if\bar{\nabla}_jf\right)\ ,\end{aligned}$$ where $\bar R_{ij}$ denotes the Ricci tensor associated to the conformal metric $\bar g_{ij}$, $\bar{\bigtriangleup}_1(A,B)\equiv \bar g^{ij} \partial_i A \partial_j B$, $\bar{\bigtriangleup}_2(A)\equiv \bar g^{ij}\bar{\nabla}_i\bar{\nabla}_j A$ represent the Beltrami operators, and the scalar $W$ is defined from the vorticity $\Omega_{ij}=f^{1/2}\left(\partial_i\varphi_j-\partial_j\varphi_i\right)$ as follows[^2] $$\hat\Omega_j=f^{-1}\partial_jW ,$$ where $\hat \Omega_j=\hat g_{jk}\frac 12 \hat \epsilon^{kil}\Omega_{il}$, and the derivatives of $W$ for the case of our metric (\[metric\]) are $\dot W=f^2J^{-1}\omega^{\prime}$, $W^{\prime}=-f^2J^{-1}\dot \omega$.
Finally, Geroch [@geroch] introduced the complex function $E=f+iW$, the [*Ernst potential*]{} (\[ernst\]), in terms of which the above shown Einstein equations (\[1\])-(\[6\]) can be written as follows $$\begin{aligned}
& & \left(E+E^{\star}\right)\bar{\bigtriangleup}_2E-2\bar{\bigtriangleup}_1E=0 \nonumber\\
& & \bar R_{ij}=\left(E+E^{\star}\right)^{-2}\left(\bar{\nabla}_iE\bar\nabla_jE^{\star}+\bar{\nabla}_jE\bar\nabla_iE^{\star}\right) \ .
\label{ernst}\end{aligned}$$
The static case
===============
If we suppose that $\omega=0$, the system of equations (\[1\])-(\[6\]) is reduced to: $$\sigma ^{\prime \prime}+\ddot \sigma+\frac{\dot J}{J}\dot
\sigma+\frac{J^\prime}{J}\sigma ^\prime=0\label{W1}$$ $$(\sigma ^\prime)^2+\dot \sigma ^2+\beta ^{\prime \prime}+\ddot
\beta=0 \label{W2}$$ $$2\dot \sigma \sigma ^\prime+\frac{\dot
J^\prime}{J}=\frac{J^\prime}{J}\dot \beta+\frac{\dot J}{J}\beta
^\prime\label{W3}$$ $$(\sigma ^\prime)^2-\dot \sigma ^2+\frac{J^{\prime
\prime}}{J}=\frac{J^\prime}{J}\beta ^\prime-\frac{\dot J}{J}\dot
\beta \label{W4}$$ $$\ddot J+ J^{\prime \prime}=0\label{W5} \ .$$
A solution of the equations
---------------------------
We proceed now to rewrite the above equations by using the following complex variables: $$u=\rho+iz \quad , \quad \bar u=\rho-iz \quad ,$$ and the equations (\[W1\]) and (\[W5\]) turn out to be the following: $$2J\sigma _{u\bar u}+J_{u}\sigma _{\bar u}+J_{\bar u}\sigma _{u}=0
\label{C2}$$ $$J_{\bar u u}=0\label{C3} \ .$$
A general solution of the equation (\[C3\]) is given by $$J(u, \bar u)=J_1(u)+J_2(\bar u) \ ,
\label{jota}$$ and therefore, the equations for the metric functions $\sigma$ and $\beta$ can be solved, in separated variables, as follows: $$\displaystyle{\sigma(u,\bar u)=\frac{c}{\sqrt{(aJ_1(u)+b)(aJ_2(\bar u)-b)}}}
\label{sigma}$$
$$\displaystyle{\beta(u,\bar u)=-\frac 18 a^2J^2\sigma^4+\frac 12 \ln\left(J_u J_{\bar u}\right)+\alpha} \ ,
\label{beta}$$
where $\alpha,a,b,c$ are arbitrary constants and the subindices denote derivation with respect to the corresponding variable.
The asymptotical conditions
---------------------------
Let us now consider the appropriated behaviour of these solutions; since we want to describe the gravitational field of isolated compact bodies, the metric is required to be asymptotically flat, i.e., in a neighborhood of infinity the line element must resembles the Minkowski metric $$ds^2=-dt^2+d\rho^2+dz^2+\rho^2 d \varphi^2 \quad ,$$ and therefore, the metric functions must fulfill the following asymptotical conditions: $$\lim_{\rho\rightarrow \infty} J= \rho \label{condijota}$$ $$\lim_{\rho\rightarrow \infty} \sigma = 0 \label{condisigma}$$ $$\lim_{\rho\rightarrow \infty} \beta = 0 \label{condibeta} \ .$$
Furthermore, a [*regularity condition*]{} is required on the $\varphi$ coordinate to have the standard periodicity $2 \pi$ [@kramer]: $$\lim_{\rho\rightarrow 0}\left[\frac{l_{\mu}l^{\mu}}{4 l} \right]=1 \ ,
\label{rc}$$ where $l\equiv g_{33}=e^{-2\sigma}J^2$ denotes the length of the spacelike Killing vector $\partial_{\varphi}$ that represents the axial symmetry. This regularity condition for the solution (\[jota\]) leads to the following limit: $$\lim_{\rho \rightarrow 0}\left(J_{\rho}^2+J_z^2\right)e^{-2 \beta} =1
\label{rc2}$$ if the following additional condition holds: $$\lim_{\rho \rightarrow 0}J=\rho=0 \label{otramas} \ .$$
Let us choose the functions $J_1(u)$ and $J_2(\bar u)$ in such a way that the function $J(u,\bar u)$, $$J(u,\bar u)=\frac 12 u+f_1(u)+\frac 12 \bar u+f_2(\bar u)\label{efes}$$ fulfills the asymptotical condition (\[condijota\]) iff the arbitrary functions $f_1(u)$ and $f_2(\bar u)$ goes to zero a infinity. Since we look for a real function $J$, the imaginary terms of the complex functions $f_1(u)$ and $f_2(\bar u)$ must be equal except for their signs, and hence we shall consider these functions to be complex conjugated as follows: $$f_1(u)=v(\rho,z)+i w(\rho,z) \quad , \quad f_2(\bar u)=v(\rho,z)-i w(\rho,z) \label{efes2} \ ,$$ where $v(\rho,z)$ and $w(\rho,z)$ are harmonic real functions that verify the Cauchy-Riemman conditions,i.e, $$v_{\rho \rho}+v_{zz}=0 \quad, \quad v_{\rho}=w_{z} \quad , \quad v_z=-w_{\rho}
\label{cr}$$ since $J(u,\bar u)$ (as well as $f_1(u)$) is a holomorphic function, solution of the equation (\[C3\]).
Hence, the asymptotical conditions (\[condijota\]),(\[condisigma\]) are fulfilled whenever $$\displaystyle{\lim_{\rho\rightarrow\infty}v=0} ,\label{uvecero}$$ and the condition (\[condibeta\]) for the metric function $\beta$ requires that $$\lim_{\rho\rightarrow \infty} \left(J_{\rho}^2+J_z^2\right)=4e^{-2\alpha} \label{infini} \ .$$ Finally, with respect to the regularity condition, since we impose to the function $v(\rho, z)$ to be zero at the symmetry axis (\[otramas\]), then the equation (\[rc\]) leads to the following limit: $$\lim_{\rho\rightarrow 0} 4 e^{-2 \alpha} =1
\label{zero}$$ since $\displaystyle{J_uJ_{\bar u}=\frac 14 (J_{\rho}^2+J_z^2)}$. Therefore, this condition (\[zero\]) implies that $\alpha=\ln 2$ and the equation (\[infini\]) turns out to be $$\lim_{\rho \rightarrow\infty}\left(J_{\rho}^2+J_z^2\right)=1 \label{infini2}.$$
Hence, we can hold the following theorem:
[**Theorem 1**]{}
The following metric functions: $$\begin{aligned}
&J(\rho,z)=\rho+2v(\rho,z) \nonumber\\
&\displaystyle{\sigma(\rho,z)=\frac{c}{\sqrt{(aJ_1+b)(aJ_2-b)}}}\nonumber\\
&\displaystyle{\beta(\rho,z)=-\frac 18 a^2J^2\sigma^4+\frac 12 \ln\left(J_{\rho}^2+J_z^2\right)} \ ,
\label{teorem}\end{aligned}$$ where $J_1(\rho,z)=\displaystyle{\frac12(\rho+iz)+v+iw}$, $J_2(\rho,z)=\displaystyle{\frac12(\rho-iz)+v-iw}$ and $a$, $b$, $c$ are arbitrary constants, determine a static and axisymmetric vacuum solution of the Einstein equations, with the appropriate asymptotical and boundary behaviour, if the function $v(\rho,z)$ is a solution of the following Cauchy-Newman problem: $$\begin{aligned}
v_{\rho\rho}+v_{zz}=0 \quad , \quad &v(\rho,z)\big|_{\rho\rightarrow 0,\infty}=0 \nonumber\\
&v_{\rho}(\rho,z)\big|_{\rho\rightarrow \infty}=0 \nonumber\\
&v_{z}(\rho,z)\big|_{\rho\rightarrow \infty}=0 \ ,
\label{cauchynewman}\end{aligned}$$ with $w(\rho,z)$ being a solution of the Cauchy-Riemman conditions: $w_z=v_{\rho}$, $w_{\rho}=-v_z$.
[**Proof**]{}: The asymptotical conditions (\[condijota\]-\[condibeta\]) and the regularity condition (\[rc\]) are fulfilled by the metric functions (\[jota\]-\[beta\]) if we use the equations (\[efes\]-\[cr\]) and we choose a function $v(\rho,z)$ that satisfies the condition (\[uvecero\]) and goes to zero at the symmetry axis. Since $J_{\rho}^2+J_z^2=1+4(v_{\rho}+v_{\rho}^2+v_z^2)$, the conditions (\[cauchynewman\]) lead to verify the equation (\[infini2\]) and it allows us to conclude the proof.
A family of solutions
---------------------
A solution of the Cauchy-Newman problem (\[cauchynewman\]) is given by the following series: $$\displaystyle{ v(\rho,z)=\sum_{n=0}^{\infty}\frac{a_{2n+1}}{r^{2n+1}}\cos\left[(2n+1)\theta_c\right]+
\frac{b_{2n}}{r^{2n}}\sin\left[(2n)\theta_c\right]} \label{familia} \ ,$$ where $r\equiv\sqrt{\rho^2+z^2}$ and $\theta_c$ being the complementary angle $\displaystyle{\theta_c\equiv\frac{\pi}{2}-\theta}$ of the standard polar angle $\theta$, and defined by[^3] $\displaystyle{\theta_c=\delta\equiv\arctan \frac{z}{\rho}}$.
This family of functions is obtained from the following considerations: the successive negative powers of the complex variable $u$ provide a series of holomorphic functions $F_k(u)=u^{-k}$ solutions of the harmonic equation (\[C3\]), and therefore, their real and imaginary parts are real functions fulfilling the equation (\[W5\]) in the variables $(\rho,z)$. And furthermore, if we take alternatively the real part of the functions $F_k(u)$ for even $k$ and the imaginary part of those functions with odd order $k$, we get a set of real functions which verify the Cauchy-Newman conditions (\[cauchynewman\]).
Therefore, according to (\[teorem\]) and (\[familia\]), the metric function $J(\rho,z)$ can be written as follows: $$\displaystyle{ J(\rho,z)=\rho\left[1+2\sum_{n=0}^{\infty}\frac{a_{2n+1}}{r^{2n+2}}(-1)^n\frac{\sin\left((2n+1)\theta\right)}{\sin\theta}+
\frac{b_{2n}}{r^{2n+1}}(-1)^{n+1}\frac{\sin\left((2n)\theta\right)}{\sin\theta}\right]} \label{jota1} \ .$$
And, by developing the trigonometric relations we can write the following expression: $$\displaystyle{ J(\rho,z)=\rho\left[1+2\sum_{n=0}^{\infty}\frac{a_{2n+1}}{r^{2n+2}}C^{(1)}_{2n}(\Omega)+
\frac{b_{2n}}{r^{2n+1}}C^{(1)}_{2n-1}(\Omega)\right]} \label{jota2} \ ,$$ with $\Omega\equiv\cos\theta$, and $C^{(1)}_n(\Omega)$ are the Gegenbauer polynomials of degree $n$, since we know that $$\begin{aligned}
&\cos\left[(2n+1)\theta_c\right]=(-1)^n\sin\left[(2n+1)\theta\right]= C^{(1)}_{2n}(\Omega) \sin\theta\nonumber\\
&\sin\left[(2n)\theta_c\right]=(-1)^{n+1}\sin\left[(2n)\theta\right]= C^{(1)}_{2n-1}(\Omega) \sin\theta\label{gegen} \ .\end{aligned}$$
Therefore, accordingly to equations (\[gegen\]) we can write the function $v(\rho,z)$ as follows: $$v(\rho,z)=\sin\theta \sum_{n=0}^{\infty}\frac{h_n}{r^{n+1}}C^{(1)}_n(\Omega) \label{lauve} \ ,$$ for any set of arbitrary parameters $h_n$, and the metric function $J$ as follows[^4]: $$J(\rho,z)=\rho \left[1+2\sum_{n=0}^{\infty}\frac{h_n}{r^{n+2}}C^{(1)}_n(\Omega)\right] \label{lajota} \ .$$
From the equation (\[teorem\]) we have that the metric function $\sigma(\rho,z)$ is given by the following expression: $$\sigma(\rho,z)=c \left[a^2(\frac14 r^2+v^2+w^2+\rho v+z w-\frac{b^2}{a^2})-iab(z+2w)\right]^{-1/2} \ , \label{sigma1}$$ which can be simplified as follows: $$\sigma(\rho,z)=\frac{2c}{a}\left[J^2+(z+2w)^2-\left(\frac {2b}{a}\right)^2-i\frac{4b}{a}(z+2w)\right]^{-1/2}
\label{sigma2} .$$ The function $w(\rho,z)$ is obtained by integration of a quadrature from the equation (\[familia\]), and this is the result: $$w(\rho,z)=2\sum_{n=0}^{\infty}\frac{h_n}{r^{n+1}}\left[C^{(1)}_{n-1}(\Omega)-\Omega C^{(1)}_n(\Omega)\right].$$ Finally, the metric function $\beta(\rho,z)$ is given by the following expression: $$\displaystyle{\beta(\rho,z)=-2\frac{c^4}{a^2}\frac{J^2}{\left[J^2+(z+2w)^2\right]^2}+\frac 12\ln\left[1+4(v_{\rho}+v_{\rho}^2+v_z^2)\right]} \label{beta1} \ ,$$ where the derivatives of the function $v(\rho,z)$ with respect to $\rho$ and $z$ are given by the following expressions: $$\begin{aligned}
&\displaystyle{v_{\rho}=\sum_{n=0}^{\infty}\frac{h_n
(n+1)}{r^{n+2}}\left[(2\Omega^2-1)C^{(1)}_n(\Omega)-\Omega
C^{(1)}_{n-1}(\Omega)\right]} \nonumber\\
&\displaystyle{v_z=\sin\theta \sum_{n=0}^{\infty}\frac{h_n(n+1)}{r^{n+2}}\left[-2\Omega C^{(1)}_n(\Omega)+C^{(1)}_{n-1}(\Omega)\right]} \label{derilauve} \ .\end{aligned}$$
The Weyl limit
--------------
The Weyl family of solutions [@weyl] are written in a special system of isotropic coordinates $\{\hat \rho, \hat z\}$ namely the canonical Weyl coordinates. This choice of coordinates makes able to fix the metric function $J$ of the general line element (\[metric\]) in such a way that it equals the Weyl coordinate $\hat \rho$, Hence, the problem of searching for solutions is reduced to solve only one differential equation for the metric function $\sigma$ (\[W1\]).
If we impose this condition on the metric function $J$, i.e., we take $J(\rho,z)=\rho$, we are equivalently considering a special choice of coordinates and $\{\rho,z\}$ become the canonical Weyl coordinates. Therefore, by taking $v(\rho,z)=0$, and consequently $$J_1=\frac12 u \quad , \quad J_2=\frac 12 \bar u ,$$ we have that $$\sigma(u,\bar u)=\frac{2c}{\sqrt{(au+2b)(a\bar u-2b)}}=\frac{2c}{a r\sqrt{1-2 \Omega \lambda +\lambda^2}}\label{unasigma} \ ,$$ where $\displaystyle{\lambda\equiv \frac{2bi}{a r}}$. By taking into account the generator function of the Legendre polynomials $P_n(\Omega)$ we can conclude that $$\sigma(r,\Omega)=\frac {2c}{a} \sum_{n=0}^{\infty}\frac{(i 2b/a)^n}{r^{n+1}}P_n(\Omega) \label{weyl} \ ,$$ and the metric function $\beta(\rho,z)$ is given by the following expression: $$\beta(\rho,z)=-\frac 18 a^2 \rho^2 \sigma^4 \ .$$
In conclusion, these metric functions represent a family of static and axisymmetric vacuum solutions that verify the following asymptotical conditions: $$\lim_{\rho\rightarrow \infty} \beta= 0 \quad , \quad \lim_{\rho\rightarrow \infty} \sigma = 0 , \label{weylsigmaybeta}$$ and the regularity condition for this case, i.e., $$\lim_{\rho\rightarrow 0} \beta = 0 . \label{weylbeta}$$
The coefficients $a^W_n$ of this solution (\[weyl\]) in the Weyl representation[^5] are given by the expression $\displaystyle{a^W_n\equiv\frac{2c}{a}\left(\frac{i2b}{a}\right)^n}$, and these particular coefficients are not arbitrary because they depend on only three parameters, and so (\[weyl\]) does not represent the whole Weyl family but only those solutions with that particular value of their set of coefficients $a^W_n$.
Since (\[unasigma\]) is a solution of the linear equation (\[C2\]) for $\sigma$ , the following expression is also a solution $$\sigma(u,\bar u)=\sum_{k=0}^{\infty}\frac{2c_k}{\sqrt{(a_ku+2b_k)(a_k\bar u-2b_k)}}\label{sumadesigmas} ,$$ and hence, the corresponding Weyl coefficients are given by a set of arbitrary parameters: $$a^W_n =\sum_{k=0}^{\infty}\frac{2c_k}{a_k}\left(\frac{i2b_k}{a_k}\right)^n =\sum_{k=0}^{\infty}B_k^nC_k \ , \label{algebracond}$$ with $B_k\equiv i2b_k/a_k$ and $C_k\equiv 2c_k/a_k$. And therefore, we generate another solutions by means of this linear combination.
In order to make a complete and detailed analysis of the results, we are going to show now that the solution previously obtained (\[sigma2\]),(\[beta1\]) belongs to the Weyl family for any analytic functions $v(\rho,z)$ and $w(\rho,z)$ considered, by means of performing explicitly a change of coordinates.
It can be seen that a change of coordinates exists that allow us to recover the Weyl family of solutions from our solution. The change of coordinates is given by the following choice: $$\hat \rho=\rho+2 v(\rho,z) \ ,
\label{rotilde}$$ $v(\rho,z)$ being any analytic function, and the coordinate $\hat z$ must be a solution of the Cauchy-Riemman equations: $\hat z_z=-\hat\rho_{\rho}$, $\hat z_{\rho}=\hat \rho_z$, and therefore both $v(\rho,z)$ and $w(\rho,z)$ satisfying the Cauchy-Newman problem (\[cauchynewman\]) can be used to define the change of coordinates, i.e., $$\hat z=-\left(z+2w(\rho,z)+\mu\right) \ ,
\label{zetatilde}$$ for an arbitrary constant $\mu$.
By substituting (\[rotilde\]), (\[zetatilde\]) in (\[sigma2\]) we conclude that $$\sigma(\hat\rho,\hat z)=\displaystyle{\frac{2c/a}{\sqrt{\hat\rho^2+(\hat z+\mu)^2+4pi(\hat z+\mu)-4p^2}}}\label{esta} \ ,$$ where $p\equiv b/a$ and the previous expression (\[esta\]) can be expanded as follows: $$\sigma(\hat \rho,\hat z)=(2c/a) \sum_{n=0}^{\infty} \left[-(\mu+2pi)\right]^n \frac{P_n(\hat\Omega)}{\hat r^{n+1}} \ ,$$ where $\hat \Omega \equiv \hat z/\hat r$, $\hat r$ being the Weyl radial coordinate.
Conclusion
==========
We have found a solution of the static and axisymmetric vacuum field equations in a system of coordinates different from the canonical Weyl coordinates. We recall that the use of non-isotropic coordinates implies the determination of three metric functions, in contrast with the Weyl representation whereby one of these metric functions is chosen a priori. We write explicitly in some system of coordinates all the metric functions and we prove that the asymptotical conditions of the solution are satisfied. The family of solutions obtained depends on an analytic function that is a solution of certain Cauchy-Newman problem. It is proved that the canonical expression of the solution in Weyl coordinates is obtained when the third metric function $J$ is equal to the standard Weyl form.
The solutions obtained are not new, in the sense that they belong to the Weyl family. The advantage of its present form is that they do not depend on a set of Weyl coefficients. From this solution we can construct some other family of static and axisymmetric solutions by making linear combinations.
There exists a relationship between the gauge used to obtain the solution and the explicit form of the metric functions. The freedom of using different analytic functions in this generalization of coordinates may be used to look for the specific representation that make possible to write the Pure Multipole Solutions in a compact form.
Acknowledgments
===============
This work was partially supported by the Spanish Ministerio de Ciencia e Innovación under Research Project No. FIS 2009-07238, and the Consejería de Educación of the Junta de Castilla y León under the Research Project Grupo de Excelencia GR234.
We also wish to thank the support of the Fundación Samuel Solórzano Barruso (University of Salamanca) with the project FS/8-2009.
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[^1]: E.T.S. Ingeniería Industrial de Béjar. Phone: +34 923 408080 Ext 2263. Also at +34 923 294400 Ext 1527. e-mail address: jlhp@usal.es
[^2]: Let us remind that the existence of the scalar $W$ is derived from one of the Einstein vacuum equation (\[2\]), $\hat\nabla_i\hat\Omega^{ij}-2\Lambda_i\hat\Omega^{ij}=0$, in terms of the so-called [*gravitational fields*]{} $\Lambda_i$, $\Omega_{ij}$ (aceleration and vorticity)
[^3]: Let us note that the angle $\delta$ is exactly equal to $\theta_c$ since $\cot\theta=\tan\theta_c$
[^4]: In fact, it is easy to verify that this expression is the general solution, in separated variables, of the equation (\[W5\]).
[^5]: All the solutions are described by the following metric function: $\displaystyle{\sigma(\hat \rho,\hat z)=\sum_{n=0}^{\infty}\frac{a_n^W}{\hat r^{n+1}}P_n(\hat z / \hat r)}$, with $\hat r \equiv \left(\hat \rho^2+\hat z^2\right)^{1/2}$
|
---
abstract: 'Complementary correlations can reveal the genuine quantum correlations present in a composite quantum system. Here we investigate the relation between complementary correlations and other aspects of genuine quantum correlations. We show that for a certain class of states quantum correlations revealed through complementary correlations become equal to entanglement and discord. We also provide a necessary and sufficient condition for entanglement distribution with separable Bell diagonal states in terms of complementary correlations.'
author:
- Prasenjit Deb
title: Complementary correlations and entanglement distribution
---
Introduction
============
Quantum mechanical systems consisting of more than one subsystem can have both quantum and classical correlations or only classical correlations between themselves[@Vedral].There are different aspects of quantum correlations, such as entanglement[@Schrodinger; @Horodecki], discord[@Zurek], measurement-induced disturbances[@luo3] etc.The presence of quantum correlations make the quantum correlated states useful for quantum information processing tasks such as *teleportation*[@Tele],*dense-coding*[@Dense], *remote state preparation*[@RSP], *quantum cryptography*[@chuang] etc.So it is very important to characterize and quantify such non-classical correlations. Though entanglement is the best studied form of quantum correlations, in the last few years discord has also received a lot of attention.
#### {#section .unnumbered}
Apart from entanglement and discord, the quantum correlations can also be revealed by measuring the correlations between measurement outcomes of complementary observables present on both sides of a bipartite state[@Lorenzo], say $\rho^{AB}$. Though complementary correlation measures can be linked to mutual information, Pearson correlation coefficient and the sum of conditional probabilities[@Lorenzo], we are mainly interested on mutual information based measure. In this article we show that for certain class of states the quantum correlations measured through complementary correlation is exactly equal to entanglement and discord.
#### {#section-1 .unnumbered}
On the other hand, entanglement distribution with seperable states is a quantum phenomena which depicts the bizzare feature of quantum correlations.First introduced by Cubitt$\it{et.al.}$[@cubitt], this phenomena has got a lot of attention in the last few years.The central idea of entanglement distribution scheme in Ref.[@cubitt] is that Alice can create entanglement between herself and Bob by sending a qubit in a seperable state.At the beginning of the protocol Alice and Bob take a state $\rho_{ABC}$ which is shared between them as $\rho_{AC\lvert B}$. Alice then applies a controlled NOT (CNOT) on her qubits $A$ and $C$ (where $A$ is the control qubit) and sends qubit $C$ to Bob. Upon receiving qubit $C$ from Alice, Bob performs CNOT on qubits $B$ and $C$ taking $B$ as the control qubit. At the end of the protocol entanglement is generated in the bipartition $\rho_{A\lvert BC}$. During the whole process qubit $C$ remains separable from the labs of Alice and Bob, that is bipartitioning $\rho_{C\lvert AB}$ is seperable.
#### {#section-2 .unnumbered}
Though Alice and Bob can create entanglement from a seperable state shared by them, the amount of entanglement gain is always upper bounded which simply means that arbitrary amount of entanglement can not be generated. More precisely to say, the non-classical correlations of the carrier (qubit $C$) with parties( qubits $A$ and $B$) bound the amount of distributed entanglement[@dagmar; @chuan].
#### {#section-3 .unnumbered}
The protocols for entanglement distribution described in Ref[@dagmar; @chuan] have one common point. All of those protocols assume that initially there is some quantum correlations between ancilla qubit $C$ and the laboratories. But here we are interested with such a protocol where ancilla qubit is uncorrelated with rest of the system, as also mentioned in Ref[@kay]. Keeping such a protocol in mind if we think EDSS as a quantum task in which entanglement is activated from other form of quantum correlations, then seperable states are the useful resources in the task and ancilla qubit helps in the distribution of entanglement. At this point one might be tempted to ask the question- *What are the necessary conditions a seperable state must satisfy to be useful for entanglement distribution*? The answer to this question has been provided in [@dagmar1], where it was shown that entanglement distribution with seperable states is possible if the rank of the states are at least 3. We aim here to find the answer of the above question through a different approach. More specifically, imposing conditions on quantum correlations present in the seperable state. We consider complementary correlations as the measure of quantum correlations and derive necessary and sufficient condition for EDSS.
Complementary Correlations
==========================
Two quantum mechanical observables are called complementary if knowledge of the measured value any one observable implies maximal uncertainty of the measured value of another. To explain more precisely, let $\mathcal{A}$ and $\mathcal{B}$ are two non-degenerate observables. The spectral representation of these observables are respectively $\mathcal{A}= \Sigma_i~ f(a_i)\lvert a_i\rangle\langle a_i\lvert$ and $\mathcal{B}= \Sigma_j~ g(b_j)\lvert b_j\rangle\langle b_j\lvert$, where$\lvert a_i\rangle$ and $\lvert b_j\rangle$ are the eigenstates of $\mathcal{A}$ and $\mathcal{B}$ respectively. $f$ and $g$ are two arbitrary bijective functions. Now, the observables $\mathcal{A}$ and $\mathcal{B}$ are complementary if $$\lvert\langle a_i\lvert b_j\rangle\lvert = \frac{1}{d},~~~~~ \forall i,j$$ where, $d$ is the dimension of the Hilbert space of the quantum system. The complementary observables with the above definition identify two mutually unbiased bases(MUBs)[@som]. For example, in 2-dimensional Hilbert space the Pauli spinors $\sigma_x$,$\sigma_y$ and $\sigma_z$ are the three complementary observables, the eigenvectors of which form the mutually unbiased bases $\{\lvert0\rangle,\lvert1\rangle\}$, $\{\frac{\lvert0\rangle+\lvert1\rangle}{\sqrt{2}}, \frac{\lvert0\rangle-\lvert1\rangle }{\sqrt{2}}\}$ and $\{\frac{\lvert0\rangle+i\lvert1\rangle}{\sqrt{2}}, \frac{\lvert0\rangle-i\lvert1\rangle }{\sqrt{2}}\}$
#### {#section-4 .unnumbered}
Now consider Alice and Bob are two parties holding quantum systems $A$ and $B$ respectively, $\{\mathcal{A}_i\}$ and $\{\mathcal{B}_i\}$ are respectively the set of complementary observables for Alice’s and Bob’s systems. The correlations between the measurement outcomes of complementary observables $\mathcal{A}_i$’s and $\mathcal{B}_i$’s can be expressed in terms mutual information as[@Lorenzo] $$I(\mathcal{A}_i:\mathcal{B}_i)\equiv H(\mathcal{A}_i) -H(\mathcal{A}_i\lvert \mathcal{B}_i)$$ where, $H(\mathcal{A}_i)$ is the Shanon entropy of the probabilities of the measurement outcomes of observable $\mathcal{A}_i$ of Alice’s system and $H(\mathcal{A}_i\lvert \mathcal{B}_i)$ is the conditional entropy, conditioning being done on Bob’s side. If $\mathcal{A}_1$,$\mathcal{A}_2$ and $\mathcal{B}_1$,$\mathcal{B}_2$ are complementary observables on Alice’s and Bob’s side, then the bipartite quantum state shared between Alice and Bob is maximally entangled if and only if $I(\mathcal{A}_1:\mathcal{B}_1)$ +$ I(\mathcal{A}_2:\mathcal{B}_2)= 2 \log_2 d$[@Lorenzo],where $d$ is the dimension of the Hilbert space of Alice’s or Bob’s system.
#### {#section-5 .unnumbered}
We begin with a generic two-qubit state [@Horodecki1] which can be represented as $$\tau=\frac{1}{4}(\mathbf{1}_2\otimes\mathbf{1}_2+\mathbf{a}\cdot\boldsymbol{\sigma}\otimes\mathbf{1}_2 +\mathbf{1}_2
\otimes \mathbf{b}\cdot\boldsymbol{\sigma} +\sum _{m,n=1}^{3}c_{nm}\sigma_n\otimes\sigma_m )\label{2qubit}$$ where, $\mathbf{1}_2$ represents the identity operator, $\mathbf{a}$ and $\mathbf{b}$ are the local Bloch vectors for each subsystem and $\{\sigma_n\}_{n=1}^{3}$ are the standard Pauli spinors $\sigma_x$, $\sigma_y$ and $\sigma_z$. The coefficients $c_{nm} =\mbox{Tr}(\tau\sigma_n\otimes\sigma_m)$ form the elements of a $3\times3$ matrix $\mathcal{T}$. From[@Horodecki1; @luo2] it is easy to show that $\tau$ is locally unitary equivalent to $$\gamma = \frac{1}{4}(\mathbf{1}_2\otimes\mathbf{1}_2+ \mathbf{a}\cdot\boldsymbol{\sigma}\otimes\mathbf{1}_2+
\mathbf{1}_2\otimes \mathbf{b}\cdot\boldsymbol{\sigma}+\sum _{n=1}^{3} c_{n}\sigma_n\otimes\sigma_n ).$$ For our purpose we will consider the states with maximally mixed marginals,that is, the states of the form $$\rho = \frac{1}{4}(\mathbf{1}_2\otimes\mathbf{1}_2+\sum _{n=1}^{3}c_{n}\sigma_n\otimes\sigma_n),\label{bell-diag}$$ As both the subsystems are qubit, the eigen vectors of Pauli spinors form the mutually unbiased bases on Bob’s side, which are: $$\begin{aligned}
\{\lvert b^j_1\rangle\lvert j=1,2\} &=&\{\lvert0\rangle,\lvert1\rangle\}\nonumber\\
\{\lvert b^j_2\rangle\lvert j=1,2\} &=&\{\frac{\lvert0\rangle+\lvert1\rangle}{\sqrt{2}}, \frac{\lvert0\rangle-\lvert1\rangle }{\sqrt{2}}\}\nonumber\\
\{\lvert b^j_3\rangle\lvert j=1,2\} &=&\{\frac{\lvert0\rangle+i\lvert1\rangle}{\sqrt{2}}, \frac{\lvert0\rangle-i\lvert1\rangle }{\sqrt{2}}\}\end{aligned}$$ Our primary goal is to measure quantum correlations by measuring the complementary correlations of the state $\rho$. Let $$\{\Pi_{\{\lvert b^j_i\rangle\lvert j=1,2\}} = \lvert b^j_i\rangle \langle b^j_i\lvert: i=1,2,3;j=1,2\}$$ be the local projective measurements on Bob’s side. Interestingly, each measurement is related to other through some unitary $V\in U(2)$, that is, $$\Pi_{\{\lvert b^j_2\rangle\lvert j=1,2\}} = V\Pi_{\{\lvert b^j_1\rangle\}}V^{\dagger}$$ and so on. Here $V$ has an explicit form as $$V= tI+i\vec{y}\vec{\sigma}$$ where, $t\in\mathcal{R}$, $\vec{y}=(y_1,y_2,y_3)\in\mathcal{R}^3$. Now, using the same techniques as in Ref.[@luo2], we find the complementary correlations as $$\begin{aligned}
\chi\{\rho\lvert\{\Pi_{\{\lvert b^j_i\rangle\lvert j=1,2\}}\}\}&=&S(\Sigma_i~ p_i\rho_i^A)-\Sigma_i~p_iS(\rho_i^A)\nonumber\\
&=&\frac{1+\theta}{2}\log(1+\theta)\nonumber\\
&&+\frac{1-\theta}{2}\log(1-\theta)\label{maxholevo}\end{aligned}$$ where, $\theta\equiv\theta(t,y_i,c_i)$. The maximum of the quantity in Eqn.([\[maxholevo\]]{}) will give the value of classical correlation of the state. Let, for $\theta=\lvert c_1\lvert$ the quantity in Eqn.([\[maxholevo\]]{}) becomes maximum. So the classical correlation of the state wil be $$\mathcal{C}(\rho)= \frac{1+c_1}{2}\log(1+c_1)+\frac{1-c_1}{2}\log(1-c_1)\label{cc}$$ A little algebra shows that $\theta=\lvert c_1\lvert$ actually refer to measurement on $\{\lvert b^j_2\rangle\lvert j=1,2\}=\{\frac{\lvert0\rangle+\lvert1\rangle}{\sqrt{2}}, \frac{\lvert0\rangle-\lvert1\rangle}{\sqrt{2}} \}$ basis. In other words, we can say that the correlations between measurement outcomes of the observable $\sigma_x$ on both sides will yield the classical correlation of the state $\rho$. The correlations between measurements outcomes of the observables $\sigma_z$ will be $$\mathcal{Q}_1=\frac{1+c_3}{2}\log(1+c_3)+\frac{1-c_3}{2}\log(1-c_3)\label{com corr}$$ The discord $(\mathcal{D})$ of the state $\rho$ can be found from Ref.[@luo2] and comparison of that with $\mathcal{Q}_1$ clearly reveals the relation $$\mathcal{Q}_1\leq\mathcal{D}$$
If the total correlations or mutual information($\mathcal{I}$) of the state $\rho$ is calculated, then it is clear that $$\mathcal{Q}_1 + \mathcal{C} < \mathcal{I}$$
#### {#section-6 .unnumbered}
Now we consider the states for which $c_1=1$, $c_2=-c_3$ and $\lvert c_3\lvert\leq 1$. The states with such parametrization are of the form $$\rho= \frac{1+c_3}{2}\lvert\Psi^{+}\rangle\langle\Psi^{+}\lvert+\frac{1-c_3}{2}\lvert\Phi^{+}\rangle\langle\Phi^{+}\lvert \label{st}$$ where, $\lvert\Phi^{+}\rangle= \frac{1}{\sqrt{2}}(\lvert 00\rangle+\lvert 11\rangle)$ and $\lvert\Psi^{+}\rangle= \frac{1}{\sqrt{2}}(\lvert 01\rangle+\lvert 10\rangle)$ The discord and relative entropy of entanglement of such states are found to be $$\mathcal{D}= \mathcal{E}_r= \mathcal{Q}_1$$ and the classical correlation is exactly the same as that in Eqn.([\[cc\]]{}). Hence, for the states described in Eqn.([\[st\]]{}) quantum correlations revealed through complementary correlation is equal to other existing measure of non-classical correlations such as discord and relative entropy of entanglement, and for such states $$\mathcal{Q}_1 + \mathcal{C} = \mathcal{I}$$
#### {#section-7 .unnumbered}
If one considers the Werner state[@Werner], then it will not be hard to prove that $\mathcal{Q}_1\leq\mathcal{D}$.
#### {#section-8 .unnumbered}
Again consider a two-qubit state $\rho^{AB}$. If any of the complementary correlations $I(\sigma_x^A:\sigma_x^B)$,$ I(\sigma_y^A:\sigma_y^B)$ and $I(\sigma_z^A:\sigma_z^B)$ of the state is zero, then what can be infered about discord and entanglement of $\rho^{AB}$? For any generic two qubit state it is hard to infer about discord and entanglement from complementary correlations but for Bell-diagonal states complementary correlations serve the purpose.
A necessary condition for non-zero entanglement of Bell-diagonal states
-----------------------------------------------------------------------
*Bell-diagonal states will have non-zero entanglement if all the complementary correlations are non-zero.*\
\
Proof:- Bell diagonal states are represented as $\rho_{AB} = \frac{1}{4}(\mathbf{1}_2\otimes\mathbf{1}_2+\sum_ {n=1}^{3}c_{n}\sigma_n\otimes\sigma_n)$. Let $\lambda(\rho_{AB})$ denote the spectrum of state $\rho_{AB}$.The partial transpose of $\rho_{AB}$ on qubit $A$ be $\rho^A_{AB}$. Now, let us consider that $$\mathcal{I}(\sigma_{(y)A}:\sigma_{(y)B})= 0$$ which implies $$\lvert c_2 \lvert=0$$ Under such a condition $\lambda(\rho^A_{AB})=\lambda(\rho_{AB})\geq 0$ and PPT condition[@ppt] confirms that the state will be seperable and hence has zero-entanglement. Similarly, considering any other complementary correlation to be equal to zero it can be proved that the state $\rho_{AB}$ will have zero-entanglement.
#### {#section-9 .unnumbered}
The above mentioned condition is necessary but not sufficient.
Complementary correlations and discord
--------------------------------------
Again consider two Bell diagonal states of the form in Eqn.(\[bell-diag\]) with $c_1=0.5,c_2=0.25,c_3=0.25$ and $c_1=0.5,c_2=0,c_3=0.25$ respectively. The discord of those states when calculated shows the relation $\mathcal{D}_{c_2=0.25}>\mathcal{D}_{c_2=0}$. Therefore, it is clear that Bell diagonal states for which all the complementary correlations are non-zero will have more discord than that of those for which $I(\sigma_y^A:\sigma_y^B)= 0$ .
#### {#section-10 .unnumbered}
For classically correlated state of the form $$\rho_{cc}=\frac{1}{2}(\lvert 00\rangle\langle00\lvert+\lvert 11\rangle\langle11\lvert)$$ $I(\sigma_x^A:\sigma_x^B)= 0$, $I(\sigma_y^A:\sigma_y^B)= 0$ and $I(\sigma_z^A:\sigma_z^B)= \mathcal{C}(\rho_{cc})$
#### {#section-11 .unnumbered}
From the above examples it is well understood that complementary correlations are very useful in revealing the genuine quantum correlations.
Necessary and sufficient condition for entanglement distribution with seperable Bell-diagonal states
====================================================================================================
Separable Bell diagonal states will be useful resource for entanglement distribution *iff* correlations exist between measurement outcomes of all the complementary observables present on both side. The proof of the statement is as follows:
#### {#section-12 .unnumbered}
Let $\rho_{AB}$ is a Bell-diagonal state as represented in Eqn.(\[bell-diag\]) and $\lambda(\rho_{ABC})$ denote the spectrum of state $\rho_{ABC}$, where, $\rho_{ABC}= U_{AC}\rho_{AB}\otimes \rho_C U_{AC}^ \dagger$ and $U_{AC}$ is the unitary applied by Alice on qubits $A$ and $C$. The partial transpose of $\rho_{ABC}$ on qubit $A$ be $\rho^A_{ABC}$. Entanglement distribution is said to be successful if in the bipartition $A\lvert BC$, $\lambda(\rho^A_{ABC}) < 0$[@boundentanglement].
#### {#section-13 .unnumbered}
Now let $\mathcal{I}(\sigma_{(y)A}:\sigma_{(y)B})=0$, which implies $\frac{(1+c_2)}{2}\log_2(1+c_2)+\frac{(1-c_2)}{2}\log_2(1-c_2)=0$ and hence $ \lvert c_2\lvert =0$. Under such a condition $\lambda(\rho^A_{ABC}) <0$ does not hold [@kay], i.e., no distillable entanglement is present between Alice and Bob.
#### {#section-14 .unnumbered}
Similarly if we consider $\mathcal{I}(\sigma_{(x)A}:\sigma_{(x)B})=0$ or $\mathcal{I}(\sigma_{(z)A}:\sigma_{(z)B})=0$ ( equivalently $\lvert c_1 \lvert =0$ or $\lvert c_3\lvert=0$) then $\lambda(\rho^A_{ABC})= \lambda(\rho_{ABC})$. Now, as $\rho_{ABC}$ is a well defined desity matrix representing a 3-qubit state, $\lambda(\rho_{ABC}) \geq 0$. Hence, there will be no distillable entanglement in the bipartition $A\lvert BC$.
Conclusion {#sec4}
==========
We have shown that for a certain class of states the genuine quantum correlations can be measured through complementary correlations. For such states the non-classical correlations measured through complementary correlations are exactly equal to the entanglement and discord. Considering Bell-diagonal states we have emphasized that simultaneous existence of correlations in all the mutually unbiased bases(equivalently $\mathcal{I}(\sigma_{(x)A}:\sigma_{(x)B})\neq0$ $\mathcal{I}(\sigma_{(y)A}:\sigma_{(y)B})\neq0$, $\mathcal{I}(\sigma_{(z)A}:\sigma_{(z)B})\neq0$) is necessary for non-zero entanglement. Moreover, for such states if correlations in any complementary base vanish then the quantum discord will decrease. Though we have considered a very special class of states and tried to investigate how complementary correlations can provide insights about discord and entanglement, we aim to generalize the results for any bipartite 2-qubit states.
#### {#section-15 .unnumbered}
We have also provided a necessary and sufficient condition for entanglement distribution with seperable Bell-diagonal states. A seperable Bell-diagonal state will be a useful resource for EDSS *iff* all the complementary correlations are non-zero. The condition for EDSS provided by us seems to be equivalent to that mentioned in Ref[@kay] but our approach is entirely different. Infact, if the coefficients $c_is$ are ordered like $\lvert c_1\lvert \geq \lvert c_3\lvert \geq \lvert c_2\lvert$ and it is assumed that $\mathcal{I}(\sigma_{(x)A}:\sigma_{(x)B})=0$, then $\lvert c_2\lvert=0$ automatically. So, the necessary condition $\lvert c_2\lvert \neq 0$ for EDSS with Bell-diagonal states[@kay] follows from the condition shown in this paper. Entanglement distribution with seperable states depends on the quantum correlations present in such states and we hope that our results will shed light on this fact.
Acknowledgements
================
This work is funded through INSPIRE-Fellowship Scheme by Department of Science and Technology, Govt. of India. The author acknowledges Professor Alastair Kay for providing useful insights.
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---
abstract: 'We demonstrate the analogue of electromagnetically induced transparency in a room temperature cavity optomechanics setup formed by a thin semitransparent membrane within a Fabry-Pérot cavity. Due to destructive interference, a weak probe field is completely reflected by the cavity when the pump beam is resonant with the motional red sideband of the cavity. Under this condition we infer a significant slowing down of light of hundreds of microseconds, which is easily tuned by shifting the membrane along the cavity axis. We also observe the associated phenomenon of electromagnetically induced amplification which occurs due to constructive interference when the pump is resonant with the blue sideband.'
author:
- 'M. Karuza'
- 'C. Biancofiore'
- 'M. Bawaj'
- 'C. Molinelli'
- 'M. Galassi'
- 'R. Natali'
- 'P. Tombesi'
- 'G. Di Giuseppe'
- 'D. Vitali'
bibliography:
- 'optomechanics-eit.bib'
title: 'Optomechanically induced transparency in membrane-in-the-middle setup at room temperature'
---
Cavity optomechanics is currently a very active field of investigation owing to the disparate possibilities offered by the ability to manipulate the state and dynamics of nanomechanical resonator with light, and at the same time of controlling light by tailoring its interaction with one (or more) mechanical resonances [@Kippenberg2007; @Genes2009; @Marquardt2009; @Favero2009; @Aspelmeyer2012]. A notable example of this kind of light beam control is provided by the optomechanical analogue of electromagnetically induced transparency (EIT) [@Arimondo1996; @Fleischhauer2005], the so called optomechanically induced transparency (OMIT), which has been recently demonstrated both in optical [@Weis2010; @Safavi-Naeini2011] and microwave domains [@Teufel2011; @Massel2012]. In EIT an intense control field (pump) modifies the optical response of an opaque medium making it transparent in a narrow bandwidth; the concomitant steep variation of the refractive index induces a significant slowing down of the group velocity of a probe beam [@Hau1999], which can be used to delay, stop, store and retrieve both classical [@Phillips2001; @Liu2001] and quantum information [@Fleischhauer2005] encoded in a light field. In OMIT, the internal resonance of the medium is replaced by a dipole-like interaction of optical and mechanical degrees of freedom which occurs when the pump is tuned to the lower motional sideband of the cavity resonance. EIT has been first observed in atomic gases and more recently in a variety of solid-state systems such as quantum wells, dots and nitrogen–vacancy centers [@Phillips2003; @Santori2006; @Xu2008]. OMIT may offer various advantages with respect to these latter implementations: i) it does not rely on naturally occurring resonances and could therefore be applied to previously inaccessible wavelength regions; ii) a single optomechanical element can already achieve unity contrast, which in the atomic case is only possible within the setting of cavity quantum electrodynamics [@Mucke2010]; iii) one can achieve significant optical delay times, since they are limited only by the mechanical resonance lifetime of the optomechanical system.
With the exception of some results shown in Ref. [@Safavi-Naeini2011], previous OMIT demonstrations have been carried out in a cryogenic setup; here we show OMIT and also the associated phenomenon of electromagnetically induced amplification [@Massel2011] in a room temperature optomechanical setup consisting of a thin semitransparent membrane within a high-finesse optical Fabry-Pérot (FP) cavity [@Thompson2008; @Wilson2009]. Our setup involves free space optics rather than guided optics as in Refs. [@Weis2010; @Safavi-Naeini2011], and it operates at lower frequencies (hundreds of kHz), with respect to the MHz-GHz regime of Refs. [@Weis2010; @Safavi-Naeini2011], allowing us to attain significantly longer delay times, up to $1$ ms. Moreover, in Refs. [@Weis2010; @Safavi-Naeini2011], the optical and mechanical modes are localized within the same structure, while in the present setup the mechanical element is separated and independent from the cavity mode, enabling the study of a larger variety of optomechanical configurations with micromechanical resonators with different material and structural properties. While in the previous demonstrations of OMIT [@Weis2010; @Safavi-Naeini2011] the interference between a probe beam and a strong pump beam results in a “transparency” frequency window, *i.e.* the probe beam is transmitted through the tapered optical fiber coupled to the resonator, in our system it leads to an “opacity” frequency window, *i.e.* the probe is completely reflected by the cavity even if in resonance with it. Furthermore the OMIT transparency window and the optical delay can be tuned in a simple way by properly shifting the membrane along the cavity axis.
*The experimental setup*. The optical power of a laser beam at $\lambda=1\,064\,\mathrm{nm}$ produced by a Nd:YAG laser (Innolight) was distributed between a probe ($\omega_\mathrm{p}$) and a pump beam ($\omega_{\mathsmaller{\mathrm{L}}}$) by means of a cascade of a half wave plate (HWP$_1$) and a polarizing beam splitter (PBS$_1$), as shown in the setup in Fig. 1 (see also Ref. [@Karuza2012]).
![Schematic description of the experimental setup. []{data-label="fig:1"}](Fig_Setup){width="45.00000%"}
The probe beam power was $100\,\mu\mathrm{W}$, while the rest, that was about $200\,\mathrm{mW}$, was fed into the pump beam optical line: at the end only a small fraction of it was used. Two cascaded acousto-optical modulators (AOMs) were used to obtain controlled frequency detuning from the probe beam in the range $0$ to $40\,\mathrm{Mhz}$, although only detunings up to $500\,\mathrm{kHz}$ have been used. The pump beam intensity is controlled by the modulation amplitude of the electrical signal used to drive AOM$_2$. After the AOMs and an optical isolator (OFR$_2$) the pump beam was mode-matched to the FP optical cavity by means of two lenses (L$_1$ and L$_3$). Before being injected in the FP cavity the pump beam was combined with the probe beam by polarization multiplexing of the fields on PBS$_2$. The cavity was $L \approx 93\,\mathrm{mm}$ long and consisted of two equal dielectric mirrors, each with a radius of curvature $R = 10\,\mathrm{cm}$. The measured value of the empty cavity finesse was $\mathcal{F} \approx 60\,000$, consistent with the mirror’s nominal reflectivity. Halfway between the mirrors a thin stoichiometric silicon nitride membrane was mounted on series of piezo-motor driven optical mounts that control the angular alignment as well as the linear positioning with respect to the optical axis. The membrane was a commercial $1\,\mathrm{mm}\times 1\,\mathrm{mm}$ ${\rm Si_3 N_4}$ stoichiometric x-ray window (Norcada), with nominal thickness $L_{\mathrm{d}} =50\,\mathrm{nm}$, and index of refraction $n_{\mathsmaller{\mathrm{R}}} \approx 2$, supported on a $200\,\mu\mathrm{m}$ $\mathrm{Si}$ frame. It has been chosen due to its high mechanical quality factor and very low optical absorption at $\lambda = 1\,064\,\mathrm{nm}$ [@Zwickl2008]. Its optical properties were also experimentally verified, yielding an intensity reflection coefficient $\mathcal{R} \approx 0.18$, and an imaginary part of the index of refraction $n_{\mathsmaller{\mathrm{I}}} \approx 2 \times 10^{-6}$. In order to avoid the deterioration of the mechanical properties of the membrane and optical properties of the FP cavity, the cavity was mounted inside a vacuum chamber which was evacuated by a turbo-molecular pump down to $10^{-5}\,\mathrm{mbar}$. The probe beam light reflected from the cavity was observed by a photodiode (PD$_2$) whose output signal is amplified and fed into a frequency locking loop (described in Ref. [@Karuza2012]) and a spectrum analyzer where the membrane’s mechanical motion was monitored. *Langevin equation description*. The pump drives the system in a steady state characterized by the driven TEM$_{00}$ mode (with photon annihilation operator $\hat{a}$) in an intense coherent state with amplitude $\alpha_{\mathrm{s}}$, and the membrane deformed by radiation pressure. We choose the detection bandwidth in order to observe the fundamental membrane vibrational mode with resonance frequency $\Omega_{\mathrm{m}}/2\pi \approx 355.6$ kHz and quality factor $Q=\omega_{\mathrm{m}}/\gamma_{\mathrm{m}}\approx122\,000$, which we describe as a harmonic oscillator with effective mass $m$, and with dimensionless position $\hat{q}$ and momentum $\hat{p}$ satisfying the commutation rule $\left[\hat{q},\hat{p}\right]=\mathrm{i}$ [@Biancofiore2011; @Karuza2012]. Under these conditions, dynamical effects are associated with the fluctuations $\bigl(\delta \hat{q},\delta \hat{p}\bigr)$ of the vibrational mode around its steady state $\bigl(q_{\mathrm{s}},p_{\mathrm{s}}=0\bigr)$, and with the cavity mode fluctuations $ \delta \hat{a}$ around $\alpha_{\mathrm{s}}$. This dynamics are well described by the following linearized Langevin equations [@Biancofiore2011; @Karuza2012]
\[lle\] $$\begin{aligned}
\delta \dot{\hat{q}}& =&\Omega _{\mathrm{m}}\delta \hat{p}, \label{lle1}\\
\delta \dot{\hat{p}}& =&-\left[\Omega _{\mathrm{m}}+\partial_q^2 \omega(q_{\mathrm{s}})|\alpha_{\mathrm{s}}|^2\right]\delta \hat{q}-\gamma _{\mathrm{m}}\delta \hat{p} \nonumber \\
&&-\partial_q \omega(q_{\mathrm{s}})\alpha_{\mathrm{s}} \left(\delta \hat{a} +\delta \hat{a}^{\dagger}\right) +\hat{\xi}, \label{lle2}\\
\delta \dot{\hat{a}}& =&-\left(\kappa_0+\kappa_2+\mathrm{i} \Delta \right) \delta \hat{a}-\mathrm{i} \partial_q \omega(q_{\mathrm{s}})\alpha_{\mathrm{s}} \delta \hat{q} \label{lle3} \nonumber \\
&& +
\sqrt{2\kappa_0}\hat{a}_0^{\mathrm{in}}+\sqrt{2\kappa_2}\hat{a}_2^{\mathrm{in}}+\sqrt{2\kappa_0} s_{\mathrm{p}} \mathrm{e}^{-\mathrm{i}\Omega t},\end{aligned}$$
where we have adopted a frame rotating at the pump frequency $\omega_{\mathsmaller{\mathrm{L}}}$, and we have chosen the phase reference of the cavity field so that $\alpha _{\mathrm{s}}$ is real and positive. $\kappa_0 $ and $\kappa_2 $ denote the cavity decay rates through the input and back mirror respectively, $\hat{a}_0^{\mathrm{in}}$ and $\hat{a}_2^{\mathrm{in}}$ are the corresponding vacuum optical input white noises [@Gardiner2000], $\Delta = \omega(q_{\mathrm{s}})-\omega_{\mathsmaller{\mathrm{L}}}$ is the cavity detuning, and $\hat{\xi}$ is the thermal stochastic force. Optomechanical coupling is provided through the position-dependent cavity mode frequency, $\omega(\hat{q})=\omega_0+{\operatorname{Re}}\bigl\{\delta\omega\left[z_0(\hat{q})\right]\bigr\}$, where $\omega_0$ is the frequency in the absence of the membrane, and ${\operatorname{Re}}\bigl\{\delta\omega\left[z_0(\hat{q})\right]\bigr\}$ is the frequency shift caused by the insertion of the membrane. This shift depends on the membrane position along the cavity axis $z_0(\hat{q}) = z_0+x_0 \Theta \hat{q}$, where $z_0$ is the membrane center-of-mass position along the cavity axis, $\Theta$ is the transverse overlap integral between the optical mode and the vibrational mode [@Biancofiore2011], and $x_0 = \sqrt{\hbar/m \Omega_\mathrm{m}}$. Radiation pressure coupling is described by the first order derivative term $\partial_q \omega(q_{\mathrm{s}})$, but, as shown in Ref. [@Karuza2012], also the second-order term $\partial_q^2 \omega(q_{\mathrm{s}})$ has to be included in Eq. (\[lle2\]) since it accounts for an observable mechanical frequency shift which is typical for the membrane-in-the-middle setup and usually negligible in other cavity optomechanical devices.
*Optomechanically induced transparency*. The last term of Eq. (\[lle3\]) describes the additional weak probe field of frequency $\omega_{\mathrm{p}} = \omega_{\mathsmaller{\mathrm{L}}}+\Omega$ and amplitude $s_{\mathrm{p}}$ which, together with the intense pump, induces a modulation at frequency $\Omega$ of the radiation pressure force acting on the membrane. When this modulation is close to the mechanical resonance frequency $\Omega_{\mathrm{m}}$, the vibrational mode is excited, giving rise to Stokes- and anti-Stokes scattering of light from the strong pump field. If the latter is tuned to the red sideband of the cavity, Stokes scattering is suppressed and only the anti-Stokes field at $ \omega_{\mathsmaller{\mathrm{L}}}+\Omega_{\mathrm{m}}$ builds up within the cavity. However when $\Omega \approx \Omega_{\mathrm{m}} \approx \Delta$, also the probe beam is resonant with the cavity, but destructive interference with the anti-Stokes field suppresses its build-up and as a result the probe beam is *perfectly reflected* by the coupled cavity-membrane system [@Weis2010; @Agarwal2010]. This OMIT phenomenon is well described by the classical limit of Eqs. (\[lle\]), in which the fluctuation operators are replaced by classical variables. The probe modulates in time the coupled optomechanical system and therefore it is reasonable to assume as trial solution of Eqs. (\[lle\]), $ \delta a = A_+ \mathrm{e}^{\mathrm{i}\Omega t}+ A_- \mathrm{e}^{-\mathrm{i}\Omega t}$, and $ \delta q = X \mathrm{e}^{-\mathrm{i}\Omega t} + \mathrm{c.c.} $. The resulting amplitudes are given by $$\label{amen}
A_\pm = \frac{\sqrt{2\kappa_0} s_{\mathrm{p}}}{\kappa_{\mathsmaller{\mathrm{T}}}+\mathrm{i} (\Delta-\Omega)}\left[\delta_{\pm 1,-1}+\mathrm{i}\frac{G^2 \chi_{{\rm eff}}(\mp\Omega)/2}{\kappa_{\mathsmaller{\mathrm{T}}}+\mathrm{i} (\Delta\pm\Omega)}\right],$$ $ X = \sqrt{\kappa_0} s_{\mathrm{p}} G \chi_{{\rm eff}}(\Omega)/\left[\kappa_{\mathsmaller{\mathrm{T}}}+\mathrm{i} (\Delta-\Omega)\right]$, where $\kappa_{\mathsmaller{\mathrm{T}}} = \kappa_0+\kappa_2$ is the total cavity decay rate, we have introduced the effective optomechanical coupling $ G = -\sqrt{2}\partial_q \omega(q_{\mathrm{s}})\alpha_{\mathrm{s}} $ [@Genes2009] given by $$G = -2\left(\frac{\partial\omega}{\partial z_0}\right)\Theta \sqrt{\frac{\mathcal{P}\kappa_0 }{m \Omega_{\mathrm{m}} \omega_{\mathsmaller{\mathrm{L}}}\left(\kappa_{\mathsmaller{\mathrm{T}}}^{2}+\Delta ^{2}\right) }}, \label{optoc}$$ with ${\mathcal P}$ the pump input power, and $$\chi _{\rm eff}(\omega ) = \Omega_{\mathrm{m}}\Biggl[\tilde{\Omega}_\mathrm{m}^{2}-\omega^{2}-\mathrm{i}\omega \gamma _\mathrm{m}-\frac{G^2\Delta\Omega _\mathrm{m}}{\left(\kappa_{\mathsmaller{\mathrm{T}}} -\mathrm{i}\omega
\right)^{2}+\Delta ^{2}}\Biggr]^{-1},\label{chieffD}$$is the mechanical susceptibility modified by the optomechanical coupling, with $
\tilde{\Omega}_\mathrm{m}^{2} = \Omega_\mathrm{m}^{2}+h \Omega_\mathrm{m},\quad h = \partial_{q}^2 \omega(q_\mathrm{s})|\alpha_\mathrm{s}|^2$, the square of the mechanical frequency modified by the second order contribution to the expansion of $\omega(\hat{q})$.
The output field transmitted by the cavity is given by $$\label{transm}
a_2^{\mathrm{out}}= \sqrt{2\kappa_2}\left(\alpha_{\mathrm{s}}+A_- \mathrm{e}^{-\mathrm{i}\Omega t} +A_+ \mathrm{e}^{\mathrm{i}\Omega t}\right);$$ as discussed above, in our setup OMIT manifests itself as a complete reflection of the probe beam by the cavity, even if at resonance. This happens when $A_-=0$, which is realized when the probe is resonant with the cavity and with the blue sideband of the pump, $\Omega\approx \Delta \approx \Omega_{\rm m}\approx \tilde{\Omega}_{\rm m}$, which is analogous to the two-photon resonant condition of usual EIT [@Weis2010; @Safavi-Naeini2011; @Arimondo1996; @Fleischhauer2005; @Agarwal2010]. In such a case, in fact, $A_- \propto 1+\mathrm{i}G^2 \chi_{\rm eff}(\Delta)/2\kappa_{\mathsmaller{\mathrm{T}}} \approx 0,$ where the latter condition is realized when the *cooperativity* $C= G^2/2\kappa_{\mathsmaller{\mathrm{T}}}\gamma_{\rm m}$ is sufficiently large, $C \gg 1$, and we are in the resolved sideband regime $\kappa_{\mathsmaller{\mathrm{T}}} \ll \Omega_{\rm m}$, conditions which are both met in our experiment.
In Refs. [@Weis2010; @Safavi-Naeini2011; @Teufel2011; @Massel2012] OMIT is shown by measuring the probe transmission as a function of $\Omega$. Here we show its occurrence in a slightly different way, by measuring the intensity and the phase shift of the beat at frequency $\Omega$ between the transmitted pump and probe fields, $A_{{\rm beat}}$. Using Eq. (\[transm\]) and neglecting the field oscillating at $-\Omega$ which is well out of resonance, one gets that the beat amplitude at frequency $\Omega$ of the transmitted field is given by $ A_{{\rm beat}} = 2\kappa_2 \alpha_{\mathrm{s}} A_- $, namely, $$\label{eq:beatamp}
A_{{\rm beat}} = \frac{4\kappa_2 \kappa_0 |s_{\mathrm{p}}|}{\kappa_{\mathsmaller{\mathrm{T}}}} \sqrt{\frac{{\cal P}}{\hbar \omega_{\mathsmaller{\mathrm{L}}} \left(\kappa_{\mathsmaller{\mathrm{T}}}^2+\Delta^2\right)}}\left[1+ \mathrm{i} \frac{G^2 \chi_{{\rm eff}}(\Delta)}{2\kappa_{\mathsmaller{\mathrm{T}}}}\right],$$ where the phase of $A_{\rm beat}$ is referred to the phase of the probe $s_{\mathrm{p}}$, and we have put $\Omega = \Delta$ in Eq. (\[eq:beatamp\]) because we have taken the weak probe to be always resonant with the cavity.
The behavior of the measured beat amplitude is shown in Fig. 2, where its phase and modulus are plotted vs the pump-probe detuning $\Omega$, which is kept equal to the cavity-pump detuning $\Delta$. The data refer to an incident pump power ${\mathcal P}\approx3$ mW, and we have independently measured a total cavity rate $\kappa_{\mathsmaller{\mathrm{T}}} \approx 85$ KHz, and an effective mass $m\approx45$ ng. Both plots are in good agreement with the theoretical prediction of Eq. (\[eq:beatamp\]) for an effective optomechanical coupling $|G|=9.4 \times 10^{-3} \Omega_{\rm m}$ (full blue line), corresponding to a membrane shifted by $z_0=4 $ nm along the cavity axis with respect to a field node.
![(Color online) Phase shift with respect to the probe (upper panel) and modulus (lower panel) of the beat between the transmitted pump and probe beams vs the pump-probe detuning, which is kept equal to the cavity-pump detuning $\Delta$. The blue full line refers to the theoretical prediction of Eq. (\[eq:beatamp\]) with parameters given in the text.[]{data-label="fig:omit"}](FIG_OMIT_Qm_122000_50mV){width=".45\textwidth"}
Fig. 2a shows the phase shift acquired by the probe beam during its transmission through the optomechanical cavity. The derivative of such a phase shift gives the group advance due to causality-preserving superluminal effects which a probe pulse spectrally contained within the transparency window would accumulate in its transmission through the cavity. From the fitting curve of Fig. 2a we infer a maximum signal time advance $\tau^{\mathsmaller{\mathrm{T}}} \approx -108$ ms, which is very close to the theoretical maximum time advance achievable at $\Omega = \Delta = \Omega_{\mathrm{m}}$ [@Safavi-Naeini2011] $\tau^{\mathsmaller{\mathrm{T}}}_{\rm max} =-2C/[\gamma_{\mathrm{m}}(1+C)]$, which is $-109$ ms in our case where $C = 160$. The reflected field is instead delayed, and from the corresponding expression for the maximum time delay $\tau^{\mathsmaller{\mathrm{R}}}_{\rm max}= 2 \eta C/[\gamma_{\mathrm{m}}(1+C)(1-\eta+C)]$ ($\eta=2\kappa_0/\kappa_{\mathsmaller{\mathrm{T}}}\approx 1$), we can also infer a group delay of the reflected probe field $\tau^{\mathsmaller{\mathrm{R}}}\approx 670$ $\mu$s.
In Fig. 2b the “transparency” frequency window in which the probe is *completely reflected* by the interference associated with the optomechanical interaction is evident. The width of the transparency window is related to the effective mechanical damping $\gamma_{\mathrm{m}}^{\rm eff}$, which is approximately given by $\gamma_{\mathrm{m}}^{\rm eff}\approx \gamma_{\mathrm{m}}(1+C)$ around the resonant condition $\Omega_{\rm m}=\Delta=\Omega$ we are considering [@Weis2010; @Safavi-Naeini2011] \[see also Eq. (\[eq:beatamp\])\], and therefore increases for increasing cooperativity. This is illustrated in Fig. 3, where the modulus of the beat amplitude vs $\Delta=\Omega$ is plotted for different positions shifts $z_0$ of the membrane from a field node: $z_0=5$ nm (red circles), $z_0=7$ nm (light green up-pointing triangles), $z_0=15$ nm (blue squares), $z_0=21$ nm (orange down-pointing triangles), corresponding to increasing values of the coupling, $|G|/\Omega_{\mathrm{m}} =1.0 \times 10^{-2}$, $1.4 \times 10^{-2}$, $3.1 \times 10^{-2}$, $4.2 \times 10^{-2}$, respectively. The other parameters are the same as those of Fig. 2 except for the mechanical quality factor, which was smaller, $Q=24\,000$, due to the lower quality of the vacuum in the chamber. The data are in very good agreement with the prediction of Eq. (\[eq:beatamp\]) (full lines). The inset in Fig. 3 explicitly shows how the EIT bandwidth can be tuned with membrane position $z_0$, at fixed input laser power.
![(Color online) Modulus of the beat between the transmitted pump and probe beams vs the pump-probe detuning, which is kept equal to the cavity-pump detuning $\Delta$ for different membrane shifts $z_0$ with respect to a cavity node, as explained in the text. The full lines refer to the theoretical prediction of Eq. (\[eq:beatamp\]). The inset shows the width of the EIT window versus the membrane position $z_0$, with respect to a field node, at fixed input laser power.[]{data-label="fig:omit2"}](Fig_3_new_MOD_2){width=".4833\textwidth"}
The results show that thermal noise does not have any relevant effect on the EIT window, even if the experiment is carried out at room temperature, at very large mean thermal phonon number $n_{\mathrm{th}} \approx 10^8 \gg C$. There is however an important limitation which occurs in this high temperature limit: the setup can be used to delay and store light pulses carrying only *classical* states but not *quantum* states. In fact only pulses with bandwidth narrower than the EIT window $\approx \gamma_\mathrm{m} C$ can be delayed and stored; at the same time a quantum state is decohered at the thermal decoherence rate $\gamma_\mathrm{m} n_{\mathrm{th}}$, and therefore it can be safely stored only if $n_{\mathrm{th}} < C$.
*Optomechanically induced amplification*. We have then investigated the situation where the pump is resonant with the *blue* sideband of the cavity mode, *i.e.*, when $\Delta=-\Omega_{\rm m}$. In such a case, the probe beam *constructively* interferes with the Stokes sideband of the pump beam which is resonant with the cavity, and may be amplified in transmission within a very narrow frequency window. This is the optomechanical analogue of electromagnetically induced amplification [@Safavi-Naeini2011; @Massel2011], demonstrated in the unresolved sideband limit in [@Verlot2010; @McRae2012; @Li2012], and closely related to the electromagnetically induced absorption observed in atomic gases [@Lezama1999]. The latter consists in the decrease of the total power at the output of the medium for increasing pump power, and may occur only when the medium internal losses are larger than the external losses. In our case internal losses are negligible, $\kappa_{\mathsmaller{\mathrm{T}}}=\kappa_0 + \kappa_2 = \kappa_{\rm ext}$, and we may only observe amplification. This can be seen using Eq. (\[eq:beatamp\]) for deriving the probe transmission $t_{\rm p}$, which at the blue sideband resonance $\Omega = \Delta \approx -\Omega_{\rm m}$ reads $ t_{\rm p} = \eta'/(1-C)$, where $\eta'=2\sqrt{\kappa_0 \kappa_2}/\kappa_{\mathsmaller{\mathrm{T}}}$. $t_{\rm p} $ gives the amplifier gain and therefore the probe is amplified in transmission when $C> 1-\eta'$, which is practically always satisfied because in our setup $\kappa_0 \approx \kappa_2 \approx \kappa_{\mathsmaller{\mathrm{T}}}/2$. The amplification bandwidth is narrow and given by the effective mechanical damping in this blue sideband driving condition $\gamma_{\mathrm{m}}^{\rm eff}\approx \gamma_{\rm m}(1-C)$. The amplification of the transmitted probe beam is well visible in Fig. 4, where the modulus of the beam amplitude is plotted vs $\Delta$, but now around the condition $\Delta=-\Omega_{\rm m}$. The full line corresponds to the prediction of Eq. (\[eq:beatamp\]). Fig. 4 refers to an input power ${\cal P}\approx50$ $\mu$W, a membrane shifted by $z_0\approx5$ nm from a node, corresponding to a coupling $|G|/\Omega_{\mathrm{m}} \approx 10^{-3}$, and a quality factor $Q\approx24\,000$, yielding in this case $C \approx 0.32$.
[1000]{}
![(Color online) Modulus of the beat between the transmitted pump and probe beams vs the pump-probe detuning, in the case of optomechanically induced amplification. The full line refers to the theoretical prediction of Eq. (\[eq:beatamp\]). See text for parameters.[]{data-label="fig:omia"}](FIG_OMIA_Qm_24000){width=".45\textwidth"}
The system is stable as long as $\gamma_{\mathrm{m}}^{\rm eff} >0$, *i.e.*, only if $C<1$. In this regime the system is the optomechanical analogue of a parametric oscillator below threshold. The system has been studied even at larger cooperativity and the nonlinear amplification process controlled by membrane position along the optical axis has been observed. At large cooperativity few mW of pump power have been transferred to the cavity resonance. This process, where the mechanical resonator starts to oscillate with a nonzero amplitude, has been theoretically discussed in [@Marquardt2006; @Rodrigues2010] and experimentally demonstrated in [@Kippenberg2005; @Metzger2008].
*Acknowledgments*. This work has been supported by the European Commission (ITN-Marie Curie project cQOM).
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abstract: 'Let $p$ be an odd prime number. In this paper, we construct $2(2p-3)$ classes of codes over the ring $R=\Bbb F_p+u\Bbb F_p,u^2=0$, which are associated with down-sets. We compute the Lee weight distributions of the $2(2p-3)$ classes of codes when the down-sets are generated by a single maximal element. Moreover, by using the Gray map of the linear codes over $R$, we find out $2(p-1)$ classes of $p$-ary distance optimal linear codes. Two classes of them are attained the Griesmer bound with equality as well.'
address:
- 'Department of Mathematics, Ewha Womans University, 52, Ewhayeodae-gil, Seodaemun-gu, Seoul, 03760, South Korea.'
- 'Konkuk University, Glocal Campus, 268 Chungwon-daero Chungju-si Chungcheongbuk-do 27478, South Korea'
author:
- Yansheng Wu
- Jong Yoon Hyun
title: 'Few-weight codes over $\Bbb F_p+u\Bbb F_p$ associated with down sets and their distance optimal Gray image'
---
Introduction
============
Let $p$ be a prime number and $\Bbb F_p$ the finite field of order $p$. An $[n, k, d]$ linear code $\mathcal{C}$ of length $n$ over $\Bbb F_p$ is a $k$-dimensional subspace of $\Bbb F_p^n$ with minimum Hamming distance $d$. The dual $\mathcal{C}^{\perp}$ of $\mathcal{C}$ is defined by $\{x\in\mathbb{F}^n_p:x\cdot c=0 \text{ for all }c\in\mathcal{C}\}$, where $x\cdot c=x_1c_1+\cdots+x_ny_n\in\mathbb{F}_p$. An $ [n,k,d]$ code $\mathcal{C}$ is called distance optimal if no $[n,k,d+1]$ code exists, see [@HP Chapter 2]. It is well-known [@G] that $n\geq\sum_{i=0}^{k-1}\left\lceil\frac{d}{p^i}\right\rceil$, called the Griesmer bound, for any $[n,k,d]$ linear code over $\mathbb{F}_p$. It follows that a linear code over $\mathbb{F}_p$ satisfying the Griesmer bound with equality is distance optimal. We say that a linear code is optimal if it attains the Griesmer bound with equality.
Denote $A_i$ by the number of codewords in $\mathcal C$ with Hamming weight $i$. The weight enumerator of $\mathcal C$ is defined by $1+A_1z+A_2z^2+\cdots+A_nz^n.$ The sequence $(1, A_1, A_2, \ldots, A_n)$ is called the weight distribution of $\mathcal C$. A code $\mathcal{C}$ is $t$-weight if the number of nonzero $A_{i}$ in the sequence $(A_1, A_2, \ldots, A_n)$ is equal to $t$.
Let $\Bbb F_{q}$ be the finite field of order $q$, where $q$ is a power of a prime $p$. Let $D=\{d_{1}, d_{2}, \ldots, d_{n}\}\subseteq \Bbb F_{w}$, where $w$ is a power of $q$. A linear code of length $n$ over $\Bbb F_{q}$ is defined by $$\mathcal{C}_{D}= \{({\operatorname{Tr}}_{w/q}(xd_{1}), \ldots, {\operatorname{Tr}}_{w/q}(xd_{n})) : x\in \Bbb F_{w}\},$$ where ${\operatorname{Tr}}_{w/q}$ is the trace function from $\Bbb F_{w}$ to $\Bbb F_{q}$. This generic construction was first introduced by Ding et al. [@D1; @DN]. Many known few-weight linear codes could be produced by selecting a proper defining set $D$, see [@HY1; @KY; @LYL; @LM2; @Y]. The construction method by Ding [*et al*]{}. can be generalized as follows: Let $R$ be a finite commutative ring, $R_m$ be an extension of $R$ of degree $m$ and $R_m^*$ be the multiplicative group of units of $R_m$. A trace code over $R$ with a defining set $L=\{l_1,l_2,\ldots, l_n\} \subseteq R_m^*$ is defined by $$\mathcal {C}_L=\{{\operatorname{Tr}}(xl_1), {\operatorname{Tr}}(xl_2), \ldots, {\operatorname{Tr}}(xl_n):x\in R_m\},$$ where ${\operatorname{Tr}}(\cdot)$ is a linear function from $R_m$ to $R$.
To derive few-weight linear codes, we choose specific commutative rings and their extensions. Now let $R=\Bbb F_q+u\Bbb F_q, u^2=0$, and $\mathcal {R}=\Bbb F_w+u\Bbb F_w$. The Lee weight distribution of a trace code $\mathcal{C}_L$ have been investigated in some literature.
\(1) When $R=\Bbb F_p+u\Bbb F_p, u^2=0$, $\mathcal {R}=\Bbb F_{p^m}+u\Bbb F_{p^m}$ and $L=\mathcal{Q}+u\Bbb F_{p^m}$, where $\mathcal{Q}$ is the set of all square elements of $\Bbb F_{p^m}^*$, the code $C_L$ is a two-weight or three-weight code, see [@S2].
\(2) When $R=\Bbb F_p+u\Bbb F_p, u^2=u$, $\mathcal {R}=\Bbb F_{p^m}+u\Bbb F_{p^m}$ and $L=\mathcal{Q}+u\Bbb F_{p^m}^*$ or $\Bbb F_{p^m}^*+u\Bbb F_{p^m}^*$, where $\mathcal{Q}$ is the set of all square elements of $\Bbb F_{p^m}^*$, the code $C_L$ is a two-weight or few-weight code, see [@S3].
\(3) When $R=\Bbb F_q+u\Bbb F_q, u^2=0$, $\mathcal {R}=\Bbb F_{q^m}+u\Bbb F_{q^m}$ and $L=C_0^{(e,q^m)}+u\Bbb F_{p^m}$, where $e$ is a divisor of $q-1$ and $C_0^{(e,q^m)}$ is the cyclotomic class of order $e$, the code $C_L$ is a two-weight or few-weight code, see [@LM].
In this paper, we study the following linear codes defined in $(1.1)$ right below. Let $L$ be a subset of $\Bbb F_p^m+u\Bbb F_p^m,u^2=0$. A code $\mathcal {C}_{L}$ over $\Bbb F_p+u\Bbb F_p$ is defined by $$\mathcal {C}_{L}=\{c_{L}(\mathbf{a})=(\langle \mathbf{a}, \mathbf{l}\rangle )_{\mathbf{l}\in L}:\mathbf{a}\in \Bbb F_p^m+u \Bbb F_p^m\},$$ where $\langle \cdot, \cdot\rangle$ is the standard inner product on $ \Bbb F_p^m+u \Bbb F_p^m$. Notice that if $\mathbf{x}=\mathbf{a}+u\mathbf{b}$ and $\mathbf{y}=\mathbf{c}+u\mathbf{d}$ for $\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}\in\mathbb{F}^m_p$, then $\langle \mathbf{x},\mathbf{y} \rangle=\mathbf{a}\cdot\mathbf{c}+u(\mathbf{a}\cdot\mathbf{d}+\mathbf{b}\cdot\mathbf{c})$.
[Let $L=\{(1,0),(1,0)+(1,0)u,(1,0)+(0,1)u,(1,0)+(1,1)u\}$ be a subset of $\Bbb F_2^2+u\Bbb F_2^2,u^2=0$. Then $$\mathcal{C}_L=\{(a_1+b_1u,a_1+(a_1+b_1)u,a_1+(a_2+b_1)u,a_1+(a_1+a_2+b_1)u):
a_i,b_i\in\mathbb{F}_2,i=1,2\}.$$ ]{}
One of the important problems in coding theory is to find the $[n, k,d]$ linear codes over $\mathbb{F}_p$ having the highest minimum distance for given $n$ and $k$. In [@HKN], the authors constructed some infinite families of distance optimal linear codes over $\mathbb{F}_p$ from down-sets. The aim of this paper is to construct the few-weight codes $\mathcal{C}_L$ over $R=\mathbb{F}_p+u\mathbb{F}_p, u^2=0$ and find out the distance optimal linear codes over $\mathbb{F}_p$ from the Gray image of $\mathcal{C}_L$, where $L$’s are subsets of $R^m$ associated with down sets of $\mathbb{F}^m_p$ generated by a single maximal element.
The rest of this paper is organized as follows. In Section 2, we recall basic concepts and introduce some known results. In Sections 3, we determine the Lee weight distribution of $2(2p-3)$ classes of codes (Theorem 3.1-3.4). In Section 4, by using the Gray map, we obtain $2(p-1)$ classes of distance optimal linear codes (Theorem 4.1, 4.2), from which we obtain Table 5 of distance optimal linear codes with small dimension. Two classes of them are attained the Griesmer bound with equality as well. In Section 5, we conclude the paper.
Preliminaries
=============
Firstly, the Lee weight defined on $R^m=(\Bbb F_p+u\Bbb F_p)^m, u^2=0$ and the Gray map from $R^m$ to $\mathbb{F}^{2m}_p$ is introduced. Next, we define a down-set of $\mathbb{F}^m_p$ by endowing a partial order on $\mathbb{F}^m_p$.
In the remainder of this paper, we always assume that $R=\Bbb F_p+u\Bbb F_p$, where $u^2=0$.
Lee weight and Gray map
-----------------------
$~$ By a code of length $m$ over $R$, we mean a subset of $R^m$. A linear code of $\mathcal{C} $ of length $m$ over $R$ is an $R$-submodule of $R^m$. The inner product between $\mathbf{x}=(x_1,x_2,\ldots, x_m)$ and $\mathbf{y}=(y_1,y_2, \ldots, y_m)\in R^m$ is defined by $\langle \mathbf{x},\mathbf{y} \rangle=\sum_{i=1}^mx_iy_i\in R$. Notice that if $\mathbf{x}=\mathbf{a}+u\mathbf{b}$ and $\mathbf{y}=\mathbf{c}+u\mathbf{d}$ for $\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}\in\mathbb{F}^m_p$, then $\langle \mathbf{x},\mathbf{y} \rangle=\mathbf{a}\cdot\mathbf{c}+u(\mathbf{a}\cdot\mathbf{d}+\mathbf{b}\cdot\mathbf{c})$.
The Gray map $\hat{\phi}$ from $R$ to $\Bbb F_p^2$ is defined by $$\hat{\phi}: R\to \Bbb F_p^2, a+ub\mapsto (b,a+b),~a,b\in \Bbb F_p$$ This leads to the Gray map $\phi$ naturally from $R^m$ to $\Bbb F_p^{2m}$ as follows: $$\phi: R^m\to \Bbb F_p^{2m},~\mathbf{x}=\mathbf{a}+u\mathbf{b}\mapsto (\mathbf{b}, \mathbf{a}+\mathbf{b}).$$
The Hamming weight of a vector $\mathbf{a}$ of length $m$ over $\Bbb F_p$ is defined to be the number of nonzero entries in the vector ${a}$. The Lee weight of a vector $\mathbf{x}=\mathbf{a}+\mathbf{b}u$ of length $m$ over $R$ is defined to be the Hamming weight of its Gray image as follows: $$w_L(\mathbf{x})=w_L(\mathbf{a}+u\mathbf{b})=w_H(\mathbf{b})+w_H(\mathbf{a}+\mathbf{b}).$$ The Lee distance $d_L(\mathbf{x},\mathbf{y})$ of between two vectors $\mathbf{x,y}\in R^m$ is defined as $w_L(\mathbf{x-y})$. It is easy to check that the Gray map $\phi$ is an isometry from $(R^m, d_L)$ to $(\Bbb F_p^{2m}, d_H)$, where $d_H$ denotes the Hamming distance. Obviously, if $\mathcal{C}$ is a $\mathbb{F}_p$-submodule of $R^m$ with parameters $(n,p^k,d)$, then $\phi(\mathcal{C})$ is a linear code over $\mathbb{F}_p$ with parameters $[2n,k,d]$.
[We continue Example 1.1 to illustrate the Gray image of $\mathcal{C}_L$. $$\begin{aligned}
\phi(\mathcal{C}_L)=\{(b_1,a_1+b_1,a_1+b_1,b_1,a_2+b_1,a_1+a_2+b_1,\\
a_1+a_2+b_1,a_2+b_2):a_i,b_i\in\mathbb{F}_2,i=1,2\}. \end{aligned}$$ We point out that the minimum distance of $\phi(\mathcal{C}_L)^{\perp}$ is two. ]{}
Down-sets
---------
$~$ Let $v=(v_1, \ldots, v_m)$ and $w=(w_1, \ldots, w_m)$ are two vectors in $\Bbb F_p^m$. We endowed with a partial order on $\Bbb F_p^m$ as follows: $v\preceq w$ if and only if $v_i\le w_i$ for all $i\in [m]=\{1,\ldots, m\}$. We say that a subset $\Delta$ of $\Bbb F_p^m$ is a down-set if $w\in \Delta$ and $v\preceq w$ imply $v\in \Delta$. Then $(\Bbb F_p^m, \preceq)$ forms a completes lattice, where the join and the meet of two vectors $v$ and $w$ in $\Bbb F_p^m$ are respectively defined by $v \lor w=(\max\{v_1, w_1\}, \ldots,\max\{v_m, w_m\} )$ and $v\land w=(\min\{v_1, w_1\}, \ldots,\min\{v_m, w_m\} )$. An element $v\in \Delta$ is maximal if $v\preceq w$ and $w\in \Delta $ imply $v=w$. It is readily verified that every down-sets $\Delta $ of $\Bbb F_p^m$ is generated by the set of maximal elements of $\Delta$, i.e., $\Delta=\langle v(1), \ldots, v(t)\rangle$, where $\{v(1), \ldots, v(t)\}$ is the set of maximal element of $\Delta$.
The Lee weight distributions
=============================
Hereafter, we assume that $p$ is an odd prime number. Let $\Delta$ be a down-set of $\Bbb F_p^m$ and $ L=\Delta^c+u\Bbb F_p^m$, where $\Delta^c=\Bbb F_p^m\backslash \Delta$. Recall from $(1.1)$ that $$\mathcal {C}_{L}=\{c_{L}(\mathbf{y})=(\langle \mathbf{y}, \mathbf{l}\rangle )_{\mathbf{l}\in L}:\mathbf{y}\in \Bbb F_p^m+u \Bbb F_p^m\}\\
=\{(\mathbf{a}\cdot\mathbf{c}+u(\mathbf{a}\cdot\mathbf{d}+\mathbf{b}\cdot\mathbf{c}))_{\mathbf{c}\in\Delta^c,\mathbf{d}\in\mathbb{F}^m_p}:\mathbf{a},
\mathbf{b}\in\mathbb{F}^m_p\}.$$ The length of the code $\mathcal{C}_{L}$ is then $|L|$. Notice that $\mathcal{C}_L$ is not linear but $\mathbb{F}_p$-submodule of $R^m$. The Gray image $\phi(\mathcal{C}_L)$ of $\mathcal{C}_L$ is $$\phi(\mathcal{C}_L)=\{(\mathbf{a}\cdot\mathbf{d}+\mathbf{b}\cdot\mathbf{c},\mathbf{a}\cdot\mathbf{d}+\mathbf{b}\cdot\mathbf{c}+\mathbf{a}\cdot\mathbf{c})_{\mathbf{c}\in\Delta^c,\mathbf{d}\in\mathbb{F}^m_p}:\mathbf{a},
\mathbf{b}\in\mathbb{F}^m_p\}.$$
Assume that $\mathbf{a}={}\alpha+u{\beta}$, $\mathbf{l}_1={t_1}+u{y}$, and $\mathbf{l}_2={t_2}+u{y}$, where ${\alpha}=(\alpha_1, \ldots, \alpha_m),$ ${\beta}=(\beta_1, \ldots, \beta_m),$ ${y}=(y_1, \ldots, y_m)$ $\in \Bbb F_p^m$, ${t_1}\in \Delta$, and ${t_2}\in \Delta^c$ without expressing in the bold face. If $\mathbf{a}=\mathbf{0}$, then $w_L(c_{L}(\mathbf{a}))=w_L(c_{L^c}(\mathbf{a}))=0$. Next we assume that $\mathbf{a}\neq \mathbf{0}$. Then the Lee weight of the codeword $c_{L^c}(\mathbf{a})$ of $\mathcal{C}_{L^c}$ becomes that $$\begin{aligned}
&&w_L(c_{L^c}(\mathbf{a}))
=w_L(({\alpha}\cdot{t_1}+u({\alpha}\cdot{y}+{\beta}\cdot{t_1}))_{{t_1}\in \Delta, {y}\in \Bbb F_p^m})\nonumber\\
&=&w_H(({\alpha}\cdot{y}+{\alpha}\cdot{t_1})_{{t_1}\in \Delta, {y}\in \Bbb F_p^m})+w_H((({\alpha}+{\beta})\cdot{t_1}+\alpha\cdot{y})_{{t_1}\in \Delta, {y}\in \Bbb F_p^m})\nonumber\\
&=&2|L^c|-\frac1p\sum_{x\in\Bbb F_p}\sum_{t_1\in \Delta}\sum_{{y}\in \Bbb F_p^m}\zeta_p^{(\alpha\cdot{y}+\beta\cdot{t_1})x}-\frac1p\sum_{x\in\Bbb F_p}\sum_{t_1\in \Delta}\sum_{{y}\in \Bbb F_p^m}\zeta_p^{((\alpha+\beta)\cdot{t_1}+\alpha\cdot{y})x}\nonumber\\
&=&2|L^c|(1-\frac 1p)-\frac1p\sum_{x\in \Bbb F_p^*}\sum_{t_1\in \Delta}\zeta_p^{\beta\cdot{t_1}x}\sum_{{y}\in \Bbb F_p^m}\zeta_p^{\alpha\cdot{xy}}
-\frac1p \sum_{x\in \Bbb F_p^*}\sum_{t_1\in \Delta}\zeta_p^{(\alpha+\beta)\cdot{t_1}x}\sum_{{y}\in \Bbb F_p^m}\zeta_p^{\alpha\cdot{xy}}\nonumber\\
&=&2|L^c|(1-\frac 1p)-p^{m-1}\delta_{0,\alpha}(\sum_{x\in \Bbb F_p^*}\sum_{t_1\in \Delta}\zeta_p^{\beta\cdot{t_1}x}
+ \sum_{x\in \Bbb F_p^*}\sum_{t_1\in \Delta}\zeta_p^{(\alpha+\beta)\cdot{t_1}x}),\end{aligned}$$ where $\delta$ is the Kronecker delta function.
Similarly, the Lee weight of the codeword $c_{L}(\mathbf{a})$ of $\mathcal{C}_{L}$ becomes that $$\begin{aligned}
&&w_L(c_{L}(\mathbf{a}))
=w_H((\alpha\cdot{y}+\beta\cdot{t_2})_{{t_2}\in \Delta^c, {y}\in \Bbb F_p^m})+w_H(((\alpha+\beta)\cdot{t_2}+\alpha\cdot{y})_{{t_2}\in \Delta^c, {y}\in \Bbb F_p^m})\nonumber\\
&=&2|L|(1-\frac 1p)-\frac1p\sum_{x\in \Bbb F_p^*}\sum_{t_2\in \Delta^c}\zeta_p^{\beta\cdot{t_2}x}\sum_{{y}\in \Bbb F_p^m}\zeta_p^{\alpha\cdot{xy}}
-\frac1p \sum_{x\in \Bbb F_p^*}\sum_{t_2\in \Delta}\zeta_p^{(\alpha+\beta)\cdot{t_2}x}\sum_{{y}\in \Bbb F_p^m}\zeta_p^{\alpha\cdot{xy}}\nonumber\\
&=&2|L|(1-\frac 1p)-p^{m-1}\delta_{0,\alpha}(\sum_{x\in \Bbb F_p^*}\sum_{t_2\in \Delta^c}\zeta_p^{\beta\cdot{t_2}x}
+ \sum_{x\in \Bbb F_p^*}\sum_{t_2\in \Delta^c}\zeta_p^{(\alpha+\beta)\cdot{t_2}x}).\end{aligned}$$
Note that $|L^c|+|L|=p^{2m}$. Then $$w_L(c_{L^c}(\mathbf{a}))+w_L(c_{L}(\mathbf{a}))=2p^{2m-1}(p-1)-p^{2m-1}(p-1)\delta_{0,\alpha}(\delta_{0,\beta}+\delta_{0,\alpha+\beta}).$$
By using $(3.1)$, $(3.2)$ and $(3.3)$, we give the Lee weight distribution of the code $\mathcal{C}_{L}$ in the case that the down-set is generated by a single maximal element.
We start with down sets of the simplest forms to determine their Lee weight distributions.
Let $m\ge 2$ be a positive integer and $p$ be an odd prime number. Let $\Delta=\langle (r, 0, \ldots, 0) \rangle$ be a down-set of $\Bbb F_p$ for $r=1, \ldots, p-1$. Then the code $\mathcal{C}_{L}$ has length $p^{m}(p^m-r-1)$, size $p^{2m}$, and its Lee weight distribution is given by Table 1.
to 0.6 [Lee Weight]{}&[Frequency]{}\
$0$&$1$\
$2p^{2m-1}(p-1)$& $p^{m-1}-1$\
$2p^{m}(p^m-p^{m-1}-r)$& $p^{m-1}(p-1)$\
$2p^{m-1}(p-1)(p^m-r-1)$& $p^m(p^m-1)$\
It is easy to check that the length of the code $\mathcal{C}_{L}$ is $|L|=p^{m}(p^m-r-1)$. Let $\mathbf{a}=\alpha+u\beta$ for $\alpha=(\alpha_1,\ldots,\alpha_m)$ and $\beta=(\beta_1,\ldots,\beta_m)\in\mathbb{F}^m_p$. If $\alpha\neq 0$, then $c_{L}(\mathbf{a})=2|L|(1-\frac 1p)=2p^{m-1}(p-1)(p^m-r-1)$. If $\alpha=0$, then by Eq. (3.1), we have $$\begin{aligned}
c_{L^c}(\mathbf{a})&=&2p^{m-1}(p-1)(r+1)-2p^{m-1}\sum_{x\in \Bbb F_p^*}\sum_{t_1\in \Delta}\zeta_p^{\beta\cdot{t_1}x}\\
&=&2p^{m-1}(p-1)(r+1)-2p^{m-1}\sum_{x\in \Bbb F_p^*}(1+\zeta^{\beta_1x}+\cdots+\zeta^{\beta_1xr})\\
&=&\left\{
\begin{array}{ll}
0, &\mbox{ if $ \beta_1=0$},\\
2p^mr, &\mbox{ if $\beta_1\neq 0$}.\\
\end{array}\right.\end{aligned}$$ By Eq. (3.3), we have $$\begin{aligned}
c_{L}(\mathbf{a})&=& 2p^{2m-1}(p-1)-c_{L^c}(\mathbf{a})\\
&=&\left\{
\begin{array}{ll}
2p^{2m-1}(p-1), &\mbox{ if $ \beta_1=0$},\\
2p^m(p^m-p^{m-1}-r), &\mbox{ if $\beta_1\neq 0$}.\\
\end{array}\right.\end{aligned}$$ The frequency of each codeword of the codes should be computed by the vector $\mathbf{a}$.
Let $m\ge 3$ be a positive integer and $p$ be an odd prime number. Let $\Delta=\langle (p-1, r, 0,\ldots, 0) \rangle$ be a down-set of $\Bbb F_p$ for $r=1, \ldots, p-1$. Then the code $\mathcal{C}_{L}$ has length $p^{m}(p^m-p(r+1))$, size $p^{2m}$, and its Lee weight distribution is given by Table 2.
to 0.7 [Lee Weight]{}&[Frequency]{}\
$0$&$1$\
$2p^{2m-1}(p-1)$&$p^{m-2}-1$\
$2p^{m+1}(p^{m-1}-p^{m-2}-r)$& $p^{m-2}(p-1)$\
$2p^{m}(p-1)(p^{m-1}-r-1)$& $p^{2m}-p^{m-1}$\
It is easy to check that the length of the code $\mathcal{C}_{L}$ is $|L|=p^{m}(p^m-p(r+1))$. Let $\mathbf{a}=\alpha+u\beta$ for $\alpha=(\alpha_1,\ldots,\alpha_m)$ and $\beta=(\beta_1,\ldots,\beta_m)\in\mathbb{F}^m_p$. If $\alpha\neq 0$, then $c_{L}(\mathbf{a})=2|L|(1-\frac 1p)=2p^{m-1}(p-1)(p^m-p(r+1))$. If $\alpha=0$, then by Eq. (3.1), we have $$\begin{aligned}
c_{L^c}(\mathbf{a})&=&2p^{m}(p-1)(r+1)-2p^{m-1}\sum_{x\in \Bbb F_p^*}\sum_{t_1\in \Delta}\zeta_p^{\beta\cdot{t_1}x}\\
&=&2p^{m}(p-1)(r+1)-2p^{m-1}\sum_{x\in \Bbb F_p^*}\sum_{y\in \Bbb F_p}\zeta_p^{\beta_1xy}\sum_{z=0}^{r}\zeta_p^{\beta_2xz}\\
&=&\left\{
\begin{array}{llll}
0, &\mbox{ if $\beta_1=\beta_2= 0$},\\
2p^{m+1}r, &\mbox{ if $\beta_1=0$} \mbox{ and } \mbox{ $\beta_2\neq0$},\\
2p^{m}(p-1)(r+1), &\mbox{ if $\beta_1\neq 0$} .
\end{array}\right.\end{aligned}$$ By Eq. (3.3), we have $$\begin{aligned}
c_{L}(\mathbf{a})&=& 2p^{2m-1}(p-1)-c_{L_1}(\mathbf{a})\\
&=&\left\{
\begin{array}{llll}
2p^{2m-1}(p-1), &\mbox{ if $\beta_1=\beta_2= 0$},\\
2p^{m}(p^m-p^{m-1}-r), &\mbox{ if $\beta_1=0$} \mbox{ and } \mbox{ $\beta_2\neq0$},\\
2p^{m}(p-1)(p^{m-1}-r-1), &\mbox{ if $\beta_1\neq 0$} .
\end{array}\right.\end{aligned}$$ The frequency of each codeword of the codes should be computed by the vector $\mathbf{a}$.
Let $m\ge 3$ be a positive integer and $p$ be an odd prime number. Let $\Delta=\langle (p-2, r, 0,\ldots, 0) \rangle$ be a down-set of $\Bbb F_p$ for $r=1, \ldots, p-2$. Then the code $\mathcal{C}_{L}$ has length $p^{m}(p^m-(p-1)(r+1))$, size $p^{2m}$, and its Lee weight distribution is given by Table 3.
to 0.8 [Lee Weight]{}&[Frequency]{}\
$0$&$1$\
$2p^{2m-1}(p-1)$&$p^{m-2}-1$\
$2p^m(p-1)(p^{m-1}-r)$& $p^{m-2}(p-1)$\
$2p^{2m-1}(p-1)-2p^{m}(p-2)(r+1)$& $p^{m-2}(p-1)(p-r)$\
$2p^{2m-1}(p-1)-2p^{m}(pr+p-2r-1)$& $p^{m-2}r(p-1)$\
$2p^{m-1}(p-1)(p^m-(p-1)(r+1))$& $p^{m}(p^m-1)$\
It is easy to check that the length of the code $\mathcal{C}_{L}$ is $|L|=p^{m}(p^m-(p-1)(r+1))$. Let $\mathbf{a}=\alpha+u\beta$ for $\alpha=(\alpha_1,\ldots,\alpha_m)$ and $\beta=(\beta_1,\ldots,\beta_m)\in\mathbb{F}^m_p$. If $\alpha\neq 0$, then $c_{L}(\mathbf{a})=2|L|(1-\frac 1p)=2p^{m-1}(p-1)(p^m-(p-1)(r+1))$. Note that $\sum_{x\in \Bbb F_p^*}\sum_{t_1\in \Delta}\zeta_p^{\beta{t_1}x}$ has been determined in [@HKN Theorem 4.11]. Table 3 follows from Eq. (3.3).
Let $m\ge 3$ be a positive integer and $p$ be an odd prime number. Let $\Delta=\langle (p-3, r, 0,\ldots, 0) \rangle$ be a down-set of $\Bbb F_p$ for $r=1, \ldots, p-2$. Then the code $\mathcal{C}_{L}$ has length $p^{m}(p^m-(p-2)(r+1))$, size $p^{2m}$, and its Lee weight distribution is given by Table 4.
to 0.9 [Lee Weight]{}&[Frequency]{}\
$0$&$1$\
$2p^{2m-1}(p-1)$&$p^{m-2}-1$\
$2p^{2m-1}(p-1)-2p^m(p-2)r$& $p^{m-2}(p-1)$\
$2p^{2m-1}(p-1)-2p^{m}(p-3)(r+1)$& $p^{m-2}(p-1)(\frac{p+1}{2}-r+\lfloor \frac r 2\rfloor)$\
$2p^{2m-1}(p-1)-2p^{m}(pr+p-3r-2)$& $p^{m-2}(p-1)(p-1-2\lfloor \frac r 2\rfloor)$\
$2p^{2m-1}(p-1)-2p^{m}(pr+p-3r-1)$& $p^{m-2}(p-1)(\lfloor \frac r 2\rfloor+r-\frac{p-1}{2})$\
$2p^{m-1}(p-1)(p^m-(p-2)(r+1))$& $p^{m}(p^m-1)$\
It is easy to check that the length of the code $\mathcal{C}_{L}$ is $|L|=p^{m}(p^m-(p-2)(r+1))$. Let $\mathbf{a}=\alpha+u\beta$ for $\alpha=(\alpha_1,\ldots,\alpha_m)$ and $\beta=(\beta_1,\ldots,\beta_m)\in\mathbb{F}^m_p$. If $\alpha\neq 0$, then $c_{L}(\mathbf{a})=2|L|(1-\frac 1p)=2p^{m-1}(p-1)(p^m-(p-2)(r+1))$. Note that $\sum_{x\in \Bbb F_p^*}\sum_{t_1\in \Delta}\zeta_p^{\beta{t_1}x}$ has been determined in [@HKN Theorem 4.14]. Table 4 follows from Eq. (3.3).
Optimal codes and examples
==========================
Recall that the Gray map $\phi$ is an isometry from $(R^m, d_L)$ to $(\Bbb F_p^{2m}, d_H)$.
Let $m\ge 3$ be a positive integer and $p$ be an odd prime number. Let $L=\Delta^c+u\mathbb{F}^m_p,u^2=0$ for $\Delta=\langle(r,0,\ldots,0)\rangle$, $r=1,2,\ldots,p-1$. Then the Gray image $\phi(\mathcal{C}_{L})$ of $\mathcal{C}_{L}$ is a distance optimal linear code and the minimum distance of $\phi(\mathcal{C}_{L})^{\perp}$ is two. In particular, if $p<2r+2$ and $2(r+1)(p-1)<p^2$, i.e., $r=
\frac{p-1}{2}$, then $\phi(\mathcal{C}_{L})$ meets the Griesmer bound with equality.
By Theorem 3.1, the code $\phi(\mathcal{C}_{L})$ has the following parameters: $$n=2p^m(p^m-r-1),~ k=2m,~ d=2p^{m-1}(p-1)(p^m-r-1).$$ Firstly, we show that $\phi(\mathcal{C}_{L})$ is distance optimal. Assume to the contrary that there is an $[n,k,d+1]$ code. We see that $$\begin{aligned}
&&\sum_{i=0}^{2m-1}\bigg\lceil {\frac{2p^{m-1}(p-1)(p^m-r-1)+1}{p^i}} \bigg\rceil\nonumber\\
&=&\sum_{i=0}^{m-1}\bigg\lceil {\frac{2p^{m-1}(p-1)(p^m-r-1)+1}{p^i}} \bigg\rceil\\
&+&\sum_{i=m}^{2m-1}\bigg\lceil {\frac{2p^{2m-1}(p-1)}{p^i}
+\frac{1-2p^{m-1}(p-1)(r+1)}{p^i}} \bigg\rceil\nonumber\\
&=&2(p^m-1)(p^m-r-1)+m+\sum_{i=0}^{m-1}2p^i(p-1)+\sum_{i=1}^{m}\bigg\lceil{\frac{1}{p^{m+i}}-\frac{2(r+1)(p-1)}{p^i}} \bigg\rceil.\end{aligned}$$ Since $1\le r\le p-1$, we have $1\le \bigg\lceil {\frac{2(r+1)}{p}} \bigg\rceil\le 2$ and $0<\frac{2(r+1)(p-1)}{p^i} <2,~(i=2,3,\ldots,m),$ so that $$\bigg\lceil{\frac{1}{p^{m}}-\frac{2(r+1)(p-1)}{p}} \bigg\rceil
=-2(r+1)+\bigg\lceil{\frac{1}{p^{m}}+\frac{2(r+1)}{p}} \bigg\rceil
=-2(r+1)+2=-2r$$ and $$\bigg\lceil{\frac{1}{p^{m+i}}-\frac{2(r+1)(p-1)}{p^i}} \bigg\rceil\geq-1~(i=2,3,\ldots,m).$$ It follows that $$\sum_{i=0}^{m-1}\bigg\lceil{\frac{1}{p^{m+i}}-\frac{2(r+1)(p-1)}{p^i}} \bigg\rceil\geq -2r-(m-1),$$ and so $$\begin{gathered}
\sum_{i=0}^{2m-1}\bigg\lceil {\frac{2p^{m-1}(p-1)(p^m-r-1)+1}{p^i}} \bigg\rceil \ge 2(p^m-1)(p^m-r-1)+m+2(p^m-1)\\-2r-(m-1)=2p^m(p^m-r-1)+1,\end{gathered}$$ which contracts to the Griesmer bound.
Secondly, we show that the minimum distance $d^{\perp}$ of $\phi(\mathcal{C}_L)^{\perp}$ is two. Assume that $d^{\perp}\geq3$. By the sphere packing bound and $r\leq p-1$, we have $$p^{2m}\geq|\phi(\mathcal{C}_L)^{\perp}|(1+2p^m(p^m-r-1)(p-1))
>2p^m(p^m-(p-1)-1)(p-1),$$ equivalently, $(p^{m-1}-1)(2p-1)<1,$ which is a contradiction. We now claim that there is no codeword in $\phi(\mathcal{C}_L)^{\perp}$ whose Hamming weight is one. Assume to the contrary that there is a codeword in $\phi(\mathcal{C}_L)^{\perp}$ whose Hamming weight is one. Then the $i$th coordinate position of any codeword in $\phi(\mathcal{C}_L)$ for some $i$ is the zero, and so for fixed $\mathbf{c}\in\Delta^c$ and $d\in\mathbb{F}^m_p$, we have either $\mathbf{a}\cdot\mathbf{d}+\mathbf{b}\cdot\mathbf{c}=0$ or $\mathbf{a}\cdot\mathbf{d}+\mathbf{b}\cdot\mathbf{c}
+\mathbf{a}\cdot\mathbf{c}=0$ for all $\mathbf{a},\mathbf{b}\in\mathbb{F}^m_p$. In any case, we derive that $\mathbf{c}=\mathbf{0}$, which is a contradiction with $\mathbf{c}\in\Delta^c$.
It remains to prove the third part. We see that $$\begin{aligned}
&&\sum_{i=0}^{2m-1}\bigg\lceil {\frac{2p^{m-1}(p-1)(p^m-r-1)}{p^i}} \bigg\rceil\nonumber\\
&=&\sum_{i=0}^{m-1}\bigg\lceil {\frac{2p^{m-1}(p-1)(p^m-r-1)}{p^i}} \bigg\rceil+\sum_{i=m}^{2m-1}\bigg\lceil {\frac{2p^{2m-1}(p-1)}{p^i}-\frac{2p^{m-1}(p-1)(r+1)}{p^i}} \bigg\rceil\nonumber\\
&=&2(p^m-1)(p^m-r-1)+\sum_{i=0}^{m-1}2p^i(p-1)-\sum_{i=1}^m\bigg\lfloor{\frac{2(r+1)(p-1)}{p^i}} \bigg\rfloor.\end{aligned}$$ Since $p<2r+2$ and $2(r+1)(p-1)<p^2$, we have that $$\bigg\lfloor {\frac{2(r+1)(p-1)}{p}} \bigg\rfloor=
\bigg\lfloor {2(r+1)-\frac{2(r+1)}{p}} \bigg\rfloor
=2(r+1)-\bigg\lceil\frac{2(r+1)}{p}\bigg\rceil=2(r+1)-2$$ and $$\bigg\lfloor {\frac{2(r+1)(p-1)}{p^i}}\bigg\rfloor=0~ (i=2,,3,\ldots,m).$$ It follows that $$\sum_{i=1}^m\bigg\lfloor{\frac{2(r+1)(p-1)}{p^i}} \bigg\rfloor
=(-2(r+1)+2)+0=-2r,$$ and so $$\begin{gathered}
\sum_{i=0}^{2m-1}\bigg\lceil {\frac{2p^{m-1}(p-1)(p^m-r-1)}{p^i}}
\bigg\rceil=2(p^m-1)(p^m-r-1)+2(p^m-1)-2r\\
=2p^m(p^m-r-1),\end{gathered}$$ which shows that if $p<2r+2$ and $2(r+1)(p-1)<p^2$, then $\phi(\mathcal{C}_{L})$ meets the Griesmer bound with equality.
This completes the proof.
We have the following theorem in a similar computation of Theorem $4.1$ with $n=2p^{m}(p^m-p(r+1)),~k=2m,~d=2p^m(p-1)(p^m-r-1)$.
Let $m\ge 3$ be a positive integer and $p$ be an odd prime number. Let $L=\Delta^c+u\mathbb{F}^m_p,u^2=0$ for $\Delta=\langle(p-1,r,0,\ldots,0)\rangle$, $r=1,2,\ldots,p-1$. Then the Gray image $\phi(\mathcal{C}_{L})$ of $\mathcal{C}_{L}$ is a distance optimal linear code and the minimum distance of $\phi(\mathcal{C}_{L})^{\perp}$ is two. In particular, if $p<2r+2$ and $2(r+1)(p-1)<p^2$, i.e., $r=
\frac{p-1}{2}$, then $\phi(\mathcal{C}_{L})$ meets the Griesmer bound with equality.
We point out that in the database of Grassl [@G1], he provides a complete list of distance optimal $[n,k]$ codes with small lengths when $p\in\{3,5,7\}$, and we see that most of the distance optimal linear codes are unknown when $n\geq31$ and $k\geq 8$, where the upper bound of $n$ is restrictive and relies on $p$. We have constructed $2(p-1)$ classes of distance optimal linear codes in Theorems 4.1 and 4.2. It is believed that our distance optimal linear codes include new codes although we can not compare with the parameters in Grassl’s table [@G1] because our code length is large.
Table 5 is obtained by using Theorem 4.1 and 4.2, and $*$ stands for a linear code attaining the Griesmer bound with equality.
[$$\begin{array}{|c||c|c|c|c||c|c|c|c|} \hline
p & n & k & d & \mbox{Optimality} & n & k & d & \mbox{Optimality}\\
\hline
3 & 1350 & 6 &900& \mbox{Optimal*} & 12798 & 8 & 8532 & \mbox{Optimal*} \\
3 & 1296 & 6 & 864 & \mbox{Distance optimal} & 12636 & 8 & 8428 &\mbox{Distance optimal}\\
3 & 1134 & 6 & 756 &\mbox{Optimal*} & 12150 & 8 & 8100&\mbox{Optimal*} \\
3 & 972 & 6 & 648 &\mbox{Distance optimal} & 11664 & 8 & 7776 &\mbox{Distance optimal}\\
5 & 30750 & 6 & 24600 &\mbox{Distance-optimal}& 778750 & 8 &623000& \mbox{Distance optimal} \\
5 & 30500 & 6 & 24400 & \mbox{Optimal*} & 777500 & 8 &622000& \mbox{Optimal*} \\
5 & 30250 & 6 &24200&\mbox{Distance optimal} & 776250& 8 &621000&\mbox{Distance optimal} \\
5 & 30000 & 6 &24000 & \mbox{Distance optimal}&775000 & 8 &620000&\mbox{Distance optimal} \\
5 & 28750 & 6 & 23000& \mbox{Distance optimal} &768750 & 8 &615000& \mbox{Distance optimal}\\
5 &27500 & 6 &22000& \mbox{Optimal*} & 762500& 8 &610000& \mbox{Optimal*}\\
5 &26250 & 6 &21000&\mbox{Distance optimal} & 756250& 8 &605000& \mbox{Distance optimal} \\
5 & 25000 & 6 &20000& \mbox{Distance optimal} &750000 & 8 &600000& \mbox{Distance optimal} \\
7 & 233926 & 6 &200508&\mbox{Distance optimal} &11519998 & 8 &9874284& \mbox{Distance optimal}\\
7 & 233240 & 6 &199920&\mbox{Distance optimal} &11515196 & 8 &9870168&\mbox{Distance optimal} \\
7 & 232554 & 6 &199332& \mbox{Optimal*} & 11510394 & 8 &9866052& \mbox{Optimal*} \\
7 &231868 & 6 &198744& \mbox{Distance optimal} &11505592 & 8 &9861936& \mbox{Distance optimal}\\
7&231182 & 6 &198156& \mbox{Distance optimal} &11500790 & 8 &9857820&\mbox{Distance optimal} \\
7 &230496 & 6 &197568&\mbox{Distance optimal} & 11495988& 8 &9853704& \mbox{Distance optimal} \\
7 & 225694 & 6 &193452&\mbox{Distance optimal} &11462374 & 8 &9824892 & \mbox{Distance optimal} \\
7 & 220892 & 6 &189336&\mbox{Distance optimal} &11428760 & 8 &9796080& \mbox{Distance optimal} \\
7 &216090 & 6 &185220 & \mbox{Optimal*} &11595146 & 8 &9767268&\mbox{Optimal*} \\
7 &211288 & 6 &181104&\mbox{Distance optimal} & 11361532& 8 &9738456& \mbox{Distance optimal} \\
7 &206486 & 6 &176988 &\mbox{Distance optimal} &11327918 & 8 &9709644&\mbox{Distanceoptimal} \\
7 &210684 & 6 &172872&\mbox{Distance optimal} & 11294304& 8 &9680832&\mbox{Distance optimal} \\
\hline
\end{array}$$ ]{}
The following are two numeral examples.
[Let $m=3, p=3$, and $r=1$ in Theorem 3.3. Then $\phi(\mathcal{C}_{L})$ is a four-weight ternary code of parameters $[1242, 6, 810]$ with the weight enumerator $$1+6z^{810}+702z^{828}+18z^{864}+2z^{972}.$$ ]{}
[Let $m=3, p=5$, and $r=2$ in Theorem 3.4. Then $\phi(\mathcal{C}_{L})$ is a five-weight pentary code of parameters $[29000, 6, 23000]$ with the weight enumerator $$1+20z^{23000}+15500z^{23200}+40z^{23250}+60z^{23500}+4z^{25000}.$$ ]{}
Concluding remarks
==================
The main contributions of this paper are the following
- Construction of the linear codes $\mathcal{C}_L$ over $\Bbb F_p+u\Bbb F_p$, where $u^2=0$ and $p$ is an odd prime number, which is defined in Eq. (1.1) associated with down-sets;
- Determination of the Lee weight distributions of the codes $\mathcal{C}_L$ over $\Bbb F_p+u\Bbb F_p$ in the case that down-sets are all generated by a single maximal element (Theorems 3.1, 3.2, 3.3 and 3.4);
- Some infinite families of $p$-ary optimal linear codes from the Gray image of the codes $\mathcal{C}_L$ over $\Bbb F_p+u\Bbb F_p$ (Theorems 4.1 and 4.2).
Finally, we would inform the reader that in Section 3 we just determined the Lee weight distributions of the codes $\mathcal{C}_L$ when the down-sets are generated by a single maximal element.The reader is cordially invited to consider the general cases.
Acknowledgments {#acknowledgments .unnumbered}
===============
The first author was supported by Brain Korea 21 plus Mathematical Science Team for Global Women Leaders at Ewha Womans University. The second author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (NRF-2017R1D1A1B05030707)
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---
bibliography:
- 'spectralNeutron.bib'
title: 'Wavelength-resolved Neutron Tomography for Crystalline Materials'
---
|
---
abstract: 'We prove that the action of a reductive complex Lie group on a Kähler manifold can be linearized in the neighbourhood of a fixed point, provided that the restriction of the action to some compact real form of the group is Hamiltonian with respect to the Kähler form.'
address:
- 'University of Pennsylvania, Department of Mathematics, Philadelphia, Pennsylvania 19104'
- 'University of Pennsylvania, Department of Mathematics, Philadelphia, Pennsylvania 19104'
author:
- Eugene Lerman
- Reyer Sjamaar
date: October 1992
title: Reductive Group Actions on Kähler Manifolds
---
\[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Corollary]{} \[theorem\][Conjecture]{} \[theorem\][Claim]{} \[theorem\][Addendum]{}
\[theorem\][Definition]{}
\[theorem\][Remark]{} \[theorem\][Example]{}
[^1] [^2]
Much attention has been given to the problem of “dividing out” a space by a group of transformations in such diverse areas of mathematics as classical mechanics and algebraic geometry. In classical mechanics one typically deals with a phase space $M$, a surface $L\subset M$ defined as a level set of a number of integrals of motion, plus a group $G$ of canonical transformations leaving the motion on $L$ invariant. The “reduced” phase space is then the space $L/G$. Modern treatments of this construction have been given by Meyer [@me:sy] and Marsden and Weinstein [@ma:re].
Algebraic geometers like to study complex-projective varieties $M$ acted upon by groups of projective transformations that are usually not compact. Often, though, their groups are reductive, i.e., they are complexifications $G{^{\bold C}}$ of compact real Lie groups $G$. A good notion of a quotient in this category has been defined by Mumford [@mu:ge]. It consists in choosing a subset $M{^{\mathrm s\mathrm s}}$ of “semistable” points of $M$, where the action is “good”, and then forming a quotient $M{^{\mathrm s\mathrm s}}{/\kern-.7ex/}G{^{\bold C}}$. This quotient is smaller than the naive, set-theoretical quotient $M{^{\mathrm s\mathrm s}}/G{^{\bold C}}$, which is not a Hausdorff space owing to the fact that the orbits of the non-compact group $G{^{\bold C}}$ are not closed. The space $M{^{\mathrm s\mathrm s}}{/\kern-.7ex/}G{^{\bold C}}$ is basically the set of [*closed*]{} orbits in $M{^{\mathrm s\mathrm s}}$.
Complex-projective manifolds are not classical phase spaces, but they have very special symplectic forms, obtained by restricting the Fubini-Study form on the ambient projective space. This raises the question whether there is any relation between the symplectic and the algebro-geometric quotients of such spaces. Guillemin, Sternberg, Mumford and others have made the important observation that for projective manifolds the notions of a quotient in symplectic and algebraic geometry are essentially the same. Kempf and Ness [@ke:le] have proved an analogous result for affine varieties. See Bott [@bo:mo] for an account of this story. On the quantum level this correspondence leads to integral formulae for multiplicities of representations; see Guillemin and Sternberg [@gu:ge].
We have sought to generalize these multiplicity formulae to the case where the reduced phase space $L/G$ is not a manifold, but a singular space. In the process we had to investigate more closely the correspondence between quotients in mechanics and in algebraic geometry, and the properties of actions of reductive complex Lie groups on Kähler manifolds. In particular, we had to address the issue of the existence of slices for such actions. The problem of linearizing the action near a fixed point and the construction of slices has been solved by Luna [@lu:sl] for affine varieties and by Snow [@sn:re] for Stein spaces. The difficulty of the problem lies in the fact that an action of a reductive group $G{^{\bold C}}$ is never proper, unless it is locally free. One therefore faces the challenge of controlling the behaviour of the action “at infinity in the group”. It seems unlikely to us that this can be done in general without imposing some restrictions on the space $M$.
In this note we present an alternative proof of a theorem of Koras [@ko:li] to the effect that one can linearize a $G{^{\bold C}}$-action at a fixed point on a Kähler manifold $M$. The advantage of our proof is that it can be generalized to show the existence of slices at certain points of $M$, namely the totally real points. This will be the subject of a forthcoming paper [@le:ho]. As a foretaste we will at the end of this paper spell out a little example given by Luna, showing that slices do not exist at arbitrary points of $M$. We are also preparing a paper on multiplicity formulae for singular reduced phase spaces. The result to be discussed here is the following:
\[theorem:linear\] Let $M$ be a Kähler manifold and let $G{^{\bold C}}$ act holomorphically on $M$. Assume the action of the compact real form $G$ is Hamiltonian. Let $m$ be any fixed point of the $G{^{\bold C}}$-action. Then the action of $G{^{\bold C}}$ can be linearized in a neighbourhood of $m$, i.e., there exist a $G{^{\bold C}}$-invariant open neighbourhood $U$ of $m$ in $M$, a $G{^{\bold C}}$-invariant open neighbourhood $U'$ of the origin $0$ in the tangent space $T_mM$ and a biholomorphic $G{^{\bold C}}$-equivariant map $U\to U'$.
We find ourselves unable to understand part of Koras’ proof of this theorem. In particular, we fail to see a justification for his application of the curve selection lemma. Our proof will follow the same strategy as Koras’, using a momentum map for the action of the compact group $G$ and an analytic continuation argument based on Weyl’s unitary trick. But at the crucial point, instead of invoking the curve selection lemma, we will rely on an “interpolation” argument, deforming the Kähler metric in the neighbourhood of a fixed point.
Let us now explain the statement of the theorem and introduce some notation. We denote the complex structure on $M$ by $J$, the Kähler metric by $ds^2$ and the Kähler form ${\operatorname{Im}}ds^2$ by $\omega$. Then ${\operatorname{Re}}ds^2=\omega(\cdot,J\cdot)$ is the corresponding Riemannian metric. We may assume without loss of generality that $ds^2$ is invariant under the compact group $G$. So the transformations on $M$ defined by $G$ are holomorphic and they are isometries with respect to the Kähler metric. Saying that the action of $G$ is Hamiltonian means that for all $\xi$ in the Lie algebra $\frak g$ of $G$ the vector field $\xi_M$ on $M$ induced by $\xi$ is Hamiltonian. In this case we have a [*momentum map*]{} $\Phi$ from $M$ to the dual $\frak
g^*$ of the Lie algebra of $G$ with the property that $$d\Phi^\xi=\iota_{\xi_M}\omega$$ for all $\xi$. Here $\Phi^\xi$ is the $\xi$-th component of $\Phi$, defined by $\Phi^\xi(m)=\bigl(\Phi(m)\bigr)(\xi)$. Because its components are Hamiltonian functions, a momentum map is uniquely determined up to additive constants. After averaging with respect to the given action on $M$ and the coadjoint action on $\frak g^*$ we may assume that the map $\Phi$ is $G$-equivariant. It is easy to give sufficient conditions for the existence of a momentum map, e.g., the first Betti number of $M$ is zero, or the Kähler form $\omega$ is exact. (See e.g. [@gu:sy; @we:le].) More surprisingly, by a theorem of Frankel [@fr:fi] a momentum map always exists if the action has at least one fixed point and $M$ is [*compact*]{}.
If $M$ is $\bold C^n$ with the standard Hermitian structure and the standard symplectic form $\Omega$, then a momentum map is given by the formula $$\label{equation:quadratic}
\Phi_{\bold C^n}^\xi(v)=1/2\,\Omega\bigl(\xi_{\bold C^n}(v),v\bigr),$$ where $\xi _{\bold C^n}$ denotes the image of $\xi \in\frak g$ in the Lie algebra $\frak s\frak p({\bold C^n},\Omega)$, and $v\in\bold
C^n$. In this setting the statement of the theorem is of course a tautology, but we will need (\[equation:quadratic\]) further on.
The decomposition of the complexified Lie algebra $\frak g{^{\bold C}}=\frak
g\otimes\bold C$ into a direct sum $\frak g{^{\bold C}}=\frak g\oplus{\sqrt{-1}}\,\frak
g$ gives rise to the Cartan decomposition $G{^{\bold C}}=G\exp{\sqrt{-1}}\,\frak
g$. (“Polar coordinates” on the group.) The restriction of the exponential map $\exp\colon{\sqrt{-1}}\,\frak g\to G{^{\bold C}}$ is a diffeomorphism onto its image. This means that every element $g$ of $G{^{\bold C}}$ can be uniquely decomposed into a product $g=k\exp{\sqrt{-1}}\,\xi$, with $k\in G$ and $\xi\in\frak g$.
Because $G{^{\bold C}}$ acts holomorphically on $M$, for any $\xi$ in $\frak g$ the vector field $({\sqrt{-1}}\,\xi)_M$ induced by ${\sqrt{-1}}\,\xi$ is equal to $J\xi_M$. It follows easily from the definition of a momentum map that $J\xi_M$ is equal to the gradient vector field (with respect to the Riemannian metric ${\operatorname{Re}}ds^2$) of the $\xi$-th component of the momentum map, $$\label{equation:grad}
({\sqrt{-1}}\,\xi)_M=J\xi_M={\operatorname{grad}}\Phi^\xi.$$ This elementary observation will enable us to gain control over the behaviour of the action “at infinity in the group”. For one thing, it implies that the trajectory $\gamma(t)$ of ${\operatorname{grad}}\Phi^\xi$ through a point $x$ in $M$ is given by $\gamma(t)=\exp({\sqrt{-1}}\,t\xi) x$, which does not depend on the choice of the Kähler metric and the momentum map.
The proof is in three steps. In the first step we reduce the statement of the theorem to a statement about the trajectories of the gradient vector fields $({\sqrt{-1}}\,\xi)_M={\operatorname{grad}}\Phi^\xi$. In the second step we consider the case where the Kähler metric $ds^2$ is flat in some neighbourhood of $m$. The third step consists in showing that an arbitrary metric $ds^2$ can always be deformed to a metric which is flat close to $m$ and which is still compatible with all the relevant data.
[*Step*]{} 1. The tangent space $T_mM$ at $m$ is a Hermitian vector space, which we shall identify with standard $\bold C^n$. Then the value of the Kähler form $\omega$ at $M$ is the standard symplectic form $\Omega$ on $\bold C^n$. The tangent action of $G{^{\bold C}}$ defines a linear representation $G{^{\bold C}}\to{\operatorname{GL}}(n,\bold C)$, the restriction of which to $G$ is a unitary representation $G\to{\operatorname{U}}(n)$. Let $\phi\colon
O\to\bold C^n$ be a local holomorphic coordinate on $M$ with $\phi(m)=0$ and $d\phi(m)={\operatorname{id}}_{\bold C^n}$, where $O$ is a small $G$-invariant open neighbourhood of $m$ such that $O'=\phi(O)$ is a ball about the origin in $\bold C^n$. Then the pullback of the form $\Omega$ is equal to $\omega$ at the point $m$. After averaging over $G$ and shrinking $O$ if necessary we may assume that $\phi$ is $G$-equivariant. Let $\psi\colon O'\to O$ be the inverse of $\phi$.
We claim that if $O$ is sufficiently small $\phi$ and $\psi$ can be uniquely extended to holomorphic $G{^{\bold C}}$-equivariant maps $\phi{^{\bold C}}\colon U\to U'$ and $\psi{^{\bold C}}\colon U'\to U$, where $U=G{^{\bold C}}O$ and $U'=G{^{\bold C}}O'$. If this can be done, it is clear that $\phi{^{\bold C}}\circ\psi{^{\bold C}}={\operatorname{id}}_{U'}$ and $\psi{^{\bold C}}\circ\phi{^{\bold C}}={\operatorname{id}}_{U}$, so then the theorem will be proved.
Following Heinzner [@he:ge] we shall call an open $G$-invariant subset $A$ of a $G{^{\bold C}}$-space $X$ [*orbitally convex*]{} with respect to the $G{^{\bold C}}$-action if for all $x$ in $U$ and all $\xi$ in $\frak g$ the intersection of the curve $\{\,\exp({\sqrt{-1}}\,t\xi)x:t\in\bold R\,\}$ with $A$ is connected.
\[proposition:orbit\] Let $X$ and $Y$ be complex manifolds acted upon by $G{^{\bold C}}$. If $A$ is an orbitally convex open subset of $X$ and $f\colon A\to Y$ is a $G$-equivariant holomorphic map, then $f$ can be uniquely extended to a $G{^{\bold C}}$-equivariant holomorphic map $f{^{\bold C}}\colon G{^{\bold C}}A\to Y$.
Consequently, if the image $f(A)$ is open and orbitally convex in $Y$ and $f\colon A\to f(A)$ is biholomorphic, then the extension $f{^{\bold C}}\colon G{^{\bold C}}A\to Y$ is biholomorphic onto the open subset $G{^{\bold C}}\cdot f(A)$.
The only way to extend $f$ equivariantly is by putting $f{^{\bold C}}\bigl(g\exp({\sqrt{-1}}\,\xi)x\bigr)=g\exp({\sqrt{-1}}\,\xi)f(x)$ for all $x$ in $A$, $g$ in $G$ and $\xi$ in $\frak g$. We have to check this is well-defined.
Let $x\in A$ and $\xi\in\frak g$ be such that $\exp({\sqrt{-1}}\,\xi)x\in A$. Then by assumption $\exp({\sqrt{-1}}\,t\xi)x\in A$ for all $t$ between 0 and 1. So $f\bigl(\exp({\sqrt{-1}}\,t\xi)x\bigr)$ is well-defined for $0\leq t\leq
1$. Define the curves $\alpha(t)$ and $\beta(t)$ in $Y$ by $\alpha(t)=f\bigl(\exp({\sqrt{-1}}\,t\xi)x\bigr)$ and $\beta(t)=
\exp({\sqrt{-1}}\,t\xi)f(x)$ for $0\leq t\leq 1$. Then $\alpha(t)$ and $\beta(t)$ are integral curves of the vector fields $f_*({\sqrt{-1}}\,\xi)_X$ and $({\sqrt{-1}}\,\xi)_Y$ respectively, both with the same initial value $f(x)$. Now since $f$ is $G$-equivariant we have $f_*\xi_X=\xi_Y$, and, because $f$ is also holomorphic, $f_*({\sqrt{-1}}\,\xi)_X= f_*(J\xi_X)=
Jf_*\xi_X= J\xi_Y= ({\sqrt{-1}}\,\xi)_Y$. Hence $\alpha(t)=\beta(t)$, in other words $f\bigl(\exp({\sqrt{-1}}\,t\xi)x\bigr)=\exp({\sqrt{-1}}\,t\xi)f(x)$ for $0\leq
t\leq 1$.
It follows that for all $x$ in $A$ and all $\xi$ in $\frak g$ such that $\exp({\sqrt{-1}}\,\xi)x$ is in $A$ we have $f\bigl(\exp({\sqrt{-1}}\,\xi)x\bigr)=\exp({\sqrt{-1}}\,\xi)f(x)$. It is easy to deduce from this that $f{^{\bold C}}$ is well-defined.
Finally observe that if the image $f(A)$ is open and orbitally convex in $Y$ and $f\colon A\to f(A)$ is biholomorphic, then the inverse $f^{-1}$ also has a holomorphic extension $(f^{-1}){^{\bold C}}\colon G{^{\bold C}}f(A)\to G{^{\bold C}}A$, and by uniqueness this must be the inverse of $f{^{\bold C}}$.
This proposition reduces the proof of Theorem \[theorem:linear\] to showing that $O\subset M$ and $O'\subset\bold C^n$ are orbitally convex.
[*Step*]{} 2. We will now show that $O$ is orbitally convex if the Kähler metric is flat on an open subset containing $O$. Since $\bold C^n$ is flat, this covers the case of $O'\subset\bold C^n$, so in particular we will have shown $O'$ is orbitally convex.
The argument is a variation on a convexity argument of Kempf and Ness [@ke:le]. The assumption that the Kähler metric is flat on an open subset containing $O$ implies that the holomorphic coordinate map $\phi\colon O\to\bold C^n$ is a Kähler isometry onto the open ball $O'$ about the origin. After choosing appropriate constants we may assume that $\Phi(m)=0$. Then $\psi^*\Phi\colon O'\to\frak g$ is equal to the quadratic momentum map given by (\[equation:quadratic\]). (Recall $\psi=\phi^{-1}$.) In order not to overburden the notation we will in the proof of the following lemma identify $O$ with $O'$ and $\Phi$ with the momentum map (\[equation:quadratic\]). By $R(x)$ we denote the Riemannian distance of $x\in O$ to $m$.
\[lemma:angle\] Assume the Kähler metric is flat in the neighbourhood of $O$. Then for all $\xi\in\frak g$ and $v\in O$ the momentum function $\Phi^\xi$ measures the inner product of the outward pointing normal ${\operatorname{grad}}R^2$ to the metric sphere of radius $R$ about $m$ and the vector field $J\xi_O={\operatorname{grad}}\Phi^\xi$, as follows: $$\label{equation:angle}
\bigl\langle{\operatorname{grad}}R^2(v),{\operatorname{grad}}\Phi^\xi(v)\bigr\rangle=4\Phi^\xi(v).$$
It follows that $O$ is orbitally convex with respect to the $G{^{\bold C}}$-action.
Let $\delta(t)=\exp({\sqrt{-1}}\,t\xi)v$ denote the gradient trajectory of $\Phi^\xi$ through $v$. On one hand, $$\frac{d}{dt}R^2\bigl(\delta(t)\bigr) = \bigl\langle{\operatorname{grad}}R^2\bigl(\delta(t)\bigr),\delta'(t)\bigr\rangle = \bigl\langle{\operatorname{grad}}R^2\bigl(\delta(t)\bigr),{\operatorname{grad}}\Phi^\xi\bigl(\delta(t)\bigr)\bigr\rangle.$$ On the other hand, $$\begin{aligned}
\frac{d}{dt}R^2\bigl(\delta(t)\bigr)&=
\frac{d}{dt}\bigl\|\delta(t)\bigr\|^2
= \frac{d}{dt}\bigl\langle\delta(t),\delta(t)\bigr\rangle =
2\bigl\langle\delta'(t),\delta(t)\bigr\rangle = \\
&=
2\bigl\langle({\sqrt{-1}}\,\xi)_O\bigl(\delta(t)\bigr),\delta(t)\bigr\rangle =
2\Omega\bigl(\xi_O\bigl(\delta(t)\bigr),\delta(t)\bigr) =
4\Phi^\xi\bigl(\delta(t)\bigr),\end{aligned}$$ where we have used (\[equation:quadratic\]) and (\[equation:grad\]). Taking $t=0$ yields (\[equation:angle\]).
Now (\[equation:angle\]) implies that the curve $\delta(t)$ can only enter $O$ at a point $p$ in the boundary $\partial O$ for which $\Phi^\xi(p)\leq0$ and leave it at a point $q\in\partial O$ where $\Phi^\xi(q)\geq0$. But $\delta(t)$ is also a gradient curve of the function $\Phi^\xi$ and so $\Phi^\xi$ is increasing along $\delta(t)$ for all $t\in\bold R$. If $\delta(t)$ is not constant, $\Phi^\xi\bigl(\delta(t)\bigr)$ is strictly increasing. Therefore, if $\delta(t)$ leaves the ball $O$ at some point, it can never sneak back in. Consequently $\{\,\delta(t):t\in\bold R\,\}\cap O$ is connected. If $\delta(t)$ is constant it is trivially true that $\{\,\delta(t):t\in\bold R\,\}\cap O$ is connected.
[*Step*]{} 3. We claim there exists a Kähler metric $d\tilde s^2$ on $M$ with the following properties: It is compatible with the holomorphic structure $J$ and $G$-equivariant, the $G$-action is Hamiltonian with respect to $d\tilde s^2$, and $d\tilde s^2$ is flat in a neighbourhood of $m$. Together with the result of step 2 this will conclude the proof of Theorem \[theorem:linear\].
Let $u$ be a Kähler potential for $ds^2$ in a neighbourhood of $m$, i.e., $$\omega={\sqrt{-1}}\,\partial\bar\partial u.$$ Consider the Taylor expansion $u(x)\sim\sum_{k,l}c_{kl}x^k\bar x^l$ of $u$ about $m$, where $x$ is the system of holomorphic coordinates defined by the map $\phi\colon O\to\bold C^n$, and where $k$ and $l$ denote multi-indices. The value of $\omega$ at $m$ is the standard symplectic form on $\bold C^n$, so $\partial^2u(m)/\partial
x_\alpha\partial\bar x_\beta=\delta_{\alpha\beta}$. Also, deleting the holomorphic terms $\sum_{k}c_{k0}x^k$ and the antiholomorphic terms $\sum_{l}c_{0l}\bar x^l$ up to any finite order does not affect $\partial\bar\partial u$, so we may assume $$u(x)=\|x\|^2+\cal R(x),$$ with remainder term $\cal R(x)$ of order $O\bigl(\|x\|^3\bigr)$. Let $\chi\colon\bold R\to[0,1]$ be a smooth function with $\chi(t)=0$ for $t\leq1$ and $\chi(t)=1$ for $t\geq2$. Pick $\lambda>0$ and put $$\tilde u(x)=\|x\|^2+ \chi\bigl(\|x\|^2/\lambda^2\bigr)\cal R(x).$$ Let $O(r)$ be the ball $\{\,x\in O:\|x\|^2<r\,\}$ and define a smooth two-form $\tilde\omega$ on $M$ by $$\tilde\omega=\left\{
\begin{array}{ll}
{\sqrt{-1}}\,\partial\bar\partial\tilde u&\mbox{ on }
O(3\lambda^2) \\
\omega&\mbox{ on }M-O(2\lambda^2).
\end{array}\right.$$ Then on $O(\lambda^2)$ the form $\tilde\omega$ is equal to the standard symplectic form $\Omega$ and so is $G$-invariant there. On $O(3\lambda^2)$ we have $\tilde\omega-\omega=
\sum_{\alpha,\beta=1}^nf_{\alpha\beta}\,dx_\alpha\wedge d\bar x_\beta$ with $$f_{\alpha\beta}(x) = \frac{\partial^2}{\partial
x_\alpha\partial\bar
x_\beta}\bigl(\chi\bigl(\|x\|^2/\lambda^2\bigr)-1\bigr)\cal R(x).$$ By carrying out the differentiation and using $\chi(t)=1$ for $t\geq2$ and $\cal R(x)=O\bigl(\|x\|^3\bigr)$ one can check in a straightforward manner that for every ${\varepsilon}>0$ there exists $\Lambda>0$ such that for all $\lambda<\Lambda$ and for all $x\in
O(3\lambda^2)$ one has $|f_{\alpha\beta}(x)|<{\varepsilon}$. In other words, $\tilde\omega$ can be made arbitrarily close to $\omega$ uniformly on $M$. It follows that for ${\varepsilon}$ and $\lambda$ small enough the symmetric bilinear form $\tilde\omega(\cdot,J\cdot)$ is positive-definite, and therefore $\tilde\omega$ is the imaginary part of a Kähler metric $d\tilde s^2$. By construction $d\tilde s^2$ is equal to $ds^2$ on $M-O(2\lambda^2)$ and flat on $O(\lambda^2)$. After averaging over $G$ we may assume $d\tilde s^2$ is $G$-invariant everywhere. (This does not destroy the flatness on $O(\lambda^2)$, because $d\tilde s^2$ was already invariant there.)
It remains to be proved that the $G$-action is Hamiltonian with respect to $\tilde\omega$. By the Poincaré Lemma we can find an $\tilde\omega$-momentum map $\tilde\Phi\colon O(3\lambda^2)\to\frak
g^*$. On the set $O(3\lambda^2)-O(2\lambda^2)$ we have $\tilde\omega=\omega$, so there $\tilde\Phi$ differs by a locally constant function $c$ from the $\omega$-momentum map $\Phi$. Since $O(3\lambda^2)-O(2\lambda^2)$ is connected, $c$ is a constant. Shifting $\Phi$ by this constant we can paste $\Phi$ and $\tilde\Phi$ together to obtain a global $\tilde\omega$-momentum map for the $G$-action.
In [@le:ho] we will show how to construct slices for a $G{^{\bold C}}$-action on a Kähler manifold at points $m$ for which the real orbit $Gm$ is [*totally real*]{}. (This includes the case of a fixed point). Let us recall the definition of a slice.
A [*slice*]{} at $m$ for the $G{^{\bold C}}$-action is a locally closed analytic subspace $S$ of $M$ with the following properties:
1. $m\in S$;
2. the saturation $G{^{\bold C}}S$ of $S$ is open in $M$;
3. $S$ is invariant under the action of the stabilizer $(G{^{\bold C}})_m$;
4. \[bundle\] the natural $G{^{\bold C}}$-equivariant map from the associated bundle $G{^{\bold C}}\times_{(G{^{\bold C}})_m}S$ into $M$, which sends an equivalence class $[g,y]$ to the point $gy$, is an analytic isomorphism onto $G{^{\bold C}}S$.
It follows from (\[bundle\]) that if a slice at $m$ exists, then for all $y$ in a $G{^{\bold C}}$-invariant neighbourhood of $m$ the stabilizer $(G{^{\bold C}})_y$ is conjugate to a subgroup of $(G{^{\bold C}})_m$. Furthermore, a slice has to be nonsingular at $m$ and transverse to the orbit $G{^{\bold C}}m$. Theorem \[theorem:linear\] clearly shows that slices exist at fixed points, where $(G{^{\bold C}})_m$ is equal to $G{^{\bold C}}$.
The basic idea of the proof of the slice theorem in [@le:ho] is the same as that of Theorem \[theorem:linear\]: We show that a totally real $G$-orbit has sufficiently small orbitally convex neighbourhoods and then use analytic continuation. The main technical difficulty is to make the interpolation argument of step 3 work globally along a totally real orbit. This involves a closer scrutiny of the second-order operator $\partial\bar\partial$.
The following example shows that slices need not exist everywhere, even at points $m$ where the action is free, that is, $(G{^{\bold C}})_m=1$.
Let $V$ be the vector space of homogeneous cubic polynomials in two indeterminates $x$ and $y$. Let $G$ be the group ${\operatorname{SU}}(2)$; then $G{^{\bold C}}={\operatorname{SL}}(2,\bold C)$. Both groups act by linear substitutions in the two variables. If we declare the monomials $x^3$, $x^2y$, $xy^2$ and $y^3$ to be an orthonormal basis, we obtain a $G$-invariant Hermitian inner product on $V$. Every polynomial in $V$ can be written as a product of three linear factors, which are unique up to scalar multiples. So, obviously, every element of $G{^{\bold C}}$ fixing the polynomial has to permute these factors (up to scalars). We can distinguish between four types of polynomials according to the number of distinct factors in the factorization, and it is easy to compute their stabilizers. (See table \[table:stabilizer\].) The points of type I form clearly a dense subset of $V$.
[*number of distinct factors*]{} $m$ $(G{^{\bold C}})_m$
----- ---------------------------------- --------------- ------------------------------------
I 3 $(x+y)(x-y)y$ $\bold Z/3$
II 2 $x^2y$ 1
III 1 $y^3$ $\bold Z/3$-extension of $\bold C$
IV 0 0 $G{^{\bold C}}$
: []{data-label="table:stabilizer"}
Consider the point $m=x^2y$. The action is free at $m$, so the orbit $G{^{\bold C}}m$ is a copy of the three-dimensional ${\operatorname{SL}}(2,\bold C)$ embedded in the four-dimensional $V$. A straightforward computation shows the orthogonal complement to $G{^{\bold C}}m$ at $m$ is the one-dimensional subspace generated by $y^3$. We can define a holomorphic $G{^{\bold C}}$-equivariant map $\phi$ from the associated bundle $E=G{^{\bold C}}\times_{(G{^{\bold C}})_m}\bold C={\operatorname{SL}}(2,\bold C)\times\bold C$ into $V$ by putting $\phi\bigl([g,\eta]\bigr)=g\cdot(x^2y+\eta y^3)$. The differential of $\phi$ is bijective at all points of the zero section of $E$ and the restriction of $\phi$ to the zero section is an embedding. Therefore $\phi$ is an open immersion in a neighbourhood of the zero section. Does there exist a $G{^{\bold C}}$-invariant neighbourhood $U$ of the zero section of $E$ such that the restriction of $\phi$ to $U$ is biholomorphic onto its image? We could draw this conclusion if we knew that $\phi$ was injective on a $G{^{\bold C}}$-invariant neighbourhood of the zero section. But there are points of type I arbitrarily close to $m$, whose stabilizer consists of three elements and so cannot be conjugate to a subgroup of $(G{^{\bold C}})_m=1$. Hence there cannot possibly exist a slice at $m$, and therefore the map $\phi$ cannot be injective in any $G{^{\bold C}}$-invariant neighbourhood of the zero section of $E$. The reason is that $\phi$ plays havoc with the fibres of $E$. Let us write $\eta=-{\varepsilon}^2$; then $\phi\bigl([g,\eta]\bigr)= gm_{\varepsilon}$ with $m_{\varepsilon}=(x+{\varepsilon}y)(x-{\varepsilon}y)y$. Let $U$ be any ${\operatorname{SL}}(2,\bold
C)$-invariant neighbourhood of the zero section in $E$. Then $U$ contains an invariant open neighbourhood of the form $E_\Delta={\operatorname{SL}}(2,\bold C)\times\Delta$, where $\Delta$ is a small disc about the origin in $\bold C$. Choose ${\varepsilon}$ such that ${\varepsilon}^2\in\Delta$. The stabilizer $(G{^{\bold C}})_{m_{\varepsilon}}$ is generated by the following matrix of order three: $$A_{\varepsilon}=
\begin{pmatrix}
-1/2&3{\varepsilon}/2\\-1/2{\varepsilon}&-1/2
\end{pmatrix},$$ and we have $$A_{\varepsilon}m= -\frac{1}{8{\varepsilon}}x^3+\frac{5}{8}x^2y-
\frac{3{\varepsilon}}{8}xy^2-\frac{9{\varepsilon}^2}{8}y^3.$$ So $A_{\varepsilon}$ moves $m$ way out along its orbit while leaving $m_{\varepsilon}$ fixed. We conclude that the images of the fibres $\{I\}\times\Delta\subset E_\Delta$ and $\{A_{\varepsilon}\}\times\Delta\subset
E_\Delta$ under the map $\phi$ intersect at the point $m_{\varepsilon}$.
We are grateful to Eugenio Calabi for showing us how to glue Kähler potentials.
[10]{}
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[^1]: This work was partially supported by NSF grant DMS92-03398.
[^2]: The final version will appear elsewhere.
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---
abstract: 'Frobenius manifold structures on the spaces of abelian integrals were constructed by I. Krichever. We use ${\EuScript D}$-modules, deformation theory, and homological algebra to give a coordinate-free description of these structures. It turns out that the tangent sheaf multiplication has a cohomological origin, while the Levi–Civita connection is related to 1-dimensional isomonodromic deformations.'
address: |
Mathematics & Statistics\
Lederlee Graduate Research Tower\
University of Massachusetts\
710 North Pleasant Street\
Amherst, MA 01003–9305
author:
- 'Roman M. Fedorov'
title: |
Frobenius manifold structures\
on the spaces of abelian integrals
---
Introduction
============
Frobenius manifolds are manifolds with a flat metric and a multiplication in the tangent sheaf, subject to some constraints. Frobenius manifolds come from physics, where they govern deformations of certain topological quantum field theories. In mathematics they arise in two different situations, corresponding to A-models and B-models in physics. In an A-model one counts rational curves on a variety; this is also known as Gromov–Witten invariants. The generating function for these invariants is the potential for the corresponding Frobenius manifold. A-models are beyond the scope of our paper.
In a B-model one studies complex analytic deformations of a certain structure. The best known examples are extended moduli spaces of Calabi–Yau manifolds [@KonBar] and the unfoldings of hypersurface singularities [@Saito; @Sabbah]. We would like to mention that Frobenius structures are important for mirror symmetry: if two varieties are mirror dual to each other, then the A-model Frobenius manifold, corresponding to the first variety, is isomorphic to the B-model Frobenius manifold, corresponding to the second.
Moduli spaces of abelian integrals
----------------------------------
Examples of Frobenius manifolds are furnished by Hurwitz spaces. Roughly speaking, Hurwitz spaces parameterize pairs $(X,f)$, where $X$ is a smooth complete algebraic curve, $f:X\to{\mathbb P}^1$. Dubrovin constructs twisted Frobenius structures on Hurwitz spaces [@Dubrovin].
Our main object is the following deformation of a Hurwitz space: a space of pairs $(X,f)$, where $f$ is a *multi-valued* meromorphic function such that $df$ is a single-valued 1-form with prescribed periods and residues. If the periods and residues are equal to zero, then this space reduces to a Hurwitz space. Our spaces will be called *spaces of abelian integrals*.
Krichever constructs in [@Krichever1; @Krichever2] Frobenius structures on the universal covers of the spaces of abelian integrals. Our main goal is to give a coordinate-free geometric description of these Frobenius structures. Our approach is based on a ${\EuScript D}$-module push-forward (also known as twisted de Rham complex, see [@Sabbah §I.3.3]). It turns out that these structures of Frobenius manifolds have a nice interpretation: the tangent sheaf multiplication has a cohomological origin, similar to that of [@KonBar]. The metric and the Levi–Civita connection are closely related to 1-dimensional isomonodromic deformations. (This is not directly related to isomonodromic deformations used to describe the semi-simple Frobenius manifolds.)
We would like to mention that a similar approach has been applied to hypersurface singularities by Saito [@Saito]. However, there are difficulties specific to the higher genus case.
Our interest in this type of Frobenius manifolds is partly due to the fact that in this case we get a *family* of Frobenius manifolds parameterized by the periods of abelian integrals. Another interesting feature is that this example is related neither to Fano varieties, nor to Calabi–Yau varieties but depends on the global geometry of a curve. We hope that our technique can be useful in higher dimensions as well and may lead to a uniform treatment of B-model Frobenius manifolds.
To simplify notation, we shall assume that $df$ has a single pole (the only principal difference is that in this case we do not need to worry about the residues).
Whitham equation
----------------
Our other motivation to study the Frobenius manifold structures on the spaces of abelian integrals is their relation to Whitham equation.
It is now well known that starting from an algebraic curve one can construct special solutions of KP equations. This construction is due to Krichever who has also built a deformation theory of such solutions. Roughly speaking, we deform the curve and try to construct a solution of KP that depends on a small parameter ${\varepsilon}$. In other words, we incorporate a “slow” flow along the moduli space of curves. Such deformations are governed by Whitham equations. This deformation theory has recently been a subject of active study.
We shall give a simple explanation and derivation of Whitham equations in [@FedorovWhitham] based on the geometry of Picard varieties. The current paper may be viewed as the first part of the “Whitham project”.
My interest in the subject was attracted by papers [@Krichever2; @Losev]. I would like to thank D. Arinkin, V. Ginzburg, A. Losev, and I. Mirković for numerous discussions. Part of this work was done, while the author visited Max Planck Institute in Bonn, and I would like to thank it for warm hospitality.
Preliminaries and the main construction
=======================================
Pencils of connections
----------------------
\[pencil\] Let $\pi:{\mathbf M}\times{\mathbb P}^1\to{\mathbf M}$ be the natural projection, where ${\mathbf M}$ is a manifold. By a *pencil of connections* on ${\mathbf M}$ we mean a pair $({\mathcal{W}},\nabla)$, where ${\mathcal{W}}\to{\mathbf M}$ is a vector bundle, $\nabla$ is a relative flat connection on ${\mathcal{V}}=\pi^*{\mathcal{W}}$ along ${\mathbf M}$ with a simple pole along ${\mathbf M}\times\{0\}$.
There is a natural way to construct twisted Frobenius manifold structures on dense open subsets of ${\mathbf M}$ starting from a pencil of connections, provided this pencil of connections satisfies some non-degeneracy condition. This will be explained in detail in §\[WDVV\].
Main objects
------------
Consider a smooth complete algebraic curve $X$ of genus $g$ over ${\mathbb C}$, let $p\in X$. Denote by $(\hat X,\hat p)$ the maximal abelian cover of $(X,p)$.
An *abelian integral* on $(X,p)$ is a meromorphic function $f$ on $\hat X$ such that $df$ descends to a meromorphic differential on $X$. We define *periods* of $f$ to be those of $df$.
Consider the moduli space ${\mathbf A}_{g,n}$ of triples $(X,p,f)$, where $(X,p)$ is as above, $f$ is an abelian integral with a single pole of order $n$ at $p$ (in other words, $df$ is a meromorphic form on $X$ with the only pole at $p$ of order $n+1$).
The periods of $f$ give a linear map $H_1(X,{\mathbb{Z}})\to{\mathbb C}$. One can identify groups $H_1(X,{\mathbb{Z}})$ locally over the moduli space of curves using the Gauss–Manin connection, therefore the periods give rise to a foliation on ${\mathbf A}_{g,n}$. Let us fix one of the leaves and denote it by ${\mathbf A}$. Thus ${\mathbf A}$ parameterizes abelian integrals with prescribed periods. Assume that $n\ge1$.
Let $\hat{\mathbf A}$ be the moduli space of quadruples $(X,p,f,\Delta)$, where $(X,p,f)\in{\mathbf A}$, $\Delta$ is a subgroup of $H_1(X,{\mathbb{Z}})$ maximal isotropic with respect to the intersection form. The elements of $\Delta$ will be called *$a$-cycles*. Clearly, $\hat{\mathbf A}$ is a cover of ${\mathbf A}$. We shall construct a pencil of connections on $\hat{\mathbf A}$, giving rise to a twisted Frobenius structure on the smooth locus of $\hat{\mathbf A}$.
${\mathbf A}$ is not algebraic, so we shall work in analytic category. However, most of our constructions can be done algebraically. For example, one can construct an algebraic Frobenius structure on the formal completion of ${\mathbf A}$ at a point.
Notation
--------
The following notation will be fixed throughout the paper. Denote by ${\mathbf X}$ the universal curve over ${\mathbf A}$, let ${\varphi}:{\mathbf X}\to{\mathbf A}$ be the natural projection. We denote by $d_X$ the relative differential ${\mathcal O}_{\mathbf X}\to\Omega_{{\mathbf X}/{\mathbf A}}$. We denote by $d$ the usual (absolute) differential.
There is a natural section of ${\varphi}$ corresponding to $p\in X$, denote it by $\tilde p$; we can also view it as a divisor on ${\mathbf X}$. For any integer number $k$ set ${\mathcal O}(k)={\mathcal O}_{\mathbf X}(k\tilde p)$, $\Omega(k)=\Omega_{{\mathbf X}/{\mathbf A}}\otimes{\mathcal O}(k)$, ${\mathcal T}(k)={\mathcal T}_{{\mathbf X}/{\mathbf A}}\otimes{\mathcal O}(k)$.
We have a “universal” multi-valued function on ${\mathbf X}$. We denote it again by $f$, hopefully it will not lead to a confusion. Set $\omega=d_Xf$.
Main construction {#MainConstruction}
-----------------
The fact that the periods of $f$ are fixed shows that $df$ is a single-valued 1-form on ${\mathbf X}$. Thus $$\nabla=d+\frac{df}z$$ is a family of flat connections on ${\mathcal O}_{\mathbf X}$ parameterized by $z\in{\mathbb P}^1\setminus0$. We can view $\nabla$ as a relative ${\EuScript D}$-module on ${\mathbf X}\times({\mathbb P}^1\setminus0)$. The idea is that we get a pencil of connections on ${\mathbf A}$ by taking the push-forward of $\nabla$ along ${\varphi}\times{\mathop{\mathrm{Id}}\nolimits}_{{\mathbb P}^1}$. There are two problems we shall have to go around:
[[) ]{}]{}Our ${\EuScript D}$-module is not defined at $z=0$, and the push-forward is not coherent near $z=\infty$. Thus some regularization is needed.
[[) ]{}]{}We get a vector bundle on ${\mathbf A}\times{\mathbb P}^1$, whose restriction to $\{m\}\times{\mathbb P}^1$ is not trivial, thus it is not a pencil of connections. We shall make some modification along ${\mathbf A}\times\{\infty\}$, this is where we need the additional structure of $a$-cycles.
We shall denote ${\varphi}\times{\mathop{\mathrm{Id}}\nolimits}_{{\mathbb P}^1}$, ${\varphi}\times\mathrm{\mathop{\mathrm{Id}}\nolimits}_{{\mathbb P}^1\setminus0}$ etc. again by ${\varphi}$ for brevity.
[[) ]{}]{}\[FirstRemark\] We can start with a connection $d_X+\frac\omega{z}$ along the fibers of ${\varphi}$. Then the condition that the periods of $\omega$ are constant is exactly the isomonodromic condition for this connection. Thus it can be extended to an absolute connection. See §\[isomonodromy\] for more on isomonodromic deformation.
[[) ]{}]{}If $f$ has zero periods (i.e. $f$ is a meromorphic function on $X$), then one can extend $\nabla$ to an absolute flat meromorphic connection on ${\mathbf X}\times{\mathbb P}^1$. In this case we get a Frobenius manifold with Euler field (i.e. with a conformal structure). For details see, e.g. [@Sabbah].
[[) ]{}]{}This ${\EuScript D}$-module push-forward can be viewed as taking the cohomology fiberwise, with the connection on cohomology being the Gauss–Manin connection. One can also think about this as about a de Rham complex, twisted by $e^f$, see, e.g. [@Sabbah §I.3.3].
Organization of the paper
-------------------------
In the next section we shall make the ideas above precise, thus constructing a pencil of connections on $\hat{\mathbf A}$. In §\[IdenStr\] we prove that the pencil of connections gives rise to Frobenius structures on some open subsets of $\hat{\mathbf A}$ and calculate these Frobenius structures explicitly. Finally, we present the relation between our construction and that of [@Krichever1; @Krichever2].
The precise construction
========================
First, we consider the complex: $${\mathcal O}(-1)\xrightarrow{d_X+\frac\omega{z}}\Omega(n).$$ We always place the leftmost term in degree zero. We can view this complex as a complex of sheaves on ${\mathbf X}$, depending on a parameter $z\in{\mathbb P}^1\setminus0$. We can also view it as a complex of ${\mathbb C}[1/z]$-modules: ${\mathcal O}(-1)\otimes{\mathbb C}[1/z]\xrightarrow{d_X+\omega/z}\Omega(n)\otimes{\mathbb C}[1/z]$.
To regularize the complex at $z=0$ we shall patch it on ${\mathbb P}^1\setminus\{0,\infty\}$ with the complex ${\mathcal O}(-1)\xrightarrow{zd_X+\omega}\Omega(n)$, using the diagram $$\label{patch}
\begin{CD}
{\mathcal O}(-1)@>zd_X+\omega>>\Omega(n)\\
@V=VV @V\times\frac1zVV\\
{\mathcal O}(-1)@>d_X+\frac\omega z>>\Omega(n)\\
\end{CD}$$ This is an isomorphism of complexes if $z\ne0,\infty$. This construction gives a complex $$\label{global}
{\mathcal O}(-1)\boxtimes{\mathcal O}_{{\mathbb P}^1}\xrightarrow{d_X+\frac\omega z}
\Omega(n)\boxtimes{\mathcal O}_{{\mathbb P}^1}(1).$$ We shall denoted this complex by ${\mathcal D^\bullet}$ and its restriction to ${\mathbf X}\times\{z\}$ by ${\mathcal D^\bullet}_z$.
Note that the map is ${\mathcal O}_{{\mathbf A}\times{\mathbb P}^1}$-linear. We shall view this complex as an object in the derived category of complexes of sheaves of ${\mathcal O}_{{\mathbf A}\times{\mathbb P}^1}$-modules on ${\mathbf X}\times{\mathbb P}^1$. We shall abuse the language by saying “complex of sheaves”, where we really mean “complex of sheaves up to a quasi-isomorphism”. The push-forward ${\varphi}_*{\mathcal D^\bullet}$ is a complex of coherent sheaves (because ${\varphi}$ is proper).
Relation to ${\EuScript D}$-modules {#Dmodules}
-----------------------------------
The connection $d+\frac{df}z$ equips ${\mathcal O}(\infty)$ with a structure of $z$-dependant ${\EuScript D}_{{\mathbf X}}$-module ($z\ne0$). Here ${\mathcal O}(\infty)$ is the sheaf of meromorphic functions on ${\mathbf X}$ with poles on $\tilde p$ only.
To calculate its ${\varphi}$-push-forward we have to consider the corresponding relative de Rham complex: $$\label{Dmod}
{\mathcal O}(\infty)\xrightarrow{d_X+\frac\omega z}\Omega(\infty).$$ Unfortunately the sheaves of cohomology groups are not coherent near $z=\infty$. To circumvent this problem we take a subcomplex of coherent sheaves: $${\mathcal O}(k)\xrightarrow{d_X+\frac\omega z}\Omega(k+n+1).$$ One easily checks that it is quasi-isomorphic to (\[Dmod\]) at $z\ne\infty$. It follows that ${\varphi}_*{\mathcal D^\bullet}$ has a ${\EuScript D}_{\mathbf X}$-module structure for $z\ne0,\infty$. We shall calculate this structure and see that it extends to $z=\infty$ and has a simple pole at $z=0$.
Our choice of $k=-1$ is imposed by the fact that in this case the push-forward is *locally free* at $z=\infty$, as we shall see shortly. If, for example, we take $k=0$, the push-forward would be $$\hat{\mathcal{V}}\oplus{\mathcal S}\oplus{\mathcal S}[1],$$ where $\hat{\mathcal{V}}$ is a vector bundle, ${\mathcal S}={\mathcal O}_{z=\infty}$ is a sheaf of sky-scrappers, $[1]$ is the shift of grading.
Study of ${\varphi}_*{\mathcal D^\bullet}$ in the direction of ${\mathbb P}^1$
------------------------------------------------------------------------------
Now we choose a point $m=(X,p,f_0)\in{\mathbf A}$ and study the restriction of ${\varphi}_*{\mathcal D^\bullet}$ to $\{m\}\times{\mathbb P}^1$. We denote this restriction again by ${\varphi}_*{\mathcal D^\bullet}$ for brevity. In this subsection ${\mathcal O}(k)$ stands for ${\mathcal O}_X(kp)$ and $\Omega(k)=\Omega_X(kp)$.
\[structure\] ${\varphi}_*{\mathcal D^\bullet}$ is isomorphic to the vector bundle $$\left(\oplus_1^g{\mathcal O}_{{\mathbb P}^1}\right)\oplus
\left(\oplus_1^{g+n-1}{\mathcal O}_{{\mathbb P}^1}(1)\right)$$ placed in degree 1.
Let us show first that ${\varphi}_*{\mathcal D^\bullet}$ is a vector bundle concentrated in degree 1. To this end we fix $z\ne0$ and consider an exact sequence: $$\label{exact}
0\to\Omega(n)[1]\to{\mathcal D^\bullet}_z\to{\mathcal O}(-1)\to0.$$ Let us write the corresponding exact sequence of hypercohomology: $$\small{0\to{\mathbb H}^0({\mathcal D^\bullet}_z)\to0\to
H^0(\Omega(n))\to{\mathbb H}^1({\mathcal D^\bullet}_z)\to H^1({\mathcal O}(-1))\to0\to{\mathbb H}^2({\mathcal D^\bullet}_z)\to0.}$$ We see that the hypercohomology of ${\mathcal D^\bullet}_z$ are concentrated in degree 1 and the dimension does not depend on $z$ (it is equal to $2g+n-1$). A similar sequence for the upper complex in (\[patch\]) shows that a similar statement is valid near $z=0$. Now it follows that ${\varphi}_*{\mathcal D^\bullet}$ is a locally free sheaf in degree 1, i.e. a vector bundle.
Evaluating the global sections of (\[global\]) along ${\mathbb P}^1$ first, we come to the following presentation of the global sections of ${\varphi}_*{\mathcal D^\bullet}$: $${\mathbb H}^1({\mathcal O}(-1)\xrightarrow{(d_X,\times\omega)}\Omega(n)\oplus\Omega(n)).$$ Using an exact sequence, similar to (\[exact\]), one checks easily that the dimension of the space of global sections of ${\varphi}_*{\mathcal D^\bullet}$ is equal to $3g+2n-2=\dim{\varphi}_*{\mathcal D^\bullet}+(g+n-1)$. The lemma will be proved if we show that (i) the global sections generate the fiber of ${\varphi}_*{\mathcal D^\bullet}$ at $z=0$ and (ii) that there are no global sections, vanishing at both $z=0$ and $z=\infty$. Indeed, every vector bundle on ${\mathbb P}^1$ is isomorphic to $\oplus_i{\mathcal O}_{{\mathbb P}^1}(m_i)$. Now (i) shows that $m_i\ge0$ for all $i$, and (ii) shows that $m_i\le1$ for all $i$.
It follows from the base change that the following map of complexes gives rise to the map from the space of global sections to ${\varphi}_*{\mathcal D^\bullet}|_{z=0}$: $$\label{restriction}
\begin{CD}
{\mathcal O}(-1)@>(d_X,\times\omega)>>\Omega(n)\oplus\Omega(n)\\
@VVV @V(0,1)VV\\
{\mathcal O}(-1)@>\omega>>\Omega(n).
\end{CD}$$ The second hypercohomology group of the kernel of this map is isomorphic to $H^1(\Omega(n))=0$. Hence the induced map on hypercohomology groups is surjective, and (i) is satisfied.
We also see that the kernel of the map from the space of global sections to ${\varphi}_*{\mathcal D^\bullet}|_{z=0}$ is given by the global sections of the *first* $\Omega(n)$ summand.
Now, one writes a map of complexes, analogous to (\[restriction\]) but generating a map from the space of global sections to ${\varphi}_*{\mathcal D^\bullet}|_{z=\infty}$ and checks that its kernel is given by the global sections of the *second* $\Omega(n)$ summand. Since these subspaces of global sections do not intersect, (ii) follows.
Let us denote by $Conn^0_{X,p,n}$ the space of degree zero line bundles on $X$ with a connection such that the connection has a pole of order at most $n$ at $p$ and no other poles. Denote by ${\lefteqn{\widetilde{\phantom{Conn}}}}Conn^0_{X,p,n}$ the universal cover of $Conn^0_{X,p,n}$.
\[isomorhism\] There is a natural isomorphism $$\label{identif}
{\varphi}_*{\mathcal D^\bullet}|_{z=\infty}\approx{\lefteqn{\widetilde{\phantom{Conn}}}}Conn^0_{X,p,n}.$$
By the base change LHS is given by ${\mathbb H}^1({\mathcal O}(-1)\xrightarrow
d\Omega(n))$. It is easy to see that the natural inclusion of the above complex into ${\mathcal O}\xrightarrow d\Omega(n)$ induces an isomorphism in the first cohomology groups. Further, ${\mathbb H}^1({\mathcal O}\xrightarrow d\Omega(n))$ is identified with the tangent space to $Conn^0_{X,p,n}$ at zero. The latter space is identified with the universal cover of $Conn^0_{X,p,n}$.
Below we always assume the identification (\[identif\]).
Improving ${\varphi}_*{\mathcal D^\bullet}$
-------------------------------------------
Recall that in order to give rise to a pencil of connections, ${\varphi}_*{\mathcal D^\bullet}$ has to be isomorphic to $\pi^*{\mathcal{W}}$, where ${\mathcal{W}}$ is a vector bundle on ${\mathbf A}$. This is impossible, since the restriction of ${\varphi}_*{\mathcal D^\bullet}$ to $\{m\}\times{\mathbb P}^1$ is not a trivial bundle (see Lemma \[structure\]). This is easy to cure if $g=0$: just twist by ${\mathcal O}_{{\mathbb P}^1}(-1)$. If $g>0$ we need to choose a trivial subbundle in the restriction of ${\varphi}_*{\mathcal D^\bullet}$ to every $m\in{\mathbf A}$.
Let us now recall that we have chosen a space $\Delta$ of $a$-cycles on $X$. Note that every degree zero line bundle has a unique non-singular connection with trivial $a$-monodromy. This gives a splitting of the standard exact sequence $$\label{UsualExact}
0\to H^0(X,\Omega(n))\to Conn^0_{X,p,n}\to Pic_0(X)\to0.$$ Thus, we have a canonical splitting $$\label{splitting}
{\varphi}_*{\mathcal D^\bullet}|_{z=\infty}=H^0(X,\Omega(n))\oplus H^1(X,{\mathcal O}).$$ We define ${\mathcal{V}}$ as a subsheaf of ${\varphi}_*{\mathcal D^\bullet}$ whose sections are sections of ${\varphi}_*{\mathcal D^\bullet}$ *vanishing along* $H^0(X,\Omega(n))$. In other words, the sections of ${\mathcal{V}}$ are sections $s$ of ${\varphi}_*{\mathcal D^\bullet}$ such that for any covector in $(H^0(X,\Omega(n)))^*$ its extension to ${\varphi}_*{\mathcal D^\bullet}|_{z=\infty}$ vanishes on $s$.
Globalization
-------------
So far we were working with a fixed quadruple $(X,p,f_0,\Delta)$. Let us now globalize the picture to $\hat{\mathbf A}$. First we get a vector bundle ${\varphi}_*{\mathcal D^\bullet}$ on ${\mathbf A}\times{\mathbb P}^1$ with connection along ${\mathbf A}$ with a pole along ${\mathbf A}\times\{0\}$. The restriction of this connection to ${\mathbf A}\times\{z\}$ we denote by $\nabla_z$. Modifying ${\varphi}_*{\mathcal D^\bullet}$ at $z=\infty$ as above we get a vector bundle ${\mathcal{V}}$ on $\hat{\mathbf A}\times{\mathbb P}^1$, its restriction to any point of $\hat{\mathbf A}$ being a trivial bundle on ${\mathbb P}^1$. Thus ${\mathcal{V}}={\varphi}^*{\mathcal{W}}$ for some vector bundle ${\mathcal{W}}$ on $\hat{\mathbf A}$.
\[identifications\] $({\mathcal{W}},\nabla)$ is a pencil of connections on $\hat{\mathbf A}$ in the sense of Definition \[pencil\].
The bundle ${\mathcal{V}}$ is equipped with a relative connection $\nabla_z$ for $z\ne0,\infty$. It is clear that this connection has at most simple pole along $\hat{\mathbf A}\times\{0\}$. In Lemma \[residue\] we shall calculate its residue explicitly and shall see that it is not equal to zero. We define $\nabla_\infty$ on ${\mathcal{V}}$ as $\lim_{z\to\infty}\nabla_z$. In Lemma \[SplitIsoConnection\] we shall calculate this limit (and see that it exists).
Identifying structures {#IdenStr}
======================
We shall see shortly that Frobenius structures come from $\nabla_\infty$ and the residue of the pencil of connections. Then we shall calculate these parts of the pencil explicitly. It will yield Theorem \[identifications\]. It will also follow that this pencil of connections gives rise to a Frobenius manifold structures on some open subsets of $\hat{\mathbf A}$.
From pencils of connections to Frobenius manifolds and WDVV equation {#WDVV}
--------------------------------------------------------------------
We shall not give a precise definition of a Frobenius manifold. However, below we explain how to get all the basic structures, starting from a pencil of connections.
Let $({\mathcal{W}},\nabla)$ be a pencil of connections on a manifold ${\mathbf M}$. One interprets it as a family of flat connections on ${\mathcal{W}}$, parameterized by ${\mathbb P}^1\setminus0$. Our condition on the pole at zero implies that this family is of the form $\nabla_\infty+\Phi/z$, where $z$ is the standard coordinate on ${\mathbb P}^1$, $\nabla_\infty$ is the restriction of $\nabla$ to ${\mathbf M}\times\{\infty\}$, $\Phi$ is a Higgs field (this explains the origin of the word ‘pencil’ in our definition).
Assume now that there exists a *primitive section*, i.e. a section $\rho$ of ${\mathcal{W}}$ such that $\nabla_\infty\rho=0$ and $\Phi(\rho,\cdot):{\mathcal T}{\mathbf M}\to{\mathcal{W}}$ is an isomorphism. One uses this isomorphism to carry $\nabla_\infty$ and $\Phi$ to ${\mathcal T}{\mathbf M}$. The former becomes a flat structure $\tilde\nabla$ on ${\mathbf M}$ (it is automatically without torsion). The latter becomes a commutative associative multiplication in the tangent sheaf. We denote this multiplication by $\circ$. The equation $\Phi(\rho,e)=\rho$ defines a unit for $\circ$. One can show that $\circ$ does not depend on the choice of $\rho$ (while $\tilde\nabla$ does depend).
The last ingredient needed to equip ${\mathbf M}$ with a structure of a Frobenius manifold is a bilinear product compatible with $\tilde\nabla$ and $\circ$. In other words, $\tilde\nabla$ is the Levi–Civita connection for this metric, while the multiplication operators are symmetric. These structures altogether make ${\mathcal T}{\mathbf M}$ into a sheaf of *Frobenius algebras*, so that ${\mathbf M}$ becomes a *Frobenius manifold*.
Giving such a metric is equivalent to a $\nabla$-flat symmetric non-degenerate pairing $\langle\cdot,\cdot\rangle:a^*{\mathcal{V}}\otimes{\mathcal{V}}\to{\mathcal O}_{{\mathbf M}\times{\mathbb P}^1}$, where $a:z\mapsto-z$ is an involution on ${\mathbb P}^1$. Indeed, if $s_1$ and $s_2$ are sections of ${\mathcal{W}}$, then we have $$\left\langle\left(\nabla_\infty+\frac\Phi z\right)s_1,s_2\right\rangle+
\left\langle s_1,\left(\nabla_\infty-\frac\Phi
z\right)s_2\right\rangle=d\left\langle s_1,s_2\right\rangle.$$ This is equivalent to $\langle\nabla_\infty s_1,s_2\rangle+ \langle
s_1,\nabla_\infty s_2\rangle=d\langle s_1,s_2\rangle$ and $ \langle\Phi
s_1,s_2\rangle=\langle s_1,\Phi s_2\rangle$.
One then chooses flat coordinates $t_A$ on ${\mathbf M}$ and shows that there exists locally on ${\mathbf M}$ a *potential* $F$ such that $$\langle\partial_A\circ\partial_B,\partial_C\rangle=
\tilde\nabla_{\partial_A}\tilde\nabla_{\partial_B}\tilde\nabla_{\partial_C}F.$$ The associativity condition for $\circ$ transforms into WDVV equation for $F$.
### Constructing primitive sections
Let $m$ be a point of ${\mathbf M}$, $\rho_m\in{\mathcal{W}}_m$ be such that $\Phi(\rho_m,\cdot):{\mathcal T}_m{\mathbf M}\to{\mathcal{W}}_m$ is an isomorphism. Then we can extend $\rho_m$ to a $\nabla_\infty$-flat section $\rho$ of ${\mathcal{W}}$. Unfortunately, $\rho$ is defined on a universal cover $\tilde{\mathbf M}$ of ${\mathbf M}$, thus it gives rise to a Frobenius manifold structure on the open subset of $\tilde{\mathbf M}$, where $\Phi(\rho,\cdot)$ is an isomorphism. One easily checks that $\tilde\nabla$ and $\circ$ descend to ${\mathbf M}$. However, the metric does not descend to ${\mathbf M}$. We call it a twisted Frobenius structure on ${\mathbf M}$.
Isomonodromic deformations {#isomonodromy}
--------------------------
We are going to identify $\nabla_\infty$ with some isomonodromic deformation. Thus we shall need some generalities on isomonodromy. For more detail we refer to [@Babelon]. Note that there is a standard relation between isomonodromy and Frobenius manifolds, see [@Sabbah], but it will not be used in our paper.
Let ${\mathbf Y}\to{\mathbf M}$ be a holomorphic fiber bundle, equipped with a family of meromorphic flat connections on fibers (i.e. a relative connection). This family is said to be *isomonodromic* if it can be extended to a flat absolute meromorphic connection on ${\mathbf Y}$. The family of connections with regular singularities is isomonodromic if and only if the monodromy does not change.
### Universal isomonodromy for line bundles {#universal}
We shall need a baby version of isomonodromy, namely, isomonodromy for *line bundles* with connections. In this case a family of meromorphic connections is isomonodromic if and only if the monodromy does not change, even if connections have irregular singularities.
Let ${\mathbf M}_{g,n}$ be the moduli space of triples $(X,p,x)$, where $X$ is a curve (which we assume smooth complete over ${\mathbb C}$), $p\in X$, $x$ is a coordinate to order $n$ at $p$. Let $Conn_n\to{\mathbf M}_{g,n}$ be the moduli space of line bundles with connections with a pole of order at most $n$ at $p$ and no other poles. One defines the *universal isomonodromic connection* on this fibration by the following requirements: a family is isomonodromic if (1) monodromy representation is constant (2) $x$-expansion of the polar part at $p$ does not change. The existence and uniqueness of such a connection is a standard fact in the theory of isomonodromic deformations (it is easy for line bundles though).
Calculating ${\varphi}_*{\mathcal D^\bullet}$ {#calc}
---------------------------------------------
We start with the Čech calculation of ${\varphi}_*{\mathcal D^\bullet}$. Let $m=(X,p,f_0)\in{\mathbf A}$. Consider the following cover of $X$: $X=\dot X\cup{}D$, where $\dot
X=X\setminus p$, $D$ is the formal neighbourhood of $p$. Set $\dot D=\dot
X\cap D$. Let $\bar{\mathbf A}$ be the formal completion of ${\mathbf A}$ at $m$, $\bar{\mathbf X}$ be the restriction of ${\mathbf X}$ to $\bar{\mathbf A}$. Since affine schemes have no infinitesimal deformations, $\bar{\mathbf X}$ can be covered by $\dot X\times\bar{\mathbf A}$ and $D\times\bar{\mathbf A}$. To calculate ${\varphi}_*{\mathcal D^\bullet}$ we shall use the Čech resolution of ${\mathcal D^\bullet}$: $$\begin{CD}
0 @>>>{\mathcal O}(-1) @>d_X+\frac\omega z>>
\Omega(n) @>>>0 @>>>0\\
@VVV @VVV @VVV @VVV @VVV\\
0 @>>>C^0({\mathcal O}(-1))
@>>>\genfrac{}{}{0pt}{}{C^1({\mathcal O}(-1))\oplus}{C^0(\Omega(n))} @>>>
C^1(\Omega(n)) @>>> 0
\end{CD}$$ Consider a section of $R^1{\varphi}_*{\mathcal D^\bullet}$ over $\bar{\mathbf A}$. It is represented by a family of 1-cocycles $(\alpha_1({\varepsilon}),\alpha_2({\varepsilon}),s({\varepsilon}))$. Precisely, $\alpha_1({\varepsilon})$ is a relative 1-form on $\dot
X\times\bar{\mathbf A}\to\bar{\mathbf A}$, $\alpha_2({\varepsilon})$ is a relative 1-form on $D\times\bar{\mathbf A}\to\bar{\mathbf A}$ with a pole of order at most $n$ along $\{p\}\times\bar{\mathbf A}$, $s({\varepsilon})\in{\mathcal O}_{\dot D\times\bar{\mathbf A}}$. The cocycle condition is $$\label{cocycle}
\alpha_1-\alpha_2=d_Xs+\frac{\omega s}z.$$ We extend $d_X+\frac\omega z$ to an absolute connection $\nabla=d+\frac{df}z$ (compare with remark (\[FirstRemark\]) after §\[MainConstruction\]).
We want to describe the Gauss–Manin connection on the first hypercohomology sheaf directly. To this end we need to assign to $(\alpha_1({\varepsilon}),\alpha_2({\varepsilon}),s({\varepsilon}))$ a cocycle with values in $\Omega_{\bar{\mathbf A}}$.
First, we define a map $$\Psi_1:\wedge^2\Omega_{\dot X\times\bar{\mathbf A}}\to
{\varphi}^*\Omega_{\bar{\mathbf A}}\otimes\Omega_{\dot X\times\bar{\mathbf A}/\bar{\mathbf A}}.$$ It is defined as follows: there is a natural surjective map ${\varphi}^*\Omega_{\bar{\mathbf A}}\otimes\Omega_{\dot
X\times\bar{\mathbf A}}\to\wedge^2\Omega_{\dot X\times\bar{\mathbf A}}$ (recall that $\dim
X=1$). Thus we can lift a section of $\wedge^2\Omega_{\dot X\times\bar{\mathbf A}}$ to ${\varphi}^*\Omega_{\bar{\mathbf A}}\otimes\Omega_{\dot X\times\bar{\mathbf A}}$ and then project it to ${\varphi}^*\Omega_{\bar{\mathbf A}}\otimes\Omega_{\dot X\times\bar{\mathbf A}/\bar{\mathbf A}}$. The lift is defined up to an element of the second symmetric power of $\Omega_{\bar{\mathbf A}}$, thus the projection does not depend on the lift. Similarly, we can define maps $$\begin{split}
\Psi_2:\wedge^2\Omega_{D\times\bar{\mathbf A}}\to
{\varphi}^*\Omega_{\bar{\mathbf A}}\otimes\Omega_{D\times\bar{\mathbf A}/\bar{\mathbf A}}.\\
\Psi_{12}:\wedge^2\Omega_{\dot D\times\bar{\mathbf A}}\to
{\varphi}^*\Omega_{\bar{\mathbf A}}\otimes\Omega_{\dot D\times\bar{\mathbf A}/\bar{\mathbf A}}.
\end{split}$$ Now let us lift $\alpha_i$ to an absolute 1-form $\tilde\alpha_i$ such that $\tilde\alpha_2$ has a pole of order at most $n$ on $\tilde p$ and set $$\nabla\alpha_i=\Psi_i(\nabla\tilde\alpha_i).$$ It follows from (\[cocycle\]) that $$\label{beta}
\tilde\alpha_1-\tilde\alpha_2=\nabla s+\beta,$$ where $\beta$ is a section of ${\varphi}^*\Omega_{\bar{\mathbf A}}$. The Gauss–Manin connection is given by $$\label{GaussManin}
\nabla(\alpha_1,\alpha_2,s)=(\nabla\alpha_1,\nabla\alpha_2,\beta).$$
[[) ]{}]{}\[One\] The 1-cochain (\[GaussManin\]) is a cocycle for ${\mathcal D^\bullet}\otimes{\mathcal O}(n+1)$.\
[[) ]{}]{}\[Two\] If we change the lifts $\alpha_i\rightsquigarrow\tilde\alpha_i$, then (\[GaussManin\]) changes by a coboundary.\
[[) ]{}]{}\[Three\] $\nabla$ satisfies the Leibnitz rule with respect to multiplication of cocycles by functions on $\bar{\mathbf A}$.
We have a well-defined relative differential $d_X:{\varphi}^*\Omega_{\bar{\mathbf A}}\to
{\varphi}^*\Omega_{\bar{\mathbf A}}\otimes\Omega_{X\times\bar{\mathbf A}/\bar{\mathbf A}}$. The proof of (\[One\]) and (\[Two\]) is based on the following: *if $\gamma$ is a section of ${\varphi}^*\Omega_{\bar{\mathbf A}}$, then $\Psi_i(\nabla\gamma)=d_X\gamma+\frac1z\gamma\otimes\omega$.* The proof of this statement is left to the reader, let us prove (\[One\]). It follows from (\[beta\]) that $\nabla\tilde\alpha_1-\nabla\tilde\alpha_2=\nabla\beta$. Now let us apply $\Psi_{12}$ to both sides, it gives $\nabla\alpha_1-\nabla\alpha_2=d_X\beta+\frac1z\beta\otimes\omega$ and the cocycle condition follows.
We leave the proof of (\[Two\]) and (\[Three\]) to the reader.
It follows that $\nabla$ descends to a connection on ${\varphi}_*{\mathcal D^\bullet}$. A little difficulty is that we get a cocycle of ${\mathcal D^\bullet}\otimes{\mathcal O}(n+1)$ instead of that of ${\varphi}_*{\mathcal D^\bullet}$. Fortunately, there is a natural quasi-isomorphism (compare with §\[Dmodules\]) $$\label{quasiisomorphism}
{\mathcal D^\bullet}\hookrightarrow{\mathcal D^\bullet}\otimes{\mathcal O}(n+1)$$ for $z\ne\infty$. For $z=\infty$ one needs to extend the cocycle to a neighbourhood of $z=\infty$ first, apply $\nabla$, and then pass to the limit as $z\to\infty$ to evaluate $\nabla_\infty$. In the next subsection we shall calculate this limit (and see that it exists).
Identifying $\nabla_\infty$
---------------------------
Recall that in §\[universal\] we defined a fibration $Conn_n\to{\mathbf M}_{g,n}$. Let $Conn_n^0\to{\mathbf M}_{g,n}$ be its part corresponding to degree zero line bundles and let ${\lefteqn{\widetilde{\phantom{Conn}}}}Conn^0_n\to{\mathbf M}_{g,n}$ be its relative universal cover. In other words, $$\label{}
{\lefteqn{\widetilde{\phantom{Conn}}}}Conn^0_n=\bigsqcup_{(X,p)}{\lefteqn{\widetilde{\phantom{Conn}}}}Conn^0_{X,p,n}$$ Consider the diagram $$\label{pullback}
\begin{CD}
{\varphi}_*{\mathcal D^\bullet}|_{\bar{\mathbf A}, z=\infty}@>>> {\lefteqn{\widetilde{\phantom{Conn}}}}Conn^0_n\\
@VVV @VVV\\
\bar{\mathbf A}@>>> {\mathbf M}_{g,n}
\end{CD}$$ The lower map is given in the following way: $f$ in the neighbourhood of $p$ gives a polar part of order $n$. This gives a coordinate to order $n$. In other words, the coordinate is $x=f^{-\frac1n}$. Precisely, this coordinate is defined up to a multiplication by a root of unity. We fix such a choice (this is why we restrict this map to $\bar{\mathbf A}$). It follows from Lemma \[isomorhism\] that this is a pull-back diagram. It follows from §\[universal\], the right fibration in (\[pullback\]) is equipped with isomonodromic connection.
The upper map respects connections. Thus $\nabla_\infty$ is the pull-back of the isomonodromic connection.
Consider an isomonodromic family of connections on $\bar{\mathbf A}$ given by a family of cocycles $(\alpha_1({\varepsilon}),\alpha_2({\varepsilon}),\exp(s({\varepsilon})))$ as in §\[calc\] (now we have $z=\infty$). Here $\exp(s({\varepsilon}))$ is a cocycle defining the line bundle. It has a single-valued logarithm because the bundle has degree zero. We need to show that this family is $\nabla_\infty$-flat. The cocycle condition (\[cocycle\]) becomes $$\alpha_1-\alpha_2=d_Xs.$$ Since the family is isomonodromic the forms $\alpha_i$ can be extended to absolute *closed* forms $\tilde\alpha_i$. The condition that the $x$-expansion of the polar part does not change in the family can be written in the following way: $$\label{Alpha}
\tilde\alpha_2({\varepsilon})=h(x^{-1})dx+\gamma({\varepsilon},x),$$ where $h$ is a polynomial with constant coefficients, $\gamma$ is a 1-form regular on $\tilde p$. A simple calculation in local coordinates shows that, changing the lift $\alpha_2\rightsquigarrow\tilde\alpha_2$ if necessary, we can assume that $\gamma/x$ has a logarithmic pole on $\tilde p$.
Since $f=x^{-n}$, we have $h(x^{-1})dx\wedge df=0$. Now we want to extend the cocycle to the neighbourhood of $z=\infty$. To this end we write $\omega
s=\sigma_1-\sigma_2$, where $\sigma_1$ is a relative 1-form on $\dot
X\times\bar{\mathbf A}$, $\sigma_2$ is a relative 1-form on $D\times\bar{\mathbf A}$ with at most simple pole. This is always possible, since $H^1(X,\Omega(1))=0$. It is easy to see that $(\alpha_1+\frac{\sigma_1}z,\alpha_2+\frac{\sigma_2}z,s)$ satisfies the cocycle condition (\[cocycle\]).
We extend $\sigma_i$ to an absolute form $\tilde\sigma_i$. It is easy to see that we can have $\tilde\sigma_2\wedge df=0$. Using (\[Alpha\]) we get: $$\nabla\left(\tilde\alpha_2+\frac{\tilde\sigma_2}z\right)=
\left(d+\frac{df}z\right)\left(\tilde\alpha_2+\frac{\tilde\sigma_2}z\right)=
\frac{df}z\wedge\gamma+\frac{d\tilde\sigma_2}z={\mathcal O}\left(\frac1z\right).$$ A similar (but easier) argument, shows that $\nabla\left(\tilde\alpha_1+\frac{\tilde\sigma_1}z\right)=O(1/z)$. Applying $\Psi$ we see that $$\label{notapriori}
\nabla(\alpha_1,\alpha_2,s)=(0,0,\beta)+{\mathcal O}\left(\frac1z\right),$$ where $d_X\beta=0$. Note also that since $\gamma/x$ has logarithmic pole on $\tilde p$, (\[notapriori\]) is a cocycle of ${\mathcal D^\bullet}$ (a priori it is a cocycle of ${\mathcal D^\bullet}(n)$). Therefor we do not need to invert the quasi-isomorphism (\[quasiisomorphism\]) (which could have altered the behavior at $z=\infty$). Taking limit as $z\to\infty$ we get a cocycle of the form $(0,0,\beta)$. Such a cocycle is a coboundary, so the family $(\alpha_1,\alpha_2,s)$ is a flat section of $\nabla_\infty$.
We have calculated $\nabla_\infty$ on ${\varphi}_*{\mathcal D^\bullet}$. Now we want to understand $\nabla_\infty$ on ${\mathcal{V}}$. Recall that the choice of $a$-cycles gives a splitting (\[splitting\]) of the exact sequence (\[UsualExact\]).
\[SplitIsoConnection\] (a) $\nabla_\infty$ respects this splitting; in particular it preserves the space of trivial bundles.\
(b) The restriction of $\nabla_\infty$ to the space of non-singular connections with trivial $a$-monodromy is the cover of isomonodromic connection.\
(c) The restriction of $\nabla_\infty$ to the space of trivial bundles is described as follows: a family $({\mathcal O},d_X+\rho)$ is flat if and only if the $a$-periods of $\rho$ are constant and the $x$-expansion of the polar part is constant.
\(b) Follows from the previous lemma because we do not modify ${\varphi}_*{\mathcal D^\bullet}$ along this subspace.\
(a) By (b) it is enough to prove that the space of trivial bundles is preserved. It follows from the fact that the sections of ${\mathcal{V}}$ corresponding to trivial bundles at $z=\infty$ correspond to the sections of ${\varphi}_*{\mathcal D^\bullet}$ that vanish at $z=\infty$.\
(c) Let $\mathcal C=({\mathcal O},d_X+\rho)$ be a family with constant $a$-monodromy and a constant $x$-expansion of the polar part. Let $\mathcal C'$ be a section of ${\varphi}_*{\mathcal D^\bullet}|_{z=\infty}$ covering a family of non-singular connections such that its $a$-monodromy is trivial and the $b$-monodromy is reciprocal to that of $\mathcal C$. Then the projection of $\mathcal
C''=\mathcal C\otimes\mathcal C'$ to $Conn_n$ is isomonodromic in the sense of §\[universal\]. Hence (using Lemma \[isomorhism\]) it can be extended to a $\nabla$-flat section $\tilde{\mathcal C}''$ of ${\varphi}_*{\mathcal D^\bullet}$ in the neighbourhood of $z=\infty$. Then $\tilde{\mathcal C}''/z$ is also $\nabla$-flat. It can be viewed as a section of ${\mathcal{V}}$, equal to $({\mathcal O},d_X+\rho)$ at $z=\infty$. It remains to recall that $\nabla_\infty$ for ${\mathcal{V}}$ is defined as $\lim_{z\to\infty}\nabla_z$.
Deformation theory
------------------
Consider the complex $$\begin{CD}
0\to{\mathcal T}(-1)\xrightarrow{\times\omega}{\mathcal O}(n)\to0 \\
\end{CD}$$ It governs deformations of a triple $m=(X,p,f_0)$, where only deformations of $f_0$ that preserve periods are allowed. Thus it is the deformation complex of $\hat{\mathbf A}$, we denote it by ${\mathcal K^\bullet}$. Let us present more details. Consider a cocycle $(h_1,h_2,\tau)$ of ${\mathcal K^\bullet}$, where $h_1\in{\mathcal O}_{\dot X}$, $h_2\in{\mathcal O}_D(n)$, $\tau\in{\mathcal T}(\dot D)$. Then $\tau$ represents a class of $H^1(X,{\mathcal T})$, thus it gives rise to an infinitesimal family $\hat
X\to{\mathop{\mathrm{Spec}}}{\mathbb C}[{\varepsilon}]/{\varepsilon}^2$. A function $\hat f$ on $\hat X$ is given by the conditions: $\hat f|_{\dot
X\times{\mathop{\mathrm{Spec}}}{\mathbb C}[{\varepsilon}]/{\varepsilon}^2}=f_0+{\varepsilon}h_1$, $\hat
f|_{D\times{\mathop{\mathrm{Spec}}}{\mathbb C}[{\varepsilon}]/{\varepsilon}^2}=f_0+{\varepsilon}h_2$. The periods of $\hat f$ are constant in the family because $h_1$ is a single-valued function on $\dot X$. Thus we have assigned a vector of ${\mathcal T}_m\hat{\mathbf A}$ to $(h_1,h_2,\tau)$. We leave it to the reader to check that it indeed gives an isomorphism of $R^1{\varphi}_*({\mathcal K^\bullet})$ with the tangent sheaf of $\hat{\mathbf A}$.
${\mathcal K^\bullet}$ is actually a dg-Lie algebra. This gives a relation of our work to [@KonBar], where a similar complex is used (in higher dimensional situation). See also [@Merkulov].
Residue of the pencil and multiplication in the tangent sheaf
-------------------------------------------------------------
Consider the residue of $\nabla$ at $z=0$, denote it by $\Phi$. We want to give a cohomological interpretation of $\Phi$. Consider a morphism of complexes: $$\label{CD}
\begin{CD}
{\mathcal T}(-2) @>\times\omega\oplus\times\omega>> {\mathcal O}(n-1)\oplus{\mathcal O}(n-1)
@>(\times\omega)-(\times\omega)>> \Omega(2n)\\
@VVV @V-VV @VV=V \\
0@>>>{\mathcal O}(n-1) @>\times\omega>>\Omega(2n)
\end{CD}$$ The upper complex is ${\mathcal K^\bullet}\otimes({\mathcal O}(-1)\xrightarrow{\times\omega}\Omega(n))$, while the lower complex is naturally quasi-isomorphic to ${\mathcal O}(-1)\xrightarrow{\times\omega}\Omega(n)$. Thus we get a map $${\mathcal T}\hat{\mathbf A}\otimes R^1{\varphi}_*({\mathcal O}(-1)\xrightarrow{\times\omega}\Omega(n))\to
R^1{\varphi}_*({\mathcal O}(-1)\xrightarrow{\times\omega}\Omega(n)).$$
\[residue\] This map coincides with $\Phi$.
Let us again fix $m=(X,p,f_0,\Delta)\in\hat{\mathbf A}$ and take a Čech cocycle $(\alpha_1,\alpha_2,s)$ of ${\mathcal D^\bullet}|_{m,z=0}$. We have $\alpha_1\in\Omega_{\dot
X}$, $\alpha_2\in\Omega_D(n)$, $s\in{\mathcal O}_{\dot D}$. Let us also take a cocycle $(h_1,h_2,\tau)$ of ${\mathcal K^\bullet}$, where $h_1\in{\mathcal O}_{\dot X}$, $h_2\in{\mathcal O}_D(n)$, $\tau\in{\mathcal T}(\dot D)$. This cocycle represents some $\xi\in{\mathcal T}_m\hat{\mathbf A}$. The product of cocycles is given by the following cocycle: $(h_1\alpha_1,h_2\alpha_2,\tau\alpha_1+h_2s)$. The cocycle condition for $(\alpha_1,\alpha_2,s)$ is given by $\alpha_1-\alpha_2=\omega s$, hence $s$ is uniquely determined by $\alpha_1$ and $\alpha_2$. Again, we view $\bar{\mathbf X}$ as glued from $\dot X\times\bar{\mathbf A}$ and $D\times\bar{\mathbf A}$. We can extend $\alpha_i$ and $h_i$ to families $\tilde\alpha_i$ and $\tilde h_i$ over $\bar{\mathbf A}$ using this direct product structure. Similarly we can lift $\xi$ to a vector field $\tilde\xi_1$ along $\dot X\times\{m\}$ and to $\tilde\xi_2$ along $D\times\{m\}$. In particular we have $\langle\tilde\xi_i,\tilde\alpha_i\rangle=0$.
Unwinding the definition of $\nabla$ we see that $\Phi$ is given on cocycles by the formula: $$\Phi(\xi,(\alpha_1,\alpha_2,s))= (\langle\xi,\Psi_1(df\wedge\tilde\alpha_1)\rangle,
\langle\xi,\Psi_2(df\wedge\tilde\alpha_2)\rangle,?).$$ Since the last element of the cocycle is uniquely determined by the others, it is enough to prove that $$\langle\xi,\Psi_i(df\wedge\tilde\alpha_i)\rangle=h_i\alpha_i.$$ Using the identification of cohomology of ${\mathcal K^\bullet}$ with the tangent space of $\hat{\mathbf A}$ one finds that $\langle\tilde\xi_i,df\rangle=\partial_{\tilde\xi_i}(f)=h_i$. It follows that $$\langle\xi,\Psi_i(df\wedge\tilde\alpha_i)\rangle=
\langle\tilde\xi_i,df\wedge\tilde\alpha_i\rangle=
\langle\tilde\xi_i,df\rangle\tilde\alpha_i-\langle\tilde\xi_i,\tilde\alpha_i\rangle
df=h_i\tilde\alpha_i.$$ The first equality is a “consistency property” of $\Psi_i$; the proof is left to the reader.
Notice that ${\mathcal K^\bullet}$ is exact except at zeros of $\omega=d_Xf_0$, therefore for fixed $m\in\hat{\mathbf A}$ the complex is naturally quasi-isomorphic to $\oplus_s{\mathcal O}_{q_s}$, where $q_s$ are zeros of $\omega$ (if $\omega$ has multiple zeroes, they should be viewed as schemes with nilpotents). Similarly, ${\mathcal D^\bullet}|_{z=0}$ is quasi-isomorphic to $\oplus_s(\Omega_X)_{q_s}$.
It follows easily from the lemma above that under this identification $\Phi$ becomes the componentwise multiplication. Thus $\Phi(\rho_m,\cdot)$ is an isomorphism for a generic element of ${\mathcal{W}}_m$. It follows that taking different primitive sections we can construct twisted Frobenius structures on open subsets covering the smooth locus of $\hat{\mathbf A}$.
The tangent sheaf multiplication also has a cohomological interpretation, namely, there is a natural map ${\mathcal K^\bullet}\otimes{\mathcal K^\bullet}\to{\mathcal K^\bullet}\otimes{\mathcal O}(n)$, similar to (\[CD\]).
The induced map on cohomology coincides with $\circ$. In particular $\circ$ does not depend on the choice of a primitive section.
Follows from associativity of cohomological multiplication.
One would like to have a canonical choice of a primitive section, this will be discussed in §\[PrimitiveSections\].
Metric
------
We are going to construct a sesquilinear metric on ${\mathcal{V}}$. We start with a map $${\mathcal D^\bullet}(-n-1)\otimes
a^*{\mathcal D^\bullet}\to({\mathcal O}(-n-1)\xrightarrow{d}\Omega)\boxtimes{\mathcal O}_{{\mathbb P}^1}(1).$$ This is defined by the following commutative diagram: $$\begin{CD}
{\mathcal O}(-n-1)@>(d+\frac\omega z,d-\frac\omega z)>>
(\Omega\oplus\Omega)\boxtimes{\mathcal O}_{{\mathbb P}^1}(1)
@>(d+\frac\omega z,-d+\frac\omega z)>> \Omega^2(n+1)\boxtimes{\mathcal O}_{{\mathbb P}^1}(2)\\
@V=VV @V+VV @VVV \\
{\mathcal O}(-n-1)@>d>>\Omega\boxtimes{\mathcal O}_{{\mathbb P}^1}(1) @>>>0
\end{CD}$$ It gives rise to a $\nabla$-flat map: $\langle\cdot,\cdot\rangle:{\varphi}_*{\mathcal D^\bullet}\otimes a^*{\varphi}_*{\mathcal D^\bullet}\to{\mathcal O}_{{\mathbb P}^1}(1)$, we leave it to the reader to check that it is symmetric and non-degenerate. We need to prove that this metric restricts to a metric ${\mathcal{V}}\otimes
a^*{\mathcal{V}}\to{\mathcal O}_{{\mathbb P}^1}$. Let $s_1$, $s_2$ be sections of ${\mathcal{V}}$. We must show that $\langle s_1,s_2\rangle|_{z=\infty}=0$. Recalling the splitting (\[splitting\]) and the definition of ${\mathcal{V}}$ we see that it is enough to check this condition in the case when $$s_i|_{z=\infty}\in H^1(X,{\mathcal O})\subset{\varphi}_*{\mathcal D^\bullet}|_{z=\infty}.$$ To this end we notice that ${\mathbb H}^1({\mathcal O}\xrightarrow{d}\Omega)=H^1(X,{\mathbb C})$ and that the restriction of $\langle\cdot,\cdot\rangle|_{z=\infty}$ to this subspace coincides with the usual intersection form on $H^1(X,{\mathbb C})$. It remains to recall that $\Delta$ is isotropic.
Choice of primitive sections {#PrimitiveSections}
----------------------------
Let $k$ be an integer such that $2\le k\le n$. Consider a section $\rho_k$ of $\Omega(k)$ with the following properties\
[[) ]{}]{}Its polar part at $\tilde p$ is of the form $x^{-k}dx$, where $x=f^{-1/n}$.\
[[) ]{}]{}The $a$-periods of $\rho$ are zero.
This form is defined locally up to a multiplication by an $n$-th root of unity. It follows from Lemma \[SplitIsoConnection\] that $({\mathcal O},d_X+\rho)$ is a flat section of ${\mathcal{V}}|_{z=\infty}$. This is a primitive section on the open set $\hat{\mathbf A}_k\subset\hat{\mathbf A}$, where $\rho_k$ and $\omega$ have no common zeroes. Thus it gives rise to a connection on $\hat{\mathbf A}_k$, which depends only on $k$. The metric is defined up to multiplication by a root of unity.
It is curious that if $n$ is even, then $\rho_{n/2}$ is defined up to sign, so the corresponding metric is also defined canonically.
Unfortunately, we do not know how to prove that $\hat{\mathbf A}_k$ is always non-empty. We have the following partial results:\
[[) ]{}]{}For a generic leaf ${\mathbf A}\subset{\mathbf A}_{g,n}$ all the sets $\hat{\mathbf A}_k$ are non-empty.\
[[) ]{}]{}If $n$ is large enough, then all the sets $\hat{\mathbf A}_k$ are non-empty.\
Relation with the construction of Krichever
-------------------------------------------
The above Frobenius structure is equivalent to that of [@Krichever1; @Krichever2]. To give a bridge between these papers and ours we present here a different point of view on ${\mathcal{V}}$.
Fix $m=(X,p,f_0,\Delta)\in\hat{\mathbf A}$ and fix some set of disjoint closed curves $a_i$ ($i=1,\ldots,g$) on $X$ representing $a$-cycles. Assume that $p\notin\cup a_i$. Let $\chi$ be a 1-form on $X$ with the only pole of order at most $n$ at $p$ and *jumps* $\lambda_i\omega$ along $a_i$, where $\lambda_i\in{\mathbb C}$.
For every $z\in{\mathbb P}^1$ there is a natural isomorphism between the set of such forms with jumps and ${\mathcal{V}}|_{m,z}$.
Let us first consider $z\ne\infty$. The idea is that a form with jumps can be, in some sense viewed as a Čech cocycle of ${\mathcal D^\bullet}$.
Precisely, we take a cover ${\mathcal U}_j$ of $X$ such that there is a unique $j$ with $p\in{\mathcal U}_j$. Let $(\alpha_j,s_{jk})$ be a cocycle, representing an element of ${\mathbb H}^1(X,{\mathcal D^\bullet})$ so that $\alpha_j\in\Gamma({\mathcal U}_j,\Omega(n))$, $s_{jk}\in\Gamma({\mathcal U}_j\cap{\mathcal U}_k,{\mathcal O})$. The cocycle conditions are $$\alpha_j-\alpha_k=d_Xs_{jk}+\frac\omega z s_{jk},\qquad
s_{jk}+s_{kl}+s_{lj}=0,\qquad s_{jk}=-s_{kj}.$$ Hence $s_{jk}$ represents an element of $H^1(X,{\mathcal O})$. It is easy to check that for every such a cocycle $s_{jk}$ we can find holomorphic functions $h_j$ *with constant jumps along $a_i$’s* such that $s_{jk}=h_j-h_k$. This functions are unique if we require $h_j(p)=0$, if $p\in{\mathcal U}_j$. It follows that $$\alpha_j-d_Xh_j-\frac\omega zh_j$$ patch together to a 1-form $\alpha$ with jumps along $a_i$. We leave to the reader to check that this map from the cohomology group to forms with jumps is an isomorphism.
For $z=\infty$ one gives an isomorphism by the following conditions\
[[) ]{}]{}If $\lambda_i=0$ for all $i$, then $\chi$ corresponds to $({\mathcal O},d+\chi)$.\
[[) ]{}]{}If $\chi$ has no pole at $p$, then the corresponding local system is given by the transition functions $\exp(\lambda_i)$ (in particular its $a$-monodromy is trivial). This map to the space of connections on $X$ is lifted to a map to its universal cover by requiring that $\chi=0$ goes to the trivial local system.
The reader should compare this description with Lemma \[SplitIsoConnection\]. It follows that in terms of forms with jumps $\nabla_\infty$ can be described very easily: a family is flat if $a$-periods do not change, $\lambda_i$’s do not change and the $x$-expansion of $\chi$ at $p$ does not change.
[1]{}
Babelon, Olivier; Bernard, Denis; Talon, Michel. *Introduction to classical integrable systems*. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 2003. xii+602 pp. ISBN: 0-521-82267-X.
Barannikov, Sergey; Kontsevich, Maxim. *Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields*. Internat. Math. Res. Notices 1998, no. 4, 201–215.
Dubrovin, Boris. *Geometry of 2d topological field theories*. Integrable systems and quantum groups (Montecatini Terme, 1993), 120–348, Lecture Notes in Math., 1620, Springer, Berlin, 1996.
Fedorov, Roman. *Deformations of algebro-geometric solutions of KP equations and Whitham equations*. In preparation.
Hertling, Claus; Manin, Yuri. *Unfoldings of meromorphic connections and a construction of Frobenius manifolds*. Frobenius manifolds, 113–144, Aspects Math., E36, Vieweg, Wiesbaden, 2004.
Krichever, Igor. *The $\tau$-function of the universal Whitham hierarchy, matrix models and topological field theories.* Comm. Pure Appl. Math. 47 (1994), no. 4, 437–475.
Krichever, Igor. *Baker-Akhiezer Functions and Integrable systems*. Integrability: the Seiberg-Witten and Whitham equations (Edinburgh, 1998), 1–22, Gordon and Breach, Amsterdam, 2000.
Losev, Andrey. *Hodge strings and elements of K. Saito’s theory of primitive form*. Topological field theory, primitive forms and related topics (Kyoto, 1996), 305–335, Progr. Math., 160, Birkhäuser Boston, Boston, MA, 1998.
Merkulov, Sergey. *Operads, deformation theory and $F$-manifolds*. Frobenius manifolds, 213–251, Aspects Math., E36, Vieweg, Wiesbaden, 2004.
Sabbah, Claude. *Frobenius manifolds: isomonodromic deformations and infinitesimal period mappings*. Exposition. Math. 16 (1998), no. 1, 1–57.
Saito, Kyoji. *Period mapping associated to a primitive form*. Publ. Res. Inst. Math. Sci. 19 (1983), no. 3, 1231–1264.
|
---
abstract: 'We use a model where the cosmological term can be related to the chiral gauge anomaly of a possible quantum scenario of the initial evolution of the universe. We show that this term is compatible with the Friedmann behavior of the present universe.'
author:
- 'M. Novello'
- 'J. Barcelos-Neto'
- 'J.M. Salim'
title: 'A model for time-dependent cosmological constant and its consistency with the present Friedmann universe'
---
Introduction
============
The idea of a varying cosmological constant is not new but has been systematically examined in the last two decades [@Overduin]. In particular, with the advent of the inflation mechanism and with the difficulties related to the graceful exit problem [@Kaloper], various authors were forced to consider $\Lambda$ as ground state of a scalar theory [@Ellis]. However, we may say that the first concrete mechanism for a varying $\Lambda$ was proposed by Dolgov [@Dolgov] based on a nominimally coupled scalar field. A sequence of other papers come afterwards, but all of them also based on scalar fields [@Ozer].
In a previous work [@Novello] we have presented a different model for a time dependent cosmological “constant" based on gauge fields where its origin was related to a possible quantum scenario of the initial evolution of the universe. This was achieved by showing that the chiral gauge anomaly could be conveniently adapted in order to generate a cosmological [*constant*]{}. In this way, the presence of this term today would be a reminiscence of that initial quantum behavior of the universe.
In the present paper, we are going to consider a particular scenario where the geometry is initially Bianchi-like (spatially homogeneous but anisotropic) [@Belinsky]. We show that the evolution obtained from the Einstein equations in the presence of the cosmological term (as well as the Maxwell one) leads to an asymptotic solution that is compatible with the Friedmann universe [@Weinberg; @Novello2].
Our paper is organized as follow. In Sec. II we review the general features of the model. The application is done in Sec. III. In Sec. IV we deal with the solution of the particular Einstein equations. We left Sec. V for some concluding remarks and introduce an Appendix to show some details of the calculations.
Review of the model
===================
Let us consider an action with the following general form [@Novello]
$$S_\Lambda=\int d^4x\,\sqrt{-g}\,\,Y(\cal G)
\label{2.1}$$
where $g$ is the determinant of the metric tensor and $Y$ is some function of an invariant quantity $\cal G$ which is constructed in terms of a gauge field strength $G_{\mu\nu}^{\rm a}$ and its dual $^\ast G_{\mu\nu}^{\rm a}=\frac{1}{2}\,\eta_{\mu\nu\rho\lambda}
\,G^{{\rm a}\rho\lambda}$ as
$${\cal G}=^\ast G^{a\mu\nu}G^a_{\mu\nu}
\label{2.1a}$$
We are using the following definition for $\eta_{\mu\nu\rho\lambda}$
$$\eta_{\mu\nu\rho\lambda}=\sqrt{-g}\,\epsilon_{\mu\nu\rho\lambda}
\label{2.2}$$
Consequently,
$$\eta^{\mu\nu\rho\lambda}=-\frac{1}{\sqrt{-g}}
\,\epsilon^{\mu\nu\rho\lambda}
\label{2.3}$$
where $\epsilon_{\mu\nu\rho\lambda}$ and $\epsilon^{\mu\nu\rho\lambda}$ are the usual Levi-Civita tensor densities ($\epsilon_{0123}=1$).
The variation of $S_\Lambda$ with respect the metric tensor leads to
$$\begin{aligned}
\frac{\delta S_\Lambda}{\delta g^{\mu\nu}}
&=&\int d^4x\,\biggl(\frac{\delta\sqrt{-g}}{\delta g^{\mu\nu}}\,Y
+\sqrt{-g}\,\frac{dY}{d{\cal G}}\,
\frac{\delta{\cal G}}{\delta g^{\mu\nu}}\biggr)
\nonumber\\
&=&\int d^4x\biggl(-\frac{1}{2}\,\sqrt{-g}\,g_{\mu\nu}\,Y
+\sqrt{-g}\,\frac{dY}{d{\cal G}}\,
\frac{1}{2}{\cal G}\,g_{\mu\nu}\biggr)
\nonumber\\
&=&\frac{1}{2}\int d^4x\,\sqrt{-g}\,
\Bigl(\frac{dY}{d{\cal G}}\,{\cal G}-Y\Bigr)\,g_{\mu\nu}
\label{2.4}\end{aligned}$$
We then observe that the action $S_\Lambda$ contributes to the energy-momentum tensor with a term that is proportional to the metric tensor. This is interpreted in the Einstein General Relativity theory as a spacetime dependent cosmological [*constant*]{},
$$\Lambda=\frac{dY}{d{\cal G}}\,{\cal G}-Y
\label{2.5}$$
For a question of simplicity, we shall consider the field strength of the Maxwell electromagnetic theory, where there is a natural realization of this model coming from the chiral gauge anomaly. For instance, from the path integral formalism of quantum matter and gauge fields in a classical curved background one obtains to the following effective action[@Birrell; @Novello3]
$$\begin{aligned}
S_{\rm eff}&=&\frac{\theta}{4}\int d^4x\,\sqrt{-g}\,\alpha(x)\,
\eta^{\mu\nu\rho\lambda}F_{\mu\nu}F_{\rho\lambda}
\nonumber\\
&=&\frac{\theta}{4}\int d^4x\,\sqrt{-g}\,\alpha(x)\,{\cal G}
\label{2.6}\end{aligned}$$
where $\theta$ is a constant and $\alpha(x)$ is the gauge parameter related to the chiral gauge transformation.
The manner in which the effective action above is presented it is not appopriated to be directly related to $S_\Lambda$. This is so because since it is linear in $\cal G$ and cannot generate a cosmological term $\Lambda$, as can be verified in Eq. (\[2.5\]). However, this problem can be circumvented by conveniently taking the generic function $\alpha(x)$ of the action (\[2.6\]) as ${\cal G}^p$, where $p$ is, at first, any rational number [^1]. In this way, the cosmological action $S_\Lambda$ turns to be
$$S_\Lambda=\frac{\theta}{4}\int d^4x\,\sqrt{-g}\,{\cal G}\,^{p+1}
\label{2.7}$$
We observe that for $p=-1$ we have an actual cosmological constant in the Einstein equation.
The next natural step is to consider this idea in some cosmological model in order to see the way that the cosmological term, coming from Eq. (\[2.7\]), modifies the dynamics of the Einstein equation.
Application of the model
========================
Let us start from the general action
$$\begin{aligned}
&&S=\int d^4x\,\sqrt{-g}\,\Bigl[\frac{1}{2\kappa}R
-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
\nonumber\\
&&\phantom{S=\int d^4x\,\sqrt{-g}\,\Bigl[\kappa R}
+\frac{1}{4}\theta(F_{\mu\nu}F^{\ast\mu\nu})^{p+1}
\nonumber\\
&&\phantom{S=\int d^4x\,\sqrt{-g}\,\Bigl[\kappa R}
+i\bar\psi\gamma^\mu(\nabla_\mu-ieA_\mu)\psi\Bigr]
\label{3.1}\end{aligned}$$
The equations of motion $\delta S/\delta g^{\mu\nu}=0$, $\delta
S/\delta A^\mu=0$, and $\delta S/\delta\psi=0$ lead respectively to
$$\begin{aligned}
&&G_{\mu\nu}=-F_{\mu\alpha}F^\alpha\,_\nu
-\frac{1}{4}\,g_{\mu\nu}\,F_{\alpha\beta}F^{\alpha\beta}
\nonumber\\
&&\phantom{G_{\mu\nu}=}
-\frac{p\theta}{2}g_{\mu\nu}(F_{\alpha\beta}F^{\ast\alpha\beta})^{p+1}
+T_{\mu\nu}^\psi
\label{3.2}\\
&&F^{\mu\nu}_{\phantom{\mu\nu};\nu}
-(p+1)\theta\bigl[(F_{\alpha\beta}F^{\ast\alpha\beta})^p
\bigr]_{,\nu}\,F^{\ast\mu\nu}
\nonumber\\
&&\phantom{G_{\mu\nu}}
=e\,\bar\psi\gamma^\mu\psi
\label{3.3}\\
&&\gamma^\mu\bigl(\nabla_\mu-ieA_\mu\bigr)\,\psi=0
\label{3.4}\end{aligned}$$
where $G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ is the Einstein tensor and the constant $\kappa$ of (\[3.1\]) was taken $1$. In the obtainment of Eq. (\[3.3\]) it was used the (Bianchi) identity
$$F^{\ast\mu\nu}_{\phantom{\ast\mu\nu};\nu}=0
\label{3.5}$$
We shall consider that $F^{\mu\nu}=F^{\mu\nu}(t)$. So, taking $\nu=0$ in Eq. (3.3), as well as $\mu=0$, we get
$$\begin{aligned}
&&\bar\psi\gamma^0\psi=\psi^\dagger\psi=0
\nonumber\\
&\Rightarrow&\psi=0
\label{3.6}\end{aligned}$$
From Eq. (\[3.5\]) we also have that only $F^{\ast i0}$ can be obtained. Using the Bianchi-like metric
$$ds^2=dt^2-a^2(t)dx^2-b^2(t)(dy^2+dz^2)
\label{3.7}$$
and just taking that $F^{\ast10}\neq0$, we have
$$\begin{aligned}
&&F^{\ast01}_{\phantom{\ast01};0}=0
\nonumber\\
&\Rightarrow&\sqrt{-g}\,F^{\ast01}_{\phantom{\ast01};0}=0
\nonumber\\
&\Rightarrow&\bigl(\sqrt{-g}\,F^{\ast01}\bigr)_{,0}=0
\nonumber\\
&\Rightarrow&\sqrt{-g}\,F^{\ast01}=constant
\nonumber\\
&\Rightarrow&F^{\ast01}=\frac{B_0}{ab^2}
\label{3.8}\end{aligned}$$
where, in the last step, the [*constant*]{} was identified as $B_0$ and we have used the metric given by (\[3.7\]). From this result and using Eq. (\[2.3\]) we directly infer that
$$F_{23}=-B_0
\label{3.9}$$
However, the solution for $F^{10}$ depends on $p$. In fact, taking $\nu=0$ and $\mu=1$ in Eq. (\[3.3\]) and using the results given by (\[3.6\]) and (\[3.8\]), we obtain
$$ab^2\,F^{10}+(p+1)\theta\,B_0^{p+1}\,
\Bigl(\frac{2aF^{10}}{b^2}\Bigr)^p=E_0
\label{3.10}$$
where $E_0$ was chosen to identify the constant that appears in the solution of the corresponding differential equation.
Our goal from now on is to look for if there exists some value of $p$ that renders a consistent solution for the Einstein equations (\[3.2\]) having in mind the Friedmann behavior of the present universe. This will be the subject of next section. We conclude the present section by emphasizing the importance of the Maxwell term, $-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$, into the initial action (\[3.1\]). Without it, the only possible solution of (\[3.3\]) would just be an actually constant. Indeed, taking $\psi=0$ and $\nu=
0$ in Eq. (\[3.3\]), we get
$$F^{i0}_{\phantom{i0};0}
-(p+1)\theta\bigl[(F_{\alpha\beta}F^{\ast\alpha\beta})^p\bigr]_{,0}\,
F^{\ast i0}=0
\label{3.11}$$
where the first term of the expression above comes from the Maxwell term. We then observe that without it we would obtain that $(F_{\alpha\beta}F^{\ast\alpha\beta})^p$ should be a constant for $p\neq-1$ \[for $p=-1$, the cosmological term is also a constant according to(\[2.7\])\].
Particular solution of the Einstein equation
============================================
We have seen that to avoid an actual cosmological constant and the trivial case without any cosmological term, $p$ cannot have the values zero and minus one, respectively (and the Maxwell term has to be present in the Lagrangian). In order to have a general view of the physical behavior of the Einstein solution with the value of $p$, let us initially choose $p=1$. This corresponds to one of the simplest solution of Eq. (\[3.10\]), that is
$$F^{10}=\frac{E_0b^2}{a(b^4+4\theta\,B_0^2)}
\label{4.1}$$
Combining this result with the ones given by (\[3.6\]), (\[3.8\]), and (\[3.9\]), we obtain that the Einstein equation (\[3.2\]) leads to
$$\begin{aligned}
2\,\frac{\dot a\dot b}{ab}
+\Bigl(\frac{\dot b}{b}\Bigr)^2
&=&-\,\frac{1}{2}\Bigl[\frac{B_0^2}{b^4}
+\frac{E_0^2b^4}{(b^4+2\theta B_0)^2}\Bigr]
\nonumber\\
&&+\frac{2\theta E_0^2B_0^2}{(b^4+2\theta B_0)^2}
\label{4.2}\\
2\,\frac{\ddot b}{b}
+\Bigl(\frac{\dot b}{b}\Bigr)^2
&=&-\,\frac{1}{2}\Bigl[\frac{B_0^2}{b^4}
+\frac{E_0^2b^4}{(b^4+2\theta B_0)^2}\Bigr]
\nonumber\\
&&-\frac{2\theta E_0^2B_0^2}{(b^4+2\theta B_0)^2}
\label{4.3}\\
\frac{\ddot b}{b}+\frac{\ddot a}{a}+\frac{\dot a\dot b}{ab}
&=&\frac{1}{2}\Bigl[\frac{B_0^2}{b^4}
+\frac{E_0^2b^4}{(b^4+2\theta B_0)^2}\Bigr]
\nonumber\\
&&-\frac{2\theta E_0^2B_0^2}{(b^4+2\theta B_0)^2}
\label{4.4}\end{aligned}$$
We observe that the term related to the cosmological [*constant*]{} has a behavior of $b^{-8}$ when $b$ goes to infinity (the behavior of the radiation term is with $b^{-4}$). Consequently, the cosmological term should disappear before the radiation era, what is not consistent with the Friedmann universe, where $a=b$ (notice that it is the Maxwell term which does not permit to have $a=b$ in the last two equations).
From the above results, it is not difficult to conclude that possible values of $p$ that should be compatible with this asymptotic behavior must stay between zero and minus one. However, for these values of $p$, the solution of Eq. (\[3.10\]) is not so direct as in the previous case. For example, taking $p=-1/2$, where Eq. (\[3.10\]) becomes a cubic equation with one real root that is given by (see Appendix A)
$$F^{10}=\frac{B_0}{2a}\Bigl(\frac{\theta}{2bB_0}\Bigr)^\frac{2}{3}\,
\biggl\{\biggl[1-\Bigl(1-\frac{32E_0^3}{27\theta^2B_0b^4}\Bigr)
^\frac{1}{2}
\biggr]^\frac{1}{3}
+\biggl[1+\Bigl(1-\frac{32E_0^3}{27\theta^2B_0b^4}\Bigr)^\frac{1}{2}
\biggr]^\frac{1}{3}\biggr\}^2
\label{4.5}$$
For $\theta\neq0$, the terms $32E_0^3/27\theta^2b^4B_0$ is $\ll1 $ as $b\rightarrow\infty$. Making appropriate expansions in the above relation, we obtain the following asymptotic and simpler solution for $F^{10}$,
$$F^{10}=\frac{1}{2a}\,\Bigl(\frac{\theta^2B_0}{b^2}\Bigr)^\frac{1}{3}
\label{4.6}$$
With this approximation, the Einstein equations read
$$\begin{aligned}
2\,\frac{\dot a\dot b}{ab}
+\Bigl(\frac{\dot b}{b}\Bigr)^2
&=&-\,\frac{1}{2}\Bigl[\frac{B_0^2}{b^4}
+\frac{\theta}{4b}
\Bigl(\frac{\theta B_0^2}{b}\Bigr)^\frac{1}{3}\Bigr]
\nonumber\\
&&+\frac{\theta}{4\sqrt2\,b}
\Bigl(\frac{\theta B_0^2}{b}\Bigr)^\frac{1}{3}
\label{4.7}\\
2\,\frac{\ddot b}{b}
+\Bigl(\frac{\dot b}{b}\Bigr)^2
&=&-\,\frac{1}{2}\Bigl[\frac{B_0^2}{b^4}
+\frac{\theta}{4b}
\Bigl(\frac{\theta B_0^2}{b}\Bigr)^\frac{1}{3}\Bigr]
\nonumber\\
&&-\frac{\theta}{4\sqrt2\,b}
\Bigl(\frac{\theta B_0^2}{b}\Bigr)^\frac{1}{3}
\label{4.8}\\
\frac{\ddot b}{b}+\frac{\ddot a}{a}+\frac{\dot a\dot b}{ab}
&=&\frac{1}{2}\Bigl[\frac{B_0^2}{b^4}
+\frac{\theta}{4b}
\Bigl(\frac{\theta B_0^2}{b}\Bigr)^\frac{1}{3}\Bigr]
\nonumber\\
&&-\frac{\theta}{4\sqrt2\,b}
\Bigl(\frac{\theta B_0^2}{b}\Bigr)^\frac{1}{3}
\label{4.9}\end{aligned}$$
Now, the cosmological term behaves as $b^{-\frac{4}{3}}$, that decay slower than the previous $b^{-4}$ of the radiation term. However, there is also a term with the behavior of $b^{-\frac{4}{3}}$ in the Maxwell counterpart and this last term avoids the obtainment of a Friedmann behavior we are looking for.
From this analysis, we see that the solution of (\[3.10\]) leads to a behavior for the cosmological term that is lower than $b^{-4}$ for $-1<p<0$. However, there is also a term with this behavior in the Maxwell counterpart. What may happen is that, depending on the value of $p$, one term may dominate relatively to the other. To see if this actually occurs we need to know the solution of (\[3.10\]) for a general $p$ (between zero and minus one). Of course, this is not an easy task. However, we observe that the asymptotic solution for $F^{10}$ given by (\[4.6\]) could have been directly inferred from (\[3.10\]) by discarding the constant term $E_0$. It is then easily seen that any asymptotic solution for any $p$ between zero and minus one can be obtained in the same way. The result is
$$F^{10}=\frac{1}{2a}\biggl[-2(1+p)\theta\,
\Bigl(\frac{B_0}{b^2}\Bigr)^{1+p}\biggr]^\frac{1}{1-p}
\label{4.10}$$
which is in agreement with (\[4.6\]) if one takes $p=-1/2$. We should not worry about the minus sign inside the $\frac{1}{1-p}$-root of equation above because $F^{10}$ always appears squared in the calculations of significant quantities. Now, the obtainment of the Einstein equations is just a matter of algebraic work. The result is (we conveniently put the two terms with the same behavior for higher $b$ together)
$$\begin{aligned}
&&2\,\frac{\dot a\dot b}{ab}
+\Bigl(\frac{\dot b}{b}\Bigr)^2
=-\frac{B_0^2}{2b^4}
-\Bigl(\frac{p2^{-p}}{1+p}+1\Bigr)\,
\biggl[2^\frac{3p-1}{2}(1+p)\theta\,
\Bigl(\frac{B_0}{b^2}\Bigr)^{1+p}\biggr]^\frac{2}{1-p}
\label{4.11}\\
&&2\,\frac{\ddot b}{b}
+\Bigl(\frac{\dot b}{b}\Bigr)^2
=-\frac{B_0^2}{2b^4}
-\Bigl(\frac{p2^{-p}}{1+p}+1\Bigr)\,
\biggl[2^\frac{3p-1}{2}(1+p)\theta\,
\Bigl(\frac{B_0}{b^2}\Bigr)^{1+p}\biggr]^\frac{2}{1-p}
\label{4.12}\\
&&\frac{\ddot b}{b}+\frac{\ddot a}{a}+\frac{\dot a\dot b}{ab}
=\frac{B_0^2}{2b^4}
-\Bigl(\frac{p2^{-p}}{1+p}-1\Bigr)\,
\biggl[2^\frac{3p-1}{2}(1+p)\theta\,
\Bigl(\frac{B_0}{b^2}\Bigr)^{1+p}\biggr]^\frac{2}{1-p}
\label{4.13}\\\end{aligned}$$
where in the terms that appear $(\frac{p2^{-p}}{1+p}+1)$ or $(\frac{p2^{-p}}{1+p}-1)$, the first part comes from the cosmological term and the other one from the Maxwell counterpart. We then observe that as $p$ is closed to minus one the cosmological term is more and more dominant and, consequently, the solution tends more and more to the Friedmann scenario. In the limit case of $p=-1$, the Maxwell term disappears, and just remains an actual cosmological constant as it had already been pointed out in the beginning.
Conclusion
==========
In this paper we have analyzed further the recent proposal[@Novello] of a cosmological scenario in which the cosmological “constant" is spacetime dependent and whose origin is related to a primordial era, supposed dominated by quantum effects.
In order to see if this model is actually compatible with the observable universe, where the anisotropy rate is considerable low \[like Friedmann-Robertson-Walker (FRD)\], we leave a parameter free ($p$) in the theory. We conclude that it can vary from minus one to zero, where these limits mean an actual cosmological constant and no cosmological term, respectively. We have show that as $p$ is closed to minus one as the solution is compatible with the Friedmann scenario.
A next natural step in this research line is to look for the solution of the Einstein equations (\[4.11\]) - (\[4.13\]). We are presently work in this problem and possible results shall be reported elsewhere [@Novello4].
This work is supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq (Brazilian Research Agency). One of us, J.B.-N. has also the support of PRONEX 66.2002/1998-9.
Solution of (\[3.10\]) for p=-1/2
=================================
The Eq. (\[3.10\]) for $p=-1/2$ reads
$$ab^2\,F^{10}+\frac{\theta b}{2}\,\sqrt\frac{B_0}{2aF^{10}}=E_0
\label{A.1}$$
This expression can be written in a cubic equation whose general form is
$$u^3+a_1u^2+a_2u+a_3=0
\label{A.2}$$
where
$$\begin{aligned}
u&=&\sqrt\frac{2aF^{10}}{B_0}
\nonumber\\
a_1&=&0
\nonumber\\
a_2&=&-\,\frac{2E_0}{B_0b^2}
\nonumber\\
a_3&=&\frac{\theta}{B_0b}
\label{A.3}\end{aligned}$$
From the discriminant relation
$$D=Q^3+R^2
\label{A.4}$$
where
$$\begin{aligned}
&&Q=\frac{3a_2-a_1^2}{9}=-\frac{2E_0}{3B_0b^2}
\label{A.5}\\
&&R=\frac{9a_1a_2-27a_3-2a_1^3}{54}=-\frac{\theta}{2B_0b}
\label{A.6}\end{aligned}$$
we obtain
$$D=-\frac{8E_0^3}{27B_0^3b^6}+\frac{\theta^2}{4B_0^2b^2}
\label{A.7}$$
We observe that in the region for higher $b$, $D$ is positive (for $\theta\neq0$). Consequently, just one root is real for this region and it is given by
$$u=S+T-\frac{a_1}{3}
\label{A.8}$$
where
$$\begin{aligned}
S&=&\sqrt[3]{R+\sqrt D}
\nonumber\\
T&=&\sqrt[3]{R-\sqrt D}
\label{A.9}\end{aligned}$$
The combination of (\[A.6\]) - (\[A.9\]) leads to the solution given by (\[4.5\]).
[100]{} For a detailed study concerning the cosmological constant as well as the possibility of its dependence on time see J.M. Overduin and F.I. Cooperstock, Phys. Rev. [**D58**]{} (1998) 043506 and references therein. N. Kaloper, R. Madden, and K.A. Olive, Nucl. Phys. [**B452**]{} (1995) 677; Phys. Lett. [**B371**]{} (1996) 34; R. Brustein and R. Madden, Phys. Lett. [**B410**]{} (1997) 110 See for example G.F.R. Ellis, D.C. Roberts, D. Solomons, and P.K.S. Dunsby, Phys. Rev. [**D62**]{} (2000) 084004 and references therein. See A.D. Dolgov, in [*The very early universe*]{}, edited by G.W. Gibbons, S.W. Hawking, and S.T.C. Siklos (Cambridge University Press, Cambridge, England, 1983) and references therein. We mention, among others, M. Özer and M.O. Taha, Phys. Lett. [**B171**]{} (1986) 363; O. Bertolami, Nuovo Cimento [**B93**]{} (1986) 36; M.O. Calvão, H.P. Oliveira, D. Pavón, and J.M. Salim, Phys. Rev. [**D45**]{} (a992) 3869. M. Novello, J. Barcelos-Neto and J.M. Salim, Class. Quantum Grav. [**18**]{} (2001) 1261. See, for example, V.A. Belinsky, E.M. Lifshitz, and I.M. Khalatnikov, Adv. Phys. [**19**]{} (1970) 525, and references therein. S. Weinberg, Rev. Mod. Phys. [**61**]{} (1989) 1. For other isotropization mechanisms, see M. Novello and S.L.S. Duque, Physica [**A168**]{} (1990) 1073, and references therein. N.D. Birrell and P.C.W. Davies, [*Quantum fields in curved space*]{}, Cambridge (1982). See reference [@Novello] for details. M. Novello, J. Barcelos-Neto and J.M. Salim, work in progress.
[^1]: The way of circumventing this problem here is slightly different of the original paper [@Novello]. There, we have redefined the gauge field in order to incorporate a nonlinearity of $\cal G$.
|
---
abstract: 'We study deviations from the perturbative asymptotic behaviour in the running QCD coupling by analysing non-perturbative measurements of ${\alpha_s}(p)$ at low momenta ($p \approx 2 \ {{\rm GeV}}$) as obtained from the lattice three-gluon vertex. Our exploratory study provides some evidence for power corrections to the perturbative running proportional to $1/p^2$.'
address:
- 'Dipartimento di Fisica, Università di Parma and INFN, Gruppo Collegato di Parma, Parma, Italy'
- |
Dept. of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, U.K.\
(UKQCD Collaboration)
- 'Institut de Physique Nucléaire Théorique, Université de Liège au Sart Tilmann, B-4000 Liège, Belgique'
author:
- 'G. Burgio, F. Di Renzo$^{\rm b}$, C. Parrinello, C. Pittori'
title: 'The running QCD coupling in the pre-asymptotic region'
---
\#1\#2\#3[Ann. Phys. (NY) \#1 (19\#3) \#2]{} \#1\#2\#3[Ann. Rev. Nucl. Part. Sci. \#1 (19\#3) \#2]{} \#1[Cambridge preprint Cavendish–HEP–\#1]{} \#1\#2\#3[Computer Phys. Comm. \#1 (19\#3) \#2]{} \#1\#2\#3[ibid. \#1 (19\#3) \#2]{} [Nucl.Phys. ]{}\#1\#2\#3[Nucl. Phys. B\#1 (19\#3) \#2]{} [Phys.Lett. ]{}\#1\#2\#3[Phys. Lett. \#1B (19\#3) \#2]{} [Phys.Rev. ]{}\#1\#2\#3[Phys. Rev. D \#1 (19\#3) \#2]{} \#1\#2\#3[Phys. Rep. \#1 (19\#3) \#2]{} \#1\#2\#3[Phys. Rev. Lett. \#1 (19\#3) \#2]{} \#1\#2\#3[Rev. Mod. Phys. \#1 (19\#3) \#2]{} \#1\#2\#3[Sov. J. Nucl. Phys. \#1 (19\#3) \#2]{} \#1\#2\#3[Zeit. Phys. C\#1 (19\#3) \#2]{}
INTRODUCTION {#sec:intro}
============
The standard procedure to parametrise non-perturbative QCD effects in terms of power corrections to perturbative results is based on the Operator Product Expansion (OPE). In this framework, the powers involved in the expansion are uniquely fixed by the symmetries and the dimension of the relevant operator product. The above picture has recently been challenged [@Akhoury; @etc; @Ceccobeppe], when it was pointed out that power corrections which are not [*a priori*]{} expected from OPE may in fact appear in physical observables. Such terms may arise from (UV-subleading) power corrections to ${\alpha_s}(p)$, corresponding to non-analytical contributions to the $\beta$-function. Clearly, the existence of OPE-independent power corrections, if demonstrated, would have a major impact on our understanding of non-perturbative QCD effects and would affect QCD predictions for several processes. For example, $\frac{\Lambda^2}{p^2}$ contributions may be relevant for the analysis of $\tau$ decays [@alt; @Akhoury].
It would be highly desirable to develop a theoretical framework where the occurrence of these effects is demonstrated and estimates are obtained from first principles QCD calculations. The results in [@Lepage; @Ceccobeppe] can be considered as a first step in this direction: some evidence for an unexpected $\frac{\Lambda^2}{Q^2}$ contribution to the gluon condensate was found by means of lattice calculations.
The aim of the present work is to test a method to detect the presence of power corrections in the running QCD coupling. Non-perturbative lattice estimates of the coupling at low momenta are compared with perturbative formulae. The final goal is to investigate the conjecture that OPE-independent power corrections to physical observables are linked to power terms in the running coupling. Although at this stage our work is exploratory in nature and further simulations will be required to obtain a conclusive answer, our analysis provides some preliminary evidence for power corrections to ${\alpha_s}(p)$ for a particular definition of the coupling.
The paper is organised as follows: in Section \[sec:relev\] we briefly review some theoretical arguments in support of power corrections to ${\alpha_s}(p)$, illustrating the special role that may be played by $\frac{\Lambda^2}{p^2}$ terms. In Section \[sec:lattice\] we analyse the lattice data and present some preliminary evidence for power corrections. Finally, in Section \[sec:conc\] we draw our conclusions.
WHY POWER CORRECTIONS?
======================
\[sec:relev\]
Power corrections to ${\alpha_s}(p)$ can be shown to arise naturally in many physical schemes [@pino; @maclep]. Such corrections cannot be excluded [*a priori*]{} in any renormalisation scheme. Clearly, the non-perturbative nature of such effects makes it very hard to assess their dependence on the renormalisation scheme. A term of order ${\Lambda^2}/{p^2}$ is a strong candidate for a power correction to ${\alpha_s}(p)$. To see why, consider the interaction of two heavy quarks in the static limit and in the one-gluon-exchange approximation (for a more detailed discussion see [@zakEQ]). The static potential $V(r) $ can be written as $$\protect\label{HQP}
V(r) \, \propto \, \alpha_s \ \int d^3p \,
\frac{\exp^{i \vec{p} \cdot \vec{r}}}{|\vec{p}|^2}.$$ If one inserts in the above formula a running coupling of the form $\alpha_s(p^2) \approx {\Lambda^2}/{p^2}$, this results in a linearly confining potential. Similarly, consider the “force" definition of the running coupling: $$\alpha_{q\bar{q}}(Q) = \frac{3}{4} r^2 \frac{dV}{dr} \;\;\;\;\;
(Q = \frac{1}{r}),$$ where again $V(r)$ represents the static interquark potential. A linear confinement term in $V(r)$ generates a $1/Q^2$ contribution to the coupling, whose order of magnitude is given by the string tension. This can be interpreted as a clue for the existence of a $\frac{\Lambda^2}{p^2}$ contribution, providing an estimate for its expected order of magnitude, at least in one (physically sound) scheme. Finally, power corrections to ${\alpha_s}(p)$ also emerge if one assumes that the singularities appearing in the perturbative formulae for the running coupling are “removed" by non-perturbative effects [@RedBog].
ANALYSIS AND RESULTS
====================
\[sec:lattice\]
We shall compare non-perturbative lattice data for ${\alpha_s}(p)$ with simple models where a power correction term is added to the perturbative formula at a given order. The first problem is the possible interplay between power corrections and our ignorance about higher orders of perturbation theory. In particular, for the scheme that we will consider, the three-loop coefficient of the $\beta$-function is not known. Knowledge of such a coefficient would allow a more reliable comparison of our estimates for the $\Lambda$ parameter in our scheme with lattice determinations of $\Lambda$ in a different scheme, for which the three-loop result is available [@Lusch]. In fact, although matching the $\Lambda$ parameter between different schemes only requires a one-loop computation, the reliability of such a comparison rests on the assumption that the value of $\Lambda$ in each scheme is fairly stable with respect to the inclusion of higher order terms in the definition of $\Lambda$. In practice, when working at two- or three-loop order, the value of $\Lambda$ is still quite sensitive to the order of the calculation. Even within such limitations, we will argue that it is possible to estimate the impact of three-loop effects and that a description with power corrections seems relevant even at that order.
Choice of the coupling
----------------------
We need to measure ${\alpha_s}(p)$ at low momenta (where power-like terms may be sizeable) and in a relatively wide momentum range. For this purpose, the best method is one where ${\alpha_s}(p)$ can be measured for several momentum values from a single Monte Carlo data set. One suitable method is the determination of the coupling from the renormalised lattice three-gluon vertex function [@io; @cpcp]. By varying the renormalisation scale $p$, one can determine ${\alpha_s}(p)$ for different momenta from a single simulation. Obviously the renormalisation scale must be chosen in a range such that finite volume effects and discretisation errors are both under control. The numerical results for ${\alpha_s}(p)$ used in this work were obtained by applying such a method on a sample of 150 Monte Carlo configurations on a $16^4$ lattice at $\beta=6.0$. The calculation was performed in the Landau gauge. For full details of the method we refer the reader to Ref. [@cpcp], where such results were first presented. In order to detect violations of rotational invariance, different combinations of lattice vectors have sometimes been used for a fixed value of $p^2$, which accounts for the graphical “splitting" of some data points.
Two-loop analysis
-----------------
At the two-loop level, we consider the following formula:
$$\begin{aligned}
\protect\label{2lp}
\alpha_s(p) \, = \, \frac{1}{b_0 \, \log(p^2/\Lambda_{2l}^2)} \, - \,
\frac{b_1}{b_0}
\frac{\log(\log(p^2/\Lambda_{2l}^2))}{(b_0 \,
\log(p^2/\Lambda_{2l}^2))^2}
\, \nonumber \\ + \, c_{2l} \, \frac{\Lambda_{2l}^2}{p^2}
\qquad \qquad \qquad \qquad \qquad \qquad \end{aligned}$$
By fitting the data to (\[2lp\]) we obtain two estimates for ($\Lambda_{2l}$,$c_{2l}$), namely ($0.84(1)$,$0.31(3)$) and ($0.73(1)$,$0.99(7)$), with comparable values for $\chi^2_{dof} \leq 1.8$. The momentum range for the fit corresponds to $p \sim 1.8 - 3$ [[GeV]{}]{}. We take the first set of values as our best estimate of the parameters as the corresponding value of $\Lambda_{2l}$ is close to what is obtained from a “pure" two-loop fit, i.e. $\Lambda_{2l}$ is stable with respect to the introduction of power corrections. This choice will be supported also by independent considerations at the three-loop level. In summary, a two-loop description with power corrections based on (\[2lp\]) fits well the data in a consistent momentum range. Our best fit is shown in Figure 1.
\[fig:2loopot\]
Three-loop analysis
-------------------
A major obstacle for a three-loop analysis is our ignorance of the first non-universal coefficient $b_2$ of the perturbative $\beta$-function. In order to gain insight, we perform a two-parameter fit to the “pure" three-loop formula, taking $\Lambda_{3l}$ and the unknown coefficient $b_2$ as the fitting parameters. We call $b_2^{eff}$ the fit estimate for $b_2$. We obtain $\Lambda_{3l} =
0.72(1)$, $b_2^{eff} = 1.3(1)$, with $\chi^2_{dof} \approx 1.8$ (see dashed curve in Fig. 2). The momentum range where we obtain the best description of the data is $p \sim 2 - 3$ [[GeV]{}]{}. Our result for $\Lambda_{3l}$ provides (via perturbative matching) an estimate for $\Lambda_{{\overline{MS}}}$ which is in very good agreement with the estimate in [@Lusch], which was obtained from the computation of the $\Lambda$ parameter in a completely different scheme. Although our estimate depends on the extra parameter $b_2^{eff}$, the agreement between the two results is remarkable.
So far, the success of the “pure" three-loop fit suggests that the power term in the two-loop formula merely provides an effective description of three-loop effects. However, it turns out that there is room for a power correction even at the three-loop level. To see this, we consider a three-loop formula with a power correction:
$$\begin{aligned}
\protect\label{3lp}
\alpha_s(p) = \frac{1}{b_0 \, L} \, - \,
\frac{b_1}{b_0}
\frac{\log(L)}{(b_0 \, L)^2} \qquad \qquad \qquad \nonumber \\
+ \, \frac{1}{(b_0 \, L)^3} \,
\left( \frac{b_2^{eff}}{b_0} + \frac{b_1^2}{b_0^2}
(\log^2(L) - \log(L) + 1 ) \right) \nonumber \\
+ \, c_{3l} \, \frac{\Lambda_{3l}^2}{p^2}, \qquad \qquad
\qquad \qquad \qquad \qquad \end{aligned}$$
\[fig:3loopot\]
where $L = \log(p^2/\Lambda_{3l}^2)$ and $b_2^{eff}$ is again to be determined from a fit. Fitting the data to (\[3lp\]), we obtain $\Lambda_{3l} = 0.72(1)$, $b_2^{eff} = 1.0(1)$ and $c_3 = 0.41(2)$, with $\chi^2_{dof} \approx 1.8$, in a momentum range $1.8 \, {{\rm GeV}}< p < 3 \, {{\rm GeV}}$ (see Fig. 2). We note that the value for $\Lambda_{3l}$ is fully consistent with the previous determination from the “pure" three-loop description. The value for $b_2^{eff}$ is also reasonably stable with respect to the previous determination. By comparing results from fits to (\[2lp\]) and (\[3lp\]), it emerges that $$c_2 \Lambda_{2l}^2 = 0.22(2) \, {{\rm GeV}}^2 \sim c_3 \Lambda_{3l}^2 = 0.21(2)
\, {{\rm GeV}}^2.
\protect\label{eq:tuttotiene}$$ In other words, the power terms providing the best fit to (\[2lp\]) and (\[3lp\]) are numerically the same, so that there seems to be no interplay between the indetermination connected to the perturbative terms and the power correction term, within the precision of our data. We take this fact as an indication that a description in terms of power corrections is still relevant at the three-loop level. Notice that the numerical value of the power correction is comparable to the standard estimate for the string tension.
One could object that at the two-loop level we had chosen between two sets of values for ($\Lambda_{2l}$,$c_{2l}$), and that our choice is crucial for the validity of (\[eq:tuttotiene\]). An [*a posteriori*]{} justification for our choice is obtained from the following test: we plot a few values for ${\alpha_s}(p)$ as generated by the “pure” three-loop formula for $\Lambda_{3l}= 0.72$ and $b_2 = 1.0$. Then, by fitting such points to the “pure” two-loop formula, one gets $\Lambda_{2l}
\approx 0.84$, i.e. the value for which (\[eq:tuttotiene\]) holds. Again, the above test seems to confirm that perturbative and non-perturbative contributions do not mix in our formulae when upgrading from a two-loop to a three-loop description, thus suggesting that a genuine $\frac{\Lambda^2}{p^2}$ correction is present in the data.
CONCLUSIONS
===========
\[sec:conc\]
We have discussed an exploratory investigation of power corrections in the running QCD coupling ${\alpha_s}(p)$ by comparing non-perturbative lattice results with theoretical models. Some evidence was found for $1/p^2$ corrections, whose size would be consistent with what is suggested by simple arguments from the static potential.
Our results need to be further tested by the analysis of a larger data set and by a study of the dependence of the fit coefficient on the ultraviolet and infrared lattice cutoff. A very delicate issue is the assessment of the scheme dependendence of our results. In particular, we note that the definition of the coupling that we adopted is [*a priori*]{} gauge-dependent. This point will be the focus of our future work.
ACKNOWLEDGEMENTS
================
We thank B. Alles, H. Panagopoulos and D. G. Richards for allowing us to use data files containing the results of Ref. [@cpcp]. C. Parrinello acknowledges the support of PPARC through an Advanced Fellowship. C. Pittori thanks J. Cugnon and the “Institut de Physique de l’Université de Liège au Sart Tilman" and acknowledges the partial support of IISN. We thank C. Michael for stimulating discussions.
[9]{} R. Akhoury and V.I. Zakharov, hep-ph/9705318. G.Grunberg, hep-ph/9705290, hep-ph/9705460. G Burgio, F. Di Renzo, G. Marchesini and E. Onofri, [Phys.Lett. ]{}[422]{}[219]{}[98]{}. G. Altarelli, et al, . G.P. Lepage and P. Mackenzie, Nucl. Phys. Proc. Suppl. 20 (1991) 173. Yu.L. Dokshitzer, G. Marchesini and B.R. Webber, [Nucl.Phys. ]{}[469]{}[93]{}[96]{} S.J. Brodsky, G.P. Lepage and P.B. Mackenzie, [Phys.Rev. ]{}[28]{} [228]{} [83]{}. R. Akhoury and V.I. Zakharov, hep-ph/9710487. See, for example, P. Redmond, [Phys.Rev. ]{}[112]{}[1404]{}[58]{}. S. Capitani et al, Nucl. Phys. Proc. Suppl. 63 (1998) 153. C. Parrinello, Phys. Rev. D50 (1994) 4247. B. Allés et al, [Nucl.Phys. ]{}[502]{}[325]{}[97]{}
|
---
abstract: 'We present an alternative proof of Perron’s theorem, which is probabilistic in nature. It rests on the representation of the Perron eigenvector as a functional of the trajectory of an auxiliary Markov chain.'
author:
- |
Raphaël Cerf Joseba Dalmau\
[DMA, [É]{}cole Normale Supérieure CMAP, Ecole Polytechnique]{}
bibliography:
- 'pamm.bib'
title: 'A probabilistic proof of Perron’s theorem'
---
In 1907, Oskar Perron proved the following theorem.
Let $A$ be a square matrix with posivite entries. Then the matrix $A$ admits a positive eigenvalue $\lambda$ such that: i) to $\lambda$ is associated an eigenvector $\mu$ whose components are all positive;
ii\) if $\alpha$ is another eigenvalue of $A$, possibly complex, then $|\alpha|< \lambda$;
iii\) any other eigenvector associated to $\lambda$ is a multiple of $\mu$.
This theorem was subsequently generalized by Frobenius in his work on non–negative matrices in 1912, leading to the so–called Perron–Frobenius theorem [@SE]. A myriad of mathematical models involve non–negative matrices and their powers, thereby calling for the use of the Perron–Frobenius theorem. Mathematicians have developped generalizations in several directions, notably in infinite dimensions (for infinite matrices [@VJ], for non–negative kernels in arbitrary spaces [@AN]) and a whole Perron–Frobenius theory has emerged. Hawkins wrote an historical account on the initial developpement of this theory [@HA]. MacCluer [@MC] describes several applications of Perron’s theorem and review the different proofs that have been found over the years. The original proof of Perron rested on an induction over the size the matrix. A few years later Perron found a proof involving the resolvent of the matrix. A nowadays popular proof, which is found in most textbooks, is due to Wielandt and it rests on a miraculous max–min functional.
We present here an alternative proof of Perron’s theorem, which is probabilistic in nature. It rests on an auxiliary Markov chain, and the representation of the Perron eigenvector as a functional of the trajectory of this Markov chain [@CD]. This formula generalizes the well–known formula for the invariant probability measure of a finite state Markov chain. To ease the exposition, we restrict ourselves to the Perron theorem, and we work with matrices whose entries are all positive. However our proof can be readily extended to primitive matrices, thereby yielding the classical Perron–Frobenius theorem. Our proof might seem lengthy compared to other proofs, yet it is completely self–contained and it requires only classical results of basic algebra and power series.
We introduce next some notation in order to define the auxiliary Markov chain. Let $d$ be a positive integer. Throughout the text, we consider a square matrix $A=(A(i,j))_{1\leq i,j\leq d}$ of size $d\times d$ with positive entries. For $i\in\{\,1,\dots,d\,\}$, we denote by $S(i)$ the sum of the entries on the $i$–th row of $A$, i.e., $$\forall i\in
\{\,1,\dots,d\,\}\qquad S(i)\,=\,\sum_{j=1}^{d}A(i,j)\,,$$ and we create a new matrix $M=(M(i,j))_{1\leq i,j\leq d}$ by setting $$\forall i,j\in
\{\,1,\dots,d\,\}\qquad
M(i,j)\,=\,\frac{A(i,j)}{S(i)}\,.$$ Obviously, the sum of each row of $M$ is now equal to one, i.e., $M$ is stochastic, and we think of it as the transition matrix of a Markov chain. So, let $(X_n)_{n\in\N}$ be a Markov chain with state space $\lbrace\,1,\dots,d\,\rbrace$ and transition matrix $M$. Let us fix $i\in\{\,1,\dots,d\,\}$. We denote by $E_i$ the expectation of the Markov chain issued from $i$ and we introduce the time $\tau_i$ of the first return of the chain to $i$, defined by $$\tau_i\,=\,\inf\,\big\{\,n\geq 1:X_n=i\,\big\}\,.$$ Finally, we define a function $\phi_i$ by setting $$\forall \lambda\geq 0\qquad
\phi_i(\lambda)\,=\,
E_i\Bigg(\lambda^{-\tau_i}\prod_{n=0}^{\tau_i-1}S(X_n)
\Bigg)
\,.$$ The quantity in the expectation is non–negative, so the function $\phi_i$ is well defined and it might take infinite values.
\[regf\] The function $\phi_i$ is continuous, decreasing on $\R^+$ and $$\lim_{\lambda\to 0}\phi_i(\lambda)\,=\,+\infty\,,\qquad
\lim_{\lambda\to +\infty}\phi_i(\lambda)\,=\,0\,.$$
In fact, the function $\phi_i$ can be written as a power series in the variable $1/\lambda$, as follows: $$\begin{gathered}
\phi_i(\lambda)\,=\,
\sum_{k=1}^{\infty}\frac{1}{\lambda^k}
E_i\Bigg(
\Big(
1_{\{\tau_i=k\}}
\prod_{n=0}^{k-1}S(X_n)\Big)\Bigg)\,=\,\cr
\sum_{k=1}^{\infty}\frac{1}{\lambda^k}
\kern-3pt
\sum_{i_1,\dots,i_{k-1}\neq i}
\kern-7pt
S(i)S(i_1)\cdots S(i_{k-1})
P\big(X_1=i_1,\dots,X_{k-1}=i_{k-1},X_k=i\big)\cr
\,=\,
\sum_{k=1}^{\infty}\frac{1}{\lambda^k}
\sum_{i_1,\dots,i_{k-1}\neq i}
S(i)M(i,i_1)\cdots S(i_{k-1})M(i_{k-1},i)\cr
\,=\,
\sum_{k=1}^{\infty}\frac{1}{\lambda^k}
\sum_{i_1,\dots,i_{k-1}\neq i}
A(i,i_1)\cdots A(i_{k-1},i)\,.\end{gathered}$$ Since $A$ has positive entries, the series contains non vanishing terms, and this implies that $\phi_i$ is decreasing and tends to $\infty$ as $\lambda$ goes to $0$. Let $R$ be the radius of the convergence circle of this series. From classical results on powers series, we know that $\phi_i(\lambda)$ is continuous for $\lambda>R$. To prove that $\phi_i$ is continuous, we have to show that $\phi_i(R)=+\infty$. Let $B$ be the matrix obtained from $A$ by removing the $i$–th row and the $i$–th column and let $\gamma_1,\dots,\gamma_{d-1}$ be its eigenvalues (possibly complex), arranged so that $|\gamma_1|\geq\cdots
\geq |\gamma_d|$. Let $m$ (respectively $M$) be the minimum (respectively the maximum) of the entries of $A$. For any $k\geq 1$, we have $$\begin{gathered}
\sum_{i_1,\dots,i_{k-1}\neq i}
A(i,i_1)\cdots A(i_{k-1},i)\,\geq\,
\frac{m^2}{M}
\sum_{i_1,\dots,i_{k-1}\neq i}
A(i_1,i_2)\cdots A(i_{k-1},i_1)
\cr
\,=\,
\frac{m^2}{M} \text{trace}(B^k)
\,=\,
\frac{m^2}{M} \Big(\gamma_1^k+\cdots+\gamma_{d-1}^k\Big)
\,.\end{gathered}$$ Although the eigenvalues $\gamma_1,\dots,\gamma_{d-1}$ might be complex numbers, the trace of $B^k$ is a positive real number. We can also a prove a similar inequality in the reverse direction, and we conclude that the power series defining $\phi_i$ converges if and only if the series $$\sum_{k=1}^{\infty}\frac{1}{\lambda^k}
\Big(\gamma_1^k+\cdots+\gamma_{d-1}^k\Big)$$ converges. This is certainly the case if $|\lambda|>|\gamma_1|$, therefore $R\leq|\gamma_1|$. Let us define, for $n\geq 1$, $$S_n(\lambda)\,=\,
\sum_{k=1}^{n}\frac{1}{\lambda^k}
\Big(\gamma_1^k+\cdots+\gamma_{d-1}^k\Big)\,.$$ We shall rely on the following result on geometric series.
\[geo\] Let $z$ be a complex number such that $|z|\leq 1$. Then $$\lim_{n\to\infty}\,\,
\frac{1}{n}\big({z+\cdots+z^n}\big)\,=\,
\begin{cases}
\quad0&\quad\text{if}\quad z\neq 1\,,\\
\quad1&\quad\text{if}\quad z=1\,.
\end{cases}$$
For $z=1$, the result is obvious. For $z\neq 1$, we compute $$\frac{1}{n}\big({z+\cdots+z^n}\big)\,=\,
\frac{z-z^{n+1}}{n(1-z)}\,,$$ and we observe that this quantity goes to $0$ when $n$ goes to $\infty$.
Lemma \[geo\] implies that, for $\lambda$ a complex number such that $|\lambda|=|\gamma_1|$, $$\lim_{n\to\infty}\,\,
\frac{1}{n}S_n(\lambda)\,=\,
\card\big\{\,j:1\leq j\leq d,\,\lambda=\gamma_j\,\big\}\,.$$ This implies in particular that $\big|S_n(\gamma_1)\big|$ goes to $\infty$ with $n$. Observing that $\big|S_n(\gamma_1)\big|
\leq
S_n(|\gamma_1|)$, we conclude that $$\phi_i(|\gamma_1|)\,=\,\lim_{n\to\infty}\,\,
S_n(|\gamma_1|)\,=\,+\infty\,.$$ Therefore $R=|\gamma_1|$ and moreover $\phi_i(R)=+\infty$.
Proposition \[regf\] implies that $\phi_i$ is one to one from $]R,+\infty[$ onto $]0,+\infty[$, thus there exists a unique positive real number $\lambda_i$ such that $\phi_i(\lambda_i)=1$. The next result is the key to our proof of the Perron–Frobenius theorem. We define a vector $\mu_i$ by setting $$j{1,…,d}\_i(j)= E\_i(\_[n=0]{}\^[\_i-1]{}( 1\_[{X\_n=j}]{} \_i\^[-n]{}\_[k=0]{}\^[n-1]{}S(X\_k) ) ) . $$
The value $\lambda_i$ is an eigenvalue of $A$ and the vector $\mu_i$ is an associated left eigenvector whose components are all positive and finite.
Let us note $E_i,\tau_i,\lambda_i,\mu_i$ simply by $E,\tau,\lambda,\mu$. Let us compute $$\begin{aligned}
\sum_{j=1}^d
\mu(j) &A(j,k)\,=\,
\sum_{j=1}^d
\mu(j) S(j)M(j,k)\cr
&\,{=}\,\sum_{j=1}^d\sum_{n\geq 0}
E\Bigg(1_{\{\tau>n\}}
\lambda^{-n}
\Big(
\prod_{t=0}^{n-1}S(X_t)
\Big)
1_{\{X_n=j\}}
f(j)M(j,k)
\Bigg)\cr
&=\,\sum_{j=1}^d\sum_{n\geq 0}
E\Bigg(1_{\{\tau>n\}}
\lambda^{-n}
\Big(
\prod_{t=0}^{n}S(X_t) \Big)
1_{\{X_n=j\}}
1_{\{X_{n+1}=k\}}
\Bigg)\cr
&=\,
E\Bigg(\sum_{n=0}^{\tau-1}
1_{\{X_{n+1}=k\}}
\lambda^{-n}
\Big(
\prod_{t=0}^{n}S(X_t)
\Big)
\Bigg)\cr
&=\,
\lambda\,E\Bigg(\sum_{n=1}^{\tau}
1_{\{X_{n}=k\}}
\lambda^{-n}
\Big(
\prod_{t=0}^{n-1}S(X_t)
\Big)
\Bigg)\,.\end{aligned}$$ Suppose that $k\neq i$. Then the term in the last sum vanishes for $n=0$ or $n=\tau$, and we obtain $$\sum_{j=1}^d
\mu(j) A(j,k)\,=\,
\lambda \mu(k)\,.$$ For $k=i$, we obtain, noticing that $\mu(i)=1$, $$\sum_{j=1}^d
\mu(j) A(j,i)\,=\,
\lambda\,E\Bigg(
\lambda^{-\tau}
\prod_{t=0}^{\tau-1}S(X_t)
\Bigg)
\,=\,
\lambda\,\phi_i(\lambda)\,=\,
\lambda\,\mu(i)\,.$$ Thus we have proved that $\mu A=\lambda\mu$. Since $\mu(i)=1$, these equations imply that $\mu(1),\dots,\mu(d)$ are all positive and finite.
\[regu\] Let $\alpha$ be an eigenvalue of $A$, possibly complex, and let $\nu$ be an associated left eigenvector. Let $i \in\{\,1,\dots,d\,\}$ be such that $\nu(i)\neq 0$. Either $\nu$ and $\mu_i$ are proportional (in which case $\alpha=\lambda_i$) or $|\alpha|<\lambda_i$.
Let $\alpha,\nu$ and $i$ be as in the statement of the proposition. We suppose that $\alpha\neq 0$, otherwise there is nothing to prove. Let $\nu$ be an associated left eigenvector. We have $$\forall k\in\{\,1,\dots,d\,\}\qquad
\nu(k)
\,=\,
\frac{1}{\alpha}
\sum_{j=1}^d
\nu(j) A(j,k)
\,.$$ Let us focus on the equation for $k=i$. We divide by $\nu(i)$ (which is assumed to be non zero) and we isolate the term $j=i$ in the sum to obtain $$1\,=\,
\frac{1}{\alpha}
A(i,i)+
\frac{1}{\alpha}
\sum_{j\neq i}
\frac{\nu(j)}{\nu(i)} A(j,i)
\,.$$ We expand $\nu(j)$ in the above equation as a sum, and we get $$\begin{gathered}
1
\,=\,
\frac{1}{\alpha}
A(i,i)+
\frac{1}{\alpha^2}
\sum_{j\neq i}
\sum_{j'}
\frac{\nu(j')}{\nu(i)}
A(j',j)A(j,i)
\cr
\,=\,
\frac{1}{\alpha}
A(i,i)+
\frac{1}{\alpha^2}
\sum_{j\neq i}
A(i,j)A(j,i)
+
\frac{1}{\alpha^2}
\sum_{j\neq i}
\sum_{j'\neq i}
\frac{\nu(j')}{\nu(i)}
A(j',j)A(j,i)
\,.\end{gathered}$$ Iterating $n$ times this procedure, we get $$\begin{gathered}
1
\,=\,
\frac{1}{\alpha}
A(i,i)+\cdots+
\frac{1}{\alpha^{n+1}}
\sum_{i_1,\dots,i_{n}\neq i}
A(i,i_1)
A(i_1,i_2)\cdots
A(i_{n},i)
\cr
+
\frac{1}{\alpha^{n+1}}
\sum_{i_0,i_1,\dots,i_{n}\neq i}
\frac{\nu(i_0)}{\nu(i)}
A(i_0,i_1)
A(i_1,i_2)\cdots
A(i_{n},i)
\,.\end{gathered}$$ If $\phi_i(|\alpha|)=+\infty$, then it follows from proposition \[regf\] and the defintion of $\lambda_i$ that $|\alpha|<\lambda_i$ and we are done. From now onwards, we suppose that $\phi_i(|\alpha|)<+\infty$. In the proof of proposition \[regf\], we worked out a power series expansion of $\phi_i$. The convergence of this series at $|\alpha|$ implies in particular that the general term of this series goes to $0$, hence $$\lim_{n\to\infty}\,\,
\frac{1}{\alpha^{n+1}}
\sum_{i_1,\dots,i_{n}\neq i}
A(i,i_1)
A(i_1,i_2)\cdots
A(i_{n},i)
\,=\,0
\,.$$ Let $m$ (respectively $M$) be the minimum (respectively the maximum) of the entries of $A$. For any $i_0\neq i$, we have $$\begin{gathered}
\sum_{i_1,\dots,i_{n}\neq i}
A(i_0,i_1)
A(i_1,i_2)\cdots
A(i_{n},i)
\,\leq\,
\cr
\frac{M}{m}
\sum_{i_1,\dots,i_{n}\neq i}
A(i,i_1)
A(i_1,i_2)\cdots
A(i_{n},i)\,.\end{gathered}$$ It follows that, for any $n\geq 1$, $$\begin{gathered}
\frac{1}{|\alpha|^{n+1}}
\sum_{i_0,i_1,\dots,i_{n}\neq i}
\frac{\nu(i_0)}{\nu(i)}
A(i_0,i_1)
A(i_1,i_2)\cdots
A(i_{n},i)
\,\leq\,\cr
\frac{Md
\max_{1\leq j\leq d}|\nu(j)|
}{m|\nu(i)|}
\frac{1}{|\alpha|^{n+1}}
\sum_{i_1,\dots,i_{n}\neq i}
A(i,i_1)
A(i_1,i_2)\cdots
A(i_{n},i)\end{gathered}$$ and we conclude from the previous inequality that $$\lim_{n\to\infty}\,\,
\frac{1}{\alpha^{n+1}}
\sum_{i_0,i_1,\dots,i_{n}\neq i}
\frac{\nu(i_0)}{\nu(i)}
A(i_0,i_1)
A(i_1,i_2)\cdots
A(i_{n},i)
\,=\,0\,.$$ We send now $n$ to $\infty$ in the identity and we get $$1
\,=\,
\frac{1}{\alpha}
A(i,i)+\sum_{n=1}^{+\infty}
\frac{1}{\alpha^{n+1}}
\sum_{i_1,\dots,i_{n}\neq i}
A(i,i_1)
A(i_1,i_2)\cdots
A(i_{n},i)
\,.$$ Recall that $\alpha$ might be complex. Taking the modulus, we conclude that $\phi_i(|\alpha|)\geq 1$, and since $\phi_i$ is decreasing, then $|\alpha|\leq\lambda_i$. It remains to examine the case $|\alpha|=\lambda_i$. We suppose that the eigenvector $\nu$ associated to $\alpha$ is normalized so that $\nu(i)=1$. We denote by $|\nu|$ the vector whose coordinates are the modulus of the coordinates of $\nu$, i.e., $|\nu|(j)=
|\nu(j)|$ for $1\leq j\leq d$. Since $\nu A=\alpha\nu$ and the entries of $A$ are positive, then $$\forall k\in\{\,1,\dots,d\,\}\qquad
|\nu|(k)
\,\leq\,
\frac{1}{\lambda}
\sum_{j=1}^d
|\nu|(j) A(j,k)
\,.$$ Starting from this inequality, we proceed as previously, that is, we isolate the term corresponding to $j=i$ in the sum, we bound from above the term $|\nu(j)|$ for $j\neq i$, we iterate the procedure $n$ times. We check that the ultimate term goes to $0$ when we send $n$ to $\infty$, as we get the inequality $$\forall k\in\{\,1,\dots,d\,\}\qquad
|\nu|(k)
\,\leq\,
\mu_i(k)\,.$$ For $k\in\{\,1,\dots,d\,\}$, we have $${\lambda}
|\nu|(k)\,=\,
\Big|
\sum_{j=1}^d
\nu(j) A(j,k)
\Big|
\,\leq\,
\sum_{j=1}^d
|\nu|(j) A(j,k)
\,.$$ It follows that $$\sum_{k=1}^d
\big(\mu_i(k)
-|\nu|(k)\big)
A(k,i)\,\leq\,
\lambda\,\big(\mu(i)-\nu(i)\big)\,=\,0\,.$$ This equation implies that $\mu_i
=
|\nu|$ and that all the intermediate inequalities were in fact equalities. Since all the entries of $A$ are positive and $\nu(i)=1$, then necessarily all the components of $\nu$ are non–negative real numbers and $\nu=\mu_i$ and $\alpha=\lambda$.
The $\lambda_i$’s are positive eigenvalues of $A$, the eigenvectors $\mu_i$ have positive coordinates, thus proposition \[regu\] readily implies the following result.
The values $\lambda_1,\dots,\lambda_d$ are all equal. Their common value $\lambda$ is a simple eigenvalue of $A$. The eigenvectors $\mu_1,\dots,\mu_d$ are proportional.
Finally, we normalize these eigenvectors by imposing that the sum of the components is equal to $1$, thereby getting a probability distribution.
The left Perron–Frobenius eigenvector $\mu$ of $A$ is given by $$\forall i\in\{\,1,\dots,d\,\}\qquad
\mu(i)\,=\,\frac{1}{
\displaystyle
E_i\Bigg(\sum_{n=0}^{\tau_i-1}\Big(
\lambda^{-n}\prod_{t=0}^{n-1}S(X_t)
\Big)
\Bigg)
}\,.$$
This formula (already proved in [@CD]) is a generalization of the classical formula for the invariant probability measure of a Markov chain. Indeed, in the particular case where $A$ is stochastic, $S$ is constant equal to 1, $\lambda$ is also equal to 1, the formula of the corollary becomes the well–known formula $$\forall i\in\{\,1,\dots,d\,\}\qquad
\mu(i)\,=\,\frac{1}{
\displaystyle
E_i(\tau_i)
}\,.$$
|
---
abstract: 'We study moduli spaces of rational weighted stable tropical curves, and their connections with the classical Hassett spaces. Given a vector $w$ of weights, the moduli space of tropical $w$-stable curves can be given the structure of a balanced fan if and only if $w$ has only heavy and light entries. In this case, we can express the moduli space as the Bergman fan of a graphic matroid. Furthermore, we realize the tropical moduli space as a geometric tropicalization, and as a Berkovich skeleton, of the classical moduli space. This builds on previous work of Tevelev, Gibney and Maclagan, and Abramovich, Caporaso, and Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fiber products of unweighted spaces, and explore parallels with the algebraic world.'
address:
- 'Renzo Cavalieri, Department of Mathematics, Colorado State University'
- 'Simon Hampe, Hannah Markwig, Fachrichtung Mathematik, Universität des Saarlandes'
- 'Dhruv Ranganathan, Department of Mathematics, Yale University'
author:
- 'Renzo Cavalieri, Simon Hampe, Hannah Markwig, Dhruv Ranganathan'
bibliography:
- 'bibliographie.bib'
title: Moduli spaces of rational weighted stable curves and tropical geometry
---
[^1]
Introduction
============
Main results {#subsec-wsc}
------------
Let $w=(w_1,\ldots,w_n)\in\Q^n$ be a vector of weights satisfying $0< w_i\leq 1$ for all $i$ and $\sum w_i>2$. Let $C$ be a tree of $\mathbb{P}^1$’s, with $n$ smooth marked points $p_1,\ldots,p_n$. Marks are not necessarily distinct: a subset of marks is allowed to coincide if and only if the sum of the corresponding weights is $\leq 1$. The curve $C$ is *$w$-stable*, if for each twig $T$ of $C$, the sum $$\sum_{i\; ; \;p_i\in T}w_i+\#\textnormal{nodes}>2.$$ Hassett [@Has03] introduces these spaces in the context of the log minimal model program and proves that there is a smooth projective scheme representing the moduli problem of weighted stable curves, denoted $\overline{M}_{0,w}$.
If $w=(1^n)$, then $\overline{M}_{0,w}=\overline{M}_{0,n}$ is the well-known moduli space of stable curves; in this case, the connection with the space of leaf labelled metric trees in the form of [*geometric tropicalization*]{} and [*tropical compactification*]{} is very strong, see Section \[cnm\].
In this paper, we study tropical analogues of moduli spaces of rational weighted stable curves and their relation to the algebro-geometric spaces. We introduce [*tropical rational weighted stable curves*]{} in the natural way, by defining the combinatorial type of a $w$-stable tropical curve to be the dual graph of a $w$-stable curve, keeping track of the weights on the marked ends (see Definition \[tws\]). Parameter spaces for tropical rational weighted stable curves carry the structure of abstract cone complexes, with graph contractions giving rise to natural gluing between cones.
We address the following questions:
(A) For which values of $w$ can the cone complex $M^{\trop}_{0,w}$ be given the structure of a balanced fan embedded into a vector space?
(B) When $M^{\trop}_{0,w}$ is a balanced fan, can it be realized as a topicalization of the classical moduli spaces of $w$-weighted stable curves?
(C) In the above cases, can the toric variety associated to the fan $M^{\trop}_{0,w}$ be used to define the $w$-stable compactification of the locus of nonsingular marked curves $M_{0,w}$?
These questions are addressed completely. We begin by observing that, for any fixed $n$, the notion of stability is governed by a finite set of first degree inequalities in the weights. The parameter space for the weights is subdivided into polyhedral chambers. For any $w,w'$ in the same chamber, $M^{\trop}_{0,w}$ is canonically isomorphic to $M^{\trop}_{0,w'}$.
A weight vector $w$ has only *heavy and light* weights if it is in the same chamber as $(1^f, \epsilon^t)$, where $\epsilon$ is a small enough value, i.e. $t\cdot \epsilon<1$ (see Definition \[lhw\]). The following answers Question (A).
The cone complex $M^{\trop}_{0,w}$ can be given the structure of a balanced fan in a vector space if and only if $w$ has only heavy and light entries.
The following result generalizes work of Ardila and Klivans [@AK06] to the weighted case case. A more precise statement may be found in the main body of the text.
The moduli space $M^{\trop}_{0,w}$ with heavy/light weights is the Bergman fan of a graphic matroid.
Finally, we answer Questions (B) and (C).
There exists a toric variety $X(\Delta)$ with torus $T$, and an embedding $M_{0,w}\hookrightarrow T$, such that
1. The topicalization of $M_{0,w}$ with respect to the given embedding is $M_{0,w}^{\trop}$.
2. The fan $\Delta$ is naturally identified with $M_{0,w}^{\trop}$.
3. The closure of $M_{0,w}$ in $X(\Delta)$ is the Hassett compactification $\overline M_{0,w}$.
In Section \[sec: skeleta\] we explore the connection with Berkovich skeletons, in the spirit of the results obtained in [@ACP12; @U]. This establishes a compatibility between the geometric tropicalization approach employed in this text, and the perspective of skeletons of analytic spaces.
A natural variation of weighted stable curves, also considered by Hassett, arises by allowing the weight vector $w$ to have zero entries. In this case, the moduli spaces are described by fibre products of the universal curve of $M_{0,w^+}$, where $w^+$ is the subcollection of positive entries of $w$. We discuss a tropical analogue of this situation in Theorem \[thm-tropfibre\] and Corollary \[cor-fibreheavylight\].
The main results rely on a careful study of the classical and combinatorial *reduction morphisms* from the spaces $\overline M_{0,n}$ and $M_{0,n}^{\trop}$ to the weighted spaces.
Context and motivation {#cnm}
----------------------
Tropical geometry has become a successful tool in algebraic geometry, with exciting applications in the study of enumerative geometry, moduli spaces, and the general study of algebraic curves. Tropical enumerative geometry began with Mikhalkin’s celebrated Correspondence Theorem relating numbers of plane curves satisfying point conditions with their tropical counterparts [@Mi03]. The results sparked substantial interest in tropical moduli spaces.
The space $M^{\trop}_{0,n}$ is a polyhedral complex parametrizing leaf-labeled metric trees. It was first studied in connection with phylogenetics [@BHV01], and later, in relation with the geometry of the tropical Grassmanian [@SS04a]. The cone complex $M^{\trop}_{0,n}$ can be embedded into a vector space, and given the structure of a balanced fan, by assigning weight $1$ to each top dimensional cone.
In [@GKM07; @Mi07], $M_{0,n}^{\trop}$ is discussed in analogy with the classical moduli space, largely with the goal of understanding its tropical intersection theory. However, the connection between the algebraic and tropical moduli spaces runs much deeper. Building on previous work of Tevelev [@Tev07], Gibney and Maclagan [@GM07] exhibit $M_{0,n}^{\trop}$ as a *tropicalization*. They find an embedding of $M_{0,n}$ (the locus of smooth curves) into a torus, such that the tropicalization of $ M_{0,n}$ is a balanced fan $\Sigma$, such that $\Sigma\cong M^{\trop}_{0,n}$. The closure of $M_{0,n}$ in the toric variety $X(\Sigma)$ is $\overline M_{0,n}$.
The theory of tropical compactification was developed by Hacking, Keel, and Tevelev [@Tev07; @HKT09] with the goal of studying compactifications of spaces of del Pezzo surfaces. The philosophy is that the features of a suitable compactification are inherent in the tropicalization of a subvariety of a torus, i.e. “the tropicalization knows a good compactification”. The $M_{0,n}$ case is particularly nice, since the tropicalization has an intrinsic modular interpretation.
With the above connection between the algebraic $\overline{M}_{0,n}$ and the tropical $\mk{n}$ established, there is a manifest combinatorial relationship. The orbit–cone correspondence of the ambient toric variety induces an order reversing bijection between dimension $k$ strata in the classical moduli space, and codimension $k$ cones in the tropical moduli space. In fact, using techniques from toric intersection theory, Katz [@Kat09 Section 7], shows that the intersection theory on $\overline M_{0,n}$ can be related to the toric intersection theory on the ambient toric variety, and hence to the tropical intersection theory on $M^{\trop}_{0,n}$.
With the spaces of rational weighted stable curves, we exhibit a new class of moduli spaces, having a natural tropical modular skeleton. It seems reasonable to expect that intersection numbers on the heavy/light spaces $\overline M_{0,w}$ can be computed tropically, following work of Katz.
The spaces $\overline{\mathcal M}_{g,w}$ for higher genus are defined analogously. In general however, such spaces do not admit nice embeddings to toric varieties. Nonetheless, the moduli space of tropical $w$-stable curves can be viewed as a skeleton of the analytification of $\overline{\mathcal M}_{g,w}$. Since this paper first appeared, Ulirsch has exhibited this connection [@U]. In genus $0$, we observe that the compactification of $\overline M_{0,w}$ can be obtained combinatorially. It is intriguing to ask whether there exists a similar relationship between ${\mathcal M}_{g,w}$ and $\overline{\mathcal M}_{g,w}$.
Structure of the paper
----------------------
Section 2 is devoted to projections of $\mk{n}$ and their relation to Bergman fans with the nested set subdivision of a building set defined in terms of graphs. We start with a subsection reviewing the necessary preliminaries, subdivided in a part about matroids, Bergman fans and nested sets and a part about basics in tropical geometry and the moduli space of abstract $n$-marked tropical curves. Subsection 2.2 contains original work. The main results are Theorem \[moduli\_cor\_mainresult\], stating that a certain projection of $\mk{n}$ is the Bergman fan of a graphic matroid in a nested set subdivision and Theorem \[thm-heavyandlight\] that states that this projection is an embedding of $\mk{w}$ as a balanced fan if $w$ contains only heavy and light points. If $w$ does not only contain heavy and light points, then we cannot embed $\mk{w}$ as a balanced fan. This is stated in Theorem \[thm-heavyandlight\].
In Section \[sec: trop-wsc\] we explore the tropicalization of the algebraic moduli spaces $\overline{M}_{0,w}$. We start with a subsection reviewing the preliminaries in the context of the tropicalization of $\overline{M}_{0,n}$. Subsection 3.2 contains our main result of this section, Theorem \[thm-tropicalizing\], which states that in the case of heavy and light points, we can embed $\overline{M}_{0,w}$ into a toric variety defined by $\mk{w}$ (which exists as a balanced fan because of our results of Section 2) and the tropicalization of the open part in the torus is canonically isomorphic to the tropical moduli space. We also discuss the situation where we do not only have heavy and light points. In Section 3.3, we consider a special case: the Losev–Manin spaces, with exactly two heavy and only light points otherwise. They are toric varieties, which also follows from Theorem \[thm-tropicalizing\], since the corresponding tropical spaces are just subdivisions of $\R^{n-3}$. In Section 3.4 we consider the connection to Berkovich skeletons.
In Section \[section\_fibre\], we construct the moduli spaces of heavy/light weighted tropical curves as fibre products of spaces of unweighted tropical curves. The motivating idea here is to replace the light points with weight $0$ points. The resulting classical spaces, studied by Hassett, are singular. The heavy/light spaces can be obtained as desingularizations of the weight $0$ spaces. The weight $0$ spaces, in turn, may be expressed as fibre products of spaces of unweighted curves. The main results of this section rely on several general structure results for fibre products of Bergman fans associated to graphic matroids.
**Acknowledgements.** The authors are grateful to Martin Ulirsch for discussing his related work. The first author is grateful for the support from NSF grant DMS-1101549, NSF RTG grant 1159964. The second and third author were partially supported by DFG-grant 4797/1-2, the third author in addition by DFG-grant 4797/3-2. The fourth author acknowledges several helpful conversations with Dustin Cartwright, Dave Jensen, Sam Payne, and Tif Shen.
Tropical moduli spaces of rational weighted curves
==================================================
Let $C$ be a rational abstract tropical curve. That is, $C$ is a leaf-labelled metric tree. Edges adjacent to leaves are called *ends*, and are metrized as $[0,\infty]$. The other edges are called *bounded edges* and have finite length.
Let $w=(w_1,\ldots,w_n)\in\Q^n$ be a vector of weights satisfying $0< w_i\leq 1$ for all $i$.
\[tws\] Let $V$ be a vertex of $C$ and assume that there are $l$ bounded edges adjacent to $V$ as well as the ends with markings $i\in I$ for a subset $I\subset [n]$. We say that $C$ is *$w$-stable* at $V$ if $\sum_{i\in I}w_i+l>2$. We say that $C$ is *$w$-stable* if it is $w$-stable at every vertex. We define $M^{\trop}_{0,w}$ to be the set of all tropical rational $w$-stable curves.
The set $M^{\trop}_{0,w}$ can be given the structure of an abstract cone complex by gluing the cones corresponding to a combinatorial type (i.e. a tree without metrization data) in the way dictated by the underlying tropical curves. As such, for any $w$, $M^{\trop}_{0,w}$ is a sub-cone complex of $M^{\trop}_{0,n}$.
The definition generalizes the definition of abstract $n$-marked rational tropical curve: if we set $w=(1^n)$, then the stability condition is that every vertex is at least $3$-valent.
To efficiently state our results, we make the following definition.
\[lhw\] Let $w\in \Q^n$. Let $i \in [n]$.
- We call $i$ *heavy in $w$*, if for all $j \neq i$ we have $w_i + w_j > 1$.
- We call $i$ *small in $w$*, if $w_i + w_j >1$ implies that $j$ is heavy in $w$.
If, in addition, the total weight of the small points is less than $1$, we say that they are *light*.
We will often consider weight vectors that are *heavy/light* (resp. *heavy/small*), meaning that each entry of $w$ is either heavy or light (resp. heavy or small).
Tropical moduli spaces of rational curves as Bergman fans
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### Matroids, Bergman fans, and nested sets
Matroids abstract the concept of linear independence of subsets of a set of vectors. Important examples are matroids of point configurations defined by the usual linear independence and matroids of graphs, where dependence is defined in terms of cycles. For a detailed introduction to matroids, see for instance [@Kat14].
To any matroid $M$ on a ground set $E(M)$ we associate a polyhedral fan, the *Bergman fan* $B(M) \subseteq \R^{\abs{E(M)}}$ in the following manner: $$B(M) := \{w \in \R^{\abs{E(M)}}; M_w \textnormal{ is loop-free}\},$$ where $M_w$ is the matroid on $E(M)$ whose bases are all bases $B$ of $M$ of minimal $w$-weight $\sum_{i \in B} w_i$. Ardila and Klivans showed in [@AK06] that $B(M)$ is a polyhedral cone complex that coincides with the order complex of the lattice of flats of $M$. More precisely, for each chain of flats in $M$ $$\curly{F} = \emptyset \subsetneq F_1 \subsetneq \dots \subsetneq F_r = E,$$ we let $C_{\curly{F}}$ be the cone in $\R^{\abs{E}}$ spanned by rays $v_{F_1},\dots,v_{F_{r-1}}$, with lineality space $v_{F_r}$. Here $v_F = - \sum_{i \in F} e_i$, where $e_i$ is the $i$-th standard basis vector. The collection of these cones forms a fan whose support is $B(M)$. We call this particular polyhedral structure the *chains-of-flats* subdivision of $B(M$).
A useful equivalent definition is the following [@FS05 Proposition 2.5]. $$B(M) := \{w;\; \max\{w_i; i \in C\} \textnormal{ is attained at least twice for all circuits }C\}.$$
Feichtner and Sturmfels demonstrate multiple polyhedral structures that can be placed on this fan using the theory of *building sets*.
Let $\curly{F}$ be the lattice of flats of a matroid $M$. For two flats $F,F' \in \curly{F}$ we write $[F,F'] := \{G \in \curly{F}: F \subseteq G \subseteq F'\}$.
A *building set* for $\curly{F}$ is a subset $\curly{G}$ of $\curly{F} \wo \{\emptyset\}$ such that the following holds:
For any $F \in \curly{F} \wo \{\emptyset\}$, let $\{G_1,\dots,G_k\}$ be the maximal elements of $\curly{G}$ contained in $F$. Then there is an isomorphism of partially ordered sets: $$\varphi_F: \prod_{j=1}^k [\emptyset,G_j] \to [\emptyset,F],$$ where the $j$-th component of $\varphi_F$ is the inclusion $[\emptyset,G_j] \subseteq [\emptyset,F]$.
A subset $\curly{S}$ of a building set $\curly{G}$ is called *nested*, if for any set of incomparable elements $F_1,\dots,F_l$ in $\curly{S}$ with $l \geq 2$, the join $F_1 \vee \dots \vee F_l$ is not an element of $\curly{G}$.
The nested sets of a building set $\curly{G}$ form an abstract simplicial complex (a subset of a nested set is a nested set). We can assign to each flat $F$ the vector $v_F \in \R^{\abs{E}}$ defined above, and accordingly a cone for each nested set of $\curly{G}$. It has been shown in [@FS05]\*[Theorem 4.1]{} that this produces a polyhedral fan whose support is $B(M)$.
Note that each Bergman fan contains the linear space $L$, spanned by the vector $(1,\dots,1)$. It is standard to quotient by this lineality space, and study the resulting space $$B'(M) := B(M) / L.$$
It is a well-known fact that the set of Bergman fans is closed under cartesian products. In fact, if $M,M'$ are matroids on ground sets $E,E'$, then $$B(M) \times B(M') = B(M \oplus M'),$$ where $M \oplus M'$ is the matroid on $E \amalg E'$, whose bases are disjoint unions of bases of $M$ and $M'$.
### Facts about tropical geometry and $M^{\trop}_{0,n}$ {#prelim-trop}
A *tropical fan* $(X,\omega)$ is a rational pure $d$-dimensional polyhedral fan in $\R^n$, with a *weight function* $\omega: X^{(d)} := \{\sigma \in X; \dim \sigma = d\} \to \Z_{>0}$, fulfilling the *balancing condition*.
We consider two balanced fans $(X,\omega),(X',\omega')$ to be *equivalent*, if they have a common refinement: if there exists a balanced fan $(X'',\omega'')$, whose cones are contained in cones of $X$ and $X'$ respectively; and $\omega''$ is compatible with $\omega$ and $\omega'$ in the natural way.
In this paper, we consider balanced fans that arise as Bergman fans of matroids. Given a matroid $M$, $B(M)$, equipped with the chain-of-flats subdivision is a fan. It is balanced with weight function identically $1$.
It is also known (see for example [@FR10]\*[Lemma 2.4]{}), that $(B(M), \omega \equiv 1)$ is *irreducible*, i.e. any balanced fan of the same dimension, which is contained in $B(M)$, must be equal to $B(M)$ as a set and its weight function must be an integer multiple of $w$.
A *tropical morphism* between balanced fans $X \subset \R^n,Y \subset \R^m$ is a map of fans, respecting the weight function, and the integral structure on $X$ and $Y$. More explicitly, it is a map $f: X \to Y$ mapping cones to cones, induced by a linear map $\Z^n \to \Z^m$. We say that $f$ is an *isomorphism*, if it is bijective and respects the weights of $X$ and $Y$.
It is often necessary to understand the local structure of a fan, near a given cone.
Let $(X,\omega)$ be a tropical fan and $\tau$ a cone of $X$. We define the local picture of $X$ around $\tau$ to be the weighted fan $$\Star_X(\tau) := (\{\Pi(\sigma); \tau \leq \sigma; \sigma \textnormal{ a cone of } X\},\omega_\Star),$$where $\Pi: \R^n \to \R^n / V_\tau$ is the residue map and the weight function is defined by $\omega_\Star: \Pi(\sigma) \mapsto \omega(\sigma)$. It is easy to see that $\Star_X(\tau)$ is a tropical fan (with respect to the lattice $\Z^n/ \Lambda_\tau$).
If $p$ is a point in the support of $X$, we also consider $$\Star_X(p) := (\{\sigma - p; p \in \sigma; \sigma \textnormal{ a cone of } X\},\omega_\Star),$$ where $\omega_\Star$ is defined in the same way and $\sigma - p = \{ a - p; a \in \sigma\}$.
We now briefly discuss some properties of the tropical moduli space $\mk{n}$ and its embedding as a balanced fan. For more details, see e.g. [@GKM07; @Mi07; @SS04a].
For any abstract $n$-marked tropical curve $C$, let $\dist(i,j)$ be the distance in $C$ of the vertices which leaves $i$ and $j$ are attached to. It has been shown that the vector $$d(C) = (\dist(i,j))_{i <j} \in \R^{\binom{n}{2}} / \Phi(\R^n)$$ identifies $C$ uniquely, where $\Phi: \R^n \to \R^{\binom{n}{2}}, x \mapsto (x_i + x_j)_{i < j}$.
Let $C$ be an abstract $n$-marked tropical curve with a single edge, inducing a partition or *split* on the leaves $[n] = I \amalg I^c$. We denote the corresponding ray spanned by $d(C)$ by $v_I = v_{I^c}$. A $d$-dimensional cone of $\mk{n}$ corresponds to a combinatorial type of curve with $d$ bounded edges. If these edges introduce splits $I_1,\dots,I_d$, the cone is spanned by rays $v_{I_1},\dots,v_{I_d}$. The fan $\mk{n}$ can be balanced with all weights equal to $1$.
The fan structure described above, with one cone for each combinatorial type is the coarsest fan structure that can be defined on $\mk{n}$. We call it the *combinatorial subdivision*.
It has been shown in [@AK06] and [@FR10], that $$\mk{n} \cong B'(K_{n-1}),$$ where $K_{n-1}$ is the complete graph on $n-1$ vertices. For later use, we now describe the explicit isomorphism between $\mk{n}$ and $B'(K_{n-1})$. That is, we describe the image of a vector $v_F$, when $F$ is any flat (see Figure \[fig\_flat\_to\_curve\] for an example):
For convenience, throughout the rest of this paper, we label the vertices of $K_{n-1}$ by $2,\ldots, n$.
A flat $F$ of $K_{n-1}$ is a union of complete subgraphs on disjoint vertex sets $V_1,\dots,V_t$. Denote by $C_F$ the tropical $n$-marked curve constructed in the following manner: Attach $t$ bounded edges $e_1,\dots,e_t$ of length 1 to a common vertex $v$. To the end of $e_i$, attach leaves $\{j; j \in V_i\}$. Then attach all leaves $\{j; j \in [n] \wo \bigcup_{i=1}^t V_i\}$ to the vertex $v$. Then $$v_F \mapsto C_F.$$
Conversely, if we pick a ray $v_I$ of $\mk{n}$ with $1 \notin I$, it corresponds to the flat $F_I$, which is the complete graph on vertices in $I$. We will see in Example \[ex\_mn\_nested\_subdiv\] how the combinatorial subdivision of $\mk{n}$ can be expressed as a nested set subdivision of $B'(K_{n-1})$.
;
Tropical moduli spaces of rational weighted curves as Bergman fans {#section_moduli}
------------------------------------------------------------------
Let $w$ be a weight vector. To obtain the cone complex $\mk{w}$, we wish to contract unstable rays and their adjacent cones. It is a natural thought to use a projection of $\mk{n}$ embedded as a fan contracting these rays; we denote this projection by $\pr_w$. It turns out however that $\pr_w$ may contract too many cones: this is the case if and only if $w$ does not only have heavy and light entries. The projection $\pr_w(\mk{n})$ is still an interesting balanced fan which we can understand in terms of the Bergman fan of a graphic matroid. Moreover, as we see in Section \[sec: skeleta\], the image of this projection can be realized as the Berkovich skeleton of the classical moduli space $\overline M_{0,w}$ with respect to an appropriate toroidal structure.
Our first goal is to gain a clearer understanding of the role of the complete graph $K_{n-1}$ in the $M_{0,n}^{\trop}$ case, and how it generalizes to the weighted case.
An $n$-marked trivalent rational tropical curve is $w$-stable if all vertices with exactly two leaves $i$ and $j$ attached fulfill the condition $w_i + w_j > 1$. Consequently, two weight vectors $w,w'$ produce the same set of stable combinatorial types of trivalent curves, if $w_i + w_j > 1 \iff w_i' + w_j' > 1$ for all pairs $i,j$. It is therefore reasonable to expect that a graph encoding these conditions will be meaningful towards understanding spaces of weighted stable tropical curves.
Let $w $ be a weight vector. We define the *total weight graph* $G_t(w)$ to be the graph on vertices $\{1,\dots,n\}$ where two vertices $i,j$ are connected by an edge, if and only if $w_i + w_j > 1$.
The notions of heavy and small can then be expressed in terms of this graph: Let $i \in [n]$.
- $i$ is heavy if $i$ is connected to all other vertices in $G_t(w)$.
- $i$ is small if $i$ is only connected to heavy vertices.
In parallel with the $M_{0,n}^{\trop}$ case, we seek to have a graph on $n-1$ vertices. This is obtained by deleting any heavy vertex.
The *reduced weight graph* is the graph obtained from $G_t(w)$ by deleting any single heavy vertex.
Of course, dropping different heavy vertices we obtain isomorphic graphs, since all heavy vertices are incident to all other vertices. We will see in Corollary \[moduli\_cor\_needstable\] that if there is no heavy weight, $\pr_w(\mk{n})$ does not have the “expected dimension” (i.e. the dimension of the classical space which is $n-3$). Henceforth, we assume that $w_1 = 1$, and the reduced weight graph is constructed by deleting the vertex $1$ from $G_t(w)$.
\[rem-prw\] There is a natural projection morphism induced by the fact that $G(w)$ is a subgraph of the complete graph $K_{n-1}$: $$\widetilde \pr_w: \mk{n}\cong B'(K_{n-1}) \to B'(G(w)),$$ which forgets the coordinates corresponding to edges not lying in $G(w)$. We can see that $\widetilde \pr_w$ is precisely the projection morphism $\pr_w$ discussed above.
A ray $v_I$ with $1 \notin I$ which is contracted because it corresponds to an unstable curve comes from a flat $F_I$ given by the induced subgraph on the vertices in $I$. The fact that $v_I$ is unstable means that $\sum_{i \in I}w_i\leq 1$ which implies $w_i+w_j\leq 1$ for all $i,j\in I$, so no edge connecting two vertices in $I$ belongs to $G(w)$. Vice versa, if no edge connecting two vertices in $I$ belongs to $G(w)$, then $w_i+w_j\leq 1$ for all $i,j\in I$ and hence $v_{\{i,j\}}$ corresponds to an unstable curve and is contracted by $\pr_w$ for all $i,j\in I$. But then $v_I=\sum_{\{i,j\}\subset I} v_{\{i,j\}}$ (see [@KM07], Lemma 2.6) is also contracted.
We first relate the projection $\pr_w(\mk{w})$ to the nested set subdivision induced by the *building set of 1-connected flats* on the graph $G(w)$.
Let $G$ be a (connected) graph and $M_G$ the corresponding matroid. We define a *building set of 1-connected flats*: $$\curly{G}_G := \{F \in \curly{F}(M_G); G_{\mid F} \textnormal{ is a connected graph}\},$$ where $G_{\mid F}$ is the restriction of $G$ to the edges contained in $F$.
The above definition depends not only on the matroid, but on the presentation of this matroid as a graphic matroid. Recall that two non-isomorphic graphs $G$, $G'$ may yield the same matroid. In this case $\curly{G}_G$, $\curly{G}_{G'}$ might be different building sets. We also warn the reader that the matroidal notion for *connected sets* differs from the above definition. A connnected set of a graphic matroid $M(G)$ is a set whose underlying graph is 2-connected. To see that $\curly{G}_G$ is a building set, let $F$ be a flat of $M_G$. The maximal elements of $\curly{G}_G$ contained in $F$ are exactly the connected components $G_1,\dots,G_k$ of the subgraph $G_{\mid F}$. Any flat $F' \subseteq F$ can also be partitioned into its connected components, which in turn must be subsets of the connected components of $F$. We see that there is an isomorphism $$\prod_{j=1}^k [\emptyset,G_j] \cong [\emptyset,F].$$
We are now ready to state the first main result of this section.
\[moduli\_cor\_mainresult\] Let $w $ be a weight vector and assume $w$ has at least two heavy entries. Then $\pr_w(\mk{n})=B'(G(w))$. Furthermore, the combinatorial types of curves in $\pr_w(\mk{n})$ correspond to the cones of $B'(G(w))$ in the nested set subdivision with respect to $\curly{G}_{G(w)}$, the building set of 1-connected flats.
Before proceeding to the proof, we consider some examples.
\[ex\_mn\_nested\_subdiv\] Let $w = (1,\dots,1)$. Since no ray of $\mk{n}$ becomes unstable, $\pr_w$ is the identity map. In this case, $G_t(w) = K_n$, the complete graph on $n$ vertices, and the reduced graph is $G(w) = K_{n-1}$. We already observed that the Bergman fan corresponding to this graph is $B'(K_{n-1})\cong \mk{n}=\pr_w(\mk{n})$. Theorem \[moduli\_cor\_mainresult\] tells us that the combinatorial subdivision of $\mk{n}$ corresponds to the nested set subdivision of $B'(K_{n-1})$ with respect to $\curly{G}_{K_{n-1}}$. This is seen as follows. It is well known that the combinatorial subdivision is the coarsest possible polyhedral structure on $\mk{n}$. Feichtner and Sturmfels showed in [@FS05]\*[Theorem 5.3]{} that it is obtained as the nested set subdivision with respect to the building set of connected flats. In the case of graphic matroids, this means choosing all flats whose underlying graph is 2-connected. However in this particular case, flats are disjoint unions of complete graphs, so they are 1-connected if and only if they are 2-connected.
The following example demonstrates that $\pr_w(\mk{n})=B'(G(w))$ may not be the embedding of the cone complex $\mk{w}$ as a fan:
\[moduli\_ex\_badtypes\] Let $w = (1,1,3/4, 3/4,1/4)$. The reduced weight graph is a $K_3$ with an additional edge attached to it (connecting the remaining 1 and the vertex with weight 1/4). The corresponding Bergman fan is $B'(G(w))\cong \mk{4} \times \R$.
The 1-connected flats of $G(w)$ are depicted in Figure \[fig\_connected\_flats\_example\]. A nested set is formed either by a chain or two incomparable flats whose join is not connected, i.e. by two vertex-disjoint flats. Only $F_3, F_4$ are vertex-disjoint, so all other nested sets are formed by chains. Hence we obtain the following 8 cones: $$\begin{aligned}
\sigma_1 &:= \textnormal{cone}(v_{F_3},v_{F_4}) & \sigma_5 &:= \textnormal{cone}(v_{F_2},v_{F_7}) \\
\sigma_2 &:= \textnormal{cone}(v_{F_1},v_{F_5}) & \sigma_6 &:= \textnormal{cone}(v_{F_3},v_{F_5}) \\
\sigma_3 &:= \textnormal{cone}(v_{F_1},v_{F_6}) & \sigma_7 &:= \textnormal{cone}(v_{F_4},v_{F_6}) \\
\sigma_4 &:= \textnormal{cone}(v_{F_2},v_{F_5}) & \sigma_8 &:= \textnormal{cone}(v_{F_4},v_{F_7})\end{aligned}$$
(0,0) – (1,-1) – (0,-1) – (0,0); (0,0) – (1,0); (0,0) circle (2pt) node\[left\][$w_2 = 1$]{}; (1,-1) circle (2pt) node\[right\][$w_3 = \frac{3}{4}$]{}; (0,-1) circle (2pt) node\[left\][$w_4 = \frac{3}{4}$]{}; (1,0) circle (2pt) node\[above\][$w_5 = \frac{1}{4}$]{};
\
[|l | c | c | c | c|]{} Rank 1 & $F_1$ & $F_2$ & $F_3$ & $F_4$\
&
(0,0.3) circle (2pt); (0,0) circle (2pt) ; (1,-1) circle (2pt) ; (0,-1) circle (2pt) ; (1,0) circle (2pt) ; (0,0) – (1,-1);
&
(0,0.3) circle (2pt); (0,0) circle (2pt) ; (1,-1) circle (2pt) ; (0,-1) circle (2pt) ; (1,0) circle (2pt) ; (0,0) – (0,-1);
&
(0,0.3) circle (2pt); (0,0) circle (2pt) ; (1,-1) circle (2pt) ; (0,-1) circle (2pt) ; (1,0) circle (2pt) ; (0,-1) – (1,-1);
&
(0,0.3) circle (2pt); (0,0) circle (2pt) ; (1,-1) circle (2pt) ; (0,-1) circle (2pt) ; (1,0) circle (2pt) ; (0,0) – (1,0);
\
Rank 2 & $F_5$ & $F_6$ & $F_7$ &\
&
(0,0.3) circle (2pt); (0,0) circle (2pt) ; (1,-1) circle (2pt) ; (0,-1) circle (2pt) ; (1,0) circle (2pt) ; (0,0) – (1,-1) – (0,-1) – (0,0);
&
(0,0.3) circle (2pt); (0,0) circle (2pt) ; (1,-1) circle (2pt) ; (0,-1) circle (2pt) ; (1,0) circle (2pt) ; (1,0) – (0,0) – (1,-1);
&
(0,0.3) circle (2pt); (0,0) circle (2pt) ; (1,-1) circle (2pt) ; (0,-1) circle (2pt) ; (1,0) circle (2pt) ; (1,0) – (0,0) – (0,-1);
&\
Figure \[fig\_nested\_set\_combinatorial\] shows how this can be interpreted as the projection of $\mk{5}$. The latter is the fan over the Petersen graph with 10 rays $v_{ij}$, $i\neq j\subset [5]$ and 15 cones spanned by $v_{\{i,j\}}$ and $v_{\{k,l\}}$ if $\{i,j\}\cap \{k,l\}=\emptyset$. The projection $\pr_w$ contracts the two rays $v_{\{3,5\}}$ and $v_{\{4,5\}}$, since the corresponding tropical curves are not $w$-stable. The balancing condition around the ray $v_{\{1,2\}}$ is given by the relation $v_{\{1,2\}}=v_{\{3,4\}}+v_{\{3,5\}}+v_{\{4,5\}}$, so we have $\pr_w(v_{\{1,2\}})=\pr_w(v_{\{3,4\}})$, and the 2-dimensional cone spanned by $v_{\{1,2\}}$ and $v_{\{3,4\}}$, even though it corresponds to curves which are $w$-stable, is mapped to a ray. Thus, the fan $\pr_w(\mk{n})$ is not an embedding of the abstract cone complex $\mk{w}$. We can read off the projection from the graph $G(t)$ interpreted as a subgraph of the complete graph on 4 vertices $K_4$: $\pr_w(v_{\{1,2\}})=\pr_w(v_{\{3,4\}})=v_{F_3}$, $\pr_w(v_{\{1,3\}})=v_{F_7}$, $\pr_w(v_{\{1,4\}})=v_{F_6}$, $\pr_w(v_{\{1,5\}})=v_{F_5}$, $\pr_w(v_{\{2,3\}})=v_{F_1}$, $\pr_w(v_{\{2,4\}})=v_{F_2}$, $\pr_w(v_{\{2,5\}})=v_{F_4}$.
(0,0,0) – (0,0,1) node\[above left\] [$v_{F_5}$]{} – (0,-2.5,1) – (0,-2.5,0) node\[below left\][$v_{F_3}$]{} – (0,0,0); (0,0,0) – (0,0,-1) – (0,-2.5,-1) – (0,-2.5,0) – (0,0,0);
(0,0,0) – (0,0,1) – (-2,0,1) node\[above left\][$v_{F_1}$]{}– (0,0,0); (0,0,0) – (-2,0,1) – (-2,0,-1) node\[below left\][$v_{F_6}$]{} – (0,0,0); (0,0,0) – (-2,0,-1) – (0,0,-1) node\[below right\][$v_{F_4}$]{}– (0,0,0);
(0,0,0) – (1.5,1.5,1) node\[above left\][$v_{F_2}$]{} – (0,0,1) – (0,0,0); (0,0,0) – (1.5,1.5,1) – (2,2,-1) node\[above right\][$v_{F_7}$]{}– (0,0,0); (0,0,0) – (2,2,-1) – (0,0,-1) – (0,0,0);
(0,-1.6,0.9) node\[above\][ 1]{}– (0,-1.6,0.1) node\[above\][2]{}; (0,-1.6,0.7) – (0,-1.4,0.7) node\[right\][5]{}; (0,-1.6,0.5) – (0,-2,0.5); (0,-2,0.5) – (0,-2.3,0.7) node\[left\][3]{}; (0,-2,0.5) – (0,-2.3,0.3) node\[right\][4]{}; (0,-1.6,-0.9) node\[above\][ 2]{}– (0,-1.6,-0.1) node\[above\][1]{}; (0,-1.6,-0.7) – (0,-1.4,-0.7) node\[right\][5]{}; (0,-1.6,-0.5) – (0,-2,-0.5); (0,-2,-0.5) – (0,-2.3,-0.7) node\[right\][4]{}; (0,-2,-0.5) – (0,-2.3,-0.3) node\[left\][3]{};
(-0.3,0,0.3) node\[right\][2]{}– (-0.3,0,0.8)node\[left\][1]{}; (-0.3,0,0.7) – (-0.2,0,0.7) node\[right\][5]{}; (-0.3,0,0.5) – (-0.6,0,0.5) node\[left\][3]{}; (-0.3,0,0.6) – (-0.6,0,0.6) node\[left\][4]{};
(-1.2,0,0.3) node\[left\][1]{}– (-1.2,0,-0.3) node\[left\][2]{}; (-1.2,0,0) – (-1,0,0) node\[right\][5]{}; (-1.2,0,0.05) – (-1.5,0,0.05) node\[left\][4]{}; (-1.2,0,-0.07) – (-1.5,0,-0.07) node\[left\][3]{};
(-0.3,0,-0.3) node\[left\][1]{}– (-0.3,0,-0.8)node\[left\][2]{}; (-0.3,0,-0.7) – (-0.2,0,-0.7) node\[right\][5]{}; (-0.3,0,-0.5) – (-0.6,0,-0.5) node\[left\][4]{}; (-0.3,0,-0.6) – (-0.6,0,-0.6) node\[left\][3]{};
(0.3,0.3,0.9) node\[below\][1]{}– (0.3,0.3,0.4) node\[right\][2]{}; (0.3,0.3,0.7) – (0.15,0.15,0.7) node\[left\][3]{}; (0.3,0.3,0.6) – (0.15,0.15,0.6) node\[below\][4]{}; (0.3,0.3,0.8) –(0.4,0.4,0.8) node\[right\][5]{};
(1.2,1.2,0.3) node\[left\][1]{}– (1.32,1.32,-0.3) node\[right\][2]{}; (1.24,1.24,0.1) – (1.04,1.04,0.1) node\[left\][3]{}; (1.28,1.28,-0.1) – (1.08,1.08,-0.1) node\[below\][4]{}; (1.26,1.26,0) – (1.36,1.36,0) node\[right\][5]{};
(0.3,0.3,-0.8) node\[right\][2]{}– (0.3,0.3,-0.3) node\[below\][1]{}; (0.3,0.3,-0.6) – (0.15,0.15,-0.6) node\[below\][4]{}; (0.3,0.3,-0.5) – (0.15,0.15,-0.5) node\[left\][3]{}; (0.3,0.3,-0.7) –(0.4,0.4,-0.7) node\[right\][5]{};
This suggests an interpretation of the combinatorics of $B'(G(w))$ as follows (see Figure \[fig\_nested\_set\_combinatorial\]). Let $C$ be an element in $\mk{4}$, i.e. a four-marked tropical curve with labels $\{1,\dots,4\}$ and weights $(1,1,3/4,3/4)$. We can interpret the additional $\R$-coordinate as placing the leaf $5$ with weight 1/4 somewhere along the subgraph consisting of leaves 1 and 2 and the edge between them, if it exists. The subdivision of the cones of $\mk{4} \times \R$ given by the building set of $1$-connected flats is then obtained by subdividing a cone if the attached leaf $5$ is at a trivalent vertex. We will see in Section \[section\_fibre\] that we can always interpret $B'(G(w))$ in this fashion.
We have seen that $\pr_w(\mk{n})$ is not an embedding of $\mk{w}$ as a balanced fan, since it is “missing” the cone $\sigma$ spanned by $v_{\{1,2\}}$ and $v_{\{3,4\}}$. The codimension $1$ type $v_{\{3,4,5\}}$ is $w$-stable, but it adjacent to only $1$ $w$-stable maximal cell, namely $\sigma$. We obtain a “univalent” codimension one face. There is no way to embed this codimension one face and its adjacent cones into a vector space as a balanced fan (c.f. Figure \[moduli\_fig\_onlyonestable\]).
;
In the following definition, we make characterize the cones that obstruct a balanced embedding of $\mk{w}$. We will see that such cones correspond exactly to the top-dimensional cones of $\mk{n}$ on which $\pr_w$ is not injective (see Lemma \[moduli\_thm\_noninjective\]).
Let $n \geq 4$ and $w$ be a weight vector. We consider $\mk{n}$ in its combinatorial subdivision.
- We denote by $\curly{U}_w^0$ the collection of all top-dimensional cones $\sigma$ of $\mk{n}$ such that the corresponding combinatorial type is not $w$-stable.
- We recursively define $\curly{U}_w^{k+1}$: A top-dimensional cone $\sigma \notin \bigcup_{j=0}^k \curly{U}_w^j$ is in $\curly{U}_w^{k+1}$ if and only if it has a codimension one face $\tau$ such that all other top-dimensional cones $\sigma'$ neighboring $\tau$ lie in $\bigcup_{j=0}^k \curly{U}_w^j$.
Finally we set $\curly{U}_w := \bigcup_{k \geq 0} \curly{U}_w^k$. We call curves lying in the support of $\curly{U}_w$ *inherited $w$-unstable*.
\[moduli\_lemma\_nested\] Let $\sigma$ be a top-dimensional cone of $\mk{n}$ with rays $v_{I_1},\dots,v_{I_{n-3}}$ and assume $1 \notin I_j$ for all $j$. Let $F_{I_k}$ be the flat of $K_{n-1}$ corresponding to the complete graph on vertices in $I_k$. Then $$\{F_{I_1} \cap G(w),\dots,F_{I_{n-3}} \cap G(w)\}$$ is a nested set with respect to the building set $\curly{G}_{G(w)}$ of 1-connected flats in $G(w)$.
Let $C_\sigma$ be the combinatorial type associated to $\sigma$. Then $I_1,\dots,I_k$ are the leaf splits induced by the bounded edges of $C_\sigma$. Since we assumed $1 \notin I_j$ for all $j$, we have that any two incomparable $I_i,I_j$ are already disjoint. In particular, any two incomparable flats $F_{I_i} \cap G(w), F_{I_j} \cap G(w)$ must be vertex-disjoint. Now notice that in any graph, the join of two vertex-disjoint flats is just the union. As this is not a connected graph, the claim follows.
\[moduli\_lemma\_wheninjective\] Let $\sigma$ be a top-dimensional cone of $\mk{n}$ such that $\pr_w$ is not injective on $\sigma$. Then one of the following holds:
- $\sigma$ has a ray $r$ such that $\pr_w(r) = 0$.
- $\sigma$ has rays $r,s$ such that $\pr_w(r) = \pr_w(s)$.
Assume $\sigma$ has rays $v_{I_1},\dots,v_{I_{n-3}}$, which are mapped to non-zero, distinct elements $v_{F_{I_j} \cap G(w)}$. By Lemma \[moduli\_lemma\_nested\], the flats $F_{I_j} \cap G(w)$ form a nested set. It is well-known that nested set subdivisions are simplicial, so the corresponding vectors must be linearly independent. In particular $\pr_w$ must be injective on $\sigma$.
\[moduli\_thm\_noninjective\] Let $n \geq 4$ and $w $ a weight vector as before. Let $\sigma$ be a top-dimensional cone of $\mk{n}$ in its combinatorial subdivision. Then $\sigma \in \curly{U}_w$ if and only if $\pr_w$ is not injective on $\sigma$.
First, let $\sigma \in \curly{U}_w = \bigcup_{k \geq 0} \curly{U}_w^k$. We will prove that $\pr_w$ is not injective on $\sigma$ by induction on $k$.
Assume $\sigma \in \curly{U}_w^0$. Then the combinatorial type of $\sigma$ is not $w$-stable, so $\sigma$ has a ray of the form $v_{\{i,j\}}$ with $w_i + w_j \leq 1$. This is equal to the ray $v_{F_{ij}}$, where $F_{ij}$ is the flat of $K_{n-1}$ consisting only of the edge between nodes $i$ and $j$. This edge does not exist in $G(w)$, so $\pr_w(v_{F_{ij}}) = 0$.
Consider a cone $\sigma \in \curly{U}_w^{k+1}$. That is, there is a codimension one cone $\tau$ such that the top-dimensional cones adjacent to $\tau$ are $\sigma, \sigma', \sigma''$ with $\sigma', \sigma'' \in \curly{U}_w^k$. By induction, we may suppose that $\pr_w$ is not injective on $\sigma'$ or $\sigma''$. The projection morphism induces a morphism on the local fan $\Star_{\mk{n}}(\tau)$, which is a tropical line, i.e. a one-dimensional balanced fan with three rays. The rays corresponding to $\sigma'$ and $\sigma''$ are mapped to 0, so by linearity of the local morphism, the ray corresponding to $\sigma$ must also be mapped to 0. Hence $\pr_w$ is not injective on $\sigma$.
Suppose that $\pr_w$ is not injective on a cone $\sigma$ with rays $v_{I_1},\dots,v_{I_{n-3}}$. By Lemma \[moduli\_lemma\_wheninjective\], we notice that either one ray of $\sigma$ is mapped to 0, or two rays are mapped to the same element.
In the former case, if $\pr_w(v_{I_j}) = 0$ for some $j$, then $w_a + w_b \leq 1$ for all $a,b \in I_j$. But this implies that $\sigma \in \curly{U}_w^0$. We now consider the latter possibility, where two rays are mapped to the same ray, say $\pr_w(v_{I_i}) = \pr_w(v_{I_j})$ for some $i \neq j$. We may assume that $1 \notin I_i,I_j$. Then we can also assume that $I_j \subset I_i$, since otherwise $I_i \cap I_j = \emptyset$ and thus $\pr_w(v_{I_i}) = \pr_w(v_{I_j}) = 0$. Now $I_i$ and $I_j$ correspond to two edges. Assume these edges do not share a vertex. Then there must be a chain of edges connecting them, corresponding to splits $I_j = J_1 \subset \dots \subset J_t = I_i$. As $\pr_w(I_i) = \pr_w(I_j)$, we must have $w_k + w_l \leq 1$ for all $k \in I_i \wo I_j, l \in I_j$. In particular $\pr_w(J_s) = \pr_w(I_j)$ for all $s = 1,\dots,t$. Hence we can assume that the edges corresponding to $I_i$ and $I_j$ share a common vertex. Denote these edges by $e_i$ and $e_j$ and the vertex by $v_{ij}:= e_i \cap e_j$.
First let us assume that $\abs{I_i} - \abs{I_j} = 1$, i.e. there is an additional leaf $l$ at $v_{ij}$. We prove that this cone is inherited unstable by an induction on $\abs{I_j}$. We start with $\abs{I_j} = 2$, i.e. $I_j = \{a,b\}$. This implies $w_l + w_a, w_l + w_b \leq 1$. We obtain a codimension one type $C_\tau$ by contracting the edge $e_j$. The two other adjacent top-dimensional types besides $C_\sigma$ have rays $v_{\{l,i\}}$ and $v_{\{l,j\}}$ respectively, both of which are mapped to 0. In particular, $\sigma$ lies in $\curly{U}_w^1$. Now assume $\abs{I_j} > 2$. The vertex of $e_j$ which is not $v_{ij}$ is also trivalent, i.e. we have a partition of $I_j$ into two edge splits $I_j', I_j''$. Again, we obtain a codimension one type $C_\tau$ by contracting $e_j$. We obtain an adjacent top-dimensional type $C_{\sigma'}$ replacing the rays $v_{I_i},v_{I_j}$ with rays $v_{I_j' \cup \{l\}}, v_{I_i}$. We now argue by induction that $\sigma' \in \curly{U}_w$. The exact same argument works for the third maximal cone $\sigma'' > \tau$, which finally implies $\sigma\in \curly{U}_w$ as required.
If $I_j' = \{a\}$, then $w_l + w_a \leq 1$ and $\sigma' \in \curly{U}_w^0$. If $\abs{I_j'} \geq 2$, then $\pr_w(v_{I_j'}) = \pr_w(v_{I_j' \cup \{l\}})$ (as $l$ is not connected to any element of $I_j$ in $G(w)$). As $\abs{I_j'} < \abs{I_j}$ and $\abs{I_j'\cup \{l\}}- \abs{I_j'}=1$, we can apply induction to see that $\sigma' \in \curly{U}_w$ (see also Figure \[moduli\_fig\_induction\]).
A similar induction argument works in the case that $\abs{I_i} - \abs{I_j} > 1$.
;
\[moduli\_cor\_needstable\] Let $w $ be a weight vector. Then $\pr_w$ contracts all top-dimensional cones of $\mk{n}$, if and only if $w$ does not have at least two heavy entries.
First, assume $w$ has no heavy entry. Choose $j \in [n]$, such that $w_j$ is minimal. Then $w_j + w_k \leq 1 $ for all $k \neq j$: If we assume $w_j + w_k > 1$ for some $k$, then $w_k + w_l > 1$ for all $ l\neq k$ and $k$ is heavy in $w$.
In particular, in any trivalent $w$-stable combinatorial type the leaf $j$ can only be at a vertex with two bounded edges $e,e'$ corresponding to rays $v_I, v_{I \cup \{j\}}$. But both rays are mapped to the same image under $\pr_w$.
Now suppose $w_1$ is the only heavy weight. Again, choose $j$ such that $w_j$ is minimal. As before, we must have $w_j + w_k \leq 1$ for all $k \neq 1,j$. Let $C$ be a trivalent, $w$-stable curve. If leaf $j$ is attached to a vertex with two bounded edges, these edges correspond to rays $v_I, v_{I \cup \{j\}}$ with $1 \notin I$, which are mapped to the same image. In particular, the cone corresponding to $C$ lies in $\curly{U}_w$ by Lemma \[moduli\_thm\_noninjective\]. Hence $C$ must have a bounded edge corresponding to the ray $v_{\{1,j\}}$. If we contract this edge, we obtain a curve corresponding to a codimension one cone. But any other adjacent top-dimensional types are in $\curly{U}_w$: It either has a ray $v_{\{j,k\}}, k \neq j$, which makes it unstable, or leaf $j$ is adjacent to two bounded edges. The corresponding cone lies in $\curly{U}_w$ by our previous argument. Hence the cone of $C$ lies in $\curly{U}_w$ as well and thus is contracted under $\pr_w$ by Lemma \[moduli\_thm\_noninjective\].
Conversely, assume $w$ has heavy entries $i$ and $j$. Then we can easily construct a trivalent curve whose cone does not lie in $\curly{U}_w$: Let $\{i_1,\dots,i_{n-2}\} = [n]\wo \{i,j\}$ be in some arbitrary but fixed order. Then it is easy to see that $\pr_w$ is injective on the cone corresponding to the “caterpillar tree” with edges $v_{\{i,i_1\}},v_{\{i,i_1,i_2\}}, \dots,v_{\{i,\dots,i_{n-3}\}}$.
We now have the necessary ingredients to prove Theorem \[moduli\_cor\_mainresult\].
The dimension of $B'(G(w))$ equals $\textnormal{rank}(G(w)) -1$. By assumption, $G(w)$ is a connected graph, so its rank is just the number of its vertices minus one. In total, we obtain $\dim B'(G(w)) = n-3 = \dim \mk{n}$. By Corollary \[moduli\_cor\_needstable\], the dimension of the image of $\pr_w$ is also $n-3$. Since $B'(G(w))$ is irreducible, $\pr_w$ is surjective. Using Lemma \[moduli\_lemma\_nested\], it is easy to see that two different top-dimensional cones on which $\pr_w$ is injective are mapped to distinct cones of the nested set subdivision of $B'(G(w))$.
We now study balanced fan structures on $\mk{w}$.
\[prop-nocontract\] Let $w $ be a weight vector with at least two heavy entries. Then $\curly{U}_w = \curly{U}_w^0$ if and only if every $i \in [n]$ is either heavy or small in $w$.
First, assume that $\curly{U}_w = \curly{U}_w^0$. Assume there is an entry $i$, which is neither small nor heavy in $w$. Hence, there must be a $j$, such that $w_i + w_j > 1$, but $j$ is not heavy (in particular, it is also not small in $w$). It follows that there must be a $k \neq i,j$, such that $w_i + w_k, w_j + w_k \leq 1$.
For now assume $n \geq 5$. Let $a,b$ be two heavy entries of $w$ and fix an order $\{i_1,\dots,i_{n-5}\}$ on $[n] \wo \{i,j,k,a,b\}$. Now let $\sigma$ be the cone with rays $$v_{\{i,j\}}, v_{\{i,j,k\}},v_{\{i,j,k,i_1\}}\dots,v_{\{i,j,k,i_1,\dots,i_{n-5}\}}$$ (see also Figure \[moduli\_fig\_stillunstable\]). The corresponding combinatorial type is clearly $w$-stable, but $\pr_w(v_{\{i,j\}}) = \pr_w(v_{\{i,j,k\}})$. This is a contradiction.
(0,0) – (2,0); (2,0) – (3,0); (3,0) – (5,0); (0,0) – (-0.5,0.3) node\[left\][$a$]{}; (0,0) – (-0.5,-0.3) node\[left\][$b$]{}; (1,0) – (1,-0.5) node\[below\][$i_{n-5}$]{}; (4,0) – (4,-0.5) node\[below\][$k$]{}; (5,0) – (5.5,0.3) node\[right\][$i$]{}; (5,0) – (5.5,-0.3) node\[right\][$j$]{};
If $n = 4$, then only the cone spanned by $v_{\{i,j\}}$ corresponds to a $w$-stable type, so by definition it must lie in $\curly{U}_w$.
To see the converse, we can assume that $w$ is of the form $(1^f, \epsilon^t)$ with $f \geq 2$. It suffices to show that $\curly{U}_w^1 = \emptyset$, i.e. after removing all $w$-unstable types, all remaining codimension one combinatorial types still have at least two resolutions. This is clear and the claim follows immediately.
\[thm-heavyandlight\] Let $w$ be heavy/light, with at least two heavy entries. The cone complex underlying $\pr_w(\mk{n})=B'(G(w))$ is If $w$ is not heavy/light, then there does not exist a balanced embedding of $\mk{w}$ into a vector space.
Suppose $w$ is a weight vector with only heavy and light entries. It follows from Lemma \[moduli\_thm\_noninjective\] that $\pr_w$ is injective on all cones which are not in $\curly{U}_w$. We can deduce from Proposition \[prop-nocontract\] that $\curly{U}_w=\curly{U}_w^0$. In other words, $\pr_w$ contracts only the top-dimensional cones we want to contract to pass from $\mk{n}$ to $\mk{w}$. The fact that the small points are light guarantees that $\mk{w}$ is pure dimensional. To see this, consider a cone whose top dimensional faces are contracted. Such a cone is spanned by vectors $v_{I_j}$, and there must exist at least one vector such that all $k\in I_j$ are light. Analyzing the projections, we see that $B'(G(w))$ is identified with $M^{\trop}_{0,w}$.
Consider a weight vector $w$ that is not of heavy/light type. We first deal with the case where $w$ has heavy and small points. Recall $i$ is small if $w_i+w_j>1$ implies that $j$ is heavy. In this situation, there is a subset $I$ of size at least three of small points satisfying $\sum_{k\in I}w_k>1$, but for any subset $I_0\subsetneq I$, $\sum_{k\in I_0}w_k<1$. Observe that there is a cone $\tau$ of $\mk{w}$ of maximal possible dimension containing the ray $v_I$ such that all its higher-dimensional faces of $\tau$ in $\mk{n}$ are not $w$-stable.
Notice that since we have at least two heavy points, the top-dimensional cone of $\mk{n}$ corresponding to the caterpillar tree with the two heavy weights on the two sides as in the proof of Corollary \[moduli\_cor\_needstable\] is $w$-stable, and hence is also a top dimensional cone of $\mk{w}$. We can now see that $\mk{w}$ is not pure dimensional, and cannot be embedded as a balanced fan. Assume now that $w$ has not only heavy and small points. By Proposition \[prop-nocontract\], $\curly{U}_w\neq \curly{U}_w^0$, so there are cones of codimension one with only one adjacent top-dimensional cone. These cones cannot be embedded in a balanced way.
Tropicalizing spaces of rational weighted stable curves {#sec: trop-wsc}
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Throughout, we work in the “constant coefficient” case, i.e. over $\mathbb C$ with the trivial valuation. We have seen that for a vector of heavy and light weights, we obtain a fan structure for the tropical moduli space $\mk{w}$. In this section, we show that this fan yields a toric variety in which we can embed $\overline{M}_{0,w}$, and the tropicalization of the open part living inside the torus can be given a canonical fan structure, making it isomorphic to $\mk{w}$. If we have not only heavy and light weights, then we still have a map from $\overline{M}_{0,w}$ to the toric variety defined by $\pr_w(\mk{n})$, but it contracts some boundary strata.
Geometric tropicalization for $\overline M_{0,n}$
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In [@GM07 Example 3.1], the locus of smooth curves $M_{0,n}$ is identified with the quotient of an open set of the Grassmannian, denoted $G^0(2,n)$, by a an $n-1$ dimensional torus. The open set $G^0(2,n)$ corresponds to the $2$ planes that do not pass through the intersection of a pair of coordinate planes.
The Grassmannian $G(2,n)$ embeds into $\P^{\binom{n}{2}-1}$ via the Plücker embedding, associating to a $2$-plane given by the data of a $2\times n$ matrix (after a choice of basis) its minors. This embedding carries the open part $G^0(2,n)$ to points in the torus of $\P^{\binom{n}{2}-1}$. As a consequence, $M_{0,n}$ is embedded into the torus $(T^{\binom{n}{2}}/T)/T^{n-1}\cong T^{\binom{n}{2}-n}$ using the Plücker embedding.
Note that the action of the $T^{n}$ torus corresponds precisely to the lineality space $\Phi(\R^n)$ that we quotient by embedding $\mk{n}$ into $\R^{\binom{n}{2}-n}$ (see Section \[prelim-trop\]). Comparing coordinates on the algebraic and tropical sides, we can effectively neglect the action of $T^{n}$ on one side and the lineality space on the other. Furthermore, the Plücker coordinates give us the distance coordinates in tropical geometry directly.
Keeping with the discussion in previous sections, we recall that the tropical moduli space $M_{0,n}^{\trop}$ comes with an embedding in a vector space, and a natural fan structure. Fix this fan structure. Then we have the following result, due to Gibney and Maclagan [@GM07 Theorem 5.7], as well as Tevelev [@Tev07 Theorem 5.5]
Consider the embedding of $M_{0,n}$ into the torus $T^{\binom{n}{2}-n}$ described above. The closure of $M_{0,n}$ in the toric variety $X(M_{0,n}^{\trop})$ is the compactification $\overline M_{0,n}$. Furthermore, the tropicalization of $M_{0,n}$ in this torus is $\mk{n}$.
The essential ingredient in the proof of this result is the understanding of the combinatorial structure of $M^{\trop}_{0,n}$ and in particular its relationship to the boundary stratification of the classical moduli space $\overline M_{0,n}$. The results provides a beautiful explanation of the various analogies and combinatorial dualities between the tropical and classical moduli space in question. In Theorem \[thm-tropicalizing\], we obtain analogous results for spaces of weighted stable curves.
The result can be obtained with the help of a technique that is now known as *geometric tropicalization* – initially used to study compactifications of subvarieties of tori in [@Tev07]. The technique was elaborated upon and applied to understand compactifications of moduli space of del Pezzo surfaces in [@HKT09]. An accessible introduction to the topic can be found in [@Cue11; @MS09].
Geometric tropicalization starts with a variety $X$ together with a simple normal crossing boundary divisor $\Delta$ (such as $\overline{M}_{0,n}$ with its usual boundary). When the complement $U$ of $D$ in $X$ has many invertible functions, it admits a map to a torus: $$\iota: U\to T.$$ In ideal situations (and indeed, in our situation) this map is an embedding. The map $\iota$ may then be used to produce a map from the dual intersection complex $\Sigma$ of $(X, D)$ to the vector space of one parameter subgroups of $T$, thus furnishing a fan structure on $\Sigma$.
In [@Cue11], Cueto equips the top dimensional cones of $\Sigma$ with a weight function which produces a balanced fan. This fan furnishes a toric variety with dense torus $T$, in which we may consider the closure of $\iota(U)$. Geometric tropicalization studies the relationship between $X$ and $\overline {\iota(U)}$.
Geometric tropicalization for spaces of weighted stable curves
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We assume that $w$ is a weight vector with only heavy and light weights, that there are at least $2$ heavy weights, and without loss of generality, that the heavy entries are the first two weights. We have seen already that the assumption of having at least two heavy weights makes sense in tropical geometry, as without it, the tropical moduli space is of incorrect dimension. On the algebraic side, this requirement is natural, because else we would have an empty locus of smooth curves.
We apply the geometric tropicalization techniques discussed above in the context of heavy/light moduli spaces of rational pointed curves. Note that the compactification of $M_{0,n}$ to $\overline{M}_{0,w}$ is *not*, in general, simple normal crossing. However, the following observation is important, and its proof is identical to the $\overline M_{0,n}$ case.
Let $M_{0,w}$ denote the locus of smooth curves in $\overline M_{0,w}$. Then, the boundary $\overline M_{0,w}\setminus M_{0,w}$ is a divisor with simple normal crossings.
In other words, we consider the “interior” of the moduli space $\overline M_{0,w}$ as not only points of $M_{0,n}$ but also loci of marked curves where the underlying curve is smooth, but the markings are not distinct. We warn the reader that the Hassett spaces $\overline{\mathcal{M}}_{g,w}$ are usually *not* toroidal compactifications of $\mathcal{M}_{g,w}$, however, the locus of non-smooth curves is often a divisor with simple normal crossings. See the paper by Ulirsch [@U].
\[rem: m0w-boundary\] The inclusion $M_{0,w}\hookrightarrow \overline M_{0,w}$ induces a stratification into locally closed strata, which agrees with the stratification by dual graph: the codimension $k$ strata of $\overline M_{0,w}$ are the loci of curves, with fixed dual graph, having $k$ nodes. Locally analytically near a stratum $S$, there is a collection of monomial coordinates on $\overline M_{0,w}$ given by the deformation parameters for the nodes of the curve(s) parametrized by $S$.
Let $\Pr_w$ be the projection from the torus $T^{\binom{n}{2}}/T^n$ dropping all the Plücker coordinates indexed by $i\neq j\in [n]$ for which $w_i=w_j=\epsilon$.
\[lem-prw\] The tropicalization of the map $\Pr_w$ agrees with the projection $\pr_w$ from $\R^{\binom{n}{2}-n}$ (see Remark \[rem-prw\]).
By [@KM07], Lemma 2.3 and 2.4, the vectors $v_I$ where $I$ is a two-element subset not containing $1$ and not equal to $\{2,3\}$ form a basis of $\R^{\binom{n}{2}}/\Phi(\R^n) $. A ray $v_I$ can be expressed in terms of the basis vectors using [@KM07], Lemma 2.6, which tells us that $v_I$ equals the sum of all $v_S$ where $S\subset I$ is a two-element subset (we assume without restriction that $1\notin I$), and the fact that $-v_{\{2,3\}}$ equals the sum of our basis vectors above. The tropicalization of the map $\Pr_w$ contracts the vectors $v_{\{i,j\}}$ (which equal $-2e_{ij}$ modulo the lineality space, being the images of $-e_{ij}$ in $B'(K_{n-1})$) for all $i,j$ such that $w_i=w_j=\epsilon$, and with these, it also contracts all rays of the form $v_I$ with $w_i=\epsilon$ for all $i\in I$, since we can express the latter in terms of the $v_{\{i,j\}}$ by the above. Thus, it equals the map $\pr_w$.
\[lem-embed\] The open part $M_{0,w}$ can be embedded into the torus $\Pr_w( T^{\binom{n}{2}}/T^n)$ using the Plücker coordinates.
Let us compare the open part $M_{0,w}$ to $M_{0,n}$: now points which are both light are allowed to collide. In the $2\times n$ matrix describing a collection of $n$ points (resp. a two-plane in $G(2,n)$) this means that two columns can now coincide (up to nonzero multiple), leading to a zero minor. However, these are exactly the minors we project away with $\Pr_w$, so using the remaining Plücker coordinates, we embed $M_{0,w}$ into the torus $\Pr_w( T^{\binom{n}{2}}/T^n)$.
\[lem-geomtrop\] The geometric tropicalization of $\overline{M}_{0,w}$ using the embedding in Lemma \[lem-embed\] is identified with $\pr_w(\mk{n})$.
It is straightforward to see that the dual intersection complex of $\overline{M}_{0,w}$ is canonically identified (as a cone complex) with $\mk{w}$. We know already by Theorem \[thm-heavyandlight\] that the latter is the cone complex underlying the fan $\pr_w(\mk{n})$. Thus, it remains only to check that the divisorial valuations of the boundary divisors for the remaining Plücker coordinate functions yield the rays of this fan. This is an easy consequence of Lemma \[lem-prw\].
Finally, we must check that the weight function for the geometric tropicalization, as given in [@Cue11]\*[Theorem 2.5]{}, is identically $1$, thus matching the weight on the Bergman fan $\pr_w(\mk{n})=B'(G(w))$. This again follows from the analogous fact for $\mk{n}$: the rays of a top-dimensional cone $\sigma$ span the lattice $\Lambda_\sigma$ of this cone, and the corresponding boundary divisors of $\overline{M}_{0,n}$ intersect in a point of multiplicity one.
\[thm-tropicalizing\] Let $w$ be heavy/light. Embed $M_{0,w}$ as described in Lemma \[lem-embed\] into the torus $\Pr_w( T^{\binom{n}{2}}/T^n)$, and compactify it in the toric variety defined by $\pr_w(\mk{n})=\mk{w}$, then we obtain $\overline{M}_{0,w}$. Furthermore, the tropicalization of $M_{0,w}$ in these coordinates equals $\pr_w(\mk{n})=\mk{w}$.
We wish to show that the map $\overline M_{0,w} \to X(M_{0,w}^{\trop})$ is an embedding. According to [@HKT09 Lemma 2.6 (4), Theorem 2.10], this occurs when the following two conditions hold. Let $S$ be a stratum, let $M_{S}$ be $\mathcal O^*(S)/k^*$ and $M^S_{M_{0,w}}$ be the sublattice of $\mathcal O^*(M_{0,w})/k^*$ generated by units having zero valuation on $S$.
1. For each boundary divisor $D$ containing $S$, there is a unit $u\in \mathcal O^*(M_{0,w})$ with valuation $1$ on $D$ and valuation $0$ on other boundary divisors containing $S$.
2. $S$ is very affine and the restriction map $M^S_{M_{0,w}}\to M_S$ is surjective.
For (1) observe that for each boundary divisor $D$, we can choose an appropriate forgetful morphism of $M_{0,4}$, informally a *cross ratio* map, as is done in [@Tev07 Section 5]. It is straightforward to check that these functions have valuation $1$ on $D$ and valuation $0$ on any other boundary divisor.
We see that all the strata $S$ are very affine in this case. Recall that the divisors containing a given stratum $S$ are precisely those rays of $M_{0,w}^{\trop}$ which are contained in the cone $\sigma_S$ corresponding to $S$. It follows immediately now from the discussion of the boundary stratification of $M_{0,w}$ in Remark \[rem: m0w-boundary\] that the restriction map is surjective.
If we drop the condition of having only heavy and light points, many of the statements discussed here are still true. The geometric tropicalization of $\overline{M}_{0,w}$ using the embedding in Lemma \[lem-embed\] still equals $\pr_w(\mk{n})$, however we know already that not all cones are mapped injectively in this case. As a result, the underlying abstract cone complex of $\pr_w(\mk{n})$ is not $\mk{w}$. On the algebraic side, this is reflected by the fact that we still have a map from $\overline{M}_{0,w}$ to the toric variety defined by $\pr_w(\mk{n})$, but it does not map all boundary strata injectively.
Allowing edge lengths to become infinite, analogously to [@ACP12] we obtain an extended cone complex $\overline M^{\trop}_{0,w}$. The arguments above also show that for $w$ heavy/light, the extended tropicalization of $\overline M_{0,w}$ inside the toric variety $X(M_{0,w}^{trop})$ can be identified with $\overline M^{\trop}_{0,w}$.
Extended example: Losev–Manin spaces {#sec: LosevManin}
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Let $w$ be the weight vector $(1,1,\epsilon,\cdots,\epsilon)$ for $\epsilon$ light. The space $\overline{M}_{0,w}$ is called the *Losev–Manin* moduli space and parametrizes chains of projective lines with $n$ marked points, where $n = \ell(w)$. These spaces were introduced and studied in [@losevmanin] and play a role, for instance, in the theory of relative stable maps, as a target for branch morphisms.
The Losev–Manin moduli spaces are toric varieties themselves, and as a result the situation simplifies considerably in this case. In fact, there is some beautiful combinatorics that arises in this situation. See [@Bat11] for a proof of the following proposition. See also [@Kap93; @losevmanin]. Let $w$ be heavy/light, as above.
Let $X_n$ be the toric variety obtained by blowing up $\mathbb P^{n-3}$ at all torus invariant subvarieties up to codimension $2$ in order of decreasing codimension. The Losev–Manin moduli space $\overline M_{0,w}$ is isomorphic to $X_n$.
The polytope associated to $X_n$ is the permutahedron, and the fan $\Sigma(X_n)$ is the normal fan over the permutahedron.
In the special case of Losev–Manin, the *modular boundary*, i.e. the complement of $M_{0,n}$, is not normal crossing. However, the locus of non-smooth curves coincides with the toric boundary (the complement of the big torus), which is simple normal crossing.
Note that since the tropical moduli space $\pr_w(\mk{n})=\mk{w}$ is a complete fan, the embedding of $\overline{M}_{0,w}$ into the corresponding toric variety is surjective. The fact that Losev–Manin spaces are toric can thus also be derived from Theorem \[thm-tropicalizing\].
Let us discuss some aspects of this fan more closely. We can use Theorem \[moduli\_cor\_mainresult\] and \[thm-heavyandlight\] and study $B'(G(w))=\pr_w(\mk{n})=\mk{w}$. The graph $G(w)$ is a star graph, i.e. it consists of $t$ edges meeting in a single vertex. The matroid of this graph is $U_{t,t}$, so we see that $\mk{w} \cong \R^{t-1}$. Furthermore, the subdivision of $\R^{t-1}$ is the nested set subdivision with respect to the 1-connected flats of $G(w)$. However, all flats of $G(w)$ are 1-connected, so the subdivision is actually the chains-of-flats-subdivision of $U_{t,t}$.
We can also describe the tropical curves we parametrize more concretely: Any $w$-stable rational curve is a so-called “caterpillar tree” (Figure \[geom\_fig\_losevmanin\]): It consists of a single chain of edges with the “heavy” leaves at either end and the remaining leaves distributed at will along the chain of edges.
(0,0) – (1,0); (1,0) – (3,0); (3,0) – (4,0); (0,0) – (-0.5,0.3) node\[left\][$1$]{}; (0,0) – (-0.5,-0.3) node\[left\][$\epsilon$]{}; (1,0) – (1,-0.5) node\[below\][$\epsilon$]{}; (2,0) – (2,-0.5) node\[below\][$\epsilon$]{}; (3,0) – (3,-0.5) node\[below\][$\epsilon$]{}; (4,0) – (4.5,0.3) node\[right\][$1$]{}; (4,0) – (4.5,-0.3) node\[right\][$\epsilon$]{};
We can identify each such curve through its vector of leaf distances $$(\dist(l_1,l_3),\dots,\dist(l_1,l_n)) \in \R^t.$$ In turn, each element of $\R^t$ can be considered such a distance vector, if we set its smallest entry to 0. So again, we obtain as parameter space $\R^t / (1,\dots,1) \cong \R^{t-1}$. A canonical subdivision is dictated by the combinatorial types, more precisely, we obtain a top-dimensional cone for each of the $(n-2)!$ orderings on the leaves $l_3,\dots,l_n$. One can easily check that this is the same as the chains-of-flats subdivision of $U_{t,t}$.
Spaces of weighted stable curves and Berkovich skeletons {#sec: skeleta}
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We continue to work over trivially valued $\mathbb C$. Let $X$ be a proper normal variety. Let $U\hookrightarrow X$ be given by the open complement of a normal crossing divisor $D$. The pair $(X,D)$ carries the structure of a toroidal embedding, in the sense of [@KKMSD]. Associated to any toroidal embedding is an extended cone complex $\overline\Sigma(X)$. Thuillier [@Thu07] realizes this cone complex as a skeleton of the Berkovich analytic space $X^{an}$. Builiding on this, Abramovich, Caporaso, and Payne [@ACP12] identify the tropical moduli spaces $\overline M_{g,n}^{\trop}$ with the skeleton of the Berkovich analytification of $\overline{\mathcal{M}}_{g,n}$. They use this formalism to study a functorial tropicalization for this moduli space.
We now extend this to the present situation. To state the results most cleanly, it is convenient to work with the extended cone complex $\overline M^{\trop}_{0,w}$, obtained by allowing edge lengths to become infinite, identical to the $\overline{\mathcal{M}}^{\trop}_{g,n}$ case.
We define a “set theoretic” tropicalization map $$\trop: \overline M_{0,w}^{an}\to \overline M_{0,w}^{\trop},$$ as follows. Let $p\in \overline M^{an}_{0,w}$. Such a point $p$ can be represented by a stable curve $[C]$ over a valued extension $K$ of $\mathbb C$. Since $\overline M_{0,w}$ is proper, this extends to curve over the valuation ring $R$ of $K$. Define $\trop(p)$ to be the dual graph $\Gamma_C$ of the special fiber of $[C]$. The edges of $\Gamma_C$ correspond to nodes in the special fiber. Such a node has a defining equation $$xy = f,$$ where $f\in R$. We assign the corresponding edge length equal to $val(f)$. Note that if nodes appear in the generic fiber, then the defining equation is locally $xy = 0$, and the corresponding edge has length $\infty$.
Let $w$ be heavy/light as before. Recall from the previous section that the complement of the locus of smooth curves in $\overline M_{0,w}$ is a divisor with simple normal crossings. We have the following result.
The (extended) cone complex $\overline M_{0,w}^{\trop}$ is identified with the skeleton of $\overline M_{0,w}^{an}$. Furthermore, there exists a section of the tropicalization map $\trop: \overline M^{an}_{0,w}\to \overline M^{\trop}_{0,w}$, $$s: \overline M^{\trop}_{0,w}\to \overline M^{an}_{0,w},$$ which realizes the tropicalization as a skeleton of the Berkovich space. Furthermore, there is a canonical strong deformation retract from $\overline M^{an}_{0,w}$ onto $\overline M^{\trop}_{0,w}$.
The proof is essentially the same as the corresponding statement for $\overline M_{g,n}$, so we merely provide a sketch. Consider a $0$-stratum of $\overline M_{0,w}$, with respect to the previously described toroidal structure. Let $[C]$ be the $w$-stable curve parametrized by this stratum. The deformation parameters of the nodes of $[C]$ form a system of local coordinates near $[C]\in \overline M_{0,w}$. This furnishes a formal neighborhood of $[C]$ isomorphic to a formal $\mathbb A^N$, where $N$ is the number of nodes of $[C]$. The valuations of these deformation parameters yield coordinates on the top dimensional cone of $M_{0,w}^{\trop}$ corresponding to this zero stratum. However, these are naturally identified with coordinates on the top dimensional cone of $M_{0,w}^{\trop}$ parametrizing tropical $w$-stable curves with underlying combinatorial type given by the dual graph $\Gamma_C$ of $[C]$. For higher dimensional strata, the deformation parameters form a subset of the coordinates of a formal local affine space, which map to lower dimensional cones in the skeleton. The fact that the set theoretic tropicalization map agrees with Thuillier’s “projection to the skeleton” map is standard and follows from analogous arguments in [@ACP12 Section 6].
Suppose that $w$ is not necessary heavy/light. Let $D$ be the boundary divisor of $\overline M_{0,w}$ given by the union of divisors corresponding to the rays of $\textnormal{pr}_w(M_{0,n}^{\trop})$. The analysis carried out in Section \[section\_moduli\] characterizes the boundary intersections of components of $D$. More precisely, two irreducible boundary divisors $D_i$ and $D_j$ intersect precisely when the corresponding rays of $\textnormal{pr}_w(M^{\trop}_{0,n})$ span a $2$-dimensional cone $\sigma$. The situation generalizes in the natural way for manifold intersections. Furthermore, the combinatorial type of graphs parametrized by $\sigma$ are dual to the universal curve over $D_i\cap D_j$. Consequently, the boundary divisor $D$ is simple normal crossing, and identical arguments as above yield the following.
The cone complex $\textnormal{pr}_w(\overline M_{0,n})$ is identified with the Thuillier skeleton of $\overline M_{0,w}^{an}$, with the toroidal structure coming from the inclusion of the complement of the divisor $D$ above.
We return now to the $w$ heavy/light case. The Hassett spaces admit natural tautological morphisms, known as reduction maps (c.f. Lemma \[lem-prw\]). Given two weight data $w = (w_j)$ and $w' = (w'_j)$ such that $w_i\geq w'_i$ for all $i$, there exists a natural birational morphism $$\rho_{w,w'}: \overline M_{0,w}\to \overline M_{0,w'},$$ obtained by collapsing components of curves that become unstable under the weights $w'$. In particular, there always exists a reduction map $$\rho_w:\overline M_{0,n}\to \overline M_{0,w}.$$
The map $\rho_w$ is compatible with the tropical projection maps $pr_w$. More precisely, in the notation above, $\textnormal{pr}_w = \trop\circ \rho_w$.
Recall that the formation of skeletons is functorial for toroidal morphisms. Since $\rho_w$ is birational, it suffices to check that for any point $[C]\in \overline M_{0,n}$, there exist formal local toric charts around $x$ and $\rho_w(x)$, such that the monomial coordinates on the target pullback to monomial coordinates on the source. We take the charts to be the ones given by the deformation parameters of $[C]$. Note that the inverse image of a node of $\rho_w([C])$ is a single node of $[C]$. If $\zeta$ is the deformation parameter at the node of $\rho_w([C])$, notice that $\rho_w^*\zeta$ is simply $\widetilde \zeta$ where $\widetilde \zeta$ is the deformation parameter of the corresponding node of $[C]$. The morphism is clearly toric and dominant in the local charts, and the result follows.
Spaces of rational weighted stable curves as fibre products {#section_fibre}
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In this section, we express the projections $\pr_w(\mk{n})$ in terms of fibre products. We use the equality $\pr_w(\mk{n})=B'(G(w))$ from Theorem \[moduli\_cor\_mainresult\] and study general properties of fibre products of Bergman fans. If $w$ has only heavy and light points, the tropical description as fibre products matches the analogous algebraic description nicely.
### Hassett spaces with weight $0$ points {#sec: alg-fibre-products}
Natural from the perspective of Mori chamber decompositions and the log minimal model program, Hassett considers “zero weight” variations on the moduli problem for weighted stable curves. That is, we consider $w = (w_1,\ldots, w_n)$ with $0\leq w_j\leq 1$, and $\sum w_i>2$.
The resulting moduli space $\overline{M}_{0,w}$ can be described as follows. Let $w^+$ be the vector of weights containing the positive entries of $w$, and assume that there are $t$ entries equal to $0$ in $w$. Following Hassett [@Has03 Section 2], the moduli space $\overline M_{0,w}$ is identified with the $t$-fold fibre product of the universal curve $\mathcal C_{0,w^+}$ of $\overline M_{0,w^+}$ over $\overline M_{0,w^+}$, $$\overline M_{0,w} \cong \mathcal C_{0,w^+}\times_{\overline M_{0,w^+}}\cdots \times_{\overline M_{0,w^+}}\mathcal C_{0,w^+}$$
The special case when all positive weights are equal to $1$ allows the universal family to also be identified with the moduli space of curves with one more point of arbitrary weight. In particular, we see that
$$\overline M(1^f,0^t)\cong\overline{M}_{0,f+1}\times_{\overline{M}_{0,f}}\ldots \times_{\overline{M}_{0,f}}\overline{M}_{0,f+1}.$$
Replacing the $0$ weights with $\epsilon$ weights, we obtain a birational morphism $$\overline M(1^f, \epsilon^t)\to \overline M(1^f,0^t),$$ which is a desingularization of $\overline M(1^f,0^t)$. The exceptional loci are described explicitly in [@Has03 Corollary 3.5]. Informally, $\overline M(1^f,0^t)$ contains a locus parametrizing curves in which multiple $0$-weight points can collide with nodes. We now investigate the tropical analogue
### Notions from tropical geometry and graph theory
Let $f: X \to Y := B(M)$ be a morphism from a tropical fan to a Bergman fan. Assume there are rational functions $\varphi_1,\dots,\varphi_r$ on $Y$ and $C := \varphi_1 \cdot \dots \cdot \varphi_r \cdot Y$ (for an in-depth discussion of rational functions and divisors, see for example [@AR07]). Then we define the *pull-back* of $C$ along $Y$ to be $$f^*C := (\varphi_1 \circ f) \cdot \dots (\varphi_r \circ f) \cdot X.$$
One could of course make this definition for an arbitrary target variety $Y$. However, in this case the pull-back may depend on the choice of rational functions $\varphi_i$. That this is not the case for Bergman fans was shown in [@FR10]\*[Example 8.2]{}.
We will need this definition in the case, where $Y = B(N) \times B(N) = B(N \oplus N)$ for some matroid $N$ and $C = \Delta_Y := \{(x,x); x \in Y\}$ is the diagonal of $Y$. It was shown in [@FR10]\*[Corollary 4.2]{} that there are rational function $\varphi_1,\dots,\varphi_r, r = \textnormal{rank}(N)$ on $B(N) \times B(N)$ such that $\Delta_{B(N)} = \varphi_1 \cdot \dots \varphi_r \cdot (B(N) \times B(N))$.
Let $f:B(M) \to B(N), g:B(M') \to B(N)$ be morphisms of Bergman fans. Then we define their *fibre product* $$B(M) \times_{B(N)} B(M') := (f,g)^*(\Delta_{B(N)}).$$ Now assume we have morphisms $f': B'(M) \to B'(N), g': B'(M') \to B'(N)$. Both induce morphisms $f: B(M) \to B(N), g: B(M') \to B(N)$ and the corresponding fibre product contains a lineality space $L$ generated by $(1,\dots,1)$. Hence we can define $$B'(M) \times_{B'(N)} B'(M') := \left(B(M) \times_{B(N)} B(M') \right) / L.$$
Tropical fibre products were first defined in [@FH11] in the more general context of *smooth* tropical varieties. However, as the definition is a bit more involved and requires notions from intersection theory, we will restrict ourselves to fibre products of Bergman fans.
All examples of fibre products that we consider in this paper will be “nice” in the sense that they are themselves Bergman fans and their support is equal to the set-theoretic fibre product $\{(x,y) \in B(M) \times B(M'); f(x) = g(y)\}$. However, both statements are false in general: The fibre product need not be a Bergman fan (in fact, it need not even be isomorphic to one!). Also, it is in general strictly contained in the set-theoretic fibre product. In fact, the latter may very well be a cone complex that is not pure or has the wrong dimension.
The tropical fibre product, however, always has the “correct” dimension due to its intersection-theoretic definition: $$\dim \left(B(M) \times_{B(N)} B(M')\right) = \textnormal{rank}(M) + \textnormal{rank}(M') - \textnormal{rank}(N).$$ This follows from the fact that we apply $\textnormal{rank}(N)$ many rational functions to the tropical fan $B(M) \times B(M')$.
In order to discuss tropical fibre products of Bergman fans of graphic matroids, we use the following terminology from graph theory.
Let $G$ be a graph on vertices $V$. We call $G$ a *split graph*, if there is a partition $V = L \amalg S$, such that:
- Restricted to $L$, $G$ is the complete graph on $L$.
- Restricted to $S$, $G$ is the empty graph.
We call $G$ a *chordal graph*, if for any cycle $C$ of length $\abs{C} > 3$ there exists an edge $e \notin C$ joining two vertices of $C$. We will call $e$ a *chord* of $C$.
We have the following elementary observation:
Every split graph is also chordal.
Spaces of rational weighted stable tropical curves as fibre products
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We are now ready to apply the graph terminology just introduced to our graphs $G(w)$, in order to study their Bergman fans as fibre products.
\[geom\_lemma\_split\] Let $w$ be a weight vector. Then $G(w)$ is a split graph.
We partition the vertex set $V := \{2,\dots,n\}$ into $\{i \in V: w_i > \frac{1}{2}\}$ and $\{i \in V: w_i \leq \frac{1}{2}\}$. The statement is now clear.
Let $G_0, G_1,G_2$ be graphs and assume $G_1 $ and $G_2$ both contain a subgraph $G_0$. We then denote by $G_1 \times_{G_0} G_2$ the graph obtained by gluing $G_1,G_2$ along these subgraphs.
\[geom\_prop\_fibreproduct\] Let $G_1,G_2$ be connected graphs, both containing a connected subgraph isomorphic to some $G_0$. If $G_1 \times_{G_0} G_2$ is a chordal graph, then $$B'(G_1) \times_{B'(G_0)} B'(G_2) \cong B'(G_1 \times_{G_0} G_2).$$ Furthermore, the support of the left hand side is the set-theoretic fibre product $$S(G_1 \times_{G_0} G_2) := \{ (v^1,v^2); v^i \in B'(G_i) \textnormal{ and } v_e^1 = v_e^2 \textnormal{ for all } e \in G_0\}.$$
First of all note that, since all graphs are connected, we have $$\begin{aligned}
\textnormal{rank}(G_1 \times_{G_0} G_2) &= \abs{V(G_1 \times_{G_0} G_2)} -1\\
&= \abs{V(G_1)} + \abs{V(G_2)} - \abs{V(G_0)} - 1\\
&= \textnormal{rank}(G_1) + \textnormal{rank}(G_2) - \textnormal{rank}(G_0)
\end{aligned}$$ In particular, both spaces have the same dimension. We must show that the linear map $$\begin{aligned}
i: S(G_1 \times_{G_0} G_2) &\to B'(G_1 \times_{G_0} G_2) \\
( (v_e)_{e \in G_1}, (v_e)_{e \in G_2}) &\mapsto (v_e)_{e \in G_1 \times_{G_0} G_2}\end{aligned}$$ is an embedding. More precisely, we only have to show that it is well-defined, i.e that its image lies in $B'(G_1 \times_{G_0} G_2)$.
So assume $v^1 \in B'(G_1), v^2 \in B'(G_2)$ and for all edges $e$ in $G_0$ we have $v_e^1 = v_e^2$. Let $v := i(v^1,v^2)$. We want to show that for any circuit $C$ of $G_1 \times_{G_0} G_2$, the maximum of $\{v_e, e \in C\}$ is attained at least twice. If $C$ lies in $G_i$, this is clear, as $v^i \in B'(G_i)$, so it already attains the maximum twice.
Assume $C$ does not lie in $G_i$. We will prove the claim by induction on the cycle length of $C$. Note that by construction, any two consecutive edges $e,e'$ of $C$ that do not lie in $G_0$ must be elements of the same $G_i$ (we just glued $G_1$ and $G_2$ along edges in $G_0$).
If $C$ has length 3, then $C$ must already lie in one of the $G_i$, so we are done. If $\abs{C} > 3$, by assumption we can find a chord $c$ of $C$, subdividing $C$ into two cycles $C',C''$ of smaller length. By induction, the maxima $\max\{v_e; e\in C'\}, \max\{v_e;e \in C''\}$ are assumed twice. If both maxima are assumed away from $c$, the maximum over $C$ is also attained twice. If one of the cycle attains its maximum on $c$, then either the other cycle attains its maximum away from $c$ and it is bigger, so that the maximum over $C$ is again attained twice, or the other cycle also attains its maximum on $c$, and they are equal. In that case the maximum is attained on $v_c,v_{e'},v_{e''}$ for edges $e' \in C'\wo \{c\}, e'' \in C''\wo\{c\}$. In either case, the maximum is attained twice on $C$.
As $B'(G_1 \times_{G_0} G_2)$ is an irreducible tropical variety containing $i(S(G_1 \times_{G_0} G_2))$ and has the same dimension as $B'(G_1) \times_{B'(G_0)} B'(G_2)$, we see that we must have $$i(\abs{B'(G_1) \times_{B'(G_0)} B'(G_2)}) =i( S(G_1 \times_{G_0} G_2)) = \abs{B'(G_1 \times_{G_0} G_2)},$$ which implies that $\abs{B'(G_1) \times_{B'(G_0)} B'(G_2)} = S(G_1 \times_{G_0} G_2)$.
In particular $B'(G_1) \times_{B'(G_0)} B'(G_2)$ is isomorphic to a multiple of $B'(G_1 \times_{G_0} G_2)$. It remains to show that the weights of the fibre product (or equivalently: at least one weight) are 1. But this follows from Lemma \[intro\_lemma\_fibreweight\].
\[intro\_lemma\_fibreweight\] Let $M_1,M_2$ be matroids on ground sets $E_1,E_2$, where $E_0 := E_1 \cap E_2 \neq \emptyset$. Assume $M_0 = M_{1 \mid E_0} = M_{2 \mid E_0}$ is the common restriction of both matroids. Let $p_i: B(M_i) \to B(M_0)$ be the corresponding projections of Bergman fans. If the support of the fibre product is the set-theoretic fibre product $$\abs{B(M_1) \times_{B(M_0)} B(M_2)} = \{ (a_1,a_2) \in B(M_1) \times B(M_2); p_1(a_1) = p_2(a_2)\},$$ then the fibre product has weight 1 on each top-dimensional cone.
Assume $B(M_1),B(M_2), B(M_0)$ are all equipped with the polyhedral structure defined by their chains of flats. In particular $p_i$ maps cones of $B(M_i)$ to cones of $B(M_0)$. Then the set-theoretic fibre product has a natural polyhedral structure through the conewise fibre product: $$\curly{P} := \{ \sigma_1 \times_\tau \sigma_2; \sigma_i \textnormal{ a cone of } B(M_i) \textnormal{ and } p_i(\sigma_i) = \tau\},$$ where $\sigma_1 \times_\tau \sigma_2 = \{(a_1,a_2); a_i \in \sigma_i; p_1(a_1) = p_2(a_2)$.
In particular, a top-dimensional cone $\sigma$ of $\curly{P}$ is of the form $\sigma_1 \times_\tau \sigma_2$, where $\sigma_1,\sigma_2,\tau$ are all top-dimensional cones of their respective fans and an interior point $q$ of $\sigma$ is mapped to an interior point of $\tau$. To compute the weight of $\sigma$, we can look at the local morphism $$\Star_{B(M_1) \times B(M_2)}(q) \to \Star_{B(M_0) \times B(M_0)}( (p_1,p_2)(q))$$ induced by the combined projections $(p_1,p_2): B(M_1) \times B(M_2) \to B(M_0) \times B(M_0)$. Since $q$ is an interior point of $\sigma$ and $(p_1,p_2)(q)$ is an interior point of $\tau$, this ist just a projection map of linear spaces $$\R^{r_1 + r_2} \to \R^{2r_0},$$ where $r_i = \textnormal{rank}(M_i)$. Applying the diagonal functions $\varphi$ defined in [@FR10] cuts out the diagonal $\Delta_{\R^{r_0}}$ with weight 1 on the right hand side. Pulling this back via a linear projection yields again a linear space with weight 1. This concludes the proof.
We give examples demonstrating the need for the chordal and connected hypotheses in Proposition \[geom\_prop\_fibreproduct\] (see also Figure \[moduli\_fig\_needchordal\]).
1. First, let $G_0$ be the graph on four vertices $\{1,2,3,4\}$ with two disjoint edges $e = \{1,2\}$ and $e' = \{3,4\}$. Let $G_1$ and $G_2$ be the graphs obtained by adding edge $e_1 = \{1,3\}$ and $e_2 = \{2,4\}$, respectively. The graph $G_1 \times_{G_0} G_2$ is the square, which has rank 3. However, the fibre product should have dimension $\textnormal{rank}(G_1) + \textnormal{rank}(G_2) - \textnormal{rank}(G_0) = 3 + 3 - 2 = 4$.
2. We choose our graphs as depicted in Figure \[moduli\_fig\_needchordal\] on the right hand side. Note that $G_1 \times_{G_0} G_2$ is not chordal (the outer cycle has no chord). We now explain why the fibre product is not isomorphic to the Bergman fan of this graph.
Observe that locally around each codimension $1$ face $\tau$ in the chains-of-flats subdivision, $\Star_{B'(G_1 \times_{G_0} G_2)}(\tau)$ is isomorphic to either an actual line or a tropical line (in particular, there are at most three top-dimensional cones at each codimension one face).
To compute the fibre product, we note first that the matroid of $G_1 = G_2$ is $U_{3,4}$. Its Bergman fan $B'(U_{3,4})$ is the standard tropical hyperplane $H$ in $\R^3$. The matroid of $G_0$ is $U_{2,2}$, so $B'(G_0) \cong \R$ and we can assume that the map $B'(G_i) \cong H \to \R \cong B'(G_0) $ is the projection to the first coordinate.
Choose the ray $r$ of $H$ generated by $-e_2$, which is mapped to 0 under this projection and consider the local picture $\Star_{H}(r) \times_{\R} \Star_{H}(r)$. It is easy to check that this local fibre product consists of four rays. The global fibre product has a codimension one face with four adjacent top-dimensional cones, and cannot be isomorphic to $B'(G_1 \times_{G_0} G_2)$.
;
Let $w$ be a weight vector. Recall that we can make $G(w)$ a split graph by subdividing its vertices into $$\begin{aligned}
L(w) &:= \{i \in \{2,\dots,n\}, w_i > 1/2\} \\
S(w) &:= \{i \in \{2,\dots,n\},w_i \leq 1/2\}\end{aligned}$$ We define the following graphs:
- $K(w)$ is the restriction of $G(w)$ to $L(w)$ (i.e. it is the complete graph on $L(w)$).
- For $i \in S(w)$, we let $G_i(w)$ be the restriction of $G(w)$ to $L(w) \cup \{i\}$.
The graphs $G_i(w)$ share the common subgraph $K(w)$, and gluing them together gives us back our weight graph: $$G(w) = G_{i_1}(w) \times_{K(w)} \dots \times_{K(w)} G_{i_s}(w),$$ where $S(w) = \{i_1,\dots,i_s\}$. Also, recall from Lemma \[geom\_lemma\_split\], that $G(w)$ (and similarly, all intermediate glued graphs) is a split graph, so we can apply Proposition \[geom\_prop\_fibreproduct\]. However, to obtain a deeper geometric understanding, we first want to study the spaces $B'(G_i(w))$.
\[geom\_prop\_single\_degree\] Assume $i \in S(w)$ has vertex degree 1. Then $$B'(G_i(w)) \cong \mk{\abs{L(w)}+1} \times \R$$
By assumption $G_i(w)$ is just $K(w)$ with an additional edge attached. Hence its matroid is just the direct sum of the matroid of $K(w)$ and a coloop. This implies the claim.
This has a natural geometric interpretation that is very similar to Example \[moduli\_ex\_badtypes\]. The fact that vertex $i$ is only connected to one vertex $j$ of $K(w)$ means that there are exactly two leaves, namely 1 and $j$, with which $i$ is compatible. More precisely, $w_i + w_j, w_i + w_1 > 1$, but $w_i + w_k \leq 1$ for all other $k$. Hence we can only place leaf $i$ along the “line” spanned by leaves $j$ and $1$ (All other choices would produce inherited unstable types).
\[geom\_prop\_higher\_degree\] Assume $i \in S(w)$ has vertex degree $d > 1$. Then $$B'(G_i(w)) \cong \mk{\abs{L(w)}+1} \times_{\mk{d+1}} \mk{d+2}.$$
We can write $G_i(w)$ as a glued graph: Let $K_i$ be the restriction of $G_i(w)$ to vertices $j \neq i$ connected to $i$. In particular it is a complete graph on $d$ vertices. If we add vertex $i$, we obtain $K_i'$, the complete graph on $d+1$ vertices. Now $K(w)$ and $K_i'$ share the common subgraph $K_i$ and $$G_i(w) = K(w) \times_{K_i} K_i'.$$ Since $G_i(w)$ is again a split graph and hence chordal, the claim follows.
\[thm-tropfibre\] Let $w $ be a weight vector. Let $D_1 := \{i \in S(w): \deg(i) = 1\}$ and assume $S(w) \wo D_1 = \{i_1,\dots i_s\}$. Let $$\curly{M} := \prod_{j=1}^s \mk{\deg(i_j)+1} \ \ \ \curly{M}' := \prod_{j=1}^s \mk{\deg(i_j)+2}$$ Then we have $$\pr_w(\mk{n}) \cong \R^{\abs{D_1}} \times \left( \mk{\abs{L}+1} \times_{\curly{M}} \curly{M}'\right),$$ where the maps from $\mk{\abs{L}+1}$ and $\curly{M}'$ to $\curly{M}$ are the natural tuples of forgetful maps (If $s = 0$, we set $\mk{\abs{L}+1} \times_{\curly{M}} \curly{M} := \mk{\abs{L}+1}$).
Assume $D_1 = \{k_1,\dots,k_t\}$. By Proposition \[geom\_prop\_fibreproduct\] we know that $$\pr_w(\mk{n}) \cong B'(G_{k_1}(w)) \times_{B'(K(w))} \dots \times_{B'(K(w))} B'(G_{i_s}(w))$$ and that the support of the latter is the set-theoretic fibre product. The claim now follows from Propositions \[geom\_prop\_single\_degree\] and \[geom\_prop\_higher\_degree\].
\[cor-fibreheavylight\] Let $w = (1^f,\epsilon^t)$, where $f \geq 3, t \geq 2$. Then $$\mk{w} \cong \underbrace{\mk{f+1} \times_{\mk{f}} \dots \times_{\mk{f}} \mk{f+1}}_{t \textnormal{ times}}.$$
The last result might seem somewhat incongruous at first, as classically, we have $$\overline M(1^f,0^t)\cong\overline{M}_{0,f+1}\times_{\overline{M}_{0,f}}\ldots \times_{\overline{M}_{0,f}}\overline{M}_{0,f+1}$$ and that $\overline{M}(1^f,\epsilon^t)$ is a blowup of this. In fact, the tropical phenomena are analogous.
The tropical fibre product has a canonical polyhedral structure $\curly{P}$ consisting of taking conewise fibre products of cones of $\mk{f+1}$ in its coarse subdivision. Points in the same cone correspond to $n$-marked tropical curves that
- have the same combinatorial “base curve” $C$ when forgetting all light ends;
- have their light ends placed on the same edges or ends of $C$.
In other words, this is not the polyhedral structure that was assigned to $\mk{w}$. Whenever multiple light ends are put on the same edge of the base curve, we need to subdivide. More precisely, if $\sigma$ is a cone in $\curly{P}$ corresponding to a base curve with $d$ bounded edges and $t$ light ends placed on its edges and leafs, then we can identify $\relint(\sigma)$ with $\R^{d+t}$. Under this identification, the subdivision is a product of $\R^k$ together with tropical Losev–Manin spaces. Recall from Section \[sec: LosevManin\] that $\mk{(1^2,\epsilon^r)} \cong \R^{r-1}$ parametrizes ways to position $r$ labelled ends relative to each other on a line. For each bounded edge with at least one light end, we get a factor of $\R^2 \times \mk{(1^2,\epsilon^r)}$, where $r$ is the number of light ends placed on this edge. In similar fashion, we obtain a factor of $\R \times \mk{(1^2,\epsilon^r)}$ for each end with at least one light end placed on it. Finally, we get a factor of $\R$ for each bounded edge with no light ends.
[^1]: *2010 Mathematics Subject Classification:* 14T05, 14D22, 14H10
|
---
abstract: 'The ’exact subgraph’ approach was recently introduced as a hierarchical scheme to get increasingly tight semidefinite programming relaxations of several NP-hard graph optimization problems. Solving these relaxations is a computational challenge because of the potentially large number of violated subgraph constraints. We introduce a computational framework for these relaxations designed to cope with these difficulties. We suggest a partial Lagrangian dual, and exploit the fact that its evaluation decomposes into two independent subproblems. This opens the way to use the bundle method from non-smooth optimization to minimize the dual function. Computational experiments on the Max-Cut, stable set and coloring problem show the efficiency of this approach.'
author:
- Elisabeth Gaar
- Franz Rendl
bibliography:
- 'papers.bib'
title: 'A Bundle Approach for SDPs with Exact Subgraph Constraints[^1]'
---
Introduction
============
The study of NP-hard problems has led to the introduction of various hierarchies of relaxations, which typically involve several levels. Moving from one level to the next the relaxations get increasingly tighter and ultimately the exact optimum may be reached, but the computational effort grows accordingly.
Among the most prominent hierarchies are the polyhedral ones from Boros, Crama and Hammer [@BorosCramaHammer] as well as the ones from Sherali and Adams [@SheraliAdamsHierarchy], Lovász and Schrijver [@LovaszSchrijverHierarchy] and Lasserre [@LasserreHierarchy] which are based on semidefinite programming (SDP). Even though on the starting level they have a simple SDP relaxation, already the first nontrivial level in the hierarchy requires the solution of SDPs in matrices of order $\binom{n}{2}$ and on level $k$ the matrix order is $n^{O(k)}$. Hence they are considered mainly as theoretical tools and from a practical point of view these hierarchies are of limited use.
Not all hierarchies are of this type. In [@BorosCramaHammer], a polyhedral hierarchy for the Max-Cut problem is introduced which maintains $\binom{n}{2}$ variables in all levels, with a growing number of constraints. More recently, Adams, Anjos, Rendl and Wiegele [@AARW] introduced a hierarchy of SDP relaxations which act in the space of symmetric $n \times n$ matrices and at level $k$ of the hierarchy all submatrices of order $k$ have to be ’exact’ in a well-defined sense, i.e. they have to fulfill an *exact subgraph constraint* (ESC).
It is the main purpose of this paper to describe an efficient way to optimize over level $k$ of this hierarchy for small values of $k$, e.g. $k\leqslant 6$, and demonstrate the efficiency of our approach for the Max-Cut, stable set and coloring problem.
Maintaining $\binom{n}{k}$ possible ESCs in an SDP in matrices of order $n$ is computationally infeasible even for $k=2$ or $k=3$, because each ESC creates roughly $\binom{k}{2}$ additional equality constraints and at most $2^k$ additional linear variables.
We suggest the following ideas to overcome this difficulty. First we proceed iteratively, and in each iteration we include only (a few hundred of) the most violated ESCs. More importantly, we propose to solve the dual of the resulting SDP. The structure of this SDP with ESCs admits a reformulation of the dual in the form of a non-smooth convex minimization problem with attractive features. First, any dual solution yields a valid bound for our relaxations, so it is not necessary to carry out the minimization to optimality. Secondly, the dual function evaluation decomposes into two independent problems. The first one is simply a sum of max-terms (one for each subgraph constraint), and the second one consists in solving a ’basic’ SDP, independent of the ESCs. The optimizer for this second problem also yields a subgradient of the objective function. With this information at hand we suggest to use the bundle method from non-smooth convex optimization. It provides an effective machinery to get close to a minimizer in few iterations.
As a result we are able to get near optimal solutions where all ESCs for small values of $k$ ($k \leqslant 6$) are satisfied up to a small error tolerance. Our computational results demonstrate the practical potential of this approach.
We finish this introductory section with some notation. We denote the vector of all-ones of size $n$ with ${\mathbbm{1}_{n}}$ and ${{\Delta}}_n = \{x \in {{\mathbb R}}^{n}_{+}: \sum_{i=1}^{n}x_{i} = 1\}$. If the dimension is clear from the context we may omit the index and write ${\mathbbm{1}_{}}$ and ${{\Delta}}$. Furthermore let $N = \{1, 2, \dots, n\}$. A graph $G$ on $n$ vertices has vertex set $N$ and edge set $E$ and $\overline{G}$ is its complement graph. ${{\mathcal S}_{n}}$ is the set of $n$-dimensional symmetric matrices.
The Problems and their Semidefinite Relaxations {#sec:DefProblemsBasicRel}
===============================================
In the Max-Cut problem a symmetric matrix $L \in
{{\mathcal S}_{n}}$ is given and $c \in \{-1,1 \}^{n}$ which maximizes $c^{{T}}Lc$ should be determined. If the matrix $L$ corresponds to the Laplacian matrix of a (edge-weighted undirected) graph $G$, this is equivalent to finding a bisection of the vertices of $G$ such that the total weight of the edges joining the two bisection blocks is maximized. Such an edge set is also called a *cut* in $G$.
Bisections of $N$ can be expressed as $c \in \{-1,1 \}^{n}$ where the two bisection blocks correspond to the entries in $c$ of the same sign. Given $c \in \{-1,1\}^{n}$ we call $C=cc^{{ {{T}} }}$ a *cut matrix*. The convex hull of all cut matrices (of order $n$) is denoted by $\operatorname{CUT}_{n}$ or simply $\operatorname{CUT}$ if the dimension is clear. Since $c^{{ {{T}} }}Lc = \langle L, cc^{{ {{T}} }} \rangle$ Max-Cut can also be written as the following (intractable) linear program $$z_{mc} = \max \{ \langle L, X\rangle:~ X \in \operatorname{CUT}\}.$$ $\operatorname{CUT}$ is contained in the spectrahedron $
{{\mathcal X}^{E}}= \left\{ X \in {{\mathcal S}_{n}} : \operatorname{diag}(X) =
{\mathbbm{1}_{n}},\, X \succcurlyeq 0 \right\},
$ hence $$\label{relaxation mc}
\max \left\{\langle L,X\rangle :~ X \in {{\mathcal X}^{E}}\right\}$$ is a basic semidefinite relaxation for Max-Cut. This model is well-known, attributed to Schrijver and was introduced in a dual form by Delorme and Poljak [@DelormePoljak]. It can be solved in polynomial time to a fixed prescribed precision and solving this relaxation for $n=1000$ takes only a few seconds.
It is well-known that the Max-Cut problem is NP-hard. On the positive side, Goemans and Williamson [@GoemansWilliamson] show that one can find a cut in a graph with nonnegative edge weights of value at least 0.878$z_{mc}$ in polynomial time.
In the stable set problem the input is an unweighted graph $G$. We call a set of vertices *stable*, if no two vertices are adjacent. Moreover we call a vector $s \in \{0,1 \}^n$ a *stable set vector* if it is the incidence vector of a stable set. The convex hull of all stable set vectors of $G$ is denoted with $\operatorname{STAB}(G)$. In the stable set problem we want to determine the *stability number $\alpha(G)$*, which denotes the cardinality of a largest stable set in $G$, hence $
\alpha(G) = \max\left\{ {\mathbbm{1}_{}}^{{ {{T}} }}s:~ s \in \operatorname{STAB}(G) \right\}.
$ Furthermore we denote with $\operatorname{STAB}^{2}(G) =
\operatorname{conv}\left\{ ss^{{ {{T}} }}:~ s \in \operatorname{STAB}(G) \right\}$ the convex hull of all *stable set matrices* $ss^{{ {{T}} }}$. Then with the arguments of Gaar [@elli-diss] it is easy to check that $
\alpha(G) = \max\{ \operatorname{trace}(X):~ X \in \operatorname{STAB}^2(G) \}.
$ Furthermore $\operatorname{STAB}^{2}(G)$ is contained in the following spectrahedron $${{\mathcal X}^{S}}= \left\{ X \in {{\mathcal S}_{n}} :~
X_{ij}=0 \quad
\forall \{i,j\} \in E,~
x = \operatorname{diag}(X),~
\left(
\begin{array}{cc}
1 & x^{{ {{T}} }} \\
x & X
\end{array}
\right)
\succcurlyeq 0 \right\},$$ which is known as the *theta body* in the literature. Therefore $$\label{relaxation ss}
\vartheta(G)= \max \left\{ \operatorname{trace}(X):~ X \in {{\mathcal X}^{S}}\right\}$$ is a relaxation of the stable set problem. The Lov[á]{}sz theta function $\vartheta(G)$ was introduced in a seminal paper by Lov[á]{}sz [@LovaszStart]. We refer to Gr[ö]{}tschel, Lov[á]{}sz and Schrijver [@OurUsedFormOfLovasTheta] for a comprehensive analysis of $\vartheta(G)$.
Determining $\alpha(G)$ is again NP-hard. Contrary to Max-Cut, which has a polynomial time .878-approximation, for every $\varepsilon>0$ there can be no polynomial time algorithm that approximates $\alpha(G)$ within a factor better than $O(n^{1-\varepsilon})$ unless $P=NP$, see H[å]{}stad [@stableSetNotApproximable].
The coloring problem for a given graph $G$ consists in determining the *chromatic number* $\chi(G)$, which is the smallest $t$ such that $N$ can be partitioned into $t$ stable sets. Let $S=(s_{1}, \ldots, s_{k})$ be a matrix where each column is a stable set vector and these stable sets partition $V$ into $k$ sets. Let us call such matrices $S$ [*stable-set partition matrices*]{} (SSPM). The $n \times n$ matrix $X=SS^{T}$ is called [*coloring matrix*]{}. The convex hull of the set of all coloring matrices of $G$ is denoted by $\operatorname{COL}(G)$. We also need the *extended coloring polytope* $${\operatorname{COL}^{\varepsilon}}(G) = \operatorname{conv}\left\{
\left(
\begin{array}{cc} k & {\mathbbm{1}_{}}^{{ {{T}} }}\\{\mathbbm{1}_{}} & X
\end{array}\right)
=
\sum_{i=1}^{k}
\binom{1}{s_{i}}
\binom{1}{s_{i}}^{{ {{T}} }}
:
\begin{array}{c}
S = (s_{1}, \ldots, s_{k}) \text{ is a} \\
\text{SSPM of } G,~ X = SS^{{ {{T}} }}
\end{array}
\right\}.$$
The difficult set ${\operatorname{COL}^{\varepsilon}}$ can be relaxed to the easier spectrahedron ${{\mathcal X}^{C}}$ $${{\mathcal X}^{C}}= \left\{
\left(
\begin{array}{cc} t & {\mathbbm{1}_{}}^{{ {{T}} }}\\{\mathbbm{1}_{}} & X
\end{array}\right) \succcurlyeq 0:~ \operatorname{diag}(X)={\mathbbm{1}_{n}}, ~X_{ij}=0 ~\forall \{i,j\} \in E
\right\}$$ and we can consider the semidefinite program $$\label{relaxation col}
t^{*}(G) = \min \left\{t:~
\left(
\begin{array}{cc} t & {\mathbbm{1}_{}}^{{ {{T}} }}\\{\mathbbm{1}_{}} & X
\end{array}\right) \in {{\mathcal X}^{C}}\right\}.$$ Obviously $t^{*}(G) \leqslant \chi(G)$ holds because the SSPM $S$ consisting of $\chi(G)$ stable sets yields a feasible coloring matrix $X=SS^{{ {{T}} }}$ with objective function value $\chi(G)$. It is in fact a consequence of conic duality that $t^{*}(G)= \vartheta(\overline{G})$ holds.
It is NP-hard to find $\chi(G)$, to find a 4-coloring of a 3-colorable graph [@GuruswamiKhanna] and to color a $k$-colorable graph with $O(k^{\frac{\log k}{25}})$ colors for sufficiently large $k$, [@Khot].
Exact Subgraph Hierarchy {#sec: ESC}
========================
In this section we will discuss how to systematically tighten the relaxations , and with ’exactness conditions’ imposed on small subgraphs. We obtained these relaxations by relaxing the feasible regions $\operatorname{CUT}$, $\operatorname{STAB}^{2}$ and $\operatorname{COL}$ of the integer problem to simple spectrahedral sets. Now we will use small subgraphs to get closer to original feasible regions again.
For $I \subseteq N$ we denote with ${X_{I}}$ the principal submatrix of $X$ corresponding to the rows and columns in $I$. Furthermore let ${G_{I}}$ be the induced subgraph of $G$ on the set of vertices $I$ and let ${k_{I}}=|I|$ be the cardinality of $I$.
We first look at the exact subgraph relaxations for Max-Cut. The *exact subgraph constraint* (ESC) on $I \subseteq N$, introduced in [@AARW] by Adams, Anjos, Rendl and Wiegele, requires that the matrix ${X_{I}}$ corresponding to the subgraph ${G_{I}}$ lies in the convex hull of the cut matrices of ${G_{I}}$, that is $${X_{I}}\in \operatorname{CUT}_{|I|}.$$ In this case we say that $X$ is *exact* on $I$. Now we want the ESCs to be fulfilled not only for one but for a certain selection of subgraphs. We denote with $J$ the set of subgraphs which we require to be exact and get the following SDP relaxation with ESCs for Max-Cut. $$\label{relaxation mc with esc}
\max \{\langle L,X\rangle:~
X \in {{\mathcal X}^{E}},~ {X_{I}}\in \operatorname{CUT}_{|I|} ~ \forall I \in J \}$$
We proceed analogously for the stable set problem in a graph $G$. The ESC of a subgraph ${G_{I}}$ for the stable set problem requires that $
{X_{I}}\in \operatorname{STAB}^{2}(G_{I})
$ holds and the SDP with ESCs for the stable set problem is $$\label{relaxation ss with esc}
\max \{ \operatorname{trace}(X):~ X \in {{\mathcal X}^{S}},~
{X_{I}}\in \operatorname{STAB}^{2}({G_{I}}) ~ \forall I \in J \}.$$
Turning to the coloring problem, we analogously impose additional constraints of the form $
{X_{I}}\in \operatorname{COL}(G_I)
$ to obtain the SDP with ESCs $$\label{relaxation col with esc}
\min \left\{t:~
\left(
\begin{array}{cc} t & {\mathbbm{1}_{}}^{{ {{T}} }}\\ {\mathbbm{1}_{}} & X
\end{array}\right) \in {{\mathcal X}^{C}},~
{X_{I}}\in \operatorname{COL}({G_{I}}) ~ \forall I \in J
\right\}.$$
Note that in the case of the stable set and the coloring problem the polytopes $\operatorname{STAB}^{2}({G_{I}})$ and $\operatorname{COL}({G_{I}})$ depend on the subgraph ${G_{I}}$, whereas in Max-Cut the polytope $\operatorname{CUT}_{|I|}$ only depends on the number of vertices of the subgraph.
From a theoretical point of view, we obtain the $k$-th level of the exact subgraph hierarchy of [@AARW] if we use $J = \{I \subseteq N:~ |I| = k\}$ in the relaxations , and respectively. We denote the corresponding objective function values with ${z_{mc}^{k}}$, ${z_{ss}^{k}}$ and ${z_{c}^{k}}$. So the $k$-th level of the hierarchy is obtained by forcing all subgraphs on $k$ vertices to be exact in the basic SDP relaxation.
In the case of the stable set and the Max-Cut problem we have ${z_{ss}^{n}} = \alpha(G)$ (see [@elli-diss]) and ${z_{mc}^{n}} = z_{mc}$. For coloring ${z_{c}^{n}} \leqslant \chi(G)$ holds. Let ${z_{c\varepsilon}^{k}}$ be the resulting value if we add the inequalities $t \geqslant \sum_{i=1}^{{t_{I}}}[{\lambda_{I}}]_{i}|S^{I}_{i}|$ where $|S^{I}_{i}|$ is the number of colors used for the SSPM $S^{I}_{i}$ and ${\lambda_{I}}\in {{\Delta}}_{{t_{I}}}$ is a variable for the convex combination for each subgraph $I$ to the SDP for ${z_{c}^{k}}$. Then ${z_{c\varepsilon}^{n}} = \chi(G)$ holds. Since the focus of this paper are computational results we are interested only in the computational results we omit the details and further theoretical investigations.
An important feature of this hierarchy is that the size of the matrix variable remains $n$ or $n+1$ on all levels of the hierarchy and only more linear variables and constraints (enforcing the ESCs, hence representing convex hull conditions) are added on higher levels. So it is possible to approximate ${z_{mc}^{k}}$, ${z_{ss}^{k}}$ and ${z_{c}^{k}}$ by forcing only some subgraphs of order $k$ to be exact. This is our key ingredient to computationally obtain tight bounds on $z_{mc}$, $\alpha(G)$ and $\chi(G)$.
From a practical point of view solving the relaxations , and with standard interior point (IP) solvers like SDPT3 [@SDPT32] or MOSEK [@mosek] is very time consuming. In we list computation times (in seconds) for one specific Max-Cut and one specific stable set instance. We vary the number of ESCs for subgraphs of order $3$, $4$ and $5$, so we solve and for different $J$. We choose $J$ such that the total number of equality constraints induced by the convex hull formulation of the ESCs $b$ ranges between 6000 and 15000. Since the matrix order $n$ is fixed to $n=100$, the overall computation time depends essentially on the number of constraints, independent of the specific form of the objective function. Aside from the ESC constraints, we have $n$ additional equations for Max-Cut and $n+m+1$ additional equations for the stable set problem. Here $m$ denotes the number of edges of the graph. We have $m=722$ in the example graph. Clearly the running times get huge for a large number of ESC. Furthermore MATLAB requires 12 Gigabyte of memory for $b=15000$, showing also memory limitations.
Note that it is argued in [@AARW] that ${z_{mc}^{4}}={z_{mc}^{3}}$, so we omit subgraphs of order ${k_{I}}=4$ for Max-Cut. This is because in the back of our minds our final algorithm to determine the best possible bounds first includes ESCs of size $k$, starting for example with $k=3$. As soon as we do not find violated ESCs of size $k$ anymore, we repeat this for size $k+1$.
Partial Lagrangian Dual {#sec: lagrangianDual}
=======================
To summarize we are interested in solving relaxations , and with a potentially large number of ESCs, where using interior point solvers is too time consuming. In this section we will first establish a unified formulation of the relaxations , and . Then we will build the partial Lagrangian dual of this formulation, where only the ESCs are dualized. This model will be particularly amenable for the bundle method, because it will be straightforward to obtain a subgradient of the model when evaluating it at a certain point.
In order to unify the notation for the three problems observe that the ESCs ${X_{I}}\in \operatorname{CUT}_{|I|}$, ${X_{I}}\in \operatorname{STAB}^{2}({G_{I}})$ and ${X_{I}}\in \operatorname{COL}({G_{I}})$ can be represented as $$\label{esc in conv hull}
{X_{I}}= \sum_{i=1}^{{t_{I}}} \lambda_{i}C^{I}_{i},\quad \lambda \in {{\Delta}}_{{t_{I}}},$$ where $C^{I}_{i}$ is the $i$-th cut, stable set or coloring matrix of the subgraph ${G_{I}}$ and ${t_{I}}$ is their total number.
A formal description of ESC in requires some additional notation. First we introduce the projection ${{\mathcal P}_I}\colon {{\mathcal S}_{n}} \mapsto {{\mathcal S}_{{k_{I}}}}$, mapping $X$ to the submatrix ${X_{I}}$. Second we define a map ${{\mathcal A}_I}\colon {{\mathcal S}_{{k_{I}}}} \mapsto {{\mathbb R}}^{{t_{I}}}$, such that its adjoint map ${{\mathcal A}_I^{{\top}}}\colon {{\mathbb R}}^{{t_{I}}} \mapsto {{\mathcal S}_{{k_{I}}}}$ is given by ${{\mathcal A}_I^{{\top}}}(\lambda)=\sum_{i=1}^{{t_{I}}}{\lambda_i C_i^{I}}$ and produces a linear combination of the cut, stable set or coloring matrices. Thus we can rewrite as $$\label{esc mc withAP}
{{\mathcal A}_I^{{\top}}}({\lambda_{I}}) - {{\mathcal P}_I}(X) = 0, \quad {\lambda_{I}}\in {{\Delta}}_{{t_{I}}}.$$
The left-hand side of the matrix equality is a symmetric matrix, of which some entries (depending on which problem we consider) are zero for sure, so we do not have to include all ${k_{I}}\times {k_{I}}$ equality constraints into the SDP. Let ${b_{I}}$ be the number of equality constraints we have to include. Note that ${b_{I}}= \binom{{k_{I}}}{2}$, ${b_{I}}= \binom{{k_{I}}+1}{2} - m_I$ and ${b_{I}}= \binom{{k_{I}}}{2} - m_I$ for the Max-Cut, stable set and coloring problem respectively, if $m_I$ denotes the number of edges of ${G_{I}}$. This is because in the case of the stable set problem we also have to include equations for the entries of the main diagonal contrary to Max-Cut and the coloring problem. Then we define a linear map ${{\mathcal M}_I}\colon {{\mathbb R}}^{{b_{I}}} \mapsto {{\mathcal S}_{{k_{I}}}}$ such that the adjoint operator ${{\mathcal M}_I^{{\top}}}\colon {{\mathcal S}_{{k_{I}}}} \mapsto {{\mathbb R}}^{{b_{I}}}$ extracts the ${b_{I}}$ positions, for which we have to include the equality constraints, into a vector. So eventually we can rephrase equivalently as $$\begin{aligned}
{{\mathcal M}_I^{{\top}}}({{\mathcal A}_I^{{\top}}}({\lambda_{I}}) - {{\mathcal P}_I}(X)) = 0, \quad {\lambda_{I}}\in {{\Delta}}_{{t_{I}}},\end{aligned}$$ which are ${b_{I}}+1$ equalities and ${t_{I}}$ inequalities. In consequence all three relaxations , and have the generic form $$\label{sdp}
z = \max \{ \langle C, \widehat{X} \rangle:~
\widehat{X} \in {{\mathcal X}},~ {\lambda_{I}}\in {{\Delta}}_{{t_{I}}},~ {{\mathcal M}_I^{{\top}}}({{\mathcal A}_I^{{\top}}}({\lambda_{I}}) - {{\mathcal P}_I}(X)) = 0 ~ \forall I \in J\},$$ where $C$, ${{\mathcal X}}$, ${{\mathcal A}_I}$, ${{\mathcal M}_I}$ and ${b_{I}}$ have to be defined problem specific. Furthermore $\widehat{X} = X$ in the case of Max-Cut and stable set and $\widehat{X} = \left(
\begin{array}{cc} t & {\mathbbm{1}_{}}^{{ {{T}} }}\\ {\mathbbm{1}_{}} & X
\end{array}\right)$ for coloring, but for the sake of understandability we will just use $X$ in the following.
The key idea to get a handle on problem is to consider the partial Lagrangian dual where the ESCs (without the constrains ${\lambda_{I}}\in {{\Delta}}_{{t_{I}}}$) are dualized. We introduce a vector of multipliers ${y_{I}}$ of size ${b_{I}}$ for each $I$ and collect them in ${y}= ({y_{I}})_{I\in J}$ and also collect ${\lambda}= ({\lambda_{I}})_{I\in J}$. The Lagrangian function becomes $${{\mathcal L}}(X,{\lambda},{y}) = \langle C, X \rangle + \sum_{I \in J}{\langle {y_{I}}, {{\mathcal M}_I^{{\top}}}({{\mathcal A}_I^{{\top}}}({\lambda_{I}}) - {{\mathcal P}_I}(X)) \rangle}
$$ and standard duality arguments (Rockafellar [@RockafellarConvexAnalysis Corollary 37.3.2]) yield $$\label{lagrangianDual}
z = \min_{{y}} \max_{\substack{X \in {{\mathcal X}}\\ {\lambda_{I}}\in {{\Delta}}_{{t_{I}}}}} {{\mathcal L}}(X,{\lambda},{y}).$$ For a fixed set of multipliers ${y}$ the inner maximization becomes $$\max_{\substack{X \in {{\mathcal X}}\\ {\lambda_{I}}\in {{\Delta}}_{{t_{I}}}}} \left\langle C - \sum_{I\in J}{{{\mathcal P}_I^{{\top}}}{{\mathcal M}_I}({y_{I}})}, X \right\rangle + \sum_{I\in J}{\langle {{\mathcal A}_I}{{\mathcal M}_I}({y_{I}}), {\lambda_{I}}\rangle}.$$
This maximization is interesting in at least two aspects. First, it is separable in the sense that the first term depends only on $X$ and the second one only on the separate ${\lambda_{I}}$. Moreover, if we denote the linear map ${{\mathcal A}_I}{{\mathcal M}_I}({y_{I}})\colon {{\mathbb R}}^{{b_{I}}} \mapsto {{\mathbb R}}^{{t_{I}}}$ with ${{\mathcal D}_I}$, the second term has an explicit solution, namely $$\begin{aligned}
\label{solutionMaxTerm}
\max_{{\lambda_{I}}\in {{\Delta}}_{{t_{I}}}}\langle {{\mathcal D}_I}({y_{I}}), {\lambda_{I}}\rangle = \max_{1 \leqslant i \leqslant {t_{I}}}\left[{{\mathcal D}_I}({y_{I}})\right]_{i}.\end{aligned}$$
In order to consider the first term in more detail, we define the following function. Let ${b}= \sum_{I\in J}{b_{I}}$ be the dimension of ${y}$. Then $h\colon {{\mathbb R}}^{{b}} \to {{\mathbb R}}$ is defined as $$\label{def:hy}
h({y})= \max_{X \in {{\mathcal X}}} \left\langle C - \sum_{I\in J}{{{\mathcal P}_I^{{\top}}}{{\mathcal M}_I}({y_{I}})}, X \right\rangle = \left\langle C - \sum_{I\in J}{{{\mathcal P}_I^{{\top}}}{{\mathcal M}_I}({y_{I}})}, X^{\ast} \right\rangle,$$ where $X^{\ast}$ is a maximizer over the set ${{\mathcal X}}$ for $y$ fixed. Note that $h({y})$ is convex but non-smooth, but shows that $g_I= -{{\mathcal M}_I}^T {{\mathcal P}_I}(X^{\ast})$ is a subgradient of $h$ with respect to ${y_{I}}$. By combining and we can reformulate the partial Lagrangian dual to $$\label{primal compact for bundle}
z = \min_{{y}} \left\{ h({y}) + \sum_{I\in J}{\max_{1 \leqslant i \leqslant {t_{I}}}\left[{{\mathcal D}_I}({y_{I}})\right]_i}\right\}.$$
The formulation of the original relaxations , and fits perfectly into the bundle method setting described by Frangioni and Gorgone in [@EasyBundle], hence we suggest to approach this problem using the bundle method.
Solving with the Bundle Method {#sec:bundle}
==============================
The bundle method is an iterative procedure for minimizing a convex non-smooth function and firstly maintains the *current center* ${\overline{y}}$, which represents the current estimate to the optimal solution, throughout the iterations. Secondly it maintains the bundle of the form ${\mathcal{B}}= \{
({y_{1}}, {h_{1}},{g_{1}},{X_{1}}), \dots,
({y_{{r}}}, {h_{{r}}},{g_{{r}}},{X_{{r}}}) \}$. Here $y_1, \ldots, y_r$ are the points which we use to set up our subgradient model. Moreover $h_i = h(y_i)$, $g_i$ is a subgradient of $h$ at $y_i$ and $X_i$ is a maximizer of $h$ at $y_i$ as in .
At the start we select $y_1={\overline{y}}=0$ and evaluate $h$ at ${\overline{y}}$, which yields the bundle ${\mathcal{B}}=\{(y_1,g_1, h_1,X_1)\}$. A general iteration consists of the two steps determining the new *trial point* and evaluating the *oracle*. For determining a new trial point ${\widetilde{y}}$ the subgradient information of the bundle ${\mathcal{B}}$ translates into the subgradient model $ \label{subgradientModel}
h(y) \geqslant {h_{j}} + \langle {g_{j}},y-{y_{j}}\rangle$ for all $j = 1$, …, ${r}$. It is common to introduce $
{e_{j}} = h({\overline{y}}) - {h_{j}} - \langle {g_{j}},{\overline{y}}-{y_{j}}\rangle$ for $j = 1,$ …, ${r}$ and with ${\overline{h}}= h({\overline{y}})$ the subgradient model becomes $$\begin{aligned}
\label{subgrMaxModel}
h(y) \geqslant \max_{1\leqslant j \leqslant r} \left\{{\overline{h}}- {e_{j}} + \langle {g_{j}},y-{\overline{y}}\rangle \right\}.\end{aligned}$$ The right-hand side above is convex, piecewise linear and minorizes $h$. In each iteration of the bundle method we minimize the right-hand side of instead of $h$, but ensure that we do not move too far from ${\overline{y}}$ by adding a penalty term of the form $
\frac{1}{2}{\mu}{\left\lVert y-{\overline{y}}\right\rVert}^{2}
$ for a parameter ${\mu}\in {{\mathbb R}}_{+}$ to the objective function. With the auxiliary variables $w \in {{\mathbb R}}$ and ${v_{I}}\in {{\mathbb R}}$ for all $I \in J$ to model the maximum terms and with $v = ({v_{I}})_{I \in J} \in {{\mathbb R}}^{q}$ and ${q}= |J|$ we end up with $$\begin{aligned}
\label{bundleProblemPiO}
\min_{y,w,v} \quad w &+ \sum_{I \in J} {v_{I}}+ \frac{1}{2}{\mu}{\left\lVert y-{\overline{y}}\right\rVert}^{2}\\ \nonumber
st \quad w &\geqslant {\overline{h}}- {e_{j}} + \langle {g_{j}},y-{\overline{y}}\rangle && \forall j = 1,\dots,{r}\\
\nonumber
{v_{I}}&\geqslant \left[{{\mathcal D}_I}({y_{I}})\right]_{i} && \forall i = 1,\dots,{t_{I}}\quad \forall I \in J.\end{aligned}$$ This is a convex quadratic problem in $1+{q}+{b}$ variables with ${r}+\sum_{I \in J}{t_{I}}$ linear inequality constraints. Its solution $({\widetilde{y}},{\widetilde{w}},{\widetilde{v}})$ includes the new trial point ${\widetilde{y}}$. Problems of this type can be solved efficiently in various ways, see [@elli-diss] for further details. In our implementation we view as a rotated second order cone program with one second-order cone constraint and solve it with MOSEK.
The second step in each bundle iteration is to evaluate the dual function $h$ at ${\widetilde{y}}$. In our case determining $h({\widetilde{y}})$ means solving the basic SDP relaxation as introduced in Section \[sec:DefProblemsBasicRel\] with a modified objective function. Hence in the case of Max-Cut the oracle can be evaluated very quickly, whereas evaluating the oracle is computationally more expensive for the stable set and the coloring problem.
The bundle iteration finishes by deciding whether ${\widetilde{y}}$ becomes the new center (serious step, roughly speaking if the increase of the objective function is good) or not (null step). In either case the new point is included in the bundle, some other elements of the bundle are possibly removed, the bundle parameter ${\mu}$ is updated and a new iteration starts.
Computational Results and Conclusions
=====================================
We close with a small sample of computational results and start with comparing our bundle method with interior point methods. In our context we are mostly interested to improve the upper bounds quickly, so we do not run the bundle method described in Section \[sec:bundle\] until we reach a minimizer, but stop after a fixed number of iterations, say $30$. In Table \[MC and SS solution times exact\] one sees that the running times decrease drastically if we use the bundle method. For $b\approx 15000$ it takes the bundle method only around $8\%$ of the MOSEK running time to get as close as $95\%$ to the optimal value, which is sufficient for our purposes. One sees that our bundle method scales much better for increasing $|J|$.
If we are given a graph and want to get an approximation on ${z_{mc}^{k}}$, ${z_{ss}^{k}}$ and ${z_{c}^{k}}$, then we iteratively perform a fixed number, say 30, iterations of the bundle method and then update the set $J$. We denote the exact subgraph bounds (ESB) obtained in this way with ${s_{mc}^{k}}$, ${s_{ss}^{k}}$ and ${s_{c}^{k}}$.
For the sake of brevity we will only outline how to determine $J$ heuristically, see [@elli-diss] for details. Let $X^\ast$ be the current solution of , or . We use the fact that the inner product of ${X_{I}}^\ast$ and particular matrices of size ${k_{I}}$ is potentially small whenever ${X_{I}}^\ast$ is not in $\operatorname{STAB}^2({G_{I}})$. Minimizing this inner product over all subgraphs of order ${k_{I}}$ would yield a quadratic assignment problem, so we repeatedly use a local search heuristic for fixed particular matrices in order to obtain potential subgraphs. Then we calculate the projection distances from ${X_{I}}^\ast$ to $\operatorname{STAB}^2({G_{I}})$ for all these subgraphs and include those in $J$ which have the largest distances and hence are violated most.
Finally we present several computational results for obtained ESBs. Note that we refrain from comparing the running times of our bundle method with the running time of inter point methods, because interior point methods would reach their limit very soon. Hence the bounds presented can only be obtained with our methods in reasonable time.
When considering Max-Cut the graphs in Table \[MC table 1\] are from the Biq Mac library [@BiqMacHomepage] with $n=100$ vertices. The edge density is 10%, 50% and 90%. The first 3 instances have positive weights and the remaining 3 have also some negative weights. The column labeled $3$ provides the deviation (in %) of the ESB with $k=3$ from $z_{mc}$. Thus if $p$ is the value in the column labeled $3$, then ${s_{mc}^{3}} = (1 + p/100)z_{mc}$. The columns labeled 5 and 7 are to be understood in a similar way for $k=5$ and $k=7$. We note that the improvement of the bound from column 3 to column 7 is quite substantial in all cases. We also point out that the relative gap is much larger if also negative edge weights are present.
In Table \[MC table 2\] we look at graphs from the Beasley collection [@BiqMacHomepage] with $n=250$. These instances were used by Rendl, Rinaldi and Wiegele [@RendlRinaldiWiegele] in a Branch-and-Bound setting. We only consider the ’hardest’ instances from [@RendlRinaldiWiegele] where the Branch-and-Bound tree has more than 200 nodes. The table provides the gap at the root node and also the number of nodes in the Branch-and-Bound tree as reported in [@RendlRinaldiWiegele]. The column 7-gap contains the gap after solving our new relaxation with ESCs up to size $k=7$. We find it remarkable that the first instance is solved to optimality and the gap in the second instance is reduced by 75 % compared to the original gap. This implies that using our ESBs would expectedly reduce the very high number of required Branch-And-Bound nodes tremendously.
We conclude that for Max-Cut our ESB constitute a substantial improvement compared to the previously used strongest bounds based on SDP with triangle inequalities. These correspond to the column 3-gap.
For the calculations for the stable set and the coloring problem all instances are chosen in such a way that $\vartheta(G)$ does not coincide and is not very close to $\alpha(G)$ and $\chi(G)$ respectively.
The instances for the stable set problem are taken partly from the DIMACS challenge [@DIMACS1992] with some additional instances from [@elli-diss] with $n$ ranging from 26 to 200. Table \[tab: ttf ss\] contains the new bounds. Here the starting point is the relaxation $\vartheta(G)$. We carry out 10 cycles of adding ESCs. In each cycle we add at most 200 ESCs, so in the final round we have no more than 2000 ESCs. The column heading indicates the order of the subgraphs. Here the improvement of the bounds is smaller than in the Max-Cut case, but we see that including larger subgraphs leads to much tighter bounds. In Table \[tab: proj dist ss\] we show that our approach also reduces the largest found projection distance over all subgraphs ${G_{I}}$ of ${X_{I}}$ to the corresponding $\operatorname{STAB}^2({G_{I}})$ in the course of the cycles. This indicates that the violation of the subgraphs decreases over the cycles and less and less subgraphs do not fulfill the ESCs. For example the value $0.000$ for the graph spin5 for ${s_{ss}^{2}}$ at the end of the cycles means that we did not find a violated subgraph of order $2$ anymore.
Results for a selection of coloring instances from [@COLInst] are provided in Tables \[tab: ttf col\] and \[tab: proj dist col\]. As in the stable set case there is only little improvement using small subgraphs ($k=2$ or 3). The inclusion of larger subgraphs ($k=6$) shows the potential of the exact subgraph approach.
Summarizing, we offer the following conclusions from these preliminary computational results.
$\bullet$ Our computational approach based on the partial Lagrangian dual is very efficient in handling also a large number of ESCs. The dual function evaluation separates the SDP part from the ESCs and therefore opens the way for large-scale computations. The minimization of the dual function is carried out as a convex quadratic optimization problem without any SDP constraints, and therefore is also suitable for a large number of ESCs.
$\bullet$ On the practical side we consider the small ESCs for Max-Cut a promising new way to tighten bounds for this problem. It will be a promising new project to explore these bounds also in a Branch-and-Bound setting.
$\bullet$ Our computational results for stable set and coloring confirm the theoretical hardness results for these problems. Here the improvement of the relaxations is small for $k\leqslant 3$ but including larger subgraphs yields a noticeable improvement of the bounds. It will be a challenge to extend our approach to larger subgraphs.
Tables
======
-- ------ ----- ----- ------- -------- -------- -------- --------- ------- -------
$b$
$3$ $4$ $5$ MOSEK SDPT3 oracle overall time value
2000 0 6000 18.37 49.22 1.01 6.05 32.93 97.20
2000 300 9000 55.24 134.78 1.18 9.33 16.90 95.02
4000 0 12000 104.56 289.78 1.71 11.13 10.64 93.66
3000 600 15000 184.43 525.85 1.56 14.83 8.04 94.54
1050 0 0 5914 23.54 79.25 7.86 10.65 45.22 98.25
1050 212 63 8719 50.11 174.33 10.61 16.52 32.96 97.89
2100 0 0 11780 126.40 388.07 7.43 12.27 9.71 93.65
1575 318 212 14653 241.29 648.83 10.79 20.21 8.38 94.44
-- ------ ----- ----- ------- -------- -------- -------- --------- ------- -------
: The running times for one Max-Cut and one stable set instance with different fixed sets of ESCs. The graphs of order $n=100$ are from the Erdős-Rényi model.
\[MC and SS solution times exact\]
name 3 5 7 $z_{mc}$
------------ ---------- --------- --------- ----------
pw01-100.1 0.40 0.00 0.00 2060
pw05-100.1 0.90 0.51 0.39 8045
pw09-100.1 0.58 0.38 0.31 13417
w01-100.1 0.13 0.00 0.00 719
w05-100.1 3.91 1.41 0.85 1606
w09-100.1 8.06 5.66 5.09 2096
: The deviation of the ESB to $z_{mc}$ for several Max-Cut instances.[]{data-label="MC table 1"}
name BBnodes root gap 7-gap $z_{mc}$
------------ --------- ---------- ------- ----------
beas-250-6 223 1.02 0.00 41014
beas-250-8 4553 2.19 0.49 35726
: The gap of the ESB to $z_{mc}$ for two Max-Cut instances.[]{data-label="MC table 2"}
name $n$ $m$ $\vartheta(G)$ ${s_{ss}^{2}}$ ${s_{ss}^{3}}$ ${s_{ss}^{4}}$ ${s_{ss}^{5}}$ ${s_{ss}^{6}}$ $\alpha(G)$
------------------ ----- ------- ---------------- ------------------ ------------------ ------------------ ------------------ ------------------ -------------
CubicVT26\_5 26 39 11.82 11.82 11.00 10.98 10.54 10.46 10
Circulant47\_030 47 282 14.30 14.30 13.61 13.21 13.24 13.14 13
G\_50\_0\_5 50 308 13.56 13.46 13.13 12.96 12.82 12.67 12
hamming6\_4 64 1312 5.33 4.00 4.00 4.00 4.00 4.00 4
spin5 125 375 55.90 55.90 50.42 50.17 50.00 50.00 50
keller4 171 5100 14.01 13.70 13.54 13.50 13.49 13.49 11
sanr200\_0\_9 200 2037 49.27 49.04 48.94 48.86 48.78 48.75 42
c\_fat200\_5 200 11427 60.35 60.34 58.00 58.00 58.00 58.00 58
: Tighten $\vartheta(G)$ towards $\alpha(G)$ for several instances for 10 cycles.
\[tab: ttf ss\]
--------------- ----- ---------------- --------------- --------------- ---------------- --------------- ---------------
name $n$ ${s_{c}^{2}} $ ${s_{c}^{4}}$ ${s_{c}^{6}}$ ${s_{c}^{2}} $ ${s_{c}^{4}}$ ${s_{c}^{6}}$
CubicVT26\_5 26 0.000 0.102 0.193 0.000 0.029 0.013
G\_50\_0\_5 50 0.087 0.093 0.118 0.000 0.013 0.024
spin5 125 0.000 0.084 0.269 0.000 0.046 0.006
sanr200\_0\_9 200 0.044 0.062 0.107 0.072 0.028 0.020
--------------- ----- ---------------- --------------- --------------- ---------------- --------------- ---------------
: Maximum found projection distance of ${X_{I}}$ to $\operatorname{STAB}^{2}({G_{I}})$ for the computations of .
\[tab: proj dist ss\]
name $n$ $m$ $\vartheta(G)$ ${s_{c}^{2}}$ ${s_{c}^{3}}$ ${s_{c}^{4}}$ ${s_{c}^{5}}$ ${s_{c}^{6}}$ $\chi(G) \leqslant$
---------------- ----- ------- ---------------- --------------- --------------- --------------- --------------- --------------- ---------------------
myciel4 23 71 2.53 2.53 2.90 2.91 3.28 3.29 5
myciel5 47 236 2.64 2.64 3.05 3.09 3.45 3.45 6
mug88\_1 88 146 3.00 3.00 3.00 3.00 3.00 3.00 4
1\_FullIns\_4 93 593 3.12 3.12 3.25 3.37 3.80 3.80 5
myciel6 95 755 2.73 2.73 3.02 3.09 3.57 3.51 7
myciel7 191 2360 2.82 2.82 3.02 3.08 3.63 3.50 8
2\_FullIns\_4 212 1621 4.06 4.06 4.32 4.38 4.66 4.68 6
flat300\_26\_0 300 21633 16.99 17.04 17.12 17.10 17.12 17.12 26
: Tighten $\vartheta(G)$ towards $\chi(G)$ for several instances for 10 cycles.
\[tab: ttf col\]
---------------- ----- ---------------- --------------- --------------- ---------------- --------------- ---------------
name $n$ ${s_{c}^{2}} $ ${s_{c}^{4}}$ ${s_{c}^{6}}$ ${s_{c}^{2}} $ ${s_{c}^{4}}$ ${s_{c}^{6}}$
myciel4 23 0.000 0.365 0.760 0.000 0.000 0.000
1\_FullIns\_4 93 0.009 0.349 0.629 0.000 0.158 0.203
myciel7 191 0.000 0.356 0.621 0.000 0.207 0.272
flat300\_26\_0 300 0.127 0.279 0.360 0.143 0.142 0.091
---------------- ----- ---------------- --------------- --------------- ---------------- --------------- ---------------
: Maximum found projection distance of ${X_{I}}$ to $\operatorname{COL}({G_{I}})$ for the computations of .
\[tab: proj dist col\]
[^1]: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 764759 and the Austrian Science Fund (FWF): I 3199-N31 and P 28008-N35. We thank three anonymous referees for their constructive comments which substantially helped to improve the presentation of our material.
|
---
abstract: 'For the automatic evaluation of Generative Question Answering (genQA) systems, it is essential to assess the *correctness* of the generated answers. However, n-gram similarity metrics, which are widely used to compare generated texts and references, are prone to misjudge fact-based assessments. Moreover, there is a lack of benchmark datasets to measure the quality of metrics in terms of the *correctness*. To study a better metric for genQA, we collect high-quality human judgments of correctness on two standard genQA datasets. Using our human-evaluation datasets, we show that existing metrics based on n-gram similarity do not correlate with human judgments. To alleviate this problem, we propose a new metric for evaluating the correctness of genQA. Specifically, the new metric assigns different weights on each token via keyphrase prediction, thereby judging whether a predicted answer sentence captures the key meaning of the human judge’s ground-truth. Our proposed metric shows a significantly higher correlation with human judgment than widely used existing metrics.'
author:
- |
Hwanhee Lee$^{1}$, Seunghyun Yoon$^{1}$, Franck Dernoncourt$^{2}$\
**Doo Soon Kim$^{2}$, Trung Bui$^{2}$, Joongbo Shin$^{1}$ Kyomin Jung$^{1}$\
$^{1}$Dept. of Electrical and Computer Engineering, Seoul National University, Seoul, Korea\
$^{2}$Adobe Research, San Jose, CA, USA\
[{wanted1007, mysmilesh, jbshin, kjung}@snu.ac.kr]{}\
[{franck.dernoncourt, dkim, bui}@adobe.com]{}\
**
bibliography:
- 'emnlp2020.bib'
title: 'KPQA: A Metric for Generative Question Answering Using Word Weights'
---
Introduction
============
Question answering (QA) system has received consistent attention in the natural language processing community. Recently, research on QA systems has reached the stage of *generating* answers, called genQA, rather than to extract the answer from the context for a given question. [@yin2015neural; @bauer2018commonsense; @nishida2019multi; @bi2019incorporating] However, a bottleneck in developing genQA models is that there is no proper automatic metrics to evaluate generated answer. [@chen2019evaluating]
In evaluating the genQA model, it is essential to consider whether the generated response correctly contains vital information to answer the question. There exist several n-gram similarity metrics such as BLEU [@papineni-etal-2002-bleu] and ROUGE-L [@lin-2004-rouge], measuring word overlaps between generated response and ground-truth answer; however, these metrics are insufficient to evaluate the genQA system. [@yang2018adaptations; @chen2019evaluating]. For instance, in the example in Fig. \[fig\_ex\] from the AVSD [@alamri2019audio] dataset, the predicted answer receives a high score by BLEU-1 and ROUGE-L (0.833 for both metrics) due to many overlaps of words with those in the ground-truth. However, humans, whom we hired to evaluate the answer, assign a lower correctness score of 0.266 in the scale from 0 to 1 due to the mismatch of critical information. To answer the question correctly for this example, we expect the prediction to include the *kitchen* in it since the question is asking about the specific place, *what room*. We find that existing metrics often fail to capture the *correctness* of the prediction considering the key information in the question as in this example, since they evenly consider the importance of each word when evaluating answers. Also, we find another limitation that there is no proper benchmark dataset to evaluate automatic evaluation metrics for assessing the correctness of generated response in genQA. To fill this gap, we firstly collect human evaluations for the correctness of generated answers obtained from state-of-the-art models on two standard genQA datasets. By designing careful instructions and filtering noisy annotations, we create high-quality datasets for evaluating the correctness in the genQA domain. With the proposed datasets, we confirm that existing metrics are poorly correlated with human judgments in the preliminary experiment. To overcome the shortcomings of previous metrics, we develop a novel keyphrase predictor, which computes the importance weight of each word in both predicted answer and ground-truth answer when evaluating its correctness. By integrating the output from the keyphrase predictor, we propose a *KPQA*-metric, which assigns high weight to an important word when assessing correctness. Our *KPQA*-metric is computed in two steps: (1) Given a {question, generated answer, reference answer}, we compute importance weights for each question-answer pair {question, generated answer} and {question, ground-truth answer} using a pre-trained keyword prediction model; (2) By using the importance weights, we then compute a weighted similarity score by integrating it into existing metrics. Our approach can be easily integrated into most of the existing metrics, including n-gram similarity metrics and recently proposed BERTScore [@zhang2019bertscore]. We evaluate the proposed method on four datasets: MS-MARCO [@nguyen2016ms], AVSD [@alamri2019audio], NarrativeQA [@kovcisky2018narrativeqa] and SemEval [@ostermann2018semeval]. To evaluate our proposed method, we newly collect human judgments for MS-MARCO and AVSD from the hired annotators. For NarrativeQA and SemEval dataset, we use the data from [@chen2019evaluating], who also studied metrics for QA. Our experimental results show that the proposed metric has significantly higher correlations with human judgments than those of the previous metrics for all of the four datasets. Also, our importance weighting mechanism has strong interpretability since the importance weights show where to focus as visualized in Figure \[fig\_kpqa\_all\]. We will release the human-annotated benchmark dataset and pre-trained models to compute *KPQA*-metric for the research community.
Preliminary: Automated Text Evaluation Metrics
==============================================
We briefly review the current automated text evaluation metrics which have been used for evaluating the genQA systems.
**BLEU** is a popular evaluation metric for generated text based on $n$-gram precision. BLEU scores a candidate by counting the number present in the reference among the $n$-gram of the candidate. In general, $n$ is varied from 1 to 4, and the scores for varying $n$ are aggregated with a geometric mean. In this work, we look at BLEU-1 and BLEU-4, where $n\,{=}\,1$ and $n\,{=}\,4$, respectively.
**ROUGE** is a set of evaluation metrics used for automatic text generation such as summarization and machine translation. Typically, most studies used ROUGE-L, which is a F-measure based on the L longest common subsequence between a candidate and the reference. Unlike BLEU, ROUGE-L has the advantage of not requiring the predefined number of contiguous sequence $n$.
**METEOR** [@banerjee2005meteor] is an F1 score of a set of unigram alignments. METEOR has a unique property that it considers stemmed words, synonyms, and paraphrases, as well as the standard exact word matches. **BERTScore** is a recently proposed text evaluation metric using pre-trained representations from BERT [@devlin2019bert]. BERTScore firstly computes the contextual embeddings for given references and predictions independently with BERT, and then computes pairwise cosine similarity scores.
Collecting Human judgements
===========================
To evaluate the evaluation metrics, we collect human judgement scores for two of genQA datasets. The human scores can be used to measure the correlation between human judgements and evaluation metrics. .
Datasets {#sec:dataset}
--------
Recently, @chen2019evaluating introduced human judgements for genQA in two datasets, NarrativeQA [@kovcisky2018narrativeqa] and SemEval 2018 Task 11 [@ostermann2018semeval]. We find that the average lengths of the answer sentence are 4.7 and 2.5 for NarrativeQa and SemEval 2018 Task, respectively, as shown in Table \[anslen\]. These short answers cannot be representative of genQA, because the answers could be long and may deliver complex meaning. To fill the gap, we collect human judgements of correctness for model predictions on two genQA datasets, MS-MARCO [@nguyen2016ms] and AVSD [@alamri2019audio]. We argue that MS-MARCO and AVSD, which have longer answers than NarrativeQA and SemEval 2018 Task 11, are more suitable for studying the metrics for general form of genQA.
Collecting Human judgement of Answer Correctness {#sec:collect}
------------------------------------------------
We first obtain the model predictions by training QA models for the target datasets and generating the answers for the test sets. In the experiments, two best performing models are employed for each dataset, UniLM [@dong2019unified] and MTN [@le2019multimodal] for MS-MARCO and AVSD, respectively. We further provide detailed information on these two models in Appendix \[appendix:model\]. To prepare the test set, we randomly select 300 samples from the development set of MS-MARCO and 300 samples in the test set for AVSD. The generated responses are evaluated by humans to annotate the correctness of the predictions compared to ground-truth answers. In the following section, we will describe the procedure of collecting human judgements.
**Instructions to Annotators** We hire the workers from Amazon Mechanical Turk (MTurk) to rate the correctness of the generated answers from the models we trained. We assign ten workers for each sample to get reliable data. The instructions are shown in Fig. \[fig\_guide\]. We request the workers to annotate the correctness using a 5-point Likert scale [@wiki:Likert_scale], where 1 means completely wrong, and 5 means completely correct. **Filtering Noisy Workers** Some workers did not follow the instructions, producing poor-quality judgements. To solve this problem, we filter noisy responses using z-score as in [@jung2011improving]. We first compute the z-score among ten responses for each sample. Then, we consider the responses whose z-score is higher than 1 as noise and remove them up to five in the order of the z-score. As a result, all of the samples have at least five annotations after removing the noisy responses. The average number of annotators is shown in Table \[inter\_agree\]. We use the average score of the annotators for each sample as a ground-truth evaluation score to assess the quality of the evaluation metric. **Inter-Annotator Agreement** The final dataset is further validated with Krippendorff’s alpha [@krippendorff2011computing], a statistical measure of inter-rater agreement for multiple annotators. We observe that the Krippendorff’s $\alpha$ is higher than 0.8 for both datasets, as shown in Table \[inter\_agree\]. These coefficient numbers indicate a “near-perfect" agreement according to one of the general guidelines [@landis1977measurement] for kappa-like measures.
Proposed Metric for Evaluating genQA {#sec:importance}
====================================
To build a better metric for the genQA, we first propose a Keyphrase Predictor for Question Answering (KPQA). By considering the question, the KPQA assigns different weights to each token in the answer sentence in such a way that salient tokens receive a high value. We integrate the KPQA to existing metrics to make them evaluate the correctness as well.
Keyphrase Predictor for Question Answering
------------------------------------------
For genQA, we observe that each word has different importance when assessing a generated answer. As shown in Fig. \[fig\_ex\], there exist keywords or key-phrases that are considered significant while evaluating the correctness of the answer. Also, articles such as “*a*" and “*the*" are mostly irrelevant to the correctness of the answer. To predict the importance of each word we introduce Keyphrase Predictor for Question Answering (KPQA). As shown in Fig. \[fig\_kpqa\_all\], KPQA is a BERT-based [@devlin2019bert] classifier that can predict salient tokens in the answer sentence depending on the question. We regard it as a multi-class classification task where each token is a single class. To train KPQA, we first prepare extractive QA datasets such as SQuAD [@rajpurkar2016squad], GQA [@hudson2019gqa] and FSVQA [@shin2016color], which consist of {passage, question, answer-span}. These datasets are transformed into pairs of {answer-sentence, question, answer-span}. The answer-sentence is extracted from the passage so that it contains answer-span in it. The question and answer-sentence are concatenated and fed into KPQA to consider the question while classifying the salient tokens in the answer-sentence.
KPQA Metric
-----------
Since KPQA’s training process allows KPQA to find essential words in the answer sentence to a given question, we use pre-trained KPQA to get the importance weights that are useful for evaluating the correctness of generated answers in genQA. We describe how we combine these weights to existing metrics to derive the *KPQA*-metric. We first compute the importance weights for a given question *Q* = ($q_1$, ..., $q_l$), predicted answer *X* = ($x_1$, ..., $x_m$) and reference answer *X* = ($\hat{x}_1$, ..., $\hat{x}_n$) using pre-trained KPQA. We provide each pair {question, generated answer} and {question, ground-truth answer} to pre-trained KPQA and get the output of the softmax layer after \[SEP\]. We define these parts as KeyPhrase Weight, KPW as shown in Fig. \[fig\_kpqa\_all\]. We note that $\text{KPW}^{(Q, \mathbf{\textit{X}} )}$ = ($w_1$, ..., $w_m$) is a importance weight of prediction $X$ given question *Q*. These weights reflect the importance of each token for evaluating the correctness of a given full-sentence answer $X$ for the given question *Q*. We set KPW to 0 for tokens that are stopwords, such as “*a*" or “*the*", to ignore them. The list of the stopwords and other implementations details are in Appendix \[appendix:stopword\]. We then compute *KPQA*-metric by incorporating KPW to several existing metrics modifying the precision and recall to compute weighted similarity.
**ROUGE-L-KPQA** For instance, we derive ROUGE-L-KPQA, which is a modified version of ROUGE-L using KPW to compute weighted precision($P_{LCS}^{KPQA}$), recall($R_{LCS}^{KPQA}$) and F1($F1_{LCS}^{KPQA}$), as follows: $$\begin{aligned}
P_{LCS}^{KPQA} = \frac{\Sigma_{i=1}^{u} LCS^{KPQA}(\hat{x}_i, \mathbf{\textit{X}}) }
{\Sigma_{i=1}^{m} \text{KPW}_i^{(Q, \mathbf{\textit{X}} )}},
\end{aligned}$$ $$\begin{aligned}
R_{LCS}^{KPQA} = \frac{\Sigma_{i=1}^{u} LCS^{KPQA}(\hat{x}_i, \mathbf{\textit{X}}) }
{\Sigma_{i=1}^{n} \text{KPW}_i^{(Q, \mathbf{\textit{ \^{X}}})}},
\end{aligned}$$$$\begin{aligned}
F_{LCS}^{KPQA} = \frac{(1 +\beta^2)R_{LCS}^{KPQA} P_{LCS}^{KPQA} }
{R_{LCS}^{KPQA} + \beta^2 P_{LCS}^{KPQA}},\\
\end{aligned}$$where LCS is the Longest Common Subsequence between a prediction and a reference. The $LCS^{KPQA}(\hat{x}_i, \mathbf{\textit{X}})$ is defined as follows: $$LCS^{KPQA}(\hat{x}_i, \mathbf{\textit{X}})= \Sigma_{i=1}^{m} I_i \cdot \text{KPW}_i^{(Q,\mathbf{\textit{ \^{X}}})},\\$$ where $I_i$ is an indicator function which is 1 if each word is in the LCS and 0 otherwise.\
**BERTScore-KPQA** Similarly deriving ROUGE-L-KPQA, we compute BERTScore-KPQA using KPW. We first compute contextual embedding $\mathbf{x}$ for a prediction *X* and $\mathbf{\hat{x}}$ for a reference *X* using the BERT model. Then, we compute weighted precision($P_{BERT}^{KPQA}$), recall($R_{BERT}^{KPQA}$) and F1($F1_{BERT}^{KPQA}$) with contextual embedding and KPW of each token as follows:
$$\begin{aligned}
P_{BERT}^{KPQA} = \frac{\Sigma_{i=1}^{m} \text{KPW}_i^{(Q, \mathbf{\textit{X}} )} \cdot \text{max}_{\hat{x}_j\in \hat{x}} \mathbf{x_i}^T\mathbf{\hat{x}_j} }
{\Sigma_{i=1}^{m} \text{KPW}_i^{(Q, \mathbf{\textit{X}} )}}\\
\end{aligned}$$
$$\begin{aligned}
R_{BERT}^{KPQA} = \frac{\Sigma_{i=1}^{n} \text{KPW}_i^{(Q, \mathbf{\textit{\^X}} )} \cdot \text{max}_{x_j\in x} \mathbf{x_i}^T\mathbf{\hat{x}_j} }
{\Sigma_{i=1}^{n} \text{KPW}_i^{(Q, \mathbf{\textit{ \^{X}}})}}\\
\end{aligned}$$
$$\begin{aligned}
F1_{BERT}^{KPQA} = 2\cdot \frac{P_{BERT}^{KPQA}\cdot R_{BERT}^{KPQA}}{P_{BERT}^{KPQA}+R_{BERT}^{KPQA}}
\end{aligned}$$
Experimental Results {#sec:experiments}
====================
Implementation Details
----------------------
**Keyphrase Predictor** We train the single keyphrase predictor with various datasets (SQuAD [@rajpurkar2016squad], GQA [@hudson2019gqa], FSVQA [@shin2016color]) to build a general keyphrase extractor for question answering. For the SQuAD dataset, we select the sentence that include a short answer span as a full-sentence answer. For GQA dataset and FSVQA dataset, both of the datasets provide full-sentence answers and short answers. We construct the trainset for KPQA by randomly extracting 75k samples from each of the three datasets to balance the number of samples for each dataset. We then train a model for two epochs on the combined dataset. The performance of our keyword predictor in the development set, which is randomly extracted 10k samples from each of three datasets, is shown in Table \[kp\_acc\].
**BERTScore** For BERTScore we use *bert-large-uncased-whole-word-masking-finetuned-squad*, (24 layers, 1024 hidden units, 16 heads) from [@Wolf2019HuggingFacesTS] which is a BERT model fine-tuned on QA dataset SQuAD. We observe that computing BERTScore through this BERT model shows slightly higher correlation with human judgements than the BERT model without fine tuning. We use the first layer of it after the word embedding layer to compute the embedding. We experiment among different layers and found that the first hidden layer yielded the best result.
Evaluation Methods for Metrics
------------------------------
To compare the performance of various existing metrics and our metric, we use the Pearson coefficient and Spearman coefficient. We compute these correlation coefficients with human judgements of correctness. We test with MS-MARCO, AVSD which we collected human judgements and also for the datasets Narrative QA and SemEval from [@chen2019evaluating].
Performance Evaluation
----------------------
Table \[comparison\] shows the correlation scores for the baseline metrics and KPQA-augmented ones for multiple datasets. The correlation between human judgement and most of the existing metrics such as BLEU or ROUGE-L is very low, especially for the MS-MARCO dataset, which has longer and more abstractive answers than the other three datasets. Hence, most of the widely used n-gram similarity metrics are inadequate to evaluate the correctness of the answer for genQA especially for the datasets that have abstractive answer. We also observe a higher correlation score for our proposed *KPQA*-metric, BERTScore-KPQA and ROUGE-L-KPQA, compared to existing metrics including original BERTScore and ROUGE-L. We observe that this gap of correlation is especially higher for MS-MARCO dataset and we argue that our proposed metric is especially more effective in evaluating abstractive answers than existing metrics.
Comparison with IDF
-------------------
The next best metric after our proposed metric is the original BERTScore, which uses contextual embeddings and adopt IDF (Inverse Document Frequency) based importance weighting. When computing the original BERTScore, we compute the IDF dictionary with reference texts and adopt importance weighting with IDF as in [@zhang2019bertscore]. One of our *KPQA*-metrics, BERTScore-KPQA, which uses KPW as importance weights instead of IDF, outperforms original BERTScore with a significant gap as shown in Table \[comparison\]. By comparing BERTScore-KPQA and BERTScore, we show that our importance weighting method using KPQA is more effective than IDF for evaluating correctness. Since IDF is dependent on the frequency, it assigns a lower weight to some important words that frequently occur in the reference sentence. Hence, IDF based importance weighting might not be helpful for some cases. On the other hand, our KPW is computed utilizing questions so that it can assign the weights to words in the context of the questions in the generated answer, and this leads to better correlation with human evaluation.
Ablation Study
--------------
To validate the effectiveness of our KPW, we perform several ablation studies and present results in Table \[ablation\]. For the results in the second row, we substitute KPW in BERTScore-KPQA with uniform weight whose weight for each token is all set to one. By doing this, we can see the effect of our importance weighting by KPW that is conditioned on the question. We can observe that performance of *KPQA*-metric is higher than uniform weights. This gap is especially higher for the MS-MARCO dataset, where the average number of tokens is longer than other datasets. In the third row in Table \[ablation\], we can see the effect of stopword removal. Since our KPW already assigns lower weights to unnecessary words, the effect of setting the weight of stopwords to zero to remove them is slight and even negative for SemEval dataset. But it is usually effective to make stopwords’ weight zero so that it cannot be used completely in other three datasets.
Related Work {#sec:relatedwork}
============
One important next steps for current QA systems are the systems that can generate long answers in natural language for given question and context. Following this interests, several generative (abstractive) QA datasets [@nguyen2016ms; @he2017dureader; @kovcisky2018narrativeqa], where the answer is not necessarily in the passage, were recently released. Since the task is to generate natural language for the given question, the QA system is often trained with seq2seq [@sutskever2014sequence] objective similarly to other natural generation tasks such as neural machine translation. Hence, researchers often use n-gram based similarity metrics such as BLEU to evaluate the genQA systems, following other natural language generation tasks. However, most of these n-gram metrics including BLEU are originally developed for evaluating machine translation and previous works [@liu2016not; @nema2018towards; @kryscinski2019neural] showed that these metrics have poor correlation with human judgements in other language generation tasks such as dialog systems. Like other text generation systems, it is difficult to assess the performance through n-gram metrics for genQA. For genQA, n-gram similarity metrics can give high scores to the generated answer that is incorrect but contains a lot of unnecessary words in the reference answer. Previous works [@yang2018adaptations; @chen2019evaluating] pointed out these problems and studied the automated metrics in evaluating QA systems. Inspired by these works, we focus on studying and developing the evaluation metrics for genQA datasets that have more abstract and diverse answers. We analyze the problem of using existing n-gram similarity metrics across multiple genQA datasets and propose alternative metrics for genQA.
Conclusion {#sec:conclusion}
==========
In this work, we study and improve the metrics for evaluating the correctness of answers in genQA, where the task is to generate an abstractive, free-form answer given a question and a context. We collected large-scale human judgements on two genQA datasets to compare the correlation with existing metrics. We show that existing metrics have a lower correlation with human judgement in the two datasets. We observe that previous n-gram-based similarity metrics cannot consider the importance weight of the words in the sentence. Based on this observation, we propose a new metric that can assign a weight to a word depending on the importance of evaluating the correctness. To compute this weight, we train KPQA that can predict the importance of words in a generated answer conditioned on the question. By adopting this weights for maximum similarity matching in several existing metrics, we propose *KPQA-metric*. Our new metric has a dramatically higher correlation with human judgements than existing metrics.
\[sec:appendix\]
Datasets {#appendix:dataset}
========
**MS-MARCO** MS-MARCO [@nguyen2016ms] is a large-scale machine reading comprehension dataset that provides ten candidate passages for each question. The model should consider the relevance of the passages for the given question and answer the question. One of the main features of this dataset is that it contains free-form answers that are abstractive. MS-MARCO provides two tasks, Natural Language Generation (NLG) task and QA task. For the NLG task, the model should generate an abstractive summary of the passages for given questions, which is a well-formed answer rather than an answer span in the passage. Although the QA task also provides some abstractive answers, most of the answers are short and do not contain the context or rationale of the question. Hence, we use the NLG subset of MS-MARCO dataset as a genQA dataset to study the metrics for genQA.
**Audio Visual Scene-aware Dialog (AVSD)** To study more general metrics for genQA, we also use a multimodal genQA dataset for our work. Audio Visual Scene-aware Dialog (AVSD) [@alamri2019audio] is a multimodal dialogue dataset composed of QA pair about Charades videos. Although the name of the dataset contains dialog, all of the dialog pairs are composed of questions answering about a video. The task of this dataset is to generate an answer for a question about a given video, audio, and the history of previous turns in the dialog. In other words, this task is to generate a free-form answer for a given multimodal context, which can be considered as a kind of genQA.
Models {#appendix:model}
======
To investigate the algorithms for automatic metrics, we gather pairs of a sentence, {answer candidate, *true*-answer}. Note that each sentence is in natural language form. Collecting high-quality answer candidates for a given context and question is an essential step; thus, we choose models for each dataset from the latest research in the literature. We describe the models to generate the answer for two datasets we use in our work. For each dataset, we train the model that shows the highest performance in each dataset.
**UniLM** Since the publicly available code for state-of-the-art is not available for the NLG setting in MS-MARCO, we train the model with UniLM [@dong2019unified], which is a state-of-the-arts seq2seq model based on pre-trained representations from BERT [@devlin2019bert]. UniLM, which stands for unified language model pretraining, is a pre-trained transformer network that can be easily fine-tuned for NLU and NLG. UNiLM achieves higher performance for various NLG tasks, such as abstractive summarization and question generation. We fine-tune UniLM for genQA similar to the way fine-tuning UniLM to NLG, where source sequences are each question and paragraphs, the target sequence is an answer. We add \[SEP\] tokens between the question and each paragraph. Then, we fine-tune UniLM for three epochs with this setting.
**MTN** For AVSD, we train the multimodal transformer network and is a transformer encoder-decoder framework (MTN) [@le2019multimodal], which is a state-of-the-art model for this task. MTN employs multimodal attention blocks to fuse multiple modalities such as text, video, and audio. We use the publicly available code to train the model with the trainset of this dataset. After training, we generate the answers for the testset released in the DSTC7 workshop [@alamri2018audio].
Stopwords Removal {#appendix:stopword}
=================
Some of the words such as articles are commonly useless for measuring the correctness of the generated answer and even harm the evaluation by the unnecessary increase in the score. For the right example in Figure \[fig\_ex\], the exact match in *be* and *a* results in a higher ROUGE or BLEU, although the answer is incorrect. Hence, we try to filter them when evaluating the correctness. We construct a stopword list based on the stopwords list for English in the NLTK library [@loper2002nltk]. We exclude tokens such as “no” and “not” in the original list since those words might be important words in genQA. By using our stopwords list, we filter stopwords in the generated sentence by setting their weights to zero when we calculate the metric. Then, we can give more focus to the remain words that have a higher possibility to be content words.
|
---
abstract: |
Uncountably many mutually non-isomorphic product systems (that is, continuous tensor products of Hilbert spaces) of types $ II_0 $ and $
III $ are constructed by probabilistic means (random sets and off-white noises), answering four questions of W. Arveson.
author:
- Boris Tsirelson
title: 'Non-Isomorphic Product Systems'
---
|
---
abstract: 'Elemental bismuth provides a rare opportunity to explore the fate of a three-dimensional gas of highly mobile electrons confined to their lowest Landau level. Coulomb interaction, neglected in the band picture, is expected to become significant in this extreme quantum limit with poorly understood consequences. Here, we present a study of the angular-dependent Nernst effect in bismuth, which establishes the existence of ultraquantum field scales on top of its complex single-particle spectrum. Each time a Landau level crosses the Fermi level, the Nernst response sharply peaks. All such peaks are resolved by the experiment and their complex angular-dependence is in very good agreement with the theory. Beyond the quantum limit, we resolve additional Nernst peaks signaling a cascade of additional Landau sub-levels caused by electron interaction.'
author:
- 'Huan Yang$^{1}$, Benoît Fauqué$^{1,2}$,Liam Malone$^{3}$, Arlei B. Antunes$^{3}$, Zengwei Zhu$^{1,4}$, Ctirad Uher$^{5}$ and Kamran Behnia$^{1}$'
date: 'June 17, 2010'
title: Phase diagram of bismuth in the extreme quantum limit
---
Seventy years ago, an intensive study of the angular dependence of the de Haas-van Alphen effect in elemental bismuth[@shoenberg] led to the first experimental determination of the Fermi Surface in any metal. The deduced structure consists of one hole ellipsoid aligned perpendicular to the plane in which lay obliquely three slightly tilted electron ellipsoids (Fig.1a). Later investigations, carried out during the following decades, led to a quantitative description of the size, the orientation, and the position of these four ellipsoids[@smith; @bhargava; @edelman; @liu]. When the magnetic field is aligned along the highest symmetry axis, known as the trigonal axis, a field of 9 T allows one to attain the quantum limit putting carriers in their lowest Landau level.
The Nernst response in this configuration has been recently found to sharply peak each time a Landau level empties, generating giant quantum oscillations[@behnia2; @behnia1]. This particular Nernst profile, absent in two-dimensional systems, has been recently observed in bulk graphite [@zhu]. It points to electronic degrees of freedom *along* the magnetic field as a source of transverse in-plane thermoelectric response. In contrast to its two-dimensional counterpart[@jonson], the theoretical description of the Nernst effect in a three-dimensional system in the vicinity of the quantum limit has only recently begun to be explored[@bergman].
The extension of the Nernst measurements to fields as high as 33 T led to the observation of three Nernst peaks above 9 T[@behnia1], where no more Landau level crossing were expected. The latter observation raised the issue of collective effects in a three-dimensional electron gas at high magnetic field where Coulomb interaction and associated many-body instabilities are expected to become significant[@halperin] and the adequacy of the band picture, which treats electrons as non-interacting entities, remains an open question. Several recent theoretical studies explored the limits on the survival of the Fractional Quantum Hall Effect in the presence of finite interlayer coupling[@burnell; @levin; @alicea2].
Subsequent experimental work found that in strong magnetic field, both bismuth[@li; @fauque1; @fauque2] and Bi$_{1-x}$Sb$_{x}$[@banerjee] are far from featureless. Transport and thermodynamic coefficients were found to display unexpected anomalies beyond the quantum limit. In particular, angular-dependent torque magnetometry measurements detected a field scale quite distinct from the Nernst anomalies and identified this field scale as a phase transition implying the massive Dirac electrons of the three electron ellipsoids[@li]. However, two independent set of theoretical calculations[@alicea; @sharlai] found that the one-particle spectrum of bismuth for a field oriented close to the trigonal axis is not trivial. Charge neutrality in a compensated system implies strict equal density of holes and electrons as the magnetic field is swept. The intricate topology of the Fermi surface and the large Zeeman energy conspire to generate a complex phase diagram. According to these calculations[@alicea; @sharlai], the Landau level crossing of the three electron ellipsoids occur at a field which sharply shifts with angle as the field is slightly tilted off the trigonal axis. The field scale found by torque magnetometry[@li] follows the theoretical angular dependence of the field scale associated with the intersection of the Landau level of an electron pocket and the Fermi level. On the other hand, according to the same theoretical calculations, if the field is strictly oriented along the trigonal axis no other Landau level crossing is expected beyond 10 T. Thus, the torque anomalies were expected in the one-particle picture[@alicea; @sharlai], but not the three high-field anomalies observed in the Nernst response[@behnia1]. It was suggested[@sharlai] that they could be attributed to the Landau levels of the electron ellipsoids assuming a small misalignment of a few degrees. Due to the absence of *in situ* orientation of the crystal in the Nernst experiment[@behnia1], this possibility could not be ruled out.
In this paper, we present the first study of angular-resolved Nernst effect in bismuth and map the angular variation of the Nernst peaks across the quantum limit. Both sets of high-field anomalies found in the two previous studies[@behnia1; @li] are present and easily distinguishable in our data. According to our findings, the one-particle picture is relevant yet insufficient. The Nernst peaks caused by the emptying of a Landau level, closely follow the theoretically expected angular path. On the other hand, there are three unexpected Nernst peaks which cannot be associated with any Landau level. According to the non-interacting picture, the Nernst peaks occur at a field when the Fermi energy is located *between* Landau levels and no sharp feature in the density of states is expected. This result points to collective effects enhancing entropy per carrier at three particular magnetic fields between 9 T and 28 T.
Results
=======
Low field measurements
----------------------
Fig. 1 presents the result of an angular-dependent study performed at T=0.49 K and up to 12 T. As previously reported[@behnia2], the Nernst response shows quantum oscillations with a period (0.147 T$^{-1}$) matching the cross section of the hole ellipsoid. In this configuration, the Nernst response, like other transport properties such as resistivity[@bompadre] and the Hall effect[@fauque1] is dominated by holes. The lower mobility of carriers traveling perpendicular to the trigonal axis on an electron ellipsoid[@hartman] makes them less visible in a transport experiment. On the other hand, torque magnetometry[@li; @fauque1], a probe of anisotropic magnetization is dominated by the electron ellipsoids. The latter are much more anisotropic than the hole ellipsoid and their diamagnetic response is enhanced by their Dirac-like dispersion.
As the field was tilted in the (trigonal, binary) plane of the crystal, the main peaks barely moved. But the structure visible between large peaks rapidly evolves with tilt angle, $\theta_{1}$, as seen by arrows tracking one of the smaller Nernst peaks. Fig. 1c presents a color map of the Nernst response in the (B, $\theta_{1}$) plane together with the position of the smaller Nernst peaks. One can clearly distinguish between two different field scales. The first are quasi-horizontal field scales, caused by the passage of hole landau levels as previously identified[@behnia2; @sharlai]. The second group of field scales are less prominent in the data and present a very sharp angular variation. Their coordinates in the (B, $\theta_{1}$) plane coincides with the torque anomalies in the same field window[@li] and we thus deduce that they are by the passage of the Landau levels of the electron pockets in agreement with the predictions of both sets of theoretical calculations[@sharlai; @alicea]. In particular, the electron spectrum calculated by Alicea and Balents [@alicea] appears to be in quantitative agreement with our data. Note the presence of a small Nernst peak (marked by a green arrow), which rapidly disappears as the field is tilted off the trigonal axis. This field scale is the only feature of our data to be absent in the torque data in this field window.
High field measurements
-----------------------
Fig. 2 presents the Nernst data taken with a rotating set-up at T= 1.5 K in a DC resistive magnet up to 28 T. As the field is tilted off the trigonal axis, the structure of the Nernst response above 9 T evolves. Panel b of the same figure presents a color map of the Nernst response in the (B, $\theta_{1}$) plane, which clearly exposes the strongly anisotropic field scales discovered by Li *et al.* in their angular-dependent torque magnetometry experiments[@li]. These field scales trace quasi-vertical lines in the (B, $\theta_{1}$) plane. In addition to these two lines, there are three additional field scales above 9 T, which cross the $\theta_{1}=0$ axis close to the position of the three anomalies detected in the earlier Nernst experiment with a field nominally along the trigonal axis, but with no *in situ* control of orientation[@behnia1]. The detection of these two distinct field scales by the same experiment definitely rules out the misalignment scenario proposed[@sharlai] as an explanation for the unexpected Nernst anomalies[@behnia1]. In this spectrum, the two quasi-vertical lines correspond to the passage of an electron ellipsoid sub-level (labeled 0$^{+}_{e}$) through the chemical potential as the field is tilted[@alicea; @sharlai]. First detected by torque magnetometry[@li], they are also visible in angular-dependent magnetoresistance data[@fauque1].
We also used a two-axis rotation set-up, which allowed us to delimit the triangular around the trigonal axis formed by the intersection of each of the three 0$^{+}_{e}$ sub-levels with the Fermi level. As seen in Fig. 3, even at 28 T, this triangle has a finite size allowing us to identify the trigonal axis. We checked the presence of the three ultraquantum Nernst peaks in such a configuration with a virtually perfect alignment of the magnetic field. Their angular dependence of these peaks is much weaker that the two quasi-vertical lines which delimit the central region. This may suggest an explanation for their absence in the torque data[@li]. There is no torque when the field is strictly aligned along the trigonal axis. When the field is tilted, the torque response, which is proportional to the anisotropy of the magnetic susceptibility, is dominated by the more anisotropic field scales. As pointed out by Li and co-workers[@li], in a torque study sweeping $\theta_{1}$, the ellipsoid $e3$ of Fig.1a, for example, is invisible because of its negligible contribution to anisotropic magnetization.
A striking feature of the Fig. 2b and Fig. 3b, is the absence of mirror symmetry between positive and negative $\theta_{1}$. In Fig. 2b, the line crossing $\theta_1$=0 at 18 T does not respect the reflection symmetry of the crystal. Moreover, in Fig. 3b, the legs of the triangle delimiting the central region around the trigonal axis do not display the same intensity. It is unlikely that these features arise because of a residual imperfection in controlling the field orientation, in particular in the latter case, as the two axis set-up scans a solid angle.
Fig. 4 presents the data obtained at different temperatures when the field is aligned along the trigonal axis. The thermal evolution of the quantum oscillations confirms the identification of Nernst peaks according to their angular dependence. The hole peaks are more robust than the electron peaks as the sample is warmed up. At T=4.3 K, while the electron peaks are already smeared out, hole peaks persist. The persistence of the hole peaks up to higher temperatures is in agreement with their lighter effective mass in this configuration ($m^{\|}_{h}\simeq 0.064 m_{0}$ for holes and $m^{\|}_{e}\simeq 0.26 m_{0}$[@smith]). The lower panel presents the high-field Nernst signal as a function of the inverse of the magnetic field, with the magnetic field aligned along the trigonal axis with our two-axis set-up. The three ultraquantum anomalies fade away with warming almost at the same temperature as the low-field peaks clearly identified as those associated with electron ellipsoids. At T=5 K, the hole anomalies are still present, but the ultraquantum peaks are all wiped out. This observation points to electron ellipsoids as the source of the three high-field anomalies. This conclusion is confirmed by the size of the latter matched to electron and hole peaks below the quantum limit.
Discussion
==========
Controlling the orientation of the magnetic field with sub-degree accuracy allows us to confirm the complex theoretical one-particle spectrum of bismuth at high magnetic fields[@alicea; @sharlai] in its basic lines. For the first time, Nernst peaks associated with the very anisotropic electron ellipsoids can be clearly identified thanks to their sharp angular variation, which is in very good agreement with theory[@sharlai; @alicea] as well as the torque data by Li and co-workers[@li]. The second and more important conclusion of this study is that the three high-field Nernst peaks reported previously are not one of these anomalies caused by the intersection of a Landau level and the chemical potential.
It has been recently pointed out that when the field is tilted off the trigonal axis, the large electron-phonon coupling can lead to redistribution of carriers between pockets and sharp transport features may arise even in the one-particle picture[@littlewood]. Therefore, the presence of the anomalous Nernst peaks with a virtually perfect alignment between the magnetic field and the trigonal axis is particularly significant. As seen in Fig. 4, the Nernst peaks resolved in this configuration smoothly vanish with warming and there is no detectable critical temperature. Moreover, their field position does not shift with temperature. These peaks are not signatures of a thermodynamic phase transition such as the one which occurs in graphite, believed to be a field-induced Density-Wave (DW) transition[@yaguchi]. Bismuth, in contrast with graphite, keeps its metallicity up to a magnetic field as large as 55 T[@fauque2].
We conclude that at magnetic fields of 12.5 T, 18.2 T and 25.7 T, for which, according to the one-particle picture, the Fermi level is between one occupied and one empty Landau level of all three electron pockets, the Nernst signal peaks as if there was a Landau level crossing. This is also the case of the peak at 6.9 T (green symbols in Fig.1 and Fig.4a). We also note that the large Nernst peak of 38 T, resolved previously and associated with the hole pocket[@fauque2], occurs when according to the theoretical one-particle spectrum, the chemical potential is between the occupied 0$^{-}_{h}$ and the empty 1$^{-}_{h}$ Landau level. What occurs at these fields is not a thermodynamical phase transition associated to a symmetry breaking order parameter, but a topological[@wen] one, which occurs each time a Landau (sub-)level intersects the Fermi level[@blanter].
When the bottom of a Landau level intersects the Fermi level, there is a sharp enhancement in entropy per carrier and thus a Nernst peak[@zhu; @bergman]. In the presence of Zeeman coupling, the degeneracy associated with spin degrees of freedom is lifted and sub-levels emerge. The challenge for theory is to find a mechanism to produce additional Landau sub-levels and leading to Nernst peaks at particular magnetic fields. Since the three electron valleys are degenerate when the field is along the trigonal axis, the valley degrees of freedom are a natural direction to look. However, as long as the three valleys are strictly identical, a field along the trigonal axis is not expected to lift this degeneracy. Very recently, a new theoretical scenario invoking spontaneous valley polarization as a result of electron interaction has emerged and may prove to be relevant to our results[@abanin]. In particular, this scenario would provide a natural explanation for the absence of rotational symmetry in our high-field data.
In summary, the study of the angular-dependent Nernst response fine tunes the challenge addressed to theory. First of all, the complex theoretical one-particle spectrum of bismuth at high magnetic fields[@alicea; @sharlai] is confirmed in its basic lines. Nernst peaks associated with the very anisotropic electron ellipsoids are clearly detected and are in very good agreement with both calculations[@alicea; @sharlai] and the torque data[@li]. More importantly, *none* of the three high-field Nernst peaks reported previously[@behnia1] are caused by the intersection of a Landau level and the Fermi level in the one-particle spectrum. Finally, the temperature dependence of the anomalous Nernst peaks clearly links them to the electron ellipsoids. The origin of these additional Landau sub-levels, unexpected in the non-interacting picture, remains an open question.
Methods
=======
Samples
-------
In total, five bismuth single crystals with a length between 2 to 5 mm and a thickness between 0.5 to 2.1 mm were studied . Their Residual Resistivity Ratio (RRR: that is the change in resistance from room temperature to 4.2 K) of these samples ranged from 40 to 120. In all five crystals, for a field aligned with sub-degree accuracy along the trigonal axis, the three ultraquantum Nernst peaks were found to occur at the same magnetic field (within a window of 1 T). The anomalies were more pronounced in the samples with the highest RRR).
Angular dependent Nernst measurements
-------------------------------------
The Nernst coefficient was measured with a standard one-heater-two-thermometer set-up. Both the heater and the thermometers were RuO$_{2}$ chips attached to the sample through silver wires. The set-up was rotated using a piezoelectric linear positioner provided by the Attocube company (www.attocube.de). A one-axis rotation set-up (allowing a rotation window of 20 degrees) was used to measure the angular-dependent Nernst effect in a dilution refrigerator down to 0.18 K and in presence of a superconducting magnet up to 12 T. The same set-up was then used to study the Nernst effect and its angular-dependence in a DC resistive magnet of LNCMI-Grenoble up to 28 T and down to 1.5 K.
A second set-up allowing rotation in two perpendicular planes, albeit in an angular window restricted to 5 to 7 degrees, was used to obtain the data presented in Fig. 3 and Fig. 4.
The angle between the magnetic field and the sample was determined using a Hall sensor attached to the rotating stage. In the case of the two-axis set-up, we used two perpendicular Hall sensors to determine the two angles. The relative accuracy of angular determination was about 0.05 degree.
**Acknowledgements:** We thank J. Alicea, L. Balents, A. Millis and Y. Sharlai for stimulating discussions. This work is supported by Agence Nationale de Recherche (ANR-08-BLAN-0121-02) as part of DELICE and by EuroMagNET II under the EU contract number 228043. B.F. is a Newton International Fellow. Z.Z. acknowledges a scholarship granted by China Scholarship Council.
**Authors contribution:** H.Y., B.F., L.M., A.B.A and K.B carried out the high-field measurements. H.Y., B.F. and Z.Z. assisted by K.B. performed the low-field measurements. C.U. made the bismuth single crystals used in this study. All the authors participated in the analysis and discussion of the results. K.B. wrote the paper.
**Conflict of interest statement:** The authors declare no conflicting financial interests.
**Corresponding author** Correspondence should be sent to K. B. (kamran.behnia@espci.fr)
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abstract: 'The massively parallel nature of biological information processing plays an important role for its superiority to human-engineered computing devices. In particular, it may hold the key to overcoming the von Neumann bottleneck that limits contemporary computer architectures. Physical-model neuromorphic devices seek to replicate not only this inherent parallelism, but also aspects of its microscopic dynamics in analog circuits emulating neurons and synapses. However, these machines require network models that are not only adept at solving particular tasks, but that can also cope with the inherent imperfections of analog substrates. We present a spiking network model that performs Bayesian inference through sampling on the BrainScaleS neuromorphic platform, where we use it for generative and discriminative computations on visual data. By illustrating its functionality on this platform, we implicitly demonstrate its robustness to various substrate-specific distortive effects, as well as its accelerated capability for computation. These results showcase the advantages of brain-inspired physical computation and provide important building blocks for large-scale neuromorphic applications.'
author:
- |
**Akos F. Kungl$^{1}$, Sebastian Schmitt$^{1}$, Johann Klähn$^{1}$, Paul Müller$^{1}$, Andreas Baumbach$^{1}$, Dominik Dold$^{1}$,**\
**Alexander Kugele$^{1}$, Eric Müller$^{1}$, Christoph Koke$^{1}$, Mitja Kleider$^{1}$, Christian Mauch$^{1}$, Oliver Breitwieser$^{1}$,**\
**Luziwei Leng$^{1}$, Nico Gürtler$^{1}$, Maurice Güttler$^{1}$, Dan Husmann$^{1}$, Kai Husmann$^{1}$, Andreas Hartel$^{1}$,**\
**Vitali Karasenko$^{1}$, Andreas Grübl$^{1}$, Johannes Schemmel$^{1}$, Karlheinz Meier$^{1}$, Mihai A. Petrovici$^{1,2}$**\
$^{1}$Kirchhoff-Institute for Physics, Heidelberg University, 69120 Heidelberg, Germany\
$^{2}$Department of Physiology, University of Bern, 3012 Bern, Switzerland
bibliography:
- 'bib.bib'
date: 2019 November 14th
title: Accelerated physical emulation of Bayesian inference in spiking neural networks
---
This article is bound to be published in Frontiers in Neuromorphic Engineering:\
<https://www.frontiersin.org/articles/10.3389/fnins.2019.01201>.\
Please cite as:\
Kungl A. F. et al. (2019) Accelerated Physical Emulation of Bayesian Inference in Spiking Neural Networks. *Front. Neurosci. 13:1201. doi: 10.3389/fnins.2019.01201*
Introduction
============
The aggressive pursuit of Moore’s law in conventional computing architectures is slowly but surely nearing its end [@waldrop2016chips], with difficult-to-overcome physical effects, such as heat production and quantum uncertainty, representing the main limiting factor. The so-called von Neumann bottleneck between processing and memory units represents the main cause, as it effectively limits the speed of these largely serial computation devices. The most promising solutions come in the form of massively parallel devices, many of which are based on brain-inspired computing paradigms [@indiveri2011neuromorphic; @furber2016large], each with its own advantages and drawbacks.
Among the various approaches to such neuromorphic computing, one class of devices is dedicated to the physical emulation of cortical circuits: not only do they instantiate neurons and synapses that operate in parallel and independently of each other, but these units are actually represented by distinct circuits that emulate the dynamics of their biological archetypes [@mead1990neuromorphic; @indiveri2006vlsi; @schemmel2010wafer; @jo2010nanoscale; @pfeil2013six; @qiao2015reconfigurable; @chang2016demonstration; @wunderlich2018demonstrating]. Some important advantages of this approach lie in their reduced power consumption and enhanced speed compared to conventional simulations of biological neuronal networks, which represent direct payoffs of replacing the resource-intensive numerical calculation of neuro-synaptic dynamics with the physics of the devices themselves.
However, such computation with analog dynamics, without the convenience of binarization, as used in digital devices, has a downside of its own: variability in the manufacturing process (fixed pattern noise) and temporal noise both lead to reduced controllability of the circuit dynamics. Additionally, one relinquishes much of the freedom permitted by conventional algorithms and simulations, as one is confined by the dynamics and parameter ranges cast into the silicon substrate. The main challenge of exploiting these systems therefore lies in designing performant network models using the available components while maintaining a degree of robustness towards the substrate-induced distortions. Just like for the devices themselves, inspiration for such models often comes from neuroscience, as the brain needs to meet similar demands.
With accumulating experimental evidence [@berkes2011spontaneous; @pouget2013probabilistic; @orban2016neural; @haefner2016perceptual], the view of the brain itself as an analytical computation device has shifted. The stochastic nature of neural activity in vivo is being increasingly regarded as an explicit computational resource rather than a nuisance that needs to be dealt with by sophisticated error-correcting mechanisms or by averaging over populations. Under the assumption that stochastic brain dynamics reflect an ongoing process of Bayesian inference in continuous time, the output variability of single neurons can be interpreted as a representation of uncertainty. Theories of neural sampling [@buesing2011neural; @hennequin2014fast; @aitchison2016hamiltonian; @petrovici2016stochastic; @kutschireiter2017nonlinear] provide an analytical framework for embedding this type of computation in spiking neural networks.
In this paper we describe the realization of neural sampling with networks of leaky integrate-and-fire neurons [@petrovici2016stochastic] on the BrainScaleS accelerated neuromorphic platform [@schemmel2010wafer]. With appropriate training, the variability of the analog components can be naturally compensated and incorporated into a functional network structure, while the network’s ongoing dynamics make explicit use of the analog substrate’s intrinsic acceleration for Bayesian inference (\[sec:experimentalSetup\]). We demonstrate sampling from low-dimensional target probability distributions with randomly chosen parameters (\[sec:distr\]) as well as inference in high-dimensional spaces constrained by real-world data, by solving associated classification and constraint satisfaction problems (pattern completion, \[sec:datasets\]). All network components are fully contained on the neuromorphic substrate, with external inputs only used for sensory evidence (visual data). Our work thereby contributes to the search for novel paradigms of information processing that can directly benefit from the features of neuro-inspired physical model systems.
Methods
=======
The BrainScaleS system {#sec:bss}
----------------------
![ Photograph of a fully assembled wafer module of the BrainScaleS system (dimensions: ). One module hosts 384 HICANN chips on 48 reticles, with 512 physical neurons per chip and 220 synapse circuits per neuron. The wafer itself lies at the center of the module and is itself not visible. FPGAs are responsible for I/O and experiment control. Support PCBs provide power supply for the on-wafer circuits as well as access to neuron membrane voltages. The connectors for inter-wafer (sockets resembling USB-A) and off-wafer/host connectivity (Gigabit-Ethernet sockets) are distributed over all four edges of the main PCB. Mechanical stability is provided by an aluminum frame. The wafer itself is composed of 48 reticles (e.g., red rectangle), each containing 8 HICANN chips (e.g., black rectangle, enlarged in C). Inter-reticle connectivity is added in a post-processing step. **(C)** On a single HICANN chip, the largest area is occupied by the two synapse matrices which instantiate connections to the neurons positioned in the neuron array. [ **(D-E)** Postsynaptic potentials (PSPs) measured on 100 different neuron membranes using the same parameter settings before **(D)** and after **(E)** calibration. The insets show the height-normalized PSPs. The calibration serves two purposes. First, it provides a translation rule between the desired neuron parameters and the technical parameters set on the hardware. In this case, it brings the time constants $\tau_\mathrm{mem}$ and $\tau_\mathrm{syn}$ close to the target of , as evidenced by the small spread of the normalized PSPs. Second, in the absence of such a translation rule, it sets the circuits to their correct working points. Here, this happens for the synaptic weights: after calibration, PSP heights are, on average closer to the target working point of , but they remain highly diverse due to the variability of the substrate. For more details see [@schmitt2017neuromorphic]. ]{} The PSPs are averaged over 375 presynaptic spikes and smoothed with a Savitzky-Golay filter [@savitzky1964smoothing] to eliminate readout noise. The time-constants are given in the biological domain, but they are $10^4$ faster on the system. []{data-label="fig:wafer_module"}](fig1.pdf){width="\textwidth"}
BrainScaleS [@schemmel2010wafer] is a mixed-signal neuromorphic system, realized in CMOS technology, that emulates networks of spiking neurons. Each BrainScaleS wafer module consists of a silicon wafer with 384 HICANN (High Input Count Analog Neural Network) chips, see \[fig:wafer\_module\] A. On each chip, 512 analog circuits emulate the adaptive exponential integrate-and-fire (AdEx) model [@brette2005adaptive; @millner2010AdEx] of spiking neurons with conductance-based synapses. The dynamics evolve with an acceleration factor of $10^4$ with respect to biological time, i.e., all specific time constants (synaptic, membrane, adaptation) are approximately $10^4$ times smaller than typical corresponding values found in biology [@schemmel2010wafer; @petrovici2014characterization]. To preserve compatibility with related literature [@petrovici2016stochastic; @schmitt2017neuromorphic; @leng2018spiking; @dold2018stochasticity], [ we refer to system parameters in the biological domain unless specified otherwise, e.g., a membrane time constant given as is actually accelerated to on the chip.]{}
The parameters of the neuron circuits are stored in analog memory cells (floating gates) with resolution, and the synaptic weights are stored in SRAM [@schemmel2010wafer]. [ The analog memory cells are similar to the ones in [@lande1996analog], and they are described in [@loock2006evaluierung] and [@millner2012development].]{}
Spike events are transported digitally and can reach all other neurons on the wafer with the help of an additional redistribution layer that instantiates an on-wafer circuit-switched network [@zoschkeguettler2017rdlembedding] (\[fig:wafer\_module\] B).
Because of mismatch effects (fixed-pattern noise) inherent to the substrate, the response to incoming stimuli varies from neuron to neuron (\[fig:wafer\_module\] D). In order to bring all neurons into the desired regime and reduce the neuron-to-neuron response variability, we employ a standard calibration procedure that is performed only once, during the commissioning of the system [@schmitt2017neuromorphic; @petrovici2017robustness]. Nevertheless, even after calibration, a significant degree of diversity persists (\[fig:wafer\_module\] E). The emulation of functional networks that do not rely on population averaging therefore requires appropriate training algorithms (\[sec:datasets\]).
Sampling with leaky integrate-and-fire neurons
----------------------------------------------
The theory of sampling with leaky integrate-and-fire neurons [@petrovici2016stochastic] describes a mapping between the dynamics of a population of neurons with conductance-based synapses (equations given in \[table:network\]) and a Markov-chain Monte Carlo sampling process from an underlying probability distribution over binary random variables (RVs). Each neuron in such a sampling network corresponds to one of these RVs: if the $k$-th neuron has spiked in the recent past and is currently refractory, then it is considered to be in the *on-state* $z_k=1$, otherwise it is in the *off-state* $z_k=0$ (\[fig:liftheory\] A, B). With appropriate synaptic parameters, such a network can approximately sample from a Boltzmann distribution defined by $$\begin{aligned}
p^*({\boldsymbol}z) &= \frac{1}{Z} \exp \left ( \frac{1}{2} {\boldsymbol}z^T {\boldsymbol}W {\boldsymbol}z + {\boldsymbol}z^T {\boldsymbol}b \right) \; ,\end{aligned}$$ where $Z$ is the partition sum, ${\boldsymbol}W$ a symmetric, zero-diagonal effective weight matrix and $b_i$ the effective bias of the $i$-th neuron.
![ **Sampling with leaky integrate-and-fire (LIF) neurons**. **(A)** Schematic of a spiking sampling network (SSN) with 5 neurons. Each line represents two reciprocal synaptic connections with equal weights. **(B)** Example membrane potentials of three neurons in the network. Following a spike, the refractory mechanism effectively clamps the membrane potential to the reset value for a duration ${\tau_\mathrm{ref}}$. During this time, the RV corresponding to that neuron is in the state $z=1$ [(marked in green)]{}. At any point in time, the state sampled by the network can therefore be constructed directly from its output spikes and the refractory time $\tau_\mathrm{ref}$ of the neurons. [ **(C)** Probability distribution sampled by an SSN with three neurons as compared to the target distribution. ]{} **(D)** Based on this framework [@petrovici2016stochastic], hierarchical sampling networks can be built, which can be trained on real-world data. [Each line represents a reciprocal connection (two synapses) between the connected neurons.]{} []{data-label="fig:liftheory"}](fig2.pdf){width="65.00000%"}
In the original model, each neuron receives excitatory and inhibitory Poisson input. [ This plays two important roles: it transforms a deterministic LIF neuron into a stochastic firing unit and induces a high-conductance state, with an effective membrane time constant that is much smaller than other time constants in the system: $\tau_\mathrm{eff} \gg \tau_\mathrm{syn}, \tau_\mathrm{ref}$ - [see, e.g., @destexhe2003high; @petrovici2016form], which symmetrizes the neural activation function, as explained in the following. The activation function of an LIF neuron without noise features a sharp onset, but only a slow converge to its maximum value, hence being highly asymmetric around the point of activity. Background Poisson noise smears out the onset of the activation function, while the reduced membrane time constant accelerates the convergence to the maximum, making the activation function more symmetric and thus more similar to a logistic function, which is a prerequisite for this form of sampling. For the explicit derivation see [@petrovici2016stochastic] and [@petrovici2016form]. ]{} A mapping of this activation function to the abovementioned logistic function $1/[1 + \exp(-x)]$ provides the translation from the dimensionless weights and biases of the target distribution to the corresponding biological parameters of the spiking network [@petrovici2016form].
Although different in their dynamics, such sampling spiking networks (SSNs) function similarly to (deep) Boltzmann machines [@hinton1984boltzmann], which makes them applicable to the same class of machine learning problems [@leng2018spiking]. Training can be done using an approximation of wake-sleep algorithm [@hinton1995wake; @hinton2012practical], which implements maximum-likelihood learning on the training set: $$\begin{aligned}
\label{eq:wake_sleep}
\Delta b_i &= \eta(\langle z_i \rangle^* - \langle z_i \rangle) \; , \\
\Delta W_{ij} &= \eta (\langle z_i z_j \rangle^* - \langle z_i z_j\rangle) \; ,\end{aligned}$$ where $\langle \cdot \rangle$ and $\langle \cdot \rangle^*$ represent averages over the sampled (model or sleep phase) and target (data or wake phase) distribution, respectively, and $\eta$ is the learning rate.
In order to enable a fully-contained neuromorphic emulation on the BrainScaleS system, the original model had to be modified. The changes in the network structure, noise generation mechanism and learning algorithm are described in \[sec:experimentalSetup\].
For low-dimensional, fully specified target distributions, we used the Kullback-Leibler divergence [DKL, @kullback1951information] as a measure of discrepancy between the sampled (${p}$) and the target (${p^*}$) distributions: $${D_\mathrm{KL}}({p}\parallel {p^*}) = - \sum_{z_i \in \Omega} {p}(z_i) \ln \left ( \frac{{p}(z_i)}{{p^*}(z_i)} \right )$$ This was done in part to preserve comparability with previous studies [@buesing2011neural; @petrovici2015sampling; @petrovici2016stochastic], but also because the DKL is the natural loss function for maximum likelihood learning. For visual datasets, we used the error rate (ratio of misclassified images in the test set) for discriminative tasks and the mean squared error (MSE) between reconstruction and original image for pattern completion tasks. The MSE is defined as $$\text{MSE} = \frac{1}{N_\text{pixels}} \sum_{k=1}^{N_\text{pixels}}\left ( {z^\mathrm{data}}_k - {z^\mathrm{recon}}_k \right)^2 \; ,$$ where ${z^\mathrm{data}}_k$ is the reference data value, ${z^\mathrm{recon}}_k$ is the model reconstruction and the sum goes over the $N_\text{pixels}$ pixels to be reconstructed by the SSN.
Experimental setup {#sec:experimentalSetup}
------------------
The physical emulation of a network model on an analog neuromorphic substrate is not as straightforward as a software simulation, as it needs to comply with the constraints imposed by the emulating device. Often, it may be tempting to fine-tune the hardware to a specific configuration that fits one particular network, e.g., by selecting specific neuron and synapse circuits that operate optimally given a particular set of network parameters, or by manually tweaking individual hardware parameters after the network has been mapped and trained on the substrate. Here, we explicitly refrained from any such interventions in order to guarantee the robustness and scalability of our results.
All experiments were carried out on a single module of the BrainScaleS system using a subset of the available HICANN chips. The network setup was specified in the BrainScaleS-specific implementation of PyNN [@davison2009pynn] and the standard calibration [@schmitt2017neuromorphic] was used to set the analog parameters. The full setup consisted of two main parts: the SSN and the source of stochasticity.
![**Experimental setup.** Each sampling unit is instantiated by a pair of neurons on the hardware. The bias neuron is configured with a suprathreshold leak potential and generates a regular spike train that impinges on the sampling neuron , thereby serving as a bias, controlled by ${w_\mathrm{b}}$. **(A)** As a benchmark, we provided each sampling neuron with private, off-substrate Poisson spike sources. **(B)** Alternatively, in order to reduce the I/O load, the noise was generated by a random network (RN). The RN consisted of randomly connected inhibitory neurons with $E_\mathrm{leak}>V_\mathrm{thresh}$. Connections were randomly assigned, such that each sampling neuron received a fixed number of excitatory and inhibitory presynaptic partners (\[table:network\]). **(C)** Exemplary activation function (mean firing frequency) of a single sampling neuron with Poisson noise and with an RN as a function of the bias weight. The standard deviation of the the trial-to-trial variability is on the order of for both activation functions, hence the error bars are to small to be shown. The inset shows the membrane trace of the corresponding bias neuron. **(D-E)** [The figures show histograms over all neurons in a sampling network on a calibrated BrainScaleS system.]{} The width $s$ and the midpoint $w^0_\mathrm{b}$ of the activation functions with Poisson noise and with an RN are calculated by fitting the logistic function $\langle \nu \rangle = \nu_0/\{1+\exp[-({w_\mathrm{b}}-w^0_\mathrm{b})/s]\}$ to the data. []{data-label="fig:networkSetup"}](fig3.pdf){width="\textwidth"}
In the original sampling model [@petrovici2016stochastic], in order to affect biases, the wake-sleep algorithm (\[eq:wake\_sleep\]) requires access to at least one reversal potential (${E_\mathrm{l}}$, ${E_\mathrm{exc}}$, or ${E_\mathrm{inh}}$), which are all controlled by analog memory cells. Given that rewriting analog memory cells is both less precise and slower than rewriting the SRAM cells controlling the synaptic weights, we modified our SSNs to implement biases by means of synaptic weights. To this end, we replaced individual sampling neurons by sampling units, each realized using two hardware neurons (\[fig:networkSetup\] A, B). Like in the original model, a sampling neuron was set up to encode the corresponding binary RV. Each sampling neuron was accompanied by a bias neuron set up with a suprathreshold leak potential that ensured regular firing (\[fig:networkSetup\] C, inset). Each bias neuron projected to its target sampling neuron with both an excitatory and an inhibitory synapse (with independent weights), thus inducing a controllable offset of the sampling neuron’s average membrane potential. Because excitatory and inhibitory inputs are routed through different circuits for each neuron, two types of synapses were required to allow the sign of the effective bias to change during training. For larger networks, in order to optimize the allocation of hardware resources, we shared the use of bias neurons among multiple sampling neurons (connected via distinct synapses). Similarly, in order to allow sign switches during training, connections between sampling neurons were implemented by pairs of synapses (one excitatory and one inhibitory) as well.
The dynamics of the sampling neurons were rendered stochastic in two different ways. The first setup served as a benchmark and represented a straightforward implementation of the theoretical model from [@petrovici2016stochastic], with Poisson noise generated on the host computer and fed in during the experiment (\[fig:networkSetup\] A). In the second setup, we used the spiking activity of a sparse recurrent random network (RN) of inhibitory neurons, instantiated on the same wafer, as a source of noise (\[fig:networkSetup\] B). For a more detailed study of sampling-based Bayesian inference with noise generated by deterministic networks, we refer to [@jordan2017stochastic]. The mutual inhibition ensured a relatively constant (sub)population firing rate with suitable random statistics that can replace the ideal Poisson noise in our application. Projections from the RN to the SSN were chosen as random and sparse; this resulted in weak, but non-zero shared-input correlations. The remaining correlations are compensated by appropriate training; the Hebbian learning rule (\[eq:wake\_sleep\]) changes the weights and biases in the network such that they cancel the input correlations induced by the RN activity [@bytschok2017spike; @dold2018stochasticity]. Hence, the same plasticity rule simultaneously addresses three issues: the learning procedure itself, the compensation of analog variability in neuronal excitability, and the compensation of cross-correlations in the input coming from the background network. This allowed the hardware-emulated RN to replace the Poisson noise required by the theoretical model.
With these noise-generating mechanisms, the activation function of the neurons, defined by the firing rate as a function of the bias weight ${w_\mathrm{b}}$, took on an approximately logistic shape, as required by the sampling model (\[fig:networkSetup\] C). Due mainly to the variability of the hardware circuits [and the precision of the analog parameters]{}, the exact shape of this activation function varied significantly between neurons (\[fig:networkSetup\] D-E). Effectively, this means that initial weights and biases were set randomly, but also that the effective learning rates were different for each neuron. However, as we show below, this did not prevent the training procedure from converging to a good solution. This robustness with respect to substrate variability represents an important result of this work. The used neuron parameters are shown in \[table:neuron\] and a summary of the used networks is given in \[table:netParam\]. Our largest experiment a network of 609 neurons with 208 sampling neurons, 1 bias neuron and 400 neurons in the RN (\[table:netParam\] C) used hardware resources on 28 HICANN chips distributed over 7 reticles. Each of these functional neurons was realized by combining four of the 512 neuronal compartments (“denmems”) available on each HICANN, in order to reduce variability in their leak potentials and membrane time constants; for details see [@schemmel2010wafer].
To train the networks on a neuromorphic substrate without embedded plasticity, we used a training concept often referred to as in-the-loop training [@schmuker2014neuromorphic; @esser2016cover; @schmitt2017neuromorphic]. With the setup discussed above, the only parameters changed during training were digital, namely the synaptic weights between sampling neurons and the weights between bias and sampling neurons. This allowed us to work with a fixed set of analog parameters, which significantly amplified the precision and speed of reconfiguration during learning, as compared to having used the analog storage instead. The updates of the digital parameters (synaptic weights) were calculated on the host computer based on the wake-sleep algorithm (\[eq:wake\_sleep\]) but using the spiking activity measured on the hardware. During the iterative procedure, the values of the weights were saved and updated as a double precision floating point variable, followed by (deterministic) discretization in order to comply with the single-synapse weight resolution of . The learning parameters are given in \[table:learnParam\]. Clamping (i.e. forcing neurons into state 1 or 0 with strong excitatory or inhibitory input) was done by injecting regular spike trains with frequency from the host through 5 synapses simultaneously, excitatory for $z_k=1$ and inhibitory for $z_k=0$. These multapses (multiple synapses connecting two neurons) were needed to exceed the upper limit of single synaptic weights and thus ensure proper clamping.
Results
=======
Learning to approximate a target distribution {#sec:distr}
---------------------------------------------
The experiments described in this section serve as a general benchmark for the ability of our hardware-emulated SSNs and the associated training algorithm to approximate fully specified target Boltzmann distributions. The viability of our proposal to simultaneously embed deterministic RNs as sources of pseudo-stochasticity is tested by comparing the sampling accuracy of RN-driven SSNs to the case where noise is injected from the host as perfectly uncorrelated Poisson spike trains.
Target distributions ${p^*}$ over 5 RVs were chosen by sampling weights and biases from a Beta distribution centered around zero: $b_i, w_{ji} \sim 2[\mathrm{Beta}(0.5,0.5) - 0.5]$. Similarly to previous studies [@petrovici2016stochastic; @jordan2017stochastic], by giving preference to larger absolute values of the target distribution’s parameters, we thereby increased the probability of instantiating rougher, more interesting energy landscapes. The initial weights and biases of the network were sampled from a uniform distribution over the possible hardware weights. Due to the small size of the state space, the “wake” component of the wake-sleep updates could be calculated analytically as $\langle z_i z_j \rangle = {p^*}(z_i = 1, z_j = 1)$ and $\langle z_i \rangle = {p^*}(z_i=1)$ by explicit marginalization of the target distribution over non-relevant RVs.
For training, we used 500 iterations with sampling time per iteration. Afterwards, the parameter configuration that produced the lowest ${D_\mathrm{KL}}({p}\parallel {p^*})$ was tested in a longer () experiment. To study the ability of the trained networks to perform Bayesian inference, we clamped two of the five neurons to fixed values $(z_1, z_2) = (0,1)$ and compared the sampled conditional distribution to the target conditional distribution. Results for one of these target distributions are shown in \[fig:proofOfConcept\].
![**Emulated SSNs sampling from target Boltzmann distributions.** Sampled distributions are depicted in blue for setups with Poisson noise and in orange for setups using RNs. Target distributions shown in dark yellow. Data was gathered from 150 runs with random initializations. Median values are shown as dark colors and interquartile ranges as either light colors or error bars. **(A)** Improvement of sampled distributions during training. The observed variability after convergence (during the plateau) is not due to noise in the system, but rather a consequence of the weight discretization: when the ideal (target) weights lie approximately mid-way between two consecutive integer values on the hardware, training leads to oscillations between these values. The parameter configuration showing the best performance during a training run – which, due to the abovementioned oscillations, was not necessarily the one in the final iteration – was chosen as the end result of the training phase. Averages of these results are shown as dashed lines. **(B)** Convergence of sampled distributions for the trained SSNs. **(C)** and **(D)** Sampled joint and marginal distributions of the trained SSNs after , respectively. **(E)** Consistency of training results for different target distributions using Poisson noise. Here, we show a representative selection of 6 distributions with 10 independent runs per distribution. The box highlighted in blue corresponds to the target distribution used in the other panels of \[fig:proofOfConcept\]. [ The data is plotted following the traditional box-and-whiskers scheme: the orange line represents the median, the box represents the interquartile range, the whiskers represent the full data range and the $\times$ represent the far outliers. **(F)** Target distributions corresponding to the last five box-and-whiskers plots in **(E)**.]{} **(G)** Convergence of conditional distributions for the trained SSNs. **(H)** and **(I)** Sampled conditional joint and marginal distributions of the trained SSNs after , respectively. []{data-label="fig:proofOfConcept"}](fig4.pdf){width="\textwidth"}
On average, with Poisson noise, the training showed fast convergence during the first 20 iterations, followed by fine-tuning and full convergence within 200 iterations. As expected, the convergence of the setups using RNs was significantly slower due to the need to overcome the additional background correlations, but they were still able to achieve similar performance (\[fig:proofOfConcept\] A).
In both setups, during the test run, the trained SSNs converged to the target distribution following an almost identical power law, which indicates similar mixing properties (\[fig:proofOfConcept\] B). For longer sampling durations ($\gg\SI{10e3}{ms}$), the systematic deviations from the target distributions become visible and the ${D_\mathrm{KL}}({p}\parallel {p^*})$ reaches the same plateau at approximately ${D_\mathrm{KL}}({p}\parallel {p^*}) \approx \num{2e-2}$ as observed during training. C and D respectively show the sampled joint and marginal distributions after convergence. These observations remained consistent across a set of 20 different target distributions (see \[fig:proofOfConcept\] E for a representative selection).
Similar observations hold for the inference experiments. Due to the smaller state space, convergence happened faster (\[fig:proofOfConcept\] E). The corresponding joint and marginal distributions are shown in \[fig:proofOfConcept\] F and G, respectively. The lower accuracy of these distributions is mainly because of the asymmetry of the effective synaptic weights caused by the variability of the substrate, towards which the learning algorithm is agnostic. The training took wall-clock time, including the pure experiment runtime, the initialization of the hardware and the calculation of the updates on the host computer (total turn-over time of the training). This corresponds to a speed-up factor of 100 compared to the equivalent of biological real time. While the nominal $10^4$ speed-up remained intact for the emulation of network dynamics, the total speed-up factor was reduced due to the overhead imposed by network (re)configuration and I/O between the host and the neuromorphic substrate.
![ **Emulated SSNs sampling from different target Boltzmann distributions.** The figure shows the results of experiments identical to the ones in \[sec:distr\] for 20 different target distributions with 10 repetitions for each sample. We show the ${D_\mathrm{KL}}({p}\parallel {p^*})$ of the test-run after training for **(A)** the joint distributions with Poisson noise, **(B)** the inference experiment with Poisson noise, **(C)** the joint distributions with a random background network and **(C)** the inference experiment with a random background network. The data is plotted following the traditional box-and-whiskers scheme: the orange line represents the median, the box represents the interquartile range, the whiskers represent the full data range and the $\times$ represent the far outliers. In each subplot the leftmost data (highlighted in red) corresponds to the distribution shown in \[fig:proofOfConcept\]. []{data-label="fig:checkTargets"}](fig5.pdf){width="\textwidth"}
We carried out the same experiments as described previously with 20 different samples for the weights and the biases of the target distribution. In \[fig:checkTargets\] we show the final DKLs after training to represent a target distribution both with Poisson noise and with the activity of a random network. The experiments were repeated 10 times for each sample. Median learning results remained consistent across target distributions, with the variability reflecting the difficulty of the problem (discrepancies between LIF and Glauber dynamics become more pronounced for larger weights and biases). Variability across trials for the same target distribution is due to the trial-to-trial variability of the analog parameter storage (floating gates), due to the inherent stochasticity in the learning procedure (sampling accuracy in an update step), as well as due to systematic discrepancies between the effective pre-post and post-pre interaction strengths between sampling units, which are themselves a consequence of the aforementioned floating gate variability.
Learning from data {#sec:datasets}
------------------
In order to obtain models of labeled data, we trained hierarchical SSNs analogously to restricted Boltzmann machines (RBMs). Here, we used two different datasets: a reduced version of the MNIST [@lecun1998gradient] and the fashion MNIST [@xiao2017online] datasets, which we abbreviate as rMNIST and rFMNIST in the following. The images were first reduced with nearest-neighbor resampling (`misc.imresize` function in the SciPy library [@jones2014scipy]) and then binarized around the median gray value over each image. We used all images from the original datasets (approx. 6000 per class) from 4 classes (0, 1, 4, 7) for rMNIST and 3 classes (**T**-shirts, **Tr**ousers, **S**neakers) for rFMNIST (\[fig:datasets\] A-B). The emulated SSNs consisted of 3 layers, with 144 visible, 60 hidden and either 4 label units for rMNIST or 3 for rFMNIST.
![ **Behavior of hierarchical SSNs trained on data.** Top row: rMNIST; middle row: rFMNIST; bottom row: exemplary setups for the partial occlusion scenarios. **(A-B)** Exemplary images from the rMNIST (A) and rFMNIST (B) datasets used for training and comparison to their MNIST and FMNIST originals. **(C-D)** Training with the hardware in the loop after translation of pre-trained parameters. Confusion matrices after training shown as insets. Performance of the reference RBMs shown as dashed brown lines. Results are given as median and interquartile values over 10 test runs. **(E-F)** Pattern completion and **(G-H)** error ratio of the inferred label for partially occluded images (blue: patch; red: salt&pepper). Solid lines represent median values and shaded areas show interquartile ranges over 250 test images per class. Performance of the reference RBMs shown as dashed lines. As a reference, we also show the error ratio of the SNNs on unoccluded images in (G) and (H). **(I)** Snapshots of the pattern completion experiments: O - original image, C - clamped image (red and blue pixels are occluded), R - response of the visible layer, L - response of the label layer. **(J)** Exemplary temporal evolution of a pattern completion experiment with patch occlusion. For better visualization of the activity in the visible layer in (J) and (I), we smoothed out its discretized response to obtain grayscale pixel values, by convolving its state vector with a box filter of width. []{data-label="fig:datasets"}](fig6.pdf){width="\textwidth"}
Pre-training was done on simulated classical RBMs using the CAST algorithm [@salakhutdinov2010learning]. The pre-training provided a starting point for training on the hardware in order to accelerate the convergence of the in-the-loop training procedure. We use the performance of these RBMs in software simulations using Gibbs sampling as a reference for the results obtained with the hardware-emulated SSNs. After pre-training, we mapped these RBMs to approximately equivalent SSNs on the hardware, using an empirical translation factor based on an average activation function (\[fig:networkSetup\] C) to calculate the initial hardware synaptic weights from weights and biases of the RBMs. Especially for rMNIST, this resulted in a significant deterioration of the classification performance (\[fig:datasets\] C). After mapping, we continued training using the wake-sleep algorithm, with the hardware in the loop. While in the previous task it was possible to calculate the data term explicitly, it now had to be sampled as well. In order to ensure proper clamping, the synapses from the hidden to the label layer and from the hidden layer to the visible layer were turned off during the wake phase.
The SSNs were tested for both their discriminative and their generative properties. For classification, the visible layer was clamped to images from the test set (black pixels correspond to $z_k=1$ and white pixels to $z_k=0$). Each image was presented for 500 biological milliseconds, which corresponds to wall-clock time. The neuron in the label layer with the highest firing rate was interpreted as the label predicted by the model. The spiking activity of the neurons was read out directly from the hardware, without additional off-chip post-processing. For both datasets, training was able to restore the performance lost in the translation of the abstract RBM to the hardware-emulated SSN. The emulated SSNs achieved error rates of $4.45^{+0.12}_{-0.36}\%$ on rMNIST and $3.32^{+0.27}_{-0.04}\%$ on rFMNIST. These values are close to the ones obtained by the reference RBMs: $3.89^{+0.10}_{-0.02}\%$ on rMNIST and $2.645^{+0.002}_{-0.010}\%$ on rFMNIST (\[fig:datasets\] C-D, confusion matrices shown as insets).
The gross wall-clock time needed to classify the 4125 images in the rMNIST test set was ( per image, $210\times$ speed-up). For the 3000 images in the rFMNIST test set, the emulation ran for ( per image; $160\times$ speed-up). This subsumes the runtime of the BrainScaleS software stack, hardware configuration and the network emulation. The runtime of the software-stack includes the translation from a PyNN-based network description to a corresponding hardware configuration. As before, the difference between the nominal acceleration factor and the effective speed-up stems from the I/O and initialization overhead of the hardware system.
To test the generative properties of our emulated SSNs, we set up two scenarios requiring them to perform pattern completion. For each class, 250 incomplete images were presented as inputs to the visible layer. For each image, of visible neurons received no input, with the occlusion following two different schemes: salt&pepper (upper row in \[fig:datasets\] I) and patch (lower row in \[fig:datasets\] I). Each image was presented for . In order to remove any initialization bias resulting from preceding images, random input was applied to the visible layer between consecutive images.
Reconstruction accuracy was measured using the mean squared error (MSE) between the reconstructed and original occluded pixels. For binary images, as in our case, the MSE reflects the average ratio of mis-reconstructed to total reconstructed pixels. Simultaneously, we also recorded the classification accuracy on the partially occluded images. After stimulus onset, the MSE converged from chance level ($\approx \SI{50}{\percent}$) to its minimum ($\approx \SI{10}{\percent}$) within (\[fig:datasets\] E-F). Given an average refractory period of $\approx\SI{10}{ms}$ (\[fig:networkSetup\] C), this suggests that the network was able to react to the input with no more than 5 spikes per neuron. For all studied scenarios, the reconstruction performance of the emulated SSNs closely matched the one achieved by the reference RBMs. Examples of image reconstruction are shown in \[fig:datasets\] I-J for both datasets and occlusion scenarios. The classification performance deteriorated only slightly compared to non-occluded images and also remained close to the performance of the reference RBMs (\[fig:datasets\] G-H). The temporal evolution of the classification error closely followed that of the MSE.
As a further test of the generative abilities of our hardware-emulated SSNs, we recorded the images produced by the visible layer during guided dreaming. In this task, the visible and hidden layers of the SSN evolved freely without external input, while the label layer was periodically clamped with external input such that exactly one of the label neurons was active at any time (enforced one-hot coding). In a perfect model, this would cause the visible layer to sample only from configurations compatible with the hidden layer, i.e., from images corresponding to that particular class. Between the clamping of consecutive labels, we injected random input to visible layer to facilitate the changing of the image. The SSNs were able to generate varied and recognizable pictures, within the limits imposed by the low resolution of the visible layer (\[fig:tsne\]). For rMNIST, all used classes appeared in correct correspondence to the clamped label. For rFMNIST, images from the class “Sneakers” were not always triggered by the corresponding guidance from the label layer, suggesting that the learned modes in the energy landscape are too deep, and sneakers too dissimilar to T-shirts and Trousers, to allow good mixing during guided dreaming.
![ **Generated images during guided dreaming.** The visible state space, along with the position of the generated images within it, was projected to two dimensions using t-SNE [@maaten2008visualizing]. The thin lines connect consecutive samples. **(A)** rMNIST; **(B)** rFMNIST. []{data-label="fig:tsne"}](fig7.pdf){width="\textwidth"}
Discussion
==========
This manuscript presents the first scalable demonstration of sampling-based probabilistic inference with spiking networks on a highly accelerated analog neuromorphic substrate. We trained fully connected spiking networks to sample from target distributions and hierarchical spiking networks as discriminative and generative models of higher-dimensional input data. Despite the inherent variability of the analog substrate, we were able to achieve performance levels comparable to those of software simulations in several benchmark tasks, while maintaining a significant overall acceleration factor compared to systems that operate in biological real time. Importantly, by co-embedding the generation of stochasticity within the same substrate, we demonstrated the viability of a fully embedded neural sampling model with significantly reduced demands on off-substrate I/O bandwidth. Having a fully embedded implementation allows the runtime of the experiments to scale as $\mathcal{O}(1)$ with the size of the emulated network; this is inherent to the nature of physical emulation, for which wall-clock runtime only depends on the emulated time in the biological reference frame. In the following, we address the limitations of our study, point out links to related work and discuss its implications within the greater context of computational neuroscience and bio-inspired AI.
Limitations and constraints
---------------------------
The most notable limitation imposed by the current commissioning state of the BrainScaleS system was on the size of the emulated SSNs. At the time of writing, [due to limited software flexibility, system assembly and substrate yield, the usable hardware real-estate was reduced to a patchy and non-contiguous area]{}, thereby strongly limiting the maximum connectivity between different locations within this area. In order to limit synapse loss to small values (below ), we restricted ourselves to using a small but contiguous functioning area of the wafer, which in turn limited the maximum size of our SSNs and noise-generating RNs. Ongoing improvements in post-production and assembly, as well as in the mapping and routing software, are expected to enhance on-wafer connectivity and thereby automatically increase the size of emulable networks, as the architecture of our SSNs scales naturally to such an increase in hardware resources.
To a lesser extent, the sampling accuracy was also affected by the limited precision of hardware parameter control. The writing of analog parameters exhibits significant trial-to-trial variability; in any given trial, this leads to a heterogeneous substrate, which is known to reduce the sampling accuracy [@probst2015probabilistic]. Most of this variability is compensated during learning, but the resolution of the synaptic weights and the imperfect symmetry in the effective weight matrix due to analog variability of the synaptic circuits ultimately limit the ability of the SSN to approximate target distributions. This leads to the “jumping” behavior of the ${D_\mathrm{KL}}({p}\parallel {p^*})$ in the final stages of learning (\[fig:proofOfConcept\] A). In smaller networks, synaptic weight resolution is a critical performance modifier [@petrovici2017robustness]. However, the penalty imposed by a limited synaptic weight resolution is known to decrease for larger deep networks with more and larger hidden layers, both spiking and non-spiking [@courbariaux2015binaryconnect; @petrovici2017pattern]. Furthermore, the successor system [BrainScaleS-2, @aamir2016highly] is designed with a 6-bit weight resolution.
[ In the setup we used shared bias neurons for several neurons in the sampling network. This helped us save hardware resources, thus allowing the emulation of larger functional networks. Such bias neuron sharing is expected to introduce some small amount of temporal correlations between the sampling neurons. However, this effect was too small to observe in our experiments for several reasons. First, the high firing rate of the bias neurons helped smooth out the bias voltage induced into the sampling neurons. Second, the different delays and spike timing jitter on the hardware reduces such cross-correlations. Third, other dominant limitations overshadow the effect of shared bias neurons. In any case, the used training procedure inherently compensates for excess cross-correlations, thus effectively removing any distortions to the target distribution that this effect might introduce [@bytschok2017spike; @dold2018stochasticity]. ]{}
In the current setup, our SSNs displayed limited mixing abilities. During guided dreaming, images from one of the learned classes were more difficult to generate (\[fig:tsne\]). Restricted mixing due to deep modes in the energy landscape carved out by contrastive learning is a well-known problem for classical Boltzmann machines, which is usually alleviated by computationally costly annealing techniques [@salakhutdinov2010learning; @desjardins2010parallel; @bengio2013better]. However, the fully-commissioned BrainScaleS system will feature embedded short-term synaptic plasticity [@schemmel2010wafer], which has been shown to promote mixing in spiking networks [@leng2018spiking] while operating purely locally, at the level of individual synapses.
Currently, the execution speed of emulation runs is dominated by the I/O overhead, which in turn is mostly spent on setting up the experiment. This leads to the classification (\[sec:datasets\]) of one image taking , whereas the pure network runtime is merely $\SI{50}{\mu s}$. A streamlining of the software layer that performs this setup is expected to significantly reduce this discrepancy.
The synaptic learning rule was local and Hebbian, but updates were calculated on a host computer using an iterative in-the-loop training procedure, which required repeated stopping, evaluation and restart of the emulation, thereby reducing the nominal acceleration factor of $10^4$ by two orders of magnitude. By utilizing on-chip plasticity, as available, for example, on the BrainScaleS-2 successor system [@friedmann2017demonstrating; @wunderlich2018demonstrating], this laborious procedure becomes obsolete and the accelerated nature of the substrate can be exploited to its fullest extent.
Relation to other work
----------------------
This study builds upon a series of theoretical and experimental studies of sampling-based probabilistic inference using the dynamics of biological neurons. The inclusion of refractory times was first considered in [@buesing2011neural]. An extension to networks of leaky integrate-and-fire neurons and a theoretical framework for their dynamics and statistics followed in [@petrovici2013stochastic] and [@petrovici2016stochastic]. The compensation of shared-input correlations through inhibitory feedback and learning was discussed in [@jordan2017stochastic], [@bytschok2017spike] and [@dold2018stochasticity], inspired by the early study of asynchronous irregular firing in [@brunel2000dynamics] and by preceding correlation studies in theoretical [@tetzlaff2012decorrelation] and experimental [@pfeil2016effect] work.
Previous small-scale studies of sampling on accelerated mixed-signal neuromorphic hardware include [@petrovici2015sampling; @petrovici2017robustness; @petrovici2017pattern]. An implementation of sampling with spiking neurons and its application to the MNIST dataset was shown in [@pedroni2016mapping] using the fully digital, real-time TrueNorth neuromorphic chip [@merolla2014million].
We stress two important differences between [@pedroni2016mapping] and this work. First, the nature of the neuromorphic substrate: the TrueNorth system is fully digital and calculates neuronal state updates numerically, in contrast to the physical-model paradigm instantiated by BrainScaleS. In this sense, TrueNorth emulations are significantly closer to classical computer simulations on parallel machines: updates of dynamical variables are precise and robustness to variability is not an issue; [ however TrueNorth typically runs in biological real time [@merolla2014million; @akopyan2015truenorth],]{} which is 10.000 times slower than BrainScaleS. Second, the nature of neuron dynamics: the neuron model used in [@pedroni2016mapping] is an intrinsically stochastic unit that sums its weighted inputs, thus remaining very close to classical Gibbs sampling and Boltzmann machines, while our approach considers multiple additional aspects of its biological archetype (exponential synaptic kernels, leaky membranes, deterministic firing, stochasticity through synaptic background, shared-input correlations etc.). Moreover, our approach uses fewer hardware neuron units to represent a sampling unit, enabling a more parsimonious utilization of the neuromorphic substrate.
Conclusion
----------
In this work we showed how sampling-based Bayesian inference using hierarchical spiking networks can be robustly implemented on a physical model system despite inherent variability and imperfections. Underlying neuron and synapse dynamics are deterministic and close to their biological archetypes, but with much shorter time constants, hence the intrinsic acceleration factor of $10^4$ with respect to biology. The entire architecture – sampling network plus background random network – was fully deterministic and entirely contained on the neuromorphic substrate, with external communication used only to represent input patterns and labels. Considering the deterministic nature of neurons in vitro [@mainen1995reliability; @reinagel2002precise; @toups2012multiple], such an architecture also represents a plausible model for neural sampling in cortex [ [@jordan2017stochastic; @dold2018stochasticity].]{}
We demonstrated sampling from arbitrary Boltzmann distributions over binary random variables, as well as generative and discriminative properties of networks trained with visual data. The framework can be extended to sampling from arbitrary probability distributions over binary random variables, as it was shown in software simulations [@probst2015probabilistic]. For such networks, the two abovementioned computational tasks (pattern completion and classification) happen simultaneously, as they both require the calculation of conditional distributions, which is carried out implicitly by the network dynamics. Both during learning and for the subsequent inference tasks, the setup benefitted significantly from the fast intrinsic dynamics of the substrate, achieving a net speedup of compared to biology.
We view these results as a contribution to the nascent, but expanding field of applications for biologically inspired physical-model systems. They demonstrate the feasibility of such devices for solving problems in machine learning, as well as for studying biological phenomena. Importantly, they explicitly addresses the search for robust computational models that are able to harness the strengths of these systems, most importantly their speed and energy efficiency. The proposed architecture scales naturally to substrates with more neuronal real-estate and can be used for a wide array of tasks that can be mapped to a Bayesian formulation, such as constraint satisfaction problems [@jonke2016solving; @fonseca2017using], prediction of temporal sequences [@sutskever2007learning], movement planning [@taylor2009factored; @alemi2015affect], simulation of solid-state systems [@edwards1975theory] and quantum many-body problems [@carleo2017solving; @czischek2018quenches].
Conflict of Interest Statement {#conflict-of-interest-statement .unnumbered}
==============================
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Author Contributions {#author-contributions .unnumbered}
====================
Akos F. Kungl (AFK), Andreas Baumbach (AB), Dominik Dold (DD), Luziwei Leng (LL), Sebastian Schmitt (SS), Paul Müller (PM) and Mihai A. Petrovici (MAP) designed the study. AFK conducted the experiments and the evaluations. Nico Gürtler (NG) contributed to the evaluations and provided software support for the evaluation. AFK wrote the initial manuscript. Eric Müller (ECM), Christian Mauch (CM), Johann Klähn (JK), Sebastian Schmitt (SS), Kai Husmann (KH) and Oliver Breitwieser (OB) supported experiment realization; ECM coordinated the software development for the neuromorphic systems. Alexander Kugele (AK), Christoph Koke (CK) and Mitja Kleider (MK) contributed with characterization, calibration testing and debugging of the system. Andreas Grübl (AG), Dan Husmann (DH), Maurice Güttler (MG) were responsible for system assembly. AG did the digital front- and back-end implementation. Vitali Karasenko (VK) provided FPGA firmware and supported system commissioning. Johannes Schemmel (JS) is the architect and lead designer of the neuromorphic platform. MAP, Karlheinz Meier (KM), JS, SS and ECM provided conceptual and scientific advice. All authors contributed to the final manuscript.
Funding {#funding .unnumbered}
=======
The work leading to these results has received funding from the European Union Seventh Framework Programme (FP7) under grant agreement No \#604102, the EU’s Horizon 2020 research and innovation programme under grant agreements No \#720270 and \#785907 (Human Brain Project, HBP), the EU’s research project BrainScaleS \#269921 and the Heidelberg Graduate School of Fundamental Physics. We acknowledge financial support by Deutsche Forschungsgemeinschaft within the funding programme Open Access Publishing, by the Baden-Württemberg Ministry of Science, Research and the Arts and by Ruprecht-Karls-Universität Heidelberg. We owe particular gratitude to the sustained support of our research by the Manfred Stärk Foundation.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Johannes Bill for many fruitful discussions.
Supplemental Data {#supplemental-data .unnumbered}
=================
Two videos can be found in the online version of the article that exemplify fully-embedded sampling from a target distribution and fully-embedded pattern completion on the BrainScaleS-1.
Tables {#tables .unnumbered}
======
Type
-------------------------- ----------------------------------------------------- --
Subthreshold dynamics
Reset and refractoriness
Spiking
neuron emits a spike with timestamp $t_\mathrm{sp}$
Synapse dynamics
: **Description of the neuron and synapse model.** The variables are described including their numerical values in the experiment in \[table:neuron\].
\[table:network\]
**A**
---------------------------------- -------------------------------- ------------------------------------------------------
*Name* *Value* *Description*
$V_\mathrm{reset}$ reset potential
$E_\mathrm{leak}$ resting potential
$V_\mathrm{thresh}$ threshold potential
$E_\mathrm{inh}$ inhibitory reversal potential
$E_\mathrm{exc}$ excitatory reversal potential
$\tau_\mathrm{ref}$ refractory time
$\tau_\mathrm{mem}$ ca. membrane time constant$^{*}$
$C_\mathrm{mem}$ membrane capacity
$\tau_\mathrm{syn}^\mathrm{exc}$ excitatory synaptic time constant
$\tau_\mathrm{syn}^\mathrm{inh}$ inhibitory synaptic time constant
**B**
*Name* *Value* *Description*
$V_\mathrm{reset}$ reset potential
$E_\mathrm{leak}$ resting potential
$V_\mathrm{thresh}$ threshold potential
$E_\mathrm{inh}$ inhibitory reversal potential
$E_\mathrm{exc}$ excitatory reversal potential
$\tau_\mathrm{ref}$ refractory time
$\tau_\mathrm{mem}$ ca. membrane time constant$^{*}$
$C_\mathrm{mem}$ membrane capacity
$\tau_\mathrm{syn}^\mathrm{exc}$ excitatory synaptic time constant
$\tau_\mathrm{syn}^\mathrm{inh}$ inhibitory synaptic time constant
**C**
*Name* *Value* *Description (all analog)*
$V_\mathrm{reset}$ reset potential
$E_\mathrm{leak}$ resting potential
$V_\mathrm{thresh}$ threshold potential
$E_\mathrm{inh}$ inhibitory reversal potential
$E_\mathrm{exc}$ excitatory reversal potential
$\tau_\mathrm{ref}$ refractory time
$\tau_\mathrm{mem}$ ca. membrane time constant$^{*}$
$C_\mathrm{mem}$ membrane capacity
$\tau_\mathrm{syn}^\mathrm{exc}$ excitatory synaptic time constant
$\tau_\mathrm{syn}^\mathrm{inh}$ inhibitory synaptic time constant
**D**
*Name* *Value* *Description*
$w_\mathrm{bias}$ \[0,15\] synaptic bias weight in hardware values (digital)
$w_\mathrm{network}$ \[0,15\] synaptic network weight in hardware values (digital)
$d$ on the order of (uncalibrated) synaptic delay, estimated in [@schemmel2010wafer]
: **Neuron parameters.** Parameters of the network setup specified in \[table:network\]. The analog parameters are shown as specified in the software setup and not as realized on the hardware. For details on the calibration procedure see, e.g., [@schmitt2017neuromorphic]. *Legend:* $^{*}$ the calibration of the membrane time constant was not available at the time of this work, and the corresponding technical parameter was set to the smallest available value instead (fastest possible membrane dynamics for each neuron).
\[table:neuron\]
*Experiment* *Learning rate* *Momentum factor* *minibatch-size* *Initial $(\mathbf{W},\mathbf{b})$*
------------------------------------- ----------------- ------------------- ------------------ -------------------------------------
target distribution, Poisson 1.0 0.6 - $\mathcal{U}(-15,15)$
target distribution, random network 0.5 0.6 - $\mathcal{U}(-15,15)$
rMNIST 0.4 0.6 7/class pre-trained
rFMNIST 0.4 0.6 7/class pre-trained
: **Parameters for learning.** We did not carry any systematic hyper-parameter optimization. Note that the used learning parameters in the experiments in \[sec:distr\] are not directly comparable because the different statistics of the background noise (Poisson or random network) correspond to different effective learning rates.
\[table:learnParam\]
**A**
--------------------------------------- ----------------- --------------------------------------------------------------
*Name* *Value* *Description*
$N_\mathrm{s}$ 5 number of sampling neurons
$N_\mathrm{b}$ 1 number of bias neurons
$N_\mathrm{r}$ 0 number of random neurons
$K_\mathrm{RN}$ - within-population in-degree of neurons in the random network
$K_\mathrm{noise}$ - in-degree of sampling neurons from the random network
$w_\mathrm{RN}$ - synaptic weights in the random network
in hardware units
$\nu^{\mathrm{e/i}}_\mathrm{Poisson}$ Poisson frequency to sampling neurons per synapse type
**B**
*Name* *Value* *Description*
$N_\mathrm{s}$ 5 number of sampling neurons
$N_\mathrm{b}$ 1 number of bias neurons
$N_\mathrm{r}$ 200 number of random neurons
$K_\mathrm{RN}$ 20 within-population in-degree of neurons in the random network
$K_\mathrm{noise}$ 15 in-degree of sampling neurons from the random network
$w_\mathrm{RN}$ 10 synaptic weights in the random network
in hardware units
$\nu^{\mathrm{e/i}}_\mathrm{Poisson}$ - Poisson frequency to sampling neurons per synapse type
**C**
*Name* *Value* *Description*
$N_\mathrm{s}$ $\{ 207,208 \}$ number of sampling neurons, { rFMNIST, rMNIST }
$N_\mathrm{b}$ 1 number of bias neurons
$N_\mathrm{r}$ 400 number of random neurons
$K_\mathrm{RN}$ 20 within-population in-degree of neurons in the random network
$K_\mathrm{noise}$ 15 in-degree of sampling neurons from the random network
$w_\mathrm{RN}$ 10 synaptic weights in the random network
in hardware units
$\nu^{\mathrm{e/i}}_\mathrm{Poisson}$ - Poisson frequency to sampling neurons per synapse type
: **Network parameters.** Parameters are shown for the three different cases described in the manuscript: **A** Target Boltzmann distribution, Poisson noise. **B** Target Boltzmann distribution, random network for stochasticity. **C** Learning from data, random network for stochasticity. [ Note that the *in-degree*, sometimes also referred to as a *fan-in factor*, represents a neuron’s number of pre-synaptic partners coming from some specific population.]{}
\[table:netParam\]
|
---
abstract: 'A mapping between operators on the Hilbert space of $N$-dimensional quantum system and the Wigner quasiprobability distributions defined on the symplectic flag manifold is discussed. The Wigner quasiprobability distribution is constructed as a dual pairing between the density matrix and the Stratonovich-Weyl kernel. It is shown that the moduli space of the Stratonovich-Weyl kernel is given by an intersection of the coadjoint orbit space of the $SU(N)$ group and a unit $(N-2)$-dimensional sphere. The general consideration is exemplified by a detailed description of the moduli space of 2, 3 and 4-dimensional systems.'
author:
- 'Vahagn Abgaryan[^1]'
- 'Arsen Khvedelidze[^2]'
- 'Astghik Torosyan[^3]'
date:
title: '**On moduli space of the Wigner quasiprobability distributions for $N$-dimensional quantum systems**'
---
Introduction
============
According to the postulates of the quantum theory, the fundamental description of a physical system is provided by the density operator [@vonNeumann1932] $$\varrho =\sum_{k}p_k |\psi_k\rangle\langle \psi_k|\,,$$ which represents the quantum statistical ensemble $\{p_k, |\psi_k\rangle \}\,, $ i.e., a set consisting of vectors $|\psi_k\rangle \in \mathcal{H}$ of the Hilbert space $\mathcal{H}$ and their probabilities $p_k $ with a sum equal to one, $\sum_k p_k= 1\,. $ The density operator $\varrho$ determines the expectation value $ \mathbb{E}(\hat{{A}})$ of a Hermitian operator $\hat{A}$ acting on $\mathcal{H}\,,$ $$\label{eq:quantObser}
\mathbb{E}(\hat{{A}}) =\mbox{Tr}\left[\hat{{A}}\varrho \right], \quad\mbox{with}\quad
\mbox{Tr}\left[\varrho\right] =1\,.$$ The latter is assigned to a physical observable associated with the operator $\hat{A}$. On the other hand, an ensemble of a classical mechanical system is characterized by a probability distribution function $\rho(q,p)\,,$ i.e., the density of the probability to find the system in a state localized in the vicinity of a phase space point with coordinates $q$ and $p$. Correspondingly, the statistical average, i.e., the expectation value $\mathbb{E}(A)$ of a physical quantity described by the function $A(q,p)$ on a phase space is given by the following convolution: $$\label{eq:classObser}
\mathbb{E}(A)=\int\mathrm{d}\Omega\,A(q,p)\,\rho(q,p), \quad\mbox{with}\quad \int\mathrm{d}\Omega\,\rho(q,p)= 1\,,$$ where $\mathrm{d}\Omega $ denotes the normalized volume form of a classical phase space.
Aiming to collate two representations of observables, the classical (\[eq:classObser\]) and the quantum (\[eq:quantObser\]), the so-called Weyl–Wigner invertible mapping between Hilbert space operators and functions on a phase space has been introduced in the early stages of the development of quantum mechanics [@Weyl1928]-[@Moyal1949]. The primary elements of this map are two notions: the *symbol of operator*, i.e., a function $A_W(q,p)$ corresponding to the operator $A$, and the *quasi-distribution function* $W(q,p)$ defined over a phase space. As a result, the quantum analogue of the statistical average (\[eq:classObser\]) reads $$\mathbb{E}(\hat{{A}})= \int\mathrm{d}\Omega\,A_W(q,p)\,W(q,p), \quad\mbox{with}\quad \int\mathrm{d}\Omega\,W(q,p)= 1\,.$$ However, even a quick-look at this attempt to build a bridge between classical and quantum statistical pictures shows a lack of their equivalence. Indeed, one can point out the following observations:
- Because of Heisenberg’s uncertainty principle, the function $W(q,p)$ has negative values for certain quantum states. Hence it is not a true probability density and is referred to as quasiprobability distribution.
- Dirac’s quantization rule based on the canonical commutator relations makes the interplay between operators and their symbols highly sophisticated. Replacement of canonical variables by their quantum counterparts in expressions of functions over the phase-space faces an ambiguity of ordering of the corresponding canonical operators. [^4]
In spite of both flaws, Wigner functions or other formulated quasiprobability distributions, such as Husimi [@Husimi1940] and Glauber-Sudarshan [@Sudarshan1963; @Glauber1963] representations, remain today an important tool for understanding of interrelations between quantum and classical statistical descriptions [@HilleryOConnellScullyWigner1984]. Moreover, nowadays one can see a growing interest to phase space formulation of quantum mechanics based on the method of quasiprobability distributions for finite dimensional systems (see e.g.[@RoweSandersGuise]-[@KlimovedeGuise2] and references therein). The latter is coming from needs of diverse applications in quantum optics [@ScullyZubairy1997] and also in quantum information and communications [@BengtssonZyczkowki]. Such an intense usage of quasi-distributions again raises an issue of understanding of the above mentioned shortcomings.[^5]
In the present note, a problem of construction of quasiprobability distribution functions for generic $N$-level systems is studied within a purely algebraic approach. The basic mathematical objects in this approach are: a special unitary group $G=SU(N)$, its Lie algebra $\mathfrak{g}=\mathfrak{su}(N)$: $$\mathfrak{su}(N) =\{X \in M(N, \mathbb{C}) \ |\ X=-X^\dagger\,,\quad \mbox{tr}X=0\}\,,$$ and its dual space $\mathfrak{g}^*=\mathfrak{su}(N)^* \,.$ [^6] It is well known that the universal covering algebra $\mathfrak{U}(\mathfrak{su}(N))$ of the Lie algebra $\mathfrak{su}(N)$ is an arena of the basic objects of N-level quantum system. Particularly, a state space $\varrho \in \mathfrak{P}_N\,$ is defined as the space of *positive semidefinite* $N\times N$ Hermitian matrices $H_N$ with a unit trace: $$\label{eq:StateSpace}
\mathfrak{P}_N=\{ X \in H_N \ |\ X \geq 0\,, \quad \mbox{tr}\left( X \right) = 1 \}\,.$$ Every state described by the density matrix $\varrho \in \mathfrak{P}_N\,$ is in correspondence with some element of the Lie algebra $\mathfrak{su}(N)$: $$\label{eq:DMsuN}
\varrho = \frac{1}{N}\mathbb{I}_N + \frac{1}{N}\,\imath\, \mathfrak{su}(N)\,.$$ In order to build up the Wigner function, apart from the quantum state space $\mathfrak{P}_N\,,$ the notion of its dual $\mathfrak{P}^\ast_N$ is required. Every point of the dual space determines the Stratonovich-Weyl (SW) kernel [@Stratonovich; @BrifMann1999]. As it was shown recently in [@KhA2018], the space $\mathfrak{P}^\ast_N\,$ can be defined as follows: $$\label{eq:SWspace}
\mathfrak{P}^\ast_N=\{ X \in H_N \ |\ \mbox{tr}\left( X \right) = 1\,,
\quad
\mbox{tr}\left( X^2 \right) = N
\}\,.$$ It turns out that the dual pairing (\[eq:DPLie\]) of a density matrix $\varrho \in \mathfrak{P}_N$ and SW kernel $\Delta(\Omega_N) \in \mathfrak{P}^\ast_N$: $$\label{eq:WignerFunction}
W_\varrho(\Omega_N) = \mbox{tr}\left[\varrho
\,\Delta(\Omega_N)\right]\,$$ enables us with the proper Wigner function which satisfies all the Stratonovich-Weyl postulates [@Stratonovich; @BrifMann1999]. Taking into account a unit trace condition, SW kernel $\Delta(\Omega_N)$ can be related to the dual of $\mathfrak{su}(N)$: $$\label{eq:SWsuN}
\Delta(\Omega_N) = \frac{1}{N}\mathbb{I}_N + \kappa\,\frac{1}{N}\,\imath\, \mathfrak{su}(N)^\ast \,,$$ where $\kappa =\sqrt{{N(N^2-1)}/{2}}\,$ is a normalization constant. From representations (\[eq:DMsuN\]) and (\[eq:SWsuN\]) it follows that all nontrivial information comes from pairing between traceless parts of a density matrix and SW kernel. In the subsequent sections, after a short overview of the Stratonovich-Weyl postulates, algebraic and geometric aspects of the dual space $\mathfrak{P}^\ast_N$ are discussed. In particular, we establish interrelation between the Wigner functions and the coadjoint orbits [@Kirillov] $\mathcal{O}_{\boldsymbol{r}}$ of $SU(N)$: $$\mathcal{O}_{\boldsymbol{r}} =\{UDU^\dagger\,: U\in SU(N)\}\,,$$ where $\boldsymbol{r}$ denotes $N$-tuple of real numbers $\boldsymbol{r}={r_1, r_2, \dots, r_N}$ which are elements of the diagonal matrix $D=\mbox{diag}||r_1, r_2, \dots, r_N|| $ ordered as $r_1\geq r_2\geq \dots \geq r_N\,$ and summed up to zero, $\sum_{i=1}^N r_i=0
\,.$ It is then proved that $$\label{eq:WFCoAdj}
W_\varrho(\Omega_N)-\frac{I}{N}\,:\ \mathfrak{P}_N \times \mathcal{O}_{\boldsymbol{r}}\bigl |_{\sum r_i^2=N/(N-1)}\quad \to \quad \mathbb{R}\,.$$ Furthermore, in order to describe in unitary invariant way an ambiguity of the Wigner function, we introduce the *moduli space* $\mathcal{P}_N$ of SW kernel as the following coset: $$\label{eq:ModuliOrbit}
\mathcal{P}_N:= \frac{\mathcal{O}_{\boldsymbol{r}}}{SU(N)}
\Bigg |_{\sum r^2_i={N}/(N-1)}\,.$$ The moduli space geometrically represents intersections of the orbit space of the $SU(N)$ group coadjoint action with an ($N-2$)-dimensional sphere. Finalizing our note, we give few examples of the moduli space of the Wigner functions for low-level quantum systems, for a qubit (N=2), qutrit (N=3) and quatrit (N=4).
Constructing the Wigner function
================================
Below we give a brief summary of the Wigner quasiprobability distribution construction starting from the basic Stratonovich-Weyl postulates and reformulating them into a set of algebraic constraints on a spectrum of SW kernels $\Delta(\Omega_N)\,.$
[$\bullet$ [**The Stratonovich-Weyl principles**]{} $\bullet$]{} Following to Brif and Mann [@BrifMann1999], the postulates known as the Stratonovich-Weyl correspondence can be written as the following constraints on the kernel $\Delta(\Omega_N)$:
1. [**Reconstruction**]{}: a state $\varrho$ is reconstructed from the WF (\[eq:WignerFunction\]) via the integral over a phase space: $$\label{eq:DMWigner}
\varrho =\int_{\Omega_N} \mathrm{d}\Omega_N\, \Delta(\Omega_N) W_\varrho(\Omega_N) \,;$$
2. [**Hermicity**]{}: $$\Delta(\Omega_N)= \Delta(\Omega_N)^\dagger\,;$$
3. [**Finite Norm**]{}: a state norm is given by the integral of the Wigner distribution: $$\mbox{tr}[ \varrho ]= \int_{\Omega_N}
\mathrm{d}\Omega_N W_\varrho(\Omega_N)\,,
\qquad
\int_{\Omega_N} \mathrm{d}\Omega_N\,\Delta(\Omega_N) = 1\,;$$
4. [**Covariance**]{}: the unitary transformations $\varrho^\prime = U(\alpha)\varrho U^\dagger(\alpha)$ induce the kernel change: $$\Delta(\Omega^\prime_N) =U(\alpha)^\dagger\Delta(\Omega_N)U(\alpha)\,.$$
#### Algebraic master equation for SW kernel
The above given axioms allow derivation of algebraic equations for SW kernel of $N$- level quantum systems. With this goal, following the paper [@KhA2018], we accomplish next steps:
- [**Identification of phase-space $\Omega_N$ with complex flag manifold**]{}.\
Hereinafter, a phase-space $\Omega_N$ will be identified with a complex flag manifold, $\Omega_N =\mathbb{F}^N_{d_1,d_2, \dots, d_s}\,.$ The latter emerges as follows: supposing that a spectrum of SW kernel $\Delta(\Omega_N)\,$ consists of real eigenvalues with the algebraic multiplicity $k_i$, i.e., the isotropy group $H$ of the kernel is $$\label{eq:isotropyH}
\nonumber H={U(k_1)\times U(k_2) \times U(k_{s+1})}\,,$$ one can see that the phase space $\Omega_N$ can be realized as a coset space $U(N)/H$, the complex flag manifold $
\mathbb{F}^N_{d_1,d_2, \dots, d_s}\,,
$ where $(d_1, d_2, \dots, d_s)$ is a sequence of positive integers with sum $N $, such that $k_1=d_1$ and $k_{i+1}=d_{i+1}-d_i$ with $d_{s+1}=N\,.$ Furthermore, since the flag manifold represents a coadjoin orbit of $SU(N)$, its symplectic structure is given by the corresponding Kirillov-Kostant-Souriau symplectic 2-form [@Kirillov].
- [**Enlarging of phase-space $\Omega_N$ to $SU(N)$ group manifold.**]{}\
Owing to the unitary symmetry of $N$-dimensional quantum system, we can relate a measure $\mathrm{d}\Omega_N $ on the symplectic space $\Omega_N$ with the normalized Haar measure $\mathrm{d}\mu_{SU(N)}$ on the $SU(N)$ group manifold: $$\mathrm{d}\Omega_N = C_N^{-1}{\mathrm{d}\mu_{SU(N)}}/{\mathrm{d}\mu_H}\,.$$ Here $C_N$ is a real normalization constant, $\mathrm{d}\mu_{H}$ is the Haar measure on the isotropy group $H$ induced by the embedding, $H \subset SU(N)\,.$ Noting that the integrand in (\[eq:DMWigner\]) is a function of the coset variables only, the reconstruction integral can be extended to the whole group $SU(N)$, $$\label{eq:reconstovergroup}
\varrho = Z_N^{-1}\int_{SU(N)} \mathrm{d}\mu_{SU(N)}\, \Delta(\Omega_N) W_\varrho(\Omega_N) \,,$$ where the normalization constant $Z_N^{-1}= C_N^{-1}/\mbox{vol}(H)\,$ includes the factor $\mbox{vol}(H)$ which is the volume of the isotropy group $H$.
- [**Derivation of algebraic equations for SW kernel.**]{}\
Relations (\[eq:WignerFunction\]) and (\[eq:reconstovergroup\]) imply the integral identity $$\label{eq:RhoIdentity}
\varrho=Z_N^{-1} \int_{SU(N)} \mathrm{d}\mu_{SU(N)}\, \Delta(\Omega_N)\,
\mbox{tr}\left[\varrho\Delta(\Omega_N)\right]\,.$$ Substituting the singular value decomposition for SW kernel into (\[eq:RhoIdentity\]) and evaluating the integral using the Weingarten formula [@Weingarten; @Colins2003; @ColinsSniady2006]: $$\begin{aligned}
\nonumber\int{d\mu}U_{i_1 j_1}U_{i_2 j_2}\bar{U}_{k_1 l_1}\bar{U}_{k_2 l_2}
=\frac{1}{N^2-1}\left(\delta_{i_1 k_1}\delta_{i_2 k_2}\delta_{j_1 l_1}\delta_{j_2 l_2}+\delta_{i_1 k_2}\delta_{i_2 k_1}\delta_{j_1 l_2}\delta_{j_2 l_1}\right)-\\
\nonumber
\frac{1}{N(N^2-1)}\left(\delta_{i_1 k_1}\delta_{i_2 k_2} \delta_{j_1 l_2}\delta_{j_2 l_1}+\delta_{i_1 k_2} \delta_{i_2 k_1}\delta_{j_1 l_1}\delta_{j_2 l_2}\right),\end{aligned}$$ we derive the equations: $$\left(\mbox{tr}[\Delta(\Omega_N)]\right)^2=Z_N N\,,\quad
\mbox{tr}[\Delta(\Omega_N)^2]=Z_N N^2\,.$$
- [**Normalization of SW kernel.**]{}\
The constant $Z_N$ in the equation (\[eq:reconstovergroup\]) can be determined with the aid of the so-called standardization condition, $$Z_N^{-1}\int \mathrm{d}\mu_{SU(N)} W_A(\Omega_N)= \mbox{tr}[A]\,.$$ Fixing the normalization constant $Z_N$, we finally arrive at the **“master equations”** for SW kernel: $$\label{eq:master}
\mbox{tr}\left[\Delta(\Omega_N)\right]=1\,, \qquad
\mbox{tr}[\Delta(\Omega_N)^2] = N\,.$$
Moduli space: reckoning up solutions to the “master equations”
===============================================================
Classifying solutions to the master equations (\[eq:master\]), we arrive at the notion of a *“moduli space”* as the space $\mathcal{P}_N\,,$ points of which are associated with the unitary equivalent admissible SW kernel of $N\--$dimensional quantum system. Analysis of eq. (\[eq:master\]) solutions space displays the following properties of the moduli space $\mathcal{P}_N$:
1. $\mbox{dim}\left(\mathcal{P}_N(\boldsymbol{\nu})\right) = N-2\,, $ i.e., a maximal number of continuous parameters $\boldsymbol{\nu}$ characterizing the solution $\Delta(\Omega_N\,|\, \boldsymbol{\nu})$ is $N-2\,;$
2. geometrically, $\mathcal{P}_N$ is represented as an intersection of an ($N-2$)-dimensional sphere $\mathbb{S}_{N-2}$ with the orbit space $\mathfrak{su}(N)^\ast\slash SU(N)$ of $SU(N)$ action on a dual space $\mathfrak{su}(N)^\ast$: $$\mathcal{P}_N \cong \mathbb{S}_{N-2}\,\bigcap\,\frac{\mathfrak{su}(N)^\ast}{SU(N)}\,\,.$$
In order to become convinced in above statements, consider the singular value decomposition of SW kernel and assume that the kernel is generic with all eigenvalues distinct. [^7] Using the orthonormal basis $\{\lambda_1, \lambda_2, \dots,\lambda_{N^2-1}\}$ of $\mathfrak{su}(N)$, the SVD decomposition reads: $$\label{eq:SWkernelexp}
\Delta(\Omega_N|\boldsymbol{\nu})=\frac{1}{N}U(\Omega_N)\left[I+\kappa\sum_{\lambda\in H }\mu_s(\boldsymbol{\nu})\lambda_s\right]U(\Omega_N)^\dagger,$$ where $\kappa=\sqrt{{N(N^2-1)}/{2}}$, and $ H$ is the Cartan subalgebra $H \in \mathfrak{su}(N)\,.$
From the master equation (\[eq:master\]) it follows that the coefficients $\mu_s(\boldsymbol{\nu})$ live on an ($N-2$)-dimensional sphere $\mathbb{S}_{N-2}(1)$ of radius one: $$\label{eq:moduliWF}
\sum_{s=2}^{N}\mu^2_{s^{2}-1}(\boldsymbol{\nu}) = 1\,.$$ A generic SW kernel can be parameterized by $N-2$ spherical angles. The parameter $(\boldsymbol{\nu})$ introduced in order to label members of the family of the Wigner functions can be associated with a point on $\mathbb{S}_{N-2}(1)\,$. More precisely, a one-to-one correspondence between points on this sphere and unitary non-equivalent SW kernels occurs only for a certain subspace of $\mathbb{S}_{N-2}(1)$. This subspace $\mathcal{P}_N({\boldsymbol{\nu}}) \subset \mathbb{S}_{N-2}(1)\,$ represents the moduli space of SW kernel. Its geometry is determined by the $\Delta(\Omega_N\,|\,\boldsymbol{\nu})$ eigenvalues ordering. The chosen descending order of the eigenvalues restricts the range of spherical angles parameterizing (\[eq:moduliWF\]) and cuts out the moduli space of $\Delta(\Omega_N\,|\,\boldsymbol{\nu})$ in the form of a spherical polyhedron. Details of SW kernels parametrization in terms of spherical angles are given in the Appendix \[AppendixA\].
The Wigner function as dual pairing between $\varrho$ and $\Delta$
==================================================================
As soon as the space of all possible SW kernels is known, the construction of the Wigner function reduces to a computation of pairing (\[eq:WignerFunction\]). Using the $\mathfrak{su}(N)$ expansions (\[eq:DMsuN\]) for a density matrix $\varrho_\xi$ of $N$-level system characterized by ($N^2-1$)-dimensional Bloch vector $\boldsymbol{\xi}$, $$\label{eq:rhoN}
\nonumber \varrho_{\xi}=\frac{1}{N}\left(I+\sqrt{\frac{N\left(N-1\right)}{2}}\left(\boldsymbol{\xi},\boldsymbol{\lambda}\right)\right)\,,$$ and SW kernel decomposition (\[eq:SWkernelexp\]), we arrive at the general representation for the WF: $$\label{eq:WFCartan}
W^{(\boldsymbol{\nu})}_{\boldsymbol{\xi}} (\theta_1,\theta_2, \dots, \theta_d)=\frac{1}{N}\left[1 + \frac{N^2-1}{\sqrt{N+1}}\,(\boldsymbol{n}, \boldsymbol{\xi})\right]\,,$$ where ($N^2-1$)-dimensional vector $\boldsymbol{n}$ is given by a linear combination of $N-1$ orthonormal vectors $\boldsymbol{n}^{(s^{2}-1)}$ with coefficients $\mu_{s^2-1}(\boldsymbol{\nu})\,,$ $s=2,3,\dots, N\,,$ $$\begin{aligned}
\nonumber \boldsymbol{n} = \mu_3
\boldsymbol{n}^{(3)} + \mu_8 \boldsymbol{n}^{(8)}+\dots+
\mu_{N^{2}-1}\boldsymbol{n}^{(N^{2}-1)}\,. \end{aligned}$$ The vectors $\boldsymbol{n}^{(s^{2}-1)}$ are determined by the Cartan subalgebra $\lambda_{s^2-1} \in H$: $$\nonumber \boldsymbol{n}^{(s^2-1)}_\mu = \frac{1}{2}\,\mbox{tr}\left( U\lambda_{s^2-1}U^\dagger\lambda_\mu \right)\,, \quad
s=2,3,\dots,N\,.$$ As it was mentioned in the Introduction, the number of the symplectic coordinates $\vartheta_1, \vartheta_2, \dots , \vartheta_d$ of the Wigner function depends on the isotropy group of SW kernel (cf. details in [@KhA2018]).
Examples
========
Below we present an explicit parametrization for a moduli space of a few low-dimensional quantum systems, including a single qubit, qutrit and quatrit.
The moduli space of a single qubit SW kernel
--------------------------------------------
For a 2-level quantum system, a qubit, the master equations (\[eq:master\]) determine the spectrum (up to permutation) of 2-dimensional SW kernel uniquely: $$\label{eq:Deltaqubit}
\Delta^{(2)}(\Omega_2)=
\frac{1}{2}
U(\Omega_2)
\begin{pmatrix}
1+\sqrt{3} & 0 \\
0 & 1-\sqrt{3}
\end{pmatrix}
U(\Omega_2)^\dagger\,,$$ with $U(\Omega_2) \in SU(2)/ U(1)\,.$ Its connection to the structure of the coadjoint orbits of $SU(2)$ is straightforward. There are two types of the coadjoint orbits of $SU(2)$:
1. 2-dimensional regular orbits $\mathcal{O}_{\boldsymbol{r}}$, $$\mathcal{O}_{\{r, - r\}} =\left\lbrace\, U
\begin{pmatrix}
r & 0 \\
0 & -r
\end{pmatrix}U^\dagger\,, \, U\in SU(2)\,
\right\rbrace\,,$$ defined for an ordered 2-tuple, $\boldsymbol{r}=\{r,-r\}\,,$ $r >0\,.$ They are isomorphic to a 2-dimensional sphere $\mathbb{S}_2(r)$ with the radius given by the value of the $SU(2)$ invariant: $$r^2=-\det\left(\mathcal{O}_{\boldsymbol{r}}\right)\,;$$
2. zero-dimensional orbit, point $ r=0\,.$
Identifying these orbits with the traceless part of SW kernel $\Delta^{(2)}-\frac{1}{2}\mathbb{I}$ and taking into account the expression (\[eq:Deltaqubit\]), we get convinced that $$r^2=
\frac{4}{3}\,\mbox{tr}
\left[\left(\Delta^{(2)}-\frac{1}{2}\,\mathbb{I}\right)^2\right]
=2\,.$$ Thus, the moduli space of SW kernel of a qubit represents the single point, $r^2=2\,,$ from the set of equivalence classes of the regular $SU(2)$ orbits, $[\mathcal{O}_{\boldsymbol{r}}] \cong \mathfrak{su}(2)/U(1)$.
[![The cone representing the orbit space of $SU(3)$. The interior of the cone represents $\dim{\mathcal{O}}=4$ orbits. The apex corresponds to a zero-dimensional orbit, while other points on the ordinate $(\mu_8=0)$ and on the positive ray $\mu_8=\mu_3/\sqrt{3}$ also determine $\dim{\mathcal{O}}=4$ orbits. The intersection of the cone with a unit circle gives an arc which is the moduli space of a qutrit SW kernel. The point $C$ with $
\cos(\zeta_{C}) = (-1+3\sqrt{5})/{8}\,
$ describes the singular SW kernel. []{data-label="Fig:Qutrit_Kernel_Mod"}](qutrit_orbits "fig:"){width="0.4\linewidth"}]{}
The moduli space of a single qutrit SW kernel
---------------------------------------------
The master equations (\[eq:master\]) determine two lowest-degree polynomial $SU(N)$ invariants of SW kernel $\Delta(\Omega_3)\,,$ the linear and the quadratic ones. For the case of a 3-dimensional quantum system, a qutrit, the third algebraically independent polynomial $SU(3)$ invariant remains unfixed, thus allowing a one-parametric family of a qutrit SW kernels to exist.
Following the normalization convention (\[eq:SWkernelexp\]), let us write down the SVD decomposition of a qutrit SW kernel in the following form: $$\label{eq:QutrutSWEigen}
\Delta(\Omega_3)=U(\Omega_3)PU(\Omega_3)^\dagger= U(\Omega_3)\left[\frac{1}{3}\mathbb{I}+
\frac{2}{\sqrt{3}}\,
\begin{pmatrix}
r_1 & 0 & 0 \\
0 & r_2 & 0 \\
0 & 0 &r_3
\end{pmatrix}
\right]
U(\Omega_3)^\dagger\,,$$ where $U(\Omega_3) \in SU(3)$, and a 3-tuple $\boldsymbol{r}=\{r_1, r_2, r_3 \} $ parameterizes a traceless diagonal part of the SVD decomposition of SW kernel, $
r_1+r_2+r_3=0\,.
$ Expanding $P$ over the Gell-Mann basis elements $\lambda_3=\mbox{diag}||1, -1, 0||$ and $\lambda_8=\frac{1}{\sqrt{3}}\mbox{diag}||1,1,-2||$ of the $SU(3)$ Cartan subalgebra, $$\label{eq:diagonalSW}
P=\frac{1}{3}\mathbb{I}+
\frac{2}{\sqrt{3}}\left(\mu_3\lambda_3 +\mu_8\lambda_8\right)\,,$$ we find: $$\begin{aligned}
\label{eq:EigeMu}
r_1=\mu_3+\frac{1}{\sqrt{3}}\mu_8\,,\quad \
r_2=-\mu_3+\frac{1}{\sqrt{3}}\mu_8\,,\quad\
r_3=-\frac{2}{\sqrt{3}}\mu_8\,.\end{aligned}$$ From these relations it follows that the chosen decreasing order of parameters $r_1\geq r_2\geq r_3\,$ determines on the $(\mu_3, \mu_8)$-plane the 2-dimensional polyhedral cone $C_2({\pi}/{3})$ with the apex angle $\pi/3$: $$C_2({\pi}/{3})=\left\lbrace \boldsymbol{x} \in \mathbb{R}^2\,\ \bigg|\ \,
\begin{pmatrix}
1 & 0 \\
\frac{-1}{\sqrt{3}} & 1
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2
\end{pmatrix}
\geq 0\,
\right\rbrace\,.$$ [$\bullet$ [**The SU(3) orbits**]{} $\bullet$]{} The cone $C_2({\pi}/{3})$ represents the orbit space of $SU(3)$ group action on $\mathfrak{su}(3)$ algebra. Next we identify this algebra times $\imath $ with the traceless part of $\Delta(\Omega_3)$ and classify SW kernel in accordance to the corresponding coadjoint orbits. In order to realize this program, let us consider the tangent space to the $SU(3)$ orbits. It is spanned by the linearly independent vectors built of the commutators: $
t_k = [\lambda_k,\Delta], \ \lambda_k \in \mathfrak{su}(3)\,.
$ The number of independent vectors $t_k$ determines the dimensionality of the orbits via the rank of the $ 8\times 8$ Gram matrix: $$\label{eq:Gram}
\mathcal{G}_{kl}(\Delta^{(3)}) = \frac{1}{2}\, \mbox{tr}\,(t_k t_l)\,, \qquad
k,l =1, 2, \dots, 8\,.$$ Since the rank of the Gram matrix (\[eq:Gram\]) is $SU(3)$ invariant, one can calculate it for the diagonal representative of SW kernel (\[eq:diagonalSW\]). The straightforward computations give $$\mathcal{G}(\Delta^{(3)})=\frac{4}{3}
\mbox{diag}\,||\, g_1\,, g_1\,, 0\,, g_2\,, g_2\,, g_3\,, g_3\,, 0\,||\,,$$ where $g_1 =4\mu_3^2$, $g_2 =\frac{1}{\sqrt{3}}(\mu_3+\sqrt{3}\mu_8)^2$, $g_3 =\frac{1}{\sqrt{3}}(\mu_3-\sqrt{3}\mu_8)^2$. From these expressions it follows that there are three types of $SU(3)$ orbits which can be classified according to their symmetry and dimensions:
1. $\underline{\dim(\mathcal{O}_{\boldsymbol{r}})=6\,.}$ These *regular orbits* abbreviated as $\mathcal{O}(123)$ (or simply $123$) are labeled by a 3-tuple $\boldsymbol{r}$ with $r_1 > r_2 > r_3\,$ and have the isotropy group $H_{(123)}$ isomorphic to a 2-dimensional torus, $H_{(123)}\cong\mathbb{T}^2$. They are in one-to-one correspondence with the interior points of the cone $C_2({\pi}/{3})$ in Fig.\[Fig:Qutrit\_Kernel\_Mod\].
2. $\underline{\dim(\mathcal{O}_{\boldsymbol{r}})=4\,.}$ These *degenerate orbits* represent two subfamilies with degenerate 3-tuples $\boldsymbol{r}$: either $r_1 = r_2 > r_3\,$ or $r_1 >r_2 = r_3\,.$ Following V.I.Arnold [@Arnold], we denote them as $1|23$ and $12|3$ respectively. Geometrically, the equivalence class $[\mathcal{O}]$ of degenerate orbits represents the boundary lines in the $SU(3)$ orbit space: $$\begin{aligned}
\nonumber
&\mathcal{O}(1|23)\mapsto 1|23:&\{\boldsymbol{x}\in C_2({\pi}/{3}) |\ x_2 = 0\ \}\,,\\
&\mathcal{O}(12|3)\mapsto 12|3:&
\{\boldsymbol{x}\in C_2({\pi}/{3}) |\ x_2=x_1/\sqrt{3}\ \}\,.
\nonumber\end{aligned}$$ Both classes, up to conjugacy in $SU(3)$ have the same isotropy group: $$H_{(12|3)} \cong H_{(1|23)}=\left\lbrace \ h \in \left[
\begin{array}{c|c}
e^{i\alpha}g & 0 \\
\hline
0 & e^{-i\alpha}
\end{array}
\right]\
\bigg|\ g \in SU(2)\ \right\rbrace\,.$$
3. $\underline{\dim(\mathcal{O}_{0})=0\,.}$ One orbit $\mathcal{O}_{0}\,,$ a single point $(0,0)\,$ which is stationary under the $SU(3)$ group action.
[$\bullet$ [**The parametrization of a qutrit SW kernels**]{} $\bullet$]{} We are now in a position to describe the moduli space of a qutrit as a certain subspace of the $SU(3)$ orbit space. Indeed, taking into account that the second order master equation (\[eq:master\]) describes a circle of radius one centered at the origin of $(\mu_3, \mu_8)$-plane, we convinced that the moduli space of a qutrit SW kernel represents the arc depicted in Fig. \[Fig:Qutrit\_Kernel\_Mod\]. More precisely, based on the above classification of the $SU(3)$ orbits, we treat a qutrit moduli space as the union of two strata:
- The regular stratum corresponding to the regular $SU(3)$-orbits. Geometrically it is the arc $\wideparen{AB}/\{A,B\}$ with its endpoints $A$ and $B$ excluded. The corresponding Wigner functions have a 6-dimensional support and 1-dimensional family of SW kernels, the spectrum of which can be written as: $$\label{eq:qutritSWnuparamet}
\mbox{spec}\left(\Delta^{(3)}(\nu)\right)= \left\{\frac{1-\nu+\delta}{2}\,,\,\frac{1-\nu-\delta}{2}\,,\,\nu\right\}\,,$$ where $\delta=\sqrt{(1+\nu) (5-3\nu)}\,$ and $\nu \in (-1/3, -1)\,.$ The parameter $\nu$ is related to the apex angle $\zeta$ of the cone $C_2(\pi/3)$: [^8] $$\nu=\frac{1}{3}-\frac{4}{3}\cos(\zeta)\,, \quad \zeta \in [0,\ {\pi}/{3}]\,.$$
- The end points $A$ and $B$ of the arc $\wideparen{AB}$ correspond to two degenerate SW kernels, with $\nu=-1$ and $\nu=-\frac{1}{3}$ respectively, $$\nonumber
\mbox{spec}\left(\Delta^{(3)}({-1})\right)=
\left\{1, 1, -1\right\},\quad
\mbox{spec}\left(\Delta^{(3)}({-\frac{1}{3}})\right)=
\frac{1}{3}\left\{5,-1,-1\right\}\,.$$
It is necessary to point out that the kernel $\Delta^{(3)}({-1})$ was found by Luis [@Luis2008].
[$\bullet$ [**The singular SW kernels of qutrit**]{} $\bullet$]{} Apart from the above categorization of SW kernels, we distinguish the *singular kernels* which have at least one zero eigenvalue. From the expression (\[eq:qutritSWnuparamet\]) it follows that for a qutrit case among three zeros of the determinant $\det(\Delta^{(3)})=\nu(\nu^2-\nu-1)\,$ only one, $\nu=(1-\sqrt{5})/{2}\,,$ is admissible: [^9] $$\nonumber
\mbox{spec}\left(\Delta_{(103)}\right) =
\left\{\frac{1+\sqrt{5}}{2}\,, \, 0\,, \, \frac{1-\sqrt{5}}{2}\right\}\,.$$
The moduli space of a single quatrit SW kernel
==============================================
The master equations (\[eq:master\]) for a four-level system, a quatrit, determine a 2-parametric family of SW kernels. We start, similarly to a qutrit case, with the SVD decomposition of a quatrit SW kernel: $$\label{eq:QuatritSWEigen}
\Delta(\Omega_4)=U(\Omega_4)PU(\Omega_4)^\dagger= U(\Omega_4)\left[\frac{1}{4}\mathbb{I}+
\frac{\sqrt{30}}{4}\,
\begin{pmatrix}
r_1 & 0 & 0 & 0\\
0 & r_2 & 0 &0\\
0 & 0 &r_3 & 0\\
0 & 0 & 0 & r_4
\end{pmatrix}
\right]
U(\Omega_4)^\dagger\,,
\nonumber$$ with $U(\Omega_4) \in SU(4)$ and a 4-tuple $\boldsymbol{r}=\{r_1, r_2, r_3, r_4 \}\,,$ such that $r_1+r_2+r_3 +r_4 =0\,.
$ These parameters expressions in terms of expansion coefficients of $P$ over the Gell-Mann basis elements $\lambda_3=\mbox{diag}||1, -1, 0, 0||$, $\lambda_8=\frac{1}{\sqrt{3}}\mbox{diag}||1,1,-2,0||$ and $\lambda_{15}=\frac{1}{\sqrt{3}}\mbox{diag}||1,1,1,-3||$ of the $SU(3)$ Cartan subalgebra, $$\label{eq:diagonalSW4}
P=\frac{1}{4}\mathbb{I}+
\frac{\sqrt{30}}{4}\left(\mu_3\lambda_3 +\mu_8\lambda_8 +\mu_{15}\lambda_{15}\right)\,,$$ read: $$\begin{aligned}
\label{eq:EigeMu4}
&&r_1=\mu_3+\frac{1}{\sqrt{3}}\mu_8+
\frac{1}{\sqrt{6}}\mu_{15}\,,\quad \
r_2=-\mu_3+\frac{1}{\sqrt{3}}\mu_8+
\frac{1}{\sqrt{6}}\mu_{15}\,,\\
&&r_3=-\frac{2}{\sqrt{3}}\mu_8 +
\frac{1}{\sqrt{6}}\mu_{15}\,, \qquad \quad
r_4=-\frac{3}{\sqrt{6}}\mu_{15}\,.\end{aligned}$$ Due to the order $r_1\geq r_2\geq r_3\geq r_4$, expansion coefficients $\mu_3, \mu_8$ and $ \mu_{15}$ belong to a 3-dimensional polyhedral cone $C_3\left(\pi/6\right)$ with the apex angle $\pi/6$: $$C_3\left({\pi}/{6}\right)=\left\lbrace \boldsymbol{x} \in \mathbb{R}^3\,\ \bigg|\ \,
\begin{pmatrix}
1 & 0 & 0 \\
\frac{-1}{\sqrt{3}} & 1 & 0 \\
0& \frac{-1}{\sqrt{2}} & 1
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
\geq 0\,
\right\rbrace\,.$$
[$\bullet$ [**The SU(4) orbits**]{} $\bullet$]{} The cone $C_3\left({\pi}/{6}\right)$ represents the $SU(4)$ orbit space. The calculated for a diagonal representative $15\times15$ Gram matrix $$\label{eq:Gramquatrit}
\mathcal{G}(\Delta^{(4)})=\frac{5}{2}\,\mbox{diag}||g_1\,, g_1\,, 0\,, g_2\,, g_2\,, g_3\,, g_3\,, 0\,, g_4\,, g_4\,, g_5\,, g_5\,, g_6\,, g_6\,, 0||\,,$$ where $$\begin{aligned}
&& g_1 = 3 \mu _3^2\,, \quad
g_2 = \frac{3}{4} \left(\mu _3+\sqrt{3} \mu _8\right)^2\,, \quad
g_3 = \frac{3}{4} \left(\mu _3-\sqrt{3} \mu _8\right)^2\,, \nonumber\\
&& g_4 = \frac{1}{8} \left(\sqrt{6} \mu _3+\sqrt{2} \mu _8+4 \mu _{15}\right)^2\,, \quad
g_5 = \frac{1}{8} \left(-\sqrt{6} \mu _3+\sqrt{2} \mu _8+4 \mu _{15}\right)^2\,, \nonumber\\
&& g_6 = \left(\mu _8-{\sqrt{2}}\mu _{15}\right)^2\,.
\nonumber\end{aligned}$$ Analysis of zeros of the Gram matrix (\[eq:Gramquatrit\]) shows the following pattern of the regular and degenerate $SU(4)$ orbits.
- $\underline{\dim(\mathcal{O}_{\boldsymbol{r}})=12\,.}$ The regular orbits have a maximal dimension owing to the smallest isotropy group: $H_{(1234)}=\mathbb{T}^3\in SU(4)\,.$ The equivalent class of the regular orbits represents an interior of the cone $C_3\left({\pi}/{6}\right)$;
- The degenerate orbits are divided into subclasses:
1. $\underline{\dim(\mathcal{O}_{\boldsymbol{r}})=10\,.}$ The equivalence class of these orbits is one of the following faces of the cone $C_3\left({\pi}/{6}\right)$: $$\begin{aligned}
&\mathcal{O}(1|234)\mapsto 1|234: &\{\boldsymbol{x}\in C_3\left({\pi}/{6}\right)\ |\ x_1 = 0\ \}\,,\\
&\mathcal{O}(12|34)\mapsto 12|34:&
\{\boldsymbol{x}\in C_3\left({\pi}/{6}\right)\ |\ x_1=-\sqrt{3}x_2 \}\,, \\
&\mathcal{O}(123|4)\mapsto 123|4:&
\{\boldsymbol{x}\in C_3\left({\pi}/{6}\right)\ |\ x_2= +\sqrt{2}x_3 \}\,. \end{aligned}$$ All the above orbits have the same isotropy group (up to SU(4)-conjugation):
$
H_{(1|234)}=\left\lbrace \ h \in \left[
\begin{array}{c|c|c}
e^{i\alpha}g & 0 &0\\
\hline
0 & e^{i\beta}&0\\
\hline
0 &0 &e^{i\gamma}
\end{array}
\right]\
\bigg|\ g \in SU(2)\,,\ \alpha+\beta+\gamma=0\ \right\rbrace\,.
$
The dimension of this stratum is in agreement with the dimension of the corresponding isotropy group, $$\dim(\mathcal{O}_{\boldsymbol{r}})= \dim(SU(4))-\dim(H_{\boldsymbol{r}})=15-(3+2)=10\,.$$
2. $\underline{\dim(\mathcal{O}_{\boldsymbol{r}})=8\,.}$ The equivalence class of these orbits is the following edge of the cone $C_3\left({\pi}/{6}\right)$: $$\begin{aligned}
&\mathcal{O}(1|23|4)\mapsto 1|23|4: &\{\boldsymbol{x}\in C_3\left({\pi}/{6}\right)\ |\ x_1 = 0\,,\ x_2=\sqrt{2}x_3\ \}\,.\end{aligned}$$ The 7-dimensional isotropy group is: $$H_{(1|23|4)}=\left\lbrace \ h \in \left[
\begin{array}{c|c}
e^{i\alpha}g & 0 \\
\hline
0 & e^{-i\alpha}g^\prime
\end{array}
\right]\
\bigg|\ g, g^\prime \in SU(2)\ \right\rbrace\,.$$
3. $\underline{\dim(\mathcal{O}_{\boldsymbol{r}})=6\,.}$ The equivalence class of these orbits is one of the following edges of the cone $C_3\left({\pi}/{6}\right)$: $$\begin{aligned}
&\mathcal{O}(1|2|34)\mapsto 1|2|34: &\{\boldsymbol{x}\in C_3\left({\pi}/{6}\right)\ |\ x_1 = 0\,,\ x_2=0\ \}\,,\\
&\mathcal{O}(12|3|4)\mapsto 12|3|4: &\{\boldsymbol{x}\in C_3\left({\pi}/{6}\right)\ |\ x_1 =\sqrt{3}x_2\,,\ x_2=\sqrt{2}x_3\ \}\,.\end{aligned}$$ Both classes have the same up to conjugacy 9-dimensional isotropy group: $$H_{(1|2|34)}=\left\lbrace \ h \in \left[
\begin{array}{c|c}
e^{i\alpha}g & 0 \\
\hline
0 & e^{-i\alpha}
\end{array}
\right]\
\bigg|\ g \in SU(3)\ \right\rbrace\,.$$
4. $\underline{\dim(\mathcal{O}_{\boldsymbol{r}})=0\,.}$The apex of cone $C_3\left({\pi}/{6}\right)$ with the stability group $SU(4)\,.$
{width="60.00000%"}
[\[Fig:QbQbKernelSimplex\]]{}
[$\bullet$ [**The parametrization of a quatrit SW kernels**]{} $\bullet$]{} Now we are ready to enumerate all SW kernels for a quatrit according to the above given classification of the $SU(4)$ orbits:
1. : $$\label{eq:quatritSWregular}
\mbox{spec}\left(\Delta^{(4)}(\nu_1,\nu_2)\right)=
\left\{\frac{1-\nu_1-\nu_2+\delta}{2}\,, \frac{1-\nu_1-\nu_2-\delta}{2}\,, \nu_1\,, \nu_2 \right\}\,,$$ where $\delta=\sqrt{7+2\nu_{1}-3\nu_{1}^2+2\nu_{2}-2\nu_{1}\nu_{2}-3\nu_{2}^2}\,.$
2. :
1. A family of SW kernels of 1|234 type: (red in color) $$\mbox{spec}\left(\Delta_{(1|234)}\right)=
\left\{\frac{1-\nu}{3}+\frac{1}{6}\delta_1\,, \frac{1-\nu}{3}+
\frac{1}{6}\delta_1\,, \nu\,,
\frac{1-\nu-\delta_1}{3} \right\}\,,$$ where $\delta_1=\sqrt{22+4\nu-8\nu^2}$ and $\nu\in \big (\frac{1}{4}\left(1-\sqrt{15}\right),\frac{1}{4} \left(1+\sqrt{5}\right)\big)$;
2. A family of SW kernels of 12|34 type: (green in color) $$\label{eq:quatritSW12|34}
\mbox{spec}\left(\Delta_{(12|34)}\right)=
\left\{\frac{1-2\nu+\delta_2}{2}\,, \nu\,, \nu\,, \frac{1-2\nu-\delta_2}{2} \right\}\,,$$ where $\delta_2=\sqrt{7+4\nu-8\nu^2}$ and $\nu \in \big (\frac{1}{4} \left(1-\sqrt{5}\right),\frac{1}{4} \left(1+\sqrt{5}\right)\big)$;
3. A family of SW kernels of 123|4 type: (blue in color) $$\label{eq:quatritSW123|4}
\mbox{spec}\left(\Delta_{(123|4)}\right)=
\left\{\frac{1-2\nu+\delta_2}{2}\,, \frac{1-2\nu-\delta_2}{2}\,, \nu\,, \nu \right\}\,,$$ where $\nu \in \big (\frac{1}{4} \left(1-\sqrt{15}\right),\frac{1}{4} \left(1-\sqrt{5}\right)\big)$.
3. :
1. SW kernel of 1|2|34 type: (brown point) $$\label{eq:quatritSW1|2|34}
\mbox{spec}\left(\Delta_{(1|2|34)}\right)=
\left\{\frac{1+\sqrt{5}}{4}\,, \frac{1+\sqrt{5}}{4}\,, \frac{1+\sqrt{5}}{4}\,, \frac{1-3\sqrt{5}}{4} \right\}\,;$$
2. SW kernel of 12|3|4 type: (black point) $$\label{eq:quatritSW12|3|4}
\mbox{spec}\left(\Delta_{(12|3|4)}\right)=
\left\{\frac{1+3\sqrt{5}}{4}\,, \frac{1-\sqrt{5}}{4}\,, \frac{1-\sqrt{5}}{4}\,, \frac{1-\sqrt{5}}{4} \right\}\,.$$
4. :
SW kernel of 1|23|4 type: (purple point) $$\label{eq:quatritSW1|23|4}
\mbox{spec}\left(\Delta_{(1|23|4)}\right)=
\left\{\frac{1+\sqrt{15}}{4}\,,\frac{1+\sqrt{15}}{4}\,, \frac{1-\sqrt{15}}{4}\,, \frac{1-\sqrt{15}}{4} \right\}\,.$$
All the above categories of SW kernels of a quatrit are depicted in Fig.\[Fig:QbQbKernelSimplex\]. The interior of a curvilinear triangle $ABC$ on $(\nu_1, \nu_2)$-plane corresponds to the regular SW kernels. The boundary lines of the domain describe the double degeneracy cases:
1. SW kernel of type $12|34$–side $AB/\{A,B\}$ (green in color) with both end points $A$ and $B$ excluded: $$AB/\{A,B\}:\
\nu_2=\frac{1}{2}-\nu_1-\frac{1}{2}\sqrt{7+4\nu_1-8\nu_1^2}\,, \qquad
\nu_1 \in \left(\frac{1-\sqrt{5}}{4},\frac{1+\sqrt{5}}{4}\right)\,;$$
2. SW kernel of type $1|234$–side $CB/\{C,B\}$ (red in color) without end points: $$CB/\{C,B\}:\ \nu_2=\frac{1}{3}-\frac{1}{3}\nu_1-\frac{1}{3}\sqrt{22+4\nu_1-8\nu_1^2}\,, \qquad
\nu_1 \in \left(\frac{1-\sqrt{15}}{4},\frac{1+\sqrt{5}}{4}\right)\,;$$
3. SW kernel of type $123|4$–side $AC/\{A,C\}$ (blue in color) without end points: $$\nu_2=\nu_1\,, \quad \nu_1 \in \left(\frac{1-\sqrt{15}}{4},
\frac{1-\sqrt{5}}{4}\right)\,.$$
The vertexes $A$ and $B$ describe a quatrit kernels with a triple degeneracy:
1. SW kernel of 12|3|4 type – point A: $\nu_1=\frac{1-\sqrt{5}}{4}\,, \nu_2=\frac{1-\sqrt{5}}{4}$;
2. SW kernel of 1|2|34 type – point B: $\nu_1=\frac{1+\sqrt{5}}{4}\,, \nu_2=\frac{1-3 \sqrt{5}}{4}$,
while the vertex $C$ corresponds to a quatrit kernel with two double degeneracy of 1|23|4 type: $\nu_1=\nu_2=\frac{1-\sqrt{15}}{4}\,$.
![A quatrit moduli space represented by the M[ö]{}bius spherical triangle $(2,3,3)$ on a unit sphere.[]{data-label="Fig:MobiusTriangle"}](quatrit_orbits_on_sphere_with_axes){width="0.8\linewidth"}
[$\bullet$ [**The singular SW kernels of a quatrit** ]{}$\bullet$]{} Among the above described SW kernels one can distinguish a set of special elements with a vanishing determinant. These singular quatrit kernels of are listed below in the accordance with the increasing singularity of the determinant:
- SW kernels with a simple root of the determinant:
1. 1-parameter family of $1204$ type, $\frac{1}{3} \left(1-\sqrt{22}\right)\leq \nu <\frac{1}{2} \left(1-\sqrt{7}\right)$, $$\label{eq:quatritSWSing1204}
\mbox{spec}\left(\Delta_{(1204)}\right)=
\left\{\frac{1-\nu+\sqrt{7+2\nu-3\nu^2}}{2}\,,\frac{1-\nu-\sqrt{7+2\nu-3\nu^2}}{2}\,,0\,,\nu \right\}\,,$$
2. 1-parameter family of $1034$ type, $\frac{1}{6} \left(2-\sqrt{22}\right)\leq \nu < 0$, $$\label{eq:quatritSWSing1034}
\mbox{spec}\left(\Delta_{(1034)}\right)=
\left\{\frac{1-\nu+\sqrt{7+2\nu-3\nu^2}}{2}\,, 0\,, \nu\,, \frac{1-\nu-\sqrt{7+2\nu-3\nu^2}}{2}\right\},$$
- SW kernel with double zero of determinant: $$\label{eq:quatritSWSing1004}
\mbox{spec}\left(\Delta_{(1004)}\right)=
\left\{\frac{1+\sqrt{7}}{2}\,, 0\,, 0\,,\frac{1-\sqrt{7}}{2}\right\}\,.$$
[$\bullet$ [**A quatrit moduli space as the M[ö]{}bius spherical triangle**]{} $\bullet$]{} As it was mentioned before, the spectrum of $\Delta^{(4)}(\nu_1,\nu_2)$ is in correspondence with points on a unit 2-sphere associated with expansion coefficients $\mu_3\,,\mu_8\,$ and $\mu_{15}$: $$\mu_3^2(\boldsymbol{\nu})+\mu_8^2(\boldsymbol{\nu})+\mu_{15}^2(\boldsymbol{\nu}) =1\,,$$ which satisfy the inequalities: $$\label{eq:quatritIN}
\nonumber \mu_3\geq 0\,, \quad
\mu_8\geq \frac{\mu_3}{\sqrt{3}}\,, \quad
\mu_{15}\geq \frac{\mu_8}{\sqrt{2}}\,.$$ Geometrically these constraints define one out of 24 possible spherical triangles with angles $(\pi/2\,, {\pi}/{3}\,, {\pi}/{3})$ that tessellate a unit sphere. Repeated reflections in the sides of the triangles will tile a sphere exactly once. In accordance with Girard’s theorem, a spherical excess of a triangle determines a solid angle: $\pi/2 + \pi/3 + \pi/3 - \pi = 4\pi/24\,.$ Relation between “flat” representation of a quatrit moduli space (Fig.\[Fig:QbQbKernelSimplex\]) and its spherical realization (Fig.\[Fig:MobiusTriangle\]) is demonstrated by the projection pattern in Fig. \[Fig:ProjectionQuatrit\].
![ Mapping of the tiling of $\mathbb{S}_{2}(1)$ sphere by the M[ö]{}bius triangles $(2,3,3)$ onto a subset of the plane $(\nu_1,\nu_{2})$. The dashed lines represent the degeneracies of the spectrum.](quatrit_orbits_on_elipse){width="60.00000%"}
[\[Fig:ProjectionQuatrit\]]{}
Concluding remark {#concluding-remark .unnumbered}
=================
The master equations (\[eq:master\]) for kernels of the Wigner functions determine the first and second degrees polynomial $SU(N)$ invariants of $N\--$dimensional system. The remaining $N-2$ algebraically independent invariants parameterize the moduli space of SW kernels. In the present article we establish relation between this moduli space and the orbit space of $SU(N)$ group. Next important issue is to clarify the role these unitary invariant moduli parameters play in dynamics of classical and quantum systems. With this aim in the forthcoming publication, a detailed analysis of the Kirillov-Kostant-Souriau symplectic 2-form for the whole family of the Wigner functions will be given.
Parametrization of the moduli space $\mathcal{P}_N({\boldsymbol{\nu}})$ {#AppendixA}
=======================================================================
As it was mentioned in the main text, the Stratonovich-Weyl kernel can be parameterized by $N-2$ spherical angles. Each member of the Wigner functions family can be associated with a point of subspace $\mathcal{P}_N({\boldsymbol{\nu}}) \subset \mathbb{S}_{N-2}(1)$, which is determined by the ordering of the eigenvalues of the Stratonovich-Weyl kernel. In order to define the $\mathcal{P}_N({\boldsymbol{\nu}})$ corresponding to the descending ordering and by means of using kernel decomposition in Gell-Mann bases, let us represent the spectrum of the Stratonovich-Weyl kernel in the following form: $$\begin{aligned}
\nonumber &\pi_{1}&=\frac{1}{N}\left(1+\sqrt{2}\, \kappa\sum_{s=2}^{N}\frac{\mu_{s^2-1}}{\sqrt{s\left(s-1\right)}}\right)\,,
\\
\nonumber&\vdots&
\\
\nonumber &\pi_{i}&=\frac{1}{N}\left(1+\sqrt{2}\, \kappa\sum_{s=i+1}^{N}\frac{\mu_{s^2-1}}{\sqrt{s\left(s-1\right)}}-\kappa \sqrt{\frac{2\left(i-1\right)}{i}}\mu_{i^{2}-1}\right)\,,
\\
\nonumber &\vdots&
\\
\nonumber&\pi_{N}&=\frac{1}{N}\left(1-\frac{N^{2}-1}{\sqrt{N+1}}\mu_{N^{2}-1}\right)\,.\end{aligned}$$ Introducing the conventional parametrization for a unit sphere $\mathbb{S}_{N-2}(1)$ in terms of spherical $N-2$ angles: $$\begin{array}{lll}
&\mu_{3}=\sin{\psi_{1}}\cdots\sin{\psi_{N-2}}\,,
\\
&\mu_{8}=\sin{\psi_{1}}\cdots\sin{\psi_{N-3}}\cos{\psi_{N-2}}\,,
\\
&\vdots&
\\
&\mu_{i^{2}-1}=\sin{\psi_{1}}\cdots\sin{\psi_{N-i}}\cos{\psi_{N-i+1}}\,,
\\
&\vdots&
\\
&\mu_{N^{2}-1}=\cos{\psi_{1}}\,,
\\&&\\
&\text{with }\quad \psi_{i}\in\left[0,\pi\right],\; i=\overline{1\,, N-3} \quad \text{and }\quad \psi_{N-2}\in\left[0,2\pi\right)\,,
\end{array}$$ and demanding the descending order of the eigenvalues, we obtain the following constraints on the coefficients $\mu_{i}$: $$\begin{aligned}
\label{eq:ineqmu1}
&&\mu_{3}\geq0\,,\\
&&\mu_{\left(i+1\right)^{2}-1}\geq\sqrt{\frac{i-1}{i+1}}\,\mu_{i^{2}-1}\,,\quad i=\overline{2\,,N-1}\,.
\label{eq:ineqmu2}\end{aligned}$$
Let us introduce the following notations:
$$\begin{array}{lcll}
&\mathcal{P}_{1}&=&\big\{\psi_{1}=0\big\}\,,\\&&&\\
&\mathcal{P}_{2}^{(k)}&=&\begin{cases}
\sin{\psi_{N-k}}=0\\
\sin{\psi_{N-(k+1)}}\cos{\psi_{N-k}}>0
\\
\cot{\psi_{N-i}}\geq\sqrt{\frac{i-1}{i+1}}\,\cos{\psi_{N-i+1}}\,\\
0<\psi_{i-k}<\pi\,,\quad i=\overline{k+1\,, N-1}\,,
\end{cases}
\end{array}$$
$$\begin{array}{lcll}
&\mathcal{P}_{3}&=& \begin{cases}
\sin{\psi_{N-2}}>0\\
\cos{\psi_{N-2}}\geq\frac{1}{\sqrt{3}}\sin{\psi_{N-2}}\\
\cot{\psi_{N-i}}\geq\sqrt{\frac{i-1}{i+1}}\,\cos{\psi_{N-i+1}}\,\\
0<\psi_{i-2}<\pi\,,\quad i=\overline{3\,,N-1}\\\
0<\psi_{N-2}<2\pi\,.
\end{cases}
\end{array}$$
In the introduced notations substitution of expressions for $\mu_i$ in terms of the spherical angles $\psi_{i}$ into (\[eq:ineqmu1\]) and (\[eq:ineqmu2\]) shows: if $k=2\,,\cdots\,N-2$ is the biggest natural number for which $\sin{\psi_{N-k}}=0$, if there is any, then the simplex is described by the restrictions $\mathcal{P}_{2}^{(k)}\subset\mathbb{S}_{N-\left(k+1\right)}(1)$ (these are some of ($N-\left(k+1\right)$)-dimensional boundaries of the simplex); otherwise, if there is no such $k$, then the restrictions are $\mathcal{P}_{3}$. Hence, the simplex will be completely defined by $$\mathcal{P}= \mathcal{P}_{1}\cup \left(\bigcup_{k=2}^{N-2}\mathcal{P}_{2}^{\left(k\right)}\right)\cup\mathcal{P}_{3}\,.$$
Partially reducing the set of inequalities for $\mathcal{P}_{3}$, we get: $$\mathcal{P}_{3}=\begin{cases}
0<\psi_{N-2}\leq\frac{\pi}{3}\\
0<\psi_{i-2}<\pi\,, \quad i=\overline{3\,,N-1}\\
\cot{\psi_{N-i}}\geq\sqrt{\frac{i-1}{i+1}}\cos{\psi_{N-i+1}}\,.
\end{cases}$$
[6]{} Johann v. Neumann, *Mathematische Grundlagen der Quantenmechanik*, Verlang von Julius Springer, Berlin, 1932. H. Weyl, *Gruppentheorie und Quantenmechanik*, Hirzel-Verlag, Leipzig, 1928. E. P. Wigner, *On the quantum correction for thermodynamic equilibrium*, Phys. Rev. **40**, 749-759, 1932. K. Husimi, *Some formal properties of the density matrix*, Proc. Phys. Math. Soc. Japan, **22**, 264-314, 1940. H. J. Groenewold *On the principles of elementary quantum mechanics*, Physica **12**, 405-460, 1946. J. E. Moyal, *Quantum mechanics as a statistical theory*, Mathematical Proceedings of the Cambridge Philosophical Society **45** 99-124, 1949. G. Dito and D. Sternheimer, *Deformation quantization: genesis, developments and metamorphoses*, arXiv: https://arxiv.org/abs/math/0201168 (2002). E. C. G. Sudarshan, *Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams*, Phys. Rev. Lett. **10**, 277–279, 1963. R.J.Glauber, *Coherent and incoherent states of the radiation field*, Phys. Rev. **131**, 2766–2788, 1963. M. Hillery, R. F. O’Connell, M. O. Scully and E. P. Wigner, *Distribution functions in physics: Fundamentals*, Phys. Rep. **106**, 121-167, 1984. D. J. Rowe, B. C. Sanders and H. de Guise, *Representations of the Weyl group and Wigner functions for SU(3)*, J. Math. Phys. **40**, 3604, 1999. S. Chumakov, A. Klimov and K. B. Wolf, *Connection between two Wigner functions for spin systems*, Phys. Rev. **A 61**, 034101, 2000. M. A. Alonso, G. S. Pogosyan, and K.B.Wolf, *Wigner functions for curved spaces*, J. Math. Phys. **43**, 5857, 2002. A. I. Lvovsky and M. G. Raymer, *Continuous-variable optical quantum-state tomography*, Rev. Mod. Phys. [**81**]{}, 299-332, 2009. A.B. Klimov and H. de Guise, *General approach to $\mathfrak{GU}(n)$ quasi-distribution functions*, J. Phys. A. **43**, 402001, 2010. I. Rigas, L. L. Sánchez-Soto, A. B. Klimov, J. Rehacek and Z. Hradil, *Orbital angular momentum in phase space*, Ann. Phys. [**326**]{}, 426-439, 2011. T. Tilma, M. J. Everitt, J. H. Samson, W. J. Munro and K. Nemoto, *Wigner functions for arbitrary quantum systems*, Phys. Rev. Lett. **117**, 180401, 2016. A. B. Klimov, J. L. Romero, and H. de Guise, *Generalized SU(2) covariant Wigner functions and some of their applications*, J. Phys. A **50**, 323001, 2017. M. O. Scully and M. S. Zubairy, *Quantum Optics*, Cambridge University Press, Cambridge, 2006. I. Bengtsson and K. Życzkowski, *Geometry of Quantum States. An Introduction to Quantum Entanglement*, Cambridge University Press, Cambridge, 2nd Edition, 2017. R. P. Feynman, *Negative Probaility*, In Basil J. Hiley & D. Peat (eds.), Quantum Implications: Essays in Honour of David Bohm. Methuen., 235-248, 1987. R.L. Stratonovich, *On distributions in representation space*, Soviet Physics JETP **4**, 6, 891-898, 1957. C. Briff and A. Mann, *Phase-space formulation of quantum mechanics and quantum-state reconstruction for physical systems with Lie-group symmetries*, Phys.Rev. **59**, 971, 1999. A. Khvedelidze and V. Abgaryan, *On the Family of Wigner Functions for [$N$]{}-Level Quantum System*, arXiv: https://arxiv.org/abs/1708.05981 (2018). A. A. Kirillov, *Lectures on the Orbit Method*, Graduate Studies in Mathematics, **64**, American Mathematical Society, 2004. D. Weingarten, *Asymptotic behavior of group integrals in the limit of infinite rank*, J.of Mat. Phys. **19**, 999, 1978. B. Collins, *Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability*, Int. Math. Res. Not. **17**, 953, 2003. B. Collins and P. Sniady, *Integration with respect to the Haar measure on unitary, orthogonal and symplectic group*, Comm. Math. Phys. **264**, 3, 773, 2006. A. Luis, *A SU(3) Wigner function for three-dimensional systems*, J. Phys. A [**41**]{}, 495302, 2008. V. I. Arnold, *Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect*, Selecta Mathematica, New Series **1**, No.1, 1995.
[^1]: vahagnab@googlemail.com
[^2]: akhved@jinr.ru
[^3]: astghik@jinr.ru
[^4]: According to Weyl’s rule of quantization [@Weyl1928], any classical observable $A(\boldsymbol{p},\boldsymbol{q})$, i.e., a function on the phase space $\mathbb{R}^{2n}$ with a standard canonical symplectic structure, is associated with an operator $\hat{A}_\omega$ on the Hilbert space $L^2(\mathbb{R}^n)\,$ constructed as the *“Weyl quantum Fourier transform"*: $$\label{eq:WeylTransform}
A \mapsto \hat{A}_\omega=\int_{\mathbb{R}^{2n}} \mathrm{d}\Omega(\omega )\, \tilde{A}(\boldsymbol{u}, \boldsymbol{v})\exp{\frac{\imath}{\hbar}\left( \boldsymbol{u} \boldsymbol{\hat{P}}+\boldsymbol{v} \boldsymbol{\hat{Q}} \right)}
\,,\qquad \mathrm{d}\Omega=\omega(\boldsymbol{u},\boldsymbol{v})\,
\mathrm{d}\boldsymbol{u} \mathrm{d}\boldsymbol{v}\,,$$ where $\hat{P}$ and $\hat{Q}$ are operators on $L^2(\mathbb{R}^n)$ obeying canonical commutator relations, $\tilde{A}(\boldsymbol{u}, \boldsymbol{v})$ is Fourier transform of ${A}(\boldsymbol{u}, \boldsymbol{v})$, and the integration measure $\mathrm{d}\Omega$ is defined by a weight function $\omega(\boldsymbol{u}, \boldsymbol{v})\,.$ Different choice of $\omega(\boldsymbol{u}, \boldsymbol{v})\,$ is a source of various orderings of non-commutative operators $\hat{P}$ and $\hat{Q}$. For example, the factor $\omega(\boldsymbol{u}, \boldsymbol{v})=\exp\left(-\frac{\imath}{2}\,\boldsymbol{u}\boldsymbol{v}\right)$ corresponds to a standard ordering of polynomials in mathematical literature when writing first the position coordinate $Q\,,$ then the momentum $P\,.$ The so-called normal ordering is related to the weight $\omega(\boldsymbol{u}, \boldsymbol{v})=\exp\left(-\frac{1}{4}\,(\boldsymbol{u}^2 + \boldsymbol{v}^2)\right)\,,$ while the original Weyl, or symmetric, order complies with $\omega(\boldsymbol{u}, \boldsymbol{v})=1\,.$ The inverse formula that maps the operator to its symbol belongs to Wigner [@Wigner1932]. For a unit weight factor case, $\omega = 1\,,$ the inverse formula reads: $$A(\boldsymbol{u}, \boldsymbol{v })=
\frac{1}{(2\pi\hbar)^n}\mbox{tr}\left[\hat{A}_1
\exp{-\frac{\imath}{\hbar}\left( \boldsymbol{u} \boldsymbol{\hat{P}}+\boldsymbol{v} \boldsymbol{\hat{Q}}\right)}\right]\,.$$ A further elaboration of Weyl’s quantization scheme leads to the non-commutative formulation of mechanics [@Moyal1949] and finally to the development of the so-called deformation quantization, cf. [@DitoSternheimer].
[^5]: History going back to Dirac’s idea on negative energies teaches us to pay more attention to a “nonsense" of negative probabilities. In this context it is the best to afford the following words by R.Feynman: *“It is that a situation for which a negative probability is calculated is impossible, not in the sense that the chance for it happening is zero, but rather in the sense that the assumed conditions of preparation or verification are experimentally unattainable” [@Feynman1987].*
[^6]: Since $\mathfrak{g}$ is a linear space over the real field $\mathbb{R}$, one can define a bilinear map $
\langle .\,,\, . \rangle \, : \mathfrak{g}^*\times\mathfrak{g} \to \mathbb{R}\,,
$ and identify algebra with its dual. The conventional inner product on $\mathfrak{g}$, $$\label{eq:DPLie}
\langle A\,,\, B \rangle : =\mbox{tr}\left(A^\dagger B\right)\,, \qquad A, B \in \mathfrak{su}(N)\,,$$ enables to set up a duality pairing and to realize an isomorphism between $\mathfrak{g}$ and $\mathfrak{g}^*\,.$
[^7]: In this case, the isotropy group of SW kernel is isomorphic to ($N-1$)-dimensional torus $\mathbb{T}^{N-1}=\lbrace g\in SU(N): g\--\mbox{diagonal}\rbrace\,.$
[^8]: The apex angle $\zeta$ determines the value of a 3-rd order polynomial SU(3)-invariant: $$\cos(3\zeta)=-\frac{27}{16}\,\det\left(\Delta^{(3)}-\frac{1}{3}\mathbb{I}\right)=-\frac{27}{16}\det\left(\Delta^{(3)}\right)-
\frac{11}{16}\,.$$
[^9]: Traces of powers of this *“golden ratio”* kernel are given by the so-called *Lucas numbers*: $$\mbox{tr}\left(\Delta_{(103)}\right)^2=3\,, \qquad
\mbox{tr}\left(\Delta_{(103)}\right)^3=4\,, \qquad \ldots \,, \qquad
\mbox{tr}\left(\Delta_{(103)}\right)^n= L_n\,.$$
|
---
abstract: 'We classify affine varieties with an action of a connected, reductive algebraic group such that the group is isomorphic to an open orbit in the variety. This is accomplished by associating a set of one-parameter subgroups of the group to the variety, characterizing such sets, and proving that sets of this type correspond to affine embeddings of the group. Applications of this classification to the existence of morphisms are then given.'
address: |
Department of Mathematics and Computer Science\
Hillsdale College\
33 East College Street\
Hillsdale, MI 49242
author:
- David Murphy
title: Affine embeddings of a reductive group
---
Introduction {#S:intro}
============
A basic problem in algebraic geometry is the study of algebraic actions. Even actions with a dense orbit are not well understood. Let $G$ be an algebraic group. A *quasihomogeneous variety* is a normal $G$-variety possessing an open orbit isomorphic to $G/H$, for some closed subgroup $H$ of $G$. Toric varieties are an important family of quasihomogeneous varieties, where $G$ is an algebraic torus and $H$ is its trivial subgroup. A partial classification of quasihomogeneous varieties was obtained in the important paper of Luna and Vust [@LV]. Their classification seeks to generalize that of toric varieties, but it is only feasible in special cases. In this paper, we solve the equivariant classification problem for one case not covered in [@LV], namely the classification of affine quasihomogeneous varieties in which $H$ is trivial.
A *$G$-embedding* is a normal $G$-variety $X$ that contains an open orbit $\Omega$ isomorphic to $G$. The closed subvariety $\partial X = X - \Omega$ is called the *boundary* of $X$. Since $\Omega$ is an open $G$-orbit, $\partial X$ is a $G$-stable divisor of $X$ unless $\Omega = X$.The irreducible components of $\partial X$ are $G$-stable prime divisors of $X$. Any toric variety is a $T$-embedding, in our terminology. Similarly, the wonderful compactification of an adjoint group $G$ defined by De Concini and Procesi [@CP1] is a $G$-embedding. All of these examples have both a left and a right $G$-action. Yet our definition of a $G$-embedding allows the consideration of $G$-varieties with only a left action of the group. Hence, our definition of $G$-embeddings includes all of the biequivariant compactifications already in the literature [@BCP], [@B2], [@CP1], [@TE], [@Tim2] and many more. In this paper, we study affine $G$-embeddings and relate our results with those of Brion [@B2] in the case of a biequivariant affine $G$-embedding in Section \[SS:biequivariance\].
Suppose $X$ is an affine $G$-embedding and $x_0 \in X$ is a closed point in the open orbit $\Omega$. In Section \[S:onePS\], we define a set $\Gamma(X,x_0)$ of one-parameter subgroups of $G$ associated to the embedding $X$ and base point $x_0$. Properties of such sets are collected in Proposition \[P:4properties\], which are then used in Theorem \[T:A\_X\] to prove that $X$, as a $G$-embedding with basepoint $x_0$, is determined by its set $\Gamma(X,x_0)$. Therefore, we turn our attention to the classification of such sets in Sections \[SS:1skeleton\] through \[SS:classification\]. We prove that sets $\Gamma(X,x_0)$ arising from affine $G$-embeddings are strongly convex lattice cones, in the sense of Definition \[D:scrpc\], and that any strongly convex lattice cone determines an affine $G$-embedding in Theorem \[T:classification\]. This generalizes the classification of affine toric varieties by strongly convex rational polyhedral cones in [@TE].
Lastly, we explore the functoriality of our classification in Sections \[SS:morphisms\] and \[SS:biequivariance\]. Specifically, Proposition \[P:morphisms\] states that equivariant morphisms of affine $G$-embeddings correspond to the inclusion of their associated cones, up to conjugation, analogous to the result for toric varieties. Moreover, our classification reveals when an affine $G$-embedding $X$ has not only a left, but also a right $G$-action compatible with the identification of $G$ with the open orbit in $X$ (Proposition \[P:biequivariance\]). Using this, we define a biequivariant resolution (Definition \[D:biresolution\]) of an arbitrary affine $G$-embedding and prove its universal property in Theorem \[T:universalbi\] of Section \[SS:biequivariance\].
Notation {#notation .unnumbered}
--------
We will always work over a ground field $k$, which we assume to be algebraically closed and of characteristic zero. All algebraic groups are assumed to be linear and defined over $k$, and will be denoted by letters such as $G$ and $H$. In particular, $G$ will refer to a connected, reductive algebraic group defined over $k$. The symbol $T$ will always denote an algebraic torus, whether abstract or as a subgroup of $G$. For an algebraic group $H$, $\mathfrak{X}_*(H)$ will denote its set of one-parameter subgroups and $\mathfrak{X}^*(H)$ will denote the group of characters of $H$.
Acknowledgments {#acknowledgments .unnumbered}
---------------
I wish to express my sincere thanks to my advisors, Robert Fossum and William Haboush, for their encouragement and guidance as I conducted this work.
One-parameter subgroups {#S:onePS}
=======================
Our primary method for describing and classifying affine $G$-embeddings $X$ is to make use of one-parameter subgroups of $G$. We are interested in the limits, when they exist, of the one-parameter subgroups of $G$ in $X$. One-parameter subgroups and their limits have been employed in a number of applications, including the Hilbert-Mumford criterion of stability [@GIT], the construction of the spherical building of the group $G$ [@GIT] and the Bialynicki-Birula decomposition of a smooth projective $T$-variety [@BB]. For our purposes, we will show that an affine $G$-embedding $X$ is determined by the set of one-parameter subgroups $\gamma$ of $G$ such that $\limit \gamma(t) x_0$ exists in $X$.
A *one-parameter subgroup* of $G$ is a homomorphism of algebraic groups $\gamma : \mathbb{G}_m \to G$, which thus corresponds to a map $\gamma^\circ : k[G] \to k[t,t^{-1}]$. Let $\mathfrak{X}_*(G)$ denote the set of one-parameter subgroups of $G$. The group $G$ acts on $\mathfrak{X}_*(G)$ by conjugation, $g \bullet \gamma : t \mapsto g \gamma(t) g^{-1}$. We will denote the trivial one-parameter subgroup $t \mapsto e$ by $\varepsilon$. Each one-parameter subgroup $\gamma \in \mathfrak{X}_*(G)$ determines a subgroup $$\label{E:P(gamma)}
P(\gamma) = \{ g \in G : \gamma(t) g \gamma(t^{-1}) \in G_{k[[t]]} \}$$ of $G$, which is parabolic if $G$ is reductive [@GIT]. In fact, every parabolic subgroup of a reductive group $G$ is of the form $P(\gamma)$ for some one-parameter subgroup $\gamma$ of $G$ [@Springer]. We define an equivalence relation on the set of non-trivial one-parameter subgroups of $G$ by $$\label{E:sim1ps}
\gamma_1 \sim \gamma_2 \text{ if and only if } \gamma_2(t^{n_2}) = g \gamma_1(t^{n_1}) g^{-1}$$ for positive integers $n_1,n_2$ and an element $g \in P(\gamma_1)$, for all $t \in k^\times$. Then the quotient $(\mathfrak{X}_*(G) - \{ \varepsilon \})/\sim$ is isomorphic to the spherical building of $G$ [@GIT], [@Tits2].
Every parabolic subgroup $P$ of $G$ defines a subset $\Delta_P(G) = \{ \gamma \in \mathfrak{X}_*(G) : P(\gamma) \supseteq P \}$ of $\mathfrak{X}_*(G)$. Clearly $\gamma \in \Delta_{P(\gamma)}(G)$ for all $\gamma \in \mathfrak{X}_*(G)$, so $\mathfrak{X}_*(G) = \bigcup \Delta_P(G)$ where the union is over all parabolic subgroups of $G$. In the spherical building of $G$, the images of the sets $\Delta_P(G)$ are simplices and constitute a “triangulation” of the building [@GIT].
The inclusion of $k[t,t^{-1}]$ in $k((t))$ allows us to view $\mathfrak{X}_*(G)$ as a subset of $G_{k((t))} = \hom_k(k[G],k((t)))$, the set of $k((t))$-points of $G$. Let $\langle\gamma\rangle \in G_{k((t))}$ denote the point corresponding to the one-parameter subgroup $\gamma$. The group $G_{k((t))}$ contains the subgroup $G_{k[[t]]}$, which consists of all $k((t))$-points of $G$ that have a specialization in $G$ as $t \to 0$. The group $G_{k((t))}$ is the disjoint union of the double cosets of $G_{k[[t]]}$, as described by the Iwahori decomposition:
\[T:Iwahori\] Let $G$ be a reductive algebraic group over $k$. Every double coset of $G_{k((t))}$ with respect to the subgroup $G_{k[[t]]}$ is represented by a point of the type $\langle \gamma \rangle$, for some one-parameter subgroup $\gamma$ of $G$. That is, $$\label{E:Iwahori}
G_{k((t))} = \bigcup_{\gamma \in \mathfrak{X}_*(G)}
G_{k[[t]]} \langle \gamma \rangle G_{k[[t]]}$$ Furthermore, each double coset is represented by a unique dominant one-parameter subgroup.
This decomposition enables us to replace $k((t))$-points of $G$ with one-parameter subgroups.
Limits of one-parameter subgroups {#SS:limits}
=================================
Let $X$ be a $G$-variety. Each point $x$ of $X$ determines a morphism $\psi_x : G \to X$ by $\psi_x(g) = g \cdot x$. For a point $x_0 \in X$ and a one-parameter subgroup $\gamma$ of $G$, we say $\limit \gamma(t) x_0$ exists in $X$ if $\psi_{x_0} \circ \gamma : \mathbb{G}_m \to X$ extends to a morphism $\tilde{\gamma} : \mathbb{A}^1 \to X$. In this case, $\limit \gamma(t) x_0$ is defined to be $\tilde{\gamma}(0)$. That is, the composition of $\psi_{x_0}^\circ : k[X] \to k[G]$ with $\gamma^\circ : k[G] \to k[t,t^{-1}]$ factors through $k[t]$, and the limit, $\limit \gamma(t)x_0$, is the $k$-point of $X$ corresponding to the composite $k[X] \to k[t] \to k$ sending $t \to 0$. This is described by the diagrams: $$\xymatrix{
\mathbb{G}_m
\ar[r]^\subset
\ar[d]_\gamma
&
\mathbb{A}^1
\ar@{-->}[d]^{\tilde{\gamma}_0}
&&
k[X]
\ar[r]^{\psi_{x_0}^\circ}
\ar@{-->}[d]_{\tilde{\gamma}_0^\circ}
&
k[G]
\ar[d]^{\gamma^\circ}
\\
G
\ar[r]_{\psi_{x_0}}
&
X
&&
k[t]
\ar[r]_\subset
&
k[t,t^{-1}].
}$$ Similarly, if $\lambda$ is a $k((t))$-point of $G$, then $\limit \lambda(t) x_0$ exists in $X$ means $\lambda^\circ|_{k[X]} : k[X] \to k[[t]]$.
The following lemma is used frequently hereafter.
\[L:product\_limit\] Suppose $\lambda \in G_{k((t))}$ and $\alpha \in G_{k[[t]]}$, so that $\alpha$ has specialization $\alpha_0 \in G_k$. Let $X$ be an affine $G$-embedding with base point $x_0$. Then $\limit [ \lambda(t) x_0 ]$ exists in $X$ if and only if $\limit [ \alpha(t) \lambda(t) x_0 ]$ exists, in which case $$\label{E:product_limit}
\limit [ \alpha(t) \lambda(t) x_0 ] =
\alpha_0 \cdot \limit [\lambda(t) x_0].$$
The proof is straightforward.
\[R:bi-limits\] Suppose $X$ is a biequivariant $G$-embedding, so $G$ has both a left and a right action on $X$ and $G$ may be identified with an open subvariety $\Omega$ which is stable for both actions. Then we could amplify Lemma \[L:product\_limit\] as follows: If $\alpha,\beta \in G_{k[[t]]}$ and $\lambda \in G_{k((t))}$, then $\limit \lambda(t) x_0 \in X$ if and only if $\limit [ \alpha(t)
\lambda(t) \beta(t) x_0 ] \in X$, in which case $$\limit [ \alpha(t) \lambda(t) \beta(t) x_0 ] =
\alpha_0 \cdot [ \limit \lambda(t) x_0 ] \cdot \beta'_0,$$ where $\beta(t) \cdot x_0 = x_0 \cdot \beta'(t)$ for some $\beta' \in G_{k[[t]]}$ and $\alpha_0,\beta'_0 \in G_k$ denote the specializations of $\alpha,\beta'$, respectively. However, we must be careful, for $\limit [\alpha(t)\lambda(t)
\beta(t) x_0]$ does not have to equal $\limit [\alpha_0 \lambda(t) \beta_0
x_0 ]$, where $\alpha_0,\beta_0$ are the specializations of $\alpha,\beta$ [@Kempf].
\[T:one-ps\] Let $X$ be an affine $G$-variety. Suppose that $Y$ is a closed $G$-stable subvariety of $X$ and that $x_0 \in X$ is a closed point such that the closure of the orbit $Gx_0$ intersects $Y$. Then there is a one-parameter subgroup $\gamma$ of $G$ such that $\limit \gamma(t) x_0 \in Y$.
One-parameter subgroups of an affine $G$-embedding {#SS:Gamma(X)}
==================================================
\[D:Gamma(X)\] Given a $G$-variety $X$ and a point $x_0 \in X$, define $$\label{E:Gamma(X)}
\Gamma(X,x_0) :=
\{ \gamma \in \mathfrak{X}_*(G) : \limit \gamma(t) x_0
\text{ exists in } X \}.$$
We will be interested in the structure of such sets of one-parameter subgroups when $X$ is an affine $G$-variety and the orbit of $x_0$ in $X$ is open and isomorphic to $G$. We call such an $x_0 \in X$ a base point. Before we proceed, we make some immediate observations about such sets.
\[P:4properties\] Let $G$ be a connected reductive group. Suppose $X$ is an affine $G$-embedding and $x_0 \in X$ is a base point.
1. If $x'_0 = hx_0$, then $\Gamma(X,x'_0) = h \Gamma(X,x_0) h^{-1}$.
2. If $\gamma \in \Gamma(X,x_0)$ and $\gamma \neq \varepsilon$, then $\gamma^{-1} \not\in \Gamma(X,x_0)$.
3. If $T$ is any torus of $G$, then $\overline{Tx_0} \cong \overline{T}_\sigma$, where $\sigma = \Gamma(X,x_0) \cap \mathfrak{X}_*(T)$ is a strongly convex rational polyhedral cone in $\mathfrak{X}_*(T)$ [@TE].
4. If $\gamma \in \Gamma(X,x_0)$ and $p \in P(\gamma)$, then $p \gamma(t) p^{-1}
\in \Gamma(X,x_0)$, and moreover $$\label{E:GammaPs}
\Gamma(X,x_0) = \bigcup_{P \subset G}
P \bullet (\Gamma(X,x_0) \cap \Delta_P(G))$$ where the union is taken over all parabolic subgroups $P$ of $G$.
5. The image of $\Gamma(X,x_0)$ in the spherical building is convex ([@GIT], Definition 2.10).
First, $\Gamma(X,x_0)$ depends on the base point $x_0$ as follows. Suppose $x'_0 \in \Omega$, so $x'_0 = h \cdot x_0$ for some unique $h \in G$ (because $G \to \Omega$ is an isomorphism). Then $\Gamma(X,x'_0) = h\Gamma(X,x_0)h^{-1}$ for if $\gamma \in \Gamma(X,x_0)$ (that is, if $\limit \gamma(t) x_0
\in X$), then $\limit (h\gamma(t)h^{-1}) x'_0 = \limit
h \gamma(t) h^{-1} h x_0 = \limit h\gamma(t) x_0 = h \limit
\gamma(t) x_0$, which exists in $X$. Therefore $h\Gamma(X,x_0)h^{-1} \subset
\Gamma(X,x'_0)$. By symmetry, since $x_0 = h^{-1} x'_0$, $h^{-1} \Gamma(X,x'_0)
h \subset \Gamma(X,x_0)$, so $\Gamma(X,x'_0) \subset h \Gamma(X,x_0) h^{-1}$. Hence, $$\label{E:Gamma(X,hx)}
\Gamma(X,h \cdot x_0) = h\, \Gamma(X,x_0)\, h^{-1}$$ for any $h \in G$.
Second, as $X$ is affine, if $\gamma \in \Gamma(X,x_0)$ and $\gamma$ is not the trivial one-parameter subgroup $\varepsilon : t \mapsto e$, then $\gamma^{-1}
\not\in \Gamma(X,x_0)$. Otherwise, if both $\limit \gamma(t) x_0$ and $\limit \gamma^{-1}(t) x_0$ exist in $X$, then the composition $\psi_{x_0}
\circ \gamma : \mathbb{G}_m \to X$ extends to a morphism $\tilde{\gamma} :
\mathbb{P}^1 \to X$, which must therefore be constant, so $\gamma = \varepsilon$.
Third, if $T$ is any torus of $G$, then $\overline{T x_0} \cong T_\sigma$, where $\sigma \subset \mathfrak{X}_*(T)$ is the strongly convex lattice cone $\Gamma(X,x_0) \cap \mathfrak{X}_*(T)$ from toric geometry.
Now suppose $\gamma \in \Gamma(X,x_0)$ and $p \in P(\gamma)$. Then $p \cdot \gamma \cdot p^{-1}$ also belongs to $\Gamma(X,x_0)$, for $\limit
[ (p \gamma(t) p^{-1}) x_0 ] = \limit [ p ( \gamma(t) p^{-1} \gamma(t^{-1}) )
\gamma(t) x_0 ] = p [\limit \gamma(t) p^{-1} \gamma(t^{-1}) ] [ \limit
\gamma(t) x_0 ]$ exists in $X$ by Lemma \[L:product\_limit\] and the definition of $P(\gamma)$. Therefore, it is clear that $\Gamma(X,x_0) = \bigcup_{P \subset G} P \bullet (\Gamma(X,x_0) \cap \Delta_P(G))$, where the union is taken over all parabolic subgroups $P$ of $G$.
Lastly, if $\delta_1,\delta_2 \in (\Gamma(X,x_0) - \{ \varepsilon \})/\sim$, then there are one-parameter subgroups $\gamma_1,\gamma_2 \in \Gamma(X,x_0)$ such that $\delta_i = [\gamma_i]$ is the equivalence class of $\gamma_i$. The one-parameter subgroups determine parabolic subgroups $P(\gamma_1)$ and $P(\gamma_2)$, whose intersection contains a maximal torus $T$ of $G$. Then $\gamma_1$ and $\gamma_2$ are equivalent to one-parameter subgroups $\gamma_1',\gamma_2' \in \mathfrak{X}_*(T)$ and $\delta_i =
[\gamma_i']$. By part 4, $\gamma_1', \gamma_2' \in \Gamma(X,x_0)$ as well. Then $\gamma_1',\gamma_2' \in \Gamma(X,x_0) \cap \mathfrak{X}_*(T)$, which is the strongly convex rational polyhedral cone associated to the toric variety $\overline{T} \subset X$ by part 3. As strongly convex rational polyhedral cones are convex, the line in $\mathfrak{X}_*(T)$ joining $\gamma_1$ and $\gamma_2$ is contained in $\Gamma(X,x_0)
\cap \mathfrak{X}_*(T)$, and hence the line in the spherical building joining $\delta_1$ and $\delta_2$ is contained in the image of $\Gamma(X,x_0)$, so this image is semi-convex. It is convex by part 2, which implies no pair of antipodal points of the building can belong to the image of $\Gamma(X,x_0)$.
Each $\gamma \in \mathfrak{X}_*(G)$ may be viewed as a $k((t))$-point of $G$. In [@LV], a $G$-stable valuation $v_\lambda$ is associated to every $\lambda
\in G_{k((t))}$ in the following way. As $\lambda$ is a $k((t))$-point of $G$, we obtain a dominant morphism $$\xymatrix{
G \times \spec k((t)) \ar[r]^>>>>>>{1 \times \lambda}
&
G \times G \ar[r]^>>>>>\mu
&
G.
}$$ This morphism induces an injection of fields $i_\lambda : k(G) \to
\text{Frac}(k(G) \otimes_k k((t))) \to k(G)((t))$. Then $v_t \circ i_\lambda : k(G)^\times
\to \mathbb{Z}$ is a valuation of $k(G)$, where $v_t : k(G)((t))^\times \to
\mathbb{Z}$ is the standard valuation associated to the order of $t$. We define $v_\lambda = \dfrac{1}{n_\lambda} (v_t \circ i_\lambda)$, where $n_\lambda \in \mathbb{Z}$ is the largest positive number such that $(v_t \circ i_\lambda)(k(G)^\times) \subset n_\lambda \mathbb{Z}$ (except when $\lambda = \varepsilon$, in which case $v_\varepsilon(f) = 0$ or $\infty$ as $f(e) \neq 0$ or $= 0$, respectively). This is $G$-stable by left translations, i.e., $v_\lambda(s \cdot f) = v_\lambda(f)$ for all $s \in G$, since $i_\lambda$ is clearly equivariant and $k(G)[[t]]$ is obviously stable for left translations by $G$ in $k(G)((t))$. We include some of the properties of these valuations that are proven in [@LV] in the following lemma.
\[L:v\_gamma\]
1. Let $\gamma$ be a one-parameter subgroup of $G$. For each $f \in k(G)$, there is an open subset $U \subset G$, depending only on $f$, such that $$\label{E:inf}
v_\gamma(f) = \inf_{s \in U} v_t(f(s \cdot \gamma(t)))$$
2. Let $\gamma_1,\gamma_2$ be one-parameter subgroups of $G$. Then $v_{\gamma_1} = v_{\gamma_2}$ if and only if $\gamma_1 \sim \gamma_2$.
Part 1 is Lemma 4.11.1 in [@LV], where $U = \{ s \in G : f(s) \neq 0 \}$. The second part is the result of Propositions 3.3 and 5.4 in [@LV].
The sets of one-parameter subgroups $\Gamma(X,x_0)$ described in Definition \[D:Gamma(X)\] are significant for the following reasons. The first result, which will serve as our foundation for the classification theorem in Section \[SS:classification\], is a uniqueness theorem which shows that an affine $G$-embedding $X$ with base point $x_0$ is determined by the set $\Gamma(X,x_0)$. The second demonstrates that the prime divisors on the boundary of an affine $G$-embedding correspond to equivalence classes of edges of the set $\Gamma(X,x_0)$.
\[T:A\_X\] Let $G$ be a connected reductive group. If $X$ is an affine $G$-embedding with base point $x_0$, then $X \cong \spec A_{\Gamma(X,x_0)}$, where $$\label{E:A_X}
A_{\Gamma(X,x_0)} := \{ f \in k[G] : v_\gamma(f) \geq 0
\text{ for all } \gamma \in \Gamma(X,x_0) \}.$$
The base point $x_0$ defines a morphism $\psi_{x_0} : g \mapsto g \cdot x_0$ from $G$ to $X$. As both $G$ and $X$ are affine, $\psi_{x_0}$ corresponds to a homomorphism $\psi_{x_0}^\circ : k[X] \to k[G]$, which is injective since the image of $G$ is open in $X$ and $X$ is irreducible. The image of $\psi_{x_0}^\circ$ lies in the subalgebra $A_{\Gamma(X,x_0)}$ since every $\gamma \in \Gamma(X,x_0)$ extends to a morphism $\tilde{\gamma} : \mathbb{A}^1 \to X$ so that $f(g \cdot \tilde{\gamma}(0))$ exists, which implies that $v_\gamma(f) = \inf_{s \in G_f} v_t(f(s \cdot \gamma(t)))
\geq 0$ for all $f \in k[X]$. We claim that $k[X] \cong A_{\Gamma(X,x_0)}$. It suffices to show that every $f \in A_{\Gamma(X,x_0)}$ extends to a regular function on $X$ to prove that $k[X] \to A_{\Gamma(X,x_0)}$ is surjective and hence that $k[X] \cong
A_{\Gamma(X,x_0)}$.
Suppose not and assume that $f \in A_{\Gamma(X,x_0)}$ is not in the image of $k[X]$. Then $f$ fails to extend to a regular function on $X$, but it is defined on $\Omega = Gx_0$ by $f(g \cdot x_0) := f(g)$. Let $P$ be the divisor of poles of $f$ in $X$. Then $P$ is closed, has codimension one in $X$, and $P \subseteq \partial X$. Let $D$ be an irreducible component of $\partial X$ and hence a closed subvariety of $X$. Since $\partial X$ is $G$-stable and $G$ is connected (and so is irreducible), $D$ is a $G$-stable prime divisor since $e_G \in G$ fixes the generic point of the irreducible subvariety $D$. Therefore, Theorem \[T:one-ps\] provides a one-parameter subgroup $\gamma_D$ of $G$ with $\limit \gamma_D(t) x_0 \in D$, so $\gamma_D \in \Gamma(X,x_0)$. Yet $f \in A_{\Gamma(X,x_0)}$ implies that $v_{\gamma_D}(f) \geq 0$, so $f$ must be defined on the orbit $G [\limit \gamma_D(t) x_0 ] \subset D$. Therefore, $P \cap D$ is a closed subset of $D$ not equal to $D$, since $P$ does not contain $G [\limit \gamma_D(t) x_0 ] \subset D$ as $v_{\gamma_D}(f) \geq 0$. Thus the codimension of $P$ in $X$, which is equal to the minimum of the codimensions of the $P \cap D$ as $D$ ranges over the irreducible components of $\partial X$, is at least $2$. This is a contradiction. Hence every $f \in A_{\Gamma(X,x_0)}$ extends to a regular function on $X$, so is in the image of $\psi_{x_0}^\circ$. Therefore, $\psi_{x_0}^\circ : k[X] \to
A_{\Gamma(X,x_0)}$ is an isomorphism, so $X \cong \spec A_{\Gamma(X,x_0)}$ as claimed. Furthermore, the selected base point $x_0 \in X$ corresponds to the maximal ideal $\mathfrak{m}_{x_0} = (\psi_{x_0}^\circ)^{-1}(\mathfrak{m}_e \cap A_{\Gamma(X,x_0)})$ of $k[X]$ as $\psi_{x_0}^\circ$ identifies $f \in A_{\Gamma(X,x_0)}$ with the unique extension of the function $f(g \cdot x_0) := f(g)$ to $X$.
\[C:right-translation\] Let $X$ be an affine $G$-embedding with base point $x_0$. If $x'_0 = hx_0$ is another base point, then $$\label{E:right-translation}
A_{\Gamma(X,h \cdot x_0)} = r_h(A_{\Gamma(X,x_0)}),$$ where $r_h$ denotes right translation by $h$ in $k[G]$, $r_h(f)(x) = f(xh)$.
Suppose $X$ is an affine $G$-embedding. We have shown in (\[E:Gamma(X,hx)\]) that the set $\Gamma(X,x_0)$ is determined by $X$ only up to conjugation, as any other base point is of the form $h \cdot x_0$ for a unique element $h \in G$ and $\Gamma(X,h \cdot x_0) = h \Gamma(X,x_0) h^{-1}$. We now show that $A_{\Gamma(X,h \cdot x_0)} = r_h(A_{\Gamma(X,x_0)})$. Suppose $f \in r_h(A_{\Gamma(X,x_0)})$. Then $f = r_h(f')$ for some $f' \in
A_{\Gamma(X,x_0)}$, which means that $v_\gamma(f') \geq 0$ for all $\gamma \in
\Gamma(X,x_0)$. If $\gamma' \in \Gamma(X,h \cdot x_0) = h \Gamma(X,x_0) h^{-1}$, write $\gamma' = h \bullet \gamma$ for $\gamma \in \Gamma(X,x_0)$. Using formula \[E:inf\], one can easily show that $v_{h \bullet \gamma}(f)
= v_\gamma(f') \geq 0$. Therefore, $f \in A_{\Gamma(X,h \cdot x_0)}$, so $r_h(A_{\Gamma(X,x_0)})
\subset A_{\Gamma(X,h \cdot x_0)}$. Likewise, $r_{h^{-1}}(A_{\Gamma(X,h \cdot x_0)}) \subset A_{\Gamma(X,x_0)}$, so $r_h(A_{\Gamma(X,x_0)}) = A_{\Gamma(X,h \cdot x_0)}$ as claimed.
By Theorem \[T:A\_X\] and Corollary \[C:right-translation\], the classification of affine $G$-embeddings is equivalent to the characterization of such subsets of $\mathfrak{X}_*(G)$ that are obtained from affine $G$-embeddings. In order to classify admissible subsets $\Gamma$, we will explore the properties of the sets $\Gamma(X,x_0)$ for arbitrary affine $G$-embeddings $X$ with choice of base point $x_0$ in the following sections.
The one-skeleton of $\Gamma(X,x_0)$ {#SS:1skeleton}
===================================
If $\sigma$ is a strongly convex rational polyhedral cone in $\mathfrak{X}_*(T)_\mathbb{R}$, let $\sigma(1)$ denote the set of rays of $\sigma$, $\sigma(1) = \{ \tau < \sigma : \dim \tau = 1 \}$. This is called the one-skeleton of the cone. By Proposition \[P:4properties\], for each maximal torus $T$ of $G$, $\Gamma(X,x_0) \cap \mathfrak{X}_*(T)$ is a strongly convex rational polyhedral cone in $\mathfrak{X}_*(T)$.
\[D:one-skeleton\] Let $X$ be an affine $G$-embedding with base point $x_0$. The *one-skeleton* of the set $\Gamma(X,x_0)$ is $$\label{E:Gamma(1)}
\Gamma_1(X,x_0) := \bigcup_T \, [\Gamma(X,x_0) \cap \mathfrak{X}_*(T)](1),$$ which is the set of extremal rays of $\Gamma(X,x_0)$.
Recall the equivalence relation on one-parameter subgroups defined in Equation \[E:sim1ps\]. Equivalence classes under this relation are now given a geometric interpretation.
\[P:Gamma(1)\] There is a bijection between $\Gamma_1(X,x_0)/\sim$ and the finite set of prime divisors of $X$ contained in $\partial X$.
Write $X = \Omega \cup D_1 \cup D_2 \cup \cdots \cup D_r$, where $D_1,\dots,D_r$ are the irreducible components of $\partial X$. Thus each $D_i$ is a $G$-stable prime divisors of $X$.
Suppose that $\rho \in \Gamma_1(X,x_0)$. Then there is a maximal torus $T$ of $G$ such that $\rho \in [\Gamma(X,x_0) \cap \mathfrak{X}_*(T)](1)$ is a ray of the strongly convex rational polyhedral cone $\Gamma(X,x_0) \cap
\mathfrak{X}_*(T)$ corresponding to $\overline{T} \subset X$. From the description of $T$-stable divisors in toric varieties in [@Fulton], the ray $\rho$ corresponds to a prime divisor $D_\rho$ in $\overline{T}$. Thus $D_\rho$ is an irreducible $T$-stable subvariety of $\overline{T}$, so $\overline{G \cdot D_\rho}$ is an irreducible $G$-stable subvariety of $X$, as both $G$ and $D_\rho$ are irreducible. Moreover, $\overline{G \cdot D_\rho}$ is contained in $\partial X$, so there is a $G$-stable prime divisor $D_i$ of $X$ such that $\overline{G \cdot D_\rho} \subseteq D_i \subset X$. However, $\text{codim}_X(\overline{G \cdot D_\rho}) = \dim X - \dim
\overline{G \cdot D_\rho} = \dim G - ( \dim G + \dim D_\rho - \dim T ) =
\dim T - \dim D_\rho = \text{codim}_{\overline{T}}(D_\rho) = 1$. Thus, as $\overline{G
\cdot D_\rho}$ is irreducible and of codimension one in $X$, and $\overline{G \cdot
D_\rho} \subset D_i \subsetneq X$, where $D_i$ is also irreducible and of codimension one, we conclude that $\overline{G \cdot D_\rho} = D_i$. Therefore, there is a map $\varphi : \Gamma_1(X,x_0) \to \{ D_1,D_2,\dots,D_r \}$ given by $\varphi(\rho) = \overline{G \cdot D_\rho}$.
The map $\varphi$ is surjective, as any of the prime divisors $D_i$ of $X$ are closed $G$-subvarieties, and thus contain the limit point of some one-parameter subgroup $\gamma \in \Gamma(X,x_0)$ by Theorem \[T:one-ps\]. This $\gamma$ is a one-parameter subgroup of some maximal torus $T$ of $G$, so $\overline{T} \cap D_i \neq \emptyset$ is a $T$-stable divisor of $\overline{T}$. Hence, there is a prime divisor $D_\rho$ of $\overline{T}$ corresponding to a ray $\rho \in \Gamma(X,x_0) \cap \mathfrak{X}_*(T)$ such that $D_\rho \subset
\overline{T} \cap D_i$. Thus $\overline{G \cdot D_\rho} \subset D_i$, from which we conclude $\overline{G \cdot D_\rho} = D_i$ as before.
Furthermore, $\varphi$ respects the equivalence relation $\sim$ described in (\[E:sim1ps\]), for if $\gamma_\rho$ denotes the first lattice point in $\mathfrak{X}_*(T)$ along $\rho$ and $\gamma_{\rho_1} \sim \gamma_{\rho_2}$, then $\limit \gamma_{\rho_1}(t) x_0$ and $\limit \gamma_{\rho_2}(t) x_0$ belong to the same $G$-orbit in $X$, and hence to the same prime divisor $D$ of $X$, as $D$ is $G$-stable. Therefore, $\overline{G \cdot D_{\rho_1}} = D = \overline{G
\cdot D_{\rho_2}}$, from the discussion above. Hence $\varphi$ induces a well-defined map $\widetilde{\varphi} : \Gamma_1(X,x_0)/\sim\, \to
\{ D_1,D_2,\dots,D_r \}$, which is surjective.
We prove that $\widetilde{\varphi}$ is an injection. Let $\rho \in \Gamma_1(X,x_0)$ and let $\gamma_\rho$ denote the first lattice point in $\mathfrak{X}_*(G)$ along $\rho$ as above. The ideal $\Gamma(X,\mathcal{O}(\overline{G \cdot D_\rho}))
= \{ f \in k[X] : f = 0 \text{ on } \overline{G \cdot D_\rho} \}$ is equal to $k[X] \cap \mathfrak{m}_{v_{\gamma_\rho}}$, where $\mathfrak{m}_{v_{\gamma_\rho}} = \{ f \in k(G) : v_{\gamma_\rho}(f) > 0 \}$ is the maximal ideal of the valuation ring $\mathcal{O}_{v_{\gamma_\rho}} =
\{ f \in k(G) : v_{\gamma_\rho}(f) \geq 0 \}$. For $D_\rho =
\overline{T \cdot z_\rho}$, where $z_\rho = \limit \gamma_\rho(t) x_0 \in
\overline{T}$, so that $\overline{G \cdot D_\rho} = \overline{G \cdot z_\rho}$ as well. Thus, if $f \in k[X] \cap \mathfrak{m}_{v_{\gamma_\rho}}$, then $f(z_\rho) = f(\limit \gamma_\rho(t) x_0) = \limit t^{v_{\gamma_\rho}(f)} u = 0$, where $u \in k[X] \cap \mathcal{O}_{v_{\gamma_\rho}}^\times$, since $v_{\gamma_\rho}(f) > 0$. Therefore $k[X] \cap \mathfrak{m}_{v_{\gamma_\rho}}
\subset \Gamma(X,\mathcal{O}(\overline{G \cdot D_\rho}))$, where both are prime ideals in $k[X]$ and the latter is of height one. Hence they are equal. Therefore, $v_{\gamma_\rho}$ is the valuation of the prime divisor $\overline{G \cdot D_\rho}$ in $X$. Suppose $\rho_1,\rho_2
\in \Gamma_1(X,x_0)$ such that $\gamma_{\rho_1} \not\sim \gamma_{\rho_2}$. Then $v_{\gamma_{\rho_1}} \neq v_{\gamma_{\rho_2}}$, so $\overline{G \cdot D_{\rho_1}} \neq \overline{G \cdot D_{\rho_2}}$. Hence $\Gamma_1(X,x_0)/\sim\, \to \{D_1,D_2,\dots,D_r\}$ is injective. Therefore, $\Gamma_1(X,x_0)/\sim\, \to \{ D_1,D_2,\dots,D_r \}$ is a bijection.
Kempf states {#SS:Gstates}
============
Our sets $\Gamma(X,x_0)$ of one-parameter subgroups associated to an affine $G$-embedding are identical to sets arising in geometric invariant theory as studied by Mumford [@GIT], Kempf [@Kempf] and Rousseau [@Rousseau]. In [@Kempf], Kempf describes such sets in terms of bounded, admissible states as follows.
\[D:Kempfstate\] A *state* $\Xi$ is an assignment of a nonempty subset $\Xi(R) \subset
\mathfrak{X}^*(R)$ to each torus $R$ of $G$ so that the image of $\Xi(R_2)$ in $\mathfrak{X}^*(R_1)$ under the restriction map $\mathfrak{X}^*(R_2) \to
\mathfrak{X}^*(R_1)$ is equal to $\Xi(R_1)$ whenever $R_1 \subset R_2$ are tori of $G$.
For each $k$-point $g$ of $G$, we have maps $g_! : \mathfrak{X}^*(g^{-1}Rg)
\to \mathfrak{X}^*(R)$ defined by $(g_! \chi)(r) = \chi(g^{-1}rg)$ for each torus $R$ of $G$. We define the [*conjugate state*]{} $g_*\Xi$ by the formula $(g_*\Xi)(R) = g_!\Xi(g^{-1}Rg)$ for each torus $R$ of $G$. With this notion in mind, we say that a state $\Xi$ is *bounded* if, for each torus $R$ of $G$, $\bigcup_{g \in G_k} g_*\Xi(R)$ is a finite set of characters of $R$.
Finally, any state defines a function $\mu(\Xi)$ on $\mathfrak{X}_*(G)$ by $$\label{E:mu(Xi)}
\mu(\Xi,\gamma) =
\min_{\chi \in \Xi(\gamma(\mathbb{G}_m))} \langle \chi,\gamma \rangle$$ called the *numerical function* of $\Xi$. We say $\Xi$ is *admissible* if its numerical function satisfies $\mu(\Xi,\gamma) =
\mu(\Xi,p \bullet \gamma)$ for all $p \in P(\gamma)$, where $P(\gamma)$ is the parabolic subgroup of $G$ associated to $\gamma$. We will refer to bounded admissible states as *Kempf states*.
\[R:restrictedstate\] Suppose $H$ is a closed subgroup of a group $G$ and that $\Xi$ is a Kempf state for $G$. Then $\Res^G_H \Xi$, which assigns to any torus $R$ of $H$ the set of characters $\Xi(R)$ (for $R$ is also a torus of $G$), is a Kempf state for $H$, as all of the compatibility conditions are clearly inherited from $G$.
With these terms defined, we return to the situation of an affine $G$-scheme $X$. For each closed $G$-subscheme $Y$ of $X$, define $\Gamma_Y(X,x_0)$ to be the set $\{ \gamma \in \Gamma(X,x_0) : \limit \gamma(t) \cdot x_0
\text{ exists in } Y \}$.
\[T:gammastate\] Let $x_0$ be a $k$-point of an affine $G$-variety $X$. Let $Y$ be a closed $G$-subvariety of $X$ not containing $x_0$. Then there are Kempf states $\Xi_{X,x_0}$ and $\Upsilon^Y_{X,x_0}$ such that
1. $\Gamma(X,x_0) = \{ \gamma \in \mathfrak{X}_*(G) :
\mu(\Xi_{X,x_0},\gamma) \geq 0 \}$,
2. $\Gamma_Y(X,x_0) = \{ \gamma \in \Gamma(X,x_0) :
\mu(\Upsilon^Y_{X,x_0},\gamma) > 0 \}$.
While not including the proof, we indicate the construction of the Kempf state $\Xi_{X,x_0}$ given in [@Kempf]. By the embedding theorem (Lemma 1.1 in [@Kempf]), there is a $G$-representation $V$ and an equivariant closed embedding $X \hookrightarrow V$. Identify $X$ with its image in $V$. Since $\psi_{x_0} : G \to X$ is an isomorphism onto the open orbit $\Omega \subset X$, we may ensure $x_0$ is not zero in $V$. As $X$ is a closed $G$-subvariety of $V$, $\Gamma(X,x_0)
= \Gamma(V,x_0)$, so we may assume $X = V$ is a $G$-representation. We define the state $\Xi_{V,x_0}$ of $x_0$ in the representation $V$ as follows. Let $R$ be a torus of $G$. Let $V = \bigoplus_{\chi \in
\mathfrak{X}^*(R)} V_\chi$ be the eigendecomposition of $V$ with respect to the torus $R$ and let $\text{proj}_{V_\chi}(x_0)$ be the projection of $x_0$ on the weight space $V_\chi$. Set $$\Xi_{V,x_0}(R) =
\{ \chi \in \mathfrak{X}^*(R) : \text{proj}_{V_\chi}(x_0) \neq 0 \},$$ for each torus $R$ in $G$. Then $\Xi_{V,x_0}$ is the Kempf state associated to the set $\Gamma(X,x_0) = \Gamma(V,x_0)$.
Given a Kempf state $\Xi$, define the set $\Xi^\vee \subset \mathfrak{X}_*(G)$ by $$\label{E:XiV}
\Xi^\vee := \{ \gamma \in \mathfrak{X}_*(G) : \mu(\Xi,\gamma) \geq 0 \}.$$ By Theorem \[T:gammastate\], every collection of one-parameter subgroups $\Gamma(X,x_0)$ arising from an affine $G$-embedding is of the form $\Xi^\vee$ for some Kempf state $\Xi$. Furthermore, observe that $\sum g \cdot v_\chi$ is the eigendecomposition of $g \cdot x_0$ with respect to $R$ whenever $\sum v_\chi$ is the eigendecomposition of $x_0$ with respect to $T = g^{-1}Rg$, so $$\label{E:g*Xi}
\Xi_{V,g \cdot x_0}(R) = g_!(\Xi_{V,x_0}(g^{-1}Rg)) = g_*\Xi_{V,x_0}(R)$$ for all $g \in G_k$. Thus, if $\Gamma(X,x_0) = \Xi^\vee$, then $g \bullet \Gamma(X,x_0) = \Gamma(X,g \cdot x_0) = (g_*\Xi)^\vee$.
Therefore, given an affine $G$-embedding $X$ and a choice of base point $x_0 \in \Omega$, we obtain a Kempf state $\Xi_{X,x_0}$ such that $\Gamma(X,x_0) = \{ \gamma \in \mathfrak{X}_*(G) : \mu(\Xi_{X,x_0},\gamma) \geq 0 \}$. For each closed $G$-subvariety $Y$ of $X$, we obtain another Kempf state $\Upsilon^Y_{X,x_0}$ such that $\Gamma_Y(X,x_0) = \{ \gamma \in \Gamma(X,x_0) :
\mu(\Upsilon^Y_{X,x_0},\gamma) > 0 \}$.
Strongly convex lattice cones {#SS:cones}
=============================
One-parameter subgroups $\gamma_1, \gamma_2 \in \mathfrak{X}_*(G)$ are equivalent, as defined in Equation \[E:sim1ps\], if and only if, for all $t \in k^\times$, $$\gamma_2(t^{n_2}) = g \gamma_1(t^{n_1}) g^{-1}$$ for some positive integers $n_1,n_2$ and some element $g \in P(\gamma_1)$. In this case we write $\gamma_1 \sim \gamma_2$.
\[D:scrpc\] We say a subset $\Gamma \subset \mathfrak{X}_*(G)$ is *saturated* (with respect to the equivalence relation (\[E:sim1ps\]) of one-parameter subgroups) if, whenever $\gamma_1 \sim \gamma_2$ and $\gamma_1 \in \Gamma$, then $\gamma_2 \in \Gamma$. $\Gamma$ is called a *lattice cone* of one-parameter subgroups of $G$ if $\Gamma$ is saturated and the quotient $\Gamma_1/\sim$ of the one-skeleton of $\Gamma$ (\[E:Gamma(1)\]) is a finite set. A lattice cone $\Gamma$ is called a *convex lattice cone* if there is a Kempf state $\Xi$ such that $\Gamma = \Xi^\vee$. Additionally, $\Gamma$ is a *strongly convex lattice cone* if it is a convex lattice cone and $\gamma,\gamma^{-1} \in \Gamma$ if and only if $\gamma$ is the trivial one-parameter subgroup of $G$.
The strong convexity condition implies that the elements of $\Xi(R)$ generate $\mathfrak{X}^*(R)$ as a group for all tori $R$ in $G$.
Our terminology is compatible with that of toric geometry. If $\Gamma$ is a strongly convex lattice cone as above, then each $\Gamma \cap \mathfrak{X}_*(T)$ is one in the sense of toric geometry. For if $\Gamma$ is a strongly convex lattice cone, $\Gamma = \{ \gamma \in \mathfrak{X}_*(G) : \mu(\Xi,\gamma) \geq 0 \}$ for some Kempf state $\Xi$. Thus each $\Gamma \cap \mathfrak{X}_*(T)$ is the intersection of finitely many half-spaces, $\Gamma \cap \mathfrak{X}_*(T) = \bigcap_{\chi \in \Xi(T)} \{ v \in \mathfrak{X}_*(T) : \langle \chi,v \rangle \geq 0 \}$, so it is a convex lattice cone in $\mathfrak{X}_*(T)$. Strong convexity follows as the same condition is required of $\Gamma$.
\[L:Gammais\] If $X$ is an affine $G$-embedding with base point $x_0 \in \Omega$, then the set $\Gamma(X,x_0) = \{ \gamma \in \mathfrak{X}_*(G) : \limit \gamma(t) x_0 \text{ exists in } X \}$ is a strongly convex lattice cone.
Suppose $X$ is an affine $G$-embedding and $x_0 \in X$ is a base point. Then $\Gamma(X,x_0) = \{ \gamma \in \mathfrak{X}_*(G) : \limit \gamma(t)x_0 \text{ exists in } X \}$. By Theorem \[T:gammastate\], $\Gamma(X,x_0) = \Xi^\vee$ for some Kempf state. This, together with Proposition \[P:4properties\].d, implies that $\Gamma(X,x_0)$ is saturated. Moreover, $\Gamma_1(X,x_0)/\sim$ is finite by Proposition \[P:Gamma(1)\], so $\Gamma(X,x_0)$ is a convex lattice cone. Therefore, it is a strongly convex lattice cone by Proposition \[P:4properties\].b.
The classification of affine $G$-embeddings {#SS:classification}
===========================================
Suppose $\gamma \in \mathfrak{X}_*(G)$. Recall that $v_\gamma$ is the valuation $\dfrac{1}{n_\gamma}(v_t \circ i_\gamma)$ of $k(G)$, where $i_\gamma : k(G) \to k(G)((t))$ is the injection of fields filling the commutative diagram $$\xymatrix{
k[G] \ar[r]^>>>>>{\mu^\circ} \ar[d]^\subset
&
k[G] \otimes k[G] \ar[rr]^{id_{k[G]} \otimes \gamma^\circ}
&&
k[G] \otimes k((t)) \ar[d]^\subset
\\
k(G) \ar[rrr]_{i_\gamma}
&&&
k(G)((t))
}$$ $v_t : k(G)((t))^\times \to \mathbb{Z}$ is the standard valuation, and $n_\gamma$ is the largest positive integer such that $(v_t \circ i_\gamma)(k(G)^\times) \subset n_\gamma\mathbb{Z}$. Let $\mathcal{O}_{v_\gamma} = \{ f \in k(G) : v_\gamma(f) \geq 0 \}$ be the valuation ring in $k(G)$ associated to $v_\gamma$.
\[L:A(W)\] Let $T$ be a maximal torus of $G$. If $\gamma_1,\gamma_2 \in \mathfrak{X}_*(T) \subset \mathfrak{X}_*(G)$, then $$\label{E:O1capO2inO12}
(k[G] \cap \mathcal{O}_{v_{\gamma_1}}) \cap (k[G] \cap \mathcal{O}_{v_{\gamma_2}})
\subseteq k[G] \cap \mathcal{O}_{v_{\gamma_1+\gamma_2}}.$$
Suppose $\gamma_1,\gamma_2$ are one-parameter subgroups of a maximal torus $T$ of $G$. Then $\gamma_1 + \gamma_2 \in \mathfrak{X}_*(T)$ is also a one-parameter subgroup of $G$ contained in $T$. The valuations $v_{\gamma_1},v_{\gamma_2}$ and $v_{\gamma_1+\gamma_2}$ are obtained from the homomorphisms $i_{\gamma_1},i_{\gamma_2}$ and $i_{\gamma_1+\gamma_2}$, so it is enough to prove that $i_{\gamma_1+\gamma_2}(k[G] \cap \mathcal{O}_{v_{\gamma_1}} \cap \mathcal{O}_{v_{\gamma_2}}) \subset k(G)[[t]]$ to prove the lemma. Yet $$\xymatrix{
k[G] \cap \mathcal{O}_{v_{\gamma_i}} \ar[rrr] \ar[d]^\subset
&&&
k[G] \otimes k[[t]] \ar[d]^\subset
\\
k[G] \ar[r]_{\mu^\circ}
&
k[G] \otimes k[G] \ar[rr]_{id_{k[G]} \otimes \gamma_i^\circ}
&&
k[G] \otimes k((t))
}$$ for $i=1,2$ and $$\xymatrix{
k[G] \ar[r]^{\mu^\circ}
&
k[G] \otimes k[G] \ar[rr]^{id_{k[G]} \otimes (\gamma_1+\gamma_2)^\circ}
\ar[d]_{id_{k[G]} \otimes \mu^\circ}
&&
k[G] \otimes k((t))
\\
&
k[G] \otimes k[G] \otimes k[G] \ar[rr]_{id_{k[G]} \otimes \gamma_1^\circ \otimes \gamma_2^\circ}
&&
k[G] \otimes k((t)) \otimes k((t)) \ar[u]_{id_{k[G]} \otimes m_{23}}
}$$ implies that $i_{\gamma_1 + \gamma_2}(k[G] \cap \mathcal{O}_{v_{\gamma_1}} \cap \mathcal{O}_{v_{\gamma_2}}) \subset k[G] \otimes k[[t]]$. Thus, $(k[G] \cap \mathcal{O}_{v_{\gamma_1}}) \cap (k[G] \cap \mathcal{O}_{v_{\gamma_2}}) \subseteq k[G] \cap \mathcal{O}_{v_{\gamma_1+\gamma_2}}$ as claimed.
\[T:classification\] Affine $G$-embeddings are classified by strongly convex lattice cones of one-parameter subgroups in $\mathfrak{X}_*(G)$, up to conjugation. Conjugation corresponds to the change of base point in the embedding.
Assume $X$ is an affine $G$-embedding. The selection of a base point $x_0$ in $X$ uniquely determines a set $\Gamma(X,x_0) \subset \mathfrak{X}_*(G)$, which is a strongly convex lattice cone by Lemma \[L:Gammais\]. By Theorem \[T:A\_X\], the set $\Gamma(X,x_0)$ determines the affine $G$-embedding $X$ and the selected base point $x_0$ via an isomorphism $\psi_{x_0} : k[X] \cong A_{\Gamma(X,x_0)}$ such that $\mathfrak{m}_{x_0} = \psi_{x_0}^{-1}(\mathfrak{m}_e \cap A_{\Gamma(X,x_0)})$. By Corollary \[C:right-translation\] and formula (\[E:Gamma(X,hx)\]), the selection of a different base point $h \cdot x_0$ is equivalent to conjugating the cone $\Gamma(X,h \cdot x_0) = h \Gamma(X,x_0) h^{-1}$ and a corresponding right translation of the algebra $A_{\Gamma(X,h \cdot x_0)} = r_h(A_{\Gamma(X,x_0)})$. Thus each affine $G$-embedding determines a strongly convex lattice cone, modulo conjugation in $\mathfrak{X}_*(G)$, which in turn recovers the variety up to isomorphism.
Conversely, we prove that every strongly convex lattice cone $\Gamma \subset
\mathfrak{X}_*(G)$ determines an affine $G$-embedding $X_\Gamma$ such that $\Gamma(X_\Gamma,x_0) = \Gamma$ for some choice of base point $x_0 \in X_\Gamma$. Assume $\Gamma$ is a strongly convex lattice cone, so $\Gamma = \Xi^\vee$ for some Kempf state $\Xi$. Define $A_\Gamma$ as in (\[E:A\_X\]) to be the subalgebra of $k[G]$ given by $$A_\Gamma = \{ f \in k[G] : v_\gamma(f) \geq 0, \text{ for all } \gamma
\in \Gamma \} = k[G] \cap \bigcap_{\gamma \in \Gamma} \mathcal{O}_{v_\gamma},$$ where $\mathcal{O}_v$ is the valuation subring $\{ f \in k(G) : v(f) \geq 0 \}$ of $k(G)$. Using this second description, since $k[G]$ and all of the valuation rings $\mathcal{O}_{v_\gamma}$ are integrally closed in $k(G)$, their intersection $A_\Gamma$ is an integrally closed domain. Furthermore, $A_\Gamma$ is left-invariant, since $k[G]$ is and $v_\gamma(g \cdot f) = v_\gamma(f)$ for all $f \in k(G)$ and $g \in G$ implies that the valuation rings $\mathcal{O}_{v_\gamma}$ are $G$-stable as well. Hence $A_\Gamma$ is an integrally closed left-invariant subalgebra of $k[G]$. It remains to prove that $A_\Gamma$ is finitely generated over $k$ and that the variety $X_\Gamma = \spec A_\Gamma$ contains $G$ as an open orbit.
Let $\gamma_1,\dots,\gamma_m$ be a set of representatives for the equivalence classes in $\Gamma_1/\sim$, which is a finite set as $\Gamma$ is a lattice cone. By Proposition \[P:Gamma(1)\] and Lemma \[L:A(W)\], $$A_\Gamma
= k[G] \cap \bigcap_{\gamma \in \Gamma} \mathcal{O}_{v_\gamma}
= \bigcap_{\gamma \in \Gamma} (k[G] \cap \mathcal{O}_{v_\gamma})
= \bigcap_{i=1}^m (k[G] \cap \mathcal{O}_{v_{\gamma_i}})$$ since any $\gamma \in \Gamma$ belongs to some $\mathfrak{X}_*(T)$, in which case it is a sum of elements from $[\Gamma \cap \mathfrak{X}_*(T)](1)$. Let $f_1,\dots,f_n$ be a finite set of generators for the algebra $k[G]$ over $k$, so $k[G] = k[f_1,\dots,f_n]$. By Gordan’s Lemma, the set $$C = \{ (c_\ell) \in \mathbb{Z}^n : c_1,\dots,c_n \geq 0,
\sum_{j=1}^n c_j v_{\gamma_i}(f_j) \geq 0 \text{ for all } i = 1,\dots,m \}$$ is finitely generated. Thus the “monomials” $f_1^{c_1} f_2^{c_2} \cdots f_n^{c_n}$ in the generators $f_1,\dots,f_n$ of $k[G]$ associated to the generators $(c_1,\dots,c_n)$ of $C$ admit a finite set of generators of $A_\Gamma$, as any element of $A_\Gamma$ is an algebraic combination of such monomials.
Therefore $X_\Gamma = \spec A_\Gamma$ is a normal affine $G$-variety with an open $G$-orbit, since $A_\Gamma \subset k[G]$ implies $f : G \to X_\Gamma$ is dominant, so $f(G)$ is an open orbit in $X_\Gamma$ ([@Springer], Theorem 1.9.5). It only remains to show that $f(G)$ is isomorphic to $G$ as an orbit in $X_\Gamma$. Consider, for each torus $R$ of $G$, the image of $A_\Gamma$ under the homomorphism $k[G] \to k[R]$. By construction of $A_\Gamma = \{ f \in k[G] : v_\gamma(f) \geq 0 \text{ for all } \gamma
\in \Gamma \}$ and the decomposition $A_\Gamma = \bigoplus_{\chi \in \Gamma^\vee(R)}
A^R_\chi$, where $A^R_\chi = \{ f \in A : f(xr) = \chi(r)f(x) \text{ for all } r \in R \}$, it is evident that the image of $A_\Gamma$ in $k[R]$ is the $k$-monoid algebra $k[\Gamma^\vee(R)]$. Hence we have the following commutative diagrams: $$\xymatrix{
A_\Gamma \ar[rr]^\subset \ar[d]_{\text{onto}}
&&
k[G] \ar[d]^{\text{onto}}
&
X_\Gamma
&&
G \ar[ll]_f
\\
k[\Gamma^\vee(R)] \ar[rr]_{\subset}
&&
k[R] = k[\mathfrak{X}^*(R)]
&
R_{\Gamma^\vee(R)} \ar[u]^{\text{closed}}
&&
R \ar[u]_{\text{closed}} \ar[ll]^{\text{open}}
}$$ Since $\Gamma$ is strongly convex, $\Gamma^\vee(R)$ is not contained in any hyperplane, so the monoid $\Gamma^\vee(R)$ generates $\mathfrak{X}^*(R)$ as a group. This implies that $R$ is isomorphic to an open subset of the closed subvariety $R_{\Gamma^\vee(R)} =
\spec k[\Gamma^\vee(R)]$ in $X_\Gamma$, so that $f(R) \cong R$ for every torus $R$ of $G$. Thus $f(\bigcup_T T) = \bigcup_T f(T) \cong \bigcup_T T \subset X_\Gamma$. Yet $\bigcup_T T$ contains a dense open subset $\mathcal{O}$ of $G$ ([@Springer], Theorem 6.4.5 and Corollary 7.6.4) and $f|_{\bigcup_T T}$ is an isomorphism, so $\mathcal{O} \cong f(\mathcal{O})$ is open in $X_\Gamma$. Thus $f : G \to X_\Gamma$ is a birational morphism [@Hart]. As $f(G)$ is an open orbit in $X_\Gamma$ containing an open subset birationally equivalent to $G$, $f(G)$ is isomorphic to $G$ and $f : G \to X_\Gamma$ is an affine $G$-embedding.
Finally, consider $\Gamma(X_\Gamma,f(e)) = \{ \gamma \in \mathfrak{X}_*(G) :
\limit \gamma(t) f(e) \text{ exists in } X_\Gamma \} = \{ \gamma \in
\mathfrak{X}_*(G) : \gamma^\circ(A_\Gamma) \subset k[t] \}$, that is, the dual map $\gamma^\circ : A_\Gamma \subset k[G] \to k[t,t^{-1}]$ factors through $k[t]$. Thus, if $\gamma \in \Gamma$, so $v_\gamma(f) \geq 0$ for all $f \in A_\Gamma$, then $\gamma^\circ(A_\Gamma) \subset k[t]$, and hence $\Gamma \subset \Gamma(X_\Gamma,f(e))$. Now let $\gamma \in \Gamma(X_\Gamma,f(e))$. Suppose $\gamma$ is a one-parameter subgroup of a torus $T$ of $G$. Then $\limit \gamma(t) f(e)$ exists in the toric variety $\overline{T} \subset X_\Gamma$, so the classification of toric varieties implies that $\gamma \in \Gamma \cap \mathfrak{X}_*(T)$, and hence that $\Gamma(X_\Gamma,f(e)) \subset \Gamma$. Therefore, $X_\Gamma = \spec A_\Gamma$ is a normal affine $G$-embedding such that $\Gamma(X_\Gamma,f(e)) = \Gamma$, which completes our proof.
In the next sections, we discuss how our classification in Theorem \[T:classification\] also describes equivariant morphisms between affine $G$-embeddings. In particular, we show that equivariant maps between affine $G$-embeddings correspond to inclusions of the associated strongly convex lattice cones and conversely in Section \[SS:morphisms\]. After that, we characterize biequivariant affine $G$-embeddings in terms of their cones and, using this result, construct the *biequivariant resolution* of an affine $G$-embedding in Section \[SS:biequivariance\]. We remark here that Proposition \[P:biequivariance\] below may be seen as an affine version of Brion’s classification [@B98] of regular $G$-compactifications.
Equivariant morphisms between affine $G$-embeddings {#SS:morphisms}
===================================================
Suppose $f : X \to Y$ is an equivariant morphism between affine $G$-embeddings $X$ and $Y$. By equivariance, if $x_0 \in X$ is a base point for $X$, then $y_0 =
f(x_0)$ is a base point for $Y$. Moreover, if $\gamma$ is a one-parameter subgroup of $G$ such that $\limit \gamma(t) x_0 = x_\gamma$ exists in $X$, then $\limit \gamma(t) y_0$ exists in $Y$ and is equal to $f(x_\gamma)$ since $f$ is continuous and $f(\gamma(t) x_0) = \gamma(t) f(x_0) = \gamma(t) y_0$ for all $t \neq 0$. Therefore, there is an inclusion $\Gamma(X,x_0) \subset
\Gamma(Y,f(x_0))$ whenever there exists an equivariant morphism $f : X \to Y$ of affine $G$-embeddings. Moreover, $f$ is the morphism dual to the inclusion of the subalgebras $A_{\Gamma(X,x_0)} \subset A_{\Gamma(Y,f(x_0))}$ in $k[G]$, because $f(g \cdot x_0) = g \cdot f(x_0)$ and $Gx_0 = \Omega_X$ is open in $X$ implies that $f$ is uniquely determined by its value at $x_0$.
\[R:nonaffinemorphism\] Suppose $X$ and $Y$ are $G$-embeddings, not necessarily affine. If $f : X \to Y$ is an equivariant morphism and if $x_0$ is a base point for $X$ (i.e., the orbit $Gx_0$ is isomorphic to $G$ as $G$-varieties), then $f(x_0)$ will be a base point for $Y$ and we can define the sets $\Gamma(X,x_0)$ and $\Gamma(Y,f(x_0))$ in the same manner as for affine $G$-embeddings. By the same argument as above, the existence of the equivariant morphism $f : X \to Y$ implies that $\Gamma(X,x_0)
\subset \Gamma(Y,f(x_0))$.
Conversely, suppose $\Gamma(X,x_0) \subset \Gamma(Y,y_0)$ for two affine $G$-embeddings $X$ and $Y$. By Theorem \[T:A\_X\], $X = \spec A_{\Gamma(X,x_0)}$ and $Y =
\spec A_{\Gamma(Y,y_0)}$. The definition of $A_\Gamma = \{ f \in k[G] : v_\gamma(f)
\geq 0 \text{ for all } \gamma \in \Gamma \}$ implies that $A_{\Gamma(Y,y_0)}
\subseteq A_{\Gamma(X,x_0)}$, so there is a corresponding equivariant morphism of $G$-embeddings $X \to Y$ sending $x_0 \mapsto y_0$. However, by Corollary \[C:right-translation\], the subalgebra of $k[G]$ isomorphic to $k[X_\Gamma]$ is only determined up to right translations, which correspond to conjugates of the cone $\Gamma$. Thus,
\[P:morphisms\] Suppose $X_1,X_2$ are affine $G$-embeddings and $\Gamma_1,\Gamma_2$ are strongly convex lattice cones.
1. If $x_1 \in X_1$ is a base point and $f : X_1 \to X_2$ is an equivariant morphism of $G$-embeddings, then $\Gamma(X_1,x_1) \subset \Gamma(X_2,f(x_1))$ and $f$ is the morphism recovered from the inclusion: $$\xymatrix{
k[X_1] \ar[r]^\cong_{\psi_{x_1}^\circ}
&
A_{\Gamma(X_1,x_1)}
\\
k[X_2] \ar[r]^\cong_{\psi_{f(x_1)}^\circ} \ar[u]^{f^\circ}
&
A_{\Gamma(X_2,x_2)} \ar[u]_\subset
}$$
2. If there is an element $h \in G$ such that $\Gamma_1 \subset h \Gamma_2 h^{-1}$, then there is an equivariant morphism $X_{\Gamma_1} \to X_{\Gamma_2}$ sending the base point $x_1 \in X_{\Gamma_1}$ to $h \cdot x_2 \in X_{\Gamma_2}$, where $\mathfrak{m}_{x_i} = A_{\Gamma_i} \cap \mathfrak{m}_e$ for $i=1,2$.
We have already proven part 1 of this Lemma in the discussion prior to Remark \[R:nonaffinemorphism\].
To prove part 2, by Theorem \[T:classification\], we know that $\Gamma_1,\Gamma_2$ correspond to affine $G$-embeddings $X_1 = \spec A_{\Gamma_1}$ and $X_2 =
\spec A_{\Gamma_2}$, respectively, such that $\Gamma_i = \Gamma(X_i,x_i)$ for $i = 1,2$, where $x_i$ is the base point of $X_i$ corresponding to the maximal ideal $\mathfrak{m}_e
\cap A_{\Gamma_i}$. Now suppose there is an element $h \in G$ such that $\Gamma_1 \subset
h \Gamma_2 h^{-1}$ as in the statement of the lemma. We know that $h \Gamma_2 h^{-1} =
h \Gamma(X_2,x_2) h^{-1} = \Gamma(X,h \cdot x_2)$ by formula (\[E:Gamma(X,hx)\]). Hence we have $\Gamma(X_1,x_1) \subset \Gamma(X_2,h \cdot x_2)$, so that $A_{\Gamma(X_1,x_1)} \supset A_{\Gamma(X_2,h \cdot x_2)}$ and thus $X_1 \to X_2$ exists, is equivariant, and maps $x_1 \mapsto h \cdot x_2$ as claimed.
\[EG:Gamma(GbarT)inGamma(X)\] If $X$ is an affine $G$-embedding and $T$ is a maximal torus of $G$ whose closure in $X$ is denoted $\overline{T}$, then $G \times^T \overline{T}$ is an affine $G$-embedding and we have an equivariant morphism $G \times^T \overline{T} \to X$. Let $\Gamma = \Gamma(X,x_0)$ and $\sigma = \Gamma(X,x_0) \cap \mathfrak{X}_*(T)$. Then $$\Gamma(G \times^T \overline{T},[e,x_0])
= \bigcup_{P \supset T} P \bullet (\sigma \cap \Delta_P(T)) ,$$ where the union is taken over all parabolic subgroups of $G$ containing $T$ and $\Delta_P(T)
= \{ \gamma \in \mathfrak{X}_*(T) : P(\gamma) \supseteq P \}$. There are only finitely many parabolic subgroups $P$ containing $T$, as there are only a finite number of Borel subgroups containing $T$ (in one-to-one correspondence with the elements of the Weyl group $W(T,G)$, \[[@Springer], Corollary 6.4.12\]), and, for each Borel subgroup $B$, there are only finitely many parabolic subgroups $P \supset B$ (indexed by subsets of the basis of positive roots of $T$ relative to $B$, \[[@Springer], Theorem 8.4.3\]). In contrast, $\Gamma = \bigcup_{Q \subset G} Q \bullet (\Gamma \cap \Delta_Q(G))$, where the union is taken over the set of all parabolic subgroups $Q$ of $G$ (there are infinitely many, as there are infinitely many Borel subgroups each corresponding to cosets in $G/B$, which is projective) and $\Delta_Q(G) = \{ \gamma \in \mathfrak{X}_*(G) : P(\gamma) \supseteq Q \}$. Clearly, for each $P \supset T$, $$\sigma \cap \Delta_P(T)
= (\Gamma \cap \mathfrak{X}_*(T)) \cap \Delta_P(T)
= \Gamma \cap \Delta_P(T)
= \Gamma \cap \Delta_P(G)$$ since $\Delta_P(G) \subset \mathfrak{X}_*(T')$ for all maximal tori $T'$ contained in $P$. Therefore, $\Gamma(G \times^T \overline{T},[e,x_0])$ is a finite union of the “parabolic components” of $\Gamma(X,x_0)$, so $\Gamma(G \times^T \overline{T},[e,x_0])$ is a finite polysimplicial subcomplex of $\Gamma(X,x_0)$, as the $\Delta_Q(G)$ provide a triangulation of $\mathfrak{X}_*(G)$ [@GIT].
Biequivariant resolutions {#SS:biequivariance}
=========================
In this section, we show that every affine $G$-embedding $X$ canonically determines a $(G \times G)$-equivariant affine $G$-embedding $X_G$ together with a left-$G$-equivariant morphism $X_G \to X$, which we call the biequivariant resolution of $X$. As we will be working with varieties some of which only have a left action and others both a left and a right action, we will be careful to specify how $G$ acts on varieties discussed in this section.
Our first tool is the following proposition, which allows us to detect when an affine $G$-embedding also has a right-$G$-action $X \times G \to X$ extending the multiplication in $G$.
\[P:biequivariance\] An affine $G$-embedding $X$ will have both a left and a right $G$-action, and thus be a biequivariant $G$-embedding, if and only if the associated strongly convex lattice cone $\Gamma(X,x_0)$, for any choice of base point $x_0 \in X$, is $G$-stable for the conjugation action of $G$ on $\mathfrak{X}_*(G)$.
Suppose that $X$ is a $(G \times G)$-equivariant affine $G$-embedding and let $x \in X$ be a base point. Then $X$ is determined by the strongly convex lattice cone $\Gamma(X,x)
= \{ \gamma \in \mathfrak{X}_*(G) : \limit \gamma(t) x \text{ exists in } X \}$. Let $h \in G$ and recall that $h \Gamma(X,x) h^{-1} = \Gamma(X,h \cdot x)$ by formula (\[E:Gamma(X,hx)\]). Thus it is enough to show that $\gamma \in \Gamma$ if and only if $\gamma \in \Gamma(X,h \cdot x)$. Assume $\limit \gamma(t) x$ exists in $X$. Then consider $\limit [ \gamma(t) \cdot hx ] = \limit [ \gamma(t) \cdot xh' ]
= [ \limit \gamma(t) \cdot x ] \cdot h'$, for some $h' \in G$, and this limit exists in $X$ by Remark \[R:bi-limits\], since there is a right $G$-action on $X$. Thus $\Gamma(X,x) \subset h \Gamma(X,x) h^{-1}$. Now assume that $\gamma' \in
\Gamma(X,h \cdot x)$. Then, the same argument implies $\gamma' \in \Gamma(X,x)$ using Remark \[R:bi-limits\]. Therefore, for every $h \in G$, $\Gamma(X,x) = h \Gamma(X,x)
h^{-1}$. Thus $\Gamma(X,x)$ is $G$-stable for the conjugation action of $G$ on $\mathfrak{X}_*(G)$ for any choice of base point $x \in X$.
Now assume that $\Gamma$ is a strongly convex lattice cone which is $G$-stable for the conjugation action of $G$ on $\mathfrak{X}_*(G)$. Theorem \[T:classification\] implies that $\Gamma = \Gamma(X_\Gamma,x_0)$ for some base point $x_0 \in X_\Gamma =
\spec A_\Gamma$. Then, by Corollary \[C:right-translation\], for any $h \in G$ we have $r_h(A_{\Gamma(X_\Gamma,x_0)}) = A_{\Gamma(X_\Gamma,h \cdot x_0)} =
A_{h \Gamma(X_\Gamma,x_0) h^{-1}} = A_\Gamma$. Hence $A_\Gamma$ is a left- and right-$G$-invariant subalgebra of $k[G]$, so there is a right $G$-action on $X_\Gamma = \spec A_\Gamma$, which extends the multiplication of $G$ by [@Springer], Proposition 2.3.6. Thus $X_\Gamma$ is a $(G \times G)$-equivariant affine $G$-embedding.
\[C:biequivariance\] If $X$ is a biequivariant affine $G$-embedding and $T$ is any maximal torus of $G$, then the closure of $T$ in $X$ determines $X$ completely.
Let $X$ be a biequivariant affine $G$-embedding with lattice cone $\Gamma = \Gamma(X,x_0)$ for some choice of base point in $X$. Let $T$ be any maximal torus of $G$. Consider $\overline{T} \subset X$, which is an affine toric variety for $T$. Let $\sigma \subset
\mathfrak{X}_*(T)$ be the cone of one-parameter subgroups of $T$ with specializations in $\overline{T}$. Then $\sigma = \Gamma \cap \mathfrak{X}_*(T)$, by definition of $\Gamma$. If $T'$ is any other maximal torus of $G$, then $T' = gTg^{-1}$ for some $g \in G$ and $\Gamma \cap \mathfrak{X}_*(T') = g[g^{-1}(\Gamma \cap \mathfrak{X}_*(T'))g]g^{-1} =
g[ g^{-1} \Gamma g \cap g^{-1} \mathfrak{X}_*(T') g ] g^{-1} = g[ \Gamma \cap
\mathfrak{X}_*(T) ]g^{-1} = g \sigma g^{-1}$. Thus $\Gamma = \bigcup_{g \in G} g \sigma
g^{-1}$ is determined by the cone $\sigma$, so $X$ is determined by its toric subvariety $\overline{T} = T_\sigma$.
We note the similarity between our classification of biequivariant affine $G$-embeddings and Brion’s classification of regular $G$-compactifications. In [@B98], Brion classified regular compactifications of a group $G$ by the $W(T,G)$-invariant fan associated to the closure of any maximal torus $T$ of $G$ in the compactification, demonstrating that regular compactifications of $G$ are completely determined by any of the associated toric subvarieties. Likewise, Corollary \[C:biequivariance\] implies that if an affine $G$-embedding $X$ is $(G \times G)$-equivariant and $T$ is a maximal torus of $G$, then the cone for $\overline{T}$ recovers $X$. Therefore, we may think of Proposition \[P:biequivariance\] and Corollary \[C:biequivariance\] as an affine version of Brion’s classification.
\[R:basepointfree\] Suppose $X$ is a $(G \times G)$-equivariant affine $G$-embedding and suppose that $x_0 \in
X$ is a base point. By Proposition \[P:biequivariance\] above, the set $\Gamma(X,x_0)$ is stable under the conjugation action of $G$. However, by formula (\[E:Gamma(X,hx)\]), we conclude that $\Gamma(X,x_0) = h\Gamma(X,x_0)h^{-1} = \Gamma(X,h \cdot x_0)$ for all $h \in G$. Therefore, the strongly convex lattice cone associated to a biequivariant affine $G$-embedding is independent of the choice of base point.
We use these observations to construct the biequivariant resolution of an affine $G$-embedding $X$. Suppose $X$ is an affine $G$-embedding, $x_0 \in X$ is a base point, and $\Gamma = \Gamma(X,x_0)$ is the corresponding strongly convex lattice cone. Then there is a unique maximal $G$-stable subset $$\label{E:Gamma^G}
\Gamma^G := \bigcap_{h \in G} h \Gamma h^{-1}$$ of $\Gamma$. In fact,
\[L:Gamma\^G\] If $\Gamma$ is a strongly convex lattice cone in $\mathfrak{X}_*(G)$, then $\Gamma^G$ is a strongly convex lattice cone.
Let $\Gamma$ be a strongly convex lattice cone. Let $\Xi$ be a Kempf state such that $\Gamma = \Xi^\vee$. Define $\Gamma^G = \bigcap_{h
\in G} h \Gamma h^{-1}$ as in (\[E:Gamma\^G\]). We claim that $G_*\Xi : R \mapsto \bigcup_{g \in G} g_*\Xi(R)$ is a Kempf state and that $\Gamma^G = (G_*\Xi)^\vee$.
First, $G_*\Xi$ is a state, for each $g_*\Xi$ is a state implies that $G_*\Xi(R)$ is nonempty for all $R$ and that if $S \supset R$, then $G_*\Xi(R) =
\bigcup_{g \in G} g_*\Xi(R) = \bigcup_{g \in G} \Res^S_R[g_*\Xi(S)] =
\Res^S_R\left[\bigcup_{g \in G} g_*\Xi(S) \right] = \Res^S_R[ G_*\Xi(S) ]$. Next, $G_*\Xi$ is bounded, for if $h \in G$, then $h_*[G_*\Xi](R) = \bigcup_{g \in G}
h_*[g_*\Xi(R)] = \bigcup_{g \in G} (hg)_*\Xi(R) = G_*\Xi(R)$, so $\bigcup_{h \in G}
h_*[G_*\Xi](R) = G_*\Xi(R) = \bigcup_{g \in G} g_*\Xi(R)$ is a finite subset of $\mathfrak{X}^*(R)$ for each $R$, since $\Xi$ is a bounded state. Consider the numerical function $\mu(G_*\Xi)$. If $\gamma \in
\mathfrak{X}_*(G)$, then $\mu(G_*\Xi,\gamma) = \min_{\chi \in
G_*\Xi(\gamma)} \langle \chi,\gamma \rangle = \min_{\chi \in \bigcup
g_*\Xi(\gamma)} \langle \chi,\gamma \rangle = \min_{g \in G}
\min_{\chi \in g_*\Xi(\gamma)} \langle \chi,\gamma \rangle =
\min_{g \in G} \mu(g_*\Xi,\gamma)$. Therefore, $$\label{E:muG*Xi}
\mu(G_*\Xi,\gamma) = \min_{g \in G} \mu(g_*\Xi,\gamma).$$ Each $g_*\Xi$ is a Kempf state, hence admissible. Hence, if $\gamma$ is a one-parameter subgroup of $G$ and $p \in P(\gamma)$ belongs to the parabolic subgroup associated to $\gamma$, then $\mu(G_*\Xi,p
\bullet \gamma) = \min_{g \in G} \mu(g_*\Xi,p \bullet \gamma) =
\min_{g \in G} \mu(g_*\Xi,\gamma) = \mu(G_*\Xi,\gamma)$. Thus $G_*\Xi$ is admissible, so $G_*\Xi$ is a Kempf state. Moreover, (\[E:muG\*Xi\]) implies $\gamma \in (G_*\Xi)^\vee$ if and only if $\mu(g_*\Xi,\gamma) \geq 0$ for all $g \in G$. But $\{ \gamma \in
\mathfrak{X}_*(G) : \mu(g_*\Xi,\gamma) \geq 0 \} = (g_*\Xi)^\vee = g
(\Xi^\vee) g^{-1} = g \Gamma g^{-1}$ by (\[E:g\*Xi\]), so $(G_*\Xi)^\vee = \Gamma^G$ as claimed.
Thus $\Gamma^G = \{ \gamma \in \mathfrak{X}_*(G) : \mu(G_*\Xi,\gamma)
\geq 0 \}$, which implies $\gamma \in \Gamma^G$ if and only if $\gamma^n \in \Gamma^G$ for every positive integer $n > 0$, as $\mu(G_*\Xi,\gamma^n) = n \mu(G_*\Xi,\gamma)$. We use this condition to demonstrate that $\Gamma^G$ is saturated with respect to the equivalence relation $\sim$. Suppose $\gamma \in \Gamma^G$ and $\delta \sim \gamma$. Then there are positive integers $m,n > 0$ and an element $g \in P(\gamma)$ such that $\delta^m = g \gamma^n g^{-1}$. Yet $\gamma \in \Gamma^G$ implies that $\gamma^n \in \Gamma^G$, for $n
> 0$. Hence $\delta^m = g \gamma^n g^{-1} \in g \Gamma^G g^{-1} = g
\left(\bigcap_{h \in G} h \Gamma h^{-1} \right) g^{-1} = \Gamma^G$. Since $m > 0$, this implies that $\delta \in \Gamma^G$ as well, so that $\Gamma^G$ is saturated.
To prove that $\Gamma^G$ is a convex lattice cone, it is left to show that the set $\Gamma^G_1/\sim$ is finite. This is clear since $\Gamma^G_1 \subset \Gamma_1$ and $\Gamma$ is a convex lattice cone.
Finally, as $\Gamma^G \subset \Gamma$, it is clear that $\gamma,\gamma^{-1} \in \Gamma^G$ implies that $\gamma = \varepsilon$ is the trivial one-parameter subgroup of $G$. Furthermore, $\varepsilon \in h \Gamma h^{-1}$ for all $h \in G$, so $\varepsilon
\in \Gamma^G$. Thus $\gamma, \gamma^{-1} \in \Gamma^G$ if and only if $\gamma = \varepsilon$. Therefore, $\Gamma^G$ is a strongly convex lattice cone in $\mathfrak{X}_*(G)$.
Let $X$ be an affine $G$-embedding and select a base point $x_0 \in X$. Let $\Gamma = \Gamma(X,x_0)$ and define $\Gamma^G = \bigcap_{h \in G} h \Gamma h^{-1}$, which is a strongly convex lattice cone. Then $X_G := \spec A_{\Gamma^G}$ is a $(G \times G)$-equivariant affine $G$-embedding by Theorem \[T:classification\] and Proposition \[P:biequivariance\], and there is a left-$G$-equivariant morphism $\beta_{(X,x_0)} : X_G \to X$ corresponding to the inclusion $A_{\Gamma(X,x_0)}
\subset A_{\Gamma^G}$ of subalgebras in $k[G]$.
\[D:biresolution\] We call $X_G$ together with the left-$G$-equivariant morphism $\beta_{(X,x_0)} : X_G \to X$ the *biequivariant resolution* of $X$.
We make a few remarks about the biequivariant resolution of an affine $G$-embedding before we prove its universal property. First, if $X$ is already a $(G \times
G)$-equivariant affine $G$-embedding, then $\Gamma = \Gamma^G$, so $X_G = X$. However, it is possible in some cases that $X_G = G$ even when $X \neq G$.
Second, while the cone $\Gamma^G$ is canonical, the morphism $\beta_{(X,x_0)} :
X_G \to X$ is only defined up to right translation, which corresponds to a different choice of base point $x \in X$ as follows: $r_h(\beta_{(X,x_0)}) = \beta_{(X,h \cdot
x_0)} : X_G \to X$. This is true because the identification of $k[X]$ with a subalgebra of $k[G]$ is determined by the selection of a base point and changing the base point corresponds to right translation of the subalgebra in $k[G]$ by Corollary \[C:right-translation\]. Then the morphism $\beta_{(X,h \cdot x_0)} : X_G
\to X$ is defined by the inclusion of $A_{\Gamma(X,h \cdot x_0)} = r_h(A_{\Gamma(X,x_0)})
\subset A_{\Gamma^G}$, from which it is clear that $\beta_{(X,h \cdot x_0)} =
r_h(\beta_{(X,x_0)})$.
\[T:universalbi\] Suppose $X$ is a left-$G$-equivariant affine $G$-embedding, $x_0 \in X$ is a base point, $Y$ is any $(G \times G)$-equivariant affine $G$-embedding, and $\varphi : Y \to X$ is a left-$G$-equivariant morphism. Then there is a unique $(G \times G)$-equivariant morphism $\varphi_{(X_G,x_0)} : Y \to X_G$ such that $\varphi = \beta_{(X,x_0)} \circ
\varphi_{(X_G,x_0)}$. That is, the biequivariant resolution of $X$ satisfies the following diagram: $$\xymatrix{
Y \ar[rr]^{\forall \, \varphi} \ar@{-->}[dr]_{\exists! \, \varphi_{(X_G,x_0)}}
&&
X
\\
&
X_G \ar[ur]_{\beta_{(X,x_0)}}
}$$ If $x_1 = h \cdot x_0$ is another base point for $X$, then $\beta_{(X,x_1)} =
r_h(\beta_{(X,x_0)})$ and $\varphi_{(X_G,x_1)} = r_{h^{-1}}(\varphi_{(X_G,x_0)})$.
Let $X$ be an affine $G$-embedding and identify $k[X]$ with the left-invariant subalgebra $A_{\Gamma(X,x_0)}$ of $k[G]$ by selecting a base point $x_0 \in X$. Suppose that $Y$ is a $(G \times G)$-equivariant affine $G$-embedding and that $\varphi : Y \to X$ is a left-$G$-equivariant morphism from $Y$ to $X$. By equivariance, there is a unique element $y_0 \in Y$ such that $\varphi(y_0) = x_0$, since $\varphi|_{\Omega_Y} :
\Omega_Y \to \Omega_X$ is an isomorphism with $\Omega_Y \cong \Omega_X \cong G$. Then the cone $\Gamma(Y,y_0)$ is $G$-stable by Proposition \[P:biequivariance\] and is a subset of $\Gamma(X,x_0)$ by Proposition \[P:morphisms\]. Using $y_0$, identify $k[Y]$ with the $(G \times G)$-invariant subalgebra $A_{\Gamma(Y,y_0)}$ of $k[G]$. By Corollary \[C:right-translation\], $A_{\Gamma(Y,h \cdot y_0)} = r_h(A_{\Gamma(Y,y_0)})
= A_{\Gamma(Y,y_0)}$, for all $h \in G$, so the algebra $A_{\Gamma(Y,y_0)}$ is independent of the choice of base point. Hence it is the only subalgebra of $k[G]$ which is isomorphic to $k[Y]$ as a $(G \times G)$-algebra by Theorem \[T:classification\].
Clearly $\Gamma(Y,y_0) = \bigcap_{h \in G} h \Gamma(Y,y_0) h^{-1} \subset
\bigcap_{h \in G} h \Gamma(X,x_0) h^{-1} = \Gamma(X,x_0)^G \subset \Gamma(X,x_0)$, so that $A_{\Gamma(X,x_0)} \subset A_{\Gamma(X,x_0)^G} \subset A_{\Gamma(Y,y_0)}$, as indicated in the diagram below. Moreover, as $\Gamma(X,x_0)^G$ is $G$-stable, applying Corollary \[C:right-translation\] again shows that $A_{\Gamma(X,x_0)^G}$ is a uniquely determined subalgebra of $k[G]$ which is independent of the choice of base point $x_0 \in X$ made above. Thus we have the following diagram of algebras: $$\xymatrix{
&
k[G]
&
\\
k[Y] \ar[r]^\cong_{\psi_{y_0}^\circ}
&
A_{\Gamma(Y,y_0)} \ar[u]_\subset
&
\\
&
&
A_{\Gamma(X,x_0)^G} = k[X_G] \ar[ul]_\subset
\\
k[X] \ar[uu]^{\varphi^\circ} \ar[r]^\cong_{\psi_{x_0}^\circ}
&
A_{\Gamma(X,x_0)} \ar[uu]^\subset \ar[ur]_\subset
}$$ The morphism $\varphi_{(X_G,x_0)} : Y \to X_G$ corresponds to the restriction of the homomorphism $(\psi_{y_0}^\circ)^{-1}$ from $k[X_G] = A_{\Gamma(X,x_0)^G} \to k[Y]$, while $\beta_{(X,x_0)} : X_G \to X$ is dual to the composition of $\psi_{x_0}^\circ :
k[X] \to A_{\Gamma(X,x_0)}$ followed by the canonical inclusion $A_{\Gamma(X,x_0)}
\subset A_{\Gamma(X,x_0)^G}$. Then it is clear that $\varphi$ factors as $\varphi =
\beta_{(X,x_0)} \circ \varphi_{(X_G,x_0)}$.
Now suppose that $x_1 = h \cdot x_0$ is another base point in $X$. Then $h \cdot y_0$ is the unique element of $Y$ such that $\varphi(y) = x_1$, for $\varphi(h \cdot y_0)
= h \cdot \varphi(y_0) = h \cdot x_0$. Then $\psi_{x_1}^\circ$ is an isomorphism from $k[X]$ to $A_{\Gamma(X,h \cdot x_0)} = A_{h \Gamma(X,x_0) h^{-1}} =
r_h(A_{\Gamma(X,x_0)}) \subset k[G]$. Yet $\Gamma(Y,h \cdot y_0) = \Gamma(Y,y_0)
\subset \Gamma(X,x_0)^G = \Gamma(X,h \cdot x_0)^G \subset \Gamma(X,h \cdot x_0)$, so we have $\varphi^\circ = (\psi_{h \cdot y_0}^\circ)^{-1} \circ \psi_{h \cdot x_0}^\circ
: k[X] \to k[Y]$. As above, this factors through the inclusions $A_{\Gamma(X,h \cdot
x_0)} \subset k[X_G] = A_{\Gamma(X,h \cdot x_0)^G} \subset A_{\Gamma(Y,h \cdot y_0)}$, giving morphisms $\varphi_{(X_G,h \cdot x_0)} : Y \to X_G$ and $\beta_{(X,h \cdot x_0)}
: X_G \to X$ corresponding to $(\psi_{h \cdot y_0}^\circ)^{-1} =
[r_h(\psi_{y_0}^\circ)]^{-1} = r_{h^{-1}}[(\psi_{y_0}^\circ)^{-1}]$ and $\psi_{h \cdot x_0}^\circ = r_h(\psi_{x_0}^\circ)$, respectively. Hence $\varphi_{(X_G,h \cdot x_0)} = r_{h^{-1}}(\varphi_{(X_G,x_0)})$ and $\beta_{(X,h \cdot x_0)} = r_h(\beta_{(X,x_0)})$.
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|
---
abstract: |
Exact solutions for nonexpanding impulsive waves in a background with nonzero cosmological constant are constructed using a “cut and paste” method. These solutions are presented using a unified approach which covers the cases of de Sitter, anti-de Sitter and Minkowski backgrounds. The metrics are presented in continuous and distributional forms, both of which are conformal to the corresponding metrics for impulsive [*pp*]{}-waves, and for which the limit as $\Lambda\to0$ can be made explicitly.\
\
PACS: 04.20.Jb; 04.30.Nk\
Keywords: Impulsive gravitational waves, (anti-)de Sitter space.
author:
- |
J. Podolský[^1]\
\
Department of Theoretical Physics, Charles University,\
V Holešovičkách 2, 18000 Prague 8, Czech Republic.\
\
and J. B. Griffiths[^2]\
\
Department of Mathematical Sciences, Loughborough University\
Loughborough, Leics. LE11 3TU, U.K.\
title: Nonexpanding impulsive gravitational waves with an arbitrary cosmological constant
---
Penrose [@Pen72] has presented a geometrical method for the construction of plane (nonexpanding) and spherical (expanding) impulsive gravitational waves in a Minkowski background by cutting the space-time along a null hypersurface and then re-attaching the two pieces with a suitable warp. The first case leads to impulsive [*pp*]{}-waves. The second case describing expanding impulsive waves [@GlePul89]–[@Hogan93] has been extended to backgrounds with a nonzero cosmological constant $\Lambda$ [@Hogan92]. However, the method has not previously been explicitly used for the construction of nonexpanding waves in backgrounds with $\Lambda\ne0$, although such solutions are already known [@HotTan93]–[@Podol98]. The purpose of the present letter is to derive these solutions using the Penrose method. Remarkably, this approach leads to a convenient unified representation of the whole family of solutions in which it is possible to set $\Lambda=0$ explicitly.
Let us first recall the line element for a space-time of constant curvature in the manifestly conformally flat form $$\d s_0^2= {2\d u\,\d v -2\d\zeta\d\bar\zeta
\over [1-{1\over6}\Lambda(uv-\zeta\bar\zeta)]^2},
\label{deS}$$ where $\Lambda$ is the cosmological constant. This is de Sitter space when $\Lambda>0$, anti-de Sitter space when $\Lambda<0$ and Minkowski space when $\Lambda=0$. As is well known, the (anti-)de Sitter space-time can be represented as the 4-dimensional hyperboloid $${Z_0}^2-{Z_1}^2-{Z_2}^2-{Z_3}^2-\epsilon{Z_4}^2=-\epsilon a^2, \qquad
a^2={3\over\epsilon\Lambda},$$ in the flat 5-dimensional space $\d s_0^2=\d{Z_0}^2-\d{Z_1}^2-\d{Z_2}^2-\d{Z_3}^2-\epsilon\d{Z_4}^2$, where $\epsilon=1$ for $\Lambda>0$ and $\epsilon=-1$ for $\Lambda<0$. The coordinates of the metric (\[deS\]) form a suitable parameterization of the hyperboloid in which $$\begin{aligned}
Z_0 &=& {\textstyle{1\over\sqrt2}(u+v)
\left[1-{1\over6}\Lambda(uv-\zeta\bar\zeta)\right]^{-1}}, \nonumber\\
Z_1 &=& {\textstyle{1\over\sqrt2}(u-v)
\left[1-{1\over6}\Lambda(uv-\zeta\bar\zeta)\right]^{-1}}, \nonumber\\
Z_2 &=& {\textstyle{1\over\sqrt2}(\zeta+\bar\zeta)
\left[1-{1\over6}\Lambda(uv-\zeta\bar\zeta)\right]^{-1}}, \label{Zcoords}\\
Z_3 &=& {\textstyle -i{1\over\sqrt2}(\zeta-\bar\zeta)
\left[1-{1\over6}\Lambda(uv-\zeta\bar\zeta)\right]^{-1}}, \nonumber\\
Z_4 &=& \sqrt{3\over|\Lambda|}\>
\left[{1+{1\over6}\Lambda(uv-\zeta\bar\zeta)
\over1-{1\over6}\Lambda(uv-\zeta\bar\zeta)}\right]. \nonumber
\end{aligned}$$ Inversely, this is given by $u={1\over\sqrt2}(Y_0+Y_1)$, $v={1\over\sqrt2}(Y_0-Y_1)$, $\zeta={1\over\sqrt2}(Y_2+iY_3)$ where $Y_\alpha=2a\,Z_\alpha/(Z_4+a)$ with $\alpha=0,1,2,3$. It may be observed that these coordinates cover the complete hyperboloid, although there is a coordinate singularity along the section $Z_4=-a$.
Let us now consider the transformation $u=U$, $v=V+H+UH_{Z}H_{\bar Z}$, $\zeta=Z+UH_{\bar Z}$ applied to the line element (\[deS\]), where $H=H(Z,\bar Z)$ is an arbitrary real function. This results in the metric $$\d s_0^2= {2\d U\,\d V
-2|\d Z+U(H_{Z\bar Z}\d Z+H_{\bar Z\bar Z}\d\bar Z)|^2
\over [1-{1\over6}\Lambda(UV-Z\bar Z+UG)]^2},
\label{deS2}$$ where $G=H-ZH_Z-\bar ZH_{\bar Z}$.
Following Penrose’s “cut and paste” method [@Pen72], we may now take the line element (\[deS\]) with $u=U$, $v=V$ and $\zeta=Z$ for $U<0$ and combine this with (\[deS2\]) for $U>0$. The resulting line element $$\d s^2= {2\d U\,\d V
-2|\d Z+U\Theta(U)(H_{Z\bar Z}\d Z+H_{\bar Z\bar Z}\d\bar Z)|^2
\over [1-{1\over6}\Lambda(UV-Z\bar Z+U\Theta(U)G)]^2},
\label{cont}$$ where $\Theta(U)$ is the Heaviside step function, is continuous across the null hypersurface $U=0=u$. However, the discontinuity in the derivatives of the metric yields impulsive components in the curvature tensor proportional to the Dirac $\delta$-function. These are interpreted as impulsive waves in de Sitter, anti-de Sitter or Minkowski backgrounds. For $\Lambda=0$, this reduces to the well known Rosen form for impulsive [*pp*]{}-waves [@PodVes98], [@KunSte99].
We may observe from (\[deS\]) that the geometry of the wavefront $u=0$ is described by the 2-metric $$\d\sigma^2= -{2\,\d\zeta\,\d\bar\zeta
\over [1+{1\over6}\,\Lambda\,\zeta\,\bar\zeta]^2}, \label{2surface}$$ which is a 2-dimensional space of constant gaussian curvature $K=\Lambda/3$. When $\Lambda=0$ the impulsive wave surface is a plane, for $\Lambda>0$ it is a sphere, while for $\Lambda<0$ it is a hyperboloid. For $\Lambda\ne0$, the geometry of these surfaces has been described in detail in [@PodGri97].
Now, the explicit form of the complete transformation formed by combining those above is given by $$\begin{aligned}
u&=&U, \nonumber\\
v&=&V+H\,\Theta(U)+U\,\Theta(U)\,H_{Z}H_{\bar Z}, \label{trans}\\
\zeta&=&Z+U\,\Theta(U)\,H_{\bar Z}. \nonumber
\end{aligned}$$ This is discontinuous at $u=0$ in such a way that $$(u=0,v,\zeta,\bar\zeta)_{M^-}
=(u=0,v-H(\zeta,\bar\zeta),\zeta,\bar\zeta)_{M^+}\>,$$ which is exactly the Penrose junction condition for reattaching the two halves of the space-time $M^-(u<0)$ and $M^+(u>0)$ with a “warp”.
Significantly, the transformation (\[trans\]), taking into account the terms which arise from the derivatives of $\Theta(u)$, relates the continuous form of the impulsive wave metric (\[cont\]), not to the initial metric (\[deS\]), but to the following metric which also includes an impulsive component explicitly located on the wavefront $u=0$ $$\d s^2= {2\d u\,\d v -2\d\zeta\,\d\bar\zeta
-2H(\zeta,\bar\zeta)\,\delta(u)\,\d u^2
\over [1-{1\over6}\Lambda(uv-\zeta\bar\zeta)]^2}.
\label{confpp}$$ This represents a nonexpanding impulsive wave in any background space-time of constant curvature.
The above result is well known for impulsive waves in a Minkowski background, where it is exactly the standard Brinkmann form for a general impulsive [*pp*]{}-wave. However, the above form has not previously been given explicitly for the case when $\Lambda\ne0$. For $\Lambda<0$, an equivalent form (in $d$-dimensions) has been used in [@HorItz99].
It may be observed that the metric (\[confpp\]) is conformal to the general impulsive [*pp*]{}-wave. In this context, we recall that Siklos [@Siklos85] has proved that Einstein spaces conformal to [*pp*]{}-waves only occur when $\Lambda<0$. However, it would appear that the [*impulsive*]{} case is a counter-example to this result.
In fact, the metric (\[confpp\]) is a suitable parameterisation of a class of nonexpanding impulsive wave solutions described in a 5-dimensional formalism in [@PodGri98]: $$\d s^2= \d{Z_0}^2 -\d{Z_1}^2 -\d{Z_2}^2 -\d{Z_3}^2
-\epsilon\d{Z_4}^2 -\tilde H(Z_2,Z_3,Z_4)\delta(Z_0+Z_1)(\d Z_0+\d Z_1)^2.$$ Indeed, using (\[Zcoords\]), we obtain exactly (\[confpp\]) where $$H(\zeta,\bar\zeta) ={\textstyle{1\over\sqrt2}\,
\left(1+{1\over6}\,\Lambda\,\zeta\,\bar\zeta\right)\, \tilde
H(\zeta,\bar\zeta)}.
\label{Htrans}$$ In this relation, the parameterisation (\[Zcoords\]), restricted to the impulsive wave surface $u=0$, is used to express the arguments of $\tilde H$ in terms of $\zeta$ and $\bar\zeta$ only.
The above solutions can describe impulsive gravitational waves or impulses of null matter. Using the tetrad frame $\ell^\mu=\Omega\,\delta_2^\mu$, $m^\mu=\Omega{1\over\sqrt2}(\delta_3^\mu+i\delta_4^\mu)$, $n^\mu=\Omega\,(\delta_1^\mu+H\delta(u)\delta_2^\mu)$ where $\Omega=1-{1\over6}\Lambda(uv-\zeta\bar\zeta)$, the nonzero components of the Weyl and Ricci tensors are $$\begin{aligned}
\Psi_4 &=& -{\textstyle \left(1+{1\over6}\Lambda\,\zeta\,\bar\zeta\right)^2
H_{\zeta\zeta} \,\delta(u), } \nonumber\\
\Phi_{22} &=& -{\textstyle \left(1+{1\over6}\Lambda\,\zeta\,\bar\zeta\right)
\left[ \left(1+{1\over6}\Lambda\,\zeta\,\bar\zeta\right) H_{\zeta\bar\zeta}
+{1\over6}\Lambda \left(H-\zeta H_\zeta-\bar\zeta H_{\bar\zeta}\right)
\right] \delta(u). }
\nonumber
\end{aligned}$$ With (\[Htrans\]) the vacuum field equations $\Phi_{22}=0$ can then be expressed as $$\left(1+{\textstyle{1\over6}}\,\Lambda\,\zeta\,\bar\zeta\right)^2 \tilde
H_{\zeta\bar\zeta} +{\textstyle{1\over3}}\,\Lambda\,\tilde H=0,
\label{vacuum}$$ which is simply $(\Delta+{2\over3}\,\Lambda)\tilde H=0$, where $\Delta$ is the Laplacian operator on a 2-dimensional impulsive wave surface (\[2surface\]). This generalises the well known vacuum field equations for impulsive waves in a Minkowski background. A general solution of this equation (\[vacuum\]) is $$\tilde H(\zeta,\bar\zeta) =(f_\zeta+\bar f_{\bar\zeta})
-{\Lambda\over3}\,{\bar\zeta f+\zeta\bar f\over
\left(1+{1\over6}\,\Lambda\,\zeta\,\bar\zeta\right)},$$ where $f(\zeta)$ is an arbitrary function of $\zeta$. Thus, the general vacuum solution for the metric (\[confpp\]) is given by $\sqrt2\,H(\zeta,\bar\zeta)= (1+{1\over6}\,\Lambda\,\zeta\,\bar\zeta)
(f_\zeta+\bar f_{\bar\zeta}) -{1\over3}\,\Lambda\,(\bar\zeta f+\zeta\bar f)$. This describes an impulsive gravitational wave in which $\Psi_4=-{1\over\sqrt2} (1+{1\over6}\,\Lambda\,\zeta\,\bar\zeta)^3
f_{\zeta\zeta\zeta} \,\delta(u)$. The space-time is conformally flat everywhere when the function $f$ is at most quadratic in $\zeta$. However, solutions of (\[vacuum\]) necessarily contain singularities. These are located on the wavefronts and may be considered as null sources of the impulsive gravitational waves.
For example, the Aichelburg–Sexl solution [@AicSex71] obtained by boosting the Schwarzschild metric is given by $f_0={1\over2}\zeta(\log\zeta-1)$. Similarly, the Hotta–Tanaka solution [@HotTan93] obtained by boosting the Schwarzschild–(anti-)de Sitter metric is given by $f_0={1\over2}\zeta(\log\zeta+{1\over2}\log{1\over6}|\Lambda|)$. For these $$\begin{aligned}
\Lambda=0: \qquad &&\sqrt2\,H_0 ={\textstyle{1\over2}}\log\zeta\bar\zeta,
\nonumber\\
\Lambda\ne0: \qquad &&\sqrt2\,H_0
={\textstyle{1\over2}(1-{1\over6}\Lambda\zeta\bar\zeta)
\log({1\over6}|\Lambda|\zeta\bar\zeta) +(1+{1\over6}\Lambda\zeta\bar\zeta)}.
\nonumber
\end{aligned}$$ In the second expression, the limit as $\Lambda\to0$ differs from the first only by a constant term that can be transformed away. These solutions describe gravitational waves generated by null monopole point particles in backgrounds of constant curvature. Further explicit solutions representing impulsive gravitational waves generated by null particles with arbitrary multipole structure in these backgrounds have been given previously in [@GriPod97] and [@PodGri98]. For dipole and quadrupole sources, the solutions are given by $f_1=(1-{1\over6}\Lambda\zeta^2)\log\zeta
+\log{1\over6}|\Lambda|$ and $f_2=-\zeta^{-1}+{1\over36}\,\Lambda^2\,\zeta^3$, for which $$\begin{aligned}
&&\sqrt2\,H_1=\left({1\over\zeta}+{1\over\bar\zeta}\right)
(1-[{\textstyle{1\over6}\Lambda\zeta\bar\zeta]^2)
-{1\over3}\Lambda(\zeta+\bar\zeta)
\log({1\over6}|\Lambda|\zeta\bar\zeta)}, \nonumber\\
&&\sqrt2\,H_2=\left({1\over\zeta^2}+{1\over\bar\zeta^2}\right)
(1+{\textstyle{1\over6}}\Lambda\zeta\bar\zeta)^3, \nonumber
\end{aligned}$$ respectively. It may be noted that, for these and higher multipole terms, the limit for the metric components as $\Lambda\to0$ can be performed explicitly. This further demonstrates the suitability of these coordinates for discussing the above class of impulsive solutions in a unified way.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by a visiting fellowship from the Royal Society and, in part, by the grant GACR-202/99/0261 of the Czech Republic.
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[^1]: E–mail: [Podolsky@mbox.troja.mff.cuni.cz]{}
[^2]: E–mail: [J.B.Griffiths@Lboro.ac.uk]{}
|
---
abstract: 'We investigated the long-term spin properties of the anomalous X-ray pulsar (AXP) by performing a temporal analysis of archival [*RXTE*]{} observations spanning about 5.2 yr from 2006 September to 2011 December. We identified two peculiar timing anomalies within $\sim$1 yr of each other: a glitch with $\Delta{\nu}$/$\nu$$\sim$4.8 $\times$ 10$^{-6}$ near MJD 54303; and an anti-glitch with $\Delta{\nu}$/$\nu$$\sim$ $-$5.8 $\times$ 10$^{-7}$ near MJD 54656. The glitch that we identified, which is the fourth glitch seen in this source in the 13 yr of RXTE monitoring, is similar to the last two detected glitches. The anti-glitch from , however, is the first to be identified. The amplitude of the anti-glitch was comparable with that recently observed in AXP . We found no significant variations in the pulsed X-ray output of the source during either the glitch or the anti-glitch. We discuss our results in relation to the standard pulsar glitch mechanisms for the glitch, and to plausible magnetospheric scenarios for the anti-glitch.'
author:
- |
S. Şaşmaz Muş$^{1}$[^1], Berk Ayd[i]{}n$^{1}$, and Ersin Göğüş$^{1}$\
$^{1}$Sabanci University, Orhanli- Tuzla, İstanbul, 34956, Turkey
title: 'A glitch and an anti–glitch in the anomalous X-ray pulsar '
---
\[firstpage\]
pulsars: individual () $-$ stars: neutron $-$ X-rays: stars
Introduction
============
Anomalous X-ray pulsars (AXPs), along with soft gamma-ray repeaters (SGRs) are extremely magnetized neutron stars (magnetars) powered by the decay of their strong magnetic fields. These sources exhibit numerous unusual characteristics, such as, relatively slow rotation speeds, high spin-down rates, bright persistent X-ray emission and, for most of them, episodic bursts seen in X-rays . The dipole magnetic field strengths, inferred from their spin periods and spin down rates are indeed extremely high, which are sufficient to account for their observed unique properties [@dunthomp92].
Long term spin behaviour of magnetars usually do not follow a secular trend, likely due to large magnetic torques along with episodic wind outflow that could take place in strongly magnetized environments [@thompson00]. Additionally, sudden increase in the angular velocity (i.e., glitches) has been observed from several AXPs. However, glitch events in AXPs have peculiar differences compared to the properties of glitches from rotation powered pulsars: e1048 is one of the most variable AXP both in timing and radiative behaviour as several short energetic burst and flare events were observed [@gavriil02; @tam08; @dib09; @gavriil06]. Its 2007 flare event was coincident with a large glitch [@dib09]. Similarly, went into an active period, exhibiting six bursts and a glitch event which was over-recovered, causing the neutron star to rotate slower than the pre-glitch level [@gavriil11]. Another AXP showing coincident burst and glitch event is [@krimm06; @israel07b; @muno07; @woods11]. On the other hand, and have experienced glitches, but there has been no evidence of accompanying radiative enhancements in these sources [@kaspi00; @kaspi03b; @dosso03; @israel07a; @dib08; @ssmeg13].
In , an outburst has been associated with a glitch [@kaspi03a; @woods04]. More interestingly, this source exhibited a sudden decrease in its angular velocity, namely an anti-glitch [@archibald13a], within two weeks of a hard X-ray burst [@foley12]. The sudden spin-down trend in this AXP occurred in conjunction with a doubled persistent source flux, as reported by [@archibald13a]. These authors also report that the observed anti-glitch was followed by either a glitch event $\sim$90 d later or by a second anti-glitch $\sim$50 d apart which was suggested as a more plausible explanation based on a Bayesian approach [@hu13]. Unlike glitches observed from isolated neutron stars, which are generally attributed to an internal mechanism [e.g, @anderson75; @alpar77], the anti-glitch event was explained in terms of external effects, including magnetospheric processes [@lyutikov13; @tong14; @katz14] or the accretion of orbiting objects [@katz14; @ouyed13; @huang14].
has a pulse period of $\sim$11.8 s. It is a bright persistent X-ray source with an emission spectrum extending into hard X-rays, up to about 150 keV [@kuiper04]. It is also a source of several short energetic magnetar bursts [@kumarsafi10; @lin11; @collazzi13; @palshin13]. Three glitches have been observed from with amplitudes ranging from $\sim$10$^{-7}$ Hz to 1.2$\times$10$^{-6}$ Hz [@dib08]. Note that these events were radiatively silent, i.e, there was no significant variations of the radiative behaviour of the source associated with these timing anomalies [@dib08; @zhu10]. The persistent X-ray emission of has remained constant during the duration of energetic bursts [@lin11; @archibald13b].
Here, we present the results of long-term timing analysis of using Rossi X-ray Timing Explorer ([*RXTE*]{}) observations spanning $\sim$5.5 years. In the following section, we introduce these [*RXTE*]{} observations and our data analysis scheme. In §\[sect:results\], we present the results of our detailed temporal investigations, and report on the discovery of an anti-glitch and additional glitch from this source. We then discuss our findings in §\[sect:discuss\].
Parameters Name Segment 0 Segment 1$^{b}$ Segment 2$^{b}$ Segment 3 Segment 4 Segment 5
-------------------------------------------- ----------------- ---------------------- ---------------------- ----------------- ----------------- ----------------
Range (MJD) 53829 $-$ 54076 54126 $-$ 54431 54492 $-$ 54807 54860 $-$ 55168 55223 $-$ 55538 55588$-$ 55903
Epoch (MJD) 53823.9694 54125.967 54491.992 54860.107 55223.090 55587.871
Number of TOAs 19 26 24 22 29 24
$\nu$ (Hz) 0.084868766(5) [*0.084861458(8)*]{} [*0.084852295(8)*]{} 0.084843233(2) 0.084834325(4) 0.084824920(3)
$\dot{\nu}$ ($10^{-13}$ Hz s$^{-1}$) $-$2.82(1) [*$-$3.92(2)*]{} [*$-$2.72(2)*]{} $-$2.834(1) $-$2.883(7) $-$2.943(6)
$\ddot{\nu}$ ($10^{-22}$ Hz s$^{-2}$) $-$5.6(9) [*187(4)*]{} [*$-$23(4)*]{} $-$ $-$9.0(5) $-$4.1(4)
$d^{3}\nu/dt^{3}$ ($10^{-28}$ Hz s$^{-3}$) $-$ [*$-$12.4(3)*]{} [*1.5(3)*]{} $-$ $-$ $-$
rms (phase) 0.0143 [*0.1184*]{} [*0.0210*]{} 0.0160 0.0230 0.0226
$\chi$${^2}$/DOF 16/15 [*1508.6/21*]{} [*37.8/19*]{} 19.1/19 46.6/25 40.6/20
\
$^{a}$ Values in parentheses are the uncertainties in the last digits of their associated measurements.\
$^{b}$ These spin parameters yield unacceptable fits to data but listed here to illustrate the inadequacy of the polynomial model.
\[tab:tablemain\]
Observations and Data Processing {#sect:analysis}
================================
has been observed with [*RXTE*]{} in 279 occasions over a time span of $\sim$13 years from 1999 February to 2011 December . Data covering the first $\sim$7.6 years have already been investigated by [@dib08]. Here, we investigated 137 [*RXTE*]{} observations performed from 2006 September 19 to 2011 December 8 for the first time. Additionally, we also included the last 12 (2006 April 3 - 2006 September 5) [*RXTE*]{} pointings in the sample of [@dib08] in order to link our long term timing results with their extensive coverage. Exposure times of these 149 pointings were between $\sim$1.2 ks and $\sim$9.6 ks with a mean of $\sim$ 4.6 ks and spacing between successive pointings varied between 0.04 and 61 days with an average of 14 days.
We employed data collected with the Proportional Counter Array (PCA), that was an array of nearly identical five proportional counter units (PCUs) operated optimally in the energy range of 2$-$30 keV [@jahoda06]. We first filtered each observation for occasional bursts, data anomalies and instrumental rate spikes by screening their light curves in the 2-30 keV band with the 31.25 ms time resolution. We then converted the arrival times of the remaining events to the Solar system barycenter. In order to maximize the signal-to-noise ratio for timing analysis of , we selected events in the 2$-$11 keV energy range recorded at the top Xenon layer of each operating PCU in GoodXenon mode, as was also done by [@dib08].
Data Analysis & Results {#sect:results}
=======================
In order to undertake coherent timing analysis, we grouped all 149 observations into six segments intercepted with observational interruptions in between due to Solar constraints. In addition, we merged observations together if the time spacing between them was less than 0.1 d. In this grouping scheme, segment 0 has 12 observations which are the last [*RXTE*]{} pointings used by [@dib08]. We first generated a high signal-to-noise ratio pulse profile template using a subset of observations (typically 5-6), that were performed in the beginning of each segment. We obtained the pulse profile for each observation by folding light curves with a nominal spin frequency. We then cross-correlated the pulse profiles with the template and measured their phase shifts with respect to the template. Finally, we fitted the phase shifts with a polynomial of the following form:
$$\phi(t) = \phi_{0}(t_{0})+\nu_{0}(t-t_{0})+ \frac{1}{2}\dot{\nu_{0}}(t-t_{0})^{2} + ...
\label{eq:taylorphase}$$
where t$_{0}$ is the epoch time. We found that all segments, except for Segment 1 and Segment 2, are fitted well with polynomials of the third order or lower. We present the results of polynomial model fits to each segment in Table \[tab:tablemain\]. We note that the spin frequency and spin-down rate of Segment 0 are consistent with those reported by [@dib08]. In Table \[tab:tablemain\], we also list root-mean-square (rms) fluctuations of the resulting phase residuals, which are presented in the top panel of Figure \[fig:resfreqpcount\]. A fourth-order polynomial fit to pulse arrival times in Segment 1 yields extremely large fit statistics ($\chi^{2}$ of 1508.6 for a degrees of freedom (DOF) of 21; see Figure \[fig:resfreqpcount\] top panel and Figure 2 panel b). We therefore identified this segment to search for glitch(es), as we describe in detail below.
We fitted the pulse arrival times of Segment 2 with a fourth-order polynomial but obtain unacceptable fit statistics ($\chi^{2}$/DOF = 37.8/19). We note the fact that there are systematic variations of the phase residuals around MJD 54650. Higher order polynomial fit to this segment results in lower fit statistics, whereas the errors of derived spin parameters become large. For these reasons, we also investigated this segment for searching timing anomalies.
We found that arrival times of Segment 3 can be represented with a second-order polynomial, that is the lowest order we obtained among our investigation span of 5.5 yr. We also find that the frequency derivatives within this segment remain constant (see the middle panel in Figure \[fig:resfreqpcount\]). For Segments 4 and 5, we obtained an adequate fit with a third-order polynomial. However, the rms phase residual fluctuations for these segments are larger than other segments, namely Segments 0 and 3. Note the important fact that energetic short duration bursts from this magnetar were observed during Segments 4 and 5, as they are denoted with vertical solid lines in Figure \[fig:resfreqpcount\].
In order to determine whether the large fluctuations in the phase residuals of Segments 1, 2, 4 and 5 are due to a sudden change in the spin frequency of the source, we fitted phase shifts of these segments using MPFITFUN routine [@markwardt09] with a model involving a quadratic polynomial and a sudden change of spin frequency (i.e., glitch). The corresponding spin trend of this model is: $$\nu(t) = \nu_{0}(t) + \Delta{\nu} + \Delta{\dot{\nu}}(t - t_{g})
\label{eq:glt}$$ Here $t_{g}$ is the time of the glitch. $\Delta{\nu}$ is the change in the frequency at the time of the glitch. $\Delta{\dot{\nu}}$ is the frequency derivative change after the glitch and $\nu_{0}(t)$ is pre-glitch frequency evolution.
In Segment 1, we found that a model involving a glitch provides statistically significant improvement in fitting the phase shifts ($\chi^{2}$/DOF = 31.6/21). We, therefore, conclude that there is a glitch at MJD $\sim$54303 with an amplitude of $\Delta\nu$ $\sim$4$\times$10$^{-7}$ Hz. This is the 4th glitch observed from . We present the parameters for Glitch 4 in Table \[tab:glitch4\]. Note that the amplitude of this glitch is on the order of last two glitches observed from this source [see @dib08]. We also present the phase residuals of the polynomial model and glitch model for comparison in Figure 2.
![(a) Spin frequency evolution of the source during Segment 1. (b) Phase residuals after the subtraction of a fourth order polynomial model from the data. (c) Phase residuals after subtracting the glitch model. (d) Pulsed count rates of the source in the 2$-$10 keV band obtained using sets of observations spanning mostly over 40$-$50 d to ensure significant pulsed flux measurement.[]{data-label="fig:glitch4"}](./f2.ps){width="8.5cm"}
---------------------------------------------- ----------------
Range (MJD) 54126$-$54431
Epoch (MJD) 54125.967
Number of TOAs 26
$\nu$ (Hz) 0.084861236(2)
$\dot{\nu}$ ($10^{-13}$ Hz s$^{-1}$) $-$2.965(3)
$t_{g}$ (MJD) 54303(3)
$\Delta{\nu}$ ($10^{-8}$ Hz) 40.7(6)
$\Delta{\dot{\nu}}$ ($10^{-15}$ Hz s$^{-1}$) 1.2(7)
rms (phase) 0.0174
$\chi$${^2}$/DOF 31.6/21
---------------------------------------------- ----------------
: Parameters for Glitch 4$^{a}$[]{data-label="tab:glitch4"}
\
$^{a}$ Values in parentheses are the uncertainties\
in the last digits of their associated measurements.
---------------------------------------------- ----------------
Range (MJD) 54492$-$54807
Epoch (MJD) 54491.992
Number of TOAs 24
$\nu$ (Hz) 0.084852317(2)
$\dot{\nu}$ ($10^{-13}$ Hz s$^{-1}$) $-$2.833(3)
$t_{g}$ (MJD) 54656.0
$\Delta{\nu}$ ($10^{-8}$ Hz) $-$4.9(6)
$\Delta{\dot{\nu}}$ ($10^{-15}$ Hz s$^{-1}$) $-$1.8(6)
rms (phase) 0.0139
$\chi$${^2}$/DOF 16.3/19
---------------------------------------------- ----------------
: Parameters for Anti-glitch$^{a}$
\
$^{a}$ Values in parentheses are the uncertainties\
in the last digits of their associated measurements.
\[tab:antiglitch\]
As we noted earlier, pulse phase modeling of the Segment 2 with a fourth order polynomial yields an unacceptable fit statistics (see panel c of Figure \[fig:antiglitch\]), while increasing the order of polynomial results in unconstrained spin parameters. For this reason, we also modeled the phases of this segment with a glitch model. We find that an ordinary glitch model fit does not improve the fit statistics. However, a glitch with a negative amplitude (i.e., an anti-glitch) of $\Delta\nu$ $\sim$$-$5$\times$10$^{-8}$ Hz provides a significant improvement in fit statistics ($\chi^{2}$/DOF = 16.3/19); a $\Delta\chi^2$ of 21.5 for the same number of DOF as the polynomial model fit. Based on this, we conclude that exhibited an anti-glitch near MJD 54656, that is $\sim$1 yr after Glitch 4. We list the parameters of the anti-glitch model fit in Table \[tab:antiglitch\]. Note that fit results suggesting the anti-glitch epoch between MJD 54645 and 54662 yield similar fit statistics, indicating that the anti-glitch occurred in this time interval. In the top panel of Figure \[fig:antiglitch\] we present the frequency evolution of the source in this segment. To show the sudden deviation of the spin frequency, we fit a linear trend to the frequencies prior to MJD $\sim$54650 and extrapolated this fit to the rest of this segment (see panel b in Figure \[fig:antiglitch\]). We found that the spin-down rate before MJD 54630 and after MJD 54670 are consistent with one another ($-$2.84(1)$\times$10$^{-13}$ and $-$2.86(1)$\times$10$^{-13}$ Hz s$^{-1}$, respectively). The average spin-down rate during the $\sim$40 d in between is about $-$3$\times$10$^{-13}$ Hz s$^{-1}$. We also present the phase residuals of the anti-glitch involving model in the panel d of Figure \[fig:antiglitch\].
![(a) Spin frequency evolution of the source during Segment 2. The solid line is the spin-down trend obtained by fitting the observations before MJD $\sim$54630, and extrapolated onwards. (b) Residuals of the model and its extrapolation in (a). (c) Phase residuals after the subtraction of a fourth order polynomial fit presented in Table \[tab:tablemain\]. (d) Phase residuals after the subtraction the glitch model presented in Table \[tab:antiglitch\] (e) The pulsed count rates in 2$-$10 keV averaged over $\sim$40 d.](./f3.ps "fig:"){width="8.5cm"} \[fig:antiglitch\]
Fits to the pulse arrival times of Segment 4 and 5 with models involving a glitch or anti-glitch did not yield any improvement in the fit statistics. We therefore conclude that the system exhibits higher level of timing noise, likely related to the emission of numerous energetic bursts during these episodes (see Figure \[fig:resfreqpcount\]).
Finally, in order to search for radiative variabilities we performed pulsed count rate analysis as explained in [@ssmeg13]. We found that pulsed count rate of the source is constant over $\sim$5.5 yr of [*RXTE*]{} observations (See, third panel of Figure \[fig:resfreqpcount\]).
Discussion and Conclusions {#sect:discuss}
==========================
was observed $\sim$13 yr by [*RXTE*]{}. Here we performed timing analysis of using $\sim$5 yr of [*RXTE*]{} observations. Previous $\sim$7.6 yr has been analysed by [@dib08] and 3 glitches have been reported. The largest glitch observed from this source has an amplitude of $\sim$1.2$\times$10$^{-6}$ Hz with a recovery timescale of 43 days and fractional increase of 0.1 in its spin-down rate. Consequent two glitches have amplitudes on the order of $\sim$10$^{-7}$ Hz without any observable exponential recovery. There is no evidence of accompanying radiative enhancements during all three glitch epochs.
Through our detailed investigations of [*RXTE*]{} monitoring of spanning over five years, we have identified two timing events separated by $\sim$1 yr: a glitch and an ’anti-glitch’. The glitch event has occurred at MJD $\sim$54303 with an amplitude of $\Delta\nu$$\sim$4$\times$10$^{-7}$ Hz without any observable exponential recovery. Note that this is the fourth glitch identified from this source: one of the earlier three has an amplitude of $\sim$10$^{-6}$ Hz with a recovery timescale of 43 days [@dib08], while the other two were at similar amplitudes and showed no exponential recovery. Similar to the three earlier glitch episodes in , we found no associated radiative enhancement deduced from the pulsed X-ray flux measurement (bottom panels of Figure \[fig:resfreqpcount\] and 2).
The second event, an anti-glitch seen from for the first time, has occurred at MJD $\sim$54656 with an amplitude of $\Delta\nu$$\sim$$-$5$\times$10$^{-8}$ Hz. The amplitude of the only other anti-glitch event observed from [@archibald13a] is strikingly similar. We found that the source experienced an elevated spin-down rate over about 40 days, and $\dot{\nu}$ returned back to the pre-anti-glitch level after MJD 54670. Note that the average spin-down rate in the elevated regime is about $-$3$\times$$10^{-13}$ Hz s$^{-1}$, which is similar to the $\dot{\nu}$ values exhibited by the source over $\sim$150 d following an energetic burst on MJD 55322.6 (see the middle panel of Figure \[fig:resfreqpcount\]). Both pulsed X-ray emission (see the bottom panel of Figure \[fig:antiglitch\]) and the 0.5-10 keV flux [see figure 3 of @lin11] of the source remain constant during the anti-glitch episode, similar to the case for the fourth glitch.
According to the standard pulsar glitch models [see for example @anderson75; @alpar77], a faster rotating superfluid transfers angular momentum to the crust which results in positive increment in the observed spin frequency of the neutron star. [@alpar94] statistically determined that superfluid vortex unpinning model with a constant fractional vortex density, which is the fraction of vortex density involved in the glitch event, provides the most plausible explanation for glitches observed from rotation powered pulsars. In this model, the number of glitches in a given time span that a pulsar would experience is related to the constant fractional vortex density, the ratio of the spin-down rate and frequency of the pulsar. From previous observations and glitches observed from we determined the fractional vortex density as 2.90$\times$10$^{-4}$. Note that this is consistent with fractional vortex density for [@ssmeg13]. Using this parameter for and an average value of the ratio of the spin-down rate to the frequency yield the expected number of glitches from this magnetar during the entire 13 yr of [*RXTE*]{} observation span as 4.5. With the fourth glitch we uncovered in this study, our results are in agreement with the expectations of the vortex unpinning model.
Recent observation of the anti-glitch from implies a neutron star superfluid interior rotating slower than the crust within the context of standard pulsar glitch models [@archibald13a; @anderson75]. It was suggested that the rotation of the superfluid can be slowed down by crustal deformations due to magnetic stresses in highly magnetized sources [@thompson00; @duncan13]. [@thompson00] suggest that and might have had episodes of accelerated spin-down due to the fact that they are older than their associated supernovae remnants as inferred by their characteristic ages. In [@thompson00] they also propose a particle outflow scheme to account for the sudden spin-down behaviour of SGR 1900+14 [@woods99], in conjunction with its August 27 giant flare. However, there was no indication of particle outflow from around the time of the anti-glitch [@archibald13a]. For this reason, the particle outflow scenario was discarded. Alternatively, there are already a couple of magnetospheric models suggested to understand the origin of the anti-glitch: [@lyutikov13] proposed that the sudden spin-down and in general variable spin-down trends are caused by the changes in torque due to transient opening of a small region of the twisted magnetosphere during the X-ray burst. [@tong13; @tong14] have applied the wind braking scenario [@michel69; @harding99; @thompson00] to magnetars in the case that the rotational energy of the star is mainly extracted via a constant particle wind from the star. In this model an anti-glitch corresponds to an enhanced state of the particle wind [@tong14]. Both partial magnetospheric opening and wind breaking models require radiative enhancements accompanying the anti-glitch, which was the case for as its X-ray flux increased by at least a factor of two in association with the anti-glitch [@archibald13a]. Our results, however, place an indirect constraint on both models since we found no observable variations in the pulsed X-ray emission from at the time of its anti-glitch. Nevertheless, occasional detection of energetic bursts from indicate that its magnetosphere is active and not all but some X-ray bursts may lead to a magnetospheric rearrangement that could lead to the episodic rapid spin-down as prescribed by [@lyutikov13].
Note the fact that we became aware of the paper by [@dib14] during the review stage of our paper. They fit a very wide data segment (spanning more than 1200 d and including our suggested anti-glitch) with a fifth-order polynomial, and find an rms of 0.041, which is much larger than that for any other segment. Even with the fifth-order polynomial fit, large fluctuations in the phase residuals are clearly visible (see panel (c) around MJD 54600 of figure 1c in [@dib14]). They make no emphasis on this significant timing anomaly, which we interpret as an anti-glitch.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank M. Ali Alpar for helpful comments. SŞM acknowledges support through the national graduate fellowship program of The Scientific and Technological Research Council of Turkey (TÜBİTAK).
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[^1]: E-mail: sinemsm@sabanciuniv.edu
|
---
abstract: 'We are interested in exploring the possibility and benefits of structure learning for deep models. As the first step, this paper investigates the matter for *Restricted Boltzmann Machines (RBMs)*. We conduct the study with Replicated Softmax, a variant of RBMs for unsupervised text analysis. We present a method for learning what we call *Sparse Boltzmann Machines*, where each hidden unit is connected to a subset of the visible units instead of all of them. Empirical results show that the method yields models with significantly improved model fit and interpretability as compared with RBMs where each hidden unit is connected to all visible units.'
author:
- |
Zhourong Chen[^1] Nevin L. ZhangDit-Yan Yeung Peixian Chen\
Hong Kong University of Science and Technology\
{zchenbb,lzhang,dyyeung,pchenac}@cse.ust.hk
bibliography:
- 'paper.bib'
title: |
Sparse Boltzmann Machines\
with Structure Learning as Applied to Text Analysis
---
Introduction
============
Deep learning has achieved great successes in recent years. It has produced superior results in a range of applications, including image classification [@krizhevsky2012imagenet], speech recognition [@hinton2012deepsppech; @mikolov2011strategies], language translation [@sutskever2014sequence] and so on. It is now time to ask whether it is possible and beneficial to learn structures for deep models.
To learn the structure of a deep model, we need to determine the number of hidden layers and the number of hidden units at each layer. More importantly, we need to determine the connections between neighboring layers. This implies that we need to talk about sparse models where neighboring layers are not fully connected.
Sparseness is desirable and full connectivity is unnecessary. In fact, [@NIPS2015_5784] have shown that many weak connections in the fully connected layers of *Convolutional Neural Networks (CNNs)* [@Lecun98gradient-basedlearning] can be pruned without incurring any accuracy loss. The convolutional layers of CNNs are sparse, and the fact is considered one of the key factors that lead to the success of CNNs. Moreover, it is well known that overfitting is a serious problem in deep models. One method to address the problem is dropout [@Srivastava:2014:DSW:2627435.2670313], which randomly drops out units (while keeping full connectivity) during training. The possibility of randomly dropping connections has also been explored in [@wan2013regularization]. Sparseness offers an interesting alternative. It amounts to deterministically drop out connections.
How can one learn sparse deep models? One method is to first learn a fully connected model and then prune weak connections [@NIPS2015_5784]. The drawbacks of this method are that it is computationally wasteful and doesn’t provide a way to determine the number of hidden units. We would like to develop a method that determines the number of hidden units and the connections between units automatically. The key intuition is that a hidden unit should be connected to a group of strongly correlated units at the level below. This idea is used in convolutional layers of CNNs, where a unit is connected to pixels in a small patch of an image. In image analysis, spatial proximity implies strong correlation.
To apply the intuition to applications other than image analysis, we need to identify groups of strongly correlated variables for which latent variables should be introduced. *Hierarchical Latent Tree Analysis (HLTA)* (Liu et al 2014, Chen et al 2016) offers a plausible solution. HLTA first partitions all the variables into groups such that the variables in each group are strongly correlated and the correlations can be properly modelled using a single latent variable. It introduces a latent variable for each group. Then it converts the latent variables into observed variables via data completion and repeats the process to produce a hierarchy. The output of HLTA is a hierarchical latent tree model where the observed variables are at the bottom and there are multiple layers of latent variables on top. To obtain a non-tree sparse deep model, we propose to use the tree model as a skeleton and introduce additional connections to model the residual correlations not captured by the tree.
In this paper, we fully develop and test the idea in the context of RBMs, which have a single layer of hidden units and are building blocks of Deep Belief Networks and Deep Boltzmann Machines. The target domain is unsupervised text analysis. We present an algorithm for learning what we call Sparse Boltzmann Machines. Empirically, we show that the full-connectivity restriction of RBMs can easily lead to overfitting, and that Sparse Boltzmann Machines are effective in avoiding overfitting. We also demonstrate that Sparse Boltzmann Machines are more interpretable than RBMs.
Related Works
=============
The concept of sparse RBMs were first mentioned in [@lee2008sparse]. The authors use sparse RBMs to build sparse Deep Belief Networks and extract some interesting features. However, in their paper, sparse RBMs were not defined from the perspective of sparse connections but sparse hidden unit activations. And it was achieved by adding a regularization term to the objective function when training the parameters. There is no structure learning.
Network pruning is also a potential way to optimize the structure of a neural network. Biased weight decay was the early approach to pruning. Later, Optimal Brain Damage [@Cun90optimalbrain] and Optimal Brain Surgeon [@Hassibi93secondorder] suggested that magnitude-based pruning may not be the best strategies and they proposed pruning methods based on the Hessian of the loss function. With respect to deep neural networks, [@NIPS2015_5784] proposed to compress a network through a three-step process: train, prune connections, and retrain. We call it redundancy pruning. In contrast, [@Srinivas2015] proposed to prune redundant neurons directly. They all reduced the number of parameters vastly with slight or even no performance loss. The drawback of network pruning is that the original networks should be large enough and hence some computation would be wasted on those unnecessary parameters during pre-training.
Restricted Boltzmann Machines
=============================
{width="4cm"}
\[fig.rbm\]
An *Restricted Boltzmann Machine (RBM)* [@Smolensky:1986:IPD:104279.104290] is a two-layer undirected graphical model with a layer of $K$ visible units $\{v^1, \ldots, v^K\}$ and a layer of $F$ hidden units $\{h_1, \ldots, h_F\}$. The two layers are fully connected to each other, while there are no connections between units at the same layer. An example is shown in Figure \[fig.rbm\]. In the simplest case, all the units are assumed to be binary. An energy function is defined over all the units as follows:
$$E({\bf v}, {\bf h}) = - \sum_{j=1}^F\sum_{k=1}^K W^k_jh_jv^k - \sum_{k=1}^K v^kb^k - \sum_{j=1}^F h_ja_j$$
where $a_j$ and $b^k$ are bias parameters for the hidden and visible units respectively, while $W^k_j$ is the connection weight between hidden unit $h_j$ and visible unit $v^k$. The energy function defines a joint probability over **v** and **h** as follows: $$P(\textbf{v},\textbf{h}) = exp(-E(\textbf{v},\textbf{h}))/ Z$$ where
$Z=\sum_{\textbf{v}',\textbf{h}'}exp(-E(\textbf{v}',\textbf{h}'))$
is a normalization term called the partition function. An important property of RBM is that the conditional distributions $P(\textbf{h}|\textbf{v})$ and $P(\textbf{v}|\textbf{h})$ factorize as below:
$$P(\textbf{h}|\textbf{v}) = \prod_{j}P(h_{j}|\textbf{v}) \hspace{20pt} P(\textbf{v}|\textbf{h}) = \prod_{k}P(v^{k}|\textbf{h})$$
$$P(h_j=1|{\bf v}) = \sigma(a_j +\sum_{k=1}^K W^k_jv^k )$$ $$P(v^k=1|{\bf h}) = \sigma(b^k +\sum_{j=1}^F W^k_jh_j )$$
where $\sigma(x) = 1/(1+e^{-x})$ is the logistic function. The model parameters of an RBM are learned using the Contrastive Divergence algorithm [@Hinton:02], which maximizes the data likelihood via stochastic gradient descent.
In [@NIPS2009_3856], RBM was used for topic modeling and the proposed model was called Replicated Softmax. Suppose the vocabulary size is $K$. Let us represent a document with $D$ tokens as a binary matrix $\cal{U}$ of size $K * D$ with $u_i^k=1$ if the $i^{th}$ token is the $k^{th}$ word in the vocabulary. The energy function of document $\cal{U}$ and hidden units $\mathbf{h}$ is defined as follows:
$$\begin{aligned}
E({\cal U}, {\bf h}) = - \sum_{j=1}^F\sum_{k=1}^K W^k_jh_j\hat{u}^k - \sum_{k=1}^K \hat{u}^kb^k - D\sum_{j=1}^F h_ja_j\end{aligned}$$
where $\hat{u}^k = \sum_{i=1}^D u_i^k$ denotes the count for the $k^{th}$ word. The conditional probabilities $P(h_j=1|{\cal U})$ can be calculated as:
$$\begin{aligned}
P(h_j=1|{\cal U}) = \sigma(Da_j + \sum_{k=1}^{K}W_j^k\hat{u}^k) \label{equ.conditional}\end{aligned}$$
The motivation behind Replicated Softmax is to properly model word counts in documents of varying lengths through weight sharing. It was shown to generalize better than *Latent Dirichlet Allocation (LDA)* [@Blei:2003:LDA:944919.944937] in terms of log-probability on held-out documents and accuracy on retrieval tasks. In this paper, we will use Replicated Softmax for text analysis.
[c]{}
\[fig.sbm\]
[c]{}
Sparse Boltzmann Machines
=========================
In this section, we will propose our new models, *Sparse Boltzmann Machines (SBMs)*. An SBM is a two-layer undirected graphical model with a layer of $K$ visible units $\{v^1, \ldots, v^K\}$ and a layer of $F$ hidden units $\{h_1, \ldots, h_F\}$. The hidden units in SBMs are directly linked up to form a tree structure, while each hidden unit is also individually connected to a subset of the visible units. See Figure \[fig.sbm\] for an example SBM. In SBM, the number of hidden units and the connectivities are both learned from data.
One technical difference between SBMs and RBMs is that there are direct connections among the hidden units in SBMs. We call them hidden connections. The reason why we introduce the hidden connections into our models is that, the hidden connections provide a way to relate a hidden unit to a visible unit without a direct connection. For example, in Figure \[fig.sbm\], hidden unit $h_1$ is not directly connected to visible unit $v_4$. However, the existence of the hidden connection between $h_1$ and $h_2$ introduces a path connecting $h_1$ and $v_4$, which can help us to better model the correlation between the two units. This is crucial in reducing the number of connections between hidden units and visible units. To avoid the connections among the hidden units becoming too dense, we restrict them to form a tree structure.
![An example HLTM from [@DBLP:conf/aaai/ChenZPC16].](HLTM.pdf){width="8.5cm"}
\[fig.HLTM\]
Parameter Learning
------------------
SBMs also can be extended for text analysis as RBMs are extended to Replicated Softmax. Here we will introduce SBMs in the context of Replicated Softmax and use the same notations in the previous section. Let $\cal G$ be a graph representing the model structure. Edge $(j, k)$ belongs to $\cal G$ if and only if there is a link between the visible unit $v^k$ and hidden unit $h_j$. Edge $(j, l)$ belongs to $\cal G$ if and only if there is a link between the hidden unit $h_j$ and hidden unit $h_l$. Also let $W_{jl}$ be the weight on the connection between hidden unit $h_j$ and hidden unit $h_l$. Then the energy function of an SBM for document $\cal U$ and hidden units $\mathbf{h}$ is as below:
$$\begin{aligned}
\begin{aligned}
E({\cal U}, {\bf h}) = & - {\sum_{(j,k) \in \cal G}}W^k_jh_j\hat{u}^k - \sum_{k=1}^K\hat{u}^kb^k \\ & - D\sum_{j=1}^F h_ja_j - D{\sum_{(j,l) \in \cal G}}W_{jl}h_jh_l.
\end{aligned}\end{aligned}$$
Similar to Replicated Softmax, our model defines the joint distribution as:
$$\begin{aligned}
P({\cal U}, {\bf h}) = \frac{1}{Z} \exp(-E({\cal U}, {\bf h})), \mbox{ }\end{aligned}$$
where
$Z=\sum_{\cal U'}\sum_{\bf h} \exp(-E({\cal U'}, {\bf h}))$
. Note that the summation over $\cal U'$ is done over all the possible documents with the same length as $\cal U$.
Let $\bar{\cal U} =\{{\cal U}_n\}_{n=1}^N$ be a collection of $N$ documents with potentially different lengths $D_1$, …, $D_N$. We assume that $P(\bar{\cal U}) = \prod_{n=1}^N P({\cal U}_n)$, where $P({\cal U}_n) = \sum_{\bf h}P({\cal U}_n, \bf h)$. The objective of training an SBM for $\bar{\cal U}$ is to maximize the log-likelihood of the documents $\log P(\bar{\cal U})$. We maximize the objective function via stochastic gradient descent. The partial derivatives of $\log P(\bar{\cal U})$ w.r.t the parameters $W_j^k$, $b^k$ and $a_j$ remain the same as in Replicated Softmax:
$$\begin{aligned}
\begin{aligned}
\frac{\partial \log P (\bar{\cal U})}{\partial W_j^k} = \sum_{n=1}^N (E_{P(h_j|{\cal U}_n)}[h_j\hat{u}^k_{n}] - E_{P({\cal U}, {\bf h})}[h_j\hat{u}^k])
\end{aligned}\end{aligned}$$
$$\begin{aligned}
\frac{\partial \log P (\bar{\cal U})}{\partial b^k}
&=& \sum_{n=1}^N (\hat{u}^k_n - E_{P({\cal U})}[\hat{u}^k]) \\
\frac{\partial \log P (\bar{\cal U})}{\partial a_j}
&=& \sum_{n=1}^N D_{n}(E_{P(h_j|{\cal U}_{n})}[h_j] - E_{P(h_j)}[h_j])\end{aligned}$$
while the partial derivative of $\log P(\bar{\cal U})$ w.r.t the new parameter $W_{jl}$ for fixed $j$ and $l$ is:
$$\begin{aligned}
\frac{\partial \log P(\bar{\cal U})}{\partial W_{jl}} = \sum_{n=1}^N D_{n}(E_{P(\mathbf h|{\cal U}_{n})}[h_jh_l] - E_{P({\mathbf h})}[h_jh_l])\end{aligned}$$
The first terms in these partial derivatives require the computation of the conditional probabilities $P(h_j|{\cal U}_{n})$ and $P(\mathbf{h}|{\cal U}_{n})$. In Replicated Softmax, $P(h_j|{\cal U}_{n})$ can be calculated using Equation (\[equ.conditional\]). While in SBMs, due to the connections between hidden units, $P(\mathbf{h}|{\cal U}_{n})$ no longer factorizes and hence Equation (\[equ.conditional\]) cannot be applied. Nevertheless, since the hidden units in Sparse Boltzmann Machines are linked as a tree structure, we can easily compute the value of $P(h_j|{\cal U}_{n})$ and $P(\mathbf{h}|{\cal U}_{n})$ by conducting message propagation [@38136] in the model.
The second terms in these derivatives require taking an expectation with respect to the distribution defined by the model, which is intractable. Thus as in Replicated Softmax, we adopt the Contrastive Divergence algorithm to approximate the second terms by running Gibbs sampling chains in the model. Specifically, the Gibbs chains are initialized at the training data and run for $T$ full steps to draw samples from the model. In SBMs, given a document $\cal U$ and the value of all the other hidden units $\mathbf{h}_{-j}$, the conditional probability to sample a hidden unit $h_j$ becomes:
$$\begin{aligned}
\begin{aligned}
P(h_j=1|{\cal U}, \mathbf{h}_{-j}) = \sigma(& \sum_{(j,k) \in \cal G} W^k_j\hat{u}^k + Da_j +\\
& D \sum_{(j,l) \in \cal G}W_{jl}h_l +D \sum_{(l,j) \in \cal G}W_{lj}h_l )
\end{aligned}\end{aligned}$$
while the conditional probability to sample an visible unit remains the same as in Replicated Softmax.
Structure Learning
------------------
We regard SBMs as a method to model correlations among the visible units. Learning an SBM hence amounts to building a latent structure to explain the correlations. Recently, [@DBLP:conf/pkdd/LiuZC14] and [@DBLP:conf/aaai/ChenZPC16] proposed a method, called HLTA, for learning a *Hierarchical Latent Tree Model (HLTM)* from data. Our structure learning algorithm for SBMs is built upon their work. We expand the tree model from HLTA to obtain the structure of an SBM.
HLTA learns a tree model ${\cal T}$ with a layer of observed variables at the bottom and multiple layers of latent variables. Note that the visible units and hidden units in SBMs are called observed variables and latent variables in HLTM respectively. The left panel in Figure \[fig.structure\_process\] and Figure \[fig.HLTM\] illustrate example models that HLTA produces. Each latent variable in the model is connected to a set of highly-correlated variables in the layer below. The number of latent variables at each layer is determined automatically by the algorithm. The number $L$ of latent layers in ${\cal T}$ is controllable. In this paper, we set $L=2$. Let $H_{l}$ be the $l^{th}$ latent layer in ${\cal T}$. Also let $\mathbf{V}_Z$ be the set of observed variables which are located in the subtree rooted at latent variable $Z$ in ${\cal T}$.
To build the structure of an SBM from ${\cal T}$, we first remove all the latent layers except the top layer $H_{L}$. Then we connect each latent variable $Z$ in $H_{L}$ to the set of observed variables $\mathbf{V}_Z$. We use the resulting structure as a skeleton ${\cal T'}$ of the corresponding SBM. This is illustrated in Figure \[fig.structure\_process\], where the hidden units $h_1$, $h_2$ in SBM correspond to $Z_{21}$, $Z_{22}$ in ${\cal T}$ respectively. Note that the skeleton is still a tree structure, where each node has only one parent.
As to remove the tree-structure constraint, we conduct an expansion step to increase the number of “fan-out” connections for each hidden unit in ${\cal T'}$. The key question is how to determine the new set of visible units that a hidden unit should be connected to. We introduce our method using $Z_{21}$ (correspondingly $h_1$ in ${\cal T'}$) and $v_7$ in Figure \[fig.structure\_process\] as an example. To determine whether $Z_{21}$ should also be connected to $v_7$, we consider the empirical conditional mutual information $I(Z_{21}, v_7|Z_{22},\bar{\cal U})$, where $Z_{22}$ is the root of the subtree that $v_7$ is in. To estimate the value, we first estimate the empirical joint distribution $\hat{p}(Z_{21}, Z_{22}, v_7)$. We go through all the documents and compute $p(Z_{21},Z_{22}|{\cal U}_{n})$ for each document ${\cal U}_{n}$ in $\bar{\cal U}$ by conducting inference in ${\cal T}$. Then we collect the statistics of $Z_{21},Z_{22}$ and $v_7$ to get $\hat{p}(Z_{21}, Z_{22}, v_7)$. After that, $I(Z_{21}, v_7|Z_{22},\bar{\cal U})$ can be estimated as:
$$\begin{aligned}
\begin{aligned}
&I(Z_{21}, v_7|Z_{22},\bar{\cal U}) = \\
&\sum_{Z_{22}}\hat{p}(Z_{22})\sum_{v_7}\sum_{Z_{21}}\hat{p}(Z_{21}, v_7|Z_{22})log \frac{\hat{p}(Z_{21}, v_7|Z_{22})}{\hat{p}(Z_{21}|Z_{22})\hat{p}(v_7|Z_{22})}.
\end{aligned}\end{aligned}$$
All the distributions in the above formula can be derived from the joint distribution $\hat{p}(Z_{21}, Z_{22}, v_7)$.
If the correlation between $Z_{21}$ and $v_7$ is properly modeled in ${\cal T}$, the two variables should be conditionally independent given $Z_{22}$, and hence $I(Z_{21}, v_7|Z_{22},\bar{\cal U})$ should be zero. Therefore, if $I(Z_{21}, v_7|Z_{22}, \bar{\cal U})$ is not 0, then we can conclude that the correlation between $Z_{21}$ and $v_7$ is not properly modeled in the model, and the model needs to be expanded by adding new connections between the two variables.
Our algorithm, called *SBM-SFC (SBM-Structure from Correlation)*, is given in Algorithm 1. It considers the latent variables one at a time. For a given latent variable $Z$ (suppose the corresponding hidden unit in ${\cal T'}$ is $h$), it computes the conditional mutual information between $Z$ and each unconnected observed variable, and sorts the observed variables in descending order with respect to the conditional mutual information. Then in ${\cal T'}$, it connects hidden unit $h$ to the visible units corresponding to the top $M$ observed variables with the highest conditional mutual information. $M$ is a predefined parameter, which normally is set to the value such that each hidden unit is connected to $0.2*K$ hidden units. After the above expansion step is done for each hidden unit in ${\cal T'}$, the whole structure of an SBM is determined.
Inputs:
: $\mathcal{T}$—Graph of an HLTM, $\bar{\cal U}$—Collection of training documents, $M$—Number of new connections for each hidden unit.
Outputs:
: Graph $\mathcal{T'}$ of a corresponding SBM.
$\mathcal{T'} \gets \emptyset,$ $H_L \gets$ graph of the top latent layer in $\mathcal{T}$ $V \gets$ observed variables in $\mathcal{T}$ $\mathcal{T'}.add\textunderscore graph(H_L), \mathcal{T'}.add\textunderscore units(V)$
$V_Z \gets$ observed variables in subtree rooted at variable $Z$ $\mathcal{T'}.add\textunderscore edges(Z, V_Z),I \gets \emptyset$ $Z' \gets$ root of the subtree containing $V'$ $I_{Z, V'} \gets I(Z, V'|Z',\bar{\cal U})$ $I.add(I_{Z, V'})$ $I \gets$ sort($I$, ‘descend’) $\mathcal{T'}.add\textunderscore edge(V', Z)$ $\mathcal{T'}$
Experiments
===========
In this section we test the performance of our Sparse Boltzmann Machines on three text datasets of different scales: NIPS proceeding papers[^2], CiteULike articles[^3], and New York Times dataset[^4]. Experimental results show that SBMs perform consistently well over the three datasets in terms of model generalizability, and SBMs always give much better interpretability.
--------------- ------------ --------- ------------ --------- ------------ -----------
Validation Test Validation Test Validation Test
RS$^*$ 518 547 591 636 1,865 1,809
RS$^+$ 505 538 795 913 2,129 1,985
RS$^{+}$ SFC 532 551 632 668 2,021 1,910
RS$^+$ Pruned 542 565 **534** **584** 1,697 1,608
SBM-SFC **476** **488** 545 597 **1,624** **1,583**
--------------- ------------ --------- ------------ --------- ------------ -----------
\[table.scores\]
Datasets
--------
NIPS proceeding papers consist of 1,740 NIPS papers published from 1987 to 1999. We randomly sample 1,640 papers as training data, 50 as validation data and the remaining 50 as test data. We pre-process the data and choose 1,000 most frequent words throughout the corpus. In this way each document is represented as a vector of 1,000 dimensions, with each element being the number of times the word appears in current document.
CiteULike article collection contains 16,980 articles. Similarly, we randomly divide it into training data with 12,000 articles, validation data with 1,000 articles and test data with 3,980 articles. 2,000 words with highest average TF-IDF values are chosen to represent the articles.
The New York Times dataset includes 300,000 documents, among which we randomly pick 290,000 documents for training, 1,000 for validation and 9,000 for testing. 10,000 words with highest average TF-IDF values are chosen to represent the documents.
Training
--------
We divide the training data into mini-batches for training. The batch sizes of dataset NIPS, CiteULike and New York Times are 10, 100 and 1,000 respectively. Model parameters are updated after each mini-batch. Assuming that going through all the mini-batches counts as one epoch, we set the maximum number of training epochs to 50. And we train all the models using the Contrastive Divergence algorithm with $T=10$ full Gibbs steps.
As for RBM-based Replicated Softmax, we determine the optimal number of hidden units over the validation data with 10 units as the step size. While for Sparse Boltzmann Machines, we firstly train a two-layer HLTM and then increase the number of connections such that every hidden unit is connected to 20% of the visible units that are most correlated. A mask matrix is applied to the connection matrix after each parameter update so as to force the sparse connectivity. The numbers of hidden units automatically determined by our algorithm are 112, 194 and 326 for dataset NIPS, CiteULike and New York Times respectively.
Evaluations
-----------
The log-probability on held-out data is used to gauge the generalization performance of Replicated Softmax and Sparse Boltzmann Machines. As exactly computing these value is intractable (due to the partition function), [*Annealed Importance Sampling (AIS)*]{} [@Neal:2001:AIS:599243.599401; @salakhutdinov2008quantitative] was used in [@NIPS2009_3856] to estimate the partition function of Replicated Softmax. We extend AIS to Sparse Boltzmann Machines in our experiments. In AIS, we use 500 “inverse temperatures” $\beta_k$ spaced uniformly from 0 to 0.5, 3,000 $\beta_k$ spaced uniformly from 0.5 to 0.9, and 6,500 $\beta_k$ spaced uniformly from 0.9 to 1.0, with a total of 10,000 intermediate distributions. The estimates are averaged over 100 AIS runs for each held-out document. Then we calculate the average per-word perplexity as $exp(-\frac{1}{N}\sum_{n=1}^{N}\frac{1}{D_{n}}logP({\cal U}_n))$. A smaller score indicates better generalization performance. Due to the high computation cost, we follow the experiments in [@NIPS2009_3856] and randomly sample 50 documents from the validation data to calculate the score. While for test, we use all the 50 test documents in NIPS dataset, and randomly sample 500 documents from test data in CiteULike and New York Times datasets.
Results
-------
### Overfitting of Fully-Connected RBMs
We first empirically show that, the fully-connected structure in Replicated Softmax can easily lead to overfitting once the number of hidden units (and hence the number of parameters) gets too large. Figure \[fig.fc\] depicts the average perplexity scores over validation data for Replicated Softmax with different number of hidden units after 30 epochs. We can see that, the optimal numbers of hidden units for the three datasets are 110, 60 and 120 respectively. After that, the performances of the models get worse when the numbers of hidden units gradually increase. Therefore, selecting a proper number of hidden units is crucial to Replicated Softmax since they are very likely to overfit the training data.
[c]{}
\[fig.fc\]
### Generalizability of Sparse Boltzmann Machines and Replicated Softmax
In this part, we compare the generalization performance of Sparse Boltzmann Machines with Replicated Softmax. We denote our method as [*SBM-SFC*]{}. Two variants of Replicated Softmax included in comparison are [*RS$^*$*]{} and [*RS$^+$*]{}. [*RS$^*$*]{} trains Replicated Softmax with the optimal number of hidden units. [*RS$^+$*]{} produces Replicated Softmax with the same number of hidden units as [*SBM-SFC*]{}. Since this number is normally larger than the optimal number, we denote the method as [*RS$^+$*]{}. As we can see in Table \[table.scores\], [*SBM-SFC*]{} consistently outperforms [*RS$^*$*]{} and [*RS$^+$*]{} over the three datasets. This confirms that Replicated Softmax with full connectivity is prone to overfitting. It also shows that SBMs can lead to better model fit than fully connected RBMs. This is true even when the number of hidden units in RBMs is optimized through held-out validation. Moreover, the poor performance of [*RS$^+$*]{} shows that the performance gain of [*SBM-SFC*]{} cannot be attributed to the larger number of hidden units.
### Comparisons with Redundancy Pruning
We also compare our method with the redundancy pruning method which produces Replicated Softmax with sparse connections [@NIPS2015_5784]. We denote the method as [*RS$^+$ Pruned*]{}. It starts from a fully trained model, produced by [*RS$^+$*]{}, and prunes the connections gradually until the number of connections is reduced to be the same as the model by [*SBM-SFC*]{}. For each hidden unit, it prunes the set of connections with the smallest absolute weight value. Then it retrains the pruned model for 1 epoch, and conducts pruning again. The pruning and retraining process is repeated until the desired sparsity is reached. In our experiments, the pruning process took 80, 40 and 40 epochs on the three datasets respectively. As shown in Table \[table.scores\], [*SBM-SFC*]{} achieves comparable model fit as [*RS$^+$ Pruned*]{}. It shows that our structure learning algorithm is effective and can ease the overfitting problem of fully connected structure as well as the pruning method does. Our method has three advantages over [*RS$^+$ Pruned*]{}. First, the iterative pruning process of [*RS$^+$ Pruned*]{} is computationally expensive. Second, it does not offer a way to determine the number of hidden units. One can do this using held-out validation, but that would be computationally prohibitive. Third, as will be seen later, the models produced by [*RS$^+$ Pruned*]{} are not as interpretable as those obtained by our method.
### Necessity of Hidden Connections
In SBMs, we impose a tree structure among the hidden units. Is this necessary? To answer the question, we compare [*SBM-SFC*]{} with a method for Replicated Softmax denoted as [*RS$^{+}$ SFC*]{}. The model produced by [*RS$^{+}$ SFC*]{} is the same as that by [*SBM-SFC*]{}, except that there are no connections among the hidden units. As we can see in Table \[table.scores\], [*SBM-SFC*]{} always performs better than [*RS$^{+}$ SFC*]{}. This supports our conjecture that the hidden connections are necessary in our models. The result is not surprising. In a multiple layer model, units at a layer are connected via units at higher layers. In a two layer model, there are no higher layers. Hence it is natural to connect the second-layer units directly. To generalize our work to multiple layers, we will need to add connections only among the hidden units at the top layer.
[l cc cc cc]{} & & &\
RS$^+$ & 0.1102 & 0.1499 & 0.1407\
RS$^+$ Pruned & 0.1006 & 0.1449 & 0.1420\
SBM-SFC & **0.1235** & **0.1725** & **0.1433**\
\[table.similarity\]
### Interpretability of Sparse Boltzmann Machines and Replicated Softmax
Next we compare the interpretability of Sparse Boltzmann Machines and Replicated Softmax. Here is how we interpret hidden units. For each hidden unit, we sort the words in descending order of the absolute value of the connection weights between the words and the hidden unit. The top 10 words with the highest absolute weights are chosen to characterize the hidden unit. We propose to measure the “interpretability” of a hidden unit by considering how similar pairs of words in the top-10 list are. The similarity between two words is determined using a word2vec model [@mikolov2013efficient; @DBLP:conf/nips/MikolovSCCD13] trained on part of the Google News datasets [^5], where each word is mapped to a high dimensional vector. The similarity between two words is defined as the consine similarity of the two corresponding vectors. High similarity suggests that the two words appear in similar contexts. Let ${\cal L}$ be the list of words representing a hidden unit. We define the [*compactness*]{} of ${\cal L}$ to be the average similarity between pairs of words in ${\cal L}$. We also call it the [*interpretability score*]{} of the hidden unit. Note that some of the words in ${\cal L}$ might not be in the vocabulary of the word2vec model we use. This happens infrequently. When it does, the words are simply skipped.
-- -------------------------------------------------------
spike neuron pruning weight rules
pixel pca image pixels images
markov likelihood conditional posterior probabilities
models model modeling causal modelling
ancestral species selection duplication evolution
network networks connected topology connectivity
china beijing south\_africa mexican chinese
george\_bush laura\_bush bill\_clinton tournament jew
gene patient doctor medical physician
-- -------------------------------------------------------
: Characterizations of selected hidden units in models produced by [*SBM-SFC*]{}. Only top 5 words are listed.
\[table:topics\]
Suppose there are $F$ hidden units in a model. Let $C_1, ... C_F$ be the interpretability scores of hidden units. We define the [*interpretability score*]{} of the model as: $Q = \frac{1}{F}\sum_{f=1}^{F}C_f$. Obviously the score depends heavily on the number of hidden units.
Table \[table.similarity\] reports the interpretability scores of the models produced by [*RS$^+$*]{}, [*RS$^+$ Pruned*]{} and [*SBM-SFC*]{}. The models all have the same number of hidden units and hence their interpertability scores are comparable. [*SBM-SFC*]{} consistently performs the best over the three datasets, showing superior coherency and compactness in the characterizations of the hidden units and thus better model interpretability. Table \[table:topics\] shows the characterizations of selected hidden units in the models produced by [*SBM-SFC*]{}. They are clearly meaningful.
Conclusions
===========
Overfitting in deep models is caused not only by excessive amount of hidden units, but also excessive amount of connections. In this paper we have developed, for models with a single hidden layer, a method to determine the number of hidden units and the connections among the units. The models obtained by the method are significantly better, in terms of held-out likelihood, than RBMs where the hidden and observed units are fully connected. This is true even when the number of hidden units in RBMs is optimized by held-out validation. In comparison with redundancy pruning, our method is more efficient and is able to determine the number of hidden units. Moreover, it produces more interpretable models. In the future, we will generalize the structure learning method to models with multiple hidden layers.
[^1]: Corresponding authors.
[^2]: Available at http://www.cs.nyu.edu/ roweis/data.html
[^3]: Available at http://www.wanghao.in/data/ctrsrdatasets.rar
[^4]: Available at http://archive.ics.uci.edu/ml/datasets/Bag+of+Words
[^5]: https://code.google.com/archive/p/word2vec/
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abstract: 'We investigate the net force on a rigid Casimir cavity generated by vacuum fluctuations of electromagnetic field in three cases, de Sitter spacetime, de Sitter spacetime with weak gravitational field and Schwarzschild-de Sitter spacetime. In de Sitter spacetime the resulting net force follows the square inverse law but unfortunately it is too weak to be measurable due to the large universe radius. By introducing a weak gravitational field into the de Sitter spacetime, we find the net force now can be splited into two parts, one is the gravitational force due to the induced effective mass between the two plates, the other one is generated by the metric structure of de Sitter spacetime. In order to investigate the vacuum fluctuation force on the rigid cavity under strong gravitational field, we perform the similar analysis in Schwarzschild-de Sitter spacetime, results are obtained in three different limits. The most interesting one is when the cavity gets closer to the horizon of a blackhole, square inverse law is recovered and the repulsive force due to negative energy/mass of the cavity now has an observable strength. More important the force changes from being repulsive to attractive when the cavity crosses the event horizon, so that the energy/mass of the cavity switches the sign which suggests the unusual time direction inside the event horizon.'
author:
- Xiang Chen
title: 'Vacuum fluctuation force on a rigid Casimir cavity in de Sitter and Schwarzschild-de Sitter spacetime'
---
Introduction
============
Since Casimir discovered that there exists an attractive force between two parallel plates of infinite extent by considering the zero-point energy of the electromagnetic mode structure, “Casimir effect" has become a very popular research field. In Casimir’s original paper, the magnitude of such a kind of force in Minkowski spacetime is shown as [@Casimir] $$F(r)=\frac{\pi^2\hbar c A}{240 d^4} \label{casimir}$$ with $A$ the area of each plate and $d$ the separation. Without putting the parallel plates, the vacuum has a uniform energy density by assigning a zero-point energy of $\frac{1}{2}\hbar \omega$ to each electromagnetic mode which can be thought as a virtual photon since they are not excited and then are not on shell. But when two parallel plates are set up with a separation, the energy density of vacuum is not longer uniform and leads to a lower energy density within the two plates than that of outside which results a net force of attraction. This effect makes people realize that zero-point fluctuation is real and has the directly physical sequence. Thorough studies have gone through both the theoretical and experimental side, detailed reviews are referred to[@SKLamoreaux; @VV; @FCapasso; @KMilton; @Lamoreaux2005].\
That Casimir energy gravitates is another part of recent theoretical discovery [@SAFulling; @KAMilton] and the vacuum fluctuation force on rigid Casimir cavity in weak gravitation field has also been studied in [@GBimonte2006; @GBimonte2007; @GBimonte2008; @GBimonte2008-2; @GBimonte2009] including spin-0 and spin-1 cases. An experiment has been suggested in [@ECalloni]to measure the Casimir force in Schwarzschild metric in order to show whether the virtual quanta from vacuum fluctuation satisfies principle of equivalence.\
The above studies all embed weak gravitational field into Minkowski spacetime background, however as the astronomical evidence shows that we are actually living in an accelerated expansion universe, this gives a central place to the de Sitter geometry in cosmology[@NStraumann]. On geometrical side, de Sitter spacetime and Minkowski spacetime both stem from the class of Lorentzian manifolds. Being maximally symmetric, they admit kinematical symmetry groups having ten generators[@AEinstein] and implies a constant curvature. In Minkowski spacetime, the constant is zero while in de Sitter spacetime it is a nonzero constant(either positive or negative depends on convention). Although sharing the same features on geometrical side, they have a very different interpretation on physical side. Casimir effect in de Sitter space may be a clue to solve dark energy problem, a central and yet unsolved problem in fundamental physics and cosmology.\
The effective lagrangian and energy-momentum tensor in de Sitter space is calculated for a scalar field in [@Dowker1976-1; @Dowker1976] wherein a general zeta-function method is developed. In [@Miao], the Casimir energy of photon field in a static de Sitter space is calculated and it is proportional to the size of horizon with the same form as the holographic dark energy. In [@MRSetare] the Casimir stress on two parallel plates in de Sitter space is found for massless scalar field by applying Robin boundary conditions on the plates. Result shows that false vacuum is formed between the two parallel plates and true vacuum formed outside, while the total Casimir force leads to an attraction of the plates, which is opposite to the result in [@Miao] for photon field implying a repulsive Casimir force. Besides, the force from the boundary term is in fourth power of inverse distance between the plates for massless scalar field. Fermionic Casimir effect has been studied in de Sitter spacetime [@Saharian2009; @Saharian2011] as well as energy-momentum tensor and Casimir force for massive scalar field.\
Motivated by these studies we evaluate the vacuum fluctuation force, i.e. the total casimir force on a Casimir cavity in de Sitter spacetime, de Sitter spacetime with a weak gravitational field and Schwarzschild-de Sitter spacetime, in order to find out how spacetime background affects the virtual quanta and then leads to the difference in the vacuum fluctuation force. Such a difference may open a way to verify which universe we are actually living in.
The de Sitter Spacetime {#desitter}
=======================
In order to show the structure of the de Sitter spacetime, we start with a flat, five-dimensional spacetime $M_5$ with metric [@Miltutin] $$\begin{aligned}
\label{metric}
ds^2=-(dx^0)^2+(dx^1)^2+(dx^2)^2+(dx^3)^2-(dx^5)^2\end{aligned}$$ and set the convention $\eta_{a,b}=(-,+,+,+,-) (a,b=0,1,2,3,5)$ and $ (\mu,\nu=0,1,2,3) $. Next embed a hypersphere $H_4$ with ‘radius’ $a$ (the radius of universe) into this five-dimensional spacetime $M_5$ which is $$\begin{aligned}
-\eta_{\mu\nu}x^{\mu}x^{\nu}+(x^5)^2=a^2 \label{H4}\end{aligned}$$ This hypersphere $H_4$ is then called the de Sitter spacetime, a maximally symmetric subspace of $M_5$. From Eq.(\[H4\]) we have $ (dx^5)^2=(\eta_{\mu\nu}x^{\mu}dx^{\nu})^2/(x^5)^2$ and substitute it into Eq.(\[metric\]). The line element on $H_4$ now becomes $$\begin{aligned}
ds^2=\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\frac{(\eta_{\mu\nu}x^{\mu}dx^{\nu})^2}{a^2+\eta_{mn}x^mx^n}\end{aligned}$$ For any maximally symmetric space, the curvature is the same at each point. So that the metric, Christoffel connection and curvature tensor can be found by consider the vicinity of $x^{\mu}=0$ only, $$\begin{aligned}
g_{\mu\nu}=\eta_{\mu\nu}+\frac{x_{\mu} x_{\nu}}{a^2}, \,\,\, \Gamma^{\mu}_{\nu\rho}=\frac{1}{a^2}x^{\mu} \eta_{\nu\rho}, \, \, \, R_{\mu\nu\rho\sigma}=\frac{1}{a^2}(\eta_{\mu\rho}\eta_{\nu\sigma}-\eta_{\mu\sigma}\eta_{\nu\rho})\end{aligned}$$ and $$g^{\mu\nu}=\eta^{\mu\nu}-\frac{x^{\mu}x^{\nu}}{a^2}\label{gtmunu}$$ where $x^{\mu}=(-x_0,x_1,x_2,x_3) $ after dropping the higher order term. The Ricci tensor is found to be proportional to the metric $$R_{\mu\nu}=\frac{3}{a^2}g_{\mu\nu}\label{Ricci}$$ leading to a constant scalar curvature $$R=12/a^2 \label{scalarR}$$ These are key ingredients in Einstein field equation with cosmological constant term [@Einstein1916], $$R_{\mu\nu}+\Lambda g_{\mu\nu}-\frac{1}{2}R g_{\mu\nu}=\frac{8\pi G}{c^4} T_{\mu\nu} \label{EFE}$$ where $\Lambda $ is the cosmological constant, $G$ is Newton’s gravitational constant, $c$ is the speed of light in vacuum and $T_{\mu\nu}$ the stress-energy tensor or energy-momentum tensor. Substitute Eq.(\[Ricci\])(\[scalarR\]) into Eq.(\[EFE\]), we can see immediately that de Sitter space is actually a vacuum solution to Einstein equation if the positive cosmological constant is set as $$\Lambda =\frac{3}{a^2}$$ The cosmological constant term can be treated as a part of energy-momentum tensor and put on the right side of Einstein equation, more detailed discussion can be found in[@Weinberg1989].
Vacuum fluctuation force in de Sitter space
============================================
Suppose there are two identical parallel plates with proper area $A$ and proper separation $d$, one is placed at the origin $(0,0,0)$ and the other one is at $(0,0,d)$ with normal vector of the plates in $z$ direction.\
In Minkowski spacetime, following the analysis of [@LSBrown], the regularized energy-momentum tensor $\langle T^{\mu\nu}\rangle $ of quantum electrodynamics is $$\langle T^{\mu\nu}\rangle =\frac{\pi^2 \hbar c}{180 d^4}(\frac{1}{4}g^{\mu\nu}- \hat{z}^{\mu}\hat{z}^{\nu})\label{emtensor}$$ here $g^{\mu\nu}$ is the space-time metric and $\hat{z}^{\mu}=(0,0,0,1)$ is the unit space-like 4-vector in z-direction which is orthogonal to the surface of two plates. This is the energy-momentum tensor of matter, without including the gravitational fields even if we generate this formula to curved space-time. While in de Sitter spacetime(3+1 dimension), the enery-momentum tensor for a conformally coupled massless($\xi=1/6$) scalar field between two plates is given in[@Elizalde] $$\begin{aligned}
\langle T^{\mu}_{\nu}\rangle &=& \langle T^{\mu}_{\nu}\rangle_{dS}+\frac{e^{-4t/a^2}}{6\pi^{2}}\text{diag}(1,1, 1, -3)
\times \int_0^{\infty} dx \frac{x^3}{c_1(x)c_2(x)e^{2dx}-1} \label{desittensor}\end{aligned}$$ where $ \langle T^{\mu}_{\nu}\rangle_{dS}$ is the renormalized vacuum expected value of energy-momentum tensor without plates. It was shown in [@Bunch1978] that $ \langle T^{\mu}_{\nu}\rangle_{dS}$ is proportional to the metric tensor of de Sitter spacetime which is $$\begin{aligned}
\langle T^{\mu\nu}\rangle_{dS}= -\frac{1}{8\pi}(\frac{1}{6}-\xi) R g^{\mu\nu}\end{aligned}$$ In Eq.(\[desittensor\]), for Dirichlet boundary condition $c_i(x)=-1$ while for Neumann boundary condition $c_i(x)=1$ with $i=1,2$ represents the plate 1 and plate 2 respectively. For a conformally coupled massless scalar field with $\xi=1/6$, $\langle T^{\mu}_{\nu}\rangle_{dS}$ vanishes. Notice that $$\begin{aligned}
\text{diag}(1,1,1,-3)=4(\frac{1}{4}g^{\mu}_{\nu}- \hat{z}^{\mu}\hat{z}_{\nu})\end{aligned}$$ Therefore, the energy-momentum tensor for conformally coupled massless scalar field can be rewritten as $$\begin{aligned}
\langle T^{\mu\nu}\rangle &=& K(\frac{1}{4}g^{\mu\nu}- \hat{z}^{\mu}\hat{z}^{\nu})\label{simtensor}\end{aligned}$$ where $$K=\frac{2e^{-4t/a^2}}{3\pi^{2}} \int_0^{\infty} dx \frac{x^3}{c_1(x)c_2(x)e^{2dx}-1} \nonumber$$ Since according to [@Elizalde], the electromagnetic field in $D=3$ is conformally invariant so that the Casimir problem dealing with two perfectly conducting parallel plates can be reduced to the corresponding problem with two scalar modes with Dirichlet boundary conditions and Neumann boundary conditions. Therefore the energy-momentum tensor for electromagnetic field in de Sitter space would be twice of the one expressed in (\[simtensor\]), which is $$K_{em}=2K = \left\{
\begin{array}{lr}
\frac{\pi^2 \hbar c e^{-4t/a^2}}{180d^4} & : c_1(x)=c_2(x)=1, -1 \\
-\frac{7\pi^2 \hbar c e^{-4t/a^2}}{1440d^4} & : c_1(x)=-c_2(x)=1, -1
\end{array}\label{Kem}
\right.$$ Here $c_1(x)=c_2(x)=1,-1$ represents that Dirichlet BCs or Neumann BCs are used for both plates, while for $c_1(x)=-c_2(x)=1,-1$, Dirichlet BCs is used for one plate and Neumann BCs used for the other. Recall the age of universe $t\sim 10^{17} s$ and the radius $a\sim 10^{26}$, we can safely replace $e^{-4t/a^2}$ by 1 and omit it in the rest of our paper. The energy-momentum tensor of electromagnetic field can then be simplified as $$\begin{aligned}
\langle T^{\mu \nu}\rangle_{em} &=& K_{em} (\frac{1}{4}g^{\mu\nu}- \hat{z}^{\mu}\hat{z}^{\nu})\label{desitemtensor}\end{aligned}$$
It is worth to notice that the energy-momentum tensor of electromagnetic field here does not has vanishing divergence, $$\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}g^{\mu\nu}+\nabla_{\mu} (\hat{z}^{\mu}\hat{z}^{\nu})=\nabla_{\mu} (\hat{z}^{\mu}\hat{z}^{\nu}) \ne 0$$ This is not consistent with the usual covariant conservation of energy-momentum tensor for isolated system[@Dewitt1975]. A further consideration reveals that the Casimir cavity contains two parts, one is the vacuum fluctuation within the two plates including the energy-momentum tensor from Maxwell action, gauge-breaking term and ghost action(Eq.(2.8) in [@GBimonte2006]), the energy-momentum tensor of which is covariant conserved. The other part, i.e. $\hat{z}^{\mu}\hat{z}^{\nu}$ is the boundary set by the two plates, the divergence of which vanishes in Minkowski spacetime, but no longer in curved spacetime, and a net force will be induced on this system which is, $$\begin{aligned}
\nabla_{\mu}T^{\mu\nu}=f^{\nu}.\end{aligned}$$ However the two-plate cavity is an isolated system in curved spacetime, there should be no external force exerted on it besides the gravitation force which can be treated as a spacetime background. Therefore we have to modify this boundary term in order to have a vanishing divergence so that the covariant conservation of the energy-momentum holds, and the new boundary term should recover the boundary term $z^{\mu}z^{\nu}$ when going from general spacetime back to Minkowski spacetime. To meet the first requirement, we have to give up the terms like $\eta^{\mu\nu},n^{\mu}n^{\nu}, \eta^{\mu\rho}n_{\rho}n^{\nu}, n^{\mu}n_{\rho}\eta^{\rho\nu}$ with $n^{\mu}$ the unit direction vector, and the only terms left will be $g^{\mu\nu},g^{\mu 3}g^{\nu 3}$ due to the metric compatibility condition $$\begin{aligned}
\nabla_{\mu}g^{\nu\rho}=0\label{compatibility}.\end{aligned}$$ To meet the second requirement, we have to discard $g^{\mu\nu}$ since it reduces to $\eta^{\mu\nu}$ instead of $z^{\mu}z^{\nu}$ in Minkowski spacetime. So the only term that meets both requirement is $g^{\mu 3}g^{\nu 3}$ and it is easy to check that $g^{\mu 3}g^{\nu 3}\rightarrow \eta^{\mu 3}\eta^{\nu 3}=\hat{z}^{\mu}\hat{z}^{\nu}$ when going from general curved spacetime to Minkowski spacetime. Therefore we can rewrite the original energy-momentum in (\[desitemtensor\]) as $$\begin{aligned}
\langle T^{\mu \nu}\rangle_{em} &=& K_{em} (\frac{1}{4}g^{\mu\nu}- g^{\mu 3}g^{\nu 3})\label{newemtensor}.\end{aligned}$$ Now it is easy to check that this new energy-momentum satisfies the covariant conservation, i.e. $\nabla_{\mu}T^{\mu\nu}=0$ according to the metric compatibility condition shown in Eq.(\[compatibility\]). The correct form of this formula as the energy-momentum tensor in general curved space-time can be obtained from a more general consideration. The regularized and renormalized energy-momentum tensor of spin-1 field in curved spacetime was discussed in [@Christensen1978], where the energy-momentum tensor for Maxwell field, gauge fixed field and ghost field were calculated via point-split method. For massless photons, the renormalized energy-momentum tensor can be expressed as (see Eq.(5.4)in [@Christensen1978]) $$\begin{aligned}
\langle T^{\mu \nu}\rangle_{\text{Maxwell}}+\langle T^{\mu \nu}\rangle_{\text{gauge}}+\langle T^{\mu \nu}\rangle_{\text{ghost}}&=&\lim_{x^{'}\rightarrow x}
\frac{1}{4}[\tensor{G}{_\lambda^{\lambda \mu\nu}}+\tensor{G}{_\lambda^{\lambda \nu\mu}}+\tensor{G}{^{\mu\nu\lambda}_\lambda}+\tensor{G}{^{\nu\mu\lambda}_\lambda}\nonumber\\
&&\hspace{7ex}-\tensor{G}{^{\lambda\mu\nu}_\lambda}-\tensor{G}{^{\lambda\nu\mu}_\lambda}-\tensor{G}{^{\mu\lambda}_\lambda^\nu}
-\tensor{G}{^{\nu\lambda}_\lambda^\mu}\nonumber\\
&&\hspace{7ex}-(\tensor{G}{_\lambda^\lambda_\rho^\rho}-\tensor{G}{_{\lambda\rho}^{\rho\lambda}})g^{\mu\nu}]\propto g^{\mu\nu}\end{aligned}$$ which gives the same form as the first term in (\[newemtensor\]) to the first order of metric. Here $G$ represents the Green’s function. This also shows that the first term of (\[newemtensor\]) arises from the contribution of Maxwell field, gauge fixed field and ghost field. While the second term arises from the boundary, and we find that only in that form it is able to satisfy the covariant conservation and reduce to the second term in (\[emtensor\]) when going back from curved spacetime to Minkowski spacetime.
Following the formula given in (\[newemtensor\]) and use the metric of de-sitter space in (\[gtmunu\]), we find the explicit form of energy-momentum tensor in de Sitter space (all the calculations below are accurate to order $(x/a)^2$) $$\begin{aligned}
\langle T^{\mu\nu}\rangle_0=\frac{K_{em}}{4}\left[\begin{array}{cccc}
-(1+\frac{x_0^2}{a^2}) & \frac{x_0x_1}{a^2} & \frac{x_0x_2}{a^2} &-\frac{3x_0x_3}{a^2} \\
\frac{x_0x_1}{a^2} & 1-\frac{x_1^2}{a^2} & -\frac{x_1x_2}{a^2} & \frac{3x_1x_3}{a^2} \\
\frac{x_0x_2}{a^2} & -\frac{x_1x_2}{a^2} &1-\frac{x_2^2}{a^2} & \frac{3x_2x_3}{a^2} \\
\frac{-3x_0x_3}{a^2} & \frac{3x_1x_3}{a^2} & \frac{3x_2x_3}{4a^2} &-3+\frac{7x_3^2}{a^2}
\end{array} \right] \label{Tmunu}\end{aligned}$$ where $\langle \rangle_0 $ denotes the energy-momentum for vacuum. It is easy to check the trace of energy-momentum tensor vanishes $$T^{\mu}_{\mu}=g_{\mu\nu}T^{\mu\nu}=0$$ the vanishing trace reflects the invariance of scale transformation, i.e. conformally invariant. If $a\rightarrow \infty $, we recover the energy-momentum in Minkowski spacetime [@LSBrown] $$T^{\mu\nu}=\frac{K_{em}}{4}(-1,1,1,-3)$$
To find the vacuum fluctuation force density we will however take the expression derived in [@CMoller] $$\begin{aligned}
f_{\mu}=-\frac{1}{\sqrt{-g}}\frac{\partial}{\partial x^{\nu}}(\sqrt{-g}T^{\nu}_{\mu})+\frac{1}{2}\frac{\partial g_{\rho\sigma}}{\partial x^{\mu}} T^{\rho\sigma}\label{fmu}\end{aligned}$$ where $$g=det(g_{\mu\nu})=(x_0^2-x_1^2-x_2^2-x_3^2-a^2)/a^2$$ Substituting (\[Tmunu\]) into (\[fmu\]) we find the density of force $$\begin{aligned}
f_{\mu} & =& f_{1\mu}+f_{2\mu}\nonumber\\
& = & \frac{K_{em}}{4a^2}(x_0,-x_1,-x_2,-9x_3)+\frac{K_{em}}{4a^2}(-x_0,x_1,x_2,-3 x_3)\nonumber\\
&=&(0,0,0,\frac{-3K_{em} x_3}{a^2})\label{fmu1}\end{aligned}$$ where $f_{1\mu}, f_{2\mu}$ refer to the first and second term of Eq.(\[fmu\]) respectively. This shows that after the cancelation between $f_{1\mu}$ and $f_{2\mu}$, only the third component(which is z direction) has a non-vanishing value as we expect. And the non-vanishing value is linear with $x_3 $ ($x_3=z$), Integrate the force density Eq.(\[fmu1\]) over the volumn, we find the $z$ component of the net force on rigid cavity, $$F_z=A \int_0^d\sqrt{-g} f_z dz=\frac{-3 K_{em} d^2}{2a^2}= \left\{
\begin{array}{lr}
-\frac{\pi^2 A}{120 a^2}\frac{\hbar c}{d^2} & : c_1(x)=c_2(x)=1, -1 \\
\frac{7\pi^2 A}{960 a^2}\frac{\hbar c}{d^2} & : c_1(x)=-c_2(x)=1, -1
\end{array} \label{integral}
\right.$$ here $A$ is the area of each plate. Obviously, the force is attractive when BCs is $c_1(x)=c_2(x)=1,-1$, i.e. Dirichlet or Neumann BCs on each plate, and is repulsive when BCs is $c_1(x)=-c_2(x)=1,-1$, i.e. Dirichlet on one plate and Neumann BCs on the other. This is consistent with [@Elizalde] in scalar field case. Notice that the force here is proportional to $d^{-2}$, following the square inverse law instead of usually being inverse proportional to fourth power of the separation between two plates, like the expression in (\[casimir\]). However due to its dependence of $a^{-2}$ and the very large radius $a$ of our universe, this force is too small to be measured and can actually be neglected, so that there is no significant net Casimir force on the rigid cavity. This result is consistent with the calculation in Eq.(67) of [@Elizalde], where the two plates have the same magnitude of pressure but with opposite direction, leading to a zero net force on such a two-plate system.
Vacuum Fluctuation Force with Weak Gravitation Field in de-Sitter Spacetime
===========================================================================
In the previous analysis, we consider the vacuum fluctuation from only de-sitter background without explicit gravitational field. If however the two parallel plates are located somewhere above the earth, the gravitational field at that point, though weak, may have some negligible effect on the quantum vacuum fluctuations of electromagnetic field in the region between the two plates. Assuming the gravitational field is in negative $z$-direction, the metric of space-time now becomes (similar way in[@CWMisner]) $$g_{\mu\nu}=\eta_{\mu\nu}+\frac{x_{\mu}x_{\nu}}{a^2}-2 A^{\rho}x_{\rho}\delta_{\mu 0}\delta_{\nu 0}\label{gmunu2}$$ with $A^{\mu}=(0,0,0,g/c^2) $, $g$ is the magnitude of earth’s gravitational acceleration. Since the gravitational field is weak, we only keep the first-order corrections to the line element and ignore the higher order[@ECalloni], $$\langle T^{\mu\nu}\rangle_0=\frac{K_{em}}{4}\left[\begin{array}{cccc}
-(1+\frac{x_0^2}{a^2})+2A_3x_3 & \frac{x_0x_1}{a^2} & \frac{x_0x_2}{a^2} &-\frac{3x_0x_3}{a^2} \\
\frac{x_0x_1}{a^2} & 1-\frac{x_1^2}{a^2} & -\frac{x_1x_2}{a^2} & \frac{3x_1x_3}{a^2} \\
\frac{x_0x_2}{a^2} & -\frac{x_1x_2}{a^2} &1-\frac{x_2^2}{a^2} & \frac{3x_2x_3}{a^2} \\
-\frac{3x_0x_3}{a^2} & \frac{3x_1x_3}{a^2} & \frac{3x_2x_3}{a^2} &-3+7\frac{x_3^2}{a^2} \label{Tmunu2} \end{array} \right].$$ Following the same path as before, we find the density of force for weak gravitational field in de-sitter space, with the first term and second term in (\[fmu\]) given as $$\begin{aligned}
f_{1\mu} &=& \frac{K_{em}}{4a^2}\bigg(-x_0(1-3A_3x_3),\,\,-x_1(1+5A_3x_3),\,\,-x_2(1+5A_3x_3),\nonumber\\
& & \hspace{10ex}3A_3(x_0^2-x_1^2-x_2^2+a^2-3)-10A_3 x_3^2\bigg)\\
f_{2\mu}& = & \frac{K_{em}}{4a^2}\bigg(x_0(1+A_3x_3),\,\,x_1(1+A_3x_3),\,\,x_2(1+A_3x_3), A_3(x_0^2+x_3^2+a^2)-3x_3\bigg)\end{aligned}$$ Add the two term together we have the net Casimir force density on the two plates system, $$\begin{aligned}
f_{\mu} & =& f_{1\mu}+f_{2\mu}\nonumber\\
&=&\frac{K_{em}}{a^2}\bigg(x_0(A_3x_3),\,\,3x_1(A_3x_3)/2,\,\,3x_2(A_3x_3)/2,\nonumber\\
& &\hspace{10ex} A_3(a^2+x_0^2-3x_1^2/4-3x_2^2/4-9x_3^2/4)-3x_3\bigg)\nonumber\\
&\simeq &\bigg(0,\,\,0,\,\,0,\,\,K_{em} A_3-\frac{3K_{em} x_3}{a^2}\bigg)+\mathcal{O}((x/a)^2)\label{fmu2}\end{aligned}$$ In the second to the last line we only keep the linear term of space-time coordinates and omit the higher order terms since they are small and negligible. Integrate (\[fmu2\]) over the volume as we show in(\[integral\]) to find the net force that exerts on the two plates cavity, $$F_z=K_{em} A(A_3d-\frac{3d^2}{2a^2})=F_{g}+F_{d}\label{netF1}$$ where $$F_g=K_{em}A A_3 d= \left\{
\begin{array}{lr}
\frac{\pi^2\hbar A }{180 c d^3}g & : c_1(x)=c_2(x)=1, -1 \\
-\frac{7\pi^2\hbar c A}{1440 c^2 d^3} g & : c_1(x)=-c_2(x)=1, -1
\end{array}
\right.$$ $$F_d=-K_{em}A\frac{3d^2}{2a^2}= \left\{
\begin{array}{lr}
-\frac{\pi^2\hbar c A }{120 a^2 d^2} & : c_1(x)=c_2(x)=1, -1 \\
\frac{7\pi^2\hbar c A }{960 a^2 d^2} & : c_1(x)=-c_2(x)=1, -1
\end{array}
\right.$$
Here $F_g$ as the first part of the Casimir force is proportional to the gravitational field strength $g$, therefore it can be considered as the gravitational force with effective mass $ \frac{-\pi^2\hbar A}{180 c d^3}$ which is in full agreement with Eq.(8) in [@ECalloni] and Eq.(5.4) in [@GBimonte2006] or $\frac{7\pi^2\hbar c A}{1440 c^2 d^3}$ depend on BCs. When both plates are in the same BCs, i.e. Dirichlet or Neumann BCs, the force $F_g$ orientates opposite to the gravitational field, so that the effective mass is considered to be negative. When two plates are in different BCs, the force $F_g$ follows the direction of gravitational field, so the effective mass should be positive. The negative and positive effective mass reflects that the energy density between the two plates is smaller or larger than the outside respectively.
We also notice that the gravitational force is proportional to $d^{-3}$ and $\frac{1}{4}$ of $F_g$ is from the energy density $T^{00}$, while the rest is from the third component of energy-momentum tensor. According to the argument presented in [@ECalloni], the net force that can be measured in experiment with BCs Dirichlet or Neumann for both plates is, $$\begin{aligned}
F=\frac{1}{4}F_g=\frac{\pi^2\hbar A}{720 c d^3}\end{aligned}$$ This reproduces Eq.(4.13) in [@GBimonte2007] and Eq.(9) in [@ECalloni] for a weak gravitational field in Minkowski space-time. The second part of Casimir force in (\[netF1\]) is $F_d$, which has no dependence on the gravitation field $g$, implying that this force is purely from the de-sitter spacetime. And $F_d$ is inverse proportional to the square of radius $a$ as well as the square of the separation $d$ between the two plates. Though the separation $d$ can be made very small in experiment, the radius of universe $a$ is so large that $a^2d^2$ is still large and can be ignored, we may expect $F_g$ to dominate the Casimir force $F_z$. To confirm this, we calculate the critical value of $d$ which is $d_c=\frac{2 g a^2}{3 c^2}\approx 10^{35} m $ so that $F_g=F_d$ by taking $a \approx 10^{26}$ m and $ g\approx 10$ m$^2$/s. The critical value here is even larger than the present radius of our universe. Such an unreachable distance implies the first term dominates and the second term is negligible. The Casimir force arises from de-Sitter space being negligible implies that the structure of our universe can not be determined by observing vacuum fluctuation of quantum electrodynamics in weak gravitational field. In order to see how things change in strong gravitational field, we will adopt the more general metric form i.e. Schwarzschild-de Sitter spacetime that can deal with strong gravitational field and perform similar analysis in the next section.
Vacuum fluctuation force in Schwarzschild-de Sitter spacetime
=============================================================
The static Schwarzschild-(anti-)de Sitter metric is expressed in spherical coordinates as[@Gibbons1977; @Bousso1998; @Podolsky1999; @Cardoso2001; @Cardoso2002] in signature (-, +,+,+), $$ds^2=-(1-f)dt^2+\frac{1}{1-f}dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2) \label{schwarz-desitter}$$ with $f=(\frac{2GM}{c^2 r}+\frac{\Lambda r^2}{3}) $ where $M$ is the mass of the black hole or any spherical source of gravitational field. When $\Lambda=0$, (\[schwarz-desitter\]) reduces to Schwarzschild metric and when $M=0$ it reduces to de(anti-de) Sitter space. As shown in [@Mackay] the metric tensor can be written in Cartesian coordinates $x=r\sin{\theta}\cos{\phi}, y=r\sin{\theta}\sin{\phi} $ and $z=r\cos{\theta}$ as $$\begin{aligned}
g_{\mu\nu}=\left[\begin{array}{cccc}
-(1-f) & 0 & 0 & 0 \\
0 & 1+\frac{f x^2}{r^2(1-f)} & \frac{f x y}{r^2(1-f)} & \frac{f x z}{r^2(1-f)} \\
0 & \frac{f x y}{r^2(1-f)} & 1+\frac{f y^2}{r^2(1-f)} & \frac{f y z}{r^2(1-f)} \\
0 & \frac{f x z}{r^2(1-f)} & \frac{f y z}{r^2(1-f)} & 1+ \frac{f z^2}{r^2(1-f)}
\end{array}\right]
\label{s-dsitgmunu}\end{aligned}$$ combine with Eq.(\[emtensor\]) we have energy-momentum tensor $$\begin{aligned}
T_{\mu\nu}=\frac{K_{em}}{12} \left[\begin{array}{cccc}
-\frac{3}{1 - 2 \frac{G M}{c^2 r} - \Lambda r^2/3}, & 0 & 0 & 0 \nonumber\\
0 & 3 -\frac{6 G M x_1^2}{c^2 r^3} - \Lambda x_1^2& - x_1 x_2(\frac{6 G M x_1^2}{c^2 r^3} + \Lambda) &
- x_1 x_3 (\frac{6 G M x_1^2}{c^2 r^3} + \Lambda) \nonumber\\
0 & - x_1 x_2 (\frac{6 G M x_1^2}{c^2 r^3} + \Lambda) & 3 -x_2^2 (\frac{6 G M x_1^2}{c^2 r^3} + \Lambda) &
- x_2 x_3(\frac{6 G M x_1^2}{c^2 r^3} + \Lambda) \nonumber\\
0 & - x_1 x_3 (\frac{6 G M x_1^2}{c^2 r^3} + \Lambda) & - x_2 x_3 (\frac{6 G M x_1^2}{c^2 r^3} + \Lambda) &
-9 - x_3^2 (\frac{6 G M x_1^2}{c^2 r^3} + \Lambda)
\end{array}\right]\\
\label{Sch-de-tensor}\end{aligned}$$ It is worth to note here that the direct evaluation of the regularized and renormalized energy-momentum tensor in Schwarzschild-de sitter spacetime with boundary remains a challenging task due to its complication in calculation, where the point-split method should be used for this purpose. Further evaluation will be considered in our future research.
Now substitute the energy-momentum tensor in Eq.(\[Sch-de-tensor\]) into Eq.(\[fmu\]) we finally obtain the Casimir force density $$\begin{aligned}
f_{\mu} &=&
K_{em} \bigg\{-\frac{27 x_3}{r^4}\bigg[8 A_3^3 r^3 +
4 A_3^2 r^2 (-2 + 3 \epsilon)+
A_3r (3 - 8 \epsilon + 6 \epsilon^2)+ (-2 + \epsilon) \epsilon^2\bigg]\times \nonumber\\
&& \,\, \, \times (0, x_1x_3, x_2x_3, r^2+x_3^2) +\frac{81 x_3}{r^2}(0, 0, 0, A_3 r-\epsilon )\bigg\}/(-3 + 6 A_3 r + 3 \epsilon)^3\end{aligned}$$ where we define $A_3=\frac{GM}{c^2 r^2}, \epsilon=\frac{r^2}{a^2}$ and $a$ is related to $\Lambda $ through $\Lambda =3/a^2$. The total force is obtained by integrate out the force density $f_{\mu}$, let the origin be in the center of the lower plate with length $L_1$ and width $L_2$, the two parallel plates are located at $(0,0,r)$ and $(0,0,r+d)$. We then have $$\begin{aligned}
F &=& \int_r^{r+d}\int_{-L_2/2}^{L_2/2}\int_{-L_1/2}^{L_1/2} f_{\mu} dx_1 dx_2 dx_3 = (0, 0, 0, F_z) \nonumber\\
F_z &=&\frac{K_{em} L_1 L_2}{(1- \delta)^5}\bigg\{ \frac{d}{r}\bigg[\frac{3 r^2}{a^2}+\frac{3}{2}\delta -7 \delta^2 + \frac{23}{2}\delta^3 -
8 \delta^4 +2 \delta^5 -\frac{r^2}{2a^2} (19\delta - 17 \delta^2 + 4 \delta^3)\bigg] \nonumber\\
&&+\frac{d^2}{r^2}\bigg[\frac{3r^2}{2a^2}-\frac{3}{2}\delta + \frac{27}{4} \delta^2 -\frac{37}{4} \delta^3 + 5 \delta^4 -
\delta^5- \frac{r^2}{2a^2}(15 \delta - 24 \delta^2 + 6 \delta^3)\bigg]\bigg\}
\label{totforce}\end{aligned}$$ here $ \delta = 2A_3r$ and far less than 1 when $r \gg r_c=\frac{2GM}{c^2} $ with $r_c$ is the horizon of a black hole, but it gets close to 1 when the Casimir cavity is near the horizon.
Now we consider several limits and for simplicity we only consider the BCs with both plates are in Dirichlet or Neumann. For the BCs with one plate in Dirichlet and the other in Neumann, the results are similar except for a factor difference including a negative sign.
(1)$ M \rightarrow 0 $ i.e. $ \delta \rightarrow 0 $, the Schwarzschild-de Sitter spacetime then reduces to pure de Sitter spacetime, Eq.(\[totforce\]) becomes $$F_z = K_{em} A \frac{3d^2}{2a^2}= \frac{\pi^2 A}{120 a^2}\frac{\hbar c}{d^2}$$ which is the same as Eq.(\[integral\]) in pure de Sitter space under the BCs we have specified before.
\(2) $ a \rightarrow \infty $, this reduces Schwarzschild-de Sitter spacetime to Schwarzschild spactime, and Eq.(\[totforce\]) becomes $$F_z =\frac{K_{em} A}{(1- \delta)^5}\bigg\{ \frac{d}{r}(\frac{3}{2}\delta -7 \delta^2 + \frac{23}{2}\delta^3 -
8 \delta^4 +2 \delta^5)+\frac{d^2}{r^2}(-\frac{3}{2}\delta + \frac{27}{4} \delta^2 -\frac{37}{4} \delta^3 + 5 \delta^4 -
\delta^5)\bigg\}$$ For weak gravitation field limit, $\delta$ is small and only its first power should be kept so that $$F_z =K_{em} A ( \frac{3d}{2r}\delta -\frac{3d^2}{2r^2})\delta \simeq K_{em} A \frac{3d}{2r}\delta = 3K_{em} A d A_3=\frac{\pi^2\hbar c A}{60 c^2 d^3}g
\label{netF2}$$ which is three times larger as Eq.(\[netF1\]) where we add uniform weak gravitation field into de Sitter space. At first sight, this seems to be inconsistent. However if we carefully compare the metric of spacetime in (\[gmunu2\]) and (\[schwarz-desitter\]), since in (\[gmunu2\]) $g_{\mu\nu}=\eta_{\mu\nu}-2 A^{\rho}x_{\rho}\delta_{\mu 0}\delta_{\nu 0}$ as $a \rightarrow \infty$, while in (\[schwarz-desitter\]) due to the extra contribution from $g_{rr}=1/(1-f) $ in Schwarzschild-de Sitter metric(isotropic but not uniform) which is different in metric (\[gmunu2\]) where $g_{rr}=1 $, so that to the first order the metric in (\[schwarz-desitter\]) can be written as[@GBimonte2008] $$\begin{aligned}
ds^2=-(1+2A_{\rho}x_{\rho})dt^2+(1-2A_{\rho}x_{\rho})d\vec{r}^2\end{aligned}$$ where the first term is exactly the same as that in (\[gmunu2\]), but the second term contains an extra contribution which is $-2A_{\rho}x_{\rho}$, it is this extra term that leads to a factor of 3 in (\[netF2\]). The essential difference lies in that we embed gravitational field in Minkowski spacetime in the last section, while here we are dealing with the Schwarzschild spacetime background when taking $a \rightarrow 0$.
(3). For strong gravitation field limit, we consider the two parallel plates located close to the horizon of a black hole that is $r\rightarrow r_c^{+}$ so that $\delta \rightarrow 1^{-}$, now we have $$F_z =\lim_{\delta \rightarrow 1^{-}}\frac{K_{em} A}{(1- \delta)^5}\frac{3d^2}{a^2}=\lim_{\delta \rightarrow 1^{-}}\frac{1}{(1- \delta)^5}\frac{\pi^2\hbar c A}{60 a^2 d^2}$$ As $(1-\delta)$ gets close to $0$, the net Casimir force on the two parallel plates system can be large enough to be observable and its direction is outwards which is similar to the case in[@GBimonte2008], acting as a repulsive force between the plates and blackhole. The repulsive force is due to the fact that energy/mass of the cavity is negative. But following the equivalent principle, the cavity will experience a free falling and fall into the blackhole. The interesting part here is that the sign of the force will change when the cavity passes through the event horizon, i.e. it changes from repulsive to attractive, therefore the energy/mass of the cavity changes from negative to positive when crossing the event horizon. As energy and time are conjugate variables in quantum mechanics, the changes of the sign in energy may suggest the unusual direction of time flow inside the event horizon.
Conclusions
===========
We have considered three cases of the vacuum fluctuation force on a rigid Casimir cavity with two parallel plates separated by a distance $d$ in de Sitter spacetime. In the first and simplest one, we only consider de Sitter space. By investigating on the energy-momentum tensor of electromagnetic field in de Sitter spacetime, we find out that there exists a net force on the rigid cavity which satisfies square-inverse law though too small to be measurable. Such a force is attractive when two plates are both in Dirichlet BCs or Neumann BCs, and it is repulsive when one plate is in Dirichlet BCs while the other one is in Neumann BCs. In the second case, we introduce a weak uniform gravitational field in de Sitter spacetime, and find that the vacuum fluctuation force on the two parallel plates has two decoupled parts to the order of $x^2$, one($F_g$) is proportional to the weak gravitational field strength $g$ and has little dependence on the radius of universe, indicating that it is generated by the weak gravitational field acting on the effective mass induced by the difference in energy density between the region inside the two plates and the region outside the two plates. The effective mass is negative if we apply Dirichlet or Neumann BCs on both plates, and positive if one plate is applied Dirichlet BCs and the other one is applied Neumann BCs. The other part ($F_d$) of the force is from the vacuum fluctuation due to de Sitter spacetime structure only and is inverse proportional to the square of radius of universe, but independent on the weak gravitational field. Due to the large radius of the present universe, the first part $F_g$ dominates this vacuum fluctuation force and it is measurable with a large area of each plate and a small gap between them even though the gravitational field strength as small as that on the earth.
Finally we study the vacuum fluctuation force in Schwarzschild-de Sitter spacetime, a more general spacetime structure so that strong gravitation field can be included. After the final result is obtained we take several limits, in zero gravitational field limit we recover the result in pure de Sitter space; in Schwarzschild spacetime limit with weak gravitation field we can reproduce the Casimir force in Minkowski spacetime with gravitation field except for a factor difference; in strong gravitational field limit the net Casimir force on rigid cavity near event horizon is found to be inverse proportional to the square of the separation between two parallel plates and the square of radius of universe, again follows the square inverse law. The force is repulsive and becomes larger as the cavity gets closer to the event horizon of blackhole. When the cavity crosses the event horizon, the repulsive force switches into attractive force, leading to the change from negative energy to positive energy within the cavity. This implies the unusual direction of time flowing inside the event horizon.
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---
abstract: 'Inductive and coinductive structures are everywhere in mathematics and computer science. The induction principle is well known and fully exploited to reason about inductive structures like natural numbers and finite lists. To prove theorems about coinductive structures such as infinite streams and infinite trees we can appeal to bisimulation or the coinduction principle. Pure inductive and coinductive types however are not the only data structures we are interested to reason about. In this paper we present a calculus to prove theorems about mutually defined inductive and coinductive data types. Derivations are carried out in an infinitary sequent calculus for first order intuitionistic multiplicative additive linear logic with fixed points. We enforce a condition on these derivations to ensure their cut elimination property and thus validity. Our calculus is designed to reason about linear properties but we also allow appealing to first order theories such as arithmetic, by adding an adjoint downgrade modality. We show the strength of our calculus by proving several theorems on (mutual) inductive and coinductive data types.'
author:
- Farzaneh Derakhshan
- Frank Pfenning
title: 'Circular Proofs in First-Order Linear Logic with Least and Greatest Fixed Points'
---
<ccs2012> <concept> <concept\_id>10011007.10011006.10011008</concept\_id> <concept\_desc>Software and its engineering General programming languages</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003456.10003457.10003521.10003525</concept\_id> <concept\_desc>Social and professional topics History of programming languages</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
Introduction
============
The induction principle is well known and presented in the literature in many different contexts. Computer scientists use this principle to reason about inductive data types such as natural numbers and finite lists. To show properties of coinductive data types, e.g. streams and infinite trees, a dual principle of coinduction is needed. In the literature bisimulation has been used effectively to prove equality of structures defined as greatest fixed points. To prove properties other than equality for coinductive data types one needs to use the somewhat less familiar coinduction principle[@brandt1998coinductive; @hermida1998structural; @niqui2009coinductive]. Kozen and Silva established a practical proof principle to produce sound proofs by coinduction [@kozen2017practical]. However for data types mutually defined by induction and coinduction these separate principles are insufficient. One recent approach in type theory integrates induction and coinduction by pattern and copattern matching and explicit well-founded induction on ordinals[@abel2016well], following a number of earlier representations of induction and coinduction in type theory [@abel2013wellfounded]. Here, we pursue a different line of research in *linear logic* with fixed points. In this paper we introduce a sequent calculus to reason about linear predicates defined as nested least and greatest fixed points. Instead of applying induction and coinduction principles directly, we follow the approach of Brotherston et al. [@brotherston2005cyclic] to allow circularity in derivations. We use cyclic reasoning in the context of first order intuitionistic multiplicative additive linear logic extended with least and greatest fixed points. To ensure soundness of the proofs we impose a validity condition on our derivations. Fortier et al. introduced an infinitary sequent calculus for propositional singleton logic with fixed points, where antecedent and succedent consist of exactly one formula [@Fortier13csl; @santocanale2002calculus]. Adding circularity comes with the cost of losing the cut elimination property. To recover this property they introduced a guard condition that ensures soundness of possibly infinite derivations. They provide a cut elimination algorithm and show its productivity on derivations satisfying their guard condition. Fortier and Santocanale’s result has been generalized by Baelde et al. [@baelde2016infinitary; @doumane2017infinitary] for propositional MALL with fixed points.
In this paper, we extend Fortier et al.’s results to *first order* multiplicative additive linear logic with fixed points. Our notion of validity is adapted from its counterpart in their system. We introduce a similar cut elimination algorithm and prove its local termination on valid derivations with a dual approach. It is worth mentioning that our calculus is essentially different from the finitary one introduced by Baelde for the first order MALL with fixed points [@baelde2007least] since we allow for circularity.
We will show with several examples that our calculus is strong enough to prove many (mutual) inductive and coinductive theorems. To make the examples concise we may use pattern matching for defining inductive predicates [@brotherston2005cyclic; @rosu2017matching]. Our underlying system is designed to reason about linear structures. However, some properties of linear structures rely on first order non-linear theories such as theory of arithmetic or order theory. To be able to prove these properties as well we extend our calculus by mixing linear and structural formulas. Our approach is to use a restricted version of the adjoint logic presented by Pfenning et al. [@benton1994mixed; @pfenning2015polarized]. The restriction is that only linear formulas can depend on non-linear ones and not vice versa. Thus we only add the *downgrade* operator ($\downarrow$) that embeds a nonlinear formula into a linear one to our language. In this way we can isolate the reasoning about nonlinear properties to the pure structural part, and use any sound nonlinear theory in a modular way.
In summary, the main contributions of this paper is to introduce an infinitary sequent calculus for first order multiplicative additive linear logic with (mutual) least and greatest fixed points. We provide a validity condition on derivations that ensures the cut elimination property. Our calculus is a tool to reason about a rich signature of mutually defined inductive and coinductive predicates and also allows using nonlinear first order theories. We show its strength by providing several examples including properties defined as nested least and greatest fixed points.
First order intuitionistic linear logic with fixed points
=========================================================
The syntax of formulas in the first order intuitionistic multiplicative additive linear logic with fixed points ($\mathit{FIMALL}^{\infty}_{\mu,\nu}$) follows the grammar [$$\begin{array}{lcl}
A & ::= & 1 \mid 0 \mid \top \mid A \otimes A \mid A \multimap A \mid A \oplus A \mid A\,\&\, A \\
& & \mid \exists x.\, A(x) \mid \forall x.\, A(x) \mid s=t \mid T(\overline{t})
\end{array}$$]{}where $s,t$ stand for terms[^1] and $x,y$ for term variables. $T(\overline{t})$ is a predicate variable defined using least and greatest fixed points in a *signature* $\Sigma$. [$$\Sigma ::= \cdot \mid \Sigma, T(\overline{x})=^{i}_{\mu} A \mid \Sigma, T(\overline{x})=^{i}_\nu A$$]{} The subscript $a$ of a fixed point $T(\overline{x})=^i_{a}$ determines its polarity. If $a=\mu$, then predicate $T(\overline{x})$ is of positive polarity and if $a=\nu$ it is of negative polarity. We represent inductively defined predicates (e.g., the property of being a natural number) as fixed points with positive polarity and coinductively defined predicates (e.g., the lexicographic order on streams) as fixed points with negative polarity. Here we restrict $\Sigma$ to the definitions in which each predicate occurs only in positive (variant) or negative(contravariant) positions, i.e. we do not allow mixed positions[@REYNOLDS81A; @pierce2002types].
The superscript $i\in \mathbb{N}$ is the relative priority of $T(\overline{x})$ in the signature $\Sigma$ with the condition that if $ T_1(\overline{x})=^{i}_{a} A, T_2(\overline{x})=^{i}_b B \in \Sigma$, then $a=b$. Similar to prior work ([@Fortier13csl; @derakhshan2019circular]) we use priority on predicates to define the validity condition on infinite derivations.
\[Nat-pred\] Let signature $\Sigma_1$ be [$$\begin{array}{lcl}
\mathtt{Nat}(x)&=^1_{\mu} &(\exists y. (x=\mathsf{s} y)\, \otimes \, \mathtt{Nat}(y))\, \oplus\, ((x=\mathsf{z})\, \otimes\, 1)\\
\mathtt{Even}(x)&=^2_{\mu}& (\exists y. (x=\mathsf{s}y)\, \otimes\, \mathtt{Odd}(y)) \oplus ((x=\mathsf{z})\, \otimes\, 1)\\
\mathtt{Odd}(x)&=^2_{\mu}& (\exists y. (x=\mathsf{s}y)\, \otimes\, \mathtt{Even}(y))
\end{array}$$]{}where positive predicates $\mathtt{Nat}$, $\mathtt{Even}$, and $\mathtt{Odd}$ refer to the properties of being natural, even, and odd numbers respectively. We interpret it as $\mathtt{Nat}$ having a higher priority relative to $\mathtt{Even}$ and $\mathtt{Odd}$.
A judgment in $\mathit{FIMALL}^{\infty}_{\mu,\nu}$ is of the form $\Gamma \vdash_{\Sigma} A$ where $\Gamma$ is a set of formulas and $\Sigma$ is the signature. We omit $\Sigma$ from the judgments, since it never changes throughout a proof. The infinitary sequent calculus for this logic is given in Figure \[fig:rules-1\], in which we generalize $\oplus$ and $\&$ to be $n$-ary connectives $\oplus\{l_j:A_j\}_{j \in I}$ and $\&\{l_i:A_i\}_{j \in I}$. The binary disjunction and conjunction are defined as $A\oplus B=\oplus \{\pi_1:A,\pi_2:B\}$ and $A\& B=\& \{\pi_1:A,\pi_2:B\}$. Constants $0$ and $\top$ defined as the nullary version of these connectives: $0=\oplus\{\}$ and $\top=\&\{\}$.
$$\begin{array}{cc}
\infer[\mathtt{fwd}]{A \vdash A}{} & \infer[\mathtt{Cut}]{\Gamma, \Gamma' \vdash C}{\Gamma \vdash A & \Gamma' , A \vdash C} \\[1em]
\infer[1R]{\cdot \vdash 1}{} & \infer[1L]{\Gamma, 1 \vdash C}{ \Gamma \vdash C} \\[1em]
\infer[\otimes R]{\Gamma, \Gamma' \vdash A_1 \otimes A_2}{\Gamma \vdash A_1 & \Gamma' \vdash A_2}&\infer[\otimes L]{\Gamma, A_1 \otimes A_2 \vdash B}{\Gamma, A_1, A_2\vdash B }\\[1em]
\infer[\multimap R]{\Gamma \vdash A_1 \multimap A_2}{\Gamma , A_1 \vdash A_2}&\infer[\multimap L]{\Gamma, \Gamma', A_1 \multimap A_2 \vdash B}{\Gamma \vdash A_1 & \Gamma', A_2 \vdash B}\\[1em]
\infer[\oplus R]{\Gamma \vdash \oplus \{l_i: A_i\}_{i \in I}}{\Gamma \vdash A_k & k \in I}&\infer[\oplus L]{\Gamma, \oplus\{l_i:A_i\}_{i\in I} \vdash B}{\Gamma, A_i\vdash B & \forall i \in I}\\[1em]
\infer[\& R]{\Gamma \vdash \& \{l_i: A_i\}_{i \in I}}{\Gamma \vdash A_i & \forall i \in I}&\infer[\& L]{\Gamma, \&\{l_i:A_i\}_{i\in I} \vdash B}{\Gamma, A_k\vdash B & k \in I}\\[1em]
\infer[\exists R]{\Gamma \vdash \exists x. P(x)}{\Gamma \vdash P(t)}&\infer[\exists L]{\Gamma, \exists x. P(x) \vdash B}{\Gamma, P(x)\vdash B }\\[1em]
\infer[\forall R]{\Gamma \vdash \forall x. P(x)}{\Gamma \vdash P(x)}&\infer[\forall L]{\Gamma, \forall x. P(x) \vdash B}{\Gamma, P(t)\vdash B }\\[1em]
\infer[\mu_{T} R]{\Gamma \vdash T(\overline{t})}{\Gamma \vdash [\overline{t}/\overline{x}]A & T(\overline{x})=_{\mu} A } &\infer[\mu_{T} L]{\Gamma, T(\overline{t}) \vdash B}{\Gamma, [\overline{t}/\overline{x}]A\vdash B & T(\overline{x})=_{\mu} A }\\[1em]
\infer[\nu_{T} R]{\Gamma \vdash T(\overline{t})}{\Gamma \vdash [\overline{t}/\overline{x}]A & T(\overline{x})=_{\nu}A } &\infer[\nu_{T} L]{\Gamma, T(\overline{t}) \vdash B}{\Gamma, [\overline{t}/\overline{x}]A\vdash B & T(\overline{x})=_{\nu} A }\\[1em]
\infer[= R]{\cdot \vdash s=s}{}&\infer[= L]{\Gamma, s=t \vdash B}{\Gamma[\theta] \vdash B[\theta] & \theta \in \mathtt{mgu}(t,s) }
\end{array}$$
\[Odd1\] Consider signature $\Sigma_1$ and predicates $\mathtt{Even}$ and $\mathtt{Odd}$ defined in Example \[Nat-pred\]. The following derivation is a finite proof of one ($\mathsf{s}\mathsf{z}$) being an odd number. [$$\infer[\mu_{\mathtt{Odd}}R]{\cdot \vdash \mathtt{Odd}(\mathsf{s}\mathsf{z})}{\infer[\exists R]{ \cdot \vdash \exists y. (\mathsf{s}\mathsf{z}=\mathsf{s}y)\, \otimes\, \mathtt{Even}(y) }{ \infer[\otimes R]{\cdot \vdash (\mathsf{s}\mathsf{z}=\mathsf{s}\mathsf{z})\, \otimes\, \mathtt{Even}(\mathsf{z})}{\infer[=R]{\cdot \vdash \mathsf{s}\mathsf{z}=\mathsf{s}\mathsf{z}}{} & \infer[\mu_{\mathtt{Even}R}]{\cdot \vdash \mathtt{Even}(\mathsf{z})}{\infer[\oplus R]{\cdot \vdash(\exists y. (x=\mathsf{s}y)\, \otimes\, \mathtt{Odd}(y)) \oplus ((\mathsf{z}=\mathsf{z})\, \otimes\, 1)}{\infer[\otimes R]{\cdot \vdash (\mathsf{z}=\mathsf{z})\, \otimes\, 1}{\infer[=R]{\cdot \vdash \mathsf{z}=\mathsf{z}}{} & \infer[1R]{\cdot \vdash 1}{}}}}}}}$$]{}
The calculus in Figure \[fig:rules-1\] is infinitary, meaning that it allows building infinite derivations as well. The infinite derivations we are interested in, are those we can represent in a finite way. A *circular derivation* is the finite representation of an infinite one in which we can identify each open subgoal with an identical interior judgement. In the first order context we may need to use a substitution rule right before a circular edge to make the subgoal and interior judgment exactly identical [@brotherston2005cyclic]: [$$\qquad \qquad \infer[\mathtt{subst}_{\theta}]{\Gamma[\theta] \vdash B[\theta]}{\Gamma\vdash B}$$]{}We can transform a circular derivation to its underlying infinite derivation in a productive way by deleting the $\mathtt{subst}_{\theta}$ rule and the circular edge. We need to instantiate the derivation to which the circular edge pointed with substitution $\theta$. This instantiation exists and does not change the structure of derivation by Lemma \[lem:subst\] in the Appendix.
\[Oddsx\] Consider Signature $\Sigma_1$ and predicates $\mathtt{Nat}$, $\mathtt{Even}$, and $\mathtt{Odd}$ defined in Example \[Nat-pred\]. Figure \[fig:ex-evenodd\] represents a circular derivation for $\mathtt{Even}(x) \vdash \mathtt{Odd}(\mathsf{s}\,x)$. $\Pi$ is the finite derivation given in Example \[Odd1\].
We can interpret the proof in Example \[Oddsx\] as an inductive proof where its circular edge corresponds to applying the induction hypothesis. In the next two examples we represent two coinductive proofs in our circular calculus. Both examples are adapted from @kozen2017practical.
\[bisim\] Define $\Sigma_2$ to consist of a single predicate with negative polarity [$\sim(x,y)=^1_{\nu} (\mathsf{hd}\, x = \mathsf{hd}\, y) \, \& \sim(\mathsf{tl}\, x, \mathsf{tl}\, y). $]{}\
Predicate $\sim(x,y)$ can be read as a bisimulation between streams $x$ and $y$. We present a circular derivation for $\sim$ being symmetric in Figure \[fig:bisim\].
We can reason about the properties of stream operations in our calculus as well. Consider three operations $\mathsf{merge}$, $\mathsf{split}_1$ and $\mathsf{split}_2$. Operation $\mathsf{merge}$ receives two streams and merge them into a single stream by alternatively outputting an element of each. Operations $\mathsf{split}_1$ and $\mathsf{split}_2$ receive a stream $x$ as an input and return the odd and even elements of it, respectively. We define these operations as negative predicates in our language. Define signature $\Sigma_3$ as [$$\begin{array}{lcl}
\mathtt{Merge}(x,y,z)&=^1_{\nu}& (\mathsf{hd} \, z = \mathsf{hd} \, x\, \&\, \mathtt{Merge}\, (y , \mathsf{tl}\, x, \mathsf{tl}\, z))\\
\mathtt{Split}_1(x,y)&=^1_{\nu}& (\mathsf{hd} \, y = \mathsf{hd} \, x\, \&\, \mathtt{Split}_2 ( \mathsf{tl} \, x, \mathsf{tl}\, y))\\
\mathtt{Split}_2(x,y)&=^1_{\nu}& (1\, \&\, \mathtt{Split}_1 ( \mathsf{tl} \, x, y))\\
\end{array}$$]{}The derivation given in Figure \[fig:streamrev\] shows that operations $\mathsf{merge}$ and $\mathsf{split}_i$ are inverses: Split a stream $x$ into two streams $y_1$ and $y_2$ using $\mathsf{split}_1$ and $\mathsf{split}_2$, respectively, then merge $y_1$ and $y_2$. The result is $x$.
Pattern Matching
================
It may not be feasible to present a large piece of derivation fully in the calculus of Figure \[fig:rules-1\]. For the sake of brevity, we may represent predicates of positive polarity in the signature using pattern matching and build equivalent derivations based on that signature [@rosu2017matching; @brotherston2005cyclic]. In all the examples we use pattern matching for it should be clear how to transform the signature and derivations into our main logical system.
\[pattern\] Redefine predicates $\mathtt{Even}$, $\mathtt{Odd}$, and $\mathtt{Nat}$ in Example \[Nat-pred\] by pattern matching in Signature $\Sigma'_1$ as: [$$\begin{array}{lcl lcl}
\mathtt{Nat}(\mathsf{z}) & =^1_{\mu} & 1 & \qquad
\mathtt{Nat}(sy) & =^1_{\mu} & \mathtt{Nat}(y)\\
\mathtt{Odd}(\mathsf{z}) & =^1_{\mu} & 0& \qquad
\mathtt{Odd}(sy) & =^1_{\mu} & \mathtt{Even}(y) \\
\mathtt{Even}(\mathsf{z})& =^1_{\mu} & 1 & \qquad
\mathtt{Even}(sy) &=^1_{\mu} & \mathtt{Odd}(y)
\end{array}$$]{}The circular derivation in Example \[Oddsx\] can be simplified in the following way: [$$\begin{array}{cc}
\textit{[1]}\quad \infer[\mu L]{\dagger \, \mathtt{Even}(\mathsf{z}) \vdash \mathtt{Odd(s\,\mathsf{z})}}{\infer[1 L]{1 \vdash \mathtt{Odd}(s\,\mathsf{z})}{\infer[\mu R]{\cdot \vdash \mathtt{Odd}(s \, \mathsf{z})}{\infer[\mu R]{\cdot \vdash \mathtt{Even}(\mathsf{z})}{\infer[1 R]{\cdot \vdash 1}{}}}}}
&
\textit{[2]}\quad \infer[\mu L]{\dagger \, \mathtt{Even}(s \, x) \vdash \mathtt{Odd(s\,s \, x)}}{\infer[\mu R]{\mathtt{Odd}(x) \vdash \mathtt{Odd}(s\, s\,x)}{\deduce{\mathtt{Odd}(x) \vdash \mathtt{Even}(s \, x)}{\star}}} \\
\textit{[3]}\quad \infer[\mu L]{\star\, \mathtt{Odd}(z) \vdash \mathtt{Even(s\,z)}}{\infer[0 L]{0 \vdash \mathtt{Odd}(z)}{}} & \textit{[4]}\quad
\infer[\mu L]{\star\, \mathtt{Odd}(s \, x) \vdash \mathtt{Even(s\,s \, x)}}{\infer[\mu R]{\mathtt{Even}(x) \vdash \mathtt{Even}(s\, s\,x)}{\deduce{\mathtt{Even}(x) \vdash \mathtt{Odd}(s \, x)}{\dagger}}}
\end{array}$$]{} By the definition of signature $\Sigma'_1$, the pattern of $x$ in $\mathtt{Odd}(x)$ is either of the form $\mathsf{s}\, y$ or $\mathsf{z}$. At the subgoal marked with $\star$ in subderivation $2$, we form a branch similar to the $\oplus \, L$ rule to cover all possible patterns of $x$; we continue with subderivations $3$ and $4$. With the same reasoning at the subgoal marked with $\dagger$ in the subderivation $4$ we form a branch with subderivations $1$ and $2$.
A major contribution of this paper is to give a criterion for validity of theorems proved by simultaneous induction and coinduction. In the next example we see an interplay between positive and negative fixed points in the derivation. Define predicate $\mathtt{run}(x,t)$ to represent computation of a stream processor, where $x$ is the list of operations we want to compute. Operations in $x$ can be either a $\mathit{skip}$ or a $\mathit{put}\langle x \rangle$. Operation $\mathit{skip}$ simply skips one step and does not contribute to the output stream $t$. Operation $\mathit{put}\langle x \rangle$ puts element $\mathsf{z}$ as the head of the output stream $t$ and inserts a new list of operations $x$ to the original list of operations. After computing $\mathit{skip}$ the length of remaining operations in $x$ goes down by one. So we can define $\mathtt{run}(\mathit{skip};x, t)$ inductively as a positive predicate. $\mathit{put}\langle x \rangle$ increases the length of the operations, but produces an element of the output stream. So $\mathtt{run}(\mathit{put}\langle x \rangle;y,t)$ needs to be defined as a negative predicate rather than a positive one.
Define the signature $\Sigma_4$ to be [$$\begin{array}{lcl}
\mathtt{run}(\cdot \, , t) &=^1_{\mu}& 1 \\
\mathtt{run}(skip;x, t) &=^1_{\mu}& \mathtt{run} (x, t) \\
\mathtt{run}( put \langle x \rangle;y, t ) &=^1_{\mu}& \, \mathtt{nrun}\, (x, y, t) \\
\mathtt{nrun}( x, y, t) &=^2_{\nu}& \mathtt{hd}\, t= \mathsf{z}\, \&\, \mathtt{run} (x;y, \mathtt{tl}\, t)
\end{array}$$]{}The equivalent signature without pattern matching is [$$\begin{array}{lll}
\mathtt{run}(x,t)& =^1_{\mu} & \oplus\{\mathsf{e}:x=\cdot\, \otimes 1,\\ && \quad \mathsf{s}: \exists x'. x=skip;x' \otimes \mathtt{run}(x',t),\\ && \quad \mathsf{p}: \exists x'. \exists y. x= \mathit{put}(x');y \otimes \mathtt{nrun}(x',y,t) \}\\
\mathtt{nrun}( x, y, t) &=^2_{\nu}& \mathtt{hd}\, t= \mathsf{z}\, \&\, \mathtt{run} (x;y, \mathtt{tl}\, t)
\end{array}$$]{}Here we define $\mathtt{run}( put \langle x \rangle;y, t )$ in two steps to follow the rules of definition by pattern matching. We can abbreviate this definition to one step as: [$$\mathtt{run}( put \langle x \rangle;y, t)=^2_{\nu}\mathtt{hd}\, t= \mathsf{z}\, \&\, \mathtt{run} (x;y, \mathtt{tl}\, t)$$]{} We want to prove that a run of any list of operations $x$ produces a (possibly infinite) list of elements $\mathsf{z}$.
[$$\begin{array}{lcl}
\mathtt{zlist} (t) &=^1_{\mu}& 1 \oplus \mathtt{ztream}(t) \\
\mathtt{ztream}(t) &=^2_{\nu}& \mathtt{hd}\, t= \mathsf{z}\, \&\, \mathtt{zlist}\, (\mathtt{tl}\, t) \\
\end{array}$$]{}We give circular derivations for both $(\dagger)\, \mathtt{run}(x, t) \vdash \mathtt{zlist}(t)$ and $(\star)\, \mathtt{nrun}(x,y, t) \vdash \mathtt{ztream}(t)$ in Figure \[fig:run\] to show the interplay between coinductive and inductive predicates.
[$$\infer[\nu_{\mathtt{ztream}} R]{\star\, \mathtt{nrun}\, (x, y, t)\vdash \mathtt{ztream}(t)}{\infer[\& R]{\mathtt{nrun}\, (x, y, t)\vdash \mathsf{hd}\,t= \mathsf{z}\, \& \, \mathtt{zlist}(\mathsf{tl}\, t)}{\infer[\nu_{\mathtt{nrun} }L]{\mathtt{nrun}\, (x, y, t)\vdash \mathsf{hd}\,t= \mathsf{z}}{\infer[\&L]{\mathtt{hd}\, t= \mathsf{z}\, \&\, \mathtt{run} (x;y, \mathtt{tl}\, t)\vdash \mathsf{hd}\,t= \mathsf{z}}{\infer[\mathtt{ID}]{\mathtt{hd}\, t= \mathsf{z} \vdash \mathsf{hd}\,t= \mathsf{z}}{}}} & \infer[\nu_{\mathtt{nrun}} L]{\mathtt{nrun}\, (x, y, t)\vdash \mathtt{zlist}(\mathsf{tl}\, t)}{\infer[\& L]{\mathtt{hd}\, t= \mathsf{z}\, \&\, \mathtt{run} (x;y, \mathtt{tl}\, t) \vdash \mathtt{zlist}(\mathsf{tl}\, t)}{\deduce{ \mathtt{run} (x;y, \mathtt{tl}\, t) \vdash \mathtt{zlist}(\mathsf{tl}\, t)}{\dagger}} }}}$$]{}
A Validity Condition
====================
Adding fixed point rules to the calculus comes with the price of losing the cut elimination property. Infinite derivations in this calculus do not necessarily enjoy the cut elimination property and thus are called *pre-proofs* instead of *proofs*. We introduce a validity condition on derivations such that the cut elimination property holds for the derivations satisfying it. Our condition is adapted from the Guard condition introduced by @Fortier13csl for singleton logic. We annotate formulas with position variables $\mathbf{x},\mathbf{y},\mathbf{z}$ and track their generations $\alpha, \beta$ to capture evolution of a formula in a derivation. With this annotation we can keep track of behaviour of any particular formula throughout the whole derivation. Our validity condition requires that at least one formula in every infinite branch behaves in a way that justifies validity of that branch.
A basic judgment in the annotated calculus is of the form $\Delta \vdash_{\Omega} \mathbf{z}^\beta:C$ where $\Delta= \cdot \mid \mathbf{x}^\alpha:A, \Delta$. The set $\Omega$ keeps the relation between different generation of position variables in a derivation. We will use the set $\Omega$ to define our validity condition. Figure \[fig:rules-2\] shows the calculus annotated with position variable generations and their relations. A new generation of a position variable is introduced when a fixed point rule applies on it. The relation of a new generation to its priors is determined by the role of the rule that introduces it in (co)induction. $\mu L$ rule breaks down an inductive antecedent and $\nu R$ produces a coinductive information. They both take a step toward termination/productivity of the proof: we put the new generation of the position variable they introduce to be less than the prior ones in the given priority. Their counterpart rules $\mu R$ and $\nu L$, however, do not contribute to termination/productivity. They break the relation between the new generation and its prior ones for the given priority.
In the $\mathit{Cut}$ rule we introduce a fresh position variable of generation zero, $\mathbf{w}^0$. Since it refers to appearance of a new formula, we put it to be incomparable to other position variables. We do not consider $\mathbf{w}^0$ as a continuation of $\mathbf{z}^\beta$ in the rule $\otimes R$ either; we need to restrict left branching on succedent position variables to prove Theorem \[thm:main\].[^2] The fresh position variable $\mathbf{w}^0$ introduced in $\multimap R$ (resp. $\multimap L$) rule switches its polarity from right to left (resp. left to right) so it cannot be equal to $\mathbf{z}^\beta$ (resp. $\mathbf{y}^\alpha$).[^3] As none of the above reasons hold for $\mathbf{w}^0$ in $\otimes L$, we keep its relation with $\mathbf{y}^\alpha$ in $\Omega$.
For a given signature $\Sigma$, define *snapshot* of an annotated position variable $\mathbf{x}^\alpha$ as a list $\mathsf{snap}(\mathbf{x}^\alpha)=[\mathbf{x}^\alpha_i]_{i <n}$, where $n$ is the maximum priority in $\Sigma$.
The list $\mathsf{snap}(\mathbf{x}^\alpha)$ stores the information of the fixed point unfolding rules applied on previous generations of position variable $\mathbf{x}^\alpha:A$ in a derivation.
\[snapex\] For signature $\Sigma_1$ defined in Example \[Nat-pred\]: [$$\begin{array}{lcl}
\mathtt{Nat}(x)&=^1_{\mu} &(\exists y. (x=\mathsf{s} y)\, \otimes \, \mathtt{Nat}(y))\, \oplus\, ((x=\mathsf{z})\, \otimes\, 1)\\
\mathtt{Even}(x)&=^2_{\mu}& (\exists y. (x=\mathsf{s}y)\, \otimes\, \mathtt{Odd}(y)) \oplus ((x=\mathsf{z})\, \otimes\, 1)\\
\mathtt{Odd}(x)&=^2_{\mu}& (\exists y. (x=\mathsf{s}y)\, \otimes\, \mathtt{Even}(y))
\end{array}$$]{}and position variables $\mathbf{x}^\alpha$ and $\mathbf{z}^\beta$ in the judgment [$\mathbf{x}^\alpha: \mathtt{Odd}(x) \vdash \mathbf{z}^\beta : \mathtt{Even}(\mathsf{s}x)$]{} we have [$\mathsf{snap}(\mathbf{x}^\alpha)=[\mathbf{x}^\alpha_i]_{i <2}= [\mathbf{x}_1^\alpha, \mathbf{x}_2^\alpha]$]{} and [$\mathsf{snap}(\mathbf{z}^\beta)=[\mathbf{z}^\beta_i]_{i <2}= [\mathbf{z}_1^\beta, \mathbf{z}_2^\beta]$.]{}
Having the relation between annotated position variables in $\Omega$, we can define a partial order on snapshots of annotated position variables. We write [$$\mathsf{snap}(\mathbf{x}^\alpha)=[\mathbf{x}^\alpha_1\cdots \mathbf{x}^\alpha_n]<_{\Omega}[\mathbf{z}^\beta_1\cdots \mathbf{z}^\beta_n]= \mathsf{snap}(\mathbf{z}^\beta)$$]{}if the list [$[\mathbf{x}^\alpha_1\cdots \mathbf{x}^\alpha_n]$]{} is less than [$[\mathbf{z}^\beta_1\cdots \mathbf{z}^\beta_n]$]{} by the lexicographic order defined by the transitive closure of the relations in $\Omega$.
Let [$\Omega=\{\mathbf{x}_1^\alpha = \mathbf{z}_1^{\beta}, \mathbf{x}_2^\alpha < \mathbf{z}_2^\gamma, \mathbf{z}_2^\gamma <\mathbf{z}_2^\beta\}.$]{} For $\mathsf{snap}(\mathbf{x}^\alpha)$ and $\mathsf{snap}(\mathbf{z}^\beta)$ defined over signature $\Sigma_1$ in Example \[snapex\], we have [$\mathsf{snap}(\mathbf{x}^\alpha)=[\mathbf{x}^\alpha_1, \mathbf{x}^\alpha_2]<_{\Omega}[\mathbf{z}^\beta_1, \mathbf{z}^\beta_2]= \mathsf{snap}(\mathbf{z}^\beta).$]{}
We adapt the definitions of left $\mu$-trace and right $\nu$-trace from Fortier and Santocanale to our own settings.
\[def:mu\] An infinite branch of a derivation is a *left $\mu$-trace* if for infinitely many position variables $\mathbf{x1}^{\alpha_1}, \mathbf{x2}^{\alpha_2}, \cdots$ appearing as antecedents of judgments in the branch we can form an infinite chain of inequalities [$$\mathsf{snap}(\mathbf{x1}^{\alpha_1})>_{\Omega_1}\mathsf{snap}(\mathbf{x2}^{\alpha_2})>_{\Omega_2}\cdots.$$]{}Dually, an infinite branch of a derivation is a *right $\nu$-trace* if for infinitely many position variables $\mathbf{y1}^{\beta_1}, \mathbf{y2}^{\beta_2}, \cdots$ appearing as the succedents of judgments in the branch, we can form an infinite chain of inequalities [$$\mathsf{snap}(\mathbf{y1}^{\beta_1})>_{\Omega_1}\mathsf{snap}(\mathbf{y2}^{\beta_2})>_{\Omega_2}\cdots$$]{}.
An infinite derivation is a *valid proof* if each of its infinite branches is either a left $\mu$-trace or a right $\nu$-trace. A circular *proof* has a valid underlying infinite derivation.
We can rewrite derivation of Example \[Oddsx\] in the annotated calculus as in Figure \[fig:ex-evenoddannot\]. To check the validity of this derivation, it is enough to observe that [$$\mathsf{snap}(\mathbf{x}^2)=[\mathbf{x}^2_1,\mathbf{x}^2_2] <_{\Omega_6}[\mathbf{x}^0_1,\mathbf{x}^0_2]=\mathsf{snap}(\mathbf{x}^0).$$]{}
(2)[ [ $
\infer[\mu_{\mathtt{Even}}]{\mathbf{x}^0:\mathtt{Even}(x) \vdash_{\emptyset} \mathbf{y}^0:\mathtt{Odd}(s(x)) }{\infer[\oplus L]{ \mathbf{x}^1:(\exists y. (x=\mathtt{s}y)\, \otimes\, \mathtt{Odd}(y)) \oplus ((x=0)\, \otimes\, 1) \vdash_{\Omega_1} \mathbf{y}^0:\mathtt{Odd}(s(x))}{\infer[\exists L]{\mathbf{x}^1:\exists y. (x=\mathtt{s}y)\, \otimes\, \mathtt{Odd}(y) \vdash_{\Omega_1} \mathbf{y}^0:\mathtt{Odd}(s(x)) }{\infer[\otimes L]{\mathbf{x}^1:(x=\mathtt{s}y)\, \otimes\, \mathtt{Odd}(y) \vdash_{\Omega_1} \mathbf{y}^0:\mathtt{Odd}(s(x))}{\infer[= L]{\mathbf{w}^0:(x=\mathtt{s}y), \mathbf{x}^1:\mathtt{Odd}(y) \vdash_{\Omega_2} \mathbf{y}^0:\mathtt{Odd}(s(x))}{\infer[\mu_{\mathtt{Odd}R}]{\mathbf{x}^1:\mathtt{Odd}(y) \vdash_{\Omega_2} \mathbf{y}^0:\mathtt{Odd}(s(sy))}{\infer[\exists R]{\mathbf{x}^1:\mathtt{Odd}(y) \vdash_{\Omega_3} \mathbf{y}^1:\exists z. (ssy=\mathtt{s}z)\, \otimes\, \mathtt{Even}(z)}{\infer[\otimes R]{\mathbf{x}^1:\mathtt{Odd}(y) \vdash_{\Omega_3} \mathbf{y}^1: (ssy=\mathtt{s}sy)\, \otimes\, \mathtt{Even}(sy)}{\infer[=R]{\cdot \vdash_{\Omega_3} \mathbf{z}^0:(ssy=\mathtt{s}sy)}{} & \infer[\mu_{\mathtt{Odd}L}]{\mathbf{x}^1:\mathtt{Odd}(y) \vdash_{\Omega_3} \mathbf{y}^1:\mathtt{Even}(sy)}{ \infer[\exists L]{\mathbf{x}^2:\exists z. (y=\mathtt{s}z)\, \otimes\, \mathtt{Even}(z) \vdash_{\Omega_4} \mathbf{y}^1:\mathtt{Even}(sy)}{\infer[\otimes L]{\mathbf{x}^2:(y=\mathtt{s}z)\, \otimes\, \mathtt{Even}(z) \vdash_{\Omega_4} \mathbf{y}^1:\mathtt{Even}(sy)}{\infer[=L]{v^0:(y=\mathtt{s}z), \mathbf{x}^2:\mathtt{Even}(z) \vdash_{\Omega_5} \mathbf{y}^1:\mathtt{Even}(sy)}{\infer[\mu_{\mathtt{Even} R}]{\mathbf{x}^2:\mathtt{Even}(z) \vdash_{\Omega_5} \mathbf{y}^1:\mathtt{Even}(ssz)}{\infer[\oplus R ]{\mathbf{x}^2:\mathtt{Even}(z) \vdash_{\Omega_6} \mathbf{y}^2:(\exists y. (ssz=\mathtt{s}y)\, \otimes\, \mathtt{Odd}(y)) \oplus ((ssz=0)\, \otimes\, 1)}{\infer[\exists R]{\mathbf{x}^2:\mathtt{Even}(z) \vdash_{\Omega_6} \mathbf{y}^2:(\exists y. (ssz=\mathtt{s}y)\, \otimes\, \mathtt{Odd}(y))}{\infer[\otimes R]{\mathbf{x}^2:\mathtt{Even}(z) \vdash_{\Omega_6} \mathbf{y}^2:(ssz=\mathtt{s}sz)\, \otimes\, \mathtt{Odd}(sz)}{\infer[=R]{\cdot \vdash_{\Omega_6} \mathbf{u}^0:ssz=ssz}{} & \infer[\mathtt{Subst_{[z/x]}}]{\mathbf{x}^2:\mathtt{Even}(z) \vdash_{\Omega_6} \mathbf{y}^2:\mathtt{Odd}(sz)}{\mathbf{x}^2:\mathtt{Even}(x) \vdash_{\Omega_6} \mathbf{y}^2:\mathtt{Odd}(sx)} }}} }}}}}}} }}} } & \cdots } }
$]{} ]{}; (4.8,4.55).. controls (7,7) and (15,-6) .. (2.9,-4.6);
[$\Omega_1=\{\mathbf{x}^1_{2}<\mathbf{x}^0_2, \mathbf{x}^1_1=\mathbf{x}^0_1\},$ $\Omega_2=\Omega_1 \cup \{\mathbf{w}^0_{2}=\mathbf{x}^1_2, \mathbf{w}^0_1=\mathbf{x}^1_1\},$ $\Omega_3=\Omega_2 \cup \{\mathbf{y}^1_1=\mathbf{y}^0_1\},$ $\Omega_4=\Omega_3 \cup \{\mathbf{x}^2_{2}<\mathbf{x}^1_2, \mathbf{x}^2_1=\mathbf{x}^1_1\},$ $\Omega_5=\Omega_4 \cup \{v^0_{2}=\mathbf{x}^2_2, v^0_1=\mathbf{x}^2_1\},$ and $\Omega_6=\Omega_5 \cup \{\mathbf{y}^2_1=\mathbf{y}^1_1\}.$]{}
Since the annotation of position variables is straightforward, for the sake of conciseness, we present future examples as circular derivations in the calculus of Figure \[fig:rules-1\]. We also use pattern matching whenever possible. All derivations presented in this paper are valid by this definition. We leave it to the reader to check their validity.
A productive cut elimination algorithm
======================================
@Fortier13csl introduced a cut elimination algorithm for infinite pre-proofs in singleton logic with fixed points. They proved that for infinite proofs satisfying their guard condition the algorithm is productive. In this section we adapt their cut elimination algorithm to $\mathit{FIMALL}^{\infty}_{\mu,\nu}$ and prove its productivity for valid derivations. The algorithm receives an infinite proof as an input and outputs a cut-free infinite proof. Since we are dealing with infinite derivations, to make the algorithm productive we need to push every cut away from the root with a lazy strategy (BFS). With this strategy we may need to permute two consecutive cuts which results into a loop. To overcome this problem, similar to Fortier and Santocanale and also @baelde2007least we generalize binary cuts to $n$-ary cuts using the notion of a *branching tape* the prior notion of *tape*.
\[def:tape\] A *branching tape* $\mathcal{C}$ is a finite list of sequents $\Delta \vdash \mathbf{w}^\beta: A$[^4], such that
- Every two judgments $\Delta \vdash \mathbf{w}^\beta: A$ and $\Delta' \vdash \mathbf{w}'^{\beta'}: A'$ on the tape share at most one position variable $\mathbf{z}^\alpha:B$. If they share such position variable, we call them connected. Moreover, assuming that $\Delta \vdash \mathbf{w}^\beta: A$ appears before $\Delta' \vdash \mathbf{w}'^{\beta'}: A'$ on the list, we have $\mathbf{z}^\alpha:B\in \Delta'$ and $\mathbf{z}^\alpha:B= \mathbf{w}^\beta:A$.
- Each position variable $\mathbf{z}^\beta$ appears at most twice in a tape and if it appears more than once it connects two judgments.
- Every tape is connected and acyclic.
The *conclusion* $\mathsf{conc}_{\mathcal{M}}$ of a branching tape $\mathcal{M}$ is a sequent $\Delta \vdash \mathbf{x}^\alpha:A$ such that
- there is a sequent $\Delta' \vdash \mathbf{x}^\alpha:A$ in the tape that $\mathbf{x}^\alpha:A$ does not connect it to any other sequent in the tape.
- For every $\mathbf{y}^\beta:B\in \Delta$ there is a sequent $\Delta', \mathbf{y}^\beta:B \vdash \mathbf{z}^\gamma:C$ on the tape such that $\mathbf{y}^\beta:B$ does not connect it to any other sequent in the tape.
We call $\Delta$ the set of *leftmost formulas* of $\mathcal{M}$: $\mathsf{lft}(\mathcal{M})$. And $x^\alpha:A$ is the *rightmost formula* of tape $\mathcal{M}$: $\mathsf{rgt}(\mathcal{M})$.
The conclusion of a branching tape always exists and is unique. An $n$-ary cut is a rule formed from a tape $\mathcal{M}$ and its conclusion $\mathsf{conc}_{\mathcal{M}}$:$\infer[nCut]{\mathsf{conc}_{\mathcal{M}}}{\mathcal{M}}$
We generalize Fortier and Santaconale’s set of primitive operations to account for $\mathit{FIMALL}^{\infty}_{\mu,\nu}$. They closely resemble the reduction rules given by @doumane2017infinitary. Figure \[fig:Reductions\] depicts a few interesting Internal and External reductions [^5].
[ $$\begin{array}{ccc}
\infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1& \infer[\multimap R]{ \Delta' \vdash \mathbf{z}^{\beta}:A_1 \multimap A_2}{\Delta', \mathbf{u}^0: A_1 \vdash \mathbf{z}^{\beta}:A_2}& \mathcal{C}_2 & \infer[\multimap L]{\Delta'', z^{\beta}:A_1 \multimap A_2 \vdash \mathbf{w}^\alpha:B}{\Delta_1''\vdash \mathbf{u}^{0}:A_1 & \Delta_2'', \mathbf{z}^{\beta}: A_2 \vdash \mathbf{w}^\alpha:B } & \mathcal{C}_3 } & \xRightarrow{\mathsf{Reduce}}&
\hspace*{10em} \\
\multicolumn{3}{r}{
\infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1& \mathcal{C}_2 & \Delta_1''\vdash \mathbf{u}^{0}:A_1 & \Delta', \mathbf{u}^0: A_1 \vdash \mathbf{z}^{\beta}:A_2 & \Delta_2'', \mathbf{z}^{\beta}: A_2 \vdash \mathbf{w}^\alpha:B & \mathcal{C}_3 }}\\
\end{array}$$]{} $${\small
\begin{array}{lcl}
\infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1 & \infer[\mu R]{ \Delta' \vdash \mathbf{z}^{\beta}:T(\overline{t})}{ \Delta'\vdash \mathbf{z}^{\beta+1}:[\overline{t}/\overline{x}]A & T(\overline{x})=_{\mu}A }& \mathcal{C}_2 &\infer[\mu L]{\Delta'', \mathbf{z}^{\beta}:T(\overline{t}) \vdash \mathbf{w}^\alpha:B}{\Delta'', \mathbf{z}^{\beta+1}: [\overline{t}/\overline{x}]A \vdash \mathbf{w}^\alpha:B & T(\overline{x})=_{\mu}A} & \mathcal{C}_3 }&
\xRightarrow{\mathsf{Reduce}}&
\hspace*{10em} \\
\multicolumn{3}{r}{
\infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1 & \Delta'\vdash \mathbf{z}^{\beta+1}:[\overline{t}/\overline{x}]A &\mathcal{C}_2 &\Delta'', \mathbf{z}^{\beta+1}: [\overline{t}/\overline{x}]A \vdash \mathbf{w}^\alpha:B& \mathcal{C}_3 }}\\
\end{array}
}$$ $${\small
\begin{array}{lcl}
\infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1 & \infer[\exists R]{ \Delta' \vdash \mathbf{z}^{\beta}:\exists x. P(x)}{ \Delta'\vdash \mathbf{z}^{\beta}:P(t) }& \mathcal{C}_2 &\infer[\exists L]{\Delta'', \mathbf{z}^{\beta}:\exists x. P(x) \vdash \mathbf{w}^\alpha:B}{\deduce{\Delta'', \mathbf{z}^{\beta}: P(x) \vdash \mathbf{w}^\alpha:B}{\Pi'} } & \mathcal{C}_3 }&
\xRightarrow{\mathsf{Reduce}}&
\hspace*{20em} \\
\multicolumn{3}{r}{
\infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1 & \Delta'\vdash \mathbf{z}^{\beta}:P(t) &\mathcal{C}_2 &\deduce{\Delta'', \mathbf{z}^{\beta}: P(t) \vdash \mathbf{w}^\alpha:B}{\Pi'[t/x]}& \mathcal{C}_3 }}\\
\infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1 & \infer[=R]{ \cdot \vdash \mathbf{z}^{\beta}:s=s}{}&\mathcal{C}_2 &\infer[=L]{\Delta'', \mathbf{z}^{\beta}:s=s \vdash \mathbf{w}^\alpha:B }{\Delta'' \vdash \mathbf{w}^\alpha:B} & \mathcal{C}_3 }& \xRightarrow{\mathsf{Reduce}} &
\infer[\mathit{nCut}]{\Delta \vdash \mathbf{v}:C}{\mathcal{C}_1 & \mathcal{C}_2 & \Delta'' \vdash \mathbf{w}^\alpha:B & \mathcal{C}_3 }\\
\end{array}
}$$ $${
\begin{array}{lcl}
\infer[\mathit{nCut}]{\Delta_1,\Delta_2 \vdash \mathbf{z}^\beta:A_1 \otimes A_2}{\mathcal{C}& \infer[\otimes R]{ \Delta_1', \Delta'_2 \vdash \mathbf{z}^{\beta}:A_1 \otimes A_2}{\Delta'_1\vdash \mathbf{u}^{0}:A_1 & \Delta'_2\vdash \mathbf{z}^{\beta}:A_2} }& \xRightarrow{\mathsf{RFLIP}} & \infer[\otimes R]{\Delta_1, \Delta_2 \vdash \mathbf{z}^\beta:A_1 \otimes A_2}{\infer[nCut]{\Delta_1\vdash \mathbf{u}^{0}:A_1}{ \mathcal{C}_{\Delta'_1} & \Delta'_1\vdash \mathbf{u}^{0}:A_1} & \infer[\mathit{nCut}]{\Delta_2\vdash \mathbf{z}^{\beta}:A_2}{\mathcal{C}_{\Delta'_2} & \Delta'_2\vdash \mathbf{z}^{\beta}:A_2 } }\\\\
\infer[\mathit{nCut}]{\Delta, \mathbf{z}^\beta: A_1 \otimes A_2 \vdash \mathbf{v}:C}{\mathcal{C}_1 & \infer[\otimes L]{ \Delta',\mathbf{z}^\beta: A_1 \otimes A_2 \vdash \mathbf{w}^{\alpha}:B}{\Delta', \mathbf{u}^0: A_1, \mathbf{z}^\beta: A_2 \vdash \mathbf{w}^{\alpha}:B} & \mathcal{C}_2 }& \xRightarrow{\mathsf{LFLIP}} & \infer[\otimes L]{\Delta, \mathbf{z}^\beta: A_1 \otimes A_2 \vdash \mathbf{v}:C}{ \infer[\mathit{nCut}]{\Delta, \mathbf{u}^0: A_1 , \mathbf{z}^\beta:A_2 \vdash \mathbf{u}^{0}:A_1}{ \mathcal{C}_1 & \Delta', \mathbf{u}^0: A_1, \mathbf{z}^\beta: A_2 \vdash \mathbf{w}^{\alpha}:B & \mathcal{C}_2}}\\\\
\infer[\mathit{nCut}]{\Delta, \mathbf{z}^\beta:s=t \vdash \mathbf{w}^\alpha:B}{ \mathcal{C}_1 & \infer[=L]{\Delta',\mathbf{z}^\beta:s=t \vdash \mathbf{w}^\alpha:B}{\Delta'[\theta] \vdash \mathbf{w}^\alpha:B'[\theta] & \theta \in \mathtt{mgu}(t,s)}& \mathcal{C}_2} & \xRightarrow{\mathsf{LFLIP}} &
\infer[=L]{\Delta, \mathbf{z}^\beta:s=t \vdash \mathbf{w}^\alpha:B}{ \infer[\mathit{nCut}]{\Delta[\theta] \vdash \mathbf{w}^\alpha:B[\theta] }{\mathcal{C}_1[\theta] & \Delta'[\theta] \vdash \mathbf{w}^\alpha:B'[\theta] & \mathcal{C}_2[\theta] } & \theta \in \mathtt{mgu}(t,s) }\\\\
\infer[\mathit{nCut}]{\Delta \vdash \mathbf{z}^\beta:C }{\mathcal{C}_1 & \infer[ID]{\mathbf{x}^\alpha:A \vdash \mathbf{w}^\gamma:A }{} & \mathcal{C}_2 } & \xRightarrow{\mathsf{ID-Elim}} & \infer[\mathit{nCut}]{\Delta \vdash \mathbf{z}^\beta:C}{\mathcal{C}_1 & \mathcal{C}_2[\mathbf{x}^\alpha/\mathbf{w}^\gamma] }
\end{array} }$$
Our cut elimination algorithm is given as Algorithms \[algorithm\] and \[alg:treat\]. The output of the algorithm is a tree labelled by $\{0,1\}$. For each node $w \in \{0,1\}^*$ of the tree it also identifies the corresponding sequent, $\mathit{s}(w)$, and the rule applied on the node, $\mathit{r}(w)$. The algorithm *Treat* reduces the sequence in a branching tape with internal reductions until either a left rule is applied on one of its leftmost formulas or a right rule is applied on its rightmost formula. While this condition holds, the algorithm applies a *flip* rule on a leftmost/rightmost formula of the tape. The flipping step is always productive since it pushes a cut one step up. It suffices to show that the treating part is terminating to prove productivity of the algorithm.
Initialization: $\Lambda \leftarrow \emptyset; Q \leftarrow [(\epsilon, [v])]$; $v$ is the root sequent. $\rho(s)$ is the rule applied on formula annotated with position variable $s$, it can either be an $\mathsf{ID}$, $\mathsf{Cut}$, a $L$ rule, or a $R$ rule. $\mathsf{lft}(M)$ and $\mathsf{rgt}(M)$ are defined in Definition \[def:tape\]. The $\mathsf{FLip}$ rules will return a rule that they permuted down, the sequent corresponding to that, and a list $\mathit{List}$ of one or two tapes.\
Initialization: $M$ is a branching tape. $i$ and $j$ in $\mathsf{Reduce}(M,i,j)$ are the index of the two sequents in tape on which the reduction rules are applied. Similarly $i$ in $\mathsf{Merge}(M,i)$ and $\mathsf{idElim}$ is the index of the sequent in the tape on which the corresponding rule is applied. $\rho'(i)$ is the rule applied on the $i$-th sequent of the tape, it can either be an $\mathsf{ID}$, $\mathsf{Cut}$, a $L$ rule, or a $R$ rule.\
\[thm:main\] For every input tape $M$, computation of $\mathit{Treat}(M)$ halts.
By a proof similar to FS, except that we use $\mu$-threads instead of $\nu$-threads to show that $\mathit{Treat}(M)$ does not have an infinite computation tree. Assume for the sake of contradiction that $\mathit{Treat}(M)$ has an infinite computation tree $\Psi$. We prove the following three contradictory statements, where Here $<$ and $\bigwedge$ are defined according to the lexicographic order on the tree $\Psi$.
1. The greatest infinite branch of $\Psi$ is a $\mu$-branch.
2. Let $E$ be a nonempty collection of $\mu$-branches and let $\gamma=\bigwedge E$. Then $\gamma$ is a $\mu$-branch.
3. If $\beta$ is a $\mu$-branch, then there exists another $\mu$-branch $\beta' < \beta$.
The complete proof is given in the Appendix.
Adding finite structural derivations
====================================
In the proof of theorems about linear structures, we may want to appeal to some first order (structural) theories such as arithmetic. To allow a restricted form of such structural (that is, nonlinear) reasoning in our system we add nonlinear formulas $A_s$ to the linear system as [$$A_l::= \cdots \mid {\downarrow }A_s$$]{}labeling all linear propositions as $A_l$. We do not prescribe the exact syntax for nonlinear proposition since our development is parametric in those (subject to a few assumptions) and may vary with the particular application.
Nonlinear judgments are of the form $\Psi \Vdash A_s$. The only restriction we put on the calculus for pure structural judgments is to have the cut elimination property. We generalize linear judgments to $\Psi; \Delta \vdash_{\Omega} x^\alpha:A_l$ where $\Psi$ is a structural context (i.e., it allows weakening and contraction) and $\Delta$ a linear context. Judgments of the rules in Figure \[fig:rules-2\] are refined by adding an extra structural context $\Psi$ to them. This change is straightforward since none of these rules affect the structural context $\Psi$. We add the following three rules to Figure \[fig:rules-2\] to connect linear and structural reasoning. [$$\begin{array}{ccc}
\infer[\downarrow L]{\Psi; \Delta, x^\alpha: \downarrow A_s \vdash_{\Omega} \mathbf{z}^\beta: B_l}{\Psi, A_s; \Delta \vdash_{\Omega} \mathbf{z}^\beta:B_l} & \infer[\downarrow R]{\Psi; \cdot \vdash_{\Omega} x^\alpha: \downarrow B_s}{\Psi \Vdash B_s}\end{array}$$ $$\infer[\mathsf{Cut_{sl}}]{\Psi; \Gamma \vdash_{\Omega} B_l }{\Psi \Vdash A_s & \Psi, A_s; \Gamma \vdash_{\Omega} B_l}$$]{}In both $\downarrow L$ and $\downarrow R$ rules, when position variable $x^\alpha$ becomes structural we simply delete it. The $\mathsf{Cut}_{sl}$ rule does not create any fresh position variable either. So our validity condition remains intact after adding the structural component.
There are three different cut rules in this system: [$$\begin{array}{cc}
\infer[\mathsf{Cut}_{ss}]{\Psi \Vdash A_s}{ \Psi \Vdash B_s & \Psi, B_s \Vdash A_s} & \infer[\mathsf{Cut_{sl}}]{\Psi; \Gamma \vdash_{\Omega} B_l }{\Psi \Vdash A_s & \Psi, A_s; \Gamma \vdash_{\Omega} B_l}
\end{array}$$ $$\infer[\mathsf{Cut}_{ll}]{\Psi;\Delta, \Delta' \vdash_{\Omega} \mathbf{z}^\beta:C}{\Psi;\Delta \vdash_{\Omega} \mathbf{w}^0:A & \Psi;\Delta', \mathbf{w}^0:A \vdash_{\Omega} \mathbf{z}^\beta:C}$$]{}By assumption, $\mathsf{Cut}_{ss}$ can be eliminated. It is enough to eliminate $\mathsf{Cut}_{ll}$ and $\mathsf{Cut}_{sl}$ rules in a productive way. We define a generalized $n$-ary cut to account for the two latter two cut rules.
A *generalized branching tape* $\mathcal{M}_{\Psi}$ is of the form $\mathcal{S}_{\Psi} \mid \mathcal{C}_{\Psi}$ where $\mathcal{C}_{\Psi}$ is a branching tape by Definition \[def:tape\] and $\mathcal{S}_{\Psi}$ is a set of structural judgments $\Psi \Vdash A_s$. For each structural judgment $\Psi \Vdash A_s \in \mathcal{S}$, formula $A_s$ appears in the structural context of at least one judgment in $\mathcal{C}_{\Psi}$. For a judgment $\Psi'; \Delta \vdash B_l$ in $\mathcal{C}_{\Psi}$, we have $\Psi'=\Psi, \Phi$, where all formulas in $\Phi$ are the succedent of a structural judgment in $\mathcal{S}_{\Psi}$.
The generalized n-ary cut rule is $$\infer[\mathit{nCut}]{\Psi; \Delta \vdash \mathbf{x}^\alpha:B_l}{\mathcal{M}_{\Psi}}$$ Where $\Delta \vdash \mathbf{x}^\alpha: B_l$ is the conclusion of the linear part of the tape as defined in Definition \[def:tape\]. We add one reduction step, a rule to merge a $\mathsf{Cut}_{sl}$ rule to the tape, and two flips to the set of primitive operations (Figure \[fig:str-Reductions\]). We keep all the primitive operations for the purely linear system. To preserve the invariants of the (generalized) branching tape in some of the operations we silently remove the structural sequents in which their succedent is not used in any linear judgment. In the $\mathsf{Reduce}$, $\mathsf{Merge{-}Cut}$, and $\mathsf{RFlip}$ rules we know that $\Psi_1=\Psi, \Phi$, where all elements in $\Phi$ appear as a succedent in $S_{\Psi}$. By cut elimination for pure linear judgments we get a proof for $\Psi \Vdash A_s$. In $\mathsf{LFlip}$ rule we use admissibility of Weakening for the structural context: We can prove coinductively that if there is a derivation for $\Psi; \Delta \vdash \mathbf{x}^\alpha: A_l$ in our calculus, there is also a derivation for $\Psi, B_s; \Delta \vdash \mathbf{x}^\alpha:A_1$ with the same structure.
$${\small
\begin{array}{lcl}
\infer[\mathit{nCut}]{\Psi; \Delta\vdash \mathbf{v}:C}{ \mathcal{S}_{\Psi} \mid {\mathcal{C}_1}_{\Psi} & \infer[\mathsf{Cut}_{sl}]{ \Psi_1; \Delta', \mathbf{z}^\beta: B \vdash \mathbf{w}^\alpha:B }{\Psi_1 \Vdash A_s & \Psi_1, A_s; \Delta' \vdash \mathbf{w}^\alpha:B } & {\mathcal{C}_2}_{\Psi} } & \xRightarrow{\mathsf{Merge{-}Cut}} &
\infer[\mathit{nCut}]{\Psi; \Delta \vdash \mathbf{v}:C}{ \mathcal{S}_{\Psi}, \Psi \Vdash A_s \mid {\mathcal{C}_1}_{\Psi} & \Psi_1, A_s; \Delta' \vdash \mathbf{w}^\alpha:B & {\mathcal{C}_2}_{\Psi} }\\\\
\infer[\mathit{nCut}]{\Psi; \cdot \vdash \mathbf{z}^\beta: \downarrow A_s}{\mathcal{S}_{\Psi} &\mid \cdot & \infer[\downarrow R]{ \Psi_1; \cdot \vdash \mathbf{z}^{\beta}:\downarrow A_s}{\Psi_1\Vdash A_s} }& \xRightarrow{\mathsf{RFLip}} & \infer[\downarrow R]{\Psi; \cdot \vdash \mathbf{z}^\beta:\downarrow A_s }{ {\Psi\Vdash A_s}}\\\\
\infer[\mathit{nCut}]{\Psi; \Delta, \mathbf{z}^\beta:\downarrow A_s \vdash \mathbf{v}:C}{ \mathcal{S}_{\Psi} \mid {\mathcal{C}_1}_{\Psi} & \infer[\downarrow L]{\Psi_1; \Delta', \mathbf{z}^\beta:\downarrow A_s \vdash \mathbf{w}^\alpha:B }{\Psi_1, A_s; \Delta' \vdash \mathbf{w}^\alpha:B } & {\mathcal{C}_2}_{\Psi} } & \xRightarrow{\mathsf{LFlip}} &
\infer[\downarrow L]{\Psi; \Delta, \mathbf{z}^\beta:\downarrow A_s \vdash \mathbf{v}:C}{\infer[\mathit{nCut}]{\Psi, A_s; \Delta \vdash \mathbf{v}:C}{\mathcal{S}_{\Psi, A_s} \mid {\mathcal{C}_1}_{\Psi, A_s} & \Psi_1, A_s; \Delta' \vdash \mathbf{w}^\alpha:B &{\mathcal{C}_2}_{\Psi, A_s}}}
\end{array} }$$
The next example shows how to use a structural context to prove a property of infinite streams.
We define the lexicographic order on streams [@kozen2017practical] in signature $\Sigma_5$ as a negative predicate [$$\begin{array}{lcl}
x \preceq y & =^1_{\nu} & \downarrow(\mathsf{hd} x < \mathsf{hd} y) \oplus (\downarrow (\mathsf{hd}x = \mathsf{hd} y)\, \& \, \mathsf{tl} x \preceq \mathsf{tl} y). \\
\end{array}$$]{}where the relation $<$ is a transitive partial order on the elements of streams. Our goal is to show that relation $\preceq$ is transitive by using the (structural) first order theory of orders ($\mathbb{O}$). In Figure \[fig:transt\] we show two branches of this proof, the rest of the proof can be completed in a similar way.
We can define even and odd predicates alternatively using structural arithmetic formulas. In the next example we show how these alternative definitions can be deduced from the ones we introduced in Example \[Nat-pred\].
\[ex:evenalt\] Define Signature $\Sigma_6$ to be [$$\begin{array}{lcl lcl}
\mathtt{Odd}(\mathsf{z}) & =^1_{\mu} & 0 & \qquad
\mathtt{Odd}(sy) & =^1_{\mu} & \mathtt{Even}(y) \\
\mathtt{Even}(\mathsf{z})& =^1_{\mu} & 1 & \qquad
\mathtt{Even}(sy) &=^1_{\mu} & \mathtt{Odd}(y)
\end{array}$$]{}[$$\begin{array}{lcllcl}
\mathtt{E}(x)& =^2_{\mu}& \exists y.\, \downarrow (x=2y) & \quad
\mathtt{O}(x)&=^2_{\mu}& \exists y.\, \downarrow (x=2y+1)
\end{array}$$]{} Put $\mathbb{P}$ to be the rules of arithmetic. We present circular derivations for $\star \; \mathbb{P};\mathtt{Even}(x) \vdash \mathtt{E}(x)$ and $\dagger \; \mathbb{P};\mathtt{Odd}(x) \vdash \mathtt{O}(x)$ in Figure \[fig:evenalt\]. These derivations satisfy the validity condition since in every infinite branch infinitely many $\mu_{\mathtt{Odd}}L$ and $\mu_{\mathtt{Even}}L$ rules are applied on the antecedent.
Conclusion
==========
In this paper we introduced an infinitary sequent calculus for first order intuitionistic multiplicative additive linear logic with fixed points. This system is mainly designed for linear reasoning but we also allow appealing to first order theories such as arithmetic, by adding an adjoint downgrade modality. Inspired by the work of @Fortier13csl we provide an algorithm to identify valid proofs among all infinite derivations. We have provided several examples to show the strength of calculus in proving theorems about mutually inductive and coinductive data types.
One of our main motivations for introducing this calculus was to have a system for reasoning about programs behaviour. In particular we want to use this calculus to give a direct proof for the strong progress property of locally valid binary session typed processes [@derakhshan2019circular]. The importance of a direct proof other than its elegance is that it can be adapted for a more general validity condition on processes without the need to prove cut elimination productivity for their underlying derivations.
The connection to the type theoretic approach by Abel et al [@abel2016well] is an interesting item for future research. A first step in this general direction was taken by Sprenger and Dam [@sprenger2003structure] who justify cyclic inductive proofs using inflationary iteration.
Appendix
========
\[lem:subst\] For a valid derivation [$$\infer{\Delta \vdash \mathbf{w}^\alpha:A}{\Pi}$$]{} in the infinite system and substitution $\theta$, there is a valid derivation for [$$\infer{\Delta[\theta] \vdash \mathbf{w}^\alpha:A[\theta]}{\Pi[\theta]}$$]{} Where $\Pi[\theta]$ is the whole derivation $\Pi$ or a prefix of it instantiated by $\theta$.
The proof is by coinduction on the structure of [$$\Delta \vdash \mathbf{w}^\alpha:A.$$]{} The only interesting case is where we get to the $= L$ rule. [$$\infer[=L]{\Gamma, s=t \vdash B}{ \deduce{\Gamma[\theta']\vdash B[\theta']}{\Pi'}& \theta' \in \mathsf{mgu}(t,s)}$$]{} If the set [$\mathsf{mgu}(t[\theta],s[\theta])$]{}is empty then so is [$\Pi[\theta]$]{}. Otherwise if $\eta$ is the single element of [$ \mathsf{mgu}(t[\theta],s[\theta])$]{}, then for some substitution $\lambda$ we have [$\theta\eta=\theta'\lambda,$]{} and we can form the rest of derivation for substitution $\lambda$ as [$\Pi'[\lambda]$]{} coinductively.
[\[thm:main\]]{} For every input tape $M$, computation of $\mathit{Treat}(M)$ halts.
We show that $\mathit{Treat}(M)$ does not have an infinite computation tree. Assume for the sake of contradiction that $\mathit{Treat}(M)$ has an infinite computation tree and gets into an infinite loop. We follow the proof by @Fortier13csl(FS) closely.
Put $M_i$ for $i \ge 1$ to be the branching tape in memory before the $i$-th turn of the loop, with $M_1=M$. We build the full trace $T$ of the algorithm with essentially the same transition rules as in FS. In our algorithm the sequents subject to reduction may not be next to each other. The $\mathsf{Reduce}$ function needs to receive two indices in the tape to find the sequents for reduction. All reductions except those corresponding to $\otimes$ and $\multimap$ are non-branching($\mathit{nb}$) and their transition rules are quite similar to the one introduced by FS.
- If $M_{n+1}= \mathsf{Reduce}_{\mathit{nb}}(M_{n},i,j)$ then
- $(n,k)\rightarrow^\bot (n+1,k)$ for $k\not\in\{i,j\}$,
- $(n,i) \rightarrow^0 (n+1, i)$,
- $(n,j)\rightarrow^{0} (n+1,j)$.
.
The reductions corresponding to $\otimes$ and $\multimap$, however, produce a branch and need to be defined separately:
- If $M_{n+1}= \mathsf{Reduce}_{\otimes}(M_{n},i,j)$ then
- $(n,k)\rightarrow^\bot (n+1,k)$ for $k<i$,
- $(n,i) \rightarrow^1 (n+1, i)$ and $(n,i) \rightarrow^2 (n+1, i+1)$,
- $(n,j)\rightarrow^{0} (n+1,j+1)$,
- $(n,k)\rightarrow^{\bot} (n+1,k+1)$ for $i<k<j$ or $k>j$.
.
- If $M_{n+1}= \mathsf{Reduce}_{\multimap}(M_{n},i,j)$ then
- $(n,k)\rightarrow^\bot (n+1,k)$ for $k<i$,
- $(n,i) \rightarrow^0 (n+1, j)$,
- $(n,k)\rightarrow^\bot (n+1,k-1)$ for $i<k<j$,
- $(n,j) \rightarrow^1 (n+1, j-1)$ and $(n,j) \rightarrow^2 (n+1, j+1)$,
- $(n,k)\rightarrow^{\bot} (n+1,k+1)$ for $k<j$.
Transitions labelled by $\bot$ mean that the sequent has not evolved by a reduction rule, while other labels show that the sequent is evolved into one or two (in the case of branching rules) new sequents in the next tape. We get the real trace $\Psi$ by collapsing the transitions labelled by $\bot$. $\Psi$ is an infinite, finitely branching labelled tree with prefix order $\sqsubseteq$ and lexicographical order $<$. A branch in $\Psi$ is a maximal path. The set of all branches of $\Psi$ ordered lexicographically forms a complete lattice.
An infinite branch is a $\mu$- branch (resp. $\nu$-branch) if its corresponding derivation is a $\mu$- trace (resp. $\nu$-trace). By our validity condition $\Psi$ satisfies the property that a $\nu$-branch can only admit finitely many branches on its right side (it may include cuts, $\otimes$, or $\multimap$ reductions).
We prove the following three contradictory statements dual to FS:
1. The greatest infinite branch of $\Psi$ is a $\mu$-branch:
The greatest infinite branch of $\Psi$ exists by Konig’s lemma and is either a $\mu$- or a $\nu$- branch. Assume it is a $\nu$- branch. Then either it forms infinitely many branches on its right or there is an infinite branch greater than it. In both cases we can form a contradiction.
2. Let $E$ be a nonempty collection of $\mu$-branches. Then $\gamma=\bigwedge E$ is a $\mu$-branch:
If $\gamma \in E$ then it is trivially true. Otherwise, by the way we constructed $\Psi$, it means that $\gamma$ has infinitely many branches on its right and thus cannot be a $\nu$ branch.
3. If $\beta$ is a $\mu$-branch, then there exists another $\mu$-branch $\beta' < \beta$:
$\beta$ is a $\mu$-branch so for infinitely many position variables $\mathbf{x1}^{\alpha_1}, \mathbf{x2}^{\alpha_2}, \cdots$ on the antecedents of $\beta$ we can form an infinite chain of inequalities [$$\mathsf{snap}(\mathbf{x1}^{\alpha_1})>_{\Omega_1^\beta}\mathsf{snap}(\mathbf{x2}^{\alpha_2})>_{\Omega_2^\beta}\cdots.$$]{} There are two possibilities here:
1. There is an infinite branch $\beta'<\beta$ with infinitely many position variables $\mathbf{xi}^{\alpha_{i}}, \mathbf{x\{i+1\}}^{\alpha_{i+1}}, \cdots$ as its succedents. Note that these position variables connect sequents in $\beta$ to the sequents in $\beta'$ infinitely many times. So every $\mu/\nu L$ rule in $\beta$ reduces with a $\mu/\nu R$ rule in $\beta'$. This means that a $\mu R$ rule with priority $i$ is applied on the succedent of $\beta'$ infinitely often but no priority $j<i$ has an infinitely many $\nu R$ rule in $\beta'$.
2. There is an infinite branch $\beta'<\beta$ with infinitely many branches on its right.
In both cases $\beta'$ cannot be a $\nu$-branch and thus is a $\mu$-branch.
Items (i)-(iii) form a contradiction. We can form the nonempty collection $E$ of all $\mu$- branches in $\Psi$ by (i). By (ii) we get $(\gamma= \bigwedge E)\in E$, which forms a contradiction with (iii).
[19]{}
\#1 \#1[\#1]{}\#1 \#1 \#1 \#1 \#1[\#1]{} \#1[\#1]{}
[abel2016well]{} . . ().
[abel2013wellfounded]{} . . In , Vol. . ACM, .
[baelde2016infinitary]{} . . ().
[baelde2007least]{} . . In . Springer, .
[benton1994mixed]{} . . In . Springer, .
[brandt1998coinductive]{} . . , (), .
[brotherston2005cyclic]{} . . In . Springer, .
[derakhshan2019circular]{} . . ().
[doumane2017infinitary]{} . **. .
[Fortier13csl]{} . . In , (Ed.). , , .
[hermida1998structural]{} . . , (), .
[kozen2017practical]{} . . , (), .
[niqui2009coinductive]{} . . In . .
[pfenning2015polarized]{} . . In . Springer, .
[pierce2002types]{} . . .
[REYNOLDS81A]{} . . In , (Eds.). , , .
[rosu2017matching]{} . . ().
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[^1]: We do not specify a grammar for terms; all terms are of the only type $U$.
[^2]: This restriction aligns with the computational interpretation of linear logic as session types.
[^3]: One can maintain their relation despite the polarity change by introducing shifts in the language. We reserve this for a further work.
[^4]: For brevity we elide the set $\Omega$ in the judgments.
[^5]: $\mathcal{C}_{\Delta_1'}$ in the fourth operation of Figure \[fig:Reductions\] is a subset of the tape $\mathcal{C}$ connected to $\Delta'_1$. By definition of tape, two sets $\mathcal{C}_{\Delta_1'}$ and $\mathcal{C}_{\Delta_2'}$ partition $\mathcal{C}$.
|
---
abstract: 'A model-independent global search for new physics has been performed at the CDF experiment. This search examines nearly $400$ final states, looking for discrepancies between the observed data and the standard model expectation in populations, kinematic shapes, and the tails of the summed transverse momentum distributions. A new approach also searches in approximately $5000$ mass variables looking for ‘bumps’ that may indicate resonant production of new particles. The results of this global search for new physics in $2 \invfb$ are presented. In addition, a model-independent search for deviations from the Standard Model prediction is performed in $e^+p$ and $e^-p$ collisions at HERA II using all H1 data recorded during the second running phase. This corresponds to integrated luminosities of $178 \invpb$ and $159 \invpb$ for $e^+p$ and $e^-p$ collisions, respectively. A statistical algorithm is used to search for deviations in the distributions of the scalar sum of transverse momenta or invariant mass of final state particles, and to quantify their significance.'
author:
- 'A. Soha (on behalf of the CDF and H1 Collaborations)'
title: General Searches for New Physics
---
INTRODUCTION
============
In stark contrast to most searches for new physics, which optimize for a particular model or signature, the general searches presented here are model-independent and include many final state particle combinations in an effort to be highly inclusive. The CDF global search results will be presented first, followed by the H1 general search results.
GLOBAL SEARCH AT CDF
====================
The model-independent global search for new high-$\pt$ physics in $p \overline{p}$ collisions at the Tevatron using CDF has three components [@cdf_two_prd]: [Vista]{} examines populations and kinematic features of the high-$\pt$ data; the [Bump Hunter]{} [@cdf_thesis] searches for resonances in invariant mass combinations; and [Sleuth]{} looks for excesses at high sum-$\pt$ ($\Sigma \pt$).
The CDF results use data corresponding to a luminosity of $2 \invfb$, acquired through inclusive high-$\pt$ electron, muon, photon, and jet triggers. Standard criteria are imposed to identify electrons, muons, taus, photons, jets, $b$-jets, and missing transverse energy (${\mbox{${E\!\!\!\!/_T}$}}$), all with thresholds equivalent to $\pt > 17 {\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c}}\xspace}$. Events are further selected to meet offline requirements such as $\Et(e) > 25 {\ensuremath{\mathrm{\,Ge\kern -0.1em V}}\xspace}$, $\pt(\mu) > 25 {\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c}}\xspace}$, or $\Et(\gamma) > 60 {\ensuremath{\mathrm{\,Ge\kern -0.1em V}}\xspace}$ [@cdf_one_prd]. Approximately $4.3$ million events are partitioned into $399$ exclusive final states, and new categories are created as needed.
The strategy is to use Monte Carlo event generators such as [Pythia]{} and [MadEvent]{} to represent the Standard Model (SM), and to pass the resulting events through a GEANT-based simulation of the CDF detector response. The simulation is then used in a global fit to the CDF data, to extract $43$ corrections factors. The fit is performed simultaneously to all final states and is subjected to external constraints. The correction factors, which include corrections to leading order theory cross sections, object reconstruction efficiencies, and mis-identification rates, are then used to improve the SM prediction. The three components ([Vista]{}, [Bump Hunter]{}, and [Sleuth]{}) of the global comparison between the data and SM prediction are performed, and the procedure is iterated by feeding information back into the simulation and correction factors until there is either a clear case for new physics or all discrepancies have known sources.
Population and Kinematic Distribution Results {#sec:vista}
---------------------------------------------
The [Vista]{} comparison of final state populations between data and SM predictions is shown in Fig. \[fig:cdf\_vista\_pop\]. The histogram shows the Poisson probability that the SM population in a final state would fluctuate above or below the observed population in data, expressed in units of standard deviations. The plotted probabilities do not include a trials factor, which accounts for the large number of final states that are examined and reduces the significance of each observed discrepancy. The greatest observed discrepancy is in the final state $be^{\pm}{\mbox{${E\!\!\!\!/_T}$}}$ where $817.7 \pm 9.2$ events are expected and $690$ events are observed, for a discrepancy of $-4.3 \sigma$ before the trials factor and $-2.7 \sigma$ after including the trials factor. Therefore, no population shows a significant discrepancy.
\
[Vista]{} also automatically produces and examines $19650$ kinematic distributions. The results are summarized in Fig. \[fig:cdf\_vista\_shapes\], where the histogram shows the Kilmogorov-Smirnov probability that the distributions in the data and SM prediction are consistent, expressed in units of standard deviations. The trials factor due to examining thousands of distributions has not yet been accounted for in the plot. Distributions are considered discrepant if they disagree by more than $5\sigma$ (approximately $>3\sigma$ after including the trials factor). The $555$ distributions that meet this criteria are examined more closely. It turns out that $81\%$ of the discrepancies can be explained by a deficiency in modeling soft jet emission in QCD parton showering. An additional $16\%$ are due to inadequate modeling of the transverse boost of the colliding system and $3\%$ are due to residual crudeness in the correction factor model, mostly from using simplified $\pt$-dependencies in fake rate correction factors. Therefore, there are no claims for new physics based on the kinematic distribution comparisons.
Bump Hunter Results
-------------------
A new resonance might appear as a bump in an invariant mass distribution. The CDF [Bump Hunter]{} uses the final states from [Vista]{} to form all invariant mass combinations and perform a comparison between data and SM backgrounds. A search window of $2 \Delta M$, where $\Delta M$ is the expected detector mass resolution, is scanned across each invariant mass distribution. A candidate bump must have at least five data events and side-bands that are in better agreement than the central search window. Pseudo-experiments are then used to estimate the significance of any qualifying bumps. The results are shown in Fig. \[fig:cdf\_bump\_hunter\], which shows the probability for a corresponding bump from pseudo-data to have a larger significance than the one found in data, cast in terms of standard deviations. Of the $5036$ scanned distributions, $2316$ have qualifying bumps. The visible shift in the histogram is caused by local deficiencies in the SM prediction, but does not invalidate the method since the shift makes it more likely that a bump surpasses the threshold for further study. The threshold for further investigation is $5\sigma$, which corresponds to $3 \sigma$ after including the trials factor for $5036$ mass distributions. There is one bump beyond this threshold, in a final state with four jets and low $\Sigma \pt$, but it is found to be due to the same soft jet modeling problem mentioned in section \[sec:vista\]. Hence, no new physics is found by the [Bump Hunter]{} in $2 \invfb$.
Search at High Sum-
-------------------
[Sleuth]{} assumes that new physics will appear as an excess, and that the excess will be at high $\Sigma \pt$ and in one final state. The $\Sigma \pt$ is the scalar sum of the $\pt$ of the individual objects, unclustered energy, and ${\mbox{${E\!\!\!\!/_T}$}}$. For each final state, the $\Sigma \pt$ distribution is scanned, and the one-sided region with the most significant excess of data is selected. The significance, $\mathcal{P}$, is determined as the fraction of pseudo-experiments that find a region at least as discrepant as the one observed in data. The final state with the largest discrepancy is $e^\pm\mu^\pm$, with $\mathcal{P}=0.00055$, corresponding to $3.26\sigma$ before including a trials factor. It is found that $8\%$ of experiments like CDF would find an excess at least as large as this most discrepant [Sleuth]{} final state. There are no claims of new physics using [Sleuth]{} with $2 \invfb$.
GENERAL SEARCH AT H1
====================
The global search for new physics at H1 uses data corresponding to luminosities of $178 \invpb$ and $159 \invpb$ from $e^+p$ and $e^-p$ collisions, respectively [@H1_update; @H1_117]. Isolated electrons, muons, photons, jets, and neutrinos are included if they have $\pt > 20 {\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c}}\xspace}$ and a polar angle satisfying $10^\circ < \theta < 140^\circ$. The event selection requires exclusive final states with two or more objects, and events are classified by the number and types of objects. This procedure examines all combinations and finds that $23$ final states are populated. Simulations are used for all contributing SM processes, including photoproduction, deep-inelastic scattering, QED Compton scattering, electroweak production, and QCD. The resulting predictions are used in comparisons to the event yields, $\Sigma \pt$, and invariant mass distributions found in data. A statistical algorithm is then employed to identify the largest deviations and evaluate the associated probabilities.
Event Yields, Sum-, and Invariant Mass
--------------------------------------
The event yield comparisons show good agreement in all final states, in both the $e^-p$ and $e^+p$ data, as shown in Fig. \[fig:H1\_pop\]. The $\Sigma \pt$ and invariant mass distribution comparisons are conducted by finding regions of greatest deviation between data and the SM expectation for each final state. All groups of neighboring $5 {\ensuremath{\mathrm{\,Ge\kern -0.1em V}}\xspace}$ bins are tested. A measure, p, is determined as the probability for a positive or negative fluctuation of the SM expectation to be at least as large as that observed in data. The procedure accounts for Poisson statistical errors and Gaussian systematic uncertainties. For each final state, the region with the smallest p-value is selected. The results for the invariant mass comparisons using $e^-p$ data are shown in Fig. \[fig:H1\_mAll\_minus\] and the results for the $\Sigma \pt$ comparison using $e^+p$ data are shown in Fig. \[fig:H1\_sumpt\_plus\].
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Significance
------------
The significance of each deviation in the H1 $\Sigma \pt$ and invariant mass comparison is evaluated using pseudo-data. The method determines the probability, $\hat{\rm P}$, to observe a region with a p-value less than the smallest p-value seen in data. Calculating this $\hat{\rm P}$ allows for the comparison of deviations across different final state categories. Fig. \[fig:H1\_phat\_a\] shows the $-\log_{10} \hat{\rm P}$ distribution for the invariant mass comparison using $e^-p$ data, while Fig. \[fig:H1\_phat\_b\] shows the $-\log_{10} \hat{\rm P}$ distribution for the $\Sigma \pt$ comparison using $e^+p$ data. Note that a $5\sigma$ discrepancy would correspond to a value of $-\log_{10} \hat{\rm P}$ between $5$ and $6$. No such significant discrepancies between data and SM expectations are observed. The largest deviation is in the $\mu j \nu$ final state category.
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CONCLUSIONS
===========
The CDF and H1 general searches for new physics have probed large datasets for indications of new physics in population and kinematic distributions, using a large number of final states. These searches provide broad views of the high-$\pt$ data samples and demonstrate understanding of the detectors and SM simulation. They do not rule out all sources of new physics, thus leaving open the possibility for future discoveries.
The author wishes to thank the 2008 ICHEP organizers and hosts, as well as the CDF and H1 collaborations and funding sources.
[9]{} T. Aaltonen [*et al.*]{} (CDF Collaboration), Submitted to Phys. Rev. D (2008), arXiv:0809.3781.
G. Choudalakis, Ph.D. thesis, Massachusetts Institute of Technology (2008), arXiv:0805.3954.
T. Aaltonen [*et al.*]{} (CDF Collaboration), Phys. Rev. D [**78**]{}, 012002 (2008).
E. Sauvan [*et al.*]{} (H1 Collaboration), “A General Search for New Phenomena at HERA”, Proc. of 15th Int. Workshop on Deep-Inelastic Scattering and Related Subjects, Munich, April 2007, arXiv:0706.4110.
A. Aktas [*et al.*]{} (H1 Collaboration), Phys. Lett. B [**602**]{}, 14 (2004), arXiv:hep-ex/0408044.
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abstract: 'In this comment it is pointed out that the electron velocity of the same order as observed in graphene had been measured in GaAs submicron devices long ago. Particle- antiparticle asymmetry related with electron and hole effective masses in graphene seems puzzling as hole in a condensed matter system cannot be treated as anti-electron. It is argued that there should be a universal electrodynamics for QHE and superconductivity. In this context attention is drawn to the new approach based on massless electron and the interpretation that magnetic field represents angular momentum of the photon fluid. Measurement of electron velocity in graphene and GaAs in parallel is suggested for testing the massless electrodynamics.'
author:
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S. C. Tiwari\
Institute of Natural Philosophy\
c/o 1 Kusum Kutir Mahamanapuri,Varanasi 221005, India
title: 'Comment on “Cyclotron resonance study of the electron and hole velocity in graphene monolayers”'
---
Sciencewatch (CERN Courier May 2007) highlighted the reported observation of QHE in graphene at room temperature. A phrase ’the fact that its electrons act in many ways like massless Dirac fermions’ attracted my attention as I thought massless electron model proposed more than two decades ago, and comprehensively discussed in a monograph [@1] may have found an experimental validation in graphene. A careful reading of the paper [@2] shows that ’exceptionally high electron velocities’ of the order of $1.117x10^8$ cm/sec in graphene are, in fact, in the range observed in GaAs ballistic transport and velocity overshoot [@3]. The concluding remark in [@2] that the band structure of monolayer graphene is not well understood at present is quite reasonable. However the one on particle-antiparticle asymmetry is puzzling. Note that effective masses of electron and hole differ considerably in general; a hole in condensed matter system cannot be treated as an antiparticle of electron.
Though it is well known, perhaps it is useful to remind ourselves that Dirac in his famous work first identified antiparticle of electron to be proton-the only known positively charged elementary particle at that time. Weyl- a mathematician argued that symmetry on mathematical grounds demanded the mass of antiparticle equal to that of electron-later discovered as positron. Secondly the linear dispersiom relation by itself, the approximate relation derived by Wallace for graphite, does not mean that hole could be treated as antiparticle. A massive (complex) Klein-Gordon field does not have linear relation between energy and momentum yet it has antiparticle, and photon has a linear relation yet there is no antiphoton (in the standard QFT).
In a more general context I would like to raise the following question: Is there a universal electrodynamics for QHE and superconductivity? First we note an important fact: quantized Hall resistance and magnetic flux quantum involve only fundamental constants (c,h, and e) and do not depend on the material properties or size of the samples. There exist compelling arguments discussed in detail in Sec. 9.2 of [@1] indicating a universal electrodynamics assuming massless electron and the interpretation that magnetic field is angular momentum flux of photon fluid. The measurement of electron velocity in superconductor and QHE device has been advocated in the book. That the problem of Cooper pair mass is nontrivial has been underscored by Mishonov [@5] quoting de Gennes remark from Tinkham’s book and Ginzburg’s private communication. Amongst three characteristics (zero resistance, Meissner effect, and flux quantization) the explanation of zero resistance in BCS theory seems unsatisfactory. Feynman says [@5] : “There is no resistance because all the electrons are collectively in the same state. In the ordinary flow of current you knock one electron or the other out of the regular flow, gradually deteriorating the general momentum. But here to get one electron away from what all the others are doing is very hard because of the tendency of all Bose particles to go in the same state. A current once started, just keeps on going forever.” This is hardly an explanation, further persistent current is not quite the same as zero resistance. Weinberg [@6] seems to have addressed the problem better- he remarks that the absence of resistance (other than that in closed rings for persistent current) can be understood considering time-dependent effects. Relating the voltage with the time derivative of a postulated Goldstone boson and defining a suitable Hamiltonian of the system he offers an explanation for zero resistance. The role of Cooper pairs, however remains obscure.
In a new approach field-free filamentary tubular structures in condensed matter system have been envisaged in which electrons move with the velocity of light [@1]. Few collisions could result in quantized average drift velocity. This idea has been used to understand QHE and superconducting electrodynamics in [@1]. A simple model for the ballistic transport [@7] found remarkable support in new experiments at the IBM as noted by Heiblum [@8] :“... new experiments, not yet published, show that ballistic transport through GaAs regions wider than previously reported (about 800 A wide), result with... . These results are qualitatively in agreement with Tiwari’s comment”.
Recall that the eigenvalue of velocity operator in Dirac’s theory is equal to $\pm c$ ; localizing electron wavepacket in a region smaller than electron’s Compton wavelength leads to a mixing of negative and positive frequency states- the zitterbewegung of Schroedinger. Point electron theory of Dirac conflicts with the proposed extended structure [@1; @9] and electron motion with light velocity is considered unphysical. In our model electron is an extended spatio-temporal 2+1 dimensional object, and has velocity in field-free region (or rather intrinsic) equal to c normal to the 2-spatial plane. Inertia is interpreted as a consequence of collisions or viscosity of the space, see modified Schroedinger equation derived using this idea in PLA (1988) [@9].
Obviously testing the radical idea of massless electron would have far reaching implications on fundamental physics. I suggest time-of-flight technique used in GaAs [@10] be explored for graphene. It seems InAs semiconductor and Nb superconductor have transparent interface i.e. no Schottky barrier [@11]. In that case long superconductor could be used to investigate the velocity of electron/ Cooper pairs. In conclusion, studies on electron transport in GaAs and graphene carried out in parallel might give useful new physics. For GaAs properties we refer to [@12].
The Library facility at Banaras Hindu University is acknowledged.
[99]{} S. C. Tiwari, Rebirth of the Electron: Electromagnetism - An unorthodox new approach to fundamental problems in physics, IONP Studies in Natural Philosophy Vol. 2, published at http://www.lulu.com/content/439658 . R. S. Deacon et al, arXiv: 0704.0410 v2 \[cond-mat.mes-hall\]. M. Heiblum et al, Direct observation of ballistic transport in GaAs, Phys. Rev. Lett. 55, 2200 (1985). T. M. Mishonov, How to measure the Cooper pair mass using plasmons in low dimensional superconducting structures, ICTP Preprint IC/90/123 (1990). R. P. Feynman, The Feynman Lectures on Physics Vol. 3 (Addison-Wesley 1965) Ch. 21 p. 21-8. S. Weinberg, The Quantum Theory of Fields, Vol. II (C.U.P. 1996) pp332-352. S. C. Tiwari, Nonequilibrium transport in submicron GaAs devices, Int. J. Electronics, 62, 27 (1987). M. Heiblum, Unpublished Note (1986). S. C. Tiwari, The nature of electronic charge, Found. Phys. Lett. 19, 51 (2006) also arXiv: physics/0408053; Relativity, entanglement and the physical reality of the photon, J. Optics B:Semiclass. and Quantum 4, S39 (2002); Derivation of the Hamiltonian form of the Klein-Gordon equation from the Schroedinger-Furth quantum diffusion theory: comments, Phys. Lett. A 133, 279 (1988); and arXiv: physics/0406136. J. G. Ruch and G. S. Kino, Transport properties of GaAs, Phys. Rev. 174, 921 (1968). A. Dimoulas et al, Phys. Rev. Lett 74, 602 (1995) Gallium Arsenide Materials, Devices and Circuits. Ed. by M. J. Howes and D. V. Morgan (John Wiley 1985).
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abstract: 'We investigate the spherically symmetric 1D ablation problem. We show that the parabolic heat equation fails to describe the approach to steady state in infinite space. The hyperbolic equation shows an approach to steady state with a time constant given by the thermal relaxation time. However the infinite geometry is rather unphysical and gives rise to a so-called zero mode. Therefore we also consider the finite problem with a large boundary at constant temperature. Then both equations show approach to steady state, but only the hyperbolic equation seems to be physically correct for small times.'
author:
- |
Günter Scharf [^1]\
Physics Institute, University of Zürich,\
Winterthurerstr. 190 , CH-8057 Zürich, Switzerland
date:
title: Approach to steady state in the heat equation and in the hyperbolic heat transfer equation
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Introduction
============
It is well known that the usual heat equation $$\ro c{\d T\over\d t}=\kappa\triangle T+Q\eqno(1.1)$$ has the defect that heat can propagate instantaneously through space. In (1.1) $T$ is temperature, $t$ is time, $\triangle$ is the Laplace operator and $Q$ describes the heat generation. The constant $\kappa$ is the thermal conductivity, $\ro$ the mass density and $c$ the specific heat. Still the equation is widely used to simulate heat transfer problems (\[1\] and references given there). The reason for this causality defect is that the heat equation is first order in time. A second defect related to this is the restriction of the initial value problem. In the Cauchy problem for (1.1) only the temperature $T$ at time $t=0$ can be specified. But in experiments also the temporal derivative $\d T/\d t$ at $t=0$ must be adjusted to the experimental situation. This is possible in the hyperbolic heat transfer equation which is second order in time.
It is our aim to compare the two equations in an analytically solvable case of some practical interest. In the next section the 1D ablation problem in infinite space is solved for the heat equation and in sect.3 for the hyperbolic heat transfer equation. In sect.4 some special functions are discussed which appear in the solutions. In sect.5 we study the problem on a finite spherical volume. This is important because it turns out that the infinite problem cannot be viewed as the limit of the finite problem. So for real applications only this finite problem is relevant.
The 1D ablation problem according to the bio-heat equation
==========================================================
Let us consider a spherical electrode of radius $r_0$ in an infinite medium with electrical conductivity $\si$. Assuming the second dispersive electrode at infinity with potential $V=0$ the potential in the medium is given by the simple solution of Laplace’s equation $$V(r)={V_0r_0\over r}\eqno(2.1)$$ where $V_0$ is the applied potential on the ablation electrode. The corresponding electric field strength is $$E_r=-{\d V\over\d r}={V_0r_o\over r^2}\eqno(2.2)$$ and the heat generation $$Q=\si{(V_0r_0)^2\over r^4}.\eqno(2.3)$$ With this constant heating we want to calculate the transient temperature $T(t,r)$ as solution of equation (1.1) which becomes $${\d T\over\d t}={a\over r^2}{\d\over\d r}\B(r^2{\d T\over\d r}\B)+{\beta\over r^4},\quad a={\kappa\over\ro c},$$ where $\beta$ is the constant appearing in (2.3) divided by $\ro c$ and $a$ the thermal diffusivity.
We write the solution as $$T(t,r)=T_1(r)+T_2(t,r),\eqno(2.4)$$ where $T_1(r)$ is the steady state solution satisfying $${d\over dr}\B(r^2{dT_1\over dr}\B)=-{\beta\over ar^2}.\eqno(2.5)$$ Assuming a temperature $T_\infty$ at infinity we get $$T_1(r)=T_\infty+{C_1\over r}-{b\over 2r^2}.\quad b={\beta\over a}\eqno(2.6)$$ The integration constant $C_1$ is fixed by the assumption of no heat flux at the electrode $${dT_1\over dr}\vert_{r_0}=0,\eqno(2.7)$$ which is reasonable for a small electrode. This gives the following steady state solution $$T_1(r)=T_\infty+{b\over r_0 r}-{b\over 2r^2}.\eqno(2.8)$$ The maximal temperature is found at the electrode, of course $$T_1(r_0)=T_\infty+{b\over 2r_0^2}.\eqno(2.9)$$
The remaining homogeneous equation for $T_2$ is solved by separation of variables $$T_2(t,r)=T_3(t)T_4(r).\eqno(2.10)$$ Then we have $${1\over a}{\dot T_3\over T_3}={T_4''\over T_4}+{2\over r}{T_4'\over T_4}={\rm const.}=-\,k^2$$ where the dot means time derivative and the prime radial derivative. This yields $$T_3(t)=C_3e^{-\,k^2at}\eqno(2.11)$$ and $$r^2T_4''+2rT_4'+\,k^2r^2T_4=0.\eqno(2.12)$$ This last equation is a special case of Bessel’s equation with the solution $$T_4(r)={1\over r}\B(C_4\sin\,k r+C_5\cos\,k r\B).\eqno(2.13)$$ Now the general solution of our problem is given by $$T(t,r)=T_\infty+{b\over r_0 r}-{b\over 2r^2}+{1\over r}\int\limits_0^\infty d\,k\,e^{-\,k^2at}\B(f_1(\,k)\sin\,k r+$$ $$+f_2(\,k)\cos\,k r\B).\eqno(2.14)$$
The unknown functions $f_1$ and $f_2$ in (2.14) are determined by the initial condition $$T(0,r)=T_\infty,\eqno(2.15)$$ $${b\over 2r^2}-{b\over r_0r}={1\over r}\int\limits_0^\infty[f_1(\,k)\sin\,k r+f_2(\,k)\cos\,k r]d\,k.\eqno(2.16)$$ Here the l.h.side is only defined for $r>r_0$. To use the theorems on the [*real*]{} Fourier transform we continue the functions to $0<r<r_0$. The even function $b/2r^2$ is continued by the constant $b/2r_0^2$. The inverse Fourier transform then yields $$f_1(\,k)={b\over\pi}\int\limits_{r_0}^\infty{\sin\,k r\over r}dr+{b\over\pi r_0^2}\int\limits_0^{r_0}r\sin\,k r\,dr=$$ $$={b\over 2}-{b\over\pi}{\rm Si}(\,k r_0)+{b\over\pi\,k^2r_0^2}(\sin\,k r_0-\,k r_0\cos\,k r_0)=$$ $$={b\over 2}+O(\,k r_0).\eqno(2.17)$$ Here [Si]{} is the sine integral. For small electrode radius $r_0$ this gives the following contribution to (2.14) $$\int\limits_0^\infty d\,k e^{-\,k^2at}f_1(\,k)\sin\,k r={b\over 4\sqrt{at}}\int\limits_0^\infty dy\,e^{-y^2/4}\sin{ry\over 2\sqrt{at}}=$$ $$={b\over 2\sqrt{at}}D_+\B({r\over 2\sqrt{at}}\B).\eqno(2.18)$$ Here $D_+(z)$ is the Dawson function (see Wikipedia and references given there).
The function $f_2(\,k)$ in (2.16) must clearly degenerate to a delta-distribution $$f_2(\,k)=-{2b\over r_0}\delta(\,k).\eqno(2.19)$$ This contribution then cancels the $1/r$-term in (2.14) which is necessary for $t=0$. Then the final result is $$T(t,r)=T_\infty-{b\over 2r^2}+{b\over 2\sqrt{at}}D_+\B({r\over 2\sqrt{at}}\B)+O(r_0).\eqno(2.20)$$ The Dawson function has the following asymptotic expansion $$D_+(x)={2\over 2x}+{1\over 4x^3}+{3\over 8x^5}+\ldots$$ for $x\gg 1$. Using this in (2.20), the first term cancels the negative contribution so that the result for small times seems to be correct. It shows the expected rise of the temperature above $T_\infty$. However the Dawson function has a maximum at $x=0.924...$, $D(x)=0.541...$ and for smaller $x$ it decreases to 0. As a consequence for large $t$ the temperature (2.20) falls below $T_\infty$ which is completely wrong ! The reason for this disaster is simple: The heat equation is first order in time. Therefore only one condition, namely the initial condition at $t=0$ is at our disposal. The hyperbolic heat equation in the next section is second order in time, then we have two free constants of integration, so that we can get the right behavior of the solution at $t=0$ [*and*]{} $t=\infty$.
Approach to steady state in the hyperbolic heat equation
========================================================
According to Cattaneo \[2\] and Vernotte \[3\] a better description of heat transfer is obtained by assuming a time delay $\tau$ between heat flux $q$ and temperature gradient $$q(t+\tau,x)=-\kappa \nabla T(t.x).\eqno(3.1)$$ Expanding up to first order in $\tau$ we have $$q+\tau{\d q\over\d t}=-\kappa\nabla T$$ and $$\nabla q=-\tau{\d\over \d t}\nabla q-\nabla \kappa\nabla t$$ which is substituted into the energy conservation equation $$-\nabla q(t,x)+Q(t,x)=\ro c{\d T(t,x)\over\d t}.\eqno(3.2)$$ For constant thermal conductivity $k$ this gives the hyperbolic heat equation $${\d^2 T\over\d t^2}+{1\over\tau}{\d T\over\d t}-{\kappa\over\tau\ro c}\triangle T={1\over\ro c}{\d Q\over\d t}+{Q\over\tau\ro c}.\eqno(3.3)$$
For our 1D ablation problem this equation assumes the following form $${\kappa\over\tau\ro c}{1\over r^2}{\d\over\d r}\B(r^2{\d\over\d r}T\B)+{d\over r^4}={1\over\tau}{\d T\over\d t}+{\d^2T\over\d t^2}.\eqno(3.4)$$ With another boundary condition this problem has been studied in \[4\]. These authors use the method of Laplace transform which becomes very complicated. By the method of the previous section we get the solution in much simpler form. We do not include a switching of the heat generation $Q$ by means of a Heaviside step function as in \[4\]. This would give an additional singular term $u_1(r)\delta(t)$. Then we have a so-called generalized Cauchy problem in the sense of distributions, which has been treated by Vladimirov \[7\]. The $\delta$-term then fixes the initial condition at $t=0$ as $${\d T(t,r)\over\d t}\B\vert_{t=0}=u_1(r).$$ But in \[4\] the simple initial condition $${\d T(t,r)\over\d t}\B\vert_{t=0}=0\eqno(3.5)$$ was used, this is a certain inconsistency. We consider the classical Cauchy problem where we have two initial conditions at $t=0$ for free.
As before we write the solution in the form (2.4) where the steady state solution $T_1(r)$ satisfies the equation $${d\over dr}\B(r^2{dT_1\over dr}\B)=-{\beta\over ar^2}.\eqno(3.6)$$ We have the same steady state solution (2.8) $$T_1(r)=T_\infty+{b\over r_0 r}-{b\over 2r^2}.\eqno(3.7)$$ But the homogeneous equation now reads $${1\over r^2}\B(2r{\d T_2\over\d r}+r^2{\d ^2T_2\over\d r^2}\B)={1\over a}{\d T_2\over\d t}+\eps{\d ^2T_2\over\d t^2}\eqno(3.8)$$ with $$a={\ro c\over\kappa},\quad \eps=a\tau.\eqno(3.9)$$ For $\eps=0$ we are back at the parabolic heat equation.
Again equation (3.8) is solved by separating the variables $$T_2(t,r)=T_3(t)T_4(r)\eqno(3.10)$$ which yields $${1\over a}{\dot T_3\over T_3}+\eps{\ddot T_3\over T_3}={T_4''\over T_4}+{2\over r}{T_4'\over T_4}=-\,k^2.\eqno(3.11)$$ The equation for $T_4$ is the same as before (2.12), but for $T_3$ we now have the second order equation $$\eps\ddot T_3+{1\over a}\dot T_3=-\,k^2T_3.\eqno(3.12)$$ It has two exponential fundamental solutions $$T_\pm (t)=C_\pm e^{\omega_\pm t}\eqno(3.13)$$ where $\omega_\pm$ are the two solutions of the quadratic equation $$\eps\omega^2+{1\over a}\omega+\,k^2=0.$$ We have two negative roots $$\omega_\pm=-{1\over 2a\eps}\pm\sqrt{{1\over 4a^2\eps^2}-{\,k^2\over\eps}}=-{1\over 2\tau}\pm\sqrt{{1\over 4\tau^2}-{a\,k^2\over\tau}},\eqno(3.14)$$ Then the general solution with the same boundary condition (2.7) as in the last section is given by $$T(t,r)=T_\infty+{b\over r_0r}-{b\over 2r^2}+{1\over r}\int\limits_0^\infty d\,k e^{\omega_+t}\B[f_1(\,k)\sin \,k r+$$ $$+f_2(\,k)\cos\,k r\B]+{1\over r}\int\limits_0^\infty d\,k e^{\omega_-t}\B[g_1(\,k)\sin \,k r
+g_2(\,k)\cos\,k r\B].\eqno(3.15)$$
To satisfy the initial condition $T=T_\infty$ we must again compensate the second term $\beta/r_0r$ by some contribution from the integrals for $\,k =0$. For $\,k=0$ we have $$\omega_+=0,\quad \omega_-=-{1\over\tau}.\eqno(3.16)$$ In the first case with $f_2(k)\sim\delta(\,k)$ we are in the same situation as in the last section and get no approach to steady state. So we take $f_2=0$. But now we can choose $$g_2(\,k)=-{b\over r_0}\delta(\,k)\eqno(3.17)$$ and have the desired compensation for $t=0$. But for $t\to\infty$ this term goes to 0 because $\omega_-$ (3.16) gives an exponential fall off $\sim\exp -t/\tau$. That means we obtain the correct steady state as far as the $1/r$ term is concerned.
Regarding the $1/r^2$ term we must determine $f_1$ and $g_1$ such that $${b\over r}=\int\limits_0^\infty d\,k\,[f_1(\,k)+g_1(\,k)]\sin\,k r.\eqno(3.18)$$ As before (2.17) this gives $$f_1(\,k)+g_1(\,k)={b\over 2}+O(\,k r_0).\eqno(3.19)$$ To determine $f_1$ and $g_1$ separately we need a second initial condition. Preliminary experiments show that the above condition (3.5) is physically correct, so we assume it. The condition (3.5) implies $$\omega_+f_1+\omega_- g_1=0$$ so that $$f_1(\,k)={b\over 2}{\omega_-\over\omega_--\omega_+},\quad
g_1(\,k)={b\over 2}{\omega_+\over\omega_+-\omega_-}.\eqno(3.20)$$ Inserting the roots (3.14) we obtain the following final result $$T(t,r)=T_\infty+{b\over r_0r}\B(1-e^{-t/\tau}\B)-{b\over 2r^2}+$$ $$+{b\over 4r}\int\limits_0^\infty d\,k\B[e^{\omega_+t}\B(1+(1-4a\tau\,k^2)^{-1/2}\B)+e^{\omega_-t}\B(1-(1-4a\tau\,k^2)^{-1/2}\B)\B]\sin\,k r.\eqno(3.21)$$ However, we note that this total solution does not satisfy the initial condition (3.5) of vanishing temporal derivative. We shall return to this point in sect.5.
The integral in (3.21) must be split at $$\,k={1\over 2\sqrt{a\tau}}=\,k_0\eqno(3.22)$$ because the roots (3.14) become complex for $k>k_0$. For small $\tau$ only the integral $$I_1=\int\limits_0^{\,k_0}d\,k\B(1+(1-4a\tau\,k^2)^{-1/2}\B)e^{\omega_+t}\sin\,k r\eqno(3.23)$$ is important. Extending the upper limit to infinity and expanding the square root we get $$I_1=\int\limits_0^\infty d\,k(2+2a\tau\,k^2)e^{-at\,k^2}\sin\,k r=$$ $$=2(1-\tau\d_t)\int\limits_0^\infty d\,k\,e^{-at\,k^2}\sin\,k r.\eqno(3.24)$$ This gives the Dawson function $D_+$ (2.18) again, up to a correction $O(\tau)$ $$I_1=2(1-\tau\d_t){1\over\sqrt{at}}D_+\B({r\over 2\sqrt{at}}\B).\eqno(3.25)$$ So for small thermal relaxation time $\tau$ we recover the term in the solution (2.20) of the usual heat equation.
For large $\tau$ the integral over $[\,k_0,\infty]$ gives the leading contribution. Since we have two complex conjugate roots (3.14) we obtain a real part $$I_2=\int\limits_{k_0}^\infty=2{\rm Re}\int\limits_{k_0}^\infty d\,k\B(1-i(4a\tau\,k^2-1)^{-1/2}\B)e^{-{t\over 2\tau}(1-
\sqrt{4a\tau\,k^2-1}}\sin\,k r=$$ $$=2e^{-t/2\tau}\int\limits_{\,k_0}^\infty d\,k\,\B[\cos\B({t\over 2\tau}\sqrt{4a\tau\,k^2-1}\B)+{\sin ({t\over 2\tau}\sqrt{4a\tau\,k^2-1})\over\sqrt{4a\tau\,k^2-1}}\B]\sin\,k r.\eqno(3.26)$$ Here the periodic time dependence indicates the appearance of thermal waves \[4\] which, however, are damped with a time constant $2\tau$. This damping is essential for the approach to steady state The integral (3.26) is investigated in the next section.
Some special functions
======================
According to (3.21-23) we must calculate the integrals $$I^\pm_1=\int\limits_0^{k_0}dk\B(1\pm(1-4a\tau k^2)^{-1/2}\B)e^{\omega_\pm t}\sin kr.\eqno(4.1)$$ With the new integration variable $$x=2\sqrt{a\tau}k\eqno(4.2)$$ we get the dimensionless form $$I_1^\pm={1\over 2\sqrt{a\tau}}\int\limits_0^1 dx\B(1\pm{1\over\sqrt{1-x^2}}\B)e^{-{t\over 2\tau}(1\mp\sqrt{1-x^2})}\sin\B({rx\over 2\sqrt{a\tau}}
\B)=$$ $$={1\over 2\sqrt{a\tau}}I_1^\pm(s,u)\eqno(4.3)$$ where $$s={t\over 2\tau},\quad u={r\over 2\sqrt{a\tau}}\eqno(4.4)$$ and $$I_1^\pm(s,u)=\int\limits_0^1 dx\B(1\pm{1\over\sqrt{1-x^2}}\B)e^{-s\pm s\sqrt{1-x^2}}\sin ux=$$ $$=\pm e^{-s}(\d_s+1)S_1(\pm s,u).\eqno(4.5)$$ The remaining integral $$S_1(s,u)=\int\limits_0^1 dx\,e^{s\sqrt{1-x^2}}{\sin ux\over\sqrt{1-x^2}}\eqno(4.6)$$ can be easily calculated by numerical integration, together with its derivatives. But it seems not possible to write it in terms of known special functions.
With the substitution $y=\sqrt{1-x^2}$ we get the form $$S_1(s,u)=\int\limits_0^1 dy\,{\sin(u\sqrt{1-x^2})\over\sqrt{1-x^2}}e^{sy}\eqno(4.7)$$ and $${\d S_1\over\d u}=\int\limits_0^1 dy\,s^{sy}\cos(u\sqrt{1-y^2}).\eqno(4.8)$$ On the other hand $${\d S_1\over\d s}=\int\limits_0^1 dx\,e^{s\sqrt{1-x^2}}\sin ux=\int\limits_0^1 dy\,{y\over\sqrt{1-y^2}}\sin(u\sqrt{1-y^2})e^{sy}=$$ $$=-{1\over u}\int\limits_0^1{d\over dy}(\cos u\sqrt{1-y^2})e^{sy}dy\eqno(4.9)$$ allows partial integration which brings us back to (4.8), so that we get the following differential equation for $S_1$: $$u{\d S_1\over\d s}=s{\d S_1\over\\d u}+\cos u-e^s.\eqno(4.10)$$
For the approach to steady state we need a bound of $S_1(s,u)$ for large $s$. Such a bound is obtained by means of the confluent hypergeometric function \[5\] $$\vert S_1(s,u)\vert<\int\limits_0^1{e^{sx}\over\sqrt{1-x}}dx=2 M(1,{3\over 2},s).\eqno(4.11)$$ Using the asymptotic behavior of $M(a,b,s)$ \[5\] we get for positive $s>0$ $$\vert S_1(s,u)\vert <\sqrt{\pi}e^s s^{-1/2}(1+O(s^{-1})).\eqno(4.12)$$ The exponential factor is cancelled in (4.5) so that we find a slow approach to steady state with $s^{-1/2}$. The other term with negative $s$ behaves better $$\vert S_1(-s,u)\vert < 2M(1.{3\over 2},-s)<{1\over s}+O(s^{-2}).\eqno(4.13)$$ This leads to an exponential decrease in (4.5).
To calculate the thermal wave integral (3.26) we introduce the second special function $$S_2(s,u)=\int\limits_1^\infty{dx\over\sqrt{x^2-1}}\sin(s\sqrt{x^2-1})\sin ux.\eqno(4.14)$$ This is obtained from (3.26) with the substitution (4.2) again. With the new integration variable $y=\sqrt{x^2-1}$ we get a Fourier - sine integral $$S_2(s,u)=\int\limits_0^\infty{dy\over\sqrt{y^2+1}}\sin(u\sqrt{y^2+1})\sin sy.\eqno(4.15)$$ The same integral with two cosine or one sine and one cosine function can be expressed by Bessel functions \[6\], but the integral (4.15) cannot. This might indicate that it is a new special function. As it stands the integral is not well suited for numerical integration. We get a better form by using the Euler substitution $$\sqrt{y^2+1}=yx+1\eqno(4.16)$$ in $$J_2^\pm=\int\limits_0^\infty{dy\over\sqrt{y^2+1}}\cos (u\sqrt{y^2+1}\pm sy).\eqno(4.17)$$ This follows from (4.15) by simple trigonometric formulas. With the substitution (4.16) we find $$J_2^\pm(s,u)=2\int\limits_0^1 dx\,{1\over 1-x^2}\cos\B({ux^2\pm 2sx+u\over 1-x^2}\B)\eqno(4.18)$$ which can be easily calculated by numerical integration. For the special case $u=s$ we obtain $$J_2^\pm(s,s)=2\int\limits_0^1{dx\over 1-x^2}\cos\B(s{1+x\over 1-x}\B)=$$ $$=\int\limits_1^\infty dz\,{\cos sz\over z}=-{\rm Ci}(s),\eqno(4.19)$$ which is the cosine-integral \[5\]
To obtain a bound for $J_2^\pm$ we use partial integration again: $$J_2^\pm=\int\limits_0^\infty{dy\over uy\pm s\sqrt{y^2+1}}{d\over dy}\sin(u\sqrt{y^2+1\pm sy})=$$ $$=\mp{\sin u\over s}+\int\limits_0^\infty{\sin(u\sqrt{y^2+1}\pm sy)\over (uy\pm s\sqrt{y^2+1})^2}\B(u\pm{sy\over\sqrt{y^2+1}}\B)\, dy.\eqno(4.20)$$ This decreases as $1/s$ for fixed $u$ or $r$.
Finite spherical geometry
=========================
In the results of the previous sections the zero-mode $k=0$ (3.16) has played an important role. This mode only appears in the infinite system. To be physically relevant we must check whether the infinite system can be considered as a limit of a large finite system. For this purpose let us assume a large spherical boundary of radius $r_1$ which is kept at a constant temperature $T_{01}$, $r_0$ is the radius of the electrode as before. We require the two boundary conditions $${\d T\over\d r}(t,r_0)=0,\quad T(t,r_1)=T_{01}.\eqno(5.1)$$ The steady state solution satisfying these conditions is now given by $$T_1(r)=T_{01}-{b\over r_0r_1}+{b\over 2r_1^2}+{b\over r_0r}-{b\over 2r^2}.\eqno(5.2)$$ At the electrode we have the higher temperature $$T_1(r_0)=T_{01}+{b\over 2}\B({1\over r_0}-{1\over r_1}\B)^2.$$
The remaining time dependent solution $T_2(t,r)$ is again factorized $=T_3(t)T_4(r)$ (2.10) where $T_4$ is the solution of $$r^2T_4''+2rT_4'=-k^2r^2T_4\eqno(5.3)$$ with the boundary conditions $${\d T_4\over\d r}(r_0)=0,\quad T_4(r_1)=0.\eqno(5.4)$$ We transform the equation (5.3) into a selfadjoint form with the substitution $$y(r)=rT_4(r)\eqno(5.5)$$ yielding $$-y''=k^2 y\eqno(5.6)$$ and the boundary conditions $$y(r_1)=0,\quad r_0y'(r_0)-y(r_0)=0.\eqno(5.7)$$ This is a simple standard Sturm-Liouville eigenvalue problem \[8\] in the Hilbert space $L^2([r_0,r_1])$. It has a discrete spectrum of eigenvalues $k_n$ in contrast to the infinite problem in the previous sections. The number $n=0,1,2,\ldots$ gives the number of zeros of the eigenfunctions $y_n$.
The first boundary condition (5.7) is immediately satisfied by $$y_n(r)=\sin k_n(r_1-r)\eqno(5.8)$$ and the second condition gives the transcendental equation $$\tan k_n(r_1-r_0)=-k_nr_0.\eqno(5.9)$$ This equation can easily be discussed graphically. It seems as if $k=0$ were the lowest eigenvalue, but a glance to (5.8) shows that this is not the case because $y_n=0$. To get the eigenvalues analytically we put $$k_n(r_1-r_0)=(2n+1){\pi\over 2}+\delta.\eqno(5.10)$$ Then (5.9) leads to $$\tan k_n(r_1-r_0)=-\cot\delta=-{1\over\delta}+{\delta\over 3}+\ldots =-(2n+1){\pi\over 2}{r_0\over r_1-r_0}$$ which for large $n$ gives $$\delta={2(r_1-r_0)\over (2n+1)\pi r_0}$$ so that $$k_n={(2n+1)\pi\over 2(r_1-r_0)}+{2\over (2n+1)\pi r_0}+O\B({1\over n^2}\B).\eqno(5.11)$$ In the infinite volume limit $r_1\to\infty$ the first term goes to 0, but the second term does not. There remains a finite gap between $k=0$ and the lowest eigenvalue $k_0$. That means the zero-mode of the previous sections is exceptional and not physical. The reason is that the boundary condition (5.7) at $r_0$ is not fulfilled for all $t$ in the infinite problem.
For the selfadjoint eigenvalue problem we have expansion and completeness theorems \[8\]. The eigenfunctions $y_n$ for different $n$ are orthogonal and complete in $L^2([r_0,r_1])$. To normalize them we compute $$\int\limits_{r_0}^{r_1}\sin^2k_n(r-r_1)dr={r_1-r_0\over 2}+O\B({1\over n^2}\B)$$ so that $$\fii_n(r)=\B(\sqrt{{2\over r_1-r_0}}+O({1\over n^2})\B)\sin k_n(r_1-r)\eqno(5.12)$$ is a complete orthonormal system. The general solution of the finite ablation problem for the hyperbolic heat equation is now given by $$T(t,r)=T_{01}-{b\over r_0r_1}+{b\over 2r_1^2}+{b\over r_0r}-{b\over 2r^2}+$$ $$+{1\over r}\sum_{n=0}^\infty\B[a_n\fii_n(r)e^{\omega_n^+t}+b_n\fii_n(r)e^{\omega_n^-t}\B].\eqno(5.13)$$ Here the two roots (3.14) appear again with $k=k_n$. For the parabolic equation we have only the terms with $\omega_n^+=-ak_n^2$. To satisfy the initial condition $$T(0,r)=T_{01}\eqno(5.14)$$ we must expand the function $${b\over 2r}-{b\over r_0}+r\B({b\over r_0r_1}-{b\over 2r_1^2}\B)=\sum_n(a_n+b_n)\fii_n(r)$$ into a Fourier series. To do so we need the $L^2$-scalar products of $\fii_n$ with the functions 1, $r$, $1/r$ which can easily be calculated. The Fourier coefficients $a_n+b_n$ are of the order $1/n$ which gives a slow convergence of the series. For large $t$ the exponential factors in (5.13) give a rapid convergence.
For the hyperbolic equation we again require the second initial condition $${\d T(t,r)\over\d t}\B\vert_{t=0}=0\eqno(5.15)$$ which yields $$b_n=-{\omega_n^+\over\omega_n^-}a_n.\eqno(5.16)$$ In the parabolic case this condition is violated. This seems to be in contradiction to experiments. The reason for condition (5.15) is the finite propagation speed of the heat which follows from the characteristics of the hyperbolic equation (3.8) \[7\] \[9\]. The characteristics are given by $${dr\over dt}=\pm{1\over\sqrt{\eps}}=\pm\sqrt{{\kappa\over\ro c\tau}}.\eqno(5.17)$$
Conclusions
===========
The approach to steady state for large times is determined by the exponential term $\exp(\omega t)$ with frequency $\omega$ closest to 0 in (5.13). Leaving aside unrealistically large $\tau$, this is given by $$\omega_0^+=-ak_0^2\approx -a\B({\pi\over 2(r_1-r_0)}+{2\over\pi r_0}\B)$$ where $\tau$ has cancelled. Therefore, in contrast to the infinite geometry in sect.2-4, both equations show the same approach to steady state in the finite system.
For small times the solutions of the two equations differ considerably. One reason for this is the different initial condition (5.15). A second interesting difference has been observed by Lopez Molina et al.\[4\]. The temperature $T$ at fixed radius $r$ shows “cuspidal-type” singularities as a function of time $t$. These are discontinuities in the derivative $\d_t T(t,r)$. The origin of this phenomenon is the change of branch in the characteristic frequencies $\omega_n^\pm$ which is connected with the appearance of thermal waves. Indeed, according to (3.26) we must consider $$F(t)={\rm Re}\B(e^{\sqrt{1-x^2}t}\B)=\cases{e^{\sqrt{1-x^2}t}&if $x<1$\cr \cos(\sqrt{x^2-1}t)&if $x>1.$\cr}$$ This function is continuous at $t=0$, but $\d_t F(t)$ makes a jump. The theory of characteristics \[9\] implies that such a discontinuity travels through the medium with the velocity (5.17).
González-Suárez A, Berjano E, Guerra JM, Gerardo-Giorda L, Computational Modeling of Open-Irrigated Electrodes for Radiofrequency Cardiac Ablation Including Blood Motion-Saline Flow Interaction,PloS ONE 11(3): e0150356, doi:10.1371/journal.pone.0150356
Cattaneo C.R. (1958), Sur une forme de equation de la chaleur eliminant le paradoxe d’une propagation instantanee, Comptes Rendus 247 (4) 431
Vernotte P. (1958),Les paradoxes de la theorie continue de l’equation de la chaleur, Comptes Rendus 246 (22) 3154
Lopez Molina JA,Rivera MJ, Trujillo M, Berjano EJ, (2008) Effect of the thermal wave in radiofrequency ablation modeling: an analytic study, Phys. Med. Biol. 53, 1447-1462
Abramowitz M, Stegun IA, Handbook of mathematical functions, Dover Publications 1965
Erdelyi A, Oberhettinger MF, Tricomi FG, Tables of integral transforms, vol.1 and 2, McGraw-Hill, New York
Vladimirov WS, Equations of mathematical physics; Moscow, Mir Publishers, 1984
Coddington E.A., Levinson N., Theory of ordinary differential equations, McGraw-Hill Book Company, Inc. 1955
Courant R, Hilbert D, Methods of mathematical physics,vol.II, Wiley, New York 1989
[^1]: e-mail: scharf@physik.uzh.ch
|
---
abstract: 'The profile of a cylindrical shell in the post–buckling regime of deformation is calculated exactly, for any load and axial deformation, and is shown to be a Jacobi elliptic sine function. Cylindrical symmetry is assumed from the beginning. Results shed light on why old tires are so effective as dock and tugboat bumpers from long ago to nowadays, proving to be safe for berthing even the largest freighters. As no technical analysis of this capacity has been published yet, it seems opportune to apply the mathematical solution to explain this long lasting enigma. The reaction force and stored energy of an axially compressed cylindrical shell, as a tire tread is, exhibit ideal behaviour for a safety shock energy absorber. Most energy absorption takes place after buckling, which bends the cylindrical mantle and increases its circumference, inducing strong tensile forces that contribute to stabilize the buckled structure.'
address: 'Facultad de Ingeniería, Universidad de Talca, Campus Los Niches, Camino a los Niches Km 1, Curicó, Chile'
author:
- Miguel Lagos
title: |
Analytic solution for the buckling of a short cylindrical shell.\
Why old tires are still being preferred as dock bumpers in harbours
---
thin films,plastic deformation,grain boundary sliding,modeling 62.20.mq,62.20.D-
Introduction {#introd}
============
There is a great deal of work done on the critical conditions for the buckling of narrow or thin walled solids of a variety of shapes, and on the subsequent evolution of the collapsing element [@Roarks]. Buckling is generally the controlling failure mode of thin shell structures and long narrow structural elements, hence the study of the elastic instabilities of shells is a most important subject of materials mechanics and engineering design. There is an extensive list of books, review articles and conference proceedings providing a wealth of information on the strength, mechanical stability and buckling behavior of thin shell structures [@Teng]. However, either explicitly or not, most of these studies have associated the idea of buckling as a catastrophic collapse of the critically strained thin walled or narrow element. Also, the mathematical treatments are often intended to be as general as possible [@Hunt; @Paschero; @Pinna; @Simitnes; @Wullschleger], which precludes the obtention of closed–form practical formulas of use in specific practical situations, and able of being tabulated in manuals [@Roarks].
This communication deals with a mechanical model for a very particular technical application in which buckling is not an undesired effect, but a highly beneficial one. The use of old tires as dock bumpers has been an old tradition in most harbours. Although they are now being replaced by specially fabricated marine fenders, ship captains appreciate the behaviour of old tires in docking because, for instance, they provide a characteristic smooth absorption of the ship kinetic energy with total absence of spring back at the end of the operation. The question of why old tires are still being preferred as dock protections, even for the docking of large cargo vessels, is an enigma whose precise solution is the subject of what follows.
The reaction force and energy absorption effect of an axially compressed tire is attributed entirely to the elastic deformation of the tread, which is modeled here as a cylindrical shell whose radius $R$ is greater than its axial length $L_0$, both magnitudes measured in the unstrained condition, as shown schematically in Fig. \[Fig1\]. In opposition to small tires, whose treads are often initially slightly barreled, big ones are essentially cylindrical when unloaded. Though small, the shell thickness $e$ is assumed large enough to preclude the formation of patterns of complex geometry upon deformation. The applied opposed forces of strength $P$ are parallel to the symmetry $z$–axis and are uniformly distributed along the edges of the cylindrical mantle. It is also assumed that the edges of the elastic cylinder are not deformed because they rest on rigid plane surfaces and the friction forces are strong enough to prevent sliding. Hooke’s law is assumed to hold in a first approach. Although rubber elastic properties exhibit deviations from linear response, the main conclusions will not be much affected by this.
Instead of dealing with the model in the most general way, facing all the complex deformation modes a cylindrical shell may display [@Hunt; @Paschero; @Pinna; @Simitnes; @Wullschleger], the mathematical approach is kept as simple as possible by allowing only barrel shaped deformations. The equilibrium equations of the model incorporate from the very beginning just the deformation modes one can observe in an axially compressed tire, and discards any other because the main objective is to understand the ability of tires as shock energy absorbers.
The mechanical analysis shows that, in a short first stage, the cylindrical shell undergoes a uniform compressive strain along the $z$–axis, conserving strictly its cylindrical shape and opposing a reaction force proportional to the strain up to a maximal load $P_B$. The maximum strain reached in this first deformation regime is indicated in Fig. \[Fig1\](a) by a discontinuous line. Compression beyond this limit makes the shell to start buckling, and the cylindrical mantle progressively acquires a barrel shape, as in Fig. \[Fig1\](b). In this second regime the reaction force diminishes monotonically with strain, but conserving always a significant strength. The development of the barrel profile involves stretching of the tread circumference, which demands strong tensile forces that contribute to stabilize the buckled structure. As a shock energy absorber, the system exhibits the advantages of having a maximum reaction force, which can be designed to warrant no damage to the colliding bodies, together with a large energy absorption capability, particularly in the buckled strain regime.
![\[Fig1\] Model for the tread of a pneumatic tire as a cylindrical shell. (a) Unstrained condition (solid lines) and critical axial strain for buckling (discontinuous lines). (b) Buckled strain regime by the action of compressive axial forces homogeneously distributed on the shell edges.](Fig1.pdf){width="7cm"}
Buckled strain regime of the cylindrical shell of finite thickness {#buckling}
==================================================================
Equilibrium of the internal forces {#forces}
----------------------------------
Fig. \[Fig2\] shows the cylindrical shell in the strain regime in which it adopts a barrel shaped profile by effect of the axial load $P$ (equivalent to the reaction force at equilibrium, or quasi–equilibrium). The cylindrical coordinate system has its $z$–axis along the main symmetry axis and the origin $O$ at the center of the deformed shell, which extends from $z=-l/2$ to $z=l/2$. Distance $r$ is the radius of the shell for a given $z$, so that function $r(z)-R$ determines the profile of the strained shell. The azimuthal coordinate $\phi$ is the usual one in cylindrical coordinates, and is not shown in Fig. \[Fig2\] for the sake of clarity.
![\[Fig2\] The strained cylindrical shell, showing schematically an elementary sector defined by the coordinates $r$, $\phi$ and $z$ ($\phi$ is implicit), and by the variations $\Delta\phi$ and $\Delta z$.](Fig2.pdf){width="7cm"}
The equilibrium equations are derived from the analysis of the forces operating on an elementary sector of the shell, schematically represented in Fig. \[Fig2\]. The method is preferred over simply writing the equations of the theory of elasticity and applying boundary conditions because is simpler and provides better physical insight. The advantages of the procedure are discussed in the final subsection of this section.
Fig. \[Fig3\] displays a diagram of the material element and the forces applied on it. Forces in the plane $\phi =\text{constant}$, containing the $z$–axis, are compressive forces of strength $F_c(z+\Delta z)$ and $F_c(z)$. The forces in the plane $z=\text{constant}$, normal to the $z$–axis, are tensile forces because stretch the shell contour in its plane, and their strength is denoted $F_\phi(z)$. Solutions with cylindrical symmetry were implicitly assumed because no dependence of the forces on $\phi$ was considered.
![\[Fig3\] The elementary sector of Fig. \[Fig2\] and the compressive and tensile forces $F_c$ and $F_\phi$ exerted on it.](Fig3.pdf){width="3.5cm"}
Fig. \[Fig4\](a) is a projection on the plane $z=\text{constant}$ of the system of Fig. \[Fig3\] showing the forces operating in this plane, and Fig. \[Fig4\](b) shows the vector composition of them. Hence, the tensile forces give a sum of strength
$$2F_\phi(z)\,\sin (\Delta\phi /2)
\rightarrow F_\phi(z)\Delta\phi
\;\,\text{if}\;\,\Delta\phi\approx 0,
\label{Ec1}$$
contained in the plane $z=\text{constant}$ and pointing towards the central $z$–axis.
![\[Fig4\] (a) The graphic scheme of Fig. \[Fig3\] projected on the plane $z=\text{constant}$. (b) Force diagram in the plane $z=\text{constant}$.](Fig4.pdf){width="5.3cm"}
![\[Fig5\] Forces in the plane $\phi =\text{constant}$. Compressive forces $F_c(z+\Delta z)$ and $F_c(z)$ are tangent to the curve $r(z)-R$ representing the shell profile, and subtend angles $\theta +\Delta\theta$ and $\theta$ with the $z$–axis, respectively. The force $F_\phi(z)\Delta\phi$ is obtained in Fig. \[Fig4\] from the tensile forces in the plane $z=\text{constant}$. The discontinuous line represents the shell profile.](Fig5.pdf){width="5.3cm"}
Forces in the plane $\phi =\text{constant}$ are shown in Fig. \[Fig5\]. They are the two compresive forces of strength $F_c(z+\Delta z)$ and $F_c(z)$ operating in the two opposite edges, and the third one is the resultant of the tensile forces in the plane $z=\text{constant}$, whose strength is $F_\phi(z)\Delta\phi$. The former two forces are tangent to the curve $r(z)-R$ determining the shell profile in the corresponding application points, and subtend angles $\theta +\Delta\theta$ and $\theta$ with the $z$–axis, respectively.
Equilibrium demands
$$-F_c(z+\Delta z)\cos (\theta +\Delta\theta)+F_c(z)\cos\theta=0,
\label{Ec2}$$
$$F_c(z+\Delta z)\,\sin (\theta +\Delta\theta)
-F_c(z)\,\sin\theta=-F_\phi\Delta\phi .
\label{Ec3}$$
From Eq. (\[Ec2\]) one can infer that the $z$–component of the forces does not depend on $z$, and hence they can be identified with the external force $P\Delta\phi /(2\pi)$ applied on the edge of the shell fringe defined by $\Delta\phi$. The argument is graphically shown in Fig. \[Fig6\] and yields
$$F_c(z+\Delta z)\cos (\theta +\Delta\theta)
=F_c(z)\cos\theta
=\frac{P}{2\pi}\Delta\phi.
\label{Ec4}$$
![\[Fig6\] Equilibrium of a shell fringe of finite size in the $z$–direction. The $z$–component $F_c(z)\cos\theta$ of the force is $P\Delta\phi /(2\pi)$ for any $z$ in $[-l/2,l/2]$.](Fig6.pdf){width="3cm"}
Differential equation for the shell profile function $r(z)-R$ {#ecuation}
-------------------------------------------------------------
Recalling Eqs. (\[Ec4\]) one can divide the first term in the right hand side of Eq. (\[Ec3\]) by $F_c(z+\Delta z)\cos (\theta
+\Delta\theta)$, the second one by $F_c(z)\cos\theta$, and the right hand side by $P\Delta\phi /(2\pi)$. It gives
$$\tan(\theta +\Delta\theta)-\tan\theta
=-\frac{2\pi F_\phi}{P} .
\label{Ec5}$$
Force $F_\phi$ can be expressed as
$$F_\phi =\sigma_{\phi\phi} e\Delta s ,
\label{Ec6}$$
where $\sigma_{\phi\phi}$ is the normal stress in the azimuthal direction, $e$ is the shell thickness and $\Delta s$ the length of the shell element shown in Fig. \[Fig5\], and whose value is
$$\Delta s=\sqrt{1+\left(\frac{dr}{dz}\right)^2}\,\Delta z.
\label{Ec7}$$
.
Replacing $\tan\theta =dr/dz$ and combining Eqs. (\[Ec5\]), (\[Ec6\]) and (\[Ec7\]) one obtains that, in the limit $\Delta
z\rightarrow 0$,
$$\frac{d^2r}{dz^2}
=-\frac{2\pi e\sigma_{\phi\phi}}{P}
\sqrt{1+\left(\frac{dr}{dz}\right)^2}.
\label{Ec8}$$
On the other hand, the normal stresses $\sigma_{rr}$, $\sigma_{\phi\phi}$ and $\sigma_{zz}$ satisfy Hooke’s law
$$\varepsilon_{\phi\phi}=\frac{r-R}{R}
=\frac{1}{E}(\sigma_{\phi\phi}-\nu\sigma_{rr}-\nu\sigma_{zz}),
\label{Ec9}$$
where $E$ and $\nu$ are the Young coefficient and Poisson ratio of the material the shell is made of. Substituting
$$\sigma_{zz}=-\frac{P}{2\pi re}\cos\theta ,
\label{Ec10}$$
$\cos\theta =[1+(dr/dz)^2]^{-1/2}$ and
$$\sigma_{rr}=-\frac{F_\phi \Delta\phi}{r\Delta\phi\Delta z}
=-\sigma_{\phi\phi}\frac{e}{r}\sqrt{1+\bigg(\frac{dr}{dz}\bigg)^2}
\label{Ec11}$$
in Eq. (\[Ec9\]), and solving for $\sigma_{\phi\phi}$, it is obtained that
$$\sigma_{\phi\phi}=\frac{1}{1+\nu\dfrac{e}{r}
\sqrt{1+\bigg(\dfrac{dr}{dz}\bigg)^2}}\left(E\,\frac{r-R}{R}
-\frac{\nu P}{2\pi er\sqrt{1+\bigg(\dfrac{dr}{dz}\bigg)^2}}\right).
\label{Ec12}$$
Combining Eqs. (\[Ec12\]) and (\[Ec8\]) one finally obtains
$$\frac{d^2r}{dz^2}
=-\dfrac{\dfrac{2\pi eE}{RP}(r-R)\sqrt{1+\bigg(\dfrac{dr}{dz}\bigg)^2}
-\dfrac{\nu}{r}}{1+\nu\dfrac{e}{r}
\sqrt{1+\bigg(\dfrac{dr}{dz}\bigg)^2}}\, .
\label{Ec13}$$
The differential equation (\[Ec13\]) gives the profile $r(z)$ of the cylindrical shell deformed by the axial load $P$ when the proper physical conditions are imposed to the solutions. The most evident conclusion one can infer from Eq. (\[Ec13\]) is that the load $P$ cannot be null. Hence it corresponds to a deformation regime occurring for high enough loads.
The unloaded cylindrical shell has a length $L$ in the axial direction, which reduces to a smaller length $l$ by the applied load $P$. The length reduction originates from the elastic compressive strain of the deformed mantle and by the shape deformation itself projected in the $z$–direction. If the elastic strain can be neglected when compared with the effect of the shell deformation, the length of the curved path adopted by the deformed shell will conserve its original value $L$. Hence the physical solutions of Eq. (\[Ec13\]) must satisfy
$$L=\int_{-l /2}^{l /2}dz\,\sqrt{1+\left(\frac{dr}{dz}\right)^2}.
\label{Ec14}$$
Present theoretical approach versus theory of elasticity {#approach}
--------------------------------------------------------
The equation for the shell profile was derived in the preceding subsections from analysing the equilibrium of the forces exerted on a representative elementary sector of the material medium, introducing from the beginning the symmetry constraints. For our present purposes, this procedure is much simpler and practical than the more standard approach of readily introducing the system boundary conditions into the general formalism of the theory of elasticity. This is because the latter approach oblige us to face complex problems we are not interested in. In the theory of elasticity the free surfaces of the shell are defined as surfaces where the stresses vanish, and the theory furnishes the equations for calculating the detailed structure of the stresses occurring in between. A recent paper by Zozulya and Zhang [@Zozulya] provides a good example of this kind of precise calculation of the deformation of cylindrical shells, together with a detailed account of the stress fields inside the material. The cost paid for such a complete solution is the introduction of numerical methods from the very start.
However, it is not necessary here to know about the precise profiles of the outer and inner surfaces of the shell, or how vary the stresses inside. The forces $F_c(z)$ and $F_\phi (z)$ in Fig. \[Fig3\] account for the integrated effect of these stresses, and their equilibrium condition proves to be enough for determining the equation for the mean shell profile, which has the advantage of being closed–form.
The profile equation in the buckling regime of deformation and its solution {#buckling}
===========================================================================
The trivial solution $r=\text{constant}$ reduces Eq. (\[Ec13\]) to
$$E\frac{r-R}{R}=\nu\frac{P}{2\pi re} \qquad (r=\text{constant}),
\label{Ec15}$$
which can be interpreted as the Poisson effect on the shell circular perimeter $2\pi r$ accompanying the homogeneous axial strain produced by the applied compressive stress $P/(2\pi re)$. This uniform solution, which is expected to be stable up to a critical load $P_B$, preserves the cylindrical shape and produces only small geometrical variations, even for big loads, because $E$ is usually very large (0.01–500 GPa). The buckled non trivial solutions of Eq. (\[Ec13\]) involve much larger deformations. In the post–buckling regime $r$ assumes values in the interval $R<r\le R+L/2$ when $l$ varies from $l=L$ to $l=0$. The second term $\nu /r$ in the numerator of the right hand side of Eq. (\[Ec13\]) is comparable with the first one only when $|r-R|/R\ll 1$. As long as $r$ departs from $R$ beyond the range of the purely elastic distortions, the term $\nu /r$ becomes negligibly small when compared with the one proportional to $E$. Hence it is advisable to distinguish between elastic strains and geometric changes and write
$$\frac{d^2r}{dz^2}
=-\frac{2\pi eE}{PR}(r-R)\sqrt{1+\left(\frac{dr}{dz}\right)^2}
\quad\text{(buckling regime)}.
\label{Ec16}$$
Ec. (\[Ec16\]) also assumes a thin enough shell to neglect the second term in the denominator of the right hand side of Ec. (\[Ec13\]). Although this non-linear equation is not listed in the specialized treatises on elliptic functions and integrals [@Gradshteyn; @Byrd], it will be shown next that its exact solution is a Jacobi elliptic sine function, which holds for any load and deformation state, including the limit in which the shell has been completely flattened by the applied force.
Defining
$$y(z)=r(z)-R, \quad p=\frac{2\pi eE}{PR}, \quad R=\text{constant},
\label{Ec17}$$
Eq. (\[Ec16\]) reads
$$y''=-py\sqrt{1+y'^2},
\label{Ec18}$$
and the substitution
$$y(z)=\frac{1}{\sqrt{p}}\, u(\sqrt{p}\, z)
\label{Ec19}$$
turns it into
$$u''=-u\sqrt{1+u'^2}.
\label{Ec20}$$
Multiplying both sides of this equation by $u'$ yields the integrable form
$$\frac{u''u'}{\sqrt{1+u'^2}}=-uu',
\label{Ec21}$$
which can be solved to give
$$\sqrt{1+u'^2}=-\frac{1}{2}u^2+\frac{1}{2}u_0^2+1,
\label{Ec22}$$
where $u_0=u(0)$. Because of the symmetry with respect to the origin, $u(\sqrt{p}\, z)$ must have its maximum at $z=0$ and the integration constant was chosen so that $u'(0)=0$. Denoting now
$$\zeta =\sqrt{p}\, z, \quad v(\zeta)=\frac{u(\zeta)}{u_0},
\quad k^2=\frac{(u_0/2)^2}{1+(u_0/2)^2}\, ,
\label{Ec23}$$
Eq. (\[Ec22\]) can be rewritten as
$$\frac{dv}{d\zeta}=
\sqrt{(1-v^2)\bigg(1+\frac{u_0^2}{4}-\frac{u_0^2}{4}\, v^2\bigg)}\, ,
\label{Ec24}$$
or
$$\sqrt{1-k^2}\,\frac{dv}{d\zeta}=\sqrt{(1-v^2)(1-k^2v^2)}\, .
\label{Ec25}$$
Inverting this equation and integrating with respect to $v$ one has
$$\frac{1}{\sqrt{1-k^2}}(\zeta +\zeta_1)
=\int_{\displaystyle\, 0}^{\displaystyle\, v}
\frac{dv}{\sqrt{(1-v^2)(1-k^2v^2)}}=F(v,k),
\label{Ec26}$$
where $F(v,k)$ is the incomplete elliptic integral of the first kind with modulus $k$, ($0\le k\le 1$) [@Gradshteyn; @Byrd], and $\zeta_1$ is an integration constant. The same symbol was used for the integration variable and the upper integration limit to simplify the notation.
The inverse function of the incomplete elliptic integral $F(v,k)$ is known as the Jacobi elliptic sine function sn. Hence Eq. (\[Ec26\]) is equivalent to
$$v=\text{sn }\bigg(\frac{\zeta +\zeta_1}{\sqrt{1-k^2}}\, ,k\bigg)
=\text{sn }\bigg(\sqrt{\frac{p}{1-k^2}}\, (z+z_1),k\bigg).
\label{Ec27}$$
Function $\text{sn }(x,k)$ takes values in the interval $[-1,1]$ and is periodic in $x$ with period $4K(k)$, where $K(k)=F(1,k)$ is the complete elliptic integral of the first kind. Also, $\text{sn }(0,k)=
\text{sn }(2K,k)=0$ and $\text{sn }(K,k)=1$. The Jacobi sine function is symmetric with respect to $x=K$ and has a maximum there.
Therefore, the solution satisfying the boundary conditions $y(\pm l/2)=0$ is such that
$$\sqrt{\frac{p}{1-k^2}}=\frac{2K(k)}{l}
\quad\text{and}\quad z_1=\frac{l}{2}.
\label{Ec28}$$
Taking $\sqrt{p}$ and $z_1$ from these equations, and recalling the third of Eqs. (\[Ec23\]), which gives $u_0/2=k/\sqrt{1-k^2}$, one has that the profile of the buckled cylindrical shell is given by
$$y(z)=r(z)-R=\frac{k}{(1-k^2)K(k)}\,\text{sn }\bigg[\frac{2K(k)}{l}
\bigg(z+\frac{l}{2}\bigg),k\bigg].
\label{Ec29}$$
Combining the first Eq. (\[Ec28\]) with the definition (\[Ec17\]) of the constant $p$, a relation between the applied force $P$ and $k$ follows
$$P=\frac{\pi eEl^2}{2R(1-k^2)K^2(k)}.
\label{Ec30}$$
Therefore, it rests just to determine the meaning of $k$ to have a complete solution of our problem. In the next subsection it will be shown that $k^2$ is essentially the relative axial deformation $\varepsilon =(L-l)/L$ of the cylindrical shell. The elliptic integral $K(k)$ can be calculated quite easily from the defining integral or the series
$$K(k)=\frac{\pi}{2}\bigg[1
+\sum_{n=1}^\infty\bigg(\frac{(2n-1)!!}{2^nn!}\bigg)^2k^{2n}\bigg].
\label{Ec31}$$
![\[Fig7\] The Jacobi elliptic sine function (solid lines) for three values of the modulus $k$, compared with the trigonometric sine function (circles). For $k<0.7$ the two functions are very similar. At $k>0.7$ the elliptic function is sensibly broader and goes to a square wave in the limit $k=1$. Physically, $k=1$ corresponds to the situation in which the cylindrical mantle has been completely flattened by the applied force $P$ and $l=0$.](Fig7.pdf){width="6cm"}
Fig. \[Fig7\] shows the Jacobi sine function $\text{sn }(2Kx,k)$ for three values of $k$. The graphs makes apparent that the Jacobi elliptic function goes from $\sin (\pi x)$ to a square wave when the modulus $k$ goes from 0 to 1. The latter situation corresponds to the final collapse of the cylindrical mantle, when $l=0$.
The equation for the modulus $k$ {#modulus}
--------------------------------
When $R=\text{constant}$ one has $r'(z)=y'(z)=u'(\sqrt{p}\, z)$, hence $(1+r'^2)^{1/2}$ can be replaced by $(1+u'^2)^{1/2}$ in the general condition expressed by Eq. (\[Ec14\]). Making this and then substituting Eq. (\[Ec22\]),
$$L=\int_{-l/2}^{l/2}\sqrt{1+u'^2}\, dz=-\frac{1}{2}\int_{-l/2}^{l/2}
u^2(\sqrt{p}\, z)\, dz+\frac{1}{2}u_0^2l+l .
\label{Ec32}$$
Recalling now
$$u(\sqrt{p}\, z)=
u_0\,\text{sn }\bigg[\frac{2K}{l}\bigg(z+\frac{l}{2}\bigg),k\bigg],
\quad u_0=\frac{2k}{\sqrt{1+k^2}},
\label{Ec33}$$
Eq. (\[Ec32\]) can be written as
$$\frac{L-l}{l}=\frac{2k^2}{1-k^2}\left( 1-\frac{1}{2K(k)}\int_0^{2K(k)}
\text{ sn}^2\, (\zeta ,k)\, d\zeta\right).
\label{Ec34}$$
The integral appearing in this equation has been solved in terms of the elliptic functions and one has the mathematical identity [@Gradshteyn]
$$\int\,\text{sn}^2\,\zeta\, d\zeta
=\frac{1}{k^2}[u-E(\text{am }\zeta,k)],
\label{Ec35}$$
where $E(\phi ,k)$, $0\le\phi\le\pi/2$, is the second kind incomplete elliptic integral with modulus $k$. In the usual notation of the theory of elliptic integrals the amplitude am means $\text{am }\zeta=
\arcsin(\text{sn }\zeta)$. Care must be taken in replacing properly the integration limits in the indefinite integral (\[Ec35\]) because $0\le\text{am }\zeta\le\pi/2$, and $\text{am }K=\pi/2$. Therefore, $2K$ is outside the domain in which the amplitude function is defined. To overcome this difficulty one can take advantage of the symmetry of $\text{sn }(x,k)$ with respect to $x=K$ and write
$$\int_0^{2K}\,\text{sn}^2\,\zeta\, d\zeta
=2\int_0^{K}\,\text{sn}^2\,\zeta\, d\zeta
=\frac{2}{k^2}[K-E(\pi/2,k)]
\label{Ec36}$$
replacing $\text{am }K=\pi/2$. $E(\pi/2,k)=E(k)$ is the complete elliptic integral of the second kind, modulus $k$, defined by the integral [@Gradshteyn]
$$E(k)=\int_0^{\pi/2}\sqrt{1-k^2\sin^2\phi}\, d\phi,
\label{Ec37}$$
or the series
$$E(k)=\frac{\pi}{2}\bigg[ 1-\sum_{n=1}^\infty
\bigg(\frac{(2n-1)!!}{2^nn!}\bigg)^2\frac{k^{2n}}{2n+1}\bigg].
\label{Ec38}$$
The term $D(k)=(1/k^2)[K(k)-E(k)]$ can be evaluated either by combining the power series (\[Ec31\]) and (\[Ec38\]) as
$$D(k)=\frac{K(k)-E(k)}{k^2}
=\frac{\pi}{2}\sum_{n=0}^\infty\frac{2(n+1)}{2n+1}
\bigg(\frac{(2n+1)!!}{2^{n+1}(n+1)!}\bigg)^2k^{2n},
\label{Ec39}$$
or solving the integral
$$D(k)=\int_0^{\pi/2}\frac{\sin^2\phi\, d\phi}{\sqrt{1-k^2\sin^2\phi}}\, ,
\label{Ec40}$$
which follows directly from the definitions of $K(k)$ and $E(k)$.
Eq. (\[Ec34\]) then becomes
$$\frac{\varepsilon}{1-\varepsilon}=
\frac{2k^2}{1-k^2}\left( 1-\frac{D(k)}{K(k)}\right),
\quad \varepsilon =\frac{L-l}{L},
\label{Ec41}$$
which determines exactly the modulus $k$ as a function of only the relative axial compression $\varepsilon =(L-l)/L$. The function $k=f(\varepsilon)$ does not depend on the characteristics of the cylindrical shell and is shown in Fig. \[Fig8\]. The amazingly simple approximate rule
$$k^2\approx\varepsilon
\label{Ec42}$$
holds over the whole range of $\varepsilon$ with a 5% maximum error.
![\[Fig8\] The squared modulus $k^2$ and the relative axial compression $\varepsilon =(L-l)/L$, as given by Eq. (\[Ec41\]).](Fig8.pdf){width="7cm"}
Critical load and reaction force {#critical}
================================
The deformation regime studied in the preceding sections takes place for a constant non–vanishing axial load $P$, given by Eq. (\[Ec30\]). This is because it was assumed from the very beginning that loading curves the cylindrical shell profile and Eqs. (\[Ec29\]), (\[Ec30\]) and (\[Ec41\]) hold at buckling and in the post–buckling regime of deformation. The variable $\varepsilon$ measures only the deformation induced relative displacement of the edges along the axial direction, and the much smaller elastic distortions have been neglected. The force
$$P=\frac{\pi eEL^2(1-\varepsilon)^2}{2R(1-k^2)K^2(k)}
\label{Ec43}$$
has an absolute maximum $P_B$ at $\varepsilon =k=0$. Recalling $K(0)=\pi/2$, the maximal reaction force reads
$$P_B=\frac{2eEL^2}{\pi R},
\label{Ec44}$$
and one can write
$$\frac{P(\varepsilon)}{P_B}
=\frac{\pi^2}{4}\frac{(1-\varepsilon)^2}{(1-k^2)K^2(k)}.
\label{Ec45}$$
![\[Fig9\] Reaction force $P$ relative to the maximum $P_B$ as a function of axial compression $\varepsilon =(L-l)/L$ , as given by Eq. (\[Ec45\]).](Fig9.pdf){width="7cm"}
Fig. \[Fig9\] shows $P/P_B$, which according to Eqs. (\[Ec45\]) and (\[Ec41\]) is an universal function of $\varepsilon$. Fig. \[Fig10\] displays the energy stored by the shell in the post–buckling deformation regime, obtained integrating Eq. (\[Ec45\]) with respect to $\varepsilon$.
![\[Fig10\] Energy absorbed by the cylindrical shell in the post–buckling deformation regime, in units of $P_BL$ and for any relative axial compression $\varepsilon$.](Fig10.pdf){width="7cm"}
Homogeneous and buckled deformation regimes
===========================================
The problem we are interested in is a deformation in which the cylindrical shell is gradually compressed along its main symmetry axis. In a first stage the compression is elastic and conserves the cylindrical shape. The relative displacement $x$ of the edges is given by Hookes’s law
$$\frac{P}{2\pi Re}=E\frac{x}{L_0},
\label{Ec46}$$
where $L_0$ is the original length of the unstrained cylinder. Once the elastic distortion $x$ takes a critical value $x_B$, such that the applied force $P$ reaches the threshold strength $P_B$, the cylinder buckles and Eq. (\[Ec43\]) starts holding, instead of Eq. (\[Ec46\]). At the critical deformation $x_B$ the surface just starts to acquire the barrel shape, the two deformation regimes coexist and Eqs. (\[Ec43\]) and (\[Ec46\]) are both valid for $L=L_0-x_B$ and $\varepsilon =0$ (notice that $L$ stands for the the shell length when buckling is just initiated). Thus the axial deformation $x_B$ at which the buckling regime sets in can be obtained from combining Eqs. (\[Ec44\]) and (\[Ec46\]) to eliminate $P_B$. This yields
$$\frac{x_B}{L_0}=
1+\left(\frac{\pi R}{\sqrt{2}L_0}\right)^2
-\frac{\pi R}{\sqrt{2}L_0}
\sqrt{\left(\frac{\pi R}{\sqrt{2}L_0}\right)^2+2},
\label{Ec47}$$
which is the equation for the critical elastic strain $x_B/L_0$. The critical load is
$$P_B=2\pi ReE\,\bigg[ 1+\left(\frac{\pi R}{\sqrt{2}L_0}\right)^2
-\frac{\pi R}{\sqrt{2}L_0}
\sqrt{\left(\frac{\pi R}{\sqrt{2}L_0}\right)^2+2}\bigg].
\label{Ec48}$$
Conclusions relative to the marine application {#example}
==============================================
To illustrate why vehicle old tires are functional as dock dampers in the light of the preceding equations, consider a 27”$\times$ 47” tire, originally designed for heavy mining trucks. The dimensions of the tread are $L_0=0.75\,\text{m}$, $R=1.25\,\text{m}$ and $e=0.10\,\text{m}$. Young’s effective modulus for tire rubber with added reinforcements can be estimated as $E=10\text{ MPa}$. The purpose of this calculation is fundamentally illustrative, to acquire a feeling of the magnitudes involved. Replacing these constants in Eqs. (\[Ec47\]) and (\[Ec48\]) one has that
$$x_B=2.55\,\text{cm}\quad P_B=267\,\text{kN}
=27.3\,\text{ton}.
\label{Ec49}$$
![\[Fig11\] Axial reaction force $P$ exerted by a tire with tread dimensions $L_0=0.75\,\text{m}$, $R=1.25\,\text{m}$ and $e=0.10\,\text{m}$, and effective Young’s modulus $E=10\,\text{MPa}$, as a function of the axial compression $x$. The broken line shows the behaviour of a spring or any other strictly elastic medium chosen to yield a maximum force comparable with the tire.](Fig11.pdf){width="7.5cm"}
Thus, when the tire is progresively compressed, the reaction force grows linearly with strain up to the critical strength $P_B=267\,\text{kN}$ at $x_B=2.55\,\text{cm}$, with the tread conserving its cylindrical shape. As compression continues, the tread curves and the radius turns dependent on $z$ according to Eq. (\[Ec29\]). Fig. \[Fig11\] shows the behaviour of the reaction $P$ of the strained tire in the two regimes, the linear and the buckled one (solid line). In the latter regime force $P$ obeys Eq. (\[Ec45\]) with $L=L_0-x_B$. The broken line is the force exerted by a strictly elastic system, like a spring matrix, introduced here for the sake of comparison.
![\[Fig12\] The solid curve represents the energy absorbed by the tire of Fig. \[Fig11\] when compressed in a distance $x$. The broken curve is the same magnitude for the strictly elastic system. The smaller capacity of the latter for compressions in the range between 0 and 40 cm is apparent.](Fig12.pdf){width="7.5cm"}
The stopping power of the tire when used as a dock bumper device is given by the area $W(x)$ under the curve of the reaction force. The solid line in Fig. \[Fig12\] shows the curve of the energy stored $W(x)$ by the tire as a function of the compressive displacement $x$. The broken line displays the same magnitude for the purely elastic shock energy absorber.
Figs. \[Fig11\] and \[Fig12\] show the features that make tires good dock bumpers:\
(a) The reaction force has a well defined maximum $P_B$, which ensures that the energy absorption device will not damage the colliding structures.\
(b) Once the maximum reaction is reached, the reaction $P$ decreases smoothly with the displacement $x$, and energy storage $W(x)$ keeps high. This is the physical magnitude determining the stopping power of the device.\
(c) Function $W(x)$ increases monotonically and rapidly with $x$, particularly for small values of $x$. This means that the stopping action of the axially compressed tire rapidly increases and keeps high at any strain, even for small ones. The superiority of the cylindrical shell (solid line) over a strictly elastic system (broken line) is shown quite dramatically in Fig. \[Fig12\]. The latter reaches an acceptable stopping ability at the cost of increasing the reaction force to a dangerous level.\
(d) As $x$ takes relatively large values, the reaction force $P$ of the cylindrical bumper decreases to a fraction of its initial strength, reducing this way its ability to produce rebound. The elastic bumper produces the opposite effect.
Think for instance of a ship of mass $m=55,500\,\text{ton}$ being pushed by the tugs to approach normally the dock at a speed $v=0.24\,\text{knot}=0.1234\,\text{m/s}$ (a rather high speed). The kinetic energy of the ship is then $423\,\text{kJ}$. Assuming a safe tire compression of 0.4 m, the graph in Fig. \[Fig12\] shows that each tire can store $W(0.40\,\text{m})=72.5\,\text{kJ}$, and then a minimum of six units would be necessary for safely docking the vessel. On the other hand, the ship side is specified to support a maximal load of $147\,\text{kN/m}^2$. The peak pressure the tires can exert on the ship side is $P_B/(\pi R^2)=54.4\,\text{kN/m}^2$, which is close to 1/3 the specified maximum.
[30]{}
Young, W., Budynas, R.G., Budynas, R., Sadegh, A.: Roark’s Formulas for Stress and Strain, 8th. edition. McGraw–Hill, New York (2011). Teng, J.G.: Buckling of thin shells: Recent advances and trends. Appl. Mech. Rev. [**49**]{} 263–274 (1996). Hunt, G.W., Lord, G.J., Peletier, M.A.: Cylindrical shell buckling: a characterization of localization and periodicity. Discrete and Continuous Dynamical Systems–Series B [**4**]{} 505–518 (2003). Paschero, M., Hyer, M.W.: Axial buckling of an orthotropic circular cylinder: Application to orthogrid concept. Int. J. Sol. Structures [**46**]{} 2151–2171 (2009). Pinna, R., Ronalds, B.F.: Buckling and postbuckling of cylindrical shells with one end pinned and the other end free. Thin-Walled Structures [**41**]{} 507-527 (2003). Simitses, G.J.: Buckling and postbuckling of imperfect cylindrical shells: A rewiew. Appl. Mech. Rev. [**39**]{} 1517–1524 (1986). Wullschleger, L., Meyer-Piening, H. R.: Buckling of geometrically imperfect cylindrical shells–definition of a buckling load. Int. J. Non-Linear Mech. [**37**]{} 645–657 (2002). Zozulya, V.V., Zhang, Ch.: A high order theory for functionally graded axisymmetric cylindrical shells. Int. J. Mech. Sci. [**60**]{} 12–22 (2012). Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edition. Academic Press, Burlington–San Diego–London (2007). Byrd, P.F., Friedman, M. D.: Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edition. Springer–Verlag, New York–Heidelberg–Berlin (1971).
|
---
author:
- 'J.-B. Delisle'
- 'N. Hara[^1]'
- 'D. Ségransan'
bibliography:
- 'fap.bib'
subtitle: 'I. Statistical significance of periodogram peaks'
title: Efficient modeling of correlated noise
---
Introduction {#sec:intro}
============
Detecting periodic signals in unevenly spaced time series is a common problem in astronomy, which is encountered, for instance, when searching for binaries or exoplanet companions in radial velocity, astrometric, or photometric time series. The Lomb-Scargle (LS) periodogram [@lomb_leastsquares_1976; @scargle_studies_1982] is a classical and efficient tool to search for sinusoidal signals. The principle of the LS periodogram is to scan a wide range of frequencies and to compare a linear sinusoidal model at a given frequency with a constant model, called the base model. A widespread variant of the LS periodogram, called the generalized LS periodogram (GLS), was proposed by @zechmeister_generalised_2009 where the constant is adjusted for at each frequency [see also @ferraz-mello_estimation_1981]. Once the periodogram is computed, the false alarm probability (FAP) criterion is often used to determine whether or not it supports the detection of a periodic signal. The FAP is often estimated using numerical methods such as bootstrapping or Monte Carlo. These methods can be computationally intensive, especially at low FAP levels. Indeed, estimating a FAP level of $P$ numerically requires at least the computation of $10/P$ periodograms.
Several analytical formula have been proposed for the FAP [@scargle_studies_1982; @horne_prescription_1986], but have been subsequently contested [@koen_significance_1990]. The periodogram framework was generalized in a series of works to handle more complex models [@baluev_assessing_2008; @baluev_accounting_2009; @baluev_detecting_2013; @baluev_vonmises_2013; @baluev_keplerian_2015], where rigorous and sharp analytical approximations of the FAP were provided based on the so-called Rice formula and previous works by @davies_hypothesis_1977 [@davies_hypothesis_1987; @davies_hypothesis_2002]. These works allow a fast and rigorous estimation of the periodogram FAP in the presence of white noise. However, the white noise assumption is often incorrect for astronomical time series. Indeed, several sources of correlated noise, such as the astronomical source itself, Earth’s atmosphere, or instrumental systematics, could contaminate the measurements.
For instance, stellar variability has a huge impact on the detection of low-mass exoplanets using high-precision radial velocity time series. Indeed, at a precision better than 1 m/s, stellar variability affects the measurements on timescales ranging from a few minutes (for stellar oscillation p-modes), to hours and days (for stellar granulation and super-granulation), and even up to the star rotation period (for the effect of spots and plages). In this context, low-mass exoplanet detection becomes a challenge [e.g., @queloz_planet_2001] and proper tools have to be developed to treat correlated noise properly and to compute reliable periodograms and FAPs. @sulis_using_2016 provide an analytical FAP estimate for periodograms normalized by the power density spectrum of the noise, in the limit of low aliasing and in the case of evenly sampled data. This is however, not the case of most radial velocity datasets. @baluev_planetpack_2013 provides a “suggestive generalization” to the correlated noise case of the FAP formula obtained in @baluev_assessing_2008, but advocates against the use of this formula since it has not been proved rigorously.
In this article, we extend the work of @baluev_assessing_2008 to account for correlated noise. In Sect. \[sec:corrperio\] we define a general periodogram for an arbitrary covariance of the noise and provide an analytical approximation of the corresponding FAP, which we validate against Monte Carlo simulations. In Sect. \[sec:wrongnoise\] we provide a method to explore the sensitivity of the periodogram to the noise model. We discuss our results in Sect. \[sec:conclusion\].
Significance of periodogram peaks with correlated noise {#sec:corrperio}
=======================================================
In this section, we present a method to assess the significance of periodogram peaks (FAP) in the correlated noise case. We first give a general definition of the periodogram in Sect. \[sec:periodogram\]. We then provide formulas to compute the corresponding FAP in Sect. \[sec:fap\]. Finally, we compare this analytical FAP with the results of Monte Carlo simulations in Sect. \[sec:montecarlo\].
General linear periodogram {#sec:periodogram}
--------------------------
We extend the general definition of least squares periodograms by @baluev_assessing_2008 to the correlated noise case. Following @baluev_assessing_2008, we compare the $\chi^2$ of the residuals of a linear base model $\H$ of $p$ parameters with enlarged linear models $\K$ of $p+d$ parameters, parameterized by the frequency $\nu$. The base model $\H$ is written as $$\H\ :\quad m_\H(\theta_\H) = \varphi_\H\theta_\H,
\label{eq:mh}$$ where $\theta_\H$ is the vector of size $p$ of the model parameters, $\varphi_\H$ is a $n\times p$ matrix, and $n$ the number of points in the time series. The columns of $\varphi_H$ are thus explanatory time series that are scaled by the linear parameters $\theta_\H$. For instance, in the case of a radial velocity time series with two different instruments and a linear drift, the linear base model could be chosen as $$m_\H = \gamma_1 \delta_1(t) + \gamma_2 \delta_2(t) + \alpha (t-\mathrm{epoch}),$$ where $\gamma_1$ and $\gamma_2$ are the velocity offsets of both instruments and $\alpha$ is the slope of the linear drift. The function $\delta_1(t)$ (respectively, $\delta_2(t)$) is equal to one for the points taken by instrument 1 (respectively, instrument 2) and zero otherwise. The matrix $\varphi_\H$ would thus be a $n\times 3$ matrix whose columns would be the three explanatory time series $\varphi_\H = (\delta_1(t), \delta_2(t), (t-\mathrm{epoch}))$ and the vector of parameters would be $\theta_\H = (\gamma_1, \gamma_2, \alpha)$.
The enlarged model $\K(\nu)$ is written as $$\K(\nu)\ :\quad m_\K(\nu, \theta_\K) = \varphi_\K(\nu)\theta_\K,
\label{eq:mk}$$ where $\theta_\K = (\theta_\H, \theta)$ is the vector of size $p+d$ of the parameters and $\varphi_\K(\nu) = (\varphi_H, \varphi(\nu))$ is a $n\times (p+d)$ matrix whose $p$ first columns are those of $\varphi_\H$, and whose $d$ last columns are functions of the frequency $\nu$. Typically, $d=2$, and the two additional columns are $\cos(\nu t)$ and $\sin(\nu t)$, but the theory developed by @davies_hypothesis_1977 [@davies_hypothesis_1987; @davies_hypothesis_2002] and @baluev_assessing_2008 is more general.
We denote by $\chi_\H^2$ and $\chi_\K^2(\nu)$ the $\chi^2$ of the residuals after a linear least squares fit with a covariance matrix $C$ of the models $\H$ and $\K(\nu)$, respectively. @baluev_assessing_2008 assumed the noise to be independent (diagonal covariance matrix) and Gaussian and that the uncertainties of the measurements are known precisely (at least within a common factor). In this generalization, we do not assume the noise to be independent anymore, but we still assume the noise to be Gaussian with a known covariance matrix $C$ (at least within a common factor). The covariance matrix $C$ accounts for all sources of correlated and uncorrelated noise, such as intrinsic noise from the source or subsequent contamination by the Earth’s atmosphere or by the instrument.
In the general case, the periodogram is a function $z(\nu) = f(\chi_\H^2, \chi_\K^2(\nu))$. A general linear periodogram $z$ is thus defined by the models $\H$ and $\K(\nu)$ and the function $f$. In the following, we consider the four definitions of the periodogram proposed by @baluev_assessing_2008: $$\begin{aligned}
\label{eq:defperio}
z_0(\nu) = \frac{1}{2}\left(\chi^2_\H-\chi^2_\K(\nu)\right), \qquad &
z_1(\nu) = \frac{n_\H}{2} \frac{\chi_\H^2 - \chi_\K^2(\nu)}{\chi_\H^2},\nonumber\\
z_2(\nu) = \frac{n_\K}{2} \frac{\chi_\H^2 - \chi_\K^2(\nu)}{\chi_\K^2(\nu)}, \qquad &
z_3(\nu) = \frac{n_\K }{2} \ln \frac{\chi_\H^2}{\chi_\K^2(\nu)},\end{aligned}$$ where $n_\H = n - p$ and $n_\K = n - (p + d)$.
The widespread GLS periodogram [see @ferraz-mello_estimation_1981; @zechmeister_generalised_2009] is very close to the definition $z_1$ of the periodogram. Indeed, we have $$\label{eq:GLS}
z_\mathrm{GLS} = \frac{\chi_\H^2 - \chi_\K^2(\nu)}{\chi_\H^2} = \frac{2}{n_\H} z_1(\nu),$$ and all the results obtained for $z_1$ are also valid for the GLS.
Once the periodogram is computed, it is useful to compute the $p$-value of the highest peak, or FAP, defined as $\mathrm{Pr}\{\max_\nu z(\nu) \geqslant Z\ |\ \H\}$, where $Z$ is the value of the maximum peak of the periodogram computed on the data.
False alarm probability for periodograms with correlated noise {#sec:fap}
--------------------------------------------------------------
$z(\nu)$ $\mathrm{FAP_{single}}(Z)$ $\tau(Z, \nu_\mathrm{max})$, approximately
------------ ---------------------------------------------------- -------------------------------------------------------------------------------------
$z_0(\nu)$ $\expo{-Z}$ $W\expo{-Z}\sqrt{Z}$
$z_1(\nu)$ $\left(1-\frac{2Z}{n_\H}\right)^\frac{n_\K}{2}$ $\gamma_\H W\left(1-\frac{2Z}{n_\H}\right)^\frac{n_\K-1}{2}\sqrt{Z}$
$z_2(\nu)$ $\left(1+\frac{2Z}{n_\K}\right)^{-\frac{n_\K}{2}}$ $\gamma_\K W\left(1+\frac{2Z}{n_\K}\right)^{-\frac{n_\K}{2}}\sqrt{Z}$
$z_3(\nu)$ $\expo{-Z}$ $\gamma_\K W\expo{-Z\left(1-\frac{1}{2 n_\K}\right)}\sqrt{n_\K\sinh\frac{Z}{n_\K}}$
: False alarm probability for different definitions (see Eq. (\[eq:defperio\])) of the periodogram power by @baluev_assessing_2008 in the case $d=2$.
\[tab:power\]
In this section, we provide analytical approximations of the FAP for the definitions of the periodogram of Eq. (\[eq:defperio\]). Their precise derivation is provided in Appendix \[sec:fapcomputation\].
The model $\H$ is defined as in Eq. (\[eq:mh\]), where the $n \times p$ matrix $\varphi_\H$ is user defined; for instance, it might include offsets and drifts. The model $\K$ (eq. (\[eq:mk\])) is the horizontal concatenation of $\varphi_\H$ and the two column vectors $\cos\nu t$ and $\sin\nu t$ ($\varphi_\K(\nu) = (\varphi_\H, \cos\nu t, \sin\nu t)$).
The periodogram is computed in the range of frequencies $]0, \nu_\mathrm{max}]$. The FAP is approximated by [see @baluev_assessing_2008] $$\label{eq:FAPmax}
\mathrm{FAP_{max}}(Z, \nu_\mathrm{max}) \approx 1 - \left(1-\mathrm{FAP_{single}}(Z)\right)\expo{-\tau(Z, \nu_\mathrm{max})},$$ where analytical expressions for $\mathrm{FAP_{single}}$ and $\tau(Z, \nu_\mathrm{max})$ are given in Table \[tab:power\]. These expressions depend on the rescaled frequency bandwidth $W$ defined as $$\label{eq:defW}
W = \frac{\nu_\mathrm{max}}{2\pi} T_\mathrm{eff},$$ where $T_\mathrm{eff}$ is the effective time series length, which we approximate by (see Appendix \[sec:analyticalfap\]) $$\label{eq:Teffestimate}
T_\mathrm{eff} \approx \sqrt{4\pi}
\sqrt{\frac{\varmean{\Pi\hadprod\operatorname{sinc}\nu_\mathrm{max}\Delta}}{\varmean{\operatorname{sinc}\nu_\mathrm{max}\Delta}}
- \left(\frac{\varmean{\Sigma\hadprod\operatorname{sinc}\nu_\mathrm{max}\Delta}}{2{\varmean{\operatorname{sinc}\nu_\mathrm{max}\Delta}}}\right)^2}.$$ The $n \times n$ matrices $\Sigma$, $\Delta$, and $\Pi$ are defined as $$\begin{aligned}
\Sigma_{i,j} & = t_i + t_j,\nonumber \\
\Delta_{i,j} & = t_i - t_j,\nonumber \\
\Pi_{i,j} & = t_i t_j,\end{aligned}$$ and for two $n\times n$ matrices $X$ and $Y$, $X \hadprod Y$ is the Hadamard (or element-wise) product $$(X \hadprod Y)_{i,j} = X_{i,j} Y_{i,j},$$ and $\varmean{X}$ is defined as $$\varmean{X} = \sum_{i,j} C^{-1}_{i,j} X_{i,j}.$$ The expression of the effective time series length found by @baluev_assessing_2008 in the white noise case can be derived from Eq. (\[eq:Teffestimate\]). Indeed in this case (diagonal covariance matrix $C$), Eq. (\[eq:Teffestimate\]) is simplified as $$\label{eq:Teffwhite}
T_\mathrm{eff} \approx \sqrt{4\pi\left(\mean{t^2}-\mean{t}^2\right)},$$ where $\mean{t}$, and $\mean{t^2}$ are weighted means with weights $C^{-1}_{i,i}/\sum_j C^{-1}_{j,j}$.
In Appendix \[sec:analyticalfap\], we additionally provide approximations of $T_\mathrm{eff}$ in the low and high frequency limit, that is, $\nu_\mathrm{max}\Delta \ll 1$ (Eq. (\[eq:Tefflow\])) and $\nu_\mathrm{max}\Delta \gg 1$ (Eq. (\[eq:Teffhigh\])).
Comparison of analytical FAP with Monte Carlo simulations {#sec:montecarlo}
---------------------------------------------------------
obs. jit. daily exp. monthly exp.
------------------------------ ------ ------ ------------ --------------
$\sigma_\mathrm{jit.}$ (m/s) – 1 – –
$\sigma_\mathrm{exp.}$ (m/s) – – 1 1
$\tau_\mathrm{exp.}$ (d) – – 1 30
: Values of the covariance matrix parameters (see Eq. (\[eq:Cij\])) for the four noise models used in our study of the radial velocity time series.
\[tab:noiseparams\]
![Comparison between analytical and numerical estimations of the FAP for four types of covariance matrix (see Sect. \[sec:montecarlo\]) and using the HARPS time series of . The periodogram power is computed following the definition $z_1$ of Eq. (\[eq:defperio\]). The expectation of $z_1$ is one (see Eq. (\[eq:expectperiotrue\])).[]{data-label="fig:compFAP_HD136352"}](HD136352_FAP_all){width="0.75\linewidth"}
We validated our analytical estimate of the FAP (Eqs. (\[eq:FAPmax\])-(\[eq:Teffestimate\]), Table \[tab:power\]) by comparing it with Monte Carlo simulations. The Monte Carlo simulations are performed by generating a large set of random time series following the same distribution (same covariance matrix). We used the times of observation and error bars of the HARPS radial velocities of [@udry_harps_2019] to obtain a realistic temporal sampling and realistic covariance matrices. The HARPS radial velocities of consist of 648 points taken over almost 11 years (2004-2015) and spread over 238 distinct nights (about 2.7 points per night).
Our method is general and does not require a particular shape for the covariance matrix. However, for illustration purposes, we assume the covariance matrix to follow $$\label{eq:Cij}
C_{i,j} = \delta_{i,j} (\sigma_i^2 + \sigma_\mathrm{jit.}^2) + \sigma_\mathrm{exp.}^2 \expo{-|t_i-t_j|/\tau_\mathrm{exp.}}$$ and vary the values of the parameters ($\sigma_\mathrm{jit.}$, $\sigma_\mathrm{exp.}$, $\tau_\mathrm{exp.}$) to define four different noise models:
1. obs. (white noise): a diagonal matrix with observational error bars;
2. jit. (white noise): a diagonal matrix with observational error bars plus a jitter of 1 m/s;
3. daily exp. (correlated noise): observational error bars on the diagonal, plus an exponential decay of 1 m/s with a timescale of 1 d;
4. monthly exp. (correlated noise): the same as daily exp. but with a timescale of 30 d.
The values of the parameters ($\sigma_\mathrm{jit.}$, $\sigma_\mathrm{exp.}$, $\tau_\mathrm{exp.}$) used for each noise model are summarized in Table \[tab:noiseparams\]. In the context of radial velocity time series, jitter terms might model both intrinsic noise from the star and instrumental noise, while exponential decay terms are often used to account for stellar noise (e.g., granulation and oscillation).
For a given covariance matrix $C$, we generated a synthetic radial velocity time series by randomly sampling from a normal distribution with covariance matrix $C$. We generate $10^6$ such random time series and compute a periodogram (with the correct covariance matrix) for each time series. The periodograms are computed in the range $]0,\frac{2\pi}{P_\mathrm{min}}]$ where $P_\mathrm{min}=0.9~\mathrm{d}$, and with an instrumental offset $\gamma$ adjusted for each frequency ($p=1$, $n_\H=n-1$, $n_\K=n-3$). Then, the distribution of the maximum of these periodograms allows us to estimate numerically the FAP.
The comparison between the analytical and numerical FAP is shown in Fig. \[fig:compFAP\_HD136352\]. As explained by @baluev_assessing_2008, the analytical formula of the FAP is an upper bound that asymptotically (for low FAP levels) converges to the exact FAP. This is indeed what we observe in Fig. \[fig:compFAP\_HD136352\]. For all the covariance matrices, the analytical and numerical estimates agree very well for $\mathrm{FAP}\lesssim 0.1$, and the analytical formula overestimates the FAP for $\mathrm{FAP} \gtrsim 0.1$
Since we used $10^6$ samples for the numerical estimation of the FAP, we could not reliably explore FAP levels below $10^{-4}$ owing to small numbers statistics. However, the analytical and numerical estimates agree very well down to $10^{-4}$, and the analytical approximation is expected to be even more accurate for lower FAP levels.
To sum up, the analytical estimate provides a very good approximation of the FAP in the range of most interest ($\mathrm{FAP}\lesssim 0.1$), and is conservative for higher FAP levels. Moreover, this analytical estimate is much faster to compute than Monte Carlo simulations (or other numerical methods), especially for low FAP levels. These properties make it very convenient to use in practical applications.
Sensitivity of periodogram to noise model {#sec:wrongnoise}
=========================================
{width="\linewidth"}
The FAP formula obtained in Sect. \[sec:fap\] provides an efficient and robust way to assess the significance of periodogram peaks when the covariance matrix is known. However, this is not the case in general, and we can often only make educated guesses about the shape of the covariance matrix. It is therefore necessary to explore the sensitivity of the periodogram to the noise model. To do so, we computed the expectation of the periodogram computed with an incorrect noise model. The analytical derivation of the periodogram expectation is described in Appendix \[sec:wrongnoiseformula\]. In this section, we illustrate its use by exploring the effect of an incorrect noise model on the periodogram and its associated FAP.
We assume that the actual covariance matrix of the noise is $C$, while the periodogram is computed with an incorrect covariance matrix $V$. We used the same dataset (HARPS radial velocities of ) and the same noise models as in Sect. \[sec:montecarlo\]. We first chose a noise model among the four models of Sect. \[sec:montecarlo\] (obs., jit., daily exp., and monthly exp.), which we considered the correct noise model (covariance matrix $C$). We then chose the incorrect noise model (covariance matrix $V$) among the three other models. The study is done with the definition $z_1$ of the periodogram (see Eq. (\[eq:defperio\])). We first computed the periodogram expectation following the analytical expression of Eq. (\[eq:expectperioO1\]) and we then estimated the FAP corresponding to the highest peak of this mean periodogram using the analytical formula of Eq. (\[eq:FAPmax\]) with the incorrect covariance matrix ($V$).
The results are shown in Fig. \[fig:perio\_HD136352\] for all pairs of noise models. We observe in Fig. \[fig:perio\_HD136352\] that adding a jitter term (jit.) does not affect the results much compared to the model with the observational error bars alone (obs.), and vice versa. This is not surprising since all the data points were taken with the same instruments (HARPS), and thus have very similar error bars. Therefore, adding a common jitter term to all error bars does not much affect the relative weight of each measurement. Moreover, the definition $z_1$ of the periodogram (Eq. (\[eq:defperio\])) is not sensitive to the multiplication of all error bars by a common factor.
Using a correlated noise model (daily or monthly exp.) while the true noise is uncorrelated (obs. or jit.) remains conservative on the whole range of frequencies ($\E(z_1(\nu)) \simeq 1$ for all $\nu$). However, the periodogram level is very low at long periods, which means that the detection capability at long periods is strongly reduced. On the other hand, using an uncorrelated noise model (obs. or jit.) while the true noise model is correlated (daily or monthly exp.) can lead to spurious detections with very low FAP levels (down to $3.22\times 10^{-6}$, see Fig. \[fig:perio\_HD136352\]). Underestimating the correlation timescale (using daily instead of monthly exp.) has a similar (but weaker) effect as using an uncorrelated model instead of a correlated model. Finally overestimating the correlation timescale (using monthly instead of daily exp.) reduces the capability to detect long periods, and might lead to spurious detections of short periods. In the two latter cases (underestimation and overestimation of the correlation timescale), the FAP remains very high (close to 1, i.e., non-significant detection).
Overall these results are not surprising but illustrate the possibility to investigate the sensitivity of the periodogram and its FAP with respect to the noise model using the formula for the periodogram expectation (Eq. (\[eq:expectperioO1\])).
Conclusions {#sec:conclusion}
===========
We present a generalization of the analytical estimate of @baluev_assessing_2008 (which was restricted to the white noise case) to the correlated noise case (see Sect. \[sec:fap\]). We show that the “suggestive generalization” of @baluev_planetpack_2013 is valid in the low frequency limit (see Eq. (\[eq:Tefflow\])), but we find a more general expression (Eq. (\[eq:Teffestimate\])) that is valid for all frequencies. We validate our analytical estimate against Monte Carlo simulations (see Sect. \[sec:montecarlo\]) and show that this analytical criterion is very efficient and accurate, provided that the covariance matrix of the noise is known (at least within a common factor).
In most cases, however, astronomical time series are contaminated by sources of correlated noise that are difficult to characterize fully, which results in an approximate modeling of the covariance matrix. We illustrate the sensitivity of the periodogram to the noise model, by deriving the expectation of a periodogram computed with an incorrect covariance matrix (see Sect. \[sec:wrongnoise\]). This method allows us to visualize which parts of the periodogram are the most affected by a change in the noise model. For instance, we observe, as expected, that overestimating the correlation timescale of the noise tends to reduce the detection capability at long periods strongly, while underestimating this timescale can lead to spurious detections. Another way to visualize the sensitivity of the periodogram to the noise model would be to compute several periodograms on the same data with various noise models. Both approaches are complementary to better understand the features observed in the periodograms.
Several methods can be used to obtain a more realistic covariance matrix. First, a likelihood maximization can be performed to adjust some noise model parameters. This maximization can be performed once, before computing the least squares periodogram (with a fixed noise model). It can also be performed for each frequency by computing a likelihood periodogram instead of a least squares periodogram. In the white noise case, @baluev_accounting_2009 proposed a FAP estimate for a likelihood periodogram with a free error term (jitter) added in quadrature to the nominal error bars and adjusted for at each frequency. A generalization to the correlated noise case could probably also be achieved for this likelihood periodogram, but is beyond the scope of this article. Finally, a Bayesian approach could be used to compare different models (signal + noise), but with a much higher computational cost.
We thank the anonymous referee for his/her useful comments. We acknowledge financial support from the Swiss National Science Foundation (SNSF). This work has, in part, been carried out within the framework of the National Centre for Competence in Research PlanetS supported by SNSF.
Computation of the FAP in the correlated noise case {#sec:fapcomputation}
===================================================
In this appendix, we extend the method of @baluev_assessing_2008 to obtain analytical FAP estimates in the correlated noise case. The main idea allowing the analytical approximation of the FAP with correlated noise is to perform a change of coordinates that yields independent Gaussian random variables. Then, the method described by @baluev_assessing_2008 can be applied on these new variables. The change of variables is described in Sect. \[sec:changevar\] and the derivation of the FAP estimate in Sect. \[sec:analyticalfap\].
Change of random variables {#sec:changevar}
--------------------------
Let us assume that the covariance matrix $C$ of the noise is known (at least within a common factor). Then, under the null hypothesis (i.e., assuming the base model is correct), the time series is written as $$\label{eq:yH}
y = \varphi_\H \theta_{\H,0} + \epsilon,$$ where $\theta_{\H,0}$ is the true value of the parameters and the noise $\epsilon$ is Gaussian with zero mean and covariance $C$.
For a linear model $\varphi_m$ and parameters $\theta_m$ ($m=\H$ or $\K$), the $\chi^2$ is defined as $$\begin{aligned}
\chi^2(\theta_m) & = (y-\varphi_m\theta_m)\t C^{-1} (y-\varphi_m\theta_m)\nonumber \\
& = (\varphi_\H \theta_{\H,0} - \varphi_m\theta_m + \epsilon)\t C^{-1}
(\varphi_\H \theta_{\H,0} - \varphi_m\theta_m + \epsilon).\end{aligned}$$ The least squares estimate of the parameters is written as $$\begin{aligned}
\hat{\theta}_m & = \left(\varphi_m\t C^{-1} \varphi_m\right)^{-1} \varphi_m\t C^{-1} y\nonumber \\
& = \theta_{\H,0} + \left(\varphi_m\t C^{-1} \varphi_m\right)^{-1} \varphi_m\t C^{-1} \epsilon,\end{aligned}$$ and the minimum $\chi^2$ is thus $$\begin{aligned}
\chi^2_m & = \min_{\theta_m} \chi^2(\theta_m) = \chi^2(\hat{\theta}_m) = (y-\varphi_m\hat{\theta})\t C^{-1} (y-\varphi_m\hat{\theta})\nonumber \\
& = y\t \left(C^{-1} - C^{-1} \varphi_m \left(\varphi_m\t C^{-1} \varphi_m\right)^{-1} \varphi_m\t C^{-1}\right) y\nonumber \\
& = \epsilon\t \left(C^{-1} - C^{-1} \varphi_m \left(\varphi_m\t C^{-1} \varphi_m\right)^{-1} \varphi_m\t C^{-1}\right) \epsilon.\end{aligned}$$ Let us now perform the following change of coordinates: $$\begin{aligned}
\label{eq:changevar}
z & = L^{-1} y,\nonumber \\
\eta & = L^{-1} \epsilon,\nonumber \\
\psi_m & = L^{-1} \varphi_m,\end{aligned}$$ where $C=L L\t$ is the Cholesky decomposition of the covariance matrix. Since we assumed $\epsilon$ to follow a Gaussian law with zero mean and covariance $C$, $\eta$ follows a Gaussian law with zero mean and covariance $\id$. The random variables $\eta$ are thus independent Gaussian variables. In these new variables, the $\chi^2$ is simply rewritten as $$\chi^2(\theta) = (z-\psi_m\theta_m)\t (z-\psi_m\theta_m),$$ the least squares estimate of the parameters is rewritten as $$\hat{\theta}_m = \theta_{\H,0} + \left(\psi_m\t \psi_m\right)^{-1} \psi_m\t \eta,$$ and the minimum $\chi^2$ as $$\chi^2_m = \eta\t \left(\id - \psi_m \left(\psi_m\t \psi_m\right)^{-1} \psi_m\t\right) \eta,$$ which follows a $\chi^2$ law with $n_m$ degrees of freedom ($n_\H = n-p$, $n_\K = n-(p+d)$). Therefore, the initial problem of analyzing a time series $y$ with covariance matrix $C$, base model $\varphi_\H$, and enlarged models $\varphi_\K = (\varphi_H, \varphi(\nu))$ is equivalent to analyzing the time series $z$, with covariance matrix $\id$, base model $\psi_\H$, and enlarged model $\psi_\K = (\psi_H, \psi(\nu))$. However, if $\varphi(\nu)$ was the sine and cosine at frequency $\nu$, this is no longer the case for $\psi(\nu)$. Nevertheless, the theory of @baluev_assessing_2008 is very general, and does not assume a particular shape for this matrix, except for the final application to the least squares periodogram. We thus follow the method proposed by @baluev_assessing_2008, and only change the hypothesis on the shape of the enlarged model matrix.
Analytical FAP estimate {#sec:analyticalfap}
-----------------------
The FAP can be bounded by [see @baluev_assessing_2008 Eq. (5)] $$\mathrm{FAP_{max}}(Z, \nu_\mathrm{max}) \leq \mathrm{FAP_{single}}(Z) + \tau(Z, \nu_\mathrm{max}),$$ and approximated by [see @baluev_assessing_2008 Eq. (6)] $$\mathrm{FAP_{max}}(Z, \nu_\mathrm{max}) \approx 1 - \left(1-\mathrm{FAP_{single}}(Z)\right)\expo{-\tau(Z, \nu_\mathrm{max})},$$ where $Z$ is the maximum periodogram power, $\mathrm{FAP_{single}}(Z)$ is the FAP in the case in which the frequency $\nu$ of the putative additional signal is fixed, $\tau(Z, \nu_\mathrm{max})$ is the expectation of the number of up-crossings of the level $Z$ by the periodogram [see @baluev_assessing_2008].
Computing $\mathrm{FAP_{single}}(Z)$ and $\tau(Z, \nu_\mathrm{max})$ requires us to specify the definition of the periodogram $z(\nu)$. @baluev_assessing_2008 proposed several definitions and derived the corresponding formulas for $\mathrm{FAP_{single}}(Z)$ and $\tau(Z, \nu_\mathrm{max})$. These results are summarized in Table \[tab:power\] for $d=2$. The general case ($d$ not necessarily equal to 2 and other definitions of $z(\nu)$) is provided in @baluev_assessing_2008, Appendix B.
For the definitions of the periodogram of Eq. (\[eq:defperio\]) and assuming $d=2$, the only quantity left to compute is the factor $W$, which is the rescaled frequency bandwidth, defined as [see @baluev_assessing_2008] $$W = \frac{A(\nu_\mathrm{max})}{2\pi^{3/2}},$$ where $$\begin{aligned}
\label{eq:Anumax}
A(\nu_\mathrm{max}) & = \int_{0}^{\nu_\mathrm{max}}\d\nu \int_{x^2<1} \frac{\sqrt{x\t M(\nu) x}}{x\t x} \d x\nonumber \\
& \leq 2\pi \int_{0}^{\nu_\mathrm{max}} \sqrt{\frac{\operatorname{tr}(M(\nu))}{2}} \d \nu.\end{aligned}$$ The $2\times 2$ matrix $M(\nu)$ is defined as follows (with $x' = \partial x/\partial\nu$): $$\begin{aligned}
\label{eq:defM}
& Q = \psi\t \psi = \varphi\t C^{-1} \varphi, \qquad
S = \psi\t \psi' = \varphi\t C^{-1} \varphi',\nonumber \\
& R = \psi\primet \psi' = \varphi\primet C^{-1} \varphi',\nonumber \\
& Q_\H = \psi_\H\t \psi = \varphi_\H\t C^{-1} \varphi, \qquad
S_\H = \psi_\H\t \psi' = \varphi_\H\t C^{-1} \varphi',\nonumber \\
& Q_{\H,\H} = \psi_\H\t \psi_\H = \varphi_\H\t C^{-1} \varphi_\H,\nonumber \\
& \tilde{Q} = Q - Q_\H\t Q_{\H,\H}^{-1} Q_\H, \qquad
\tilde{S} = S - Q_\H\t Q_{\H,\H}^{-1} S_\H,\nonumber \\
& \tilde{R} = R - S_\H\t Q_{\H,\H}^{-1} S_\H,\nonumber \\
& M = \tilde{Q}^{-1}\left(\tilde{R} - \tilde{S}\t\tilde{Q}^{-1}\tilde{S}\right).\end{aligned}$$ @baluev_assessing_2008 also defined the effective time series length as $$\label{eq:defTeff}
T_\mathrm{eff} = \frac{A(\nu_\mathrm{max})}{\sqrt{\pi}\nu_\mathrm{max}},$$ such that $$W = \frac{\nu_\mathrm{max}}{2\pi} T_\mathrm{eff}.$$ From Eqs. (\[eq:Anumax\]) and (\[eq:defTeff\]), we obtain $$\label{eq:Teff}
T_\mathrm{eff} = \frac{1}{\sqrt{\pi}}\mean{\int_{x^2<1} \frac{\sqrt{x\t M(\nu) x}}{x\t x} \d x}
\leq \mean{\sqrt{2\pi\operatorname{tr}(M(\nu))}},$$ where $\mean{x}$ is the mean of $x(\nu)$ over the frequency range $]0,\nu_\mathrm{max}]$. As noted by @baluev_assessing_2008, the inequality in Eqs. (\[eq:Anumax\]) and (\[eq:Teff\]) is very sharp in practical applications. In particular, it saturates (i.e., becomes an equality) when the eigenvalues of $M(\nu)$ are equal. This expression can be evaluated numerically by sampling the frequency over the interval $]0,\nu_\mathrm{max}]$, and computing $M(\nu)$ according to Eq. (\[eq:defM\]) for each frequency $\nu$. The cost of evaluating $T_\mathrm{eff}$ is of the same order of magnitude as computing the periodogram itself. It is therefore much more efficient than performing Monte Carlo simulations. However, this cost is not negligible compared to the periodogram cost and analytical approximations might be useful.
We now specify the expression of $T_\mathrm{eff}$ for $\varphi = (\cos(\nu t), \sin(\nu t))$. Replacing $\varphi$ in the definitions of $Q$, $S$, and $R$, we find $$\begin{aligned}
Q & = \frac{1}{2}\left(\arraycolsep=2.0pt\begin{array}{cc}
\varmean{\cos\nu\Sigma + \cos\nu\Delta} &
\varmean{\sin\nu\Sigma} \\
\varmean{\sin\nu\Sigma} &
\varmean{\cos\nu\Delta - \cos\nu\Sigma}
\end{array}\right),\nonumber \\
S & = \frac{1}{4}\left(\arraycolsep=2.0pt\begin{array}{cc}
-\varmean{\Sigma\hadprod\sin\nu\Sigma + \Delta\hadprod\sin\nu\Delta} &
\varmean{\Sigma\hadprod(\cos\nu\Sigma + \cos\nu\Delta)} \\
\varmean{\Sigma\hadprod(\cos\nu\Sigma - \cos\nu\Delta)} &
\varmean{\Sigma\hadprod\sin\nu\Sigma - \Delta\hadprod\sin\nu\Delta}
\end{array}\right),\nonumber \\
R & = \frac{1}{2}\left(\arraycolsep=2.0pt\begin{array}{cc}
\varmean{\Pi\hadprod(\cos\nu\Delta - \cos\nu\Sigma)} &
-\varmean{\Pi\hadprod\sin\nu\Sigma} \\
-\varmean{\Pi\hadprod\sin\nu\Sigma} &
\varmean{\Pi\hadprod(\cos\nu\Sigma + \cos\nu\Delta)}
\end{array}\right),\end{aligned}$$ where $\,\hadprod\,$ denotes the Hadamard (or elementwise) product, $$\begin{aligned}
\varmean{X} & = \sum_{i,j} C^{-1}_{i,j} X_{i,j},\nonumber \\
\Sigma_{i,j} & = t_i + t_j,\nonumber \\
\Delta_{i,j} & = t_i - t_j,\nonumber \\
\Pi_{i,j} & = t_i t_j.\end{aligned}$$ We follow @baluev_assessing_2008 and neglect aliasing effects. In this approximation all the terms containing sine or cosine of $\nu \Sigma$ average out. The terms containing $\sin\nu\Delta$ can also be neglected. Indeed, in the low frequency limit ($\nu\Delta \ll 1$), the terms in $\sin\nu\Delta$ vanish, while in the high frequency limit ($\nu\Delta \gg 1$), the terms in $\sin\nu\Delta$ average out. We thus obtain $$\begin{aligned}
\label{eq:QSRnoalias}
Q & \approx \frac{1}{2}\varmean{\cos\nu\Delta} \id,\nonumber \\
S & \approx \frac{1}{4}\varmean{\Sigma\hadprod\cos\nu\Delta} J,\nonumber \\
R & \approx \frac{1}{2} \varmean{\Pi\hadprod\cos\nu\Delta} \id,\end{aligned}$$ where $J$ is the antisymmetric matrix $$J=\left(\begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}\right).$$ As in @baluev_assessing_2008, we additionally assume that $\psi(\nu)$ is orthogonal to $\psi_\H$ for all $\nu$. As a consequence, $\tilde{Q}=Q$, $\tilde{S}=S$, $\tilde{R}=R$, and $$\label{eq:Mortho}
M(\nu) = Q^{-1}\left(R - S\t Q^{-1}S\right).$$ Replacing Eq. (\[eq:QSRnoalias\]) in Eq. (\[eq:Mortho\]), we find $$M(\nu) \approx \left(
\frac{\varmean{\Pi\hadprod\cos\nu\Delta}}{\varmean{\cos\nu\Delta}}
- \left(\frac{\varmean{\Sigma\hadprod\cos\nu\Delta}}{2\varmean{\cos\nu\Delta}}\right)^2\right)\id.$$
The two eigenvalues of $M$ are thus equal in this approximation, and we can approximate $T_\mathrm{eff}$ with (see Eq. (\[eq:Teff\])) $$\label{eq:Teff2}
T_\mathrm{eff} \approx \sqrt{4\pi}\mean{\sqrt{\frac{\varmean{\Pi\hadprod\cos\nu\Delta}}{\varmean{\cos\nu\Delta}}
- \left(\frac{\varmean{\Sigma\hadprod\cos\nu\Delta}}{2{\varmean{\cos\nu\Delta}}}\right)^2}}.$$ In the low frequency limit $\nu_\mathrm{max}\Delta \ll 1$, the cosines can all be replaced by 1, and we obtain $$\begin{aligned}
\label{eq:Tefflow}
T_\mathrm{eff,\,low} & \approx \sqrt{4\pi}\sqrt{\frac{\varmean{\Pi}}{\varmean{1}}
- \left(\frac{\varmean{\Sigma}}{2\varmean{1}}\right)^2}\nonumber\\
& \approx \sqrt{4\pi} \sqrt{\frac{t\t C^{-1} t}{u\t C^{-1}u} - \left(\frac{u\t C^{-1} t}{u\t C^{-1}u}\right)^2},\end{aligned}$$ where $u$ is the vector of size $n$ filled with ones. This expression was proposed by @baluev_planetpack_2013 as a “suggestive generalization” of the results of @baluev_assessing_2008 to correlated noise. However, @baluev_planetpack_2013 advocates against its use since it was not proved rigorously. Moreover, this expression is not valid in the general case but only in the low frequency limit.
In the high frequency limit $\nu_\mathrm{max}\Delta \gg 1$, the cosines average out in Eq. (\[eq:Teff2\]), except on the diagonal (where they are equal to 1), and we find $$\label{eq:Teffhigh}
T_\mathrm{eff,\,high} \approx \sqrt{4\pi}
\sqrt{\frac{w\t(t\hadprod t)}{w\t u} - \left(\frac{w\t t}{w\t u}\right)^2},$$ where $w = \operatorname{diag}(C^{-1})$.
Finally, for any frequency $\nu_\mathrm{max}$, we can approximate (at first order) the integral of Eq. (\[eq:Teff2\]) by replacing each sum $\varmean{X(\nu)}$ by its average $\mean{\varmean{X}}$ over the range $]0,\nu_\mathrm{max}]$. We have $$\begin{aligned}
\mean{\varmean{\cos\nu\Delta}} & = \varmean{\operatorname{sinc}\nu_\mathrm{max}\Delta},\nonumber \\
\mean{\varmean{\Sigma\hadprod\cos\nu\Delta}} & = \varmean{\Sigma\hadprod\operatorname{sinc}\nu_\mathrm{max}\Delta},\nonumber \\
\mean{\varmean{\Pi\hadprod\cos\nu\Delta}} & = \varmean{\Pi\hadprod\operatorname{sinc}\nu_\mathrm{max}\Delta},\end{aligned}$$ and thus $$\label{eq:Teff3}
T_\mathrm{eff} \approx \sqrt{4\pi}
\sqrt{\frac{\varmean{\Pi\hadprod\operatorname{sinc}\nu_\mathrm{max}\Delta}}{\varmean{\operatorname{sinc}\nu_\mathrm{max}\Delta}}
- \left(\frac{\varmean{\Sigma\hadprod\operatorname{sinc}\nu_\mathrm{max}\Delta}}{2{\varmean{\operatorname{sinc}\nu_\mathrm{max}\Delta}}}\right)^2}.$$ Equations (\[eq:Tefflow\]) and (\[eq:Teffhigh\]) can also be derived directly from Eq. (\[eq:Teff3\]) in the low frequency and high frequency approximations. Moreover, as explained in Sect. \[sec:fap\], the expression found by @baluev_assessing_2008 in the white noise case can also be derived from Eq. \[eq:Teff3\] by using the fact that the covariance matrix is diagonal.
In practical applications, all estimations of $T_\mathrm{eff}$ (numerical evaluation of Eqs. (\[eq:Teff\]) or (\[eq:Teff2\]), or directly using Eqs. (\[eq:Tefflow\]), (\[eq:Teffhigh\]), and (\[eq:Teff3\])) yield similar results. Moreover, as noted by @baluev_assessing_2008 in the case of independent Gaussian noise, $T_\mathrm{eff}$ is often of the same order of magnitude as the total time span of the time series ($T_\mathrm{span} = \max(t)-\min(t)$).
Periodogram expectation {#sec:wrongnoiseformula}
=======================
In this appendix, we show how to obtain an analytical estimate of the expectation of a periodogram computed with an incorrect noise model. We assume that the actual covariance matrix $C$ of the noise is not known and that the $\chi^2$ and periodograms are computed using an incorrect covariance matrix $V$. Under the null hypothesis (model $\H$), the time series still follows Eq. (\[eq:yH\]) but the $\chi^2$ becomes $$\begin{aligned}
\chi^2(\theta_m) & = (y-\varphi_m\theta_m)\t V^{-1} (y-\varphi_m\theta_m)\nonumber \\
& = (\varphi_\H \theta_{\H,0} - \varphi_m\theta_m + \epsilon)\t V^{-1}
(\varphi_\H \theta_{\H,0} - \varphi_m\theta_m + \epsilon).\end{aligned}$$ The least squares estimate of the parameters becomes $$\hat{\theta}_m = \theta_{\H,0} + \left(\varphi_m\t V^{-1} \varphi_m\right)^{-1} \varphi_m\t V^{-1} \epsilon,$$ and the minimum $\chi^2$ is thus $$\label{eq:chi2eps}
\chi^2_m = \epsilon\t \left(V^{-1} - V^{-1} \varphi_m \left(\varphi_m\t V^{-1} \varphi_m\right)^{-1} \varphi_m\t V^{-1}\right) \epsilon.$$ We introduce a change of coordinates that is slightly different from Sect. \[sec:changevar\] (Eq. (\[eq:changevar\])), i.e., $$\begin{aligned}
\eta & = L^{-1} \epsilon,\nonumber\\
\zeta_m & = M^{-1} \varphi_m,\nonumber\\
N & = M^{-1} L,\end{aligned}$$ where $C=L L\t$ and $V = M M\t$ are the Cholesky decompositions of the true and assumed covariance matrices, and $\eta$ is a vector of independent, centered, and reduced Gaussian random variables. In these coordinates, the minimum $\chi^2$ (Eq. (\[eq:chi2eps\])) is rewritten as $$\chi^2_m = \eta\t N\t \left(\id - P_{V,m}\right) N \eta,$$ where $$P_{V,m} = \zeta_m \left(\zeta_m\t\zeta_m\right)^{-1} \zeta_m\t$$ is the projection matrix on the subspace of $\R^n$ defined by the vectors of $\zeta_m$. The expectation of the minimum $\chi^2$ with the wrong covariance matrix $V$ is thus $$\begin{aligned}
\label{eq:mum}
\mu_m &= \E(\chi^2_m) = \operatorname{tr}\left(N\t \left(\id - P_{V,m}\right) N\right)\nonumber\\
&= \operatorname{tr}\left(\left(\id - P_{V,m}\right) C_V\right),\end{aligned}$$ where $C_V = N N\t = M^{-1} C M\mt$. In the case $V = C$, we have $C_V = \id$, and we deduce $$\begin{aligned}
\mu_\H &= n_\H,\nonumber\\
\mu_\K &= n_\K.\end{aligned}$$
First order formula
-------------------
At first order, the expectation of the periodogram can be obtained by replacing $\chi^2_\H$ (respectively, $\chi^2_\K$) by its expectation $\mu_\H$ (respectively, $\mu_\K$) in the definition of the periodogram (Eq. (\[eq:defperio\])). We find $$\begin{aligned}
\label{eq:expectperioO1}
\E(z_0(\nu)) = \frac{1}{2}\left(\mu_\H-\mu_\K(\nu)\right), \qquad &
\E(z_1(\nu)) \approx \frac{n_\H}{2}\frac{\mu_\H-\mu_\K(\nu)}{\mu_\H},\nonumber\\
\E(z_2(\nu)) \approx \frac{n_\K}{2}\frac{\mu_\H-\mu_\K(\nu)}{\mu_\K(\nu)}, \qquad &
\E(z_3(\nu)) \approx \frac{n_\K}{2}\ln\frac{\mu_\H}{\mu_\K(\nu)}.\end{aligned}$$ In the case $V=C$ (the actual covariance matrix is known), we deduce $$\label{eq:expectperiotrue}
\E(z_i(\nu)) \approx \frac{d}{2},$$ for $i =0,\dots,3$, and for all frequencies $\nu$. However, in the case $V\neq C$ (wrong noise model), the periodogram expectation can significantly depart from $d/2$ and depends on the frequency $\nu$.
Higher order formulas
---------------------
Higher order estimates can also be obtained by developing Eq. (\[eq:defperio\]) in power series of $\chi^2_m-\mu_m$ and computing higher order momenta of $\chi^2_\H$, $\chi^2_\K$. We provide more details in the following, however, the first order formula already yields very accurate results, and we thus adopt it in our study.
We introduce the random variables $X_m = \chi^2_m - \mu_m$ ($m=\H,\ \K$), which we assume to be small with respect to $\mu_m$. We then develop the periodogram power of Eq. (\[eq:defperio\]) in power series of $X_m$. For instance, at second order we obtain $$\begin{aligned}
\label{eq:devperio}
z_0(\nu) &= \frac{1}{2}\left(\mu_\H-\mu_\K(\nu) + X_\H - X_\K(\nu)\right),\nonumber\\
z_1(\nu) &= \frac{n_\H}{2} \left(1-\frac{\mu_\K(\nu)}{\mu_\H}
\left(1+\frac{X_\K(\nu)}{\mu_\K(\nu)}
-\frac{X_\H}{\mu_\H}
\right.\right.\nonumber\\
&\hspace{2.7cm}\left.\left.
-\frac{X_\H X_\K(\nu)}{\mu_\H\mu_\K(\nu)}
+\left(\frac{X_\H}{\mu_\H}\right)^2
\right)\right) + \O\left(X^3\right),\nonumber\\
z_2(\nu) &= \frac{n_\K}{2} \left(-1+\frac{\mu_\H}{\mu_\K(\nu)}
\left(1+\frac{X_\H}{\mu_\H}
-\frac{X_\K(\nu)}{\mu_\K(\nu)}
\right.\right.\nonumber\\
&\hspace{2.9cm}\left.\left.
-\frac{X_\H X_\K(\nu)}{\mu_\H\mu_\K(\nu)}
+\left(\frac{X_\K(\nu)}{\mu_\K(\nu)}\right)^2
\right)\right) + \O\left(X^3\right),\nonumber\\
z_3(\nu) &= \frac{n_\K }{2} \left(
\ln\frac{\mu_\H}{\mu_\K(\nu)}
+ \frac{X_\H}{\mu_\H} - \frac{X_\K(\nu)}{\mu_\K(\nu)}
\right.\nonumber\\
&\hspace{1.0cm}\left.
- \frac{1}{2}\left(\frac{X_\H}{\mu_\H}\right)^2
+ \frac{1}{2}\left(\frac{X_\K(\nu)}{\mu_\K(\nu)}\right)^2
\right) + \O\left(X^3\right).\end{aligned}$$ The expectation of the periodogram is thus (at second order) $$\begin{aligned}
\label{eq:expectperioO2}
\E(z_0(\nu)) &= \frac{1}{2}\left(\mu_\H-\mu_\K(\nu)\right),\nonumber\\
\E(z_1(\nu)) &\approx \frac{n_\H}{2}\left(1-\frac{\mu_\K(\nu)}{\mu_\H}
+ \frac{\operatorname{cov}(\chi^2_\H,\chi^2_\K(\nu))}{\mu_\H^2} - \frac{\operatorname{var}(\chi^2_\H) \mu_\K(\nu)}{\mu_\H^3} \right),\nonumber\\
\E(z_2(\nu)) &\approx \frac{n_\K}{2}\left(\frac{\mu_\H}{\mu_\K(\nu)} - 1
+ \frac{\operatorname{var}(\chi^2_\K(\nu)) \mu_\H}{\mu_\K(\nu)^3} - \frac{\operatorname{cov}(\chi^2_\H,\chi^2_\K(\nu))}{\mu_\K(\nu)^2}\right),\nonumber\\
\E(z_3(\nu)) &\approx \frac{n_\K}{2}\left(\ln\frac{\mu_\H}{\mu_\K(\nu)}
+ \frac{\operatorname{var}(\chi^2_\K(\nu))}{2\mu_\K(\nu)^2} - \frac{\operatorname{var}(\chi^2_\H)}{2\mu_\H^2}\right),\end{aligned}$$ where $\mu_\H$, $\mu_\K(\nu)$ are computed according to Eq. (\[eq:mum\]), and $$\begin{aligned}
\label{eq:varm}
\operatorname{cov}(\chi^2_m, \chi^2_{m'})
& = \operatorname{tr}\left(N\t \left(\id - P_{V,m}\right) N N\t \left(\id - P_{V,m'}\right) N\right)\nonumber\\
& = \operatorname{tr}\left(\left(\id - P_{V,m}\right) C_V \left(\id - P_{V,m'}\right) C_V\right).\end{aligned}$$
[^1]: NCCR CHEOPS fellow
|
---
abstract: 'We solve the Sp(N) Heisenberg and SU(N) Hubbard-Heisenberg models on the anisotropic triangular lattice in the large-N limit. These two models may describe respectively the magnetic and electronic properties of the family of layered organic materials $\kappa$-(BEDT-TTF)$_2$X. The Heisenberg model is also relevant to the frustrated antiferromagnet, Cs$_2$CuCl$_4$. We find rich phase diagrams for each model. The Sp(N) antiferromagnet is shown to have five different phases as a function of the size of the spin and the degree of anisotropy of the triangular lattice. The effects of fluctuations at finite-N are also discussed. For parameters relevant to Cs$_2$CuCl$_4$ the ground state either exhibits incommensurate spin order, or is in a quantum disordered phase with deconfined spin-1/2 excitations and topological order. The SU(N) Hubbard-Heisenberg model exhibits an insulating dimer phase, an insulating box phase, a semi-metallic staggered flux phase (SFP), and a metallic uniform phase. The uniform and SFP phases exhibit a pseudogap. A metal-insulator transition occurs at intermediate values of the interaction strength.'
address:
- 'Department of Physics, Brown University, Providence, RI 02912-1843, USA'
- 'Department of Physics, University of Queensland, Brisbane 4072, Australia'
author:
- 'C. H. Chung and J. B. Marston'
- 'Ross H. McKenzie'
title: 'Large-N solutions of the Heisenberg and Hubbard-Heisenberg models on the anisotropic triangular lattice: application to Cs$_2$CuCl$_4$ and to the layered organic superconductors $\kappa$-(BEDT-TTF)$_2$X'
---
Introduction {#sec:intro}
============
The family of layered organic superconductors $\kappa$-(BEDT-TTF)$_2$X has attracted much experimental and theoretical interest[@ishiguro; @wosnitza]. There are many similarities to the high-$T_{c}$ cuprates, including unconventional metallic properties and competition between antiferromagnetism and superconductivity[@mckenzie]. The materials have a rich phase diagram as a function of pressure and temperature. At low pressures and temperatures there is an insulating antiferromagnetic ordered phase; as the temperature is increased a transition occurs to an insulating paramagnetic state. A first-order metal-insulator transition separates the paramagnetic insulating phase from a metallic phase; it is induced by increasing the pressure[@ito; @komatsu]. The metallic phase exhibits various temperature dependences which are different from that of conventional metals. For example, measurements of the magnetic susceptibility and NMR Knight shift are consistent with a weak pseudogap in the density of states[@desoto; @mayaffre].
The main purpose of this paper is to attempt to describe the magnetic ordering, the metal-insulator transition, and the unconventional metallic properties of these materials with two simplified models. We model the magnetic ordering in the insulating phase with the quantum Heisenberg antiferromagnet (HAF). We model the metallic phase as well as the metal-insulator transition with a hybrid Hubbard-Heisenberg model. To substitute for the lack of a small expansion parameter in either model, we enlarge the symmetry group from the physical SU(2) $\cong$ Sp(1) spin symmetry to Sp(N) (symplectic group) for the Heisenberg model[@subir91; @sachdev0] and to SU(N) for the Hubbard-Heisenberg model[@affleck; @marston]. We then solve these models in the large-N limit and treat $1/N$ as our systematic expansion parameter. In section \[sec:models\] we briefly summarize the physical SU(2) Heisenberg and Hubbard-Heisenberg model on the anisotropic triangular lattice. In section \[sec:spn\] we review the large-N theory of the Sp(N) quantum Heisenberg model. Based on the large-N solution of this model, we present the magnetic phase diagram in the parameter space of quantum fluctuation $n_b/N$ (where $n_b$ is the number of bosons in the Schwinger boson representation of the spin) and the magnetic frustration $J_2/J_1$. We discuss the effects of finite-N fluctuations on the Sp(N) magnetic phases with short-range order (SRO). We also discuss how our results are relevant to understanding recent neutron scattering experiments on Cs$_2$CuCl$_4$. In section \[sec:sun\] we review the large-N theory of the SU(N) Hubbard-Heisenberg model. We present the phase diagram based on the large-N solution in the parameter space of the dimensionless ratio of the nearest-neighbor exchange to the hopping constant $J_1/t_1$ and the dimensionless anisotropy ratio $J_2/(J_1+J_2)$. Away from the two nested limits $J_1 = 0$ and $J_2 = 0$ we find a metal-insulator transition which occurs at finite critical value of $J_1/t_1$. We also find that the density of states in the metallic state is suppressed at low temperatures, in qualitative agreement with the unconventional metallic properties seen in experiments. We conclude in section \[sec:discuss\] with a brief review our results.
Heisenberg and Hubbard-Heisenberg Models on the Anisotropic Triangular Lattice {#sec:models}
==============================================================================
Based on a wide range of experimental results and quantum chemistry calculations of the Coulomb repulsion between two electrons on the BEDT-TTF molecules it was argued in reference that the $\kappa$-(BEDT-TTF)$_2$X family are strongly correlated electron systems which can be described by a half-filled Hubbard model on the anisotropic triangular lattice. The Hubbard Hamiltonian is: $$\begin{aligned}
H &=&
-t_1 \sum_{<{\bf{ij}}>}[c^{\dagger \sigma}_{\bf{i}}
c_{{\bf{j}}\sigma}+ H.c.] - t_2\sum_{<<{\bf{ij}}>>}
[c^{\dagger \sigma}_{\bf{i}} c_{{\bf{j}} \sigma}+ H.c.]
\nonumber \\
&+&
\frac{U}{2} \sum_{\bf{i}}(c^{\dagger \sigma}_{\bf{i}}c_{{\bf{i}} \sigma}-1)^2.\end{aligned}$$ Here $c_{{\bf{i}}\sigma}$ is the electron destruction operator on site ${\bf{i}}$ and there is an implicit sum over pairs of raised and lowered spin indices $\sigma = \uparrow, \downarrow$. Matrix element $t_1$ is the nearest-neighbor hopping amplitude, and $t_2$ is the next-nearest-neighbor hopping along only one of the two diagonals of the square lattice as shown in figure \[lattice\]. The sum $<{\bf{ij}}>$ runs over pairs of nearest-neighbor sites and $<<{\bf{ij}}>>$ runs for next-nearest-neighbors.
= 8cm
Physical insight can be attained by considering the Hubbard model for different values of the ratio $U/t$. In the limit of large $U/t$ the Hubbard model at half-filling is insulating and the spin degrees of freedom are described by a spin-1/2 Heisenberg antiferromagnet[@auerbach]: $$\begin{aligned}
H &=&
J_1\sum_{<{\bf{ij}}>} \vec{S}_{\bf{i}}\cdot \vec{S}_{\bf{j}} +
J_2\sum_{<<{\bf{ij}}>>} \vec{S}_{\bf{i}}\cdot \vec{S}_{\bf{j}}, \end{aligned}$$ where $\vec{S}_{\bf{i}}$ is the spin operator on site $\bf{i}$, and the exchange couplings $J_1 = 4t_1^2/U$ and $J_2 = 4t_2^2/U$. Competition between $J_1$ and $J_2$ leads to magnetic frustration. Using parameters from quantum chemistry calculations[@fortu; @okuno; @castet] it was estimated in reference that $J_2/J_1 \sim 0.3$ to $1$ for the $\kappa$-(BEDT-TTF)$_2$X family. Hence, magnetic frustration is important. The frustrated Heisenberg Hamiltonian interpolates between the square-lattice ($J_2 = 0$) and the linear chain ($J_1 = 0$). Much is known about these two limiting cases. Additional insight[@trumper; @merino] comes by considering different values of the ratio $J_2/J_1$. At $J_2 = 0$, the square lattice limit, there is long-range Néel order with a magnetic moment of approximately $0.6 \mu_B$; see reference [@auerbach]. If $J_2$ is small but nonzero, the magnetic moment will be reduced by magnetic frustration. At $J_2/J_1$ around $0.5$, quantum fluctuations combined with frustration should destroy the Néel ordered state. As $J_2/J_1$ is further increased, the system may exhibit spiral long range order[@merino]. At $J_1 = J_2$, the lattice is equivalent to the isotropic triangular lattice. Anderson suggested in 1973 that the ground state could be a spin liquid without long range order[@anderson]. However, subsequent numerical work indicated that there is long-range-order with ordering vector $\vec{q}=(2\pi/3, 2\pi/3)$[@sorella]. Finally, at $J_1 = 0$ the model reduces to decoupled Heisenberg spin-1/2 chains which cannot sustain long range spin order[@auerbach] as per the quantum Mermin-Wagner theorem. For $J_1$ small but non-zero, the system consists of spin-1/2 chains weakly coupled in a zigzag fashion. The case of two such weakly coupled zigzag spin chains was studied by Okamoto and Nomura[@okamoto] and by White and Affleck[@white] who showed that there is a spin gap in the spectrum $\Delta \sim e^{-{\rm const} \times J_2/J_1}$, and the ground state exhibits dimerization and incommensurate spiral correlations. Although our system consists of an infinite number of weakly coupled spin chains instead of two chains, we find similar behavior.
As $U/t$ decreases, charge fluctuations from electron hopping become significant. Competition between hopping and Coulomb repulsion leads to a transition from the insulator to a metal. We use the hybrid Hubbard-Heisenberg Hamiltonian[@affleck; @marston] with independent parameters $t$, $J$, and $U$ (where in general $J \neq 4t^2/U$) to describe some aspects of the transition: $$\begin{aligned}
H &=&
\sum_{<{\bf{ij}}>}[-t_1 (c^{\dagger \sigma}_{\bf{i}}
c_{{\bf{j}}\sigma} + H.c.) + J_1 (\vec{S}_{\bf{i}} \cdot \vec{S}_{\bf{j}}
- \frac{1}{4} n_{\bf i} n_{\bf j})]
\nonumber \\
&+&
\sum_{<<{\bf{ij}}>>}[-t_2 (c^{\dagger \sigma}_{\bf{i}} c_{{\bf{j}}\sigma}+ H.c.) +
J_2 (\vec{S}_{\bf{i}} \cdot \vec{S}_{\bf{j}} - \frac{1}{4} n_{\bf i} n_{\bf j})]
\nonumber \\
&+&
\frac{U}{2}\sum_{\bf{i}}(c^{\dagger \sigma}_{\bf{i}}c_{{\bf{i}}\sigma}-1)^2\ .
\label{HSU2}\end{aligned}$$ Here $n_{\bf i} \equiv c_{\bf i}^{\dagger \sigma} c_{\bf {i} \sigma}$ is the number of fermions at site $\bf{i}$. The Hamiltonian reduces to the Hubbard model when $J_1=J_2=0$ and the Heisenberg model for $t_1, t_2 \to 0$. In the square lattice limit and at half-filling, perfect nesting drives the system to an antiferromagnetic insulator at arbitrarily small value of the interactions. As diagonal hopping $t_2$ is turned on, however, nesting of the Fermi surface is no longer perfect and the metal-insulator transition (MIT) occurs at non-zero critical interaction strength.
Sp(N) Heisenberg Model {#sec:spn}
======================
We first focus on the spin degree of freedom, appropriate to the insulating phase of the layered organic materials, by solving the Heisenberg model in a large-N limit. Our choice of the large-N generalization of the physical SU(2) spin-$1/2$ problem is dictated by the desire to find an exactly solvable model which has both long-range ordered (LRO) and short-range ordered (SRO) phases. This leads us to symmetric (bosonic) SU(N) or Sp(N) generalizations. The former can only be applied to bipartite lattices. The latter approach has been applied to the antiferromagnetic Heisenberg model on the square lattice with first-, second-, and third-neighbor coupling (the $J_1 - J_2 - J_3$ model)[@subir91], the isotropic triangular lattice[@subir92], and the kagomé lattice[@subir92]. As the anisotropic triangular lattice is not bipartite, we must choose the Sp(N) generalization[@subir91].
Brief Review of the Approach
----------------------------
To ascertain the likely phase diagram of the frustrated Heisenberg antiferromagnet, we consider the Sp(N) symplectic group generalization of the physical SU(2)$\cong$ Sp(1) antiferromagnet[@subir91; @subir92]. The model can be exactly solved in the $N \rightarrow \infty$ limit. Both LRO and SRO can arise if we use the symmetric (bosonic) representations of Sp(N). We begin with the bosonic description of the SU(2) HAF, where it can be shown that apart from an additive constant, the Hamiltonian may be written in terms of spin singlet bond operators: $$H = -\frac{1}{2}\sum_{\bf{ij}} J_{\bf{ij}}
(\varepsilon_{\alpha\beta}
b_{\bf{i}}^{\dagger \alpha} b_{\bf{j}}^{\dagger \beta})
(\varepsilon^{\gamma \delta} b_{\bf{i} \gamma} b_{\bf{j} \delta})
\label{Hsp1}$$ where we have used the bosonic representation for spin operator $$\begin{aligned}
\vec{S}_{\bf i} = \frac{1}{2} b_{\bf i}^{\dagger \alpha}
\vec{\sigma}_{\alpha}^{\beta} b_{{\bf i} \beta}\ ,\end{aligned}$$ where $\alpha = \uparrow, \downarrow$ labels the two possible spins of each boson. The antisymmetric tensor, $\varepsilon_{\alpha\beta}$ is as usual defined to be: $$\begin{aligned}
\varepsilon = \left(
\begin{array}{cc}
0&1\\
-1&0
\end{array} \right)\ .\end{aligned}$$ We enforce the constraint $n_b = b_{\bf{i}}^{\dagger \alpha} b_{\bf{i} \alpha} = 2S$ to fix the number of bosons, and hence the total spin, on each site. The SU(2) spin singlet bond creation operator $\varepsilon_{\alpha\beta} b_{\bf{i}}^{\dagger \alpha}
b_{\bf{j}}^{\dagger \beta}$ may now be generalized to the Sp(N)-invariant form ${\mathcal J}_{\alpha\beta} b_{\bf{i}}^{\dagger \alpha} b_{\bf{j}}^{\dagger \beta}$. Global Sp(N) rotations may be implemented with $2N \times 2N$ unitary matrices $U$: $$\begin{aligned}
b &\to& U b
\nonumber \\
U^{\dagger} {\mathcal J} U &=& {\mathcal J}.\end{aligned}$$ Here $${\mathcal J} =
\left( \begin{array}{cccccc}
& 1 & & & & \\
-1 & & & & & \\
& & & 1 & & \\
& & -1 & & & \\
& & & & \ddots & \\
& & & & & \ddots
\end{array} \right)$$ generalizes the $\epsilon$ tensor to $N > 1$, and $b_{\bf{i}\alpha}$ with $\alpha = 1 \ldots 2N$ is the Sp(N) boson destruction operator. The Hamiltonian of the Sp(N) HAF may then be written: $$H = -\sum_{\bf{ij}} \frac{J_{\bf{i j}}}{2N}
\left({\mathcal J}_{\alpha \beta}
b_{\bf{i}}^{\dagger \alpha} b_{\bf{j}}^{\dagger \beta} \right)
\left( {\mathcal J}^{\gamma \delta} b_{\bf{i} \gamma} b_{\bf{j} \delta}
\right)
\label{Hspn}$$ where again the Greek indices run over $1 \ldots 2N$, and the constraint $b_{\bf{i}}^{\dagger \alpha} b_{\bf{i} \alpha} = n_b$ is imposed at every site of the lattice. We have rescaled $J_{\bf{i j}}/2 \rightarrow
J_{\bf{i j}}/ 2N$ to make the Hamiltonian of $O(N)$. For fixed $J_1/J_2$ we have a two-parameter ($n_b$, $N$) family of models with the ratio $\kappa \equiv n_b / N$ determining the strength of the quantum fluctuations. In the physical Sp(1) limit, $\kappa = 1$ corresponds to spin-1/2. At large $\kappa$ (equivalent to the large-spin limit of the physical SU(2) model) quantum effects are small and the ground states break global Sp(N) spin-rotational symmetry. LRO then corresponds to Bose condensation which we quantify by defining $$b_{\bf{i} m \sigma} \equiv \left ( \begin{array}{c}
\sqrt{N} x_{{\bf i} \sigma}\\
\tilde{b}_{{\bf i} \tilde{m} \sigma}
\end{array} \right )\ .
\label{bx}$$ Here the spin-quantization axis has been fixed by introducing the paired-index notation $\alpha \equiv (m, \sigma)$ with $m = 1 \ldots N$, $\sigma = \uparrow,\downarrow$, and $\tilde{m}=2 \ldots N$. The c-number spinors $x_\sigma$, when non-zero, quantify the Bose condensate fraction, and are given by $\langle b_{{\bf i} m \sigma} \rangle = \sqrt{N} \delta^1_m x_{\bf{i} \sigma}$. For sufficiently small $\kappa$, however, quantum fluctuations overwhelm the tendency to order and there can be only magnetic SRO.
Upon inserting equation \[bx\] into equation \[Hspn\], decoupling the quartic boson terms within the corresponding functional integral by introducing complex-valued Hubbard-Stratonovich fields $Q_{\bf{ij}}$ directed along the lattice links, enforcing the constraint on the number of bosons on each site with Lagrange multipliers $\lambda_{\bf{i}}$, and finally integrating out the $b$-fields, we obtain an effective action proportional to $N$. Thus, as $N \to \infty$, the effective action may be replaced by its saddle-point value. At the saddle-point, the $Q_{\bf{ij}}$, $\lambda_{\bf{i}}$, and $x_{\bf{i} \sigma}$ fields are expected to be time-independent, so the effective action can be put into correspondence with a suitable mean-field Hamiltonian. The Hamiltonian may be diagonalized by a Bogoliubov transformation, yielding a total ground state energy $E_{MF}$ given by[@subir92] $$\begin{aligned}
\frac{E_{MF}[Q, \lambda]}{N}
&=& \sum_{\bf{i>j}}\left(
\frac{J_{\bf{ij}}}{2}{|Q_{\bf{ij}}|}^{2} -
\frac{J_{\bf{ij}}}{2}Q_{\bf{ij}}\varepsilon^{\sigma\sigma^{\prime}}
x_{\bf{i} \sigma} x_{\bf{j} \sigma^{\prime}} + H.c.\right)
\nonumber \\
&-&
\sum_{\bf{i}}\lambda_{\bf{i}}\left(1+\frac{n_b}{N}
- |x_{\bf{i} \sigma}|^{2} \right) + \sum_{\bf k} \omega_{\bf k}
\label{EMF}\end{aligned}$$ where $\omega_{\bf k}$ are eigenenergies obtained from diagonalizing the mean-field Hamiltonian. Finding the ground state of the Sp(N) HAF now reduces to the problem of minimizing $E_{MF}$ with respect to the variables $Q_{{\bf{ij}}}$ and $x_{\bf{i} \sigma}$, subject to the Lagrange-multiplier constraints $$\frac{\partial E_{MF}[Q, \lambda]}{\partial \lambda_{\bf i}} = 0\ .
\label{lagrange}$$
It is essential to note that the action possesses local U(1) gauge symmetry under local phase rotations by angle $\theta_i(\tau)$: $$\begin{aligned}
b_{{\bf i}\alpha} \to b_{{\bf i}\alpha}~ e^{-i \theta_{\bf{i}}(\tau)}
\nonumber \\
Q_{\bf{ij}} \to Q_{\bf{ij}}~ e^{i \theta_{\bf{i}}(\tau) + i\theta_{\bf{j}}(\tau)}
\nonumber \\
\lambda_{\bf{i}} \to \lambda_{\bf{i}} + \frac{\partial \theta_{\bf{i}}}
{\partial \tau}\ .
\label{U(1)}\end{aligned}$$ This symmetry reminds us that the representation of spin operators in terms of the underlying bosons is redundant as the phase of each boson fields can be shifted by an arbitrary amount without affecting the spin degree of freedom. Two gauge invariant quantities of particular importance for our classification of the phases are $|Q_{\bf {ij}}|$, and $\sum_{\bf {i}} \lambda_{\bf i}$.
Extrema of $E_{MF}$ are found numerically with the simplex-annealing method[@NRC]. We work with a lattice of $40 \times 40 $ sites and check that this is sufficiently large to accurately represent the thermodynamic limit. Constraint equation \[lagrange\] is tricky, however, as $E_{MF}[\lambda_{\bf i}]$ is neither a minimum, nor a maximum with respect to the $\lambda_{\bf i}$ directions at the saddle point. The problem is solved by decomposing $\lambda_{\bf{i}}$ into its mean value $\bar{\lambda}$ and the deviations from the mean, $\delta \lambda_{\bf i} \equiv \lambda_{\bf i} - \bar{\lambda}$. As $\bar{\lambda}$ is gauge-invariant, we solve the constraint equation \[lagrange\] separately for it by applying Newton’s method. The resulting $E_{MF}[\bar{\lambda}, \delta \lambda_{\bf i}]$ can then be maximized in the remaining $\delta \lambda_{\bf i}$ directions. Individual values of $\delta \lambda_{\bf{i}}$ are in general non-zero, but by definition it must be the case that $\sum_{\bf i} \delta\lambda_{\bf{i}} = 0$.
Phase Diagram of the Sp(N) Heisenberg antiferromagnet
-----------------------------------------------------
To make progress in actually solving the model, we now make the assumption that spontaneous symmetry breaking, if it occurs, does not lead to a unit cell larger than 4 sites. Our choice of unit cell is shown in figure \[4sitespn\].
= 6.5cm
The $2 \times 2$ unit cell requires 12 different complex-valued Q-fields (8 on the square links, and 4 along the diagonal links) and 4 different $\lambda$ fields and $x$ spinors at each of the four sites. However, we have checked that at every saddle-point in the SRO region of the phase diagram ($x_{{\bf i} \sigma} = 0$) each of the 8 Q-fields on the horizontal and vertical links take the same value. Likewise, the 4 Q-fields on the diagonals are all equal, as well as all 4 $\lambda$-fields. Thus the $2 \times 2$-site unit cell can be reduced to only a single site per unit cell as shown in figure \[1sitespn\]. We expect this to also hold in the LRO phases, in accord with previous work on the Sp(N) model[@subir92].
= 5.5cm
The zero-temperature phase diagram is a function of two variables: $J_2/J_1$ and $\kappa$. The various saddle points can be classified in several ways. Both SRO and LRO phases may be characterized in terms of an ordering wavevector $\vec{q}$ via the relation $\vec{q} = 2 \vec{k}_{min}$, where $\vec{k}_{min}$ is the wavevector at which the bosonic spinon energy spectrum has a minimum. The spin structure factor $S(\bf{q})$ peaks at this wavevector[@subir92]. LRO is signaled by non-zero spin condensate $x_{\bf{i} \sigma}$, which we assume occurs at only one wavevector, $\vec{k}_{min}$; that is, $x_{\bf{j} \uparrow} = x \exp(i \vec{k}_{min} \cdot \vec{r}_{\bf j})$ and $x_{\bf{j} \downarrow} = -i x \exp(-i \vec{k}_{min} \cdot \vec{r}_{\bf j})$. The phases may be further classified[@subir91] according to the particular value of $\vec{q}$. There can be commensurate collinear ordering tendencies where the spins rotate with a period that is commensurate with the underlining lattice. Alternatively, there may be incommensurate coplanar ordering tendencies where the spins rotate in a two dimensional spin space with a period that is not commensurate with the lattice.
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The phase diagram is shown in figure \[spnphase\]. Note that the general shape of the phase diagram is qualitatively consistent with the finding of spin wave calculations[@merino; @trumper] that quantum fluctuations are largest for $J_2 / J_1 \sim 0.5$ and large $J_2/J_1$. For large enough values of $\kappa$, the ground state has magnetic LRO. As the magnetic LRO phases break Sp(N) symmetry, there are gapless Goldstone spin-wave modes. As a check on the calculation, we note that in the $\kappa \rightarrow \infty$ limit there is a transition between the Néel ordered and incommensurate $(q, q)$ LRO phase at $J_2 / J_1 = 0.5$ in agreement with the classical large-spin limit. At smaller values of $\kappa$ there are quantum disordered phases that preserve global Sp(N) symmetry. In the $N \rightarrow \infty$ limit these are rather featureless spin liquids with gapped, free, spin-1/2 bosonic excitations (spinons). Finite-N fluctuations, however, induce qualitative changes to the commensurate SRO phases (see below). In the limiting case of the nearest-neighbor square lattice, $J_2 = 0$, we reproduce the previously obtained result[@subir91] that Néel order arises for $\kappa > 0.4$. In the opposite limit of decoupled one dimensional spin chains, $J_1 = 0$, the ground state is in a disordered phase at all values of $\kappa$. There are five phases in all, three commensurate and two incommensurate, as detailed in the following two subsections.
### Commensurate phases
At small to moderate $J_2/J_1$ there are two phases with $Q_{1x} = Q_{1y} \neq 0$, and $Q_2 = 0.$ The eigenspectrum $\omega_{\bf k}$ has its minimum at ${\bf k}= \pm(\pi/2, \pi/2)$, with the implication that the spin-spin correlation function peaks at $\vec{q} = (\pi, \pi)$. Néel LRO with $x_{\bf{i} \sigma} \neq 0$ appears when $\kappa$ is sufficiently large. The boundary between LRO and SRO phases is independent of $J_2/J_1$ except at one end of the boundary, but this is expected to be an artifact of the large-N limit[@subir92]. Finite-N corrections should bend this horizontal phase boundary.
At large $J_2/J_1$ and small $\kappa$ the ground state is a disordered state characterized by $Q_{1x} = Q_{1y} = 0$ and $Q_2 \neq 0$. The chains decouple from one another exactly in the large-N limit, but at any finite-N the chains will be coupled by the fluctuations about the saddle point. Also, $x_{\bf{i} \sigma} = 0$ as it must, by the Mermin-Wagner theorem. The Sp(N) solution does not properly describe the physics of completely decoupled spin-$1/2$ chains. All spin excitations are gapped, and there is dimerization at finite-N (see below). This behavior is in marked contrast to the gapless, undimerized ground state of the physical SU(2) spin-$1/2$ nearest-neighbor Heisenberg chain[@auerbach].
### Incommensurate phases
At intermediate values of $J_2/J_1$, there are two incommensurate phases with $Q_{1x} = Q_{1y} \neq 0$, and $Q_2 \neq 0$. The ordering wavevector $\vec{q} = (q, q)$ with $q$ varying continuously from $\pi$ to $\pi/2$, a sign of helical spin order in a given plane. The inverse $\kappa_c^{-1}$ of the critical $\kappa$ separating LRO from SRO peaks near the isotropic triangular point $J_2/J_1 = 1$, where it agrees in value to that reported in a prior study of the triangular lattice[@subir92], and then decreases with a long tail as $J_2/J_1$ increases. As the one-dimensional limit of decoupled chains is approached, ${\kappa}_c \rightarrow \infty$. Again this accords with the Mermin-Wagner theorem.
All the phase transitions are continuous except for the transitions between the $(\pi, \pi)$ LRO phase and the $(q, q)$ SRO phase which is first order. Fluctuations at finite-N, however, modify the mean-field results[@subir91]. The modifications are only quantitative for the LRO phases, and for the incommensurate SRO phase. In particular, instantons in the U(1) gauge field have little effect in the incommensurate SRO phase which is characterized by non-zero $Q_2$. The $Q_2$ fields carry charge $\pm 2$, so when they acquire a non-zero expectation value, this is equivalent to the condensation of a charge-2 Higgs field. Fradkin and Shenker[@fradkin] showed some time ago that a Higgs condensate in 2+1 dimensions quenches the confining U(1) gauge force. Singly-charged spinons are therefore deconfined, instead the instantons which carry magnetic flux are confined, and no dimerization is induced[@subir91]. The relevant non-linear sigma model which describes the transition from an ordered incommensurate phase to a quantum disordered phase has been studied by Chubukov, Sachdev, and Senthil[@chubukov].
In the case of the commensurate $(\pi, \pi)$ SRO and the decoupled chain phases, instantons [*do*]{} alter the states qualitatively. The Berry’s phases associated with the instantons lead to columnar spin-Peierls order[@subir90; @subir91] (equivalent to dimer order) as indicated in figure \[spnphase\]. The dimerization pattern induced in the decoupled chain phase is similar to that found by White and Affleck[@white] for a pair of chains with zigzag coupling. Furthermore, spinon excitations are confined into pairs by the U(1) gauge force.
Note that the deconfined spinons in the $(q, q)$ SRO phase are qualitatively different than the spinons found in the limit of completely decoupled chains, $J_1 = 0$. The $(q, q)$ SRO phase exhibits true 2-dimensional fractionalization, in contrast to the decoupled one-dimensional chains. The spinons are massive, again in contrast to those found in one-dimensional half-odd-integer spin chains. The transition from the dimerized chain phase at small $J_1/J_2$, which has confined spinons, to the deconfined incommensurate phase at larger $J_1/J_2$ is described by a 2+1 dimensional Z$_2$ gauge theory[@subir00]. In fact the $(q, q)$ SRO phase is similar to a resonating valence bond (RVB) state recently found on the isotropic triangular lattice quantum dimer model[@moessner]. The phase has “topological” order; consequently when the lattice is placed on a torus (that is, when periodic boundary conditions are imposed), the ground state becomes four-fold degenerate in the thermodynamic limit[@topological; @cms].
Physical Spin-$1/2$ Limit
-------------------------
It is interesting to examine in more detail the physical spin-$1/2$ limit corresponding to $\kappa = 1$. In figure \[spinhalf\] we plot the ordering wavevector $q$ as a function of the ratio $J_2/(J_1 + J_2)$. Note that quantum fluctuations cause (i) the Néel phase to be stable for larger values of $J_2/J_1$ than classically, and (ii) the wave vector associated with the incommensurate phases deviates from the classical value. Similar behavior was also found in studies based on a series expansion[@zhang], and slave bosons including fluctuations about the saddle point[@ceccatto].
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Commensurate $q = \pi$ Néel order persists up to $J_2/(J_1+J_2) = 0.369$, and this is followed by the incommensurate LRO phase for $0.369 < J_2/(J_1+J_2) < 0.880$. Then a tiny sliver of the incommensurate SRO phase arises for $0.880 < J_2/(J_1+J_2) < 0.886$. Finally there is the decoupled chain phase for $0.886 < J_2/(J_1+J_2) \leq 1$. A strikingly similar phase diagram has been obtained by the series expansion method[@zhang]. A comparison between the Sp(N) and series expansion results is shown in figure \[spincom\]. Both methods suggest that there exists a narrow SRO region between Néel and incommensurate ordered phases, though in the Sp(N) case the region does not extend all the way up to $\kappa = 1$. Possibly finite-$N$ corrections to this large-N result could change the phase diagram quantitatively such that the $(q, q)$ SRO phase persists up to $\kappa = 1$ in agreement with the series expansion results. Another difference is that the Sp(N) result shows no dimerization in this narrow SRO region (due to the above-mentioned Higgs mechanism), while the series expansion indicates possible dimer order.
= 9cm
Similar results have also been obtained in a weak-coupling renormalization group (RG) treatment of the Hubbard model on the anisotropic triangular lattice[@tsai]. For the special case of a pure square lattice (with next-nearest-neighbor hopping $t_2 = 0$) at half-filling, the antiferromagnetic (AF) couplings diverge much faster during the RG transformations than couplings in the BCS sector, indicating a tendency towards magnetic LRO. As $t_2$ is turned on, BCS and AF instabilities begin to compete. For sufficiently large $t_2$, a crossover occurs to a $d_{x^2-y^2}$ BCS instability, suggesting that the system is now in a magnetic SRO state. Further increasing $t_2$ to reach the isotropic triangular lattice ($t_1 = t_2$) there are indications that long-range AF order re-enters. Finally, for $t_2 \gg t_1$ LRO tendencies are again eliminated, this time by the strong one-dimensional fluctuations. It is remarkable that the same sequence of ordering and disordering tendencies – LRO to SRO to LRO to SRO – occurs in all three approximate solutions.
Application To Materials
------------------------
The region of the phase diagram intermediate between the square lattice and the isotropic triangular lattice is relevant to the insulating phase of the organic $\kappa$-(BEDT-TTF)$_2$X materials. The expected range in the spin exchange interaction[@mckenzie] is $J_2/J_1 \sim 0.3$ to $1$. Depending on the precise ratio $J_2/J_1$, our phase diagram indicates that these materials could be in the Néel ordered phase, the $(q, q)$ LRO phase, or possibly the paramagnetic $(q, q)$ SRO phase (see above). In fact antiferromagnetic ordering with a magnetic moment of $0.4$ to $1 \times \mu_B$ per dimer has been seen in the splitting of proton NMR lines in the $\kappa$-(BEDT-TTF)$_2$Cu\[N(CN)$_2$\]Cl compound[@kanoda]. It is conceivable that a quantum phase transition from the Néel ordered phase to the paramagnetic $(q, q)$ SRO phase or to the $(q, q)$ LRO phase can be induced by changing the anion X.
Coldea [*et al.*]{}[@coldea] have recently performed a comprehensive neutron scattering study of Cs$_2$CuCl$_4$. They suggest that this material is described by the spin-1/2 version of our model with $J_2/J_1 = 2.5$ and $J_2 = 0.37$ meV. The measured incommensurability with respect to Néel order, $\pi - q$, is reduced below the classical value by a factor $0.47$, consistent with series results[@zhang] (our Sp(N) solution shows a smaller reduction). Maps of the excitation spectra show that the observed dispersion is renormalized upward in energy by a factor of $1.67$, which can be compared to theoretical values of $1.18$ for the square lattice and $1.57$ for decoupled chains. Furthermore, the dynamical structure factor $S(\vec{q}, \omega)$ does not exhibit well defined peaks corresponding to well defined spin-1 magnon excitations. Instead there is a continuum of excitations similar to those expected and seen in completely decoupled spin-$1/2$ Heisenberg chains. In the case of a chain these excitations can be interpreted as deconfined gapless spin-1/2 spinons.
In the relevant parameter regime, $J_2/J_1$ = 2.5, the large-N Sp(N) phase diagram predicts that the ground state is spin-ordered with an incommensurate wavevector $(q, q)$. However, as noted above, finite-N corrections could move the phase boundaries so that the physical spin-1/2 limit is in fact described by the $(q, q)$ SRO phase. In this phase there is a non-zero gap to the lowest lying excitations \[which occur at wave vector $(k_{min}, k_{min}) = (q/2, q/2)$\] rather than gapless excitations. At $\kappa = 0.56$ and $J_2/J_1 = 2.5$, in the large-N limit, the system is in the $(q, q)$ SRO phase of figure \[spnphase\] and the spinon gap is approximately $0.05 J_2$. This is much smaller than the resolution of the experiment \[see Figure 2(a) in reference [@coldea]\], in which excitation energies have only been measured down to about 0.5 $J_2$. The corresponding spinon dispersion is shown in figure \[spinonSRO\]. Again we stress that the spinons in the incommensurate SRO phase are qualitatively different than those which arise in a one-dimensional chain.
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In the deconfined $(q, q)$ SRO phase, there are no true spinwave excitations, as spin rotational symmetry is unbroken. Nevertheless, when the gap to create a spinon is small, the spinwave description remains useful. For example, in neutron scattering experiments, spinons are created in pairs, as each spinon carries spin-$1/2$. So a spinwave may be viewed as an excitation composed of two spinons, though of course this spinwave does not exhibit the sharp spectral features of a true Goldstone mode. At large-N the spinons do not interact; finite-N fluctuations will lead to corrections in the combined energy and to a finite lifetime. The minimum energy of such a spinwave of momentum $\vec{q}$ is given, in the large-N limit, by: $$E_{\vec{q}} = {\rm Min}\{\omega(\vec{q}/2 + \vec{p}/2) + \omega(\vec{q}/2 - \vec{p}/2)\}
\label{twospinon}$$ where the minimization is with respect to all possible relative momenta $\vec{p}$. The resulting spinwave dispersion is plotted in figure \[spinwaveSRO\] alongside the classical result, scaled to the same value of the spin. The Sp(N) calculation shows a rather large upward renormalization in the energy scale compared to the classical calculation; this is a result of the quantum fluctuations which are retained in the large-N limit.
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In the incommensurate $(q, q)$ LRO phase ($\kappa = 1$ and $N \rightarrow \infty$) the spinon spectrum has gapless excitations, as shown in figure \[spinon\]. Apart from the absence of the small gap, the spinwave dispersion in the ordered phase is similar to the minimum energy spectrum in the disordered phase. Again there is an upward renormalization of the energy scale as shown in figure \[spinwave\], though the ratio is relatively smaller than in the more quantum $\kappa = 0.56$ case. The size of the renormalization is in good agreement with that seen in the Cs$_2$CuCl$_4$ experiment[@coldea]. It is important to note that finite-N gauge fluctuations bind spinons in the LRO phase into true spinwave excitations, with corresponding sharp spectral features. In contrast, as noted above, spectral weight is smeared out in the SRO phase. A large spread of spectral weight is seen in the neutron scattering experiments[@coldea]. But as the incommensurate SRO and LRO states are separated by a continuous phase transition (see figure \[spnphase\]), in the vicinity of the phase boundary it is difficult to distinguish the two types of excitation spectra. The spin moment in the LRO phase is small there, as is the gap in the SRO phase. Further experiments on Cs$_2$CuCl$_4$ may be needed to determine which of the two phases is actually realized in the material.
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SU(N) Hubbard-Heisenberg Model {#sec:sun}
==============================
We now turn our attention to the charge degrees of freedom by studying the hybrid Hubbard-Heisenberg model. This model should provide a reasonable description of the layered organic materials because the Hubbard interaction is comparable in size to the hopping matrix elements, $U \approx t$. Thus the stringent no-double-occupancy constraint of the popular $t-J$ model should be relaxed. In the large-N limit it is also better to work with the hybrid Hubbard-Heisenberg model than with the pure Hubbard model because the crucial spin-exchange processes are retained in the large-N limit[@marston]. In the Hubbard model these are only of order $1/N$ and therefore vanish in the mean-field description.
As there are now both charge and spin degrees of freedom, it is no longer possible to employ purely bosonic variables, in contrast to the previous section. Instead we use antisymmetric representations of the group SU(N) as the large-N generalization of the physical SU(2) system. As shown below, this generalization precludes the possibility of describing magnetic LRO or superconducting phases, at least in the large-N limit. However, as the same representation is placed on each lattice site, this large-N generalization works equally well for bipartite and non-bipartite lattices. It has been applied to the spin-1/2 Heisenberg antiferromagnet on the kagomé lattice[@zeng].
Brief Review of the Approach
----------------------------
The Hubbard-Heisenberg Hamiltonian on the anisotropic triangular lattice is specified by equation \[HSU2\]. The generalized SU(N) version is obtained[@marston] by simply letting the spin index $\sigma$ in equation \[HSU2\] run from 1 to N (where N is even): $$\begin{aligned}
H &=&
\sum_{<{\bf{ij}}>}[-t_1 (c^{\dagger\sigma}_{\bf{i}} c_{{\bf{j}}\sigma}+ H.c.) -
\frac{J_1}{N}(c^{\dagger\alpha}_{\bf{i}}
c_{{\bf{j}}\alpha}c^{\dagger\beta}_{\bf{j}}c_{{\bf{i}}\beta}+
\frac{1}{2} n_{\bf{i}}n_{\bf{j}})]
\nonumber \\
&+&
\sum_{<<{\bf{ij}}>>}[-t_2 (c^{\dagger\sigma}_{\bf{i}}
c_{{\bf{j}}\sigma} + H.c.) - \frac{J_2}{N}(c^{\dagger\alpha}_{\bf{i}}
c_{{\bf{j}}\alpha}c^{\dagger\beta}_{\bf{j}}c_{{\bf{i}}\beta}+
\frac{1}{2} n_{\bf{i}}n_{\bf{j}})]
\nonumber \\
&+&
\frac{U}{N}\sum_{\bf{i}}(c^{\dagger\sigma}_{\bf{i}}c_{{\bf{i}}\sigma} - N/2)^2\
\label{HSUN}\end{aligned}$$ where all spin indices are summed over. Here we have also rescaled the interaction strengths $J_i/2 \rightarrow J_i/N$ and $U/2 \rightarrow U/N$ to make each of the terms in the Hamiltonian of order N. At half-filling, the only case we consider here, a further simplification occurs as the term $J_{\bf{ij}}n_{\bf{i}}n_{\bf{j}}$ is simply a constant. There is no possibility of phase separation into hole-rich and hole-poor regions, nor can stripes form[@matthias], as the system is at half-filling. We drop this constant term in the following analysis.
We could also include a biquadratic spin-spin interaction of the form $\tilde{J} (\vec{S}_{\bf i} \cdot \vec{S}_{\bf j})^2$ with $\tilde{J} > 0$. In the physical SU(2) limit this term does nothing except renormalize the strength of the usual bilinear spin-spin interaction. But for $N > 2$ it suppresses dimerization[@marston], as the concentration of singlet correlations on isolated bonds is particularly costly when the biquadratic term is included. Thus there exists a family of large-N theories parameterized by the dimensionless ratio $\tilde{J}/J$, each of which has the same physical SU(2) limit. In this paper, however, we set $\tilde{J} = 0$ as we find that the phase most likely to describe the metallic regime of the organic superconductors is a uniform phase with no dimerization which is stable even at $\tilde{J} = 0$.
After passing to the functional-integral formulation in terms of Grassman fields, the quartic interactions are decoupled by a Hubbard-Stratonovich transformation which introduces real-valued auxiliary fields $\phi$ on each site and complex-valued $\chi$ fields directed along each bond: $$\begin{aligned}
\phi_{\bf{i}} =
{\it i} \frac{U}{N} (c^{* \sigma}_{\bf{i}} c_{{\bf{i}}\sigma} - N/2),\quad
\chi_{\bf{i}\bf{j}} =
\frac{J_{\bf{i}\bf{j}}}{N}c^{* \sigma}_{\bf{i}} c_{{\bf{j}}\sigma}\ .
\label{phichi}\end{aligned}$$ Clearly $\langle \phi_{\bf{i}} \rangle$ is proportional to the local charge density relative to half-filling (corresponding to $N/2$ fermions at each site) and $\chi_{\bf{i}\bf{j}}$ may be viewed as an effective hopping amplitude for the fermions between site $\bf{i}$ and $\bf{j}$. The effective action in terms of these auxiliary fields, which may be viewed as order parameters, can now be obtained by integrating out the fermions. We note that, unlike the bosonic formulation of the pure antiferromagnet, here there is no possibility of magnetic LRO as the order parameters $\chi$ and $\phi$ are global SU(N) invariants, and of course there is no possibility of Bose condensates in the fermionic antisymmetric representation of SU(N). Superconductivity likewise is not possible in the large-N limit because the order parameters are invariant under global U(1) charge symmetry rotations. In the Heisenberg limit $t\to 0$ the action is also invariant under local U(1) gauge transformations as long as the $\chi$- and $\phi$-fields transform as gauge fields: $\chi_{\bf ij}(\tau) \to e^{[i\theta_{\bf i}(\tau) -
\theta_{\bf j}(\tau)]} \chi_{\bf ij}(\tau)$, $\phi_{\bf i}(\tau ) \to \phi_{\bf i}(\tau ) -
d\theta_{\bf i}(\tau)/d\tau$. For the more general case of Hubbard-Heisenberg model, this local U(1) gauge symmetry breaks to only global U(1) gauge symmetry reflecting the conservation of total charge.
Since $S_{eff}$ has an overall factor of N, the saddle-point approximation is exact at $N \to \infty$. We expect $\phi$ and $\chi$ to be time-independent at the saddle point, so $S_{eff}$ can be written in terms of the free energy of fermions moving in a static order-parameter background: $$\begin{aligned}
S_{eff}[\phi, \chi] &=& \beta F[\phi, \chi; \mu]\ ,
\nonumber \\
{{F[\phi, \chi; \mu]}\over{N}} &=&
\sum_{<\bf{ij}>} \frac{|\chi_{\bf ij}|^2}{J_1}
+ \sum_{<<\bf{ij}>>}\frac{|\chi_{\bf ij}|^2}{J_2}
+ \sum_{\bf{i}} [\frac{1}{4U} \phi_{\bf{i}}^2-\frac{i}{2}\phi_{\bf{i}}]
\nonumber \\
&-&
\frac{1}{\beta}\sum_{\bf k} \ln \{1 + \exp[-\beta(\omega_{\bf k} - \mu)]\}\ .
\label{sunfree}\end{aligned}$$ Here the $\omega_{\bf k}$ are the eigenenergies of the mean-field Hamiltonian $H_{MF}$: $$\begin{aligned}
H_{MF} = \sum_{<\bf{ij}>}[(-t_1 + \chi_{\bf{ij}}) c^{\dagger}_{\bf{i}} c_{\bf{j}} + H.c.]
+ \sum_{<<\bf{ij}>>}[(-t_2 + \chi_{\bf{ij}}) c^{\dagger}_{\bf{i}} c_{\bf{j}} + H.c.]
+ i \sum_{\bf{i}}\phi_{\bf{i}}c^{\dagger}_{\bf{i}}c_{\bf{i}}.\end{aligned}$$ In the zero-temperature $\beta \rightarrow \infty$ limit the fermionic contribution to the free energy reduces to a sum over the occupied energy eigenvalues. The saddle point solution is found by minimizing the free energy with respect to $\chi$ fields, and maximizing it with respect to the $\phi$ fields. We carry out the minimization numerically via the simplex-annealing method[@NRC] on lattices with up to $40 \times 40$ sites.
After ascertaining the zero-temperature phase diagram we then study the effects of non-zero temperature. As the temperature is raised, $\beta \rightarrow 0$ and the last term in equation \[sunfree\] approaches $k_B \ln2$ per site reflecting the fact that each site is half-occupied. The entropy then dominates the free energy, terms linear in $\chi_{\bf ij}$ in equation \[sunfree\] disappear, and the free energy is minimized by setting $\chi_{\bf ij} = 0$. Thus as the temperature is raised, antiferromagnetic spin correlations are weakened and then eliminated altogether.
Zero-Temperature Phase Diagram of the SU(N) Hubbard-Heisenberg Model
--------------------------------------------------------------------
We again choose a $2 \times 2$ unit cell, in anticipation that translational symmetry can be broken at the saddle points. The $2 \times 2$ unit cell requires 12 different complex $\chi$-fields and 4 different real $\phi$-fields as shown in figure \[4sitesun\]. At half-filling all $\phi_{\bf i} = 0$. As expected, there is no site-centered charge density wave, and the phase diagram does not depend on the size of the Hubbard interaction $U$ (so long as it is repulsive) because fluctuations in the on-site occupancy, which are $O(\sqrt{N}) \ll N/2$ are suppressed in the large-N limit. Therefore the saddle-point solutions may be classified solely in terms of the remaining order parameter, the $\chi$-fields. For the special case $t_1 = t_2 = 0$ it is important to classify the phases in a gauge-invariant way because there are many gauge-equivalent saddle-points. In this limit there are two important gauge-invariant quantities:
\(i) The magnitude $|\chi_{\bf ij}|^2$ which is proportional to the spin-spin correlation function $\langle \vec{S}_{\bf i} \cdot \vec{S}_{\bf j} \rangle$. Modulations in $|\chi|$ signal the presence of a bond-centered dimerization.
\(ii) The plaquette operator $\Pi \equiv \chi_{12} \chi_{23} \chi_{34}
\chi_{41}$, where 1, 2, 3, and 4 are sites on the corners of a unit plaquette. By identifying the phase of $\chi$ as a spatial gauge field it is clear that the plaquette operator is gauge-invariant, and its phase measures the amount of magnetic flux penetrating the plaquette[@marston]. Different saddle points are therefore gauge equivalent if the plaquette operator has the same expectation value, even though the $\chi$-fields may be different. In the Heisenberg limit, the flux always equals to $0$ or $\pi$ (mod $2 \pi$), so a gauge can always be found such that all the $\chi$-fields are purely real.
= 7cm
Away from the pure Heisenberg limit $t_1 = t_2 = 0$ we further classify the saddle-point solutions in terms of whether or not they break time-reversal symmetry ($\hat{T}$). Finally, as there are four independent parameters ($t_1$, $t_2$, $J_1$, and $J_2$) the phase diagram lives in a three-dimensional space of their dimensionless ratios. To reduce this to a more manageable two-dimensional section, we assume that $J_1/J_2 = (t_1/t_2)^2$; then by varying $J_1$ and the two hopping matrix elements we explore a two-dimensional space. The resulting phase diagram is shown in figure \[sunphase\]. We summarize the phases which appear in the diagram in the following subsections.
= 12cm
### One-Dimensional Dimer Phase
This phase exists in the region $J_2 > J_1 > t_1$ and it exhibits spin-Peierls (= dimer) order. All $\chi_2$-fields are negative real numbers with $\chi_2^1 = \chi_2^3$, $\chi_2^2 = \chi_2^4$, and $|\chi_2^1| > |\chi_2^2|$. Also, all $\chi_1$-fields are either small negative real numbers or zero with $\chi_1^1 = \chi_1^3 = \chi_1^5 = \chi_1^7 < 0$, $\chi_1^2 = \chi_1^4 = \chi_1^6 = \chi_1^8 = 0$ and $|\chi_1^1| \ll |\chi_2^2|$. See figure \[dimer1d\] for a sketch. The system breaks up into nearly decoupled dimerized spin chains. This phase breaks preserves $\hat{T}$-symmetry as all the $\chi$-fields are real. It is insulating as there is a large gap in the energy spectrum at the Fermi energy. The phase is very similar to the decoupled chain phase of the insulating Sp(N) model; in fact the dimerization pattern is identical to one of two such possible patterns in the Sp(N) model (see figure \[spnphase\]) and echos the pattern found by White and Affleck for the two-chain zigzag model[@white]. In the extreme one-dimensional limit of $J_1 = t_1 = 0$ the SU(N) solution, like the Sp(N) solution, fails[@marston] to accurately described the physics of decoupled chains, for at half-filling, the physical one-dimensional SU(2) Hubbard model is neither dimerized nor spin-gapped. The inclusion of the biquadratic interaction suppresses dimerization[@matt] and yields a state qualitatively similar to the exact solution of the physical system, but for simplicity we do not consider such a term here.
= 6.0cm
### Box (also called “Plaquette”) Phase
This phase is also insulating and consists of isolated plaquettes with enhanced spin-spin correlations[@dombre; @cms]. See figure \[box\] for a sketch. All $\chi_1$-fields are complex with $|\chi_1^1| = |\chi_1^3| > |\chi_1^2| = |\chi_1^4| >
|\chi_1^6| = |\chi_1^8| > |\chi_1^5| =|\chi_1^7|$. The $\chi_2$ fields are small, $|\chi_2^1| = |\chi_2^2| = |\chi_2^3| = |\chi_2^4| \ll |\chi_1^5|$. The phase $\theta$ of the plaquette product $\chi_1^1 \chi_1^2 \chi_1^3 \chi_1^4$ is neither 0 nor $\pi$ in general. The box phase breaks $\hat{T}$-symmetry when $t_1 \neq 0$. As time-reversal symmetry is broken, there are real orbital currents circulating around the plaquettes[@ted] as shown in the figure. Apart from $\hat{T}$-breaking, this phase is rather similar to the $(\pi, \pi)$ SRO phase of the Sp(N) model as it is a commensurate SRO phase with a large spin gap.
= 4.5cm
### Staggered Flux Phase (SFP)
All $\chi_1$-fields are equal, with an imaginary component, and the $\chi_2$-fields are equal, real, and much smaller in magnitude than the $\chi_1$ fields. The phase of the plaquette operator differs in general from 0 or $\pi$; see figure \[flux\] for a sketch. The staggered flux phase breaks $\hat{T}$-symmetry. Like the box phase, there are real orbital currents circulating around the plaquettes[@ted], in an alternating antiferromagnetic pattern. The SFP is semi-metallic as the density of states (DOS) is small, and in fact vanishes linearly at the Fermi energy at $J_2 = 0$. Apart from $\hat{T}$-breaking this phase is rather similar to the $(\pi, \pi)$ LRO phase of the Sp(N) antiferromagnet, as the spins show quasi-long-range order with power-law decay in the spin-spin correlation function. In the limiting case of a pure Heisenberg AF on the nearest-neighbor square lattice, $t_1 = t_2 = J_2 = 0$, gauge fluctuations at sufficiently small-N are expected to drive the SFP (which can be stabilized against the box phase by the addition of the biquadratic interaction) into a $(\pi, \pi)$ Néel-ordered state[@gauge].
= 4.5cm
### Uniform Phase
All $\chi$-fields are negative real numbers and $\chi_1^1 = \chi_1^2 = \cdots = \chi_1^8$, $\chi_2^1 = \chi_2^2 = \chi_2^3 = \chi_2^4$ with $\chi_1^1 \neq \chi_2^1$ in general. See figure \[uniform\] for a sketch. This phase preserves $\hat{T}$-symmetry and since all $\chi$-fields are real, they simply renormalize hopping parameters $t_1$, $t_2$. The uniform phase is therefore a metallic Fermi liquid. Spin-spin correlations in the uniform phase decay as an inverse power law of the separation, with an incommensurate wavevector, so the uniform phase behaves similarly to that in the $(q, q)$ LRO phase of the Sp(N) model (see below).
= 4.5cm
Global Similarities Between SU(N) and Sp(N) Phase Diagrams
----------------------------------------------------------
There are some global similarities between the SU(N) phase diagram of the Hubbard-Heisenberg model (figure \[sunphase\]) and the Sp(N) phase diagram of the insulating Heisenberg antiferromagnet (figure \[spnphase\]). We have already pointed out similarities between the four phases of the SU(N) model and the decoupled chain, $(\pi, \pi)$ SRO, $(\pi, \pi)$ LRO, and $(q, q)$ LRO phases of the Sp(N) model. Apparently the dimensionless parameter $J_1/t_1$ on the vertical axis of the SU(N) phase diagram (figure \[sunphase\]) is the analog of the quantum parameter $1/\kappa$ of the Sp(N) phase diagram. The reason why these two dimensionless parameter play similar roles can be understood by considering the limit of the pure insulating antiferromagnet, corresponding to $t_1 \rightarrow 0$ and $t_2 \rightarrow 0$. In these limits, the SU(N) model can only be in the purely insulating box phase, or in the dimerized phase. The spins are always quantum disordered, and behave like the extreme quantum $\kappa \rightarrow 0$ limit of the Sp(N) model. In the opposite limit $t_1 \rightarrow \infty$ and $t_2 \rightarrow \infty$ the spin-spin correlation function decays more slowly, as an inverse power law instead of exponentially. This is as close to LRO as is possible in the large-N limit of the SU(N) model. Roughly then it corresponds to the ordered classical $\kappa \rightarrow \infty$ limit of the Sp(N) model.
Observable Properties of the SU(N) Hubbard-Heisenberg Model
-----------------------------------------------------------
We now comment on the possible relevance of our large-N solution to the electronic and magnetic properties of the organic superconductors. In reference it is estimated that $J_2/J_1 \sim 0.3$ to $1$ and $J_1/t_1 \sim 0.5$ to $2$. Figure \[sunphase\] then implies that the ground state is the uniform phase, which as noted above, is a rather ordinary Fermi liquid with no broken symmetries.
We note that in the one-dimensional limit ($t_1 = J_1 = 0$) and in the square lattice limit ($t_2 = J_2 = 0$), nesting of the Fermi surface is perfect, and the system is driven into an insulating phase no matter how large the hopping amplitudes. Away from these two extreme limits, however, there is a non-trivial metal-insulator transition line. This accords with expectations because as the Fermi surface is not perfectly nested, the metallic state is only eliminated at a nonzero value of $J_1/t_1$. Experimentally it is found that upon increasing pressure, a SIT transition from the antiferromagnetic insulator to a superconductor is seen in the $\kappa$-(BEDT-TTF)$_2$X family of materials[@kanoda; @kino]. This transition can be understood in terms of the SU(N) phase diagram as follows. As pressure increases, the effective hopping amplitudes $t_1$ and $t_2$ also increase because Coulomb blocking is less effective[@mckenzie]. The bandwidth broadens, but the spin exchange couplings $J_1$ and $J_2$ remain nearly unchanged as these are determined by the [*bare*]{}, not the effective, hopping matrix elements. Thus the ratio $J_1/t_1$ is reduced at high pressure, and the many-body correlations weaken. From the phase diagram (figure \[sunphase\]) it is apparent that the system can be driven from an insulating state into a metallic state, which presumably superconducts at sufficiently low temperature. Thus the transition to a conducting state at high pressure can be seen as being due to bandwidth broadening, as in the Brinkman-Rice picture of the MIT.
= 11cm
-------------------------- ------- ------- ------- ------- ------------- ---------------------
Symbol In Figure \[DOS\] $t_1$ $t_2$ $J_1$ $J_2$ $J_1 / t_1$ $J_2 / (J_1 + J_2)$
star 6 5.4 24 19.2 4 0.44
dot 12 10.8 24 19.2 2 0.44
triangle 24 21.6 24 19.2 1 0.44
-------------------------- ------- ------- ------- ------- ------------- ---------------------
: Parameters (in meV) for the three points in figure \[DOS\].
\[table1\]
We plot the DOS as a function of temperature in figure \[DOS\] for three points inside the uniform phase identified in table \[table1\]. The DOS decreases as the temperature decreases from room temperature down to absolute zero. Clearly the pseudogap is more prominent near the boundary between the conducting uniform phase and the semimetallic SFP and insulating box phases. The drop in the DOS could qualitatively explain the $30\%$ depression seen in the uniform susceptibility[@kanoda] (NMR experiments find a reduction of about $50\%$ as the temperature decreases from $100~ ^\circ$K to $10~ ^\circ$K[@desoto; @mayaffre]). The behavior may be understood as follows. As the temperature decreases, antiferromagnetic spin-spin correlations develop. These correlations are signaled in the mean-field theory by the link variables $\chi_{\bf ij}$ which become non-zero at low enough temperature. Electrons on neighboring sites then tend to have opposite spin, reducing Pauli-blocking, and making their kinetic energy more negative. The band widens and the density of states drops.
Discussion {#sec:discuss}
==========
We have solved the bosonic Sp(N) Heisenberg and fermionic SU(N) Hubbard-Heisenberg models on the anisotropic triangular lattice in the large-N limit. The bosonic Sp(N) representation of the Heisenberg model is useful for describing magnetic ordering transitions. It therefore may be an appropriate description of the insulating phase of the layered organic superconductors $\kappa$-(BEDT-TTF)$_2$X and of the insulating Cs$_2$CuCl$_4$ compound. The fermionic SU(N) Hubbard-Heisenberg model provides a complementary description of the charge sector, in particular the physics of metal-insulator transitions and the unconventional metallic phases. Systematic expansions in powers of $1/N$ about the large-N limit are possible for either model.
For the Sp(N) model we found five phases: (i) three commensurate phases, the $(\pi, \pi)$ LRO and SRO phases, and the decoupled chain phase and (ii) two incommensurate phases, the $(q, q)$ LRO and SRO phases. Passing from the square lattice ($J_2 = 0$) to the one-dimensional limit ($J_1 = 0$) at $\kappa = 1$ which corresponds to spin-1/2 in the physical Sp(1) limit first there is the Néel ordered phase, the incommensurate $(q, q)$ LRO phase, the incommensurate $(q, q)$ SRO phase, and finally a phase consisting of decoupled chains. These phases are similar to those obtained from a series expansion method[@zhang] and a weak coupling renormalization group technique[@tsai]. The effects of finite-N fluctuations on the saddle-point solutions were also discussed. The observed dispersion of spin excitations in the Cs$_2$CuCl$_4$ material[@coldea] can be described either as spinwaves in the incommensurate $(q, q)$ LRO phase, or in terms of pairs of spinons in the deconfined $(q, q)$ SRO phase.
The zero-temperature phase diagram of the SU(N) Hubbard-Heisenberg model in the large-N limit has a 1-D dimer phase, a box (or plaquette) phase, a staggered-flux phase, and a uniform phase. In the extreme 1-D and square lattice limits, the ground state of the half-filled Hubbard-Heisenberg model is always insulating because the nesting of the Fermi surface is perfect. But away from these two extreme limits there is a metal-insulator transition. For parameters appropriate to the $\kappa$-(BEDT-TTF)$_2$X class of materials we find that the conducting regime is described by our uniform phase, which is a rather conventional Fermi liquid with no broken symmetries. In this phase we find that the density of states at Fermi level decreases at low temperatures, due to the development of antiferromagnetic correlations. This could explain the depression seen in the uniform susceptibility of the organic superconducting materials at low temperatures. It agrees qualitatively with experiments which suggest the existence of a pseudo-gap.
We have used two models instead of one because the two large-N theories have complementary advantages and disadvantages. The bosonic Sp(N) approach is not suitable for describing electronic properties, as there are no fermions. The fermionic SU(N) approach, on the other hand, is not useful for describing magnetic ordering, as the order parameters are all SU(N) singlets. Neither mean-field approximation can describe the superconducting phase. The [*fermionic*]{} Sp(N) Hubbard-Heisenberg model does support superconducting states, but no non-superconducting metallic phases[@matthias]. Whether or not a single model can be constructed which is exact in a large-N limit and yet encompasses all the relevant phases remains an open problem.
We thank Jaime Merino, Bruce Normand, Nick Read, Shan-Wen Tsai, and Matthias Vojta for helpful discussions. We especially thank Subir Sachdev for his insightful comments about the Sp(N) phase diagram and the nature of the deconfined phases. This work was supported in part by the NSF under grant Nos. DMR-9712391 and PHY99-07949. J.B.M. thanks the Institute for Theoretical Physics at UCSB for its hospitality while he attended the ITP Program on High Temperature Superconductivity. Computational work in support of this research was carried out using double precision C++ at the Theoretical Physics Computing Facility at Brown University. Work at the University of Queensland was supported by the Australian Research Council.
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abstract: 'Using [Ly$\alpha$]{} emission line as a tracer of high redshift star forming galaxies, hundreds of [Ly$\alpha$]{} emission line galaxies (LAEs) at z > 5 have been detected. These LAEs are considered to be low mass young galaxies, critical to the re-ionization of the universe and the metal enrichment of circumgalactic medium (CGM) and intergalactic medium (IGM). It is assumed that outflows in LAEs can help ionizing photons and [Ly$\alpha$]{} photons escape out of galaxies. However we still know little about the outflows in high redshifts LAEs due to observational difficulties, especially at redshift > 5. Models of [Ly$\alpha$]{} radiative transfer predict asymmetric [Ly$\alpha$]{} line profiles with broad red wing in LAEs with outflows. Here we report a z $\sim$ 5.7 [Ly$\alpha$]{} emission line with a broad red wing extending to $>$ 1000 [km s$^{-1}$]{} relative to the peak of [Ly$\alpha$]{} line, which has been detected in only a couple of z $>$ 5 LAEs till now. If the broad red wing is ascribed to gas outflow instead of AGN activity, the outflow velocity could be larger than the escape velocity ($\sim$ 500 [km s$^{-1}$]{}) of typical halo mass of z $\sim$ 5.7 LAEs, being consistent with the picture that outflows in LAEs disperse metals to CGM and IGM.'
author:
- 'Huan Yang, JunXian Wang, Zhen-Ya Zheng, Sangeeta Malhotra, James E. Rhoads, Leopoldo Infante'
title: 'A z $\sim$ 5.7 [Ly$\alpha$]{} Emission Line with an Ultra Broad Red Wing'
---
Introduction
============
High redshift star forming galaxies can be detected by optical to near-infrared observations of rest-frame ultraviolet light. One efficient method selects Lyman break galaxies (LBGs) via the distinctive ÒstepÓ introduced into their blue ultraviolet continuum emission by neutral hydrogen absorption (Steidel et al. 1999; Shapley 2011). Another efficient method selects LAEs by their large equivalent width (EW) of [Ly$\alpha$]{} recombination line due to star formation. The standard approach to detect high redshift LAEs has been wide field narrow band imaging survey in gaps of the telluric OH-bands (Cowie & Hu 1998; Rhoads et al. 2000; Ouchi et al. 2003, 2008; Hu et al. 2004; Wang et al. 2005; Gawiser et al. 2006; Guaita et al. 2010; Tilvi et al. 2010; Nakajima et al. 2012; Cl[é]{}ment et al. 2012; Krug et al. 2012; Shibuya et al. 2012). The photometric samples could have contamination rates as large as $\sim$ 40% (e.g. Wang et al. 2009), and spectroscopy follow up is necessary to remove interlopers (Hu et al. 2004). Generally, LAEs are small, low-mass, weakly clustered, young galaxies (Gawiser et al. 2006; Kovac et al. 2007; Pirzkal et al. 2007; Malhotra et al. 2012) and considered to be a subset of LBGs with fainter continuum and smaller dust extinction (Pentericci et al. 2009; Kornei et al. 2010).
Almost every extreme star forming galaxy has gas outflows. Traced by the blue-shifted interstellar metal absorption lines in rest-frame UV or optical spectra, galactic gas outflows are common in local starburst galaxies (Heckman et al. 2000; Veilleux et al. 2005) and ubiquitous in LBGs samples at high redshifts (Pettini et al. 2001; Adelberger et al. 2003; Shapley et al. 2003; Steidel et al. 2010). It is not clear yet whether gas outflows are also prevalent in LAEs, as LAEs samples are biased to galaxies with dim stellar continuum, thus it is difficult to detect the absorption lines. Outflows in LAEs and LBGs can regulate star formation in galaxies and are assumed to help ionizing photons escape to ionize the universe and spread metal to circum-galactic medium (CGM) and intergalactic medium (IGM) (eg. Oppenheimer & Dave 2006).
Outflows are also key to [Ly$\alpha$]{} photon escape and to [Ly$\alpha$]{} emission line profiles. [Ly$\alpha$]{} is expected to be strong in star forming galaxies. However, due to the large scattering cross section of [Ly$\alpha$]{} photons by HI, [Ly$\alpha$]{} emission could be strongly altered in intensity, kinematics, and apparent spatial distribution (Charlot & Fall 1993). Outflows can help [Ly$\alpha$]{} photons escape out of local starburst galaxies (Kunth et al. 1998), resulting in P-Cygni profile of [Ly$\alpha$]{} line (Mas-Hesse et al. 2003). In z $\sim$ 2 – 3 LBGs with [Ly$\alpha$]{} emission lines, peaks of [Ly$\alpha$]{} lines are red-shifted relative to galactic systemic redshift while the low ionization interstellar metal absorption lines (tracing gas outflows) are blue-shifted, and the EWs and the velocity offsets of [Ly$\alpha$]{} lines are closely related to those of low ionization interstellar metal absorption lines (Shapley et al. 2003; Steidel et al. 2010). In z $\sim$ 4 – 6 LBGs sample, peaks of [Ly$\alpha$]{} emission lines are also red-shifted relative to the low ionization interstellar metal absorption lines (Vanzella et al. 2009; Jones et al. 2012). Profiles of [Ly$\alpha$]{} lines in LBGs are complex, varying from damped absorption to double peak emission, which can be reproduced by models of [Ly$\alpha$]{} radiative transfer in outflowing gas (Verhamme et al. 2006; Tapken et al. 2007). In z $\sim$ 1.8 – 4.5 gamma-ray burst (GRB) host galaxies with [Ly$\alpha$]{} emission, the velocity centroid of the [Ly$\alpha$]{} lines are also redshifted with respect to the galactic systemic velocity, similar to what is seen for LBGs (Milvang-Jensen et al. 2012).
Consistently, in a few z $\sim$ 2 – 3 LAEs with rest frame optical emission lines (such as \[OIII\] and H$\alpha$) detected, [Ly$\alpha$]{} lines are redshifted by $\sim$ 100 – 300 [km s$^{-1}$]{} relative to optical lines (McLinden et al. 2011; Finkelstein et al. 2011; Hashimoto et al. 2013). Furthermore, the composite spectra of eight z $\sim$ 2 – 3 LAEs (Hashimoto et al. 2013) show blue-shifted interstellar absorption lines relative to optical H$\alpha$ lines. These studies all suggest existence of outflows in LAEs.
However, in most LAEs the [Ly$\alpha$]{} emission line is the only detectable feature in spectroscopic observations. Since the [Ly$\alpha$]{} emission lines are luminous and their profiles strongly depend on the gas and dust distribution and kinematics, [Ly$\alpha$]{} lines profiles alone can also be used to trace the interstellar medium properties. Hundreds of LAEs at z > 5 have been spectroscopically confirmed by asymmetric [Ly$\alpha$]{} line profiles that show a sharp blue cutoff (Dawson et al. 2004; Kashikawa et al. 2006, 2011; Sawicki et al. 2008; Hu et al. 2010; Ouchi et al. 2010). The asymmetric profile may be caused by absorption and scattering of photons bluer than [Ly$\alpha$]{} line center by IGM at lower redshift. However, a few z > 5 LAEs show [Ly$\alpha$]{} emission lines with very broad red wing, which is ascribed to scattering of [Ly$\alpha$]{} photons from a shell of gas outflows driven by a powerful starburst (Dawson et al. 2002; Ajiki et al. 2002; Westra et al. 2005). Models of [Ly$\alpha$]{} radiative transfer in galaxies with gas outflows (eg. Verhamme et al. 2006) also predict asymmetric [Ly$\alpha$]{} profiles with broad red wings. We note that in some z > 5 LBGs with [Ly$\alpha$]{} emission lines detected, the [Ly$\alpha$]{} line profiles also show very broad red wings (eg. Vanzella et al. 2010; Curtis-Lake et al. 2012).
In this work we report a z = 5.7 [Ly$\alpha$]{} emission line with a broad red wing located at RA(J2000) = 03:35:39.335, DEC(J2000) = -27:53:15.99 (hereafter J0335) discovered in spectroscopic follow-up of narrow-band selected LAE candidates near the CDF-S field. We discuss the implications of the broad red wing for outflow in LAEs and the potential of using [Ly$\alpha$]{} as a tracer of galactic outflow at z >5. We adopt a cosmology with $H_{0} =70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_{m} = 0.3$, and $\Omega_{\Lambda} = 0.7$.
Observation and Spectra Analysis Results
========================================
J0335 was selected as a candidate z $\sim$ 5.7 LAE in a deep narrow band imaging survey in a field next to the Chandra Deep Field South (CDF-S). We obtained deep NB823 narrowband images using the Mosaic II CCD imager at the Cerro Tololo Inter-American Observatory (CTIO) 4 m V. M. Blanco telescope on 2005 September 9–11 (UT). The narrowband filter NB823 has a central wavelength $\lambda_c$ of 823 nm, and a FWHM transmission of 7.5 nm. The broadband B, V and I images used for candidate LAE seletion are from ESO Image Survey[^1] in the same field. The optical thumbnail images of J0335 are given in Fig. 1. The 5 $\sigma$ limiting Vega magnitudes in a 2.5" aperture of the B, V, I and NB823 images are 25.92, 25.15, 23.83, and 23.84 mag respectively. The galaxy J0335 is only detected in NB823 with a narrow band flux of 3.2$\pm$0.6 $\times$ 10$^{-17}$ [erg s$^{-1}$ cm$^{-2}$ ]{}. As the source is non-detected in the underlying broadband (I band), we use 1$\sigma$ (2$\sigma$) upper limit of I band flux to estimate the lower limit to equivalent width (EW) of the line. Assuming the line is [Ly$\alpha$]{} line at z $\sim$ 5.7 and simply following Malhotra & Rhoads (2002)[^2], we obtained a lower limit to the rest frame line EW of $>$ 106 Å ($>$ 48 Å).
The spectroscopic observations of J0335 were taken with the Inamori-Magellan Areal Camera & Spectrograph (IMACS; Dressler et al. 2006) in multi-slit mode on Magellan I Baade telescope on 2012 October 11. The slit width was 1.0$\arcsec$ and the typical seeing during the observation was between 0.5 and 0.7. The exposures were taken with CTIO-I band filter and 300-l/mm grism with a blaze angle of 26.7 degree, resulting in spectra coverage of 7000Å-9000Å and a dispersion of 1.25 Å pixel$^{-1}$. A total of 9000s exposure was obtained.
The science frames were bias-subtracted, internal flat corrected and wavelength calibrated with the IMACS data reduction package COSMOS[^3]. Sky subtractions were performed following the two-dimensional spectroscopy background subtraction method (Kelson 2003). We combined the clean 2-d spectra from each individual exposure, and extracted the 1-d spectrum by summing up counts in a 0.8 apertures of the 2-d spectra. Flux calibration was done using the spectroscopic standard star LTT1788.
The emission line of J0335 clearly showed an asymmetric profile with a broad red wing in both the 2-d and 1-d spectra (Fig.2). The asymmetric line profile and non-detections of other emission lines confirm the line is in fact [Ly$\alpha$]{} emission, but not [\[O[II]{}\]$\lambda$3727]{}, [\[O[III]{}\]$\lambda$5007]{} or [H$\alpha$]{}. In particular, the \[O[II]{}\]$\lambda$$\lambda$3726, 3729 doublet would be resolved at this wavelength with a spectral resolution of $\sim$ 6Å.
The observed line flux is (1.71$\pm$0.04)$\times$10$^{-17}$ [erg s$^{-1}$ cm$^{-2}$ ]{}(measured in a larger 1.6 aperture to avoid aperture loss; the error here represents only statistical uncertainties in the spectrum). The line luminosity is (6.13$\pm$0.17)$\times$10$^{42}$ erg s$^{-1}$, close to $L^*\ $(7-10$\times$10$^{42}$ erg s$^{-1}$) of the [Ly$\alpha$]{} luminosity function at z $\sim$ 5.7 (Ouchi et al. 2008; Hu et al. 2010; Kashikawa et al. 2011). To estimate spectroscopic line EW, we fit the continuum redward of the [Ly$\alpha$]{} line (8221.2 - 8800.8 Å in the observed frame) with a constant, and obtain a continuum flux of 0.021$\pm$0.005 $\times$ 10$^{-18}$ [erg s$^{-1}$ cm$^{-2}$ Å$^{-1}$ ]{}and a rest frame [Ly$\alpha$]{} line EW of 121$\pm$29 Å (without correction to IGM absorption to [Ly$\alpha$]{} line).
The [Ly$\alpha$]{} profile showed a marginally resolved narrow line core and a broad red wing. The commonly adopted approach to fit high z [Ly$\alpha$]{} profiles is to use a red continuum plus a half-gaussian profile (setting both components red ward of the line center to zero) convolved with instrument profile. For J0335 this approach can fit the narrow line core, but ignores the broad red wing. Therefore we add an extra broad Gaussian component to the fitting (Fig.2). The resulting narrow line core is centered at 8180.5$\pm$0.09 Å, with FWHM(narrow) = 6.6$\pm$0.3 Å. If we neglect the possible offset of [Ly$\alpha$]{} line to galaxy systemic redshift, the narrow line center wavelength gives a redshift of 5.7274$\pm$0.0001. As the instrument profile resolution is FWHM(instru) $\approx$ 5.9 Å, estimated by fitting sky line with Gaussian profile, the intrinsic FWHM of the narrow component is $\sim$ 3.0$^{+0.6}_{-0.8}$ Å (108 [km s$^{-1}$]{}). Since the seeing during the observation (0.5– 0.7) was smaller than the slit width (1.0), and considering that high redshift LAEs are usually compact in continuum (Malhotra et al. 2012) and in [Ly$\alpha$]{} emission (Bond et al. 2010), our spectral resolution may be better than 5.9 Å, and the intrinsic FWHM of the narrow [Ly$\alpha$]{} core could be $\sim$ 5 – 6 Å (180 – 220 [km s$^{-1}$]{}). The broad component (with signal-to-noise ratio S/N = 5.0) is centered at 8188.4$\pm$2.1 Å with FWHM(broad) = 23$\pm$4 Å (852 [km s$^{-1}$]{}). Its velocity offset relative to the narrow component is 290$\pm$77 [km s$^{-1}$]{}.
Discussion
==========
AGN or Starburst
----------------
Both AGN and starburst can ionize the surrounding gas and generate [Ly$\alpha$]{} emission. Assuming z = 5.7274, the expected [N[V]{}$\lambda$1240]{} line from an AGN is sitting on the edge of a strong sky line, and is non-detected. We can only obtain a loose 3$\sigma$ upper limit to [N[V]{}$\lambda$1240]{}/[Ly$\alpha$]{} $<$ 0.13 (typical value for narrow lines in AGNs is $\sim$ 10%, Alexandroff et al. 2013). The [Si[IV]{}$\lambda$1398]{} and [C[IV]{}$\lambda$1549]{} lines are out of the spectral range. The archived mid-IR and X-ray data are too shallow to constrain the AGN activity. However, as only $\lesssim$ 5% of high redshift LAEs are possible AGNs (Malhotra et al. 2003; Wang et al. 2004, 2009; Zheng et al. 2010, 2012, 2013), and J0335 was identified among only a couple of spectroscopically confirmed z $\sim$ 5.7 LAEs in our sample, it is unlikely to be an AGN. Although we can not securely rule out AGN activity in J0335 based on current available data, the more likely possibility is that the [Ly$\alpha$]{} emission line is due to starburst. While [Ly$\alpha$]{} luminosity is admittedly a poor indicator of star formation rate due to radiative transfer effects in galaxies, assuming the [Ly$\alpha$]{} line is totally due to star formation and taking the standard case B conversion of [Ly$\alpha$]{} to [H$\alpha$]{}, we estimate a SFR $\sim$ 6 M$_\sun$/yr (Kennicutt & Evans 2012).
Interpretations of [Ly$\alpha$]{} Profile with Broad Red Wing
-------------------------------------------------------------
### Outflow Shell Model
To interpret the [Ly$\alpha$]{} profile of LBGs and LAEs, previous studies have explored [Ly$\alpha$]{} emission line profile as a result of a thin shell of outflowing gas driven by starbursts (Verhamme et al. 2006, 2008). In their model, gas in far side of outflow can scatter [Ly$\alpha$]{} photons back to the observer’s direction, making photons red-shifted relative to galaxy systemic redshift, avoiding absorption by the material in the near side. By changing model parameters such as outflowing velocity, HI column density, velocity dispersion and dust attenuation, a diversity of profiles can be generated. The [Ly$\alpha$]{} line profile (with a broad red wing) of J0335 is qualitatively comparable to those generated by outflow shell model (Verhamme et al. 2006, 2008; Schaerer et al. 2011). In particular, an red wing extending to $>$1000 [km s$^{-1}$]{} can be generated from a very dusty, high column density outflow shell with outflow velocity $\sim$ 300 – 600 [km s$^{-1}$]{} (see Fig. 7 of Schaerer et al. 2011). However, considering the relatively low spectral resolution (FWHM=220[km s$^{-1}$]{}) here and that we are unable to determine the model parameters uniquely based on a single line profile, we do not fit the profile with radiative transfer models in this work.
However, although thin shell outflow model can successfully explain the [Ly$\alpha$]{} line profiles in many LBGs and LAEs, there are also discrepancies between the outflowing thin shell model and observations. Kulas et al. (2012) fitted a sample of z $\sim$ 2 – 3 LBGs with double peak line profiles and reported clear discrepancies between the models and data. Chonis et al. (2013) fitted high resolution [Ly$\alpha$]{} profiles in three z $\sim$ 2.4 LAEs and also found the model can’t fit the profiles well, especially for the two line profiles with a weak and highly blueshifted line peak.[^4] Furthermore the best fit models usually result in low internal velocity dispersion of the outflowing thin shell in these works. This is in contrary to the detected broad interstellar absorption line profiles, which instead suggest a large bulk velocity range of outflowing gas if we assume that the outflowing shell are of the same material responsible for the interstellar absorption line (Quider et al. 2009; Kulas et al. 2012).
### Clumpy Outflow at a Large Range of Radii
Steidel et al. (2010) considered simultaneously the profiles of [Ly$\alpha$]{} emission and low-ionization interstellar absorption lines in LBGs and suggested a scenario in which the gas outflows are clumpy, spread over a large range in radius and have gradual velocity gradients. Photons scattering from the surfaces of discrete clumps would acquire a doppler shift that reflect the velocity of the last scattered clump. So the velocity distribution and covering fraction of clumps are most responsible for the kinematics of the observed [Ly$\alpha$]{} emission line. To reproduce the profile and velocity offset of the [Ly$\alpha$]{} line ([$\Delta v_{Ly\ensuremath{\alpha}}$]{}) and the low-ionization interstellar absorption lines in a sample of z $\sim$ 2 – 3 LBGs, Steidel et al. (2010) constructed a kinematic model where optically thick gas is presented in two kinematic components: one component is at the galaxy systemic redshift and the other is outflowing with a velocity distribution. The apparent peak of [Ly$\alpha$]{} emission is modulated primarily by gas at the galaxy systemic redshift. When the velocity range spanned by the gas at the galaxy systemic redshift is broader, the [Ly$\alpha$]{} emission core is more red-shifted (larger [$\Delta v_{Ly\ensuremath{\alpha}}$]{}) and weaker.
Interestingly, an anti-correlation between the EW([Ly$\alpha$]{}) and [$\Delta v_{Ly\ensuremath{\alpha}}$]{} (the velocity offset between [Ly$\alpha$]{} emission and the low-ionization interstellar absorption lines or optical emission lines) has been detected in z $\sim$ 2 – 3 LBGs and LAEs (Adelberger et al. 2003; Shapley et al. 2003; Hashimoto et al. 2013), supporting the scenario that a smaller [Ly$\alpha$]{} velocity offset suggests less absorption by gas at the galaxy systemic redshift, thus resulting in a stronger [Ly$\alpha$]{} line. If J0335 is on that trend, its large EW([Ly$\alpha$]{}) implies weak absorption and small [$\Delta v_{Ly\ensuremath{\alpha}}$]{}. The relatively low flux of the broad wing compared to the narrow line core suggests that gas clumps with an outflow velocity ranging from zero to larger than 1000 [km s$^{-1}$]{} has a small sky coverage.
We can compare this outflow velocity with the estimated velocity required for gas to escape the dark matter halo. For LAEs at z $\sim$ 5.7 with [Ly$\alpha$]{} luminosity about $10^{42.6}~erg~s^{-1}$, the average dark matter halo mass is about 6.1$\times$10$^{11}$ M$_\sun$ (Kovac et al. 2007; Ouchi et al. 2010). For an isothermal gravitational potential that extends to a maximum radius $r_{max}$, a very rough estimation of the escape velocity at radius r is $v_{esc}(r)=\sqrt{2}~v_{c}~[1+ln(r_{max}/r)]^{\frac{1}{2}}$ (Veilleux et al. 2005). Taking $r_{max}$ = 100 kpc, r = 1 kpc and $v_{c} = \sqrt{\frac{GM_{halo}}{r_{max}}}$, we obtain an escape velocity $v_{esc}$ = 536 [km s$^{-1}$]{}. As in the equation $r_{max}$ is in square root term and $r_{max}/r$ is in the logarithmic term, changes in the assumed $r_{max}$ and/or r by a of factor 2 would only result in less than 50% changes in $v_{esc}$. The max outflow velocity in J0335 is larger than the estimated escape velocity, being consistent with the suggestion that low mass galaxies at z > 4 dominate the dissipation of heavy elements into the CGM (Martin et al. 2010) and the enrichment of IGM (Madau et al. 2001; Scannapieco et al. 2002).
Prevalence of Broad Red Wing in [Ly$\alpha$]{} Profiles of LAEs
---------------------------------------------------------------
In this subsection we discuss the prevalence of broad red wing in profiles of z $\sim$ 5 – 6 LAEs by comparing our result with published [Ly$\alpha$]{} profiles. Among the published [Ly$\alpha$]{} profiles at z = 4.5, 5.7 and 6.5 (Dawson et al. 2007; Hu et al. 2010; Ouchi et al. 2010; Kashikawa et al. 2011), we notice that a few [Ly$\alpha$]{} line profiles seem to have broad red wings comparable to our result (eg. 2HC124128+622022 in Fig.A2 of Hu et al. 2010, and J1425554+353039 in Fig.2 of Dawson et al. 2007) but a large fraction of profiles with good S/N do not show broad red wing. The reasons why only a small fraction of LAEs show broad red wing may be: 1) Some LAEs intrinsically lack gas outflows with large velocity, or the covering fraction and/or the column density of outflowing gas is too small, so the broad red wing is too weak to be detected; 2) Due to large optical depth of gas at the galaxy systemic redshift, and/or radiative transfer effect in outflowing gas, the apparent peak of [Ly$\alpha$]{} emission line shifts redward greatly, reducing the significance of broad red wing; 3) The gas outflow is anisotropic as suggested by simulations (eg. Barnes et al. 2011), so that a broad red wing can only be observed along particular directions. By studying [Ly$\alpha$]{} line profiles with high spectral resolution, good spectral S/N, and other detectable nebulae emission/absorption lines (mostly doable for LAEs at lower redshifts, Finkelstein et al. 2011; McLinden et al. 2011; Chonis et al. 2013; Guaita et al. 2013) it is possible to obtain a better understanding of the production of broad [Ly$\alpha$]{} red wing and the role outflows have in shaping [Ly$\alpha$]{} line profiles, and enable the use of the [Ly$\alpha$]{} line as a tracer of gas kinematics at higher redshifts (such as at z > 5).
![2-d and 1-d spectra of J0335. Y axis of 2-d spectra is sky position along the slit in units of arcsecond. The middle panel show the 1-d spectra (black), best fit model (blue solid line), two Gaussian components (magenta solid lines) and $\pm$1$\sigma$ error of the spectrum (grey dashed line, also plotted in the bottom panel. The position of [N[V]{}$\lambda$1240]{} line is marked with vertical dashed line.). Both fluxes and errors of 1-d spectra are in unit of 10$^{-18}$ [erg s$^{-1}$ cm$^{-2}$ Å$^{-1}$ ]{}. ](fig2.eps)
We acknowledge support from National Natural Science Foundation of China through grant 10825312 & 11233002. JXW acknowledges support from Chinese Top-notch Young Talents Program. This research uses data obtained through the Telescope Access Program (TAP), which is funded by the National Astronomical Observatories, Chinese Academy of Sciences, and the Special Fund for Astronomy from the Ministry of Finance. We would like to thank the scientists and telescope operators at Magellan telescope for their help. LI acknowledges support from CONICYT CATA Based program.
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[^1]: <http://www.eso.org/sci/activities/projects/eis/surveys/strategy_DPS.html>
[^2]: In the calculation, the IGM absorption to the broadband continuum and the Ly$\alpha$ line were ignored since they may cancel each other in the calculation of Ly$\alpha$ EW, see Malhotra & Rhoads (2002).
[^3]: <http://code.obs.carnegiescience.edu/cosmos>
[^4]: but see <http://www.nordita.org/docs/agenda/slides-alpha2013-schaerer.pdf> for a conference presentation by Daniel Schaerer in September 2013, which gave different results.
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abstract: 'We analyzed the photoluminescence intermittency generated by a single paramagnetic spin localized in an individual semiconductor quantum dot. The statistics of the photons emitted by the quantum dot reflect the quantum fluctuations of the localized spin interacting with the injected carriers. Photon correlation measurements which are reported here reveal unique signatures of these fluctuations. A phenomenological model is proposed to quantitatively describe these observations, allowing a measurement of the spin dynamics of an individual magnetic atom at zero magnetic field. These results demonstrate the existence of an efficient spin relaxation channel arising from a spin-exchange with individual carriers surrounding the quantum dot. A theoretical description of a spin-flip mechanism involving spin exchange with surrounding carriers gives relaxation times in good agreement with the measured dynamics.'
author:
- 'L. Besombes'
- 'Y. Leger'
- 'J. Bernos'
- 'H. Boukari'
- 'H. Mariette'
- 'J.P. Poizat'
- 'J. Fernández-Rossier'
- 'R. Aguado'
title: Optical probing of spin fluctuations of a single magnetic atom
---
Introduction
============
The decrease of the structure size in semiconductor electronic devices and magnetic information storage devices has dramatically reduced the number of atoms necessary to process and store bit of information. Information storage on a single magnetic atom would be an ultimate limit. The performance of such memory elements will be governed by the quantum fluctuations of the localized spins [@xiao2004]. Diluted magnetic semiconductors (DMS) systems combining high quality semiconductor structures and the magnetic properties of Mn impurities are good candidates for these ultimate spintronic devices [@Fernandez2007]. It has been shown that in a DMS with low magnetic atom concentration, the spin dynamics under magnetic field is ultimately controlled by the spin-lattice coupling [@lambe60; @kneip06]. An extrapolation of the spin dynamics measurements in bulk DMS suggests that a long spin relaxation time in the millisecond range could be expected for an isolated Mn spin [@dietl95]. However, despite the recent development of different successful techniques to address a single spin [@heinrich2004; @Kitchen2006], such dynamics has never been directly observed.
Growth and optical addressing of DMS quantum dots (QDs) containing a single magnetic atom were achieved recently [@Besombes04; @Maingault2006]. When a Mn dopant atom is included in a II-VI QD, the spin of the optically created electron-hole pair interacts with the five [*d*]{} electrons of the Mn (total spin M=5/2). This leads to a splitting of the once simple photoluminescence (PL) spectrum of an individual QD into six (2M+1) components. This splitting results from the spin structure of the confined holes which are quantized along the QDs’ growth axis with their spin component taking only the values J$_z$=$\pm$3/2. The hole-Mn exchange interaction reduces to an Ising term J$_z$.M$_z$ and shifts the emission energy of the QD, depending on the relative projection of the Mn and hole spins. As the spin state of the Mn atom fluctuates during the optical measurements, the six lines are observed simultaneously in time average PL spectrum. The intensities of the lines reflect the probability for the Mn to be in one of its six spin components and the emitted photon is a probe of the spin state of the Mn when the exciton recombines.
\[bt\] ![ (Color online) (a) Experimental time resolved PL spectra recorded on three different energy lines (labeled 1, 2 and 3) of an X-Mn complex. The inset shows the corresponding time integrated PL spectrum. (b) PL decay time calculated using the parameters T=5K, $\tau_{b}$=280ps, $\tau_{d}$=8ns, $\tau_{X-Mn}$=25ns. (c) Energy levels involved in the rate equation model described in the text displayed as a function of their total angular momentum S$_z$.[]{data-label="fig1"}](Fig1.eps "fig:"){width="2.8in"}
In this article, we show how we can use the statistics of the photons emitted by a single Mn doped QD to probe the spin dynamics of a magnetic atom interacting with both confined and free carriers. We performed correlations of the PL intensity emitted by individual lines of an isolated Mn-doped QD. In these start-stop experiments [@Couteau2004], the detection of the first photon indicates by its energy and polarisation that the Mn spin has a given orientation. The detection probability of a second photon with the same energy and polarisation is then proportional to the probability of conserving this spin state. The time evolution of this intensity correlation signal is a probe of the spin dynamics of the Mn atom. The auto-correlation signal displays a bunching effect revealing a PL intermittency. This intermittency results from fluctuations of the Mn spin. Correlation of the intensity emitted by opposite spin states of the exciton-Mn complex (namely,cross-correlation) presents an antibunching at short delays. The characteristic time of this antibunching corresponds to the spin transfer time between the two degenerated spin states. A thermalization on the exciton-Mn complex is directly evidence by the energy and temperature dependences of the correlation curves. The measured single Mn spin relaxation times are in the range of 20 ns and are strongly influenced by the injection of carriers in the vicinity of the QDs. Mn spin-flips induced by the injection of carriers in and around the QD are theoretically described. These scattering processes give relaxation times in good agreement with the measured dynamics.
Samples and experiments
=======================
Single Mn atoms are introduced in CdTe/ZnTe QDs during their growth by molecular beam epitaxy adjusting the density of Mn atoms to be roughly equal to the density of QDs [@Maingault2006]. The statistics of the photons emitted by individual Mn-doped QDs was analyzed using a combination of a low-temperature (5K) scanning optical microscope and a Hanbury Brown and Twiss (HBT) setup for photon-correlation measurements [@Couteau2004]. High refractive index hemispherical solid immersion lens were used to enhance the collection of the single dot emission. The PL was quasi-resonantly excited with a tunable CW dye laser or by picosecond pulses from a doubled optical parametric oscillator and collected through aluminum shadow masks. The circularly polarized collected light was spectrally dispersed by a 1 m monochromator before being detected in the HBT setup or by a single fast avalanche photodiode (time resolution 40 ps) for time resolved measurements. The time resolution of the HBT setup was about 500 ps. Under our experimental conditions with counts rates of a few kHz the measured photon pair distribution yields after normalization the intensity autocorrelation function g$^{(2)}$($\tau$).
Exciton-Mn spin flips.
======================
![(a) Auto-correlation function of the intensity collected in $\sigma+$ polarisation on the low energy line of the X-Mn complex (solid line) and on the overall PL spectra (dotted line). (b) Circularly polarized cross-correlation function (solid line) and auto-correlation (dotted line) on the same line as in (a) but for a larger excitation intensity. The inset shows the PL spectrum of the corresponding QD. (c) Experimental auto-correlation function and theoretical function calculated with the rate equation model described in the text with parameters T=5K, $\tau_{b}$=280ps, $\tau_{d}$=8ns, $\tau_{X-Mn}$=25ns, $\tau_{Mn}$=50ns and g=0.05x10$^{-3}$ps$^{-1}$.[]{data-label="fig2"}](Fig2.eps){width="4in"}
A signature of the Mn spin dynamics can be observed first in the PL decay of the X-Mn complex. Fig.1 presents the PL decay of three different transitions of the X-Mn complex. These transitions present a biexponential decay. The fast component corresponds to the radiative lifetime of the exciton, as already measured in non magnetic QDs. The long component is associated with the existence of the dark excitons [@smith05]. Direct recombination of the dark exciton can be observed in some Mn-doped QDs because of a slight admixture of the bright states with the dark ones induced by a valence band mixing. However, the dark excitons mainly contribute to the signal by undergoing a spin flip to become bright excitons which decays radiatively. The PL decay is then determined by both radiative decay and excitons spin-flips. The exciton decay, and particularly the amplitude of the slow component, depends strongly on the energy level observed. For the high energy lines, the slow component makes a negligible contribution to the time integrated signal. Conversely, for the low energy lines, the secondary component makes a significant contribution while the primary lifetime remains constant. In this regime, the secondary lifetime can be associated either with the dark exciton lifetime or the exciton spin flip time.
To extract these two parameters from the PL decay curves, we compare the experimental data with a rate equation model describing the time evolution of the population of the 24 X-Mn spin levels (Fig.1(c)) after the injection of a single exciton [@govorov2005]. Different spin-flips times are expected depending on wether the transitions occur with or without conservation of the energy or of the total spin. However, we consider in first approximation that the spin-flips among the X-Mn states can be described by a single characteristic time $\tau_{X-Mn}$. We consider that at finite temperature, the intraband relaxation rates $\Gamma_{\gamma\rightarrow\gamma'}$ between the different spin states of the exciton-Mn complex depend on the energy separation $E_{\gamma\gamma'}=E_{\gamma'}-E_{\gamma}$. Here we use $\Gamma_{\gamma\rightarrow\gamma'}$=1/$\tau_{X-Mn}$ if $E_{\gamma\gamma'}<0$ and $\Gamma_{\gamma\rightarrow\gamma'}$=1/$\tau_{X-Mn}e^{-E_{\gamma\gamma'}/k_BT}$ if $E_{\gamma\gamma'}>0$ [@govorov2005]. This describes a partial thermalization among the 24 levels of the X-Mn system during the lifetime of the exciton (bright or dark). In this model, we also neglect the influence of the valence band mixing on the oscillator strength [@Leger2007] and consider that all the excitonic bright (dark) states have the same lifetime $\tau_b$ ($\tau_d$). Because of an efficient hole spin-flip during the phonon assisted relaxation of the unpolarized injected carriers, we consider that the excitons with spins $\pm1$ and $\pm2$ are excited with the same probability at $t=0$. Only optical transitions conserving the Mn spin are considered. The calculation shows that changing $\tau_d$ mainly influences the characteristic time of the long decay component whereas changing $\tau_{X-Mn}$ affects also the amplitude of this component. The PL decay curves can be reproduced well by this rate equation model using $\tau_b=280ps$, $\tau_d=8ns$ and $\tau_{X-Mn}=25ns$ (Fig. 1(b)). The value of $\tau_d$ controls the decay time of the long component whereas $\tau_{X-Mn}$, larger than $\tau_d$, reproduces the emission energy dependence of the amplitude of the long component very well.
Within the X-Mn complex, the relaxation time between different spin states $\tau_{X-Mn}$ can be affected by the spin orbit, carrier-phonon and exchange interactions [@govorov2005] that affect the exciton. In QDs, the electron spin relaxation is longer than the radiative lifetime and is ultimately controlled by random fluctuations of the nuclear spins [@feng2007]. The hole spin relaxation is mainly controlled by the interaction with phonons [@woods2004] and can be faster explaining the observed partial thermalization of the X-Mn complex during the lifetime of the ground state exciton.
Fluctuations of an isolated Mn spin.
====================================
To directly observe the time fluctuations of the Mn spin interacting with the injected carriers, we analyzed the statistics of the photons emitted by a Mn-doped QD. This statistics can be deduced from an intensity correlation function of the QD emission. Fig. 2(a) shows the intensity correlation function g$^{(2)}$($\tau$) of the circularly polarized ($\sigma+$) low energy line of a Mn-doped QD compared with the correlation function obtained for the overall PL of the QD. The auto-correlation function obtained for all the photons emitted by the QD is characteristic of a single photon emitter with a dip at short delay. The width of this antibunching signal is given by the lifetime of the emitter and the generation rate of excitons and its depth is limited by the time resolution of the HBT setup. A similar experiment performed on one of the single line of the X-Mn complex still presents a reduced coincidence rate near $\tau$=0, but it is mainly characterized by a large bunching signal with a half width at half maximum (HWHM) of about 10ns. This bunching reflects an intermittency in the QD emission. This intermittency likely comes from the fluctuations of the Mn spin orientation.
To confirm this result, cross-correlation measurements were performed between different spin states of the X-Mn complex. Cross-correlation of the $\sigma+$ and $\sigma-$ photon emitted by the low energy line (fig.2(b)) shows an antibunching with g$^{(2)}$(0)=0.2 and a HWHM of about 5ns. These two different behaviors, namely the bunching of the auto-correlation signal and the antibunching of the cross-correlation signal, demonstrate unambiguously that the statistic of the QD emission is completely governed by the Mn spin fluctuations. Whereas the auto-correlation probes the time dependence of the probability for the spin of the Mn to be conserved (M$_z$ =+5/2 at $\tau$=0 in Fig.2(a)), the cross-correlation presented in Fig.2(b) is a probe of the spin transfer between +5/2 and -5/2.
![ (Color online) (a) Power dependence of the autocorrelation function of the hight energy line of an X-Mn complex. (b) Calculated power dependence of the HWHM of the autocorrelation function of the hight energy line. The parameters used in the model are: $\tau_{b}$=280ps, $\tau_{d}$=8ns, $\tau_{X-Mn}$=20ns, and $\tau_{Mn}$=40ns (plain line) or $\tau_{Mn}$=4$\mu$s (dotted line). (c) Experimental HWHM of the autocorrelation signals presented in (a).[]{data-label="fig3"}](Fig3.eps){width="2.8in"}
As observed in the time resolved PL measurements, fluctuations of the Mn spin occur during the lifetime of an exciton. However, they can also take place when the QD is empty. As the spin relaxation rate of the Mn is expected to be influenced by the presence of carriers in the QD, we have to consider two relaxation times, $\tau_{Mn}$ for an empty dot and $\tau_{X-Mn}$ for a dot occupied by an exciton. Their relative contributions to the observed effective relaxation time will depend on the generation rate of excitons. The rate equation model described previously can be extended to extract an order of magnitude of the parameters $\tau_{Mn}$ and $\tau_{X-Mn}$ from the correlation experiments. Six biexciton states are added to the 24 $(X-Mn)$+6 $(Mn)$ level system (see Fig.1(c)). A continuous generation rate $g$ is considered to populate the exciton and biexciton states. The initial state of the system is fixed on one of the six Mn ground states and one monitors the time evolution of the population of the corresponding bright X-Mn state. When normalized to one at long time, this time evolution accounts for the correlation function of the transition associated with the considered X-Mn level.
The time evolution of the correlation function calculated using this model are presented in Fig.2(c) and compared with the experimental data. At low generation rate, when the average time between two injected excitons is longer than any spin relaxation rate, $\tau_{Mn}$ and $\tau_{X-Mn}$ have distinguishable effects on the calculated correlation curves. In average, the relaxation of the Mn alone (controlled by $\tau_{Mn}$) is only perturbed by the injection of the exciton used to probe the Mn spin projection. During the lifetime of this exciton, the system relaxes with the relaxation rate $\tau_{X-Mn}$. This produces a relaxation of the Mn spin proportional to the ratio of $\tau_{X-Mn}$ and the exciton lifetime. A reduction of $\tau_{X-Mn}$ then reduces the amplitude of the bunching curve expected for a Mn alone (because of the six available spin states, the maximum amplitude of the bunching should be six) without significantly changing its width controlled by $\tau_{Mn}$. With the generation rate used in the measurements of Fig.2(a) (a generation rate of about g=0.05x10$^{-3}$ps$^{-1}$ can be deduced from the ratio of the exciton and biexciton amplitude), $\tau_{X-Mn}$ mainly affects the height of the bunching signal whereas $\tau_{Mn}$ preferentially controls its width. Then, with given values of $g$, $\tau_b$ and $\tau_d$, it is possible to extract a parameter pair ($\tau_{Mn}$,$\tau_{X-Mn}$) that reproduces the bunching and anti-bunching curves. The bright and dark excitons lifetimes were estimated from the PL decay curves and the exciton generation rate can be estimated from the relative amplitudes of the exciton and biexciton emissions [@bes05]. For the data presented in Fig.2(c), the best fit is obtained with $\tau_{X-Mn}=25ns$ and $\tau_{Mn}=50ns$. The X-Mn relaxation time obtained is consistent with the value deduced from the PL decay curves. The relaxation time of the Mn alone (empty dot) appears to be 3 orders of magnitude shorter than expected from the extrapolation of measurements in bulk dilute CdMnTe under magnetic field [@dietl95].
Carriers induced Mn spin fluctuations
=====================================
The observed Mn spin dynamics is not simply an intrinsic property of the localized Mn atom but depends on the optical excitation conditions. The power dependence of the correlation signal of the high energy transition of a X-Mn complex is presented in Fig.3(a). Increasing the excitation power significantly reduces the width of the correlation signal. This reduction has two origins: first, when carriers are injected in the QD under quasi-resonant conditions (excitation on an excited state of the QD), increasing the carrier generation rate increases the probability of finding the QD occupied by an exciton. The spin relaxation time being shorter for an occupied dot than for an empty dot, the Mn spin fluctuates faster and the width of the auto-correlation curve decreases. This effect is illustrated by the power dependence of the HWHM of the calculated and experimental correlation curves presented in Fig.3(b) and 3(c) respectively. At high generation rate, the width of the correlation signal is controlled by $\tau_{X-Mn}$ whereas at low excitation the photon statistics is ultimately determine by the spin fluctuations of the Mn alone. The width of the calculated correlation curves saturates at low excitation. This maximum width is controlled by $\tau_{Mn}$. In the experiments, this saturation is not observed due to the limit in the accessible excitation power range.
![ (color online)(a) Auto-correlation function on the low energy line of an X-Mn complex in $\sigma+$ polarization for excitation intensities P$_0$ and 3P$_0$. Theoretical curves are presented in red. A reduction of $\tau_{X-Mn}$(=15ns) and $\tau_{Mn}$(=20ns) has to be included to describe the high excitation intensity autocorrelation curve. (b) Auto-correlation function on the low energy line of an X-Mn complex in $\sigma+$ polarization for two different excitation conditions: resonant on an excited state (577.5nm) and non-resonant (514nm).[]{data-label="fig4"}](Fig4.eps){width="2.8in"}
However, this process is not sufficient to explain the observed excitation power dependence of the correlation signal in all the investigated QDs. For instance, to reproduce the power dependence presented in Fig.4(a), one has also to reduce the spin relaxation times $\tau_{Mn}$ and $\tau_{X-Mn}$ at high excitation intensity. In Fig.4(a) the best fit at high excitation power is obtained with $\tau_{Mn}=25$ and $\tau_{X-Mn}=15$ whereas at low excitation $\tau_{Mn}=50$ and $\tau_{X-Mn}=25$. This reduction of the relaxation time likely comes from the spin-spin coupling with carriers injected in the surroundings of the QD thought the background absorption observed in PLE spectra of this individual QD [@vasanelli02].
The influence of the free carriers on the spin relaxation rate is shown by the correlation signals obtained on the same X-Mn transition for two different excitation wavelengths (Fig.4(b)): resonant on an excited state (577,5nm) and non-resonant in the ZnTe barriers (514nm). These two signals are recorded with the same photon count rate, suggesting a similar occupation rate of the QD. The characteristic bunching signal observed under quasi-resonant excitation completely disappears when the excitation energy is tuned above the wetting layer absorption. As already observed in DMS quantum wells, this behavior reflects the extreme sensitivity of the localized Mn spin to the spin-spin coupling with the free carriers or the carriers relaxing in the QD [@Tyazhlov97].
![ (a) Auto-correlation function of the emission intensity of the hight (upper trace) and low (lower trace) energy lines of a X-Mn complex recorded in the same circular polarization. (b) Cross-correlation function of the emission intensity of the high and low energy line recorded in $\sigma+$ and $\sigma-$ polarization respectively for two different excitation intensities. Detail of the experimental (c) and calculated (d) cross-correlation function.[]{data-label="fig5"}](Fig5.eps){width="2.8in"}
For an isolated Mn atom, the spin relaxation $\tau_{Mn}$ comes only from the spin-lattice interaction [@scalbert88] and a long spin relaxation time is expected. As we will discuss in the next section, the Mn spin dynamics can be modified significantly by the presence of free carriers which are strongly coupled with both the magnetic atom and the phonons. These free carriers serve as a bypass channel for the slow direct spin-lattice relaxation.
Thermalization of the exciton-Mn complex.
=========================================
The X-Mn complex is also significantly coupled to the phonon bath. A partial thermalization of the X-Mn system appears directly in the amplitude of the correlation curves obtained on different energy levels of the X-Mn system (Fig.5(a)) as well as in cross-correlation measurements (Fig.5(b) and (c)). A finite temperature enhances the probability of the spin-flips involving an energy loss. This introduces a dissymmetry in the spin relaxation channels of the X-Mn complex. The consequence of this dissymmetry in the relaxation process is an energy dependence of the amplitude of the correlation signal. This is illustrated by the correlation curves obtained on the high ($|-1,-5/2\rangle$) and low energy states ($|+1,-5/2\rangle$) of a X-Mn complex (Fig.5(a)). For the high energy state, all the relaxation transitions within the X-Mn complex take place with an energy loss: The leakage probability is then maximum and the probability for this state to be re-populated by spin-flips from low energy states is very weak. A large bunching signal is then observed (Fig.5(a)). On the opposite, the low energy level can be populated by a transfer from the high energy states, and some relaxation channels involving an absorption of energy are blocked at low temperature. The associated bunching signal is weaker (Fig.5(a)).
This thermalization process directly appears if a cross-correlation of the intensity emitted by a low and a high energy levels is performed. Fig.5(c) shows the correlation of the photons emitted by $|-1,-5/2\rangle$ (high energy line) and $|+1,-5/2\rangle$ (low energy line). At low excitation intensity, this correlation signal presents a clear dissymmetry. This cross-correlation measurement probe the time dependence of the probability of finding an exciton (either $|+1\rangle$ or $|-1\rangle$) coupled with the Mn spin in the state M$_z$=-5/2. At positive time delay, g$^2(\tau)$ gives the probability to find the system in the state $|+1,-5/2\rangle$ knowing that at $\tau=0$ the Mn spin projection was M$_z$=-5/2 (detection of a photon from $|-1,-5/2\rangle$). The situation is reversed for the negative delay where a photon from $|+1,-5/2\rangle$ acts as the trigger in the start-stop measurement and g$^2(\tau)$ give the probability for the system to be in the high energy state $|-1,-5/2\rangle$. The dissymmetry in the cross-correlation curve reflects the difference in the spin relaxation channels available for the high ($|-1,-5/2\rangle$) and the low ($|+1,-5/2\rangle$) energy X-Mn states.
![ Intensity auto-correlation function of the high energy line $|-1,-5/2\rangle$ of a X-Mn complex recorded in the same excitation conditions at T=5K (a) and T=20K (b). The theoretical curves (solid line) are obtained with the same set of parameters: $\tau_{X-Mn}$=10ns, $\tau_{Mn}$=15ns,$\tau_{b}$=280ps, $\tau_{d}$=8ns and g=0.1.10$^{-3}$ps$^{-1}$.[]{data-label="fig6"}](Fig6.eps){width="2.8in"}
The dissymmetry in the relaxation processes is influenced by the excitation intensity: as observed in the PL decay measurements presented in Fig.1, the low energy bright exciton states can be efficiently populated by spin-flips from the dark exciton states reducing the effective population loss of these states and consequently reducing the amplitude of the photon bunching. Increasing the excitation intensity decreases the effective lifetime of the dark excitons because of the formation of the biexciton [@besombes05]. This opens an efficient spin relaxation channel for the low energy bright X-Mn states: once a dark exciton has been created after a spin-flip, it is quickly destroyed by the injection of a second exciton with the formation of a biexciton. It can no longer flip back to the low energy bright state. This effect stop the refilling process and consequently increases the amplitude of the bunching signal. As observed in Fig.5(b), increasing the excitation intensity decreases the difference in the amplitude of the corresponding bunching signal of the low and high energy lines.
As shown in Fig.6, these spin relaxation processes are also influenced by the lattice temperature. The effect is especially pronounced for the high energy states. Increasing the temperature allows a refilling of these levels by a transfer of population from the low energy states, decreasing the amplitude of the bunching signal.
Model for the Mn Spin relaxation
================================
In this section we discuss two Mn spin relaxation mechanisms in the optical ground state. The short ($\tau_{Mn}\simeq 20$ns) spin relaxation time inferred from our photon correlation measurements can not be accounted for by Mn-Mn spin interaction, which is considered the most efficient spin relaxation mechanism in II-VI semiconductor. Since this is a short range interaction [@Furdyna], the Mn spin relaxation time, $T_{1M}$ increases exponentially as the density of Mn goes down [@dietl95]. Thus, the spin relaxation of a single Mn atom inside a quantum dot with 20 10$^3$ atoms, if generated by Mn-Mn coupling, should be in the range of microseconds. Thus, some other spin relaxation mechanism is at play. Here we study the Mn spin relaxation due to exchange coupling to photo-carriers that occupy extended states in the wetting layer (WL). The Mn is assumed to lie in the wetting layer, at a location where both quantum dot and extended states are finite. In particular, the QD states are expected to peak in the plane that separates the wetting layer and the QD. Mn atoms located in that region are more strongly coupled to the QD exciton.
We consider two independent mechanisms: (i)exchange coupling with carriers in extended WL states and (ii) exchange coupling with carriers that scatter from the extended WL states to confined QD states, exchange with the Mn and then return to the WL. The first mechanism has been considered before in the context of Mn spin relaxation interacting with quantum well carriers [@konig2000; @Fonseca04] and is identical to Korringa mechanism for nuclear spin relaxation due to spin-flip with itinerant electrons in metals. The second mechanism involves single charge tunneling in and out of the dot. The QD confined states remains empty except for some intervals during which a single carrier tunnels in and out from the optically excited wetting layer. Once in the QD state, the carrier can give (or take away) one unit of spin to Mn. Both the Korringa and the charge fluctuation mechanisms require several tunneling and exchange events, involving different carriers, to relax the Mn spin from an initial situation where is known to take the value $M_z=+5/2$ to a final situation of thermal equilibrium, where the 6 Mn spin orientations are equally likely.
The proposed charge fluctuation induced spin relaxation mechanism is in line with our previous works[@Besombes04; @JFR04; @JFR06; @Leger06; @Fernandez2007] whose central claim is the very strong dependence of the spin properties of Mn doped quantum dots on the addition of a single carrier. It is also motivated by the observation of a weak contribution of the positive trion PL signal which implies that, under our experimental conditions, holes are captured by the QD.
Formalism
---------
We use a Bloch-Redfield master equation approach[@cohen] which tracks the dynamical evolution of the density matrix of the quantum dot ground states, corresponding to the 6 Mn spin orientations, under the influence of the reservoir of carriers. The diagonal terms of the density matrix represent the probability of finding the Mn spin in a given spin projection. Their dynammics is given by the master equation $$\frac{d p_M}{dt}= -\left(\sum_{M'} \Gamma_{M\rightarrow
M'}\right)p_M + \sum_{M} \Gamma_{M'\rightarrow M} p_{M'}$$ with the initial condition that, at $t=0$, the probability of finding the Mn spin with $M_z=+5/2$ is $p_{+5/2}=1$. The evolution of $p_{+5/2}(t)$ and $p_{-5/2}(t)$ are directly associated to the auto-correlation and cross-correlation measurements. The rates in the master equation depend on the Hamiltonian of the sytem, which is the sum of the QD Hamiltonian, the WL carrier Hamiltonian and the QD-WL coupling: ${\cal H}= {\cal H}_{0} + {\cal H}_{res} + {\cal V}$ $\Gamma_{M\rightarrow M'}$ is the scattering rate between the eigenstate $M$ and state $M'$ of ${\cal H}_0$ induced by the coupling ${\cal V}$. The rates are given by the statistical average over initial reservoir states and sum over final reservoir states of the Fermi Golden rule rate associated to ${\cal V}$ [@cohen]. They depend on the mechanism under consideration, either direct exchange or charge fluctuation. Since our experimental data indicate that hole confinement is much weaker than electron confinement, we consider that the carriers that couple to the Mn when there is no exciton in the dot are holes. Our theory results can easily be adapted for the case of electrons. The number of states $M$ in the master equation also depends on the mechanism. In the case of charge fluctuation mechanism we need to keep track of the 6 neutral states and the 12 states of the dot with one hole inside. In the case of the Korringa relaxation, only the 6 states corresponding to the “empty dot” Mn spin are included in the master equation.
The QD Hamiltonian is: $${\cal H}_{0}= J \left(\tau^z {\bf \hat M}^z +
\frac{\epsilon_h}{2}\left( \tau^{+}{\bf \hat M}^{-} +\tau^{-}{\bf
\hat M}^{+}\right)\right) +D \left({\bf \hat M}^z\right)^2
\label{H_0}$$ where $J=\beta |\psi_{QD}(\vec{r}_I)|^2n_d$ is the Mn QD hole coupling, $\beta\simeq 60 eV\AA^3$ is the Mn-hole exchange coupling constant of CdTe, $|\psi_{QD}(\vec{r})|^2$ is the QD hole envelope function, $n_d=0,1$ counts the number of holes in the dot. The Mn spin operators are ${\bf \hat M}^{z,\pm}$ and $\tau^{z,\pm}$ are the Pauli matrices acting on the hole space. We consider antiferromagnetic hole-Mn coupling. $\epsilon_h$ is the dimensionless parameter that accounts for the reduction of the spin-flip interaction due to spin-orbit coupling[@JFR06; @Fernandez2007].
Notice that the exchange coupling of the Mn to the quantum dot fermion is only relevant in the charge-fluctuation mechanism, for which $n_d$ changes between 0 and 1. For the Korringa mechanism $n_d=0$ and Mn is only exchanged coupled to carriers in extended WL states. The $D ({\bf \hat M}^z)^2$ term in ${\cal H}_{QD0}$ describes the strain induced anisotropy which has been observed in strained (Cd,Mn)Te layers [@Furdyna95]. This term is negligible when the Mn interacts with a QD hole but is not when we consider the Korringa relaxation, for which no carrier occupies the QD state. The term ${\cal H}_{\rm res}=\sum_{{\bf k},\nu} \epsilon_{{\bf k},\nu}
c^{\dagger}_{{\bf k},\nu}c_{{\bf k},\nu}$ describes the delocalized carriers in states momentum ${\bf k}$ and band index (including the spin) $\nu$.
Each of the two relaxation mechanisms considered here has its own dot-reservoir coupling. In the Korringa mechanism we assume that there is some overlap between the WL extended states and the Mn spin. Igoniring the dependence on ${\bf k}$ of the spin matrix element $\epsilon({\bf k},{\bf k}')$,the coupling reads: $${\cal V}\simeq \frac{\beta\epsilon}{2A} |\phi(z_I)|^2 \sum_{\bf
k,k'}\left({\bf \hat M}^{(+)}c^{\dagger}_{{\bf k},\Downarrow}c_{{\bf
k}',\Uparrow} +
{\bf \hat M}^{(-)}c^{\dagger}_{{\bf k}¡,\Uparrow}c_{{\bf k},\Downarrow} \right)$$ where $A$ is the area of the WL, $|\phi(z_I)|^2$ is the envelope part of the WL wave function evaluated at the Mn location, and $L$ is the WL width. $\epsilon_h$ could take different values for the QD and the WL.
In the charge-fluctuation mechanism the coupling between the reservoir and the dot is the sum of an operator that transfers one hole from the reservoir to the dot and its hermitian conjugate, which takes the hole out from the dot and transfers it to the reservoir: $${\cal V}= \sum_{\sigma} |\sigma\rangle\langle0| \sum_{{\bf k}}
V_{{\bf k}} c_{{\bf k},\sigma} + h.c. \label{tunnel}$$ The tunneling operator conserves the energy and the spin of the carrier. This kind of coupling has been considered recently to account for peculiar PL lineshapes of self assembled quantum dots in contact with electronic reservoirs [@Petroff05; @Petroff08]. The mechanism would operate analogously if the transfer from the itinerant to the localized state is inelastic, but the timescale would be longer.
Results
-------
### Mn spin relaxation due to exchange with WL carriers
We discuss first the relaxation of the Mn spin due to exchange with WL carriers, or Korringa mechanism. We assume a parabolic dispersion with effective mass $m^*$ for the holes which yields a stepwise density of states. We assume holes are in a thermal state with effective temperature $k_BT_h$ larger than the lattice and chemical potential $\mu$. The Korringa relaxation rate is proportional to the square of the density of states at the Fermi energy and the square of the exchange coupling constant. After some work we obtain the Mn spin flip rates between $M$ and $M'$, eigenstates of ${\cal H}_0=D({\bf \hat M}^z)^2$, : $$\begin{aligned}
\Gamma^{\pm}_{M\rightarrow M'}=
\left(\frac{ \beta \epsilon \eta m^*}{ \pi L\hbar^2 }\right)^2
I(y,z) |\langle M|{\bf \hat M}^{(\pm)}|M'\rangle|^2\end{aligned}$$ where we have approximated $|\phi(z_I)|^2= 2\eta/L$ , $0<\eta<1$ is a dimensionless parameter that accounts for the overalp of the WL states with the Mn, $y=\beta \mu$ is the chemical potential of the WL carriers, $z=(E_M-E_M')/k_BT_h$ is the change in energy of the Mn spin divided by the effective temperature of the WL carriers, and $I(y,z)=k_BT_h
\frac{e^y}{e^{|z|}-1}Log\left[\frac{1+e^y}{1+e^{y-|z|}}\right]$ for $z>0$ and $I(y,z)=e^{|z|}I(y,-z)$ for $z<0$.
The solution of the master equation, taking as initial condition $p_{+5/2}=1$, is shown in figure (\[relaxKO\]). In the left panel we plot the evolution of the occupations of the 6 Mn spin projections along the growth axis. The decay of $p_{+5/2}$ is accompained by a rise of the $+3/2$ state. This state is also connected with the $1/2$ state whose occupation starts to build. Interestingly, the population of the $+3/2$ states overshoots initially. The figure shows a “falling domino” effect. They evolve towards thermal equilibrium. The average magnetization $\langle M
\rangle(t) = \sum_{n=1,6} M_n p_n(t)$ decays according to $$\langle M \rangle(t) =M_0 e^{-t/T_{1M}}$$ in all the simulations performed. In the figure we take $T_h\simeq 10 K$, $\epsilon=0.1$, $\eta=1$, ${\cal D}=7\mu$eV, $m^*$=0.4, $L=20 \AA$, and $n_h=10^{11}cm^{-2}$ and we obtain $T_{1M}\simeq 8$ ns. Thus, this mechanism is consistent with the Mn spin relaxation time that we observe. In the figure (\[relaxKO\]) we see how $T_{1M}$ increases as the density of carriers decreases. The order of magnitude of $T_{1M}$ is consistent with the calculations for Mn spin relaxation induced by scattering with electrons in quantum wells [@konig2000]. There $T_{1M}\simeq 2\times 10^3$ ns is obtained in a quantum well with $L=80\AA$. Taking into account that the electron Mn coupling $\alpha$ is 4 times smaller than $\beta$ and the $L$ is 4 times bigger, there $T_{1M}$ for holes should be at least $16^2$ shorter.
\[hbt\] . (a) $p_n$ for Mn spin states. These can be considered as the [*conditional probabilities*]{} that, having observed the Mn spin at $t=0$ in the state $M_z=+5/2$, a state with spin $M_z$ is observed at time $t$. (b) $T_{1M}$ as a function of hole density (see text). (c) Average magnetization corresponding to panel (a), in logarithmic scale.](Fig7.eps "fig:"){width="2.8in"}
Thus Korringa exchange with a sufficiently large density of photoholes might account for the Mn spin relaxation when the dot is in the optical ground state. We consider now a different mechanism, involving single hole charging of the dot, motivated by the fact that the PL often shows a weak contribution of positive trions.
### Mn spin relaxation due to charge fluctuation in the dot
The elementary process for spin relaxation due to charge fluctuation is a combination of 3 steps. First, a hole tunnels into the empty dot with the Mn in the spin state $M$. Second, the hole and the Mn exchange spin, so that the Mn is now in the state $M\pm1$. Third, the hole tunnels out of the dot. Thus, in the master equation we need to include both the 6 charge neutral states of the Mn doped dot as well as the 12 states with one hole inside. The dissipative dynamics, induced by ${\cal V}$ connects neutral states with charged states ($+$) and viceversa ($-$). The charging transition rates are given by $$\Gamma^{(+)}_{n,m}= \Gamma_0 n_F\left((E_n-E_m)\right) |\langle
n|\sum_{\sigma} |0\rangle\langle \sigma ||m\rangle |^2
\label{rateplus}$$ and the decharging rate: $$\Gamma^{(-)}_{n,m}= \Gamma_0 \left(1-n_F(E_n-E_m)\right) |\langle
n|\sum_{\sigma} |0\rangle\langle \sigma ||m\rangle |^2
\label{rateminus}$$ Here $n_F(x)$ is the Fermi function which depends on the temperature and chemical potential of the electron reservoir, $E_n$ and $E_m$ are the eigenstates of the QD Hamiltonian and $\Gamma_0=\sum_k
|V_k|^2 \rho(E_f)$ is the tunneling rate for the fermion in and out of the dot. The matrix elements featured in eq. (\[rateplus\],\[rateminus\]) are strongly spin dependent and only connect states in which the Mn spin changes, at most, by one unit.
An important quantity in our simulations is the average charge, $\langle q \rangle =\sum_{n=7,18} p_n(t)$. Since we consider an initally neutral dot, this quantity goes from zero to the steady state occupation in a time scale given by $T_{1Q}\equiv(\Gamma_0)^{-1}$. In this second set of simulations we measure time in units of $T_{1Q}$. The evolution of the magnetization and the empty dot occupations on this case look very similar to those of figure (\[relaxKO\]). In particular, the magnetization decays exponentially in a time scale much longer than the charge relaxation time.
\[hbt\] . Mn spin relaxation time, originated by charge fluctuations in the dot, as a function of average charge in the dot for electrons and holes with different $\epsilon_h$ values. The straight lines are linear fits of the numerical data.](Fig8.eps "fig:"){width="2.8in"}
We have computed the Mn spin relaxation time, $T_{1M}$, as a function of the parameters of the simulation, $\Gamma_0$, $k_B T$, $J$, $\epsilon$ and $\langle q\rangle$. We have found that the crucial quantities are the average charge, the spin flip anisotropy $\epsilon_h$ and $\Gamma_0$, as shown in figure (\[summary\]). To very good approximation, we have that $\Gamma_0 T_{1M}$ depends linearly on $\langle q\rangle^{-1}$, for moderate values of $\langle
q\rangle$. Thus, this mechanism shows a strong dependence on the density of carriers in the WL, in agreement with our observations.
In figure \[summary\] we see that, in the case of tunneling electrons ($\epsilon=1$), for an average charge of $0.1$ we have $T_{1M}\simeq 100 \Gamma_0 ^{-1}$. For tunneling holes with $\epsilon_h=0.2$ and $\langle q\rangle=0.1$ we have $T_{1M}\simeq
600 \Gamma_0^{-1}$. Interestingly, these numbers are weakly dependent of the value of $J$ and on the temperature. In order to obtain an absolute number for $T^M_1$ we need an estimate of $\Gamma_0$ the charge scattering rate. An upper limit for $\Gamma_0$ is provided by the linewidth observed in the PL spectrum. Since the charge scattering would result in a broadening of the linewidth, we can infer that $\hbar \Gamma_0< $ 50$\mu$eV. Thus, using $\hbar=0.65$meV$\times$ps we have $T^M_1 > 600
\frac{\hbar}{\Gamma_0} \simeq 8 ns $ for $\langle q \rangle=0.1$. This is a lower bound for the spin scattering time, or an upper bound for the spin scattering rate. Thus, single hole tunneling events could relax the spin of the Mn in a time scale of 8ns for a dot which is charged 10 percent of the time. Agreement with the experimental result can be obtained by taking smaller average charge in the dot or smaller spin-flip interaction or smaller tunneling rate $\Gamma_0$. We note that the proposed mechanism is similar to the hole spin-relaxation mechanism proposed by Smith [*et al.*]{} [@Petroff05]. There, a spin up hole tunnels out of the dot and spin down tunnels in, resulting in an effective spin relaxation for the hole.
In summary, we have two mechanisms that account for carrier induced Mn relaxation in the dot in a time scale of 10 nanoseconds. In both cases the dot must be considered as an open system. In the Korringa relaxation mechanism the Mn spin is exposed to extended WL states. In the charge-fluctuation mechanism the occupation of the dot state fluctuates, due to its coupling to extended WL states. These two mechanisms are not exclusive and might operate at the same time. Whereas the Korringa mechanism is almost identical to that of (Cd,Mn)Te quantum wells [@konig2000], the charge-fluctuation mechanism is specific of quantum dots coupled to a continuum.
Conclusion.
===========
In conclusion, we used time resolved and photon correlation measurements to probe the spin dynamics of a single magnetic atom (Mn) interacting with photo-created carriers at zero magnetic field. Fluctuations of the localized Mn spin control the statistics of the photons emitted by a single Mn doped QD. The spin relaxation time of the Mn atom, around 20ns, appears to be particularly sensitive to the spin-spin coupling with free carriers injected in the QD or on the vicinity of the QD. The fast photo-induced spin relaxation of the Mn is dominated by spin and energy transfer from hot photo-carriers directly to the Mn atom. Minimizing the injection of free carriers by optimizing the excitation conditions would make it possible to reach longer fluctuation times. As a consequence,a determination of the intrinsic Mn spin dynamics at zero magnetic field, ultimately limited by the spin-lattice interaction, could only be performed in the absence of free or confined carriers.
This work was supported by French ANR contracts MOMES and CoSin.
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---
address: 'Department of Physics, Ohio State University, 1040 Physics Research Building, Columbus, Ohio 43210-1117'
author:
- 'B.Kilminster on behalf of the CDF and DØ Collaborations'
title: Searches for the Standard Model Higgs at the Tevatron
---
Introduction
============
The Higgs boson is the last remaining Standard Model particle to be discovered, and the one responsible for generating the $W$ and $Z$ gauge boson masses. Direct searches at the LEP experiments have excluded a Higgs boson with mass less than 114.4 GeV/c$^2$ at 95% Confidence Level (CL) in the production mode $e^+e^- \to ZH$ [@Barate:2003sz]. Experimental measurements of the top quark and $W$ boson masses provide the strongest indirect constraints on $m_H$. Considering the newest CDF and DØ combined top mass measurement of $m_t =$ 170.9 $\pm$ 1.8 GeV/$c^2$, and the newest CDF $W$ mass measurement of $m_W =$ 80.398 $\pm$ 0.025 GeV$c^2$, in addition to other precision electroweak other observables from LEP and SLD, the Higgs boson mass is expected to be less than 144 GeV/c$^2$ at 95% CL [@eweak].
The Tevatron at Fermilab provides 1.96 TeV center-of-mass energy from proton-antiproton collisions in the two multi-purpose detectors, CDF and DØ . Gluon fusion is the highest cross-section process for producing a Higgs boson, but because of high backgrounds at lower masses, this production process is sensitive to Higgs mainly for $m_H >$ 135 GeV/$c^2$, where BR($H
\to W^+W^-$) starts at 68 % and increases up to 90% for $m_H =$ 160 GeV/$c^{2}$. For $H \to W^+W^-$ in this mass range, the most sensitive final state topology is two charged leptons with large missing transvere energy ( $\MET$). For 114 $< m_H <$ 135 GeV/$c^2$, quark annhilation into an offshell $W$ or $Z$ boson, which then emits a Higgs boson, provides the best opportunity for discovery. At this mass range, the Higgs decays predominantly $H \to b \bar{b}$. CDF and DØ have previously done 1 fb$^{-1}$ searches for $H \to W^+W^- \to l^{+}\nu l^{-}\bar{\nu}$, $WH \to l\nu b\bar{b}$, $ZH \to l^+l^-b\bar{b}$, and $ZH \to \nu\bar{\nu}b\bar{b}$, where $l= e,
\mu$. Because of the small expected Higgs signals, to maximize search sensitivity over the allowed Higgs mass range, it is necessary to combine all searches from both the CDF and DØ experiments, as well as improve analysis techniques.
This proceeding outlines updates to searches in several of these channels, focusing mainly on improvements in analysis techniques. New CDF searches for $ZH \to l^+l^-b\bar{b}$ and $H \to W^+W^- \to l^{+}\bar{\nu}l^{-}\nu$ are presented for the first time, as well as two new DØ searches in the $WH
\to l\nu b\bar{b}$ channel.
$H \to W^+W^- \to l^{+}\nu l^{-}\bar{\nu}$ {#sec:HWW}
==========================================
CDF presents a new search in the $H \to W^+W^- \to l^{+}\nu l^{-}\bar{\nu}$ channel. One improvement in this analysis is the increasing of geometric lepton acceptance by defining new, less stringent lepton types for regions of the detector without complete instrumentation, such as leptons not identifiable as electrons or muons since they are not fiducial to calorimeters or muon chambers. Such lepton types were successfully used in CDFs observaton of $WZ$ production [@Abulencia:2007tu]. The number of Higgs signal events expected in the data for $m_H$ = 160 Gev/c, increases from 2.5 to 4.0 events with this new selection, as compared to the previous CDF analysis with the same dataset. The new analysis also improves upon the technique for extracting the signal from the data. The previous search had performed a likelihood fit for the Higgs signal using the distribution of $\Delta \Phi$ between the two leptons which is sensitive to the angular correlations from $WW$ produced by a scalar Higgs boson. The newest CDF measurement is done by constructing matrix element probabilties using the observed lepton four-vectors and $\MET$ for the processes $H \to WW$, $WW$, $ZZ$, $W+\gamma$, and $W+$parton. A likelihood ratio (LR) is formed for each event by dividing the signal probability by the sum of the signal and background probabilities. LRs are constructed for different processes specified as signal in order to validate background modeling. Figure \[fig:HWW\] shows the $H \to WW$ LR distribution for $m_H$ = 160 GeV/$c^2$, which is used to search for an excess consistent with Higgs signal for a range of masses. No significant excess is measured, and limits are set such that the observed (expected) upper limit is 5 (3.5) times larger than the Standard Model expected cross-section for the most sensitive Higgs mass of $m_H$ = 160 GeV/$c^2$ [@hww].
$ZH \to l^+l^-b\bar{b}$ {#sec:zhllbb}
=======================
CDF also presents a new result in the $ZH \to l^+l^-b\bar{b}$ channel. Previous results first presented at ICHEP 2006 [@zhllbb-ichep] demonstrated the use of a two-dimensional neural network trained to separate $ZH \to l^+l^-b\bar{b}$ from the dominant background of $Z + >=$ 2 jets production and $t\bar{t} \to WbWb \to l{\nu}b l{\nu}b$. The new result uses the same dataset but makes several improvements which result in improved sensitivity. One technique is in the identification of $b$ hadrons from $H
\to b\bar{b}$ using secondary vertex finders or “$b$-tagging” algorthms. The previous $ZH$ analysis selected events with $>= 1$ “$b$-tag”, using tight requirements for the secondary vertex significance (40% efficient). Since S:B is 200:1 for events with one $b$-tag (40% efficient), but 50:1 for events with two looser $b$-tags (each 50% efficient), there is an improvement in Higgs sensitivity by fitting these classes of events separately. Another new technique is to improve the resolution of the $H \to
b\bar{b}$ dijet mass distribution, which is one of the most important Neural Network inputs. Since the main cause of $\MET$ in $ZH \to l^+l^-b\bar{b}$ events is from jet energy mismeasurement, a correction is applied which corrects the leading two jets independently according to their projection onto the $\MET$ direction. The effect is to reduce the dijet mass resolution from 14% to 9% for events with two $b$-tags. With these two enhancements, the analysis improves its Higgs search sensitivity by a factor of two in terms of an effective luminosity increase as compared to the previous version of the analysis with the same dataset. For $m_H =$ 115 GeV/$c^2$, the observed (expected) upper limits for $\sigma_{ZH}$ are 16 times that of the Standard Model [@zhllbb-new].
$WH \to l \nu b\bar{b}$
=======================
DØ presents two new searches in the $WH \to l \nu b\bar{b}$ channels. The first analysis improves over previous analyses by using multiple muon triggers in order to retain 100% muon acceptance for $WH \to \mu \nu
b\bar{b}$. This results in 50% more signal than previous techniques. The $b$-tagging selection is optimized to separate events into one tight $b$-tag and two loose $b$-tags as is described in Section \[sec:zhllbb\]. But in addition, making use of a neural network $b$-tagging algorithm which uses variables in addition to the secondary vertex displacement to identify $b$-quarks, the $b$-tagging efficiency is increased to 50% for tight $b$-tags and 70% for loose $b$-tags, with misidentification rates of 0.5% and 4.5%, respectively. The discriminant used to search for the Higgs signal is the dijet invariant mass (Figure \[fig:wd-d0\]), and the combined limit from both single and double $b$-tagged events is expected (observed) to be less than 9 (10) times the Standard Model expectation [@wh-d0-jj].
The second analysis makes use of a matrix element technique similar to the one described in Section \[sec:HWW\]. This matrix element technique was developed originally in the context of the DØ single-top analyses which established 3-$\sigma$ evidence for single-top production [@Abazov:2006gd]. By fitting the matrix element discriminant for $WH$ (Figure \[fig:wd-d0\]), DØ obtains expected (observed) limits of 9 (13) times the Standard Model expectation [@wh-d0-me]. However, this analysis does not make use of the improved muon acceptance and optimized $b$-tagging used in the first analysis. Incorporating these improvements into the matrix element approach is expected to yield 30% better expected sensitivity.
Conclusions
===========
CDF and DØ are improving analysis techniques in order to make gains in Higgs sensitivity which scale much faster than increasing statistics alone. New results in $H \to WW$, $ZH$, and $WH$ rely on improved lepton acceptance and triggers, higher efficiency $b$-taggers, and dijet mass resolution improvements. The new $WH$ and $H \to WW$ results are better in their most sensitive mass ranges than the combined results of all CDF and DØ channels presented a year ago.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Culture, Sports, Science and Technology of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China; the Swiss National Science Foundation; the A.P. Sloan Foundation; the Bundesministerium für Bildung und Forschung, Germany; the Korean Science and Engineering Foundation and the Korean Research Foundation; the Particle Physics and Astronomy Research Council and the Royal Society, UK; the Institut National de Physique Nucleaire et Physique des Particules/CNRS; the Russian Foundation for Basic Research; the Comisión Interministerial de Ciencia y Tecnología, Spain; the European Community’s Human Potential Programme under contract HPRN-CT-2002-00292; and the Academy of Finland.
References {#references .unnumbered}
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LEP Electronweak Working Group, http://lepewwg.web.cern.ch/LEPEWWG, March 2007 results. A. Abulencia [*et al.*]{} \[CDF Collaboration\], arXiv:hep-ex/0702027. CDF Collaboration. CDF public note 8774, 2007. B.Kilminster \[CDF and DØ collaborations\], ICHEP 2006 proceedings, hep-ex arXiv:hep-ex/0611001. CDF Collaboration. CDF public note 8742, 2007. D0 Collaboration. D0 public note 5350, 2007. V. M. Abazov [*et al.*]{} \[D0 Collaboration\], Phys. Rev. Lett. [**98**]{}, 181802 (2007) \[arXiv:hep-ex/0612052\]. D0 Collaboration. D0 public note 5365, 2007.
|
**A new algorithm that generates the image of the attractor of a generalized iterated function system**
*Radu MICULESCU*, *Alexandru MIHAIL* and *Silviu-Aurelian URZICEANU*
**Abstract**
[We provide a new algorithm (called the grid algorithm) designed to generate the image of the attractor of a generalized iterated function system on a finite dimensional space and we compare it with the deterministic algorithm regarding generalized iterated function systems presented by P. Jaros, Ł. Maślanka and F. Strobin in \[Algorithms generating images of attractors of generalized iterated function systems, Numer. Algorithms, 73 (2016), 477-499\].]{}
*Key words and phrases:*[ ]{}generalized infinite iterated function system (GIFS), attractor, deterministic algorithm, grid algorithm
*2010 Mathematics Subject Classification:* Primary 28A80; Secondary 37C70, 41A65, 65S05, 65P99
**I. Introduction**
As part of the effort to extend the classical theory of iterated function systems due to J. Hutchinson (see \[2\]), R. Miculescu and A. Mihail introduced the concept of generalized iterated function system (see \[7\] and \[9\]) which was obtained by considering contractions from $X^{p}$ into $X$ rather than contractions from $X$ into itself (here $(X,d)$ is a metric space and $p$ is a natural number). Sufficient conditions for the existence and uniqueness of the attractor of a generalized iterated function system (for short GIFS) $\mathcal{F}=((X,d),(f_{i})_{i\in \{1,2,...,L\}})$, an upper bound for the Hausdorff–Pompeiu distance between the attractors of two such GIFSs, an upper bound for the Hausdorff–Pompeiu distance between the attractor of such a GIFS and an arbitrary compact set of $X$ have been provided and the continuous dependence of the attractor on the functions $f_{i}$ was proved. In the last years this concept has been intensively studied. Let us mention some lines of research regarding this subject:
In \[15\], F. Strobin and J. Swaczyna extended the concept of GIFS by using weaker types of generalized contractions which are similar to those introduced by F. Browder (see \[1\]) or J. Matkowski (see \[5\]). In \[14\], Strobin emphasized the fact that the set of the attractors generated by GIFSs is larger than the one generated by iterated function systems. Another related topics can be found in \[4\], \[6\], \[8\], \[10\], \[11\], \[12\], \[13\] and \[16\].
Moreover, in \[3\], Strobin and his collaborators provided four algorithms which generate images of attractors of GIFSs, one of them being the deterministic algorithm for GIFSs (a counterpart of the classical deterministic algorithm for iterated function systems). Note that in \[3\] one can also find a list of papers dealing with algorithms generating images of the attractors of iterated function systems.
In this paper we present another algorithm (called the grid algorithm) allowing to generate images of the attractors of GIFSs on finite dimensional spaces and we compare it with the deterministic algorithm for GIFSs. The deterministic algorithm for GIFSs consists in choosing a finite set of points and applying to this set each of the constitutive functions of the system obtaining in this way a new finite set of points. To each of these new points we apply again each of the constitutive functions of the system. Continuing the procedure described above we approach the attractor. The main idea of the grid algorithm is to simplify the deterministic algorithm by dividing, at each step, the space that we are working with in small pieces and to choose for each such piece just one point.
**II. Preliminaries**
Given a metric space $(X,d)$, we adopt the following notation: $$P_{cp}(X)\overset{not}{=}\{K\subseteq X\mid K\text{ }\ \text{is non-empty
and compact}\}\text{.}$$
For $K_{1},K_{2}\in P_{cp}(X)$, we consider $$d(K_{1},K_{2})\overset{def}{=}\underset{x\in K_{1}}{\sup }d(x,K_{2})\text{,}$$ where $d(x,K_{2})\overset{def}{=}\underset{y\in K_{2}}{\inf }d(x,y)$.
The function $h:P_{cp}(X)\times P_{cp}(X)\rightarrow \lbrack 0,\infty )$ given by $$h(K_{1},K_{2})=\max \{d(K_{1},K_{2}),d(K_{2},K_{1})\}\text{,}$$ for every $K_{1},K_{2}\in P_{cp}(X)$, turns out to be a metric which is called the Hausdorff-Pompeiu metric.
If $(X,d)$ is complete, then $(P_{cp}(X),h)$ is complete.
Given a metric space $(X,d)$ and $p\in \mathbb{N}^{\ast }$, by $X^{p}$ we denote the Cartesian product of $p$ copies of $X$. We endow $X^{p}$ with the maximum metric $d_{\max }$ defined by $$d_{\max }((x_{1},...,x_{p}),(y_{1},...,y_{p}))=\max
\{d(x_{1},y_{1}),...,d(x_{p},y_{p})\}\text{,}$$for all $(x_{1},...,x_{p}),(y_{1},...,y_{p})\in X^{p}$.
**Definition 2.1**. *A generalized iterated function system (of order* $p$*) is a pair* $\mathcal{F}=((X,d),(f_{i})_{i\in
\{1,2,...,L\}})$*, where* $(X,d)$ *is a metric space,* $p,L\in
\mathbb{N}^{\ast }$* and* $f_{i}:X^{p}\rightarrow X$* is contraction for each* $i\in \{1,...,L\}$*. The function* $F
_{\mathcal{F}}:(P_{cp}(X))^{p}\rightarrow P_{cp}(X)$, *described by* $$F_{\mathcal{F}}(K_{1},...,K_{p})=\underset{i\in \{1,...,L\}}{\cup }f_{i}(K_{1}\times ...\times K_{p})\text{,}$$*for all* $K_{1},...,K_{p}\in P_{cp}(X)$, *is called the fractal operator associated to* $\mathcal{F}$.
We shall use the abbreviation GIFS for a generalized iterated function system.
**Theorem 2.2** (see Theorem 3.9 from \[9\])**.** *Given a complete metric space* $(X,d)$ *and a GIFS* $\mathcal{F=}((X,d),(f_{i})_{i\in \{1,...,L\}})$ *of order* $p$*, there exists a unique* $A_{\mathcal{F}}\in P_{cp}(X)$ *such that* $$F_{\mathcal{F}}(A_{\mathcal{F}},...,A_{\mathcal{F}})=A_{\mathcal{F}}\text{\textit{.}}$$
*In addition, for every* $K_{1},...,K_{p}\in P_{cp}(X)$*, the sequence* $(K_{n})_{n}$ *defined by* $$K_{n+p}=F_{\mathcal{F}}(K_{n},...,K_{n+p-1})\text{,}$$*for every* $n\in \mathbb{N}^{\ast }$, *converges, with respect to the Hausdorff-Pompeiu metric, to* $A_{\mathcal{F}}$.
**Definition 2.3**. *In the framework of the above theorem, the set* $A_{\mathcal{F}}$ *is called the fractal generated by* $\mathcal{F}$*.*
**Remark 2.4** (see Remark 12 from \[3\]). *In the framework of the above definition, the function* $\mathcal{G}_{\mathcal{F}}:P_{cp}(X)\rightarrow P_{cp}(X)$, *described by*$$\mathcal{G}_{\mathcal{F}}(K)=F_{\mathcal{F}}(K,...,K)=\underset{i\in \{1,...,L\}}{\cup }f_{i}(K\times...\times K)\text{,}$$*for all* $K\in P_{cp}(X)$,* is a contraction on the complete metric space* $(P_{cp}(X),h)$* since it has the Lipschitz constant less of equal to* $\max \{lip(f_{1}),...,lip(f_{L})\}<1$*.*
**III.** **The presentation of the algorithms**
For $(x_{1},....,x_{M})\in \mathbb{R}^{M}$, we shall use the following notation:$$\lbrack (x_{1},....,x_{M})]=([x_{1}],....,[x_{M}])\text{,}$$where $[x]$ designates the greatest integer less than or equal to the real number $x$.
In the sequel, without loss of generality, $$\mathcal{F}=(([0,D]^{M},d),\{f_{1},...,f_{L}\})\text{,}$$where $L,M\in \mathbb{N}$ and $d$ is the euclidean distance in $\mathbb{R}^{M}$, will be a generalized iterated function system of order $p\geq 2$ (so $f_{i}:([0,D]^{M})^{p}\rightarrow \lbrack 0,D]^{M}$ for every $i\in
\{1,...,L\}$).
We shall use the following notation:
$\bullet $ $\max \{lip(f_{1}),...,lip(f_{L})\}\overset{not}{=}C<1$
$\bullet $ $\beta =pM$.
We also consider the following functions:
$\bullet $ $ F_{\mathcal{F}}:(P_{cp}([0,D]^{M}))^{p}\rightarrow
P_{cp}([0,D]^{M})$ described by$$F_{\mathcal{F}}(K_{1},...,K_{p})=f_{1}(K_{1}\times...\times K_{p})\cup
...\cup f_{L}(K_{1}\times...\times K_{p})\text{,}$$for all $K_{1},...,K_{p}\in P_{cp}([0,D]^{M})$
$\bullet $ $\mathcal{G}_{\mathcal{F}}:P_{cp}([0,D]^{M})\rightarrow
P_{cp}([0,D]^{M})$ described by$$\mathcal{G}_{\mathcal{F}}(K)= F_{\mathcal{F}}(K,...,K)\text{,}$$for every $K\in P_{cp}([0,D]^{M})$.
$\bullet $ $(n_{k})_{k\in \mathbb{N}^{\ast }}$ a sequence of natural numbers.
For a finite set $K_{0}\in P_{cp}([0,D]^{M})$, we shall use the following notations:
$\bullet $ $$A_{k}\overset{not}{=}\mathcal{G}_{\mathcal{F}}^{[k]}(K_{0})\text{,}$$where $k\in \mathbb{N}$
$\bullet $$$\overset{\sim }{A_{k}}\overset{not}{=}\{\frac{D}{n_{k}}[\frac{n_{k}}{D}f_{l}(u_{1},...,u_{p})]\mid u_{1},...,u_{p}\in \overset{\sim }{A}_{k-1}\text{, }l\in \{1,...,L\}\}\text{,}$$where $k\in \mathbb{N}^{\ast }$ and $\overset{\sim }{A_{0}}=K_{0}$
$\bullet $$$\frac{D\sqrt{M}}{n_{k}}\overset{not}{=}\varepsilon _{k}\text{,}$$where $k\in \mathbb{N}$.
Let us recall the pseudocode for the deterministic algorithm for a GIFS (see \[3\])
**Pseudocode for the deterministic algorithm for a GIFS**
Read initially defined objects: constants: $L,M$, finite set, $m$ natural number: $K_{0}\in P_{cp}([0,D]^{M})$, mappings: $f_{1},...,f_{L}$, variables: $k,D_{0}$.
Initial values: $D_{0}:=K_{0}$.
For $k$ from $1$ to $m-1$
$D_{1}:=\mathcal{G}_{\mathcal{F}}(D_{0})$
$D_{0}:=D_{1}$.
Print $D_{m}$.
Now let us present the pseudocode for our new algorithm.
**Pseudocode for the grid algorithm for a GIFS**
Read initially defined objects: constant: $L,$ $M$, finite set, $m$ natural number: $K_{0}\in P_{cp}([0,D]^{M})$, mappings: $f_{1},...,f_{L}$, sequence: $(n_{k})_{k}$, variables: $k,D_{0}$.
Initial values: $D_{0}:=K_{0}$.
For $k$ from $1$ to $m-1$
$D_{1}:=\{\frac{D}{n_{k}}[\frac{n_{k}}{D}f_{l}(u_{1},...,u_{p})]\mid u_{1},...,u_{p}\in D_{0},l\in \{1,...,L\}\}$
$D_{0}:=D_{1}$.
Print $D_{m}$.
**IV.** **The complexity of the algorithms**
By $x_{k}$ we denote the number of points computed at the step $k$ of the deterministic algorithm and by $y_{k}$ the number of points computed up to the step $k$ of the grid algorithm.
**A**. *The case of the deterministic algorithm*
We have $x_{k+1}\leq L(x_{k})^{p}$, so, with the notation $z_{k}\overset{not}{=}\ln x_{k}$, we obtain $z_{k+1}\leq \ln L+pz_{k}$ for every $k\in \mathbb{N}$. Therefore $z_{k}\leq \frac{p^{k}-1}{p-1}\ln L+p^{k}z_{0}$, i.e.$$x_{k}\leq \frac{1}{L^{\frac{1}{p-1}}}(x_{0}L^{\frac{1}{p-1}})^{p^{k}}\text{,}
\tag{1}$$for every $k\in \mathbb{N}$.
Note that, according to Remark 2.4, we have $h(A_{k},A_{\mathcal{F}})\leq
\frac{h(A_{0},A_{1})}{1-C}C^{k}$ for every $k\in \mathbb{N}$. Therefore, in order to be sure that $A_{k}$ approximates the attractor $A_{\mathcal{F}}$ with accuracy $\varepsilon \frac{h(A_{0},A_{1})}{1-C}$, we need $k>\log
_{C^{-1}}(\varepsilon ^{-1})$. Hence, based on $(1)$, the quantity $\frac{1}{L^{\frac{1}{p-1}}}(x_{0}L^{\frac{1}{p-1}})^{p^{\log _{C^{-1}}(\varepsilon
^{-1})}}=\frac{1}{L^{\frac{1}{p-1}}}(x_{0}L^{\frac{1}{p-1}})^{(\frac{1}{\varepsilon })^{\log _{C^{-1}}(p)}}$ describes the number of points that we have to compute in order to be sure that $A_{k}$ is an approximation of $A_{\mathcal{F}}$ with an error less than $\varepsilon \frac{h(A_{0},A_{1})}{1-C}
$.
**Conclusion**: *The complexity of the deterministic algorithm is described by the function* $\mathcal{C}_{c}:(0,\infty )\rightarrow
\mathbb{R}$* given by*$$\mathcal{C}_{c}(\varepsilon )=(x_{0}L^{\frac{1}{p-1}})^{(\frac{1}{\varepsilon })^{\log _{\frac{1}{C}}(p)}}\text{,}$$*for every* $\varepsilon >0$*.*
**B**. *The case of the grid algorithm*
**Remark 4.1**. Since $y_{k+1}\leq L(n_{k})^{\beta }$ for every $k\in
\mathbb{N}^{\ast }$, *up to the step* $N$*, we have to compute* $L\overset{N}{\underset{k=1}{\sum }}(n_{k})^{\beta }=L(D\sqrt{M})^{\beta }\overset{N}{\underset{k=1}{\sum }}(\frac{1}{\varepsilon _{k}})^{\beta }$ *points.*
**Remark 4.2**. *We have*$$h(\overset{\sim }{A_{k}},\mathcal{G}_{\mathcal{F}}(\overset{\sim }{A}_{k-1}))\leq \varepsilon _{k}\text{,}$$*for every* $k\in \mathbb{N}^{\ast }$.
**Remark 4.3**. *We have* $$h(\overset{\sim }{A_{0}},A_{\mathcal{F}})\leq diam([0,D]^{M})=D\sqrt{M}\text{.}$$
As the inequality$$h(\overset{\sim }{A_{k}},A_{\mathcal{F}})\leq h(\overset{\sim }{A_{k}},\mathcal{G}_{\mathcal{F}}(\overset{\sim }{A}_{k-1}))+h(\mathcal{G}_{\mathcal{F}}(\overset{\sim }{A}_{k-1}),\mathcal{G}_{\mathcal{F}}(A_{\mathcal{F}}))\leq$$$$\overset{\text{Remarks 2.4 and 4.2}}{\leq }\varepsilon _{k}+Ch(\overset{\sim
}{A}_{k-1},A_{\mathcal{F}})\text{,}$$is valid for every $k\in \mathbb{N}^{\ast }$, we get$$h(\overset{\sim }{A_{k}},A_{\mathcal{F}})\leq \varepsilon _{k}+C\varepsilon
_{k-1}+C^{2}\varepsilon _{k-2}+...+C^{k-2}\varepsilon
_{2}+C^{k-1}\varepsilon _{1}+C^{k}h(\overset{\sim }{A_{0}},A_{\mathcal{F}})\text{,}$$so, taking into account Remark 4.3, we obtain$$h(\overset{\sim }{A_{k}},A_{\mathcal{F}})\leq \varepsilon _{k}+C\varepsilon
_{k-1}+C^{2}\varepsilon _{k-2}+...+C^{k-2}\varepsilon
_{2}+C^{k-1}\varepsilon _{1}+C^{k}D\sqrt{M}\text{,}$$for every $k\in \mathbb{N}^{\ast }$. Consequently, we arrive to the following problem: given a fixed natural number $N$ and $\varepsilon >0$ such that $\frac{\varepsilon }{C^{N}}-D\sqrt{M}>0$, find the minimum of the function $f:[0,\infty )^{N}\rightarrow \lbrack 0,\infty )$, given by$$f(\varepsilon _{1},...,\varepsilon _{N})=\overset{N}{\underset{k=1}{\sum }}(\frac{1}{\varepsilon _{k}})^{\beta }\text{,}$$for every $\varepsilon _{1},...,\varepsilon _{N}\in \lbrack 0,\infty )$, with the constraint $$\varepsilon _{N}+C\varepsilon _{N-1}+C^{2}\varepsilon
_{N-2}+...+C^{N-2}\varepsilon _{2}+C^{N-1}\varepsilon _{1}+C^{N}D\sqrt{M}=\varepsilon \text{.}$$
We adopt the following notations:
$\bullet $ $t\overset{not}{=}C^{-\frac{\beta }{\beta +1}N}-1$
$\bullet $ $K_{1}\overset{not}{=}\frac{C^{\frac{1}{\beta +1}}-C}{C^{\frac{1}{\beta +1}}}=1-C^{\frac{\beta }{\beta +1}}$
$\bullet $ $K_{2}\overset{not}{=}K_{1}^{-\beta -1}$
$\bullet $ $K_{3}\overset{not}{=}K_{2}\varepsilon ^{-\beta }$
$\bullet $ $a\overset{not}{=}\frac{D\sqrt{M}}{\varepsilon }$
$\bullet $ $y\overset{not}{=}\frac{1}{C^{N}}$.
Since we are going to use the method of Lagrange multipliers, we consider the function $F=f+\lambda g$, where $\lambda \in \mathbb{R}$ and the function $g:[0,\infty )^{N}\rightarrow \lbrack 0,\infty )$ is given by$$g(\varepsilon _{1},...,\varepsilon _{N})=\varepsilon _{N}+C\varepsilon
_{N-1}+...+C^{N-2}\varepsilon _{2}+C^{N-1}\varepsilon _{1}+C^{N}D\sqrt{M}-\varepsilon \text{,}$$for every $\varepsilon _{1},...,\varepsilon _{N}\in \lbrack 0,\infty )$. The equation $\frac{\partial F}{\partial \varepsilon _{k}}=0$, i.e. $-\beta
(\varepsilon _{k})^{-\beta -1}+\lambda C^{N-k}=0$, has the solution$$\varepsilon _{k}^{0}=k_{N}C^{\frac{k}{\beta +1}}\text{,} \tag{1}$$for every $k$, where $k_{N}=\frac{1}{C^{\frac{N}{\beta +1}}}(\frac{\beta }{\lambda })^{\frac{1}{\beta +1}}$. As $g(\varepsilon _{1}^{0},...,\varepsilon
_{N}^{0})=0$, we get $$k_{N}(C^{\frac{N}{\beta +1}}+C^{1+\frac{N-1}{\beta +1}}+C^{2+\frac{N-2}{\beta +1}}+...+C^{N-2+\frac{2}{\beta +1}}+C^{N-1+\frac{1}{\beta +1}})=\varepsilon -C^{N}D\sqrt{M}\text{,}$$i.e.$$k_{N}C^{\frac{N}{\beta +1}}(1+C^{\frac{\beta }{\beta +1}}+C^{^{2\frac{\beta
}{\beta +1}}}+...+C^{^{^{(N-2)\frac{\beta }{\beta +1}}}}+C^{(N-1)\frac{\beta
}{\beta +1}})=\varepsilon -C^{N}D\sqrt{M}\text{,}$$so $k_{N}C^{\frac{N}{\beta +1}}\frac{(C^{\frac{\beta }{\beta +1}})^{N}-1}{C^{\frac{\beta }{\beta +1}}-1}=\varepsilon -C^{N}D\sqrt{M}$, which implies $k_{N}\frac{C^{N}-C^{\frac{N}{\beta +1}}}{C-C^{\frac{1}{\beta +1}}}=\frac{\varepsilon -C^{N}D\sqrt{M}}{C^{\frac{1}{\beta +1}}}$. The last equality takes the form $k_{N}\frac{C^{-\frac{\beta }{\beta +1}N}-1}{C^{\frac{1}{\beta +1}}-C}=\frac{\frac{\varepsilon }{C^{N}}-D\sqrt{M}}{C^{\frac{1}{\beta
+1}}}$. Thus we obtain$$k_{N}=\frac{K_{1}}{t}(\frac{\varepsilon }{C^{N}}-D\sqrt{M})\text{.} \tag{2}$$We have$$f(\varepsilon _{1}^{0},...,\varepsilon _{N}^{0})=\overset{N}{\underset{k=1}{\sum }}(\varepsilon _{k}^{0})^{-\beta }\overset{(1)}{=}(k_{N})^{-\beta }\overset{N}{\underset{k=1}{\sum }}C^{-\frac{\beta }{\beta +1}k}=(k_{N})^{-\beta }C^{-\frac{\beta }{\beta +1}}\frac{(C^{-\frac{\beta }{\beta +1}})^{N}-1}{C^{-\frac{\beta }{\beta +1}}-1}=$$$$\overset{(2)}{=}t^{\beta }K_{1}^{-\beta }(\frac{\varepsilon }{C^{N}}-D\sqrt{M})^{-\beta }\frac{t}{1-C^{\frac{\beta }{\beta +1}}}=t^{\beta +1}(\frac{\varepsilon }{C^{N}}-D\sqrt{M})^{-\beta }\frac{K_{1}^{-\beta }}{K_{1}}=$$$$=t^{\beta +1}(\frac{\varepsilon }{C^{N}}-D\sqrt{M})^{-\beta }K_{1}^{-\beta
-1}=K_{2}(\frac{\varepsilon }{C^{N}}-D\sqrt{M})^{-\beta }(C^{-\frac{\beta }{\beta +1}N}-1)^{\beta +1}\text{.}$$Therefore, the last equality can be written as$$f(\varepsilon _{1}^{0},...,\varepsilon _{N}^{0})=K_{3}(y^{\frac{\beta }{\beta +1}}-1)^{\beta +1}(y-a)^{-\beta }\text{.} \tag{3}$$
As the right hand side of $(3)$ gives us the optimal number of points that we have to compute, after $N$ steps, in order to approximate $A_{\mathcal{F}} $ by $\overset{\sim }{A_{k}}$ with an error not greater than $\varepsilon $, we need to find the minimum value of the function $h:(a,\infty
)\rightarrow \mathbb{R}$ given by $$h(y)=K_{3}(y^{\frac{\beta }{\beta +1}}-1)^{\beta +1}(y-a)^{-\beta }\text{,}$$for every $y\in (a,\infty )$. One can easily see that$$h^{^{\prime }}(y)=K_{3}\beta (y^{\frac{\beta }{\beta +1}}-1)^{\beta
}(y-a)^{-\beta -1}(1-ay^{-\frac{1}{\beta +1}})\text{,}$$for every $y\in (a,\infty )$. As we can suppose that $a>1$ (since we are interested in the case when $\varepsilon $ is small), $\underset{y\rightarrow \infty }{\lim }h(y)=K_{3}$ and $\underset{\underset{y>a}{y\rightarrow a}}{\lim }h(y)=\infty $, we conclude that $h$ attains its minimum at $a^{\beta +1}$ and the value of the minimum is $$h(a^{\beta +1})=K_{3}(a^{\beta }-1)^{\beta +1}(a^{\beta +1}-a)^{-\beta }=$$$$=K_{3}\frac{a^{\beta }-1}{a^{\beta }}=\varepsilon ^{-\beta }(1-C^{\frac{\beta }{\beta +1}})^{-\beta -1}\frac{(\frac{D\sqrt{M}}{\varepsilon })^{\beta
}-1}{(\frac{D\sqrt{M}}{\varepsilon })^{\beta }}\text{,}$$so$$\underset{\underset{\varepsilon >0}{\varepsilon \rightarrow 0}}{\lim }\frac{h(a^{\beta +1})}{(1-C^{\frac{\beta }{\beta +1}})^{-\beta -1}(\frac{1}{\varepsilon })^{\beta }}=1\text{.}$$
**Conclusion**: *The complexity of the grid algorithm is described by the function* $\mathcal{C}_{g}:(0,\infty )\rightarrow \mathbb{R}
$* given by*$$\mathcal{C}_{g}(\varepsilon )=(1-C^{\frac{\beta }{\beta +1}})^{-\beta -1}(\frac{1}{\varepsilon })^{pM}\text{,}$$*for every* $\varepsilon >0$*.*
In the final of this section we mention that (in order to avoid very complicated computations) we did not pay attention to the fact that the best values of $n_{k}$ and $N$ that we obtained (namely $\frac{D\sqrt{M}}{\varepsilon _{k}^{0}}$ and $(\beta +1)\frac{\ln (\frac{\varepsilon }{D\sqrt{M}})}{\ln (C)}$) may not be integers. In reality we could work with $n_{k}=[\frac{D\sqrt{M}}{\varepsilon _{k}^{0}}]+1$ and $N=[(\beta +1)\frac{\ln (\frac{\varepsilon }{D\sqrt{M}})}{\ln (C)}]+1$ without a significant change.
**V. Final remarks**
**Remark 5.1**. We have $$\underset{\underset{\varepsilon >0}{\varepsilon \rightarrow 0}}{\lim }\frac{\mathcal{C}_{g}(\varepsilon )}{\mathcal{C}_{c}(\varepsilon )}=\underset{\underset{\varepsilon >0}{\varepsilon \rightarrow 0}}{\lim }\frac{(1-C^{\frac{\beta }{\beta +1}})^{-\beta -1}(\frac{1}{\varepsilon })^{pM}}{(x_{0}L^{\frac{1}{p-1}})^{(\frac{1}{\varepsilon })^{\log _{\frac{1}{C}}(p)}}}=0\text{,}$$*so the grid algorithm is more efficient than the deterministic algorithm.*
**Remark 5.2**. As $\left\vert u-[u+\frac{1}{2}]\right\vert \leq \frac{1}{2}$, *we can improve our grid algorithm* (which is based on the inequality $\left\vert u-[u]\right\vert <1$)* in the following way:*
**Pseudocode for the improved grid algorithm for GIFS**
Read initially defined objects: constant: $L,$ $M$, finite set, $m$ natural number: $K_{0}\in P_{cp}([0,D]^{M})$, mappings: $f_{1},...,f_{L}$, sequence: $(n_{k})_{k}$, variables: $k,D_{0}$.
Initial values: $D_{0}:=K_{0}$.
For $k$ from $1$ to $m-1$
$D_{1}:=\{\frac{D}{n_{k}}[\frac{n_{k}}{D}f_{l}(u_{1},...,u_{p})+\frac{1}{2}]\mid u_{1},...,u_{p}\in D_{0},l\in
\{1,...,L\}\}$
$D_{0}:=D_{1}$.
Print $D_{m}$.
**Remark 5.3**. On the one hand, repeating the arguments used in IV, A, for the case of an iterated function system (i.e. $p=1$), we obtain that the complexity of the corresponding algorithm is* *described by the function* *$\mathcal{C}:(0,\infty )\rightarrow \mathbb{R}$* *given by$$\mathcal{C}(\varepsilon )=(\frac{1}{\varepsilon })^{\frac{\ln L}{\ln \frac{1}{C}}}\text{,}$$for every* *$\varepsilon >0$*,* so $C$ is involved at the exponent of $\frac{1}{\varepsilon }$.* *We stress upon the fact that since $\underset{\underset{C<1}{C\rightarrow 1}}{\lim }\frac{1}{\ln \frac{1}{C}}=\infty $, the closer is $C$ to $1$, the bigger is the number of points that we have to compute in order to approximate the attractor with an error less that $\varepsilon $.
On the other hand, in the rule that gives $\mathcal{C}_{g}(\varepsilon )$ the constant $C$ is involved only in the coefficient $(1-C^{\frac{\beta }{\beta +1}})^{-\beta -1}$.
Moreover, note that $\underset{\underset{C<1}{C\rightarrow 1}}{\lim }\frac{\mathcal{C}_{g}(\varepsilon )}{\mathcal{C}_{c}(\varepsilon )}=0$ for each $\varepsilon \in (0,1)$.
**VI. Examples**
In section IV we compared the algorithms with respect to a fixed preassigned error. In this section our goal is to get an optimal image (with respect to the computer that we worked with) for three examples. For this reason we chose a version of the grid algorithm for which $n_{k}=k^{2}$, the error being less than $D\sqrt{M}\left(\frac{1}{n^{2}}+C\frac{1}{(n-1)^{2}}+C^{2}\frac{1}{(n-2)^{2}}+...+C^{n}\right)$, where $n$ is the number of steps and $C$ is the contraction constant of the system.
**A**.
Consider the GIFS $\mathcal{F}=(([0,1]^{2},d),\{f_{1},f_{2},f_{3}\})$, where, for $x=(x_{1},y_{1})$ and $y=(x_{2},y_{2})$, we have $$f_{1}(x,y)=(0.2x_{1}+0.2y_{2};0.2x_{2}+0.1y_{2})$$$$f_{2}(x,y)=(0.15x_{1}+0.07x_{2}+0.4;0.15y_{1}+0.07y_{2})\text{.}$$and$$f_{3}(x,y)=(0.15y_{1}+0.07x_{2};0.15x_{1}+0.07y_{2}+0.04)\text{.}$$
Using the deterministic algorithm we get the image indicated in figure 1 and using the grid algorithm we get the image in figure 2.
The deterministic algorithm run 4 steps in 10 seconds, while the grid algorithm 8 steps in less that 10 seconds.
Figure 1 has approximately $3^{2^{4}}=43046721$ points, while figure 2 comprises around 20000 points.
**Remark 6.1**. *If we allow the deterministic algorithm to run 5 steps it needs about 90 minutes and we get a very similar image with the one in figure 2.*
\
Figure 1
{width="9.5cm"}\
Figure 2
**B**.
Consider the GIFS $\mathcal{F}=(([0,1]^{2},d),\{f_{1},f_{2}\})$, where, for $x=(x_{1},y_{1})$ and $y=(x_{2},y_{2})$, we have $$f_{1}(x,y)=(0.1x_{1}+0.16y_{1}-0.01x_{2}+0.3y_{2};-0.05y_{1}+0.15x_{2}+0.15y_{2})$$and$$f_{2}(x,y)=(0.09x_{1}-0.1y_{1}-0.15x_{2}+0.14y_{2}+0.4;0.14x_{1}+0.14y_{1}+0.14x_{2}+0.04)\text{.}$$
The deterministic algorithm yields the image in figure 3 and the grid algorithm produces the image in figure 4.
The deterministic algorithm needed 20 seconds to run 5 steps, while the grid algorithm needed about 10 minutes to run 14 steps.
Figure 3 consists of about $2^{2^{5}}=4294967296$ points, while figure 4 is built up using around $2000000$ p oints.
**Remark 6.2**. *With the aid of the computer that we utilized, the deterministic algorithm would need 42 days to run 6 steps.*
{width="8.5cm"}\
Figure 3
{width="8cm"}\
Figure 4
**C**.
Consider the GIFS $\mathcal{F}=(([0,1]^{2},d),\{f_{1},f_{2}\})$, where, for $x=(x_{1},y_{1})$ and $y=(x_{2},y_{2})$, we have $$f_{1}(x,y)=(0.5x_{1}-0.5y_{1}+0.001x_{2}+0.45;0.5x_{1}+0.5y_{1}+0.001y_{2}-0.05)$$and$$f_{2}(x,y)=(0.2x_{1}+0.01x_{2}+0.14y_{2}+0.147;0.2y_{1}+0.01y_{2}+0.105)\text{.}$$
The image in figure 5 indicates what we get running the deterministic algorithm and the image in figure 6 what we obtain using the grid algorithm.
Both algorithms ran about 2 minutes, the deterministic one running 5 steps, while the grid one 22 steps.
Figure 5 consists of about $2^{2^{5}}=4294967296$ points, while figure 6 is made up of circa $217800$ points.
**Remark 6.3**. *Even though the number of points making up figure 5 is considerably bigger that the number of points building up figure 6, one can observe that the grid algorithm produces a much better approximation of the attractor.*
\
Figure 5
{width="13cm"}\
Figure 6
**Acknowledgements.** The authors are very grateful to the reviewers whose extremely generous and valuable remarks and comments brought substantial improvements to the paper.
**References**
\[1\] F. Browder, On the convergence of successive approximations for nonlinear functional equations, Indag. Math., **30** (1968), 27–35.
\[2\] J.E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J., **30** (1981), 713-747.
\[3\] P. Jaros, Ł. Maślanka and F. Strobin, Algorithms generating images of attractors of generalized iterated function systems, Numer. Algorithms, **73** (2016), 477-499.
\[4\] Ł. Maślanka, F. Strobin, On generalized iterated function systems defined on $l^{\infty }$-sum of a metric space, J. Math. Anal. Appl., **461** (2018), 1795-1832.
\[5\] J. Matkowski, Integrable solutions of functional equations, Diss. Math., 127 (1975), 68 pp.
\[6\] R. Miculescu, Generalized iterated function systems with place dependent probabilities, Acta Appl. Math., **130** (2014), 135-150.
\[7\] A. Mihail and R. Miculescu, Applications of Fixed Point Theorems in the Theory of Generalized IFS, Fixed Point Theory Appl. Volume 2008, Article ID 312876, 11 pages doi: 10.1155/2008/312876.
\[8\] A. Mihail and R. Miculescu, A generalization of the Hutchinson measure, Mediterr. J. Math., **6** (2009), 203–213.
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\[10\] E. Oliveira, The Ergodic Theorem for a new kind of attractor of a GIFS, Chaos Solitons Fractals, **98** (2017), 63–71.
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[Radu MICULESCU]{}
[Faculty of Mathematics and Computer Science]{}
[Transilvania University of Braşov]{}
[Iuliu Maniu Street, nr. 50, 500091]{}
[Braşov, Romania]{}
[E-mail: radu.miculescu@unitbv.ro]{}
[Alexandru MIHAIL]{}
[Faculty of Mathematics and Computer Science]{}
[University of Bucharest, Romania]{}
[Academiei Street 14, 010014, Bucharest, Romania]{}
[E-mail: mihail\_alex@yahoo.com]{}
[Silviu-Aurelian URZICEANU]{}
[Faculty of Mathematics and Computer Science]{}
[University of Piteşti, Romania]{}
[Târgul din Vale 1, 110040, Piteşti, Argeş, Romania]{}
[E-mail: fmi\_silviu@yahoo.com]{}
|
---
abstract: 'The current large uncertainty on the extrapolation of the proton–proton total cross section at the LHC energy will be resolved by the precise measurement by the TOTEM experiment. Its accurate studies on the basic properties of proton–proton collisions at the maximum accelerator energy could provide a significant contribution to the understanding of cosmic ray physics.'
address: |
Istituto Nazionale di Fisica Nucleare, Sezione di Pisa,\
Largo B. Pontecorvo 3, I–56127 Pisa, Italy
author:
- 'S. Lami [^1]'
title: 'TOTEM: The experiment to measure the total proton–proton cross section at LHC'
---
INTRODUCTION
============
TOTEM is an experiment dedicated to the measurement of the proton–proton total cross section ($\sigma_{TOT}^{pp}$) at LHC, i.e. the probability that two protons interact at the center of mass energy of 14 TeV. In addition to that, it will also study the proton–proton elastic scattering and diffractive dissociation processes.
TOTEM foresees specific measurements and experimental techniques which are very different from the other ‘general purpose’ experiments at LHC. A precise ‘luminosity independent’ measurement of $\sigma_{TOT}^{pp}$ will be achievable in special beam optics runs by simultaneously measuring: 1) the elastic scattering rate at low transfer momentum, possibly as small as $t=10^{-3}$ GeV$^2$, and 2) the inelastic scattering rate with the largest possible coverage to reduce losses to few percents. The first goal requires detectors located into units mounted into the vacuum chamber of the accelerator, called Roman Pots (RPs), as the scattered protons are emitted at angles of the order of 10 $\mu$rad, therefore without leaving the beam–pipe. The latter requires the measurement of all the inelastically produced particles in the very forward direction with respect to the $pp$ collision point; this can be achieved by using tracking detector telescopes with a complete azimuthal coverage around the beam–pipe.
In the following, a brief description of the experimental apparatus is given. More details can be found in [@TDR]. I will then discuss physics issues such as the measurement of $\sigma_{TOT}^{pp}$ and the validation of hadronic shower models used in cosmic ray physics.
EXPERIMENTAL APPARATUS
======================
The experimental setup comprises Roman Pot detectors to measure the leading protons at $\pm$ 147 and 220 meters from the LHC interaction point IP5, while the inelastic telescopes T1 and T2 are located inside the CMS end–caps around the beam–pipe, as shown in Fig. \[fig:cmsbello\]. Note also the planned forward calorimeter Castor, under CMS’s responsibility, with the same acceptance of the T2 detector.
Rapidity gaps and forward particle flows could be measured by the TOTEM telescopes T1 and T2, while forward energy flows could be measured by T2 and the CMS Castor forward calorimeter.
Each RP station has two units 4 m apart, as shown in Fig. \[fig:station\]. Each unit has two vertical pots approaching the beam from the top and the bottom, where beams are usually more stable, and one lateral pot sensitive to diffractive protons. Furthermore, the overlap between the horizontal and the vertical pots (Fig. \[fig:overlap\]) will serve for measuring the relative distance of the vertical detectors. Each pot will contain 5+5 planes of Silicon detectors, their strips having orientations of $\pm 45^{\rm o}$ w.r.t. the detector edge, and a pitch of 66 $\mu$m.\
In order to optimize the measurement of microscopic proton scattering angles, the RP detector edge has to move as close to the beam as $\sim$1 mm and, therefore, the edge dead area has to be greatly minimized. A new edgeless technology of Silicon microstrips has been developed, where a current terminating structure will reduce to only 50 $\mu$m the decoupling area between edge and sensitive volume.
Strong magnetic dipoles between the RP stations will provide a powerful magnetic spectrometer. Particle momenta will be measured with an accuracy of a few parts per thousand, allowing an accurate determination of the momentum loss of quasi–elastically scattered protons in diffractive processes.
The T1 telescopes on both sides of the interaction point will cover the pseudorapidity range $3.1 < |\eta| < 4.8$. It will consist of five planes, each composed of six trapezoidal Cathode Strip Chambers (CSC). Each detector will measure three projections: one set of anode wires with a pitch of 3mm measuring the radial coordinate and two sets of cathode strips with a pitch of 5mm, rotated by $\pm 60^{\rm o}$ with respect to the wires. The radial measurement will provide level-1 trigger information and will be used for vertex reconstruction in order to suppress beam-gas background. Beam tests of final prototypes have shown a spatial resolution of 0.36mm in the radial and 0.62mm in the azimuthal coordinate.
For T2, which extends the acceptance into the range $5.2 < |\eta| < 6.7$, the Gas Electron Multiplier (GEM) technology as used successfully in COMPASS [@compassgem] has been chosen. GEMs are gas-filled detectors in which the charge amplification structure is decoupled from the charge collection and readout structure. Furthermore, they combine good spatial resolution with very high rate capability and a good resistance to radiation. The T2 telescope will be placed 13.5m away from the IP5 and the GEMs employed will have an almost semicircular shape, with an inner radius matching the beam pipe. Each half–arm of T2 will have a set of 10 aligned detector planes mounted ‘back-to-back’ on each side of the vacuum pipe (Fig. \[fig:t2\]). To avoid efficiency losses, the angular coverage of each half plane is more than 180$^{\circ}$. The read-out boards will have two separate layers with different patterns: one with 256 concentric circular strips, 80$\mu$m wide and with a pitch of 400$\mu$m, and the other with a matrix of pads varying in size from $2 \times 2\,\rm mm^{2}$ to $7 \times 7\,\rm mm^{2}$ (for a constant $\Delta \eta \times \Delta \phi = 0.06 \times 0.017 \pi$). The pad information will also provide level-1 trigger information. A final prototype has been successfully tested in the 2004 test-beam.
The read-out of all TOTEM detectors will be based on the digital VFAT chips, enhancing the system uniformity from the point of view of the data processing chain.
PHYSICS GOALS
=============
Fig. \[fig:sigmapp\] shows recent predictions [@compete] for the energy dependence of the total $pp$ cross section $\sigma_{TOT}^{pp}$ by fitting all available data. The black error band shows the statistical errors to the best fit, the closest curves near it give the sum of statistical and systematic errors to the best fit due to the ambiguity in the Tevatron data, and the highest and lowest curves show the total error bands from all models considered. For the LHC energy a value $\sigma_{TOT}^{pp}= 111.5 \pm 1.2 ^{+4.1}_{-2.1} $ mb is obtained from the best fit, while the total error band ranges in the 90–130 mb interval.
This large theoretical uncertainty is due to the current lack of a fully satisfactory theoretical explanation of the cross section in low momentum transfer collisions, and their description relies on phenomenological models. TOTEM aims to measure $\sigma_{TOT}^{pp}$ with a precision of 1% or 1 mb, therefore discriminating among the different models.
The typical instantaneous luminosity for the TOTEM $\sigma_{TOT}^{pp}$ measurement will be of the order of 10$^{28} cm^{-2}s^{-1}$. This is due to the high machine optics $\beta^*$ value - 1540 m for the 1% measurement - required in special runs in order to keep as small as possible the beam angular divergence for a precise measurement of small scattering angles. As a consequence, the beam size at the interaction point increases. Therefore, in order to avoid extra interactions between the colliding beams inside the common vacuum chamber, a small number of bunches as well as a zero crossing angle are desirable, resulting in the forementioned luminosity for an optics with $\beta^* =$ 1540 m and 43 bunches.
A special beam optics with $\beta^* =$ 90 m (and a luminosity close to 10$^{30} cm^{-2}s^{-1}$), still enabling a $\sigma_{TOT}^{pp}$ measurement with a few percent uncertainty, would also provide an excellent measurement of the momentum loss of diffractive protons, opening the studies of soft and semihard diffraction, the latter in combination with the CMS detectors.
Without an accurate measurement of the machine luminosity, the only practical way to determine $\sigma_{TOT}^{pp}$ is the ‘luminosity independent’ method which combines the total rate equation, as the sum of elastic and inelastic interactions, to the optical theorem relation between $\sigma_{TOT}^{pp}$, luminosity and the imaginary part of the forward amplitude, such that the luminosity is eliminated and $\sigma_{TOT}^{pp}$ can be written as a function of measurable rates: $${\sigma_{TOT}^{pp}} = {16\pi \over{(1+\rho^2)}} {(dN_{el}/dt)_{t=0} \over{(N_{el}+N_{inel})}}$$ where the optical point at $t=0$ has to be extrapolated from the measurement of the elastic scattering at low momentum transfers.
Let’s consider the measurement with the special $\beta^* =$ 1540 m optics. The statistical error on the extrapolation of the elastic cross section at $t=0$ is less than 0.1% already after 10 hours of data taking. The systematic error is dominated by the insufficient knowledge at very low $t$ of the functional form for the extrapolation, and it will be less than 0.5% if angles as low as 14$\mu$rad - equivalent to about $ t = 10^{-2}$ GeV$^2$ - could be measured, well within the experiment expectations. Elastic events will be selected by a double–arm trigger, with a signal from left amd right RPs, plus the collinearity of the two protons. The vertex reconstruction will help eliminating the background from beam–gas and beam halo events. In addition to that, the selection of inelastic events will include a single–arm trigger in coincidence with a leading proton in the opposite side RP for the single diffractive events. Single diffractive events with masses below 10 GeV will fail the trigger, but the resulting loss can be corrected. This is estimated to give the largest contribution to the total error on the total rate measurement which is about 0.8%. The sum of all uncertainties results in an error of about 1% on $\sigma_{TOT}^{pp}$.
An important measurement to understand the mechanism of high energy collisions is the ratio of the elastic over total cross sections, which in the past was found to increase with energy at CERN and Fermilab. Among several phenomenological models, the one by Bourelly, Soffer and Wu [@bsw] foresees for instance that, at very high energies, the effective interaction radius of the colliding hadrons increase as log$s$, $\sigma_{TOT}$ increase as log$s^2$ and $\sigma_{el}/\sigma_{TOT}$ approach 1/2. At LHC energy, the elastic cross section is supposed to be about 30 mb. To discriminate between different models it is thus important to precisely measure the elastic scattering over the largest possible $t$ region. The $t$ distribution assuming the BSW model is given in Fig. \[fig:sigmael\]. It extends over 11 orders of magnitude and has therefore to be measured with different optics settings. The exponential fall at low $t$ is followed by a diffractive structure at $\sim$1 GeV$^2$ and continues to large $t$-values where perturbative calculations suggest a power–law behaviour ($t^{-8}$). The number of events at the right side of the plot refers to a day running at two conditions (optics with special $\beta^*=$ 1540 m and injection $\beta^*=$ 18 m, respectively). The maximal detectable $t$-value due to aperture limitations in the LHC is 8 GeV$^2$ with $\beta^*=$ 18 m.
While the measurement of the total cross section and the elastic scattering can be performed using only TOTEM detectors, the integration of TOTEM into the general purpose detector CMS offers the prospect of more detailed studies of diffractive events. The TOTEM triggers, combining information from the inelastic detectors and the silicon detectors in the RPs, can be incorporated into the general CMS trigger scheme. The CMS experiment extended by the TOTEM detectors provides the largest acceptance detector ever implemented at a hadron collider.
THE VHE COSMIC RAYS CONNECTION
==============================
The aim of the TOTEM experiment is to obtain accurate information on the basic properties of proton–proton collisions at the maximum accelerator energy, thus providing a significant contribution to the understanding of very high energy cosmic ray physics.
Primary cosmic rays in the PeV (10$^{15}$ eV) energy range and above are a challenging issue in astrophysics. The LHC center of mass energy corresponds to a 100 PeV energy for a fixed target collision in the air, at the same time providing a high event rate relative to the very low rate of cosmic particles in this energy domain.
A primary cosmic ray entering the upper atmosphere experiences a nuclear interaction, with the production of nuclear fragments and $\pi$ mesons, starting an air shower with hadronic, electromagnetic and muon components. The real challenge is to determine the nature of the primary interaction and the energy and composition of the incident particle from the measurement of the shower. Several high energy hadronic interaction models are nowadays available, which predict energy flow, multiplicity and other quantities of such showers. There are large differences between the predictions of currently available models, with significant inconsistencies in the forward region.
Among the several quantities that can be measured by TOTEM and CMS, and compared with model predictions, are: energy flow, transverse energy, elastic/total cross section, fraction of diffractive events, particle multiplicity, ratio of the number of hadronic secondaries to that of leptonic secondaries, and the distribution of the inelasticity coefficient of the incident nucleon (i.e. the ratio of the energy of the most energetic outgoing particle to the energy of the incident particle, it defines the shape of the shower).
Samples of events obtained with some of the available generators (QGSjet 0.1 [@Kalmykov:1997te], SIBYLL 2.1 [@Engel:1999db], DPMJet 3 [@Roesler:2001mn], neXus 3 [@Kalmykov:1997te]) were passed through the simulation of T1, T2 and Castor detectors. Two data samples were considered: all inelastic collisions and diffractive events. Diffractive events were defined as those with a leading proton with a momentum loss $0.003 < \xi < 0.05$.
As an example, Figure \[fig:cosmic-multiplicity\] shows the predictions of the quoted Monte Carlo generators for the charged particle multiplicity. Table \[tab:cosmic-fraction-castor\] shows the fraction of diffractive events expected in Castor. Significant differences in the predictions are evident. The differences are larger for the diffractive events rather than for inelastic events. Appreciable differences are also observed for the inelasticity coefficient as well as for the energy flow. The study of the features of diffractive and inelastic events as measured in Castor and TOTEM may thus be used to validate/tune the generators [@ptdr].
CONCLUSIONS
===========
TOTEM will be ready for data taking at LHC start. The undergoing production of the final TOTEM detectors is proceeding fine and, at the time of writing, a fraction of them is under test on the CERN SPS H8 beam line. The RPs are foreseen to be installed in Spring 2007, while all the detectors will be ready by July 2007.
TOTEM precise measurements of basic properties of proton–proton collisions at LHC will provide a significant contribution to the understanding of cosmic ray physics, by discriminating among currently popular shower models.
\[tab:cosmic-fraction-castor\]
[9]{} TOTEM: Letter of Intent, CERN-LHCC 97-49; Technical Proposal, CERN-LHCC 99-7; Technical Design Report, CERN-LHCC-2004-002; and references therein. C. Altunbas et al., ‘Construction, test and commissioning of the triple-GEM tracking detector for Compass’, NIM A 490 (2002) 177. J.R. Codell et al., Phys. Rev. Lett. 89, 201801 (2002). C. Bourelly, J. Soffer and T.T. Wu, Eur. Phys. J. C28 (2003) 97. N.N. Kalmykov, S. Ostapchenko, A. Pavlov, Nucl. Phys. Proc. Suppl. 52B (1997) 17. R. Engel, T.K. Gaisser, T. Stanev and P. Lipari, ‘Air shower calculations with the new version of SIBYLL’, Prepared for 26th International Cosmic Ray Conference (ICRC99), Salt Lake City, Utah, 17-25 Aug 1999. S. Roesler, R. Engel and J. Ranft, ‘The event generator DPMJET-III at cosmic ray energies’, Prepared for 27th International Cosmic Ray Conference (ICRC2001), Hamburg, Germany, 7-15 Aug 2001. The CMS and TOTEM diffractive and forward physics working group, ‘Prospects for Diffractive and Forward Physics at the LHC’, in preparation.
[^1]: For the TOTEM Collaboration
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abstract: 'For three dimensional geometries, we consider stones (modeled as convex polyhedra) subject to weathering with planar slices of random orientation and depth successively removing material, ultimately yielding smooth and round (i.e. spherical) shapes. An exponentially decaying acceptance probability in the area exposed by a prospective slice provides a stochastically driven physical basis for the removal of material in fracture events. With a variety of quantitative measures, in steady state we find a power law decay of deviations in a toughness parameter $\gamma$ from a perfect spherical shape. We examine the time evolution of shapes for stones initially in the form of cubes as well as irregular fragments created by cleaving a regular solid many times along random fracture planes. In the case of the former, we find two sets of second order structural phase transitions with the usual hallmarks of critical behavior. The first involves the simultaneous loss of facets inherited from the parent solid, while the second transition involves a shift to a spherical profile. Nevertheless, for mono-dispersed cohorts of irregular solids, the loss of primordial facets is not simultaneous but occurs in stages. In the case of initially irregular stones, disorder obscures individual structural transitions, and relevant observables are smooth with respect to time. More broadly, we find that times for the achievement of salient structural milestones scale quadratically in $\gamma$. We use the universal dependence of variables on the fraction of the original volume remaining to calculate time dependent variables for a variety of erosion scenarios with results from a single weathering scheme such as the case in which the fracture acceptance probability depends on the relative area of the prospective new face. In this manner, We calculate time scales of interest and also obtain closed form approximate expressions which bound direct simulation results from above.'
author:
- 'D. J. Priour, Jr'
title: Time Scales for Rounding of Rocks through Stochastic Chipping
---
Introduction
============
Various types of mechanisms act to weather or transform the shapes of rocks by eroding away their volume. Many of these processes, such as collision induced fragmentation, are inherently stochastic. Salient questions related to the emerging shapes as more and more volume is removed by random interactions with neighboring stones include the degree to which rock surfaces become smooth on a macroscopic scale, as well as the extent to which the accumulating random fractures yield spherical profiles, and the rate at which mass is removed through weathering. We investigate these concerns with large scale Monte Carlo simulations in which primordial convex polyhedral rocks (i.e. proto-clasts) with a small number of faces are subject to a sequence of planar fractures which successively carve away material from the parent stones.
With a combination of quantitative measures of the deviation from a perfect spherical shape, the overall oblateness and prolateness, and smoothness in terms of how evenly the total surface area is distributed over the individual facets, we characterize the evolution of the shapes of ensembles of rocks as stochastic fractures accumulate and volume is worn away. To statistical Monte Carlo statistical error, we follow the shape trajectories of at least 1000 stones, which accumulate as many as $2 \times 10^{6}$ planar slices, yielding polyhedra with up to 3,600 facets on average.
Using a two-pronged approach, we find that stones subject to stochastically driven collisional weathering ultimately converge to spherical shapes. On the one hand, we consider a steady state scenario where observables (apart from the volume, which decreases monotonically in time) stabilize at constant values and thus in a sense equilibrate. By tuning a parameter $\gamma$ controlling the coarseness of the shapes at equilibrium (i.e. the mean relative area $\langle \Delta A \rangle /A_{\Sigma}$ of newly formed facets), we find that the equilibrium mean number $\langle n \rangle$ of planar faces scales as $\gamma$ while quantitative measures of the deviation from spherical shapes and/or a perfectly smooth surface tend systematically to zero with increasing $\gamma$. On the other hand, we also calculate the time dependence (i.e. with the number of sustained slices $N_{\mathrm{sust}}$ serving as a proxy for time) of observables in the scenario in which the latter eventually equilibrate. We use the latter to obtain corresponding quantities for a broad range of non-steady state scenarios, a technique we validate with direct Monte Carlo simulation in the context of the fixed velocity scheme, envisaged for rocks impelled by a current at a mean constant speed independent of their mass. In this way, we glean quantitative results for time scales for the attainment of milestones such as the removal of a specific fraction of the primordial volume, finding that such times vary depending on the specific erosion scenario under consideration but do not diverge, a point we underscore with approximate closed form expressions also yielding finite times while serving as upper bounds for simulation results.
Taking into consideration that in practice cohorts of primordial stones prior to weathering may be structurally diverse, we consider protoclasts in the form of irregularly shaped polyhedra to account for aggressive fragmentation events prior to the collision induced weathering. To mimic structures resulting from dramatic fragmentation processes, initial cohorts of structurally disordered shapes are generated with a sequence of random fracturing planes, invariably accepted independent of the orientation and depth of the prospective slice. With protoclast cohorts generated in this manner, we find relevant observables to vary smoothly with time. As companion calculations, we also examine structurally homogeneous primordial stones, including regular cubes, which ultimately are smoothed into spherical shapes just as occurs in the case of irregular protoclasts. However in contrast to structurally disordered stones, observables in the case of cubes exhibit two continuous phase transitions with all of the hallmarks of a second order phase transition, including power law scaling for singularities in relevant observables as we show explicitly in this work. The first transition is a structural transformation in which the primordial cube facets are sheared away, while in a subsequent transition the stones derived from regular solids revert to spherical shapes with concomitant singularities in quantities sensitive to deviations from a perfect spherical profile. A structural phase transition of the former type has been proposed in previous studies based on laboratory experiments and numerical studies [@Domokos2; @Domokos3]. We apply the tools of finite size scaling to locate the transition and calculate associated critical exponents.
In addition to regular shapes, we consider a homogeneous ensemble of geometrically irregular solids finding asynchronous transitions for the erosion of primordial facets instead of a single structural transition for the solid as a whole, though again we find a single second order transition to a round spherical shape at a later time after all of the primordial facets have been removed. Thus, only in the case of highly symmetric shapes is there a simultaneous elimination of the facets inherited from the parent solid.
In spite of the stochastic nature of the erosion scheme, we find in the regime that fragments cleaved away through random fracture events are small in volume relative to that of the stone as a whole that shapes evolve deterministically, with reproducible changes on scales small in comparison with the overall size of the stone but large relative to freshly exposed faces. As in a recent study, we find a universal dependence of observables on the remaining volume fraction [@Domokos1]. We take advantage of this characteristic to calculate time dependent observables for a variety of erosion schemes using results from the scenario of fixed relative area for typical fresh facets created by fracture events.
Aside from measures of deviation from a perfectly spherical shape, we also examine more specific macroscopic structural characteristics, including the degree to which stones are oblate or prolate as volume is chipped away. While both the former and the latter tend to zero as rocks are rounded, the oblateness measure is non-monotonic in time, initially rising and then peaking after approximately 50% of the original volume has been removed, and declining and tending to zero thereafter.
As a novel feature of the calculations in this work, no restrictions are imposed as to how many vertices and facets are truncated or removed by a sustained slice, and our calculation is compatible with the elimination of an arbitrary number of vertices per fracture. For a plausible physical basis for fracture events, measures such as the area of a newly exposed face are considered to determine if a candidate fracture plane is accepted. To set the scale of new facet areas, we implement an energy based criterion for accepting prospective slices, taking the kinetic energy input needed to cleave away a slice to be proportional to the area of the new face. In this vein, in the spirit of statistical mechanical treatments, we assume the fracture probability to be exponentially suppressed in the exposed area $\Delta A$, and given by $P(\Delta A) = e^{-\gamma \Delta A/\theta}$ where $\gamma$ is a constant related to the toughness of the material comprising the rocks and $\theta(V)$ is a volume dependent measure of the mean kinetic energy or “temperature”, with volume serving as a proxy for the mass in the case of stones of uniform composition as assumed in this manuscript. In addition, since we consider initial cohorts of stones to be mono-dispersed with respect to mass (though structurally diverse in the case of irregular protoclasts), we operate in terms of the relative volume $\tilde{v} \equiv V/V_{0}$ with $V_{0}$ the volume prior to the erosion process. With analytical arguments and direct simulation results, we show that characteristic times scale quadratically in the toughness parameter $\gamma$.
In this work to be definite we assume a power law dependence $\theta = \theta_{0} \tilde{v}^{\alpha}$ (i.e. with $\alpha > 0$), reflecting kinetic energy dependence on volume as a rock is borne along (e.g. by water currents in a river or stream as collisions chip away material). Defining $\gamma^{'} \equiv \gamma/\theta_{0}$ for the sake of brevity, the acceptance probability for a prospective fracture plane is $P(\Delta A) = e^{-\gamma^{'} \Delta A \tilde{v}^{-\alpha} }$. Assuming $\gamma^{'} \langle \Delta A \rangle \tilde{v}^{-\alpha} \sim 1$ for the mean area $\langle \Delta A \rangle$ of a new facet, one would argue on dimensional grounds that the mean number of faces is $\langle n \rangle \sim A_{\Sigma}/\langle \Delta A \rangle = \gamma^{'} \tilde{v}^{-\alpha} A_{\Sigma}$, where $A_{\Sigma}$ is the total polyhedron surface area. For the 3D case, taking the total area to scale as $\tilde{v}^{2/3}$, $\alpha_{c}$ marks a boundary between stones which in principle could become smoother over time ($\alpha > \alpha_{c}$) and shapes which become coarser ($\alpha < \alpha_{c}$) as fractures accumulate; whereas relative areas $\langle \Delta A \rangle /A_{\Sigma}$ of new facets decrease for the former, newly exposed facets encompass an increasingly large share of the surface in the case of the latter. The boundary case $\alpha = \alpha_{c}$ thus offers the possibility for the attainment of a steady state for the mean relative area of new faces, as well as other variables of interest.
Another merit of obtaining steady state or relative area results is the fact that the universal dependence of observables on volume fraction permits one to map remaining stone volumes onto time for a scheme distinct from the relative area case, an technique bolstered with results from large-scale Monte Carlo simulations. We show in this manuscript that both the former and latter are identical up to random Monte Carlo statistical error. Time scales obtained in this manner are then used to validate a closed form expression serving as an approximation and strict upper bound for times associated with the chipping away of a specific volume fraction or the structural phase transition for cohorts of regular protoclasts in which primordial facets disappear.
In Section II, we discuss principal quantitative measures of deviation from sphericity as well as key methodological elements. Section III contains results for rocks in the case of the steady state scenario, while Section IV broadens the discussion to non-steady state situations. In Section V, we calculate time scales for a range of erosion schemes, with a closed form expression obtained as an upper bound for time scales for salient structural milestones. Conclusions are found in Section VI.
Quantitative Measures and Methodology
=====================================
Two salient quantities which permit one to address in a direct manner the degree to which a shape departs from a perfect spherical profile include the configuration averaged sphericity $\phi_{3}$ and the ratio $r^{\mathrm{min}}_{\mathrm{max}}$ of the minimum and maximum distance of surface points for the center of mass. Both measures distinguish among a spherical shape and a stone which is not perfectly round. The sphericity benefits from the fact that surface area to volume ratio of a solid attains a global minimum for spheres, and is given by $\phi_{3} \equiv [6 \pi^{\frac{1}{2}} V]^{\frac{2}{3}}/A_{\Sigma}$ [@Wadell], where $A_{\Sigma}$ is the surface area and $V$ the volume. Since $\phi_{3} < 1$ (except in the case of a perfect sphere where $\phi_{3} = 1$), the complement $1 - \phi_{3}$ serves as a measure of the departure from a perfectly spherical shape. An alternative measure, sensitive to local features, is $r^{\mathrm{min}}_{\mathrm{max}} \equiv d_{\mathrm{min}}/d_{\mathrm{max}}$, the ratio of the minimum and maximum distances, respectively, of points on the rock surface from its center of mass. In the context of our calculations involving faceted objects, $d_{\mathrm{min}}$ is the distance to the closest planar facet and $d_{\mathrm{max}}$ is the distance to the farthest vertex. Again, $r^{\mathrm{min}}_{\mathrm{max}}$ peaks at unity only in the case of a perfect sphere, with $1 - r^{\mathrm{min}}_{\mathrm{max}}$ serving as a measure of the deviation from a perfect spherical shape.
Related to whether there is a systematic convergence to a spherical shape is if erosion through the stochastic chipping mechanism yields a stone with a smooth surface. Although sphericity implies smoothness, the converse is not true. In fact, in this work we find an example of a perfectly smooth non-spherical shape at the phase transition for structurally homogeneous cohorts where primordial facets disappear. Hence, as a measure of smoothness not anchored to a specific geometry, we use the Inverse Participation Ratio (IPR), where $\textrm{IPR} = \langle A_{\Sigma}^{-2} \sum_{i=1}^{N} A_{i}^{2} \rangle$ with $A_{\Sigma}$ being the total surface area, $A_{i}$ the area of the $i$th of $N$ facets, and angular brackets indicated a configurational average. The IPR tends to zero with increasing $N$ for an even distribution of the area of the polyhedron faces (i.e. for smooth shapes), but converges to a finite value if a small number of facets contain a macroscopic fraction of $A_{\Sigma}$ (i.e for rough stones), and thus serves as a quantitative measure of smoothness. Appealing to the IPR in this manner is analogous to its usage in the context of charge transport discussions to distinguish among extended and localized carrier states [@Wegner].
In this work, we simulate collisional weathering with large scale Monte Carlo simulations involving sequences of randomly oriented planar fractures, where rocks begin as regular or irregular protoclasts with a small number of faces \[i.e. 6 and a mean of 9.03(1) sides for cubic and irregular polyhedra respectively\], ultimately yielding polyhedra with as many as 3,600 facets. Although there are qualitative differences for the time dependence of observables in the cases of cube shaped and irregular protoclasts (e.g. the structural phase transition for cube shaped protoclasts not seen in ensembles of irregular shapes), both cohorts ultimately converge to spherical shapes in the long time regime. Moreover, we obtain an approximate analytical expression for time scales for salient stages in the attainment of a perfectly spherical shape. The fact that these times are finite, bounding corresponding numerical results from above is compatible with the eventual conversion of rocks into spherical shapes for a wide range of erosion scenarios.
In stochastic chipping sequences, each sustained slice exposes a new face while truncating one or more vertices. The likelihood of accepting a prospective slice hinges solely on the area $\Delta A$ and volume of the parent solid with no restriction on the number of truncated vertices, raising the possibility of the elimination of an entire face or faces if each of the constituent vertices are cleaved away. For the sake of an efficient implementation, updates of lists of vertices and planes making up the polyhedron include the addition of new features as well as the deletion of those eliminated by the most recent fracture event. Indeed, in the steady state scenario, as many vertices and facets are removed on average as accumulate with each sustained slice. Previous studies in two and three dimensional geometries have involved the truncation of a finite number of vertices or facets per fracture event [@Durian2; @Krapivsky; @Domokos2; @Domokos1]. A calculation involving two dimensional shapes allowed for the removal of an arbitrary number of vertices [@Durian1] To our knowledge, our calculations are the first to examine erosion phenomena due to a sequence of planar fractures allowing for the elimination of an arbitrary number of vertices and/or facets for three dimensional geometries.
To facilitate the examination of polyhedra with a large number of faces, a variety of measures are employed to optimize the efficiency of locating vertices of prospective facets. While one could in principle examine all possible intersections of a fracture plane with the faces comprising the stone, in general only a small subset of vertices identified in this manner populate the new face, comprising a share of the total which diminishes as the number of planar faces increases. We avoid considering spurious vertices by only examining faces which contain vertices both above and below the prospective fracture plane, as facets not meeting this condition are either sheared away altogether or are not truncated by slice at all and hence do not yield any new constituent vertices. As a further constraint, we only seek intersections of the slicing plane with pairs of faces which have an edge in common. Finally, we avoid computational costs associated with slices unlikely to be accepted by considering a sphere inscribed in the polyhedron and concentric with its center of mass, for which circular regions exposed by a fracture plane bound the area of a prospective face from below, with the corresponding acceptance probability overestimating that of the prospective new facet. In this manner, we avoid consideration of fracture planes for which the acceptance probability is no greater than $10^{-8}$, events with negligible incidence over the course of an erosion sequence. Operating in this manner, the computational burden of a sustained slice scales no more rapidly than the number of facets of the solids we consider.
The areas of individual facets and the total volume are key quantities in finding the probability of a prospective fracture event as well as many of the observables we calculate. For individual faces, the arithmetic mean of each of the member vertex coordinates provides a convenient interior point for subdividing the polygonal facet into triangles, whose areas are calculated and summed to yield the combined face area. On the other hand, to find the total volume of the stone one operates in an analogous fashion, partitioning the polyhedron into constituent tetrahedra whose combined volume is the shape total volume. To this end, we again subdivide faces into triangles; the three vertices of the latter along with an interior point (given by the mean of all of the polyhedron member vertices) define the component tetrahedron.
Complementary to the sphericity measures and the IPR are quantities which bear more specifically on the shape of the stone, namely parameters which characterize the degree to which a stone is oblate or prolate. Measures of the latter are gleaned from the principle moments of inertia of the equivalent ellipsoid, obtained by diagonalizing the moment of inertia tensor of the stone. As in the calculation of total volume, the polyhedron is partitioned into constituent tetrahedra with elements $I_{ij}$ of the moment of inertia tensor given by [@Tonon] $$I_{ij} = \frac{V_{\mathrm{tet}}}{20} \left [\delta_{ij} \sum_{k=1}^{3}(s_{k}^{2} + t_{kk}) - s_{i}s_{j} - t_{ij} \right ]
\label{eq:Eq137}$$ where $V_{\mathrm{tet}}$ is the volume of the constituent tetrahedron, while $s_{i} \equiv \sum_{l=1}^{4} x_{li}$ and $t_{ij} \equiv \sum_{l=1}^{4} x_{li} x_{lj}$ with, e.g., $x_{li}$ being the $i$th component of the $l$th vertex of the tetrahedron. Combining moments for each component tetrahedron yields the moment of inertia tensor for the shape as a whole. To find the moment of inertia relative to the center of mass, as appropriate for a *bona fide* ellipsoid, we appeal to the Steiner parallel axis theorem [@Marion] while taking advantage of the fact that the center of mass for a tetrahedron of uniform composition is the arithmetic mean of its four vertices to find the center of mass of the entire shape. In terms of semi-major axes $\{ a, b, c \}$ of the equivalent ellipsoid (i.e. with $a > b > c$), the oblateness and prolateness measures $\psi_{\mathrm{O}}$ and $\psi_{\mathrm{P}}$ are taken to be $\psi_{\mathrm{O}} = (b - c)/a$ and $\psi_{\mathrm{P}} = (a - b)/a$, respectively.
Regular and Irregular Protoclasts
---------------------------------
In selecting initial shapes, our simulation program bifurcates, including both regular and irregular polyhedra as primordial forms. In the case of the former, we examine cubes, and the benefit of beginning with regular shapes is two-fold, with one useful feature being that equilibration in steady state scenarios is more rapid for regular polyhedra than for irregular counterparts. In addition, we find in the case of cube shaped protoclasts two structural phase transitions, in the first of which all remnants of the primordial facets are cleaved away simultaneously. Subsequently, an additional second order transition signals a shift to a spherical shape.
![\[fig:Fig1\] (Color online) Sixty four sample irregular protoclasts](clast1.eps){width=".45\textwidth"}
Cohorts of initially irregular shapes, on the other hand, are envisaged as a closer geometric match to protoclasts formed through dramatic fragmentation events. To generate irregular polyhedra in this very aggressive manner, a cube is subject to a sequence of slices, with each prospective slice accepted independent of the area $\Delta A$ of the new face. Images of sample polyhedra generated in this fashion are shown in Fig. \[fig:Fig1\]. As illustrated in panel (a) of Fig. \[fig:Fig2\], with on the order of 15 sustained slices, the mean number of sides quickly converges to 9.03(1) facets on average. The traces in the main graph, plotted with respect to $N_{\mathrm{sust}}$, were obtained with cubes and tetrahedra as initial forms. The frequency distribution of sides, shown in the graph inset, converges with similar rapidity; in the inset graph, symbols indicating Monte Carlo results and the solid line being a fit to a log-normal distribution with $\sigma = 0.233$ and $\mu = 2.189$. We also exhibit in panels (b) and (c) of Fig. \[fig:Fig2\] the survival probability $f_{\mathrm{sur}}$ or mean likelihood of the persistence of a facet or portion of a facet original to the parent solid. As indicated in panel (b) of Fig. \[fig:Fig2\], $f_{\mathrm{sur}}$ is strongly suppressed after 25 sustained slices. As the semilogarithmic plot with respect to the square of the number of sustained slices in panel (c) of Fig. \[fig:Fig2\] suggests, the $f_{\mathrm{sur}}$ decay at an approximately Gaussian rate with $f_{\mathrm{sur}} \approx e^{-(N_{\mathrm{sust}}/N_{0})^{2}}$ where $N_{0} = 9.98$ for cube shaped initial forms and $N_{0} = 11.4$ for tetrahedron shaped protoclasts. In simulations involving irregular protoclasts, 100 sustained slices are imposed to remove any vestige of the seed form.
![\[fig:Fig2\] (Color online) Mean number of facets versus sustained slices versus sustained slices for cubic and tetrahedral protoclasts (square and triangular symbols respectively), with solid curves a guide to the eye. Shown in the inset is the frequency facet number distribution; circles are Monte Carlo results while the solid line is an optimized fit to a log-normal distribution. Panels (b) and (c) show the fraction of original facets remaining with solid lines a fit to a Gaussian decay in $N_{\mathrm{sust}}$.](clast2a.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig2\] (Color online) Mean number of facets versus sustained slices versus sustained slices for cubic and tetrahedral protoclasts (square and triangular symbols respectively), with solid curves a guide to the eye. Shown in the inset is the frequency facet number distribution; circles are Monte Carlo results while the solid line is an optimized fit to a log-normal distribution. Panels (b) and (c) show the fraction of original facets remaining with solid lines a fit to a Gaussian decay in $N_{\mathrm{sust}}$.](clast2b.eps "fig:"){width=".45\textwidth"}
Steady State Scenario
=====================
For the $\alpha = \alpha_{c}$ case where the acceptance probability for a prospective slice is $e^{-\gamma \Delta A/A_{s}}$ ($A_{s} = [6 \pi^{\frac{1}{2}} V]^{\frac{2}{3}}$ being the area of the volume equivalent sphere), a steady state may in principle develop with the mean number of faces $\langle n \rangle$ converging to a fixed value since $\langle \Delta A \rangle \sim A_{\Sigma}/\gamma$ per the acceptance probability relation. Images of stones sampled from the steady state ensemble appear in Fig. \[fig:Fig3\], showing the emergence of smooth spherical shapes for toughness parameters ranging from $\gamma = 10$ to $\gamma = 2000$.
A salient question related to the attainment of equilibrium is how long is needed (in terms of the number of sustained slices $N_{\mathrm{sust}}$) for an ensemble of stones to reach steady state. In the context of numerical simulations, we insist that all observables of interest remain constant with respect to doubling of $N_{\mathrm{sust}}$. For $\gamma = 100$, 3,000 sustained slices satisfies this criterion. As we now argue from geometric considerations, for $\gamma \gg 1$ (i.e. certainly for $\gamma \geq 100$) time scales for the removal of equivalent fractions of the original volume of an ensemble of rocks (and hence salient stages such as the achievement of steady state) scale quadratically with the toughness parameter $\gamma$.
Although the acceptance probability sets the scale $\langle \Delta A \rangle$ for the typical area of a new facet, fractures leaving behind faces with the same area need not cleave away the same volume. One anticipates more volume to be removed from sharply peaked features than from regions with less curvature by the same number of sustained slices [@Jerolmack1]. Thus the edges and vertices of the primordial polyhedron, are quickly worn down in the initial stages of the erosion process.
In the $\gamma \gg 1$ regime, fracture planes are constrained to be shallow, unable to penetrate deeply due to the exponential penalty on $\Delta A$ in the acceptance probability. We introduce the cleaving plane, just below the rock surface, as well as the parallel plane of tangency with a single point of contact with the surface, while superimposing Cartesian axes with the $xy$ plane coinciding with the tangent plane and the $z$ axis directed toward the interior of the stone. Hence, for the rock surface one would generically have have $f(x,y) \approx A x^{2} + 2 B xy + C y^{2}$ near its minimum at the point of tangency, where $A$, $B$, and $C$ are locally determined constants. Seeking to eliminate the diagonal term, we have in matrix form $$f(x,y) = \left [ \begin{array}{cc} x & y \end{array} \right ] \left[ \begin{array}{cc} A & B \\ B & C \end{array} \right ]
\left[ \begin{array}{c} x \\ y \end{array} \right ];~~\vec{v}_{\mathrm{plane}} \equiv \left [ \begin{array}{c} x \\ y \end{array} \right ]$$ where $\vec{v}_{\mathrm{plane}}$ is the component in the tangent plane. The eigenvalues of the symmetric matrix are: $\lambda_{1,2} = \frac{1}{2}[(A + C) \pm \sqrt{ (A - C)^{2} + 4 B^{2}} ]$ with $\lambda_{1,2} > 0$ if $AC > B^{2}$, as must be the case for the convex objects we consider. In terms of the orthonormal eigenvectors $\hat{u}_{1}$ and $\hat{u}_{2}$, one may write $\vec{v}_{\mathrm{plane}} = \alpha_{1} \hat{u}_{1} + \alpha_{2} \hat{u}_{2}$ and thus $$f(x,y) \rightarrow g(\alpha_{1},\alpha_{2}) = \lambda_{1} \alpha_{1}^{2} + \lambda_{2} \alpha_{2}^{2}$$
The cleaving plane hence exposes an elliptical region for which $z = \lambda_{1} \alpha_{1}^{2}
+ \lambda_{2} \alpha_{2}^{2}$ for a fixed depth $z$ beneath the point of tangency, where the area is $A(z) = \pi a b = \pi z/l$ with $l \equiv (\lambda_{1} \lambda_{2})^{-1/2}$ being a locally defined length scale. Thus, the depth of a typical fracture plane (i.e. with an area $\langle A \rangle$) is $D = \langle \Delta A \rangle (\pi l)^{-1}$, and the typical volume $\Delta V$ removed is $\Delta V = \int_{0}^{D} A(z) dz = \langle \Delta A \rangle^{2}/(2 \pi l) \sim \gamma^{-2} A_{\mathrm{\Sigma}}/(2 \pi l)$ since $\langle \Delta A \rangle \sim A_{\Sigma} \gamma^{-1}$ for the relative area scenario. Thus, time scales for the removal of volume scale as $\gamma^{2}$ in the steady state scenario and more broadly as the square of the toughness parameter in non-steady state schemes as well.
![\[fig:Fig3\] (Color online) Steady state shapes for various $\gamma$ values](clast3.eps){width=".45\textwidth"}
The Independent Facet Model
---------------------------
In the steady state calculations, we obtain configuration averaged observables (i.e. for at least 1000 distinct realizations) for two and a half decades of the mean number $\langle n \rangle$ of facets, finding power law decays in $\gamma$ and $\langle n \rangle$ for $\gamma \gg 1$ for all measures of the deviation from perfect spherical shapes. As a model for quantities of interest, we assume quasi-spherical shapes for cohorts of stones in steady state with properties (e.g. areas) of polyhedron faces assumed to be statistically uncorrelated; in this Independent Facet Model (IFM), characteristics of polyhedron faces are considered to statistically uncorrelated; nevertheless, we find good quantitative agreement with salient observables from Monte Carlo simulations.
For a spherical geometry, the relative area of a fresh facet is $\Delta A/A = (2 \tilde{u} - \tilde{u}^{2})/4$ where $\tilde{u} = u/R$, with $R$ being the mean radius of the stone and $u$ the distance of the planar slice below the surface of the rock. With the probabilities of new faces being exponentially suppressed in $\Delta A$, mean facet variables $\langle f(\tilde{v}) \rangle$ are given by $$\langle f( \tilde{u} ) \rangle =
\frac{\int_{0}^{1} f(\tilde{u}) e^{-\gamma(2 \tilde{u} - \tilde{u}^{2})/4} d \tilde{u}}{\int_{0}^{1} e^{-\gamma (2 \tilde{u} -
\tilde{u}^{2})/4} d \tilde{u}}
\label{eq:Eq100}$$ where the denominator plays a role analogous to that of the partition function in statistical mechanics; for the sake of obtaining closed form relations accurate to leading or next to leading order in $\gamma$ in for $\gamma \gg 1$, Eq. \[eq:Eq100\] reduces to $$\langle f(\tilde{u}) \rangle \approx \frac{1}{2} \gamma \int_{0}^{\infty} f(\tilde{u}) e^{-\gamma \tilde{u}/2} d \tilde{u}
\label{eq:Eq150}$$
![\[fig:Fig4\] (Color online) Mean number of facets with respect to $\gamma$. Symbols are Monte Carlo results, while the solid line in the main graph and inset is an analytical curve.](clast4.eps){width=".45\textwidth"}
Fig. \[fig:Fig4\] shows the mean facet number $\langle n \rangle$ with respect to $\gamma$, with symbols representing Monte Carlo results; analytical results indicated by the solid curve are in good agreement with the latter. The main graph is a plot of the ratio $\langle n \rangle/\gamma$ with respect to $\log_{10}(\gamma)$, which converges to 1.82(1) in the large $\gamma$ regime, while the inset graph is a log-log plot of $\langle n \rangle$ versus $\gamma$. A salient feature is the comparatively gradual increase in $\langle n \rangle$ for $\gamma \leq 10$ with a more rapid asymptotically linear rise thereafter.
The analytical curve is obtained assuming a truncation over time of newly exposed faces, where in terms of the mean area $\langle A_{0} \rangle$ of newly created faces, the average area of $\langle A \rangle$ over its lifetime is $\langle A \rangle = \eta \langle A_{0} \rangle$ where $\eta < 1$. The mean solid angle subtended by a facet, modeled as a circular shape (with the reduction in area taken into account) is $f_{\Omega}(\tilde{u}) = 2 \pi [1 - \sqrt{1 - \eta(2 \tilde{u} - \tilde{u}^{2})}]$ where the mean number of facets is hence $\langle n \rangle = 4 \pi/\langle f_{\Omega} (\tilde{u} ) \rangle$.
One fixes $\eta$ by appealing to the large $\gamma$ limit where $\tilde{u} \ll 1$ due to the shallowness of typical sustained slices. In this regime, one may expand the radical in $f_{\Omega}(\tilde{u})$ expression; using Equation \[eq:Eq150\], we find $\langle n \rangle \approx \gamma/\eta$, $\eta = 0.549(3)$ and $\langle n \rangle = 1.82(1) \gamma$ for $\gamma \gg 1$.
![\[fig:Fig5\] (Color online) Inverse Participation Ratio (IPR) log-log plot (main graph) and IPR with respect to the reciprocal of the mean facet number (inset). Symbols represent Monte Carlo data, while the solid curve is from an analytical model.](clast5.eps){width=".45\textwidth"}
The Inverse Participation Ratio (IPR) is plotted in Fig. \[fig:Fig5\] with the solid curve obtained in the Independent Facet Model framework and symbols representing Monte Carlo results in both the inset and the main graph with good quantitative agreement among the latter and the former. In the case of the inset, the IPR is plotted with respect to $\langle n \rangle^{-1}$, suggesting an extrapolation to zero as the number of facets becomes infinitely large and thus smooth surfaces in the large $\gamma$ limit. On the other hand, the linear decrease of IPR in the log-log curve for $\gamma \geq 10$ in the main graph is compatible with an asymptotically power law decrease (specifically as $\gamma^{-1}$ or $\langle n \rangle^{-1}$) for the Inverse Participation Ratio.
In the IFM framework, we posit that $\langle \sum_{i = 1}^{n} A_{i} \rangle$ may be replaced with $\langle A \rangle \langle n \rangle$ and $\langle \sum_{i = 1}^{n} A_{i}^{2} \rangle$ with $\langle A^{2} \rangle \langle n \rangle$, yielding $$\textrm{IPR} \approx \frac{\chi \langle f(2 \tilde{u} - \tilde{u}^{2}) \rangle }{ \langle n \rangle \langle f([2 \tilde{u} - \tilde{u}^{2}]^{2}) \rangle}$$ where $\chi = 1.32(1)$ is a dimensionless fitting parameter on the order of unity. The analytical IPR curve obtained in this manner with the mild rescaling by $\chi$ is in excellent quantitative agreement with Monte Carlo simulation results as may be seen in Fig. \[fig:Fig6\]. By appealing to Equation \[eq:Eq150\] in the $\gamma \gg 1$ regime, we find that the inverse participation ratio tends to: $\textrm{IPR} = (2 \chi \eta) \gamma^{-1} = (2 \chi) \langle n \rangle^{-1}$.
![\[fig:Fig6\] (Color online) Spherical measure complements $1 - r^{\textrm{min}}_{\textrm{max}}$ (open circles) $1 - \phi_{3}$ (filled circles) log log plot (main graph) and results plotted with respect to the reciprocal of the mean facet number (inset). The solid blue curve in the main graph is an analytical curve from the Independent Facet Model, while the magenta line fitted to the $1 - r_{\mathrm{max}}^{\mathrm{min}}$ is a power law decay scaling as $\gamma^{-\zeta}$ where $\zeta = 0.81$.](clast6.eps){width=".45\textwidth"}
Sphericity complement measures are shown in Fig. \[fig:Fig6\] for $1 - \phi_{3}$ (filled circles) and $1 - r^{\mathrm{min}}_{\mathrm{max}}$ (open circles) on a log-log scale with respect to $\gamma$ in the main graph and versus $\langle n \rangle^{-1}$ in the inset graph. From the main graph, one see that both complements shift to power law decays after a plateau qualitatively similar to that other IPR in Fig. \[fig:Fig5\], with $1 - \phi_{3}$ decreasing more rapidly than $1 - r^{\mathrm{min}}_{\mathrm{max}}$ with the asymptotic downward slope in the log-log graph appearing to be greater for the former than for the latter. The trend to zero of departures from a perfect spherical shape with increasing number of facets is also evident in the graph inset which shows $1 - \phi_{3}$ and $1 - r_{\mathrm{max}}^{\mathrm{min}}$ with respect to the reciprocal of the mean number of facets, $\langle n \rangle^{-1}$.
In the IFM framework, we obtain the correct asymptotic behavior in the case of the sphericity complement $1 - \phi_{3}$. In this vein, one begins by considering a sphere with a volume of $\frac{4}{3} \pi R^{3}$, and $\langle n \rangle$ lenticular slices are sheered away by the fracture planes; one calculates in the IFM context the mean of the ratio of the surface area of the volume equivalent sphere to the area of this solid, namely $\langle [(2/s)(1 - s)(1 - \sqrt{1 - s})]^{\frac{1}{3}} \rangle$ where $s \equiv \eta (2 \tilde{u} - \tilde{u}^{2})$. The result is the solid curve in the main graph of \[fig:Fig6\]. Although depressed relative to the $1 - \phi_{3}$ curve, the IFM result and Monte Carlo simulation results nonetheless have in common a plateau-like region for $\gamma \leq 10$ which gives way to a linear slope signaling a power law decay. In fact, from Eq. \[eq:Eq150\], the steady state sphericity complement $1 - \phi_{3}$ scales as $\frac{4}{3} \eta \gamma^{-1}$, asymptotically correct apart from a dimensionless prefactor.
Whereas $1 - r_{\mathrm{max}}^{\mathrm{min}}$ is likely sensitive to correlations among facet sizes and positions, and thus less amenable to the IFM treatment, a statistical analysis of the time dependence of the $1 - r_{\mathrm{max}}^{\mathrm{min}}$ suggests that it saturates at values proportional to $\gamma^{-\zeta}$ where $\zeta = 0.81(5)$. This power law decay is indicated by the magenta line in the main graph of Fig. \[fig:Fig6\].
![\[fig:Fig7\] (Color online) Log-log plot of ellipticity measures, including oblateness and prolateness with respect to the mean facet number $\langle n \rangle^{-1}$ Simulation results are indicated with symbols, while the red line on the right side corresponds to a power law scaling of $\gamma^{-1}$. In the inset, the ellipticity measures are plotted with respect to $\langle n \rangle^{-1}$.](clast7.eps){width=".45\textwidth"}
Measures of ellipticity, the oblateness (open symbols) and prolateness (filled symbols), are shown in Fig. \[fig:Fig7\] in a log-log plot. Both the oblateness and the prolateness decrease gradually with $\gamma$, for $\gamma \leq 10$, with the former initially flat as indicated by the blue line. Asymptotically linear on a log-log scale, the prolateness and oblateness converge in the large $\gamma$ regime. The red line highlights the power law decay of both ellipticity measures, corresponding to a $1/\gamma$ dependence.
Time Evolution of Shapes
------------------------
![\[fig:Fig8\] (Color online) Images of stones derived from a cube shaped protoclast at various stages of erosion for $\gamma = 2000$, with facets original to the parent cube shown in blue. The structural transition appears at the extreme lower right corner.](clast8.eps){width=".45\textwidth"}
### Structural Phase Transitions in Mono-dispersed Protoclasts
Although steady state shapes are smooth and spherical for $\gamma \gg 1$, the initial cohort of protoclasts are polyhedral with a relatively small number of facets. We consider regular cubes as protoclasts as well as the highly irregular initial shapes mentioned earlier. In the case of the former, we observe critical behavior as primordial facets are eroded away, and stones become smoother and rounder as additional volume is chipped away; this transition from shapes which possess facets form the parent solid and those which do not is abrupt with all of the hallmarks of a second order phase transition and is reflected in all of the variables we calculate apart from measures of deviation from a spherical shape. The latter exhibit critical behavior at a subsequent phase transition in which the stones revert to spherical shapes.
Selected images from the erosion trajectory of cube shaped protoclasts appear in Fig. \[fig:Fig8\] with the image at the lower right coinciding with the structural transition in which all primordial facets are removed; blue areas are facets or portions of facets original to the parent solid. On the other hand, due to the inherent strong structural disorder of the irregular protoclasts, the disappearance of primordial facets occurs at different times for different shapes, with a concomitant loss of any well defined phase transition for ensemble averaged observables for the cohort as a whole. In fact, by considering mono-dispersed cohorts made up of a single irregular shape, we see that the elimination of protoclast faces is in general asynchronous even for an ensemble of initially identical shapes, with a simultaneous elimination of primordial facets limited to highly symmetric shapes and being the exception rather than the rule.
![\[fig:Fig9\] (Color online) Facet survival probabilities $f_{\mathrm{sur}}$ for cube shaped protoclasts are shown in panel (a) with respect to $\tilde{\tau} = N_{\mathrm{sust}}/\gamma^{2}$;a closer view of the $f_{\mathrm{sur}}$ curves in the inset. Panel (b) shows an optimized data collapse for cube shaped protoclasts with $\varepsilon = 0.76(5)$.](clast9a.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig9\] (Color online) Facet survival probabilities $f_{\mathrm{sur}}$ for cube shaped protoclasts are shown in panel (a) with respect to $\tilde{\tau} = N_{\mathrm{sust}}/\gamma^{2}$;a closer view of the $f_{\mathrm{sur}}$ curves in the inset. Panel (b) shows an optimized data collapse for cube shaped protoclasts with $\varepsilon = 0.76(5)$.](clast9b.eps "fig:"){width=".45\textwidth"}
In statistical mechanical analyses of phase transitions, a standard practice is to specify an order parameter, a variable which is finite when the phase in question is present, and zero otherwise if one is in the bulk or thermodynamic limit. For this purpose, we use the survival probability (denoted in this work as $f_{\mathrm{sur}}$), or the ensemble averaged fraction of primordial facets remaining.
The survival probability for cube shaped protoclasts is shown in in Fig. \[fig:Fig9\] for $\gamma$ values ranging from $\gamma = 250$ to $\gamma = 2000$. With characteristic times scaling as $\gamma^{2}$, we plot the survival probability and other salient variables with respect to the scaled time, $N_{\mathrm{sust}}/\gamma^{2}$, also denoted as $\tilde{\tau}$. The mean fraction of surviving original facets initially exhibits a plateau, decreasing to zero as all of the primordial cube facets are eroded away. That the descent of the survival index sharpens as $\gamma$ increases \[with curves crossing at a common point as shown in the inset of panel (a) of Fig. \[fig:Fig9\]\] is compatible with the original facet survival probability being a viable order parameter (i.e. non-zero only when vestiges of the original protoclast faces are still present) in the context of a second order phase transition at a critical $\tilde{\tau}$ value, $\tilde{\tau}_{c}$.
Singularities in observables or their derivatives in the vicinity of $\tilde{\tau}$ are hallmarks of a second order phase transition, which we use to show that salient variables are influenced by critical behavior, and to calculate indices of interest such as $\tilde{\tau}_{c}$. In the context of our simulations, the thermodynamic limit corresponds to $\gamma \gg 1$, where the number of facets also is large, (e.g. as high as 3,600 for $\gamma = 2000$).
As is customary in the analysis of second order phase transition, we elucidate critical behavior in the structural phase transition of eroding cubes by appealing to single parameter finite size scaling theory, where one uses the data collapse phenomenon as a quantitative tool to calculate critical indices. Although in general one would plot $\gamma^{\beta} f_{\mathrm{sur}}$ with respect to $\gamma^{\varepsilon} (\tilde{\tau} - \tilde{\tau}_{c})$, the crossing of the $f_{\mathrm{sur}}$ curves for different $\gamma$ values suggests that $f_{\mathrm{sur}}$ is of zero scaling dimension, as is true for the Binder Cumulant [@Binder] used in connection with thermally driven magnetic phase transitions, for which $\beta = 0$. Accordingly, $f_{\mathrm{sur}}$ is on the ordinate axis in panel (b), with the optimal data collapse occurring for $\tau_{c} = 0.0968(2)$ and $\varepsilon = 0.76(5)$.
![\[fig:Fig10\] (Color online) Inverse Participation Ratios (IPR) for cube shaped protoclasts in panel (a) for various $\gamma$ values. Panel (b) is a plot of $\gamma \times \textrm{IPR}$ with respect to $\tilde{\tau}$ on a semi-logarithmic scale with a splaying of the curves for $\tilde{\tau} < 0.097$. Panel (c) is a data collapse plot with $\gamma \times \textrm{IPR} $ plotted with respect to $\gamma^{\Xi}(\tilde{\tau} - \tilde{\tau}_{c}$) where $\Xi = 0.30(5)$.](clast10a.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig10\] (Color online) Inverse Participation Ratios (IPR) for cube shaped protoclasts in panel (a) for various $\gamma$ values. Panel (b) is a plot of $\gamma \times \textrm{IPR}$ with respect to $\tilde{\tau}$ on a semi-logarithmic scale with a splaying of the curves for $\tilde{\tau} < 0.097$. Panel (c) is a data collapse plot with $\gamma \times \textrm{IPR} $ plotted with respect to $\gamma^{\Xi}(\tilde{\tau} - \tilde{\tau}_{c}$) where $\Xi = 0.30(5)$.](clast10b.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig10\] (Color online) Inverse Participation Ratios (IPR) for cube shaped protoclasts in panel (a) for various $\gamma$ values. Panel (b) is a plot of $\gamma \times \textrm{IPR}$ with respect to $\tilde{\tau}$ on a semi-logarithmic scale with a splaying of the curves for $\tilde{\tau} < 0.097$. Panel (c) is a data collapse plot with $\gamma \times \textrm{IPR} $ plotted with respect to $\gamma^{\Xi}(\tilde{\tau} - \tilde{\tau}_{c}$) where $\Xi = 0.30(5)$.](clast10c.eps "fig:"){width=".45\textwidth"}
The Inverse Participation Ratio (IPR), a measure of the smoothness of stones, is shown in panel (a) of Fig. \[fig:Fig10\] on a semi-logarithmic scale for cube shaped protoclasts. The regular spacing on the logarithmic scale of the asymptotically flat IPR curves for large enough $\tilde{\tau}$ is compatible with the Participation Ratio scaling as $\gamma^{-1}$ at steady state, also evident in panel (b) of Fig. \[fig:Fig10\] where $\gamma \times \textrm{IPR}$ is plotted with respect to $\tilde{\tau}$. For $\tilde{\tau} > \tilde{\tau}_{c}$, the curves for various $\gamma$ values merge onto a single flat line, splaying outward for $\tilde{\tau} < \tilde{\tau}_{c}$, suggesting singular behavior for $\tilde{\tau} = 0.0967$. In panel (c) of Fig. \[fig:Fig10\], the quantity $\gamma \times \textrm{IPR}$ is plotted with respect to $\gamma^{\Xi}(\tilde{\tau} - \tilde{\tau}_{c})$ where $\Xi = 0.30(5)$ \[and $\tilde{\tau} = 0.097(1)$ as in the case of $f_{\mathrm{sur}}$\] for various $\gamma$ values, with the overlap of the curves indicating a collapse of the entire gamut of the IPR onto a universal curve.
![\[fig:Fig11\] (Color online) Sphericity complement measures are shown for various $\gamma$ values in panel (a) in the case of cube shaped protoclasts, with continuous and broken curves representing $1 - r^{\mathrm{min}}_{\mathrm{max}}$ and $1 - \phi_{\mathrm{3}}$ respectively. Vertical gray and blue lines indicate structural phase transitions involving the removal of primordial facets for the former and a reversion of stones to spherical shapes for the latter. Panel (b) and Panel (c) display collapses of $1 - \phi_{3}$ and $1 - r_{\mathrm{max}}^{\mathrm{min}}$ onto universal curves.](clast11a.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig11\] (Color online) Sphericity complement measures are shown for various $\gamma$ values in panel (a) in the case of cube shaped protoclasts, with continuous and broken curves representing $1 - r^{\mathrm{min}}_{\mathrm{max}}$ and $1 - \phi_{\mathrm{3}}$ respectively. Vertical gray and blue lines indicate structural phase transitions involving the removal of primordial facets for the former and a reversion of stones to spherical shapes for the latter. Panel (b) and Panel (c) display collapses of $1 - \phi_{3}$ and $1 - r_{\mathrm{max}}^{\mathrm{min}}$ onto universal curves.](clast11b.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig11\] (Color online) Sphericity complement measures are shown for various $\gamma$ values in panel (a) in the case of cube shaped protoclasts, with continuous and broken curves representing $1 - r^{\mathrm{min}}_{\mathrm{max}}$ and $1 - \phi_{\mathrm{3}}$ respectively. Vertical gray and blue lines indicate structural phase transitions involving the removal of primordial facets for the former and a reversion of stones to spherical shapes for the latter. Panel (b) and Panel (c) display collapses of $1 - \phi_{3}$ and $1 - r_{\mathrm{max}}^{\mathrm{min}}$ onto universal curves.](clast11c.eps "fig:"){width=".45\textwidth"}
In Fig. \[fig:Fig11\] the complements $1 - \phi_{3}$ and $1 - r_{\mathrm{max}}^{\mathrm{min}}$ are shown for the case of cube shaped protoclasts, graphed on a semilogarithmic scale in panel (a); the gray line in the plot corresponds to the structural phase transition in which primordial facets disappear. As in the case of the IPR, both complements are asymptotically flat and evenly spaced on the logarithmic ordinate scale, and both measures exhibit singular behavior indicating a structural phase transition which does not coincide with the structural transformation in which faces original to the parent cube vanish, but occurs at a later time indicated by the vertical blue line in panel (a) of Fig. \[fig:Fig11\].
In a spirit similar to the case of the IPR, panel (b) and panel (c) of Fig. \[fig:Fig11\] show the collapse onto universal curves of of $1 - \phi_{3}$ and $1 - r_{\mathrm{max}}^{\mathrm{min}}$ respectively. In the case of the former, $\gamma (1 - \phi_{3})$ is plotted with respect to $\gamma^{\sigma_{1}}(\tilde{\tau} - \tilde{\tau}_{c})$ where $\sigma_{1} = 0.23(2)$ and $\tilde{\tau}_{c} = -.137(5)$. On the other hand, for the latter we achieve a collapse of the $r_{\mathrm{max}}^{\mathrm{min}}$ complement measure onto a universal scaling curve by plotting $\gamma^{\zeta} (1 - r_{\mathrm{max}}^{\mathrm{min}})$ with respect to $\gamma^{\sigma_{2}} (\tilde{\tau} - \tilde{\tau}_{c})$ where the collapse is optimal for $\zeta = 0.81(5)$ and $\sigma_{2} = 0.36(5)$. In this manner, we see that the stones become smooth even as their primordial facets vanish.
![\[fig:Fig12\] (Color online) Normalized mean facet numbers, plotted versus $\tilde{\tau} = N_{\mathrm{sust}}/\gamma^{2}$ are shown for cube shaped protoclasts (broken lines) and irregular protoclasts (solid lines). The inset shows a magnified view of the gray rectangular region with symbols indicating Monte Carlo simulation data points.](clast12.eps){width=".45\textwidth"}
In Fig. \[fig:Fig12\], normalized mean facet numbers ($\langle n \rangle/\gamma$) are juxtaposed for cube shaped and regular protoclasts (broken and solid lines respectively), and are well converged with respect to $\gamma$ in both cases. In addition, $\langle n \rangle/\gamma$ increases monotonically with increasing $\tilde{\tau}$ toward the same asymptotic value of 1.82, though the characteristic time constant in the case of irregular protoclasts exceeds that of the cubic counterparts. On a qualitative level, while the convergence to a normalized facet number of 1.82 is gradual for initially irregular fragments, the mean facet number curves for cube shapes protoclasts saturate for a finite $\tilde{\tau}$ value, quickly becoming level thereafter. The inset is a magnified view illustrating the increasing abruptness of change in slope near $\tilde{\tau}_{c}$ with increasing $\gamma$, singular behavior signaling second order phase transition.
![\[fig:Fig13\] (Color online) Semilogarithmic plots of volume fraction remaining $\tilde{v}$ with respect to $\tilde{\tau}$ as well as the derivatives of the latter are show in panel (a) and panel (b) respectively for cube shaped protoclasts; the vertical gray line indicates the structural transition for which primordial facets vanish. Similarly, panel (c) and panel (d) show $\tilde{v}$ on a semilogarithmic scale as well as the slope of $\log_{10} \tilde{v}$ versus for cohorts of irregular protoclasts.](clast13.eps){width=".45\textwidth"}
In Fig. \[fig:Fig13\], volume fraction remaining $\tilde{v}$ also reflects singular behavior at the structural transition for $\tilde{\tau}_{c} = 0.0968(2)$. The main graphs in panel (a) and panel (b) of Fig. \[fig:Fig13\] show $\tilde{v}$ on a semilogarithmic scale, while the inset plots display the sloe of the $\log_{10} (\tilde{v})$ curves with respect to $\tilde{\tau} = N_{\mathrm{sust}}/\gamma^{2}$. In general, as discussed previously, one anticipates the small decrease of volume fraction per fracture event to be $d \tilde{v} = -\Delta A^{2}/l$ with $l$ a length scale on the order of $\tilde{v}^{1/3}$. In the regime that stones are worn down to quasi-spherical shapes, one would expect the volume fraction decrement for a fracture event to be $\Delta \tilde{v} = -\Delta A^{2}/(12 \pi R)$, exact for the case of a truncated sphere in the limit that $r/R$ (i.e. with r the radius of the new circular facet and $R$ the sphere radius) tends to zero. Taking $\Delta A$ to be on the order of $4 \pi R^{2}/\gamma$, we see that the typical volume sheared away per normalized time increment $d \tilde{\tau} = \gamma^{-2} \Delta N_{\mathrm{sust}}$ is $d \tilde{v} \sim - \tilde{v} d \tilde{\tau}$, which when integrated leads to an exponential decay in $\tilde{\tau}$, $\tilde{v} = e^{-\Gamma \tilde{\tau}}$ where $\Gamma$ is a dimensionless constant on the order of unity.
For both cube shaped and irregular protoclasts, the mean remaining volume fraction curves appear to become asymptotically linear, behavior highlighted in panel (b) and panel (d) for cubic and irregular parent solids respectively. Common to both cases, slopes of $\log_{10} \tilde{v}$ level out at a common value, which is well converged with respect to the toughness parameter $\gamma$ in both instances. A salient distinction is a discontinuity in the slope of the volume fraction curves for cube shaped protoclasts not replicated in the case of irregular protoclasts. The latter, in which the slope abruptly becomes constant, signals pure exponential decay of a quasi-spherical shape and also coincides with the facet elimination phase transition indicated by the vertical gray line in panel (b) of Fig. \[fig:Fig13\].
A sharply defined structural phase transition in the case of regular protoclasts such as cubes, even for mono-dispersed solids, is the exception rather than the rule, as may be seen in the case of cohorts of identical irregular shapes. In this vein, we consider an irregular six sided protoclasts for which the elimination of primordial facets is not simultaneous. Instead, the facets of the original protoclast vanish at distinct times $\tau$. The parent solid is shown in the upper left corner of Fig. \[fig:Fig14\], illustrating the erosion trajectory with intermediate stages (with $\tilde{\tau}$ values in black) interspersed with images corresponding to transitions in which one or more primordial facets are removed (with $\tilde{\tau}$ values in red). Crimson regions are facets original to the parent solid, with the expanding gold areas being material exposed by stochastic fracture events.
![\[fig:Fig14\] (Color online) Sequences of structures for an irregular protoclast where $\gamma = 2000$. Images with $\tilde{\tau}$ values in red correspond to transitions involving the loss of one or more primordial facets. Red areas the latter or portions of the latter.](clast14.eps){width=".45\textwidth"}
![\[fig:Fig15\] (Color online) Survival probability for the irregular solid in Fig. \[fig:Fig14\]. Inset graphs are data collapses at structural transitions in which one or more facets original to the parent solid are cleaved away.](clast15.eps){width=".45\textwidth"}
In spite of the absence of a single structural phase transition, the loss of individual primordial facets is accompanied by singular behavior in salient variables encountered in the vicinity of a second order phase transition. The ensemble averaged facet survival probability shown in Fig. \[fig:Fig15\] for various $\gamma$ values ranging from $\gamma = 250$ to $\gamma = 2000$ exhibits abrupt transitions signaling the loss of individual facets of the original protoclast, with common crossings of the curves for each transition. While some of the latter involve the loss of a single facet, in some cases multiple surfaces from the original protoclast vanish at the same time. The first and third transitions \[for $\tilde{\tau} = 0.0248(1)$ and $\tilde{\tau} = 0.1375(5)$\] are of the former variety, while the second and fourth steps downward \[for $\tilde{\tau} = 0.0893(1)$ and $\tilde{\tau} = 0.275(1)$\] involve the simultaneous disappearance of two primordial facets. The inset graphs are data collapse plots corresponding to the four facet transitions, with the horizontal scale being $\gamma^{\varepsilon} (\tilde{\tau} - \tilde{\tau}_{c})$ as in the case of panel (b) of Fig. \[fig:Fig9\]. However, while the data collapses are optimized for $\varepsilon = 0.76(5)$ in the case of the first three facet elimination transitions, we find a departure for the transition involving the removal of the last two facets of the parent solid where instead $\varepsilon = 0.45(5)$.
![\[fig:Fig16\] (Color online) Normalized facet numbers for mono-dispersed irregular protoclasts and the slope of the log-log volume curves are depicted in panel (a) and panel (b) respectively.](clast16.eps){width=".45\textwidth"}
Signatures of the asynchronous facet removal transitions are evident in variables other than $f_{\mathrm{sur}}$, such as the mean facet number $\langle n \rangle$ and the slope the logarithm of the mean remaining volume fraction $\tilde{v}$ displayed in panel (a) and panel (b) of Fig. \[fig:Fig16\] for various $\gamma$ values. In both cases, shift in slopes of the curves are subtler than those marking the disappearance of facets original to the parent solid for cubic protoclasts, with slope changes at phase boundaries most pronounced following the fourth transition in which the final two primordial faces are worn away.
![\[fig:Fig17\] (Color online) Sphericity complement measures are show for various $\gamma$ values in the case of mono-dispersed irregular protoclasts, with continuous and broken curves representing $1 - r^{\mathrm{min}}_{\mathrm{max}}$ and $1 - \phi_{\mathrm{sph}}$ respectively in panel (a). Panel (b) and panel (c) show collapses of complements $1 - phi_{3}$ and $1 - r_{\mathrm{max}}^{\mathrm{min}}$ respectively onto universal scaling curves.](clast17a.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig17\] (Color online) Sphericity complement measures are show for various $\gamma$ values in the case of mono-dispersed irregular protoclasts, with continuous and broken curves representing $1 - r^{\mathrm{min}}_{\mathrm{max}}$ and $1 - \phi_{\mathrm{sph}}$ respectively in panel (a). Panel (b) and panel (c) show collapses of complements $1 - phi_{3}$ and $1 - r_{\mathrm{max}}^{\mathrm{min}}$ respectively onto universal scaling curves.](clast17b.eps "fig:"){width=".45\textwidth"}
Spherical deviation measures $1 - \phi_{3}$ and $1 - r_{\mathrm{max}}^{\mathrm{min}}$ are shown in the main graph of Fig. \[fig:Fig17\] on a semilogarithmic scale. In both cases, the curves plotted for different $\gamma$ values begin to diverge, in a manner qualitatively similar to the case of cube shaped protoclasts, though asymptotically flat regions do not appear on the domain of time scales $\tilde{\tau}$ accessed in the simulation. Data collapse plots onto universal scaling curves for the sphericity and $r_{\mathrm{max}}^{\mathrm{min}}$ complements are displayed in panels (b) and (c) of Fig. \[fig:Fig17\], respectively. In the case of the former, $\gamma (1 - \phi_{3})$ is plotted with respect to $\gamma^{\sigma_{1}} (\tilde{\tau} - \tilde{\tau}_{c})$ where $\sigma_{1} = 0.17(1)$ and $\tilde{\tau}_{c} = 0.57(5)$. On the other hand, for the latter we plot $\gamma^{\zeta} (1 - r_{\mathrm{max}}^{\mathrm{min}})$ with respect to $\gamma^{\sigma_{2}} (\tilde{\tau} - \tilde{\tau}_{c} )$ where $\zeta = 0.81$, $\sigma_{2} = 0.36(5)$, and $\tilde{\tau}_{c} = 0.52(5)$, compatible with the transition time obtained for the $\phi_{3}$ complement. As for cube shaped parent solids, the transition times $\tilde{\tau}_{c}$ occur later than for any of the structural transitions in which one or more primordial facets are removed for the irregular mono-dispersed cohort.
### Structural Evolution of Poly-dispersed Cohorts
![\[fig:Fig18\] (Color online) Twenty five irregular protoclasts at various erosion stages](clast18.eps){width=".45\textwidth"}
Sample stones at various stages of their erosion trajectories are shown in Fig. \[fig:Fig11\], with the irregular protoclasts in the upper left panel. As in the case of cube shaped protoclasts, individual stones are found to shed primordial facets, but in an asynchronous manner due to the structural disorder inherent in the irregular protoclasts, eliminating the possibility of a well defined structural phase transition. Thus, variables exhibit no singular behavior, converging for $\gamma \gg 1$ as in the survival indices plotted in Fig. \[fig:Fig14\], which overlap closely for $\gamma$ ranging from $\gamma = 250$ to $\gamma = 2000$.
![\[fig:Fig19\] (Color online) Sphericity complements $1 - \phi_{3}$ and $1 - r^{\mathrm{min}}_{\mathrm{max}}$ are shown in panel (a) for irregular protoclast cohorts, while $f_{\mathrm{sur}}$ and the IPR (on a semi-logarithmic scale) are displayed in panel (b) and panel (c) respectively.](clast19a.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig19\] (Color online) Sphericity complements $1 - \phi_{3}$ and $1 - r^{\mathrm{min}}_{\mathrm{max}}$ are shown in panel (a) for irregular protoclast cohorts, while $f_{\mathrm{sur}}$ and the IPR (on a semi-logarithmic scale) are displayed in panel (b) and panel (c) respectively.](clast19b.eps "fig:"){width=".45\textwidth"}
Results for the complements $1 - \phi_{3}$ and $1 - f_{\mathrm{max}}^{\mathrm{min}}$, the facet survival probability, and the Inverse Participation ratio are shown in panels (a), (b), and (c) of Fig. \[fig:Fig19\] respectively for irregular protoclasts. In the case of the measures of deviation from a spherical shape in panel (a), though the curves diverge, the separation occurs at later and later times with increasing $\gamma$, suggesting convergence with respect to the toughness parameter. The survival probability shown in panel (b) of Fig. \[fig:Fig19\] evidently converges rapidly with increasing $\gamma$ when plotted with respect to $N_{\mathrm{sust}}/\gamma^{2}$ with little difference among the curves for all $\gamma$ values shown. In contrast to the case of cube shaped parent stones, IPR curves for irregularly shaped protoclasts converge with increasing $\gamma$, and there is no finite value of $\tilde{\tau}$ where the participation ratio decays to zero in the large $\gamma$ limit. This characteristic is compatible with primordial facet removal events being spread over the entire $\tilde{\tau}$ domain due to the strong structural disorder in the cohort of poly-dispersed irregular protoclasts.
![\[fig:Fig20\] (Color online) Ellipticity measures plotted versus $\tilde{\tau}$ for various $\gamma$ values, with solid and broken lines representing oblateness and prolateness respectively. Gray regions about $\gamma = 2000$ curves indicate Monte Carlo statistical error.](clast20.eps){width=".45\textwidth"}
Oblateness and prolateness measures $\psi_{\mathrm{P}}$ and $\psi_{\mathrm{O}}$, as described in Section II, are displayed in Fig. \[fig:Fig20\] for a range of $\gamma$ values. The prolateness measure initially exceeds the oblateness measure, but eventually decreases sharply and falls below $\psi_{\mathrm{O}}$ with a continued monotonic decrease thereafter. A study of strongly disordered fragments [@Domokos4] generated stochastically found a prevalence of prolate fragments in cohorts generated with a variety of techniques. On the other hand, the disorder averaged oblateness measure actually increases initially, peaking when approximately half of the stone’s original volume has been chipped away. The variance in the $\psi_{\mathrm{O}}$ curves, while greater than for the $\psi_{\mathrm{P}}$ traces, is comparable to the Monte Carlo statistical error shade in gray in the case of the $\gamma = 2000$ curve, and thus both the oblateness and prolateness measures may be considered to be converged with respect to the toughness parameter $\gamma$. That $\psi_{\mathrm{O}}$ peaks even as $\psi_{\mathrm{P}}$ continues to decrease could predispose stones to long term oblate shapes, though much of the enhancement and maintenance of an oblate profile, not addressed in this study, may be due to specific nature of fluvial motion along a stream or river bed [@Oblate1].
![\[fig:Fig21\] (Color online) Oblateness measure $\psi_{\mathrm{O}}$ for an individual case showing non-monotonic behavior in the oblateness. The solid curve is the oblateness measure plotted with respect to time, and rectangular gray boxes enclose images of the stones at specific stages. The lower image in each case is an edge-on view of the same stone.](clast21.eps){width=".45\textwidth"}
To gain an understanding on an intuitive level as to why the time dependent oblateness measure is non-monotonic in some cases, we exhibit $\psi_{\mathrm{O}}$ as well as images of the stone at various stages in the erosion trajectory. The protoclast for the example shown in Fig. \[fig:Fig21\] begins as a blade-like shape, narrow but also thin. As fractures begin to carve away volume, the stone becomes thicker (as may be seen in the edge-on views), but appears to lose material from the extremes along the longer axis at an even greater rate. In combination, these trends contribute to an initial rise in oblateness (with the shape remaining relatively thin while becoming less oblong), ultimately peaking an declining as the stone ultimately is rounded into a spherical shape.
Non Steady State Erosion Scenarios
==================================
![\[fig:Fig22\] (Color online) Observables plotted with respect to $N_{\mathrm{sust}}/(\gamma^{'})^{2}$ for $1 - r_{\mathrm{max}}^{\mathrm{min}}$ and $1 - \phi_{3}$ in panel (a), the IPR in panel (b), the oblateness and prolateness in panel (c), and the primordial facet survival probability in panel (d).](clast22a.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig22\] (Color online) Observables plotted with respect to $N_{\mathrm{sust}}/(\gamma^{'})^{2}$ for $1 - r_{\mathrm{max}}^{\mathrm{min}}$ and $1 - \phi_{3}$ in panel (a), the IPR in panel (b), the oblateness and prolateness in panel (c), and the primordial facet survival probability in panel (d).](clast22b.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig22\] (Color online) Observables plotted with respect to $N_{\mathrm{sust}}/(\gamma^{'})^{2}$ for $1 - r_{\mathrm{max}}^{\mathrm{min}}$ and $1 - \phi_{3}$ in panel (a), the IPR in panel (b), the oblateness and prolateness in panel (c), and the primordial facet survival probability in panel (d).](clast22c.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig22\] (Color online) Observables plotted with respect to $N_{\mathrm{sust}}/(\gamma^{'})^{2}$ for $1 - r_{\mathrm{max}}^{\mathrm{min}}$ and $1 - \phi_{3}$ in panel (a), the IPR in panel (b), the oblateness and prolateness in panel (c), and the primordial facet survival probability in panel (d).](clast22d.eps "fig:"){width=".45\textwidth"}
Although we discuss a variety of non-steady state schemes, choose the constant velocity scenario (e.g. for stones borne along at constant speed in a stream or river) for direct Monte Carlo simulations. With kinetic energy scaling as $\frac{1}{2} m v^{2}$, and $m = \rho V$ in the case of stones of uniform composition, typical energy would be proportional to the volume. That the $\alpha = 1$ exponent for the fixed velocity scenario exceeds the critical $\alpha_{c} = 2/3$ for the relative area scheme implies an ever diminishing relative area with time for typical sustained slices and a concomitant rise in the mean number of facets. Nevertheless, although the relative area and fixed velocity scenarios are distinct, the universal dependence of stone profiles on fractional remaining volume [@Domokos1] dictates that key structural milestones occur at finite values of the reduced time $\tilde{\tau}$, borne out in the case of the fixed velocity scheme.
Fixed velocity erosion trajectories are calculated for irregular protoclasts with 2000 distinct realizations of disorder. A range of toughness parameter values from $\gamma^{'} = 20$ to $\gamma^{'} = 160$ are considered with the mean number of facets $\langle n \rangle$ exceeding 4000 for the latter. Representative examples of relevant variables are shown in Fig. \[fig:Fig22\] with the sphericity complements $1 - \phi_{3}$ and $1 - r_{\mathrm{max}}^{\mathrm{min}}$ and the IPR shown in panel (a) and (b) while oblateness and prolateness measures are shown in panel (c) with the primordial facet survival probability $f_{\mathrm{sur}}$ plotted in panel (d). As in the case of the time dependent relative area results, $f_{\mathrm{sur}}$ and ellipticity measures are converged with respect to the $\gamma^{'}$ parameter with minor discrepancies for the oblateness measure near its maximum likely due to Monte Carlo statistical error. Quantities such as measures of deviation from a spherical shape and the IPR do diverge on the semilogarithmic scale, but at later times with increasing $\gamma$ suggesting convergence with respect to the latter. In a qualitative sense, in comparison to results in the case of the relative area scenario, fixed velocity variables evolve more rapidly initially with structural change slowing as sustained slices accumulate. This trend is compatible with the typical area of freshly exposed facets decreasing relative to the total area with decreasing remaining volume fraction $\tilde{v}$.
![\[fig:Fig23\] (color online) Effective Gamma plots in the fixed velocity scenario are displayed for the mean facet number $\langle n \rangle$ and IPR in panel (a) and panel (b), and spherical complements $1 - \phi_{3}$ and $1 - r_{\mathrm{max}}^{\mathrm{min}}$ in panel (c) and panel (d) respectively. Symbols represent equilibrated steady state systems, while broken and solid lines are simulation data in the fixed velocity scenario.](clast23a.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig23\] (color online) Effective Gamma plots in the fixed velocity scenario are displayed for the mean facet number $\langle n \rangle$ and IPR in panel (a) and panel (b), and spherical complements $1 - \phi_{3}$ and $1 - r_{\mathrm{max}}^{\mathrm{min}}$ in panel (c) and panel (d) respectively. Symbols represent equilibrated steady state systems, while broken and solid lines are simulation data in the fixed velocity scenario.](clast23b.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig23\] (color online) Effective Gamma plots in the fixed velocity scenario are displayed for the mean facet number $\langle n \rangle$ and IPR in panel (a) and panel (b), and spherical complements $1 - \phi_{3}$ and $1 - r_{\mathrm{max}}^{\mathrm{min}}$ in panel (c) and panel (d) respectively. Symbols represent equilibrated steady state systems, while broken and solid lines are simulation data in the fixed velocity scenario.](clast23c.eps "fig:"){width=".45\textwidth"}
An additional merit of discussing the steady state scenario in the equilibrated regime is its relevance to non-steady state scenarios after a sufficient number of fractures have accumulated. We define an instantaneous or “effective” $\gamma$ with $\gamma_{\mathrm{eff}} = A_{\Sigma}/A_{\textrm{char}}$ where $A_{\textrm{char}}$ is the characteristic area exposed by a fracture event obtained by setting the exponent in the prospective slice acceptance probability to 1 and solving for the area. In the case of the fixed velocity erosion scheme, one has $\gamma_{\mathrm{eff}} = \tilde{v}^{-\frac{1}{3}} (6 \pi^{\frac{1}{2}})^{-1} \gamma^{'}$, which increases with decreasing remaining volume fraction. For representative variables of interest in Fig. \[fig:Fig23\], we show on the same plots equilibrated observables from steady state calculations and fixed velocity simulation results (solid or broken curves) plotted with respect to $\gamma_{\mathrm{eff}}$. The mean number of sites normalized with respect to $\gamma_{\mathrm{eff}}$, shown in panel (a), ultimately saturates at the large $\gamma$ equilibrium value of $1.82$. The IPR and complements $1 - \phi_{3}$ and $1 - r_{\mathrm{max}}^{\mathrm{min}}$ obtained from fixed velocity scheme simulations decrease monotonically with time in panels (b), panel (c), and panel (d) of Fig. \[fig:Fig23\], converging with steady state equilibrium results (open or filled circles). The merging of the fixed velocity simulation and equilibrium results suggests that even in non-steady state situations, stones reach a condition of quasi-equilibrium, in the sense that variables correspond closely with the steady-state counterparts calculated for $\gamma_{\mathrm{eff}}$ after enough time has elapsed.
General Scenario Variables from Relative Area Results
-----------------------------------------------------
![\[fig:Fig24\] (Color online) Variables for relative area and fixed velocity scenarios plotted with respect to volume fraction $\tilde{v}$.](clast24a.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig24\] (Color online) Variables for relative area and fixed velocity scenarios plotted with respect to volume fraction $\tilde{v}$.](clast24b.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig24\] (Color online) Variables for relative area and fixed velocity scenarios plotted with respect to volume fraction $\tilde{v}$.](clast24c.eps "fig:"){width=".45\textwidth"}
Equivalent milestones occurring at the same volume fractions remaining independent of the model (e.g. the oblateness maxima near the halfway point in the erosion of volume) is an important aspect of the deterministic time evolution of structures in spite of the stochastic nature of the collisional erosion process; in more succinct terms, graphs of relevant observables should coincide when plotted versus $\tilde{v}$ even for distinct models, a phenomenon which we see in Figure \[fig:Fig24\] for sample observables of interest. In panel (a), global measures of departure from a perfect spherical shape are plotted with respect to $\tilde{v}$, with the solid curve gleaned from relative area calculations and the broken trace corresponding to the fixed velocity scenario. Panel (b) and (c), depicting the mean facet survival probability and the Inverse Participation Ratio also show very close agreement among the relative area and constant velocity results when plotted versus the volume fraction remaining. Finally, measures of ellipticity are displayed in panel (d) of Fig. \[fig:Fig24\], with close agreement for both the prolateness and oblateness measures. This quantitative agreement of result for distinct scenarios when plotted with respect to the remaining volume fraction has been mentioned previously [@Domokos1].
The relative area scenario (i.e. $\alpha = \alpha_{c} = 2/3$) considered in the preceding section provides an avenue for predicting the time evolution of salient observables in a broad set of cases in which the number of facets does not ultimately saturate at a finite value, but continues to rise monotonically. The universal dependence of variables on the mean volume fraction $\tilde{v}$ for steady state and non-steady state scenarios (e.g. the constant velocity scheme) would permit a prediction of the time dependence of variables in a wide variety of non-steady state scenarios if one knew the relationship of $N_{\mathrm{sust}}$ and $\tilde{v}$ for the scheme under consideration. We argue here and validate with large scale simulations involving irregular protoclasts that this mapping of $\tilde{v}$ onto $N_{\mathrm{sust}}$ in greater generality may be determined from steady state results as long as $\gamma \gg 1$.
With the characteristic mean facet area $A_{\mathrm{char}}$ being on the order of $A_{\mathrm{char}} = \kappa \gamma^{-1} \tilde{v}^{2/3}$, where $\kappa \equiv
(36 \pi^{2})^{1/3}$, the volume cleaved away by a sustained slice scales as $A_{\mathrm{char}}^{2}/\tilde{r}$, where $\tilde{r}$ is as discussed earlier a length scale related to the local radii of curvature, which we take to be on the order of $\tilde{v}^{1/3}$, the cube root of the volume fraction. We obtain an equality with a volume dependent function $f(\tilde{v})$, yielding for the mean volume decrement $\langle \Delta \tilde{v} \rangle = - \kappa^{2} \gamma^{-2} f(\tilde{v}) \tilde{v} \Delta N_{\mathrm{sust}}$ where $\Delta N_{\mathrm{sust}}$ is a series of sustained slices small in the sense that $\langle \Delta \tilde{v} \rangle \ll \tilde{v}$. In the large $\gamma$ limit, one may replace $\langle \Delta \tilde{v} \rangle$ and $\Delta N_{\mathrm{sust}}$ with corresponding differential quantities, and in this regime one has $$f(\tilde{v}) = - \frac{\gamma^{2}}{\kappa^{2} \tilde{v} } \left( \frac{d \tilde{v}}{d N_{\mathrm{sust}}} \right)_{\mathrm{steady~state}}
\label{eq:Eq500}$$ which may be extracted numerically from data obtained in the context of the relative area scenario for $\gamma \gg 1$ (i.e. for $\gamma = 2000$ in this work).
The characteristic function $f(\tilde{v})$ in Eq.\[eq:Eq500\] in principle allows one access of $N_{\mathrm{sust}}(\tilde{v})$ for a broad range of erosion scenarios distinct from the steady state case by exploiting the fact that $d \tilde{v} = - f(\tilde{v}) \tilde{v}^{-1/3} A_{\mathrm{char}}(\tilde{v})^{2} d N_{\mathrm{sust}}$ Solving for $d N_{\mathrm{sust}}$ and integrating yields $$N_{\mathrm{sust}}(\tilde{v}) = \int_{\tilde{v}}^{1} \tilde{v}^{1/3} \frac{d \tilde{v}}{A_{\mathrm{char}}^{2}f(\tilde{v})}
\label{eq:Eq800}$$
![\[fig:Fig25\] (Color online) Characteristic function $f(\tilde{v})$ for cube shaped protoclasts (broken lines) and irregular protoclasts (solid lines). the inset is a magnified view of the regions boxed in pink for cubic protoclasts.](clast25.eps){width=".45\textwidth"}
Characteristic functions for cubic and irregular protoclasts are shown in Fig. \[fig:Fig25\], with solid lines representing the former and broken lines the latter. The inset graph is a magnified view of the region indicated by the pink box in the main graph for cubic protoclasts. Common to both cases is the decline from an elevated value for $\tilde{v}$ near unity at the beginning of the erosion process, with $f(\tilde{v})$ leveling out at a finite value for $\tilde{v} \ll 1$ (i.e. for later times after a large portion of the original volume has eroded). This asymptotic value, indicated in the main graph and inset plots with a solid gray line, is 0.216(1) for both cube shaped and irregular protoclasts. The elevated portion of $f(\tilde{v})$ for small times is compatible with an initially accelerated loss of volume as corners and edges give way to regions of high local curvature which shed volume comparatively rapidly.
Although $f(\tilde{v})$ for the case of irregular and cube shaped counterparts appears to converge to a common asymptotic value, the characteristic function for the case of cubic protoclasts is distinct in abruptly reaching 0.216 for a finite volume fraction $\tilde{v} = 0.5732$, where the second order phase transition to shapes lacking primordial facets occurs. The singular behavior is highlighted in the inset, which shows a magnified view of $f(\tilde{v})$ for cube shaped protoclasts with a slope discontinuity signaling the attainment of the long time value, and coinciding with the loss of primordial facets transitions indicated with the solid vertical line.
The flattening of $f(\tilde{v})$ following an initial rapid decrease provides an avenue for obtaining a closed form expression for time scales for the attainment of specific structural milestones at particular volume fractions $\tilde{v}$; given the structure of $f(\tilde{v})$, the relationship we obtain serves as an approximation as well as a rigorous upper bound for the time needed to wear stones down to a particular volume fraction.
![\[fig:Fig26\] (Color online) Primordial facet survival probabilities are show in panel (a) and panel (b) respectively; non-steady state simulation data (broken lines) for $\gamma^{'} = 160$ and calculated curves based on $\gamma = 2000$ steady state results (solid line) are plotted versus $N_{\mathrm{sust}}$ with corresponding semilogarithmic graphs in the insets of panel (a) and panel (b).](clast26a.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig26\] (Color online) Primordial facet survival probabilities are show in panel (a) and panel (b) respectively; non-steady state simulation data (broken lines) for $\gamma^{'} = 160$ and calculated curves based on $\gamma = 2000$ steady state results (solid line) are plotted versus $N_{\mathrm{sust}}$ with corresponding semilogarithmic graphs in the insets of panel (a) and panel (b).](clast26b.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig26\] (Color online) Primordial facet survival probabilities are show in panel (a) and panel (b) respectively; non-steady state simulation data (broken lines) for $\gamma^{'} = 160$ and calculated curves based on $\gamma = 2000$ steady state results (solid line) are plotted versus $N_{\mathrm{sust}}$ with corresponding semilogarithmic graphs in the insets of panel (a) and panel (b).](clast26c.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig26\] (Color online) Primordial facet survival probabilities are show in panel (a) and panel (b) respectively; non-steady state simulation data (broken lines) for $\gamma^{'} = 160$ and calculated curves based on $\gamma = 2000$ steady state results (solid line) are plotted versus $N_{\mathrm{sust}}$ with corresponding semilogarithmic graphs in the insets of panel (a) and panel (b).](clast26d.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig26\] (Color online) Primordial facet survival probabilities are show in panel (a) and panel (b) respectively; non-steady state simulation data (broken lines) for $\gamma^{'} = 160$ and calculated curves based on $\gamma = 2000$ steady state results (solid line) are plotted versus $N_{\mathrm{sust}}$ with corresponding semilogarithmic graphs in the insets of panel (a) and panel (b).](clast26e.eps "fig:"){width=".45\textwidth"}
Curves predicted based on relative area results (solid lines) and results from direct Monte Carlo calculations (broken traces) appear in panel (a) and panel (b) of Fig. \[fig:Fig25\], respectively with log-log graphs in the inset. Broadly there is excellent agreement among the predicted and directly calculated variables.
Salient Time Scales
===================
Due to the universal dependence of relevant variables on volume fraction $\tilde{v}$, to find time scales for a salient event such as a structural phase transition, one need only determine the volume fraction (e.g. in the context of the relative area scenario) for the case of interest and calculate the elapsed time in terms of sustained slices using Equation \[eq:Eq800\]. Since $f(\tilde{v})$ decreases monotonically, ultimately converting to the asymptotic value $f_{0} \equiv 0.216$, one has $f(\tilde{v}) \leq f_{0}$ and thus time scales gleaned from Equation \[eq:Eq800\] are approximately given and bounded from above by $f_{0} \int_{\tilde{v}}^{1} A_{\mathrm{char}}^{2} \tilde{v}^{\frac{1}{3}} d \tilde{v}$, often amenable to exact calculation. For an power law scaling of the kinetic energy where $A_{\mathrm{char}} = (\kappa /\gamma) \tilde{v}^{\alpha}$, we obtain $$\tilde{\tau} = \frac{3}{\kappa^{2}f_{0}(6 \alpha - 4)} [\tilde{v}^{4/3 - 2 \alpha} - 1]
\label{eq:Eq900}$$
![\[fig:Fig27\] (Color online) Time scales for selected volume fractions $\tilde{v}$ for cubic and irregular protoclasts in panel (a) and panel (b) respectively. Solid lines are calculated from direct Monte Carlo simulation results, while broken lines are closed form results bounding time scales from above. Open and filled symbols represent direct simulation results for the relative area and fixed velocity scenarios, respectively.](clast27a.eps "fig:"){width=".45\textwidth"} ![\[fig:Fig27\] (Color online) Time scales for selected volume fractions $\tilde{v}$ for cubic and irregular protoclasts in panel (a) and panel (b) respectively. Solid lines are calculated from direct Monte Carlo simulation results, while broken lines are closed form results bounding time scales from above. Open and filled symbols represent direct simulation results for the relative area and fixed velocity scenarios, respectively.](clast27b.eps "fig:"){width=".35\textwidth"}
Fig. \[fig:Fig27\] shows salient reduced time scales $\tilde{\tau}$ for a range of values of the exponent $\alpha$; solid curves are obtained from Monte Carlo simulation results in the case of the relative area scenario, while broken lines are an upper bound for time scales from Equation \[eq:Eq900\]. In panel (a) of Fig. \[fig:Fig27\], pertaining to cube shaped protoclasts, the two time scales shown include the structural phase transition involving the loss of primordial facets as well as the attainment of spherical shapes (corresponding to equilibration in the relative area scenario) for volume fractions $\tilde{v} = 0.57$ and $\tilde{v} = 0.37$, respectively. On the other hand, in panel (b) of Fig. \[fig:Fig27\], for cohorts of irregular protoclasts, the three time scales shown correspond to $\tilde{v} = 0.50$, $\tilde{v} = 0.25$, and $\tilde{v} = 0.10$. In spite of the monotonic rise in $\tilde{\tau}$ for the attainment of various volume fraction milestones, panel (a) and panel (b) of Fig. \[fig:Fig24\] do not diverge for a finite value of $\alpha$. Moreover, the upper bound in Eq. (i.e. the same upper bound for the cases of cubic and irregular protoclasts), albeit increasing with increasing $\alpha$, nevertheless remains finite for finite $\alpha$.
Conclusions
===========
In conclusion, with large scale Monte Carlo simulations, we have examined the erosion of rocks through stochastic chipping, considering polyhedral stones with as many as 3,600 facets and averaging over at least 1000 realizations of disorder. Using an energy based criterion for accepting a randomly chosen slicing plane, our calculation is unique in placing no restrictions on the number of vertices and faces sheared away with each fracture event. We have argued on theoretical grounds and verified by direct simulation that time scales for the removal of a specific amount of volume or the attainment of a particular structural milestone scale quadratically in the toughness parameter $\gamma$ which specifies the amount of energy per area associated with a fracture; the largest time scales examined in our calculations exceed a million sustained slices per erosion sequence.
Cohorts of stochastically chipped protoclasts in the form of regular polyhedra undergo a structural phase transition in which all primordial facets are abruptly lost with concomitant singularities in other relevant observables, marking a genuine second order phase transition; in a subsequent structural transformation evident in measures of sphericity, the stones revert to spherical shapes. More broadly, however, ensemble averaged observables in the case of initially identical irregular shapes are punctuated by an elimination of facets in multiple stages instead of a single event. On the other hand, cohorts of irregular protoclasts exhibit no structural phase transitions as an aggregate, with individual eliminations of primordial surfaces being blurred by structural disorder.
We find direct measures of deviation from a spherical profile (i.e. the sphericity $\phi_{3}$ and ratio of minimum and maximum center of mass distances $r^{\mathrm{min}}_{\mathrm{max}}$) decrease monotonically with accumulating sustained slices. However, the oblateness measure actually rises at the initial stages of the erosion trajectory, reaching a peak when half of the original volume has been chipped away and then decreasing. This non-monotonic behavior of the oblateness is attributed to initially blade-like protoclasts which lose material more rapidly from the ends than from the edges, briefly becoming more oblate in shape before eventually being rounded into a spherical profile.
That data from distinct erosion scenarios (i.e. relative area and fixed velocity schemes) collapse onto common curves when plotted with respect to the remaining volume fraction confirms the universal dependence of observables on the latter [@Domokos1]. This phenomenon provides an avenue for using results in the relative area case to calculate time dependent observables for arbitrary erosion schemes, a technique we validate by reproducing results gleaned in the context of the fixed velocity scenario by direct Monte Carlo calculation. In addition, one also is afforded an efficient approach for calculating characteristic times (e.g. for specific remaining volume fractions) for a range of erosion scenarios. We have obtained closed form approximate expressions which bound these time scales from above, and which are in good quantitative agreement with the latter.
We acknowledge helpful discussions with Michael Crescimanno. Calculations in this work have benefited from use of the Ohio Supercomputer facility (OSC) [@OSC].
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---
abstract: 'Recently Lin et al. proposed a method of using the underdetermined BSS (blind source separation) problem to realize image and speech encryption. In this paper, we give a cryptanalysis of this BSS-based encryption and point out that it is not secure against known/chosen-plaintext attack and chosen-ciphertext attack. In addition, there exist some other security defects: low sensitivity to part of the key and the plaintext, a ciphertext-only differential attack, divide-and-conquer (DAC) attack on part of the key. We also discuss the role of BSS in Lin et al.’s efforts towards cryptographically secure ciphers.'
author:
- 'Shujun Li[^1], Chengqing Li[^2], Kwok-Tung Lo, and Guanrong Chen, [^3]'
bibliography:
- 'IEEEabrv.bib'
- 'mypaper.bib'
title: 'Cryptanalysis of an Encryption Scheme Based on Blind Source Separation[^4]'
---
blind source separation (BSS), speech encryption, image encryption, cryptanalysis, known-plaintext attack, chosen-plaintext attack, chosen-ciphertext attack, differential attack, divide-and-conquer (DAC) attack.
Introduction
============
With the rapid development of multimedia and networking technologies, the security of multimedia data becomes more and more important in many real applications. To fulfill such an increasing demand, during past decades many encryption schemes have been proposed to protect multimedia data, including speech, images and videos [@Beker:SecureSpeech:Book1985; @Kumar:SecureCryptology:Book1997; @Nichols:SpeechCryptology:Book2002; @Furht:ImageVideoEncryption:Handbook2004; @Li:ChaosImageVideoEncryption:Handbook2004; @Uhl:ImageVideoEncryption:Book2005; @Furht:MultimediaSecurity:Book2005; @Zeng:MultimediaSecurity:Book2006; @Javidi:OpticalSecurity2005].
According to the nature of protected data, multimedia encryption schemes can be classified into two basic types: analog and digital. Most early schemes were designed to encrypt analog data in various ways: element permuting, signal masking, frequency shuffling, etc., all of which may be exerted in time domain or transform domain or both. However, due to the simplicity of the encryption procedures, almost all analog encryption schemes are not sufficiently secure against cryptographical attacks, especially those modern attacks such as known/chosen-plaintext and chosen-ciphertext attacks [@Kumar:SecureCryptology:Book1997; @Nichols:SpeechCryptology:Book2002; @BreakingNagravision:1999; @Li:AttackingPOMEA2004]. As a comparison, in digital encryption schemes, one can employ any cryptographically strong cipher, such as DES [@Schneier:AppliedCryptography96] or AES [@NIST:AES2001], to achieve a higher level of security. Besides, to achieve a higher efficiency of encryption and some special demands of multimedia encryption (such as format-compliance [@Zeng:VideoScrambling:IEEETCASVT2002] and perceptual encryption [@Li:PerceptualEncryption:2005]), many specific multimedia encryption schemes have also been developed [@Furht:ImageVideoEncryption:Handbook2004; @Li:ChaosImageVideoEncryption:Handbook2004; @Uhl:ImageVideoEncryption:Book2005]. Recent cryptanalysis work [@BreakingSFCVideoEncryption:EuroCrypt89; @Jan-Tseng:SCAN:IPL1996; @Qiao:IsRandomOrderSecure:ISCE97; @Uehara:ChosenDCTAttack:IEEEPCM2000; @Yu-Chang:SCAN:PRL2002; @Youssef:BreakingFEA-M:IEEETCE2003; @Li:AttackingFEAM:JSS2006; @Li-Zheng:CKBA:ISCAS2002; @Li-Zheng:BRIE:ICIP2002; @LiLi:AttackingCNN2004; @Li:EURASIP-JASP2005; @LiLiLiChen:ISCAS2005; @Li:JSS2006; @Li:AttackingRCES2004; @ShujunLi:AttackISWBE2006] has shown that some multimedia encryption schemes are insecure against various cryptographical attacks.
Recently Lin et al. suggested employing blind source separation (BSS) for the purpose of image and speech encryption [@Lin:BSS_IE:IEE_EL2002; @Lin:BSS_IE:ICNNSP2003; @Lin:BSS_IE:CASSET2004; @Lin:BSS_SIE:ISNN2005; @Lin:BSS_IE:ISNN2006; @Lin:BSS_SE:ICCCAS2004; @Lin:BSS_SE:IEEETCASI2006]. The basic idea is to mix multiple plaintexts (or multiple segments of the same plaintext) with a number of secret key signals, in the hope that an attacker has to solve a hard mathematical problem – the underdetermined BSS problem. In Sec. VII of [@Lin:BSS_SE:IEEETCASI2006], Lin et al. claimed that this BSS-based cipher “is immune from the attacks such as the ciphertext-only attack, the known-plaintext, and the chosen-plaintext attack", “as long as the intractability of the underdetermined BSS problem is guaranteed by the mixing matrix for encryption".
This paper re-evaluates the security of the BSS-based encryption scheme and points out that it is actually insecure against known/chosen-plaintext attack and chosen-ciphertext attack. In addition, some other security defects are also found under the ciphertext-only attacking scenario, including the low sensitivity to the mixing matrix (part of the secret key) and the plaintext, and a differential attack that works well when the matrix size is small. Based on the cryptanalytic findings, we also discuss the role of BSS in Lin et al.’s efforts towards cryptographically secure ciphers.
The rest of this paper is organized as follows. In next section we give a brief introduction to the BSS-based encryption scheme. Section \[section:Cryptanalysis\] is the main body of this paper and focuses on the cryptanalysis of the BSS-based encryption scheme. Then, the role of BSS in cryptography is discussed in Sec. \[section:Discussion\]. Finally the last section concludes this paper.
BSS-Based Encryption
====================
Blind source separation is a technique that tries to recover a set of unobserved sources or signals from observed mixtures [@Cardoso:BSS:IEEEProc1998]. Given $N$ unobserved signals ${\bm{\mathrm{s}}}_1,\cdots,{\bm{\mathrm{s}}}_N$ and a mixing matrix ${\bm{\mathrm{A}}}$ of size $N\times M$, the BSS problem is to recover ${\bm{\mathrm{s}}}_1,\cdots,{\bm{\mathrm{s}}}_N$ from $M$ observed signals ${\bm{\mathrm{x}}}_1,\cdots,{\bm{\mathrm{x}}}_M$, where $$[{\bm{\mathrm{x}}}_1,\cdots,{\bm{\mathrm{x}}}_M]^T={\bm{\mathrm{A}}}[{\bm{\mathrm{s}}}_1,\cdots,{\bm{\mathrm{s}}}_N]^T.$$ When $M\geq N$, the blind source separation is possible when ${\bm{\mathrm{A}}}$ satisfies some requirements. However, when $M<N$, this is generally impossible (whatever ${\bm{\mathrm{A}}}$ is), thus leading to the underdetermined BSS problem.
In [@Lin:BSS_IE:IEE_EL2002; @Lin:BSS_IE:ICNNSP2003; @Lin:BSS_IE:CASSET2004; @Lin:BSS_SE:ICCCAS2004; @Lin:BSS_SIE:ISNN2005; @Lin:BSS_IE:ISNN2006; @Lin:BSS_SE:IEEETCASI2006], Lin et al. introduced a number of secret key signals to make the determination of the plaintext signals become an underdetermined BSS problem in the case that the key signals are unknown. Given $P$ input plain-signals $s_1(t),\cdots,s_P(t)$ and $Q$ key signals $k_1(t),\cdots,k_Q(t)$, the encryption procedure is described as follows[^5]: $$\label{equation:encryption0}
{\bm{\mathrm{x}}}(t)=[x_1(t),\cdots,x_P(t)]^T={\bm{\mathrm{A}}}{\bm{\mathrm{s}}}_k(t),$$ where ${\bm{\mathrm{x}}}(t)$ denote $P$ cipher-signals, ${\bm{\mathrm{s}}}_k(t)=[s_1(t),\cdots,s_P(t),k_1(t),\cdots,k_Q(t)]^T$, and ${\bm{\mathrm{A}}}$ is a $P\times(P+Q)$ mixing matrix whose elements are within in $[-1,1]$. Assume that ${\bm{\mathrm{A}}}=[{\bm{\mathrm{A}}}_s,{\bm{\mathrm{A}}}_k]$, where ${\bm{\mathrm{A}}}_s$ is a $P\times P$ matrix and ${\bm{\mathrm{A}}}_k$ is a $P\times Q$ matrix. Then, the encryption procedure can be represented in an equivalent form: $$\label{equation:encryption}
{\bm{\mathrm{x}}}(t)={\bm{\mathrm{A}}}_s{\bm{\mathrm{s}}}(t)+{\bm{\mathrm{A}}}_k{\bm{\mathrm{k}}}(t),$$ where ${\bm{\mathrm{s}}}(t)=[s_1(t),\cdots,s_P(t)]^T$ and ${\bm{\mathrm{k}}}(t)=[k_1(t),\cdots,k_Q(t)]^T$. Thus, as long as ${\bm{\mathrm{A}}}_s$ is an invertible matrix, one can decrypt ${\bm{\mathrm{s}}}(t)$ as follows[^6]: $$\label{equation:decryption}
{\bm{\mathrm{s}}}(t)={\bm{\mathrm{A}}}_s^{-1}\left({\bm{\mathrm{x}}}(t)-{\bm{\mathrm{A}}}_k{\bm{\mathrm{k}}}(t)\right).$$
Different values of $Q$ was used in Lin et al.’s papers: $Q=1$ in [@Lin:BSS_IE:IEE_EL2002] and $Q=P$ in [@Lin:BSS_IE:ICNNSP2003; @Lin:BSS_IE:CASSET2004; @Lin:BSS_SE:ICCCAS2004; @Lin:BSS_SIE:ISNN2005; @Lin:BSS_IE:ISNN2006; @Lin:BSS_SE:IEEETCASI2006]. When $Q=P$, Lin et al. further set ${\bm{\mathrm{A}}}_s={\bm{\mathrm{B}}}$ and ${\bm{\mathrm{A}}}_k=\beta{\bm{\mathrm{B}}}$, where $\beta\geq 10$ for image encryption and $\beta\geq 1$ for speech encryption. In this case, the encryption procedure becomes $${\bm{\mathrm{x}}}(t)={\bm{\mathrm{B}}}\left({\bm{\mathrm{s}}}(t)+\beta{\bm{\mathrm{k}}}(t)\right),$$ and the decryption procedure becomes $${\bm{\mathrm{s}}}(t)={\bm{\mathrm{B}}}^{-1}{\bm{\mathrm{x}}}(t)-\beta{\bm{\mathrm{k}}}(t).$$
Observing Eq. (\[equation:encryption\]), one can see that the encryption procedure contains two steps:
- *Step 1*: ${\bm{\mathrm{x}}}^{(1)}(t)={\bm{\mathrm{A}}}_s{\bm{\mathrm{s}}}(t)$;
- *Step 2*: ${\bm{\mathrm{x}}}(t)={\bm{\mathrm{x}}}^{(1)}(t)+{\bm{\mathrm{A}}}_k{\bm{\mathrm{k}}}(t)$.
The first step corresponds to a substitution (block) cipher, and the second step corresponds to a additive stream cipher. From another point of view, the two steps are exchanged as follows:
- *Step 1*: ${\bm{\mathrm{x}}}^{(1)}(t)={\bm{\mathrm{s}}}(t)+{\bm{\mathrm{A}}}_s^{-1}{\bm{\mathrm{A}}}_k{\bm{\mathrm{k}}}(t)$;
- *Step 2*: ${\bm{\mathrm{x}}}(t)={\bm{\mathrm{A}}}_s{\bm{\mathrm{x}}}^{(1)}(t)$.
In any case, the BSS-based encryption scheme is always a product cipher composed by a simple block cipher and an additive stream cipher. In next section, we will show that the two sub-ciphers can be separately broken by known/chosen-plaintext attack and chosen-ciphertext attack.
In the BSS-based encryption scheme, the key signals $k_1(t),\cdots,k_Q(t)$ are as long as the plain-signals and have to be generated by a pseudo-random number generator (PRNG) with a secret seed $\mathrm{I}_0$, which serves as the secret key. In Lin et al.’s papers, it was not explicitly mentioned whether or not the mixing matrix should be used as part of the secret key. However, if the attacker knows ${\bm{\mathrm{A}}}$, the product cipher degrades to be a stream cipher. Considering ${\bm{\mathrm{x}}}^*(t)={\bm{\mathrm{A}}}_s^{-1}{\bm{\mathrm{x}}}(t)$ as the equivalent cipher-signal, the encryption procedure becomes $${\bm{\mathrm{x}}}^*(t)={\bm{\mathrm{s}}}(t)+{\bm{\mathrm{A}}}_s^{-1}{\bm{\mathrm{A}}}_k{\bm{\mathrm{k}}}(t).$$ In this case, the encryption scheme is actually independent of the underdetermined BSS problem. In addition, as we shown later in Sec. \[section:CODA\], the key signals can be totally circumvented in a ciphertext-only differential attack, so the mixing matrix ${\bm{\mathrm{A}}}$ must be kept as the secret key. Thus, in this paper we assume that the secret key consists of both $\mathrm{I}_0$ and ${\bm{\mathrm{A}}}$.
In [@Lin:BSS_IE:IEE_EL2002; @Lin:BSS_IE:ICNNSP2003; @Lin:BSS_IE:CASSET2004; @Lin:BSS_SIE:ISNN2005; @Lin:BSS_IE:ISNN2006], the BSS-based encryption scheme was mainly designed to encrypt $P$ images simultaneously, where $s_i(t)$ is the $t$-th pixel in the $i$-th image. In [@Lin:BSS_SE:ICCCAS2004; @Lin:BSS_SE:IEEETCASI2006], the encryption scheme was suggested to encrypt a single speech, each frame of which is divided into $P$ segments and $s_i(t)$ is the $t$-th sample in the $i$-th segment. This encryption scheme can also be applied for a single image, by dividing it into $P$ blocks of the same size. To facilitate the following discussion, we assume that the encryption scheme is used to encrypt a single plaintext with $P$ segments of equal size.
In Sec. VII of [@Lin:BSS_SE:IEEETCASI2006], Lin et al. claimed that the BSS-based encryption scheme is secure against most modern cryptographical attacks, including the ciphertext-only attack, the known-plaintext attack, and the chosen-plaintext attack. In next section we will show that this claim is problematic.
Cryptanalysis {#section:Cryptanalysis}
=============
Before introducing the cryptanalytic results, let us see how large the key space is. In Lin et al.’s papers, each element of ${\bm{\mathrm{A}}}$ is within the interval $[-1,1]$. Then, assuming that each element in ${\bm{\mathrm{A}}}$ has $R$ possible values[^7], the number of all possible mixing matrix ${\bm{\mathrm{A}}}$ is $R^{P(P+Q)}$. Furthermore, assuming that the bit size of $\mathrm{I}_0$ is $L$, the size of the whole key space is $R^{P(P+Q)}2^L$. When $Q=P$ and ${\bm{\mathrm{A}}}=[{\bm{\mathrm{B}}},\beta{\bm{\mathrm{B}}}]$, the size of the whole key space is $R^{P^2}2^L$. Later we will show that the real size of the key space is much smaller than this estimation, due to some essential security defects of the BSS-based encryption scheme. We will also point out that the encryption scheme under study is not secure against known/chosen-plaintext attack and chosen-ciphertext attack.
Ciphertext-Only Attack
----------------------
### Divide-and-Conquer (DAC) Attack
Rewriting Eq. (\[equation:decryption\]) in the following form: $${\bm{\mathrm{s}}}(t)={\bm{\mathrm{\hat{A}}}}{\bm{\mathrm{x}}}_k(t),$$ where ${\bm{\mathrm{x}}}_k(t)=[x_1(t),\cdots,x_P(t),k_1(t),\cdots,k_Q(t)]^T$ and $${\bm{\mathrm{\hat{A}}}}={\bm{\mathrm{A}}}_s^{-1}\left[{\bm{\mathrm{I}}},-{\bm{\mathrm{A}}}_k\right]=\left[{\bm{\mathrm{A}}}_s^{-1},-{\bm{\mathrm{A}}}_s^{-1}{\bm{\mathrm{A}}}_k\right].$$ From the above equation, to recover $x_i(t)$, one only needs to know ${\bm{\mathrm{k}}}(t)$ and the $i$-th row of ${\bm{\mathrm{\hat{A}}}}$. In other words, when the BSS-based encryption scheme is used to encrypt $P$ independent plaintexts, the $i$-th plaintext can be exactly recovered with the knowledge of $\mathrm{I}_0$ and the $i$-th row of ${\bm{\mathrm{\hat{A}}}}$. A similar result can be obtained when $P$ segments of one single plaintext is encrypted with the encryption scheme. This fact means that $P$ rows of ${\bm{\mathrm{\hat{A}}}}$ can be separately broken with a divide-and-conquer (DAC) attack. As a result, the size of the key space is reduced to be $PR^{(P+Q)}2^L$. When $Q=P$ and ${\bm{\mathrm{A}}}=[{\bm{\mathrm{B}}},\beta{\bm{\mathrm{B}}}]$, it becomes $PR^{P}2^L$.
### Low Sensitivity to ${\bm{\mathrm{A}}}$ {#section:Low_Sensitivity_A}
From the cryptographical point of view, given two distinct keys, even if their difference is the minimal value under the current finite precision, the encryption and decryption results of a good cryptosystem should still be completely different. In other words, this cryptosystem should have a very high sensitivity to the secret key [@Schneier:AppliedCryptography96]. Unfortunately, the BSS-based encryption scheme does not satisfy this security principle, because the involved matrix computation is not sufficiently sensitive to matrix mismatch. Given two matrices ${\bm{\mathrm{A}}}_1$ and ${\bm{\mathrm{A}}}_2$ of size $M\times N$, if the maximal difference of all elements is $\varepsilon$, then one can easily deduce that each element of $|{\bm{\mathrm{A}}}_1{\bm{\mathrm{s}}}(t)-{\bm{\mathrm{A}}}_2{\bm{\mathrm{s}}}(t)|$ is not greater than $N\max({\bm{\mathrm{s}}}(t))\varepsilon$. As a result, the matrix ${\bm{\mathrm{A}}}$ can be approximately guessed under a relatively large finite precision $\varepsilon$, still maintaining an acceptable quality of the recovered plaintexts. This immediately leads to a significant reduction of the size of the key space: from $PR^{(P+Q)}2^L$ to $P\lceil 2/\varepsilon\rceil^{(P+Q)}2^L$, where $\lceil 2/\varepsilon\rceil^{(P+Q)}\ll R^{(P+Q)}$.
The errors induced in inversion of ${\bm{\mathrm{A}}}$ is much harder to estimate, but conceptually they should also have an upper limit proportional to $\varepsilon$.
The above low sensitivity can be easily verified with experiments described as follows:
- *Step 1*: for a randomly-generated key $({\bm{\mathrm{A}}},\mathrm{I}_0)$, calculate the ciphertext ${\bm{\mathrm{x}}}(t)$ corresponding to a plaintext ${\bm{\mathrm{s}}}(t)$;
- *Step 2*: with another mismatched key $({\bm{\mathrm{A}}}+\varepsilon{\bm{\mathrm{R}}},\mathrm{I}_0)$, decrypt ${\bm{\mathrm{x}}}(t)$ to get ${\bm{\mathrm{\tilde{s}}}}(t)$ – an estimated version of ${\bm{\mathrm{s}}}(t)$, where $\varepsilon\in(0,1)$ and ${\bm{\mathrm{R}}}$ is a $P\times(P+Q)$ random $(1,-1)$-matrix.
For each value of $\varepsilon$, the second step was repeated for 100 times to get a mean value of the recovery error (measured in MAE – mean absolute error)[^8]. Then, we can observe the relationship between the recovery error and the value of $\varepsilon$. Figure \[figure:epsilon\_MAE\] shows the experimental results when the plaintexts are a digital image and a speech file, respectively.
![The experimental relationship between the recovery error and the value of $\varepsilon$: a) the plaintext is a digital image “Lenna" (Fig. \[figure:HVS\_error\]a); b) the plaintext is a speech file “one.wav" that corresponds to the pronunciation of the English word “one" (from Merriam-Webster Online Dictionary, http://www.m-w.com).[]{data-label="figure:epsilon_MAE"}](epsilon_MAE_Lenna "fig:"){width="\textwidth"} Legend: $\textcolor{blue}{\ast}$ – $P=Q=4$; $\textcolor{red}{\circ}$ – $P=4$ and ${\bm{\mathrm{A}}}=[{\bm{\mathrm{B}}},\beta{\bm{\mathrm{B}}}]$ ($\beta=10$).\
a)
![The experimental relationship between the recovery error and the value of $\varepsilon$: a) the plaintext is a digital image “Lenna" (Fig. \[figure:HVS\_error\]a); b) the plaintext is a speech file “one.wav" that corresponds to the pronunciation of the English word “one" (from Merriam-Webster Online Dictionary, http://www.m-w.com).[]{data-label="figure:epsilon_MAE"}](epsilon_MAE_one "fig:"){width="\textwidth"} Legend: $\textcolor{blue}{\ast}$ – $P=Q=4$; $\textcolor{red}{\circ}$ – $P=4$ and ${\bm{\mathrm{A}}}=[{\bm{\mathrm{B}}},\beta{\bm{\mathrm{B}}}]$ ($\beta=2$).\
b)
The experimental results confirms that a mismatched key can approximately recover the plaintext. Considering that humans have a good capability of resisting errors in images and speech, even relatively large errors may not be able to prevent a human attacker from recognizing the plain-image or plain-speech. Thus, the value of $\varepsilon$ may be relatively large. When $P=4$, ${\bm{\mathrm{A}}}=[{\bm{\mathrm{B}}},\beta{\bm{\mathrm{B}}}]$ and $\varepsilon=0.1$, we give two examples of such recognizable plaintexts with relatively large errors in Figs. \[figure:HAS\_error\] and \[figure:HVS\_error\].
![An example of human capability against large noises in speech. From top to bottom: the original plain-speech “one.wav", the recovered speech, the recovery error (MAE=0.164103). For reader’s sake, the recovered speech is posted online at http://www.hooklee.com/Papers/Data/BSSE/one\_MAE=0.164103.wav.[]{data-label="figure:HAS_error"}](B-sensitivity-B-0_1-one-0_164103){width="\figwidth"}
![An example of human capability against large noises in images: a) the original plain-image “Lenna"; b) the recovered image (MAE=47.6913).[]{data-label="figure:HVS_error"}](Lenna "fig:"){width="\imagewidth"} a)
![An example of human capability against large noises in images: a) the original plain-image “Lenna"; b) the recovered image (MAE=47.6913).[]{data-label="figure:HVS_error"}](B-sensitivity-B-0_1-Lenna-47_6913 "fig:"){width="\imagewidth"} b)
From the above experimental results, we can exhaustively search for an approximate version of ${\bm{\mathrm{A}}}$ under the finite precision $\varepsilon=0.01\sim 0.1$. Such an approximate version of ${\bm{\mathrm{A}}}$ is then used to roughly reveal the plaintext. Considering the searching complexity is $O\left(\varepsilon^{-(P+Q)}\right)$, such an exhaustive search is feasible when $P,Q$ is not very large[^9]. When $P=2$ and ${\bm{\mathrm{A}}}=[{\bm{\mathrm{B}}},\beta{\bm{\mathrm{B}}}]$, we carried out a large number of experiments in the following steps:
- *Step 1*: for a randomly-generated key $({\bm{\mathrm{B}}},\mathrm{I}_0)$, calculate the ciphertext ${\bm{\mathrm{x}}}(t)$ corresponding to a plaintext ${\bm{\mathrm{s}}}(t)$;
- *Step 2*: randomly generate a matrix ${\bm{\mathrm{R}}}$ (each element over the interval $[-1,1]$), and then decrypt ${\bm{\mathrm{x}}}(t)$ with the guessed key $({\bm{\mathrm{R}}},\mathrm{I}_0)$ to get ${\bm{\mathrm{\tilde{s}}}}(t)$;
- *Step 3*: repeat *Step 2* for $r$ rounds, output the recovered plaintext ${\bm{\mathrm{\tilde{s}}}}^*(t)$, every segment of which corresponds to the best recovery performance in all the $r$ rounds;
- *Step 4*: for the $i$-th segment of ${\bm{\mathrm{\tilde{s}}}}^*(t)$, find the corresponding matrix ${\bm{\mathrm{R}}}$, extract its $i$-th row of its inverse ${\bm{\mathrm{R}}}^{-1}$ to form the $i$-th row of ${\bm{\mathrm{\tilde{B}}}}^{-1}$, the inverse of an estimation of the original matrix ${\bm{\mathrm{B}}}$.
Assuming that the target finite precision is $\varepsilon>0$, the interval $[-1,1]$ is divided into $n_{\varepsilon}=\lceil
2/\varepsilon\rceil$ sub-intervals. Without loss of generality, assuming that $2/\varepsilon$ is an integer, then each sub-interval is of equal size. Thus, if the element in the random matrix ${\bm{\mathrm{R}}}$ has a uniform distribution over $[-1,1]$, the probability that $|r_{i,j}-a_{i,j}|<\varepsilon$ occurs at least one time in $r$ rounds of experiment is $p(n_{\varepsilon},r)=1-(1-1/n_{\varepsilon})^r$, where $r_{i,j}$ and $a_{i,j}$ are the $(i,j)$-th elements of ${\bm{\mathrm{R}}}$ and ${\bm{\mathrm{A}}}$, respectively. One can easily deduce that $p(n_{\varepsilon},r)$ is an increasing function with respect to $r$ and $$\begin{aligned}
p(n_{\varepsilon},n_{\varepsilon})>
\lim_{n_{\varepsilon}\to\infty}p(n_{\varepsilon},n_{\varepsilon}) &
= &
1-\lim_{n_{\varepsilon}\to\infty}(1-1/n_{\varepsilon})^{n_{\varepsilon}}\\
& = & 1-e^{-1}\approx 0.6321,\end{aligned}$$ which leads to the result that $p(n_{\varepsilon},r)>1-e^{-1}$ when $r\geq n_{\varepsilon}$. In other words, with $r\geq
n_{\varepsilon}$ experiments, it is a high-probability event that we have at least one $r_{i,j}$ “equal" to $a_{i,j}$ under the finite precision $\varepsilon$. To get an approximate estimation of the $i$-th row of ${\bm{\mathrm{A}}}$, we can see that $r=O\left(n_\varepsilon^P\right)$ rounds of experiment are needed.
Apparently, the above steps actually simulate the process of a real ciphertext-only attack that tries to reveal the plaintext and to exhaustively guess ${\bm{\mathrm{B}}}^{-1}$ (under the assumption that $\mathrm{I}_0$ has been known). Note that MAE cannot be calculated to evaluate the recovery performance in a real attack, in which one does not know the plaintext. Fortunately, exploiting the large information redundancy existing in natural images and speech, one can turn to use some other measures to reflect the recovery performance of each segment of ${\bm{\mathrm{\tilde{s}}}}(t)$. In our experiments, we use a measure called MANE (mean absolute neighboring error), which is defined as follows for the $i$-th segment of ${\bm{\mathrm{\tilde{s}}}}(t)$ $$\frac{1}{T-2}\sum_{t=2}^{T-1}\frac{|\tilde{s}_i(t)-\tilde{s}_i(t-1)|+|\tilde{s}_i(t)-\tilde{s}_i(t+1)|}{2},$$ where $T$ denotes the segment length. In Figs. \[figure:B\_search\_B\_one\] and \[figure:B\_search\_B\_Lenna\], one recovered plain-speech and two recovered plain-images are shown for demonstration. One can see that $r=O(10,000)$ (or $\varepsilon\approx 0.01$) is sufficient to get a good estimation of the plaintext.
![A recovered speech in one 50,000-round experiment of exhaustively guessing ${\bm{\mathrm{A}}}$ when $P=2$ and ${\bm{\mathrm{A}}}=[{\bm{\mathrm{B}}},\beta{\bm{\mathrm{B}}}]$. From top to bottom: the original plain-speech “one.wav", the recovered speech (MANE of each segment: 0.0469, 0.0521), the recovery error. For reader’s sake, the recovered speech is posted online at http://www.hooklee.com/Papers/Data/BSSE/one\_MANE=0.0469-0.0521.wav.[]{data-label="figure:B_search_B_one"}](B-search-B-50000-one){width="\figwidth"}
![Two recovered plain-images in our experiments of exhaustively guessing ${\bm{\mathrm{B}}}$ when $P=2$ and ${\bm{\mathrm{A}}}=[{\bm{\mathrm{B}}},\beta{\bm{\mathrm{B}}}]$: a) $r=1,000$ (MANE of each segment: 39.7491, 14.9373); b) $r=10,000$ (MANE of each segment: 16.3888, 15.1722).[]{data-label="figure:B_search_B_Lenna"}](B-search-B-1000-Lenna "fig:"){width="\imagewidth"} a)
![Two recovered plain-images in our experiments of exhaustively guessing ${\bm{\mathrm{B}}}$ when $P=2$ and ${\bm{\mathrm{A}}}=[{\bm{\mathrm{B}}},\beta{\bm{\mathrm{B}}}]$: a) $r=1,000$ (MANE of each segment: 39.7491, 14.9373); b) $r=10,000$ (MANE of each segment: 16.3888, 15.1722).[]{data-label="figure:B_search_B_Lenna"}](B-search-B-10000-Lenna "fig:"){width="\imagewidth"} b)
Note that for 2-D images the above 1-D MANE may be generalized to include more neighboring pixels, thus achieving a more accurate description of the recovery performance. In addition, multiple quality factors can be employed to further increase the efficiency of evaluation of the recovery performance.
### Low Sensitivity to ${\bm{\mathrm{k}}}(t)$
Due to the same reason of the low sensitivity to ${\bm{\mathrm{A}}}$, one can deduce that the BSS-based encryption scheme is also insensitive to the key signal ${\bm{\mathrm{k}}}(t)$. Given two key signals ${\bm{\mathrm{k}}}_1(t)$ and ${\bm{\mathrm{k}}}_2(t)$, if the maximal difference of all elements is $\varepsilon$, each element of $|{\bm{\mathrm{A}}}_k{\bm{\mathrm{k}}}_1(t)-{\bm{\mathrm{A}}}_k{\bm{\mathrm{k}}}_2(t)|$ is not greater than $Q\max(|{\bm{\mathrm{A}}}_k|)\varepsilon=Q\varepsilon$. Since ${\bm{\mathrm{k}}}(t)$ itself is not part of the secret key, but generated from $\mathrm{I}_0$, this problem does not have much negative influence on the security of the whole cryptosystem against ciphertext-only attacks.
### Low Sensitivity to Plaintext {#section:Low_Sensitivity_Plaintext}
Another cryptographical property required by a good cryptosystem is that the encryption is very sensitive to plaintext, i.e., the ciphertexts of two plaintexts with a slight difference should be much different [@Schneier:AppliedCryptography96]. However, this property does not hold for the BSS-based encryption scheme. Given two key signals ${\bm{\mathrm{s}}}_1(t)$ and ${\bm{\mathrm{s}}}_2(t)$, if the maximal difference of all elements is $\varepsilon$, each element of $|{\bm{\mathrm{A}}}_s{\bm{\mathrm{s}}}_1(t)-{\bm{\mathrm{A}}}_s{\bm{\mathrm{s}}}_2(t)|$ is not greater than $P\max(|{\bm{\mathrm{A}}}_s|)\varepsilon=P\varepsilon$. When the same secret key is used to encrypt two close-correlated plaintexts, such as a plaintext and its watermarked version, this security defect means that the exposure of one plaintext leads to the revealment of both.
### Differential Attack {#section:CODA}
Given two plaintexts ${\bm{\mathrm{s}}}^{(1)}(t)$ and ${\bm{\mathrm{s}}}^{(2)}(t)$, if they are encrypted with the same key $({\bm{\mathrm{A}}},\mathrm{I}_0)$, we can get the following formula from Eq. (\[equation:encryption\]): $$\label{equation:Delta_xt}
\Delta_{{\bm{\mathrm{x}}}}(t)={\bm{\mathrm{A}}}_s\Delta_{{\bm{\mathrm{s}}}}(t),$$ where $\Delta_{{\bm{\mathrm{x}}}}(t)={\bm{\mathrm{x}}}^{(1)}(t)-{\bm{\mathrm{x}}}^{(2)}(t)$ and $\Delta_{{\bm{\mathrm{s}}}}(t)={\bm{\mathrm{s}}}^{(1)}(t)-{\bm{\mathrm{s}}}^{(2)}(t)$. Note that ${\bm{\mathrm{A}}}_k{\bm{\mathrm{k}}}(t)$ disappears in the above equation. This means that from the differential viewpoint only ${\bm{\mathrm{A}}}_s$ is the secret key, i.e., $\mathrm{I}_0$ is removed from the key. Considering the low sensitivity of the encryption scheme to ${\bm{\mathrm{A}}}$, under finite precision $\varepsilon$ the key space becomes $O\left(P\varepsilon^{-P}\right)$, and one might exhaustively search ${\bm{\mathrm{A}}}_s$ to recover the plaintext differential as follows: $$\label{equation:Delta_st}
\Delta_{{\bm{\mathrm{s}}}}(t)={\bm{\mathrm{A}}}_s^{-1}\Delta_{{\bm{\mathrm{x}}}}(t).$$ From the obtained plaintext differential, one can get a mixed view of the two interested plaintexts, from which both plaintexts may be completely recognizable by humans. See Figs. \[figure:Differential\_Speech\] and \[figure:Differential\_Images\] for four plaintext differentials of two speech files and two images.
![Differentials of two plain-speech files. From top to bottom: the first speech “one.wav", the second speech “two.wav", the differential one-two, the differential two-one. For readers’ sake, the two differential speech files are posted online at http://www.hooklee.com/Papers/Data/BSSE/one-two.wav and http://www.hooklee.com/Papers/Data/BSSE/two-one.wav.[]{data-label="figure:Differential_Speech"}](differential_speech){width="\figwidth"}
![Differentials of two plain-images, “Lenna" and “cameraman": a) Lenna-cameraman; b) cameraman-Lenna.[]{data-label="figure:Differential_Images"}](Lenna-cameraman "fig:"){width="\imagewidth"} a)
![Differentials of two plain-images, “Lenna" and “cameraman": a) Lenna-cameraman; b) cameraman-Lenna.[]{data-label="figure:Differential_Images"}](cameraman-Lenna "fig:"){width="\imagewidth"} b)
Denoting the guessed matrix by ${\bm{\mathrm{\tilde{A}}}}_s$, we have $$\tilde{\Delta}_{{\bm{\mathrm{s}}}}(t)={\bm{\mathrm{\tilde{A}}}}_s^{-1}\Delta_{{\bm{\mathrm{x}}}}(t)=
{\bm{\mathrm{\tilde{A}}}}_s^{-1}{\bm{\mathrm{A}}}_s\Delta_{{\bm{\mathrm{s}}}}(t).$$ Apparently, if ${\bm{\mathrm{\tilde{A}}}}_s\neq{\bm{\mathrm{A}}}_s$, the obtained plaintext differential $\tilde{\Delta}_{{\bm{\mathrm{s}}}}(t)$ will have an inter-segment mixture, which may make the recognition of the two plaintexts more difficult. Fortunately, when $P$ is relatively small, such an inter-segment mixture may not be too severe to prevent the recognition of the two plaintexts by humans. More importantly, our experiments showed that humans can even be able to recognize the two plaintexts even when the mismatch between ${\bm{\mathrm{\tilde{A}}}}_s$ and ${\bm{\mathrm{A}}}_s$ is not very small. When $P=2$, $${\bm{\mathrm{A}}}_s=\left[
\begin{matrix}
0.7123 & -0.4272\\
0.1958 & 0.1295
\end{matrix}\right]\mbox{, }
{\bm{\mathrm{\tilde{A}}}}_s=\left[
\begin{matrix}
0.5914 & 0.9527\\
0.5726 & 0.1437
\end{matrix}\right],\label{equation:As_As2}$$ a plaintext differential obtained in our experiments is shown in Fig. \[figure:image-large-mismatch\]. One can see that both plain-images, “Lenna" and “cameraman", can still be roughly recognized from such a heavily mixed differential. Another obtained plain-speech differential for “one.wav" and “two.wav", is shown in Fig. \[figure:speech-large-mismatch\], from which the two English words (“one" and “two") are also perceptible.
![One obtained plain-image differential when ${\bm{\mathrm{A}}}_s$ and ${\bm{\mathrm{\tilde{A}}}}_s$ have a relatively large mismatch as shown in Eq. (\[equation:As\_As2\]).[]{data-label="figure:image-large-mismatch"}](Lenna-camera-001){width="\imagewidth"}
![One obtained plain-speech differential when ${\bm{\mathrm{A}}}_s$ and ${\bm{\mathrm{\tilde{A}}}}_s$ have a relatively large mismatch. For readers’ sake, this differential speech is posted online at http://www.hooklee.com/Papers/Data/BSSE/two-one-large-mismatch.wav.[]{data-label="figure:speech-large-mismatch"}](two-one-001){width="\figwidth"}
![A visually-optimal result obtained in 100 plain-image differentials: a) the differential; b) the negative image of the differential.[]{data-label="figure:1_in_100"}](Lenna-cameraman-061 "fig:"){width="\imagewidth"} a)
![A visually-optimal result obtained in 100 plain-image differentials: a) the differential; b) the negative image of the differential.[]{data-label="figure:1_in_100"}](Lenna-cameraman-061-n "fig:"){width="\imagewidth"} b)
In this differential attack, the quality evaluation factors (such as MANE) used in Sec. \[section:Low\_Sensitivity\_A\] is not suitable to automatically determine the best result in many plaintext differentials, because each segment of the obtained plaintext differential is also a natural signal with abundant information redundancy. Instead, one has to output all obtained differentials, and check them with naked eyes or ears to find a perceptually-optimal result with the least inter-segment mixture. Figure \[figure:1\_in\_100\] shows such a result in 100 plain-image differentials when $P=2$ and ${\bm{\mathrm{A}}}$ follows Eq. (\[equation:As\_As2\]). By checking each segment separately and combine the $P$ optimal segments together, one can further get a better result with less inter-segment mixture.
While this differential attack works well for $P=2$ as shown above, it will become infeasible when $P$ is sufficiently large, due to the following facts: 1) the inter-segment mixture is too severe; 2) the complexity of checking all $O\left(\varepsilon^{-P}\right)$ differentials is beyond humans’ capability.
Known-Plaintext Attack
----------------------
In this kind of attack, one can access to a number of plaintexts that are encrypted with the same key. Then, from Eq. (\[equation:Delta\_xt\]), with $P$ plaintext differentials, one immediately knows that the mixing matrix can be uniquely determined as follows: $${\bm{\mathrm{A}}}_s=\Delta_{{\bm{\mathrm{X}}}}(t)(\Delta_{{\bm{\mathrm{S}}}}(t))^{-1},$$ where $\Delta_{{\bm{\mathrm{S}}}}(t)$ and $\Delta_{{\bm{\mathrm{X}}}}(t)$ are $P\times P$ matrices, constructed row by row from the $P$ plaintext differentials and the corresponding ciphertext differentials, respectively. Then, ${\bm{\mathrm{A}}}_k{\bm{\mathrm{k}}}(t)$ can be further solved from any plaintext and its ciphertext: $${\bm{\mathrm{A}}}_k{\bm{\mathrm{k}}}(t)={\bm{\mathrm{x}}}(t)-{\bm{\mathrm{A}}}_s{\bm{\mathrm{s}}}(t).$$ Now, $({\bm{\mathrm{A}}}_s,{\bm{\mathrm{A}}}_k{\bm{\mathrm{k}}}(t))$ can be used to recover other plaintexts encrypted by the same key $({\bm{\mathrm{A}}},\mathrm{I}_0)$. Note that ${\bm{\mathrm{A}}}_k{\bm{\mathrm{k}}}(t)$ has a finite length determined by the maximal length of all known plaintexts, so $({\bm{\mathrm{A}}}_s,{\bm{\mathrm{A}}}_k{\bm{\mathrm{k}}}(t))$ can only recover plaintexts under this finite length.
When ${\bm{\mathrm{A}}}=[{\bm{\mathrm{B}}},\beta{\bm{\mathrm{B}}}]$, the key signals can also be determined: $${\bm{\mathrm{k}}}(t)=\frac{{\bm{\mathrm{s}}}(t)-{\bm{\mathrm{B}}}^{-1}{\bm{\mathrm{x}}}(t)}{\beta}.$$ If the PRNG used is not cryptographically strong (such as LFSR[@Schneier:AppliedCryptography96]), it may be possible to further derive the secret seed $\mathrm{I}_0$, thus completely breaking the BSS-based encryption scheme.
Note that $n$ distinct plaintexts can generate $\binom{n}{2}=n(n-1)/2$ plaintext differentials. Solving the inequality $n(n-1)/2\geq P$, one can get the number of required plaintexts to yield at least $P$ plaintext differentials: $$n\geq\left\lceil\sqrt{P-1/4}+1/2\right\rceil\approx\sqrt{P}.$$
Chosen-Plaintext/Ciphertext Attack
----------------------------------
In chosen-plaintext attack, one can freely choose a number of plaintexts and observe the corresponding ciphertexts, while in chosen-ciphertext attack, one can freely choose a number of ciphertexts and observe the corresponding plaintexts. So in these attacks, one can choose $P$ plaintext differentials easily, which means that the above differential known-plaintext attack still works in the same way.
Discussion {#section:Discussion}
==========
As we pointed out in last section, the BSS-based encryption scheme is always insecure against plaintext attack. So the secret key cannot be repeatedly used in any case. This means that the encryption scheme has to work like a common stream cipher, by changing the secret key for each distinct plaintext. However, in this case, ${\bm{\mathrm{k}}}(t)$ (equivalently, the secret seed $\mathrm{I}_0$) is enough to provide a high level of security, since ${\bm{\mathrm{k}}}(t)$ satisfies the cryptographical properties in a perfectly secure one-time-a-pad cipher (see Sec. V.B of [@Lin:BSS_SE:IEEETCASI2006]). Then, the mixing matrix ${\bm{\mathrm{A}}}$ becomes excessive.
Even when one wants to add a second defense to potential attacks by applying the BSS mixing, the low sensitivity of encryption/decryption to the mixing matrix ${\bm{\mathrm{A}}}$ (recall Sec. \[section:Low\_Sensitivity\_A\]) makes this goal less useful. As a result, with the current encryption design, the BSS model does not play a key role in the security of the scheme. The real core of the encryption scheme is the embedded PRNG that is in charge of generating the key signals masking the plaintexts.
If one wants to use the BSS-based encryption scheme with repeatedly used key, some essential modifications have to be made to reinforce the security against various attacks. Following the cryptanalytic results given in last section, we suggest adopting two coutermeasures simultaneously: 1) use a sufficiently large $P$; 2) like the design of most modern block ciphers [@Schneier:AppliedCryptography96], iterate the BSS-based encryption for many rounds to avoid the original scheme’s low sensitivity to the secret key and plaintext. It is obvious that both countermeasures will significantly influence the encryption/decryption speed of the encryption scheme. It seems doubtful if such an enhanced encryption scheme will have any advantages compared with other multiple-round block ciphers, especially AES [@NIST:AES2001] that can be optimized to run with a very high rate on PCs [@Gladman:AESCode].
Finally, it deserve mentioning that the original BSS-based encryption scheme can be used to realize **lossy** decryption, an interesting feature that may find useful in some real applications[^10]. This feature means that an encryption scheme can still (maybe roughly) recover the plaintext even when there are some errors in the ciphertexts. An typical use of this feature is that the ciphertext can be compressed with some lossy algorithms to save the required storage in local computers or the channel width for transmission. For the BSS-based encryption scheme, the lossy decryption feature is ensured by low sensitivity of decryption to ciphertext, which is due to the same reason of the low sensitivity of encryption to plaintext (recall Sec. \[section:Low\_Sensitivity\_Plaintext\]). However, keep in mind that the lossy decryption feature is induced by the low sensitivity to plaintext/ciphertext, so there is a tradeoff between this feature and security.
Conclusion
==========
This paper analyzes the security of an image/speech encryption scheme based on BSS mixing technology [@Lin:BSS_IE:IEE_EL2002; @Lin:BSS_IE:ICNNSP2003; @Lin:BSS_IE:CASSET2004; @Lin:BSS_SIE:ISNN2005; @Lin:BSS_IE:ISNN2006; @Lin:BSS_SE:ICCCAS2004; @Lin:BSS_SE:IEEETCASI2006]. It has been shown that this BSS-based encryption scheme suffers from some security defects, including its vulnerability to a ciphertext-only differential attack, known/chosen-plaintext attack and chosen-ciphertext attack. It remains an open problem how to apply BSS technology to construct cryptographically strong ciphers.
[^1]: Shujun Li and Kwok-Tung Lo are with the Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong SAR, P. R. China.
[^2]: Chengqing Li and Guanrong Chen are with the Department of Electronic Engineering, City University of Hong Kong, Kowloon Toon, Hong Kong SAR, P. R. China.
[^3]: The corresponding author is Shujun Li. Contact him via his person web site: http://www.hooklee.com.
[^4]: This research was partially supported by The Hong Kong Polytechnic University’s Postdoctoral Fellowships Program under grant no. G-YX63. The work of K.-T. Lo was supported by the Research Grants Council of the Hong Kong SAR Government under Project Number 523206 (PolyU 5232/06E).
[^5]: To achieve a clearer description of the BSS-based encryption scheme, in this paper we use some notations different from those in Lin et al.’s original papers. For example, in [@Lin:BSS_SE:IEEETCASI2006], the $i$-th key signal is denoted by $s_{ni}(t)$, while in this paper we use $k_i(t)$ to emphasize the fact that it is a **key** signal.
[^6]: In Lin et al.’s papers, it is said that the decryption procedure was achieved via BSS. However, from the cryptographical point of view, it is more convenient to denote the decryption procedure by Eq. (\[equation:decryption\]).
[^7]: The value of $R$ is determined by the finite precision under which the cryptosystem is realized. For example, if the cryptosystem is implemented with $n$-bit fixed-point arithmetic, $R=2^n$; if it is implemented with IEEE floating-point arithmetic, $R\approx 2^{31}$ (single-precision) or $R\approx 2^{63}$ (double-precision) [@IEEEStandard754:Floating-Point], where note that the sign bit of the floating-point number is always negative.
[^8]: When the plaintext is a digital image with 256 gray scales, we first calibrate each sub-image into the range $\{0,\cdots,255\}$ and then calculate the recovery error of the whole image.
[^9]: In [@Lin:BSS_IE:IEE_EL2002; @Lin:BSS_IE:ICNNSP2003; @Lin:BSS_IE:CASSET2004; @Lin:BSS_SIE:ISNN2005; @Lin:BSS_IE:ISNN2006; @Lin:BSS_SE:ICCCAS2004; @Lin:BSS_SE:IEEETCASI2006], small values are used in all examples: $P=2$ or 4 and $Q\leq P$.
[^10]: Another scheme is a matrix-based image scrambling system proposed in [@Ville:ISWBE:CASVT2004], as pointed out in [@ShujunLi:AttackISWBE2006].
|
---
abstract: 'Mathematical modeling of biological systems is crucial to effectively and efficiently developing treatments for medical conditions that plague humanity. Often, systems of ordinary differential equations are a traditional tool used to describe the spread of disease within the body. We consider the dynamics of the Human Immunodeficiency Virus (HIV) in vivo during the initial stages of infection. In particular, we examine the well-known three-component model and prove the existence, uniqueness, and boundedness of solutions. Furthermore, we prove that solutions remain biologically meaningful, i.e., are positivity preserving, and perform a thorough, local stability analysis for the equilibrium states of the system. Finally, we incorporate random coefficients within the model and obtain numerical results to predict the probability of infection given the transmission of the virus to a new individual.'
author:
- Eric Jones
- |
Peter Roemer\
Faculty Advisors: Mrinal Raghupathi
- Stephen Pankavich
title: 'Analysis and Simulation of the Three-Component Model of HIV Dynamics'
---
mathematical biology, in-host dynamics, virology, HIV, ordinary differential equations.
Introduction
============
The human body has been studied for hundreds of years, and the knowledge accumulated to date includes intricate molecular level mechanisms that determine the ways in which a virus may infect cells and replicate. This information is necessary to develop treatments for maladies ranging from Hepatitis to the Human Immunodeficiency Virus (HIV) and Acquired Immunodeficiency Syndrome (AIDS). An understanding of the accumulation of millions of these interactions across a timescale of medical relevance permits the creation of better treatment methods. As recently as fifty years ago, this could only be done via experiment and repeated trials. Today, through the advent of high speed computing, viral kinetics can be simulated using mathematical models. Hence, one can utilize our current understanding of molecular and cellular dynamics, describe them in mathematical terms, and simulate interactions within the body to examine the course of a virus from initial transmission to a long-term infection. In this paper we will consider the three-component model (3CM) for HIV [@BCN; @KW; @NB; @DS; @NM]. We will study the infection within a single human host using this system of differential equations, analyze the resulting dynamics of the model, and simulate the behavior of solutions.
Generally speaking, HIV infection without the inclusion of anti-retroviral therapy (ART) is described by a number of distinct phases [@PerelsonSIAM]. In the early stages of HIV infection, symptomatic primary infection yields high concentrations of virions within an individual’s blood or tissue. After several weeks, the flu-like symptoms disappear and the viral density then declines rapidly within several days. This corresponds to an increase in the amount of cytotoxic T lymphocytes and the subsequent appearance of anti-HIV antibodies in the blood. For years afterward, the viral concentration deviates very little from this low level and the host typically does not exhibit symptoms of HIV infection, but the concentration of CD$4^+$ T cells measured in blood slowly declines. Such a period can last as long as $10$ years. Ultimately, the viral load increases, the T-cell count drops below $200$ cells per $\mu L$ of blood, and the symptoms of AIDS appear.
Within the current study, we will focus on the initial stages of transmission and infection, detailing the quick rise of the virion population within the body and the noted decrease in the number of CD$4^+$ T-cells. Mathematical models for HIV population dynamics have been used to understand the basic mechanisms involved in the evolution of the infection at the microscopic level and to ascertain the effects of anti-retroviral therapy [@PerelsonSIAM; @PKdB]. To date, deterministic models of HIV dynamics have included at least two components: the population of uninfected CD$4^+$ T-cells and the density of virus-producing cells. However, in the construction of a model for early HIV dynamics one may consider several components as populations within the the blood or lymphatic tissue, including latently-infected CD$4^+$ T-cells, macrophages, and actively-producing infected macrophages. As previously described, the time-dependent changes in the cytotoxic T-lymphocytes which directly attack virus producing CD$4^+$ T-cells could also be included within a model, but as previously stated, this may not have significant impact within the earliest stage of infection. Similarly, effects of time delays between infection of T-cells and active production of virions, or multiple compartment models, which couple the dynamics of multiple infected regions of the body, may also play a role in later stages, but are omitted from the current study. Nevertheless, some authors have found it useful to consider simple models which include only the density of virions, the uninfected CD$4^+$ T-cell population, and one class of (actively-productive) infected CD$4^+$ T-cells. This gives rise to three-component models as in [@BCN; @KW; @NB]. We emphasize that the current article focuses only on the early period, up to a couple of months after infection, and does not study the later progression to the acquired immune deficiency syndrome (AIDS) which may follow without the use of anti-retroviral therapy.
Background and Derivation of Mathematical Model
===============================================
The general principle behind any sort of mathematical model is to examine closely the interactions between the quantities being analyzed. We begin with a small number of initial axioms, and then follow the implications thoroughly to their conclusion. As a basis for our model, we utilize a general biological understanding of HIV dynamics, including infection, replication, and clearance.
In this section we describe the three-component model that has been widely used in the study of HIV. There are many biological operators involved in the interaction between HIV and cells within the human body. The first group is a subset of the population of lymphocytes, which in turn are a type of white blood cell. This subset is known as CD$4^+$ T-cells, or helper T-cells. These T-cells have a variety of functions, including secreting substances that stimulate the immune system, in addition to acting as memory agents and regulating the immune response. In short, CD4+ T-cells detect and direct immune system responses to invading bacteria and viruses. Without them, the body significantly suffers from opportunistic infections that are greater in severity and duration than they would be if the CD4+ T-cells were otherwise not present. HIV, which refers in this case to the virus, not the disease or symptoms associated with it, is a retrovirus that infects helper T-cells. The virus, which is significantly smaller than the T-cell ($120$ nm in diameter compared with $7$ $\mu$m in diameter, respectively), breaches the cell wall and transports its RNA into the T-cell nucleus, where it may remain dormant for a time. Upon activation, the T-cell ceases its function as part of the immune system and instead produces additional copies of HIV. These infected T-cells, along with the healthy T-cells and free floating HIV, are the populations with which we are concerned, and will appear within the mathematical model.
The second aspect of the model’s creation involves incorporating the interactions between these populations and deriving their corresponding mathematical representation. Healthy T-cells are created from stem cells in the bone marrow, and mature in the thymus. While production of T-cells does decrease with the aging of the human body, we shall consider it to be a constant process for two reasons. First, there is no known method other than this production that can affect T-cell creation, and second, the time scale of interest within the model is sufficiently small to consider the T-cell production rate as constant. When considering removal of these cells, we note that T-cells do age and, in time, expire. Within the model, we assume that each T-cell functions for roughly the same amount of time, and thus, the death rate does not vary over the entire population. Instead, we assume the overall number of T-cells lost in a group over a certain period of time is proportional to the number of T-cells within the group. The other mechanism through which the population of healthy T-cells may decrease is via infection. In considering such effects, we utilize an interaction term that arises from the widely-used “mass action principle” to describe the transfer between populations when the virus infects healthy T-cells. This mass action term represents the idea that the rate of interaction, or infection, is directly proportional to the product of the participating populations, namely those of virions (or virus particles) and healthy T-cells. This completes the interactions for the healthy T-cells. We note that the only way to increase the population of infected T-cells is through HIV infection. Thus, our model will contain the same mass action term to describe the removal of the healthy T-cells and the addition of infected T-cells. Similar to the healthy T-cells, infected T-cells die-off or are cleared by the immune system at a rate proportional to the size of their current population. The virus, while produced from the infected T-cells, does not cause the destruction of infected T-cells. Thus, such a transition is not included within the model.
While the virus production rate does differ from cell to cell, we can assume that the aggregate rate is proportional to the population of infected T-cells. This is the only mechanism by which the virus can be created. In contrast, there are two manners in which the virus can be removed, or cleared, from the body. The first is through viral infection of T-cells. The act of infecting a healthy T-cell must technically remove a virus particle from the population of viruses that can infect further T-cells. However, when considering the overall population quantities, the amount of viruses lost this way is minute compared to other methods of creation and destruction. Hence, we will omit this mechanism. The second method of removal is known as viral clearance. It is the removal by the body of individual virus particles, and is performed at a rate proportional to the current amount of virus particles within the body.
We denote by $T$, $I$, and $V$, the number of healthy T-cells, infected T-cells and virions respectively. Based on the biological description above we have the following system of three ODEs:
$$\tag{3CM}
\label{3CM}
\left \{ \begin{aligned}
\frac{dT} {dt} &= \lambda - \mu T - k T V \\
\frac{dI} {dt} &= k T V - \delta I \\
\frac {dV} {dt} &= p I - c V.
\end{aligned} \right.$$
The parameters $\lambda, \mu, k, \delta, p, c$ play an important role in our later results on viral persistence. Based on biological considerations we assume that these constants are positive. Table \[tab1\] shows typical values for these constants and the corresponding units. The values are the average values of these parameters from [@SCCDHP].
Parameter Biological Process Minimum Mean Value Maximum Units
----------- ---------------------------- ---------------------- ------------------------ ---------------------- -------------------------
$\lambda$ T-cell growth rate $0.043$ $0.1089$ $0.2$ $\mu L^{-1}$ day$^{-1}$
$\mu$ T-cell death rate $0.0043$ $0.01089$ $0.02$ day$^{-1}$
$k$ Infection rate $1.9 \times 10^{-4}$ $1.179 \times 10^{-3}$ $4.8 \times 10^{-3}$ $\mu L$ day$^{-1}$
$\delta$ Infected T-cell death rate $0.13$ $0.3660$ $0.8$ day$^{-1}$
$p$ Virus production rate $98$ $1.427 \times 10^3$ $7.1 \times 10^3$ day$^{-1}$
$c$ Viral clearance rate $3$ $3$ $3$ day$^{-1}$
: Parameter values for (\[3CM\]) as observed in [@SCCDHP][]{data-label="tab1"}
The diagram (Fig \[tikz3CM\]) provides a brief visual representation of the mechanisms which govern the system of differential equations. We see each of the populations within a circle, while the arrows running to and from each circle describe their respective interaction.
= \[draw, -latex’\]
\(t) at (0, 0) [$T$]{};
\(i) at (3, -3) [$I$]{};
\(v) at (-3, -3) [$V$]{};
(vpathstart) at (-6, 0) ;
(vpathend)at (0, -6) ;
(tpathstart) at (-3, 3);
(tpathend) at (6, -6) ;
(tpathend2) at (3, 3) ;
(tpathstart) edge\[transition\] node \[auto\][[$\lambda$]{}]{}(t) (t) edge\[transition\] node\[auto\] (i) (i) edge\[transition\] node \[auto\] (tpathend) (vpathstart) edge\[transition\] node \[auto\] (v) (v) edge\[transition\] node \[auto\] (vpathend) (t) edge\[transition\] node \[auto\] (tpathend2);
Mathematical Analysis
=====================
We now answer some fundamental questions about 3CM. In particular we will show that solutions to 3CM exist for all positive time, and are unique. Later we will show that the solution converge to one of two possible steady-states and that the solutions to 3CM remain positive given positive initial conditions. This last property is important since it shows that the model is biologically relevant.
Properties of Solutions
-----------------------
The first step in examining (\[3CM\]) is to prove that a solution to the initial-value problem does, in fact, exist, and that this solution is unique.
\[T1\] Let $T_0,I_0, V_0 \in \mathbb{R}$ be given. There exists $t_0 > 0$ and continuously differentiable functions $T, I, V : [0,t_0) \to
\mathbb{R}$ such that the ordered triple $(T,I,V)$ satisfies (\[3CM\]) and $(T,I,V)(0) = (T_0,I_0,V_0)$.
To prove the result, we utilize the classical Picard-Lindelöf Theorem (cf. [@Bartle]). Since the system of ODEs is autonomous, it suffices to show that the function $f : \mathbb{R}^3
\to \mathbb{R}^3$ defined by $$f(y) = \begin{bmatrix}
\lambda - \mu y_1 - ky_1y_3 \\
ky_1y_3 - \delta y_2 \\
py_2 - c y_3
\end{bmatrix}$$ is locally Lipschitz in its $y$ argument. In fact, it is enough to notice that the Jacobian matrix $$\nabla f(y) = \begin{bmatrix}
-\mu - ky_3 & 0 & -ky_1 \\
k y_3 & - \delta & ky_1 \\
0 & p & - c
\end{bmatrix}$$ is linear in $y$ and therefore locally bounded for every $y \in \mathbb{R}^3$. Hence, $f$ has a continuous, bounded derivative on any compact subset of $\mathbb{R}^3$ and so $f$ is locally Lipschitz in $y$. By the Picard- Lindelöf Theorem, there exists a unique solution, $y(t)$, to the ordinary differential equation $y'(t)=f(y(t))$ on $[0,t_0]$ for some time $t_0 > 0$.
Additionally, we may show that for positive initial data, solutions remain positive as long as they exist. A fortunate byproduct of this result is that the solutions are also bounded.
\[T2\] Assume the initial conditions of (\[3CM\]) satisfy $T_0>0$, $I_0>0$, and $V_0>0$. If the unique solution provided by Theorem \[T1\] exists on the interval $[0,t_0]$ for some $t_0>0$, then the functions $T(t), I(t),$ and $V(t)$ will be bounded and remain positive for all $t \in [0,t_0]$.
We assume that $T(t)$, $I(t)$, and $V(t)$ initially have positive values. From the previous theorem, there exists a $t^*$ such that the solution exists on $[0,t^*]$. Let us denote by $T^*$ the largest time for which all populations remain positive, or more precisely $$T^* = \sup\{t \in [0,t^*] : T(s), I(s), V(s) > 0 \ \mbox{for all} \ s \in [0,t]\}.$$ Then on the interval $[0,T^*]$ we can make estimate the population values.
Recall that all constants in the system are positive. Using this and the positivity of solutions on $[0,T^*]$, we can place lower bounds on $\frac{dI}{dt}$ and $\frac{dV}{dt}$ since $$\frac{dI}{dt}=kTV-\delta I \geq -\delta I$$ and $$\frac{dV}{dt}=pI-cV \geq -cV.$$ Using an integrating factor, we rewrite these differential inequalities to find $$I(t) \geq I(0)e^{-\delta t} > 0$$ and $$V(t) \geq V(0)e^{- c t} >0$$ for $t \in [0,T^*]$. Similarly, we can place an upper bound on $\frac{dT}{dt}$ so that $$\frac{dT}{dt}=\lambda-\mu T -kTV \leq \lambda.$$ Solving for $T$ yields $$T(t) \leq T(0) + \lambda t \leq C_1(1+t).$$ where the constant $C_1$ satisfies $C_1\geq \max\{\lambda, T(0)\}$. We can sum the equations for $\frac{dI}{dt}$ and $\frac{dV}{dt}$ and place bounds on this sum so that $$\frac{d}{dt}(I+V)=kTV-\delta I + pI - cV \leq kTV+pI.$$ Recall that we have a bound on $T$, so we can substitute $$\frac{d}{dt}(I+V) \leq k C_1(1+t)V+pI \leq C_2(1+t)(I+V)$$ where $C_2 \geq \max\{kC_1, p\}$. Solving the differential equation yields $$(I+V)(t) \leq C_3e^{t^2}$$ for $ t\in[0,T^*]$ where $C_3 > 0$ depends upon $C_2$, $I(0)$, and $V(0)$ only. Since $I(t)$ is positive, we can place an upper bound on $V$ by $$C_3e^{t^2} \geq (I+V)(t) \geq V(t)$$ Additionally, since $V(t)$ is positive, it follows that $I(t)$ must be as well. $$C_3e^{t^2} \geq (I+V)(t) \geq I(t)$$ With these bounds in place, we can now examine $T(t)$ and bound it from below using $$\begin{aligned}
\frac{dT}{dt} & = & \lambda-\mu T -kTV \geq - \mu T - kTV \geq -\mu T - kC_3e^{t^2}T\\
& \geq & -C_4(1+e^{t^2})T\end{aligned}$$ for $t \in[0,T^*]$, where $C_4 \geq \max\{\mu, kC_3\}$. Shifting that last term to the other side of the equation yields $$\frac{dT}{dt} + C_4(1+e^{t^2})T \geq 0$$ Since we know $$\frac{d}{dt} ( T(t) + e^{C_4\int_0^t(1+e^{\tau^2}d\tau)} ) \geq 0,$$ then we find for $t \in [0,T^*]$ $$T(t) \geq T(0) e^{-C_4\int_0^t(1+e^{\tau^2}d\tau)} > 0.$$
Thus, the values of $T$, $I$, and $V$ stay strictly positive for all of $[0, T^*]$, including at time $T^*$. By continuity, there must exist a $t>T^*$ such that $T(t)$, $I(t)$, and $V(t)$ are still positive. This contradicts the definition of $T^*$, and shows that $T(t)$, $I(t)$, and $V(t)$ are strictly positive on the entire interval $[0, t^*]$. Additionally, on this same interval, all of the functions remain bounded, so the interval of existence can be extended further. In fact, the bounds on $T$, $I$, and $V$ derived above hold on any compact time interval. Thus, we may extend the time interval on which the solution exists to $[0,t_0]$ for any $t_0 > 0$ and from the above argument, the solutions remain both bounded and positive on $[0,t_0]$.
With this, we have a general idea that the model is sound, and can say with certainty that it remains biologically valid as long as it began with biologically-reasonable (i.e, positive) data. This also shows that once infected, it is entirely possible that the virus may continue to exist beneath a detectable threshold without doing any damage. Finally, we remark that the bounds obtained above ensure the global existence of solutions.
Let $T_0,I_0, V_0 > 0$ be given. Then, for any $t_0 > 0$ there exist continuously differentiable functions $T, I, V : [0,t_0]
\to \mathbb{R}$ such that the ordered triple $(T,I,V)$ satisfies (\[3CM\]) and $(T,I,V)(0) = (T_0,I_0,V_0)$.
Thus, given positive initial data and any $t_0 > 0$, we can be certain that the solution stays both positive and bounded on the interval $[0,t_0]$.
Steady States
-------------
In order to fully understand the dynamics of the three component model, it is necessary to first determine values of equilibria. An equilibrium point is a constant solution of (\[3CM\]) so that if the system begins at such a value, it will remain there for all time. In other words, the populations are unchanging so the rate of change for each population is zero. Setting $\frac {dT} {dt}$,$\frac {dI} {dt}$, and $\frac {dV} {dt}$ equal to zero and solving the resulting equations for $T$, $I$, and $V$, we find that there exist exactly two equilibria. From a biological perspective, we can categorize these points to be when the HIV virus is either extinct from the body, i.e., $I=V=0$, or when the virus persists within the body $(I\neq 0, V \neq
0)$ as $t$ grows large.
We begin by solving for the nonlinear term in (\[3CM\]) and find $kTV = \delta I$. Additionally, the final equation implies $I =
\frac{c}{p} V$. Using the latter equation to substitute for $I$, we find $$kTV = \frac{\delta c}{p} V$$ or $$V \left ( kT - \frac{\delta c}{p} \right ) = 0.$$ Thus, either $V = 0$ or $T = \frac{\delta c}{kp}$. In the former case, we find $I = 0$ and thus $T = \frac{\lambda}{\mu}$. Hence, the ordered triplet $$(T, I, V) = \left (\frac {\lambda} {\mu}
, 0 , 0 \right )$$ is an equilibrium solution known as **viral extinction**, since there are no virus particles or infected cells. We will refer to this point as $P_E$.
In the latter case, $T= \frac {c \delta}{pk}$, and substituting this value of $T$ into the first equation yields $V =\frac {p \lambda} {c
\delta} - \frac {\mu} {k}$ and further substitution shows $I = \frac
{\lambda} {\delta} - \frac {\mu c}{kp}$. Thus, a second equilibrium exists at the point $$(T, I, V) = \left (\frac {c \delta} {pk} , \frac
{\lambda} {\delta} - \frac {\mu c} {kp} , \frac {p \lambda} {c
\delta} - \frac {\mu} {k} \right ).$$ Since there are distinct presences of virus particles and infected T-cells, we refer to this point as **viral persistence** and abbreviate the point as $P_P$.
In terms of biology, we can say $P_E$ is the case in which an infection exists for a short period of time, then is removed from the body by natural means. The virus does not persist. The second case, where the system of equations tends to $P_P$, denotes that situation where the body is unable to clear the infection by itself. If this ends up being the case, than after a certain period of time, the Three Component Model loses its applicability as the infection takes a deeper hold on the body. More complex models, which consider latent infection, effects of macrophages, or cytotoxic immune response, are then required to describe the spread of HIV within the body and its development towards AIDS.
Stability Analysis
------------------
For linear ODEs, it is well-known that the stability properties depend only upon the eigenvalues of the system. However, our model (\[3CM\]) is nonlinear, and thus we must rely on linearization and a theorem of Hartman & Grobman [@HartmanGrobman] to unify the local behavior of the linear and nonlinear systems.
We will investigate the local stability properties of these equilibria by approximating the nonlinear system of differential equations (\[3CM\]) with a linear system at the points $P_E$ and $P_P$. Then, we locally perturb the system from equilibrium and examine the resulting long time behavior. This is done by linearizing the system about each equilibria, using the Jacobian for (\[3CM\]) $$J_{3CM} = \begin{bmatrix}
-kV-\mu & 0 & -kT \\
kV & - \delta & kT \\
0 & p & -c.
\end{bmatrix}$$ Then, by studying the linearized system $$\dot{z}(t) = J_{3CM}(P)
z(t)$$ we can investigate the stability of each equilibrium point $P=
P_E$ and $P=P_P$. As we will see below, this property depends only on a single number, referred to as the basic reproduction number, $R$ given by $$\label{R}
R = \frac{kp\lambda}{c\delta\mu}.$$
We now prove two theorems that demonstrate the relationship between the value of $R$ and the local asymptotically stability of equilibria. These results imply that one can simply examine the value of $R$ to determine whether viral persistence or viral extinction occurs in the limit of the system as $t \to \infty$. This is a remarkable result that allows for the estimation of the persistence of HIV upon initial infection solely by Monte-Carlo simulations by generating different values of $R$, as in [@TuckShip].
\[ExtinctStability\] The viral extinction equilibrium $P_E$ given by $$(T, I, V) = \left (\frac {\lambda} {\mu} , 0 , 0 \right )$$ is locally asymptotically stable if and only if $R \leq 1$.
We begin by computing $J_{3CM}(P_E)$ and determining its corresponding eigenvalues, since these values are known to characterize the local asymptotic behavior of the associated linear system. Specifically, if every eigenvalue possesses negative real part, then the equilibrium point will be stable. On the other hand, if one or more of the eigenvalues possess positive real part, then small perturbations from equilibrium result in magnifications of those disturbances, and the unstable manifold is nontrivial. We remark that in the rare event that $R =1$, the equilibria $P_E$ and $P_P$ are identical. Hence, we’ll focus on the case in which $R < 1$ since the asymptotic stability can been shown when $R=1$ by using a Lyapunov function [@Korob] instead of an analysis of the linearized system.
Evaluating the Jacobian at $P_E = \left (\frac{\lambda}{\mu} , 0 , 0 \right )$ results in $$J_{3CM}(P_E) = \begin{bmatrix}
- \mu & 0 & - \frac {k \lambda} {\mu} \\[4 pt]
0 & - \delta & \frac {k \lambda} {\mu} \\[4 pt]
0 & p & -c \\
\end{bmatrix}
\nonumber$$ The corresponding characteristic equation can be written as $$\det \left [x\mathbb{I} - J_{3CM}(P_E)\right ] = 0$$ or $$(x+\mu)\left ((x+\delta)(x+c)-\frac{kp\lambda}{\mu} \right) = 0.$$ Thus, $x = - \mu < 0$ is one negative eigenvalue of the system, The remaining quadratic equation is $$x^2+a_1x+a_2 = 0$$ where $ a_1 = c+\delta$ and $a_2 = c\delta - \frac{kp\lambda}{\mu}$. Thus, the other eigenvalues are $$x_\pm = \frac{-(c + \delta) \pm \sqrt{(c+\delta)^2 - 4(c\delta - \frac{kp\lambda}{\mu}})}{2}.$$ Since the first term under the square root is nonnegative, these eigenvalues have negative real part if and only if $$4\left (c\delta - \frac{kp\lambda}{\mu} \right ) > 0$$ or $$4c \delta(1 - R) > 0.$$ Since all parameters are positive, we see that all eigenvalues possess negative real part if and only if $R < 1$. Thus, in this case the origin is a locally asymptotically stable equilibrium for the system $$\dot{z}(t) = J_{3CM}(P_E) z(t).$$ Finally, by the Hartman-Grobman Theorem, the asymptotic behavior of (\[3CM\]) is equivalent to that of this linear system for local perturbations, and the result follows. For a more detailed look at the transference of stability properties from linear to nonlinear systems of ordinary differential equations, see [@Logan].
Now that $R$ has been incorporated within the stability analysis, we can rewrite the viral persistence equilibrium in terms of $R$, and notice that it possesses nonpositive population values for $R \leq 1$. Hence, it should not be surprising that solutions do not tend to this equilibrium for such values of $R$. However, as long as $R > 1$, we find that viral persistence is stable.
\[PersistStability\] The viral persistence equilibrium $P_P$ given by $$(T, I, V) = \left (\frac {\lambda}{\mu R}, \frac {\lambda}{\delta R}( R - 1), \frac {\mu} {k} (R- 1) \right )$$ is locally asymptotically stable if and only if $R > 1$.
The analysis for $P_P$ is similar to that of $P_E$. We first linearize (\[3CM\]) about $P_P$ and examine the characteristic equation. The Jacobian is slightly more complicated in this case, but it is given by $$\begin{gathered}
J_{3CM}(P_P) =
\begin{bmatrix}
-k\left (\frac {p \lambda} {c \delta} - \frac {\mu} {k} \right )-\mu && 0 && -k \left (\frac {c \delta} {pk} \right ) \\[4 pt]
k \left (\frac {p \lambda} {c \delta} - \frac {\mu} {k} \right ) && - \delta && k \left(\frac {c \delta} {pk} \right ) \\[4 pt]
0 && p && -c \\
\end{bmatrix}\\
=
\begin{bmatrix}
-\frac{k\lambda p}{c\delta} && 0 && -\frac{c\delta}{p} \\[4 pt]
\frac{k\lambda p}{c\delta}-\mu && -\delta && \frac{c\delta}{p} \\[4 pt]
0 && p && -c.
\end{bmatrix}
\end{gathered}$$ This results in the characteristic equation $$\left (x + \frac{k\lambda p}{c\delta} \right )\biggl ((x+\delta)(x+c)-c\delta)\biggr )+\frac{c\delta}{p} \left ({\frac{k\lambda p}{c\delta}-\mu} \right )p = 0
\nonumber$$ with expanded form $$x^3+a_1x^2+a_2x+a_3 = 0$$ where $$\begin{gathered}
a_1 = c+\delta +\frac{k\lambda p}{c\delta}\\
a_2 = \frac{k \lambda p}{\delta} + \frac{k \lambda p}{c}\\
a_3 = k \lambda p - c \delta \mu.
\end{gathered}$$ In the case of (\[3CM\]), it is possible to determine the signs of the solutions to this equation using a theorem of Routh and Hurwitz [@Routh; @Hurwitz]. According to the Routh-Hurwitz criteria, all roots of this cubic equation possess negative real part if and only if $a_1, a_2, a_3>0$ and $a_1a_2> a_3$. Hence, it is sufficient to show that $R>1$ if and only if the Routh-Hurwitz criteria are satisfied.
Let us first assume $R > 1$. Then, $a_3= k \lambda p - c \delta \mu = c \delta \mu (R-1) > 0$ and since all the coefficients in the system are positive, $a_1, a_2 > 0$. Additionally, $$a_1a_2=\left (c+\delta +\frac{k\lambda p}{c\delta} \right ) \left (\frac{k \lambda p}{\delta} +\frac{k \lambda p}{c} \right )>c \cdot \frac{k \lambda p}{c} = k\lambda p>k\lambda p -c\delta\mu=a_3 .$$ Thus, $R>1$ implies that $P_P$ is a locally asymptotically stable equilibrium. The other direction follows trivially since $a_3 > 0$ implies $R > 1$. Hence, this is both a necessary and sufficient condition due to the form of $a_3$, and the proof is complete.
Our analysis reveals one very important fact about the overall system: for starting values sufficiently close to equilibrium, the long term behavior depends only on the value of $R$. If $R > 1$ then the system tends towards an end state with a non-zero population of infected cells and virions (**viral persistence**), but if $R \leq 1$ then the final equilibrium is a state with no virus or infection (**viral extinction**). Figure \[illustrative\] serves as an example that illustrates solutions in the two different cases given by Theorems \[PersistStability\] and \[ExtinctStability\]. Finally, we also mention that global asymptotic stability of the equilibria can also be shown using a Lyapunov function as in [@Korob].
Unfortunately, our analysis has been restricted to a deterministic model, and hence limited to only one possible solution, whereas in reality the systems under consideration may contain vast uncertainties, especially with regards to the parameters previously described. In the next section, we incorporate chance mechanisms for population coefficients in order to estimate the probability that a persistent infection develops upon an initial viral load being transmitted to a new host.
Numerical Simulations and the Probability of Persistence
========================================================
Both mathematical and biological results support the idea that contact with HIV does not automatically imply the development of a persistent infection. Given factors such as the CD4+ T-cell growth rate, infection rate, and viral clearance rate, the theorems of the previous section display that it is possible to accurately predict the end viral state in the model. While this is very useful, it does not take into account the variability in parameter values amongst a group of individuals. To account for this, we now incorporate random coefficients for the early stages of HIV infection into the model and examine the resulting behavior. If these are introduced in a biologically meaningful fashion, we can estimate their contributions to the variability in the early time course of the viral load, which is not possible with the deterministic coefficients of the previous section. Furthermore, we can obtain predictions of the probability that HIV levels reach certain values as a function of time since initial infection. Such levels can correspond to thresholds in various tests for the detection of HIV in blood. Another study [@TuckShip] previously estimated the probability of viral persistence using the three-component model with random variable coefficients by using the results of Theorems \[ExtinctStability\] and \[PersistStability\]. In this paper, the authors assumed that each random variable, except for $c$ which was assumed constant, possessed truncated normal distributions with the mean and standard deviation given by a clinical study of $10$ infected patients [@SCCDHP]. Upon sampling from these distributions, the authors computed the value of $R$ and used this to determine the asymptotic behavior of the system as $t \to \infty$, thereby avoiding the need to directly simulate the model itself. The results of their simulations estimated that the average probability of viral extinction was approximately $1-7 \%$.
There are a few issues with this approach that we plan to remedy in the current section. First and foremost, the validity of the model declines rapidly after several months [@PerelsonSIAM; @TuckShip] (approximately $100$ days). Hence, determining viral persistence or extinction based solely on the asymptotic behavior of the system as $t \to \infty$ seems problematic. One can easily find solutions which possess large viral loads for several weeks, but eventually tend to extinction as $t
\to \infty$. Another difficulty is that for certain coefficient values, the virus population in the model may stay quite small for the first few months after transmission, but then grow steadily to a persistent steady state over large times. These possible outcomes can be seen more clearly within Figure \[fig1\]. Thus, instead of using the conditions $R \leq 1$ and $R > 1$ (which provide information only about the behavior of (\[3CM\]) in the limit as $t \to \infty$) to determine whether or not a viral load has established a persistent infection, we will formulate and utilize new conditions which possess a finite time horizon. Since many standard tests for HIV currently display a threshold of detection of $50$ virions per $\mu$L [@RP; @Rib], we will set this as the barrier for viral persistence at the end time of the model’s validity. Namely, the condition $V(100) \geq 50$ will represent viral persistence, while $V(100) < 50$ will represent extinction.
[0.5]{} ![ A log plot of the virion population is shown (left) for a choice of parameters resulting in $R = 2.03$, yet the viral load remains small, around $10^{-7}$ copies per $\mu$ L, up to time $t=60$. For larger times, however, the result of Theorem \[PersistStability\] must apply and hence the viral population rebounds and settles into equilibrium near $10$ copies per $\mu$L by $t=600$ (left inset). A log plot of the virion population is shown (right) for a choice of parameters resulting in $R = 0.74$, yet the viral load remains large, around $10^4$ copies per $\mu$L up to time $t=60$. For larger times, however, the result of Theorem \[ExtinctStability\] must apply and hence the viral population sharply decreases to zero, with values around $10^{-8}$ copies per $\mu$L by $t=600$ (right inset). []{data-label="fig1"}](Rlarge_V60small "fig:")
[0.5]{} ![ A log plot of the virion population is shown (left) for a choice of parameters resulting in $R = 2.03$, yet the viral load remains small, around $10^{-7}$ copies per $\mu$ L, up to time $t=60$. For larger times, however, the result of Theorem \[PersistStability\] must apply and hence the viral population rebounds and settles into equilibrium near $10$ copies per $\mu$L by $t=600$ (left inset). A log plot of the virion population is shown (right) for a choice of parameters resulting in $R = 0.74$, yet the viral load remains large, around $10^4$ copies per $\mu$L up to time $t=60$. For larger times, however, the result of Theorem \[ExtinctStability\] must apply and hence the viral population sharply decreases to zero, with values around $10^{-8}$ copies per $\mu$L by $t=600$ (right inset). []{data-label="fig1"}](Rsmall_V60large "fig:")
With these issues now in context, we perform a similar computational study to estimate the probability of persistence. To simulate the dynamics of the virus, we will employ a Monte-Carlo method along with a traditional Runge-Kutta solver to compute solutions of the corresponding systems of ordinary differential equations given by (\[3CM\]).
Sampling {#sec:sampling}
--------
We are interested in examining multiple definitions of viral persistence and allowing for a different parameter distribution than that of [@TuckShip]. Hence, we consider two different cases. In the first case, we sample from truncated normal distributions as in [@TuckShip] to determine the values of the random variable coefficients $\lambda,
\mu, k, \delta$ and $p$, while keeping the parameter $c=3$ constant throughout. The probability of persistence is then estimated both using the time-asymptotic definition of viral persistence (i.e. $R > 1$) and our new finite-time definition (i.e., $V(100) \geq 50$). In the second case, we investigate the influence of the distribution of parameters by sampling from uniform and triangular distributions, as previously performed for this system while considering the additional effects of viral mutation [@RFP]. Here, viral promoters (i.e., those parameters which lead to large reproductive numbers) are sampled from uniform distributions, while viral inhibitors (in this case, death rates) are sampled from triangular distributions, data for both of which are taken from Table \[tab1\]. In particular, the promoter $k$ is sampled from a uniform distribution over the interval $(1.9 \times 10^{-4}, 4.8 \times 10^{-3})$ $\mu$L/day similar to [@RFP], and $p$ is sampled from another uniform distribution over the interval $(98, 7100)$ day$^{-1}$. As in [@BD], we assume an asymmetric triangular distribution $\mathrm{Tri}(0.0043, 0.01089, 0.02)$ day$^{-1}$ for $\mu$, the death rate of uninfected $T$-cells, where $0.0043$ is the minimum value, $0.01089$ is the peak (occurring at the mean recorded value in Table \[tab1\]) and $0.02$ is the maximum value. The growth parameter $\lambda$ is set to be $10\mu$ as in [@TuckShip], and the viral clearance rate $c$ is held constant at $3$ as in [@SCCDHP; @TuckShip]. Finally, the death rate of infected $T$-cells, $\delta$ is sampled from another triangular distribution $\mathrm{Tri}(0.13, 0.366, 0.8)$ day$^{-1}$. Then, as before, the probability of persistence is estimated separately using the $R$ definition of viral persistence and our finite-time horizon.
Of course, because our condition depends upon solving the ODEs until a specific time, our results may now have a strong dependence on initial conditions. Thus, to generate a suitable range of initial data, we estimate the probabilities of persistence and extinction considering a variety of initial conditions. In particular, we choose values of $T_0$ between $100-1000$ cells/$\mu$L, initial viral loads $V_0$ between $100-500$ virions/$\mu$L, and fix the initial infected $T$-cell population at $0$. These values are obtained from similar initial conditions of previous studies [@PerelsonSIAM; @PKdB].
Results
-------
![Sample paths of the healthy T-cell, infected T-cell, and virion populations[]{data-label="samplepaths"}](ode_sims)
According to the described computational methods, $500,000$ simulations were conducted to determine the probability of persistence of a specific infection. Sample paths of associated trials are displayed in Figure \[samplepaths\] and the results are summarized within Table \[table:rHorizons\]. Notice that the proportion of trials within our simulation that resulted in $R \leq 1$ is much smaller - by a factor of ten or twenty - than those which resulted in viral populations less than $50$ virions per $\mu$L approximately three months after the initial infection. Hence, a standard mathematical definition of extinction, namely $\lim_{t \to \infty} V(t) = 0$ which (by Theorem \[ExtinctStability\]) is equivalent to $R \leq 1$, appears to be insufficient to accurately describe the behavior of the viral population on the timescales of biological relevance, specifically during the time period up to a few months after initial infection. Instead, a more precise determination of viral extinction can be made by measuring whether the virus population will remain below the current detectable threshold of $50$ virions per $\mu$L, and under this measure, the probability of extinction is much larger. As can be seen by Table \[table:rHorizons\], the probability of extinction by the former measure is only around $1-2\%$, while the probability jumps to approximately $8-15\%$ under the latter notion that we have proposed. In some types of transmission, such as passage of the disease from a mother to an unborn child or through needlesticks, this percentage is much closer to current estimates of the probability of extinction after initial infection than the criteria $R\leq 1$.
Parameter Distribution $\mathbb{P} \left (R \leq 1 \right )$ $\mathbb{P} \left ( V(60)<50 \frac{\mbox{copies}}{\mu \mbox{L}} \right )$
------------------------ --------------------------------------- ---------------------------------------------------------------------------
Truncated 0.0136 0.1510
Normal
Uniform & 0.0046 0.0859
Triangular
: Probabilities of virion extinction, using finite-time and time-asymptotic definitions of extinction, as well as different parameter distributions (truncated normal and uniform/triangular, respectively). A total of $500,000$ trials were performed for each case; for the time-asymptotic probabilities, initial conditions were varied over 50 combinations, with $V(0)$ varying from 100-500 $\mu L^{-1}$ and $T(0)$ varying from 100-1000 $\mu L^{-1}$, and $I(0)$ set at 0. Hence, for each initial condition pair $(T(0),V(0))$, $10,000$ trials were performed.[]{data-label="table:rHorizons"}
Based on the results of our simulations, which have been consolidated in Table \[table:rHorizons\], the probability of virus extinction using the finite time definition is significantly less than the extinction probability of the time-asymptotic definition by more than an order of magnitude. This result marks the distinction between the finite- and infinite-time results, and also alludes to the eventual breakdown of the model for $t > 100$ as predicted results for (\[3CM\]) stray from clinical results [@PerelsonSIAM; @TuckShip].
![ Average values of $\lambda$ over $100,000$ draws using a truncated normal distribution (left), a uniform distribution (center), and a triangular distribution (right).[]{data-label="randomDraws"}](sanity-check-lambda-draws){width="\textwidth"}
While the probability of viral extinction associated with the finite-time definition remained ten times higher than the corresponding asymptotic-time probability for each distribution, the extinction probabilities across the two distributions also varied by almost a factor of two. This demonstrates the system’s sensitivity to variations within the range of currently accepted values. In Figure \[randomDraws\] the difference between parameter distributions is shown, providing a description of how variations in parameter values can prompt a change in the model behavior.
Lastly, it was expected that the initial conditions for T-cell and virus populations would have a significant impact on the probability of virus extinction, but based on our simulations this is not the case. As seen in Figures \[truncAvg\] and \[triAvg\], for both distributions the initial conditions played little role: for each distribution, when averaged over the initial T-cell and initial virion conditions, the probability of persistence varied less than 5$\%$ from the mean value (averaged over all initial conditions). In addition, there is no clear pattern in the residuals of the probability of persistence of each initial condition relative to the overall probability, which suggests that the initial conditions have little to no effect to the disease’s end behavior. This same phenomena occurs in the asymptotic-time horizon definition of persistence, where the initial conditions are irrelevant to the probability of persistence of the virus.
Conclusions
===========
Three main accomplishments are noted within the current article. First, a local mathematical analysis was performed, and theorems were proved to justify the viability and utility of the three-component model. Second, the local asymptotic stability of steady states were proved and used to define mathematical notions for viral persistence and extinction. These conditions then inspire ODE-free simulations of the in-host viral dynamics merely be sampling from distributions describing the variation of parameters amongst a population of exposed individuals. Finally, we proposed alternative definitions for the notions of viral persistence and extinction, which possess a finite time horizon, rather than depending only upon the asymptotic limit as $t \to \infty$. Simulations were performed to measure the differences in these criteria and their dependence on the probability distribution of parameters. We note that the end viral population, $V(100)$, could be less than the threshold of detectability permitted by modern science much more often than predicted by the associated value of the reproduction ration $R$. This indicates that the methods used in [@TuckShip] are not as accurate as could be hoped for. Hence, it should be clear that a full simulation of the model is required to obtain accurate results regarding the probability of developing a persistent infection.
With full simulation of the ordinary differential equations, we discovered that viral persistence occurred at a rate of roughly ten times that suggested in [@TuckShip]. This proves rather conclusively the finite time and asymptotic limits conditions while related, do not yield identical predictions, and simulation is necessary to realize the full implications of the model. In reality, persistence has been estimated to occur at a rate of up to $90\%$, if the virus is transmitted by blood transfusion, or around $1\%$ if transmitted via sexual intercourse [@Gray]. While [@TuckShip] arrived at an average probability of persistence around $93-99\%$ and we approximated this figure to be around $85-92\%$, recall that these coefficients were drawn from a biased population of HIV-infected individuals because the data arose from people known to have already developed a persistent HIV infection. Additionally, the previous estimates already account for the probability of transmission within them, while our estimates of persistence and those of [@TuckShip] specifically assume that transmission has occurred within a new host.
Stochastic models, such as in [@TuckLC], that utilize Brownian motion to incorporate additional random effects stemming from factors outside of parameter estimation could provide additional improvement to our estimates of persistence, and this could be the goal of a future project. Due to the lack of mathematical tools to analyze stochastic differential equations, though, one would be forced to resort to a more computational framework, rather than a mathematical or asymptotic analysis, in order to glean information from the model. With the development of more descriptive models and more advanced analytical and computational tools, the probability of HIV persistence after initial infection can be estimated with even greater precision.
Acknowledgements
================
This work was completed under the direction of Prof. Mrinal Raghupathi and Prof. Stephen Pankavich, and submitted in partial fulfillment of MIDN Roemer’s Trident Scholar thesis requirement. Additionally, the authors were supported in part by National Science Foundation grants DMS-0908413 and DMS-1211667.
|
---
abstract: 'With the recent trend of applying machine learning in every aspect of human life, it is important to incorporate fairness into the core of the predictive algorithms. We address the problem of predicting the quality of public speeches while being fair with respect to sensitive attributes of the speakers, e.g. *gender* and *race*. We use the TED talks as an input repository of public speeches because it consists of speakers from a diverse community and has a wide outreach. Utilizing the theories of *Causal Models*, *Counterfactual Fairness* and state-of-the-art neural language models, we propose a mathematical framework for fair prediction of the public speaking quality. We employ grounded assumptions to construct a causal model capturing how different *attributes* affect public speaking quality. This causal model contributes in generating counterfactual data to train a *fair* predictive model. Our framework is general enough to utilize any assumption within the causal model. Experimental results show that while prediction accuracy is comparable to recent work on this dataset, our predictions are counterfactually fair with respect to a novel metric when compared to true data labels. The FairyTED setup not only allows organizers to make informed and diverse selection of speakers from the unobserved counterfactual possibilities but it also ensures that viewers and new users are not influenced by unfair and unbalanced ratings from arbitrary visitors to the [ted.com](ted.com) website when deciding to view a talk.'
author:
- |
Rupam Acharyya,^$\dagger$^ Shouman Das,^$\dagger$^ Ankani Chattoraj,^$\dagger$^ Md. Iftekhar Tanveer^^\
^$\dagger$^University of Rochester, ^^Comcast Applied AI Research\
racharyy@cs.rochester.edu, shouman.das@rochester.edu,\
achattor@ur.rochester.edu, mdiftekhar\_tanveer@comcast.com
bibliography:
- 'references.bib'
title: 'FairyTED: A Fair Rating Predictor for TED Talk Data'
---
Introduction
============
In recent times, artificial intelligence is being used in consequential decision making. Governments make use of it in criminal justice system to predict recidivism [@brennan2009evaluating; @tollenaar2013method] which affects the decision about bail, sentencing and parole. Various firms are also using machine learning algorithms to examine and filter resumes of job applicants [@nguyen2016hirability; @chen2017automated; @naim2016automated] which is crucial for the growth of a company. Machine learning algorithms are also being used to evaluate human’s social skills such as presentation performance [@Chen2017a; @Tanveer2015], essay grading [@alikaniotis2016automatic; @taghipour2016neural] etc.
To solve such decision making problems, machine learning algorithms are trained on massive datasets that are usually collected in the wild. Due to difficulties in the manual curation or adjustment over large dataset, it is likely that the data capture unwanted bias towards the underrepresented group based on race, gender or ethnicity. Such bias results in unfair decision making systems, leading to unwanted and often catastrophic consequences to human life and society. For example, the recognition rates of pedestrians in autonomous vehicles are reported to be not equally accurate for all groups of people [@wilson2019predictive]. Matthew et al. [@kay2015unequal] showed that societal bias gets reflected in the machine learning algorithms through biased dataset and causes representational harm for occupations. Face recognition has been found to be not as effective for people with different skin tones. Dark-skinned females have $43$ times higher detection error than light-skinned males [@buolamwini2018gender].
In this work, we propose a predictive framework that tackles the issue of designing a fair prediction system from biased data. As an application scenario, we choose the problem of fair rating prediction in the TED talks. TED talks cover a wide variety of topics and influence audience by educating and inspiring them. In addition, it consists of speakers from a diverse community with imbalances in the age, gender and ethnic attributes. The ratings are provided by spontaneous visitors to the TED talk website. A machine learning algorithm trained solely from the audience ratings will have a possibility of the predicted rating being biased by sensitive attributes of the speakers.
It is a challenging problem because human behavior is driven by numerous factors and hence have huge variability. It is difficult to know the way these factors interact among each other. In addition, uncovering the true interaction model may not be feasible and often expensive. Even though the sharing platforms such as YouTube, Massive Open Online Courses (MOOC), or [ted.com](ted.com) make it possible to collect a large amount of observational data, these platforms do not correct for bias and unfair ratings.
In this work, we utilize *causal models* [@pearl2009causal] to define possible dependencies between attributes of the data. We then address the issue of not knowing the true interaction model by averaging outputs of predictors across several possible causes. Further using these causal models we generate *counterfactual samples* of the sensitive attributes. These counterfactual samples are the key components in our fair prediction framework (adapted from @kusner2017counterfactual ) and help reducing bias in the ratings with respect to sensitive attributes. Finally, we introduce a novel metric to quantify the degree of fairness employed by our FairyTED pipeline. To the best of our knowledge, FairyTED is the first fair prediction pipeline for public speaking dataset and can be applied to any dataset of similar grounds. Apart from the theoretical contribution, our work also has practical implications in helping both the viewers and the organizers make informed and unbiased choices for selection of talks and speakers.
Related Works {#relatedworks}
=============
There has been a rising interest in developing fair algorithms focused to mitigate the bias arising from discriminatory preferences of attributes such as gender, ethnicity, race, etc. These can cause bias across various domain, from college admissions process to criminal justice [@bickel1975sex; @brennan2009evaluating]. Training a machine learning algorithm with an objective of getting higher prediction accuracy can be sometimes unfair towards underrepresented groups in the dataset. For example, it has been shown in [@bolukbasi2016man] that the geometry of word embedding trained with traditional machine learning algorithms reflect gender stereotypes present in our society. Since machine learning models are used to take important and sensitive decisions including credit score prediction, loan applications assessment or predicting crime scenes, a careful approach should be designed to make traditional predictive model fair. @baeza2018bias emphasized the importance of increasing awareness about fairness in web based system. @calders2010three proposed methods for designing discrimination-free bayesian classifier. @dwork2012fairness formulated fairness as an optimization problem and made use of a task specific similarity metric which describes the similarity of two individuals for the classification task. @grgic2016case defined the notion of process fairness by focusing on the process of decision making rather than outcome of the classifier. @schumann2019transfer proposed a framework with theoretical gurantees to transfer fairness in machine learning across various domains. Other related research has been done where the main focus is to quantify the unfairness in a machine learning algorithms and create a model for a certain dataset. Readers are referred to [@kamiran2009classifying; @kamishima2011fairness; @joseph2016rawlsian; @garg2019counterfactual]. For a recent complete survey see [@mehrabi2019survey]. The notion of fairness in a prediction algorithm has been defined in various ways based on researcher’s assumption of fairness [@zliobaite2015survey; @zafar2017fairness] . We follow the causal approach first introduced by @kusner2017counterfactual to address the notion of fairness in a machine learning model. For the causal framework, we have adopted the definition of [@pearl2009causal].
Preliminaries {#prelims}
=============
Causal Model Definition
-----------------------
Following general convention we define causal model as a Directed Acyclic Graph (DAG) with a set of nodes ($N$) and edges $(E \subseteq N \times N)$. Let $Pa_i = \{ n_j| (n_j,n_i) \in E\}$ denote the set of parents of node $n_i$. Adapting the conventions used in @pearl2009causal , @kusner2017counterfactual , we then define the main characteristics of the causal DAGs. Each causal DAG consists of the triple ($\mathcal{U}$, $\mathcal{V}, \mathcal{F}$) where,
- $\mathcal{U}$ denotes the set of unobserved variables in the outside world that influence observed variables of the causal models.
- $\mathcal{V}$ denotes the set of observed variables consisting of three mutually exclusive sets, 1) set of sensitive attributes $S$, 2) set of data attributes $X$ and 3) label $Y$; i.e, $\mathcal{V} = S \cup X \cup Y$.
- $\mathcal{F} = \{ F_1, F_2, \ldots F_n \}$ is a set of functions that define the relationship between $v_i \in \mathcal{V}$ and $Pa_i$ for all $i$. In other words, $v_i = F_i(Pa_i) + \eta_i$, where $\eta_i$ is a random variable drawn from a distribution and $Pa_{i} \subseteq \mathcal{V} \cup \mathcal{U}$.
Defining the set of functions $\mathcal{F}$ is crucial for generating counterfactual samples of $S$ and performing related computations (for details see @kusner2017counterfactual and @pearl2009causal ). For intuitive examples explaining the variables $(\mathcal{U} ,\mathcal{V},\mathcal{F})$ in relation to causal models, see Section \[pipeline\] and Fig. \[fig\_causal\_models\].
Counterfactual sample generation
--------------------------------
To create an augmented dataset including actual observations and counterfactual samples of $S$ we take the following steps (see chapter 4 of @pearl2009causal):
- We assume a prior distribution over $\mathcal{U}$ and infer its posterior distribution given $\mathcal{V}$.
- Intervene on sensitive attributes $S \subseteq \mathcal{V}$ to generate counterfactual samples of $S$. The counterfactual samples of $S$ are then augmented with actual observations to create augmented dataset (See Section \[pipeline\] for details).
Counterfactual fairness
-----------------------
We adapt the definition of counterfactual fairness [@russell2017worlds] and use $Y_{S \rightarrow s'}$ to denote the label of the counterfactual sample of $S$. A predictor $\hat{Y}$ of $Y$ is said to be counterfactually fair given the observed data attributes $X$ and sensitive attributes $S$ if $P(\hat{Y}_{S\leftarrow s}=y|X=x,S=s)
= P(\hat{Y}_{S\leftarrow s'}=y|X=x,S=s)$ for all $s' \neq s$ and all $y$. Intuitively, this equation ensures that the prediction probability remains unaffected by interventions on sensitive attributes $S$ when all other attributes are same. For example, if we observe that talks given by white male speakers are rated to be fascinating with a probability of 0.6 then counterfactually assigning the same talk content and other attributes to white females, say, should not change the probability from 0.6.
Data
====
**Property** **Quantity**
------------------------------- --------------
**Total number of talks** 2,383
**Total number of views** 4206,164,936
**Total length of all talks** 564.63 Hours
**Total number of ratings** 5,954,233
: TED talk Dataset Properties: Information about the TED talk videos that are used in the causal DAGs
\[tab:datasize\]
The data analyzed in our study was collected from TED talk website ([ted.com](ted.com)). We crawled the website to obtain data from 2400 videos published between 2006 and 2017, covering a wide range of topics such as cultural, social and scientific issues. This not only highlights the vast appeal and diversity of TED talks but also exhibits the importance of fair rating predictions. We have removed 17 talks from our dataset as those were not held at a public speaking set up.\
The preliminary data contains details about the total number of *views* $(V)$, the *transcripts* ($T$) used by the speaker, *ratings* ($Y$) of the videos given by the viewers, etc. The rating $Y$ for each video consists of 14 labels such as *beautiful*, *ingenious* and *confusing*. Summary of the dataset is given in Table \[tab:datasize\].
Data Annotations {#dataannot}
----------------
We use Amazon mechanical turk to collect data on the protected attributes $S$ (*race* and *gender*). Each video was annotated by $3$ turkers and we verified the inter-rater reliability using Krippendorff’s alpha [@krippendorff2011agreement] which gives an average agreement of **$93\%$**. The remaining data was manually investigated and annotated.
Data Preprocessing {#datapreproc}
------------------
- We obtain embedded transcript ($T \in \mathbb{R}^d$) using the doc2vec implementation of Gensim package [@le2014distributed] and use $d = 200$ for all reported results.
- The original view count ($V_{old} \in \mathbb{Z}$) in the data ranges over large values compared to other attributes. We use the min-max technique to normalize $V_{(old)}$ and obtain $V \in \mathbb{R}$. $$V = \frac{V_{(old)}-\min\{V_{(old)}\}_{talks}}{\max\{V_{(old)}\}_{talks}-\min\{V_{(old)}\}_{talks}}$$ We assume that, how long a video has been online gets inherently captured by the total views and does not need to be explicitly modeled.
- We denote each original rating as $Y_{(old)} = (y_{1(old)}, \cdots,$ $ y_{14(old)} ) \in \mathbb{Z}^{14}$, where $y_{i(old)}$ is the count of $i^{th}$ label from viewers. $Y_{(old)}$ is scaled w.r.t corresponding total ratings to acquire $Y \in \mathbb{R}^{14}$ as, $$y_{i} = \frac{y_{i(old)}}{\sum_{j=1}^{14} y_{j(old)}}$$
- We binarize each rating label $y_j$ by thresholding w.r.t median $m_j$ ($median\{y_j\}_{talks}$). For each $j$ the label $y_j$ then becomes $0$ or $1$, where $1$ indicates the $y_j > m_j$. We train our classifier to predict these 14 binarized rating labels. The attributes of our final dataset are shown in Table \[tab:final\_data\].
**Sensitive attributes** $S$, race and gender
-------------------------- --------------------------------
**Data attributes** $X$, transcript and view count
**Label** $Y$, rating
: Pre-processed Dataset Attributes: Final attributes from the TED talk dataset that are used to train the FairyTED classifier
\[tab:final\_data\]
Observation in Data {#observation}
===================
We used an open source tool-kit AIF360 [@aif360-oct-2018] to examine existing *bias* or *unfairness* in our preprocessed dataset w.r.t. $S$ (race and gender of the speaker). We calculated the *statistical parity difference* (SPD) and *disparate impact* (DI) [@biddle2006adverse] for each of the 14 binarized rating labels as, $$\begin{gathered}
\textrm{SPD} = \mathbb{P}(y_i = 1 | S \in \textrm{Grp 1})-\mathbb{P}(y_i=1|S \in \textrm{Grp 2})\\
\textrm{DI} = \mathbb{P}(y_i = 1 | S \in \textrm{Grp 1})/\mathbb{P}(y_i = 1 | S \in \textrm{Grp 2})\end{gathered}$$ Using these metrics we calculate the marginal probability of $y_i$ for each $i$ across various groups and observe many significant differences. Fig. \[fig\_aif360metric\] shows some examples where we compare *male* speakers with speakers of *other genders*. The difference between blue and orange bars are noticeable for rating labels marked with red blocks. We observe that talks from male speakers are rated *ingenious*, *fascinating* and *jaw dropping* with greater probability. This identifies some classic instances of bias in data arising from social norms and structures. However, not all bias observed in data are against the presumed *unprivileged* community, for example, speakers from other genders get higher probability for *courageous* label as compared to male speakers. Our goal is to remove all types of bias from data, both expected and unexpected. The counterfactual fairness is agnostic to the type of bias and aims to remove all possible unfairness in rating across all possible combinations of sensitive attributes. Moreover it can be shown that under suitable assumptions, counterfactual fairness implies group fairness (see arxiv version for details).
![Existence of bias in data: Here we show an example of biased rating in data using disparate SPD and DI metrics (see Observation in Data). In this example we compare male speakers with others and find male speakers are rated to give fascinating, ingenious and jaw dropping talks with higher probability. However contrary to expectation, other speakers had higher probability of being courageous.[]{data-label="fig_aif360metric"}](metric_barplot.pdf){width="0.95\columnwidth"}
FairyTED Pipeline {#pipeline}
=================
To achieve the goal of building a fair predictor of TED talk ratings $Y$, we execute the following steps:
1. **Show bias in real data:** We show that the viewer ratings in the TED talks is actually biased by using SPD and DI. This justifies the need to build a fair classification model for TED talk ratings.
2. **Preprocess data and define causal model:** We then define a causal model, C-DAG of the TED talk ratings, which consist of three major components: the unobserved variables $U$, the sensitive attributes $S$ and the TED talk video attributes $X = \{T,V,Y\}$ obtained by preprocessing the data (see Section \[datapreproc\]). For a specific model shown in Fig. \[fig\_causal\_models\], $U$ is the skill set and background of the speaker and $S$ is gender and race. $U$ and $S$ causally influence $T$, $V$ and $Y$. The skill set, background, gender and race of the speakers strongly influence their life experiences and hence govern the content of their talks. These also determine whether viewers choose to view their talk or not and what type of rating they get if viewed. Since other similar models can also be justified for the dataset, we ensure that our system is robust to any kind of causal model with similar setup (see Fig. \[fig\_causal\_models\]).
3. **Model average:** We consider variants of C-DAG with two intuitively possible manipulations, 1) C-DAG1: unobserved causes affecting $T$ and $V$ are independent, meaning $U$ decomposes into $U_1$ and $U_{2}$ such that $U_1$ affects $T$ and $U_{2}$ affects $V$ (e.g, skill set only influences how likely a talk will be viewed and background of the speaker influences the content of the talk, shown in Fig. \[fig\_causal\_models\](b)). 2) C-DAG2: the affect of sensitive attributes is manipulated, we consider the case where gender does not influence $T$ (Fig. \[fig\_causal\_models\](c)).
4. **Fit model parameters:** We fit the parameters of each causal model, C-DAG, C-DAG1, C-DAG2.
5. **Create augmented data with counterfactual samples:** Next from each of these fitted models, we generate counterfactual samples of $S$ (such as replacing male speaker with female speaker for a particular talk with fixed skill set and background, also see Preliminaries) and create augmented datasets ($D_{aug}, D1_{aug}, D2_{aug}$) to be used for classification.
6. **Train fair classifier:** Finally for each model we use corresponding augmented dataset to train a neural network for binary classification of each of the 14 rating labels. The loss function used to train the network has two parts: 1) the first part minimizes prediction error when compared to true data labels and 2) the second part reduces disparity between the labels of observed values of $S$ and their corresponding counterfactuals. This ensures that simply changing a male speaker to female with fixed skill set and background does not influence the rating. The prediction accuracy of the fair classifier is obtained by averaging performance across all three models.
7. **Fairness validation:** We finally validate that our classifier is counterfactually fair as compared to actual ratings provided by viewers. For this we introduce a novel metric *coefficient of probability variance*, $CV_{prob}$ that compares variability of ratings across possible instances of $S$ before and after introduction of fairness measure (e.g. hypothetically if male and female speakers were rated funny with probability 0.75 and 0.45 just due to difference in $S$ and after fairness was introduced in prediction this variability dropped, becoming 0.75 and 0.70 solely based on same content, skill set and background).
This pipeline brings together fairness measuring metrics and counterfactual fairness incorporation techniques to build a complete setup for TED talk dataset (Fig. \[fig\_model\_pipeline\]). Our setup can be applied to any language or video dataset whose feature embedding can be obtained using any state-of-the-art method. In addition, this setup can accommodate multiple causal models to ensure fairness in classifier across all possible models. Besides this, our setup also allows having nodes that cater to unobserved causes in the world.
{width=".95\textwidth"}
Causal Model {#causalmodelsub}
------------
We consider three relevant variants of causal models for TED talk data using the general definitions mentioned in preliminaries.
- C-DAG as in Fig. \[fig\_causal\_models\](a), $$\begin{aligned}
T & \sim \mathcal{N}\left( w_T ^U U + w_T ^S S,\sigma_T ^2 \mathbf{1} \right)\\
V & \sim \mathcal{N}^{[0,1]}\left( \sigma \left(w_{V} ^U U + w_{V} ^S S + w_{V} ^T T\right),\sigma_{V} ^2 \mathbf{1} \right)\\
Y & \sim Bern\left( \sigma\left(w_Y ^U U + w_Y ^S S + w_Y ^T T + w_Y ^{V} V \right)\right)\end{aligned}$$
where $\mathcal{N}^{[0,1]}$ denotes the truncated Gaussian in the domain $[0,1]$. We generate $U$ from $\mathcal{N} \left(0,\mathbf{1} \right)$ and fit $\Theta_{\textrm{C-DAG}} = \{ w_T^U, w_T^S, \sigma_T, w_{V}^U, w_{V}^S, w_{V}^T,\sigma_{V}, w_Y^U, w_Y^S, w_Y^T, w_Y^{V}, \sigma_Y\}$ using variational inference algorithm in PyMC3 [@Salvatier2016].
- We modify C-DAG such that $U$ decomposes into mutually exclusive sets $U_1$ and $U_2$ to influence $T$ and $V$ independently (Fig. \[fig\_causal\_models\](b)) giving C-DAG1, $$\begin{aligned}
T & \sim \mathcal{N}\left(w_T ^{U_1} U_1 + w_T ^S S,\sigma_T ^2 \mathbf{1} \right)\\
V & \sim \mathcal{N}^{[0,1]}\left(\sigma(w_{V} ^{U_2} U_2 + w_{V} ^S S + w_{V} ^T T),\sigma_{V} ^2 \mathbf{1} \right)\\
Y & \sim Bern\left(\sigma\left(w_Y ^{U_1} U_1 + w_Y ^S S + w_Y ^T T + w_Y ^V V \right)\right)\end{aligned}$$ Both $U_1$ and $U_2$ are drawn from $\mathcal{N} \left(0,\mathbf{1} \right)$ and we fit $\Theta_{\textrm{C-DAG1}} = \{w_T ^{U_1}, w_T^S,\sigma_T, w_{V} ^{U_2}, w_{V} ^S, w_{V} ^T,\sigma_{V}, w_Y ^{U_1}, w_Y ^S,$ $ w_Y ^T, w_Y ^{V},\sigma_Y\}$ to obtain posterior distributions over $U_1$ and $U_2$.
- In C-DAG2 $S$ from C-DAG decomposes into $S_1$ and $S_2$ to influence $T$ and $V$ as in Fig. \[fig\_causal\_models\](c). $S_1 \subset S$ consists of race only, whereas, $S_2 = S$ which includes both race and gender, $$\begin{aligned}
T & \sim \mathcal{N}\left(w_T ^U U + w_T ^{S_1} S_1,\sigma_T ^2 \mathbf{1} \right)\\
V & \sim \mathcal{N}^{[0,1]}\left(\sigma(w_{V} ^U U + w_{V} ^{S_2} S_2 + w_{V} ^T T),\sigma_{V} ^2 \mathbf{1} \right)\\
Y & \sim Bern \left(\sigma\left(w_Y ^U U + w_Y ^S S + w_Y ^T T + w_Y ^{V} V \right)\right)\end{aligned}$$ Here, $U \sim \mathcal{N} \left(0,\mathbf{1} \right)$ and we fit $\Theta_{\textrm{C-DAG2}} = \{w_T ^{U}, w_T ^{S_1},\sigma_T, w_{V} ^{U}, w_{V} ^{S_2}, w_{V} ^T,\sigma_{V}, $ $w_Y ^{U}, w_Y ^S, w_Y ^T, w_Y ^{V},\sigma_Y\}$ to obtain posterior distribution over $U$.
{width=".95\textwidth"}
From each of the three models we generate counterfactual samples of $S$.
Classifier Model {#classifier}
----------------
We train a neural network with one hidden layer of 400 nodes to predict ratings $Y$. Let $\{(s_i,x_i)\}_{i=1}^N$, $\{y^i\}_{i=1}^N$ represent the attributes and labels of the dataset where $s_i$ represents an instance of $S$ and $x_i$ represents instance of $(T, V)$. We train a classification function $g$ such that $\hat Y = g(s,x)$ using a loss function which is a combination of *prediction loss* and *unfairness loss*. We use binary cross entropy loss (BCE) to calculate the prediction error and an unfairness function $u$ to estimate the unfairness of the classifier as,
$$\label{loss_function}
\begin{split}
\mathcal{L}(g) = \frac{1}{N} \sum_{i=1}^N \bigg(\underbrace{\textrm{BCE}\left(g(s_i,x_i), y^i\right)}_{\textit{prediction loss}} \\+\underbrace{\gamma \sum_{s'\neq s_i} u(g, s',s_i, x_i)}_{\textit{unfairness loss}} \bigg)
\end{split}$$
$$u(g, s',s_i, x_i) = \frac{1}{C} \sum_{c=1}^C \max\{0, |g(s_i, x_i^c)-g(s'_i, x_i^c)|-\epsilon\}$$
where $C$ represents the number of counterfactual samples for each observed data instance and $\epsilon$ is a hyperparameter which makes sure that our predictor maintains a $(\epsilon,\delta)$- approximate counterfactual fairness ($\delta$ is a function of $\gamma$, for more details about the choice of the unfairness function, please refer to [@russell2017worlds]). We tune $\gamma$ and $\epsilon$ to obtain best results in our causal models, see Table 3.
Results
=======
Prediction Accuracy
-------------------
After training the classifier with augmented datasets for the three models, we obtain an average prediction accuracy of $69\%$ across all rating labels (Table \[accuracy\]). This accuracy is obtained by training the classifier without the unfairness measure in the loss function. The mean accuracy obtained from an unfair classifier is comparable to accuracy reported in recent studies on TED talk data [@cheng2014predicting; @tanveer2019causality]. However, note that our language model [@le2014distributed] is much simpler when compared to methods used in the cited studies. With this simple choice we emphasize the general appeal of our approach whose goal is to reduce unfair prediction in data irrespective of the embedding technique. After addition of the unfairness measure $u$ in the loss function in equation , the average prediction accuracy goes down as expected, to $67\%$ but not significantly (see section \[classifier\] for details). The hyperparameter $\gamma$ in the loss function plays a critical role in determining the trade-off between fairness in prediction and its accuracy, the smaller the value of $\gamma$, the more unfair the prediction.
Fairness Improvement
--------------------
We first show that there is a decrease in the unfairness measure of the classifier with the increase in training iterations (Fig \[result\_fig\](A)). We then verify the improvement of fairness in prediction of all 14 rating labels across possible groups of $S$. To do so we come up with a fairness comparison metric $CV_{prob}$ and compare prediction fairness before and after addition of unfairness function $u$ in the loss function in equation .
{width=".95\textwidth"}
$CV_{prob}$ Metric
------------------
We have 3 types of gender (male, female and other) and 4 types of race (White, Asian, African American and other) under $S$ giving 12 possible groups denoted by $G=\left\{ G_1, G_2,\cdots, G_{12} \right\}$ on whom counterfactual fairness is tested. We then calculate with what probability each of these 12 groups obtain a particular rating, i.e., for all $k \in \{1,\cdots,14\}$ and $i \in \{1,\cdots,12\}$, we compute $p_{i}^{k}= \mathbb{P}( y_k=1| S = G_i)$. We denote, $P^{k} = (p_1^{k},\ldots p_{12} ^{k})$. Similarly for the predicted label we compute $\hat{p}_{i}^{k}= \mathbb{P}(\hat{y}_k = 1| S = G_i)$ and denote, $\hat{P}^{k} = (\hat{p}_1^{k},\ldots, \hat{p}_{12} ^{k}).$ Note that, variability in the coordinates of $P^{k}$ is a measure of fairness across the group for the rating $y_k$. In particular, the more variable these coordinates are, the more unfair the label is. Also $CV = \frac{std}{mean}$ is a common statistical metric used to quantify variability/irregularity in a set of values. Hence we define, $CV_{prob}^k$ as $CV$ for the coordinates of $P^{k}$. Similarly, $\hat{CV}_{prob}^k$ is the $CV$ for the coordinates of $\hat{P}^{k}.$ Hence, if a predictor improves fairness in the prediction, then $\hat{CV}_{prob}^k$ should be less than $CV_{prob}^k$ as in Fig. \[result\_fig\](B). We also compared the SPD and DI for the true labels and predicted labels of the test dataset (Fig. \[result\_fig\] (C,D)).
Conclusion
==========
The FairyTED setup is applicable to any dataset with the following properties, 1) contains sensitive attributes which can cause biased predictions, 2) it is possible to define a causal model with relevant attributes of the dataset, 3) a counterfactual fairness measure can be defined on the prediction probability by using counterfactual samples of the sensitive attributes.\
We successfully identified the above properties in the TED talk dataset and removed bias/unfairness in rating prediction. The impact of this work is many-fold: 1) First, it identifies the necessity of applying counterfactual fairness on the rich and influential database of TED talk videos. 2) Second, it identifies that counterfactual fairness measure is the most relevant for TED talk videos and like datasets. This is because it allows identification of attributes in data which play a critical role in causing biased predictions for example male versus female speakers. In public speaking platforms it is expensive and implausible to test the change in rating of a talk when given by a female speaker instead of a male speaker with the same content and same skill set. Counterfactual fairness allows to hypothetically test any such example and correct for the resulting bias. 3) In most real world situations it is hard to know the true causal model for the observed dataset, and our setup can deal with this issue as long as we have some idea of possible models. 4) In rich and complicated domain such as TED talks, there can be a lot of unobserved attributes that can affect the data and our setup can take care of it by inferring a posterior distribution over the unobserved attributes. 5) Any dataset with a simple embedding scheme can use this model. One straightforward extension of this framework can be on job interview based datasets. We can include better encoding schemes such as [@devlin2018bert; @yang2019xlnet] to obtain even better prediction accuracy besides making it fair. We can also include multi-modal (e.g. audio-visual) information from the dataset to obtain rich representation and test how counterfactual fairness measure generalizes across various modes. 6) We propose an intuitive novel metric that quantifies degree of fairness employed by our setup. Besides these, information from temporal evolution can also be used to improve our framework in future work. Finally the FairyTED setup has three important social impacts: 1) It ensures that speakers get fair feedback and can improve only based on fair fallacies 2) Organizers can employ diverse speakers without worrying about degradation in rating when skill, ability, influence and content are matched. For example, suppose a talk on global warming was given by a male speaker and obtained good rating. With our setup, the organisers can choose a female speaker with comparable qualities as the male to speak on the global warming without worrying about deterioration in rating. 3) Finally, new viewers will not be biased by prevalent unwanted biased ratings from previous users and prevents propagation of unfairness over time.
Acknowledgments
================
We thank the anonymous reviewers for their valuable suggestions.
Relation between Counterfactual Fairness and Group Fairness
===========================================================
Let $R$ be the set of all races and $G$ be the set of all genders. Usually group fairness considers fairness between racial groups(e.g. White vs African American) or gender groups (e.g. Male vs Female), whereas counterfactual fairness measures fairness among finer groups (e.g. white male vs African American female). More formally, counterfactual fairness measures fairness among all (race,gender) pairs. Let, $$pr(r,g)=\mathbb{P}(\hat{Y}=y| X = x, \text{Race} = r,\text{Gender} = g)$$ Under the assumption that counterfactual fairness tries to minimize the absolute difference between $pr(r,g)$ and $pr(r',g')$ $\forall r,r' \in R $ and $g,g' \in G$. In an ideal scenario this difference is $0$ and we call this predictor ideally and counterfactually fair.
\[cf\_implication\] With causal model assumption (C-DAG,C-DAG1,C-DAG2) any ideally and counterfactually fair predictor satisfies group fairness w.r.t. each race group and each gender group, i.e.,we have, $$\mathbb{P}(\hat{Y}=y| X = x, \text{Race} = r)= \mathbb{P}(\hat{Y}=y| X = x, \text{Race}= r')$$ $\forall r,r'\in R$ and $$\mathbb{P}(\hat{Y}=y| X = x, \text{Gender} = g)$$ $$= \mathbb{P}(\hat{Y}=y| X = x, \text{Gender}= g')$$ $\forall g,g'\in G.$
(Proof of Theorem \[cf\_implication\]\] $$\begin{aligned}
& \mathbb{P}(\hat{Y}=y| X = x, \text{Race} = r)\nonumber &\\
& = \sum_{g} \mathbb{P}(\hat{Y}=y, \text{Gender} = g| X = x, \text{Race} = r)&\nonumber\\
& =\sum_{g} \mathbb{P}(\hat{Y}=y| X = x, \text{Race} = r, \text{Gender} = g)\cdot &\nonumber\\
&\quad \quad\mathbb{P}( \text{Gender} = g | \text{Race} = r)&\label{gender_race_ind}\\
& = \sum_{g} \mathbb{P}(\hat{Y}=y| X = x, \text{Race} = r, \text{Gender} = g)\cdot &\nonumber\\
&\quad \quad\mathbb{P}(\text{Gender} = g )\label{cf_def}&\\
& = \mathbb{P}(\hat{Y}=y| X = x, \text{Race} = r')&\end{aligned}$$ follows from the independence assumption between race and gender, follows from the definition of ideal counterfactually fair model.
This establishes the fact that counterfactual fairness is a stronger notion than group fairness.
|
---
abstract: 'We analyze EUV spectra of the full solar disk from the [*Cosmic Hot Interstellar Plasma Spectrometer*]{} (CHIPS) spanning a period of two years. The observations were obtained via a fortuitous off-axis light path in the 140[[–]{}]{}275 Å passband. The general appearance of the spectra remained relatively stable over the two-year time period, but did show significant variations of up to 25% between two sets of Fe lines that show peak emission at 1 MK and 2 MK. The variations occur at a measured period of 27.2 days and are caused by regions of hotter and cooler plasma rotating into, and out of, the field of view. The CHIANTI spectral code is employed to determine plasma temperatures, densities, and emission measures. A set of five isothermal plasmas fit the full disk spectra well. A 1[[–]{}]{}2 MK plasma of Fe contributes 85% of the total emission in the CHIPS passband. The standard Differential Emission Measures (DEMs) supplied with the CHIANTI package do not fit the CHIPS spectra well as they over-predict emission at temperatures below [log$_{10} T$]{}= 6.0 and above [log$_{10} T$]{}= 6.3. The results are important for cross-calibrating TIMED, SORCE, SOHO/EIT, and CDS/GIS, as well as the recently launched [*Solar Dynamics Observatory*]{}.'
author:
- 'M. M. $^{1}$, M. $^{1}$, W. $^{1}$'
bibliography:
- 'chips\_solar.bib'
title: 'EUV Spectra of the Full Solar Disk: Analysis and Results of the [*Cosmic Hot Interstellar Plasma Spectrometer*]{} (CHIPS)'
---
Introduction {#S-Introduction}
============
The solar EUV irradiance is the primary source of energy and ionization of the Earth’s upper atmosphere. Variations in the EUV flux are responsible for dramatic changes in the density of neutrals and ions, affecting satellite drag forces in near-Earth orbit by up to a factor of ten and modifying the propagation of radio frequency signals [@Fuller-Rowell04].
Solar high spectral-resolution EUV spectra were first obtained via sub-orbital rocket flights [@Behring72; @Malinovsky73; @Thomas94], and [*SkyLab*]{} [@Dere78]. Collectively, these early missions discovered over 400 emission lines from 17 elements between 50 Å and 600 Å, indicating the presence of a 1 to 2 MK plasma dominated by emission from Fe.
The [*Solar and Heliospheric Observatory/EUV Imaging Telescope*]{}(SOHO/EIT) obtains full-disk images in four $\approx$10 Å (FWHM) passbands centered at 171, 195, 285, and 304 Å. These pictures show where EUV emission originates, however, as concluded by , it is essentially impossible to accurately convert the broad-band photometry to physical units without simultaneous knowledge of the solar spectrum.
SOHO/CDS/GIS has one channel (150 to 220 Å) that overlaps CHIPS, but shows spectral “ghosts” and temporal variations in throughput that require careful calibration [@Kuin07] to provide acurate line intensities and ratios. The [*Hinode/Extreme Ultraviolet Spectrograph*]{} (EIS) is a high spectral- and spatial-resolution instrument that provides very detailed temperature and density maps, but can only see a small strip (1 or 2[$^{\prime\prime}$]{} wide by 512[$^{\prime\prime}$]{} long) of the Sun [@Young09].
Each of these instruments has its advantages and limitations. At one extreme are the high spectral- and spatial-resolution spectrographs that are not well suited for total solar irradiance; at the other are the broad-band photometers which, even if accurately calibrated, cannot provide details of the solar spectrum. Rocket flights typically provide only a four-minute snapshot, and thus cannot address temporal variations.
Great progress in determining the EUV contribution to total solar irradiance has been made by the [*Thermosphere, Ionosphere, Mesosphere, Energetics, and Dynamics*]{} (TIMED), and [*Solar Radiation and Climate Experiment*]{} (SORCE) missions [@Woods08]. The daily data provided by these instruments are well calibrated, broad-band (width 70 to 100 Å) photodiode measurements, which are used to scale reference CHIANTI [@Landi06; @Dere97] model spectra (quiet Sun, active region, coronal hole, and flare) that depend upon assumed Differential Emission Measures (DEMs) and out-of-band proxies. We refer to the SORCE XPS Level 4 Version 10 model (outlined in detail in Woods [[*et al.*]{}]{}, 2008) as XPSL4 hereafter.
Here we present full-disk solar spectra from the [*Cosmic Hot Interstellar Plasma Spectrometer*]{} (CHIPS) covering a period of two years. The data from the SORCE, TIMED/SEE, and SOHO/EIT missions complement those of CHIPS. The first two provide the total irradiance, SOHO/EIT shows where the EUV emission originates, and CHIPS provides spectra from which temperatures and densities may be determined as well as provide the spectral shape between 140 and 270 Å required to improve the calibration of the contemporaneous SORCE/XPS and TIMED/SEE photodiode measurements, and the SOHO/EIT broadband full-disk images. A recent flight of the rocket prototype [*Solar Dynamics Observatory*]{} (SDO)/[*Extreme Ultraviolet Variability Experiment*]{} (EVE) [@Woods09] provided an EUV spectrum with similar passband and resolution as CHIPS. This spectrum is utilized to cross-check the quality of the solar observations and the calibration of the CHIPS throughput. An uncalibrated, mean CHIPS spectrum was first presented by , and compared to laboratory plasmas by .
In this work we explore to what extent the solar corona can be described by a set of simple plasmas, and how these may be used in modeling the spectral shape of the solar irradiance. We present both raw and flux calibrated spectra, quantify the spectrometer’s response to off-axis solar illumination, perform CHIANTI plasma modeling, and outline the contents of a publicly available archive.
CHIPS Instrument {#S-Instrument}
================
The CHIPS satellite is a NASA University Explorer mission devoted to diffuse background spectroscopy of the interstellar medium at moderate resolution ($\lambda/ (\Delta \lambda$FWHM)$\approx 120$) in the EUV passband 90[[–]{}]{}260 Å. NASA launched CHIPS from Vandenberg AFB on 12 January 2003 as a secondary payload on a Delta II rocket into a circular polar orbit with an inclination of 94[$^{\circ}$]{} and an altitude of 590 km.
The three-axis stabilized satellite consists of a 35-kg spacecraft built bySpaceDev, Inc., and a 25-kg science instrument provided by the University of California, Berkeley. The spectrograph consists of a micro-channel plate detector placed at the focus of six varied-line-spacing cylindrically curved diffraction gratings. Mounted 8.8 mm in front of the detector face is a filter frame which holds the vacuum deposited Al, Zr, and polyamide-boron thin film ($\approx$ 1000 Å thick) filters, which suppress higher diffracted orders as well as longer-wavelength scattered and stray light. Unlike traditional imaging spectrographs, the CHIPS design does not employ a telescope to gather light. Regardless of a source’s position within the field of view, light of a particular wavelength is directed at a particular dispersion angle to the micro-channel plate detector. A single spectrum of the entire 4[$^{\circ}$]{} $\times$ 26[$^{\circ}$]{} field of view may thus be obtained simultaneously. The opto-mechanical design of the instrument is outlined in , the detector characteristics in , the calibration and in-flight performance in , and the primary mission science results in .
Each diffraction grating is equipped with a pair of adjacent, fixed, entrance slits, 0.25 and 1.0 mm wide (Figure \[F-grating\]). In front of each slit pair is a rotary mechanism that allows diffuse light to illuminate the grating through one slit or the other (or can be set to block both). Between the slits and three of the gratings are flat grazing-incidence mirrors that serve to co-align the field of view of those spectral channels with their non-mirrored counterparts (see Figure \[F-slit\_wheel\], right). During normal science observations, the instrument boresight is constrained to be far from the Sun, so that the rotary mechanism and entrance slits are shaded from direct sunlight. When several comets presented themselves as targets of opportunity [@Sasseen06], we relaxed this constraint and allowed the instrument boresight to encroach within tens of degrees of the Sun. The resulting spectra showed features that seemed not to originate from the comets, were far too bright to arise in the interstellar medium, and did not (without shifting) match the reflected solar spectrum that had been recorded during observations of the full Moon.
We ultimately discovered that in certain geometries, scattered sunlight could pass through an entrance slit, avoid the flat mirror, and strike a diffraction grating at an angle different from the nominal 14[$^{\circ}$]{} incidence. Ray-trace analysis by us and confirmed that light scattered from the back side of a baffle could follow this path to the detector, producing a spectrum shifted by $\approx$ 11 Å from the nominal wavelength solution (see Figure \[F-slit\_wheel\], left). After much experimentation with the Sun angle and slit orientation (angle of slit axis with respect to the ecliptic plane), a configuration was found that nominally showed a single channel solar spectrum. We refer to this path as the “light-leak.” The light-leak has no adverse effect on ordinary science observations. Its throughput is six orders of magnitude lower than for emission within the direct field of view, and diffuse emission comprising the main science target is so faint as to be barely detectable. The light-leak in this case provides a serendipitous way of measuring the EUV spectrum of light integrated from the solar disk.
Observations {#S-Observations}
============
The CHIPS satellite performed 1450 observations of the Sun from 03 April 2006 to 05 April 2008, a period of generally declining activity following the solar maximum of 2001. The satellite was decommissioned on 14 April 2008. Typical observations ranged from 1 to 15 minutes which yielded spectra of roughly $10^5$ total counts each. Occasionally, up to 15 such observations per day were performed. However for a variety of causes, including passages through the South Atlantic Magnetic Anomaly, passages through the electron belts at high magnetic latitudes, targets of opportunity, precession of the orbit into full sunlight, and satellite shutdowns, the cadence of observations is far from uniform.
We present raw spectra integrated over the two-year period for the upper Zr/Al filters, and the lower large Al filter in Figure \[F-rawspec\]. The brightest features are from Fe [ix]{} through Fe [xv]{}. Also evident is a continuum of in-band scattered light from the diffraction gratings superimposed with second (and possibly third) channel features (discussed in Section \[S-multiple-channels\]), which occur more frequently, and to a greater degree, in the upper Zr/Al filter. Particle background (primarily high energy electrons) is negligible as evinced by the near zero count rates in the shadows of the filter frames.
The general appearance of the daily spectra remained relatively stable. The emission lines labeled in Figures \[F-alspec\] and \[F-zrspec\] did not appear and disappear, but were visible at all times, even during the quiet Sun period of April 2008. During 2006, CHIPS observed an M-class flare (27 April 2006 15:22 UT M7.9), and just missed a second one by 24 hours (06 July 2006 08:13 UT M2.5). Spectra extracted in the hours just prior to and during the flares showed no discernible differences in line ratios, indicating that moderate intensity flares do not necessarily produce large changes in the shape of the EUV spectrum. We present a more detailed analysis in Section \[S-flare\].
Calibrations {#S-Calibrations}
============
Calibration Strategy {#S-calibration-strategy}
--------------------
Each observation is processed by our standard science pipeline and spectra extracted as described by . Because the solar data are obtained via the fortuitous light-leak, the ground-based calibrations for wavelength scale and throughput are not directly applicable. To analyze the spectra quantitatively, the instrument required an-[*ad hoc*]{} recalibration using the solar data itself. The strategy is one of iteration: as various instrumental effects become known, the entire data set is reprocessed to produce the highest-quality spectra.
Reference Spectra {#S-reference-spectra}
-----------------
The first step is to create a provisional reference spectrum for each filter set by simply adding up all the observations. Next, each individual spectrum is shifted in the dispersion direction until it best matches (by cross correlation) the provisional reference spectrum, thus placing all data onto a common dispersion grid. The shifts proved to be minor, typically 0 to 0.2 Å, demonstrating that the spectrograph is essentially free from flexure problems. Finally, all of the shifted spectra are co-added a second time to create improved reference spectra for both the Al filter and the Zr/Al filter panels.
Multiple Channel Contamination {#S-multiple-channels}
------------------------------
In spite of our efforts to best configure the satellite orientation to produce a single-channel spectrum, multiple channel contamination is still evident in the majority of pointings and is quite variable in intensity from one observation to the next. The contamination is not the result of second-order grating features, but rather, is caused by scattered light from a second (or third) slit and grating pair striking the detector (see Figure \[F-grating\]). The contamination is manifest as additional features shortward of [[Fe [ix]{} 171.073 Å]{}]{} in Figure \[F-rawspec\] in both the Zr and Al filters. The likely cause for the observed variations is that small errors in the pointing knowledge of the satellite of $\approx 1.0$[$^{\circ}$]{} lead to large changes in the illumination of the slits, with each channel responding differently with regard to wavelength scale and throughput. Efforts to decouple the channels proved futile. We found, for example, that a multi-channel spectrum is not simply the linear combination of scaled and shifted reference spectra, even if the reference spectrum is constructed from contamination-free spectra.
Fortunately, the degree of second channel contamination is in general small and is easily quantified by comparing count rates on either side of the absorption edge of the Al filter at 170 Å. The effective area of the Al filter determined by ground calibrations [@Sirk03] show that the sensitivity just shortward of 170 Å is about 13% of the 170[[–]{}]{}180 Å region. Since second-channel contamination is shifted bluewards relative to the “good” channel, any measured flux shortward of 170 Å that is greater than expected is a direct indication of contamination. We calculate a contamination fraction for each observation as the ratio of the average counts per Å between 155 Å and 167 Å divided by the average counts per Å of the four brightest iron lines [[Fe [ix]{} 171.073 Å]{}]{}, [[Fe [x]{} 174.531 Å]{}]{}, [[Fe [xi]{} 180.408 Å]{}]{}, and [[Fe [xi]{} 188.232 Å]{}]{} for the Al filter. The contamination ranges from 0 to 41% with a median value of 6%. By selecting appropriate contamination thresholds, the unwanted light may be greatly reduced, or even eliminated, albeit at the expense of rejecting a portion of the observations. For the lightcurve, line ratio, and line-intensity correlation analysis described in Section \[S-lightcurves\], we choose a threshold of 12%, which results in a rejection of about 40% of the observations. This is a compromise between accepting a small amount of unwanted light and rejecting large quantities of the data. For the CHIANTI plasma modeling (Section \[S-modeling\]), we apply a more stringent threshold of 8% when fitting the Zr filter data since they show a greater degree of contamination, and because there are Ni emission features of critical interest shortward of 170 Å.
Background Subtraction {#S-background}
----------------------
The CHIPS solar-spectra background is dominated by in-band scattering from the diffraction gratings. High-energy particles, second-order features, and out-of-band scattering from geocoronal emission lines are all negligible. Initial mean backgrounds are determined from the reference spectra by simply drawing by hand. Provisional CHIANTI model fits (specifically outlined in Section \[S-modeling\]) are performed and the backgrounds adjusted until reasonable fits are found. In Figures \[F-alspec\] and \[F-zrspec\] we present the time-averaged background-subtracted spectra with negligible second-channel contamination for the Al and Zr filters, respectively. These filtered spectra along with their respective backgrounds are saved as new reference spectra, and the steps outlined in Sections \[S-reference-spectra\] and \[S-multiple-channels\] repeated once more to create a database with all spectra on a common grid. For analysis, spectral subsets may be easily created by selections based on date, count rate, contamination fraction, line ratios, exposure time, [*etc.*]{}
Wavelength Scale {#S-wavelength-scale}
----------------
The dispersion of the CHIPS spectrograph is known to be nearly linear [@Sirk03] with an average value of 0.11 Å per pixel. Ray-trace analysis also shows that small translations of the slit position cause nearly constant shifts in wavelength of spectral features. Because the solar spectra light-leak is along a unforeseen path, the observed spectral shifts are effectively caused by a slit translation. Thus, the solar wavelength dispersion is also nearly linear, and is shifted about 11 Å bluewards relative to the ground-based wavelength solution. Twenty lines are identified in the Al and Zr/Al cleaned reference spectra using the CHIANTI V5.2.1 database [@Landi06; @Dere97] and a quadratic fit performed to determine wavelength as a function of detector [*x*]{} position. Each fit shows an RMS residual of 0.08 Å. The resolution of the reference spectral features is about 1.2 Å FWHM, or $R=140$ at 170 Å.
Analysis {#S-Analysis}
========
The CHIPS EUV spectra are for the full solar disk, and thus a melange of emission from active regions, quiet Sun, small flares, and coronal holes. Both temperature and density in these plasmas range over two orders of magnitude. In Section \[S-lightcurves\], we investigate the observed temporal variations from one observation to the next and compare the CHIPS data to the sunspot number, and the SORCE XPS L4 V10 spectral model flux (described in Section \[S-Introduction\]). In Section \[S-modeling\] we determine values for temperature and density that best describe the full-disk for time-averaged spectra.
Lightcurves, Line Ratios, and Correlations {#S-lightcurves}
------------------------------------------
The integrated Al-filter count rate is determined over the passband 170[[–]{}]{}273 Å and is plotted for the two year period as Figure \[F-lightcurves\] (upper panel). The median count rate is 447 ct s$^{-1}, +155, -135$. The large scatter from one observation to the next is greater than the variations seen in the integrated XPSL4 lightcurve (Figure \[F-lightcurves\], red line, middle panel) and is probably caused by the satellite pointing error of $\pm 1$[$^{\circ}$]{}which affects the throughput of the light-leak. Thus, any given CHIPS solar observation is only good to within a factor of $\approx$ two in absolute flux. Line ratios and correlations, however, are much more reliable since systematic uncertainties and instrumental effects cancel out.
The variations in count rate observed in Figure \[F-lightcurves\] (upper panel) may be caused by a scale factor that is constant over the passband, or may also have a wavelength dependence. If the scatter is caused by just a scale factor, then it will cancel out when taking line ratios. If, however, there is also a time-varying wavelength dependence in sensitivity, then the degree of correlation in line intensities should decrease as the separation in wavelength between a pair of lines increases.
To determine line intensities, the scale factor needed to match the spectrum in question to the reference spectrum is first determined. This scale factor is then applied to the mean background reference spectrum and each spectrum is then background subtracted. Count rates are next determined in 2 Åwide bins centered on the wavelengths of the desired features. For the correlation analysis, the individual line intensities for each observation are presented as a percentage of total count rate of the entire passband for that observation. Correlation plots are made that compare lines of both the same and different ionization states, and lines both close and far apart in wavelength. Some of the more instructive cases are presented as Figure \[F-correlations\], and fall into three distinct categories. The first group are the strongly correlated lines of Fe [ix]{} through Fe [xi]{} (dominated by [[Fe [ix]{} 171.073 Å]{}]{}) which peak in brightness at a temperature around 1 MK (Figure \[F-correlations\](a[[–]{}]{}c)). The second group are the 2 MK ionization states of Fe [xiii]{} and [xiv]{} (dominated by [[Fe [xiv]{} 211.318 Å]{}]{}) which also show strong correlation in Figure \[F-correlations\](f[[–]{}]{}h, j). Note that the first two groups are distinctly anti-correlated (Figure \[F-correlations\](e)). When the 1 MK lines are bright, the 2 MK lines are faint, and [*vice versa.*]{} The third set involves [[Fe [xii]{} 193.509 Å]{}]{} and [[Fe [xii]{} 195.119 Å]{}]{}, which correlate poorly with either of the first two groups (Figure \[F-correlations\](d,i)). Lastly, there is a set of lines (not shown in Figure \[F-correlations\]) that show bimodal distributions ([*i.e.*]{}, both positive and negative correlations simultaneously). We interpret these cases as involving blends of both high and low temperature lines ([*e.g.*]{}, Fe [xiv]{} and Si [ix]{} at 227 Å, Figure \[F-alspec\]).
If the variations in intensity between observations are caused by a change in the wavelength dependence of the throughput, then [*all*]{} lines in a given wavelength region would vary in unison regardless of ionization state. This is clearly not the case for these data. The observed variations in count rate of the CHIPS spectra when features from one ionization state are compared to another are caused by real temporal variations in plasma temperature. This effect is best illustrated by combining the count rates of a few of the strongest lines into two temperature groups: The 1 MK “cool” group consists of [[Fe [ix]{} 171.073 Å]{}]{}, [[Fe [x]{} 174.531 Å]{}]{}, [[Fe [xi]{} 180.408 Å]{}]{}, and [[Fe [xi]{} 188.232 Å]{}]{}, while the 2 MK “hot” group of [[Fe [xiii]{} 202.044 Å]{}]{}, [[Fe [xiv]{} 211.318 Å]{}]{}, [[Fe [xiv]{} 251.956 Å]{}]{}, and [[Fe [xiv]{} 264.790 Å]{}]{}. The ratio of the hot-to-cool group count rates is plotted as Figure \[F-lightcurves\] (dots, middle panel). Any uncertainty in the exposure time or throughput cancels out when taking this ratio. For comparison, the XPSL4 flux is also plotted (red line, middle panel). Evident in the hot-to-cool ratio is the 27-day mean rotation period of the Sun (manifested as 27.2-day power in a Fourier spectrum of the hot-to-cool ratio lightcurve), the general decrease in high-temperature activity, and the correlation in slope of the two satellite lightcurves ([*i.e.*]{}, when CHIPS sees an increase in temperature, the XPSL4 spectral model shows an increase in flux). However, when the same line ratios are extracted from the XPSL4 spectral model and compared to the CHIPS ratios, the correlation vanishes. This fact indicates that CHIPS is seeing a real change in plasma temperature caused by solar rotation, while the XPSL4 model is just reflecting a change in overall intensity. In the bottom panel of Figure \[F-lightcurves\] we plot the density-sensitive line ratio [[Fe [xiii]{} 203.828 Å]{}]{}/ [[Fe [xiii]{} 202.044 Å]{}]{} (which is discussed in Section \[S-modeling\]), and the International Sunspot Number (red line). Note that the slopes of the CHIPS hot-to-cool ratio, XPSL4 model, sunspot number, and Fe 203/202 ratio are all positively correlated.
To further show that the temporal variations in temperature are real, we construct two spectra filtered on times when the hot-to-cool ratio is at maximum or minimum during 2006 (a period when the XPSL4 lightcurve is relatively flat, Figure \[F-lightcurves\] (middle)). The “hot” spectrum minus the “cool” spectrum is presented as Figure \[F-hotminuscool\] and shows that only lines with a high ionization state (Fe [xii]{} to Fe [xv]{}) show a positive residual. Thus, we conclude that the observed hot-to-cool ratio temporal variations are dominated by differences in plasma temperature and not some instrumental effect, and that, because of the observed 27.2-day period, are caused by variations in emission from hotter and cooler regions rotating into, and out of, view. This behavior has also been seen by the TIMED/SEE instrument [@Woods05].
We also chose two times about 13 days apart ($\approx 1/2$ the solar rotation period) where the hot-to-cool ratio was at minimum or maximum in Figure \[F-lightcurves\] (diamonds, middle panel), retrieved the corresponding SOHO/EIT full-disk images for the 171 Å and 284 Å channels, and present them as Figure \[F-soho\]. Visual inspection of the images shows no obvious changes in the number and intensity of active regions. However, several independent data sources show an increase in solar activity during this period. The ratio of the SOHO images (284 Å / 171 Å) increased by 25%, the hot-to-cool ratio of CHIPS increased by 50%, and the CHIPS Fe [xiii]{} 203/202 ratio by 39%. The sunspot number increased from zero to around 20 during this 13-day period, and the ratio of [*Michaelson Doppler Interferometrer*]{} [@Scherrer95] images extracted for these two timesshows an increase in magnetic flux by a factor of 1.65 which implies an increase of EUV and X-ray flux [@Pevtsov03; @Fludra08]. The XPSL4 model shows an increase in total flux of 8% in the CHIPS passband. These changes are all in the same direction and indicate an increase in solar activity over the 13-day period. The positive correlation between the CHIPS and SOHO/EIT line ratios suggest the possibility of using the CHIPS spectra to help calibrate the SOHO/EIT 171 Å and 195 Å broadband images, which in turn could be used as proxies to better calibrate other instruments for time periods not covered by CHIPS. We explore these ideas further in Section \[S-Conclusions\].
Daily Averaged Spectra {#S-dayspec}
----------------------
The significant variations seen in Figure \[F-lightcurves\] are the short-period ($\approx$ 14 days) changes in the hot-to-cool line group ratio and the general decrease in intensity of the 2 MK lines over the two year span of the observations. For reference, we present an atlas of extreme-case hot and cool spectra in Figure \[F-specatlas\] that illustrates the transition from moderately active Sun to quiet Sun (compare day 103 to day 825). The hot and cool spectra were extracted during times corresponding to local maxima and minima of the hot-to-cool ratio indicated with + symbols in Figure \[F-lightcurves\] (middle panel). These spectra are not background-subtracted and so illustrate the typical quality of the CHIPS multi-channel contamination-free daily averaged spectra.
CHIANTI Modeling {#S-modeling}
----------------
The CHIPS EUV spectra are from the full solar disk, and as such are the combination of plasma features from the corona and transition region which exhibit large ranges of temperature and density. The dominance of the Fe emission lines, however, allows us to decompose the spectra into several principal components. The overriding strategy is to adequately fit the spectra with the fewest number of atomic elements, temperatures, and densities, and proceeded in this order: identify lines, the peak temperatures associated with the observed lines, and the densities dictated by specific line ratios.
Starting with just Fe, the bulk of the EUV emission is accounted for by a 1 to 2 MK plasma. In this temperature range, a few more features are accounted for by Si and Ni. The residual flux indicates the presence of O and He, but at lower temperatures of [log$_{10} T$]{}= 5.0 to 5.5. The best density diagnostic available at the CHIPS spectral resolution is the line ratio [[Fe [xiii]{} 203.828 Å]{}]{}/[[Fe [xiii]{} 202.044 Å]{}]{} whose average value is $0.34 \pm 0.05$ for the entire mission and varied in step with the hot-to-cool ratio ([*i.e.*]{}, higher densities are observed when the spectra are hotter, dots on Figure \[F-lightcurves\], middle and lower panels). This ratio gives an electron density range of [log$_{10}$]{}$N_e$ (cm$^{-3}$) = 8.5 to 8.7 which is at the low end of the typical densities determined in and around active regions by the [*Hinode*]{}/EIS [@Young09; @Warren08]. Fainter lines from other elements (such as Ne, Mg, S, Ar, and Ca seen by the [*Solar EUV Rocket Telescope and Spectrograph*]{} (SERTS): ) were searched for, but could not be identified with any certainty. The spectral resolution of CHIPS is the limiting factor, not the instrument sensitivity.
The standard Differential Emission Measure (DEM) models (quiet sun, active region, coronal hole) that come with the CHIANTI plasma package version 5.2.1 [@Landi06; @Dere97] did not fit the CHIPS spectra very well, either singly, or in combination. We instead built a set of isothermal plasmas combining the elements Fe, Ni, and Si (using the solar coronal abundances of ) at temperatures ranging from [log$_{10}$]{}$T$ = 5.8 to 6.4 at 0.1 intervals. We chose an average density of [log$_{10}$]{}$N_e$ (cm$^{-3}$) = 8.6 since the values determined from observations that ranged from 8.5 to 8.7 made insignificant differences in the model spectra. An isothermal plasma of He at [log$_{10}$]{}$T$ = 5.0, and three of O at [log$_{10}$]{}$T$ = 5.3, 5.4, and 5.5 were also constructed, all at a density [log$_{10}$]{}$N_e$ (cm$^{-3}$) = 10.0 typical of the transition region [@Doschek91]. Our model is simply the linear combination of these individual plasmas with the only free parameters being the emission measures.
Because the changes in the CHIPS spectra are large over the short time period of one-half solar rotation, but minor over longer periods, we built two time-averaged spectra corresponding to the minimum and maximum time intervals of the hot-to-cool ratio of Figure \[F-lightcurves\] (middle panel) during 2006. We fit these hot and cool spectra separately. Good fits were found requiring plasmas of [log$_{10}$]{}$T$ = 6.0, 6.1, and 6.3. Inclusion of the He plasma decreased $\chi^2$ significantly, while that of O decreased $\chi^2$ by 5 to 8%. The inclusion of the [log$_{10}$]{}$T$ = 5.8, 5.9, 6.2, and 6.4 temperature components of the Fe-Si-Ni plasma did not reduce $\chi^{2}$ significantly, and thus are not included in the final CHIANTI model fits.
Light-Leak Efficiency {#S-lightleak}
---------------------
The modeling outlined in Section \[S-modeling\] successfully reproduced the majority of observed features in the CHIPS spectra, but showed a systematic error in amplitude from one end of the spectrum to the other. This residual is an expected consequence of the wavelength dependence of the light-leak. The exact nature of the light-leak cannot be modeled directly since the angle of incidence and surface roughness of the Au and Cu/Be slit-shutter mechanisms are poorly known. However, the functional form is estimated using the Lawrence Berkeley National Laboratory X-ray Properties of Matter web utility [@Henke93]. We find a smoothly varying function that increases in reflective efficiency with wavelength, and which is well approximated by a simple exponential over the wavelength range of interest $$\label{Eq-lightleak_eff}
{\rm efficiency}(\lambda) = S \exp{(\alpha\lambda)}.$$ where $\alpha$ governs the wavelength dependence of the light-leak, and $S$ is a scale factor, which, when combined with the CHIPS nominal effective-area curves, converts count rate to physical units (determined in Section \[S-chips\_vs\_xps\]).
Our final model is thus the sum of the individual plasmas multiplied by an exponential. The model fits to the cool and hot spectra are shown as Figure \[F-hotcool\], and the corresponding emission measures presented in Table \[T-table1\]. The random errors of the emission-measure determinations are $\approx 8\%$. The total uncertainty is discussed in Section \[S-Conclusions\]. Since the He and O emissions are from the transition region [@Reeves77] and are optically thick, the CHIANTI-derived values of emission measure may match the data, but are probably only lower limits.
The CHIANTI model fits to the large Al and Zr filters produced essentially identical values for temperature, density, and light-leak efficiency. On average, the Zr data are of inferior quality suffering from multiple-channel contamination and non-uniform slit illumination. Furthermore, because the Zr passband (140[[–]{}]{}202 Å) is smaller than the Al filter, fewer lines are available to constrain the model parameters. However, the presence of Ni lines in the Zr filter spectra clearly indicate a plasma component of [log$_{10} T$]{}= 6.3, and the relative weakness of the Fe [viii]{} features at 168 Å indicate the absence of significant plasma components of [log$_{10} T$]{}= 5.8[[–]{}]{}5.9. We present only the Al-filter results in Table \[T-table1\] as they are the most reliable. A new version of CHIANTI (6.0.1) was released in October 2009. The changes, however, do not affect the main conclusions of this work.
---------- ------- ------- ------- ------- ------- ------------
He O
Spectrum 6.0 6.1 6.3 5.0 5.5 $\chi^{2}$
Cool 20.53 20.53 20.40 20.95 20.82 17.3
Hot 20.36 20.45 20.54 21.05 20.63 10.7
---------- ------- ------- ------- ------- ------- ------------
: Emission Measures$^{a}$ (cm$^{-5}$) determined from modeling of the Al-filter spectra for the different element and temperature plasma components.[]{data-label="T-table1"}
$^{a}$[log$_{10}$]{} of the emission measure and temperature are given.\
Relative uncertainties in the emission measures are typically 8%.\
Absolute uncertainties are discussed in Section \[S-Conclusions\].\
Ionization fractions used are from and\
solar corona abundances from .\
Comparison to SORCE/XPS and SDO/EVE {#S-chips_vs_xps}
-----------------------------------
To answer specific questions regarding solar irradiance, solar variability,differential-emission-measure maps, and other solar-physics problems, theCHIPS spectra must be converted to physical units. We accomplish this by determining the average scale factor $[S]$ in Equation (\[Eq-lightleak\_eff\]) by comparing the CHIPS spectra to contemporaneous SORCE/XPS Level 4 model fluxes [@Woods08etal] integrated over the passband 145-273 Å. We chose the first 370 days since 2006 because during this period the XPSL4 flux remained fairly flat (see Figure \[F-lightcurves\]). The CHIPS ground-based effective-area curves are multiplied by Equation (\[Eq-lightleak\_eff\]) ($eff(145)= 6.04 \times 10^{-7},\
eff(275) = 3.75 \times 10^{-6},\ eff_{{\rm mean}} = 1.73 \times 10^{-6}$). We show the mean-flux calibrated CHIPS spectrum as Figure \[F-chips-xps\] (top) and compare it directly to the XPSL4 spectrum extracted for the same time period. The spectral features are in general agreement, but there are several significant differences which we discuss in Section \[S-Conclusions\].
In Figure \[F-chips-xps\] (bottom) we show the EVE rocket spectrum of 14 April 2008 [@Chamberlin09; @Woods09], and compare it directly to the scaled CHIPS spectra averaged over 2[[–]{}]{}5 April 2008. The excellent agreement between the two spectra shows that both instruments are free from spurious features such as grating ghosts and second-order effects, and that the CHIPS background subtraction and light-leak efficiency determination are reasonable. The only significant difference is the intensity of the [[Fe [ix]{} 171.073 Å]{}]{} line, which we address in Section \[S-Conclusions\].
Solar Flares {#S-flare}
------------
During 2006, CHIPS observed an M-class flare, and narrowly missed a second flare by 24 hours (GOES X-ray peak times 27 April 2006 15:22UT M 7.9, 06 July 2006 08:13UT M 2.5, respectively). A baseline spectrum was extracted over a five-day period just prior to the first flare and compared to the three CHIPS observations obtained during flare. We present these spectra in Figure \[F-flare\] in their raw state which is free from any possible systematic error introduced by background subtraction. There is no evidence of a change in plasma temperature. The CHIPS observations of the second flare which were obtained 24 hours after the GOES X-ray peak also showed no differences between the pre- and post-flare spectra.
Discussion and Conclusions {#S-Conclusions}
==========================
CHIPS recorded spectra of the solar disk between 140 Å and 273 Åover a two-year period. The general appearance of the spectrum remained relatively stable. The emission lines labeled in Figures \[F-alspec\] and \[F-zrspec\] did not appear and disappear, but were visible at all times, even during moderate (M-class) flares and the quiet-Sun period of April 2008. There was, however, variation in the ratio of the hot and cool emission line groups of about $\pm$ 25%, and a variation in the 1 to 2 MK plasma density of $\pm$ 15%.
The CHIPS spectra indicate a 1 to 2 MK plasma dominated by Fe emission. Iron is also the most significant element for higher temperature spectra of active regions and flares, and for wavelengths both shorter and longer than the CHIPS passband (see the rocket-flight spectra of ; ; ; and the [*SkyLab*]{} data of ). In Table 2 we present the fraction of total intensity for each element used in the CHIANTI fits, and the temperature of peak emission. Recent theoretical work by using the Los Alamos ATOMIC plasma code concludes that Fe plays the dominant role in total radiative loss of the corona, even when it is depleted relative to the other elements.
Element Fraction (%) [log$_{10}$]{} T$_{{\rm peak}}$
--------- -------------- ---------------------------------
Fe 85.3 6.0[[–]{}]{}6.3
Si 5.7 6.1
Ni 1.8 6.3
He 4.3 5.0
O 3.0 5.5
: Fraction of total intensity by species, and temperature of peak emission between 140 Å and 273 Å.[]{data-label="T-table2"}
The CHIPS hot and cool spectra are reasonably well modeled with a small set of isothermal plasmas of just five elements, five temperatures, and two densities. The standard DEMs supplied with the CHIANTI package do not model the CHIPS spectra nearly as well. These results are in agreement with those of who, in a study of the unresolved EUV corona, conclude that above non-flaring active regions the corona is isothermal (typically between 1 and 2 MK), has an average electron density of [log$_{10}$]{}$N_e$ (cm$^{-3}$) = 8.0, and varies little in time. In a subsequent survey of 20 active regions, J. Cirtain (2010, private communication) also finds that the unresolved EUV corona component comprises 55% of the total EUV emission. We propose that the variation seen by CHIPS in plasma temperature over the solar rotation period is caused by hotter unresolved plasma associated with active regions, and cooler plasma rotating into and out of view.
The spectral shape, line ratios, temperatures, densities, and light-leak efficiency presented here are determined directly from the CHIPS data. The integrated XPSL4 flux is used only to adjust the CHIPS spectra by a single scale factor (one per observation) when converting count rate to physical flux units. Any CHIPS solar spectrum, when divided by the exposure time and the modified effective-area curves, now yield a calibrated flux (erg s$^{-1}$ cm$^{-2}$ Å$^{-1}$) to within a factor of two if raw, or to within 50% if scaled to XPSL4 [@Woods08etal; @Woods10]. However, since the XPS CHIANTI model spectrum shows several significant differences when compared to the empirical CHIPS mean solar spectrum, there are probably additional systematic uncertainties in the integrated XPSL4 flux. In particular, the XPSL4 model under-predicts He [ii]{}, and over-predicts Ni [xii]{} and Ni [xiii]{} around 160 Å. Several other spectral features in Figure \[F-chips-xps\] differ by factors of $\approx$ two. The Ni [x]{} and [xi]{} lines (seen by CHIPS shortward of 150 Å) are not yet in the CHIANTI (version 5.2.1) database. In the XPSL4 spectra, [[Fe [ix]{} 171.073 Å]{}]{} is typically twice as bright as its closest neighbors, whereas the CHIPS spectra consistently show it to be only $\approx$ 8% brighter. The XPSL4 line ratio [[Fe [xiii]{} 203.828 Å]{}]{}/[[Fe [xiii]{} 202.044 Å]{}]{} averaged 2.0 in 2006 which is about a factor of six greater than the value determined from the CHIANTI fits to CHIPS spectra for the same time period. This large discrepancy implies that the electron density used by the XPSL4 model is an order of magnitude larger than that required to model the CHIPS spectra (see Figure 20 in ). In general, features visible at temperatures $>$ [log$_{10} T$]{}= 6.3, and $<$ [log$_{10} T$]{}= 6.0, are over-estimated by the XPSL4 model. One important difference to bear in mind between the CHIPS empirical spectra and SORCE XPS Level 4 model is that XPSL4 is based on a single photodiode (Ti filter, 1[[–]{}]{}70 Å passband, ), which does not overlap the CHIPS spectral range. So, there is no surprise that there are some differences. These differences may result in systematic uncertainties in the XPSL4 model flux and can be eliminated if the SORCE XPS diode currents are used to scale a CHIANTI model that is based upon DEMs derived from empirical spectra rather than assumed DEMs and out-of-band proxies.
The agreement between CHIPS and EVE seen in Figure \[F-chips-xps\] (bottom) is excellent apart from the factor of two discrepancy at 171 Å. A likely cause for this difference may be that the resolution of the effective-area curve of one of the instruments does not match the actual instrument resolution in the region of the Al absorption edge around 169 Å. To test this idea we compare the [[Fe [ix]{} 171.073 Å]{}]{} line to the [[Fe [x]{} 174.531 Å]{}]{} line since the two lines are close together in wavelength and, hence, the difference in instrument throughput is small. Over the course of the two year CHIPS mission, the 171/174 Å line ratio averaged 1.09 $\pm$ 0.05, and 1.07 $\pm$ 0.09 for the Al and Zr filters, respectively. Because the Zr filter has no absorption edge near 171 Å(see Figure \[F-chips-xps\], inset), we believe the ratio of near-unity is quite robust. A quiet-Sun spectrum obtained in early 1997 from SOHO/GIS shows a 171/174 Å line ratio of 1.3 (see Figure 2 of ), and the full-disk rocket spectrum of 04 April 1969 (obtained during a period of some solar activity) shows a ratio 1.0 [@Malinovsky73].
The total uncertainty in absolute flux of the CHIPS spectra is the sum of the XPS absolute photometric calibration uncertainty (50%) and the systematic error introduced in scaling the CHIPS spectra to the XPSL4 model. By definition, this latter error is zero when averaged over the entire CHIPS band and over an entire solar-rotation period, but as can be seen in Figure \[F-chips-xps\] (top) is often a factor of $\approx$ two for individual features. Since the XPSL4 products are at 1.0 Å bin width, a quantitative line-by-line comparison between CHIPS and XPSL4 is currently not possible.
The CHIPS spectra can be used to help cross-calibrate TIMED/SEE,SORCE/XPS, and SOHO/EIT for periods of simultaneous observations. Then SOHO/EIT line ratios could be used to determine relative amounts of hot (2 MK) and cool (1 MK) plasma to make a more realistic solar spectrum for the SORCE/XPS CHIANTI models for periods where CHIPS data do not exist.
In summary, our primary findings are:
- Near simultaneous spectra from CHIPS and the SDO/EVE rocket show excellent agreement indicating that both instruments perform as designed, and that the CHIPS light-leak throughput is calibrated correctly.
- Full-solar disk EUV spectra obtained April 2006 through April 2008 show temporal variations of up to 25% with a period of 27.2 days. CHIANTI model plasma fits to the spectra show changes in the relative proportions of 1 MK and 2 MK plasmas, and changes in plasma density of [log$_{10}$]{}$N_e$ (cm$^{-3}$) = 8.5 to 8.7 over this period.
- The CHIPS spectra are well modeled with a simple set of five isothermal plasmas requiring five temperatures, five elements, and two densities. A 1 to 2 MK plasma of Fe accounts for 85% of the observed flux.
- The full-disk spectra are less well fit by standard CHIANTI DEMs which over-predict features above [log$_{10} T$]{}= 6.3 and below [log$_{10} T$]{}= 6.0. This is true for individual spectra as well as those averaged over many solar rotations and suggests that the true solar DEMs are not smoothly varying functions.
- The CHIPS spectra may be utilized to help cross-calibrate contemporaneous SOHO/EIT, TIMED/SEE, and SORCE/XPS broad-band data, which ultimately will improve our knowledge of the total solar irradiance.
All CHIPS solar observations will be made available through a public archive a year from the acceptance of this paper. Preliminary mean “hot” and “cool” state spectra may be found at <http://ssl.berkeley.edu/chips/archive.html>. Each observation is stored in standard FITS table format. The polyamide-B/Al and Zr/Al-filter spectra are presented on a common, uniform (0.1 Å) wavelength grid as both raw counts, and in flux calibrated units.
The FITS headers contain spacecraft parameters determined at the time of telemetry processing, plus subsequent parameters created during the standard science pipeline processing. Additional parameters unique to the solar observations are also included. Because the EUV emitting region of the Sun extends beyond the solar limb and is quite variable, the Sun is treated as a point source. Hence, flux units of (erg s$^{-1}$ cm$^{-2}$ Å$^{-1}$) are used, which are good to within a factor of two.
Finally, as a convenience to other researchers, the multiplicative conversion factor required to scale the CHIPS spectra (over the 140[[–]{}]{}273 Å passband) to the contemporaneous SORCE/XPS Level 4 data products is also included for each observation. However, if the XPSL4 products are recalibrated, then this scale factor will also change somewhat.
This work is supported by the Office of Space Sciences, National Aeronautics and Space Administration, under Grant No. NAG5-5219. We thank Brian Welsch for numerous discussions and comments on the manuscript, the anonymous referee for many helpful improvements, Michael Sholl for Figures 1 and 2, Tom Woods and the LASP Team for providing the high resolution SDO/EVE rocket spectrum, and John McDonald, Jeremy Thorsness, and Mark Lewis of the CHIPS Operations Team who kept the instrument flying for several years beyond its expected lifetime.
|
---
address:
- 'Laboratoire d’Annecy–Le–Vieux de Physique Theorique LAPP, Chemin de Bellevue, B.P. 110, F-74941, Annecy–Le–Vieux, France'
- 'Fakultät für Physik, Albert–Ludwigs–Universität Freiburg, Hermann–Herder–Strasse 3, 79104 Freiburg, Germany'
author:
- 'T. Binoth'
- 'J.J. van der Bij'
title: 'The stealthy Higgs model at future Linear Colliders [^1]'
---
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
Freiburg-THEP 99/18\
LAPTH 745-99\
August 1999\
Introduction
============
Understanding of the electroweak symmetry breaking mechanism is one of the main tasks in particle physics. The establishment of the structure of the Higgs sector would be a break-through in our knowledge about matter. So it is important to think about alternatives to the Standard Model Higgs sector especially if they lead to a dilution of the signal. The simplest possible extension is the addition of scalar fields which are singlets under the gauge group of the Standard Model. Radiative corrections to weak processes are not sensitive to the presence of singlets in the theory, because no Feynman graphs containing singlets appear at the one–loop level. Since effects at the two–loop level are below the experimental precision, the presence of a singlet sector is not ruled out by any of the LEP1 precision data. The only connection to such a hidden sector is a possible Higgs singlet coupling, leading to a nonstandard invisible Higgs decay. Whereas the invisible decay of the Higgs boson with a width comparable to the Standard Model leads to relatively sharp missing energy signals, e.g. well known from discussions on Majoron models [@valle], a strongly coupled hidden sector could lead to fast Higgs decay and thereby to wide resonances. This would disturb the signal to background ratio if necessary cuts are imposed.
To check the influence of a hidden sector we will study the coupling of a Higgs boson to an O(N) symmetric set of scalars, which is one of the simplest possibilities, introducing only a few extra parameters in the theory. The effect of the extra scalars is practically the presence of a possibly large invisible decay width of the Higgs particle. When the coupling is large enough the Higgs resonance can become wide even for a light Higgs boson. It was shown earlier that there is a range of parameters, where such a Higgs boson can be seen neither at LEP nor at the LHC [@vladimir; @lep2report; @DPRoy].
The model
=========
The scalar sector of the model consists of the usual Higgs sector coupled to a real N–component vector $\vec\varphi$ of scalar fields, denoted by Phions in the following. The lagrangian density is given by, \[definition\] [L]{} &=& - \_\^+ \^-(\^+- v\^2/2)\^2 - 1/2\_ \^ -1/2 m\^2 \^2\
&&- /(8N) (\^2 )\^2 -/(2) \^2 \^+where $\phi$ is the standard Higgs doublet. Couplings to fermions and vector bosons are the same as in the Standard Model. The ordinary Higgs field acquires the vacuum expectation value $v/\sqrt{2}$. For positive $\omega$ the $\vec\varphi$–field acquires no vacuum expectation value. After spontaneous symmetry breaking one is left with the ordinary Higgs boson, coupled to the Phions into which it decays. Also the Phions receive an induced mass from the spontaneous symmetry breaking which is suppressed by a factor $1/\sqrt{N}$. If the factor N is taken to be large, the model can be analysed with $1/N$–expansion techniques. By taking this limit the Phion mass is suppressed, whereas the decay width of the Higgs boson is not. Because the Higgs width is now depending on the Higgs Phion coupling its value is arbitrary. Therefore the main effect of the presence of the Phions is to give a possibly large invisible decay rate to the Higgs boson. The invisible decay width is given by \_H = = .The Higgs width is compared with the width in the Standard Model for various choices of the coupling $\omega$ in Fig. \[width\]. The model is different from Majoron models [@valle], since the width is not necessarily small. The model is similar to the technicolor–like model of Ref. [@chivukula].
Consistency of the model requires two conditions. One condition is the absence of a Landau pole below a certain scale $\Lambda$. The other follows from the stability of the vacuum up to a certain scale. An example of such limits is given in Fig. \[stability\], where $\kappa=0$ was taken at the scale $2m_Z$, which allows for the widest parameter range. The regions of validity up to a given scale $\Lambda$ is sandwiched between the upper–right and the lower–left contour lines in the figure. The first stem from the Landau pole, the second from instability of the vacuum at that scale.
To search for the Higgs boson there are basically two channels, one is the standard decay, which is reduced in branching ratio due to the decay into Phions. The other is the invisible decay, which rapidly becomes dominant, eventually making the Higgs resonance wide (see Fig. \[width\]). In order to give the bounds we neglect the coupling $\kappa$ as this is a small effect. We also neglect the Phion mass. (For other values of the Phion mass the bounds can be found by rescaling the decay widths with the appropriate phase space factor.)
LC bounds
=========
At a linear collider (LC) the upper limits on the couplings in the present model come essentially from the invisible decay, as the branching ratio into visible particles drops with increasing $\varphi$–Higgs coupling, whereas for the Higgs mass limits one has to consider visible decays, too. The $WW$–fusion process can not be used to look for invisible Higgs decay. One is therefore left with the Higgsstrahlung und $ZZ$–fusion reaction. For energies up to 500 GeV the Higgsstrahlungs cross section is dominant and still comparable if one multiplies with the branching ratio $B(Z\rightarrow e^+e^-,\mu^+\mu^-)$. The Higgsstrahlungs reaction is preferred, because one can tag the on-shell Z boson. Thus we only have considered reactions containing an on shell Z boson with its decay into $e^+e^-$ or $\mu^+\mu^-$. The signal cross section is the well known Higgsstrahlungs cross section modified by the non standard Higgs width due to Phion decay. With the invariant mass of the invisible Phion system, $s_I$, it reads:
\_[(e\^+e\^-Z+E/)]{} = ds\_I \_[(e\^+e\^-ZH)]{}(s\_I)
To reduce the $Z \nu\nu$ background [@mele], we used the fact that the angular distribution of the Z–boson for the signal peaks for small values of $|\cos\theta_Z|$ in contrast to the background. Thus we imposed the cut $|\cos\theta_Z|<0.7$. Because we assume the reconstruction of the on-shell Z–boson we use the kinematical relation $E_Z=(\sqrt{s}-M_Z^2+s_I)/(2\sqrt{s})$ between the Z energy and the invariant mass of the invisible system to define a second cut. Because the differential cross section $d\sigma/ds_I$ peaks at $M_H^2$, we impose the following condition on the Z energy: <E\_Z< For the choice of $\Delta_H$ a comment is in order. As long as the Higgs width is small one is allowed to use small $\Delta_H$, which reduces the background considerably keeping most of the signal events. But in the case of large $\varphi$–Higgs coupling, $\omega$, one looses valuable events. To compromise between both effects we took $\Delta_H=30 (100)$ GeV for colliders with center of mass energy of $500 (1400)$ GeV, respectively.
For the exclusion limits we assumed an integrated luminosity of $500$ ($1000$) $fb^{-1}$ for the two center of mass energies. To define the $95 \%$ confidence level we used Poisson statistics as in Ref. [@lep2report]. The result is given in Fig. \[exclu1\]. We conclude from the above that a LC with the proposed high luminosities can essentially cover the parameter range up to the theoretically allowed limit with a completely clean signal, consisting of leptons plus missing energy. Such a LC appears to be the unique machine to be sensitive to this class of models.
References {#references .unnumbered}
==========
[99]{} T. Binoth, J. J. van der Bij, 75, 17 (1997) and references therein. J. Valle et al. LEP2 Higgs Report, CERN 96-01, 350 (1996). D. Choudhury, D. P. Roy, Phys.Lett. B322, 368 (1994). O. J. P. Eboli et al., DESY 93-123C, 55 (1993). R. S. Chivukula, M. Golden, Phys. Lett. B267, 233 (1991);\
J. D. Bjorken, Int. J. Mod. Phys. A7, 4221 (1992). LEP2 Higgs Report, CERN 96-01, 350 (1996). Ambrosanio S., Mele B., Nucl. Phys. B374, 3 (1992).
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the DFG-Forschergruppe Quantenfeldtheorie, Computeralgebra und Monte Carlo Simulation, the EU grant FMRX-CT98-0194(DG12-MIHT) and by the NATO-grant CRG 970113.
[^1]: Presented at the Worldwide Study on Physics and Experiments with Future Linear $e^+e^-$ Colliders, Sitges, Spain, april,may 1999.
|
---
abstract: 'We show that the rotating generalization of Hayward’s non-singular black hole previously studied in the literature is geodesically incomplete, and that its straightforward extension leads to a singular spacetime. We present another extension, which is devoid of any curvature singularity. The obtained metric depends on three parameters and, depending on their values, yields an event horizon or not. These two regimes, named respectively *regular rotating Hayward black hole* and *naked rotating wormhole*, are studied both numerically and analytically. In preparation for the upcoming results of the Event Horizon Telescope, the images of an accretion torus around Sgr A\*, the supermassive object at the center of the Galaxy, are computed. These images contain, even in the absence of a horizon, a central faint region which bears a resemblance to the shadow of Kerr black holes and emphasizes the difficulty of claiming the existence of an event horizon from the analysis of strong-field images. The frequencies of the co- and contra-rotating orbits at the innermost stable circular orbit (ISCO) in this geometry are also computed, in the hope that quasi-periodic oscillations may permit to compare this model with Kerr’s black hole on observational grounds.'
address:
- '$^1$ AstroParticule et Cosmologie, Université Paris Diderot, CNRS, CEA, Observatoire de Paris, Sorbonne Paris Cité. Bâtiment Condorcet, 10 rue Alice Domon et Léonie Duquet, F-75205 Paris Cedex 13, France'
- '$^2$ LUTH, Observatoire de Paris, Université PSL, CNRS, Université Paris Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92190 Meudon, France'
- '$^3$ LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Univ. Paris Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France'
author:
- 'F. Lamy$^1$, E. Gourgoulhon$^2$, T. Paumard$^{3}$, F. H. Vincent$^{3}$'
bibliography:
- 'biblio.bib'
title: 'Imaging a non-singular rotating black hole at the center of the Galaxy'
---
Introduction {#s:intro}
============
With the advent of the Event Horizon Telescope [@Doeleman:2017] and the VLTI/GRAVITY instrument [@Abuter_al:2017], black hole physics is entering a new era, where observational tests of the celebrated no-hair theorem of general relativity (see e.g. [@CardosoG:2016]) are becoming feasible. This theorem states that the unique solution for a steady isolated black hole in four-dimensional vacuum general relativity is the Kerr-Newman black hole, which depends on only three parameters: the mass $M$, the reduced angular momentum $a=J/M$ and the electric charge $Q$ (see [@IonescuK:2015] for a precise mathematical statement). In the (astrophysically relevant) electrically neutral case ($Q=0$), the solution reduces to the Kerr black hole. To prepare the observational tests, it is primordial to compute observables from theoretically plausible alternatives to the Kerr black hole (see [@Berti_al:2015] for a review).
Among the numerous alternatives, a large class is constituted by the so-called *non-singular black holes*, also named *regular black holes*, namely asymptotically flat spacetimes with a black hole region (and hence an event horizon) but without any curvature singularity. This class circumvents the no-hair theorem because the metrics are not vacuum solutions of Einstein equation. The first non-singular black hole spacetime has been proposed by Bardeen in 1968 [@Bardeen:1968]. It has been mentioned by Hawking and Ellis (p. 265 of Ref. [@HawkingE:1973]), while discussing the famous Penrose singularity theorem [@Penrose:1965], since it provides an instructive counter-example: all geodesics of Bardeen’s spacetime are regular, despite it contains trapped surfaces and obeys the weak energy condition. Actually, this spacetime violates the third hypothesis of Penrose theorem in its original version [@Penrose:1965]: the existence of a Cauchy surface. It was shown later by Ayón-Beato and García [@AyonG:2000] that the Bardeen metric is a solution of Einstein’s equations with the energy-momentum tensor arising from a magnetic monopole in some nonlinear electrodynamics theory, thereby giving some physical content to the model. Another famous regular black hole metric has been proposed by Hayward in 2006 [@Hayward:2006]; it fulfills the weak energy condition as well and was shown recently to be a solution of Einstein’s equations corresponding to a magnetic monopole, as the Bardeen black hole, but in another nonlinear electrodynamics theory [@Fan:2017]. Both Bardeen and Hayward black holes are actually part of a larger class of solutions of Einstein’s equations coupled to nonlinear electrodynamics found recently by Fan and Wang (see also [@Bronnikov:2017; @Bronnikov:2001]).
The Bardeen and Hayward black holes, and more generally all solutions of the Fan-Wang class, are spherically symmetric and static outside the event horizon. Now, on astrophysical grounds, it sounds more relevant to consider rotating black holes and even rapidly rotating ones (see e.g. the spin values in Tables I and II of Ref. [@Bambi:2017]). The metrics of Bardeen and Hayward have been generalized to rotating axisymmetric metrics by Bambi and Modesto via the Newman-Janis algorithm. More generally, all metrics of the Fan-Wang class have been recently extended to rotating ones by Toshmatov et al. [@Toshmatov_et_al:2017; @Toshmatov_et_al:2018]. However, contrary to the nonrotating ones, the rotating metrics are only approximate solutions describing a magnetic monopole in some nonlinear electrodynamics theory [@RodriguesJ:2017; @Toshmatov_et_al:2018]. Another shortcoming of these spacetimes is being geodesically incomplete. Moreover, as we show below, their straightforward extension leads to *singular* black holes, i.e. to spacetimes with a curvature singularity.
In this article, we apply a prescription devised by Torres [@Torres:2017] to obtain a spacetime extended to negative values of the “radial” coordinate $r$ and representing a rotating non-singular black hole that reduces to the Hayward solution in the nonrotating limit. Moreover, we consider values in the parameter space for which the solution is trully a black hole one, that we call *regular rotating Hayward black hole*, but also those for which the solution has no event horizon. In the Kerr case, this would correspond to $a > M$ and would yield a naked singularity. In our case, the regularity of spacetime is still preserved and we obtain instead a rotating traversable wormhole configuration, which we call a *naked rotating wormhole*, to distinguish it from other rotating wormholes introduced in the literature [@Teo:1998; @ChewKK:2016; @AbdujabbarovJAS:2016].
The paper is organized as follows. In Sec. \[section 2\], we start by giving a review of Hayward’s black hole and its generalization to nonzero rotation as introduced by Bambi and Modesto . We show that this model still possesses a curvature singularity and apply Torres’ prescription to obtain a regular rotating Hayward metric, whose features are investigated in detail. Sec. \[section 3\] is devoted to the numerical study of this model, with or without horizons, using the ray-tracing code [<span style="font-variant:small-caps;">Gyoto</span>]{} [@Vincent_et_al:2011]. Finally in Sec. \[section 4\] we investigate analytically the existence of circular orbits of massive particles as well as the propagation of photons in this geometry. We give the explicit expressions for the specific energy and angular momentum of a particle on a co- or contra-rotating orbit, which differ from the expressions of Ref. [@Toshmatov_et_al:2017], and compute the frequencies of such particles at the innermost stable circular orbit (ISCO).
Metric and motivations {#section 2}
======================
Spinning up Hayward’s black hole
--------------------------------
### Hayward’s original metric
A standard example of a non-singular static black hole of mass $m$ is provided by Hayward’s metric [@Hayward:2006]: $$\label{e:Hayward_metric}
{\mathrm{d}}s^2=- \left( 1-\frac{2M(r)}{r} \right) {\mathrm{d}}t^2
+ \left( 1-\frac{2M(r)}{r} \right) ^{-1} {\mathrm{d}}r^2
+r^2 \,{\mathrm{d}}\theta^2+r^2\sin^2\theta \, {\mathrm{d}}\phi^2,$$ with $$\label{e:def_F_M}
M(r):= m \frac{r^3}{r^3+2mb^2} ,$$ where $m$ and $b$ are two constants having the dimension of a length (or a mass in the geometrized units used here). The metric (\[e:Hayward\_metric\]) reduces to Schwarzschild’s metric of mass $m$ in the limit $r \rightarrow +\infty$, where the effective mass $M$ tends to $m$. Furthermore, if $b\not=0$, we have $2M(r)/r \sim r^2/b^2$ for $r\rightarrow 0$, so that Hayward’s metric behaves as a de Sitter metric with cosmological constant $\Lambda=3/b^2$ around $r=0$, thereby avoiding any singularity.
Hayward [@Hayward:2006] interpreted the parameter $b$ as a cut-off of the order of the Planck length, i.e. a length at which general relativity is no longer valid. An alternative interpretation of $b$, allowing for macroscopic values, has been provided by Fan and Wang (see also [@Bronnikov:2017]). These authors have shown that the metric (\[e:Hayward\_metric\])-(\[e:def\_F\_M\]) can be obtained as a solution of Einstein’s equations sourced by the energy-momentum tensor of a magnetic monopole within some nonlinear electrodynamics. The parameter $b$ is then related to the amplitude of the total magnetic charge $Q_{\rm mag}$ by $$Q_{\rm mag} = \left( \frac{b m^4}{2} \right) ^{1/3} ,$$ while the nonlinear electrodynamics theory is defined by the Lagrangian density $$L = L(\mathcal{F}) := \frac{6}{b^2} \frac{(2b^2 \mathcal{F})^{3/2}}{(1+ (2b^2 \mathcal{F})^{3/4})^2},$$ $\mathcal{F}$ being the invariant $\mathcal{F}:= F_{\mu\nu} F^{\mu\nu}$ of the electromagnetic field. Note that standard (Maxwell) electrodynamics corresponds to $L(\mathcal{F}) = \mathcal{F}$.
### A first attempt to generalize Hayward’s metric to nonzero rotation
By means of the Newman-Janis algorithm , Bambi and Modesto (see also [@TorresFayos:2017]) have obtained some rotating generalization of Hayward’s metric as $$\label{metricKerrmodified}
\begin{aligned}
{\mathrm{d}}s^2=&-\left(1-\frac{2rM(r)}{\Sigma} \right){\mathrm{d}}t^2-\frac{4arM(r)\sin^2 \theta}{\Sigma}\, {\mathrm{d}}t \, {\mathrm{d}}\phi + \frac{\Sigma}{\Delta} \, {\mathrm{d}}r^2+\Sigma \, {\mathrm{d}}\theta^2 \\
&+\sin^2 \theta \left(r^2+a^2+\frac{2a^2rM(r)\sin^2\theta}{\Sigma} \right) {\mathrm{d}}\phi^2,
\end{aligned}$$ where $$\label{e:M_r_BambiModesto}
\begin{aligned}
& \Sigma :=r^2+a^2\cos^2 \theta, \quad
\Delta :=r^2-2M(r)r+a^2, \\
& M(r) :=m \displaystyle \frac{r^3}{r^3+2mb^2}.
\end{aligned}$$ In addition to the total mass $m$ and the characteristic length $b$, the new parameter with respect to Hayward’s metric (\[e:Hayward\_metric\])-(\[e:def\_F\_M\]) is the spin parameter $a$, such that the total angular momentum is $J = am$. Note that the function $M(r)$ is identical to that defined by Eq. (\[e:def\_F\_M\]) and that, except for the dependency of $M$ with respect to $r$, the line element (\[metricKerrmodified\]) is identical to that of the Kerr metric expressed in Boyer-Lindquist coordinates.
As claimed in Ref. , there is no singularity at $r=0$ as long as $b \neq 0$ (see [@TorresFayos:2017] for a rigorous proof). However, as in the Kerr case, if the above metric is limited to $r\geq 0$, it yields a spacetime that is not geodesically complete: some timelike and null geodesics stop at $r=0$ for a finite value of their affine parameter, while (i) there is no curvature singularity there and (ii) $r=0$ is not a coordinate singularity as in Minkowski’s spacetime. The last point can be seen by considering the value of the metric (\[metricKerrmodified\])-(\[e:M\_r\_BambiModesto\]) at $r=0$: $$\left. {\mathrm{d}}s^2\right| _{r=0} = - {\mathrm{d}}t^2 + \cos^2\theta \, {\mathrm{d}}r^2
+ a^2\cos^2 \theta \, {\mathrm{d}}\theta^2 +a^2 \sin^2 \theta \, {\mathrm{d}}\phi^2.$$ If $a\not=0$, this defines a regular (i.e. nondegenerate) metric, except for $\theta=\pi/2$, the vanishing of $\sin^2\theta$ at $\theta=0$ or $\pi$ reflecting only the standard coordinate singularity of spherical coordinates on the rotation axis. The regularity of the metric at $r=0$ and the unphysical ending of geodesics there leads one to extend the spacetime to negative values of $r$. In other words, we consider $$\mathscr{M} = \mathbb{R}^2\times\mathbb{S}^2$$ as the spacetime manifold, with $(t,r)$ spanning $\mathbb{R}^2$ and $(\theta,\phi)$ spanning the 2-sphere $\mathbb{S}^2$.
Now, $\mathscr{M}$ endowed with the metric (\[metricKerrmodified\])-(\[e:M\_r\_BambiModesto\]) suffers from some curvature singularity, albeit not at $r=0$. Indeed, the Ricci scalar is (see \[s:calculations\] for the computation) $$\label{Kretschmann&Ricci}
R=-\frac{24 m^2 b^2 r^2 \left(r^{3} - 4 m b^{2}\right)}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} + 2 m b^{2}\right)}^{3}} ,$$ which is singular in the entire hypersurface defined by $r=-(2mb^2)^{1/3}$. Similarly, the Kretschmann scalar $K := R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma}$ diverges at the same value of $r$ (cf. Fig. \[R&K\_plots\_H\]). We conclude that the rotating generalization (\[metricKerrmodified\])-(\[e:M\_r\_BambiModesto\]) of Hayward’s metric does not describe a regular black hole.
![\[R&K\_plots\_H\] Ricci scalar (left) (in units of $m^{-2}$) and Kretschmann scalar (right) (in units of $m^{-4}$) as functions of $r$ for the extension to $r<0$ of Bambi and Modesto ’s rotating version of Hayward’s metric with $a/m=0.9$ and $b/m=1$. Note that both scalars are diverging at $r/m=-\sqrt[3]{2}\approx -1.26$.](Ricci_scalar_Hayward.pdf "fig:"){width="49.00000%"} ![\[R&K\_plots\_H\] Ricci scalar (left) (in units of $m^{-2}$) and Kretschmann scalar (right) (in units of $m^{-4}$) as functions of $r$ for the extension to $r<0$ of Bambi and Modesto ’s rotating version of Hayward’s metric with $a/m=0.9$ and $b/m=1$. Note that both scalars are diverging at $r/m=-\sqrt[3]{2}\approx -1.26$.](Kretschmann_scalar_Hayward.pdf "fig:"){width="49.00000%"}
The rotating Hayward metric extended to $r<0$
---------------------------------------------
### Metric
Following a prescription applied by Torres [@Torres:2017] to rotating regular black boles arising from quantum gravity consideration, we define the metric tensor in all $\mathscr{M}= \mathbb{R}^2\times\mathbb{S}^2$ by $$\label{metric_improved_Hayward}
\begin{aligned}
{\mathrm{d}}s^2=&-\left(1-\frac{2rM(r)}{\Sigma} \right){\mathrm{d}}t^2-\frac{4arM(r)\sin^2 \theta}{\Sigma}\, {\mathrm{d}}t \, {\mathrm{d}}\phi + \frac{\Sigma}{\Delta} \, {\mathrm{d}}r^2+\Sigma \, {\mathrm{d}}\theta^2 \\
&+\sin^2 \theta \left(r^2+a^2+\frac{2a^2rM(r)\sin^2\theta}{\Sigma} \right) {\mathrm{d}}\phi^2,
\end{aligned}$$ with $$\label{e:M_r_Torres}
\begin{aligned}
& \Sigma :=r^2+a^2\cos^2 \theta, \quad
\Delta :=r^2-2M(r)r+a^2, \\
& M(r) :=\displaystyle m \frac{|r|^3}{|r|^3+2mb^2}.
\end{aligned}$$ The difference with Bambi-Modesto’s metric (\[metricKerrmodified\])-(\[e:M\_r\_BambiModesto\]) lies only in the replacement of $r$ by $|r|$ in the function $M(r)$. This is motivated by the expression of $M(r)$ in Torres’ work [@Torres:2017]: $$M(r)_{\rm Torres} = m \frac{|r|^3}{|r|^3 + \tilde\omega(|r| + \gamma m)},$$ where $\tilde\omega$ and $\gamma$ are two constants.
### The hypersurface $r=0$
In $\mathscr{M}= \mathbb{R}^2\times\mathbb{S}^2$, the hypersurface $r=0$ is a 3-dimensional cylinder $\mathscr{T}_0 = \mathbb{R}\times\mathbb{S}^2$, spanned by the coordinates $(t,\theta,\phi)$, which we call the *throat*, as in the Kerr case [@ONeill:1995]. The metric induced on $\mathscr{T}_0$ by the spacetime metric (\[metric\_improved\_Hayward\])-(\[e:M\_r\_Torres\]) is $$\label{e:metric_throat}
{\mathrm{d}}\sigma^2=-{\mathrm{d}}t^2+a^2\cos^2 \theta \, {\mathrm{d}}\theta^2 +a^2 \sin^2 \theta \,
{\mathrm{d}}\phi^2.$$ We may then split $\mathscr{T}_0$ into three components: $\mathscr{T}_0 = \mathscr{T}_0^+ \cup \mathscr{R} \cup \mathscr{T}_0^-$, where $\mathscr{T}_0^+$ is the Northern hemisphere $0\leq\theta<\pi/2$ times (Cartesian product) $\mathbb{R}$, $\mathscr{R}$ is the equatorial ring $\theta=\pi/2$ times $\mathbb{R}$ and $\mathscr{T}_0^-$ is the Southern hemisphere $\pi/2<\theta\leq\pi$ times $\mathbb{R}$. Introducing in $\mathscr{T}_0^+$ or $\mathscr{T}_0^-$ the coordinates $$\left\{
\begin{array}{ll}
X&=a \sin \theta \cos \phi, \\
Y&=a \sin \theta \sin \phi
\end{array}
\right.
\qquad X^2 + Y^2 \leq a^2 ,$$ the line element (\[e:metric\_throat\]) reduces to $${\mathrm{d}}\sigma^2=-{\mathrm{d}}t^2 + {\mathrm{d}}X^2+{\mathrm{d}}Y^2.$$ We recognize a 3-dimensional Minkowskian metric and conclude that, as long as $a\not = 0$, the throat $\mathscr{T}_0$ comprises two flat open disks of radius $a$ times $\mathbb{R}$: $\mathscr{T}_0^+$ and $\mathscr{T}_0^-$. Moreover, from the signature of (\[e:metric\_throat\]), it appears that the throat is timelike; it is therefore a 2-way membrane, i.e. it can be crossed by particles from the region $r>0$ to the region $r<0$, in the reverse way as well.
### Regularity
The metric (\[metric\_improved\_Hayward\])-(\[e:M\_r\_Torres\]) has no curvature singularity. This can be seen on Figs. \[R\_plot\_HT\], \[K\_plot\_HT\], \[CP\_plot\_HT\] and \[E\_plot\_HT\], where the Ricci scalar $R$, Kretschmann scalar $K$, Chern-Pontryagin scalar $CP$ and Euler scalar $E$ are plotted for $a/m=0.9$, $b/m=1$ and various values of $\theta$ (see \[s:calculations\] for details). The curvature scalars remain finite, although the Ricci scalar is discontinuous at the equatorial ring $r=0$ and $\theta=\pi/2$.
[0.75]{} ![Ricci scalar (a) (in units of $m^{-2}$) and Kretschmann scalar (b) (in units of $m^{-4}$) of the improved rotating Hayward metric (\[metric\_improved\_Hayward\])-(\[e:M\_r\_Torres\]) with $a/m=0.9$ and $b/m=1$ as a function of $r$ for $\theta=0$, $\pi/4$ and $\pi/2$.[]{data-label=""}](Ricci_scalar_HT.pdf "fig:"){width="0.8\linewidth"}
[0.75]{} ![Ricci scalar (a) (in units of $m^{-2}$) and Kretschmann scalar (b) (in units of $m^{-4}$) of the improved rotating Hayward metric (\[metric\_improved\_Hayward\])-(\[e:M\_r\_Torres\]) with $a/m=0.9$ and $b/m=1$ as a function of $r$ for $\theta=0$, $\pi/4$ and $\pi/2$.[]{data-label=""}](Kretschmann_scalar_HT.pdf "fig:"){width="0.8\linewidth"}
[0.75]{} ![Chern-Pontryagin scalar (a) and Euler scalar (b) (in units of $m^{-4}$) of the improved rotating Hayward metric (\[metric\_improved\_Hayward\])-(\[e:M\_r\_Torres\]) with $a/m=0.9$ and $b/m=1$ as a function of $r$ for $\theta=0$, $\pi/4$ and $\pi/2$.[]{data-label=""}](ChernPontryagin_scalar "fig:"){width="0.8\linewidth"}
[0.75]{} ![Chern-Pontryagin scalar (a) and Euler scalar (b) (in units of $m^{-4}$) of the improved rotating Hayward metric (\[metric\_improved\_Hayward\])-(\[e:M\_r\_Torres\]) with $a/m=0.9$ and $b/m=1$ as a function of $r$ for $\theta=0$, $\pi/4$ and $\pi/2$.[]{data-label=""}](Euler_scalar "fig:"){width="0.8\linewidth"}
### Horizons
In the context of the stationary metric (\[metric\_improved\_Hayward\]), the trapping horizons, which identify to Killing horizons, are the null hypersurfaces where the expansion of a congruence of null outgoing geodesics vanishes. This condition reduces to $$\label{presence_horizons}
\Delta = r^2-2M(r)r+a^2=0.$$ This equation admits some real solutions depending on the values of the parameters $a$ and $b$. We will call *event horizon* the outermost Killing horizon, which corresponds to the biggest value of the radial coordinate among the solutions of (\[presence\_horizons\]). The region of existence of Killing horizons is depicted on Fig. \[Horizon\_a\_b\].
![\[Horizon\_a\_b\] Region of existence of one (black line) or two (in blue) Killing horizon(s), depending on the parameters $a$ and $b$.](Horizon_a_b.pdf)
When $b=0$ one recovers the Kerr case: two horizons exist for values of $a$ ranging from $a=0$ to $a=m$, the latter value corresponding to the extremal Kerr black hole, where the two horizons coincide. The most interesting cases with horizons are the metrics well different from Kerr ($b=0$) and Hayward ($a=0$) ones, for instance the metric with $a=b=0.5\,m$. The image of such configurations, computed using the ray-tracing code[^1] [<span style="font-variant:small-caps;">Gyoto</span>]{} [@Vincent_et_al:2011], will be discussed in Sec. \[improved\_Hawyard\_horizons\].\
In the absence of horizon (hence of trapped region), the spacetime can no longer be qualified of a regular rotating black hole. That is why we call it a *naked rotating wormhole*. Indeed, the wormhole whose throat is located at $r=0$, which is also present in Kerr’s case (with a singularity), is no longer hidden by any horizon. Photons can even go through the throat and come back to the observer, as will be shown in Sec. \[Naked rotating wormhole\].
### Causality
The Kerr spacetime possesses a well-known acausal region, the Carter time machine [@Carter:1968]. In this region, the Killing vector $\eta=\partial_\phi$ is timelike, giving birth to closed timelike curves. However the whole spacetime does not become acausal thanks to the presence of an event horizon: the particles that are able to move backward in time are trapped inside the black hole.\
Considering now the rotating Hayward metric extended to $r<0$, one has to check whether $\eta$ can become timelike even in the absence of horizons, in which case the whole spacetime would be acausal. In view of (\[metric\_improved\_Hayward\]), one has $$\eta \cdot \eta=g_{\phi \phi}=\left(r^2+a^2+\frac{2a^2M(r)r\sin^2\theta}{r^2+a^2\cos^2\theta} \right) \sin^2 \theta,$$ so that $$\label{eta_timelike}
\eta \hspace{2mm} \mbox{timelike} \hspace{2mm} \Leftrightarrow (r^2+a^2)(r^2+a^2\cos^2\theta) + 2a^2M(r)r\sin^2\theta<0 .$$ The only negative contribution in the left-hand side of (\[eta\_timelike\]) comes from the second term, when $r<0$. It reaches a minimum for $\theta=\pi/2$. Fig. \[Causality\_Horizon\_a\_b\] shows that there exists a red region (region I) for which $g_{\phi \phi}<0$ while no event horizon is present. The parameters $a$ and $b$ associated with such a region thus correspond to acausal spacetimes, which we will not deal with in this paper.
![\[Causality\_Horizon\_a\_b\] Regions of existence of an event horizon (in blue) and of negative $g_{\phi \phi}$ for $\theta=\pi/2$ in the absence of horizons (in red), depending on the parameters $a$ and $b$.](Causality_Horizon_a_b.pdf)
Region II also has causality issues, but these are hidden behind an event horizon. Regions III and IV are totally free of closed timelike curves, the latter is also devoid of any event horizon and represents a naked rotating wormhole.
### Energy conditions
The existence of horizons, hence of trapped surfaces, along with the absence of singularity, questions the hypotheses of Penrose’s singularity theorem. As mentionned in Sec. \[s:intro\], both Bardeen and Hayward nonrotating metrics fulfill the weak energy condition and circumvent the original Penrose theorem [@Penrose:1965] by the lack of a Cauchy surface. In an improved version of the singularity theorem, by Hawking and Penrose (1970) [@HawkingP:1970; @HawkingE:1973], the hypothesis of existence of a Cauchy surface is relaxed, at the price of replacing the weak energy condition by the strong one. This version is still compatible with Bardeen’s and Hayward’s regular black holes because both violate the strong energy condition.
In the rotating case, it has been shown by Torres [@Torres:2017] that any metric of the type (\[metric\_improved\_Hayward\]) with $a\not=0$ violates the weak energy condition in all the region $r<0$ as soon as $M'(r)<0$ there. This is the case for our choice (\[e:M\_r\_Torres\]) for $M(r)$.
Here, we investigate the violation of the weakest of all energy conditions, the *null energy condition (NEC)*. It is the weakest condition in the sense that its violation also implies the violation of the weak, strong and dominant energy conditions. For any null vector $k^{\mu}$ the NEC reads $$T_{\mu \nu}k^{\mu}k^{\nu} \geq 0.$$ In order to compute this scalar we switch to the locally nonrotating frame which diagonalizes the metric [@Bardeen:1972]. Its basis is such that $e_{\hat{\mu}} \cdot e_{\hat{\nu}}=\eta_{\hat{\mu}\hat{\nu}}$. The dual cobasis at each point $(t,r,\theta,\phi)$ reads $$\begin{aligned}
e^{(t)}&=\sqrt{\frac{\Sigma \Delta}{A}} \,{\mathrm{d}}t, \\
e^{(r)}&=\sqrt{\frac{\Sigma}{\Delta}} \,{\mathrm{d}}r, \\
e^{(\theta)}&=\sqrt{\Sigma}\,{\mathrm{d}}\theta, \\
e^{(\phi)}&=-\frac{2M(r)ar\sin \theta}{\sqrt{\Sigma A}}\,{\mathrm{d}}t+\sqrt{\frac{A}{\Sigma}} \sin \theta \,{\mathrm{d}}\phi,
\end{aligned}$$ with $$A:=(r^2+a^2)^2-a^2\Delta \sin^2 \theta.$$ Solving Einstein’s equations “in reverse”, we obtain $T_{\hat{\mu}\hat{\nu}}k^{\hat{\mu}}k^{\hat{\nu}}=G_{\hat{\mu}\hat{\nu}}k^{\hat{\mu}}k^{\hat{\nu}}/8\pi$. This effective energy density is plotted in Fig. \[NEC\_a09\_b1\] in the case $a=0.9$, $b=1$ (see \[s:calculations\] for details). One can see that the NEC is violated from near the centre up to $r\rightarrow -\infty$.
![\[NEC\_a09\_b1\] $T_{\hat{\mu}\hat{\nu}}k^{\hat{\mu}}k^{\hat{\nu}}$ as a function of $r$ for $\theta=\pi/6,\pi/3,\pi/2$ and $a=0.9, b=1$. The NEC is violated when any of the curves goes below zero.](NEC_a09_b1.pdf)
Numerical study of the regular rotating Hayward metric with [<span style="font-variant:small-caps;">Gyoto</span>]{} {#section 3}
===================================================================================================================
Regular rotating Hayward black hole {#improved_Hawyard_horizons}
-----------------------------------
This section aims at discussing the differences in ray-traced images between the regular rotating Hayward black hole and the standard Kerr black hole.\
We now use the ray-tracing code [<span style="font-variant:small-caps;">Gyoto</span>]{}to obtain images from an accretion torus surrounding the two black holes (see \[Gyoto\_plugin\] for details). Another approach could consist in computing analytically the contour of the shadow based on the formula developed by Tsukamoto [@tsukamoto_2018], but here we opt for a numerical computation in an astrophysical context instead. The set-up is composed of an accretion torus identical to the one presented in Ref. [@Vincent_et_al:2015]. It is a magnetized optically thin torus, with angular momentum $l=4m$, inner radius $r_\mathrm{inner}=8.3m$, where we take $m$ to be the mass of Sgr A\* ($m=4.31 \times 10^6 \hspace{2mm} M_{\odot}$). The outer radius varies according to the value of the spin parameter, it is for instance $r_\mathrm{outer}=30m$ for $a=0.9$. The central temperature is of $T=5.3\times10^{10}$ K and the central electron number density $n_e = 6.3\times 10^6\,\mathrm{cm}^{-3}$. We compute the thermal synchrotron radiation emitted in the millimeter band by this torus, the ray-traced photons are observed at a frequency of 230 GHz. Such an accretion flow was shown (see [@Vincent_et_al:2015]) to reproduce well the millimeter spectral data of Sgr A\*, as well as the constraints on the size of the emitting region imposed by early EHT data [@Doeleman:2008]. The observer is located at a radial coordinate which corresponds to the distance between Earth and Sgr A\*, and at an inclination (angle between the black hole rotation axis and the line of sight) of $\theta=90^{\circ}$.\
In the case $a/m=0.5$, $b/m=0.5$, the outer horizon is located at $r_0 \approx 1.65m$. It is thus at a smaller value of the radial coordinate from the center than in the Kerr black hole case ($b/m=0$), where $r_+=m+\sqrt{m^2-a^2}\approx 1.87m$. Hence, for a given ADM mass, the black hole radius is smaller when $b \neq 0$.
[0.75]{} ![Images of an accretion torus surrounding a Kerr black hole (a) and a regular rotating Hayward black hole (b), seen from a distance of $8.31$ kpc. The field of view is $200 \hspace{1mm} \mu\mbox{as}$ and the inclination $\theta=90^{\circ}$. The specific intensity $I_{\nu}$ is plotted in CGS units, as will be the case for the following images.[]{data-label="2BH with horizons"}](far_obs_torus_SgA_Hayward-Torres_a05_b0_fov200_inclination90_resol1000 "fig:"){width="0.85\linewidth"}
[0.75]{} ![Images of an accretion torus surrounding a Kerr black hole (a) and a regular rotating Hayward black hole (b), seen from a distance of $8.31$ kpc. The field of view is $200 \hspace{1mm} \mu\mbox{as}$ and the inclination $\theta=90^{\circ}$. The specific intensity $I_{\nu}$ is plotted in CGS units, as will be the case for the following images.[]{data-label="2BH with horizons"}](far_obs_torus_SgA_Hayward-Torres_a05_b05_fov200_inclination90_resol1000 "fig:"){width="0.85\linewidth"}
The millimeter images of these two black holes are visible on Fig. \[2BH with horizons\]. On both panels, we observe the distorted primary image of the torus, that forms an Einstein ring. The very center of the image shows a thin lensing ring delineating the black hole shadow. The shadow is defined as the region in the observer’s sky comprising the directions of photons that asymptotically approach the event horizon in a backward ray-tracing computation [@VincentGourgoulhonHerdeiroRadu:2016]. The lensing ring is the secondary image of the torus. The higher-order images of the torus, hardly visible on Fig. \[2BH with horizons\], converge to the photon ring, which is the projection of the innermost photon orbit on the observer’s sky. This photon orbit marks the innermost limit an approaching photon can visit without falling into the event horizon [@VincentGourgoulhonHerdeiroRadu:2016]. The differences between the millimeter images of the two black holes appear to be indistinguishable with the naked eye. However, substracting one image from another we can distinguish the two different lensing rings (Fig. \[Difference\_a05b05\_a05b0\]). The difference of diameter between these two rings is about 2 $\mu$as ($\approx 3$%). This difference is out of reach for the observations in the foreseeable future, but we may hope that a telescope would be able to measure the radius of the lensing ring in a far future, or equivalently the area of the shadow, and could thus discriminate between the two black holes for a given ADM mass.
![\[Difference\_a05b05\_a05b0\]Difference of the images of Fig. \[2BH with horizons\]. The lensing rings of the configurations $b/m=0.5$ (black) and $b/m=0$ (white) are visible at the centre.](Difference_a05b05_a05b0)
It should be noted that we considered here a macroscopic value of $b$ ($b=0.5m$) when computing these images. The underlying assumption is that this parameter arises because of some “macroscopic” energy-momentum tensor supporting this regular geometry, similar to that arising from a nonlinear-electrodynamics magnetic monopole in the non-rotating case . Had we assumed that the singularity was resolved by using $b$ as a Planckian cut-off, the difference between the images would have been invisible to any telescope even in the far future.
Naked rotating wormhole {#Naked rotating wormhole}
-----------------------
Let us now discuss the case of geometries without horizons described by the rotating Hayward metric extended to $r<0$ (\[metric\_improved\_Hayward\]), that we call naked rotating wormholes. We will first describe the results obtained with [<span style="font-variant:small-caps;">Gyoto</span>]{}as well as their consequences, and then explain them by studying some relevant geodesics.\
The major difference in this configuration, with respect to the previous section, is the absence of horizons. Hence, the images obtained with this geometry do not contain any shadows, in the precise sense defined above. However, they do contain a central faint region showing a mixture of low-flux regions and strongly lensed contours (of increasing order from left to right, see Fig. \[no\_shadow\]). The shape of these contours highly depends on the value of the parameter $b/m$. It it is thus very important to stress that observing such strongly lensed contours, which can look like a lensing ring without a good enough resolution, does not imply the existence of an event horizon, just as in the case of boson stars [@Vincent_et_al:2016].\
The two panels of Fig. \[no\_shadow\] are remarkably similar to the images of accretion tori surrounding rotating boson stars (see the middle- and lower-right panels of Fig. 5 in Ref. [@Vincent_et_al:2016]). However, these spacetimes are completely different, the spacetimes analyzed here being naked rotating wormholes spacetimes, while boson stars are compact distributions of fundamental scalar fields. It is thus rather intriguing that such very different spacetimes lead to images that are difficult to differentiate. Further studies would be necessary in order to determine whether the distorted, hyper-lensed contours of Fig. \[no\_shadow\] are general features of spacetimes of compact object with no event horizon and no hard surface (i.e. different from neutron stars).
[0.75]{} ![ Images of an accretion torus surrounding a naked rotating wormhole with $a/m=0.9$, $b/m=0.4$ (a) and $b/m=0.7$ (b). The field of view is $200\; \mu{\rm as}$ and the inclination $\theta=90^{\circ}$.[]{data-label="no_shadow"}](far_obs_torus_SgA_Hayward-Torres_a09_b04_fov200_inclination90_resol1000 "fig:"){width="0.85\linewidth"}
[0.75]{} ![ Images of an accretion torus surrounding a naked rotating wormhole with $a/m=0.9$, $b/m=0.4$ (a) and $b/m=0.7$ (b). The field of view is $200\; \mu{\rm as}$ and the inclination $\theta=90^{\circ}$.[]{data-label="no_shadow"}](far_obs_torus_SgA_Hayward-Torres_a09_b07_fov200_inclination90_resol1000 "fig:"){width="0.85\linewidth"}
Another interesting feature appears when the accretion torus is observed from an inclination angle $\theta$ different from $90^{\circ}$. In this case, the disk $r=0$ located in the equatorial plane becomes visible. It can hardly be seen on the left of Fig. \[dark\_zone\], but a zoom clearly allows identifying a central dark ellipse[^2] on the image on the right. Its contour corresponds to the throat of the wormhole.\
The blue pixels forming this ellipse-like shape (right panel) represent geodesics coming from $r \rightarrow -\infty$; a similar distorded disk also appears in the case of naked Kerr singularities [@HiokiMaeda:2009]. These pixels are not completely black since a part of the torus, located between the throat and the observer, emits some photons directly towards the latter. However there also exists luminous (green and yellow) pixels inside the dark ellipse. All these illuminated pixels are associated with photons emitted from the torus and travelling through negative values of $r$ back to the observer. The location of this luminous feature inside the dark ellipse highly depends on the value of $b/m$, as is illustrated on the right panel of Fig. \[dark\_zone\].\
There exists a sharp contrast between the dark ellipse and this luminous feature, which can be studied in further detail if we consider three different geodesics (Fig. \[Geodesics\]).
[11cm]{} ![Images of an accretion torus surrounding a naked rotating wormhole with $a/m=0.9$, $b/m=1$ (upper row) and $a/m=0.9$, $b/m=0.5$ (lower row). The inclination is $\theta=80^{\circ}$ while the field of view is $200\; \mu{\rm as}$ (left column) or $25\; \mu{\rm as}$ (right column)[]{data-label="dark_zone"}](far_obs_torus_SgA_Hayward-Torres_a09_b1_fov200_inclination100_resol1000.pdf "fig:"){width="\linewidth"}
[11cm]{} ![Images of an accretion torus surrounding a naked rotating wormhole with $a/m=0.9$, $b/m=1$ (upper row) and $a/m=0.9$, $b/m=0.5$ (lower row). The inclination is $\theta=80^{\circ}$ while the field of view is $200\; \mu{\rm as}$ (left column) or $25\; \mu{\rm as}$ (right column)[]{data-label="dark_zone"}](far_obs_torus_SgA_Hayward-Torres_a09_b1_fov25_inclination100_resol1000.pdf "fig:"){width="\linewidth"}
[11cm]{} ![Images of an accretion torus surrounding a naked rotating wormhole with $a/m=0.9$, $b/m=1$ (upper row) and $a/m=0.9$, $b/m=0.5$ (lower row). The inclination is $\theta=80^{\circ}$ while the field of view is $200\; \mu{\rm as}$ (left column) or $25\; \mu{\rm as}$ (right column)[]{data-label="dark_zone"}](far_obs_torus_SgA_Hayward-Torres_a09_b05_fov200_inclination100_resol1000.pdf "fig:"){width="\linewidth"}
[11cm]{} ![Images of an accretion torus surrounding a naked rotating wormhole with $a/m=0.9$, $b/m=1$ (upper row) and $a/m=0.9$, $b/m=0.5$ (lower row). The inclination is $\theta=80^{\circ}$ while the field of view is $200\; \mu{\rm as}$ (left column) or $25\; \mu{\rm as}$ (right column)[]{data-label="dark_zone"}](far_obs_torus_SgA_Hayward-Torres_a09_b05_fov25_inclination100_resol1000.pdf "fig:"){width="\linewidth"}
The green geodesic of Fig. \[Geodesics\] corresponds to a luminous pixel just outside the dark ellipse. This geodesic comes from $r=+\infty$, crosses the torus on its way in, approaches the $r=0$ disk (represented by the grey sphere on Fig. \[Geodesics\]) without reaching it, and escapes to the observer (crossing a second time the torus on its way out). Similarly, the blue geodesic comes from $r=+\infty$ and crosses the torus on its way in. Contrarily to the green geodesic, it enters the wormhole throat, reaching negative values of $r$. It also reaches a turning point and comes back to $r>0$, also eventually reaching the observer after having crossed the torus a second time on its way out. Finally, the red geodesic originates from $r=-\infty$. It emerges from the throat and crosses the torus only once, on its way out to the observer.
![\[Geodesics\] The null geodesics associated with three different pixels, starting from the accretion torus for $a/m=0.9$, $b/m=1$, are plotted in a frame with $x=\mbox{e}^r \sin \theta \cos \phi $, $y=\mbox{e}^r \sin \theta \sin \phi$, $z=\mbox{e}^r \cos \theta$. A frame is drawn at the origin, where $r \rightarrow -\infty$. Geodesic I (in green) has a turning point and does not enter the grey sphere of radius $r=0$. Geodesic II (in red) has no turning point, it represents the trajectory of a photon in the central dark ellipse coming from $r \rightarrow -\infty$. Finally, geodesic III (in blue) enters the sphere of radius $r=0$ but has a turning point inside, and goes back to an observer at $r \rightarrow +\infty$.](Geodesics1)
Analytical study of the regular rotating Hayward metric {#section 4}
=======================================================
Circular orbits in the equatorial plane
---------------------------------------
### Energy and angular momentum of a massive particle
Let us consider the geodesic motion of a test particle with momentum $p$ and mass $m_0>0$, following a circular orbit in the background of the metric (\[metric\_improved\_Hayward\]). This motion occurs in the equatorial plane ($\theta=\pi/2$) due to the axisymmetry of the metric. Along with the property of stationarity, this also implies the existence of two Killing vectors $\xi=\partial_t$ and $\eta=\partial_\phi$. The energy and angular momentum of the particle read:
$$\begin{aligned}
E&=-\xi \cdot p=-p^t (g_{tt}+g_{t\phi}\Omega) \\
L&=\eta \cdot p=p^t(g_{\phi t}+g_{\phi \phi}\Omega),
\end{aligned}$$
with $\Omega := \frac{d\phi}{dt}$.\
The angular velocity $\Omega$ can be found by considering the Euler-Lagrange equation for a free particle whose Lagrangian is
$$\mathscr{L}=\frac{1}{2}g_{\mu \nu}\dot{x}^{\mu}\dot{x}^{\nu},$$
with $\dot{x}^\mu:=\frac{dx^\mu}{d \lambda}$, where $\lambda$ is an affine parameter. The Euler-Lagrange equation supplemented by the conditions for a circular orbit $\dot{r}=\ddot{r}=0$ boils down to:
$$g_{tt,r}\dot{t}^2+2g_{t\phi,r}\dot{t}\dot{\phi}+g_{\phi\phi,r}\dot{\phi}^2=0,$$
where the coma denotes a derivative with respect to the radial coordinate.\
The angular velocity of a particle on a circular co- or contra-rotating orbit is then:
$$\label{Omega}
\Omega_\pm=\frac{-g_{t\phi,r}\pm \sqrt{g_{t\phi,r}-g_{tt,r}g_{\phi\phi,r}}}{g_{\phi \phi,r}}.$$
The specific energy and angular momentum of a massive particle on a circular orbit in a stationary axisymmetric spacetime thus read:
$$\begin{aligned}
\mathcal{E_\pm}& := \frac{E_\pm}{m_0}=-\frac{g_{tt}+g_{t\phi}\Omega_\pm}{\sqrt{-(g_{tt}+2g_{t\phi}\Omega_\pm+g_{\phi\phi}\Omega_\pm^2)}} \\
\mathcal{L_\pm}& := \frac{L_\pm}{m_0}=\frac{g_{\phi t}+g_{\phi\phi}\Omega_\pm}{\sqrt{-(g_{tt}+2g_{t\phi}\Omega_\pm+g_{\phi\phi}\Omega_\pm^2)}}.
\end{aligned}$$
In the context of our metric (\[metric\_improved\_Hayward\]), we obtain in the equatorial plane ($\theta=\pi/2$):
$$\begin{aligned}
\mathcal{E_\pm}&= \frac{r^3+a^2rM'(r)-M(r)\left(a^2+2r^2\mp2ar\sqrt{A(r)} \right)}{\left(r^3-a^2M(r)+a^2rM'(r)\right)\sqrt{\frac{r^2B_\pm(r)}{\left(r^3-a^2M(r)+a^2rM'(r) \right)^2}}} \\
\mathcal{L_\pm}&=\frac{-(a^3+3ar^2)M(r)+(a^3r+ar^3)M'(r)\pm(a^2r^2+r^4+2a^2rM(r))\sqrt{A(r)}}{\left(r^3-a^2M(r)+a^2rM'(r)\right)\sqrt{\frac{r^2B_\pm(r)}{\left(r^3-a^2M(r)+a^2rM'(r) \right)^2}}},
\end{aligned}$$
where $$\begin{aligned}
A(r)&=\frac{M(r)}{r}-M'(r) \\
B_\pm(r)&=-a^2r^2M'^2(r)+r^4-3a^2M(r)^2-3(a^2r+r^3)M(r)+(3a^2r^2+r^4+4a^2rM(r))M'(r) \\
& \pm 2 \left[(a^3+3ar^2)M(r)-(a^3r+ar^3)M'(r) \right] \sqrt{A(r)}.
\end{aligned}$$ These expressions differ from Toshmatov et al. [@Toshmatov_et_al:2017] (see \[s:calculations\] for details and a comparison with the results of Bardeen et al. [@Bardeen:1972]).\
Circular orbits can therefore exist only for $A(r) \geq 0$ and $B_\pm(r)>0$. These three functions are plotted below, for $a/m=0.9$ and $b/m=1$. The regions of allowed co-rotating circular orbits are pictured in grey on Fig. \[Circular\_orbits\_a/m=09\_b/m=1\]. The region of positive $r$ goes up to $r \rightarrow +\infty$, while the one of negative values of $r$ exists only near the center. This is coherent, since from $r \rightarrow -\infty$ the metric (\[metric\_improved\_Hayward\]) behaves as a Schwarzschild metric with negative mass: the repulsive gravity does not allow circular orbits for large enough negative radii.
![\[Circular\_orbits\_a/m=09\_b/m=1\] Plot of $A(r)$, $B_+(r)$ and $B_-(r)$ in the case $a/m=0.9$, $b/m=1$. The shaded regions represent the zones where circular orbits are allowed.](Circular_orbits_a09_b1)
### Influence of the spin
Let us study how the regions of allowed circular orbits are modified when the spin varies. First of all, it should be noted that $A(r)$ does not depend on the value of the spin. Hence for a fixed $b$, e.g. $b/m=1$ like on Fig. \[Circular\_orbits\_a/m=09\_b/m=1\], the shaded regions will be modified only if $B_\pm(r)$ changes. As shown on Fig. \[Circular\_orbits\_a/m=07\_b/m=1\], decreasing the value of $a$ only widens the zone of circular orbits below $r=0$. It thus does not have any impact on the allowed circular orbits with $r>0$.
![\[Circular\_orbits\_a/m=07\_b/m=1\] Plot of $A(r)$, $B_+(r)$ and $B_-(r)$ in the case $b/m=1$, for two different values of the spin. The shaded region for the negative values of $r$ gets wider as $a$ decreases.](Circular_orbits_a08_b1)
a\) $a/m=0.8$
![\[Circular\_orbits\_a/m=07\_b/m=1\] Plot of $A(r)$, $B_+(r)$ and $B_-(r)$ in the case $b/m=1$, for two different values of the spin. The shaded region for the negative values of $r$ gets wider as $a$ decreases.](Circular_orbits_a07_b1)
b\) $a/m=0.7$
### Influence of the parameter $b$
Contrarily to the spin, the parameter $b$ has a direct influence on the region of allowed circular orbits of positive radius. Going from $b/m=1$ (Fig. \[Circular\_orbits\_a/m=09\_b/m=1\]) to $b/m=0.7$ and $b/m=0.4$ (Fig. \[Circular\_orbits\_a/m=09\_b/m=07\]), at a constant $a/m=0.9$, we observe that circular orbits can occur for smaller and smaller positive values of $r$.
Meanwhile, the region of allowed circular orbits with negative radius shrinks as $b$ decreases. This region even disappears for $b/m=0$, as one can see on Fig. \[Circular\_orbits\_a/m=09\_b/m=0\] below. In this configuration, two horizons exist and circular orbits occur only for values of the radial coordinate above the radius of the outer horizon. For $b/m=0.2$ (left of Fig. \[Circular\_orbits\_a/m=09\_b/m=0\]), some circular orbits can also occur below the radius of the inner horizon.
![\[Circular\_orbits\_a/m=09\_b/m=07\] Plot of $A(r)$, $B_+(r)$ and $B_-(r)$ in the case $a/m=0.9$ for two different values of $b$. The shaded region for the negative (resp. positive) values of $r$ gets narrower (resp. wider) as $b$ decreases.](Circular_orbits_a09_b07)
a\) $b/m=0.7$
![\[Circular\_orbits\_a/m=09\_b/m=07\] Plot of $A(r)$, $B_+(r)$ and $B_-(r)$ in the case $a/m=0.9$ for two different values of $b$. The shaded region for the negative (resp. positive) values of $r$ gets narrower (resp. wider) as $b$ decreases.](Circular_orbits_a09_b04)
b\) $b/m=0.4$
![\[Circular\_orbits\_a/m=09\_b/m=0\] Plot of $A(r)$, $B_+(r)$ and $B_-(r)$ in the case $a/m=0.9$, in the presence of two horizons (black vertical lines), for two different values of $b$.](Circular_orbits_a09_b02)
a\) $b/m=0.2$
![\[Circular\_orbits\_a/m=09\_b/m=0\] Plot of $A(r)$, $B_+(r)$ and $B_-(r)$ in the case $a/m=0.9$, in the presence of two horizons (black vertical lines), for two different values of $b$.](Circular_orbits_a09_b0)
b\) $b/m=0$
### Innermost stable circular orbit (ISCO)
The *innermost stable circular orbit (ISCO)*, which corresponds to the stable circular orbit of smaller $r$, is astrophysically relevant since it provides the highest orbital frequency possible around the central object. In particular, the ISCO frequency is involved in various models of quasi-periodic oscillations (QPO) [@remillard_x-ray_2006].
In the Kerr case, it has been shown by Carter [@Carter:1968] that the radial geodesic motion is governed by the following relation: $$\label{def_of_R}
\begin{aligned}
\Sigma \frac{dr}{d\lambda}&=\sqrt{\mathcal{R}} \\
\mathcal{R}&=\left[(r^2+a^2)E-aL \right]^2-\Delta \left[(aE-L)^2+m_0^2r^2+ \mathcal{Q} \right],
\end{aligned}$$ where $\lambda$ is an affine parameter, $m_0$ the mass of the particle, and $\mathcal{Q}$, $E$, $L$ are the three integrals of motion (respectively the Carter constant, the energy and the angular momentum of the test particle). The zeros of $\mathcal{R}$ thus represent turning points of the motion of such a test particle in Kerr’s spacetime.
Stable circular orbits are defined by the three conditions $$\label{stable_circular_orbits}
\mathcal{R}(r)=0, \quad \frac{d\mathcal{R}(r)}{dr}=0, \quad\mbox{and}\quad \frac{d^2\mathcal{R}(r)}{dr^2} \leq 0 .$$
The frequency (\[Omega\]) of a particle following a circular orbit reads: $$\label{Omega_expl}
\Omega_\pm=\frac{4 \, a b^{2} m^{2} r - a m r^{4} \pm {\left(4 \, b^{4} m^{2} + 4 \, b^{2} m r^{3} + r^{6}\right)} \sqrt{-\frac{4 \, b^{2} m^{2} r^{2} - m r^{5}}{4 \, b^{4} m^{2} + 4 \, b^{2} m r^{3} + r^{6}}}}{r^{7} - {\left(a^{2} - 4 \, b^{2}\right)} m r^{4} + 4 \, {\left(a^{2} b^{2} + b^{4}\right)} m^{2} r}$$ The ISCO values of the radius and the orbital frequency (\[Omega\_expl\]), for co-rotating and contra-rotating orbits in the equatorial plane ($\mathcal{Q}=0$), have been computed for different values of $a$ and $b$ (see \[s:calculations\] for details). The result is shown in Table \[ISCO\_a\_b\].
Null geodesics
--------------
Let us now focus on the propagation of light rays in order to understand the images that can be seen by an observer on Earth, such as the ones of Fig. \[dark\_zone\]. Due to (\[def\_of\_R\]), in which we now take $m_0=0$, the condition for the existence of a photon of energy $E$ with angular momentum $L$ and Carter’s constant $\mathcal{Q}$ is $$\label{R>0}
\left[(r^2+a^2)E-aL \right]^2-\left(r^2+a^2-2rM(r) \right) \left[(aE-L)^2+\mathcal{Q} \right] \geq 0 ,$$ with $M(r)$ given by Eq. (\[e:M\_r\_Torres\]).
In the case of a Kerr spacetime, $M(r)=m$ and Eq. (\[R>0\]) is polynomial in $r$, of degree 4. It can then be shown that a photon trajectory has at most one radial turning point in the black hole exterior . Here, due to the form (\[e:M\_r\_Torres\]) of $M(r)$, Eq. (\[R>0\]) reduces to a polynomial equation of degree 7. The phenomenology is thus much richer than in Kerr’s case. In particular, some photon trajectories can have more than one radial turning point. This is illustrated on Fig. \[R\_2\_turning\_points\] below. The central shaded region of Fig. (b) is particularly striking: photons with energy $E_0$, angular momentum $L_0$ and a Carter constant $\mathcal{Q}_0$ can oscillate back and forth between two radial turning points, around $r=0$.
![\[R\_2\_turning\_points\] Plot of $\mathcal{R}$ as a function of the radial coordinate $r/m$. The shaded regions represent the allowed regions for a photon with $E_0/m=1$, $L_0/m=2$, $\mathcal{Q}_0/m^2=-1$, in the case of a rotating Hayward black hole with $a/m=0.9$ and $b/m=0$ (a) (the black lines denote the outer and inner horizons) and of a naked rotating wormhole with $b/m=1$ (b).](R_2_turning_points_b0)
a\) $b/m=0$
![\[R\_2\_turning\_points\] Plot of $\mathcal{R}$ as a function of the radial coordinate $r/m$. The shaded regions represent the allowed regions for a photon with $E_0/m=1$, $L_0/m=2$, $\mathcal{Q}_0/m^2=-1$, in the case of a rotating Hayward black hole with $a/m=0.9$ and $b/m=0$ (a) (the black lines denote the outer and inner horizons) and of a naked rotating wormhole with $b/m=1$ (b).](R_2_turning_points)
b\) $b/m=1$
This analysis, using the inequality (\[R>0\]), also allows us to understand the behaviour of the photons travelling into the region with $r<0$ (Fig. \[dark\_zone\]) before reaching an observer on Earth. Fig. \[R\_photons\_dark\_zone\] shows the allowed region for a photon with $E_1/m=1$, $L_1/m=-2$, $\mathcal{Q}_1/m^2=-1$ in two different cases: $b/m=0$ (left) and $b/m=1$ (right), while $a/m=0.9$. One can see that in both cases, a photon going from $r>0$ to $r<0$ has a radial turning point and goes back to the region with positive radial coordinate. However, in the case $b/m=0$ where a trapped region is located between the two Killing horizons (black vertical lines), this photon cannot cross the inner horizon and thus reach the observer. When $b/m=1$ no horizon is present, which allows a photon from the accretion torus to travel towards the region $r<0$, reach a turning point and then an observer on Earth.
![\[R\_photons\_dark\_zone\] Plot of $\mathcal{R}$ as a function of the radial coordinate $r/m$. The shaded regions represent the allowed regions for a photon with $E_1/m=1$, $L_1/m=-2$, $\mathcal{Q}_1/m^2=-1$, in the case of a rotating Hayward black hole with $a/m=0.9$ and $b/m=0$ (a) (the black lines denote the outer and inner horizons) and of a naked rotating wormhole with $b/m=1$ (b).](R_photons_dark_zone_b0)
a\) $b/m=0$
![\[R\_photons\_dark\_zone\] Plot of $\mathcal{R}$ as a function of the radial coordinate $r/m$. The shaded regions represent the allowed regions for a photon with $E_1/m=1$, $L_1/m=-2$, $\mathcal{Q}_1/m^2=-1$, in the case of a rotating Hayward black hole with $a/m=0.9$ and $b/m=0$ (a) (the black lines denote the outer and inner horizons) and of a naked rotating wormhole with $b/m=1$ (b).](R_photons_dark_zone)
b\) $b/m=1$
Conclusion
==========
We have investigated the geometry of a non-singular rotating black hole, both numerically and analytically. To this end we have extended the geodesically incomplete rotating Hayward spacetime to the region $r<0$, thereby obtaining a regular rotating Hayward metric. This metric describes a *regular rotating Hayward black hole* in the presence of an event horizon, and a *naked rotating wormhole* otherwise.
The numerical study of the regular rotating Hayward black hole using the ray-tracing code [<span style="font-variant:small-caps;">Gyoto</span>]{}has shown that, at a given spin $a$, the image of an accretion torus around such a black hole possesses a smaller shadow compared to that of the Kerr black hole. This difference is however out of reach of the observations in the foreseeable future.
Some images in the horizonless case (naked rotating wormhole) have also been computed. They display a central faint region with hyper-lensed contours whose shape depends on the value of the parameter $b$. The simulations with [<span style="font-variant:small-caps;">Gyoto</span>]{}allow distinguishing very well these contours from the shadows associated with the standard Kerr case or with the rotating Hayward black hole, as can be seen by comparing Fig. \[no\_shadow\] & \[dark\_zone\] to Fig. \[fig6:a\] & \[fig6:b\]. Without any good resolution data, we must stress that distinguishing these contours from the lensing ring delineating the shadow of a Kerr black hole could be extremely challenging, as in the case of boson stars [@Vincent_et_al:2016].
Another interesting feature of the naked rotating wormhole occurs when we compute images with some inclination angle $\theta \neq \pi/2$. A dark ellipse then appears at the centre, corresponding to the image of the wormhole’s throat on the observer’s sky. This ellipse is mainly dark because there is no source in the region with $r<0$. However, a luminous feature appears in it, whose shape depends on the value of $b$: it is produced by photons emitted from the accretion torus, which have crossed the throat and made some journey in the region $r<0$ before coming back to the observer.
An analytical study of this geometry also has been performed. After giving the expressions for the specific energy and angular momentum of massive particles in co- and contra-rotating orbits in the equatorial plane, which differ from the results of Tomashtov et al. [@Toshmatov_et_al:2017], we computed the radius of the ISCO for various values of the parameters $a$ and $b$. The values of the frequency of the orbits at the ISCO, highly depending on $b$, open up the possibility of distinguishing regular rotating and Kerr black holes thanks to quasi-periodic oscillations.\
With the upcoming results of the Event Horizon Telescope, studies of alternatives to Kerr black holes happen to be particularly timely. This study of a regular rotating black hole is not designed to make a case for the existence of such an object at the center of the Galaxy, especially because it is only an approximate solution of non-linear electrodynamics. But it comes within the scope of previous works on boson stars [@Vincent_et_al:2016] or hairy black holes [@VincentGourgoulhonHerdeiroRadu:2016] aiming at better understanding the data coming from the EHT. The intriguing similarities observed between boson stars and regular black holes spur to investigate other horizonless geometries in order to make out a common pattern.
Calculation details
===================
SageMath worksheets {#s:calculations}
-------------------
Computation of geometric quantities relative to the metric (\[metric\_improved\_Hayward\])-(\[e:M\_r\_Torres\]) have been performed by means of the free computer algebra system SageMath [@SageMath], thanks to its tensor calculus part (SageManifolds [@SageManifolds]). The corresponding worksheets are available at the following url’s:
- Curvature of the naively extended rotating Hayward metric \[Eq. (\[Kretschmann&Ricci\])\]:\
[<https://cocalc.com/projects/09367c7f-3a39-4079-9d4d-cd59ebdca289/files/Rotating_Hayward_metric_curvature.ipynb>]{}
- Curvature of the regular rotating Hayward metric (\[metric\_improved\_Hayward\]) extended to the region $r<0$ \[Fig. \[R\_plot\_HT\] & \[K\_plot\_HT\]\]:\
[<https://cocalc.com/projects/09367c7f-3a39-4079-9d4d-cd59ebdca289/files/rotating_Hayward_metric_ext.ipynb>]{}
- Null energy condition in the regular rotating Hayward metric (\[metric\_improved\_Hayward\]) extended to the region $r<0$ \[Fig. \[NEC\_a09\_b1\]\]:\
[<https://cocalc.com/share/09367c7f-3a39-4079-9d4d-cd59ebdca289/Locally_nonrotating_frames_and_NEC.ipynb?viewer=share>]{}
- Expressions of the energy, angular momentum and angular velocity of a test particle in the regular rotating Hayward metric (\[metric\_improved\_Hayward\]) extended to the region $r<0$; comparison with Toshmatov et al. [@Toshmatov_et_al:2017] and Bardeen et al. [@Bardeen:1972]:\
[<https://cocalc.com/share/09367c7f-3a39-4079-9d4d-cd59ebdca289/Comparison_of_E_L_and_Omega.ipynb?viewer=share>]{}
- Stable circular orbits in the regular rotating Hayward metric (\[metric\_improved\_Hayward\]) extended to the region $r<0$ \[Table \[ISCO\_a\_b\]\]:\
[<https://cocalc.com/projects/09367c7f-3a39-4079-9d4d-cd59ebdca289/files/Stable_circular_orbits.ipynb>]{}
Gyoto plugin {#Gyoto_plugin}
------------
A Gyoto metric class has been developed inside [<span style="font-variant:small-caps;">Gyoto</span>]{}to obtain all the ray-tracing images displayed in the present paper. It encodes Hayward’s regular rotating metric (10) extended to $r < 0$, and boils down to the metric of a Kerr black hole when $b = 0$. This metric class is part of the standard distribution of Gyoto, freely available at <http://gyoto.obspm.fr>.
References {#references .unnumbered}
==========
[^1]: Freely available at <http://gyoto.obspm.fr>
[^2]: It is not necessarily an ellipse in the mathematical sense.
|
---
author:
- 'Eugenio Aulisa[^1]'
- Giacomo Capodaglio
- Guoyi Ke
title: '[[Construction of h-refined continuous finite element spaces with arbitrary hanging node configurations and applications to multigrid algorithms]{}]{}[^2]'
---
[^1]: Broadway & Boston, Department of Mathematics and Statistics, Texas Tech University, Lubbock TX 79409, USA (), (), ().
[^2]: Submitted to the editors October 2, 2017.
|
---
author:
- 'Lazaros K. Gallos'
- Panos Argyrakis
title: 'Scale-free networks resistant to intentional attacks'
---
A large number of diverse systems in society, nature and technology can be described by the concept of a network [@AB; @DM]. In a network the form of inter-relations between the system parts determines many structural and dynamic properties of the system. One such property that has received considerable attention is the robustness of a network under intentional attack [@AJB00; @Cohen01]. In the course of such an attack nodes of the network are removed in decreasing order of their degree $k$ (number of connections to other nodes). This is considered to be the most harmful type of attack on a network, since the removal of the hubs results in the largest possible damage. In fact, this vulnerability of the networks to attacks has been described as their Achilles’ heel [@AJB00], because it is generally accepted that scale-free networks are easily destroyed under intentional attacks. This removal process has many and important implications, since depending on the application, it may describe the resilience of a network, such as the Internet, or the required number of vaccinations for immunization considerations, etc.
For a scale-free network, where the probability that a node has a given number of links decays as a power-law, it has been shown that the critical percentage $p_c$ of removed nodes that results in network disintegration is very low (of the order of a few percent) [@Cohen01; @Callaway00]. It is, thus, a well-established fact, supported by exact analytic results and simulations of attacks on model and real-life networks, that a scale-free network is very vulnerable to intentional attacks (where $p_c$ is close to 0), although the same network is extremely robust under random node failures (where $p_c\simeq 1$) [@Cohen00].
In this Letter we show that there exists a large class of networks that are usually found in nature and society and have already been characterized as scale-free, but nevertheless remain robust against removal of the most connected nodes. We first present the results for real-life networks and then introduce a modified version for the degree distribution of scale-free networks, for which our analytic and simulation treatment support these findings.
![\[fig1\] Percentage of nodes, $P_{\infty}(p)$, belonging to the largest cluster after removal of a fraction $p$ of nodes, as a function of $p$. The results correspond to intentional attacks on a number of different networks (shown in the plot). ](fig1.eps)
To demonstrate this issue we performed intentional attacks and random nodes removal to many different real-life networks. The critical point was calculated via two distinct methods. In the first method, during the removal process we monitored the value of the parameter $\kappa\equiv \langle k^2\rangle / \langle k\rangle$, where $\kappa$ is the connectivity parameter, and which has been shown to be a measure of the global network connectivity [@Cohen00; @Paul05]. A value of $\kappa<2$ signifies the disintegration of a network into isolated clusters. The second method was a direct measurement of the largest cluster size. The value of $p_c$ was identified as the one where this size assumes for the first time a value close to zero. The two methods coincide only when the network is ‘random’ and uncorrelated, in the sense that there is no inherent organization (or equivalently degree-degree correlations) in the network. In a clustered network, though, where these correlations are present, such as the IMDB actors network, the two methods give different results ($p_c=0.96$ with the first method, but $p_c=0.62$ with the second). Here, we considered the $p_c$ value derived by the largest cluster size calculation. The corresponding results for the fraction of nodes $P_{\infty}(p)$ that belong to the largest cluster of the network during an intentional attack are shown in Fig. \[fig1\], as a function of the percentage $p$ of removed nodes. While the size of the spanning cluster falls rapidly in most cases (similarly to a model random network) there are some systems where this size remains significant even for larger values of $p$.
Network Intentional Random
------------------------------------ ------------- --------
Configuration model ($\gamma=2.5$) 0.055 0.99
Online community 0.04 0.90
WWW (nd.edu) 0.10 0.99
IMDB actors collaboration 0.62 0.99
HEP-TH arxiv.org citations 0.68 0.98
: Critical fraction $p_c$ for intentional attacks and random removal on different networks.[]{data-label="table1"}
In Table \[table1\] we summarize the numerical results we obtained for the critical threshold $p_c$ of the networks presented in Fig. \[fig1\]. Although many of these systems behave in a similar way to the configuration model network, there is a number of networks, such as actors collaboration and science citations, where the intentional attack requires removal of a considerable portion of the network nodes, which is of the order of 65%. In order to outline the common feature of these networks, in Fig. \[fig2\] we present their degree distribution. These distributions have a flat or rising part at low-degree nodes and only after a threshold value the distribution decays as a power-law. We will show that this feature alone is enough to render a network resistant to attacks, while the resilience to random node removal remains intact, as we have verified with simulations that show that in this case the critical threshold remains the same as in simple scale-free networks, i.e. $p_c\to 1$.
![\[fig2\] Degree distributions for IMDB actors (filled symbols) and HEP citations (open symbols). The solid line represents a typical degree distribution (Eq. \[eq1\]) that we used as a model. Inset: Percentage of nodes belonging to the scale-free part of the distribution as a function of $\gamma$. From top to bottom: $k_c=$2, 3, 5, 10, and 50. ](fig2.eps)
The analytical considerations in the current work apply to simple and random networks, where connections between nodes are completely random and the network does not include any self-loops or multiple links between two nodes. The construction of a network for our numerical calculations follows a slightly modified version of the configuration model. We start with $N$ unconnected nodes and to each node $i$ we assign a degree $k_i$ from a given distribution $P(k)$, so that each node has initially a number of unconnected links. We randomly choose two of these unconnected links. If these links belong to the same node or they belong to two nodes that are already connected we ignore this selection and randomly choose two other unconnected links. Otherwise, we establish a connection between these two nodes. We repeat this procedure until all nodes have reached their pre-assigned connectivity. The use of this method leads to a simple network (i.e. one without self-loops and multiple links) where the degree distribution follows the pre-defined $P(k)$ function. We do not impose any upper cutoff for this distribution, so that correlations between degrees are similar to those of a network with completely random connections and no upper cutoff.
We consider networks whose degree distribution is uniform for all k values up to a threshold value $k_c$, while for larger k values it decays as a power law $k^{-\gamma}$, where $\gamma$ is a parameter with typical values in the range 2-4. The exact form of the distribution, also plotted in Fig. 2 for $k_c=50$ and $\gamma=2.5$, is $$\label{eq1}
P(k) = \left\{
\begin{array}{ll}
A & 1<k<k_c \\
B k^{-\gamma} & k\geq k_c
\end{array}
\right. \,,$$ where the values for the $A$ and $B$ constants are $$A = \frac{\gamma-1}{k_c\gamma-\gamma+1} \; , \; B=k_c^{\gamma} A \,.$$ These values are derived by the requirement that the distribution is properly normalized and continuous. The fraction of the nodes that belong to the scale-free part of the distribution (i.e. nodes with $k>k_c$) is shown in the inset of Fig. \[fig2\] for different values of $k_c$ as a function of $\gamma$. We can conclude that the network retains a substantial scale-free character in practically all cases studied (note also that even for pure scale-free networks a large portion of the nodes has $k=1$).
We calculate the critical threshold $p_c$ for such a network based on ideas introduced by Cohen et al [@Cohen01] and Dorogovtsev and Mendes [@Mendes01]. We employ a continuum approximation where the degree of a node is treated as a continuous variable. Nodes are removed according to their initial degree, so that the intentional attack finally results in the disruption of the network. We consider that the degrees of the nodes for the network at criticality, i.e. just before disruption, are given by the parameter $\tilde{k}$, with corresponding averages $$\langle \tilde{k} \rangle = \int_1^{\tilde{K}} k P(k) dk \;,\; \langle \tilde{k}^2 \rangle = \int_1^{\tilde{K}} k^2 P(k) dk \,.$$ The effect of an intentional attack is to remove all nodes of a network whose degree is larger than a cutoff value $\tilde{K}$, i.e. $\tilde{k} \in [1,\tilde{K}]$. This also implies that $p_c$ equals $$\label{EQresult}
p_c = 1-\int_1^{\tilde{K}} P(k) dk = \int_{\tilde{K}}^\infty P(k) dk \,,$$ where the first form is simpler to compute when $\tilde{K}<k_c$ and the second form when $\tilde{K}>k_c$. At the same time, removal of a node leads to removing all its links to other nodes. We consider random networks with no correlations in the nodes connections, which means that a removal of a node results in removal of random links with probability $$\label{EQtildep}
\tilde{p} = \frac{\int_{\tilde{K}}^\infty kP(k) dk}{\int_1^\infty kP(k) dk} = 1-\frac{\langle \tilde{k} \rangle}{\langle k \rangle} \,.$$
It has been shown [@Cohen00; @Paul05] that a random network loses its large-scale connectivity after the removal of a critical fraction $p_c$ of nodes, according to $$\label{EQpc}
p_c = 1 - \frac{1}{\kappa-1} \,,$$ where $\kappa\equiv \langle k^2\rangle / \langle k\rangle$. This equation has been shown in Ref. [@Cohen01] to be valid for removal of either nodes or links. As explained in detail there, an intentional attack leads to the equivalent of a scale-free network with upper cutoff $\tilde{K}$ where a random fraction $\tilde{p}$ of nodes has been removed. Because of the random character of the network all the links have the same probability of being removed, and this results to a new degree disribution $\tilde{P}(k)$. This fact is then used to prove Eq. (\[EQpc\]). We can then use this equation for the network resulting after the attack, by substituting a) $p_c$ with $\tilde{p}$ from Eq. \[EQtildep\] and b) $\kappa=\langle \tilde{k}^2 \rangle / \langle \tilde{k} \rangle$. After a few trivial steps Eq. \[EQpc\] becomes $$\label{EQfinal}
\langle \tilde{k}^2 \rangle - \langle \tilde{k} \rangle = \langle k \rangle \,.$$ This formula, which is exact, has been already proven in Refs. [@Cohen01; @Mendes01].
In order to use Eq. \[EQfinal\] we need to know whether the value of $\tilde{K}$ is larger or smaller than the threshold value of the distribution $k_c$, so we consider each case separately. Calculation of the integrals involved yields $$\langle \tilde{k} \rangle = \left\{
\begin{array}{ll}
\frac{A}{2}(\tilde{K}^2-1) & \tilde{K}<k_c \\
\frac{A}{2}(k_c^2-1)+ \frac{B}{\gamma-2}(k_c^{2-\gamma}-\tilde{K}^{2-\gamma}) & \tilde{K}>k_c
\end{array}
\right. \,,$$ and $$\langle \tilde{k}^2 \rangle = \left\{
\begin{array}{ll}
\frac{A}{3}(\tilde{K}^3-1) & \tilde{K}<k_c \\
\frac{A}{3}(k_c^3-1)+ \frac{B}{\gamma-3}(k_c^{3-\gamma}-\tilde{K}^{3-\gamma}) & \tilde{K}>k_c
\end{array}
\right. \,.$$
The average value of the initial degree distribution $P(k)$ (Eq. \[eq1\]) can be approximated with the assumption that $k_{\rm max}=\infty$. However, for low $\gamma$ values this assumption does not work well and for a finite-size network we should compute the integral up to the maximum value $k_{\rm max}=K$, which can be found from the relation $\int_{k_{\rm max}}^{\infty} P(k) = 1/N $, and is given in our case by $K= ((\gamma-1)/BN)^{1/(1-\gamma)}$. This results in a correction to the average value of the unperturbed distribution, which finally becomes $$\label{EQavk}
\langle k \rangle = \frac{A}{2}(k_c^2-1)+\frac{B}{\gamma-2}k_c^{2-\gamma} - \frac{B}{\gamma-2}\left( \frac{\gamma-1}{BN} \right)^{\frac{\gamma-2}{\gamma-1}} \,.$$ The third term is important only for finite-size networks and vanishes as $N\to\infty$.
Combining Eqs. (\[EQfinal\])-(\[EQavk\]) we get $$\label{EQK}
\begin{array}{ll}
2\tilde{K}^3-3\tilde{K}^2 = \frac{3\gamma k_c^2-4\gamma+8}{\gamma-2} - \frac{6 k_c^\gamma}{\gamma-2} \left( \frac{\gamma-1}{BN} \right)^{\frac{\gamma-2}{\gamma-1}} & \tilde{K}<k_c \\
\frac{k_c}{\gamma-3}\left( \frac{\tilde{K}}{k_c} \right)^{3-\gamma} -
\frac{1}{\gamma-2}\left( \frac{\tilde{K}}{k_c} \right)^{2-\gamma} = & \\
\frac{\gamma k_c}{3(\gamma-3)} - \frac{\gamma}{\gamma-2} + \frac{2}{3k_c^2} + \frac{k_c^{\gamma-2}}{\gamma-2} \left( \frac{\gamma-1}{BN} \right)^{\frac{\gamma-2}{\gamma-1}}
& \tilde{K}>k_c
\end{array}
\,.$$ Solving the above equations for $k_c=\tilde{K}$ we can find the $\gamma$ value for which the lowest degree $\tilde{K}$ of the nodes that need to be removed switches from $\tilde{K}>k_c$ to $\tilde{K}<k_c$. This $\gamma$ value is $$\label{EQtildeeqkc}
\gamma= \frac{2k_c^3-3k_c^2+4}{k^3-3k^2+2} \,.$$
We can now compute the value of $\tilde{K}$ from Eq. \[EQK\] and substitute it to Eq. \[EQresult\], which can also be written as $$\label{EQp}
p_c = \left\{
\begin{array}{ll}
\frac{k_c\gamma-\tilde{K}(\gamma-1)}{k_c\gamma-\gamma+1} & \tilde{K}<k_c \\
\frac{A}{3}(k_c^3-1)+ \frac{B}{\gamma-3}(k_c^{3-\gamma}-\tilde{K}^{3-\gamma}) & \tilde{K}>k_c
\end{array}
\right. \,.$$
![\[fig3\] Critical fraction $p_c$ of removed nodes for networks that undergo an intentional attack, as a function of the exponent $\gamma$. From top to bottom: $k_c=$ 50, 20, 10, 5, 4, 3, 2 and 1. Thick lines represent the infinite-size numerical solution of Eqs. \[EQK\] and \[EQp\], dashed lines represent the same solution for $N=10^6$, and filled symbols are simulation results on a network of size $N=10^6$ nodes. The bottom curve for $k_c=1$ is identical to the solution for pure scale-free networks (Ref. [@Cohen01]) The empty circles denote the solution of Eq. \[EQtildeeqkc\], where the value of $\tilde{K}$ switches from $\tilde{K}>k_c$ to $\tilde{K}<k_c$. ](fig3.eps)
The numerical solution of Eqs. \[EQK\] and \[EQp\] is shown in Fig. \[fig3\] as a function of $\gamma$ for different values of the threshold value $k_c$. In the same figure we also plot results of simulations on networks that were created with the configuration model. The size of these networks was $N=10^6$ nodes and their degree distribution obeys Eq. \[eq1\]. During the attack process we removed nodes in decreasing order of their degree and monitored continuously the value of $\kappa$ until it became less than 2. The percentage of the removed nodes up to that point corresponds to the critical value $p_c$. Note that this method does not have the problems described above, since it is applied to the randomized networks created via the configuration model. We verified this statement by also comparing to the results from the largest cluster size method.
Our results for $k_c=1$ coincide with the solution provided in Ref. [@Cohen01], as can also be seen numerically from Eqs. \[EQK\] and \[EQp\]. Comparison of the curves in Fig. \[fig3\] for $k_c>1$ to the intentional attack on regular scale-free networks shows a dramatic increase in the value of $p_c$, over the entire $\gamma$ range. As the threshold value $k_c$ increases, the stability of the network is further enhanced. Even for $k_c=2$ we observe a significant influence in the resilience of the network, where $p_c$ is usually more than two times larger than for the case of $k_c=1$. For $k_c=5$ the critical fraction is already above 30%, while when $k_c=10$ the value of $p_c$ lies in the range of 50%. For even larger values, such as $k_c=50$ which as can be seen in Fig. \[fig1\] is not unusual for real-world networks, the networks exhibit a remarkable resilience to intentional attacks, with a $p_c$ value close to 70%. Notice here, that the variation of $p_c$ for $\gamma>2.5$ is almost independent of $\gamma$. Thus, the important part of the distribution for robustness is the low-degree part and in our model networks its extent in the $k$-range. On the contrary, an exponent $\gamma>2.5$ for the decaying part does not really influence the attack result. As the value of $\gamma$ approaches 2, though, the decrease in the value of $p_c$ is quite sharp, with the infinite-size result $p_c=0$ for $\gamma=2$. For finite size networks this decrease is much slower and the critical threshold remains significant.
The stability of the solution with respect to the network size $N$ is shown in Fig. \[fig4\]. The value of $p_c$ is practically not influenced by $N$ when $\gamma$ is not close to $\gamma=2$, such as $\gamma=2.5$ or larger. For these smaller $\gamma$ values the critical threshold exhibits larger variations, such as in the case of $\gamma=2.1$ presented in the plot. Even in this case, though, when the network size becomes larger than a moderate size of $N\sim 10^4$ then the critical threshold remains practically constant.
![\[fig4\] Variation of the critical threshold $p_c$ with the network size $N$ for different values of $\gamma$ and $k_c$. The effect of the size on the threshold is in general not significant, with small exceptions for $\gamma$ values close to 2. ](fig4.eps)
The explanation behind the enhanced stability can be largely attributed to the increasing average number of connections per node when the $k_c$ value increases. Although a large value for $\langle k \rangle$ means obviously an enhanced robustness for the network, we also find that the network is resilient even for very small $k_c$ values. Indeed, in real networks it is usually not easy to clarify the exact behavior of the degree distribution at very small degrees, and a difference between $k_c=1$ and $k_c=2$ or 3 can easily be unnoticed.
These findings suggest a structure that is very robust against both random failures and targeted attacks. This optimization is desirable in most cases and the structure itself, which as we have seen emerges naturally in many instances, may be used to efficiently protect a network against most attacks. On the contrary, for immunization purposes, the existence of such networks may present difficulties for efficient strategies. Even if global knowledge of the entire network structure is available, the required number of vaccinations remains very high. In such a case, it is very important to acquire as accurate information on the network structure as possible, and especially for the low-degree part, because a simple power-law decay of the degree distribution over a large degree range does not guarantee efficient immunization, if at small values of the degree this power-law decay is not obeyed.
A study for networks that offer better resilience to attacks than simple scale-free networks has been performed in Ref. [@Paul04]. The authors find that the optimal network design for optimization against both random and intentional attack is one where all nodes have the same degree $k_1$, except for a ‘central’ node with a large degree $k_2\sim N^{2/3}$. That work, though, has a different scope than ours since the authors kept in all instances the average value $\langle k \rangle$ constant, while in our work this average value is modified as we modify $k_c$.
In summary, we have studied intentional attacks on networks whose distribution is uniform for low degrees $k$ and decays as a power law for larger $k$. Such a structure is very robust against both random and intentional attacks, and outlines the importance of the low-degree nodes in the connectivity of the structure. Although hubs connect a large part of the network, it is true that they will be unavoidably removed sooner or later, depending on the removal strategy. However, it seems that the form of the distribution at low degrees is equally or more important than the existence of the hubs and may render a network vulnerable or stable against intentional attacks.
This work was supported by a NEST/PATHFINDER project DYSONET/012911 of the EC, and also by a project of the Greek GGET in conjunction with ESF in the frame of international organizations
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abstract: 'Consider a measurable space with an atomless finite vector measure. This measure defines a mapping of the $\sigma$-field into an Euclidean space. According to the Lyapunov convexity theorem, the range of this mapping is a convex compactum. Similar ranges are also defined for measurable subsets of the space. Two subsets with the same vector measure may have different ranges. We investigate the question whether, among all the subsets having the same given vector measure, there always exists a set with the maximal range of the vector measure. The answer to this question is positive for two-dimensional vector measures and negative for higher dimensions. We use the existence of maximal ranges to strengthen the Dvoretzky-Wald-Wolfowitz purification theorem for the case of two measures.'
address:
- 'Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794-3600'
- 'Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794-3600'
author:
- Peng Dai
- 'Eugene A. Feinberg'
title: On Maximal Ranges of Vector Measures for Subsets and Purification of Transition Probabilities
---
[^1]
Introduction {#s1}
============
Let $\left({X},\mathcal{F}\right)$ be a measurable space and ${\mu}=\left(\mu_{1},...,\mu_{m}\right)$, $m=1,2,\ldots,$ be a finite atomless vector measure on it. We recall that a measure $\nu$ is called atomless if for each ${Z}\in\mathcal{F},$ such that $\nu\left({Z}\right)>0$, there exists ${Z}'\in\mathcal{F}$ such that ${Z}'\subset{Z}$ and $0<\nu\left({Z}'\right)<\nu\left({Z}\right)$. A vector measure ${\mu}=\left(\mu_{1},...,\mu_{m}\right)$, is called finite and atomless if each measure $\mu_{i}$, $i=1\dots m$, is finite and atomless. For each ${Y}\in \mathcal{F}$ consider the range ${R}_{{\mu}}\left({Y}\right)=\left\{
{\mu}\left({Z}\right):{Z}\in\mathcal{F},{Z}\subseteq{Y}\right\} $ of the vector measures of all its measurable subsets $Z$. According to the Lyapunov convexity theorem [@Liapounoff:1940], the sets ${R}_{{\mu}}\left({Y}\right)$ are convex compactums in $\mathbb{R}^m.$ For a review on this theorem and its applications see [@Olech].
In this paper we study whether for any $ p\in {R}_{{\mu}}(X)$, the set of all ranges $\{{R}_{{\mu}}\left({Y}\right): \mu(Y)= p,
Y\in{\mathcal{F}}\}$ contains a maximal element. In other words, is it always true that for any $ p\in {R}_{{\mu}}(X)$ there exists a subset $Y^*\in \mathcal{F}$ such that $ \mu(Y^*)= p$ and ${R}_{{\mu}}\left({Y^*}\right)\supseteq
{R}_{{\mu}}\left({Y}\right)$ for any $Y\in \mathcal{F}$ with $\mu(Y)= p$? We show that the answer is positive when $m=2$ and negative when $m>2$. Furthermore, for $m=2$, this maximal range can be constructed by a simple geometric transformation of ${R}_{{\mu}}\left({X}\right).$
In addition to the maximal range, it is possible to consider a minimal range. For $q\in R_\mu (X)$, the set $M^* \in \mathcal{F}$ with $\mu(M^*)=q$ has minimal range corresponding to $q$ if $R_\mu(M) \supseteq R_\mu\left( M^* \right)$ for any $M\in
\mathcal{F}$ with $\mu(M)=q$. We show that a set has a maximal range corresponding to $p$ if and only if its complement has a minimal range corresponding to $\mu(X)-p$. Therefore, minimal ranges also exist for dimension two and they may not exist for higher dimensions.
Lyapunov’s theorem is relevant to purification of transition probabilities discovered by Dvoretzky, Wald and Wolfowitz [@Dvoretzky:1950; @Dvoretzky:1951] for a finite image set. Let $(A,{\mathcal{A}})$ be a measurable space and $\pi$ be a transition probability from $X$ to $A$; that is, $\pi(B|x)$ is a measurable function on $(X,{\mathcal{F}})$ for any $B\in \mathcal{A}$ and $\pi(\cdot|x)$ is a probability measure on $(A,{\mathcal{A}})$ for any $x\in X$. According to Dvoretzky, Wald and Wolfowitz [@Dvoretzky:1950; @Dvoretzky:1951], two transition probabilities $\pi_1$ and $\pi_2$ are called strongly equivalent if $$\label{eq:SETEQ}
\int_{{X}}\pi_{1}\left(B|x\right)\mu_i\left(dx\right)=
\int_{{X}}\pi_{2}\left(B|x\right)\mu_i\left(dx\right),
\qquad i=1,\dots,m, \quad B\in {\mathcal{A}}.$$
A transition probability $\pi$ is called pure if each measure $\pi(\cdot|x)$ is concentrated at one point. A pure transition probability $\pi$ is defined by a measurable mapping $\varphi:X\to
A$ such that $\pi(B|x)=I\{\varphi(x)\in B\}$ for all $B\in
\mathcal{A}.$ According to the contemporary terminology, a transition probability can be purified if for it there exists a strongly equivalent pure transition probability.
For a finite set $A$, Dvoretzky, Wald and Wolfowitz [@Dvoretzky:1950; @Dvoretzky:1951] proved that any transition probability can be purified (we recall that $\mu$ is atomless). Edwards [@Edwards:1987 Theorem 4.5] generalized this result to the case of a countable set $A$. Khan and Rath [@Khan:2009 Theorem 2] gave another proof of this generalization. Loeb and Sun [@Loeb:2006 Example 2.7] constructed an elegant example when a transition probability cannot be purified for $m=2$, $X=[0,1]$, and $A=[-1,1]$. However, purification holds for a countable set of nonatomic, finite, signed Loeb measures, when $A$ is a complete separable metric space [@Loeb:2006 Corollary 2.6]. Note that for a countable (finite or infinite) set $A$, a transition probability $\pi$ can be purified if and only if there exists a partition $\{Z^a\in {\mathcal{F}}:a\in A\}$ of $X$, such that $$\label{eq:atom}
\int_{{X}}\pi\left(a|x\right)\mu \left(dx\right) = \mu (Z^a),\qquad a\in{A}.$$ where $\mu$ is a $m$-dimensional finite atomless vector measure. Since $\int_{{X}}\pi\left(a|x\right)\mu \left(dx\right)$ are vectors in $\mathbb{R}^m$, a natural question is: under what conditions for an arbitrary set of vectors $\left\{p^a:a\in
A\right\}$ there exists a partition $\{Z^a\in {\mathcal{F}}:a\in
A\}$ of $X$ such that $p^a =\mu (Z^a)$ for each $a\in{A}$. We use the theorem on maximal ranges proved in this paper to show that for $m=2$, such partition exists if and only if (i) $\sum_{a\in
A}p^a=\mu(X)$, and (ii) $\sum_{a\in B}p^a \in R_\mu (X)$ for any finite subset $B$ of $A$. For $m=2$, the Dvoretzky-Wald-Wolfowitz theorem for a countable set $A$ [@Edwards:1987; @Khan:2009] follows from the sufficient part of this statement.
We formulate the main results in the following section, prove the existence of maximal and minimal subsets for $m=2$ in Section \[s3\], provide counterexamples when $m>2$ in Section \[s4\], and describe geometric constructions of maximal ranges in Section \[s5\]. Section \[s6\] is devoted to the proof of the theorem on the existence of a partition.
Main results {#s2}
============
\[def:sets\]
Given a measurable subset ${Y}$ of the measurable space $\left({X},\mathcal{F}\right)$ with a vector measure ${\mu}$ and a vector ${p}\in{R}_{{\mu}}\left({Y}\right)$, we define
\(a) the set of all subsets of ${Y}$ with vector measure ${p}$, $$\mathcal{S}^{{p}}_\mu\left({Y}\right)=\left\{
{Z}\in\mathcal{F}_Y:
{\mu}\left({Z}\right)={p}\right\} ,$$ where ${\mathcal{
F}}_Y=\{Z\subseteq Y: Z\in{\mathcal{F}}\};$
\(b) the union of all the ranges of all subsets of ${Y}$ with the vector measure ${p}$, $${R}^{{p}}_{\mu} \left({Y}\right)=\bigcup \limits
_{{Z}\in\mathcal{S}^{{p}}_\mu\left({Y}\right)}{R}_\mu\left({Z}\right);$$
\(c) the intersection of the $R_\mu(Y)$ with its shift by a vector $- ({ \mu(Y) -p})$, $$Q^{{p}}_{\mu}\left({Y}\right)=\left({R}_{{\mu}}\left({Y}\right)-\left\{
{\mu}\left({Y}\right)-{p}\right\}
\right)\cap{R}_{{\mu}}\left({Y}\right),$$ where ${S}_1-{S}_2=\{ {q}-{r}:{q}\in {S}_1,{r}\in{S}_2\} $ for ${S}_1,{S}_2\in \mathbb{R}^m$. In particular, ${R}_{{\mu}}\left({Y}\right)-\{{r}\}$ is a parallel shift of ${R}_{{\mu}}\left({Y}\right)$ by $-{r}$.
\[def:maxset\] For a measurable subset ${Y}\in\mathcal{F}$, the set ${Z}^{*}\in\mathcal{S}^{{p}}_\mu\left({Y}\right)$, such that ${R}_\mu\left({Z}^{*}\right)={R}^{{p}}_\mu\left({Y}\right)$, is called the maximal subset of $Y$ with the measure $p$. The set $M^*\in\mathcal{S}^{{q}}_\mu\left({Y}\right)$, such that ${R}_\mu\left({M}^{*}\right) \subseteq {R}_\mu \left({M}\right)$ for any $M\in\mathcal{S}^{{q}}_\mu\left({Y}\right)$, is called the minimal subset of $Y$ with the measure $q$.
Our first result is the following theorem.
\[thm1\] For a two-dimensional finite atomless vector measure ${\mu}=\left(\mu_{1},\mu_{2}\right)$ and for a vector ${p}\in{R}_{{\mu}}\left({X}\right)$, there exists a maximal set ${Z}^{*}\in\mathcal S_\mu^{{p}}\left({X}\right)$ and, in addition, ${R}^{{p}}_\mu\left({X}\right)=
Q^{{p}}_{\mu}\left({X}\right).$
Theorem \[thm1\] immediately implies that the set ${R}^{{p}}_\mu\left({X}\right)$, which is the union of the ranges of ${\mu}$ on ${Z}$, for all ${Z}\in\mathcal{S}^{{p}}_\mu\left({X}\right)$, is a convex compactum. Furthermore, if ${R}_{{\mu}}\left({X}\right)$ and $ p$ are given, the set ${R}^{ p}_\mu(X)$ is defined by two simple geometric operations, a shift and an intersection, since $Q^{{p}}_{\mu}\left({X}\right)$ is defined by these operations. The following theorem links the notions maximal and minimal subsets.
\[thm1.5\] The set $Z^*$ is the maximal subset of $X$ with the measure $p$, if and only if $M^*=X \setminus Z^*$ is the minimal subset of $X$ with the measure $\mu(X) - p$.
We will use Theorem \[thm1\] to prove the following theorem that, as shown in Section \[s6\], strengthens the Dvoretzky-Wald-Wolfowitz purification theorem [@Edwards:1987; @Khan:2009] for the case $m=2.$
\[thm2\] Consider a measurable space $\left({X},\mathcal{F}\right)$ with a two-dimensional finite atomless vector measure ${\mu}$, a countable set $A$, and a set of two-dimensional vectors $\left\{p^a:a\in A\right\}$. A partition $\{Z^a\in
{\mathcal{F}}:a\in A\}$ of $X$, with $p^a =\mu (Z^a)$ for all $a\in{A}$, exists if and only if (i) $\sum_{a \in A} {p^a} =
\mu(X)$ and (ii) $\sum_{a \in B} {p^a} \in R_\mu (X)$ for any finite subset $B \subset A$.
Maximal and minimal subsets {#s3}
===========================
In this section, we prove Theorems \[thm1\] and \[thm1.5\]. Recall that for a set ${S}\subseteq \mathbb{R}^m$, its reflection across a point ${c}\in \mathbb{R}^m$ is ${\textrm{Ref}}({S},{c})=\{2{c}\}- {S}$. If ${S}=\{{s}\}$ is a singleton, we shall write ${\textrm{Ref}}({s},{c})$ instead of ${\textrm{Ref}}(\{{s}\},{c})$. A set ${S} \subseteq \mathbb{R}^{m}$ is called centrally symmetric if $ {\textrm{Ref}}\left({S},{c}\right) = {S} $ for some point ${c}\in \mathbb{R}^m$ called the center of ${S}$. Any bounded centrally symmetric set has only one center.
In this section we let ${Y}\in\mathcal{F}$ be any measurable subset of $X$. Lemmas \[lem:censym\]-\[lem:censymsub\] hold for any finite atomless vector measure ${\mu}=(\mu_1,\ldots,\mu_m)$ on $(X,{\mathcal F})$, where $m=1,2,\ldots\ $.
\[lem:censym\] The set ${{R}_\mu}({Y})$ is centrally symmetric with the center $\frac{1}{2}{\mu}\left({Y}\right)$.
The proof is straightforward, and this fact was observed by Lyapunov [@Liapounoff:1940 p. 476].
\[lem:translation\] The equality ${R}_{{\mu}}\left({Y}\right)-\left\{
{\mu}\left({Y}\right)-{p}\right\}
={\textrm{Ref}}\left({R}_{{\mu}}\left({Y}\right),\frac{1}{2}{p}\right)
$ holds for any ${p}\in{R}_{{\mu}}\left({Y}\right)$.
By Lemma \[lem:censym\], ${R}_{{\mu}}\left({Y}\right)={\textrm{Ref}}\left({R}_{{\mu}}\left({Y}\right)
,\frac{1}{2}{\mu}\left({Y}\right)\right)=\{{\mu}(Y)\}-{R}_{{\mu}}\left({Y}\right)$. Therefore, ${R}_{{\mu}}\left({Y}\right)-\left\{
{\mu}\left({Y}\right)-{p}\right\}=(\{ {\mu}\left({Y}\right)\}-{R}_{{\mu}}\left({Y}\right))- \left\{
{\mu}\left({Y}\right)-{p}\right\}=\{{{p}}\}-{R}_{{\mu}}\left({Y}\right)={\textrm{Ref}}\left({R}_{{\mu}}\left({Y}\right),\frac{1}{2}{p}\right).$
\[lem:censymsub\] Each of the sets ${R}_\mu^{{p}}\left({Y}\right)$ and $Q_\mu^{{p}}\left({Y}\right)$ is centrally symmetric with the center $\frac{1}{2}{p}$.
According to Lemma \[lem:censym\], each set $ Z \in
\mathcal{S}^{{p}}_\mu(Y)$ is centrally symmetric with the center $\frac{1}{2} {p}$. Therefore, ${R}_\mu^{{p}}\left({Y}\right)$, which is the union of all the sets in $Z \in \mathcal{S}^{{p}}_\mu(Y)$, is also centrally symmetric with the center $\frac12 {p}$.
In addition, $$\begin{aligned}
{\textrm{Ref}}\left(Q^{{p}}_{\mu}\left({Y}\right),\frac{1}{2}{p}\right)
&=&{\textrm{Ref}}\left(\left({R}_{{\mu}}\left({Y}\right)-\left\{
{\mu}\left({Y}\right)-{p}\right\}
\right)\cap{R}_{{\mu}}\left({Y}\right),\frac{1}{2}{p}\right)\\
&=&{\textrm{Ref}}\left({\textrm{Ref}}\left({R}_{{\mu}}\left({Y}\right),\frac{1}{2}{p}\right)\cap{R}_{{\mu}}\left({Y}\right),\frac{1}{2}{p}\right)\\
&=&{R}_{{\mu}}\left({Y}\right)\cap{\textrm{Ref}}\left({R}_{{\mu}}\left({Y}\right),\frac{1}{2}{p}\right) \\
&=&{R}_{{\mu}}\left({Y}\right)\cap\left({R}_{{\mu}}\left({Y}\right)-\left\{{\mu}\left({Y}\right)-{p}\right\}\right)
=Q^{{p}}_{\mu}\left({Y}\right),\end{aligned}$$ where the first and last equalities follow from the definition of $Q^{{p}}_{\mu},$ the second and second to the last equalities follow from Lemma \[lem:translation\]. The third equality holds because a reflection of intersections equals the intersection of reflections and, in addition, a reflection of a reflection across the same point is the original set.
Here we present the major ideas of the proof of Theorem \[thm1\]. First, as shown later, after Theorem \[thm1\] is proven for equivalent measures $\mu_1$ and $\mu_2$, this condition can be removed. So, we make the following assumption in Lemmas \[lem:l\_q\], \[lem:L\_q\]-\[lem:weakthm\].
\[ass:abscon\] The measures $\mu_{1}$ and $\mu_{2}$ are finite, atomless, and equivalent.
Under Assumption \[ass:abscon\], let $f\left(x\right)=\frac{d\mu_{2}}{d\mu_{1}}(x)$ be a Radon-Nikodym derivative of $\mu_2$ with respect to $\mu_1$. Since $f$ is defined $\mu_1$-[*a.e.*]{}, we fix any its version. We shall frequently use notations similar to $$\left\{ f(x)<l \right\} =\left\{ x\in{X} : f(x)<l\right\}.$$
Second, under Assumption \[ass:abscon\], for any $a\in\left[0,\mu_1\left(X\right)\right]$, we denote $$\label{eq:lq}
l_{a}=\min\left\{ l\ge0:\mu_{1}\left(\left\{ f\left(x\right)\le
l\right\} \right)\ge a\right\}.$$ Observe that the minimum in (\[eq:lq\]) exists. Indeed, let $$l_{a}=\inf\left\{l\ge0:\mu_{1}\left(\left\{ f\left(x\right)\le l\right\} \right)\ge a\right\}.$$ We need to show that $\mu_{1}\left(\left\{ f\left(x\right)\le l_a\right\} \right)\ge a.$ If $l_a=\infty$ then $\mu_1 (\{f(x)\le \infty\}) = \mu_1(X) \ge a$. Let $l_a<\infty$. Consider a sequence $l^k\searrow l_a$, $k=1,2,\dots\ $. Then $\cap_{k=1}^\infty \{f(x)\le l^k\}=\{f(x)\le l_a\}$ and $\{f(x)\le
l^k\}\supseteq \{f(x)\le l^{k+1}\}$. Therefore $\mu_1(\{f(x)\le
l_a\})=\lim_{k\to\infty}\mu_1(\{f(x)\le l^k\})\ge a.$
Third, it is possible to construct the maximal set $Z^*=X\setminus M^*$, where $M^*$ can be defined explicitly. Let ${X}^{l}=\left\{f\left(x\right)=l\right\}$. If $\mu_1(X^l)=0$ for all $l\in [0,\infty)$, then the definition of $M^*$ is easier and we explain it first. In this case, there exists $a^*\in [0,\mu_1(X)]$ such that $\mu_2\left(M^*\right)=\mu_2(X)-p_2$, and $M^*$ can be defined as $$\label{eq:M_star}
M^*=\{l_{a^*} \le y<l_{a^*+(\mu_1(X)-p_1)}\}.$$ In the general situation, the number $a^*$ can be chosen to satisfy $$\nonumber \mu_2(\{l_{a^*}<y<l_{a^*+(\mu_1(X)-p_1)}\})\le\mu_2(X)-p_2\le\mu_2(\{l_{a^*}\le y\le l_{a^*+(\mu_1(X)-p_1)}\}),$$ and $$\label{eq:gM_star}
M^*=\{l_{a^*}<y<l_{a^*+(\mu_1(X)-p_1)}\}\cup Z^1\cup Z^2,$$ for $Z^i$, $i=1,2$, being some measurable subsets of $X^{l^i}$, where $l^1=l_{a^*}$ and $l^2=l_{a^*+(\mu_1(X)-p_1)}$. In particular, if $\mu_1\left(X^{l^1}\right)=0$, let $Z^1 = X^{l^1}$, and if $\mu_1\left(X^{l^2}\right)=0$, let $Z^2 = \emptyset$. If $\mu_1\left(X^{l^1}\right)=\mu_1\left(X^{l^2}\right)=0$, then (\[eq:gM\_star\]) reduces to (\[eq:M\_star\]). It is easy to show that the number of $l$ such that $\mu_1\left(X^l\right)=0$ is countable, but we do not use this fact.
The proof of Theorem \[thm1\] is based on several lemmas.
\[lem:l\_q\] Under Assumption \[ass:abscon\], the numbers $l_a$, $a\in\left[0,\mu_1(X) \right]$ have the following properties: (a) $\mu_{1}\left(\left\{ f\left(x\right)<l_{a}\right\} \right)\le a\le\mu_{1}\left(\left\{ f\left(x\right)\le l_{a}\right\} \right)$; (b) $l_a \le l_{a'}$ if $a \le a'$.
For (a), by definition, $a\le\mu_{1}\left(\left\{ f\left(x\right)\le l_{a}\right\} \right)$. To prove that $\mu_{1}\left(\left\{ f\left(x\right)<l_{a}\right\} \right)\le a$, assume that $\mu_{1}\left(\left\{ f\left(x\right)<l_{a}\right\} \right) > a $. If $l_a=0$, then $\mu_{1}\left(\left\{ f\left(x\right)<l_{a}\right\} \right) = 0 > a $ which contradicts the assumption that $a \ge 0$. If $l_a>0$, let $\epsilon_k \searrow 0$, $k=1,2,\dots\ $, be a sequence of positive numbers. Then, for $k=1,2,\dots$, $$\mu_1\left( \left\{f(x)<l_a \right\}\right) = \mu_1\left( \left\{f(x) \le l_a-\epsilon_k \right\}\right) + \mu_1\left( \left\{l_a-\epsilon_k < f(x) < l_a \right\} \right) > a.$$ Let $D_k= \left\{ l_a-\epsilon_k < f(x) < l_a \right\}$. We observe that $D_{k+1} \subseteq D_k$ and $\cap_{u=1}^\infty D_k =\emptyset$. Therefore, $\lim_{k\rightarrow \infty} \mu_1\left(D_k\right) = 0$. Thus, $\mu_1\left( f(x) \le l_a-\epsilon \right)>a$ for some $\epsilon>0$ and this contradicts (\[eq:lq\]). These contradictions imply the lemma.
For (b), assume $l_a > l_{a'}$, then $\mu_{1}\left(\left\{ f\left(x\right)\le
l_{a'}\right\} \right)\ge a' \ge a$, and this contradicts (\[eq:lq\]).
Note that for each $l\in[0,\infty)$, there exists a subfamily $$\left\{W_b\left({X^l}\right) \in \mathcal{F}_{{X^l}}: b\in\left[0,\mu_1 \left({X^l}\right)\right]\right\}$$ such that ${W}_{b}\left({X^l}\right)\subset
{W}_{b'}\left({X^l}\right)\subseteq{X^l}$ whenever $b <
b'\le\mu_{1}\left({X^l}\right)$ and $\mu_{1}\left({W}_{b}\left({X^l}\right)\right)=b$ for each $b\in\left[0,\mu_1 \left({X^l}\right)\right]$. This fact follows from Ross [@Ross:2005 Theorem 2(LT3)]. We set $W_0\left(X^l\right)=\emptyset$. From now on we fix a family of ${W}_{b}\left({X^l}\right)$ for each $l \in [0,\infty)$.
\[def:L\_q\] Under Assumption \[ass:abscon\], for each $a$, define the following set $$\label{eq:L_a}
{L}_{a}=
\left\{ f\left(x\right)<l_{a}\right\} \cup{W}_{c}\left(X^{l_a}\right),$$ where $c={a-\mu_{1}\left(\left\{ f\left(x\right)<l_{a}\right\} \right)}$.
Note that property (a) in Lemma \[lem:l\_q\] guarantees that $c \in\left[0,\mu_1 \left({X^l}\right)\right]$.
\[lem:L\_q\] Under Assumption \[ass:abscon\], the sets $L_a\in\mathcal{F}$, $a\in\left[0,\mu_1(X) \right]$, have the following properties: (a) $\mu_{1}\left({L}_{a}\right)=a$; (b) $\left\{ f\left(x\right)<l_{a}\right\} \subseteq{L}_{a}\subseteq\left\{ f\left(x\right)\le l_{a}\right\} $; (c) ${L}_{a}\subset{L}_{a'}\subseteq{X}$ if $a < a'\le\mu_{1}\left({X}\right)$.
For (a), $\mu_{1}\left({L}_{a}\right) = \mu_{1}\left(\left\{ f\left(x\right)<l_{a}\right\} \right)+\mu_{1}\left(W_{c}\left(X^{l_{a}}\right)\right)
= \mu_{1}\left(\left\{ f\left(x\right)<l_{a}\right\} \right)+a-\mu_{1}\left(\left\{ f\left(x\right)<l_{a}\right\} \right)=a.$ Property (b) follows from ${W}_{c}\left(X^{l_a}\right) \subseteq X^{l_a}$ and (\[eq:L\_a\]). For (c), if $l_a=l_{a'}$ then $c<c'$ where $c'={a'-\mu_{1}\left(\left\{ f\left(x\right)<l_{a}\right\} \right)}$, and thus $$L_a = \left\{ f\left(x\right)<l_{a}\right\} \cup{W}_{c}\left(X^{l_a}\right) \subset \left\{ f\left(x\right)<l_{a}\right\} \cup{W}_{c'}\left(X^{l_a}\right) = L_{a'}.$$ If $l_a < l_{a'}$ then $L_a\subseteq \left\{ f\left(x\right)\le l_{a}\right\} \subset \left\{ f\left(x\right)<l_{a'}\right\} \subseteq L_{a'}$.
Let ${M}_{a,d}={L}_{a+d}\setminus{L}_{a}$. For each $d\in\left[0,\mu_{1}\left({X}\right)\right]$ and each $a\in\left[0,\mu_{1}\left({X}\right)-d\right]$, denote $g_{d}\left(a\right)=\mu_{2}\left({M}_{a,d}\right)=\int_{M_{a,d}}
f(x) \mu_1 (dx)$. The function $g_d(a)$ is non-decreasing and continuous in $a\in\left[0,\mu_{1}\left({X}\right)-d\right]$ for each $d \in \left[0,\mu_1(X)\right]$. However, we will not use the fact that it is non-decreasing. So we only prove the continuity in the following lemma.
\[lem:cont\] Under Assumption \[ass:abscon\], $g_{d}\left(a\right)$ is continuous in $a\in\left[0,\mu_{1}\left({X}\right)-d\right]$ for each $d \in
\left[0,\mu_1(X)\right]$.
We show that $\mu_2(L_{a+d})$ is continuous in $a \in
\left([0,\mu_1(X)-d\right]$ for any $d \in
\left[0,\mu_1(X)\right]$. Since $g_d(a)=L_{a+d}-L_a$, this implies the lemma. Consider a sequence $\{a^k: k=1,2,\ldots\},$ where $a_k\in [0,\mu_{1}({X})-d]$. Let $a^k\nearrow a.$ Then $L_{a^k+d}\subset L_{a^{k+1}+d}\subset B\subseteq L_{a+d},$ where $B=\cup_{k=1}^\infty L_{a^k+d}.$ Therefore, $\mu_i(L_{a^k+d})\nearrow\mu_i(B)$ and $\mu_i(L_{a+d})=\mu_i(B)+\mu_i(L_{a+d}\setminus B),$ $i=1,2.$ Since $\mu_1(L_{a^k+d})=a^k+d\nearrow a+d=\mu_1(L_{a+d}),$ we have $\mu_1(B)=a+d$ and $\mu_1(L_{a+d}\setminus B)=0.$ Since $\mu_1$ and $\mu_2$ are equivalent measures, $\mu_2(L_{a+d}\setminus B)=0$ and $\mu_2(L_{a^k+d})\nearrow \mu_2(L_{a+d}).$
Now let $a^k\searrow a.$ Then $L_{a^k+d}\supset
L_{a^{k+1}+d}\supset D\supseteq L_{a+d},$ where $D=\cap_{k=1}^\infty L_{a^k+d},$ and $\mu_i(L_{a^k+d})\searrow\mu_i(D)$, $\mu_i(L_{a+d})=\mu_i(D)-\mu_i(D\setminus L_{a+d})$ for $i=1,2.$ Similar to the previous case, $\mu_1(L_{a^k+d})=a^k+d\searrow
a+d=\mu_1(L_{a+d}),$ so $\mu_1(D)=a+d,$ $\mu_2(D\setminus
L_{a+d})= \mu_1(D\setminus L_{a+d})=0$, and $\mu_2(D)=\mu_2(L_{a+d}).$ Thus, $\mu_2(L_{a^k+d})\searrow \mu_2(L_{a+d}).$
Observe that a point ${q}\in \mathbb{R}^2$ is on the upper (lower) boundary of ${R}_{{\mu}}\left({X}\right)$, if and only if ${q}\in {R}_{{\mu}}\left({X}\right)$ and $q'_{2}\le q_{2}$ ($q'_{2}\ge q_{2}$) for any ${q}'\in{R}_{{\mu}}\left({X}\right)$ with $q'_{1}=q_{1}$.
\[lem:onboundary\]\[lem:infamily\] Under Assumption \[ass:abscon\], a point ${q}\in \mathbb{R}^2$ is on the lower boundary of ${R}_{{\mu}}\left({X}\right)$ if and only if $0\le q_1\le\mu_1(X)$ and $q_2={\mu}_2\left({L}_{q_{1}}\right)$, and it is on the upper boundary of ${R}_{{\mu}}\left({X}\right)$ if and only if $0\le
q_1\le\mu_1(X)$ and $q_2={\mu}_2\left({X}\setminus{L}_{\mu_{1}\left({X}\right)-q_{1}}\right)$.
For the lower boundary, let $q_2=\mu_2(L_{q_1}).$ Since $q_1=\mu_1(L_{q_1}),$ we have $q=\mu(L_q)\in {R}_\mu(X).$ For any set ${Z}\in\mathcal{F}$ with $\mu_{1}\left({Z}\right)=q_{1}$, define disjoint sets ${Z}_{1}={Z}\setminus{L}_{q_{1}}$, ${Z}_{2}={L}_{q_{1}}\setminus{Z}$, and $M={Z} \cap {L}_{q_{1}}$. Then $Z=Z_1\cup M$, ${L}_{q_{1}}=Z_2\cup M$, and $\mu_1(Z_1)=\mu_1(Z_2)$, since $\mu_1(Z)=q_1=\mu_1(L_{q_1})$. Furthermore, $Z_1\subseteq
\left\{f(x) \ge l_{q_{1}} \right\}$ and $Z_2\subseteq \left\{ f(x) \le l_{q_{1}} \right\}$. Therefore, $$\begin{aligned}
\mu_{2}\left({Z}_{1}\right) & = & \int_{{Z}_{1}}f\left(x\right)\mu_{1}\left(dx\right)
\ge l_{q_{1}}\int_{{Z}_{1}}\mu_{1}\left(dx\right) \\
&=&l_{q_{1}}\int_{{Z}_{2}}\mu_{1}\left(dx\right)
\ge \int_{{Z}_{2}}f\left(x\right)\mu_{1}\left(dx\right)=\mu_{2}\left({Z}_{2}\right).
\end{aligned}$$ So $\mu_{2}\left({Z}\right)= \mu_{2}\left({Z_1}\right) + \mu_{2}\left({M}\right) \ge \mu_{2} \left(Z_2\right) + \mu_{2} (M) = \mu_{2} \left({L}_{q_{1}}\right)$, and thus ${q}$ is on the lower boundary of ${R}_{{\mu}}\left({X}\right)$.
If ${q}$ is on the lower boundary of ${R}_{{\mu}}\left({X}\right)$, then $q_{2}\le\mu_{2}\left({L}_{q_{1}}\right)$. Since ${q}\in
{R}_{{\mu}}\left({X}\right)$, there exists ${Z}\in\mathcal{F}$ with ${\mu}\left({Z}\right)={q}$. But, as proved above, $\mu_{2}\left({Z}\right)\ge\mu_{2}\left({L}_{q_{1}}\right)$ for any ${Z}\in\mathcal{F}$ with $\mu_{1}\left({Z}\right)=q_{1}$. Thus $q_{2}\ge\mu_{2}\left({L}_{q_{1}}\right)$. Therefore, $q_{2}=\mu_{2}\left({L}_{q_{1}}\right)$.
The statement on the upper boundary follows from the symmetry of ${R}_{{\mu}}\left({X}\right)$.
\[lem:Z\_star\]
Under Assumption \[ass:abscon\], given ${u}=\left(u_{1},u_{2}\right)\in{R}_{{\mu}}\left({X}\right)$, there exists $a^{*}\in\left[0,\mu_{1}\left({X}\right)-u_{1}\right]$ such that ${\mu}\left({M}_{a^{*},u_{1}}\right)={u}$.
Since $\mu_1(L_0)=0$ and $\mu_1$ and $\mu_2$ are equivalent, $\mu_2(L_0)=0$. Therefore, $ g_{u_{1}}(0)=\mu_2\left(L_{u_1}
\setminus L_0 \right) = \mu_2\left( L_{u_1} \right)-\mu_2\left(
L_{0} \right) = \mu_2\left( L_{u_1} \right). $ Similarly, $\mu_2\left(X \setminus L_{\mu_1(X)}\right) = 0$, because $\mu_1\left(X \setminus L_{\mu_1(X)}\right) =\mu_1\left(X\right) -
\mu_1\left(L_{\mu_1(X)}\right) = 0$. Thus $$\begin{aligned}
g_{u_1}\left(\mu_1(X)-u_1\right) &=& \mu_2\left(L_{\mu_1(X)}
\setminus L_{\mu_1(X)-u_1} \right)
= \mu_2\left(L_{\mu_1(X)}\right) - \mu_2 \left(L_{\mu_1(X)-u_1} \right) \\
&=& \mu_2\left(X\right) - \mu_2\left(X \setminus L_{\mu_1(X)}\right) - \mu_2 \left(L_{\mu_1(X)-u_1} \right) \\
% = \mu_2\left(X\right) - \left(L_{\mu_1(X)-u_1} \right)
&=& \mu_2\left(X \setminus L_{\mu_1(X)-u_1} \right).\end{aligned}$$ According to Lemma \[lem:onboundary\], the point $\left(u_1,g_{u_{1}}\left(0\right)\right)$ is on the lower boundary of the range ${R}_{{\mu}}\left({X}\right)$ and the point $\left(u_1,g_{u_{1}}\left(\mu_{1}\left({X}\right)-u_{1}\right)\right)$ is on the upper boundary of the range ${R}_{{\mu}}\left({X}\right)$. So $u_{2}\in\left[g_{u_{1}}\left(0\right),g_{u_{1}}\left(\mu_{1}\left({X}\right)-u_{1}\right)\right]$. Since $g_{u_{1}}\left(a\right)$ is continuous in $a\in\left[0,\mu_{1}\left({X}\right)-u_{1}\right]$, there exists $a^{*}$, such that $g_{u_{1}}\left(a^{*}\right)=u_{2}$. That is, $\mu_{2}\left({M}_{a^{*},u_{1}}\right)=u_{2}$. By definition, $\mu_{1}\left({M}_{a^{*},u_{1}}\right)=u_{1}$. Therefore, ${\mu}\left({M}_{a^{*},u_{1}}\right)={u}$.
Note that Lemmas \[lem:l\_q\], and \[lem:L\_q\]-\[lem:Z\_star\] hold if one replaces everywhere the set $X$ with any measurable subset $Z \in
\mathcal{F}$. In particular, expressions such as $\{f(x)<l\}$ should be replaced with $\{x\in Z: f(x)<l\}$. We define explicitly $$\label{eq:l_a(Y)}
l_{a}(Z)=\min\left\{ l\ge0:\mu_{1}\left(\left\{ x \in Z: f\left(x\right)\le
l\right\} \right)\ge a\right\}.$$ Let $Z^l = \{ x \in Z : f(x)=l \}$. As follows from Ross [@Ross:2005 Theorem 2(LT3)], for each $l\in[0,\infty)$, there exists a family $$\left\{W_b\left({Z^l}\right) \in \mathcal{F}_{{Z^l}}: b\in\left[0,\mu_1 \left({Z^l}\right)\right]\right\}$$ such that ${W}_{b}\left({Z^l}\right)\subset
{W}_{b'}\left({Z^l}\right)\subseteq{Z^l}$ whenever $b <
b'\le\mu_{1}\left({Z^l}\right)$ and $\mu_{1}\left({W}_{b}\left({Z^l}\right)\right)=b$ for each $b\in\left[0,\mu_1 \left({Z^l}\right)\right]$. Again, we fix a family of ${W}_{b}\left({Z^l}\right)$ for each $l \in [0,\infty)$ and each $Z$, and define $${L}_{a}(Z)= \left\{ x \in Z : f\left(x\right)<l_{a}\right\}
\cup{W}_{c}\left(Z^{l_a}\right),$$ where $c=a-\mu_1 (\{x\in Z : f(x) <a\})$. Note that $l_a(X)=l_a$ and $L_a(X)=L_a$, for each $a\in[0,\mu_1(X)]$. In the following two lemmas and their proofs, for a given $u\in R_\mu(X)$, we consider a point $a^*\in\left[0,\mu_1(X)-u_1\right]$ with $\mu\left(
{M}_{a^{*},u_{1}} \right) = u$ and the set $Z=X\setminus
M_{a^*,u_1}$. The existence of $a^*$ is stated in Lemma \[lem:Z\_star\]. Later it will become clear that that $Z$ is the maximal subset with the vector measure $p=\mu(X)-u$ and $M_{a^*,u_1}$ is the the minimal subset with the vector measure $p=u$.
\[lem:connection\] Let Assumption \[ass:abscon\] hold. For a given $u=\left(u_1,u_2\right)\in R_\mu(X)$, consider $a^*\in\left[0,\mu_1(X)-u_1\right]$ with $\mu\left(
{M}_{a^{*},u_{1}} \right) = u$. Then $$\label{eq:newL}
\mu_2\left(L_{a}(Z)\right)=
\begin{cases}
\mu_2\left({L}_{a}\right), & \textrm{if } a\in\left[0,a^{*}\right];\\
\mu_2\left({L}_{a+u_{1}}\setminus{M}_{a^{*},u_{1}}\right), & \textrm{if }
a\in\left(a^{*},{\mu_1}\left({X}\right)-u_{1}\right].
\end{cases}$$
First, consider the case $a\in\left[0,a^{*}\right]$. We have $Z=X
\setminus M_{a^*,u_1} \supseteq L_{a^*} \supseteq L_{a} =
\left\{f(x)<l_a\right\} \cup W_c\left(X^{l_{a}}\right)$, where $c
= a - \mu_1 \left( \left\{ f(x) < l_a \right\} \right)$. In addition, $\left\{f(x)<l_a\right\} \cup W_c\left(X^{l_{a}}\right)
\subseteq \left\{f(x) \le l_a\right\} $. Therefore $$\begin{aligned}
&&\mu_1 \left( \left\{x \in Z: f(x) \le l_a \right\} \right)
=\mu_1(Z\cap\{f(x)\le l_a\})\\ &\ge& \mu_1 \left(Z\cap( \left\{
f(x)<l_a\right\} \cup W_c(X^{l_a})) \right) = \mu_1 \left(
\left\{f(x)<l_a\right\} \cup W_c(X^{l_a}) \right) = a.\end{aligned}$$ Thus, (\[eq:l\_a(Y)\]) implies that $l_a(Z) \le l_a$. On the other hand, take an arbitrary $l<l_a$. Since $Z \subseteq X$, $$\begin{aligned}
\mu_1 \left( \left\{x \in Z: f(x) \le l \right\} \right)\le \mu_1
\left( \left\{ f(x) \le l \right\} \right)<a.\end{aligned}$$ Therefore, $l_a(Z)>l$ for all $l<l_a.$ Thus, $l_a(Z)\ge l_a.$ We conclude that $l_a(Z)=l_a.$
Denote $A=\{ f(x)<l_a\}.$ Since $Z\supseteq L_a\supseteq A$ and $l_a(Z)=l_a$, then $\{x\in Z:\, f(x)<l_a(Z)\}= A.$ By definition, each of the sets $L_a$ and $L_a(Z)$ is the union of two disjoint subsets: $L_a= A\cup W_c(X^{l_a})$ and $L_a(Z)=A\cup
W_b(Z^{l_a})$ with $c = a - \mu_1(A) = b$. Thus, since $X^{l_a}\supseteq Z^{l_a}$ and $f(x)=l_a$ when $x\in X^{l_a}$, we have $\mu_2(W_c(X^{l_a}))= \mu_2(W_c(Z^{l_a}))=l_ac.$ So, $\mu_2(L_a(Z))=\mu_2(A)+\mu_2(W_c(Z^{l_a}))=\mu_2(A)+\mu_2(W_c(X^{l_a}))=\mu_2(L_a).$
Second, consider the case $a\in\left(a^{*},\mu_1(X) - u_1
\right]$. Observe that $M_{a^*,u_1} \subseteq L_{a^*+u_1} \subset
L_{a+u_1} = \left\{f(x)<l_{a+u_1}\right\} \cup
W_c\left(X^{l_{a+u_1}}\right)$, where $c = a + u_1 - \mu_1 \left(
\left\{ f(x) < l_{a+u_1} \right\} \right)$. In addition, $\left\{f(x)<l_{a+u_1}\right\} \cup W_c\left(X^{l_{a+u_1}}\right)
\subseteq \left\{f(x) \le l_{a+u_1}\right\} $. Therefore, $$\begin{aligned}
&&\mu_1 \left( \left\{x \in Z: f(x) \le l_{a+u_1} \right\} \right) \\
&=& \mu_1 \left( \left\{ f(x) \le l_{a+u_1} \right\} \cap Z \right)
= \mu_1 \left( \left\{ f(x) \le l_{a+u_1} \right\} \setminus
M_{a^*,u_1} \right) \\
&\ge& \mu_1 \left( \left\{f(x)<l_{a+u_1}\right\} \cup W_c \left(X^{l_{a+u_1}}\right) \setminus M_{a^*,u_1} \right)
= a + u_1 - u_1 = a.\end{aligned}$$ Thus, (\[eq:l\_a(Y)\]) implies that $l_a(Z) \le l_{a+u_1}$. On the other hand, note that $M_{a^*,u_1} \subseteq \left\{f(x) \le l_a(Z)\right\}$. Indeed, since $a>a^*$, we have $l_a(Z) \ge l_{a^*} (Z) = l_{a^*}$. Assume $l_{a^*} \le l_a(Z) < l_{a^*+u_1}$, then $\{ x \in Z: f(x) \le l_a (Z) \} = \{f(x)\le l_a(Z)\} \setminus M_{a^*,u_1} = L_{a^*}$, and $a = \mu_1 (\{ x \in Z: f(x) \le l_a (Z) \}) = \mu_1 (L_{a^*}) = a^*$, which is a contradiction. Therefore, $l_a(Z) \ge l_{a^*+u_1}$ and $M_{a^*,u_1} \subseteq \left\{f(x) \le l_{a^*+u_1}\right\} \subseteq \left\{f(x) \le l_a(Z)\right\}$. Thus, $\{x\notin Z:\, f(x)\le l_a(Z)\}=\{x\in M_{{a^*},u_1}:\, f(x)\le
l_a(Z)\}=M_{{a^*},u_1}$ and $$\mu_1 \left( \left\{ f(x) \le l_a(Z) \right\} \right) = \mu_1
\left( \left\{x \in Z: f(x) \le l_a(Z) \right\} \right) +
\mu_1\left(M_{a^*,u_1}\right) \ge a+u_1,$$ where the last step follows from property (b) in Lemma \[lem:L\_q\]. Formula (\[eq:lq\]) implies that $l_a(Z) \ge l_{a+u_1}$. Therefore, $l_a(Z) = l_{a+u_1}$.
Consider again the identity $L_{a+u_1}=\{f(x)<l_{a+u_1}\}\cup
W_c(X^{l_a+u_1}),$ where the sets in the union are disjoint and $c=\left( a+u_1 \right) -\mu_1(\{f(x)<l_{a+u_1}\})$. Similarly, $L_a(Z)=\{x\in Z:\,f(x)<l_{a+u_1}\}\cup
W_b(Z^{l_a+u_1}),$ where $b=a - \mu_1(\{ x \in Z : f(x) < l_{a}(Z) \})$. Since $l_a(Z) = l_{a+u_1}$ and $\{f(x)<l_{a+u_1}\}\supset M_{a^*,u_1}$, we have $b=a - \mu_1(\{ x \in Z : f(x) < l_{a+u_1} \}) = a - \mu_1(\{ f(x) < l_{a+u_1} \} \setminus M_{a^*,u_1}) = (a + u_1) - \mu_1(\{ f(x) < l_{a+u_1} \}) = c$. Thus, $$\begin{aligned}
\mu_2(L_{a}(Z)) &=& \mu_2(\{x \in Z: f(x) < l_a(Z)\}) + \mu_2( W_b (Z^{l_a(Z)})) \\
&=& \mu_2(\{ x \in Z : f(x) < l_{a+u_1} \}) + l_{a+u_1} \mu_1( W_b (Z^{l_{a+u_1}})) \\
&=& \mu_2(\{ f(x) < l_{a+u_1} \}) - \mu_2( M_{a^*,u_1} )+ l_{a+u_1} \mu_1( W_c (X^{l_{a+u_1}}))\\
%&&+ l_{a+u_1} \left( a - \mu_1 \left( \left\{x \in Z : f(x) < l_{a+u_1} \right\} \right) \right) l_{a+u_1} \right\}\right) - \mu_2\left( M_{a^*,u_1} \right) \\
%&&+ l_{a+u_1} \left( a + u_1 - \mu_1 \left( \left\{ f(x) < l_{a+u_1} \right\} \right) \right) \\
%&=& \mu_2\left(\left\{ f(x) < l_{a+u_1} \right\}\right) + \mu_2\left( W_c \left(X^{l_{a+u_1}}\right)\right) - \mu_2\left( M_{a^*,u_1} \right) \\
&=& \mu_2({L}_{a+u_1}) -\mu_2(M_{a^*,u_1} )
=\mu_2({L}_{a+u_1} \setminus M_{a^*,u_1} ),\end{aligned}$$ where the second equality holds because $l_a(Z)=l_{a+u_1}$, $f(x)=l_{a+u_1}$ for $x\in X^{l_{a+u_1}}$, and $Z^{l_{a+u_1}}\subseteq X^{l_{a+u_1}}$ (in fact $Z^{l_{a+u_1}}=X^{l_{a+u_1}}$, but we do not use this). The third equality holds because of $\{x\in
Z:\,f(x)<l_{a+u_1}\}=\{f(x)<l_{a+u_1}\}\setminus M_{a^*,u_1}$, $\{f(x)<l_{a+u_1}\}\supset M_{a^*,u_1}$, and $b=c.$ The fourth equality follows from $l_{a+u_1} \mu_1( W_c (X^{l_{a+u_1}})) = \mu_2 ( W_c (X^{l_{a+u_1}}))$.
\[lem:boundary\] Let Assumption \[ass:abscon\] hold. For a given $u=\left(u_1,u_2\right)\in R_\mu(X)$, consider $a^*\in\left[0,\mu_1(X)-u_1\right]$ with $\mu\left(
{M}_{a^{*},u_{1}} \right) = u$. Let ${q}=\left(q_1,q_2\right)$ be on the lower (upper) boundary of ${R}_\mu \left(Z\right)$. If $q_1\in
\left[0,a^*\right]$ ($q_1 \in \left[0, \mu_1(X)-u_1 -a^*
\right)$), then ${q}$ is on the lower (upper) boundary of ${R}_{{\mu}}\left({X}\right)$ and, if $q_1 \in \left(a^*,
\mu_1(X)-u_1 \right]$ ($q_1\in
\left[\mu_1(X)-u_1-a^*,\mu_1(X)-u_1\right]$), then ${r}={\mu}\left({X}\right)-{u}-{q}$ is on the upper (lower) boundary of ${R}_{{\mu}}\left({X}\right)$.
When ${q}$ is on the lower boundary of ${R}_\mu \left(Z\right)$, according to Lemma \[lem:onboundary\], $\mu_2(L_{q_1}(Z))=q_2$. If $q_{1}\in\left[0,a^{*}\right]$, then by Lemma \[lem:connection\], $\mu_2(L_{q_1})=\mu_2(L_{q_1}(Z))=q_2$, and Lemma \[lem:onboundary\] implies that ${q}$ is on the lower boundary of ${R}_{{\mu}}\left({X}\right)$.
If $q_{1}\in\left(a^{*},{\mu}\left({X}\right)-u_{1}\right]$, then for $r=\left(r_1,r_2\right)$ $$\begin{aligned}
r_2 & = & \mu_2\left({X}\right)- u_2 - q_2
= \mu_2\left({X}\right)-\mu_2\left({M}_{a^{*},u_{1}}\right)-\mu_2\left({L}_{q_{1}}(Z)\right)\\
&=& \mu_2\left({X}\right)-\left(\mu_2\left({M}_{a^{*},u_{1}}\right)+\mu_2\left({L}_{q_{1}+u_{1}}\setminus{M}_{a^{*},u_{1}}\right)\right)
= \mu_2\left({X}\right)-\mu_2\left({L}_{q_{1}+u_{1}}\right)\\
& = & \mu_2\left({X}\setminus{L}_{q_{1}+u_{1}}\right)=\mu_2\left({X}\setminus{L}_{\mu_{1}\left({X}\right)-r_{1}}\right),\end{aligned}$$ where the first and last equalities follow from the definition of $r$, the second equality follows from Lemma \[lem:Z\_star\], the third equality follows from Lemma \[lem:connection\], and the fourth equality follows from $q_1>a^*$. According to Lemma \[lem:onboundary\], $r$ is on the upper boundary of ${R}_{{\mu}}\left({X}\right)$.
If ${q}$ is on the upper boundary of ${R}_\mu \left(Z\right)$, the, because of symmetry, $r = \mu(X) - u -q$ is on the lower boundary of ${R}_\mu \left(Z\right)$. If $q_1\in \left[\mu_1(X)-u_1-a^*,\mu_1(X)-u_1\right]$, then $\mu_1(X)-u_1-q_1 \in \left[0,a^*\right]$. From the first part of the proof, $r = \mu(X)-u-q$ is on the lower boundary of ${R}_{{\mu}}\left({X}\right)$. If $q_1\in \left[0, \mu_1(X)-u_1 -a^* \right)$, then $\mu_1(X)-u_1-q_1 \in \left(a^*, \mu_1(X)-u_1 \right]$. Again, from the first part of the proof, $\mu(X)-u-(\mu(X)-u-q_1) = q_1$ is on the upper boundary of ${R}_{{\mu}}\left({X}\right)$.
\[lem:weakthm\] Under Assumption \[ass:abscon\], for any vector ${p}\in{R}_{{\mu}}\left({X}\right)$, there exists a maximal set ${Z}^{*}\in\mathcal S_{{p}}\left({X}\right)$ and, in addition, ${R}^{{p}}_\mu\left({X}\right)=Q^{{p}}_{\mu}\left({X}\right).$
For $u=\mu(X)-p$, consider $a^*$ defined in Lemma \[lem:Z\_star\]. For $Z^*=X\setminus {M}_{a^{*},\mu_1(X)-p_{1}}$, the following three statements are true: (1) ${R}_{{\mu}}\left({Z}^{*}\right)\subseteq{R}^{{p}}_\mu\left({X}\right)$; (2) ${R}^{{p}}_\mu\left({X}\right)\subseteq Q^{{p}}_{\mu}\left({X}\right)$; (3) $Q^{{p}}_{\mu}\left({X}\right)\subseteq{R}_{{\mu}}\left({Z}^{*}\right)$.
For (1), $\mu\left(Z^{*}\right)=\mu\left({X}\setminus{M}_{a^{*},\mu_1(X)-p_{1}}\right)=\mu(X) - \mu\left({M}_{a^{*},\mu_1(X)-p_{1}}\right) = \mu(X) - \left(\mu(X) - p \right) = p$, where the second to the last equality follows from Lemma \[lem:Z\_star\]. Thus, $R_\mu \left(Z^*\right) = R_\mu ^p (X)$.
For (2), assume that there exists a vector ${q}\in{R}^{{p}}_\mu\left({X}\right)$ such that ${q}\notin Q^{{p}}_{\mu}\left({X}\right)$. Then Definition \[def:sets\] implies that either ${q}\notin{R}_{{\mu}}\left({X}\right)$ or ${q}\notin\left({R}_{{\mu}}\left({X}\right)-\left\{ {\mu}\left({X}\right)-{p}\right\} \right)$. However $q\in R_\mu ^p (X) \subseteq R_\mu (X)$. Therefore, ${q}\notin{R}_{{\mu}}\left({X}\right)-\left\{ {\mu}\left({X}\right)-{p}\right\} $, which is equivalent to $p-q \notin \left\{\mu(X) \right\} - R_\mu(X) = R_\mu(X)$, where the equality follows from Lemma \[lem:censym\]. Since ${R}^{{p}}_\mu(X) \subseteq R_\mu(X)$, we have $p-q \notin R_\mu^p(X)$. By Lemma \[lem:censymsub\], $R_\mu^p(X)$ is centrally symmetric with the center $\frac{p}{2}$. Therefore $q\notin R_\mu^p(X)$. The above contradiction implies (2).
For (3), assume ${q}\in Q^{{p}}_{\mu}\left({X}\right)$, but ${q}\notin{R}_{{\mu}}\left({Z}^{*}\right)$. By Lyapunov’s theorem $R_\mu\left(Z^*\right)$ is a convex compactum. Let ${q}^{u}=\left(q_{1},q^u_{2}\right)$ and ${q}^{l}=\left(q_{1},q^l_{2}\right)$ be the intersection points of the vertical line $\mu_{1}=q_{1}$ and the upper and lower boundaries of ${R}_{{\mu}}\left({Z}^{*}\right)$ respectively. Then one of the following must be true: $q_{2}>q^u_{2}$ or $q_{2}<q^l_{2}$. Without loss of generosity, we consider the former case. Since $q_{u}$ is on the upper boundary of ${R}_{{\mu}}\left({Z}^{*}\right)$, according to Lemma \[lem:boundary\], one of the following is true: (a) ${q}^{u}$ is on the upper boundary of ${R}_{{\mu}}\left({X}\right)$ or (b) $r={p}-{q}^{u}$ is on the lower boundary of ${R}_{{\mu}}\left({X}\right)$. For (a), $q_{2}>q^u_{2}$ implies ${q}\notin{R}_{{\mu}}\left({X}\right)$. Thus ${q}\notin Q^{{p}}_{\mu}\left({X}\right)$. This contradicts our assumption. For (b), we let $r'={p}-{q}$. Obviously, $r'_1=r_1$ and $r'_2<r_2$. This implies that $r'$ is below the lower boundary point $r$. Thus, $r' \notin{R}_{{\mu}}\left({X}\right)$ and $r' \notin Q^{{p}}_{\mu}\left({X}\right)$. But according to Lemma \[lem:censymsub\], this means ${q}\notin Q^{{p}}_{\mu}\left({X}\right)$, which contradicts to our assumption. Statement (1)-(3) imply the lemma.
Let $D$ be a two-by-two invertible matrix with positive entries, and $A\subseteq \mathbb{R}^2$. We denote by $A D$ the set $ \left\{ p D: p\in A \right\}$. For a vector measure $\mu = \left(\mu_1,\mu_2\right)$, let $\nu=\mu D$ be the vector measure $\left(\nu_1,\nu_2\right) = \left(D_{11}\mu_1 + D_{21} \mu_2 , D_{12}\mu_1 + D_{22} \mu_2 \right)$. Then the measure $\nu_1$ and $\nu_2$ are equivalent.
\[lem:mapping\] (a) $R_\mu(Y)D=R_\nu(Y)$ for all $Y \in \mathcal{F}$; (b) $R_\mu^p(X) D = R_\nu^{pD}(X)$ for all $p\in R_\mu(X)$; (c) $Q_\mu^p(X) D = Q_\nu^{pD}(X)$ for all $p\in R_\mu(X)$.
\(a) For any point $q \in R_\nu(Y) $, there exists a set $Z\in \mathcal{F}_Y$ such that $\nu(Z)=q$. Since $\mu(Z)=q D^{-1}$ and $q D^{-1} \in R_\mu(Y)$, we have $q \in R_\mu(Y) D $. For any point $q \in R_\mu(Y) D $, we have $q D^{-1} \in R_\mu(Y)$. Thus there exists a set $Z\in \mathcal{F}_Y$ such that $\mu(Z)=q D^{-1}$, and $\nu(Z)=q$. Therefore, $\nu(Z) \in R_\nu(Y)$.
\(b) For any point $q \in R_\nu^{p D}(X)$, there exist sets $Y\in\mathcal{F}$ and $Z\in\mathcal{F}_Y$ such that $\nu(Y) = p D$ and $\nu(Z) = q$. So $\mu(Y) = p$ and $\mu(Z) = q D^{-1}$. Thus, $q D^{-1} \in R_\mu^p(X)$ and therefore, $q \in R_\mu^p(X) D$. For any point $q \in R_\mu^p(X)D$, we have $q D^{-1} \in R_\mu^p(X)$. So there exist sets $Y\in\mathcal{F}$ and $Z\in\mathcal{F}_Y$ such that $\mu(Y) = p$ and $\mu(Z) = q D^{-1}$, and consequently $\nu(Y) = p D$ and $\nu(Z) = q$. Thus $q \in R_\mu^{p D}(X)$.
\(c) According to Definition \[def:sets\], $Q_\mu^p(X) D = (R_\mu(X) D-\{\mu (X)D-{p}D\})\cap{R}_{{\mu}}(X)D = (R_\nu(X)-\{\nu (X)-{p}D\})\cap{R}_{{\nu}}(X) = Q_\nu^{pD}(X)$.
According to Lemma \[lem:weakthm\], Theorem \[thm1\] holds under Assumption \[ass:abscon\] that states that $\mu_1$ and $\mu_2$ are equivalent. If $\mu_1$ and $\mu_2$ are not equivalent, consider $\nu=\mu D$. Since $\nu_1$ and $\nu_2$ are equivalent, $Q^p_\mu (X) = Q^{pD}_{\nu}(X) D^{-1} = R_\nu^{pD}(X) D^{-1} = R_\mu^p(X)$, where the first equality and the last equality is by Lemma \[lem:mapping\], and the second equality is due to Lemma \[lem:weakthm\]. Furthermore, according to Lemma \[lem:weakthm\], there exists a maximal set $Z^*$, such that $R_\nu\left(Z^*\right) = R_\nu^{pD}(X)$. Therefore, $R_\mu\left(Z^*\right) = R_\nu\left(Z^*\right) D^{-1} = R^{pD}_{\nu}(X) D^{-1} = R^p_\mu (X)$.
Now consider Theorem \[thm1.5\]. For $A\subseteq \mathbb{R}^m$ and $b\in\mathbb{R}^m$, let $A+b = \left\{a+b:a\in A \right\}$ and $A-b = A+(-b)$. Observe that $A-b=A-\{b\}$. Recall that $A\oplus B
= \bigcup_{b\in B} \left( A + b \right) $ is called the Minkowski addition, and $A \ominus B = \bigcap_{b\in B} \left(A-b\right)$ is called the Minkowski subtraction, where $A,B \subseteq
\mathbb{R}^m$.
\[lem:seesaw\] Let $A_1,A_2,B_1,B_2 \subseteq \mathbb{R}^2$ be convex and compact sets such that $A_1 \oplus B_1 = A_2 \oplus B_2 $ and $B_1 \subseteq B_2$. Then $A_2 \subseteq A_1$.
According to [@Schneider:2008 Lemma 3.1.8], if $A,B\subseteq \mathbb{R}^2$ are convex and compact sets then $(A \oplus B) \ominus B = A$. Thus if $a\in A_2$, then $a \in \left( A_2 \oplus B_2 \right) \ominus B_2$, and consequently $a \in \left( A_1 \oplus B_1 \right) \ominus B_2$. So $a \in \left( A_1 \oplus B_1 \right)-b $, for any $b\in B_2$. Since $B_1 \subseteq B_2$, we have $a \in \left( A_1 \oplus B_1 \right)-b $, for any $b\in B_1$, and thus $a \in \left( A_1 \oplus B_1 \right) \ominus B_1 = A_1$.
Now let $Z^*$ be the maximal set with the measure $p$, then $\mu\left(X\setminus Z^*\right) = \mu(X) - p $. Consider any set $M$, such that $\mu(M)=\mu(X)-p$. Obviously, $R_\mu(M) \oplus R_\mu(X \setminus M) = R_\mu\left(X \setminus Z^*\right) \oplus R_\mu\left( Z^* \right) = R_\mu(X)$. In addition, $R_\mu(X \setminus M) \subseteq R_\mu\left( Z^* \right)$ by definition. Thus according to Lemma \[lem:seesaw\], $R_\mu\left(X \setminus Z^*\right) \subseteq R_\mu(M)$.
Similarly, let $M^*=X \setminus Z^*$ be the minimal set with the measure $\mu(X)-p$, then $\mu\left(Z^*\right) = p $. Consider any set $Z$, such that $\mu(Z)=p$. Obviously, $R_\mu(Z) \oplus R_\mu{(X \setminus Z)} = R_\mu\left(Z^*\right) \oplus R_\mu\left( X \setminus Z^* \right) = R_\mu(X)$. In addition, $R_\mu(X \setminus Z^*) = R_\mu(M^*) \subseteq R_\mu\left(X \setminus Z \right)$ by definition. Thus according to Lemma \[lem:seesaw\], $R_\mu\left(Z\right) \subseteq R_\mu(Z^*)$.
With Theorem \[thm1.5\], the existence of the minimal subset ${M}^{*}\in\mathcal S_\mu^{{q}}\left({X}\right)$ immediately follows from the existence of the maximal subset ${Z}^{*}\in\mathcal S_\mu^{\mu(X) - q}\left({X}\right)$. Furthermore, ${R}_\mu\left({M}^{*}\right)= \left( {R}_\mu\left({M}^{*}\right) \oplus {R}_\mu\left({Z}^{*}\right) \right)\ominus {R}_\mu\left({Z}^{*}\right) = R_\mu(X) \ominus Q^{\mu(X) - {q}}_{\mu}\left({X}\right)$.
For a two-dimensional finite atomless vector measure ${\mu}=\left(\mu_{1},\mu_{2}\right)$ and for a vector ${q}\in{R}_{{\mu}}\left({X}\right)$, there exists a minimal set ${M}^{*}\in\mathcal S_\mu^{{q}}\left({X}\right)$. In addition, ${R}_\mu\left({M}^{*}\right)=
R_\mu(X) \ominus Q^{\mu(X) - {q}}_{\mu}\left({X}\right)$.
Counterexample for 3D measures {#s4}
==============================
In this section, we present an example of a measurable space $(X,\mathcal{F})$ endowed with a three-dimensional atomless finite measure $\nu=\left(\nu_{1},\nu_{2},\nu_{3}\right)$ and a vector $p\in R_{\nu}(X)$ such that a maximal subset of $X$ with the measure $p$ does not exist. Theorem \[thm1.5\] implies that the minimum set does not exist either in this example.
Recall that, with respect to a measure $\mu$, set $A$ and $B$ are said to be equal up to null sets (denoted by $A\simeq B$) if $\mu\left(A\setminus B\right)=\mu\left(B\setminus A\right)=0$. Also recall that $X^l=\left\{f(x)=l\right\}$.
\[lem:unique\] Let $\mu=\left(\mu_1,\mu_2\right)$ satisfy Assumption \[ass:abscon\] and let $Y \in \mathcal{F}$. If $\mu_{1}\left\{
X^{l_{\mu_{1}(Y)}}\right\} =0$ and $\mu_{2}(Y)=\mu_{2}\left(L_{\mu_{1}(Y)}\right)$, then $Y\simeq
L_{\mu_{1}(Y)}$.
Assume that $Y\simeq L_{\mu_1(Y)}$ does not hold. We define three disjoint sets ${Z}_{1}={Y}\setminus{L}_{\mu_1(Y)}$, ${Z}_{2}={L}_{\mu_1(Y)}\setminus{Y}$, and $M={Y} \cap {L}_{\mu_1(Y)}$. Observe that $Y=Z_1\cup M$ and ${L}_{\mu_1(Y)}=Z_2\cup M$. These equalities and $\mu_1(Y)=\mu_1(L_{\mu_1(Y)})$ imply $\mu_1(Z_1)=\mu_1(Z_2)$. Furthermore, $Z_1\subseteq \left\{f(x) \ge l_{\mu_1(Y)} \right\}$ and $Z_2\subseteq \left\{ f(x) < l_{\mu_1(Y)} \right\}$, because according to (\[eq:L\_a\]), ${L}_{\mu_1(Y)}=
\left\{ f\left(x\right)<l_{\mu_1(Y)}\right\}$ when $\mu_{1}\left\{ X^{l_{\mu_{1}(Y)}}\right\} =0$. Therefore, $$\begin{aligned}
\mu_{2}\left({Z}_{1}\right) &=& \int_{{Z}_{1}}f\left(x\right)\mu_{1}\left(dx\right)
\ge l_{\mu_1(Y)}\int_{{Z}_{1}}\mu_{1}\left(dx\right) \\
&=& l_{\mu_1(Y)}\int_{{Z}_{2}}\mu_{1}\left(dx\right)
> \int_{{Z}_{2}}f\left(x\right)\mu_{1}\left(dx\right)=\mu_{2}\left({Z}_{2}\right).\end{aligned}$$ So $\mu_{2}\left({Y}\right)= \mu_{2}\left({Z_1}\right) + \mu_{2}\left({M}\right) > \mu_{2} \left(Z_2\right) + \mu_{2} (M) = \mu_{2} \left({L}_{\mu_1(Y)}\right)$. This contradiction implies the proposition.
\[exa:counter\] Let $X=[0,1]$ and $\mathcal{F}$ is the Borel $\sigma$-field. Consider the three-dimensional vector measure ${\nu}\left(dx\right)=\left(\nu_{1},\nu_{2},\nu_{3}\right)\left(dx\right)=\left(1,2x,\rho\left(x\right)\right)dx$, where $$\rho \left(x\right)=\left\{ \begin{array}{ll}
4x, & \textrm{if } x\in\left[0,\frac{1}{2}\right);\\
4x-2, & \textrm{if } x\in\left[\frac{1}{2},1\right].\end{array}\right.$$ Consider the points ${p}=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$, ${q^1}=\left(\frac{1}{4},\frac{1}{16},\frac{1}{8}\right)$, and ${q^2}=\left(\frac{1}{4},\frac{5}{32},\frac{1}{16}\right)$. It is easy to show that $q^1,q^2\in R_\nu^p(X)$. Indeed let $Z^1=\left[0,\frac{1}{4}\right) \cup\left[\frac{3}{4},1\right]$, $Z^2=\left[0,\frac{1}{8}\right)\cup\left[\frac{3}{8},\frac{5}{8}\right)\cup\left[\frac{7}{8},1\right]$, $W^1=\left[0,\frac{1}{4}\right)\subseteq Z^1$, and $W^2=\left[0,\frac{1}{8}\right)\cup \left[\frac{1}{2},\frac{5}{8}\right)\subseteq Z^2$, and we have $\nu\left(Z^1\right)=\nu\left(Z^2\right)={p}$, $\nu\left(W^1\right)=q^1$, and $\nu\left(W^2\right)=q^2$. Since $Z^1$ and $Z^2$ are not equal up to a null set, Proposition \[pro:Z\] implies that there doesn’t exist a set $Z$ such that $\nu(Z)=p$ and $q^1,q^2\in R_\mu(Z)$.
\[pro:Z\] Consider the sets $X$, $Z^1$, $Z^2$, the measure $\nu$ and vectors $p$, $q^1$, $q^2$ from Example \[exa:counter\]. Let $Z \in \mathcal{S}_\nu^p(X)$. For each $i=1,2$, if $q^i \in R_\nu(Z)$, then $Z \simeq Z^i$.
Let $i =1$. Since $q^1 \in R_\nu(Z)$, there exists a set ${W^1}
\in \mathcal{F}_Z$ such that $\nu({W^1}) = q^1$. Define two-dimensional vector measure $\mu=\left(\mu_1,\mu_2\right)=\left(\nu_1,\nu_2\right)$. Then $\mu({W^1}) = \left(\frac{1}{4},\frac{1}{16}\right)$. Observe that, according to (\[eq:lq\]) and (\[eq:L\_a\]), $l_{\mu_1({W^1})}=l_{\frac{1}{4}}=\frac{1}{2}$ and $L_{\mu_1({W^1})}=L_{\frac{1}{4}}=\left[0,\frac{1}{4}\right)$. In addition, $\mu_1\left(X^{l_{\mu_1({W^1})}}\right)=0$ and $\mu_{2}({W^1})=\frac{1}{16}=\mu_{2}\left(L_{\mu_{1}({W^1})}\right)$. Therefore, according to Proposition \[lem:unique\], ${W^1}
\simeq L_{\mu_{1}({W^1})}=\left[0,\frac{1}{4}\right)$. On the other hand, let $Y={W^1}\cup (X\setminus Z)$. Since ${W^1}
\subseteq Z$, $\nu(Y)=\nu({W^1})+(\nu(X)-\nu(Z))=q^1+(\nu(X)-p)=\left(\frac{3}{4},\frac{9}{16},\frac{5}{8}\right)$, and thus, $\mu(Y)=\left(\frac{3}{4},\frac{9}{16}\right)$. Observe that, according to (\[eq:lq\]) and (\[eq:L\_a\]), $l_{\mu_1(Y)}=l_{\frac{3}{4}}=\frac{3}{2}$ and $L_{\mu_1(Y)}=L_{\frac{3}{4}}=\left[0,\frac{3}{4}\right)$. In addition, $\mu_1\left(X^{l_{\mu_1(Y)}}\right)=0$ and $\mu_{2}(Y)=\frac{9}{16}=\mu_{2}\left(L_{\mu_{1}(Y)}\right)$. Therefore, according to Proposition \[lem:unique\], $Y \simeq
L_{\mu_{1}(Y)}=L_{\frac{3}{4}}$. Above observations imply that $Z
= {W^1} \cup (X\setminus Y) \simeq L_{\frac{1}{4}} \cup
\left(X\setminus L_{\frac{3}{4}}\right) =
\left[0,\frac{1}{4}\right) \cup \left(\left[0,1\right]\setminus
\left[0,\frac{3}{4}\right)\right) = \left[0,\frac{1}{4}\right)
\cup \left[\frac{3}{4},1\right]=Z^1$.
Let $i =2$. Since $q^2 \in R_\nu(Z)$, there exists a set ${W^2}
\in \mathcal{F}_Z$ such that $\nu({W^2}) = q^2$. Define two-dimensional vector measure $\mu=\left(\mu_1,\mu_2\right)=\left(\nu_1,\nu_3\right)$. Then $\mu({W^2}) = \left(\frac{1}{4},\frac{1}{16}\right)$. Observe that, according to (\[eq:lq\]) and (\[eq:L\_a\]), $l_{\mu_1({W^2})}=l_{\frac{1}{4}}=\frac{1}{2}$ and $L_{\mu_1({W^2})}=L_{\frac{1}{4}}=\left[0,\frac{1}{8}\right) \cup
\left[\frac{1}{2},\frac{5}{8}\right)$. In addition, $\mu_1\left(X^{l_{\mu_1({W^2})}}\right)=0$ and $\mu_{2}({W^2})=\frac{1}{16}=\mu_{2}\left(L_{\mu_{1}({W^2})}\right)$. Therefore, according to Proposition \[lem:unique\], ${W^2}
\simeq L_{\mu_{1}({W^2})}=\left[0,\frac{1}{8}\right) \cup
\left[\frac{1}{2},\frac{5}{8}\right)$. On the other hand, let $Y={W^2} \cup (X\setminus Z)$. Since ${W^2} \subseteq Z$, $\nu(Y)=\nu({W^2})+(\nu(X)-\nu(Z))=q^2+(\nu(X)-p)=\left(\frac{3}{4},\frac{21}{32},\frac{9}{16}\right)$, and thus, $\mu(Y)=\left(\frac{3}{4},\frac{9}{16}\right)$. Observe that, according to (\[eq:lq\]) and (\[eq:L\_a\]), $l_{\mu_1(Y)}=l_{\frac{3}{4}}=\frac{3}{2}$ and $L_{\mu_1(Y)}=L_{\frac{3}{4}}=\left[0,\frac{3}{8}\right) \cup
\left[\frac{1}{2},\frac{7}{8}\right)$. In addition, $\mu_1\left(X^{l_{\mu_1(Y)}}\right)=0$ and $\mu_{2}(Y)=\frac{9}{16}=\mu_{2}\left(L_{\mu_{1}(Y)}\right)$. Therefore, according to Proposition \[lem:unique\], $Y \simeq
L_{\mu_{1}(Y)}=L_{\frac{3}{4}}$. Above observations imply that $Z
= {W^2} \cup (X\setminus Y) \simeq L_{\frac{1}{4}} \cup
\left(X\setminus L_{\frac{3}{4}}\right)
=\left(\left[0,\frac{1}{8}\right) \cup
\left[\frac{1}{2},\frac{5}{8}\right)\right) \cup
\left(\left[0,1\right]\setminus \left(\left[0,\frac{3}{8}\right)
\cup \left[\frac{1}{2},\frac{7}{8}\right)\right)\right) =
\left[0,\frac{1}{8}\right)\cup\left[\frac{3}{8},\frac{5}{8}\right)\cup\left[\frac{7}{8},1\right]=Z^2$.
Geometric construction of maximal ranges {#s5}
========================================
In [@Liapounoff:1940], Lyapunov commented that a subset of the two-dimensional Euclidean space $\mathbb{R}^2$ is the range of some two-dimensional finite atomless vector measure on some measurable space if and only if it satisfies the following conditions: (1) it is convex; (2) it is closed; (3) it is centrally symmetric; (4) it contains the origin. Since the geometrically constructed set $Q_\mu^p(X)$ satisfies the conditions (1)-(4), it must be the range of some two-dimensional finite atomless vector measure on some measurable space. Theorem \[thm1\] immediately tells us that it is the range of the vector measure $\mu$ on the measurable space $\left(Z^*,\mathcal{F}_{Z^*}\right)$. The second equality in Theorem \[thm1\] allows us to construct geometrically the set $R_\mu^p(x)$ by shifting the set $R_\mu(X)$ by $\left(p-\mu(X)\right)$ and intersecting the shifted set with $R_\mu(X)$.
We consider three examples with the same set $X=[0,1]$, but with different probability vector measures. Let $p=(0.7,0.8)$ in all these examples.
![Plots (a)-(c) present the maximal subsets for the vector measures described in Examples \[Exa:Singular\]-\[Exa:Linear\], respectively, with $p=(0.7,0.8)$. The area enclosed by the dashed lines is the range $R_\mu(X)$. The area enclosed by the dotted lines are obtained by parallelly shifting the dashed area by $\left(-0.3,-0.2\right)$. The shaded areas are intersection of the above two areas and represents the identical sets ${R}_\mu\left({Z}^{*}\right)$, ${R}^{{p}}_\mu\left({X}\right)$ and $Q^{{p}}_{\mu}\left({X}\right)$.](FigTot.eps "fig:")\[fig:FigTot\]
\[Exa:Singular\] Let $\mu_{1}$ and $\mu_{2}$ be singular. Then the range $R_\mu(X)$ is the unit square enclosed by the dashed lines in Fig. 1(a). The shaded area denotes the identical sets ${R}_\mu\left({Z}^{*}\right)$, ${R}^{{p}}_\mu\left({X}\right)$ and $Q^{{p}}_{\mu}\left({X}\right)$ with ${p}=\left(0.7,0.8\right)$.
\[Exa:Constant\] Consider the vector measure ${\mu}\left(dx\right)=\left(\mu_{1},\mu_{2}\right)\left(dx\right)=\left({1,f\left(x\right)}\right)dx$, where $$f\left(x\right)=\begin{cases}
\frac{1}{2}, & \textrm{if } x\in\left[0,\frac{1}{2}\right);\\
\frac{3}{2}, & \textrm{if } x\in\left[\frac{1}{2},1\right].\end{cases}$$ Then the range of $R_\mu(X)$ is the area enclosed by the dashed lines in Fig. 1(b). The shaded area denotes the three identical sets ${R}_\mu\left({Z}^{*}\right)$, ${R}^{{p}}_\mu\left({X}\right)$ and $Q^{{p}}_{\mu}\left({X}\right)$ with ${p}=\left(0.7,0.8\right)$.
\[Exa:Linear\] Let ${\mu}\left(dx\right)=\left(\mu_{1},\mu_{2}\right)\left(dx\right)=\left({1,2x}\right)dx$. Then the range $R_{\mu}(X)$ is the area enclosed by the dashed lines in Fig. 1(c). The shaded area denotes the three identical sets ${R}_\mu\left({Z}^{*}\right)$, ${R}^{{p}}_\mu\left({X}\right)$, and $Q^{{p}}_{\mu}\left({X}\right)$ when ${p}=\left(0.7,0.8\right)$.
Proof of Theorem \[thm2\] {#s6}
=========================
For any $B \subseteq A$, denote $p(B)=\sum_{a \in B}p^a$, where either $A=\{1,2,\ldots\}$ or $A=\{1,\ldots,n\}$ for some $n=1,2,\ldots\ .$
\[lem:PartialSum\] Let $\mu=\left(\mu_1,\mu_2\right)$ be a two-dimensional finite atomless measure. If $p(B) \in R_\mu(X)$ for all $B \subset A$ and $\sum_{a \in A} {p^a} = \mu (X)$, then there exists a partition $\{Z^a\in {\mathcal{F}}:a\in A\}$ of $X$, such that $p^a=\mu
(Z^a)$ for each $a\in{A}$.
Consider $p=\mu(X)-p^1$. According to Theorem \[thm1\], there exists a maximal subset $Z^*\in\mathcal{S}_\mu^{p}(X)$ and $R_\mu\left(Z^*\right) = Q_\mu^{p}(X)$. Let $Z^1 = X\setminus
Z^*$, $X^1=Z^*$, and $A^1 = A \setminus \{1\}$. Note that $p^1=\mu(Z^1)$ and $p(B) \in R_\mu\left(X^1\right)$ for all $B
\subseteq A^1$. Indeed, ${p}(B) + {p^1}=p(B\cup \{1\}) \in R_\mu
(X)$. Thus, $p(B) \in R_\mu (X) - \{(\mu(X) - p)\}$, and in addition ${p}(B) \in R_\mu (X)$. Therefore, $p(B) \in Q_\mu^{p}(X)
= R_\mu\left(X^1\right)$.
Now for $p^2 \in \left\{p^a:a \in A^1\right\}$ there exists a maximal set $Z^*\in\mathcal{S}_\mu^{p}(X^1)$, where $p=\mu(X^1)-p^2$. Let $Z^2 = X^1 \setminus Z^*$, $X^2=Z^*$, and $A^2 = A^1 \setminus \{2\}$, then $p^2=\mu(Z^2)$ and $p(B) \in
R_\mu\left(X^2\right)$ for all $B \subseteq A^2$. The repetition of this procedure generates the desired partition $\left\{Z^a\in\mathcal{F}: a \in A\right\}$.
The necessity is obvious. For the sufficiency, in view of Lemma \[lem:PartialSum\], it is sufficient to prove that condition (ii) implies $p (B) \in R_\mu (X)$ for all $B\subseteq A$. If $B$ is finite, condition (ii) implies $p(B) \in R_\mu (X)$. If $B$ is infinite, let $B = \left\{a^1,a^2,\dots \right\}$ and $B_n =
\left\{a^1,a^2,\dots a^n\right\}$, $n=1,2,\ldots\ .$ Then $p(B) =
\lim_{n \rightarrow \infty} { p(B_n)}$ and $p(B_n) \in R_\mu (X)$ for $n=1,2,\dots$, according to condition (ii). Since $R_\mu (X)$ is closed, $p(B) \in R_\mu (X)$.
Finally we show that, when $m=2$, the Dvoretzky-Wald-Wolfowitz purification theorem for a countable image set $A$ [@Edwards:1987; @Khan:2009] is a particular case of Theorem \[thm2\]. Let $p^a=\int_{{X}}\pi\left(a|x\right)\mu
\left(dx\right)$, $a\in A$. If these vectors $p^a$ satisfy conditions (i) and (ii) of Theorem \[thm2\], then Theorem \[thm2\] implies that transition probability can be purified in the case of countable $A$ and $m=2$. Indeed, for (i), obviously $\sum_{a \in A} p_a = \mu(X)$. For (ii), if $B
\subseteq A$ then $$\sum_{a \in B} {p^a}={\sum_{a \in B} {\int_{{X}}\pi\left(a|x\right)\mu \left(dx\right)}}
%=\int_{{X}}\left(\sum_{a \in B} {\pi\left(a|x\right)} \right) \mu \left(dx\right)
=\int_{{X}}\pi\left(B|x\right) \mu \left(dx\right)\in R_\mu(X),$$ where the inclusion follows from a version of Lyapunov’s theorem [@Barra:1981 p. 218].
[99]{}
J. R. Barra. [*Mathematical Basis of Statistics.*]{} Academic Press, New York, 1981.
A. Dvoretzky, A. Wald, and J. Wolfowitz. Elimination of randomization in certain problems of statistics and of the theory of games. [*Proc. Nat. Acad. Sci.*]{} [**36**]{}(1950), 256-260.
A. Dvoretzky, A. Wald, and J. Wolfowitz. Elimination of randomization in certain statistical decision procedures and zero-sum two-person games. [*Ann. Math. Stat.*]{} [**22**]{}(1951), 1-21.
D. A. Edwards. On a theorem of Dvoretsky, Wald, and Wolfowitz concerning Liapounov measures. [*Glasgow Math. J.*]{} [**29**]{}(1987), 205-220.
E. A. Feinberg and A. B. Piunovskiy. On strongly equivalent nonrandomized transition probabilities. [*Theory Probab. Appl.*]{} [**54**]{}(2010), 300-307. M. A. Khan and K. P. Rath. On games with incomplete information and the Dvoretsky-Wald-Wolfowitz theorem with countable partitions. [*J. Math. Econ.*]{} [**45**]{}(2009), 830-837.
P. Loeb and Y. Sun. Purication of measure-valued maps. [*Illinois J. of Mathematics*]{} [**50**]{}(2006), 747-762.
A. A. Lyapunov. Sur les fonctions-vecteurs complètement additives. [*Izv. Akad. Nauk SSSR Ser. Mat.*]{} [**4**]{}(1940), 465-478.
Cz. Olech, The Lyapunov theorem: its extensions and applications, [*Lecture Notes in Math.*]{} [**1446**]{}(1990), 84-103.
D. A. Ross. An Elementary Proof of Lyapunov’s Theorem. [*The American Mathematical Monthly*]{} [**112**]{}(2005), 651-653.
R. Schneider. [*Convex Bodies: The Brunn-Minkowski Theory.*]{} Cambridge University Press, Cambridge, 2008.
[^1]: This research was partially supported by NSF grants CMMI-0900206 and CMMI-0928490.
|
---
abstract: 'The adsorption of charged colloids (macroions) onto an oppositely charged planar substrate is investigated theoretically. Taking properly into account the finite size of the macroions , unusual behaviors are reported. It is found that the role of the coions (the little salt-ions carrying the same sign of charge as that of the substrate) is crucial to understand the mechanisms involved in the process of macroion adsorption. In particular, the coions can accumulate near the substrate’s surface and lead to a counter-intuitive [*surface charge amplification*]{}.'
author:
- René Messina
title: 'Macroion adsorption: The crucial role of excluded volume and coions'
---
Introduction
============
Whereas the bulk behavior of homogeneous (charged [@Loewen_AnnRevPhyChem_2000; @Levin_RepProgPhys_2002] and uncharged [@Hansen_Book_1990]) colloidal suspensions is rather well understood, the situation for its inhomogeneous counterpart, such as that emerging in an adsorption process, is less clear. Potential applications of adsorption of charged colloidal particles (macroions) can vary from technological processes such as surface coating [@Decher_1997] to biological material problems. [@Kawaguchi_Prog_Polym_Sci_2000] From a fundamental point of view, the tremendous long-ranged Coulomb interaction that sets in represents a formidable theoretical challenge. Consequently, a deeper understanding of the phenomenon of macroion adsorption is justified and needed.
On one hand, experiments [@Bu_Langmuir_2006; @Luo_Science_2006] and the well known mean field Gouy-Chapman theory [@Gouy_JPhys_1910; @Chapman_PhilMag_1913] seem to nicely agree for the ion distribution of an aqueous monovalent electrolyte near planar charged interfaces, as long as non-specific forces as well as excluded volume effects are negligible. On the other hand, if the solution contains highly multivalent and/or large sized ions, then the Gouy-Chapman theory may severely qualitatively fail. A crucial missing ingredient in this theory is the inclusion of the finite size of the macroions that can lead to non-trivial phenomena, such as substrate’s surface charge reversal, already with monovalent ions. [@Spitzer_JColIntfSci_1983; @Kjellander_JCP_1998; @Messina_EPL_2002; @LozadaCassou_JCIS_2006] The lateral macroion-macroion electrostatic correlations that are also absent in a mean field theory can be attenuated (and even become marginal) at sufficiently high salt content, in contrast to excluded volume effects. In the past, Gonz[á]{}les-Mozuelos and Medina-Noyola [@GonzalesMozuelos_JCP_1991] used integral equations to address the problem of macroions near repulsive/attractive charged walls interacting via an effective Yukawa potential. More recently, Netz [@Netz_PRE_1999] investigated thoroughly and analytically the behavior of macroions near charged interfaces, based on the Debye-Hückel (DH) theory, but ignoring its finite size. Thereby, the striking effect of surface charge amplification advocated here could not be captured by those approaches. [@GonzalesMozuelos_JCP_1991; @Netz_PRE_1999] It is only very recently, that this phenomenon was reported by Lozada-Cassou and coworkers. [@LozadaCassou_JCIS_2006; @LozadaCassou_JPCB_2004] For a size-asymmetrical electrolyte, [@LozadaCassou_JCIS_2006] surface charge amplification was identified by solving numerically the modified Gouy-Chapman (where cations and anions have different distances of closest approach to the interface). Intimately related to our work, the phenomenon of surface charge amplification in presence of macroions was also found by applying a sophisticated hypernetted chain/mean spherical approximation (HNC/MSA) integral equation. [@LozadaCassou_JPCB_2004]
In this work, we present a simple model, where the finite size of the macroions is taken into account, to reveal the mechanisms governing macroion adsorption. Although our approach is less accurate than the one used by Lozada-Cassou and coworkers, [@LozadaCassou_JPCB_2004] it presents the nice advantage to be analytical and very intuitive. The basic driving force of surface charge amplification is that (spherical) macroions tend to be surrounded by its counterions over [*its whole surface in a uniform manner*]{}. This corresponds actually to the old classical Thomson’s sphere problem. [@Thomson_PhilMag_1904] As long as the strength of the surface charge density of the oppositely charged substrate is low enough, a finite number of counterions of the macroions should stay in the vicinity of the interface (see Fig. \[fig.model\_setup\]), leading to a surface charge amplification. Our paper is organized as follows: Our (modified) DH theory is explained in Sec. \[ sec.DH-theory\]. The results are presented in Sec. \[sec.results\] and followed by a brief summary (see Sec. \[sec.conclu\]).
Debye Hückel theory \[ sec.DH-theory\]
======================================
Linearized Poisson-Boltzmann equation
-------------------------------------
Our electrostatic model (see Fig. \[fig.model\_setup\]) resembles that employed by Spitzer [@Spitzer_JColIntfSci_1983] who studied monovalent size-asymmetrical ions near a wall. The negative substrate’s surface charge density is denoted by $\sigma_0$. The macroions carry a positive central charge $+Z_me$ with $e$ representing the usual elementary charge and $Z_m$ its valency. Excluded volume effects are taken into account via the distance of closest approach $a$ to the charged interface (see Fig. \[fig.model\_setup\]). Our model corresponds somehow to the minimal correction for macroion’ size effect. Having to deal with a [*finite*]{} concentration of macroions at contact with a reservoir, whose bulk value is given by $n_m$, electroneutrality requires a bulk macroion’s counterion concentration $Z_m n_m$ (assuming monovalent point-like anions, i.e. $Z_-=-1$). Those anions will be referred to as the [*coions*]{} of the substrate. Additional (point-like) monovalent counterions ($Z_+=+1$) and coions are also considered with a bulk salt concentration $n_s$. An intuitive and widely used way to connect self-consistently the electrostatic potential, $\Psi (z)$, to the total charge density, $\rho(z) \approx \sum_{\alpha} n_{\alpha} Z_{\alpha} e \exp(-\beta Z_{\alpha} e \Psi)$ (with $\alpha$ standing for the ionic species), is provided by the so-called Poisson-Boltzmann equation that reads: $$\begin{aligned}
\label{eq:PB}
\Delta \Psi (z) & = &
-\frac{e}{\varepsilon_0 \varepsilon_r}
\left[
-2n_s \sinh(e\beta \Psi) - Z_m n_m \exp(e\beta \Psi)
\right.
\nonumber \\ && \left.
+ Z_m n_m \exp(-Z_me\beta \Psi) \Theta(z-a)
\right],\end{aligned}$$ where $\varepsilon_0$ ($\varepsilon_r$) is the vacuum (relative) permittivity, $\beta=1/(k_BT)$ is the reduced inverse temperature, and $\Theta$ is the usual step (or Heaviside) function. This non-linear differential equation can only be solved numerically. However a linearization of Eq. \[i.e., DH approximation\], valid for $|e\beta \Psi| \ll 1$ when $0<z<a$ and for $|Z_me \beta \Psi| \ll 1$ when $z>a$, permits an analytical treatment that is going to be discussed.
For the first diffuse region ($0<z<a$) made up uniquely of the little monovalent ions (see also Fig. \[fig.model\_setup\]), the DH equation is given by $\Delta u(z) = (\kappa_s^2 + \kappa_c^2)u + \kappa_c^2$ with $\kappa_s^2 \equiv 8\pi \ell_B n_s$ and $\kappa_c^2 \equiv 4\pi \ell_B Z_m n_m$, where we have introduced the Bjerrum length $\ell_B=\frac{e^2}{4\pi \varepsilon_0 \varepsilon_r k_BT}$ and the dimensionless variable $u=e \beta \Psi$. The corresponding solution reads: $$\begin{aligned}
\label{eq:u_<}
u(z) = C_1 e^{-\kappa_0 z} + C_2 e^{\kappa_0 z} - \frac{\kappa_c^2}{\kappa_0^2}
\quad (0<z<a),\end{aligned}$$ where $\kappa_0^2 \equiv \kappa_s^2 + \kappa_c^2 = 4\pi \ell_B (2n_s+Z_m n_m)$. $C_1$ and $C_2$ are integration constants that are going to be determined after having applied the suitable boundary conditions.
For the second diffuse region ($z>a$) containing all the ions (including the macroions - see also Fig. \[fig.model\_setup\]) the DH equation reads $\Delta u(z) = \kappa^2 u$ with $\kappa^2 \equiv \kappa_0^2 + Z_m\kappa_c^2 = 4\pi \ell_B [2n_s+ Z_m(Z_m+1)n_m]$. The physically sound solution with vanishing electric field at $z \to +\infty$ corresponds to: $$\begin{aligned}
\label{eq:u_>}
u(z) = C_3 e^{-\kappa z}
\quad (z>a),\end{aligned}$$ where $C_3$ is a third integration constant.
Boundary conditions
-------------------
Our electrostatic model system is completely characterized once the integration constants $C_1$, $C_2$ and $C_3$ appearing in Eqs. and are specified. To do so, we apply the three following boundary and/or matching conditions: (i) The Gauss’ law applied at the charged interface $z=0$ requires $u'(0) = \frac{2}{b}$, where $b=\frac{e}{2\pi\ell_B|\sigma_0|}$ is the so-called Gouy-Chapman length. (ii) The zero intrinsic surface charge at $z=a$ imposes the continuity of the electric displacement $\vec D = \varepsilon_0 \varepsilon_r \vec E$ and hence also that of the electric field $\vec E$ \[i.e., $u'(z \to a^-) = u'(z \to a^+)$\]. [@boundary_electroneutrality] (iii) The continuity of the electrostatic potential at $z=a$ requires $u(z \to a^-) = u(z \to a^+)$. The resulting set of three equations can be readily solved and yields: $$\begin{aligned}
\label{eq:C_1}
%\left \{
%\begin{array}{lll}
%C_1
\displaystyle
C_1 & = &
%num
\displaystyle
\frac{ -2 Q \displaystyle
%BIG fraction of C_3
- \frac{2}{\kappa_0b} e^{\kappa_0 a} + \frac{\kappa_c^2}{\kappa_0^2}}
%den
{2 \cosh(\kappa_0 a)},
\\
\nonumber \\
%C_2
\label{eq:C_2}
\displaystyle
C_2 & = &
%num
\displaystyle
\frac{ -2Q
- \frac{2}{\kappa_0b} e^{\kappa_0 a} + \frac{\kappa_c^2}{\kappa_0^2}}
%den
{2 \cosh(\kappa_0 a)}
+ \frac{2}{\kappa_0b},
\\
\nonumber \\
%C_3
\label{eq:C_3}
C_3 & = &
\displaystyle
-2e^{\kappa a}
%BIG fraction
\underbrace{
\left[ \frac{
% num
\frac{1}{\kappa_0b \cosh{(\kappa_0 a)}} + \frac{\kappa_c^2}{2\kappa_0^2} \tanh{(\kappa_0 a)}}
% den
{ \frac{\kappa}{\kappa_0} + \tanh{(\kappa_0 a)}}
\right]}_{\equiv Q}.
%\end{array}
%\right .\end{aligned}$$ Note that in the limit of point-like macroions ($\kappa_0 a \to 0$) one recovers the well-known result $C_3 \to -\frac{2}{\kappa b}$. At this stage all the relevant observables of the system can be in principle obtained within the framework of the DH theory.
Results and discussion \[sec.results\]
======================================
A pertinent quantity that is appropriate to characterize the strength of the macroion adsorption is provided by the contact potential of interaction $U_m$. The latter corresponds to the external work accomplished upon bringing a macroion from infinity ($z=+\infty$) to contact ($z=a$): $U_m=-\int_{+\infty}^{a} Z_me E_z dz$ (with $E_z=-\frac{\partial \Psi}{\partial z}$ denoting the $z$-component of the electric field). With the help of Eq. we get $\beta U_m=Z_mu(a)=Z_mC_3e^{-\kappa a}$. Using then Eq. for $C_3$, we obtain the following expression for $U_m$: $$\begin{aligned}
\label{eq:Um_tanh}
\beta U_m & = &
- \displaystyle
2Z_m
%BIG fraction
\left[ \frac{
% num
\frac{1}{\kappa_0b \cosh{(\kappa_0 a)}} + \frac{\kappa_c^2}{2\kappa_0^2} \tanh{(\kappa_0 a)}}
% den
{ \frac{\kappa}{\kappa_0} + \tanh{(\kappa_0 a)}}
\right].\end{aligned}$$ As can be seen from Eq. , $U_m$ depends on many parameters, such as $Z_m$, $b$, $a$, $\kappa$ and $\kappa_0$, [@kappa_c] making its understanding rather difficult. Nonetheless, Eqs. - suggest that more insight on the adsorption behavior can be gained by considering two limits: (i) the high screening regime ($\kappa_0a \gg 1$) and (ii) the weak screening regime ($\kappa_0a \ll 1$). Without loss of generality, we are going to explore these two limits.
High screening regime
---------------------
In the [*high screening regime*]{} ($\kappa_0 a \gg 1$), Eq. can be approximated by: $$\begin{aligned}
\label{eq:Um_high_screening}
\beta U_m & \simeq &
1 - \frac{\kappa}{\kappa_0},\end{aligned}$$ where the relation $\kappa_c^2=\frac{1}{Z_m}(\kappa^2-\kappa_0^2)$ has been used. Interestingly, in this regime, $U_m$ depends only on the [*screening strength contrast*]{} $\kappa/\kappa_0$. One can also conveniently express $U_m$ as a function of the bulk concentrations $n_m$ and $n_s$, which reads $$\begin{aligned}
\label{eq:Um_high_screening_nm_ns}
\beta U_m & \simeq &
1 - \sqrt{ \frac{2n_s+Z_m(Z_m+1)n_m}{2n_s+Z_mn_m}}.\end{aligned}$$
Equation shows that $U_m$ is even independent of the Bjerrum length $\ell_B$, which means that the effective wall-macroion attraction is entropically (or [*depletion*]{}) driven. To better understand this phenomenon, we have sketched on Fig. \[fig.u\_E\_high\_scr\] the profile of $u(z)$ as well as profile of the reduced (dimensionless) electric field $E^*(z) \equiv -\frac{b}{2}u'(z)$ \[with $E^*(z=0)=-1$\] for a set of typical parameters of charged colloidal suspensions: $a=0.5 {\rm \mu m}$, $b=10^{-3} {\rm \mu m}$ (corresponding to $|\sigma_0| \approx 0.036 {\rm Cm^{-2}}$ for an aqueous solvent), $Z_m=10^4$, $n_s=10^{-4}{\rm M}$, a (fictive) equivalent volume fraction $\phi_m \equiv n_m \frac{4}{3} \pi a^3$ set to $10^{-3}$, (i.e., $n_m \approx 3.1714 \times 10^{-12} {\rm M}$ with M standing for mole per liter units). Thereby we have $\kappa_0 a \simeq 16.4$. The profile of $E^*(z)$ reveals an [*unusual non-monotonic*]{} behavior near contact. This is a direct consequence of an accumulation of “[*excess*]{}” coions in the macroion depleted zone ($z<a$) leading to a [*weaker*]{} screening.
In order to further characterize the adsorption behavior in the high screening regime, we examine $U_m$ as predicted by Eq. and Eq. for some typical experimental values of the parameters $\kappa$ and $\kappa_0$ of charged colloidal systems. In Fig. \[fig.Um\_ns\_high\_scr\], $-\beta U_m=\beta |U_m|$ is plotted against the bulk salt concentration $n_s$ for different prescribed values of $Z_m=10^4,10^5$ and $10^6$. Only values of $\beta |U_m|$ smaller than unity are shown such as to explore its behavior where the DH approximation is valid. Figure \[fig.Um\_ns\_high\_scr\] shows that beyond a certain threshold of salt content $n_s$ that is $Z_m$-dependent, no adsorption occurs as signaled by a nearly zero value of $U_m$. At relatively low enough salt concentration, adsorption is favored where $U_m$ increases with growing $Z_m$. The latter point can be clearly understood by noticing that for typical values of the parameters of charged colloidal systems ($Z_m\frac{n_m}{2n_s} \ll 1$ and $Z_m \gg 1$) we have $\beta U_m \approx1 - \sqrt{1+Z_m^2 \frac{n_m}{2n_s}}$. Moreover, in the limit $Z_m^2 \frac{n_m}{2n_s} \ll 1$, the simple following relation $\beta U_m \approx -Z_m^2 \frac{n_m}{4n_s}$ holds.
Weak screening regime
---------------------
We now address the [*weak screening regime*]{} characterized by $\kappa_0 a \ll 1$. In this situation Eq. becomes: $$\begin{aligned}
\label{eq:Um_weak_screening_final}
\beta U_m & \simeq &
- \frac{2Z_m}{\kappa b}
\left[ 1
+ \frac{\kappa_0}{\kappa} \left\{ \frac{\kappa b}{2Z_m}
\left( \frac{\kappa^2}{\kappa_0^2} - 1\right) - 1 \right\}
\kappa_0 a \right],
\nonumber \\\end{aligned}$$ where the dispersion relation $\kappa^2 = \kappa_0^2 + Z_m \kappa_c^2$ has been used. Equation reveals that $U_m$ varies affinely with macroion size $a$. Depending on the sign of $ \frac{\kappa b}{2Z_m}\left( \frac{\kappa^2}{\kappa_0^2} - 1\right) - 1$, $U_m$ may either decrease or increase with $a$. For $\frac{\kappa b}{2Z_m} \gg 1$ we obtain the following limit behavior: $$\begin{aligned}
\label{eq:Um_weak_scr_inf_b}
\lim_{\frac{\kappa b}{2Z_m} \to \infty} \beta U_m & \approx &
- \left( 1 - \frac{\kappa_0^2}{\kappa^2} \right) \kappa a. \end{aligned}$$ In this “entropic” limit $|U_m|$ increases linearly with $a$, meaning that a finite wall-macroion attraction persists even at vanishing surface charge density $\sigma_0$ due to excluded volume effect ($a \neq 0$). Noticing that $$\begin{aligned}
\label{eq:E}
E^*(a)=\frac{\kappa b}{2Z_m} \beta U_m,\end{aligned}$$ we deduce that the strength of the electric field at contact $|E^*(a)|$ becomes larger than unity when $\frac{\kappa b}{2Z_m}$ is sufficiently large \[especially true for Eq. \]. This feature corresponds to an [*electric field (or surface charge) amplification*]{} at contact.
Let us consider some typical experimental values of parameters representative of charged [*micellar*]{} systems (see caption of Fig. \[fig.u\_E\_weak\_scr\]). Thereby we have $\kappa_0a \simeq 0.177$. To access the mechanisms of wall-macroion attraction in the weak screening regime, we have plotted the profiles of $u(z)$ and $E^*(z)$ in Fig. \[fig.u\_E\_weak\_scr\] for various values of $b$. Strikingly, in the macroion depleted zone, the electric field gets monotonically [*amplified*]{} (from $z=0$ up to contact $z=a=5 {\rm nm}$) at (very) large $b$ \[corresponding to poorly charged interfaces ($|\sigma_0| \lesssim 10^{-4} {\rm Cm^{-2}}$) - see Fig. \[fig.u\_E\_weak\_scr\]\]. This phenomenon is again due to an accumulation of “excess” coions in the macroion depleted zone, that leads here to a net surface charge that is more negative than $\sigma_0$. It is important to remark, that surface charge amplification does not necessarily involve a subsequent charge reversal, as clearly indicated in Fig. \[fig.u\_E\_weak\_scr\]. In fact this charge-amplification phenomenon was already observed, although uncommented, in computer simulations \[see Fig. 4(b) in Ref. [@Tanaka_JCP_2001] and Fig. 3 in Ref. [@Maiti_NanoLett_2006]\].
Concluding remarks \[sec.conclu\]
=================================
To summarize, we have studied analytically within the framework of the Debye-Hückel theory the (weak) electrostatic adsorption of macroions at oppositely charged planar surfaces. Taking into account the crucial role of the finite size of the macroions, our model reveals non trivial adsorption driving forces. In the strong screening regime, the wall-macroion attraction strength at contact is exclusively governed by the screening strength contrast $\kappa/\kappa_0$. In the weak screening regime, the wall-macroion attraction strength can either decrease or increase with macroion size, depending on the surface charge density of the substrate. In particular, at sufficiently small surface charge densities an effective electric filed (or equivalently an effective surface charge) amplification sets in. All these adsorption mechanisms have a common feature, namely, the accumulation of excess coions in the macroion depleted zone. The latter mechanism also explains the surface charge amplification recently reported by molecular dynamics simulations for DNA-dendrimer complexation. [@Maiti_NanoLett_2006] Our findings could be experimentally verified by employing an (extra) ultra fine dispersion of charged particle tracers (with a smaller size than the macroions and such as to nearly not modify the screening strengths $\kappa$ and $\kappa_0$), whose density profile should reveal the electric filed’ one in the macroion depleted zone.
The author thanks M. Lozada-Cassou for enlightening discussions. Financial support from DFG via LO418/12 and SFB TR6 is acknowledged.
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Note that the same result is obtained by requiring global elecroneutrality: $\int_0^{\infty} \sum_{\alpha} \rho_{\alpha}(z) dz = - \sigma_0$.
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|
---
abstract: 'Production cross sections for $t\bar t$ and $t\bar tj$ events at hadron colliders are calculated, including finite width effects and off resonance contributions for the entire decay chain, $t\to bW\to b\ell\nu$, for both top quarks. Resulting background rates to Higgs search at the CERN LHC are updated for inclusive $H\to WW$ studies and for $H\to\tau\tau$ and $H\to WW$ decays in weak boson fusion events. Finite width effects are large, increasing $t\bar t(j)$ rates by 20% or more, after typical cuts which are employed for top-background rejection.'
address: ' Department of Physics, University of Wisconsin, Madison, WI 53706, USA '
author:
- 'N. Kauer and D. Zeppenfeld\'
title: ' Finite-Width Effects in Top Quark Production at Hadron Colliders '
---
\#1[0=1=to0[$/$]{}0=1=to0[/]{}\#1-01 ]{}
Introduction {#sec:intro}
============
$t\bar t$ production [@CDFD0] is a copious source of $W$-pairs and, hence, of isolated leptons at the Tevatron and the LHC. Top quark production will be intensely studied as a signal at these colliders. In addition, it constitutes an important background for many new particle searches. Examples include the leptonic signals for cascade decays of supersymmetric particles [@SUSY] or searches for $H\to W^+W^-$ [@DittDrein; @CMS; @ATLAS_TDR_2; @Trefzger_Higgs; @KPRZ; @RZ_WW] and $H\to\tau\tau$ [@ATLAS_TDR_2; @RZH_tau; @PRZ_TauTau] decays.
Usually, $t\bar t$ production is considered in the narrow-width approximation (NWA), which effectively decouples top production and decay (see Fig. \[fig:nwa\](a)). Whenever resonant top production dominates, this approximation is well motivated and it greatly simplifies matrix element evaluation and phase space generation. The NWA is also useful for single-resonant top production as shown in Fig. \[fig:nwa\](b) [@Boos_SingleTop; @Tait_tW; @Boos_SingleTop_Followup; @OtherSingleTopRefs]. In some cases calculations have been further simplified by also treating the decaying $W$ bosons as on-shell particles.
{width="16.5cm"}\
\[fig:nwa\]
Naturally, the accuracy of these approximations needs to be tested, which requires a full calculation of off-shell effects. Restrictive selection cuts, as used for efficient suppression of $t\bar t$ backgrounds [@Trefzger_Higgs; @RZ_WW], tend to be optimized against on-shell top production which may substantially enhance the relative importance of off-shell contributions. In applying the NWA, another problem inadvertently arises: the Feynman diagrams in Figs. \[fig:nwa\](a) and \[fig:nwa\](b) have identical initial and final states. When approximate cross sections, each specialized to a particular phase space region, are added up to obtain the total rate, double-counting can occur, and interference effects in overlap regions may not be properly accounted for. One thus needs a calculation which includes both resonant and non-resonant contributions, using finite width top-quark propagators, which correctly includes interference effects between the various contributions. The purpose of this paper is to present such a calculation for $t\bar t$ and $t\bar tj$ production. In addition to merging resonant and non-resonant effects for the top quarks, we also include finite width effects for the $W$ bosons, i.e. we consider general $b\bar b e^-\bar\nu_e\mu^+\nu_\mu(j)$ final states. Finite top width effects, at the level of $t\to bW$ decays, have been considered previously for $tWb$ production [@Tait_tW]. The complete set of Feynman graphs for $pp\to bW^+\bar bW^-$ processes has been generated as well [@Boos_SingleTop_Followup]. However, the full treatment of lepton final states with spin correlations and off-shell contributions is new even for the $t\bar t$ case. To our knowledge, no previous calculation of finite width effects in $t\bar tj$ production exists.
The paper is organized as follows. In Section \[sec:method\] we discuss various methods of including finite width effects and discuss their advantages and their practicality. We adopt the “overall factor scheme” and apply it in Section \[sec:general\] to $b\bar b e^-\bar\nu_e\mu^+\nu_\mu$ production and the analogous process with one additional colored parton in the final state. While the matrix elements can, in principle, be generated with automated programs like MADGRAPH [@MADGRAPH], the proper inclusion of finite widths, preservation of electroweak and strong gauge invariance, avoidance of double counting and of divergences in extraneous phase space regions requires manual intervention. In Section \[sec:general\] we describe the content of our calculation, the various consistency tests, and other important features of the program which we have developed.
Our program has already been used to study backgrounds to $H\to WW$ decays at the LHC [@KPRZ]. We expand on this analysis in Section \[sec:application\] and use $H\to WW$ and $H\to\tau\tau$ decays, more precisely the backgrounds produced by $t\bar t$ and $t\bar tj$ production, to exemplify the size of off-shell and on-shell contributions and compare our full simulations with previous background estimates. A summary and conclusions are given in Section \[sec:summary\].
Finite-width effects and gauge invariance {#sec:method}
=========================================
The inclusion of finite width effects is needed in order to avoid the singular behavior of the tree level propagators on mass-shell, $p^2-m^2=0$. An approach that is straightforward to implement and, hence, well suited for automatic code generators like MADGRAPH/HELAS [@MADGRAPH; @HELAS] is to use a Breit-Wigner-type propagator with fixed width for all top and $W$ propagators, making no distinction between time-like and space-like momenta. For massive fermions like the top quark one simply substitutes \[eq:tprop\_naive\] (p) = = .
However, this technique, also referred to as [*fixed-width scheme*]{}, will lead to gauge-dependent matrix elements. Gauge invariance requires that the Ward identity k\_V\^= -i g \[i\^[-1]{}(p\_1) - i\^[-1]{}(p\_2)\] \[eq:ward-id\] (with $p_1=k+p_2$) be satisfied, where $V^\mu = -i g \gamma^\mu$ is the gauge boson – fermion vertex. For the top-propagator of Eq. (\[eq:tprop\_naive\]), the inverse, $\sla{S}^{-1}$, even diverges at $p^2=m^2$ and the Ward identity is violated. The naive use of Breit-Wigner propagators does not produce consistent matrix elements.
Calculating vertex functions and inverse propagators perturbatively, the Ward identity of Eq. (\[eq:ward-id\]) will be satisfied order by order. This is the basis of the so called [*fermion loop scheme*]{} for the $W$-propagator where, for a LO calculation, the imaginary part of the fermionic 1-loop corrections is included in both the vertex function and in the inverse propagator [@Baur:1995aa; @Argyres:1995ym; @Beenakker:1997kn]. For the propagator this corresponds to the Dyson resummation of the imaginary part of the $W$ vacuum polarization.
{width="16.5cm"}\
\[fig:topprop\]
A theory driven solution of the finite width problem for the top quark propagator would generalize this scheme. More specifically, a Dyson resummation of the imaginary parts of the 1-loop self-energy of the top quark, due to $bW$ intermediate states (see Fig. \[fig:topprop\]), results in the effective propagator S\_t(p) = \[eq:topprop\] Here $P_L$ is the left-chiral projector and $$\textstyle
\gamma(p^2) =
\frac{1}{64\pi}\;\frac{g^2}{m_W^2}\;\sqrt{\left[1 -
\frac{(m_W + m_b)^2}{p^2}\right]\left[1 - \frac{(m_W - m_b)^2}{p^2}\right]}\
\left[\left(1-\frac{m_W^2}{p^2}\right)(2m_W^2+p^2)+\frac{m_b^2}
{p^2}(m_W^2+m_b^2-2p^2)\right]$$ In order to satisfy the $SU(3)$ Ward identity of Eq. (\[eq:ward-id\]) one also needs to calculate the imaginary part of the $ttg$ vertex (see Fig. \[fig:ttgvert\]). We have checked by explicit calculation that this $SU(3)$ Ward identity is indeed satisfied. However, the effective $ttg$ vertex already is too complex to be displayed here.[^1] For our applications we would need to know the imaginary contributions to $ttgg$ and $ttggg$ vertices as well.
{width="16.5cm"}\
\[fig:ttgvert\]
A further complication arises from the need of electroweak gauge invariance in our calculations. Consider the simple process (or subgraph) depicted in Fig. \[fig:bbWW\]: elastic scattering of a $b$-quark and a longitudinal $W$ boson. The longitudinal polarization vectors of the two $W$’s scale like $\sqrt{\hat s}/m_W$, which leads to a rise of the two subamplitudes with the center of mass energy: \_t &\~& [s\^2s \^2 - m\_t\^2]{}\
[M]{}\_[,Z]{} &\~& [s\^2t - m\_V\^2 ]{} . As is well known for the crossed process $e^+e^-\to W^+W^-$, gauge invariance of the electroweak couplings leads to a cancellation of these leading terms and results in partial wave amplitudes which do not grow with energy and, thus, respect partial wave unitarity [@HPZH]. When using the finite width top-quark propagator of Eq. (\[eq:topprop\]), with $\gamma(\hat s)\sim \hat s$ at high energy, the width correction dominates the propagator at very high $\hat s$ and leads to ${\cal M}_t\sim {\rm const}$ at high energy, thus spoiling the gauge theory cancellations between ${\cal M}_t$ and ${\cal M}_{\gamma,Z}$: the $bW_L\to
bW_L$ scattering amplitude violates unitarity in the $J={1\over 2}$ partial wave at sufficiently high energy. The likely solution to this problem lies in adding the imaginary parts of $btW$, $btWg$ vertices etc. to the loop scheme prescription, a solution which clearly becomes too involved for practical applications.
{width="12.0cm"}\
\[fig:bbWW\]
When considering cross sections which contain contributions from both resonant and non-resonant amplitudes, \~d[PS]{}\_[if]{} |[M]{}\_[if, res. ]{} + [M]{}\_[if, nonres. ]{} |\^2 , we need an alternative which guarantees gauge invariance and unitarity while allowing the effective substitution of propagators by a Breit-Wigner form in the resonant contributions, ${\cal M}_{\scriptstyle if, res}$. In our work, we adopt the overall factor scheme [@bvz] which starts from the observation that the zero width amplitudes, as derived from unresummed Feynman rules, provide a gauge invariant expression with proper high energy behavior. Multiplying the full lowest order amplitude by an overall factor $(p^2-m^2)/(p^2-m^2+im\Gamma)$, for each resonant propagator, preserves gauge invariance but effectively replaces the tree level propagators, which are divergent on mass-shell, by finite-width Breit-Wigner propagators, \_[if (= 0)]{} = [M]{}\_[if, res. (= 0)]{} + [M]{}\_[if, nonres. (= 0)]{}
In the overall factor scheme, the cross section is thus calculated in terms of this modified amplitude. Note that the additional overall factor is close to $1$ in phase space regions where the non-resonant amplitudes yield significant contributions. Close to resonance, on the other hand, the amplitude effectively reduces to the dominant resonant terms. As we will show in Section III, this approach provides an excellent interpolation between on and off resonance regions, with ambiguities never rising beyond the 1–2% level.
A second practical solution starts from the observation that the Ward identity in Eq. (\[eq:ward-id\]) remains fulfilled if one changes constant terms in the inverse propagator. This suggests another method to restore gauge invariance, which has been dubbed the [*complex-mass scheme*]{} [@Denner:1999gp; @compMother]. Its finite-width amplitude is derived from the full lowest-order amplitude with zero-width propagators by substituting all $W,$ $Z$ and top quark masses according to m \[eq:complexmass\] This scheme has recently been used in a single top quark study for the Tevatron [@vanderHeide:2000fx]. An unphysical consequence of the universal substitution (\[eq:complexmass\]) is that space-like propagators receive imaginary contributions, or that the top-Higgs Yukawa coupling, $h_t=m_t/v$, receives an imaginary part. However, such effects are suppressed by factors $\Gamma/m \ll 1$ and would presumably not be noticeable in a LO Monte Carlo program.
In our finite width $t\bar t$ and $t\bar tj$ Monte Carlo programs, we have opted for the overall factor scheme. We expect very similar numerical results in the complex-mass scheme, which we have not implemented, however.
Cross Sections for $WW$ production {#sec:general}
==================================
Unstable particles occur in several places in top quark pair production processes: in the form of decaying top-quarks and also as the $W$ bosons which arise in their decays. In order to include off-resonance contributions for both, we are led to consider the full tree-level matrix elements for $b\bar b e^-\bar\nu_e\mu^+\nu_\mu$ final states (“$t\bar t$ production”) at ${\cal O}(\alpha_s^2\alpha^4)$, and final states with one additional parton (“$t\bar tj$ production”) at ${\cal O}(\alpha_s^3\alpha^4)$.
Matrix Elements {#subsec:matrixel}
---------------
For $pp$ or $p\bar p$ collisions, matrix elements for the following subprocesses need to be evaluated (we neglect CKM mixing):
& & gg b|b e\^-|\_e\^+\_, q|q b|b e\^-|\_e\^+\_ t|t \[eq:tt-mat\]\
& & gg b|b e\^-|\_e\^+\_g, g[\^[(]{}\^[)]{}]{}b|b e\^-|\_e\^+\_[\^[(]{}\^[)]{}]{}, q|q b|b e\^-|\_e\^+\_g t|tj . \[eq:ttj-mat\]
Representative Feynman graphs are shown in Fig. \[fig:offshell\]. Double resonant contributions include gluon radiation off initial state partons, the top quarks, and final state $b$-quarks (Figs. \[fig:offshell\](a) and (b)). An example for a single resonant graph is shown in Fig. \[fig:offshell\](c), while (d) depicts one of the non-resonant graphs. Electroweak gauge invariance of the $b\bar b\to W^+W^-$ subgraphs in (c) and (d) requires inclusion of $\gamma$ and $Z$ exchange contributions. One such contribution is shown in Fig. \[fig:offshell\](e). Others include $W$-emission off the final state lepton lines (see Fig. \[fig:ampsum\] and discussion below). Our code includes finite $b$-quark masses (set to a default value of $m_b=5$ GeV). This allows to integrate over the entire $b$-quark phase space, including the $g\to b\bar b$ splitting region. A finite $b$-quark mass necessitates new contributions, however, namely Higgs exchange diagrams like the one depicted in Fig. \[fig:offshell\](f). Our code includes all these contributions. We avoid goldstone boson exchange graphs by working in the unitary gauge for the electroweak sector.
{width="16.5cm"}\
\[fig:offshell\]
Our calculation assumes different lepton flavors in the decay of the two $W$-bosons. However, the amplitudes for this mixed lepton flavor case can also be used to obtain approximate results for same flavor processes, specifically $e^-\bar\nu_e e^+\nu_e$ and $\mu^-\bar\nu_\mu\mu^+\nu_\mu$ final states. The double- and single-resonant contributions (with respect to top) are identical for the mixed and same flavor sample. However, the latter also features $(\gamma,Z \to \ell^+\ell^-) \circ (Z \to \nu_\ell\bar\nu_\ell)$ graphs, non-resonant from the view-point of top-decay, which do not occur in the mixed flavor case. Away from the $Z$-boson mass-shell and small $\ell^+\ell^-$ invariant mass, these contributions are small, and the complete $\ell^\pm_1\ell^\mp_2$ cross section (with $\ell_{1,2} =e,\mu$) is obtained by multiplying the result presented below with a lepton-flavor factor of 4.
\[tab:amps\]
------------ ----- ------- ----- ------- ------ -------
$t\bar t$ 87 (39) 40 (16)
$t\bar tj$ 558 (258) 246 (102) 246 (102)
------------ ----- ------- ----- ------- ------ -------
: Number of Feynman graphs contributing to different subprocesses. The numbers in brackets reflect the reduction of subamplitudes illustrated in Fig. \[fig:ampsum\] (see text). The columns correspond to different initial states.
![Amplitude factorization for $\gamma$ and $Z$ decays. Explicit summation of the sketched helicity amplitude fragments leads to a significant computational reduction, as shown in Table \[tab:amps\].](ampsum.eps "fig:"){width="15.0cm"}\
\[fig:ampsum\]
The number of subamplitudes, corresponding to individual Feynman graphs, is sizable for the processes of Eqs. (\[eq:tt-mat\],\[eq:ttj-mat\]) and is listed in Table \[tab:amps\]. Constructing the code that evaluates the matrix elements was simplified by using automatically generated output of MADGRAPH [@MADGRAPH] as a starting point. However, the code had to be modified by hand in several ways. First, every Feynman graph amplitude needs to be multiplied with the proper overall factors f\_a(p) = [p\^2-m\^2\_ap\^2-m\^2\_a+im\_a\_a]{} , depending on its resonance structure. For $t\bar t$ production the factors correspond to a resonant top quark with momentum $p_b+p_{\mu^+}+p_{\nu_\mu}$ (for which we use the shorthand notation $p_{b\mu^+\nu_\mu}$) and/or a resonant anti-top quark with momentum $p_{\bar b e^-\bar\nu_e}$. For $t\bar tj$ production two additional factors $f_t(p_{b\mu^+\nu_\mu j})$ and $f_t(p_{\bar b e^-\bar\nu_e j})$ appear, corresponding to gluon emission off the final state $b$ or $\bar b$. In both cases two $W$ factors ($f_W(p_{e^-\bar\nu_e})$ and $f_W(p_{\mu^+\nu_\mu})$) and one $Z$ factor $f_Z(p_{e^-\bar\nu_e\mu^+\nu_\mu})$ need to be explicitly multiplied into amplitudes that are non-resonant with respect to these propagators. MADGRAPH generates all top, $W$ and $Z$ propagators automatically as resonant propagators, i.e. with a finite fixed width. These propagators can be viewed as the result of multiplying the zero-width tree-level propagators with the overall factor, or, alternatively, as obtained by the substitution \_a: (p\^2-m\_a\^2)\^[-1]{} (p\^2-m\_a\^2+im\_a\_a)\^[-1]{}, which we denote by the symbol $[p]_a$ in the amplitudes given below. For the first few Feynman graphs of Fig. \[fig:offshell\] these changes can be summarized as & & (\[p\_[b\^+\_]{}\]\_t, \[p\_[|b e\^-|\_e]{}\]\_t, \[p\_[|b e\^-|\_e j]{}\]\_t, \[p\_[e\^-|\_e]{}\]\_W, \[p\_[\^+\_]{}\]\_W) \* f\_t(p\_[b\^+\_j]{}) f\_Z(p\_[e\^-|\_e\^+\_]{}) ,\
& & (\[p\_[b\^+\_]{}\]\_t, \[p\_[|b e\^-|\_e j]{}\]\_t, \[p\_[e\^-|\_e]{}\]\_W, \[p\_[\^+\_]{}\]\_W) \* f\_t(p\_[|b e\^-|\_e]{}) f\_t(p\_[b\^+\_j]{}) f\_Z(p\_[e\^-|\_e\^+\_]{}) ,\
& & (\[p\_[b\^+\_]{}\]\_t, \[p\_[e\^-|\_e]{}\]\_W, \[p\_[\^+\_]{}\]\_W) \* f\_t(p\_[|b e\^-|\_e]{}) f\_t(p\_[b\^+\_j]{}) f\_t(p\_[|b e\^-|\_e j]{}) f\_Z(p\_[e\^-|\_e\^+\_]{}) ,\
& & (\[p\_[e\^-|\_e]{}\]\_W, \[p\_[\^+\_]{}\]\_W) \* f\_t(p\_[b\^+\_]{}) f\_t(p\_[|b e\^-|\_e]{}) f\_t(p\_[b\^+\_j]{}) f\_t(p\_[|b e\^-|\_e j]{}) f\_Z(p\_[e\^-|\_e\^+\_]{}) ,\
(Z)& & (\[p\_[e\^-|\_e]{}\]\_W, \[p\_[\^+\_]{}\]\_W, \[p\_[e\^-|\_e\^+\_]{}\]\_Z) \* f\_t(p\_[b\^+\_]{}) f\_t(p\_[|b e\^-|\_e]{}) f\_t(p\_[b\^+\_j]{}) f\_t(p\_[|b e\^-|\_e j]{}).
In the overall factor scheme, imaginary parts for $t$-channel top-quark propagators are absent. The top-width was eliminated by hand from the MADGRAPH output for these space-like propagators. The effect of this modification should be small, however, since $|p^2-m^2|\gg m\Gamma$ if $p^2 < 0,$ due to $\Gamma/m \ll 1$. The number of Feynman diagrams for $t\bar tj$ production is formidable, partially due to repeating sequences of subgraphs, like the ones depicted in Fig. \[fig:ampsum\]. These subgraphs were combined to effective $\gamma/Z$-currents. As shown in Table \[tab:amps\], this procedure reduces the number of sub-amplitudes, which need to be calculated, by a factor of two or more.[^2]
{width="12.5cm"}\
\[fig:hiqcd\]
Since our calculation includes full matrix elements to a high order in perturbation theory, care has to be taken to avoid overlap with other backgrounds and double-counting. Closer inspection of the matrix elements for the $t\bar t$ and $t\bar tj$ modes yields two groups of Feynman diagrams that are candidates for double-counting. They are schematically depicted in Fig. \[fig:hiqcd\]. Consider $gu\to b\bar bW^+W^-u$, a subprocess of $t\bar tj$ production, as an example. Graph (a) represents $g\to b\bar b$ splitting for a final state gluon. It constitutes a QCD radiative correction to $gu\to W^+W^-ug$ and should be counted in the QCD $WWjj$ background. For small $b\bar b$ invariant masses, graph (a) is a contribution to the gluon fragmentation, which must be counted only once. Group (b) features $g\to b\bar bg$ splitting, where the $b$-quarks and one gluon are external particles. When the on-shell gluon is in the initial state and the $\bar b$ is collinear to it, this group represents an $\alpha_s$ correction to $bu\to uW^+W^-b$ production. When both the $b$ and the $\bar b$ are collinear to the initial gluon, we are considering an $\alpha_s^2$ QCD correction to $gu\to uW^+W^-$. It is inappropriate to include either group as non-resonant contributions to $t\bar tj$ production. These splitting contributions can be eliminated either through suitable cuts or by explicitly excluding the relevant Feynman graphs. A numerical comparison shows that typical selection cuts, like for the Higgs search to be considered in Section \[sec:application\], are sufficient to render the splitting contributions negligible. Since the analyses in Refs. [@RZ_WW] and [@PRZ_TauTau] do not include QCD corrections to general $WWjj$ production, we chose to omit these graphs in our final calculations. This is permissible since the graphs of group (a) and (b) necessarily contain electroweak interactions of $u$ and $d$ quarks. In the context of top backgrounds, the primary focus is on electroweak interactions of bottom and top quarks. When CKM mixing is neglected, the gauge invariance of our amplitudes is preserved when setting all electroweak interactions of the first two quark generations to zero. This procedure eliminates the graphs of groups (a) and (b) and avoids double-counting.
When integrating over collinear $g\to b\bar b$ configurations for initial state gluons, the differential cross section receives enhancement factors of order $\log(\mu_f^2/m_b^2)$. Since $\alpha_s(m_t)\log(m_t^2/m_b^2)\approx 0.85$, one may wonder whether our perturbative leading-order calculations are still reliable, or whether a resummation of these collinear logarithms is required. The correct treatment of the collinear region would include $gb$ scattering, convoluted with the $b$-quark parton density. Then, a subtraction of the gluon splitting term would also be required to avoid double-counting [@Boos_SingleTop; @Tait_tW].
In the cases at hand, the net effect of the $b$-quark PDF contribution is small, however. For inclusive $pp\to b\bar bWW,$ only 10.4 pb are added to 622 pb (see Table 1 in Ref. [@Boos_SingleTop_Followup]) . For the SM Higgs searches via weak boson fusion, discussed in Section \[sec:application\], the tagging criteria and selection cuts are chosen such that for $t\bar t$ production both $b$-quarks are resolved, with $p_T > 20$ GeV. This avoids the collinear region. For $t\bar t$+1 jet production, the $b$-quark observed as a forward tagging jet is required to have $p_T > 20$ GeV, while the other $b$-quark has no lower transverse momentum threshold. However, these collinear regions contribute little to the cross section within typical cuts. For example, in the $H\to\tau\tau\to e^\pm\mu^\mp\sla{p}_T$ search with forward jet tagging cuts [@PRZ_TauTau], the phase space region with $p_T < 20$ GeV for the untagged $b$- (or $\bar b$-) quark contributes only 8% to the total cross section.
The smallness of these collinear effects is related to the fact that we are generating jets and $b$-quarks of $p_T>20$ GeV as explicit partons in our calculations. In order to avoid double counting, the factorization scale in the $b$-quark PDF should then be chosen as $\mu_f=20$ GeV, which mitigates the role of the collinear logarithms, $\alpha_s \log(\mu_f^2/m_b^2)$.[^3] We conclude that a special treatment of collinear effects is not required in our studies. We regularize the $b$-quark collinear region by the finite $b$-quark mass, which provides a simplified but adequate model for the $b$-quark PDF.
Differential cross sections for top production in the narrow-width approximation are independent of $m_H$. However, once all off-shell effects are included, a dependence on $m_H$ is caused by Higgs propagators that appear in the non-resonant contributions of Fig. \[fig:offshell\](f). This dependence has a negligible effect on the rates considered in this paper.
Phase Space Generator for Production and Numerical Tests {#subsec:phasespace}
--------------------------------------------------------
{width="9.0cm"}\
\[fig:phasespace\]
While a simple phase space generator proved sufficient for $t\bar t$ production, for $t\bar tj$ production with significant selection cuts a composite phase space generator that interfaces optimized mappings for double-, single- and non-resonant phase space regions becomes a necessity. These three regions are defined with the help of the two variables
n\_t = n\_[|t]{} = \[ps-vars\]
The double-resonant region is then defined by $|n_t|, |n_{\bar t}| < n_c$, the single-resonant region by $(|n_t| < n_c \wedge |n_{\bar t}| > n_c) \vee (|n_t| > n_c
\wedge |n_{\bar t}| < n_c)$ and the non-resonant region by $|n_t|, |n_{\bar t}| > n_c$ (see Fig. \[fig:phasespace\]). In our calculations the boundary parameter $n_c$ is chosen to be 8 by default (but see tests below). Since the runtime of the $t\bar tj$ program is fairly long, the code currently evaluates only the $S_t$ region and multiplies the result by two to account for $S_{\bar t}$. This symmetry assumption can easily be removed in the source code should the need arise.
To assure the correctness of the programs three tests were performed. First, the Lorentz-invariance of the modified matrix elements is highly sensitive to errors. A suitable test variable can be defined in the following way: One generates a phase space configuration and evaluates the matrix element squared, summed over all helicity combinations.[^4] Applying an arbitrary boost (with $\gamma < 5$) to all external momenta and re-evaluating the matrix element, one finally computes
|1 - |. \[eq:test-var\]
Evaluating $50\!\cdot\! 10^6$ phase space configurations (within the forward jet tagging cuts of Ref. [@PRZ_TauTau]), we found an average (maximal) test variable of $2\!\cdot\! 10^{-10}$ ($7\!\cdot\! 10^{-8}$) for $t\bar t$ and $1\!\cdot\! 10^{-10}$ ($2\!\cdot\! 10^{-5}$) for $t\bar tj$.\
The test variable is highly sensitive to errors in the matrix element: Omitting the overall factors for a single amplitude that corresponds to a single-resonant Feynman diagram, increases the average and maximal test variable by 5-6 orders of magnitude. Second, the same test variable was used to check the modifications that are necessary to implement the reduction in Fig. \[fig:ampsum\].
A more recent MADGRAPH version can be used (with minor modifications) to automatically generate the full matrix elements of Eqs. (\[eq:tt-mat\]) and (\[eq:ttj-mat\]) with 8 and 9 external particles, respectively. These matrix elements can then be compared numerically to the factorized matrix elements in our programs. All overall factors need to be set to 1 for this test and finite widths for space-like top propagators need to be restored. A single, wrong coupling constant, e.g. $g_{Zu}$ rather than $g_{Zd}$, increases the average and maximal test values by 11-13 orders of magnitude.
Finally, the phase space generation for $t\bar t$ has been tested by comparing with known cross sections for the Tevatron and the LHC. The composite phase space generation for $t\bar tj$ has been checked by moving the boundary between phase space regions, i.e. changing the value of $n_c$. For different cut sets and $n_c =$ 4, 8 and 16 top-widths, we obtained results consistent within statistical errors of less than 1%. For $t\bar tj$ one can explicitly take the narrow top-width limit and compare with the results in Ref. [@RZ_WW], for example. The programs passed all tests.
In order to achieve an accuracy of 1% in a practical time, the programs use an enhanced version[^5] of the VEGAS algorithm [@VEGAS1; @VEGAS2]. It applies importance sampling also to the summation of physical helicity combinations and separately optimizes suitably chosen combinations of subprocesses/phase space regions.
Numerical Results
-----------------
First numerical results were obtained using CTEQ4L parton distribution functions as a default, with[^6] $\alpha_s(m_Z) = 0.132$[@CTEQ4_PDF]. The renormalization and factorization scales $\mu_{\scriptstyle r,f}$ are fixed to the top mass, $m_t = 175$ GeV. In contrast to the Tevatron, top production at the LHC is dominated by the gluon-gluon channel and hence noticeably affected by uncertainties of the gluon density. To assess the impact of effects related to PDF and $\alpha_s(m_Z)$ choice, we compared CTEQ4L and CTEQ5L parton distribution functions [@CTEQ5_PDF]. Studying the basic process $pp\to t\bar t$, one obtains 622 and 510 pb for CTEQ4L and $\alpha_s(m_Z) = 0.132$ and 0.118, respectively. On the other hand, for CTEQ5L and $\alpha_s(m_Z) = 0.127$ and 0.118 one gets 560 and 487 pb, respectively, suggesting an overall uncertainty of about 10 or 20% related to the choice of PDF and $\alpha_s(m_Z)$ related to it. With selection cuts, we found similar deviations. For example, in Table \[tab:tau\_data\] below, switching from CTEQ4L to CTEQ5L leads to a 4-8% decrease in cross sections at the forward tagging cuts level.
\[tab:incl\_data\]
process full ${\cal O}(\alpha_s^2\alpha^{2,4})$ $|m_{bW}-m_t|>10/15\Gamma_t$ full - NWA rel.contr. $W^-t+W^+\bar t$ rel.contr.
--------------------------------------- ----------------------------------------- ------------------------------ ------------ ------------ --------------------- ---------------
$pp\to b\bar bWW$ 622 63.9/49.6 25 $+4\%$ $66.4^\ast/51.6^\S$ $+11.1/8.6\%$
$pp\to b\bar b\ell\bar\nu\bar\ell\nu$ 39.2 4.0/3.1 1.4 +4% 4.0 +10%
: Comparison of inclusive cross sections with and without finite top width effects. All results are given in pb. The first four columns represent our results: full $t\bar t$ cross section including off shell effects, contribution within $|m_{bW^+}-m_t| > n_c\Gamma_t$ or $|m_{\bar bW^-}-m_t| > n_c\Gamma_t$ with $n_c=$ 10 and 15, difference to the NWA result, and this difference as a fraction of the total. The $Wt$ column uses results from Table 1 in Ref. [@Boos_SingleTop_Followup] for $pp\to b\bar bWW$ and Table 1 in Ref. [@Trefzger_Higgs] for $pp\to b\bar
b\ell\bar\nu\bar\ell\nu$. The results in the second row include leptonic $\tau$ decays, but not $b$ decays. In Ref. [@Boos_SingleTop_Followup] different methods are used for $Wt$ to reduce overlap with $t\bar t$: $\ \ ^{\ast)}\ |m_{bW}-m_t|
> 10\Gamma_t \ {\rm cut}\quad ^{\S)}\ |m_{bW}-m_t| > 15\Gamma_t$ cut.
A first comparison of our full calculation with single top production cross sections in the literature is presented in Table \[tab:incl\_data\]. The two rows correspond to $b\bar bWW$ and $b\bar b\ell^+\nu\ell^-\bar\nu$ final states, i.e. in the second row off-shell-$W$ corrections are included in our calculation. In our simulation, $b\bar be^+\nu\mu^-\bar\nu$ results were multiplied by a factor 5.48 to account for all lepton flavor combinations, including leptonic $\tau^\pm$ decays. Off-shell contributions from $W^-t+W^+\bar t$ final states were calculated in Ref. [@Boos_SingleTop_Followup] with several definitions of the off-shell region. These results agree with ours at the 4% level (see second column). The agreement improves to the 1–2% level once double counting effects in the non-resonant regions ($N$-regions of Fig. \[fig:phasespace\]) are taken into account. Compared to this $W^-t+W^+\bar t$ cross section, however, the actual enhancement of the $t\bar t$ cross section due to off-shell contributions, calculated as the difference of our full result minus the cross section in the NWA (see column three), is only about half as large.
The problem can be traced to the fact that a Breit-Wigner distribution has long tails, more precisely \_[(m - n )\^2]{}\^[(m + n )\^2]{} (1 - ) i.e. for each of the two top-quark resonances, about 2% of the cross section is located outside an $n=15$ top-widths window, for example. This 4% resonant contribution, which is double-counted when combining the rate in the $|m_{bW}-m_t| > 15\Gamma_t$ region with the $t\bar t$ cross section in the narrow width approximation, accounts for the difference between our “full minus NWA” result and the estimate in terms of the $W^-t+W^+\bar t$ cross section. This means that off-shell $t\bar t$ calculations which directly take $W^-t+W^+\bar t$ cross sections as the off-shell rate, tend to produce a serious overestimate of the number of extra events.
Our method of choice to include finite-width effects involves overall factors that are expected to be small in non-resonant regions as explained in Section \[sec:method\]. In order to test this expectation in a realistic context, we compared the differential cross section obtained with overall factors with various proxies in different phase space regions (see Fig. \[fig:phasespace\]). In the non-resonant region $N$ the proxy is the tree-level matrix element with unmodified tree-level top quark propagators, because they are not singular and a good approximation to the full propagator in this region. In the single-resonant regions $S_t$ and $S_{\bar t}$ the proxy is derived from the proxy used in region $N$ by replacing the potentially singular top (in $S_t$) or anti-top (in $S_{\bar t}$) propagators with fixed-width propagators, i.e. . \[eq:fixed-width-prop\]
![Integrated transverse momentum distribution for the additional massless parton (jet) in off-shell $t\bar tj$ production: $\sigma(p_{Tj}^{\text{min}}) =\int_{p_{Tj}^{\text{min}}}^{\infty}
(d\sigma/dp_{Tj})\,dp_{Tj}$. No selection cuts are applied. For $p_{Tj}^{\rm min} = 40$ GeV one obtains $\sigma_{t\bar
tj} \approx \sigma_{t\bar t} = 39.2$ pb (dotted line, see Table \[tab:incl\_data\]).](incl_cut.ps "fig:"){width="9.0cm"}\
\[fig:ptj\_dist\]
In the double-resonant region $D$ the proxy matrix element is a subset of all Feynman diagrams, namely all diagrams with at least one time-like top propagator. Since all these top propagators are potentially resonant here, they all feature the fixed-width form of Eq. (\[eq:fixed-width-prop\]). Each region is covered by 20-28 bins. Bin sizes are adjusted so that each value roughly has the same integration error. For each bin the relative deviation from the proxy is estimated by |\_[bin]{} d\_[fac]{} - \_[bin]{} d\_[proxy]{}| / \_[bin]{} d\_[proxy]{} . . \[eq:proxy-deviation\] We studied this measure for $t\bar t$ production without cuts as well as the forward tagging cuts of Refs. [@RZ_WW] and [@PRZ_TauTau] and found similar results. In the double- and single-resonant regions the overall factor scheme deviates from the proxies by 2% or less, and the deviation drops to less than 1% in the non-resonant region. We therefore estimate the error associated with our finite-width scheme to be of ${\cal O}$(1%), which is comparable to the statistical error of our results, but completely negligible compared to missing higher order QCD corrections. These results suggest that our method to include finite-width effects provides reliable results, not only for fairly inclusive cross sections, where finite width effects are strongly suppressed, but also when complex selection cuts result in on- and off-shell contributions of similar size.
Comparing inclusive cross sections, we find $B\sigma_{t\bar t} = 39.2$ pb in Table \[tab:incl\_data\]. Including an additional jet we obtain $B\sigma_{t\bar tj}=80$ pb, when a common $p_T > 20$ GeV cut is imposed on the additional jet. This raises the question, whether the cut should typically be chosen higher, such that $\sigma_{t\bar tj} \lesssim \sigma_{t\bar t}$, i.e. $\sigma_{t\bar tj}$ is sub-dominant, as typically expected for a NLO QCD correction. The integrated $p_{Tj}$ distribution of the non-$b$ parton is shown in Fig. \[fig:ptj\_dist\]. For $p_{Tj}^{\rm min} = 40$ GeV one obtains $\sigma_{t\bar tj} \approx \sigma_{t\bar t}$. Real parton emission cross sections saturating the LO cross section at low $p_T$ of the extra parton are a sign of copious multi-jet production in actual data [@rsz].
The additional parton emission is dominated by initial state radiation. This can be inferred from the invariant mass distributions of the potential top-quark decay products, the $bW^+$ and the $bW^+j$ systems shown in Fig. \[fig:bW\_mass\]. The $m_{bW^+}$ invariant mass distribution in Fig. \[fig:bW\_mass\](a) contains 85% of the total cross section in the displayed 165-185 GeV window around the top resonance ($\pm6.4\Gamma_t$). In contrast, the same $m_{bW^+j}$ invariant mass window (insert of Fig. \[fig:bW\_mass\](b)) accounts for only 10% of the total cross section. Final state radiation is relatively unimportant in $t\bar tj$ production at the LHC.
{width="7.0cm"}
{width="7.0cm"}
\[fig:bW\_mass\]
Application to SM Higgs search at the LHC {#sec:application}
=========================================
\[tab:tdr\_data\]
$m_T$ window \[GeV\] top in NWA full off-shell rel. chg. $t\bar{t}$ $t\bar{t}+Wt$ rel. chg.
---------------------- ------------ ---------------- ----------- ------------ --------------- -----------
120–150 50 93 1.9 41 206 5.0
130–160 50 96 1.9 66 204 3.1
140–170 43 83 1.9 51 146 2.9
140–180 49 95 1.9 61 164 2.7
150–190 35 68 1.9 56 111 2.0
: Numbers of expected $pp\to b\bar b\ell\bar\nu\bar\ell\nu$ background events in the ATLAS $H\to WW^*$ Higgs search for an integrated luminosity of 30 fb$^{-1}$. Basic ATLAS TDR background suppression cuts are applied [@ATLAS_TDR_2; @Trefzger_Higgs]. ATLAS detector effects are simulated as described in Section II.F in Ref. [@RZ_WW]. The left columns show our results for the four $W\to e\nu,\mu\nu$ lepton combinations. The right columns are the corresponding results from Table 2 in Ref. [@Trefzger_Higgs]. For both categories, the relative change due to the inclusion of finite width effects is given as an enhancement factor in the third column.
An important application of $t\bar t{j}$ production as a background occurs in Higgs physics. Higgs mass limits have recently been pushed to $m_H>114.1$ GeV by the LEP experiments [@lep2higgs]. As a result, the LHC search for $H\to WW$ [@DittDrein; @CMS; @ATLAS_TDR_2; @Trefzger_Higgs; @KPRZ; @RZ_WW] and $H\to\tau\tau$ [@ATLAS_TDR_2; @RZH_tau; @PRZ_TauTau] decays in the intermediate mass range has gained even greater importance. For the $H\to WW\to \ell^\pm\ell^\mp\sla{p}_T$ decay mode, top-quark decays constitute the largest reducible background. The impact of this background has been analyzed, in the narrow-width approximation, for Higgs masses around 170 GeV [@ATLAS_TDR_2; @Trefzger_Higgs; @RZ_WW] and most recently for a light Higgs boson with $m_H\approx 115$ GeV [@KPRZ]. This background also plays a role in the $H\to\tau\tau\to
e^\pm\mu^\mp\sla{p}_T$ decay mode, which was analyzed in the narrow-width approximation in Ref. [@PRZ_TauTau]. In this section we present updated results for these analyses obtained with our parton-level Monte Carlo program that includes off-shell top and $W$ effects and takes into account the single-resonant and non-resonant contributions.[^7]
Backgrounds to Inclusive $H\to WW$ Searches
--------------------------------------------
The $t\bar t$ background calculations (without an additional jet) are most relevant for inclusive $H\to WW$ searches [@DittDrein; @CMS; @ATLAS_TDR_2], where Higgs production is dominated by the gluon fusion process. In Tables \[tab:tdr\_data\] and \[tab:opt\_data\] we compare the relative contributions from off resonant top effects for two selections of $H\to WW\to \ell^\pm\ell^\mp\sla{p}_T$ events in ATLAS, as described in Refs. [@ATLAS_TDR_2; @Trefzger_Higgs]. The selection looks for two isolated, opposite charge leptons of $p_T>20,10$ GeV within the pseudo-rapidity range $|\eta_\ell|<2.5$ and of invariant mass $m_{\ell\ell}<80$ GeV. Events must have significant missing $E_T$, $\sla E_T>40$ GeV. A small angle between the charged leptons favors $H\to WW$ decays versus backgrounds. Finally, a veto on additional jets in the central region is very effective against the $b$-quark jets in the top-quark backgrounds. The main difference between Tables \[tab:tdr\_data\] and \[tab:opt\_data\] is the definition of this veto on jet activity with $p_T>15$ GeV in the central region. It is imposed within $|\eta_j|<3.2$ in Table \[tab:tdr\_data\] and within $|\eta_j|<2.4$ in Table \[tab:opt\_data\].
![Transverse mass distribution for the low-luminosity ATLAS selection cuts of Ref. [@ATLAS_TDR_2], Section 19.2.6, and Ref. [@Trefzger_Higgs]. The dotted curve shows the distribution obtained when treating the top quarks in the narrow-width approximation while the solid curve includes all off-shell effects. ](comparisons.ps "fig:"){width="9.0cm"}\
\[fig:mT\]
In addition, the selection looks for events inside the Jacobian peak of the dilepton-$\sla E_T$ transverse mass distribution, as indicated in the first column of the two tables. Here the transverse mass is defined as m\_T(,E\_T) = . \[eq:mtrans\] Fig. \[fig:mT\] shows this transverse mass distribution underlying the ATLAS TDR analysis. Off-shell contributions raise the normalization of the $t\bar t$ background by about a factor of 2, but have little effect on the shape of the background, for $m_T \gtrsim 120$ GeV. The ATLAS analysis attempts to take (single-resonant) off-shell effects into account via on-shell $Wt$ calculations. As the comparison in Table \[tab:tdr\_data\] shows, our unified calculation indicates a lower background increase than the combined on-shell $t\bar t$ and $Wt$ calculations would suggest. This observation is consistent with the small increase due to off-resonant effects observed in Table \[tab:incl\_data\].
For a precise comparison with the ATLAS simulation, the nature of the programs (parton-level MC with full matrix elements vs. event generator with parton showers/hadronization) is currently too different. In particular our program does not allow to simulate the effect of the central jet veto on extra gluon radiation in the event. Also, we did not include any detector efficiencies. We would expect an extra suppression, by perhaps a factor of 3 to 4, of the “full off-shell” results in Tables \[tab:tdr\_data\] and \[tab:opt\_data\] due to these effects, i.e. the apparent agreement of the $t\bar t$ cross section with our NWA result in Table \[tab:tdr\_data\] is somewhat fortuitous. This suspicion is confirmed by the fairly large disagreement between our NWA or full off-shell results and the PYTHIA simulation in Table \[tab:opt\_data\].
\[tab:opt\_data\]
$m_T$ window \[GeV\] top in NWA full off-shell rel. chg. $t\bar{t}$ $t\bar{t}+Wt$ rel. chg.
---------------------- ------------ ---------------- ----------- ------------ --------------- -----------
120–150 364 536 1.5 107 346 3.2
130–160 363 544 1.5 122 306 2.5
140–170 310 465 1.5 82 215 2.6
140–180 352 528 1.5 107 254 2.4
150–190 251 374 1.5 87 168 1.9
: Same as Table \[tab:tdr\_data\], but for the optimized background suppression cuts of Ref. [@Trefzger_Higgs].
At present the source of these discrepancies is not completely understood. The rapidity distribution of $b$-quarks in our full matrix element calculation appears to be wider than in the PYTHIA generated events, potentially due to approximations in the top decay chain in PYTHIA. This makes a veto over a smaller pseudo-rapidity range less efficient. On the other hand we cannot simulate the effect of additional gluon radiation. A reanalysis of these effects, combining full matrix elements with a parton shower Monte Carlo, including hadronization, is clearly warranted, given that top-quark production constitutes about half the background to the inclusive $H\to WW$ search[@Trefzger_Higgs].
While the overall background normalization requires further study, the ratio of “full off-shell” and NWA results is expected to be robust, i.e. it will be little affected by detector efficiencies and by higher order gluon radiation. These ratios are given in the third columns, marked “relative change”, in Tables \[tab:tdr\_data\] and \[tab:opt\_data\]. Our results clearly indicate that the increase in top-backgrounds due to the inclusion of off-shell effects (the $Wt$ contribution in the ATLAS analysis) is substantially smaller than previously thought.
In a LO calculation, substantial uncertainties arise from ambiguities in the choice of factorization and renormalization scales. For the double resonant phase space configurations, $\mu_{\scriptstyle r}=\mu_{\scriptstyle f}=m_t$ or the transverse energy of the top quarks are well-motivated choices. When forcing the $b$-quarks to low $p_T$ values by a central jet veto and simultaneously enhancing the off-shell phase space regions, a smaller renormalization and/or factorization scale may be more appropriate. The impact of a lower scale on the full off-shell results in Table \[tab:tdr\_data\] is demonstrated by the scale choice \_[r,f]{} = {
[r@l]{} m\_t & |m\_[bW\^+]{}-m\_t| < 4\_t |m\_[|bW\^-]{}-m\_t| < 4\_t\
20 & |m\_[bW\^+]{}-m\_t| > 4\_t |m\_[|bW\^-]{}-m\_t| > 4\_t
. \[eq:lower-scale\] where the value of 20 GeV is motivated by the veto threshold for central jets. Resulting cross sections are about 65% higher than the rates obtained with $m_t$ as a universal scale. A NLO calculation would be needed to distinguish the virtue of either choice. At present, this variation indicates the uncertainty of our LO results.
Backgrounds in Weak Boson Fusion Studies
-----------------------------------------
In addition to the inclusive search for $H\to WW$ events, weak boson fusion (WBF) presents a very attractive search channel for $H\to WW$ and $H\to\tau\tau$ events and will likely play a crucial role in the measurement of Higgs boson couplings to fermions and gauge bosons [@knrz]. Top quark decays again form an important background in these searches and, due to the central jet veto proposed for the reduction of QCD backgrounds, off-shell effects might be important. The signal and backgrounds for $H\to WW\to e^\pm\mu^\mp\sla{p}_T$ and $H\to\tau\tau\to e^\pm\mu^\mp\sla{p}_T$ in WBF were analyzed in Refs. [@RZ_WW] and [@PRZ_TauTau], respectively, with the top-quark backgrounds determined in the NWA, however. With our new programs we are able to update these background estimates, including off-shell contributions.
Because of the two additional jets which are present in the WBF process $qq\to qqH$, the dominant top quark background arises from $t\bar tj$ events. Event selection for WBF requires two tagging jets, of $p_T>20$ GeV, which are widely separated in pseudo-rapidity ($|\eta_1 - \eta_2|>4.2$) and which have a very large dijet invariant mass, $M_{jj}>650$ GeV. By definition, the $t\bar t$ background has both $b$-quarks identified as tagging jets while in the $t\bar tj$ background exactly one $b$ or $\bar b$ is taken as a tagging jet. The two $b$-quarks rarely have a large enough dijet mass or are far enough separated to satisfy the tagging criteria, and this leaves $t\bar tj$ events as the dominant background to WBF. This result holds in both the NWA and with inclusion of off-resonant effects as is evident in both Tables \[tab:WW\_data\] and \[tab:tau\_data\], at all cut levels.
The veto of central jets of $p_T>20$ GeV is effective against the extra $b$-quark jet and off-shell contributions are only slightly enhanced after this cut (see line “$b$ veto” in the Tables). Overall, off-shell contributions are fairly modest, increasing the NWA results by about 20% (see the second last columns in the Tables). This means that our new complete calculation of off-shell effects in $t\bar tj$ production confirms the conclusions about the observability of $H\to\tau\tau$ and $H\to WW$ events reached in Refs. [@RZ_WW] and [@PRZ_TauTau]. Tables \[tab:WW\_data\] and \[tab:tau\_data\] display our updated results. Precise definitions of cuts are given in the earlier papers. A breakdown into subprocesses and phase space regions of the overall $t\bar tj$ background of 351 fb to $H\to WW\to e^\pm\mu^\mp\sla{p}_T$, after forward jet tagging cuts, is given in Table \[tab:WWbreakdown\].
\[tab:WW\_data\]
------------------------------------------- ------------ ------------- --------------- -------- ------- ------ ------ ---------------
cuts $t\bar{t}$ $t\bar{t}j$ S/B S/B
forward tagging (10)-(12) 12.4 308 $\approx$1/65 13.0 +4.4% 351 +14% $\approx$1/67
+ $b$ veto (13) 43.5 1/5.1 51.4 +18% 1/5.6
+ $M_{jj}$, angular cuts (14)-(16) 0.0551 4.67 1.1/1 0.0761 +38% 5.42 +16% 1.0/1
+ real $\tau$ rejection (17) 0.0527 4.34 1.7/1 0.0737 +40% 5.09 +17% 1.5/1
$P_{surv,20}$ (${\it\times 0.29}$) + (18) 0.0153 1.26 4.6/1 0.0214 +40% 1.48 +17% 4.2/1
+ tag ID efficiency (${\it\times 0.74}$) 0.0113 0.932 4.6/1 0.0158 +40% 1.09 +17% 4.2/1
------------------------------------------- ------------ ------------- --------------- -------- ------- ------ ------ ---------------
: $t\bar{t}(j)$ background cross sections for $H\to WW\to e^\pm\mu^\mp\sla{p}_T$ for $m_H = 160$ GeV in $pp$ collisions at $\protect\sqrt{s}=14$ TeV. Results are given for various levels of cuts and are labeled by equation numbers from Ref. [@RZ_WW]. All cross sections are given in fb. Cuts and other calculational details are described in Ref. [@RZ_WW]. The integration error is 1% or better. The signal over background ratio is also shown. Cross sections not listed here are as in Table I in Ref. [@RZ_WW].
\[tab:tau\_data\]
----------------------------------- ------------ ------------- -------- --------- ------ ------- ------ --------
cuts $t\bar{t}$ $t\bar{t}j$ S/B S/B
forward tagging (7)-(10) 13.5 357 1/1100 15.9 +17% 436 +22% 1/1100
+ $b$ veto (11) 50.1 1/550 63.6 +27% 1/550
+ $\sla{p}_T$ (12) 11.1 43.0 1/74 13.2 +19% 55.2 +28% 1/83
+ $M_{jj}$ (13) 0.593 12.9 1/32 0.712 +20% 15.8 +22% 1/34
+ non-$\tau$ reject. (14, 15, 17) 0.00303 0.257 1/5.8 0.00365 +20% 0.293 +14% 1/5.8
----------------------------------- ------------ ------------- -------- --------- ------ ------- ------ --------
: $t\bar{t}(j)$ background cross sections for $H\to\tau\tau\to e^\pm\mu^\mp\sla{p}_T$ for $M_H = 120$ GeV in $pp$ collisions at $\protect\sqrt{s}=14$ TeV. Results are given for various levels of cuts and are labeled by equation numbers from Ref. [@PRZ_TauTau]. All cross sections are given in fb. Cuts and other calculational details are described in Ref. [@PRZ_TauTau]. The integration error is 1% or better. The signal over background ratio is also shown. Cross sections not listed here are as in Table I in Ref. [@PRZ_TauTau].
\[tab:WWbreakdown\]
$D$ $S_t+S_{\bar t}$ $N$
--------- ------ ------------------ -------
$gg$ 171 28.6 1.1
$gq+qg$ 128 18.8 0.64
$qq$ 2.92 0.35 0.009
: Distribution of the $t\bar tj$ background to $H\to WW\to
e^\pm\mu^\mp\sla{p}_T$, with forward tagging cuts, among subprocesses (labeled by initial partons) and phase space regions (see Fig. \[fig:phasespace\]). The cut between double resonant ($D$), single resonant ($S_t+S_{\bar t}$), and non-resonant regions ($N$) is set at 8 top quark widths. Cross sections are given in fb.
Summary {#sec:summary}
=======
Top-quarks are a very important source of lepton backgrounds to new physics searches at the LHC: the large production cross section typical for a strong interaction process combines with a sizable branching fraction into leptons which, due to the large mass of the $W$, often survive isolation cuts. Suppression techniques for top quark backgrounds, like a veto on central jets, which is very effective against the $b$-quarks of the $t\to bW\to b\ell\nu$ decay chain, enhance the relative importance of off-resonant effects and may exacerbate errors introduced by approximate modeling of matrix elements. Severe cuts select the tails of various distributions and these phase space regions may well differ from the ones for which the models were optimized originally. General purpose Monte Carlo programs like PYTHIA[@pythia] or Herwig[@herwig] should thus be gauged against matrix element programs.
In this paper we have presented results for two new programs which allow to model the $t\bar t\to b\bar bWW \to b\bar b \ell^+\nu\ell^-\bar\nu$ decay chain at tree level, including full angular correlations of all top and $W$ decay products and with proper interpolation between double-resonant, single-resonant and non-resonant phase space regions. These full correlations are available for $t\bar t$ and $t\bar tj$ production. Electroweak and $SU(3)$ gauge invariance is maintained throughout, by employing the overall factor scheme for the Breit Wigner propagators of all unstable particles in the $t$ and $\bar t$ decay chains.
Comparing to earlier calculations of off-shell effects, via the $gb\to Wtb$ production cross sections, we find excellent agreement when avoiding the top-quark resonance for the $Wb$ system. However, some earlier combinations of $t\bar t$ and $Wtb$ cross sections have involved substantial double counting, leading to an overestimate of backgrounds in e.g. Higgs search analyses at the LHC. In addition, the full simulation of $V-A$ couplings in the $t\to Wb\to \ell\nu b$ decay chain is available with our programs and may have sizable effects in background estimates.
When considering $t\bar t$ backgrounds to Higgs searches, off-shell effects are most pronounced in inclusive $H\to WW$ analyses, where both $b$-quark jets may be vetoed. In weak boson fusion studies the additional jets in the signal selection make the background suppression due to the jet veto less severe, which diminishes the overall importance of off-shell contributions.
Our programs now allow detailed study of these off-shell effects in top pair backgrounds with zero or one additional parton in the final state.
We thank K. Jakobs, T. Trefzger and T. Plehn for useful discussions. This research was supported in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation and in part by the U. S. Department of Energy under Contract No. DE-FG02-95ER40896.
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[^1]: It should be noted that the simplifications that led to concise results for the effective $WW\gamma$ and $WW\gamma\gamma$ vertices (see Refs. [@Baur:1995aa; @Kauer_Master; @Baur:1997bn]) can not be applied when evaluating the effective $ttg$ vertex.
[^2]: Code generators have some freedom in the composition of basic elements when constructing helicity amplitudes, as explained in Section 2.7 of Ref. [@HELAS]. In rare cases, the algorithm employed by MADGRAPH is incompatible with the factorization outlined in Fig. \[fig:ampsum\]. In these cases, suitable amplitudes were composed by hand.
[^3]: As in any LO multi-parton calculation, the dangerous large logarithms are of the form $\alpha_s \log (m_t^2/p_{Tj}^2)$. An improvement of our calculation would first of all require the resummation of these contributions. Initial state collinear logs from $g\to b\bar b$ splitting are unimportant by comparison.
[^4]: Because of the massive external $b$-quarks, the matrix element for any particular helicity combination is not boost-invariant.
[^5]: The code is available at http://hepsource.org/dvegas/.
[^6]: This $\alpha_s(m_Z)$ is based on $\Lambda^{(5)}_{QCD}$ as determined in the PDF set fit.
[^7]: The analysis in Ref. [@KPRZ] already features off-shell $t\bar t$ and $t\bar tj$ background estimates calculated with the programs presented here.
|
---
abstract: 'We investigate the color-magnitude distribution in the rich cluster AC118 at $z=0.31$. The sample is selected by the photometric redshift technique, allowing to study a wide range of properties of stellar populations, and is complete in the K-band, allowing to study these properties up to a given galaxy mass. We use galaxy templates based on population synthesis models to translate the physical properties of the stellar populations - formation epoch, time-scale of star formation, and metallicity - into observed magnitudes and colors. The distributions of galaxies in color-magnitude space thus map into distributions in the space of physical parameters. This is achieved by means of a statistical procedure which constrains the photometric properties of AC118 galaxies to reproduce those of a nearby rich cluster once evolved at $z\sim0$. In this way we show that a sharp luminosity-metallicity relation is inferred without any assumption on the galaxy formation scenario (either monolithic or hierarchical). Our data exclude significant differences in star formation histories along the color-magnitude relation, and therefore confirm a pure metallicity interpretation for its origin, with an early ($z\sim5$) formation epoch for the bulk of stellar populations. The dispersion in the color-magnitude diagram implies that fainter galaxies in our sample (K$\sim18$) ceased to form stars as late as $z\sim 0.5$, in agreement with the picture that these galaxies were recently accreted into the cluster environment. The trend with redshift of the total stellar mass shows that half of the luminous mass in AC118 was already formed at $z \sim 2$, but also that 20% of the stars formed at $z<1$.'
author:
- 'P. Merluzzi'
- 'F. La Barbera'
- 'M. Massarotti and G. Busarello'
- 'M. Capaccioli'
title: 'Age, Metallicity and Star Formation History of Cluster Galaxies at $z\sim 0.3$[^1]'
---
Introduction {#INTRO}
============
The color-magnitude relation (CMR) of cluster early-type galaxies has been extensively investigated at $z < 1$ to trace their star formation history and hence to constrain their formation epoch (e.g. e.g Kodama & Arimoto 1997, hereafter KA97; Ellis et al. 1997; Gladders et al. 1998; Stanford, Eisenhardt, & Dickinson 1998; Kodama & Bower 2001, hereafter KB01; Smail et al. 2001; van Dokkum et al. 2001).
The most important observational results are: i) the slope of the CMR does not depend on redshift; ii) the optical-NIR rest-frame colors of early-type cluster members become bluer with increasing redshift; iii) the intrinsic scatter in the optical-NIR colors of early-type galaxies is small at all redshifts (e.g. Ellis et al. 1997; Stanford et al. 1998; van Dokkum et al. 1998, 2000; Kodama et al. 2001). These points lead to explain the color-magnitude (CM) sequence as a correlation between galaxy mass and metallicity, while the age of galaxies play only a marginal role, if any (e.g. KA97). Two different scenarios can successfully explain the CMR as function of redshift: the monolithic collapse (e.g. Eggen, Lynden-Bell, & Sandage 1962; Tinsley & Gunn 1976) and the hierarchical merging (e.g. Kauffmann 1996, Kauffmann & Charlot 1998). In the former, the trend of the mass-metallicity sequence is explained by the fact that the more massive galaxies retain supernova ejecta more effectively, resulting in higher metallicities and hence in redder colors for more luminous galaxies (e.g. Arimoto & Yoshii 1987; KA97). The unchanged scatter of the colors of early-type galaxies with redshift indicates either that the galaxies assembled synchronously over redshifts (at least for $z<1$) or that they stochastically formed at much earlier times (see Ellis et al. 1997). For what concerns the alternative picture, Kauffmann & Charlot (1998) claimed that the CMR can be reproduced in a hierarchical merging picture, where the more massive/metal-rich ellipticals result from mergers of massive/metal-rich progenitor disk galaxies. In both scenarios the color evolution of early-type cluster galaxies is in agreement with the passive evolution of an old stellar population formed early in the past (see also Stanford et al. 1998; Kodama et al. 1998).
Both the main evolutionary scenarios have to face with the evidence for the presence of a significant population of blue galaxies in rich cluster environments at $z\geq$ 0.2, as shown for the first time by Butcher & Oelmer (1978) and has confirmed by several photometric and spectroscopic observations (e.g. Butcher & Oelmer 1984; Couch & Newell 1984; Ellis et al. 1985; Dressler & Gunn 1982; Couch & Sharples 1987, Couch et al. 1994; Dressler et al. 1994). Taking into account a representative sample of the whole cluster population, KB01 re-investigated the photometric Butcher-Oelmer (B-O) effect in distant clusters. They found that the passive evolution of galaxy populations can reconcile the B-O effect with the tight CMR of the Coma cluster. Furthermore, KB01 found that the distribution in the color-magnitude diagrams suggests a scenario where star formation of galaxies accreted by the cluster declines on a 1 Gyr time-scale and it is not sharply truncated by interaction with the cluster environment. In this scenario, the B-O effect depends on the decline of star formation of field galaxies when they are accreted into the cluster and on the decline of the rate of accretion of new galaxies at lower redshifts.
In the present work we will apply the CM diagram to gain insight into the star formation history in the galaxy cluster AC118 at $z=0.31$. We will use population synthesis models in order to describe the observed CM distribution of galaxies in AC118 in terms of stellar populations parameters. The cluster sample is selected according to the photometric redshift technique, and is complete in K-band, avoiding biases introduced by measuring the blue wavelengths in the cluster rest-frame. The early-type galaxy population in the core of AC118 was already analyzed by Stanford et al. (1998) who found evidence in favor of the passive evolution scenario. A spectroscopic study of the cluster was performed by Couch & Sharples (1987) and Barger et al. (1996) who claimed for recent ($\lesssim 2$ Gyr) bursts of star formation.
The layout of the paper is the following. In § 2 we describe the sample of galaxies at $z\sim 0.3$. In § 3 we introduce the galaxy templates that will be used to interpret the observed photometry in terms of physical properties of stellar populations and we describe our approach. The resulting distribution of the physical parameters is analyzed in § 4, where we also discuss the origins of the CMR and the global star formation history. In § 5 we summarize the main aspects of the work and draw the conclusions. In the following we assume $\Omega_m=0.3$, $\Omega_\Lambda=0.7$ and $\mathrm{ H_0= 70~Km s^{-1} Mpc^{-1} }$. With this cosmology the age of the universe is $\mathrm{13.5~Gyr}$, and the redshift of AC118 corresponds to a look-back time of $\mathrm{3.5~Gyr}$. We verified that changing the cosmology does not affect the results of the present work.
The Sample at $z\sim0.3$
========================
The present analysis is based on VRIK photometry for a sample of galaxies in a field of $6.0 \times 6.0 ~ \mathrm{arcmin}^2$ ($1.6
\times 1.6 ~ \mathrm{Mpc}^2 $ at $z=0.31$) centered on the galaxy cluster AC118. The optical (VRI) data are taken from the catalog in Busarello et al. (2002), which also includes photometric redshifts, while the K-band photometry is described in Andreon (2001). The present sample was selected according to the following criteria: a) galaxies are cluster members according to their photometric redshifts and b) the sample is complete in the K-band. In Figure \[SELZP\] we compare the distribution of the K-band magnitudes for the 459 member galaxies from Busarello et al. (2002) with the K-band luminosity function of AC118 by Andreon (2001), obtained by statistically subtracting field counts. The Figure shows that the trend of our counts and the luminosity function of AC118 are consistent down to $\mathrm{K}=18.25$, suggesting, therefore, that the sample of member galaxies is fairly complete down to this limit. This leads to a final sample of $\mathrm{N}=252$ galaxies brighter than $\mathrm{K}=18.25$. In order to quantify the field contamination in the redshift range adopted to select cluster members (i.e. $z \in
[0.24, 0.38]$), we note that the field population at $z\sim0.3$ is dominated by late-type galaxies bluer than $\mathrm{I-K=2.0-2.5}$, and therefore, the field contaminants in our sample are expected to be brighter than $\mathrm{I\sim20.5}$. According to the Canada France Redshift Survey (Lilly et al. 1995), we expect $\sim15$ galaxies down to $\mathrm{I=20.5}$ in the cluster area and redshift range, amounting to $6\%$ of the galaxies in the final sample. Since this estimate is an upper limit of the number of field contaminants, we conclude that foreground/background contamination is not statistically relevant to the present analysis.
The color indices were measured within a fixed circular aperture of diameter $4.4''$ ($20 ~ \mathrm{kpc}$ at $z=0.31$). In the following, we will also use the $\mathrm{V\!-\!K}$ colors derived by Bower, Lucey, & Ellis (1992a) for galaxies in the Coma cluster within an aperture of $10''$, which corresponds to $\sim 7$ kpc. Since galaxies are known to have internal color gradients, a suitable comparison between different redshifts must take into account the physical size of the aperture within which galaxy colors are derived. However, as shown by Kodama et al. 1998 (see their Figure 3), the correction from the $\mathrm{\sim 7~kpc}$ aperture to the $20~\mathrm{kpc}$ aperture turns out to be negligible for the Coma galaxies.
In order to estimate the total magnitude, we used adaptive apertures of radius $\alpha \cdot r_c$, where $r_c$ is the Kron radius (see Kron 1980). We chose $\alpha=2.5$, for which the Kron magnitude is expected to enclose $94\%$ of the total flux of the object (see Bertin $\&$ Arnouts 1996), and to correct for this factor, we added to the Kron magnitudes the term $2.5 \log(0.94)$. Since the bright cluster galaxies have extended halos[^2], the estimate of the total magnitude requires a large extrapolation of the light profile. To account for this fact, it is necessary to correct the Kron magnitudes $K_c$ of the brightest galaxies. To this aim, we compared the $K_c$ values with those derived by the two-dimensional fit of the surface brightness distribution for the subsample of $\mathrm{N}=95$ galaxies analyzed in La Barbera et al. (2002). The comparison is shown in Figure \[KCKT\] as a function of $K_c$. We found that the Kron magnitude underestimates the galaxy luminosity for values of $K_c$ brighter than $\mathrm{K\sim17}$. The trend in Figure \[KCKT\] is described by the relation $K_T-K_c = 0.13 \times K_c-2.23$ ($K_c < 17.2$), that was used to correct the values of $K_c$ for each galaxy in our sample.
In Figure \[CMCCTOT\] we show the CM distributions of 1) all the galaxies in the K-band field, 2) the galaxies with available photometric redshift and 3) the $\mathrm{N}=252$ galaxies of the sample considered in the analysis.
Modelling the Evolution of Stellar Populations {#MODEL}
===============================================
Our goal is to fit the colors and magnitudes of galaxies in AC118 by imposing that their evolution at $z\sim0$ reproduces the properties of the CM diagram of a nearby galaxy cluster. To this aim, we use stellar population models at different evolutionary stages.
Since AC118 is a rich, Coma-like, cluster with high X-ray luminosity, the properties of its galaxy population have to be compared with those of galaxies in a rich cluster at $z\sim0$. The sample of galaxies at $z\sim0.3$ covers the central cluster region ($6.0 \times 6.0 ~ \mathrm{arcmin}^2$), which corresponds to an area of radius $\mathrm{\sim1~Mpc}$. In this region, rich nearby clusters are very similar in their photometric properties. The galaxy population is dominated by early-type galaxies which follow a tight CM relation (see Bower et al. 1992a, Bower, Lucey, & Ellis 1992b), with few percents of galaxies having bluer colors (see Butcher & Oemler 1978). Therefore, we can constrain the properties of the galaxies using only the overall features of the CM diagrams at $z\sim0$, without comparing the properties of AC118 galaxies with those of a specific nearby cluster.
We describe the stellar populations in terms of their formation epoch $t_0$, time scale $\tau$ of star formation and metallicity $Z$. We do not assume any [*a priori*]{} probability distribution of $
\{t_0,\tau,Z \}$, but instead we derive it by comparison of a set of model (template) galaxies with the observed CM distributions.
Galaxy templates and the constraints at $z\sim0$ {#galtemp}
------------------------------------------------
The galaxy templates were obtained by the GISSEL98 synthesis code of Bruzual & Charlot (1993). Each template is defined by the three physical parameters $t_0$, $\tau$ and $Z$. The code allows to build galaxy templates with metallicity in the range $Z=$0.0001 – 0.1 [^3] and predicts the template properties at 220 steps in age ranging from $t=1~\mathrm{Myr}$ to $t=20 ~\mathrm{Gyr}$. The star formation rate is chosen in the form $e^{-t/\tau}$, with $\tau$ in the range 0.01 – 15.0 Gyr, with a Scalo (1986) initial mass function. For each value of $ \{t_0,\tau,Z \}$ in the grid of input parameters, we computed the magnitudes in the V-, R-, I- and K-band. For all other values of $ \{t_0,\tau,Z \}$, the magnitudes were derived by interpolation.
Since the magnitudes of the GISSEL98 templates are arbitrarily normalized to one solar mass, they are defined within an additive term, and therefore they cannot be directly compared to the observed magnitudes. We derived the additive term by using the properties of the CM distribution at $\mathrm{z\sim0}$. To this aim, for each template we computed the $\mathrm{V\!-\!K}$ color at $z\sim0$ and compared the template magnitude in the K-band with that expected for a galaxy in a rich nearby cluster with the same $\mathrm{V\!-\!K}$ color. We used the ($\mathrm{K,V\!-\!K}$) CM distribution at $z\sim0$ because i) the CM relation in the ($\mathrm{K,V\!-\!K}$) plane has well known properties (Bower et al. 1992a, Bower et al. 1992b) and ii) the V- and K-band at $z\sim0$ correspond approximately to the same rest-frame of R- and K-band[^4] at $z\sim0.3$.
In order to derive the magnitude expected for a galaxy at $z\sim0$ with a given $\mathrm{V\!-\!K}$ color, we took advantage the following properties of galaxies in the Coma cluster.
[*1–Red sequence.*]{} To describe the ($\mathrm{K,V\!-\!K}$) red sequence, we used of the CM relation by Bower et al. (1992a): $$\mathrm{ V-K = (-0.07 \pm 0.013) \cdot K + 3.92 \pm 0.20} \ \ .
\label{CM_COMA}$$ The intrinsic dispersion of this relation is $\sigma_{\mathrm{V-K}}\sim0.03$ mag along the color direction.
[*2–Blue galaxies.*]{} We described the distribution of galaxies below the red sequence by using the properties of the CM diagram for the Coma cluster recently studied by Terlevich, Caldwell, & Bower (2001). The sample is complete down to $\mathrm{K\sim16}$ ($\mathrm{K\sim13}$ for E/S0 templates at $z\sim0$), that corresponds approximately to the completeness limit of AC118 evolved to $z\sim0$. In Figure 5 of Terlevich et al. (2001) we notice that most of the objects within the completeness limit follow a tight $\mathrm{U\!-\!V}$ CMR, while a small fraction of Sp/Irr galaxies ($\sim 5\%$) are located in a rectangular region with $\mathrm{14.5<V<16}$ ($\mathrm{11.5<K<13}$) and with significantly bluer colors $\mathrm{0<U\!-\!V<0.4}$ ($\mathrm{2.6<V\!-\!K<2.9}$) with respect to the red sequence.
[*3–Luminosity function.*]{} We adopted the luminosity function (LF) in the K-band for the Coma cluster. To this aim, we used the H-band LF by de Propris et al. (1998) and Andreon & Pelló (2000) for the central regions of the Coma cluster, corrected (see note \[nota1\]) by the color term $\mathrm{H\!-\!K=0.22~mag}$.
Each template describes either i) a galaxy of the red sequence, or ii) a blue (Sp/Ir) galaxy if it lies $3\sigma_{\mathrm{V-K}}$ below the CM sequence of Eq. \[CM\_COMA\]. In case i), the K magnitude is obtained by a normal deviate of width $\sigma_{\mathrm{V\!-\!K}}/b_{\mathrm{CM}}$, where $b_{\mathrm{CM}}$ is the slope of the ($\mathrm{K,V\!-\!K}$) CM relation (see point [*1*]{}), while in case ii) the magnitude is obtained by a uniform distribution with extremes K=11.5 and K=13 (see point [*2*]{}). In both cases, the magnitudes were extracted by the adopted distributions (normal or uniform), using as weighting factor the K-band LF at $z\sim0$ (see point [*3*]{}).
The magnitudes of each template at $z\sim0.3$ were corrected by taking into account the corresponding additive terms and the luminosity distance term relative to the redshift of AC118.
The fitting procedures {#FITMOD}
----------------------
For each galaxies in AC118, we derived the ‘best’ values of $\{t_0,\tau,Z \}$ by two different fitting procedures. In case [*a)*]{}, we obtained $\{t_0, \tau, Z \}$ by minimizing for each galaxy the distance of the models from the observed point in color-color space at $z \sim 0.3$, that is by minimizing the function: $$\rm \chi^2_{AC118}(t_0,\tau,Z) =
\left[(V-I)_{j}-(V-I)_{templ}\right]^2+\left[(R-K)_{j}-(R-K)_{templ}\right]^2,
\label{eqchiACa}$$ where the subscript $\mathrm{j}$ denotes the galaxies of the AC118 sample, while the subscript $\mathrm{templ}$ refers to the photometric quantities of the templates, which are functions of $t_0$, $\tau$ and $Z$. In this case, the choice of the best templates depends only on the photometric properties of galaxies at $z \sim 0.3$ without any constraint at $z\sim0$. We point out that this procedure is completely independent of the photometric properties of the local cluster.
In case [*b)*]{} the ‘best’ values of $ \{t_0,\tau,Z \}$ were obtained by minimizing the function: $$\rm \chi^2_{AC118}(t_0,\tau,Z) =
\left[K_{j}-K_{templ}\right]^2+\left[(V-I)_{j}-(V-I)_{templ}\right]^2+
\left[(R-K)_{j}-(R-K)_{templ}\right]^2.
\label{eqchiACb}$$ In this case the choice of $\{ t_0,\tau,Z \}$ is also driven by the template K-band magnitudes, which were scaled as described in the previous section. In such a way, we are constraining the template of each galaxy in AC118 to occupy the red sequence locus or the region populated by blue galaxies (points [*1*]{} and [*2*]{} of Section \[galtemp\]) when evolved to $z\sim0$. This constrains, therefore, AC118 to belong to the same evolutionary sequence of a rich nearby cluster. We point out that this procedure does not imply that the set of N=252 best fitting templates of AC118 galaxies, when evolved to $z\sim0$, reproduces a CM diagram with the same properties of that observed for a nearby rich cluster, i.e. the slope and the intrinsic scatter of the CM relation, the fraction of blue galaxies and the LF. In fact, our unique constraint is that each template is bounded by the same region of galaxies in the CM diagram at $z\sim0$. This point will be further discussed in Section \[FINE\].
To account for measurement errors, the fitting procedures were iterated by shifting colors and magnitudes of galaxies in AC118 according to their photometric uncertainties[^5]. In this way, for each iteration we obtained a distribution of ‘best’ parameters $ \{t_0,\tau,Z \}_j \ , \ j=1...N$ which describes the photometric properties of all the $\mathrm{N}=252$ galaxies in our sample at $z\sim 0.3$. Since the distributions of $\{t_0,\tau,Z \}$ coming from the different iterations are practically identical, in the following we will discuss the results by averaging the properties of the different distributions of ‘best’ parameters.
Ages, Star Formation Rates and Metallicities {#RESULTS}
============================================
In Figure \[MODOBS\] we compare the distributions in the CM space of the best fitting templates obtained according to case [*b)*]{} of Section. \[FITMOD\] with those of the sample at $z\sim0.3$ and with the CM distributions expected for a nearby rich cluster. This local sample is obtained by using the same recipe used for deriving the additive terms of galaxy templates in Section. \[galtemp\]. First, we generated a set of magnitudes according to the Coma K-band LF (see point [*3*]{} of Section. \[galtemp\]). Then we assigned to each magnitude a $\mathrm{V\!-\!K}$ color according to the CMR of the Coma cluster (see point [*1*]{} of Section. \[galtemp\]) and by imposing that the number of blue galaxies in the CM plane amounts to $5\%$ (see point [*2*]{} of Section. \[galtemp\]).
In Figure \[MODOBS\], the distributions of the model match those observed for AC118, with the exception of few points whose $\mathrm{R\!-\!K}$ colors are too red with respect to the templates. In order to address this problem, we introduced a red envelope of the CMR of AC118, defined as the locus in the plane ($\mathrm{K,R\!-\!K}$) corresponding to the reddest stellar populations among the considered templates (cfr. KB01). To this aim, we considered simple stellar populations with formation epoch equal to the age of the universe and different metallicities. It turns out that $\sim15\%$ of the galaxies in AC118 are located above the red envelope. Six objects deviate by more than $0.1~\mathrm{mag}$, while the photometric errors are not large enough to explain this difference. We will come back to this point at the end of Section 4.4.
When evolved to z$\sim$0, the CM distribution of the best models for AC118 gives a reliable description of the CM distribution of the local simulated sample: most of galaxies follow a tight CMR with slope and scatter consistent with those of the CMR of Coma, while few galaxies (4%) lie in the blue-faint area of the CM diagram. It is worth to be noticed that the derived luminosity function also matches that of the Coma cluster.
Distributions in the Parameter $\{t_0,\tau,Z \}$ Space
------------------------------------------------------
In order to analyze the allowed ranges of physical parameters, we compare the distributions of $\{t_0,\tau,Z\}$ obtained in case [*a)*]{}, by considering only the sample at $z\sim 0.3$, and in case [*b)*]{}, by considering the properties of both the distant sample and a nearby rich cluster, as discussed in the Section. \[FITMOD\].
In Figure \[CPLOT1\] we show the frequency distributions relative to case [*a)*]{} in all the planes that can be constructed from the quantities $ \{t_0,\tau,Z,K \}$, where both $K$ and $t_0$ refer to $z\sim0.3$. Figure \[CPLOT1\] clearly shows the well known age–metallicity degeneracy for which the photometric properties of older (younger) stellar populations are equivalent to those of the more metal rich (poor) ones. This is particularly evident in the upper middle and lower left panels as indicated by the elongation of the contours and from the fact that very extended regions of the parameter space are populated.
The most remarkable feature that arises from the comparison of the cases [*a)*]{} and [*b)*]{} is the segregation in the space of parameters obtained by constraining AC118 to belong to the same evolutionary sequence of the local cluster. Figure \[CPLOT2\] shows that a large fraction ($\sim70\%$) of the points are constrained to the region $ 0<\tau<3~\mathrm{Gyr}$, $5 <t_0< 9.0~\mathrm{Gyr}$ and $0.008<Z<0.03$. The constraints at $z\sim0$ produce a sharp metallicity sequence in the plane ($K$,$Z$), constraining brighter galaxies to have higher values of $Z$. It is also interesting to notice that about $20\%$ of the templates are not constrained to follow a tight luminosity–metallicity relation, but are described by higher values of the metallicity. A deeper inspection of Figure \[CPLOT2\] shows that these objects are mostly found in the region $\tau>3~\mathrm{Gyr}$ and $ t_0 > 4~\mathrm{Gyr}$ (formation redshift $z_0>0.9$). Moreover, they do not show any significant difference in their photometric properties with respect to the other points of the model.
Origins of the Color-Magnitude Relation
---------------------------------------
KA97 and Kodama et al. (1998), by means of a population synthesis code that accounts for chemical evolution in a self-consistent manner (Arimoto & Yoshii 1987), proved that the small evolution of the CMR with look-back time constrains this relation to be a metallicity-luminosity sequence. In Figure \[ZETASEQ\] (upper panel) we compare the relation between the luminosity-weighted mean stellar metallicity and the absolute V-band magnitude at $z=0$ given by KA97 (see their Table 2), with the same relation for our models. The points of the models were binned in the plane ($M_V$,$Z/Z_{\odot}$) with respect to V-band magnitudes and the biweight estimator (e.g. Beers, Flynn & Gebhardt 1990) was applied to derive the location of the peak of the metallicity distribution at a given magnitude. Absolute magnitudes were computed by a distance modulus for the Coma cluster of $34.6~\mathrm{mag}$ (see KA97). It is evident that the observed trend is fully consistent with the findings of KA97. By using a least squares analysis, we find: $$\log (Z / Z_{\odot}) = (-0.097 \pm 0.005) \cdot M_V + (-2.09 \pm 0.09).
\label{EQZSEQ}$$ In order to investigate possible variations with luminosity of the age of galaxies, we derive the relation between magnitude and formation redshift $z_0$ for the objects that lie within $3\sigma$ of the metallicity–luminosity relation. This distribution is shown in Figure \[ZETASEQ\], bottom panel. The formation epoch does not change along the sequence, and is constrained to be greater than $z=1$ at the confidence level of $90\%$.
The Scatter of the Color Magnitude Relation
-------------------------------------------
So far, we have not yet discussed the constraints set by the present analysis on the origin of the dispersion in the CMR. To this aim, we computed for each galaxy the age $t_\alpha$ at which a given fraction $\alpha$ of its stellar mass formed. The parameter $t_{\alpha}$ is given by the following combination of $t_0$ and $\tau$: $$t_{\alpha} = t_0 - \tau \cdot \ln(1-\alpha).
\label{EQTBURN}$$ In Figure \[TBURN\] we plot the mean value of $t_{\alpha}$ (expressed as redshift $z_{\alpha}$) as a function of the K-band magnitude at $z=0.31$, and the relative percentiles of $16\%$ and $84\%$ (corresponding to a $1\sigma$ interval for a normal deviate). We chose $t_{90\%}$, that corresponds to the age at which galaxies formed $90\%$ of their stellar mass, and included only the points within the metallicity sequence. For $\mathrm{K \lesssim 16.8}$, the value of $z_\alpha$ is greater than $\sim1$ for almost all the points in the model, while it decreases progressively at fainter magnitude. At $\mathrm{K\sim18}$ ($\sim K^{\star}+3$) the redshift at which some galaxies ceased to form most of their stars can be as low as $z\sim0.5$. On the contrary, the templates that lie outside the metallicity sequence have larger values of $\tau$, and therefore describe objects with a more recent star forming activity. By applying Eq. \[EQTBURN\], we find that all these objects did not complete to form their stars at $z\sim0.3$.
Global Star Formation History
-----------------------------
Finally we consider the global formation history of the stellar populations in the galaxies of AC118. In Figure \[SFRGLOB\] we show, as a function of redshift, the total (cumulative) stellar mass $M(z)$ already formed at a given epoch in cluster galaxies. The function $M(z)$ was obtained by summing the K-band luminosity-weighted mass already formed at a given redshift $z$. Most of the luminous mass ($\sim50\%$) present in the cluster at $z=0.31$ was formed at $z>2$ although star formation continued at $z<1$ for $\sim20\%$ of the stars.
As noticed in previous studies (e.g. Poggianti et al. 1999, KB01), a crucial role in estimating the star formation rate of cluster galaxies can be played by the dust absorption. To investigate this subject, we construct a simple model by assuming that all galaxies redder than the red envelope of AC118 are obscured by a uniform screen of dust. We use the differential dust extinction law introduced by Seaton (1979) and adopt a color excess value $\mathrm{E(B\!-\!V)=0.1}$. Therefore, the intrinsic magnitudes and colors of the templates are transformed according to the equations $\Delta (\mathrm{R\!-\!K})=+0.44$ and $\Delta \mathrm{K=-0.09}$. In Figure \[dustps\] we compare the CM diagram of AC118 with the corresponding distribution of templates obtained by including the dust effect in the model. As can be seen, all the galaxies of AC118 are properly represented by the model, including the red outliers of the CM envelope. These objects are described by dusty blue spirals with extensive on-going star formation activity. As a consequence, about $30\%$ (instead of $20\%$) of stars formed at $z<1$ in this model, while the cumulative mass function decreases at higher redshifts (see the dotted line in Figure \[SFRGLOB\]). We note that, if the red outliers are actually blue dusty spirals, the adopted value for the color excess corresponds to the minimum contribution of the dust.
Discussion and conclusions {#FINE}
==========================
We have studied the star formation history of galaxies in the rich cluster AC118 at redshift $z=0.31$ by constraining their photometric properties to reproduce, once evolved at $z\sim0$, those of a local rich cluster. The analysis is based on a large wavelength baseline including accurate VRIK photometry for a large sample of cluster galaxies (N=252). The sample was selected by the photometric redshift technique and is complete in the NIR, thus reducing possible biases towards objects with more recent/intense star formation activity.
One of the main current issues in the comparison of the properties of local and intermediate–redshift clusters concerns the selection criteria of the samples. Studies of the CM relation based on pure morphological selection can be biased towards the older progenitors of nearby early-type galaxies (see van Dokkum et al. 2000 for a detailed discussion). On the other side, the application of a statistical field subtraction approach requires a wide area around the cluster field to be observed, while the use of a spectroscopically selected sample at faint luminosities is very expensive in terms of observing time (but not impossible, see Abraham et al. 1996; van Dokkum et al. 2000). The main advantage of a selection based on photometric redshifts is that it allows to estimate the typical luminosity-weighted formation epoch of a stellar population irrespective of the past history of the host galaxy (such as, for example, clustering through a merging hierarchy), and it is therefore an ideal tool to define cluster membership for large samples of galaxies without any tie to a particular scenario of galaxy formation.
A more tricky point is represented by the areas of the clusters to be compared at different redshifts. In a hierarchical clustering picture, clusters of galaxies are likely to accrete a significant fraction of their population from the field even at relatively modest redshifts ($z<0.5$, see Kauffmann 1996). As a consequence, cluster richness tends to increase with time, while the population accreted at an old epoch becomes concentrated in a progressively smaller area (see e.g. KB01). On the other hand, the cores of rich nearby clusters are very similar in their photometric properties: the galaxy population is dominated by E/S0 galaxies with few Sp/Irr having bluer colors. Moreover, early-type galaxies seem to follow a universal well defined CM relation (see Bower et al. 1992a, Bower et al. 1992b). For such reasons, we have analyzed the constraints on the properties of the stellar populations of the galaxies in AC118 by imposing that their evolution at $z \sim 0$ mimics the overall distribution in the ($\mathrm{K,V\!-\!K}$) plane for a local rich cluster.
With the aim of constraining the galaxy evolution scenarios, several studies have adopted a purely parametric approach, by comparing the observed properties in the CM diagram with those predicted by models that are based on different sets of parameters and that explore different assumptions on the probability distributions of such parameters. These studies also assume that the scatter in the CM diagram arises merely from age (but see Ferreras, Charlot, & Silk 1999). Although the first epoch of star formation for the cluster early-type population seems to be constrained to high redshifts for almost all such models, further properties, as the last epoch and the spread of star formation activity, are more model dependent.
The procedure we adopted describes each galaxy of AC118 by a stellar population model, which is constrained, when evolved to $z\sim0$, to be bounded by the red sequence locus or by the region of blue galaxies of a rich nearby cluster (see Section 3.1). This is achieved by a suitable procedure which scales the magnitudes of the galaxy templates. We find that the best fitting models of AC118 galaxies are able to match both the distributions in the ($\mathrm{K,R\!-\!K}$) and ($\mathrm{R\!-\!K}$,$\mathrm{V\!-\!I}$) planes at $z\sim0.3$, and the properties of the ($\mathrm{K,V\!-\!K}$) color-magnitude distribution at $z \sim 0$, i.e. slope and intrinsic scatter of the CM sequence, fraction of blue galaxies and luminosity function. It is important to notice that such a result is not implicit in the method we used to scale the template magnitudes (see Section \[FITMOD\]).
The constraint at $z \sim 0$ largely reduces the region of input parameters available to the model. In particular, a sharp sequence arises in the metallicity-luminosity diagram (cfr. lower left panels of Figures \[CPLOT1\] and \[CPLOT2\]), for which brighter galaxies are described by higher values of $Z$. The slope of the sequence is in full agreement with that derived by KA97 in the framework of the monolithic collapse/galactic wind model. It is interesting to notice that if we adopt the luminosity–weighted mean stellar metallicity of the KA97 models, the zero-points of the relations also coincide. The main difference between the results of KA97 and those of the present work is that we do not obtain the metallicity sequence on the basis of a particular galaxy evolution scenario.
The present data seem to exclude significant variations of star formation history along the CMR, and therefore confirm a pure metallicity interpretation, in which the bulk of the populations formed at high redshift ($z \sim 5$). These results, however, do not describe the properties of all the stellar populations in AC118: we find that about $20\%$ of the points of our model do not follow any metallicity–luminosity relation, but are characterized by higher values of Z and more prolonged star formation activity ($\tau>4~\mathrm{Gyr}$). Since these objects do not show peculiar photometric properties in the colors-magnitude space, this result could be the consequence of a residual age–metallicity degeneracy. However, other possibilities can be explored. For instance, the scatter of the CM relation at a given luminosity could be partly due to the fact that more recently assembled galaxies have higher metallicity than older systems of similar luminosity (see Ferreras et al. 1999).
To study the dispersion in the CM diagram at $z\sim0.3$, we computed the epoch $t_{90\%}$ at which galaxies completed to form $90\%$ of their stars. While for $\mathrm{K<17}$ the corresponding redshift is greater than $z=1$, at faintest magnitudes ($\mathrm{K}\sim K^{\star}+3$) we find that some galaxies ceased to form stars at epochs as low as $z\sim0.5$. These results are in agreement with the general picture that fainter galaxies were more recently accreted from the field to the cluster environment and therefore ceased to form stars at later epochs (see KB01 for a wide discussion).
One half of the luminous mass present at $z\sim0.3$ formed at $z>2$, and star formation continued at $z<1$ for $\sim 20\%$ of the stars. This result changes if we are neglecting the effect of the dust obscuration in a significant fraction of cluster galaxies. To investigate this subject, we adopted a simple model in which all the galaxies redder than the CM envelope at $z\sim0.3$ are obscured by a uniform screen of dust. The introduction of this model is also supported by the presence of few galaxies of AC118 whose R-K color is too red with respect to the CMR. These objects may be accounted for as dusty galaxies with extensive on-going star formation activity (cfr. KB01). In the model with dust, the fraction of mass that forms at $z<1$ increases from $\sim 20 \%$ to $\sim 30 \%$.
The observations at ESO were collected during the guaranteed time of the Osservatorio Astronomico di Capodimonte. Michele Massarotti is partly supported by a ‘MIUR-COFIN’ grant.
Abraham, R.G., van den Bergh, S., Glazebrook, K., Ellis, R.S., Santiago, B.X., Surma, P., & Griffiths, R.E. 1996, , 107, 1 Andreon, S., & Pelló, R. 2000, , 353, 479 Andreon, S. 2001, , 547, 623 Arimoto, N., & Yoshii, Y., 1987, , 173, 23 Barger, A.J., Aragón-Salamaca, A., Ellis, R.S., Couch, W.J., Smail, I., & Sharples, R.M. 1996, , 279, 1 Bertin, E., & Arnouts, S. 1996, , 117, 393 Beers, T.C., Flynn, K., & Gebhardt, K. 1990, , 100, 32 Bower, R.G., Lucey, J.R., & Ellis, R.S. 1992a, , 254, 601 Bower, R.G., Lucey, J.R., & Ellis, R.S. 1992b, , 254, 589 Bruzual, G.A., & Charlot, S. 1993, , 405, 538 Busarello, G., Merluzzi, P., La Barbera, F., Massarotti, M., & Capaccioli, M. 2002, , 389, 787 Butcher, H., & Oelmer, A. 1978, , 219, 18 Butcher, H., & Oelmer, A. 1984, , 285, 426 Couch, W.J., & Newell, E.B. 1984, , 56, 143 Couch, W.J., & Sharples, R.M. 1987, , 229, 423 Couch, W.J., Sharples, R.M., Ellis, R.S., &, Smail, I. 1994, , 430, 121 de Propris, R., Eisenhardt, P.R., Stanford, S.A., & Dickinson, M. 1998, , 503, 45L Dressler, A., & Gunn, J.E. 1982, , 263, 533 Dressler, A., Oelmer, A., Butcher, H., & Gunn, J.E. 1994, , 430, 107 Eggen, O.J., Lynden-Bell, D., & Sandage, A.R. 1962, , 136, 748 Ellis, R.S., Couch, W.J., MacLaren, I., & Koo, D.C. 1985, , 217, 239 Ellis, R.S., Smail, I., Dressler, A., Couch, W.J., Oemler, A.Jr., Butcher, H., & Sharples, R.M. 1997, , 483, 582 Ferreras, I., Charlot, S., & Silk, J. 1999, , 521, 81 Gladders, M.D., Lopez-Cruz, O., Yee, H.K.C., & Kodama, T. 1998, , 501, 571 Kauffmann, G. 1996, , 281, 487 Kauffmann, G., & Charlot 1998, , 294, 705 Kodama, T., & Arimoto, N. 1997, , 320, 41 (KA97) Kodama, T., Arimoto, N., Barger, A.J, & Aragón-Salamanca, A. 1998, , 334, 99 Kodama, T., & Bower, G. 2001, , 321, 18 (KB01) Kodama, T., Smail, I., Nakata, F., Okamura, S., & Bower, R. G. 2001, , 562, 9L Kron, R.G. 1980, , 43, 305 La Barbera, F., Busarello, G., Merluzzi, P., Massarotti, M. & Capaccioli, M. 2002, , 571, 790 Lilly, S.J., Tresse, L., Hammer, F., Crampton, D., & Le Fèvre, O. 1995, , 455, 108 Poggianti, B.M., Smail, I., Dressler, A., Couch, W.J., Barger, A.J., Butcher, H., Ellis, R.S., & Oemler, A.Jr. 1999, , 518, 576 Scalo, J.M. 1986, Fundamentals of Cosmic Physics, 11, 1 Seaton, M.J. 1979, , 187,73 Smail, I., Kuntschner, H., Kodama, T., Smith, G.P., Packham, C., Fruchter, A.S., & Hook, R.N. 2001, , 323, 939 Stanford, S.A., Eisenhardt, P.R.M., & Dickinson, M. 1998, , 492, 461 Terlevich, A.I., Caldwell, N., & Bower, R.G. 2001, , 326, 1547 Tinsley, B.M., & Gunn, J.E. 1976, , 203, 52 van Dokkum, P.G., Franx, M., Kelson, D.D., Illingworth, G.D., Fisher, D., & Fabricant, D. 1998, ., 500, 714 van Dokkum, P.G., Franx, M., Fabricant, D., Illingworth, G.D., & Kelson, D.D. 2000, , 514, 95 van Dokkum, P.G., Stanford, S.A., Holden, B.P., Eisenhardt, P.R., Dickinson, M., &Elston, R. 2001, , 552, 101L
[^1]: Based on observations collected at European Southern Observatory (ESO n. 62.O-0369, 63.O-0257, 64.O-0236)
[^2]: High values of the Sersic index $n$.
[^3]: With intermediate values $Z= 0.0004, 0.004, 0.008, 0.02,
0.05$.
[^4]: The K-band at $z\sim0.3$ matches the H-band rest-frame. However, the $\mathrm{H\!-\!K}$ color is almost independent of the galaxy spectral type, and therefore the difference between the K- and the H-band rest-frame magnitudes is not relevant for the present analysis. \[nota1\]
[^5]: The shifts were assigned by taking into account also the correlation between the measurement errors on colors and magnitudes.
|
---
abstract: 'The protection of quantum states is challenging for non-orthogonal states especially in the presence of noises. The recent research breakthrough shows that this difficulty can be overcome by feedback control with weak measurements. However, the state-protection schemes proposed recently work optimally only for special quantum states. In this paper, **by applying different weak measurements, we extend the idea of the state-protection scheme to protect general states.** We calculate numerically the optimal parameters and discuss the performance of the scheme. Comparison between this extended scheme and the earlier scheme is also presented.'
author:
- 'Y. Yang$^1$, X. Y. Zhang$^1$, J. Ma$^{1,2}$, and X. X. Yi$^1$'
title: Extended Techniques for Feedback Control of A Single Qubit
---
In classical physics, it is possible in principle to acquire all information about the state of a classical system by precise measurements. Namely, the state of a single classical system can be precisely determined by measurements. This ensures the measurement-based classical feedback control and makes the feedback control beneficial to the manipulation of classical system.
For a quantum system, however, this is not possible: If the system is prepared in one of several non-orthogonal states, no measurement can determine determinately which state the system is really in. Furthermore, Heisenberg’s uncertainty principle imposes a fundamental limit on the amount of information obtained from a quantum system, and the act of measurement necessarily disturbs the quantum system [@1; @2; @3; @4; @5] in an unpredictable way. This means when extend the measurement-based classical control theory to quantum system, we need careful examinations of the control scheme. The extension of the classical feedback to quantum systems can be used not only in quantum control [@6; @7; @8; @9; @10; @11], but also in quantum information processing, for example in the quantum key distribution [@12] and quantum computing, as well as in other practical quantum technologies [@13].
Recent works in this field [@14; @15; @16] suggested that we can balance the information gain from a measurement and the disturbance caused by the measurement via weak measurement. To be specific, in Ref.[@15] Branczyk [*et al.*]{} investigated the use of measurement and feedback control to protect the state of a qubit. The qubit is prepared in one of two non-orthogonal states **in the $x-z$ plane of the Bloch sphere and subjected to noise.** The authors shown that, in order to optimize the performance of the state protection, one must use non-projective measurements to balance the trade-off between information gain and disturbance. **The measurement operators used in[@15] are among the $y-$ axis and the subsequent correction is a rotation about the $z-$axis**. This scheme was realized recently [@14], where the stabilization of non-orthogonal states of a qubit against dephasing was experimentally reported. It is shown that the quantum measurements applied in the experiment play an important role in the feedback control. **We should notice that the measurements used in[@14] are different to those in [@15], namely, its measurement operators are along the $z-$axis and the correction is about the $y-$ axis. Geometrically, for initial states in the $x-z$ plane, the dephasing noise can not map the initial states out of the $x-z$ plane, then all states including the initial states, the states passed the noise and measurements as well as the final stats are in the $x-z$ plane in[@14], this is the difference between [@14] and [@15] from the geometric viewpoint. We will modify the measurement operators in [@14] and use it in this paper.**
[With these knowledge in quantum information science [@17; @18; @19], one may wonder if the weak measurement used in the scheme is also the best one for the protection of general states? I.e., $M_+ =\cos(\chi/2)|0\rangle\langle 0| +
\sin(\chi/2)|1\rangle\langle1|\,, $ and $M_- =
\sin(\chi/2)|0\rangle\langle 0| +\cos(\chi/2)|1\rangle\langle 1|\,,$ are these measurements best for the protection of general states? Are there other measurements that can better the performance of the scheme for general states? In this paper, we shall shed light on this issue by introducing different measurements for the feedback control. We find that the scheme can be extended to protect general quantum states with the new weak measurement. We derive the performance and give the parameters best for the performance, a discussion on this extended scheme is also presented.]{}
Consider two non-orthogonal states that we want to protect from noise, $$\label{initial state} {\vert\psi_{\pm}\rangle}{=}
\cos\frac{\theta}{2}{\vert+\rangle}{\pm}e^{i\phi}\sin\frac{\theta}{2}{\vert-\rangle},$$ with ${\vert\pm\rangle}{=}\frac{1}{\sqrt{2}}({\vert0\rangle}\pm{\vert1\rangle})$, the corresponding density matrices are given by $\rho_\pm{=}{\mbox{$|\psi_\pm\rangle\langle \psi_\pm|$}}$. Note that $|\psi_+\rangle$ and $|\psi_-\rangle$ are non-orthogonal and are more general than the states in [@14; @15], the overlapping of the two states is independent of $\phi$, but depends on $\theta$, $\langle\psi_+|\psi_-\rangle=\cos\theta.$ In fact, ${\vert\psi_{\pm}\rangle}$ are rotated about the $x$-axis with respect to the Branczyk’s one, this may offer a chance to improve the fidelity given by the previous proposals [@14; @15] for general states of a qubit.
The qubit is subjected to dephasing noises [@14; @15]. We shall use $\{{\vert0\rangle},{\vert1\rangle}\}$ as the basis of the qubit Hilbert space, and define the Pauli operator $Z$ as $Z{\vert0\rangle}{=}{\vert0\rangle},
Z{\vert1\rangle}{=}{-}{\vert1\rangle}$, similar definitions are for Pauli matrices $X$ and $Y$. The dephasing noise can be described by a phase flip $Z$ with probability $p$ and with probability $1-p$ that the system remains unchanged. The density matrix of the qubit passed through the noisy channel is, $$\rho^{'}_{\pm}= (1-p) \rho_{\pm}+p Z \rho_{\pm} Z.\label{noise}$$
The purpose of this paper is to find better measurements and controls to send the qubit back as close as possible to its initial state. For this purpose, we use a quantum operation $\mathcal{C}$ as a map acting on the single qubit to describe the controls and measurements, $$\mathcal{C}(\rho^{\prime})=Y_{+\eta}M^{\prime}_{+}
\rho^{\prime}M_{+}^{\prime\dagger}Y_{+\eta}^{\dagger}+Y_{-\eta}M^{\prime}_{-}
\rho^{\prime}M_{-}^{\prime\dagger}Y_{-\eta}^{\dagger}.$$ The notations of $Y$ and $M^{\prime}$ will be given later. To quantify the performance of $\mathcal{C}$, we use the average fidelity $\overline{F}$ [@14; @15] between the noiseless input state and the corrected output state as a measure, $$\begin{aligned}
\overline{F}&=\tfrac{1}{2}[{\langle\psi_+\vert}\mathcal{C}(\rho'_+){\vert\psi_+\rangle}
+{\langle\psi_-\vert}\mathcal{C}(\rho'_-){\vert\psi_-\rangle}]\nonumber\\
&=\tfrac{1}{2}(F_{\psi_+}+F_{\psi_-})\,.\end{aligned}$$ This measure quantifies the performance well when ${\vert\psi_+\rangle}$ and ${\vert\psi_{-}\rangle}$ are sent into the control with equal probability.
![Illustration of the scheme. The meter qubit was entangled with the qubit (for protection) which has passed through the noise channel. After the measurement and correction, we can get a signal. Then we compare the signal with the initial state and apply a feedback control to the qubit. An average fidelity is define and used to determine the parameters in the feedback control. In the earlier scheme, the authors use a ${\vert\varphi\rangle}{=}\cos\frac{\chi}{2}{\vert+\rangle}{+}\sin\frac{\chi}{2}{\vert-\rangle}$ as the meter-qubit state, while in the present scheme a complex phase factor is introduced, i.e. the meter state is, ${\vert\varphi\rangle}{=}\cos\frac{\chi}{2}{\vert+\rangle}{+}e^{i\beta}
\sin\frac{\chi}{2}{\vert-\rangle}.$[]{data-label="FIG:1"}](yyfig1){width="8cm" height="4cm"}
To find a good control procedure, we must first find the appropriate measurement which has to have the following two features. First, it must be a weak measurement, that is, it can not completely disturb the system. Second, it has to be strength-dependent, such that we can adjust the strength of the measurement as we need. This family of weak measurements in the logical basis $\{{\vert0\rangle},{\vert1\rangle}\}$ can be written as, $$\begin{aligned}
\label{eq:M'_{+}}
M'_+ &= \cos(\chi/2)|0\rangle\langle 0| + e^{i\beta}\sin(\chi/2)|1\rangle\langle 1|\,, \\
\label{eq:M-} M'_- &= e^{i\beta}\sin(\chi/2)|0\rangle\langle 0| +
\cos(\chi/2)|1\rangle\langle 1|\,.\end{aligned}$$ In contrast to the measurements used in Ref.[@14; @15; @16], $M_+
=\cos(\chi/2)|0\rangle\langle 0| + \sin(\chi/2)|1\rangle\langle1|\,,
$ $M_- = \sin(\chi/2)|0\rangle\langle 0|
+\cos(\chi/2)|1\rangle\langle
1|\,,$ a new parameter $\beta$ was introduced in this weak measurement [@20; @21]. Here $\chi$ ranges from 0 to $\pi/2$ [@20], we can change the value of the parameter $\chi$ to adjust the strength of measurement. The corresponding positive measurement operators are given by $\Pi_\pm{=}M^{'\dag}_\pm M'_\pm
{=}[\mathbbm{1}\pm\cos{(\chi)}Z]/2$, with $\mathbbm{1}$ being the identity operator. Clearly, $\chi=0$ describes the projective measurement, while $\chi=\frac{\pi}{2}$, do nothing. At first glance, this proposal is trivial, i.e., the initial states (the state sent into protection) are rotated about $x-$axis in the Bloch sphere with respect to that in Ref.[@15], by properly choosing $\beta$, the next measurements $M_+^{\prime}$ and $M_-^{\prime}$ may send them back, then the resulting states will return to that in the earlier proposal, and the performance can not be improved. We will show later that this is not the case.
Our main task is to figure out how the parameter $\beta$ affects the results of the control, and if the parameter $\beta$ can better the performance. The correction performed in this paper is the same as that in [@14], i.e., $Y_{\pm\eta} = \exp(\pm
i\tfrac{1}{2}\eta Y)$ representing a rotation with an angle $\eta$ around the $y-$axis of the Bloch sphere. All parameters should be optimized for the performance of the control.
Straightforward calculation show that the average fidelity of the control is a function of $\theta, \phi, \eta, \chi, \beta$ and $p$, $$\begin{gathered}
\label{fertility}
\overline{F'}(\theta,p,\chi,\eta,\phi,\beta)=\\
\tfrac{1}{2}\left[1+\cos\theta\cos\chi\sin\eta+
\cos\eta\cos^{2}\phi\sin^{2}\theta\right.\\
+\tfrac{(1-2p)}{2}\sin\chi\left(2\cos\beta
(\cos\eta\cos^{2}\theta+\sin^{2}\theta\sin^{2}\phi)\right.\\
\left.\left.-\sin\beta\sin\eta\sin^{2}\theta\sin2\phi\right) \right]
\,.\end{gathered}$$ [For each $\theta,$ $\phi$ and $p$, there are an optimum measurement strength $\chi,$ correction angle $\eta,$ and measurement parameter $\beta$, which maximizes the average fidelity. First we start with $\eta$.]{} The $\eta$ which optimizes the average fidelity can be given by, $$\begin{gathered}
\label{N}
\eta_{\rm opt}(\theta, p, \chi,\phi,\beta)\\
=\arctan{\frac{\cos\theta\cos\chi-\tfrac{1}{2}(1-2p)
\sin\beta\sin^{2}\theta\sin2\phi\sin\chi}{\cos^{2}\phi\sin^{2}\theta
+(1-2p)\cos^{2}\theta\sin\chi\cos\beta}
} \,.\\\end{gathered}$$ [Substituting the optimum $\eta_{opt}$ into the average fidelity, we have,]{} $$\begin{gathered}
\label{fertility'}
\overline{F'}(\theta,p,\chi,\phi,\beta)=
\tfrac{1}{2}+\tfrac{1}{2}(1-2p)\cos\beta\sin^{2}\theta\sin^{2}\phi\sin\chi\\
+\frac{1}{2}\left[(\cos\theta\cos\chi
-\tfrac{1}{2}(1-2p)\sin\beta\sin^{2}\theta\sin2\phi\sin\chi)^{2}\right.\\
+(\cos^{2}\phi\sin^{2}\theta
\left.+(1-2p)\cos^{2}\theta\sin\chi\cos\beta)^{2}\right
]^{\tfrac{1}{2}}\,.\end{gathered}$$ We can see that when $\phi=0$, $\overline{F'}(\theta,p,\chi,\phi,\beta)$ reduces to $$\begin{aligned}
&\ &\overline{F'}|_{\phi=0}=\frac 1 2 +\frac 12\left
[\cos^2\theta\sin^2\chi\right. \nonumber\\
&\
&\left.+(\sin^2\theta+(1-2p)\cos^2\theta\sin\chi\cos\beta)^2\right
]^{\frac 12}.\end{aligned}$$ Obviously, $\beta=0$ maximize the average fidelity $\overline{F'}$, this is exactly the case discussed in Ref. [@14; @15]. So, for the initial states lying in the $xz-$ plane of the Bloch sphere, the weak measurements with $\beta=0$ already maximize the performance.
![This figure shows how much our scheme improve the performance of the state protection for general qubit states. The improvement is quantified by $\delta_{F}$, which is plotted as a function of $\theta$ and $\phi$. For different $p$, the improvement is different, as (a), (b), (c) and (d) show. (a)$p=0.10;$ (b)$p=0.20;$ (c)$p=0.30;$ (d)$p=0.40.$[]{data-label="FIG:2"}](yyfig2){width="8.5cm" height="6cm"}
To find the optimal feedback control for $\phi\neq 0$, we follow the procedure in [@14]. Here again $\theta$ and $\phi$ are related to the initial state of the qubit, while $p$ characterizes the noise and is regarded as a fixed value, $\chi$ and $\beta$ are related to the measurement procedure, $\eta$ denotes the correction parameter.
![The $\beta$ which maximizes $\delta_{F}$ as a function of $\theta$ and $\phi$ for different $p$, (a)$p=0.10;$ (b)$p=0.20;$ (c)$p=0.30;$ (d)$p=0.40.$[]{data-label="FIG:3"}](yyfig3){width="8.5cm" height="6cm"}
By the same procedure as in the earlier works, we maximize the fidelity of the control over the remaining parameters $\chi$, $\theta, \phi, \beta$ and $p$. The analytical expression for the fidelity is complicated, so we choose to find the optimal parameters by numerical simulations. As aforementioned, we have already had the relations between the average fidelity and the initial parameters $\theta$ and $\phi$. We shall use $\delta_{F}{=}F'_{opt}{-}F_{opt}$ to quantify the improved fidelity due to the parameter $\beta$, select results are presented in Fig.\[FIG:2\], where $F'_{opt}$ denotes the optimal fidelity in our paper, while $F_{opt}$ denotes that by the scheme in Ref.[@14; @15], i.e., with $\beta=0.$ The optimized $\beta$ would depend on $\theta$ and $\phi$ and is shown in Fig.\[FIG:3\].
![The fidelity difference $\delta_{F}$ versus $p$.[]{data-label="FIG:4"}](yyfig4){width="8cm" height="5cm"}
Fig.\[FIG:2\] plots the improvement of the average fidelity as a function of the original states (characterized by $\theta$ and $\phi$) with different amount of noise (characterized by $p$). We note that there are no improvement for the following cases. If $p=0$, there is no noise and so the state is not perturbed, in this case the fidelity is 1 for all original states including $\phi=0$ and the measurement strength is $\chi=\frac{\pi}{2}$ (do nothing). When $\theta=\frac{\pi}{2}$, the state $|\psi_+\rangle$ and $|\psi_-\rangle$ are orthogonal, the earlier scheme gives unit fidelity, hence there is no room to improve the performance. When $\theta=0$ the two states are equal and point along the $x-$axis, these states are also the same as that in the earlier scheme, leading to zero improvement. If $\phi=\frac{\pi}{2}$, nothing should change since the two states would interchange by this control. Finally, when $\phi=0,$ the initial states return to the earlier scheme. Fig.\[FIG:3\] shows the parameter $\beta$, which maximize the average fidelity as a function of the original states and the amount of noise $p$. As expected, non-zero maximal $\beta_{opt}$ exists. To show clearly the dependence of the improvement on the noise strength, we plot $\delta_F$ in Fig. \[FIG:4\] as a function of $p$. The maximal improvement arrived at about p=0.1800, the corresponding improvement is $\delta_{F}$=0.0102.
![Bloch vectors of the original states(blue-solid), the resulting states by our scheme (red-dashed) and the resulting states in [@15] (green-dotted). The parameters chosen are, $p=0.18, \eta=0.7913, \chi=0.8583, \beta=5.8905, \theta=1.0155,
\phi=0.8976.$ All parameters except $p$ are in units of arc. []{data-label="FIG:7"}](yyfig7){width="6.5cm" height="5.5cm"}
For developing an intuitive picture, we now take a snapshot for the states going through the control and measurement. Suppose the initial state is $|\psi_+\rangle$ with $\phi=\frac{\pi}{4}$, and let $\{|0\rangle, |1\rangle\}$ be a basis for the Hilbert space. In terms of density matrix, the initial state is $$\rho_{+}=\frac 1 2 \mathbbm{1}+\frac 1 2\cos\theta\cdot\sigma_{x}
-\frac{\sqrt{2}}{4}\sin\theta\cdot\sigma_{y}+\frac{\sqrt{2}}{4}\sin\theta\cdot\sigma_{z},$$ This state lies in the $z=-y$ plane and points along the direction with an angle $\theta$ from the $x-$axis. The state passed the noisy channel is $\rho^{'}_{+},$ $$\begin{aligned}
\rho^{'}_{+}&=&\frac 1 2\mathbbm{1}\nonumber\\
&+&(\frac 1 2-p)\cos\theta\cdot\sigma_{x} \nonumber\\
&+&\frac{\sqrt{2}}{2}(p-\frac 1 2)\sin\theta\cdot\sigma_{y}\nonumber\\
&+&\frac {\sqrt{2}} {4}\sin\theta\cdot\sigma_{z},\end{aligned}$$ we see that the $z-$component of the Bloch sphere remains unchanged, while the $x-$ and $y-$ components are shortened by $(1-2p)$ times due to the noise. The resulting state (unnormalized) immediately after the measurement is denoted by $\rho^{m}_{+},$ and it takes, $$\begin{aligned}
\rho^{m}_{+}&=&\frac 1 2
(1+\frac{\sqrt{2}}{2}\cos\chi\sin\theta)\cdot\mathbbm{1}\nonumber\\
&+&\frac 1
2(1-2p)(\cos\beta\cos\theta+\frac{\sqrt{2}}{2}\sin\beta\sin\theta)\sin\chi\cdot\sigma_{x}\nonumber\\
&+&\frac 1 2
(1-2p)(\cos\theta\sin\beta-\frac{\sqrt{2}}{2}\cos\beta\sin\theta)\sin\chi\cdot\sigma_{y}\nonumber\\
&+&\frac 1 2
(\cos\chi+\frac{\sqrt{2}}{2}\sin\theta)\cdot\sigma_{z}.\label{statem}\end{aligned}$$ Finally after the correction $Y_{+\eta}$, the unnormalized states has been mapped into,
$$\begin{aligned}
\rho^{c}_{+}&=&\frac 1 2
(1+\frac{\sqrt{2}}{2}\cos\chi\sin\theta)\cdot\mathbbm{1}\nonumber\\
&+&\frac 1 2 \left(\sin\eta(\cos\chi+\frac{\sqrt{2}}{2}\sin\theta)
+(1-2p)\cos\eta\sin\chi(\cos\beta\cos\theta+\frac{\sqrt{2}}{2}\sin\beta\sin\theta)\right )
\cdot\sigma_{x}\nonumber\\
&+&\frac 1 2
(1-2p)(\cos\theta\sin\beta-\frac{\sqrt{2}}{2}\cos\beta\sin\theta)\sin\chi\cdot\sigma_{y}\nonumber\\
&+&\frac 1 2 \left(\cos\eta(\cos\chi+\frac{\sqrt{2}}{2}\sin\theta)+
(-1+2p)\sin\eta\sin\chi(\cos\beta\cos\theta+\frac{\sqrt{2}}{2}\sin\beta\sin\theta)\right
)\cdot\sigma_{z}.\end{aligned}$$
Note that this state is also unnormalized. For a specific set of $\theta$, $\phi$ and $p$, the resulting state together with the resulting state in Ref.[@14] are illustrated in Fig. \[FIG:7\]. This shows clearly that our resulting states are more close to the initial state than that given by the proposal with $\beta=0$. As shown, the new measurements can do better than the earlier one for general quantum states. This suggests that we can apply the new set of measurements to the feedback control. Now we examine how much this new scheme improves the fidelity with respect to the schemes with measurements “do nothing” and “strong measurement” (Helstrom).
Before processing, we briefly review the two special cases of the schemes, which differ from each other at the measurements: In the zero strength measurement, $\cos\chi=0$, namely, no measurement is applied. So the state protection with this measurement is called “do nothing” (DN) control scheme; The projective measurement is applied with maximum strength ($\cos\chi=1$), with which the protection scheme had already been named as “Helstrom” (H) scheme[@22]. In fact, DN control is actually not a measurement-based control because of no application of measurement to quantum states. And H scheme is not what we need, because it makes an unnecessary correction to the system. To quantify the fidelity difference between these schemes, we define $$F_{imp}=F_{opt}^{\prime}-max\{F_{DN},F_{H}\}$$ as a measure to quantify the difference, where $F_{DN}$ is the fidelity of DN control scheme, while $F_H$ represents the fidelity of the H scheme.
![$F_{imp}$ versus $\theta$ and $\phi$ with different $p$,(a)$p=0.10$; (b)$p=0.20$; (c)$p=0.30$; (d)$p=0.40.$ This figure shows the improvement of our scheme over the DN and H schemes.[]{data-label="FIG:5"}](yyfig5){width="8cm" height="5cm"}
We have performed numerical simulations for $F_{imp}$, selective results are presented in Fig.\[FIG:5\] and Fig. \[FIG:6\]. In Fig.\[FIG:5\], we present $F_{imp}$ as a function of $\theta$ and $\phi$ for different $p$. A common feature is that $F_{imp}$ reach its maximum at around $\theta=\pi/4$ and $\phi=\pi/4$. For different $p$, the improvement in the fidelity is different.
![$F_{imp}$ as a function of $p$. In this figure, $F_{imp}$ is numerically optimized over $\theta$ and $\phi$ for each $p$. $p$ runs from 0 to 0.5, covering all possible choices.[]{data-label="FIG:6"}](yyfig6){width="8cm" height="5cm"}
To show the dependence of $F_{imp}$ on $p$ clearly, we plot the maximum $F_{imp}$ versus the parameter $p$ in Fig. \[FIG:6\] with different $\theta$ and $\phi$. As the figure shows, when $p=0.2501$, $F_{imp}$ reaches the maximum value 0.0662. Although the improvement is small, it can work under most conditions and it does improve the state protection over other schemes with different measurements[@16]. This tells that the scheme without the parameter $\beta$ is not the best scheme for state protection of general states.
[It is illustrative to view the difference between our scheme (see Fig.\[fig8\](Right top)) and the scheme (Fig.\[fig8\](Left top)) in Ref.[@14] on the Bloch sphere. In Fig.\[fig8\](Left top), we can see that the original states $|\psi_+\rangle$ and $|\psi_-\rangle$ (green) are shorten by the noise, but the $z-$component of the Bloch vector remains unchanged (pink vector on the Bloch sphere, i.e., $\rho^{\prime}_{\pm}$). The measurements lengthen the Bloch vectors(blue, i.e., $M_+^{\prime}\rho^{\prime}M_+^{\prime\dagger}$) and diminish the angle between the Bloch vector and the $z-$ axis. We should remind that the Bloch vectors remains in the $xz-$ plane in the whole process of measurements and controls, this is the core difference between the scheme in [@14] and ours. This difference offers us a room to improve the performance of the control.]{}
In our scheme, the original states are rotated about the $x-$axis with respect to the earlier scheme, see Fig.\[fig8\](Right top). The effect of the noise is not only to shorten the length of the Bloch vector of the states, but also map the Bloch vector out of the plane of the original states. When the measurement is made, two things happen, as Fig.\[fig8\] (Right top) shows. (1) The Bloch vector is lengthened, in other words, the state become more pure, see also Eq.(\[statem\]). (2) The $x$ and $y$ components of the Bloch vector is mixed, in contrast to the proposal with $\beta=0$. As a consequence, the next rotation $Y_{\pm\eta}$ about the $y-$axis may make the resulting states (red vector) more close to the original states with respect to the earlier scheme.
Both control schemes in [@14] and [@15] are optimal for depolarizing noise and states lying in the $x-z$ plane, the depolarizing noise keeps these particular states in the $x-z$ plane and maintains the trace distance between the two states. If the original states are not in the $x-z$ plane, the depolarizing noise can not maintain the trace distance between the two states and causes the plane in which the two states lie to rotate as the states pass through the depolarizing channel. The optimal control scheme will depend on the orientation of the post-noise states. **From the optimality proof in Ref. [@15], we find that one optimal scheme is to use measurement operators to prolong the Bloch vectors of the post-noise states, and the correction is to bring the post-measurement states to the initial states as close as possible. The measurements and the correction are closely connected for a high performance.** In the present scheme, the optimal scheme is to use measurement operators that can map the two post-noise states as close as possible to the cone formed by the initial states. Specifically, the Bloch vectors of the initial state, the post-noise state and the post-measurement state form three cones(see the bottom figure of Fig. \[fig8\]), these cones share an axis: the $y-$axis, which pass perpendicularly through the centers of the bases. The three cones have a common apes, i.e., the origin of the Bloch sphere. One optimal scheme is to use measurement operators that map the two post-noise states very close to the initial-state-cone. The correction is a rotation about the $y-$axis, which would rotate the post-measurement states as close as possible to the initial states. This analysis simply consider the rotations of the axes of the Bloch sphere, to have a good performance, the length of the post-measurement state should be taken into account, this makes the optimization of $\beta$ complicated.
It is worth emphasizing that the angle rotated of our initial states is $\phi$. One may suspect that when the measurement cancels this rotation and send the states back to the $x-z$ plane, i.e., $\beta=\phi$, the optimal performance can be obtained. This intuition comes from the optimality proof in Ref.[@15], however this is not true as we shall show below.
[By using the average fidelity $\overline{F'}(\theta,p,\chi,\eta,\phi,\beta)$ in Eq.(\[fertility\]), we can calculate $\frac{\partial\overline{F'}}{\partial \beta}.$ From $\frac{\partial\overline{F'}}{\partial \beta}|_{\beta=\beta_c}=0,$ $\beta_c$ follows, which maximize the average fidelity $\overline{F'}$ and takes, $$\tan\beta_c=-\frac12\frac{\sin\eta\sin^2\theta\sin2\phi}
{\cos\eta\cos^2\theta+\sin^2\theta\sin^2\phi}.$$]{} [Clearly, the $\beta$ that maximize the performance depends not only on $\phi$ and $\theta$, but also on $\eta$, namely, it connects closely with the correction $Y_{\pm\eta}$. When $\phi=0$, $\beta_c=0$, returning back to the earlier scheme. This observation can be understood as follows. We denote $U$ the rotation about the $x-$axis, which sends the initial state back to the $xz-$plane, i.e., $\rho_{\pm}=U\tilde{\rho}_{\pm}U^{\dagger}.$ Here, $\tilde{\rho}_{\pm}=\rho_{\pm}|_{\phi=0}$. Then the resulting state $\mathcal{C}(\rho^{\prime})$ can be written as, $$\mathcal{C}(\rho^{\prime})=U\left(
\tilde{Y}_{+\eta}\tilde{M}^{\prime}_{+}
\tilde{\rho}^{\prime}\tilde{M}_{+}^{\prime\dagger}\tilde{Y}_{+\eta}^{\dagger}+
\tilde{Y}_{-\eta}\tilde{M}^{\prime}_{-}
\tilde{\rho}^{\prime}\tilde{M}_{-}^{\prime\dagger}\tilde{Y}_{-\eta}^{\dagger}\right
) U^{\dagger},\label{Cnew}$$ where $\tilde{\rho}^{\prime}=(1-p)\tilde{\rho}
+p\tilde{Z}\tilde{\rho}\tilde{Z},$ and $\tilde{(...)}=U^{\dagger}(...)U.$ This suggests that when the initial states are written as the same as that in the earlier scheme, the noise, measurement and the correction all need to change. Since $X$, $Y$ and $Z$ do not commute with each other, these changes are not trivial.]{} We should emphasize that the effect of the noise given in Eq. (\[noise\]) is to spoil the off-diagonal elements of the density matrix, or to shorten the $x-$ and $y-$component of the Bloch vector for any state, not only for the states lie in the $xz-$plane, so the aim of our scheme is to protect states against the same noise as that in the earlier scheme.
In conclusion, we introduce new measurements to better the state protection for a qubit. The average fidelity is calculated and discussed. [Numerical optimizations over these parameters show that the new measurements can extend the state protection scheme from special states to general states. This scheme works for a wide range of initial states and generalize the scheme in the earlier works.]{} The construction of the new proposal has several advantages. First, the initial states are more general, namely the corresponding Bloch vectors are allowed to lie outside the $xz-$plane, this extends the range of state protection and makes the scheme more realistic. The effect of the noise is to shorten the $x$ and $y$ components of the Bloch sphere, hence the noise is of dephasing. Second, we made use of a measurement which allow us to mix the $x-$ and $y-$components of the Bloch sphere, offering a room to improve the performance of the state protection. Finally, we note that the key elements to our scheme have already been experimentally demonstrated [@14], we expect that this extension of the earlier quantum control scheme is within reach of current technologies. \
\
This work is supported by the NSF of China under Grants Nos 61078011, 10935010 and 11175032.
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abstract: 'Based on weak-coupling anisotropic BCS theory, the temperature dependence of energy gap and the specific heat are evaluated for MgB$_2$ superconductor, and the results are compared with experimental data. We show that the weak-coupling anisotropic BCS theory describes thermodynamic experimental data with high precision 3–6%.'
author:
- 'Todor M. Mishonov'
- 'Valery L. Pokrovsky'
- Hongduo Wei
title: 'Two-band BCS model describes well the thermodynamics of MgB$_2$'
---
A keen interest excited by discovery and experimental investigation of a new high $T_c$ superconductor MgB$_2$ is to a large extent associated with its dissimilarity to cuprate superconductors. The superconductivity of MgB$_2$ is definitely three-dimensional effect, whereas in cuprates it is presumably two-dimensional (2D). Nevertheless, the superconducting gap in MgB$_2$ displays strong anisotropy. The most precise tunneling measurements by Gonnelli [*et al.*]{} [@gonnelli] give the value $2.6$ for the ratio of the gaps at two conductivity bands. On the other hand, the measured gaps are the same for the tunneling in $ab$-plane and in $c$-direction, indicating that they do not depend on direction within each piece of the Fermi surface.
An important problem is how strong is the interaction in MgB$_2$. First-principles calculations [@mazin; @golubov; @shulga] indicate that electron-phonon interaction is not weak and that Eliashberg description is appropriate. However, anisotropy and interaction were shown to influence thermodynamics oppositely. For example, the anisotropy decreases the relative discontinuity of the specific heat at the transition point [@pokrovsky1; @pokrovsky2], whereas the first correction due to interaction increases it [@gelikman-kresin]. Besides, MgB$_2$ is a very hard material with high value of Debye frequency, which usually correlates with a weak coupling. Therefore it is not *a priori* clear what is more substantial in the case of MgB$_2$.
The purpose of our work is to demonstrate that the anisotropy effects are more substantial at least for thermodynamic measurements. We show that, as a matter of fact, the weak coupling anisotropic BCS theory describes all known thermodynamic experimental data including the temperature dependence of the energy gap and specific heat with a high precision 3–6%.
The main features of anisotropic weak coupling BCS model were elucidated in the early 1960s [@pokrovsky1; @pokrovsky2; @hohenberg; @markowitz; @clem], the ultimate result being the factorization of the gap [@pokrovsky1] $$\Delta(T,{\mathbf{k}})=Q(T)\chi({\mathbf{k}})\,,
\label{factor}$$ which was experimentally verified by Zavaritskii [@zavaritskii]. The function of angle $\chi({\mathbf{k}})$ is the eigenfunction of the interaction operator $V({\mathbf{k}},{\mathbf{k}}^{\prime})$ corresponding to the maximal eigenvalue $\lambda_{+}$. It satisfies linear homogeneous integral equation: $$\int V({\mathbf{k}},{\mathbf{k}}^{\prime}) \chi({\mathbf{k}}^{\prime}) \frac{d
\sigma^{\prime}} {\nu_F v_F}=\lambda_{+}\, \chi({\mathbf{k}}) \,.
\label{eigen-eq}$$ Integration in Eq. (\[eigen-eq\]) proceeds over the Fermi surface with $d\sigma=\frac{dS}{8\pi^3}$ and $dS$ being a differential area of the Fermi surface; $\nu_F=\int \frac{d
\sigma}{v_F}$ is the electron density of the state per spin at Fermi level. The function $\chi({\mathbf{k}})$ is normalized as follows: $$\langle \chi^2({\mathbf{k}})\rangle =1
\label{normalization}$$ The angular average value $\langle X\rangle$ is: $\langle
X\rangle=\int \frac{X d \sigma}{\nu_F}$. the temperature dependent factor $Q(T)$ can be found from the orthogonality condition: $$\ln\frac{Q(0)}{Q(T)} =\left< \chi^2({\mathbf{k}})
F\left(\frac{Q(T)\chi({\mathbf{k}})}{T}\right)\right>,
\label{orth-cond}$$ where $$F(x)=\int_{-\infty}^{+\infty}
\frac{du}{\sqrt{x^2+u^2}(\exp{\sqrt{x^2+u^2}}+1)}\,.
\label{fx}$$ The value $Q(0)$ is associated with the transition temperature $T_c$ by the following relationship: $$\frac{Q(0)}{T_c}=\frac{\pi}{\gamma}\exp(-\langle\chi^2({\bf k})\ln|\chi({\bf k})|\rangle) \,,
\label{Q0-Tc}$$ here $\gamma=e^{C}=1.781072\cdots$ and $C$ is Euler’s constants. The specific heat $C(T)$ reads: $$ C(T)= 2\nu_F T \frac{d}{d T} \left< \Delta_{{\mathbf{k}}}\,
G\left(\frac{\Delta_{{\mathbf{k}}}}{T}\right) \right>,
\label{heat-capacity}$$ where $G(x)=2x\int_{0}^{\infty}\cosh (2\varphi) F(x\cosh \varphi)d
\varphi$.
We now apply these formulas to MgB$_2$. The Fermi surface of MgB$_2$ has two $\sigma$-type 2D cylindrical hole sheets and two $\pi$-type three-dimensional tubular networks [@Belashchenko; @choi; @liu]. We accept a simple model introduced first by Moskalenko [@Moskalenko], in which the interaction does not depend on the momentum inside each band, but only on the band index. Thus, it can be written as $2\times 2$ Hermitian matrix $V_{ik}$ ($i,k
=\sigma$, $\pi$). The order parameter (energy gap) in each band in such a model does not depend either on the momentum within each band and can be described by a 2D vector with components $\Delta_{\sigma}$, $\Delta_{\pi}$. The validity of this simple model is supported by the tunneling measurements of the energy gap [@gonnelli]. which displays the same values for two gaps in $ab$-plane and in $c$-direction. The normalized wave function of the Cooper pairs $\chi_{{\mathbf{k}}}$ has the same property: $\chi_{\sigma}({\mathbf{k}})=\chi_{\sigma}$, $\chi_{\pi}({\mathbf{k}})=\chi_{\pi}$, where $\chi_{\sigma}$ and $\chi_{\pi}$ are two constants. We introduce an additional simplification assuming these constants to be real. Let us denote the density of states in the $\sigma$ and $\pi$ bands as $\nu_{F\sigma}$ and $\nu_{F\pi}$, respectively. Then the definition of an average value $\langle X\rangle$ for any physical value $X$, which does not change within each band reads: $$\langle X\rangle=X_{\sigma}c_{\sigma}+X_{\pi}c_{\pi}\,,
\label{twoband-average}$$ where $c_{\sigma}$ and $c_{\pi}$ are statistical weights of the bands $c_{\sigma}=\nu_{F\sigma}/\nu_{F}$ and $c_{\pi}=\nu_{F\pi}/\nu_{F}$, $\nu_{F}=\nu_{F\sigma}+\nu_{F\pi}$. The general normalization condition Eq. (\[normalization\]) for this model reads: $$\chi_{\sigma}^2 c_{\sigma}+\chi_{\pi}^2 c_{\pi}=1\,.
\label{norm-twoband}$$ Equation (\[Q0-Tc\]) can be written explicitly as follows: $$\frac{Q(0)}{T_c}=\frac{\pi}{\gamma \chi_{\mathrm{av}}} \,,
\label{Q0-Tc-twoband}$$ where $\chi_{\mathrm{av}}=\chi_{\sigma}^{\chi_{\sigma}^2
c_{\sigma}}\chi_{\pi}^{\chi_{\pi}^2 c_{\pi}}$. We assume the values $c_{\sigma}= 0.44$ and $c_{\pi}= 0.56$ as found from density-functional theory calculations in Refs. [@Belashchenko; @choi; @liu]. The second fitting parameter is $T_c$. There is no experimental discrepancy on this value, and it is commonly accepted to be $T_c\approx 39$ K. One additional fitting parameter for the two-band theory is the ratio $\delta=\chi_{\sigma}/
\chi_{\pi}$. We have extracted it from the tunneling gap measurements extrapolating them to zero temperature: $$\delta=\chi_{\sigma}/ \chi_{\pi}\approx 2.54 \,.
\label{delta}$$ Equations (\[Q0-Tc-twoband\]) and (\[delta\]) allow us to determine $\chi_{\sigma}$ and $\chi_{\pi}$ separately: $\chi_{\sigma}=\delta/\sqrt{c_{\sigma}\delta^2+c_{\pi}}=1.38$; $\chi_{\pi}=1/\sqrt{c_{\sigma}\delta^2+c_{\pi}}=0.54$. According to the weak-coupling theory, the ratio $\delta$ must be the same at any temperature. This crucial condition is satisfied in the tunneling experiment [@gonnelli] with all experimental precision.
![\[f1\] The solid curve depicts the ratio $Q(T)/Q(0)$ vs $t=T/T_c$ for the two-band model; the dashed curve is the same value for the standard (isotropic) BCS theory.](./fig1.eps){width="85mm"}
![\[f2\] The solid curve is the theoretical graph of $\Delta_{\sigma}$ vs $T/T_c$; the dashed curve is the same for $\Delta_{\pi}$; “$+$” and “$\times$” represent experimental data by Gonnelli [*et al.*]{} [@gonnelli]](./fig2.eps){width="85mm"}
For the temperature dependence of the gap in the BCS two-band model, we find from Eq. (\[orth-cond\]): $$-\ln{q}=\chi^2_{\sigma}F\left(\frac{\pi \chi_{\sigma} q}{\gamma
\chi_{\mathrm{av}} t}\right)c_{\sigma}
+\chi^2_{\pi}F\left(\frac{\pi\chi_{\pi} q}{\gamma \chi_{\mathrm{av}}
t}\right)c_{\pi}\,.
\label{q-t}$$ Here $q(t)=Q(t)/Q(0)$ and $t=T/T_c$. $F(x)$ is defined by Eq. (\[fx\]). The graph of the function $q(t)$ is shown in Fig. \[f1\] by the solid curve. The dashed curve in Fig. \[f1\] represents $q(t)$ in the isotropic single-gap model (standard BCS model). The graphs of the energy gaps $\Delta_{\sigma}=Q(t)\chi_{\sigma}$ and $\Delta_{\pi}=Q(t)\chi_{\pi}$ vs $T/T_c$ are shown in Fig. \[f2\] together with the experimental data [@gonnelli], which agree with theory within the limits of experimental uncertainty.
The specific heat in the two-band model is given by the following equation directly stemming from Eq. (\[heat-capacity\]): $$\begin{aligned}
\frac{C(T)}{C_N(T)}&=&c_{\sigma}r_{c}(y_\sigma)+c_{\pi}r_{c}(y_\pi)
\nonumber \\
&+&\frac{12}{7\zeta(3)}\frac{[c_{\sigma}\chi_{\sigma}^2
r_{a}(y_\sigma)+c_{\pi}\chi_{\pi}^2 r_{a}(y_\pi)]^2}
{c_{\sigma}\chi_{\sigma}^4 r_{b}(y_\sigma)+c_{\pi}\chi_{\pi}^4
r_{b}(y_\pi)},
\label{specialheat}\end{aligned}$$ where $C_{N}(T)=\gamma T$ is the specific heat for the normal metal; $y_{\sigma}=\frac{\pi}{2
\gamma}\frac{q}{t}\frac{\chi_{\sigma}}{\chi_{\mathrm{av}}}$, $y_{\pi}=\frac{\pi}{2
\pi}\frac{q}{t}\frac{\chi_{\pi}}{\chi_{\mathrm{av}}}$. The functions $r_i$ are defined by integrals $r_i(x)=\int_{-\infty}^{+\infty}g_i(\sqrt{x^2+y^2})dy$, $i=a$, $b$, $c$, where $g_i$ read: $$\begin{aligned}
g_a(x) &=& \frac{1}{2\cosh^2 (x)}, \nonumber \\
g_b(x) &=& \frac{\pi^2}{14 \zeta(3)}
\left( \frac{\tanh x}{x}-\frac{1}{\cosh x} \right)
\frac{1}{x^2}, \nonumber \\
g_c(x) &=& \frac{6}{\pi^2}\frac{x^2}{\cosh^2 x}.\end{aligned}$$ For technical details related to this calculation see Mishonov [*et al.*]{} [@mishonov2]; the functions $g_i$ were introduced and graphically presented in Ref. [@mishonov3]
![\[f3\] The solid curve is the theoretical graph of the specific heat for the two band MgB$_2$ vs $t=T/T_c$; the circles are the experimental data due to Bouquet [*et al.*]{} [@Bouquet]; the dashed curve is the theoretical plot of the specific heat given by the isotropic BCS theory.](./fig3.eps){width="85mm"}
The jump of the specific heat at $T_c$ reads: cf.[@pokrovsky1; @Moskalenko] $$\frac{\Delta C(T_c)}{C_N(T_c)}=\frac{12}{7\zeta(3)}
\frac{\left(\chi_{\sigma}^2c_{\sigma}+\chi_{\pi}^2c_{\pi}\right)^2}
{\chi_{\sigma}^4 c_{\sigma}+\chi_{\pi}^4 c_{\pi}}\,.
\label{jump-C}$$ For the data specified earlier, we find $\Delta
C(T_c)/C_N(T_c)=0.874$. It agrees with the high precision measurements by Bouquet [*et al.*]{} [@Bouquet] with about $3$% precision. In Fig. \[f3\] the ratio $C(T)/C_N(T)$ vs. $T/T_c$ is plotted. The solid curve is the prediction of the two-band weak coupling theory; the dots are experimental data by Bouquet [*et al.*]{} [@Bouquet], courteously sent to us by the authors. The theoretical graph $C(T)/C_{N}$ vs $T/T_c$ agrees well with the experimental data everywhere except of a range of low temperature $T/T_c \leq 0.2$. The discrepancy most probably is caused by a relatively small variation of the gap within one band. The specific heat at low temperature is proportional to $e^{-\Delta_{\mathrm{min}}/T}$, whereas the tunneling measurements give the value of the gap along the direction of the tunneling. Given the value of discrepancy, we can estimate the variation of the gap $\bar{\Delta}-\Delta_{\mathrm{min}} \sim 0.1\mbox{--}0.15\,T_c \ln
2\approx 3.3\mbox{--}4.2$ K. It is about 8–12% of the value of the smaller gap.
Another group of available experimental thermodynamic data relates to magnetic properties: the energy gaps in external magnetic field [@gonnelli2] and the dependence of the second critical field on temperature [@lyard]. The dependence of $H_{c2}$ on temperature was considered theoretically in the framework of anisotropic BCS model by two groups of authors [@kogan2; @dahm]. based on classical approach by Helfand and Wertheimer [@HW]. Unfortunately, a consistent solution of these problems at any temperature between 0 and $T_c$ requires much more detailed knowledge about the Fermi surface. For example, to reach a satisfactory convergence Miranović [*et al.*]{} [@kogan2] were forced to introduce $11$ different parameters characterizing the Fermi surface and electron interaction. It is clear, that our real knowledge of the Fermi surface is too poor for such a sophistication. Dahm and Schopohl [@dahm] applied a simplified model of the Fermi surface as consisting of a torus and cylinder characterized by 4 parameters only and assumed a plausible variational procedure introducing one more parameter. As it could be expected from the results by Miranović [*et al.*]{}, the number of parameters is too small to ensure a reasonable precision. Indeed, a satisfactory agreement with the experiment in Ref. [@dahm] is reached at the expense of a rather exotic choice of parameter. Summing up, the magnetic properties can not be described by such an elementary theory as the described above two-band BCS model and require much more sophisticated approach even in the weak coupling approximation.
Let us discuss why this simplified theory works so well. Let us start from the assumption supported by experiments that the gap does not vary within each band. The in-band isotropy of the gap could be a result of sufficiently strong in-band scattering. At the scattering time $\tau\sim 10^{-14}$ s, i.e. at the residual resistance larger than $10^{-5}\,\Omega\mathrm{cm}$, the energy gap becomes isotropic. However, the ratio of the gaps for different bands still remains bigger than 2 indicating that the inter-band scattering must be much weaker. It should be emphasized that it is the density of states which becomes isotropic, whereas the order parameter remains anisotropic unless the Ioffe-Regel limit $\tau\varepsilon_F\sim 1$ of scattering rate is reached [@PP]. The tunnelling experiment measures just the density of state.
Second question is why the weak-coupling model gives so high accuracy. Two different aspects must be enlightened. First, the separability of variables for the order parameter, even in the framework of the weak-coupling approximation, has the precision of of the weak coupling constant, i.e. $(\ln\frac{\Delta}
{\omega_D})^{-1}\sim 0.3$. For the case of the two-band model such a crude estimate can be checked more accurately by a direct solution of the nonlinear matrix equation for the energy gap. It has a following form: $$\Delta_i=\sum_j
V_{ij}c_j\Delta_j\left[\frac{1}{\lambda_+}-f(\beta\Delta_j)\right]{\Delta_j}\,,
\label{non-linear}$$ where $i,j$ take values $\sigma$, $\pi$ and $f(x)=\int_{=\infty}^{\infty}\left(\frac{\tanh u}{u}-
\frac{\tanh\sqrt{u^2+x^2}}{\sqrt{u^2+x^2}}\right)du$. Its solution can be found as a superposition of two normalized eigenstates of the corresponding linear equation: $\Delta_j=Q_{+}\Psi_{+j}+Q_{-}\Psi_{-j}$. In our calculations we used only one of them, $\Psi_{+}$ corresponding to the larger eigenvalue $\lambda_{+}$. Such an approximation is justified when the second eigenvalue $\lambda_{-}$ is much less than $\lambda_{+}$, even if $\lambda_{+}$ is not very small. Indeed, the symmetrized matrix $\tilde{V}$ with matrix elements $\tilde{V}_{ij}=\sqrt{c_ic_j}V_{ij}$ can be represented as $\tilde{V}=\lambda_{+}|+\rangle\langle
+|+\lambda_{-}|-\rangle\langle -|$. This representation shows that, at $\lambda_{-}=0$ the operator $\tilde{V}$ is separable, and the solution of non-linear equation (\[non-linear\]) is factorizable: $\Delta=Q(T)\Psi_{+}$. The equation $\lambda_{-}=0$ is equivalent to ${\rm Det}
V=V_{\sigma\sigma}V_{\pi\pi}-V_{\pi\sigma}^2=0$. Though such a fine tuning of parameters seems improbable, our numerical calculations demonstrate that the ratio $\lambda_{-}/\lambda_{+}$ and the thermal variation of of the ratio $\Delta_{\pi}/\Delta_{\sigma}$ remain small (about 3%) even at ${\rm Det} V/(c_{\sigma}V_{\sigma\sigma}+c_{\pi}V_{\pi\pi})\sim\pm
0.2$. Thus, the experimental facts seem to indicate that one of the two eigenvalues is significantly smaller than another. Such a situation occurred earlier in a band calculation for high-$T_c$-superconductors [@mishonov-band].
The second aspect mentioned in the preamble is that the BCS approximation itself has a low precision and should be substituted by the Eliashberg formalism. The numerical calculations by Golubov [*et al.*]{} [@golubov] indicate that the Eliashberg weight functions are very small in a broad range of low energy and has rather sharp peaks in the range of 800–1000 K. This is an unusual situation. Leavens and Carbotte [@carbotte] considered an extended Eliashberg weight function $\alpha^2 F(\omega )$ centered at values $\omega\sim \omega_0$ much larger than the superconducting energy gap $\Delta (0)$. They argued on the basis of numerical calculations that in this case the function $\Delta
(\omega )$ varies very weakly at $\omega <\omega_0$ and then rapidly changes sign. They even modeled $\Delta (\omega )$ by the step function. Their arguments seem to be correct for the considered case as well. Then it is obvious that by integrating in the range of high frequency, it is possible to obtain the BCS-like equations with a renormalized, not small interaction between electrons with momenta on the Fermi surface. Though such an explanation is plausible, further study of the Eliashberg equation with a model weight is highly desirable.
We are thankful to Dr. A. Junod and to Dr. R. Gonnelli for sending us original experimental data of their works. This work was supported by NSF under the grants DMR-0321572 and DMR 0103455.
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|
---
abstract: |
[[ELAN]{}]{} is a powerful language and environment for specifying and prototyping deduction systems in a language based on rewrite rules controlled by strategies. Timed automata is a class of continuous real-time models of reactive systems for which efficient model-checking algorithms have been devised. In this paper, we show that these algorithms can very easily be prototyped in the [[ELAN]{}]{} system.
This paper argues through this example that rewriting based systems relying on rules *and* strategies are a good framework to prototype, study and test rather efficiently symbolic model-checking algorithms, i.e. algorithms which involve combination of graph exploration rules, deduction rules, constraint solving techniques and decision procedures.
author:
- Emmanuel Beffara
- Olivier Bournez
- Hassen Kacem
- Claude Kirchner
title: Verification of Timed Automata Using Rewrite Rules and Strategies
---
Introduction
============
[[ELAN]{}]{} is a powerful language and environment for specifying and prototyping deduction systems in a language based on rewrite rules controlled by strategies. It offers a natural and simple framework for the combination of the computation and the deduction paradigms. The logical and semantical foundations of the [[ELAN]{}]{} system rely respectively on rewriting logic [@MeseguerTCS92] and rewriting calculus [@CirsteaKirchner-LivreFroCoS99] and are in particular described in [@ELAN-wrla98; @CirsteaThese2000].
Timed automata [@AD94] is a particular class of hybrid systems, i.e. systems consisting of a mixture of continuous evolutions and discrete transitions. They can be seen as automata augmented with clock variables, which can be reset to $0$ by guarded transitions of some special type. They have proven to be a very useful formalism for describing timed systems, for which verification and synthesis algorithms exist [@Alu98; @AD94], and are implemented in several model-checking tools such as [TIMED-COSPAN]{} [@AK95], [KRONOS]{} [@DOTY95] or [UPPAAL]{} [@LPY97].
This paper describes our experience using the [[ELAN]{}]{}system to prototype the reachability verification algorithms implemented in the model-checking tools for timed automata.
It is known that rewriting logic is a good framework for unifying the different models of discrete-time reactive systems [@MeseguerTCS92]. Rewriting logic can be extended to deal with continuous real-time models. Such an extension, called “Timed rewriting logic” has been investigated, and applied to several examples and specification languages [@KW97; @PKW96; @SK00]. In this approach the time is somehow built in the logic. Another approach is to express continuous real-time models directly in rewriting logic. This has been investigated in [@OM96; @OM99] and recently Olveczky and Meseguer have conceived “Real-Time Maude” which is a tool for simulating continuous real-time models [@OM00]. Our approach is different. First, we do not intend to conceive a tool for [*simulating*]{} real-time systems, but for [*verifying*]{} real-time systems. In other words, we do not intend to prototype real-time systems but to prototype verification algorithms for real-time systems.
Second we focus on [*Timed Automata*]{}. Since verification of hybrid systems is undecidable in the general case [@ACH+95], any verification tool must restrict to some decidable class of real-time systems, or must be authorised to diverge for some systems. Timed automata is a class of continuous real-time systems which is known to be decidable [@AD94]. Real-Time Maude falls in the second approach in the sense that the “find” strategy implemented in this tool gives only [*partially*]{} correct answers [@OM00].
The implemented model-checking algorithms for timed automata are typical examples where the combination of exploration rules, deduction rules, constraint solving and decision procedures are needed. One aim of this work is to argue and demonstrate through this example that the rewriting calculus is a natural and powerful framework to understand and formalise combinations of proving and constraint solving techniques.
Another aim is to argue the suitability advantages of using a formal tool such as [[ELAN]{}]{} to specify and prototype a model checking algorithm compared to doing it in a much cumbersome way using a conventional programming language. First this allows a clear flexibility for customisation not available in typical hard-wired model checkers, through for example programmable strategies. Second, using the efficient [[ELAN]{}]{} compiler, the performances are indeed close to dedicated optimised model-checking tools.
This paper is organised as follows. In Section \[elan\], we describe the [[ELAN]{}]{} system based on rewriting calculus. In Section \[timedautomata\], we recall what a timed automaton is. In Section \[tool\], we describe our tool for verifying reachability properties of product of timed automata. In Section \[implementation\], we discuss the implementation.
The [[ELAN]{}]{} system {#elan}
=======================
The [[ELAN]{}]{} system takes from functional programming the concept of abstract data types and the function evaluation principle based on rewriting. In [[ELAN]{}]{}, a program is a set of labelled conditional rewrite rules with local affectations $$\ell: l \Rightarrow r \mbox{ if } c \mbox{ where } w$$ Informally, rewriting a ground term $t$ consists of selecting a rule whose left-hand side (also called pattern) matches the current term ($t$), or a subterm ($t_{|\omega}$), computing a substitution $\sigma$ that gives the instantiation of rule variable ($l\sigma = t_{|\omega}$), and if instantiated condition $c$ is satisfied ($c\sigma$ reduces to $true$), applying substitution $\sigma$ enriched by local affectation $w$ to the right-hand side to build the reduced term.
In general, the normalisation of a term may not terminate, or terminate with different results corresponding to different selected rules, selected sub-terms or non-unicity of the substitution $\sigma$. So evaluation by rewriting is essentially non-deterministic and backtracking may be needed to generate all results.
One of the main originalities of the [[ELAN]{}]{} language is to provide strategies as first class objects of the language. This allows the programmer to specify in a precise and natural way the control on the rule applications. This is in contrast to many existing rewriting-based languages where the term reduction strategy is hard-wired and not accessible to the designer of an application. The strategy language offers primitives for sequential composition, iteration, deterministic and non-deterministic choices of elementary strategies that are labelled rules. From these primitives, more complex strategies can be expressed, and new strategy operators can be introduced and defined by rewrite rules.
The full [[ELAN]{}]{} system includes a preprocessor, an interpreter, a compiler, and standard libraries available through the [[ELAN]{}]{} web page[^1]. From the specific techniques developed for compiling strategy controlled rewrite systems [@Moreau-RTA00; @MoreauK-PLILP+ALP98], the [[ELAN]{}]{} compiler is able to generate code that applies up to 15 millions rewrite rules per second on typical examples where no non-determinism is involved and typically between 100 000 and one million controlled rewrite per second in presence of associative-commutative operators and non-determinism.
Timed automata {#timedautomata}
==============
A [[*clock*]{}]{} is a variable which takes value in the set ${\mathbb{R}}^+$ of non-negative real numbers. A [*clock constraint*]{} is a conjunction of constraints of type $x \# c$ or $x-y \# c$ for some clocks $x,y$, rational number $c \in {\mathbb{Q}}$ and $\#
\in \{\leq,<,=,>,\geq\}$. Let $TC(K)$ denotes the set of clock constraints over clock set $K$.
Informally, a [[*Timed Automaton*]{}]{} is a finite automaton augmented with clock variables, which can be reset to $0$ by guarded transitions.
Formally, a [[*timed automaton*]{}]{} [@AD94; @Alu98] is a $5$-tuple $\mathcal{A}=(\Sigma,L,K,I,\Delta)$ where
1. $\Sigma$ is a finite alphabet,
2. $L$ is a finite set of [[*locations*]{}]{},
3. $K$ is a finite set of clocks. A [[*state*]{}]{} is given by some location and some valuation of the clocks, i.e. by some element $(s,v)$ of $L \times
{\mathbb{R}}^{+^{|K|}}$.
4. $I$ is a function from $L$ to $TC(K)$ that labels each location $s$ by some [[*invariant*]{}]{} $I(s)$. Invariant $I(s)$ restricts the possible values of the clocks in location $s$.
5. $\Delta$ is a subset of $\Sigma \times L \times TC(K) \times \mathcal{P}(K) \times L$. [[*Transition*]{}]{} $(a,s,c,z,s') \in \Delta$ corresponds to a transition from location $s$ to location $s'$, labelled by $a$, guarded by constraint $c$ on the clocks, and which resets the clocks $k \in z$ to $0$.
Timed automaton $\mathcal{A}$ evolves according to two basic types of transitions:
1. [[*Delay transitions*]{}]{} corresponds to the elapsing of time while staying in some location: write $(s,v) \longrightarrow^{d} (s,v')$, where $d \in {\mathbb{R}}^+$, $v'=v+(d,\dots,d)$ provided for every $0 \leq e \leq d$, state $v+(e,\dots,e)$ satisfies constraint $I(s)$.
2. [[*Action transitions*]{}]{} corresponds to the execution of some transition from $\Delta$: write $(s,v) \longrightarrow^{a} (s',v')$, for $a \in
\Sigma$, provided there exists $(a,s,c,z,s') \in \Delta$ such that $v$ satisfies constraint $c$ and $v'_{k}=v_{k}$ for $k \not\in z$, $v'_{k}=0$ for $k \in z$.
A [[*trajectory*]{}]{} of $\mathcal{A}$ starting from $(s,v)$ is a sequence $$(s,v) \longrightarrow^{e_0} (s_1,v_1) \longrightarrow^{e_1} (s_2,v_2) \dots$$ for some $e_0,e_1,\dots \in \Sigma \cup {\mathbb{R}}^+$.
#### Product construction.
Timed automata can be composed by [[*synchronisation*]{}]{} [@Alu98; @AD94] (see Figure \[figproduit\] for a classical example). Intuitively, building the product of two timed automata consists in considering a timed automaton whose state space is the product of the state spaces of the automata, where transitions labelled by a same letter in the two automata occur synchronously, and others may occur asynchronously.
The tool {#tool}
========
Our verification system for timed automata is fully implemented in [[ELAN]{}]{}. It works according to the schema of Figure \[fig:schem\]. Concretely,
1. The tool takes a specification of a product of automata. The specification is given in the form of a text file containing a list of clocks, a list of locations, a list of labels, and a list of automaton descriptions. Each automaton description is in turn a list of list of locations, labels, invariants, transitions: cf Figure \[fig:spec\].
2. The specification is then parsed using the [[ELAN]{}]{} system, and compiled into an executable program. More precisely, the [[ELAN]{}]{} preprocessor, manipulates the encapsulated lists of the specification through [[ELAN]{}]{} rules in order to generate rewrite rules, which are in turn compiled by the [[ELAN]{}]{} compiler into an executable $C$ program.
3. This $C$ program tests reachability properties of the product of automata:
1. it takes as input a query of type $go(s/c, s'/c')$ where $s,s'$ are some locations of the product of automata, and $c,c'$ are some clock constraints.
2. it answers $True$ iff there is a trajectory starting from some state $(s,v)$ with $v$ satisfying clock constraint $c$ that reaches some $(s',v')$ with $v'$ satisfying clock constraint $c'$.
-------------------------------------------------------------------------------------------------------------
specification train
Clocks lower down raise up
X Y Z nil
nil Invariants
States Up : true
Far Near In After Up t1 Down t2 u0 u1 u2 t1 : Y<=1 ^ true
nil Down : true
Labels t2 : Y<=2 ^ true
app raise lower up down enter out exit nil
nil Transitions
Automata Up , lower : true, Y nil, t1 .
( t1 , down : true, nil, Down .
Locations Down, raise : true, Y nil, t2 .
Far Near In After t2 , up : true, nil, Up .
nil nil
Labels ) .
app enter out exit (
nil Locations
Invariants u0 u1 u2
Far : true nil
Near : X<=5 ^ true Labels
In : X<=5 ^ true app lower raise exit
After : X<=5 ^ true nil
nil Invariants
Transitions u0 : true
Far , app : true, X nil, Near . u1 : Z<=1 ^ true
Near , enter: X>2 ^ true, nil, In . u2 : Z<=1 ^ true
In , out : true, nil, After . nil
After, exit : true, nil, Far . Transitions
nil u0, app : true, Z nil, u1 .
) . u1, lower : Z>=1 ^ Z<=1 ^ true, nil, u0 .
( u0, exit : true, Z nil, u2 .
Locations u2, raise : true, nil, u0 .
Up t1 Down t2 nil
nil ) .
Labels nil
end
----------------------------------------------------- -------------------------------------------------------
-------------------------------------------------------------------------------------------------------------
Figure \[figexample\] shows some queries and their execution times for the $train$ $\|$ $gate$ $\|$ $controller$ system, and for the system described in [@DY95] consisting of two robots, a conveyer belt and a processing station[^2]. The execution times of system [[*KRONOS*]{}]{} are shown for comparison. Observe that they are of same magnitude.
Query Answer [Kronos]{} [Elan]{}
--------------------------------------------------------- --------- -------------- --------------
$Train$: $go(Far.Up.u0.nil/true,In.Down.u0.nil/true)$ $True$ 0.03 seconds 0.00 seconds
$Train$: $go(Far.Up.u0.nil/true,In.Up.u0.nil/true)$ $False$ 0.03 seconds 0.03 seconds
$Robots$:$go(D-Wait.G-Inspect.S-Empty.B-Mov.nil/true$ $True$ 0.04 seconds 0.03 seconds
$,D-Turn-L.G_-Inspect.S-Empty.B-Mov.nil/true)$
$Robots$: $go(D-Wait.G-Inspect.S-Empty.B-Mov.nil/true,$ $False$ 0.04 seconds 0.14 seconds
$D-Turn-R.G-Turn-L.S-Empty.B-On-S.nil/true)$
Implementation
==============
We now describe how the reachability algorithm is implemented using rewrite rules controlled by strategies. We need first to recall the basis of timed automata theory.
Region automaton
----------------
Recall that a [[*labelled transition system*]{}]{} is a tuple $(Q,\Sigma,\to)$ where $Q$ is set of states, $\Sigma$ is some finite alphabet, and $\to \in Q \times \Sigma \times Q$ is a set of transitions.
Given some timed automaton $A=(\Sigma,L,K,I,\Delta)$, denoting $(s,v) \leadsto^{a}
(s',v')$ if there exists $s''$ and $v''$ such that $(s,v) \longrightarrow^{d}
(s'',v'') \longrightarrow^{a} (s',v')$ for some $d \in {\mathbb{R}}^+$, the [[*time-abstract labelled transition system*]{}]{} associated to $A$ is the labelled transition system $S(A)=(Q_A,\Sigma,\leadsto)$ whose state space $Q_A$ is unchanged, i.e. $Q_A=L \times {\mathbb{R}}^{+^{|K|}}$, but whose transition relation is given by $\leadsto$.
Although $S(A)$ has uncountably many states, we can associate some equivalence relation ${\sim}_A$ over the state space $Q_A$ which is stable and which is of finite index [@Alu98; @AD91]. Some equivalence relation ${\sim}$ over the space of a labelled transition system $(Q,\Sigma,\to)$ is said to be [[*stable*]{}]{} iff whenever $q {\sim}u$ and $q \to^{a} q'$ there exists some $u'$ with $u \to ^{a} u'$ and $u' {\sim}u$.
The [[*quotient of $S(A)$ with respect to ${\sim}_A$*]{}]{}, denoted by $[S(A)]$, is the transition system whose state space is made of the equivalence classes of ${\sim}_A$, called [[*regions*]{}]{}, and such that there is a transition from region $\pi$ to region $\pi'$ labelled by $a$ if for some some $q \in \pi$ and $q' \in \pi'$, $q \leadsto^{a} q'$.
Since ${\sim}_A$ is of finite index, $[S(A)]$ is a finite automaton, which is called [[*the region automaton*]{}]{} of $A$ [@Alu98; @AD91].
Since ${\sim}_A$ is stable, the set of reachable states from some region $s_0$ in timed automaton $A$ is equal to the union of the reachable regions in $[S(A)]$ starting from region $s_0$ [@Alu98; @AD91]. Hence, the reachability problem for $A$ reduces to the reachability problem for finite automaton $[S(A)]$.
This is the basis of all model-checking tools for timed automata. See [@Alu98; @AD94] for details.
Manipulating regions using states with constraints
--------------------------------------------------
Computing the reachable regions in region automaton $[S(A)]$ requires to manipulate regions.
This is can be done by manipulating symbolic representations of these sets, i.e. by manipulating clock constraints [@Alu98; @AD94]. Hence, our program in [[ELAN]{}]{} manipulates terms of form $s/c$ where $s$, $c$ is a (term representation of) a state of the product of automata, and $c$ is a clock constraint. Term $s/c$, represents the (convex) set, denoted by ${[[}s/c{]]}$, of all the states $(s,v) \in Q_A$, such that $v$ satisfies constraint $c$. Such a set is called a [[*zone*]{}]{} [@Alu98; @AD94].
The heart of the reachability algorithm in [[ELAN]{}]{} is made of [ *rewrite*]{} rules which manipulate such terms through symbolic operators on constraints. Figure \[figconstraint\] shows an example of such a rule for the system of Figure \[fig:spec\]. This rewrite rule uses “Intersection” and “Reset” operators on clock constraints.
[] Post.enter(Near/c) => In /Reset(Intersection(X>2 ^ true,c), nil) end
The $Intersection$ operator transforms two clock constraints into a representation of their intersection. The $Reset$ operator transforms a constraint $c$, and a list of clocks $k$, to a constraint representing the set of states reachable from a state satisfying $c$ after the variables of $k$ are reset to $0$.
Generation of rules from the automaton specification
----------------------------------------------------
These rules on zones are generated from the description of the timed automaton given as input using the preprocessor facilities of the [[ELAN]{}]{} system.
For example, for any transition $e=(a,s,d,z,s')$ of the timed automaton, we need to generate a rewrite rule which transforms any zone $s/c$ into some zone $s'/c'$ which represents all the states reachable from $s/c$ by transition $e$. Formally, we want to rewrite $s/c$ into $s'/c'$ with $${[[}s'/c'{]]}=\{(s',v')| (s,v) \in {[[}s/c{]]}\mbox{ and } (s,v)
\longrightarrow^{a}(s',v') \}$$ This can be done by generating rewrite rule $$post.e(s/c) \Rightarrow s'/Reset(Intersection(c,d),z)$$
In order to generate this rewrite rule for any transition $e$ of the automaton specification, we just need in the [[ELAN]{}]{} system, to write the preprocessor rule of Figure \[figpreproc\].
FOR EACH A : Automaton SUCH THAT A:=(listExtract) elem(LA):{
rules for statezone
z : clockzone ;
global
FOR EACH tr : transition ;
bef, aft : state ;
label : label ;
cond : clockzone ;
zero : list[clock]
SUCH THAT tr := (listExtract) elem(lst_trans(A))
AND bef :=()tr_before(tr)
AND label:=()tr_label (tr)
AND cond :=()tr_cond (tr)
AND zero :=()tr_zero (tr)
AND aft :=()tr_after (tr) :
{
[] Post.label(bef/z) => aft / Reset(Intersection(cond,z),zero) end
}
end // of rules for statezone
Constraint representation
-------------------------
Implementing the operators on constraints requires to represent clock constraints.
One solution consists in representing clock constraints using [[*bounded differences matrices*]{}]{} [@Alu98; @Dil89]. With this representation scheme, any clock constraint has a normal form which can be computed using a Floyd-Warshal algorithm based technique [@Alu98].
This method using bounded differences matrices has been implemented as a rewrite system in our tool. That means in particular that the Floyd-Warshal based technique for computing normal forms of bounded differences matrices is implemented as rewrite rules, as well as all operators required on constraints ($Intersection$, $Reset$, $Is\_Empty?$ , $Is\_Equivalent?$, $Effect\_Of\_Time$$\_Elapsing$).
#### Constraint representation alternatives.
Other representations of constraints are possible. In particular, we have experimented the representation of clock constraints using classical logical formulas. Clock constraints are closed by quantifier elimination [@BeffaraStage2000; @LivreClarke]. Denoting by $Exists$ the operator which maps a clock constraint $c$ and a list of variable $k$ to a clock constraint logically equivalent to formula $\exists k\ c$, the above operators on constraints can all be expressed using the $Exists$ operator [@BeffaraStage2000; @LivreClarke]. For example, the $Reset$ operator can be expressed by the rewrite rule $$Reset(c,k) \Rightarrow Exists(c,k) \land k=0.$$
This as been implemented in our tool. The $Exists$ operator is computed using a technique based on Fourier-Motzkin algorithm [@BeffaraStage2000].
Exploration
-----------
Once the constraint manipulation is implemented, one interesting part is to implement the exploration of the reachable regions of the automaton. This is done by manipulating terms of the form $Transitions\_List(lsz,szs,szc)$ where $lsz$ is a list of already explored zones, $szs$ is the current zone under investigation, and $zsc$ is the objective zone.
One main originality of [[ELAN]{}]{} is the possibility to express strategies. Hence, the exploration of graph can be guided by the simple strategy language of the [[ELAN]{}]{} system.
As an example, suppose we want to explore the graph by backtracking. Rule named $SuccessStep$ of Figure \[figstep\] does one step of the graph exploration for the particular case when the objective zone is reachable by one step of the graph exploration, and rule named $NextStep$ does one step of the graph exploration for the generic case.
To explore the whole graph, we just need to iterate these rules, taking at each step the first one of the two elementary rules $SuccessStep$ and $NextStep$ which succeeds. This is easily done using the [[ELAN]{}]{} strategy language as in Figure \[fig:strategy\].
rules for bool
s,sO : state;
sz : zone;
c,cO : clockzone;
result : bool;
lsz : hashSet[statezone];
global
[SuccessStep] Transitions_List(lsz,s/c,s0/c0) => result
where sz := (Post) s/c
if sz.state== s0
if not Is_Empty?(Intersection(sz.constraint,c0))
where result:=() True
end
[NextStep] Transitions_List(lsz,s/c,s0/c0) => result
where sz := (Post) s/c
if not Is_Empty?(sz.constraint)
if not lsz.contains(sz)
where result:=(Exploration) Transitions_List(lsz.add(sz),sz,s0/c0)
end
strategies for bool
implicit
[] Exploration => first one(SuccessStep,NextStep) end
end
rules for bool
s,sO: state;
c,cO: clockzone;
result: bool;
global
[] go(s/c,s0/c0) => result
choose
try where result:=(Exploration) Transitions_List(EmptySet,s/c,s0/c0)
try where result:=()False
end
end
#### Exploration alternatives.
Of course, other exploration strategies can also be used and experimented just by modifying the above lines. Graph can easily be traversed depth first, breath first, if one prefers.
On-fly generation.
------------------
The tool implements [[*on-fly model-checking*]{}]{}. This means that the tool does not need to build the full product of the timed automata before testing reachability properties, but that the transitions of the product of timed automata are generated on-line only when needed. This is in contrast with what happens in some model-checking tools.
This is easily done, in the case of an input consisting of a product of $n$ timed automata, by using a succession of $n$ [[ELAN]{}]{} $first\_one$ strategy operators applied on named rules $ExecuteTransition\_i$ and $JumpStep\_i$ which compute on-line the transitions to apply in each automaton of the product corresponding to some possible label: cf Figure \[figonline\].
// Transcription Of The Synchronised Product
FOR EACH N : int SUCH THAT N:=()size_of_Automaton_list(LA):{
rules for statezone
{s_I : state ;}_I=1...N
{ss_J : state ;}_J=1...N
Phi, nPhi : clockzone ;
lbl : label ;
sz : statezone ;
global
{
[ExecuteTransition_i] SynTransitionOperator(lbl,{s_j.}_j=1...(i-1) s_i.{s_j.}_j=(i+1)...N nil/Phi) =>
SynTransitionOperator(lbl,{s_j.}_j=1...(i-1) ss_i.{s_j.}_j=(i+1)...N nil/nPhi)
if action(lbl,s_i,i)
where sz:=()TransitionOperator.lbl(s_i/Phi)
where ss_i:=()st(sz)
where nPhi:=()zn(sz)
end
[JumpStep_i] SynTransitionOperator(lbl,sz) => SynTransitionOperator(lbl,sz) end
}_i=1...N
[FinishSynTransition] SynTransitionOperator(lbl,sz) => sz end
end // of rules for statezone
} // End Of The Transcription Of The Synchronised Product
strategies for statezone
implicit
FOR EACH N : int SUCH THAT N:=()size_of_Automaton_list(LA) :{
[] next_sz => {first one(ExecuteTransition_I,JumpStep_I);}_I=1...N FinishSynTransition end
}
end // of strategies for statezone
Conclusion
==========
This paper presents the use of the rewrite based system [[ELAN]{}]{} to prototype model-checking algorithms for timed automata. As expected, the performance are a little bit lower than model-checking dedicated tools like KRONOS [@DOTY95] or UPPAAL [@LPY97]. However, using the specific techniques already developed for compiling strategy controlled rewrite systems implemented in the [[ELAN]{}]{} system compiler, the performances turns out to be of same magnitude.
The main advantage is the gained flexibility compared to conventional programming languages. The whole [[ELAN]{}]{} code for the described model-checkers is less than $1100$ lines (including comments). Changing the graph exploration technique, or the constraint solving algorithm, for example, turned out to require to modify only a few lines. We presented the direct and classical implementation of the reachability algorithms for timed automata. But other techniques (e.g.: partition refinement techniques [@Alu98; @ACH+95], alternative representations for constraints ) could also be experimented and would require only a few modifications of the existing [[ELAN]{}]{} code.
Furthemore, the [[ELAN]{}]{} system offers facilities, such as a strategy language which provides flexibily for customizations that are not available in typical hard-wired model checkers such as programmable strategies.
Moreover, we believe that such a work helps to understand mixture of proving and constraint manipulation techniques by studying them in the same unifying framework. In particular, it clearly helps to delimit the difference between pure computations and deductions in model-checking techniques which are very often presented as relying only on computations on constraints.
More details on the tool together with the full code can be found on web page [http://www.loria.fr/$\sim$kacem/AT]{}.
Thanks
======
The authors would like to thank all members of the PROTHEO group for their discussions. Among them, we would like to address some special thanks to Pierre-Etienne Moreau who helped us a lot through his expertise of the [[ELAN]{}]{} compiler.
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[^1]: [http://elan.loria.fr]{}
[^2]: See web page [http://www.loria.fr/$\sim$kacem/AT]{} for details.
|
---
author:
- 'Wu-Zhong Guo,'
- 'Song He,'
title: Rényi entropy of locally excited states with thermal and boundary effect in 2D CFTs
---
Introduction
============
Many kinds of observables can be defined in Quantum field theories (QFTs). When we study global or non-local structures, entanglement entropy (EE) or the entanglement Rényi entropy (RE) are very helpful quantities. For a subsystem $A$, both of them are defined as a function of the reduced density matrix $\rho_A$. The reduced density matrix $\rho_A$ can be defined from the original density matrix $\rho$ by tracing out the subsystem $B$ which is the complementary of $A$.
One may be curious about whether there is a kind of topological contribution in entanglement entropy even for gapless theories, particularly for conformal field theories (CFTs). For example, topological properties can be qualified by computing topological contributions in entanglement entropy called topological entanglement entropy [@wen]. In this paper, we focus on extracting such kind of topological quantity from both Rényi entropy and von-Neumann entropy of locally excited states in two dimensional rational CFTs with thermal effect and boundary effect.
The $n$-th Rényi entanglement entropy is defined by $S^{(n)}_A=\log\mbox{Tr}[\rho_A^n]/(1-n)$ formally. The limit $n\to 1$ coincides with the von-Neumann entropy. This is standard replica trick method to calculate the entanglement entropy. The difference of $S^{(n)}_A$ between the locally excited states and the ground states with introducing thermal and boundary effects are main interest in this paper. The difference is denoted by $\Delta S^{(n)}_A$. Replica approach calculations of $\Delta S^{(n)}_A$ for states excited by operators have been given in [@UAM; @Nozaki:2014uaa; @Nozaki:2014hna]. We will closely follow the construction in [@Nozaki:2014hna][@He:2014mwa][@Nozaki:2014uaa]with introducing thermal and boundary effect.
We would like to review the thermal effect and boundary effect respectively. There are many studies about thermal effect in 2D CFTs. For thermal states, EE is not a good measurable quantity, which is contaminated by the thermal entropy of the subregion. In high temperature limit, the EE will be dominated by thermal entropy. To reveal the quantum entanglement of system with thermal effect, one should identify the thermal contribution and other contributions of EE. Ref. [@Herzog:2012bw]conjectured universal form of correction of EE in any quantum system with mass gap. Ref.[@Cardy:2014jwa] provided the form of the coefficient of such correction in 2D CFT. To generalize studies in [@Cardy:2014jwa] to higher dimensions, Ref.[@Herzog:2014fra] and Ref.[@Herzog:2014tfa] considered thermal corrections to the entanglement entropy on spheres. On the other hand, the dynamics in 2D CFTs with a boundary have many new features comparing with 2D CFTs in full complex plane. The original works have been done by Cardy, who discussed surface critical behavior of correlation functions [@cardy1]. Ref.[@cardy2] studied the constraints on the operator content by imposing by boundary conditions and also the classification of boundary states in terms of the modular transformation. In [@cardy3][@cardylewellen], the concept of boundary operators have been introduced. Ref.[@3sp2] showed that the resulting set of boundary conditions to be complete. There are also nice correspondence called as AdS/BCFT proposed by [@Takayanagi:2011zk][@Fujita:2011fp]. The boundary effect can be also studied holographically, which is beyond the issue considered in this paper.
In 2D rational CFTs, the authors in [@He:2014mwa] get an amazing result for the locally excited states, which relates the Rényi entropy to the quantum dimension of the primary operator which is kind of topological quantity. In this paper, we generalize the previous study [@He:2014mwa] on Rényi entropy with thermal and boundary effects. Firstly, there is a simple sum rule between the thermal correction and local excitation in low temperature limit. That is to say the total Rényi entropy are summing over Rényi entropy of local excitation and the one of thermal excitation in low temperature limit. Such kind of relation is similar to the sum rule related to the Rényi entropy in [@Nozaki:2014uaa]. One can generalize the result to local excitation in pure state in 2D CFT. We make use of a different approach [@Herzog:2012bw] to obtain the thermal correction to Rényi entropy which can be reduced to [@Caputa:2014eta]. Secondly, we investigate the the Rényi entropy for states excited by local primary operators in the rational CFTs with a boundary. These boundaries introduced here do not break the conformal symmetry. Such theories are called as BCFTs. We show the time evolution of the Rényi entropy in 2D free field theory and Ising model. Then we generalize to rational CFTs with a boundary. The boundary changes the time evolution of the Rényi entropy, but does not change the maximal value of the Rényi entropy. All these cases studied in this paper show that the Rényi entropy does not depend on the choice of boundary conditions. In 2D rational CFTs with a boundary, we also show that the maximal value of the Rényi entropy always coincides with the log of quantum dimension of the primary operator during some periods of the evolution. We give the physical understanding of boundary effect, which support the quasi-particles explanation of the local excitation. The boundary behaves as an infinite potential barrier which reflects the quasi-particle moving towards it.
The layout of this paper is as follows. In section 2, we study the thermal effect on the Rényi entropy of the local excited state in low temperature limit. In section 3, we set up the local excitation in 2D CFT with a boundary and obtain the Rényi entropy of a subsystem with time evolution. We study the 2D free scalar, and Ising model as examples, then generalize the result to 2D rational CFT. In section 4, we devote to the conclusion and physical interpretations of such kinds of effects shown in this paper.
Local excitation in non-vacuum states
=====================================
In this section, we would like to study the local excitation of thermal state. We consider a system with temperature $T=1/\beta$ and assume the excitation is local at $x=-L$ by primary operator $O$ shown in fig.\[\[fig0\]\]. In this section, we just only consider the low temperature case with large $\beta$ [@Herzog:2014fra]. The subsystem $A$ is $-l<x<0$. The density matrix $\rho(t)$ is $$\begin{aligned}
\label{T1}
\rho(t)=N(t)\Big(O(\omega_2,\bar \omega_2)(\sum \Ket{n}\Bra{n} e^{-\beta E_n-2\beta \epsilon E_n})O^{\dag}(\omega_1,\bar \omega_1)\Big),\end{aligned}$$ where we have considered the real time evolution, $\epsilon$ is the ultraviolet regularization, $N(t)$ is fixed by normalization condition $tr\rho(t)=1$, $E_n$ are the energy of the excited states. The complex coordinates in $\omega$ plane are listed as follows. $$\begin{aligned}
\omega_1=i({\epsilon-it})-L, \ \ \omega_2=-i(\epsilon+it)-L, \nonumber \\
\bar \omega_1=-i(\epsilon-it)-L,\ \ \bar \omega_2=i(\epsilon+it)-L.\end{aligned}$$
=8.0 cm =5.0 cm
For later convenience, we define $$\begin{aligned}
&&\rho_0(t)=tr_B( O(\omega_2,\bar \omega_2)\Ket{0}\Bra{0}O^{\dag}(\omega_1,\bar \omega_1)),\nonumber\\
&&\rho_1(t)=tr_B( e^{-\beta E_1-2\beta \epsilon E_1} O(\omega_2,\bar \omega_2)\Ket{1}\Bra{1}O^{\dag}(\omega_1,\bar \omega_1)),\end{aligned}$$ which can be taken as the reduced density matrix related to the vacuum and first excited state respectively, where we normalize the vacuum energy to be zero, $B$ is the complementary part of subsystem $A$. In the low temperature expansion with $\beta E_1 \ll 1$ $$\begin{aligned}
\label{Expand}
\rho_A(t)=tr_B \rho(t)=\frac{\rho_0(t)+\rho_1(t)+...}{tr_A (\rho_0(t)+\rho_1(t))+...}.\end{aligned}$$ In terms of the definition of the Rényi entropy $$\begin{aligned}
\label{resultofthermal}
S^{(n)}_A&=&\frac{\log tr \rho_A(t)^n }{1-n}\nonumber \\ &&\simeq \frac{1}{1-n}\log\Big[\frac{tr(\rho_0(t)^n)}{(tr\rho_0(t))^n}(1+\frac{n tr(\rho_0(t)^{n-1}\rho_1(t))}{tr(\rho_0(t)^n)}-\frac{n tr\rho_1(t)}{tr\rho_0(t)})\Big]\nonumber \\
&&\simeq \frac{1}{1-n}\Big[\log \frac{tr(\rho_0(t)^n)}{(tr\rho_0(t))^n}+\frac{n tr(\rho_0(t)^{n-1}\rho_1(t))}{tr(\rho_0(t)^n)}-\frac{n tr\rho_1(t)}{tr\rho_0(t)}\Big].\end{aligned}$$ When there is no local excitation, i.e., the operator $O=I$, the result is the same as [@Cardy:2014jwa]. The second and third terms of the last line in (\[resultofthermal\]) involve in the coupling between the local excitation and thermal environment. In terms of the state operator correspondence, one can denote $\Ket{1}=\lim_{t\to -\infty}\psi(x,t)\Ket{0}$ [@Cardy:2014jwa]for the excited state with energy $E_1$. Here we consider the whole system has an infrared cut-off $\Lambda$, and $l/\Lambda \ll 1$. Using the path integral language, (\[resultofthermal\]) is $$\begin{aligned}
\label{ResultSncorrelation}
S^{(n)}_A&=&\frac{1}{1-n} \log \frac{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)...O(\omega_{2n},\bar \omega_{2n})\rangle_{C_n}}{(\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{C_1})^n}\nonumber \\
&+&\frac{ne^{-\beta E_1}}{1-n}\frac{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)...O(\omega_{2n},\bar \omega_{2n})\psi(-\infty)\psi(+\infty)\rangle_{C_n}}{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)...O(\omega_{2n},\bar \omega_{2n})\rangle_{C_n}}\nonumber \\
&-&\frac{n e^{-\beta E_1}}{1-n}\frac{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2) \psi(-\infty)\psi(+\infty)\rangle_{C_1}}{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{C_1}},\end{aligned}$$ where $C_n$ is the $n$-sheet cylinder with circumference $\Lambda$ and $O(\omega_1,\bar \omega_1)...O(\omega_{2n},\bar \omega_{2n})$ are the operators that are inserted in the suitable place in the $n$-sheet cylinder. The first term (\[ResultSncorrelation\]) is given by [@He:2014mwa] for the local excitation in vacuum. We will study the second and third term of (\[ResultSncorrelation\]) in detail.
The following transformation [@cag] can map the $n$-sheet cylinder to a cylinder with circumference $\Lambda$, $$\begin{aligned}
\label{thermaltransformation}
z=\Big(\frac{e^{2i\pi \omega/\Lambda}-1}{e^{2i\pi \omega/\Lambda}-e^{2i\pi l/\Lambda}}\Big)^{1/n}.\end{aligned}$$ The points $\omega=-\infty$ and $\omega=\infty$ are mapping to $z_{-\infty}=e^{-2i\pi l/\Lambda}$ and $z_{+\infty}=1$ respectively. For simplifying our analysis, we only consider the range $|\omega|\ll \Lambda$. Otherwise, the following calculation will be much more complicated. But the final statement does not change without this approximation. Thus (\[thermaltransformation\]) reduces to $$\begin{aligned}
z\simeq \Big(\frac{w}{w-l}\Big)^{1/n},\end{aligned}$$ which is same as the one that is used in [@He:2014mwa]. The points $z_1,...,z_{2n}$ are given by $$\begin{aligned}
z_{2k+1}&=&e^{2i\pi k/n}\Big(\frac{i\epsilon +t-L}{i\epsilon +t-L-l} \Big)^{1/n},\nonumber \\
z_{2k+2}&=&e^{2i\pi k/n}\Big(\frac{-i\epsilon +t-L}{-i\epsilon +t-L-l} \Big)^{1/n},\nonumber \\
\bar z_{2k+1}&=&e^{2i\pi k/n}\Big(\frac{-i\epsilon-t-L}{-i\epsilon -t-L-l} \Big)^{1/n},\nonumber \\
\bar z_{2k+2}&=&e^{2i\pi k/n}\Big(\frac{i\epsilon -t-L}{i\epsilon -t-L-l} \Big)^{1/n}.\label{transformationrule}\end{aligned}$$ In $t>l+L$ or $0<t<L$, $$\begin{aligned}
\label{factor1thermal}
z_{2k+1}-z_{2k+2}\simeq -\frac{2iL\epsilon}{n(t-L)(t-l-L)}z_{2k+1},\nonumber \\
\bar z_{2k+1}-\bar z_{2k+2}\simeq -\frac{2iL\epsilon}{n(t+L)(t-l+L)}\bar z_{2k+1}.\end{aligned}$$ In $L<t<L+l$, $$\begin{aligned}
\label{factor1therma2}
z_{2k}-z_{2k+1}\simeq -\frac{2iL\epsilon}{n(t-L)(t-l-L)}z_{2k},\nonumber \\
\bar z_{2k+1}-\bar z_{2k+2}\simeq -\frac{2iL\epsilon}{n(t+L)(t-l+L)}\bar z_{2k+1}.\end{aligned}$$ In $t>l+L$ or $0<t<L$ the second term of (\[ResultSncorrelation\]) is $$\begin{aligned}
&&\frac{n e^{-\beta E_1}}{1-n}\frac{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)...O(\omega_{2n},\bar \omega_{2n})\psi(-\infty)\psi(+\infty)\rangle_{C_n}}{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)...O(\omega_{2n},\bar \omega_{2n})\rangle_{C_n}}\nonumber \\
&=&\frac{ne^{-\beta E_1}}{1-n}\frac{\langle O^{\dag}(z_1,\bar z_1)O(z_2,\bar z_2)...O(z_{2n},\bar z_{2n})\psi^{'}(z_{-\infty})\psi^{'}(z_{+\infty})\rangle_{C_1}}{\langle O^{\dag}(z_1,\bar z_1)O(z_2,\bar z_2)...O(z_{2n},\bar z_{2n})\rangle_{C_1}}\nonumber \\
&=&\frac{ne^{-\beta E_1}}{1-n}\langle \psi^{'}(z_{-\infty})\psi^{'}(z_{+\infty})\rangle_{C_1}.\end{aligned}$$ In the above formula, we label $\psi^{'}(z,\bar z)$ as the map of the operator $\psi(\omega,\bar \omega)$[^1]. In the second step due to (\[factor1thermal\]) we have used $$\begin{aligned}
\label{Thermalcasefactorize}
&&\langle O^{\dag}(z_1,\bar z_1)O(z_2,\bar z_2)...O(z_{2n},\bar z_{2n})\psi^{'}(z_{-\infty})\psi^{'}(z_{+\infty})\rangle_{C_1}\nonumber \\
&&\simeq \langle O^{\dag}(z_1,\bar z_1)O(z_2,\bar z_2)\rangle...\langle O^{\dag}(z_{2n-1},\bar z_{2n-1}) O(z_{2n},\bar z_{2n})\rangle \langle \psi^{'}(z_{-\infty})\psi^{'}(z_{+\infty})\rangle.\nonumber\\
\end{aligned}$$
In $L<t<L+l$, the correlation function can not be factorized as (\[Thermalcasefactorize\]) directly due to (\[factor1therma2\]). Following logic in [@He:2014mwa], one can make use of $n-1$ times fusion transformation $(z_1,z_2)(z_3,z_4)...(z_{2n-1},z_{2n}) \to (z_2,z_3)(z_4,z_5)...(z_{2n},z_1)$. The second terms of (\[ResultSncorrelation\]) is still given by in $\epsilon \rightarrow 0$ $$\begin{aligned}
\frac{ne^{-\beta E_1}}{1-n}\langle \psi^{'}(z_{-\infty})\psi^{'}(z_{+\infty})\rangle_{C_1}.\end{aligned}$$ Here we assume that there are no nontrival correlation between $O$ and $\psi$ for simplifying analysis. The third term of (\[ResultSncorrelation\]) in the limit $\epsilon \to 0$ is $$\begin{aligned}
&&-\frac{ne^{-\beta E_1}}{1-n}\frac{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2) \psi(-\infty)\psi(+\infty)\rangle_{C_1}}{\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{C_1}}\nonumber \\
&=&-\frac{n}{1-n}\langle \psi(-\infty)\psi(+\infty)\rangle_{C_1}=-\frac{n}{1-n}e^{-\beta E_1}.\end{aligned}$$ The sum of the second and third term is the same as the thermal correlation [@Cardy:2014jwa] for the short interval limit. (\[ResultSncorrelation\]) is the summation over the thermal correction of the vacuum state and local excitation in the vacuum state in low temperature limit. The Rényi entropy is summation of thermal effect and local excitation. This relationship can be considered as a sum rule for Rényi entropy of low temperature thermal state with local excitation, which is different with the sum rule proposed in [@Nozaki:2014hna]. For the pure state one could also get a similar sum rule. The local excitation in thermal states have been also studied in [@Caputa:2014eta] for free boson and Ising model. With a different method [@Cardy:2014jwa], we can reproduce the Rényi entropy [@Caputa:2014eta] for short interval in the low temperature with taking limit $\epsilon \to 0$ for general 2D rational CFT.
Local excitation in 2D CFTs with a boundary
===========================================
In this section, we would like to study the Rényi entropy of locally excited state in 2D CFT with a boundary which preserves conformal symmetry. The global property of the CFT with a boundary has been discussed in literature[@cag][@Cardy1][@Cardy2]. As we know, the Rényi entropy is sensitive to correlation function. The boundary will change the correlation functions. And the boundary conditions also affect the correlation functions. It is interesting to check what will happen to the Rényi entropy for the local excitation in 2D CFT with a boundary.
Set-up of local exciation {#3.1}
-------------------------
We begin with a CFT with a boundary at $x=0$ and the CFT is living in the range $x\le 0$. We divide the this region into two parts, one part is $-l<x<0$ denoted by A and the other is complement to the region A denoted by B. The time $t$ vary from $-\infty$ to $+\infty$, and the Hamiltonian $H$ is well defined as a operator to generate the time evolution.
We assume that the local excitation of vacuum is at $x=-L$ and consider the Rényi entropy of the subsystem A. The time dependent density matrix can be written as $$\begin{aligned}
\rho(t)=N O(\omega_2,\bar \omega_2)\Ket{0}\Bra{0}O^{\dag}(\omega_1,\bar \omega_1),\end{aligned}$$ where the coordinates are, $$\begin{aligned}
\omega_1=i(\epsilon-it)-L, \ \ \omega_2=-i(\epsilon+it)-L, \nonumber \\
\bar \omega_1=-i(\epsilon-it)-L,\ \ \bar \omega_2=i(\epsilon+it)-L.\end{aligned}$$
=6.0 cm =5.0 cm
We still make use of replica trick to study the variation of the Rényi entropy of the subsystem A in this section. By definition, the variance of $n$-th Rényi entropy can be calculated as $$\begin{aligned}
\label{Mainformula}
\Delta S^{(n)}_A&=&\frac{1}{1-n} \Big[\log \langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)...O(\omega_{2n},\bar \omega_{2n})\rangle_{B_n}\nonumber \\
&-&n\log \langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{B_1}\Big],\end{aligned}$$ where $B_n$ is the $n$-sheet Riemann surface that consists of n copies of original plane $x\le 0$ with gluing together along $-l \leqslant x \leqslant 0$, $t=0$. With the following conformal transformation, $$\begin{aligned}
\label{Transformation1}
z^n=\frac{\omega+l}{\omega-l}\end{aligned}$$ the $n$-sheet Riemann surface can be mapped to a disc $|z|\leqslant 1$ which is smooth surface. The boundary x=0 corresponds to $|z|=1$. Furthermore, we can map disc to the upper half plane (UHP) $t \geqslant 0$ with the other conformal map $$\begin{aligned}
\xi=-i\frac{z+1}{z-1}\label{Transformation2}.\end{aligned}$$ After two conformal maps, one can make use of well known results of 2D CFT on UHP. Finally, the variation of the Rényi entropy (\[Mainformula\]) is $$\begin{aligned}
\label{result1}
\Delta S_{A}^{(n)}=\frac{1}{1-n}\log \Big[\prod_{k=1}^{2n} (\frac{d\omega_k}{d\xi_k})^{-h}(\frac{d\omega_k}{d\xi_k})^{-\bar h}\frac{\langle O^{\dag}(\xi_1,\bar \xi_1)O(\xi_2,\bar \xi_2)...O^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}}{(\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{B_1})^n}\Big].\nonumber\\
\\end{aligned}$$ This is the key formula in the remainder of this paper. As a simple example, we consider the 2nd Rényi entropy firstly. These coordinates on UHP can be expressed by the original spacetime coordinate as follows. $$\begin{aligned}
\label{Transformationpoint}
\xi_1=-i\frac{(\frac{\omega_1+l}{\omega_1-l})^{1/2}+1}{(\frac{\omega_1+l}{\omega_1-l})^{1/2}-1},&&
\xi_2=-i\frac{(\frac{\omega_2+l}{\omega_2-l})^{1/2}+1}{(\frac{\omega_2+l}{\omega_2-l})^{1/2}-1},\nonumber \\
\xi_3=-i\frac{-(\frac{\omega_1+l}{\omega_1-l})^{1/2}+1}{-(\frac{\omega_1+l}{\omega_1-l})^{1/2}-1},&&
\xi_4=-i\frac{-(\frac{\omega_2+l}{\omega_2-l})^{1/2}+1}{-(\frac{\omega_2+l}{\omega_2-l})^{1/2}-1},\nonumber \\
\bar \xi_1=i\frac{(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}+1}{(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}-1},&&
\bar \xi_2=i\frac{(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}+1}{(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}-1},\nonumber \\
\bar \xi_3=i\frac{-(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}+1}{-(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}-1},&&
\bar \xi_4=i\frac{-(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}+1}{-(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/2}-1}.\end{aligned}$$ One can analyze the above formula carefully in two different time regions. When $0<t<L-l$ or $t>l+L$ with $\epsilon \rightarrow 0$, $$\begin{aligned}
\label{Varianceofpoint}
\xi_1-\xi_2 \simeq \frac{2\xi_1\epsilon}{\sqrt{\omega_1+l}\sqrt{\omega_1-l}},\nonumber \\
\xi_3-\xi_4 \simeq \frac{2\xi_3\epsilon}{\sqrt{\omega_1+l}\sqrt{\omega_1-l}},\nonumber \\
\bar \xi_1-\bar \xi_2 \simeq \frac{2\bar \xi_1\epsilon}{\sqrt{\bar \omega_1-l}\sqrt{\bar \omega_1+l}},\nonumber \\
\bar \xi_3-\bar \xi_4 \simeq \frac{2\bar \xi_3\epsilon}{\sqrt{\bar \omega_1-l}\sqrt{\bar \omega_1+l}},\end{aligned}$$ and the Jacobi factor of conformal transformation, $$\begin{aligned}
\frac{d\xi_1}{d\omega_1}&\simeq& \frac{d\xi_2}{d\omega_2}\simeq{1\over 2 i\epsilon}\xi_{1 2},\text{ }\text{}\frac{d\bar{\xi}_1}{d\bar{\omega}_1}\simeq \frac{d\bar{\xi}_2}{d\bar{\omega}_2}\simeq -{1\over 2 i\epsilon}\bar{\xi}_{1 2}\label{DerivitiveGeneralzationfour12}\\
\frac{d\xi_3}{d\omega_3}&\simeq& \frac{d\xi_4}{d\omega_4}\simeq {1\over 2 i\epsilon}\xi_{34}, \text{ }\text{ }\frac{d\bar{\xi}_3}{d\bar{\omega}_3}\simeq \frac{d\bar{\xi}_4}{d\bar{\omega}_4}\simeq -{1\over 2 i\epsilon}\bar{\xi}_{34}\label{DerivitiveGeneralzationfour34}\\\end{aligned}$$ When $L-l<t<L+l$ with $\epsilon \rightarrow 0$, $$\begin{aligned}
\label{Varianceofpoitmiddletime}
\xi_4-\xi_1 \simeq \frac{2\xi_1\epsilon}{\sqrt{\omega_1+l}\sqrt{\omega_1-l}},\nonumber \\
\xi_2-\xi_3 \simeq \frac{2\xi_2\epsilon}{\sqrt{\omega_1+l}\sqrt{\omega_1-l}},\nonumber \\
\bar \xi_1-\bar \xi_2 \simeq -\frac{2\bar \xi_1\epsilon}{\sqrt{\bar \omega_1-l}\sqrt{\bar \omega_1+l}},\nonumber \\
\bar \xi_3-\bar \xi_4 \simeq -\frac{2\bar \xi_3\epsilon}{\sqrt{\bar \omega_1-l}\sqrt{\bar \omega_1+l}},\end{aligned}$$ and the Jacobi factor of conformal transformation, $$\begin{aligned}
\frac{d{\xi}_1}{d{\omega}_1}&\simeq& \frac{d{\xi}_4}{d{\omega}_4}={1\over 2 i\epsilon}{\xi}_{1 4}, \text{ }\text{ }\frac{d\bar{\xi}_1}{d\bar{\omega}_1}\simeq \frac{d\bar{\xi}_2}{d\bar{\omega}_2}\simeq-{1\over 2 i\epsilon}\bar{\xi}_{1 2}\label{DerivitiveGeneralzationfour21}\\
\frac{d\xi_2}{d\omega_2}&\simeq& \frac{d\xi_3}{d\omega_3}\simeq{1\over 2 i\epsilon}\xi_{23},\text{ }\text{ }\frac{d\bar{\xi}_3}{d\bar{\omega}_3}\simeq \frac{d\bar{\xi}_4}{d\bar{\omega}_4}\simeq -{1\over 2 i\epsilon}\bar{\xi}_{34}\label{DerivitiveGeneralzationfour22}\end{aligned}$$
2nd Rényi entropy for free boson
--------------------------------
We will focus on the following local operators in the free scalar field firstly, $$\begin{aligned}
\label{OperatorBoson}
O_1=e^{i\phi/2}, \ \ O_2=\frac{1}{\sqrt{2}}(e^{i\phi/2}+e^{-i\phi/2}).\end{aligned}$$ The time evolution of Rényi entropy for such operators have already been studied in [@He:2014mwa] in 2D CFT on the complex plane. There are two kinds of boundary conditions for 2D free scalar field theory. One is $\frac{\partial \phi}{\partial n}|_B=0$ called Neumann boundary condition, the other is $\phi|_B=0$ called Dirichlet boundary condition. Since the boundary condition is homogenous, it is invariant under the conformal transformation.\
The image method [@CFT] [^2] is an efficient way to obtain the correlation function on UHP from correlation function on the full complex plane. The two kinds of boundary conditions correspond to different parity transformation in image method. Due to the presence of boundary, there are constraints on local conformal transformation, the anti-holomorphic and the holomorphic sectors in correlation function are no longer independent. More precisely, the correlation function on the upper half plane can be expressed by holomorphic part of conformal block on the full complex plane with including the ‘images’ of the holomorphic coordinates. That is to say, $$\begin{aligned}
\langle \phi(z_1,\bar z_1)\phi(z_2,\bar z_2)...\phi(z_n,\bar z_n)\rangle_{UHP}\end{aligned}$$ equals to $$\begin{aligned}
\label{4ninR2}
\langle \phi(z_1)\bar \phi( {z_1}^*)\phi(z_2)\bar \phi( {z_2}^*)...\phi(z_n)\bar \phi(z_n^*)\rangle_{R^2},\end{aligned}$$ where $\phi$ and $\bar \phi$ refer to the holomorphic and anti-holomorphic part of the field $\phi$ [[^3]]{}. After a parity transformation the anti-holomorphic part become a holomorphic field with conformal dimension of the original anti-holomorphic part. In terms of image method, we should introduce parity transformation. For the free boson the parity transformation is $$\begin{aligned}
\label{ParityBoson}
\phi(z,\bar z)=\eta \phi(\bar z,z),\ \ \eta=\pm 1,\end{aligned}$$ $\eta=1,-1$ corresponds to the Neumann boundary condition and Dirichlet boundary condition respectively. In terms of (\[result1\]), to obtain Rényi entropy, it is necessary to know the two-point and four-point correlation function on the UHP.
### Local excitation $O_1$ {#3.2.1}
Let’s consider the operator $O_1$ firstly. Using the image method, we could get the two-point function $$\begin{aligned}
\label{twopointfunction}
\langle O_1^{\dag}(\omega_1,\bar \omega_1)O_1(\omega_2,\bar \omega_2)\rangle_{B_1}&=&\prod_{i=1}^2(\frac{d\omega_i}{d\xi_i^{'}})^{-h}(\frac{d\bar \omega_i}
{d\bar \xi_i^{'}})^{-\bar h}\langle O_1^{\dag}(\xi_1^{'},\bar \xi_1^{'})O_1(\xi_2^{'},\bar \xi_2^{'})\rangle_{UHP}\nonumber \\&=&\prod_{i=1}^2(\frac{d\omega_i}{d\xi_i^{'}})^{-h}(\frac{d\bar \omega_i}{d\bar \xi_i^{'}})^{-\bar h}\Big[\langle O_1^{\dag}(\xi_1^{'},\bar \xi_1^{'})O_1(\xi_2^{'},\bar \xi_2^{'})\tilde{O_1}^\dag (\xi_3^{'},\bar \xi_3^{'})\tilde{O_1}(\xi_4^{'},\bar \xi_4^{'})\rangle_{R^2}\Big]_{\text{holo}}\nonumber \\
&\simeq &\frac{1}{(\xi^{'}_{12}\xi^{'}_{34})^{1/4}}=\frac{1}{(4\epsilon^2)^{1/4}}.\end{aligned}$$ where $\xi_3^{'}\equiv \xi_1^{'*}$,$\xi_4\equiv \xi_2^{'*}$,$\xi_i^{'}=i\omega_i$, $h=\bar h=1/8$ and $\tilde{O_1}$ is the field with parity transformation. The subindex ‘holo’ means that we only keep the holomorphic part of the correlation and set the anti-holomorphic part to be constant [^4] which is determined by boundary condition in general. For two-point function one could normalize the field and take constant to be $1$. The 4-point correlation function could be obtained by similar procedure. $$\begin{aligned}
\label{Fourpoitfunction}
&&\langle O_1^{\dag}(\omega_1,\bar \omega_1)O_1(\omega_2,\bar \omega_2)O_1^{\dag}(\omega_3,\bar \omega_3)O_1(\omega_4,\bar \omega_4)\rangle_{B_2}\nonumber \\
&&=\prod_{i=1}^4(\frac{d\omega_i}{d\xi_i})^{-h}(\frac{d\bar \omega_i}{d\bar \xi_i})^{-\bar h}\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)O_1^{\dag}(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4)\rangle_{UHP}\nonumber \\
&&= \prod_{i=1}^4(\frac{d\omega_i}{d\xi_i})^{-h}(\frac{d\bar \omega_i}{d\bar \xi_i})^{-\bar h}\Big[\langle O_1^{\dag}(\xi_1,\bar \xi_1)...O_1(\xi_4,\bar \xi_4)\tilde{O_1}^\dag (\xi_5,\bar \xi_5)...\tilde{O_1}(\xi_8,\bar \xi_8)\rangle_{R^2}\Big]_{\text{holo}},\nonumber \\
&&\end{aligned}$$ where $\xi_5=\bar \xi_1$, $\xi_6=\bar \xi_2$, $\xi_7=\bar \xi_3$, $\xi_8=\bar \xi_4$, the operator $\tilde{O}_1=e^{i\eta \phi}$ on $R^2$ with parity transformation.
In the region $0<t<L-l$ or $t>l+L$, as (\[Varianceofpoint\]) shows, the correlation function (\[Fourpoitfunction\]) can be factorized as $$\begin{aligned}
\label{factorized}
&&\langle O_1^{\dag}(\xi_1,\bar \xi_1)...O_1(\xi_4,\bar \xi_4)\tilde{O_1}^\dag (\xi_5,\bar \xi_5)...\tilde{O_1}(\xi_8,\bar \xi_8)\rangle_{R^2}\nonumber \\
&&\propto\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)\rangle_{R^2}...\langle \tilde{O_1}^{\dag}(\xi_7,\bar \xi_7)\tilde{O_1}(\xi_8,\bar \xi_8)\rangle_{R^2}\end{aligned}$$ in $\epsilon\rightarrow 0$ limit. The image method leaves us with a constant C. To fix the constant C, we take the limit $\xi_1\to \xi_2$, $\bar \xi_1 \to \bar \xi_2$, $\xi_3\to \xi_4$ and $\bar \xi_3 \to \bar \xi_4$ in (\[Fourpoitfunction\]), one could find $$\begin{aligned}
&&\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)O_1^{\dag}(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4)\rangle_{UHP}\nonumber \\
&\simeq& \langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)\rangle_{UHP} \langle O_1^{\dag}(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4)\rangle_{UHP}.\label{blockCC}\end{aligned}$$ In terms of image method, we can also obtain $$\begin{aligned}
&&\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)O_1^{\dag}(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4)\rangle_{UHP}\nonumber \\
&=&\Big[\langle O_1^{\dag}(\xi_1,\bar \xi_1)...O_1(\xi_4,\bar \xi_4)O(\xi_2,\bar \xi_2)\tilde{O_1}^\dag (\xi_5,\bar \xi_5)...\tilde{O_1}(\xi_8,\bar \xi_8)\rangle_{R^2}\Big]_{\text{holo}}\nonumber\\
&\simeq& \frac{C }{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{1/4}},\label{blockC}\end{aligned}$$ where $C$ is constant. Comparing (\[blockCC\]) with (\[blockC\]), we can fix $C=1$ which is consistent with the normalization of the two point correlation function.
For $n=2$, the variation of the Rényi entropy (\[result1\]) is $$\begin{aligned}
\label{O1result1}
\Delta S^{(2)}_A&=&-\log \Big[\prod_{k=1}^{4} (\frac{d\omega_k}{d\xi_k})^{-h}(\frac{d\omega_k}{d\xi_k})^{-\bar h}\frac{\langle O^{\dag}(\xi_1,\bar \xi_1)O(\xi_2,\bar \xi_2)O^{\dag}(\xi_{3},\bar \xi_{3})O(\xi_{4},\bar \xi_{4})\rangle_{UHP}}{(\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{B_1})^2}\Big]\nonumber\\
&=& -\log \Big[(\frac{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{8}})^{1/8}\frac{(4\epsilon^2)^{1/2}}{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{1/4}}\Big]=0,\end{aligned}$$ where we have used (\[twopointfunction\]), (\[Fourpoitfunction\]),(\[blockC\]) and Jacobi factor (\[DerivitiveGeneralzationfour12\])(\[DerivitiveGeneralzationfour34\]).
In the other region $L-l<t<L+l$, we can not factorize the correlation function as (\[factorized\]) directly. For 2D free scalar theory, the situation becomes much simpler. The correlation function could be expressed as [@CFT] $$\begin{aligned}
\label{Bosoncorrelation}
\langle e^{i\alpha_1\phi}...e^{i\alpha_n \phi}\rangle=\prod_{i<j}[z_{ij}^{\alpha_i\alpha_j}][\bar z_{ij}^{\alpha_i\alpha_j}],\end{aligned}$$ with the neutral condition $\alpha_1+\alpha_2+...\alpha_n=0$ and $z_{ij}=z_i-z_j, \bar{z}_{ij}=\bar{z}_i-\bar{z}_j $. Thus $$\begin{aligned}
\label{8pointfunction}
&&\langle O_1^{\dag}(\xi_1,\bar \xi_1)...O_1(\xi_4,\bar \xi_4)\rangle_{UHP}
\nonumber\\
&=&\Big[\langle O_1^{\dag}(\xi_1,\bar \xi_1)...O_1(\xi_4,\bar \xi_4)O(\xi_2,\bar \xi_2)\tilde{O_1}^\dag (\xi_5,\bar \xi_5)...\tilde{O_1}(\xi_8,\bar \xi_8)\rangle_{R^2}\Big]_{\text{holo}}\nonumber\\
&\simeq& \frac{1}{(\xi_{41}\xi_{23}\xi_{56}\xi_{78})^{1/4}},\end{aligned}$$ where we have fix the constant to be 1. To get $\Delta S^{(2)}_A$ we only need to change $2\leftrightarrow 4$ in (\[O1result1\]). Using (\[Varianceofpoitmiddletime\])(\[twopointfunction\])(\[8pointfunction\]) and Jacobi factor (\[DerivitiveGeneralzationfour21\])(\[DerivitiveGeneralzationfour22\]), we get $$\begin{aligned}
\Delta S^{(2)}_A= -\log \Big[(\frac{(\xi_{14}\xi_{32}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{8}})^{1/8}\frac{(4\epsilon^2)^{1/2}}{(\xi_{14}\xi_{32}\xi_{56}\xi_{78})^{1/4}}\Big]=0,\end{aligned}$$ in $L-l<t<L+l$. To close this subsection, one more thing should be noted that $\Delta S^{(2)}_A$ does not depend on the choice of parity transformation. Actually in our above calculation we do not use the value of $\eta$, which is related to the boundary condition. The leading order of the correlation function is same. $\Delta S^{(2)}_A$ is always zero for operator $O_1$. To see the effect of the boundary, we should consider more complicated example.
### Local excitation $O_2$
$O_2$ is a linear combination between $O_1$ and $O^{\dag}_1$. We can calculate $\Delta S^{(2)}_A$ with following the logic in previous section. The two-point correlation function of $O_2$, $$\begin{aligned}
\label{2pointfunctionO2}
&&\langle O^{\dag}_2(\xi_1^{'},\bar \xi_1^{'}) O_2(\xi_2^{'},\bar \xi_2^{'})\rangle_{UHP}= \frac{1}{2}\Big[\langle O^{\dag}_1(\xi_1^{'},\bar \xi_1^{'}) O^{\dag}_1(\xi_2^{'},\bar \xi_2^{'})\rangle_{UHP} +\langle O_1(\xi_1^{'},\bar
\xi_1^{'})O_1(\xi_2^{'},\bar \xi_2^{'})\rangle_{UHP} \nonumber \\
&&+\langle O_1(\xi_1^{'},\bar \xi_1^{'}) O^{\dag}_1(\xi_2^{'},\bar \xi_2^{'})\rangle_{UHP}+\langle O^{\dag}_1(\xi_1^{'},\bar
\xi_1^{'}) O_1(\xi_2^{'},\bar \xi_2^{'})\rangle_{UHP} \Big]\nonumber \\
&\simeq& \frac{1}{(\xi_{12}^{'})^{1/4}(\xi_{34}^{'})^{1/4}}= \frac{1}{(4\epsilon^2)^{1/4}},\end{aligned}$$ in the limit $\epsilon \to 0$. Here we have used the parity transformation (\[ParityBoson\]) related to the boundary condition. We have set anti-holomorphic parts to be $1$.
The four-point correlation function on UHP is $$\begin{aligned}
\label{O2operator4pointfunction}
\langle O_2^{\dag}(\xi_1,\bar \xi_1)...O_2(\xi_4,\bar \xi_4)\rangle_{UHP}
&&=\frac{1}{4}\Big[\langle O_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\
&&+\langle O_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\
&&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\
&&+\langle O_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\
&&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\
&&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\
&&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\
&&+\langle O_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\
&&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\
&&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\
&&+\langle O_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP} \nonumber \\
&&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\
&&+\langle O_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\
&&+\langle O_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP} \nonumber \\
&&+\langle O^{\dag}_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\
&&+\langle O_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O_1(\xi_4,\bar \xi_4) \rangle_{UHP}\Big].\end{aligned}$$ We could use the result of four-point function of $O_1$ which has been studied in section (\[3.2.1\]). From (\[ParityBoson\])and (\[4ninR2\]) one could see that the four-point function is dependent on the boundary conditions or parity transformations. We will calculate $\Delta S^{(2)}_A$ with Neumann and Dirichlet boundary condition respectively.
1. For the Neumann boundary condition, i.e., $\eta=1$.
In (\[O2operator4pointfunction\]), terms containing equal number of $O_1$ and $O^{\dag}_1$ will survive due to the neutrality condition. Thus there are 6 terms making contribution to the 4-point correlation function. In the region $0<t<L-l$ or $t>L+l$, (\[O2operator4pointfunction\]) can be expressed by factorized form on the $R^2$. For example, the first term in (\[O2operator4pointfunction\]) as a leading term is $$\begin{aligned}
&&\langle O_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\
&=&\Big[\langle O_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4)O_1(\xi_5,\bar \xi_5) O^{\dag}_1(\xi_6,\bar \xi_6)O_1(\xi_7,\bar \xi_7) O^{\dag}_1(\xi_8,\bar \xi_8)\rangle\Big]_{\text{holo}}\nonumber \\
&\simeq& \Big[\langle O_1(\xi_1,\bar \xi_1)O^{\dag}_1(\xi_2,\bar \xi_2)\rangle \langle O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle \langle O_1(\xi_5,\bar \xi_5)O_1(\xi_6,\bar \xi_6)\rangle\nonumber \\
&&\langle O^{\dag}_1(\xi_7,\bar \xi_7)O^{\dag}_1(\xi_8,\bar \xi_8)\rangle\Big]_{\text{holo}}=\frac{1}{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{1/4}}\sim \frac{1}{\epsilon},\end{aligned}$$ in the limit $\epsilon \to 0$. The second terms as sub-leading term is $$\begin{aligned}
&&\langle O_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O^{\dag}_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4)\rangle_{UHP}=\Big[\langle O_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) \nonumber \\
&&O^{\dag}_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4)O_1(\xi_5,\bar \xi_5)O_1(\xi_6,\bar \xi_6) O^{\dag}_1(\xi_7,\bar \xi_7)O^{\dag}_1(\xi_8,\bar \xi_8)\rangle_{R_2}\Big]_{\text{holo}} \nonumber \\
&\simeq& (\frac{\xi_{12}\xi_{34}...\xi_{56}\xi_{78}}{\xi_{13}\xi_{14}...\xi_{57}\xi_{58}})^{1/4}\sim \epsilon,\end{aligned}$$ in the limit $\epsilon \to 0$. Totally, there are four terms that are of order $O(\epsilon^{-1})$ in (\[O2operator4pointfunction\]).
Using (\[result1\]), we get $\Delta S_A^{(n)}$ for $n=2$, $$\begin{aligned}
\Delta S^{(2)}_A&\simeq &-\log \Big[(\frac{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{8}})^{1/8} \frac{\langle O_2^{\dag}(\xi_1,\bar \xi_1)...O_2(\xi_4,\bar
\xi_4)\rangle_{UHP}}{(\langle O^{\dag}_2(\xi_1,\bar \xi_1) O_2(\xi_2,\bar \xi_2)\rangle_{UHP})^2}\Big]\nonumber \\
&=&-\log \Big[(\frac{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{8}})^{1/8}\frac{(4\epsilon^2)^{1/2}}{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{1/4}}\Big]=0,\end{aligned}$$ In $L-l<t<L+l$ with $\epsilon \to 0$, (\[Varianceofpoitmiddletime\]) shows $\xi_{14}\sim \xi_{23}\sim \epsilon$. Terms making non-vanishing contribution to 4-point correlation function will change in the limit $\epsilon \to 0$. For example the third term in (\[O2operator4pointfunction\]) as a sub-leading term is, $$\begin{aligned}
&&\langle O^{\dag}_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber \\
&=&\Big[\langle O^{\dag}_1(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2) O_1(\xi_3,\bar \xi_3)O^{\dag}_1(\xi_4,\bar \xi_4)O^{\dag}_1(\xi_5,\bar \xi_5)O_1(\xi_6,\bar \xi_6) O_1(\xi_7,\bar \xi_7)O^{\dag}_1(\xi_8,\bar \xi_8) \rangle_{R^2}\Big]_{\text{holo}}\nonumber \\
&=&(\frac{\xi_{14}\xi_{23}...\xi_{58}\xi_{67}}{\xi_{12}\xi_{34}...\xi_{56}\xi_{78}})^{1/4}\sim O(1),\end{aligned}$$ where ‘...’ stands for the terms that are $O(1)$ in the limit $\epsilon \to 0$. One could count the leading contribution term by term in (\[O2operator4pointfunction\]), there are only two terms that are of $O(\epsilon^{-1})$. Thus the variation of the Rényi entropy $\Delta S^{(n)}_{A}$ for $n=2$ is $$\begin{aligned}
\Delta S^{(2)}_A&\simeq &-\log \Big[(\frac{(\xi_{14}\xi_{32}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{8}})^{1/8} \frac{\langle O_2^{\dag}(\xi_1,\bar \xi_1)...O_2(\xi_4,\bar
\xi_4)\rangle_{UHP}}{(\langle O^{\dag}_2(\xi_1,\bar \xi_1) O_2(\xi_2,\bar \xi_2)\rangle_{UHP})^2}\Big]\nonumber \\
&=&-\log \Big[(\frac{(\xi_{14}\xi_{32}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{8}})^{1/8}\frac{(4\epsilon^2)^{1/2}}{2(\xi_{14}\xi_{23}\xi_{56}\xi_{78})^{1/4}}\Big]=\log 2,\end{aligned}$$
2. For the Dirichlet boundary condition, i.e., $\eta =-1$.\
All the terms in (\[O2operator4pointfunction\]) are non-vanishing in term of the neutrality condition. However, there are still four different terms in (\[O2operator4pointfunction\]) contributing to the leading order in the region $L+l<t$ or $t<L-l$ in $\epsilon \rightarrow 0$ limit. Thus the ratio (\[result1\]) is 1 and $\Delta S^{(2)}_A=0$. In the region $L-l<t<L+l$, there are [two]{} terms in (\[O2operator4pointfunction\]) with $\epsilon \rightarrow 0$ limit, which is the same as the situation of Neumann boundary condition. Thus the ratio (\[result1\]) is ${1\over 2}$ and $\Delta S^{(2)}_A=\log 2$. In this example we could see that $\Delta S^{(2)}_A$ does not depend on the choice of boundary conditions.
n-th Rényi Entropy for free boson
---------------------------------
In this subsection, we would like to generalize our studies to n-th Rényi entropy which involves in the the 2n-point correlation function on $B_n$. The conformal transformation (\[Transformation1\]) (\[Transformation2\]) can map $B_n$ to UHP. Finally one can calculate the 2n-point correlation function on UHP by the ‘method of images’ in terms of 4n-point correlation function on $R_2$. The points $\xi_1,\xi_2...\xi_{2n}$ on $B_n$ are $$\begin{aligned}
\label{Npointaftermapping}
&&\xi_{2k+1}=-i \frac{e^{2i\pi k/n}(\frac{\omega_1+l}{\omega_1-l})^{1/n}+1}{e^{2i\pi k/n}(\frac{\omega_1+l}{\omega_1-l})^{1/n}-1},\ \ \ \
\xi_{2k+2}=-i \frac{e^{2i\pi k/n}(\frac{\omega_2+l}{\omega_2-l})^{1/n}+1}{e^{2i\pi k/n}(\frac{\omega_2+l}{\omega_2-l})^{1/n}-1},\nonumber \\
&&\bar \xi_{2k+1}=-i \frac{e^{2i\pi k/n}(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/n}+1}{e^{2i\pi k/n}(\frac{\bar \omega_1+l}{\bar \omega_1-l})^{1/n}-1},\ \ \ \
\bar \xi_{2k+2}=-i \frac{e^{2i\pi k/n}(\frac{\bar \omega_2+l}{\bar \omega_2-l})^{1/n}+1}{e^{2i\pi k/n}(\frac{\bar \omega_2+l}{\bar \omega_2-l})^{1/n}-1},\end{aligned}$$ where $0\le k\le n-1$. In the region $t>L+l$ or $t<L-l$, one can obtain $$\begin{aligned}
\label{varianceofpoint1}
&&\xi_{2k+1}-\xi_{2k+2}\simeq \frac{8ie^{2i\pi k/n} l \epsilon}{n(\omega_1-l)^{1-1/n}(\omega_1+l)^{1-1/n}[e^{2i\pi k/n}(\omega_1-l)^{1/n}-(\omega_1+l)^{1/n}]^2}\nonumber \\
&&\bar \xi_{2k+1}-\bar \xi_{2k+2}\simeq \frac{-8ie^{2i\pi k/n} l \epsilon}{n(\bar \omega_1-l)^{1-1/n}(\bar \omega_1+l)^{1-1/n}[e^{2i\pi k/n}(\bar \omega_1-l)^{1/n}-(\bar \omega_1+l)^{1/n}]},\nonumber \\
\\end{aligned}$$ and the Jacobi factor of conformal transformation, $$\begin{aligned}
\frac{d\xi_{2k+1}}{d\omega_{2k+1}}&\simeq&\frac{d\xi_{2k+2}}{d\omega_{2k+2}}\simeq{1\over 2 i\epsilon}\xi_{2k+1, 2k+2}\label{DerivitiveGeneralzation11}\\
\frac{d\bar {\xi}_{2k+1}}{d\bar{\omega}_{2k+1}}&\simeq&\frac{d\bar {\xi}_{2k+2}}{d\bar{\omega}_{2k+2}}\simeq-{1\over 2 i\epsilon}\bar{\xi}_{2k+1, 2k+2}\label{DerivitiveGeneralzation12}\end{aligned}$$ In $L-l<t<L+l$, one could find $$\begin{aligned}
\label{varianceofpoint2}
&&\xi_{2k}-\xi_{2k+1} \simeq \frac{8 i e^{2i\pi k/n} l \epsilon }{n (-l+\omega_1)^{1-1/n}(l+\omega_1)^{1-1/n}[e^{2i\pi k/n}(\omega_1-l)^{1/n}-(\omega_1+l)^{1/n}]^2}\nonumber \\
&&\bar \xi_{2k+1}-\bar \xi_{2k} \simeq \frac{-8ie^{2i\pi k/n} l \epsilon}{n(\bar \omega_1-l)^{1-1/n}(\bar \omega_1+l)^{1-1/n}[e^{2i\pi k/n}(\bar \omega_1-l)^{1/n}-(\bar \omega_1+l)^{1/n}]}\end{aligned}$$ and the Jacobi factor of conformal transformation, $$\begin{aligned}
\frac{d\xi_{2k}}{d\omega_{2k}}&\simeq&\frac{d\xi_{2k+1}}{d\omega_{2k+1}}\simeq{1\over 2 i\epsilon}\xi_{2k, 2k+1}\label{DerivitiveGeneralzation21}\\
\frac{d\bar{\xi}_{2k}}{d\bar{\omega}_{2k}}&\simeq&\frac{d\bar{\xi}_{2k+1}}{d\bar{\omega}_{2k+1}}\simeq-{1\over 2 i\epsilon}\bar{\xi}_{2k, 2k+1}\label{DerivitiveGeneralzation22}.\end{aligned}$$
### Local excitation $O_1$ {#local-excitation-o_1}
We consider the operator $O_1$ firstly. In the region $L-l<t$ or $t>L+l$ with $\epsilon \rightarrow 0$, the 2n-point correlation function of $O_1$ is $$\begin{aligned}
\label{BosonGeneraliztion2nO1}
&&\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)...O_1(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\
&&=\Big[\langle O_1^{\dag}(\xi_1,\bar \xi_1)...O_1(\xi_{2n},\bar \xi_{2n})\tilde{O}_1^{\dag}(\xi_{2n+1},\bar \xi_{2n+1})...\tilde{O}_1(\xi_{4n},\bar \xi_{4n})\rangle_{R_2}\Big]_{\text{holo}}\nonumber \\
&&\simeq \Big[\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_{2},\bar \xi_{2})\rangle... \langle \tilde{O}_1^{\dag}(\xi_{4n-1},\bar \xi_{4n-1})\tilde{O}_1(\xi_{4n},\bar \xi_{4n})\rangle_{R_2}\Big]_{\text{holo}}\nonumber \\
&&=\frac{1}{(\xi_{12}...\xi_{2n-1,2n})^{1/4}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{1/4}},\end{aligned}$$ where $\tilde{O}_1$ is defined as $e^{-i\eta \phi/2}$ with parity transformation and $\xi_{2n+1}=\bar \xi_{1}$,...,$\xi_{4n}=\bar \xi_{2n}$. Using (\[result1\]), the variation of the $n$-th Rényi entropy can be obtained as follows. $$\begin{aligned}
\Delta S^{(n)}_{A}&=&\frac{1}{1-n}\log \Big[\prod_{k=1}^{2n} (\frac{d\omega_k}{d\xi_k})^{-h}(\frac{d\omega_k}{d\xi_k})^{-\bar h}\frac{\langle O^{\dag}(\xi_1,\bar \xi_1)O(\xi_2,\bar \xi_2)...O^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}}{(\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{B_1})^n}\Big]\nonumber\\
&=&\frac{1}{1-n}\log \Big[ \Big(\frac{(\xi_{12}...\xi_{2n-1,2n})^{2}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{2}}{(2\epsilon)^{4n}}\Big)^{1/ 8} \nonumber\\&& \frac{(4\epsilon^2)^{{n/ 4}}}{(\xi_{12}...\xi_{2n-1,2n})^{1/4}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{1/4}}\Big]=0,\end{aligned}$$ where we have used (\[twopointfunction\])(\[varianceofpoint1\])(\[BosonGeneraliztion2nO1\]), and Jacobi factor (\[DerivitiveGeneralzation11\])(\[DerivitiveGeneralzation12\]).
In the region $L-l<t<L+l$ with $\epsilon\rightarrow 0$, the $2n$-point correlation function on UHP is $$\begin{aligned}
\label{BosonGeneralizationO1two}
&&\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)...O_1(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\
&\simeq& \frac{1}{(\xi_{23}...\xi_{2n,1})^{1/4}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{1/4}}.\end{aligned}$$ Then, $$\begin{aligned}
\Delta S^{(n)}_{A}&=&\frac{1}{1-n}\log \Big[ \Big(\frac{(\xi_{23}...\xi_{2n,1})^{2}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{2}}{(2\epsilon)^{4n}}\Big)^{1/ 8} \nonumber\\&& \frac{(4\epsilon^2)^{{n/ 4}}}{(\xi_{23}...\xi_{2n,1})^{1/4}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{1/4}}\Big]=0,\end{aligned}$$ where we have made use of (\[BosonGeneralizationO1two\]) and Jacobi factor (\[DerivitiveGeneralzationfour21\])(\[DerivitiveGeneralzationfour22\]).
### Local excitation $O_2$
For operator $O_2$, there are $2^{2n}$ terms making contribution to the correlation function like the ones in (\[O2operator4pointfunction\]). Firstly let us consider the case with Neumman boundary condition. These terms with equal number of $O_1$ and $O^{\dag}_1$ can survive in the limit $\epsilon \rightarrow 0$. The $2n$-point correlation function of $O_1$ on $B_n$ can be expressed by $4n$-point correlation function on $R^2$, $$\begin{aligned}
\label{2npointfunction}
&&\langle O_1^{\dag}(\xi_1,\bar \xi_1)O_1(\xi_2,\bar \xi_2)...O_1^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O_1(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\
&=& \Big[\langle O_1^{\dag}(\xi_1,\bar \xi_1)...O_1(\xi_{2n},\bar \xi_{2n})\tilde{O}^{\dag}_1(\xi_{2n+1},\bar \xi_{2n+1})...\tilde{O}_1(\xi_{4n},\bar \xi_{4n})\rangle\Big]_{\text{holo}}.\end{aligned}$$ To be convenient, we use the symbol $+1$ referring to $O^\dag_1$ and $-1$ referring to $O_1$ in the correlation function to simplify our analysis. Then the $2n$-point correlation function on UHP can be formally written as $$\begin{aligned}
\label{Key2nBosonO22}
&&\langle O_2^{\dag}(\xi_1,\bar \xi_1)O_2(\xi_2,\bar \xi_2)...O_2^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O_2(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\
&=&\frac{1}{2^n}\sum_{\substack{i_1,i_2,...,i_{2n}=\pm1 \\ i_{2n+1}=i_1,...i_{4n}=i_{2n}} }\langle i_1,i_2,...,i_{2n},i_{2n+1}...,i_{4n} \rangle_{R^2},
\end{aligned}$$ where we have made use of the image method in the second line and $i_j=\pm 1$ stands for the operator $O_1$ or $O^{\dag}_1$ with the coordinate ($\xi_j$,$\bar \xi_j$). The constraints ${i_{2n+j}}={i_{j}}$ with $ 1\le j \le 2n$ corresponds to the Neumann boundary condition in terms of image method. For the correlation function in the Neumann boundary condition case, the non-zero terms in (\[Key2nBosonO22\]) should satisfy neural condition $$\begin{aligned}
\label{Neumannconstraint}
i_1+i_2+...+i_{2n}+...i_{4n}=0.\end{aligned}$$ In the region $t<L-l$ or $t>L+l$ with the limit $\epsilon\rightarrow 0$, due to (\[varianceofpoint1\]), the leading contribution of the 2n-point correlation function (\[Key2nBosonO22\]) are $$\begin{aligned}
\label{free1st2n}
&&\sum_{\substack {i_1+i_2=0,i_3+i_4=0 \\...i_{4n-1}+i_{4n}=0 \\ {i_{2n+1}}={i_{1}}...{i_{4n}}={i_{2n}}}} \langle i_1,i_2,...,i_{2n},i_{2n+1}...,i_{4n} \rangle_{R^2}\nonumber \\
&&\simeq 2^n \langle i_1,i_2\rangle_{R^2}...\langle i_{2n-1},i_{2n}\rangle_{R^2} \langle i_{2n+1},i_{2n+2}\rangle_{R^2} ...\langle i_{4n-1},i_{4n} \rangle_{R^2},\end{aligned}$$ there are $2^n$ terms that are leading divergence after considering the constraints, and these terms are all equal to each other. Equivalently, (\[Key2nBosonO22\]) can be written by the notation $O_{1}$ and $O^{\dag}_1 $ as $$\begin{aligned}
&&2^n \langle O_1^\dag(\xi_{1}) O_1(\xi_{2})\rangle_{R^2} ...\langle O_1^\dag(\xi_{2n-1}) O_1(\xi_{2n})\rangle_{R^2}...\langle O_1^\dag(\xi_{4n-1}) O_1(\xi_{4n})\rangle_{R^2}\nonumber \\
&=&\frac{2^n}{(\xi_{12}...\xi_{2n-1,2n})^{1/4}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{1/4}}.\end{aligned}$$
Using (\[Key2nBosonO22\]), the $2n$-point correlation function on $B_{2n}$ is $$\begin{aligned}
&&\langle O_2^{\dag}(\xi_1,\bar \xi_1)O_2(\xi_2,\bar \xi_2)...O_2^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O_2(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\
&\simeq& \frac{1}{(\xi_{12}...\xi_{2n-1,2n})^{1/4}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{1/4}},\end{aligned}$$ which is same as the $2n$-point correlation function of $O_{1}$ in $t<L-l$ or $L+l<t$. Thus the variation of the $n$-th Rényi entropy in the CFT with Neumman boundary condition is $$\begin{aligned}
\label{EarlytimeresultinO2General}
\Delta S_{A}^{(n)}=0.\end{aligned}$$ In $L-l<t<L+l$ with the limit $\epsilon\rightarrow 0$, due to (\[varianceofpoint2\]) the leading contribution of (\[Key2nBosonO22\]) should satisfy following constraints $$\begin{aligned}
\label{constraint1}
i_{2n+1}+i_{2n+2}=0,\ \ i_{2n+3}+i_{2n+4}=0,...,\ \ i_{4n-1}+i_{4n}=0.\end{aligned}$$ Combining with ${i_{2n+j}}={i_{j}}$ and (\[constraint1\]), one can obtain $$\begin{aligned}
\label{constraint2}
i_{1}+i_{2}=0,\ \ i_{3}+i_{4}=0,...,\ \ i_{2n-1}+i_{2n}=0.\end{aligned}$$ In terms of (\[varianceofpoint2\]), the leading terms of (\[Key2nBosonO22\]) should also satisfy following constraints $$\begin{aligned}
\label{constraint3}
i_{2}+i_3=0,\ \ i_{4}+i_5=0,...,\ \ i_{2n}+i_{1}=0.\end{aligned}$$ With these constraints (\[constraint1\])(\[constraint2\])(\[constraint3\]), the correlation function is $$\begin{aligned}
&&\langle O_2^{\dag}(\xi_1,\bar \xi_1)O_2(\xi_2,\bar \xi_2)...O_2^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O_2(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\
&\simeq& \frac{2}{2^{n}}\frac{1}{(\xi_{23}...\xi_{2n,1})^{1/4}(\xi_{2n+1,2n+2}...\xi_{4n-1,4n})^{1/4}}\label{freen}.\end{aligned}$$ Putting (\[freen\]) into (\[result1\]) with considering Jacobi factor (\[DerivitiveGeneralzationfour21\])(\[DerivitiveGeneralzationfour22\]), the variation of the $n$-th Rényi entropy is $$\begin{aligned}
\label{LatertimeresultO2General}
\Delta S_{A}^{(n)}=\log 2.\end{aligned}$$ In CFT with the Dirichlet boundary, the only difference with the Neumann boundary condition is the constraints $i_{2n+j}=i_{j}$ $\to$ $i_{2n+j}=-i_{j}$. One could check that the leading order correlation function is same as the Neumann boundary condition. Thus the $n$-th Rényi entropy is not dependent on the choice of boundary condition. We do not repeat here.
Rényi Entropy in Ising model
----------------------------
It is natural to ask how about the Ising model which is simplest unitary minimal model. There are three kinds of primary operators, i.e., the identity $\bm{\mathbb{I}}$, the spin operator $\bm{\sigma}$ and the energy operator $\bm{\epsilon}$. There are also two kinds of parity transformation which involve in two kinds of boundary conditions, which correspond to two different parity transformations. One of the parity transformation [@CFT] is $$\begin{aligned}
\label{parityIsing1}
\sigma(z,\bar z)\to \sigma(\bar z,z),\ \ \ \ \ \mu(z,\bar z)\to \mu(\bar z,z),\end{aligned}$$\[parityIsing2\] where $\bm{\mu}$ is the disorder operator. The other parity transformation [@CFT] is $$\begin{aligned}
\sigma(z,\bar z)\to \mu(\bar z,z),\ \ \ \ \ \mu(z,\bar z)\to \sigma(\bar z,z).\end{aligned}$$ We would like to study the local excitation by the spin operator $\bm{\sigma}$ with conformal dimension ($h=\frac{1}{16}$, $\bar h=\frac{1}{16}$) in the same setup given in section (\[3.1\]). The variation of the Rényi entropy for the subsystem $-l<x<0$ is given by (\[result1\]).
The two-point correlation function $$\begin{aligned}
&&\langle\sigma^{\dag}(\omega_1,\bar \omega_1)\sigma(\omega_2,\bar \omega_2)\rangle_{B_1}=\prod_{i=1}^2(\frac{d\omega_i}{d\xi_i^{'}})^{-h}(\frac{d\bar \omega_i}
{d\bar \xi_i^{'}})^{-\bar h}\langle \sigma^{\dag}(\xi_1^{'},\bar \xi_1^{'})\sigma(\xi_2^{'},\bar \xi_2^{'})\rangle_{UHP}\nonumber \\
&=&\prod_{i=1}^2(\frac{d\omega_i}{d\xi_i^{'}})^{-h}(\frac{d\bar \omega_i}
{d\bar \xi_i^{'}})^{-\bar h}\langle \sigma^{\dag}(\xi^{'}_1) \sigma(\xi^{'}_2) \tilde{\sigma}^{\dag}(\xi^{'}_3) \tilde{\sigma}(\xi^{'}_4) \rangle_{R_2}\end{aligned}$$ where $\xi_3^{'}\equiv \xi_1^{'*}$,$\xi_4\equiv \xi_2^{'*}$,$\xi_i^{'}=i\omega_i$, $\tilde{\sigma}$ is the field with parity transformation.
The 2-point correlation functions on UHP have already obtained in literature, e.g., [@Cardy1][@Dotsenko:1984nm][@Dotsenko:1984ad], $$\begin{aligned}
\label{IsingTwopointfunction}
&&\langle \sigma^{\dag}(\xi^{'}_1) \sigma(\xi^{'}_2) \sigma^{\dag}(\xi^{'}_3) \sigma(\xi^{'}_4) \rangle_{R_2}= (\frac{\xi^{'}_{13}\xi^{'}_{24}}{\xi^{'}_{12}\xi^{'}_{23}\xi^{'}_{14}\xi^{'}_{34}})^{\frac{1}{8}}F_{+}(x^{'}),\nonumber \\
&&\langle \sigma^{\dag}(\xi^{'}_1) \sigma(\xi^{'}_2) \mu^{\dag}(\xi^{'}_3) \mu(\xi^{'}_4) \rangle_{R_2}= (\frac{\xi^{'}_{13}\xi^{'}_{24}}{\xi^{'}_{12}\xi^{'}_{23}\xi^{'}_{14}\xi^{'}_{34}})^{\frac{1}{8}}F_{-}(x^{'}),\end{aligned}$$ with conformal blocks [@BPZ] $$\begin{aligned}
\label{F}
&&F_{+}(x^{'})=\sqrt{\sqrt{1-x^{'}}+1}+\sqrt{\sqrt{1-x^{'}}-1},\text{ }\text{ }\text{and}\nonumber \\
&&F_{-}(x^{'})=\sqrt{\sqrt{1-x^{'}}+1}-\sqrt{\sqrt{1-x^{'}}-1},\end{aligned}$$ where $x^{'}$ is the conformal cross ratio $x^{'}=\xi^{'}_{12}\xi^{'}_{34}/\xi^{'}_{13}\xi^{'}_{24}$. $F_{+}(x^{'})$ and $F_{-}(x^{'})$ correspond to different boundary conditions respectively. The leading behavior of the 2-point correlation function in $\epsilon \rightarrow 0$ is $$\begin{aligned}
\label{2pointising}
\langle\sigma^{\dag}(\omega_1,\bar \omega_1)\sigma(\omega_2,\bar \omega_2)\rangle_{B_1}\simeq \frac{\sqrt{2}}{(4\epsilon^2)^{1/8}}.\end{aligned}$$ (\[2pointising\]) for both boundary conditions.
In $t>L+l$ or $t<L-l$ with $\epsilon \rightarrow 0$, the leading behavior of 4-point correlation function is $$\begin{aligned}
\label{4pointIsing}
\langle \sigma^{\dag}(\xi_1,\bar \xi_1)\sigma(\xi_2,\bar \xi_2) \sigma^{\dag}(\xi_3,\bar \xi_3)\sigma(\xi_4,\bar \xi_4) \rangle_{UHP}&\simeq& \langle \sigma^{\dag}(\xi_1,\bar \xi_1)\sigma(\xi_2,\bar \xi_2)\rangle_{UHP}\langle \sigma^{\dag}(\xi_3,\bar \xi_3)\sigma(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber\\ &=& \frac{2}{(\xi_{12}\xi_{56})^{1/8}(\xi_{34}\xi_{78})^{1/8}},\nonumber\\\end{aligned}$$ where we have used the 2-point function of Ising model on UHP (\[IsingTwopointfunction\]). In terms of (\[result1\]), $\Delta S_{A}^{(2)}$ is $$\begin{aligned}
\label{Isingresultone1}
\Delta S_{A}^{(2)}&\simeq& -\log \Big[(\frac{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{4}})^{1/16} \frac{\langle \sigma_2^{\dag}(\xi_1,\bar \xi_1)...\sigma_2(\xi_4,\bar
\xi_4)\rangle_{UHP}}{(\langle \sigma^{\dag}_2(\xi_1,\bar \xi_1) \sigma_2(\xi_2,\bar \xi_2)\rangle_{UHP})^2}\Big]\nonumber \\
&=&-\log \Big[(\frac{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{2}}{(2\epsilon)^{4}})^{1/16} \frac{(4\epsilon^2)^{1/8}}{(\xi_{12}\xi_{34}\xi_{56}\xi_{78})^{1/8}}\Big]=0,\end{aligned}$$
In $L-l<t<L+l$ with the limit $\epsilon \rightarrow 0$, the 4-point correlation function (\[4pointIsing\]) on UHP can not be factorized directly. We also use the image method, which states that the 4-point correlation function on UHP can be expressed as linear combination of the holomorphic part of conformal blocks of the 8-point correlation function on the full complex plane, with coordinates $\xi_1...\xi_8$. As we know in 2 dimension full complex plane, there are $(n-3)$ independent cross ratios for $n$-point correlation function. In our case, there are 5 independent cross ratios. Thus the 4-point correlation function on UHP can be expressed by conformal blocks [@BPZ] $$\begin{aligned}
\label{UHP-Ising}
&&\langle \sigma^{\dag}(\xi_1,\bar \xi_1)\sigma(\xi_2,\bar \xi_2) \sigma^{\dag}(\xi_3,\bar \xi_3)\sigma(\xi_4,\bar \xi_4) \rangle_{UHP}\nonumber\\&=&(\frac{\xi_{13}\xi_{24}}{\xi_{12}\xi_{23}\xi_{14}\xi_{34}})^{\frac{1}{8}}\sum_b A^bC^b \mathcal{F}[b;x,x_1,x_2,...,x_4],\end{aligned}$$ where we define the 5 independent conformal ratios $x,x_1,...,x_4$, i.e. $x=\xi_{12} \xi_{34}/\xi_{13}\xi_{24}$, $x_1=\xi_{14} \xi_{\bar{1}\bar{2}}/\xi_{1\bar{1}}\xi_{4\bar{2}},x_2=\xi_{1\bar{1}} \xi_{\bar{2}\bar{3}}/\xi_{1\bar{2}}\xi_{\bar{1}\bar{3}},x_3=\xi_{2\bar{1}} \xi_{\bar{2}\bar{4}}/\xi_{2\bar{2}}\xi_{\bar{1}\bar{4}}, x_4=\xi_{3\bar{2}} \xi_{\bar{3}\bar{4}}/\xi_{3\bar{3}}\xi_{\bar{2}\bar{4}}$, $b$ in conformal blocks $\mathcal{F}$ runs over all the intermediate conformal families and the coefficients $A^p$ are determined by boundary conditions. Here we define $\xi_{\bar{i}}=\bar{\xi}_{i}$. The conformal blocks [@BPZ] satisfy the fusion transformation under the braiding operation [@Moore:1988uz][@Verlinde:1988sn][@Lewellen:1991tb], i.e., $$\begin{aligned}
\mathcal{F}[b;x,x_1,x_2,...,x_4]=F_{bc} \mathcal{F}[c;1-x,x_1,x_2,x_3,x_4],\end{aligned}$$ $F_{bc}$ is the fusion matrix. Making the fusion is equal to $\xi_{2}\leftrightarrow \xi_4$. In summary, the leading divergence of the correlation function is related to $c=0$ as identity as follows $$\begin{aligned}
(\ref{UHP-Ising})=\sum_{b,c}
F_{bc}(\frac{\xi_{13}\xi_{42}}{\xi_{14}\xi_{43}\xi_{12}\xi_{23}})^{\frac{1}{8}}A^b C^b \mathcal{F}[c;1-x,x_1,x_2,...,x_4].\end{aligned}$$ In terms of (\[Varianceofpoitmiddletime\]), one can find leading contribution of express (\[UHP-Ising\]) as following form in $L-l<t<L+l$ with $\epsilon \rightarrow 0$ $$\begin{aligned}
(\ref{UHP-Ising})\simeq F_{00}\langle \sigma^{\dag}(\xi_1,\bar \xi_1)\sigma(\xi_4,\bar \xi_2) \rangle_{UHP} \langle \sigma^{\dag}(\xi_3,\bar \xi_3)\sigma(\xi_2,\bar \xi_4) \rangle_{UHP},\end{aligned}$$ where $0$ stands for the identity operator. Then the variation of the Rényi entropy for $n=2$ $$\begin{aligned}
\Delta S^{(2)}_{A}=-\log F_{00}=\log \sqrt{2},\end{aligned}$$ where we have used the fact $F_{00}=\frac{1}{\sqrt{2}}$ [@Moore:1988ss] in two-dimensional Ising model. Note that the Rényi entropy does not depend on the choice of boundary condition in the Ising model, since the leading behavior of $F_{+}$ is the same as one of $F_{-}$ with $\epsilon \rightarrow 0$.
One alternative way to understand the phenomenon is to make use of diagram representation of conformal block as fig.\[\[f4-T-4\]\].
=8.5 cm =1.5 cm
Since the behavior of coordinates $\bar \xi_j$ does not change when $L-l<t<L+l$, we only need one time fusion transformation, which is different with [@He:2014mwa]. In the fig.\[\[f4-T-4\]\], $$\begin{aligned}
F_{00}[\sigma]=F_{00}\left[
\begin{array}{cc}
\sigma & \sigma \\
\sigma & \sigma \\
\end{array}
\right].\end{aligned}$$ In $L-l<t<L+l$ with $\epsilon \rightarrow 0$, the leading contribution of (\[UHP-Ising\]) originates from the above conformal block involving in identity operator. Because the fusion factor $F_{00}$ can not be canceled in the ratio (\[result1\]), $\Delta S^{(2)}_{A}=-\log F_{00}=\log \sqrt{2}$ [@Moore:1988ss].
Rényi Entropy in General Rational CFTs
--------------------------------------
In this subsection, we would like to generalize the analysis to the rational CFTs with a boundary in our previous set-up shown in subsection (\[3.1\]). In terms of (\[result1\]), we should know the 2-point correlation on $B_1$ and $2n$-point correlation fuction on UHP as usual. In generic rational CFTs with a boundary, the 2-point correlation can be expressed as $$\begin{aligned}
\label{2pointfucntiongeneral}
&&\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{B_1}=\langle O^{\dag}(\xi_1^{'},\bar \xi^{'}_1)O(\xi^{'}_2,\bar \xi^{'}_2)\rangle_{UHP}\nonumber \\
&=&\frac{1}{(\xi^{'}_{13}\xi^{'}_{24})^{2h}}\sum_b A^b C^b \mathcal{F}[b;\xi^{'}],\end{aligned}$$ where $A^b$ are constants which are determined by the boundary condition, $\xi^{'}_3$, $\xi^{'}_4$, $\xi^{'}$ are given in (\[twopointfunction\]). 4-point correlation function [@Dotsenko:1984nm][@Dotsenko:1984ad] of a primary function $O$ on the $R^2$, which can be expressed by $$\begin{aligned}
\label{genericfourpoint}
\langle O^{\dag}(z_1,\bar z_1) O(z_2,\bar z_2) O^{\dag} (z_3,\bar z_3)O (z_4,\bar z_4) \rangle_{R_2}=\sum_b\frac{1}{(z_{13}z_{24})^{2h}} C^b \mathcal{F}[b;z]\times c.c.\end{aligned}$$ In the limit $z\to 0$ $$\begin{aligned}
\label{Blocksim}
\mathcal{F}[b;z]=z^{h_b-2h}+...,\end{aligned}$$ where “..." stands for higher order terms and $z={z_{12}z_{34}\over z_{13}z_{24}}$ and $h_b$ is conformal dimension of primary class $b$.
In terms of (\[Blocksim\]), (\[2pointfucntiongeneral\]) with taking $\xi^{'}\to 0$ is $$\begin{aligned}
\langle O^{\dag}(\omega_1,\bar \omega_1)O(\omega_2,\bar \omega_2)\rangle_{B_1}&=&\frac{1}{(\xi^{'}_{13}\xi^{'}_{24})^{2h}} A^0 C^0 {\xi^{'}}^{-2h}
\simeq \frac{A^0 C^0}{(4\epsilon^2)^{2h}},\end{aligned}$$ In $t>L+l$ or $t<L-l$ with $\epsilon \rightarrow 0$, the 2n-point correlation function on UHP is $$\begin{aligned}
\label{2npointfunction}
&&\langle O^{\dag}(\xi_1,\bar \xi_1)O(\xi_2,\bar \xi_2)...O^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\
&\simeq& \langle O^{\dag}(\xi_1,\bar \xi_1)O(\xi_2,\bar \xi_2)\rangle_{UHP} ...\langle O^{\dag}(\xi_{2n-1}, \bar \xi_{2n-1})O(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}.\end{aligned}$$ The 2-point correlation on UHP is $$\begin{aligned}
\label{2pointfunctionafterfactor}
\langle O^{\dag}(\xi_{2k-1},\bar \xi_{2k-1})O(\xi_{2k},\bar \xi_{2k})\rangle_{UHP}\simeq \frac{A^0C^0}{(\xi_{2k-1,2k}\xi_{2n+2k-1,2n+2k})^{2h}},\end{aligned}$$ where $1\le k\le n$, $\xi_{2n+2k-1}\equiv \bar \xi_{2k-1}$.
Taking (\[2npointfunction\])(\[2pointfunctionafterfactor\]) (\[varianceofpoint1\])(\[2pointfucntiongeneral\])and Jacobi factor (\[DerivitiveGeneralzation11\])(\[DerivitiveGeneralzation12\])into (\[result1\]), one can obtain $$\begin{aligned}
\label{genralresult1}
\Delta S_{A}^{(n)}=0.\end{aligned}$$
In $L-l<t<l+L$ with $\epsilon \rightarrow 0$, the $2n$-point correlation function on UHP could be written as a linear combination of the holomorphic conformal blocks of the $4n$-point correlation function. We make the following fusions [@Moore:1988uz][@Verlinde:1988sn][@Lewellen:1991tb] $$\begin{aligned}
&&(\xi_1,\xi_2)(\xi_3,\xi_4)...(\xi_{4n-1},\xi_{4n})\to(\xi_2,\xi_3)(\xi_1,\xi_4)...(\xi_{4n-1},\xi_{4n})\nonumber \\
&\to& (\xi_2,\xi_3)(\xi_4,\xi_5)(\xi_1,\xi_{6})...\to...\to (\xi_2,\xi_3)...(\xi_1,\xi_{2n})...(\xi_{4n-1},\xi_{4n}),\end{aligned}$$ where $\bar \xi_{2n+i}=\xi_i$, $0\le i\le 2n$. With using $n-1$ times fusion transformation as the one shown in fig.\[\[f4-T-4\]\], we get $$\begin{aligned}
&&\langle O^{\dag}(\xi_1,\bar \xi_1)O(\xi_2,\bar \xi_2)...O^{\dag}(\xi_{2n-1},\bar \xi_{2n-1})O(\xi_{2n},\bar \xi_{2n})\rangle_{UHP}\nonumber \\
&\simeq& F_{00}[O]^{n-1} \langle O^{\dag}(\xi_1,\bar \xi_1)O(\xi_{2n},\bar \xi_2)\rangle_{UHP} ...\langle O^{\dag}(\xi_{2n-2}, \bar \xi_{2n-1})O(\xi_{2n-1},\bar \xi_{2n})\rangle_{UHP},\end{aligned}$$ We could calculate $\Delta S^n_A$ by using (\[result1\]). The variation of the $n$-Rényi entropy is $$\begin{aligned}
\label{genralresult2}
\Delta S_{A}^{(n)}=-\log F_{00}=\log d_{O},\end{aligned}$$ where $d_O$ is the quantum dimension [@Moore:1988ss] of operator $O$. (\[genralresult1\]) and (\[genralresult2\]) are same as the case in rational CFTs living on the full complex plane. We apply the fusion rule of conformal block of $2n$-point function on UHP to obtain the $\Delta S^{(n)}_A$. As we know, the $2n$-point correlation function on UHP can be expressed by linear combination of chiral part of conformal block associated with $4n$-point correlation on full plane. In this subsection, we do not make use of parity transformation containing the boundary information. The boundary data has been encoded in the coefficient of conformal block in this subsection, i.e., $A^p$ in (\[genericfourpoint\]). We could see that $\Delta S^n_A$ also does not depend on the choice of boundary.
Conclusion and Discussion
=========================
In this paper, we have investigated two kinds of effects on the locally excited states with time evolution. In the first case, we study the locally excited states with thermal effect in low temperature system. We have figured out the thermal correlation which is the same as [@Cardy:2014jwa] for the short interval limit. The Rényi entropy is equal to summation over the logarithmic of quantum dimension and thermal entropy in low temperature during the time $L-l<t<L+l$. In this paper, we just only confirm that such kind of sum rule is only true for the short interval $l$ in the low temperature limit. We make use of different approach [@Herzog:2012bw] to obtain the thermal correction to Rényi entropy which can be reduced to [@Caputa:2014eta] in low temperature. One can also calculate the Rényi entropy in the large interval[@Cardy:2014jwa] limit, the higher temperature limit[@Chen:2014hta] as well as beyond the leading order of the perturbation (\[Expand\]). We expected the sum rule relation is still hold in those cases. But one should note that we actually do not consider the back-reaction of the locally excited states to the thermal environment. When the energy of the local excitation is much lower than the thermal environment, it is safe to ignore the back-reaction. But in some special situation we expect the sum rule will break down. It is an interesting topic to consider in the future.
In the second case, we have studied the Rényi entropy of local excited states in 2 dimensional CFTs with a boundary. For 2D CFTs with a boundary, to obtain Rényi entropy can be converted to obtain the correlation function on UHP by using of conformal transformation technique. As a warm up, the Rényi entropy has been calculated with help of image method in the 2D free field theory with a boundary. The Rényi entropy is vanishing for operator $O_2$ (\[OperatorBoson\]) in $t>L+l$ or $t<L-l$ and $\log 2$ in $L-l<t<l+L$, which is the same as previous study [@He:2014mwa] in full complex plane without boundary. To confirm this fact, the Rényi entropy have been calculated in Ising model and more generic rational CFTs. Although the correlation function and conformal blocks in 2D CFTs with boundary are totally different from the ones in 2D CFT without boundary, we get a same maximal value of the Rényi entropy for the rational CFTs without a boundary[@He:2014mwa]. In generic 2D rational CFTs with a boundary, we confirm that the maximal value of Rényi entropy is the same as the one in 2D rational CFTs without boundary. [@Jackson:2014nla] also try to understand the fact which is not contract with that the left- and right-moving chiral sectors are decoupled. [@Jackson:2014nla] generalize the result in [@He:2014mwa] to irrational CFT, for example, Liouvile CFT. They found that the left-right entanglement entropy saturates the Cardy entropy. In terms of standard view, the Cardy entropy counts the microscopic entropy of actual CFT spectrum. The Cardy entropy seems to suggest two chiral sectors are decoupled. The authors in [@Jackson:2014nla] proposed a pragmatic point of view to reconcile [@He:2014mwa] with the fact that there should also be a comparably large EE between the two chiral sectors of CFT. For example, Non-chiral local operators will be left-right entangled. In BCFTs, the two chiral sectors are no longer independent. We have shown some additional examples to confirm the pragmatic perspective.
For general rational CFTs in 2D, the Rényi entropy highly relys on the conformal blocks of the theory. The $n$-point correlation functions in 2D CFTs with boundary are related to the holomorphic part of conformal blocks of the $2n$-point correlation functions on the 2D full complex plane. This relationship had been studied by the image method [@Cardy1][@Cardy2] very well. More precise relation is that an $n$-point function in the UHP, which is a function of the coordinates $(z_1,,z_n; \bar{z}_1,...,\bar{z}_n)$ behaves under conformal transformations in the same way as the holomorphic factor of a $2n$-point function in the full plane which depends on $(z_1,...,z_n; z_1^*,...,z_n^*)$, analytically continued to $z_j^*=\bar{z_j}$. In [@He:2014mwa], the time evolution of Rényi entropy highly depends on the holomorphic part of conformal block. In 2D CFTs with a boundary, the boundary changes the evolution of the Rényi entropy but does not change the value of the Rényi entropy, which is closely related to fusion constants in the bulk. Because the behavior of the ‘image’ coordinates (anti-holomorphic coordinates) does not change as the holomorphic coordinates when $L-l<t<L+l$, we get the same Rényi entropy as [@He:2014mwa].
The boundary introduced here works as the infinite potential barrier for the time evolution of the entangled quasi-particles pairs [@Jean-Marie; @Stphan][@Nozaki:2014hna][@He:2014mwa] triggered by local excitation as shown in fig.\[\[fig2\]\].
=12.0 cm =4.0 cm
The the Rényi entropy measures the entanglement between the quasi-particles generated by local excitation. After entangled pairs are created at $-L$, the two quasi-particles will propagate in two opposite directions, i.e., left-moving and right moving. When the right-moving particle enter the interval $-l<x<0$ denoted by $A$, the Rényi entropy takes maximal value due to entanglement between two entangled particles. In fig.\[\[fig2\]\], the blue wave lines mimic the entanglement of two entangled quasi-particles. When the right-moving quasi-particle reaches the boundary, the quasi-particle will be reflected by the boundary without losing energy. As the calculation in [@Caputa:2014vaa] shows the locally excited states carry the energy of $O(\epsilon^{-1})$. The conformal transformation, $z\to z+\epsilon(z)$ and $\bar z\to \bar z+\bar \epsilon(\bar z)$, should keep the boundary conformal invariant, which lead to the constraint $T=\bar T$ on the boundary. In the Cartesian coordinates, the constraint becomes $T_{xt}=0$, which means that no energy can flow across the boundary. This is main reason the quasi-particle must be reflected no matter what is the conformal invariant boundary condition. In this sense, the boundary change the time evolution of Rényi entropy.In our paper we take the scale $\epsilon$ as the minimal scale, and keep the leading order of $\epsilon$ in the calculation. So we must miss some information when $t\sim L$, i.e., the quasi-particle is close to the boundary. In this case, we should make use of bulk boundary correlation functions and boundary structure constants in BCFT to figure out the time evolution of entangled quasi-particles. The next leading order calculation of $\epsilon$ may give us more insight on this point.
0.5cm [**Acknowledgement**]{} 0.2cm We are grateful to J. L. Cardy, Mitsutoshi Fujita, Rene Meyer, Masahiro Nozaki, T. Numasawa, Noburo Shiba, Tadashi Takayanagi and K. Watanabe for useful conversations and correspondence. We thank Miao Li, Tadashi Takayanagi for their encouragement and support. W. Z. Guo is supported by Postgraduate Scholarship Program of China Scholarship Council. S.H. is supported by JSPS postdoctoral fellowship for foreign researchers and by the National Natural Science Foundation of China (No.11305235).
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[^1]: The operator may not be a primary operator , such as the energy-momentum tensor $T$. A Schwarzian derivative term related to energy momentum tensor operator [@Cardy:2014jwa][@Chen:2014unl] will present due to conformal transformation.
[^2]: In conformal field theory literature, the image method is also called double trick sometimes.
[^3]: For most primary fields, it is impossible to divide the field $\phi$ into holomorphic and anti-holomorphic part. Here we only want to express that the $n$-point correlation functions on UHP which are dependent on the holomorphic conformal blocks of $2n$-point correlation functions on the full complex plane. In 2D free field theory, the conformal blocks of the operator $O_1$ is trivial. In this paper, we will also show the image method in 2D Ising model and other generic 2D CFTs.
[^4]: We can use this rule in 2D free scalar field theory, since the conformal blocks related $O_1$ are actually trivial. We would like to appreciate communication with Cardy on this point.
|
---
abstract: 'When a plane shock hits a wedge head on, it experiences a reflection-diffraction process and then a self-similar reflected shock moves outward as the original shock moves forward in time. Experimental, computational, and asymptotic analysis has shown that various patterns of shock reflection may occur, including regular and Mach reflection. However, most of the fundamental issues for shock reflection have not been understood, including the global structure, stability, and transition of the different patterns of shock reflection. Therefore, it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, there has been no rigorous mathematical result on the global existence and structural stability of shock reflection, including the case of potential flow which is widely used in aerodynamics. Such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic-hyperbolic type, free boundary problems, and corner singularity where an elliptic degenerate curve meets a free boundary. In this paper we develop a rigorous mathematical approach to overcome these difficulties involved and establish a global theory of existence and stability for shock reflection by large-angle wedges for potential flow. The techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties.'
date: 'March 28, 2006'
title: 'Global Solutions of Shock Reflection by Large-Angle Wedges for Potential Flow'
---
Introduction
============
We are concerned with the problems of shock reflection by wedges. These problems arise not only in many important physical situations but also are fundamental in the mathematical theory of multidimensional conservation laws since their solutions are building blocks and asymptotic attractors of general solutions to the multidimensional Euler equations for compressible fluids (cf. Courant-Friedrichs [@CF], von Neumann [@Neumann], and Glimm-Majda [@GlimmMajda]; also see [@BD; @ChangChen; @GlimmK; @LaxLiu; @Morawetz2; @Serre; @VD]). When a plane shock hits a wedge head on, it experiences a reflection-diffraction process and then a self-similar reflected shock moves outward as the original shock moves forward in time. The complexity of reflection picture was first reported by Ernst Mach [@Mach] in 1878, and experimental, computational, and asymptotic analysis has shown that various patterns of shock reflection may occur, including regular and Mach reflection (cf. [@BD; @GRT; @GlimmMajda; @hunter1; @HK; @KB; @Morawetz2; @VD; @Neumann]). However, most of the fundamental issues for shock reflection have not been understood, including the global structure, stability, and transition of the different patterns of shock reflection. Therefore, it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, there has been no rigorous mathematical result on the global existence and structural stability of shock reflection, including the case of potential flow which is widely used in aerodynamics (cf. [@Bers1; @CC; @GlimmMajda; @MajdaTh; @Morawetz2]). One of the main reasons is that the problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic-hyperbolic type, free boundary problems, and corner singularity where an elliptic degenerate curve meets a free boundary. In this paper we develop a rigorous mathematical approach to overcome these difficulties involved and establish a global theory of existence and stability for shock reflection by large-angle wedges for potential flow. The techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties.
The Euler equations for potential flow consist of the conservation law of mass and the Bernoulli law for the density $\rho$ and velocity potential $\Phi$: $$\begin{aligned}
&&\partial_t\rho + { \mbox{div}}_{\bf x}(\rho\nabla_{\bf x}\Phi)=0, \label{1.1.1} \\
&&\partial_t\Phi +\frac{1}{2}|\nabla_{\bf x}\Phi|^2+i(\rho)=K,
\label{1.1.2}\end{aligned}$$ where $K$ is the Bernoulli constant determined by the incoming flow and/or boundary conditions, and $$i'(\rho)=p'(\rho)/\rho=c^2(\rho)/\rho$$ with $c(\rho)$ being the sound speed. For polytropic gas, $$p(\rho)=\kappa\rho^\gamma,\qquad
c^2(\rho)=\kappa\gamma\rho^{\gamma-1}, \qquad \gamma>1, \,\,
\kappa>0.$$ Without loss of generality, we choose $\kappa=(\gamma-1)/\gamma$ so that $$i(\rho)=\rho^{\gamma-1}, \qquad c(\rho)^2=(\gamma-1)\rho^{\gamma-1},$$ which can be achieved by the following scaling: $$({\bf x},t,K)\to (\alpha {\bf x}, \alpha^2 t, \alpha^{-2} K), \quad
\alpha^2=\kappa\gamma/(\gamma-1).$$ Equations – can written as the following nonlinear equation of second order: $$\label{nonlinear-second-order:1}
\partial_t\hat{\rho}\big(K-\partial_t\Phi-\frac{1}{2}|\nabla_{\bf x}\Phi|^2\big)
+{ \mbox{div}}_{\bf
x}\big(\hat{\rho}(K-\partial_t\Phi-\frac{1}{2}|\nabla_{\bf
x}\Phi|^2)
\nabla_{\bf x}\Phi\big)=0,$$ where $\hat{\rho}(s)=s^{1/(\gamma-1)}=i^{-1}(s)$ for $s\ge 0$.
When a plane shock in the $({\bf x},t)$–coordinates, ${\bf
x}=(x_1,x_2)\in{ {\bf R}}^2$, with left state $ (\rho,\nabla_{\bf
x}\Psi)=(\rho_1, u_1,0) $ and right state $(\rho_0, 0,0), u_1>0,
\rho_0<\rho_1$, hits a symmetric wedge $$W:=\{|x_2|< x_1 \tan\theta_w, x_1>0\}$$ head on, it experiences a reflection-diffraction process, and the reflection problem can be formulated as the following mathematical problem.
. [*Seek a solution of system – with $K=\rho_0^{\gamma-1}$, the initial condition at $t=0$: $$\label{initial-condition}
(\rho,\Phi)|_{t=0} =\begin{cases}
(\rho_0, 0) \qquad& \mbox{for}\,\, |x_2|>x_1\tan\theta_w, x_1>0,\\
(\rho_1, u_1 x_1) \qquad &\mbox{for}\,\, x_1<0,
\end{cases}$$ and the slip boundary condition along the wedge boundary $\partial
W$: $$\label{boundary-condition}
\nabla\Phi\cdot \nu|_{\partial W}=0,$$ where $\nu$ is the exterior unit normal to $\partial W$ (see Fig. [1]{}).*]{}
![Initial-boundary value problem[]{data-label="fig:IBVP"}](fig01.eps){height="2.5in" width="2.7in"}
Notice that the initial-boundary value problem – is invariant under the self-similar scaling: $$({\bf x}, t)\to (\alpha {\bf x}, \alpha t), \quad (\rho, \Phi)\to
(\rho, \Phi/\alpha) \qquad \quad\mbox{for}\quad \alpha\ne 0.$$ Thus, we seek self-similar solutions with the form $$\rho({\bf x},t)=\rho(\xi,\eta), \quad \Phi({\bf
x},t)=t\,\psi(\xi,\eta) \qquad\quad \mbox{for}\quad (\xi,\eta)={\bf
x}/t.$$ Then the pseudo-potential function $\varphi=\psi-\frac{1}{2}(\xi^2+\eta^2)$ satisfies the following Euler equations for self-similar solutions: $$\begin{aligned}
&&{ \mbox{div}}\,(\rho\, D\varphi)+2\rho=0, \label{1.1.3}\\
&&\frac{1}{2}|D\varphi|^2+\varphi+ \rho^{\gamma-1}
=\rho_0^{\gamma-1},
\label{1.1.4}\end{aligned}$$ where the divergence ${ \mbox{div}}$ and gradient $D$ are with respect to the self-similar variables $(\xi,\eta)$. This implies that the pseudo-potential function $\varphi(\xi,\eta)$ is governed by the following potential flow equation of second order: $${ \mbox{div}}\, \big(\rho(|D\varphi|^2, \varphi)D\varphi\big)
+2\rho(|D\varphi|^2, \varphi)=0 \label{1.1.5}$$ with $$\rho(|D\varphi|^2, \varphi)
=\hat{\rho}(\rho_0^{\gamma-1}-\varphi-\frac{1}{2}|D\varphi|^2).
\label{1.1.6}$$ Then we have $$\label{c-through-density-function}
c^2=c^2(|D\varphi|^2,\varphi,\rho_0^{\gamma-1})
=(\gamma-1)(\rho_0^{\gamma-1}-\frac{1}{2}|D\varphi|^2-\varphi).$$
Equation is a mixed equation of elliptic-hyperbolic type. It is elliptic if and only if $$|D\varphi| < c(|D\varphi|^2,\varphi,\rho_0^{\gamma-1}),
\label{1.1.8}$$ which is equivalent to $$|D \varphi| <c_*(\varphi, \rho_0, \gamma)
:=\sqrt{\frac{2(\gamma-1)}{\gamma+1}(\rho_0^{\gamma-1}-\varphi)}.
\label{1.1.8a}$$ Shocks are discontinuities in the pseudo-velocity $D\varphi$. That is, if $\Omega^+$ and $\Omega^-:=\Omega\setminus\overline{\Omega^+}$ are two nonempty open subsets of $\Omega\subset{ {\bf R}}^2$ and $S:=\partial\Omega^+\cap\Omega$ is a $C^1$ curve where $D\varphi$ has a jump, then $\varphi\in W^{1,1}_{loc}(\Omega)\cap
C^1(\Omega^\pm\cup S)\cap C^2(\Omega^\pm)$ is a global weak solution of (\[1.1.5\]) in $\Omega$ if and only if $\varphi$ is in $W^{1,\infty}_{loc}(\Omega)$ and satisfies equation in $\Omega^\pm$ and the Rankine-Hugoniot condition on $S$: $$\label{FBConditionSelfSim-0}
\left[\rho(|D\varphi|^2,\varphi)D\varphi\cdot\nu\right]_S=0.$$ The continuity of $\varphi$ is followed by the continuity of the tangential derivative of $\varphi$ across $S$, which is a direct corollary of irrotationality of the pseudo-velocity. The discontinuity $S$ of $D\varphi$ is called a shock if $\varphi$ further satisfies the physical entropy condition that the corresponding density function $\rho(|D\varphi|^2,\varphi)$ increases across $S$ in the pseudo-flow direction. We remark that the Rankine-Hugoniot condition with the continuity of $\varphi$ across a shock for is also fairly good approximation to the corresponding Rankine-Hugoniot conditions for the full Euler equations for shocks of small strength, since the errors are third-order in strength of the shock.
The plane incident shock solution in the $({\bf x},t)$–coordinates with states $(\rho, \nabla_{\bf x}\Psi)=(\rho_0, 0,0)$ and $(\rho_1,
u_1,0)$ corresponds to a continuous weak solution $\varphi$ of (\[1.1.5\]) in the self-similar coordinates $(\xi,\eta)$ with the following form: $$\begin{aligned}
&&\varphi_0(\xi,\eta)=-\frac{1}{2}(\xi^2+\eta^2) \qquad
\hbox{for } \,\, \xi>\xi_0,
\label{flatOrthSelfSimShock1} \\
&&\varphi_1(\xi,\eta)=-\frac{1}{2}(\xi^2+\eta^2)+ u_1(\xi-\xi_0)
\qquad
\hbox{for } \,\, \xi<\xi_0,
\label{flatOrthSelfSimShock2}\end{aligned}$$ respectively, where $$\label{shocklocation}
\xi_0=\rho_1
\sqrt{\frac{2(\rho_1^{\gamma-1}-\rho_0^{\gamma-1})}{\rho_1^2-\rho_0^2}}
=\frac{\rho_1u_1}{\rho_1-\rho_0}>0$$ is the location of the incident shock, uniquely determined by $(\rho_0,\rho_1,\gamma)$ through (\[FBConditionSelfSim-0\]). Since the problem is symmetric with respect to the axis $\eta=0$, it suffices to consider the problem in the half-plane $\eta>0$ outside the half-wedge $$\Lambda:=\{\xi\le 0,\eta>0\}\cup\{\eta>\xi \tan\theta_w,\, \xi>0\}.$$ Then the initial-boundary value problem – in the $({\bf x},
t)$–coordinates can be formulated as the following boundary value problem in the self-similar coordinates $(\xi,\eta)$.
(see Fig. 2). [*Seek a solution $\varphi$ of equation in the self-similar domain $\Lambda$ with the slip boundary condition on the wedge boundary $\partial\Lambda$: $$\label{boundary-condition-3}
D\varphi\cdot\nu|_{\partial\Lambda}=0$$ and the asymptotic boundary condition at infinity: $$\label{boundary-condition-2}
\varphi\to\bar{\varphi}=
\begin{cases} \varphi_0 \qquad\mbox{for}\,\,\,
\xi>\xi_0, \eta>\xi \tan\theta_w,\\
\varphi_1 \qquad \mbox{for}\,\,\,
\xi<\xi_0, \eta>0,
\end{cases}
\qquad \mbox{when $\xi^2+\eta^2\to \infty$},$$*]{} where (\[boundary-condition-2\]) holds in the sense that $
\displaystyle
\lim_{R\to\infty}\|\varphi-\overline{\varphi}\|_{C(\Lambda\setminus
B_R(0))}=0. $
![Boundary value problem in the unbounded domain[]{data-label="fig:BVP"}](fig02.eps){height="2.6in" width="2.8in"}
Since $\varphi_1$ does not satisfy the slip boundary condition , the solution must differ from $\varphi_1$ in $\{\xi<\xi_0\}\cap\Lambda$, thus a shock diffraction by the wedge occurs. In this paper, we first follow the von Neumann criterion to establish a local existence theory of regular shock reflection near the reflection point and show that the structure of solution is as in Fig. \[fig:RegularReflection\], when the wedge angle is large and close to $\pi/2$, in which the vertical line is the incident shock $S=\{\xi=\xi_0\}$ that hits the wedge at the point $P_0=(\xi_0, \xi_0 \tan\theta_w)$, and state (0) and state (1) ahead of and behind $S$ are given by $\varphi_0$ and $\varphi_1$ defined in and , respectively. The solutions $\varphi$ and $\varphi_1$ differ only in the domain $P_0{{P_1}}{{P_2}}{{P_3}}$ because of shock diffraction by the wedge vertex, where the curve $P_0{{P_1}}{{P_2}}$ is the reflected shock with the straight segment $P_0{{P_1}}$. State (2) behind $P_0{{P_1}}$ can be computed explicitly with the form: $$\label{state2a}
\varphi_2(\xi,\eta)=-\frac{1}{2}(\xi^2+\eta^2)+u_2(\xi-\xi_0)+
(\eta-\xi_0\tan\theta_w)u_2\tan\theta_w,$$ which satisfies $$D\varphi\cdot\nu=0 \qquad \hbox{on }\, \partial\Lambda\cap
\{\xi>0\};$$ the constant velocity $u_2$ and the angle $\theta_s$ between $P_0{{P_1}}$ and the $\xi$–axis are determined by $(\theta_w,\rho_0,\rho_1,\gamma)$ from the two algebraic equations expressing (\[FBConditionSelfSim-0\]) and continuous matching of state (1) and state (2) across $P_0{{P_1}}$, whose existence is exactly guaranteed by the condition on $(\theta_w,\rho_0,\rho_1,\gamma)$ under which regular shock reflection is expected to occur.
![Regular reflection[]{data-label="fig:RegularReflection"}](fig03.eps){height="2.6in" width="3.4in"}
We develop a rigorous mathematical approach to extend the local theory to a global theory for solutions of regular shock reflection, which converge to the unique solution of the normal shock reflection when $\theta_w$ tends to $\pi/2$. The solution $\varphi$ is pseudo-subsonic within the sonic circle for state (2) with center $(u_2, u_2\tan\theta_w)$ and radius $c_2>0$ (the sonic speed) and is pseudo-supersonic outside this circle containing the arc ${{P_1}}{{P_4}}$ in Fig. \[fig:RegularReflection\], so that $\varphi_2$ is the unique solution in the domain $P_0{{P_1}}{{P_4}}$, as argued in [@ChangChen; @Serre]. In the domain $\Omega$, the solution is expected to be pseudo-subsonic, smooth, and $C^1$-smoothly matching with state (2) across ${{P_1}}{{P_4}}$ and to satisfy $\varphi_\eta=0$ on ${{P_2}}{{P_3}}$; the transonic shock curve ${{P_1}}{{P_2}}$ matches up to second-order with $P_0{{P_1}}$ and is orthogonal to the $\xi$-axis at the point ${{P_2}}$ so that the standard reflection about the $\xi$–axis yields a global solution in the whole plane. Then the solution of Problem 2 can be shown to be the solution of Problem 1.
There exist $\theta_c=\theta_c(\rho_0,\rho_1,\gamma) \in (0,\pi/2)$ and $\alpha=\alpha(\rho_0,\rho_1,\gamma)\in (0, 1/2)$ such that, when $\theta_w\in [\theta_c,\pi/2)$, there exists a global self-similar solution $$\Phi({\bf x}, t) =t\,\varphi(\frac{\bf x}{t}) +\frac{|\bf x|^2}{2t}
\qquad\mbox{for}\,\, \frac{\bf x}{t}\in \Lambda,\, t>0$$ with $$\rho({\bf x}, t)=(\rho_0^{\gamma-1}-\Phi_t
-\frac{1}{2}|\nabla_{\bf x}\Phi|^2)^{\frac{1}{\gamma-1}}$$ of Problem 1 (equivalently, Problem 2) for shock reflection by the wedge, which satisfies that, for $(\xi,\eta)={\bf x}/t$, $$\varphi\in C^{\infty}(\Omega)\cap C^{1,\alpha}(\bar{\Omega}),$$ $$\label{phi-states-0-1-2}
\varphi=\left\{\begin{array}{ll}
\varphi_0 \qquad\mbox{for}\,\, \xi>\xi_0 \mbox{ and } \eta>\xi\tan\theta_w,\\
\varphi_1 \qquad\mbox{for}\,\, \xi<\xi_0
\mbox{ and above the reflection shock} \,\, P_0{{P_1}}{{P_2}},\\
\varphi_2 \qquad \mbox{in}\,\, P_0{{P_1}}{{P_4}},
\end{array}\right.$$ $\varphi$ is $C^{1,1}$ across the part ${{P_1}}{{P_4}}$ of the sonic circle including the endpoints ${{P_1}}$ and ${{P_4}}$, and the reflected shock $P_0{{P_1}}{{P_2}}$ is $C^2$ at ${{P_1}}$ and $C^\infty$ except ${{P_1}}$. Moreover, the solution $\varphi$ is stable with respect to the wedge angle in $W^{1,1}_{loc}(\overline
\Lambda)$ and converges in $W^{1,1}_{loc}(\overline\Lambda)$ to the solution of the normal reflection described in §\[section:4\] as $\theta_w\to\pi/2$.
One of the main difficulties for the global existence is that the ellipticity condition for (\[1.1.5\]) is hard to control, in comparison to our earlier work on steady flow [@ChenFeldman1; @ChenFeldman2]. The second difficulty is that the ellipticity degenerates at the sonic circle ${{P_1}}{{P_4}}$ (the boundary of the pseudo-subsonic flow). The third difficulty is that, on ${{P_1}}{{P_4}}$, we need to match the solution in $\Omega$ with $\varphi_2$ at least in $C^1$, that is, the two conditions on the fixed boundary ${{P_1}}{{P_4}}$: the Dirichlet and conormal conditions, which are generically overdetermined for an elliptic equation since the conditions on the other parts of boundary have been prescribed. Thus we have to prove that, if $\varphi$ satisfies (\[1.1.5\]) in $\Omega$, the Dirichlet continuity condition on the sonic circle, and the appropriate conditions on the other parts of $\partial\Omega$ derived from Problem 2, then the normal derivative $D\varphi\cdot\nu$ automatically matches with $D\varphi_2\cdot\nu$ along ${{P_1}}{{P_4}}$. We show that, in fact, this follows from the structure of elliptic degeneracy of (\[1.1.5\]) on ${{P_1}}{{P_4}}$ for the solution $\varphi$. Indeed, equation (\[1.1.5\]), written in terms of the function $u=\varphi-\varphi_2$ in the $(x,y)$–coordinates defined near ${{P_1}}{{P_4}}$ such that ${{P_1}}{{P_4}}$ becomes a segment on $\{x=0\}$, has the form: $$\label{degenerate-equation}
\big(2x-(\gamma+1)u_x\big)u_{xx}+\frac{1}{c_2^2}u_{yy}-u_x=0
\qquad\,\, \mbox{in } x>0 \mbox{ and near } x=0,$$ plus the “small” terms that are controlled by $\pi/2-\theta_w$ in appropriate norms. Equation is elliptic if $u_x<2x/(\gamma+1)$. Thus, we need to obtain the $C^{1,1}$ estimates near ${{P_1}}{{P_4}}$ to ensure $|u_x|<2x/(\gamma+1)$ which in turn implies both the ellipticity of the equation in $\Omega$ and the match of normal derivatives $D\varphi\cdot\nu=D\varphi_2\cdot\nu$ along ${{P_1}}{{P_4}}$. Taking into account the “small” terms to be added to equation , we need to make the stronger estimate $|u_x|\le 4x/\big(3(\gamma+1)\big)$ and assume that $\pi/2-\theta_w$ is appropriately small to control these additional terms. Another issue is the non-variational structure and nonlinearity of this problem which makes it hard to apply directly the approaches of Caffarelli [@Ca] and Alt-Caffarelli-Friedman [@AC; @ACF]. Moreover, the elliptic degeneracy and geometry of the problem makes it difficult to apply the hodograph transform approach in Kinderlehrer-Nirenberg [@KinderlehrerNirenberg] and Chen-Feldman [@ChenFeldman4] to fix the free boundary.
For these reasons, one of the new ingredients in our approach is to further develop the iteration scheme in [@ChenFeldman1; @ChenFeldman2] to a partially modified equation. We modify equation (\[1.1.5\]) in $\Omega$ by a proper cutoff that depends on the distance to the sonic circle, so that the original and modified equations coincide for $\varphi$ satisfying $|u_x| \le 4x/\big(3(\gamma+1)\big)$, and the modified equation ${\mathcal N}\varphi=0$ is elliptic in $\Omega$ with elliptic degeneracy on ${{P_1}}{{P_4}}$. Then we solve a free boundary problem for this modified equation: The free boundary is the curve ${{P_1}}{{P_2}}$, and the free boundary conditions on ${{P_1}}{{P_2}}$ are $\varphi=\varphi_1$ and the Rankine-Hugoniot condition (\[FBConditionSelfSim-0\]).
On each step, an “iteration free boundary” curve ${{P_1}}{{P_2}}$ is given, and a solution of the modified equation ${\mathcal N}\varphi=0$ is constructed in $\Omega$ with the boundary condition (\[FBConditionSelfSim-0\]) on ${{P_1}}{{P_2}}$, the Dirichlet condition $\varphi=\varphi_2$ on the degenerate circle ${{P_1}}{{P_4}}$, and $D\varphi\cdot\nu=0$ on ${{P_2}}{{P_3}}$ and ${{P_3}}{{P_4}}$. Then we prove that $\varphi$ is in fact $C^{1,1}$ up to the boundary ${{P_1}}{{P_4}}$, especially $|{{D}}(\varphi-\varphi_2)|\le Cx$, by using the nonlinear structure of elliptic degeneracy near ${{P_1}}{{P_4}}$ which is modeled by equation (\[degenerate-equation\]) and a scaling technique similar to Daskalopoulos-Hamilton [@DG] and Lin-Wang [@LW]. Furthermore, we modify the “iteration free boundary” curve ${{P_1}}{{P_2}}$ by using the Dirichlet condition $\varphi=\varphi_1$ on ${{P_1}}{{P_2}}$. A fixed point $\varphi$ of this iteration procedure is a solution of the free boundary problem for the modified equation. Moreover, we prove the precise gradient estimate: $|u_x|<4x/\big(3(\gamma+1)\big)$, which implies that $\varphi$ satisfies the original equation (\[1.1.5\]).
Some efforts have been made mathematically for the reflection problem via simplified models. One of these models, the unsteady transonic small-disturbance (UTSD) equation, was derived and used in Keller-Blank [@KB], Hunter-Keller [@HK], Hunter [@hunter1], Morawetz [@Morawetz2], and the references cited therein for asymptotic analysis of shock reflection. Also see Zheng [@Zheng1] for the pressure gradient equation and Canic-Keyfitz-Kim [@CKK1] for the UTSD equation and the nonlinear wave system. On the other hand, in order to deal with the reflection problem, some asymptotic methods have been also developed. Lighthill [@Lighthill1; @Lighthill2] studied shock reflection under the assumption that the wedge angle is either very small or close to $\pi/2$. Keller-Blank [@KB], Hunter-Keller [@HK], and Harabetian [@Harabetian] considered the problem under the assumption that the shock is so weak that its motion can be approximated by an acoustic wave. For a weak incident shock and a wedge with small angle in the context of potential flow, by taking the jump of the incident shock as a small parameter, the nature of the shock reflection pattern was explored in Morawetz [@Morawetz2] by a number of different scalings, a study of mixed equations, and matching the asymptotics for the different scalings. Also see Chen [@Sxchen] for a linear approximation of shock reflection when the wedge angle is close to $\pi/2$ and Serre [@Serre] for an apriori analysis of solutions of shock reflection and related discussions in the context of the Euler equations for isentropic and adiabatic fluids.
The organization of this paper is the following. In §2, we present the potential flow equation in self-similar coordinates and exhibit some basic properties of solutions to the potential flow equation. In §3, we discuss the normal reflection solution and then follow the von Neumann criterion to derive the necessary condition for the existence of regular reflection and show that the shock reflection can be regular locally when the wedge angle is large. In §4, the shock reflection problem is reformulated and reduced to a free boundary problem for a second-order nonlinear equation of mixed type in a convenient form. In §5, we develop an iteration scheme, along with an elliptic cutoff technique, to solve the free boundary problem and set up the ten detailed steps of the iteration procedure.
Finally, we complete the remaining steps in our iteration procedure in §6–§9: Step 2 for the existence of solutions of the boundary value problem to the degenerate elliptic equation via the vanishing viscosity approximation in §6; Steps 3–8 for the existence of the iteration map and its fixed point in §7; and Step 9 for the removal of the ellipticity cutoff in the iteration scheme by using appropriate comparison functions and deriving careful global estimates for some directional derivatives of the solution in §8. We complete the proof of Main Theorem in §9. Careful estimates of the solutions to both the “almost tangential derivative" and oblique derivative boundary value problems for elliptic equations are made in Appendix, which are applied in §6–§7.
Self-Similar Solutions of the Potential Flow Equation
=====================================================
In this section we present the potential flow equation in self-similar coordinates and exhibit some basic properties of solutions of the potential flow equation (also see Morawetz [@Morawetz2]).
Equation is a mixed equation of elliptic-hyperbolic type. It is elliptic if and only if holds. The hyperbolic-elliptic boundary is the pseudo-sonic curve: $|D\varphi|=c_*(\varphi,\rho_0,\gamma)$.
We first define the notion of weak solutions of –. Essentially, we require the equation to be satisfied in the distributional sense.
\[def2.1\] A function $\varphi\in W^{1,1}_{loc}(\Lambda)$ is called a weak solution of – in a self-similar domain $\Lambda$ if
1. $\rho_0^{\gamma-1}-\varphi-\frac{1}{2}|D\varphi|^2\ge 0$ a.e. in $\Lambda$;
2. $(\rho(|D\varphi|^2, \varphi),
\rho(|D\varphi|^2, \varphi)|{{D}}\varphi|)\in
(L^1_{loc}(\Lambda))^2$;
3. For every $\zeta\in C^\infty_c(\Lambda)$, $$\int_\Lambda\Big(\rho(|D\varphi|^2,
\varphi){{D}}\varphi\cdot{{D}}\zeta -2\rho(|D\varphi|^2,
\varphi)\zeta\Big) \,d{{\xi}}d{{\eta}}=0.$$
It is straightforward to verify the equivalence between time-dependent self-similar solutions and weak solutions of defined in Definition \[def2.1\] in the weak sense. It can also be verified that, if $\varphi\in C^{1,1}(\Lambda)$ (and thus $\varphi$ is twice differentiable a.e. in $\Lambda$), then $\varphi$ is a weak solution of (\[1.1.5\]) in $\Lambda$ if and only if $\varphi$ satisfies equation a.e. in $\Lambda$. Finally, it is easy to see that, if $\Lambda^+$ and $\Lambda^-=\Lambda\setminus\overline{\Lambda^+}$ are two nonempty open subsets of $\Lambda\subset{ {\bf R}}^2$ and $S=\partial \Lambda^+\cap
\Lambda$ is a $C^1$ curve where $D\varphi$ has a jump, then $\varphi\in W^{1,1}_{loc}(D)\cap C^1(\Lambda^\pm\cup S)\cap
C^{1,1}(\Lambda^\pm)$ is a weak solution of (\[1.1.5\]) in $\Lambda$ if and only if $\varphi$ is in $W^{1,\infty}_{loc}(\Lambda)$ and satisfies equation a.e. in $\Lambda^\pm$ and the Rankine-Hugoniot condition (\[FBConditionSelfSim-0\]) on $S$.
Note that, for $\varphi\in C^1(\Lambda^\pm\cup S)$, the condition $\varphi\in W^{1,\infty}_{loc}(\Lambda)$ implies $$[\varphi]_{S}=0. \label{1.1.14}$$ Furthermore, the Rankine-Hugoniot conditions imply $$[\varphi_\xi][\rho\varphi_\xi]-[\varphi_\eta][\rho\varphi_\eta]=0
\qquad \text{on } S \label{1.1.16}$$ which is a useful identity.
A discontinuity of ${{D}}\varphi$ satisfying the Rankine-Hugoniot conditions and is called a shock if it satisfies the physical entropy condition: [*The density function $\rho$ increases across a shock in the pseudo-flow direction*]{}. The entropy condition indicates that [*the normal derivative function $\varphi_\nu$ on a shock always decreases across the shock in the pseudo-flow direction*]{}.
When the density $\rho$ is constant, – imply that $\varphi$ satisfies $$\Delta \varphi +2=0,\qquad \frac{1}{2}|D\varphi|^2 +\varphi=const.$$ This implies $(\Delta\varphi)_\xi=0, (\Delta\varphi)_\eta=0,$ and $(\varphi_{\xi\xi}+1)^2+\varphi_{\xi\eta}^2=0$. Thus, we have $$\varphi_{\xi\xi}=-1, \quad \varphi_{\xi\eta}=0, \quad
\varphi_{\eta\eta}=-1,$$ which yields $$\varphi(\xi,\eta) =-\frac{1}{2}(\xi^2+\eta^2)
+a\xi +b\eta +c,
\label{1.1.11}$$ where $a, b$, and $c$ are constants.
Consider state $(0)$: $(\rho_0,u_0,v_0)=(\rho_0,0,0)$ with $\rho_0>0$ and state $(1)$: $(\rho_1,u_1,v_1)$ $=(\rho_1,u_1,0)$ with $\rho_1>\rho_0>0$ and $u_1>0$. The plane incident shock solution with state (0) and state (1) corresponds to a continuous weak solution $\varphi$ of (\[1.1.5\]) in the self-similar coordinates $(\xi,\eta)$ with form and for state (0) and state (1) respectively, where $\xi=\xi_0>0$ is the location of the incident shock.
The unit normal to the shock line is $\nu=(1,0)$. Using , we have $$u_1=\frac{\rho_1-\rho_0}{\rho_1}\xi_0>0.$$ Then implies $$\rho_1^{\gamma-1}-\rho_0^{\gamma-1}
=-\frac{1}{2}|D\varphi_1|^2-\varphi_1
=\frac{1}{2}\frac{\rho_1^2-\rho_0^2}{\rho_1^2}\xi_0^2.$$ Therefore, we have $$u_1=(\rho_1-\rho_0)
\sqrt{\frac{2(\rho_1^{\gamma-1}-\rho_0^{\gamma-1})}{\rho_1^2-\rho_0^2}},
\label{1.2.4}$$ and the location of the incident shock in the self-similar coordinates is $\xi=\xi_0>u_1$ determined by .
The von Neumann Criterion and Local Theory for Shock Reflection {#section:3}
===============================================================
In this section, we first discuss the normal reflection solution. Then we follow the von Neumann criterion to derive the necessary condition for the existence of regular reflection and show that the shock reflection can be regular locally when the wedge angle is large, that is, when $\theta_w$ is close to $\pi/2$ and, equivalently, the angle between the incident shock and the wedge $$\label{angleCloseToPiOver2}
\sigma:=\pi/2-\theta_w$$ tends to zero.
\[section:4\] In this case, the wedge angle is $\pi/2$, i.e., $\sigma=0$, and the incident shock normally reflects (see Fig. 4). The reflected shock is also a plane at $\xi=\bar{\xi}<0$, which will be defined below. Then $\bar{u}_2=\bar{v}_2=0$, state (1) has form , and state (2) has the form: $$\begin{aligned}
\varphi_2(\xi,\eta)=-\frac{1}{2}(\xi^2+\eta^2)+u_1(\bar\xi-\xi_0)\qquad\mbox{for}\,\,
\xi\in (\bar{\xi},0), \label{phi-2-a}\end{aligned}$$ where $\xi_0=\rho_1 u_1/(\rho_1-\rho_0)>0$ may be regarded to be the position of the incident shock.
![Normal reflection[]{data-label="fig:NF"}](fig04.eps){height="1.8in" width="3.0in"}
At the reflected shock $\xi=\bar{\xi}<0$, the Rankine-Hugoniot condition implies $$\label{3.14}
\bar{\xi}=-\frac{\rho_1 u_1}{\bar{\rho}_2-\rho_1}<0.$$
We use the Bernoulli law (\[1.1.4\]): $$\rho_0^{\gamma-1}=\rho_1^{\gamma-1}+\frac{1}{2}u_1^2-u_1 \xi_0
=\bar{\rho}_2^{\gamma-1}+u_1(\bar{\xi}-\xi_0)$$ to obtain $$\label{3.15}
\bar{\rho}_2^{\gamma-1}=\rho_1^{\gamma-1}+\frac{1}{2}u_1^2
+\frac{\rho_1u_1^2}{\bar{\rho}_2-\rho_1}.$$
It can be shown that there is a unique solution $\bar{\rho}_2$ of such that $$\bar{\rho}_2>\rho_1.$$ Indeed, for fixed $\gamma>1$ and $\rho_1, u_1>0$ and for $F(\bar{\rho}_2)$ that is the right-hand side of (\[3.15\]), we have $$\begin{aligned}
&&\lim_{s\to\infty}F(s)=\rho_1^{\gamma-1}+\frac{1}{2}u_1^2>\rho_1^{\gamma-1},
\quad \lim_{s\to\rho_1+}F(s)=\infty,\\
&&F'(s)=-\frac{\rho_1u_1^2}{(s-\rho_1)^2}<0 \qquad\, \mbox{ for}
\,\, s>\rho_1.\end{aligned}$$ Thus there exists a unique $\bar{\rho}_2\in(\rho_1, \infty)$ satisfying $\bar{\rho}_2^{\gamma-1}=F(\bar{\rho}_2)$, i.e., . Then the position of the reflected shock $\xi=\bar{\xi}<0$ is uniquely determined by .
Moreover, for the sonic speed $\bar{c}_2=\sqrt{(\gamma-1)\bar{\rho}_2^{\gamma-1}}$ of state (2), we have $$\label{sonic-intersect-shock-normal}
|\bar{\xi}|< \bar{c}_2.$$ This can be seen as follows. First note that $$\label{meanValTh}
\bar{\rho}_2^{\gamma-1}-\rho_1^{\gamma-1}=\beta(\bar{\rho}_2-\rho_1),$$ where $\beta=(\gamma-1)\rho_*^{\gamma-2}>0$ for some $\rho_*\in
(\rho_1,\bar{\rho}_2)$. We consider two cases, respectively.
[**Case 1.**]{} $\gamma\ge 2$. Then $$\label{meanValTh-A-1}
0<(\gamma-1)\rho_1^{\gamma-2}\le \beta\le
(\gamma-1)\bar{\rho}_2^{\gamma-2}.$$ Since $\beta>0$ and $\bar{\rho}_2>\rho_1$, we use and to find $$\bar{\rho}_2=\rho_1+\frac{u_1}{4\beta}\big(u_1+\sqrt{u_1^2+ 16\beta
\rho_1}\big),$$ and hence $$\label{meanValTh-A-xi}
\bar{\xi}=-\frac{4\beta \rho_1}{u_1+\sqrt{u_1^2+16\beta \rho_1}}.$$ Then using –(\[meanValTh-A-xi\]), $\bar{\rho}_2>\rho_1>0$, and $u_1>0$ yields $$|\bar{\xi}|=\frac{4\beta \rho_1}{u_1+\sqrt{u_1^2+16\beta \rho_1}}
<\sqrt{\beta \rho_1}\le
\sqrt{(\gamma-1)\bar{\rho}_2^{\gamma-2}\bar{\rho}_2}=\bar{c}_2.$$
[**Case 2.**]{} $1<\gamma<2$. Then, since $\bar{\rho}_2>\rho_1>0$, $$\label{meanValTh-A-2}
0<(\gamma-1)\bar{\rho}_2^{\gamma-2}\le \beta\le
(\gamma-1)\rho_1^{\gamma-2}.$$ Since $\beta>0$, (\[meanValTh-A-xi\]) holds by the calculation as in Case 1. Now we use (\[meanValTh-A-xi\])–, $\bar{\rho}_2>\rho_1>0$, $u_1>0$, and $1<\gamma<2$ to find again $$|\bar{\xi}|<\sqrt{\beta \rho_1}
\le \sqrt{(\gamma-1)\rho_1^{\gamma-1}}\le
\sqrt{(\gamma-1)\bar{\rho}_2^{\gamma-1}} =\bar{c}_2.$$ This shows that (\[sonic-intersect-shock-normal\]) holds in general.
\[section:3.3\] In this subsection, we first follow the von Neumann criterion to derive the necessary condition for the existence of regular reflection and show that, when the wedge angle is large, there exists a unique state (2) with two-shock structure at the reflected point, which is close to the solution $(\bar{\rho}_2,\bar{u}_2,\bar{v}_2)=(\bar{\rho}_2,0,0)$ of normal reflection for which $\theta_w=\pi/2$ in §3.1.
For a possible two-shock configuration satisfying the corresponding boundary condition on the wedge $\eta=\xi\tan\theta_w$, the three state functions $\varphi_j, j=0,1,2$, must be of form , , and (cf. ).
Set the reflected point $P_0=(\xi_0, \xi_0 \tan\theta_w)$ and assume that the line that coincides with the reflected shock in state (2) will intersect with the axis $\eta=0$ at the point $(\tilde{\xi},
0)$ with the angle $\theta_s$ between the line and $\eta=0$.
Note that $\varphi_1(\xi,\eta)$ is defined by (\[flatOrthSelfSimShock2\]). The continuity of $\varphi$ at $(\tilde{\xi}, 0)$ yields $$\label{state2}
\varphi_2(\xi,\eta) =-\frac{1}{2}(\xi^2+\eta^2)+u_2\xi+v_2\eta
+\big(u_1(\tilde\xi-\xi_0)-u_2\tilde{\xi}\big).$$ Furthermore, $\varphi_2$ must satisfy the slip boundary condition at $P_0$: $$\label{3.11a}
v_2=u_2 \tan\theta_w.$$
Also we have $$\label{state2.5}
\tilde{\xi}=\xi_0-\xi_0\frac{\tan \theta_w}{\tan \theta_s}.$$ The Bernoulli law (\[1.1.4\]) becomes $$\label{state2.2}
\rho_0^{\gamma-1}=\rho_2^{\gamma-1}+\frac{1}{2}(u_2^2+v_2^2)
+(u_1-u_2)\tilde{\xi}-u_1\xi_0.$$ Moreover, the continuity of $\varphi$ on the shock implies that $D(\varphi_2-\varphi_1)$ is orthogonal to the tangent direction of the reflected shock: $$\label{thetaS-uv}
(u_2-u_1, v_2)\cdot (\cos\theta_s, \sin\theta_s)=0,$$ that is, $$\label{state2.3}
u_2=u_1\frac{\cos\theta_w\cos\theta_s}{\cos(\theta_w-\theta_s)}.$$ The Rankine-Hugoniot condition (\[FBConditionSelfSim-0\]) along the reflected shock is $$[\rho\,D\varphi]\cdot (\sin\theta_s, -\cos\theta_s)=0,$$ that is, $$\label{state2.4}
\rho_1(u_1-\tilde\xi)\sin\theta_s
=\rho_2\big(u_2\frac{\sin(\theta_s-\theta_w)}{\cos\theta_w}-\tilde\xi\sin\theta_s\big).$$
Combining –, we obtain the following system for $(\rho_2, \theta_s, \tilde{\xi})$: $$\begin{aligned}
&&(\tilde{\xi}-\xi_0)\cos\theta_w
+\xi_0\sin\theta_w\cot\theta_s=0, \label{suf:1}\\
&&\rho_2^{\gamma-1}+\frac{u_1^2\cos^2\theta_s}{2\cos^2(\theta_w-\theta_s)}
+\frac{u_1\sin\theta_w\sin\theta_s}{\cos(\theta_w-\theta_s)}\tilde{\xi}
-u_1\xi_0-\rho_0^{\gamma-1}=0, \label{suf:2}\\
&&\big(u_1\cos\theta_s\tan (\theta_s-\theta_w)
-\tilde{\xi}\sin\theta_s\big)\rho_2
-\rho_1(u_1-\tilde{\xi})\sin\theta_s =0. \label{suf:3}\end{aligned}$$ The condition for solvability of this system is the necessary condition for the existence of regular shock reflection.
Now we compute the Jacobian $J$ in terms of $(\rho_2,\theta_s,
\tilde{\xi})$ at the normal reflection solution state $(\bar{\rho}_2,\frac{\pi}{2}, \bar{\xi})$ in §3.1 for state $(2)$ when $\theta_w=\pi/2$ to obtain $$J=-\xi_0\big((\gamma-1)\bar{\rho}_2^{\gamma-2}(\bar{\rho}_2-\rho_1)
-u_1 \bar{\xi}\big)< 0,$$ since $\bar{\rho}_2>\rho_1$ and $\bar{\xi}<0$. Then, by the Implicit Function Theorem, when $\theta_w$ is near $\pi/2$, there exists a unique solution $(\rho_2,\theta_s,
\tilde{\xi})$ close to $(\bar{\rho}_2,\frac{\pi}{2}, \bar{\xi})$ of system –. Moreover, $(\rho_2,\theta_s,
\tilde{\xi})$ are smooth functions of $\sigma=\pi/2-\theta_w\in
(0,\sigma_1)$ for ${\sigma}_1>0$ depending only on $\rho_0, \rho_1$, and $\gamma$. In particular, $$\begin{aligned}
\label{theta_s-close-w} |\rho_2-\bar{\rho}_2|+|\pi/2-\theta_s|+
|\tilde{\xi}-\bar{\xi}|+|c_2-\bar{c}_2| \le C\sigma,\end{aligned}$$ where $\displaystyle c_2=\sqrt{(\gamma-1)\rho_2^{\gamma-1}}$ is the sonic speed of state (2).
Reducing ${\sigma}_1>0$ if necessary, we find that, for any $\sigma\in
(0,\sigma_1)$, $$\label{xi-1-negative}
\tilde{\xi}<0$$ from (\[3.14\]) and (\[theta\_s-close-w\]). Since $\theta_w\in(\pi/2-{\sigma}_1, \pi/2)$, then $\theta_s\in(\pi/4,
3\pi/4)$ if ${\sigma}_1$ is small, which implies $\sin\theta_s>0$. We conclude from (\[suf:1\]), (\[xi-1-negative\]), and $\xi_0>0$ that $ \tan\theta_w>\tan\theta_s>0$. Thus, $$\label{theta_s<w}
\pi/4<\theta_s<\theta_w<\pi/2.$$
Now, given $\theta_w$, we define $\varphi_2$ as follows: We have shown that there exists a unique solution $(\rho_2,\theta_s,
\tilde{\xi})$ close to $(\bar{\rho}_2,\frac{\pi}{2}, \bar{\xi})$ of system –. Define $u_2$ by (\[state2.3\]), $v_2$ by (\[3.11a\]), and $\varphi_2$ by (\[state2\]). Then the shock connecting state (1) with state (2) is the straight line $S_{12}=\{({{\xi}},{{\eta}})\;:\,
\varphi_1({{\xi}},{{\eta}})=\varphi_2({{\xi}},{{\eta}})\}$, which is $\xi={{\eta}}\cot\theta_s +\tilde{\xi}$ by (\[flatOrthSelfSimShock2\]), (\[state2\]), and (\[state2.3\]). Now (\[suf:3\]) implies that the Rankine-Hugoniot condition (\[FBConditionSelfSim-0\]) holds on $S_{12}$. Moreover, (\[3.11a\]) and (\[state2.3\]) imply (\[thetaS-uv\]). Thus the solution $(\theta_s, \rho_2, u_2, v_2)$ satisfies (\[3.11a\])–(\[suf:3\]). Furthermore, (\[suf:1\]) implies that the point $P_0$ lies on $S_{12}$, and (\[suf:2\]) implies (\[state2.2\]) that is the Bernoulli law: $$\label{bernLawState2}
\rho_2^{\gamma-1}+{1\over 2}|{{D}}\varphi_2|^2+\varphi_2=
\rho_0^{\gamma-1}.$$ Thus we have established the local existence of the two-shock configuration near the reflected point so that, behind the straight reflected shock emanating from the reflection point, state (2) is pseudo-supersonic up to the sonic circle of state (2). Furthermore, this local structure is stable in the limit $\theta_w\to\pi/2$, i.e., $\sigma\to 0$.
We also notice from (\[3.11a\]) and (\[state2.3\]) with the use of (\[theta\_s-close-w\]) and (\[theta\_s<w\]) that $$\label{u2-v2-bound}
|u_2|+ |v_2|\le C \sigma.$$ Furthermore, from (\[sonic-intersect-shock-normal\]) and the continuity of $\rho_2$ and $\tilde{\xi}$ with respect to $\theta_w$ on $(\pi/2-{\sigma}_1, \pi/2]$, it follows that, if $\sigma>0$ is small, $$\label{sonic-intersect-shock}
|\tilde{\xi}|< c_2.$$
In §\[reformulProbSection\]–§\[proofSection\], we prove that this local theory for the existence of two shock configuration can be extended to a global theory for regular shock reflection.
Reformulation of the Shock Reflection Problem {#reformulProbSection}
=============================================
We first assume that $\varphi$ is a solution of the shock reflection problem in the elliptic domain ${\Omega}$ in Fig. \[fig:RegularReflection\] and that $\varphi-\varphi_2$ is small in $C^1(\overline{\Omega})$. Under such assumptions, we rewrite the equation and boundary conditions for solutions of the shock reflection problem in the elliptic region.
\[shiftCoordSection\] It is more convenient to change the coordinates in the self-similar plane by shifting the origin to the center of sonic circle of state (2). Thus we define $$({{\xi}},{{\eta}})_{new}:=({{\xi}}, {{\eta}})-(u_2,v_2).$$ For simplicity of notations, throughout this paper below, we will always work in the new coordinates without changing the notation $({{\xi}}, {{\eta}})$, and we will not emphasize this again later.
In the new shifted coordinates, the domain ${\Omega}$ is expressed as $$\label{ellipticDomain}
{\Omega}=B_{c_2}(0)\cap\{{{\eta}}>-v_2\}\cap \{f({{\eta}})<{{\xi}}<
{{\eta}}\cot\theta_w\},$$ where $f$ is the position function of the free boundary, i.e., the curved part of the reflected shock ${\Gamma_{shock}}:=\{{{\xi}}=f({{\eta}})\}$. The function $f$ in will be determined below so that $$\label{FBfunct-estimate}
\|f-l\|\le C \sigma$$ in an appropriate norm, specified later. Here $\xi=l({{\eta}})$ is the location of the reflected shock of state (2) which is a straight line, that is, $$\label{reflected-shock-s2}
l({{\eta}})={{\eta}}\cot\theta_s +\hat{{\xi}}$$ and $$\label{x1-in-shifed}
\hat{{\xi}}=\tilde{{\xi}}- u_2 +v_2\cot\theta_s<0,$$ if $\sigma=\pi/2-\theta_w>0$ is sufficiently small, since $u_2$ and $v_2$ are small and $\tilde{{\xi}}<0$ by (\[3.14\]) in this case. Also note that, since $u_2 =v_2\cot\theta_w>0$, it follows from (\[theta\_s<w\]) that $$\label{x1-in-shifed-2}
\hat{{\xi}}>\tilde{{\xi}}.$$
Another condition on $f$ comes from the fact that the curved part and straight part of the reflected shock should match at least up to first-order. Denote by $P_1=({{\xi}}_1,{{\eta}}_1)$ with ${{\eta}}_1>0$ the intersection point of the line ${{\xi}}=l({{\eta}})$ and the sonic circle ${{\xi}}^2+{{\eta}}^2=c_2^2$, i.e., $({{\xi}}_1,{{\eta}}_1)$ is the unique point for small $\sigma>0$ satisfying $$\label{coord-P4}
l({{\eta}}_1)^2+{{\eta}}_1^2=c_2^2,\qquad {{\xi}}_1=l({{\eta}}_1), \qquad {{\eta}}_1>0.$$ The existence and uniqueness of such point $({{\xi}}_1,{{\eta}}_1)$ follows from $-c_2<\tilde{\xi}<0$, which holds from (\[theta\_s<w\]), (\[sonic-intersect-shock\]), (\[x1-in-shifed\]), and the smallness of $u_2$ and $v_2$. Then $f$ satisfies $$\label{curved-straight-shock-match}
f({{\eta}}_1)=l({{\eta}}_1), \qquad f'({{\eta}}_1)=l'({{\eta}}_1)=\cot\theta_s.$$ Note also that, for small $\sigma>0$, we obtain from (\[sonic-intersect-shock\]), (\[x1-in-shifed\])–(\[x1-in-shifed-2\]), and $l'(\eta)=\cot\theta_s>0$ that $$\label{inSonicRegion-in-shifed}
-c_2<\tilde{{\xi}}<\hat{{\xi}}< {{\xi}}_1<0, \qquad c_2-|\tilde{\xi}|\ge
\frac{\bar{c}_2-|\bar{\xi}|}{2}>0.$$
Furthermore, equations (\[1.1.5\])–(\[1.1.6\]) and the Rankine-Hugoniot conditions (\[FBConditionSelfSim-0\]) and (\[1.1.14\]) on ${\Gamma_{shock}}$ do not change under the shift of coordinates. That is, we seek $\varphi$ satisfying (\[1.1.5\])–(\[1.1.6\]) in ${\Omega}$ so that the equation is elliptic on $\varphi$ and satisfying the following boundary conditions on ${\Gamma_{shock}}$: The continuity of the pseudo-potential function across the shock: $$\varphi=\varphi_1\qquad\mbox{on }\;{\Gamma_{shock}}\label{cont-accross-shock-mod-phi}$$ and the gradient jump condition: $$\rho(|{{D}}\varphi|^2,\varphi){{D}}\varphi\cdot\nu_s= \rho_1
{{D}}\varphi_1\cdot\nu_s \qquad\mbox{on }\;{\Gamma_{shock}},
\label{RH-mod-phi}$$ where $\nu_s$ is the interior unit normal to $\Omega$ on ${\Gamma_{shock}}$.
The boundary conditions on the other parts of $\partial{\Omega}$ are $$\begin{aligned}
&&\varphi=\varphi_2\qquad\mbox{on }\;{\Gamma_{sonic}}=\partial {\Omega}\cap
\partial B_{c_2}(0),
\label{condOnSonicLinePhi}
\\
&& \varphi_\nu=0\qquad\mbox{on }\;{\Gamma_{wedge}}=\partial {\Omega}\cap
\{{{\eta}}={{\xi}}\tan\theta_w\}, \label{condOnWedgePhi}
\\
&&\varphi_\nu=0\qquad\mbox{on }\;\partial {\Omega}\cap \{{{\eta}}=-v_2\}.
\label{condOnSymmtryLinePhi}\end{aligned}$$
Rewriting the background solutions in the shifted coordinates, we find $$\begin{aligned}
&&\qquad
\varphi_0({{\xi}},{{\eta}})=-\frac{1}{2}(\xi^2+\eta^2)-(u_2{{\xi}}+v_2{{\eta}})
-{1\over 2}{q_2}^2,
\label{phi-0-shifted} \\
&&\qquad \varphi_1({{\xi}},{{\eta}})=-\frac{1}{2}(\xi^2+\eta^2)
+(u_1-u_2){{\xi}}-v_2{{\eta}}-{1\over 2}{q_2}^2+u_1(u_2-{{\xi}}_0),
\label{phi-1-shifted} \\
&&\qquad \varphi_2({{\xi}},{{\eta}})=-\frac{1}{2}(\xi^2+\eta^2) -{1\over
2}{q_2}^2 +(u_1-u_2)\hat{\xi}+u_1(u_2-\xi_0),
\label{phi-2-shifted}\end{aligned}$$ where $q_2^2=u_2^2+v_2^2$.
Furthermore, substituting $\tilde{{\xi}}$ in (\[x1-in-shifed\]) into equation (\[suf:1\]) and using (\[3.11a\]) and (\[thetaS-uv\]), we find $$\rho_2\hat{{\xi}}=\rho_1\big(\hat{{\xi}}-{(u_1-u_2)^2+v_2^2\over
u_1-u_2}\big), \label{RH-states-1-2'}$$ which expresses the Rankine-Hugoniot conditions on the reflected shock of state (2) in terms of $\hat{{\xi}}$. We use this equality below.
![Regular reflection in the new coordinates[]{data-label="fig:RR"}](fig05.eps){height="2.9in" width="2.8in"}
\[equationForPsiSection\] It is convenient to study the problem in terms of the difference between our solution $\varphi$ and the function $\varphi_2$ that is a solution for state (2) given by . Thus we introduce a function $$\label{psi-definition}
\psi=\varphi-\varphi_2\qquad\mbox{in }\;{\Omega}.$$ Then it follows from (\[1.1.5\])–(\[c-through-density-function\]), (\[bernLawState2\]), and by explicit calculation that $\psi$ satisfies the following equation in ${\Omega}$: $$\begin{aligned}
\label{potent-flow-nondiv-psi-1}
&&\quad
\big(c^2(D\psi,\psi,\xi,\eta)-(\psi_{{\xi}}-{{\xi}})^2\big)\psi_{{{\xi}}{{\xi}}}
+\big(c^2(D\psi,\psi,\xi,\eta)-(\psi_{{\eta}}-{{\eta}})^2\big)\psi_{{{\eta}}{{\eta}}}
\\
&&\qquad\quad -2(\psi_{{\xi}}-{{\xi}})(\psi_{{\eta}}-{{\eta}})\psi_{{{\xi}}{{\eta}}}
=0,\nonumber\end{aligned}$$ and the expressions of the density and sound speed in ${\Omega}$ in terms of $\psi$ are $$\begin{aligned}
&&\rho({{D}}\psi,\psi,\xi,\eta) =\Big(\rho_2^{\gamma-1}
+{{\xi}}\psi_{{\xi}}+{{\eta}}\psi_{{\eta}}-\frac{1}{2}|{{D}}\psi|^2-\psi
\Big)^\frac{1}{\gamma-1},
\label{density-psi} \\
&&c^2({{D}}\psi, \psi,\xi,\eta) =c_2^2
+({\gamma-1})\Big({{\xi}}\psi_{{\xi}}+{{\eta}}\psi_{{\eta}}-\frac{1}{2}|{{D}}\psi|^2
-\psi \Big).
\label{speedOfsound-psi}\end{aligned}$$ where $\rho_2$ is the density of state (2). In the polar coordinates $(r, \theta)$ with $r=\sqrt{\xi^2+\eta^2}$, $\psi$ satisfies $$\label{potent-flow-nondiv-psi-Polar}
\big(c^2-(\psi_r-r)^2\big)\psi_{rr}
-\frac{2}{r^2}(\psi_r-r)\psi_\theta\psi_{r\theta}
+\frac{1}{r^2}(c^2-\frac{1}{r^2}\psi_\theta^2)\psi_{\theta\theta}
+\frac{c^2}{r}\psi_r +\frac{1}{r^3}(\psi_r-2r)\psi_\theta^2 =0$$ with $$\label{speedSound-expression-psi-Polar-1}
c^2=(\gamma-1)\Big(\rho_2^{\gamma-1}-\psi
+r\psi_r-\frac{1}{2}\big(\psi_r^2+\frac{1}{r^2}\psi_\theta^2\big)\Big).$$ Also, from (\[condOnSonicLinePhi\])–(\[condOnWedgePhi\]) and (\[phi-2-shifted\])–(\[psi-definition\]), we obtain $$\begin{aligned}
&&\psi=0\qquad\mbox{on }\;{\Gamma_{sonic}}=\partial {\Omega}\cap
\partial B_{c_2}(0),
\label{condOnSonicLine-Psi}
\\
&&\psi_\nu=0\qquad\mbox{on }\;{\Gamma_{wedge}}=\partial {\Omega}\cap
\{{{\eta}}={{\xi}}\tan\theta_w\}, \label{condOnWedge-Psi}
\\
&&\psi_{{\eta}}=-v_2\qquad\mbox{on }\;\partial{\Omega}\cap\{{{\eta}}=-v_2\}.
\label{condOnSymmtryLine-Psi}\end{aligned}$$
Using (\[phi-1-shifted\])–(\[phi-2-shifted\]), the Rankine-Hugoniot conditions in terms of $\psi$ take the following form: The continuity of the pseudo-potential function across (\[cont-accross-shock-mod-phi\]) is written as $$\psi-{1\over 2}{q_2}^2+\hat{{{\xi}}}(u_1-u_2)+u_1(u_2-\xi_0)
={{\xi}}(u_1-u_2)-{{\eta}}v_2 -{1\over 2}{q_2}^2+u_1(u_2-\xi_0)
\quad\mbox{on }\;{\Gamma_{shock}}, \label{cont-accross-shock-psi}$$ that is, $${{\xi}}=\frac{\psi({{\xi}},{{\eta}})+v_2{{\eta}}}{u_1-u_2}+\hat{{\xi}},
\label{cont-accross-shock-psi-resolved}$$ where $\hat{{\xi}}$ is defined by (\[x1-in-shifed\]); and the gradient jump condition (\[RH-mod-phi\]) is $$\rho({{D}}\psi,\psi)\left({{D}}\psi-
\left({{\xi}},{{\eta}}\right)\right)\cdot\nu_s =\rho_1 \left(u_1-u_2-{{\xi}},
-v_2-{{\eta}}\right)\cdot\nu_s \qquad\mbox{on }\;{\Gamma_{shock}}, \label{RH-psi}$$ where $\rho({{D}}\psi,\psi)$ is defined by (\[density-psi\]) and $\nu_s$ is the interior unit normal to ${\Omega}$ on ${\Gamma_{shock}}$. If $|(u_2, v_2, {{D}}\psi)|<u_1/50$, the unit normal $\nu_s$ can be expressed as $$\nu_s=\frac{{{D}}(\varphi_1-\varphi)}{|{{D}}(\varphi_1-\varphi)|} =
\frac{(u_1-u_2-\psi_{{\xi}},-v_2-\psi_{{\eta}})}
{\sqrt{(u_1-u_2-\psi_{{\xi}})^2+(v_2+\psi_{{\eta}})^2}},
\label{norm-to-Shock}$$ where we have used (\[phi-1-shifted\])–(\[phi-2-shifted\]) and (\[psi-definition\]) to obtain the last expression.
Now we rewrite the jump condition (\[RH-psi\]) in a more convenient form for $\psi$ satisfying (\[cont-accross-shock-mod-phi\]) when $\sigma>0$ and $\|\psi\|_{C^1(\bar{{\Omega}})}$ are sufficiently small.
We first discuss the smallness assumptions for $\sigma>0$ and $\|\psi\|_{C^1(\bar{{\Omega}})}$. By (\[1.2.4\]), (\[theta\_s-close-w\]), and (\[u2-v2-bound\]), it follows that, if $\sigma$ is small depending only on the data, then $$\label{condRewritingRH-0}
\frac{5 \bar c_2}{6}\le c_2\le \frac{6\bar c_2}{5}, \quad
\frac{5\bar \rho_2}{6}\le \rho_2\le \frac{6\bar\rho_2}{5}, \quad
\sqrt{u_2^2+v_2^2}\le {u_1\over 50}.$$ We also require that $\|\psi\|_{C^1(\bar{{\Omega}})}$ is sufficiently small so that, if (\[condRewritingRH-0\]) holds, the expressions (\[density-psi\]) and (\[norm-to-Shock\]) are well-defined in ${\Omega}$, and ${{\xi}}$ defined by the right-hand side of (\[cont-accross-shock-psi-resolved\]) satisfies $|{{\xi}}|\le 7\bar
c_2/5$ for ${{\eta}}\in (-v_2, c_2)$, which is the range of ${{\eta}}$ on ${\Gamma_{shock}}$. Since (\[condRewritingRH-0\]) holds and ${\Omega}\subset
B_{c_2}(0)$ by (\[ellipticDomain\]), it suffices to assume $$\label{condRewritingRH}
\|\psi\|_{C^1(\bar{{\Omega}})}\le \min\big(
{\bar\rho_2^{\gamma-1}\over 50(1+4\bar c_2)}, \min(1,\bar
c_2){u_1\over 50} \big)=:\delta^*.$$
For the rest of this section, we assume that (\[condRewritingRH-0\]) and (\[condRewritingRH\]) hold.
Under these conditions, we can substitute the right-hand side of (\[norm-to-Shock\]) for $\nu_s$ into (\[RH-psi\]). Thus, we rewrite (\[RH-psi\]) as $$\label{RH-psi-int1}
F({{D}}\psi, \psi, u_2, v_2, {{\xi}}, {{\eta}})=0 \qquad\mbox{on}\;{\Gamma_{shock}},$$ where, denoting $p=(p_1,p_2)\in{ {\bf R}}^2$ and $z\in{ {\bf R}}$, $$\label{RH-psi-func1}
F(p, z, u_2, v_2, {{\xi}}, {{\eta}})=\big(\tilde\rho\left(p-
\left({{\xi}},{{\eta}}\right)\right)-\rho_1 \left(u_1-u_2-{{\xi}},
-v_2-{{\eta}}\right)\big)\cdot\hat\nu$$ with $\tilde\rho:=\tilde\rho(p, z, {{\xi}}, {{\eta}})$ and $\hat\nu:=\hat\nu(p, u_2, v_2)$ defined by $$\begin{aligned}
&&\tilde\rho(p, z, {{\xi}}, {{\eta}}) =\left(\rho_2^{\gamma-1} +{{\xi}}p_1+{{\eta}}p_2-\frac{|p|^2}{2}-z \right)^{\frac{1}{\gamma-1}},
\label{RH-psi-func2}
\\
&&\hat\nu(p, u_2, v_2)=\frac{(u_1-u_2-p_1,-v_2-p_2)}
{\sqrt{(u_1-u_2-p_1)^2+(v_2+p_2)^2}}. \label{RH-psi-func3}\end{aligned}$$ From the explicit definitions of $\tilde\rho$ and $\hat\nu$, it follows from (\[condRewritingRH-0\]) that $$\tilde\rho\in C^\infty(\overline{B_{\delta^*}(0)\times (-\delta^*,
\delta^*)\times B_{2\bar c_2}(0)}), \quad \hat\nu\in
C^\infty(\overline{B_{\delta^*}(0)\times B_{u_1/50}(0)}),$$ where $B_{R}(0)$ denotes the ball in ${ {\bf R}}^2$ with center $0$ and radius $R$ and, for $k\in {\bf N}$ (the set of nonnegative integers), the $C^k$–norms of $\tilde\rho$ and $\hat\nu$ over the regions specified above are bounded by the constants depending only on $\gamma, u_1, \bar\rho_2, \bar c_2$, and $k$, that is, by §\[section:3\], the $C^k$–norms depend only on the data and $k$. Thus, $$\label{rewriteRH-reg-1}
F\in C^\infty(\overline{ B_{\delta^*}(0)\times (-\delta^*, \delta^*)
\times B_{u_1/50}(0)\times B_{2\bar c_2}(0)}),$$ with its $C^k$–norm depending only on the data and $k$.
Furthermore, since $\psi$ satisfies (\[cont-accross-shock-mod-phi\]) and hence (\[cont-accross-shock-psi-resolved\]), we can substitute the right-hand side of (\[cont-accross-shock-psi-resolved\]) for ${{\xi}}$ into (\[RH-psi-int1\]). Thus we rewrite (\[RH-psi\]) as $$\label{RH-psi-int2}
\Psi({{D}}\psi, \psi, u_2, v_2, {{\eta}})=0 \qquad\mbox{on}\;{\Gamma_{shock}},$$ where $$\label{RH-psi-func4}
\Psi(p, z, u_2, v_2, {{\eta}})=F(p, z, u_2, v_2,
(z+v_2{{\eta}})/(u_1-u_2)+\hat{{\xi}}, {{\eta}}).$$ If $\eta\in (-6\bar c_2/5, 6\bar c_2/5)$ and $|z|\le \delta^*$, then, from (\[inSonicRegion-in-shifed\]) and (\[condRewritingRH-0\])–(\[condRewritingRH\]), it follows that $\big|(z+v_2{{\eta}})/(u_1-u_2)+\hat{{\xi}}\big|\le 7\bar c_2/5$. That is, $((z+v_2{{\eta}})/(u_1-u_2)+\hat{{\xi}},\,\eta)\in B_{2\bar c_2}(0)$ if $\eta\in (-6\bar c_2/5, 6\bar c_2/5)$ and $|z|\le \delta^*$. Thus, from (\[rewriteRH-reg-1\]) and (\[RH-psi-func4\]), $\Psi\in
C^\infty(\overline{\mathcal A})$ with $\|\Psi\|_{C^k(\overline{\mathcal A})}$ depending only on the data and $k\in{\bf N}$, where ${\mathcal A}=B_{\delta^*}(0)\times
(-\delta^*, \delta^*)\times B_{u_1/50}(0)\times (-6\bar c_2/5, 6\bar
c_2/5)$.
Using the explicit expression of $\Psi$ given by (\[RH-psi-func1\])–(\[RH-psi-func3\]) and (\[RH-psi-func4\]), we calculate $$\begin{aligned}
&&\Psi((0,0),0, u_2,v_2,{{\eta}})\\
&& =- {(u_1-u_2)\rho_2\hat{{\xi}}\over\sqrt{(u_1-u_2)^2+v_2^2}}
-\rho_1\big(
\sqrt{(u_1-u_2)^2+v_2^2}-{(u_1-u_2)\hat{{\xi}}\over\sqrt{(u_1-u_2)^2+v_2^2}}
\big).\end{aligned}$$ Now, using (\[RH-states-1-2’\]), we have $$\Psi((0,0), 0,u_2,v_2,{{\eta}}) =0 \qquad\text{for any } (u_2,v_2,{{\eta}})\in
B_{u_1/50}(0)\times (-6\bar c_2/5, 6\bar c_2/5).$$ Then, denoting $p_0=z$ and ${\mathcal X}=((p_1,p_2), p_0, u_2,v_2,{{\eta}})\in
{\mathcal A}$, we have $$\label{Psi-Function-1} \Psi({\mathcal X})=\sum_{i=0}^2p_i D_{p_i}\Psi((0,0),
0, u_2,v_2,{{\eta}}) +\sum_{i,j=0}^2 p_ip_j g_{ij}({\mathcal X}),$$ where $ g_{ij}({\mathcal X})=\int_0^1(1-t)D^2_{p_i p_j}\Psi((tp_1,tp_2),
tp_0, u_2,v_2,{{\eta}})dt$ for $i,j=0,1,2$. Thus, $g_{ij}\in
C^\infty(\overline{\mathcal A})$ and $\|g_{ij}\|_{C^k(\overline{\mathcal
A})}\le\|\Psi\|_{C^{k+2}(\overline{\mathcal A})}$ depending only on the data and $k\in {\bf N}$.
Next, denoting $\rho_2'{:=}\hat\rho'(\rho_2^{\gamma-1})=\rho_2/c_2^2>0,$ we compute from the explicit expression of $\Psi$ given by (\[RH-psi-func1\])–(\[RH-psi-func3\]) and (\[RH-psi-func4\]): $$\begin{aligned}
D_{(p,z)}\Psi((0,0), 0, 0,0,{{\eta}})= \big(\rho_2'(c_2^2-{\hat{{\xi}}}^2),\;
\big(\frac{\rho_2-\rho_1}{u_1}-\rho_2'\hat{{\xi}}\big){{\eta}},\,\;
\rho_2'\hat{{\xi}}-\frac{\rho_2-\rho_1}{u_1}\big).\end{aligned}$$ Note that, for $i=0,1,2$, $$\partial_{p_i} \Psi((0,0), 0, u_2, v_2,{{\eta}})=\partial_{p_i} \Psi((0,0),0,0,0,{{\eta}})
+h_i(u_2, v_2,{{\eta}})$$ with $\|h_i\|_{C^k(\overline{B_{u_1/50}(0)\times (-6\bar c_2/5,
6\bar c_2/5)})} \le\|\Psi\|_{C^{k+2}(\overline{\mathcal A})}$ for $k\in {\bf N}$, and $|h_i(u_2, v_2,{{\eta}})|\le C(|u_2|+|v_2|)$ with $C=\|D^2\Psi\|_{C(\overline{\mathcal A})}$. Then we obtain from (\[Psi-Function-1\]) that, for all ${\mathcal X}=(p, z, u_2,
v_2,{{\eta}})\in{\mathcal A}$, $$\label{Psi-Function-2} \Psi({\mathcal X})=\rho_2'(c_2^2-\hat{{{\xi}}}^2)p_1
+\big(\frac{\rho_2-\rho_1}{u_1}-\rho_2'\hat{{{\xi}}} \big)({{\eta}}p_2-z)
+\hat E_1({\mathcal X})\cdot p +\hat E_2({\mathcal X})z,$$ where $\hat E_1\in C^\infty(\overline{\mathcal A}; { {\bf R}}^2)$ and $\hat
E_2\in C^\infty(\overline{\mathcal A})$ with $$\begin{aligned}
&&\|\hat E_i\|_{C^k(\overline{\mathcal A})}\le
\|\Psi\|_{C^{k+2}(\overline{\mathcal A})},
\qquad i=1,2, \quad k\in {\bf N}, \\
&&|\hat E_i(p,z, u_2, v_2, {{\eta}})| \le C(|p|+|z|+|u_2|+|v_2|)
\qquad\mbox{for all }(p, z, u_2, v_2, {{\eta}})\in {\mathcal A},\end{aligned}$$ for $C$ depending only on $\|D^2\Psi\|_{C(\overline{\mathcal A})}$.
From now on, we fix $(u_2, v_2)$ to be equal to the velocity of state (2) obtained in §\[section:3.3\] and write $E_i(p, z,
{{\eta}})$ for $\hat E_i(p,z,u_2, v_2, {{\eta}})$. We conclude that, if (\[condRewritingRH-0\]) holds and $\psi\in C^1({\Omega})$ satisfies (\[condRewritingRH\]), then $\psi=\varphi-\varphi_2$ satisfies (\[cont-accross-shock-mod-phi\])–(\[RH-mod-phi\]) on ${\Gamma_{shock}}$ if and only if $\psi$ satisfies conditions (\[cont-accross-shock-psi-resolved\]) on ${\Gamma_{shock}}$, $$\rho_2'(c_2^2-\hat{{{\xi}}}^2)\psi_{{\xi}}+\big(\frac{\rho_2-\rho_1}{u_1}-\rho_2'\hat{{{\xi}}}
\big)({{\eta}}\psi_{{\eta}}-\psi) +E_1({{D}}\psi,\psi, {{\eta}})\cdot {{D}}\psi
+E_2({{D}}\psi,\psi,{{\eta}})\psi =0, \label{RH-psi-2}$$ and the functions $E_i(p, z, {{\eta}}), i=1,2,$ are smooth on $$\overline{B_{\delta^*}(0)\times(-\delta^*, \delta^*) \times (-6\bar
c_2/5, 6\bar c_2/5)}$$ and satisfy that, for all $(p,z,{{\eta}})\in
B_{\delta^*}(0)\times(-\delta^*, \delta^*)\times (-6\bar c_2/5,
6\bar c_2/5)$, $$|E_i(p,z,{{\eta}})|\le C\left(|p|+|z|+ \sigma\right)
\label{RH-psi-2-error-term1}$$ and, for all $(p,z,{{\eta}})\in B_{\delta^*}(0)\times(-\delta^*,
\delta^*)\times (-6\bar c_2/5, 6\bar c_2/5)$, $$|({{D}}_{(p,z,{{\eta}})}E_i, \; D^2_{(p,z,{{\eta}})}E_i)|\le C,
\label{RH-psi-2-error-term2}$$ where we have used (\[u2-v2-bound\]) in the derivation of (\[RH-psi-2-error-term1\]) and $C$ depends only on the data.
Denote by $\nu_0$ the unit normal on the reflected shock to the region of state (2). Then $\nu_0=(\sin\theta_s, -\cos\theta_s)$ from the definition of $\theta_s$. We compute $$\begin{aligned}
\label{obliquenessRH}
&&
\big(\rho_2'(c_2^2-\hat{{{\xi}}}^2),(\frac{\rho_2-\rho_1}{u_1}-\rho_2'\hat{{{\xi}}}
){{\eta}}\big) \cdot\nu_0\\
&&=\rho_2'(c_2^2-\hat{{{\xi}}}^2)\sin\theta_s-
\big(\frac{\rho_2-\rho_1}{u_1}-\rho_2'\hat{{\xi}}\big){{\eta}}\cos\theta_s\nonumber
\\
&&\ge \frac{1}{2}\rho_2'(c_2^2-\hat{{{\xi}}}^2)>0,\nonumber\end{aligned}$$ if $\pi/2-\theta_s$ is small and ${{\eta}}\in Proj_{{\eta}}({\Gamma_{shock}})$. From (\[thetaS-uv\]) and (\[norm-to-Shock\]), we obtain $\|\nu_s-\nu_0\|_{L^\infty({\Gamma_{shock}})}\le
C\|{{D}}\psi\|_{C(\overline{\Omega})}$. Thus, if $\sigma>0$ and $\|{{D}}\psi\|_{C(\overline{\Omega})}$ are small depending only on the data, then (\[RH-psi-2\]) is an oblique derivative condition on ${\Gamma_{shock}}$.
\[sectionEqNearSonicLine\] For the shock reflection solution, equation (\[1.1.5\]) is expected to be elliptic in the domain ${\Omega}$ and degenerate on the sonic circle of state (2) which is the curve ${\Gamma_{sonic}}=\partial {\Omega}\cap \partial
B_{c_2}(0)$. Thus we consider the subdomains: $$\label{defOfSubdomains}
\begin{array}{l}
{\Omega'}:={\Omega}\cap\{({{\xi}},{{\eta}})\; : \;{ \mbox{dist}}(({{\xi}},{{\eta}}),{\Gamma_{sonic}})<2\varepsilon\},\\
{\Omega''}:={\Omega}\cap \{({{\xi}},{{\eta}})\; :
\;{ \mbox{dist}}(({{\xi}},{{\eta}}),{\Gamma_{sonic}})>\varepsilon\},
\end{array}$$ where the small constant $\varepsilon>0$ will be chosen later. Obviously, ${\Omega'}$ and ${\Omega''}$ are open subsets of ${\Omega}$, and ${\Omega}={\Omega'}\cup{\Omega''}$. Equation (\[1.1.5\]) is expected to be degenerate elliptic in ${\Omega'}$ and uniformly elliptic in ${\Omega''}$ on the solution of the shock reflection problem.
In order to display the structure of the equation near the sonic circle where the ellipticity degenerates, we introduce the new coordinates in ${\Omega'}$ which flatten ${\Gamma_{sonic}}$ and rewrite equation (\[1.1.5\]) in these new coordinates. Specifically, denoting $(r,
\theta)$ the polar coordinates in the $({{\xi}}, {{\eta}})$–plane, i.e., $({{\xi}},{{\eta}})=(r\cos\theta, r\sin\theta)$, we consider the coordinates: $$\label{coordNearSonic}
x=c_2-r, \quad y=\theta-\theta_w \qquad \,\,
\hbox{on } {\Omega'}.$$ By §\[section:3.3\], the domain ${{\mathcal D}'}$ does not contain the point $({{\xi}},{{\eta}})=(0,0)$ if $\varepsilon$ is small. Thus, the change of coordinates $({{\xi}}, {{\eta}})\to (x,y)$ is smooth and smoothly invertible on ${\Omega'}$. Moreover, it follows from the geometry of domain ${\Omega}$ especially from (\[FBfunct-estimate\])–(\[curved-straight-shock-match\]) that, if $\sigma>0$ is small, then, in the $(x,y)$–coordinates, $${\Omega'}=\{(x,y)\: : \; 0<x<2\varepsilon,\,\,
0<y<\pi+\mbox{arctan}\left({{{\eta}}(x)/f({{\eta}}(x))}\right)-\theta_w \},$$ where ${{\eta}}(x)$ is the unique solution, close to ${{\eta}}_1$, of the equation ${{\eta}}^2+f({{\eta}})^2=(c_2-x)^2$.
We write the equation for $\psi$ in the $(x,y)$–coordinates. As discussed in §\[equationForPsiSection\], $\psi$ satisfies equation (\[potent-flow-nondiv-psi-Polar\])–(\[speedSound-expression-psi-Polar-1\]) in the polar coordinates. Thus, in the $(x,y)$–coordinates in ${\Omega'}$, the equation for $\psi$ is $$\big(2x-(\gamma+1)\psi_x+O_1 \big)\psi_{xx} +O_2\psi_{xy} + ({1\over
c_2}+O_3 )\psi_{yy} -(1+O_4)\psi_{x} +O_5\psi_{y}=0,
\label{equationForPsi-sonicStruct}$$ where $$\begin{aligned}
\\
O_1({{D}}\psi,\psi,x) &=& -\frac{x^2}{c_2}+{\gamma+1\over
2c_2}(2x-\psi_x)\psi_x -{\gamma-1\over c_2}\big(\psi+{1\over
2(c_2-x)^2}\psi_y^2\big),\nonumber
\\
O_2({{D}}\psi,\psi, x)&=&-{2\over c_2(c_2-x)^2}(\psi_x+c_2-x)\psi_y,
\nonumber
\\
O_3({{D}}\psi,\psi, x) &=&{1\over c_2(c_2-x)^2}\Big(x(2c_2-x)-
(\gamma-1)(\psi+(c_2-x)\psi_x+{1\over 2}\psi_x^2) \nonumber
\\
\label{erTerms-xy-nontrunc} && \qquad\qquad\qquad
-\frac{\gamma+1}{2(c_2-x)^2}\psi_y^2\Big), \nonumber
\\
O_4({{D}}\psi,\psi, x) &=&\frac{1}{c_2-x}\Big(x- {\gamma-1\over
c_2}\big(\psi+(c_2-x)\psi_x+{1\over 2}\psi_x^2 +\frac{\psi_y^2}{2
(c_2-x)^2}\big)\Big), \nonumber
\\
O_5({{D}}\psi,\psi, x)&=&
-\frac{1}{c_2(c_2-x)^3}\big(\psi_x+2c_2-2x\big)\psi_y. \nonumber\end{aligned}$$ The terms $O_k({{D}}\psi,\psi, x)$ are small perturbations of the leading terms of equation (\[equationForPsi-sonicStruct\]) if the function $\psi$ is small in an appropriate norm considered below. In order to see this, we note the following properties: For any $(p,z,x)\in { {\bf R}}^2\times{ {\bf R}}\times (0, c_2/2)$ with $|p|<1$, $$\begin{aligned}
\label{estSmallterms}
&&|O_1(p,z,x)|\le C(|p|^2+|z|+|x|^2), \nonumber\\
&&|O_3(p,z,x)|+|O_4(p,z,x)|\le C(|p|+|z|+|x|),\\
&&|O_2(p,z,x)|+|O_5(p,z,x)|\le C(|p|+|x|+1)|p|.\nonumber\end{aligned}$$
In particular, dropping the terms $O_k$, $k=1,\dots, 5$, from equation (\[equationForPsi-sonicStruct\]), we obtain the [**transonic small disturbance equation**]{} (cf. [@Morawetz2]): $$\big(2x-(\gamma+1)\psi_x \big)\psi_{xx} + \frac{1}{c_2}\psi_{yy}
-\psi_{x} =0. \label{equation-TSD}$$
Now we write the boundary conditions on ${\Gamma_{sonic}}$, ${\Gamma_{shock}}$, and ${\Gamma_{wedge}}$ in the $(x,y)$–coordinates. Conditions (\[condOnSonicLine-Psi\]) and (\[condOnWedge-Psi\]) become $$\begin{aligned}
&&\psi=0\qquad\mbox{on }\;{\Gamma_{sonic}}=\partial {\Omega}\cap \{x=0\},
\label{condOnSonicLine-Psi-xy}
\\
&&\psi_\nu\equiv\psi_y=0\qquad\mbox{on }\;{\Gamma_{wedge}}=\partial
{\Omega}\cap \{y=0\}. \label{condOnWedge-Psi-xy}\end{aligned}$$
It remains to write condition (\[RH-psi-2\]) on ${\Gamma_{shock}}$ in the $(x,y)$–coordinates. Expressing $\psi_{{\xi}}$ and $\psi_{{\eta}}$ in the polar coordinates $(r,\theta)$ and using (\[coordNearSonic\]), we write (\[RH-psi-2\]) on ${\Gamma_{shock}}\cap\{x<2\varepsilon\}$ in the form: $$\begin{array}{l}
\left(-\rho_2'(c_2^2-\hat{{{\xi}}}^2) \cos(y+\theta_w) -
(\frac{\rho_2-\rho_1}{u_1}
-\rho_2'\hat{{{\xi}}})(c_2-x)\sin^2(y+\theta_w)\right)\psi_x
\displaystyle\\
\,\,+ \sin(y+\theta_w)
\left(-\frac{\rho_2'}{c_2-x}(c_2^2-\hat{{{\xi}}}^2)
+(\frac{\rho_2-\rho_1}{u_1}
-\rho_2'\hat{{{\xi}}})\cos(y+\theta_w)\right)\psi_y
\displaystyle\\
\,\, -\left(\frac{\rho_2-\rho_1}{u_1}-\rho_2'\hat{{{\xi}}} \right)\psi
+\tilde E_1({{D}}_{(x,y)}\psi, \psi, x,y)\cdot {{D}}_{(x,y)}\psi+
\tilde E_2({{D}}_{(x,y)}\psi, \psi, x,y)\psi=0, \displaystyle
\end{array}
\label{RH-psi-2-xy}$$ where $\tilde E_i(p,z,x,y), i=1,2,$ are smooth functions of $(p,z,x,y)\in{ {\bf R}}^2\times{ {\bf R}}\times{ {\bf R}}^2$ satisfying $$|\tilde E_i(p,z,x,y)|\le C\left(|p|+|z|+ \sigma\right)
\qquad\mbox{for }\, |p|+|z|+x\le\varepsilon_0(u_1,\bar{\rho}_2).$$
We now rewrite (\[RH-psi-2-xy\]). We note first that, in the $({{\xi}}, {{\eta}})$–coordinates, the point ${{P_1}}={\Gamma_{sonic}}\cap{\Gamma_{shock}}$ has the coordinates $({{\xi}}_1,{{\eta}}_1)$ defined by (\[coord-P4\]). Using (\[theta\_s-close-w\]), (\[theta\_s<w\]), (\[reflected-shock-s2\]), and (\[coord-P4\]), we find $$0\le |\hat{{\xi}}|- |{{\xi}}_1|\le C \sigma.$$ In the $(x,y)$–coordinates, the point ${{P_1}}$ is $(0, y_1)$, where $y_1$ satisfies $$\label{xy-xieta-at-P1}
c_2\cos(y_1+\theta_w)={{\xi}}_1,\qquad c_2\sin(y_1+\theta_w)={{\eta}}_1,$$ from (\[coord-P4\]) and (\[coordNearSonic\]). Using this and noting that the leading terms of the coefficients of (\[RH-psi-2-xy\]) near ${{P_1}}=(0, y_1)$ are the coefficients at $(x,y)=(0, y_1)$, we rewrite (\[RH-psi-2-xy\]) as follows: $$\begin{array}{l}
-\frac{\rho_2-\rho_1}{u_1c_2} {{\eta}}^2_1\psi_x
-\left(\rho_2'-\frac{\rho_2-\rho_1}{u_1
c_2^2}{{\xi}}_1\right){{\eta}}_1\psi_y -\left(\frac{\rho_2-\rho_1}{u_1}-
\rho_2'{{\xi}}_1\right)\psi
\displaystyle\\
+\hat E_1({{D}}_{(x,y)}\psi, \psi, x,y)\cdot {{D}}_{(x,y)}\psi+ \hat
E_2({{D}}_{(x,y)}\psi,\psi, x,y)\psi=0\,\,\mbox{on
}{\Gamma_{shock}}\cap\{x<2\varepsilon\}, \displaystyle
\end{array}
\label{RH-psi-3-xy}$$ where the terms $\hat E_i(p,z,x,y), i=1,2,$ satisfy $$|\hat E_i(p,z,x,y)|\le C\left(|p|+|z|+x+|y-y_1|+\sigma\right) \qquad
\label{RH-psi-2-error-term-xy-1}$$ for $(p,z,x,y)\in {\mathcal
T}:=\{(p,z,x,y)\in{ {\bf R}}^2\times{ {\bf R}}\times{ {\bf R}}^2 :\;
|p|+|z|\le\varepsilon_0(u_1,\bar{\rho}_2)\}$ and $$\label{RH-psi-2-error-term-xy-2}
\|({{D}}_{(p,z,x,y)}\hat E_i,\; D^2_{(p,z,x,y)}\hat
E_i)\|_{L^\infty( {\mathcal T})} \le C.$$
We note that the left-hand side of (\[RH-psi-3-xy\]) is obtained by expressing the left-hand side of (\[RH-psi-2\]) on ${\Gamma_{shock}}\cap\{c_2-r<2\varepsilon\}$ in the $(x,y)$–coordinates. Assume $\varepsilon<\bar c_2/4$. In this case, transformation (\[coordNearSonic\]) is smooth on $\{0<c_2-r<2\varepsilon\}$ and has nonzero Jacobian. Thus, condition (\[RH-psi-3-xy\]) is equivalent to (\[RH-psi-2\]) and hence to (\[RH-psi\]) on ${\Gamma_{shock}}\cap\{x<2\varepsilon\}$ if $\sigma>0$ is small so that (\[condRewritingRH-0\]) holds and if $\|\psi\|_{C^1(\overline{\Omega})}$ is small depending only on the data such that (\[condRewritingRH\]) is satisfied.
Iteration Scheme {#iterSchemeSection}
================
In this section, we develop an iteration scheme to solve the free boundary problem and set up the detailed steps of the iteration procedure in the shifted coordinates.
\[iteration-dom-subsect\] Fix $\theta_w<\pi/2$ close to $\pi/2$. Since our problem is a free boundary problem, the elliptic domain ${\Omega}$ of the solution is apriori unknown and thus we perform the iteration in a larger domain $$\label{ellipticDomainFull}
{{\mathcal D}}\equiv{{\mathcal D}}_{\theta_w}{:=}B_{c_2}(0)\cap\{{{\eta}}>-v_2\}\cap \{l({{\eta}})<{{\xi}}<
{{\eta}}\cos\theta_w\},$$ where $l({{\eta}})$ is defined by (\[reflected-shock-s2\]). We will construct a solution with $\Omega\subset{{\mathcal D}}$. Moreover, the reflected shock for this solution coincides with $\{{{\xi}}=l({{\eta}})\}$ outside the sonic circle, which implies $\partial {{\mathcal D}}\cap\partial
B_{c_2}(0) =\partial {\Omega}\cap\partial B_{c_2}(0)=:{\Gamma_{sonic}}$. Then we decompose ${{\mathcal D}}$ similar to (\[defOfSubdomains\]): $$\label{defOfSubdomains-iteration}
\begin{array}{l}
\displaystyle
{{\mathcal D}'}:={{\mathcal D}}\cap \{({{\xi}},{{\eta}})\; : \;{ \mbox{dist}}(({{\xi}},{{\eta}}),{\Gamma_{sonic}})<2\varepsilon\},\\
\displaystyle {{\mathcal D}''}:={{\mathcal D}}\cap \{({{\xi}},{{\eta}})\; :
\;{ \mbox{dist}}(({{\xi}},{{\eta}}),{\Gamma_{sonic}})>{\varepsilon/2}\}.
\end{array}$$ The universal constant $C>0$ in the estimates of this section depends only on the data and is independent on $\theta_w$.
We will work in the $(x,y)$–coordinates (\[coordNearSonic\]) in the domain ${{\mathcal D}}\cap\{c_2-r<\kappa_0\}$, where $\kappa_0\in (0, \bar
c_2)$ will be determined depending only on the data for the sonic speed $\bar c_2$ of state (2) for normal reflection (see §\[section:4\]). Now we determine $\kappa_0$ so that $\varphi_1-\varphi_2$ in the $(x,y)$–coordinates satisfies certain bounds independent of $\theta_w$ in ${{\mathcal D}}\cap\{c_2-r<\kappa_0\}$ if $\sigma=\pi/2-\theta_w$ is small.
We first consider the case of normal reflection $\theta_w=\pi/2$. Then, from (\[flatOrthSelfSimShock2\]) and (\[phi-2-a\]) in the $(x,y)$–coordinates (\[coordNearSonic\]) with $c_2=\bar c_2$ and $\theta_w=\pi/2$, we obtain $$\varphi_1-\varphi_2=-u_1 (\bar c_2-x)\sin y -u_1 \bar\xi \qquad
\mbox{for }\;0<x<\bar c_2, \;0<y<\pi/2.$$ Recall $\bar\xi<0$ and $|\bar\xi|<\bar c_2$ by (\[sonic-intersect-shock\]). Then, in the region ${{\mathcal D}}_0:=\{0<x<\bar c_2, \;0<y<\pi/2\}$, we have $\varphi_1-\varphi_2=0$ only on the line $$y=\hat f_{0,0}(x){:=}\arcsin\big(\frac{|\bar\xi|}{\bar c_2-x}\big)
\qquad \mbox{for }x\in (0, \bar c_2-|\bar\xi|).$$
Denote $\kappa_0{:=}{(\bar c_2-|\bar\xi|)/2}$. Then $\kappa_0\in
(0, \bar c_2)$ by (\[sonic-intersect-shock-normal\]) and depends only on the data. Now we show that there exists $\sigma_0>0$ small, depending only on the data, such that, if $\theta_w\in(\pi/2-{\sigma}_0, \pi/2)$, then $$\begin{aligned}
&&C^{-1}\le\partial_x(\varphi_1-\varphi_2),
-\partial_y(\varphi_1-\varphi_2)\le C\;\label{nondegenPolar-1}\\
&& \qquad\qquad\qquad\qquad\qquad\mbox{ on }[0,\kappa_0]\times[{\hat
f_{0,0}(0)\over 2}, {\hat f_{0,0}(\kappa_0)+{\pi/2}\over 2}],
\nonumber
\\
&&\varphi_1-\varphi_2\ge C^{-1}>0 \qquad\mbox{on }
[0,\kappa_0]\times[0, \frac{\hat f_{0,0}(0)}{2}],
\label{nondegenPolar-2}
\\
&&\varphi_1-\varphi_2\le -C^{-1}<0 \qquad\mbox{on }
[0,\kappa_0]\times\{\frac{\hat f_{0,0}(\kappa_0)+\pi/2}{2}\},
\label{nondegenPolar-3}
\end{aligned}$$ where ${\hat f_{0,0}(\kappa_0)+{\pi/2}\over 2}<{\pi/2}$.
We first prove (\[nondegenPolar-1\])–(\[nondegenPolar-3\]) in the case of normal reflection $\theta_w=\pi/2$. We compute from the explicit expressions of $\varphi_1-\varphi_2$ and $\hat f_{0,0}$ given above to obtain $$\begin{aligned}
0<\arcsin\big({2|\bar\xi|\over \bar c_2+|\bar\xi|}\big)<\hat
f_{0,0}(x)< \arcsin\big({|\bar\xi|\over \bar c_2}\big)<{\pi\over
2},\quad && C^{-1}\le \hat f_{0,0}'(x)\le C\\
&&\qquad\quad\mbox{for }x\in [0,\kappa_0],\end{aligned}$$ $
\partial_x(\varphi_1-\varphi_2)=u_1\sin y$, and $\partial_y(\varphi_1-\varphi_2)=-u_1(\bar c_2-x)\cos y$, which imply (\[nondegenPolar-1\]). Now, (\[nondegenPolar-2\]) is true since $\bar\xi=-\bar c_2\sin(\hat f_{0,0}(0))$ and thus $\varphi_1-\varphi_2=u_1\big(\bar c_2\sin(\hat f_{0,0}(0))-(\bar
c_2-x)\sin y\big)$, and (\[nondegenPolar-3\]) follows from (\[nondegenPolar-1\]) since $(\varphi_1-\varphi_2)(\kappa_0, \hat
f_{0,0}(\kappa_0))=0$ and ${(\hat f_{0,0}(\kappa_0)+{\pi/ 2})/
2}-\hat f_{0,0}(\kappa_0)\ge C^{-1}$.
Now let $\theta_w<\pi/2$. Then, from (\[thetaS-uv\])–(\[phi-2-shifted\]) and (\[coordNearSonic\]), we have $$\varphi_1-\varphi_2=-(c_2-x)\sin(y+\theta_w-\theta_s)\sqrt{(u_1-u_2)^2+v_2^2}
-(u_1-u_2)\hat\xi.$$ By §\[section:3.3\], when $\theta_w\to \pi/2$, we know that $(u_2, v_2)\to (0,0)$, $\theta_s\to\pi/2$, $\tilde\xi\to\bar\xi$, and thus, by (\[x1-in-shifed\]), we also have $\hat\xi\to\bar\xi$. This shows that, if $\sigma_0>0$ is small depending only on the data, then, for all $\theta_w\in(\pi/2-{\sigma}_0, \pi/2)$, estimates (\[nondegenPolar-1\])–(\[nondegenPolar-3\]) hold with $C$ that is equal to twice the constant $C$ from the respective estimates (\[nondegenPolar-1\])–(\[nondegenPolar-3\]) for $\theta_w=\pi/2$.
From (\[nondegenPolar-1\])–(\[nondegenPolar-3\]) for $\theta_w\in(\pi/2-{\sigma}_0, \pi/2)$ and since $${{\mathcal D}}\cap\{c_2-r<\kappa_0\} =\{\varphi_1>\varphi_2\}\cap\{0\le
x\le\kappa_0, 0\le y\le {\hat f_{0,0}(\kappa_0)+{\pi/2}\over 2} \},$$ there exists $\hat f_0:=\hat f_{0,\pi/2-\theta_w}\in
C^\infty(\overline{{ {\bf R}}_+})$ such that $$\begin{aligned}
\label{domain-in-xy-0}
&&{{\mathcal D}}\cap\{c_2-r<\kappa_0\}
=\{0< x< \kappa_0,\quad 0<y<\hat f_0(x)\},
\\
\label{domain-in-xy-funct-0} &&\hat f_0(0)=y_{{{P_1}}},\qquad
C^{-1}\le \hat f_0'(x)\le C \,\,\,\, \mbox{ on } [0,\kappa_0],
\\
&&{\hat f_{0,0}(0)/2}\le \hat f_0(0)<\hat f_0(\kappa_0)\le
({\hat f_{0,0}(\kappa_0)+{\pi/2})/2}.
\label{fbFUnctionCloseToNormalPolar}\end{aligned}$$ In fact, the line $y=\hat f_0(x)$ is the line ${{\xi}}=l({{\eta}})$ expressed in the $(x,y)$–coordinates, and thus we obtain explicitly with the use of (\[thetaS-uv\]) that $$\label{referenceFB-polar} \hat
f_0(x)=\arcsin\big(\frac{|\hat\xi|\sin\theta_s}{(c_2-x)}\big)-\theta_w+\theta_s
\qquad\mbox{on }[0,\kappa_0].$$
For the elliptic estimates, we need the Hölder norms in ${\Omega}$ weighted by the distance to the corners ${{P_2}}={\Gamma_{shock}}\cap\{{{\eta}}=-v_2\}$ and ${{P_3}}=(-u_2, -v_2)$, and with a “parabolic” scaling near the sonic circle.
More generally, we consider a subdomain ${\Omega}\subset {{\mathcal D}}$ of the form ${\Omega}:= {{\mathcal D}}\cap \{{{\xi}}\ge f({{\eta}})\}$ with $f\in C^1({ {\bf R}})$ and set the subdomains ${\Omega'}:={\Omega}\cap{{\mathcal D}'}$ and ${\Omega''}:={\Omega}\cap{{\mathcal D}''}$ defined by (\[defOfSubdomains\]). Let $\Sigma\subset\partial{\Omega''}$ be closed. We now introduce the Hölder norms in ${\Omega''}$ weighted by the distance to $\Sigma$. Denote by $X=({{\xi}},{{\eta}})$ the points of ${\Omega''}$ and set $$\delta_X:={ \mbox{dist}}(X,\Sigma),\quad \delta_{X,Y}:=\min(\delta_X,
\delta_Y) \qquad\mbox{for }\, X, Y\in {\Omega''}.$$ Then, for $k\in { {\bf R}}$, $\alpha\in (0, 1)$, and $m\in {\bf N}$, define $$\begin{aligned}
&&\qquad\,\,|u\|^{(k,\Sigma)}_{m,0,{\Omega''}} :=\sum_{0\le |\beta|\le
m} \sup_{X\in{\Omega''}}
\left(\delta_X^{\max(|\beta|+k,0)}|D^\beta u(X)|\right),
\nonumber \\
&&\qquad\,\, [u]^{(k,\Sigma)}_{m,\alpha,{\Omega''}}
:=\sum_{|\beta|=m}\sup_{X,Y\in{\Omega''}, X\ne Y}
\left(\delta_{X,Y}^{\max(m+\alpha+k,0)}
{\frac{|D^\beta u(X)-D^\beta u(Y)|}{|X-Y|^\alpha}}\right),
\label{weightNormsApp}
\\
&&\qquad\,\, \|u\|^{(k,\Sigma)}_{m,\alpha,{\Omega''}}
:=\|u\|^{(k,\Sigma)}_{m,0,{\Omega''}}
+[u]^{(k,\Sigma)}_{m,\alpha,{\Omega''}}, \nonumber\end{aligned}$$ where $D^\beta=\partial_{{{\xi}}}^{\beta_1}\partial_{{{\eta}}}^{\beta_2}$, and $\beta=(\beta_1,\beta_2)$ is a multi-index with $\beta_j\in {\bf
N}$ and $|\beta|=\beta_1+\beta_2$. We denote by $C^{(k,\Sigma)}_{m,\alpha,{\Omega''}}$ the space of functions with finite norm $\|\cdot\|^{(k,\Sigma)}_{m,\alpha,{\Omega''}}$.
If $m\ge -k\ge 1$ and $k$ is an integer, then any function $u\in
C^{(k,\Sigma)}_{m,\alpha,{\Omega''}}$ is $C^{|k|-1,1}$ up to $\Sigma$, but not necessarily $C^{|k|}$ up to $\Sigma$.
In ${\Omega'}$, the equation is degenerate elliptic, for which the Hölder norms with parabolic scaling are natural. We define the norm $\|\psi\|_{2,\alpha,{\Omega'}}^{(par)}$ as follows: Denoting $z=(x,y)$ and $\tilde z=(\tilde x,\tilde y)$ with $x, \tilde x\in(0,
2\varepsilon)$ and $$\delta^{(par)}_\alpha(z, \tilde z){:=}\left(|x-\tilde x|^2+
\min(x,\tilde x)|y-\tilde y|^2\right)^{\alpha/2},$$ then, for $u\in C^2({\Omega'})\cap C^{1,1}(\overline{\Omega'})$ written in the $(x,y)$–coordinates (\[coordNearSonic\]), we define $$\begin{aligned}
&&\qquad\,\, \|u\|^{(par)}_{2,0,{\Omega'}} :=\sum_{0\le k+l\le 2}
\sup_{z\in{\Omega'}}\left(x^{k+l/2-2}|\partial_x^k\partial_y^lu(z)|\right),
\nonumber\\
&&\qquad\,\, [u]^{(par)}_{2,\alpha,{\Omega'}} :=\sum_{k+l=2}\sup_{z,
\tilde z\in{\Omega'}, z\ne \tilde z}
\bigg(\min(x,\tilde x)^{\alpha-l/2}
\frac{|\partial_x^k\partial_y^lu(z)-\partial_x^k\partial_y^lu(\tilde z)|}
{\delta^{(par)}_\alpha(z,\tilde z)}\bigg),
\label{parabNormsApp} \\
&&\qquad\,\,\|u\|^{(par)}_{2,\alpha,{\Omega'}}
:=\|u\|^{(par)}_{2,0,{\Omega'}} +[u]^{(par)}_{2,\alpha,{\Omega'}}.
\nonumber\end{aligned}$$ To motivate this definition, especially the parabolic scaling, we consider a scaled version of the function $u(x,y)$ in the parabolic rectangles: $$\label{parabRectangles}
R_{(x,y)}=\Big\{(s,t)\;\;:\;\; |s-x|<\frac{x}{4},
|t-y|<\frac{\sqrt{x}}{4}\Big\}\cap {\Omega}\qquad\mbox{for }\,
z=(x,y)\in {\Omega'}.$$ Denote $Q_1{:=}(-1, 1)^2$. Then the rescaled rectangle (\[parabRectangles\]) is $$\label{rescaled-parabRectangles}
Q_1^{(z)}:=\Big\{(S,T)\in Q_1\; : \; (x+\frac{x}{4}S,
y+\frac{\sqrt{x}}{4}T)\in {\Omega}\Big\}.$$ Denote by $u^{(z)}(S, T)$ the following function in $Q_1^{(z)}$: $$\label{parabRescaling}
u^{(z)}(S, T):=\frac{1}{x^2}u(x+\frac{x}{4}S, y+\frac{\sqrt{x}}{4}T)
\qquad\mbox{for } (S, T)\in Q_1^{(z)}.$$ Then we have $$C^{-1}\sup_{z\in{\Omega'}\cap\{x<3\varepsilon/2\}}
\|u^{(z)}\|_{C^{2,\alpha}\big(\overline{Q_1^{(z)}}\big)}\leq
\|u\|^{(par)}_{2,\alpha,{\Omega'}}\leq
C\sup_{z\in{\Omega'}}\|u^{(z)}\|_{C^{2,\alpha}\big(\overline
{Q_1^{(z)}}\big)},$$ where $C$ depends only on the domain ${\Omega}$ and is independent of $\varepsilon\in (0, \kappa_0/2)$.
\[iterSet-Section\] We consider the wedge angle close to $\pi/2$, that is, ${\sigma}=\frac{\pi}{2}-\theta_w> 0$ is small which will be chosen below. Set $\Sigma_0:=\partial{{\mathcal D}}\cap\{{{\eta}}=-v_2\}$. Let $\varepsilon, {\sigma}> 0$ be the constants from (\[defOfSubdomains-iteration\]) and (\[angleCloseToPiOver2\]). Let $M_1, M_2 \geq 1$. We define ${{\mathcal K}}\equiv {{\mathcal K}}({\sigma},
\varepsilon, M_1, M_2)$ by $$\label{defSetK_R}
{{\mathcal K}}{:=}\bigg\{
{\phi}\in C^{1,\alpha}(\overline{{\mathcal D}})\cap C^{2}({{\mathcal D}})
\, : \, \|{\phi}\|_{2,\alpha,{{\mathcal D}'}}^{(par)} \leq M_1,
\|{\phi}\|_{2,\alpha,{{\mathcal D}''}}^{(-1-\alpha, \Sigma_0)} \leq M_2{\sigma},
{\phi}\ge 0 \mbox{ in }{{\mathcal D}}\bigg\} \quad$$ for $\alpha\in (0, 1/2)$. Then ${{\mathcal K}}$ is convex. Also, ${\phi}\in{{\mathcal K}}$ implies that $$\|{\phi}\|_{C^{1,1}(\overline{{\mathcal D}'})}\le M_1,\qquad
\|{\phi}\|_{C^{1,\alpha}(\overline{{\mathcal D}''})}\le M_2{\sigma},$$ so that ${{\mathcal K}}$ is a bounded subset in $C^{1,\alpha}(\overline{{\mathcal D}})$. Thus, ${{\mathcal K}}$ is a compact and convex subset of $C^{1,\alpha/2}(\overline{{\mathcal D}})$.
We note that the choice of constants $M_1, M_2\ge 1$ and $\varepsilon,{\sigma}>0$ below will guarantee the following property: $${\sigma}\max(M_1,M_2)+\varepsilon^{1/4} M_1+ {\sigma}M_2/\varepsilon^2 \le {\hat C}^{-1} \label{condConst-00}$$ for some sufficiently large $\hat C>1$ depending only on the data. In particular, (\[condConst-00\]) implies that ${\sigma}\le {\hat
C}^{-1}$ since $\max(M_1,M_2)\ge 1$, which implies $\pi/2-\theta_w\le{\hat C}^{-1}$ from (\[angleCloseToPiOver2\]). Thus, if we choose $\hat C$ large depending only on the data, then (\[condRewritingRH-0\]) holds. Also, for $\psi\in {{\mathcal K}}$, we have $$|({{D}}\psi, \psi)(x,y)|\le M_1x^2+M_1x \,\,\mbox{ in }\, {{\mathcal D}'},
\qquad \|\psi\|_{C^1(\bar{{{\mathcal D}''}})}\le M_2{\sigma}.$$ Furthermore, $0<x<2\varepsilon$ in ${{\mathcal D}'}$ by (\[coordNearSonic\]) and (\[defOfSubdomains-iteration\]). Now it follows from (\[condConst-00\]) that $\|\psi\|_{C^1}\le 2/{\hat C}$. Then (\[condRewritingRH\]) holds if $\hat C$ is large depending only on the data. Thus, in the rest of this paper, we always assume that (\[condRewritingRH-0\]) holds and that $\psi\in {{\mathcal K}}$ implies (\[condRewritingRH\]). Therefore, (\[RH-psi\]) is equivalent to (\[RH-psi-2-error-term1\])–(\[RH-psi-2-error-term2\]) for $\psi\in {{\mathcal K}}$.
We also note the following fact.
\[relatingNorms\] There exist $\hat C$ and $C$ depending only on the data such that, if ${\sigma}, \varepsilon>0$ and $M_1, M_2\ge 1$ in [(\[defSetK\_R\])]{} satisfy [(\[condConst-00\])]{}, then, for every ${\phi}\in{{\mathcal K}}$, $$\label{relateHolderNorms}
\|{\phi}\|_{2,\alpha,{{\mathcal D}}}^{(-1-\alpha, \Sigma_0\cup{\Gamma_{sonic}})} \le
C(M_1\varepsilon^{1-\alpha}+M_2{\sigma}).$$
In this proof, $C$ denotes a universal constant depending only on the data. We use definitions (\[weightNormsApp\])–(\[parabNormsApp\]) for the norms. We first show that $$\label{relateHolderNorms-Sing}
\|{\phi}\|_{2,\alpha,{{\mathcal D}'}}^{(-1-\alpha, {\Gamma_{sonic}})} \le
CM_1\varepsilon^{1-\alpha},$$ where $\delta_{(x,y)}:={ \mbox{dist}}((x,y), {\Gamma_{sonic}})$ in (\[weightNormsApp\]). First we show (\[relateHolderNorms-Sing\]) in the $(x,y)$–coordinates. Using (\[domain-in-xy-0\]), we have ${{\mathcal D}'}=\{0< x< 2\varepsilon,\, 0<y<\hat f_0(x)\}$ with ${\Gamma_{sonic}}=\{x=0,\; 0<y<\hat f_0(x)\}$, where $\|f_0'\|_{L^\infty((0,
2\varepsilon))}$ depends only the data, and thus ${ \mbox{dist}}((x,y),
{\Gamma_{sonic}})\le Cx$ in ${{\mathcal D}'}$. Then, since $\|{\phi}\|_{2,\alpha,{{\mathcal D}'}}^{(par)} \leq M_1$, we obtain that, for $(x,y)\in{{\mathcal D}'}$, $$\begin{aligned}
&&|{\phi}(x,y)|\le M_1x^2\le M_1\varepsilon^2, \qquad
|D{\phi}(x,y)|\le M_1x\le M_1\varepsilon, \\
&&\delta_{(x,y)}^{1-\alpha}|D^2{\phi}(x,y)|
=x^{1-\alpha}|D^2{\phi}(x,y)|\le \varepsilon^{1-\alpha}M_1.\end{aligned}$$ Furthermore, from (\[condConst-00\]) with $\hat C\ge 16$, we obtain $\varepsilon\le 1/2$. Thus, denoting $z=(x,y)$ and $\tilde
z=(\tilde x,\tilde y)$ with $x, \tilde x\in(0, 2\varepsilon)$, we have $$\begin{aligned}
\delta^{(par)}_\alpha(z, \tilde z)&{:=}& \left(|x-\tilde x|^2+
\min(x,\tilde x)|y-\tilde y|^2\right)^{\alpha/2}\\
&\le& \left(|x-\tilde x|^2+ 2\varepsilon|y-\tilde
y|^2\right)^{\alpha/2}\le |z- \tilde z|^\alpha,\end{aligned}$$ and $\min(\delta_{z},\delta_{\tilde z})=\min(x, \tilde x)$, which implies $$\begin{aligned}
\min(\delta_{z},\delta_{\tilde z}) {|D^2{\phi}(z)-D^2{\phi}(\tilde
z)|\over |z- \tilde z|^\alpha} &\le& C\varepsilon^{1-\alpha}\min(x,
\tilde x)^\alpha {|D^2{\phi}(z)-D^2{\phi}(\tilde z)|\over
\delta^{(par)}_\alpha(z, \tilde z)}\\
&\le& C\varepsilon^{1-\alpha}M_1.\end{aligned}$$ Thus we have proved (\[relateHolderNorms-Sing\]) in the $(x,y)$–coordinates. By (\[condRewritingRH-0\]) and (\[condConst-00\]), we have $\varepsilon\le c_2/50$ if $\hat C$ is large depending only on the data. Then the change $({{\xi}}, {{\eta}})\to (x,y)$ in ${{\mathcal D}'}$ and its inverse have bounded $C^3$–norms in terms of the data. Thus, (\[relateHolderNorms-Sing\]) holds in the $({{\xi}},
{{\eta}})$–coordinates.
Since ${\phi}\in{{\mathcal K}}$, then $\|{\phi}\|_{2,\alpha,{{\mathcal D}''}}^{(-1-\alpha,
\Sigma_0)} \leq M_2{\sigma}$. Thus, in order to complete the proof of (\[relatingNorms\]), it suffices to estimate $\{\min(\delta_{z},\delta_{\tilde z}) {|D^2{\phi}(z)-D^2 {\phi}(\tilde
z)|\over |z- \tilde z|^\alpha}\}$ in the case $z\in{{\mathcal D}'}\setminus{{\mathcal D}''}$ and $\tilde z\in{{\mathcal D}''}\setminus{{\mathcal D}'}$ for $\delta_z={ \mbox{dist}}(z, {\Gamma_{sonic}}\cup\Sigma_0)$. From $z\in{{\mathcal D}'}\setminus{{\mathcal D}''}$ and $\tilde z\in{{\mathcal D}''}\setminus{{\mathcal D}'}$, we obtain $0<c_2-|z|<\varepsilon/2$ and $c_2-|\tilde z|\ge
2\varepsilon$, which implies that $|z-\tilde z|\ge 3\varepsilon/2$. We have $c_2-|z|\le{ \mbox{dist}}(z,{\Gamma_{sonic}})\le C(c_2-|z|)$, where we have used (\[condRewritingRH-0\]) and (\[ellipticDomainFull\]). Thus, $\min(\delta_z, \delta_{\tilde z})\le C(c_2-|z|)\le C\varepsilon$. Also we have $|D^2{\phi}(z)|\le M_1$ by (\[parabNormsApp\]). If $\delta_{\tilde z}\ge \delta_z$, then $\delta_{\tilde z}\ge
\varepsilon/2$ and thus $|D^2{\phi}(\tilde z)|\le
(\varepsilon/2)^{-1+\alpha}M_2{\sigma}$ by (\[weightNormsApp\]). Then we have $$\min(\delta_{z},\delta_{\tilde z}) {|D^2{\phi}(z)-D^2{\phi}(\tilde
z)|\over |z- \tilde z|^\alpha} \le
C\varepsilon{M_1+(2\varepsilon)^{-1+\alpha}M_2{\sigma}\over
(3\varepsilon/2)^\alpha} \le
C\big(\varepsilon^{1-\alpha}M_1+M_2{\sigma}\big).$$ If $\delta_{\tilde z}\le \delta_z$, then ${ \mbox{dist}}(\tilde z,
\Sigma_0)\le { \mbox{dist}}(\tilde z, {\Gamma_{sonic}})$, which implies by (\[inSonicRegion-in-shifed\]) that $|z-\tilde z|\ge 1/C$ if $\varepsilon$ is sufficiently small, depending only on the data. Then $|D^2{\phi}(\tilde z)|\le \delta_{\tilde z}^{-1+\alpha}M_2{\sigma}$ and $$\min(\delta_{z},\delta_{\tilde z}) {|D^2 {\phi}(z)-D^2 {\phi}(\tilde
z)|\over |z- \tilde z|^\alpha} \le
C\big(\delta_{z}M_1+\delta_{\tilde z}\delta_{\tilde
z}^{-1+\alpha}M_2{\sigma}\big) \le C\big(\varepsilon M_1+M_2{\sigma}\big).$$
\[Constr-iter-section\] In this section, for simplicity of notations, the universal constant $C$ depends only on the data and may be different at each occurrence.
By (\[u2-v2-bound\]), it follows that, if ${\sigma}$ is sufficiently small depending on the data, then $$\label{q2-u1}
q_2\le u_1/10,$$ where $q_2=\sqrt{u_2^2+v_2^2}$. Let ${\phi}\in{{\mathcal K}}$. From (\[phi-1-shifted\])–(\[phi-2-shifted\]) and (\[q2-u1\]), it follows that $$\label{nondegeneracy}
(\varphi_1-\varphi_2-{\phi})_{{{\xi}}}({{\xi}},{{\eta}})\geq u_1/2>0\quad
\mbox{in}\;\;{{\mathcal D}}.$$ Since $\varphi_1-\varphi_2=0$ on $\{{{\xi}}=l({{\eta}})\}$ and ${\phi}\ge 0$ in ${{\mathcal D}}$, we have ${\phi}\ge \varphi_1-\varphi_2$ on $\{{{\xi}}=l({{\eta}})\}\cap\partial{{\mathcal D}}$, where $l({{\eta}})$ is defined by (\[reflected-shock-s2\]). Then there exists $f_{\phi}\in
C^{1,\alpha}({ {\bf R}})$ such that $$\label{shockPL}
\{{\phi}=\varphi_1-\varphi_2\}\cap{{\mathcal D}}=\{(f_{\phi}({{\eta}}),{{\eta}})\; :
\;{{\eta}}\in(-v_2,\eta_2)\}.$$ It follows that $f_{\phi}({{\eta}})\ge l({{\eta}})$ for all ${{\eta}}\in[-
v_2,\eta_2)$ and $$\label{OmegaPL}
\Omega^+({\phi}):=\{{{\xi}}>f_{\phi}({{\eta}})\}\cap {{\mathcal D}}=
\{{\phi}<\varphi_1-\varphi_2\}\cap {{\mathcal D}}.$$ Moreover, $\partial\Omega^+({\phi})={\Gamma_{shock}}\cup{\Gamma_{sonic}}\cup{\Gamma_{wedge}}\cup\Sigma_0$, where $$\label{shockIterDef}
\begin{array}{l}
\displaystyle
{\Gamma_{shock}}({\phi}):=\{{{\xi}}=f_{\phi}({{\eta}})\}\cap\partial\Omega^+({\phi}),\qquad
\displaystyle {\Gamma_{sonic}}:=\partial {{\mathcal D}}\cap
\partial B_{c_2}(0),\\
\displaystyle {\Gamma_{wedge}}:=\partial {{\mathcal D}}\cap
\{{{\eta}}={{\xi}}\tan\theta_w\},\qquad \displaystyle
\Sigma_0({\phi}):=\partial \Omega^+({\phi})\cap \{{{\eta}}=-v_2\}.
\end{array}$$ We denote by $P_j, 1\le j\le 4$, the corner points of $\Omega^+({\phi})$. Specifically, ${{P_2}}={\Gamma_{shock}}({\phi})\cap \Sigma_0({\phi})$ and ${{P_3}}=(-u_2, -v_2)$ are the corners on the symmetry line $\{{{\eta}}=-v_2\}$, and ${{P_1}}={\Gamma_{sonic}}\cap{\Gamma_{shock}}({\phi})$ and ${{P_4}}={\Gamma_{sonic}}\cap{\Gamma_{wedge}}$ are the corners on the sonic circle. Note that, since ${\phi}\in{{\mathcal K}}$ implies ${\phi}=0$ on ${\Gamma_{sonic}}$, it follows that ${{P_1}}$ is the intersection point $({{\xi}}_1,{{\eta}}_1)$ of the line ${{\xi}}=l({{\eta}})$ and the sonic circle ${{\xi}}^2+{{\eta}}^2=c_2^2$, where $({{\xi}}_1,{{\eta}}_1)$ is determined by (\[coord-P4\]).
We also note that $f_0=l$ for $0\in{{\mathcal K}}$. From ${\phi}\in{{\mathcal K}}$ and Lemma \[relatingNorms\] with $\alpha\in (0, 1/2)$, we obtain the following estimate of $f_{\phi}$ on the interval $(-v_2, {{\eta}}_1)$: $$\begin{aligned}
&&\|f_{\phi}-l\|_{2,\alpha,(-v_2, {{\eta}}_1)}^{(-1-\alpha, \{-v_2,
{{\eta}}_1\})} \leq C\big(M_1\varepsilon^{1/2}+M_2{\sigma}\big)\le
\varepsilon^{1/4}, \label{OmegaPL-f-higher}\end{aligned}$$ where the second inequality in (\[OmegaPL-f-higher\]) follows from (\[condConst-00\]) with sufficiently large $\hat C$.
We also work in the $(x,y)$–coordinates. Denote $\kappa{:=}\kappa_0/2$. Choosing $\hat C$ in (\[condConst-00\]) large depending only on the data, we conclude from (\[nondegenPolar-1\])–(\[nondegenPolar-3\]) that, for every ${\phi}\in{{\mathcal K}}$, there exists a function $\hat f\equiv \hat
f_{\phi}\in C^{(-2, \{0\})}_{2,\alpha,(0,\kappa)}$ such that $$\label{domain-in-rescaled-lemma}
\Omega^+({\phi})\cap\{c_2-r<\kappa\} =\{0< x< \kappa,\quad 0<y<\hat
f_{\phi}(x)\},$$ with $$\label{holder-hat-f}
\hat f_{\phi}(0)=\hat f_0(0)>0, \quad \hat f_{\phi}'>0 \mbox{ on }
(0,\kappa), \quad \|\hat f_{\phi}-\hat f_0\|^{(-1-\alpha,
\{0\})}_{2,\alpha,(0,\kappa)}\le
C\big(M_1\varepsilon^{1-\alpha}+M_2{\sigma}\big),$$ where we have used Lemma \[relatingNorms\]. More precisely, $$\begin{array}{l}
\displaystyle \sum_{k=0}^2
\sup_{x\in(0,2\varepsilon)}\big(x^{k-2}|D^k(\hat f_{\phi}-\hat
f_0)(x)|\big)
\\
\displaystyle \qquad + \sup_{x_1\ne x_2\in(0,2\varepsilon)}
\Big((\min(x_1,x_2))^\alpha\, {|(\hat f_{\phi}''-\hat f_0'')(x_1)-
(\hat f_{\phi}''-\hat f_0'')(x_2)| \over |x_1-x_2|^\alpha}\Big) \le
CM_1,
\\
\displaystyle
\|\hat f_{\phi}-\hat f_0\|_{2,\alpha,({\varepsilon/2},
\kappa)}\le CM_2{\sigma}.
\end{array}
\label{holder-hat-f-S}$$
Note that, in the $(\xi,\eta)$–coordinates, the angles $\theta_{{{P_2}}}$ and $\theta_{{{P_3}}}$ at the corners ${{P_2}}$ and ${{P_3}}$ of $\Omega^+({\phi})$ respectively satisfy $$\label{anglesCloseToPi2}
|\theta_{P_i}-\frac{\pi}{2}|\le \frac{\pi}{16}\qquad\mbox{for }i=2,3.$$ Indeed, $\theta_{{{P_3}}}=\pi/2-\theta_w$. The estimate for $\theta_{{{P_2}}}$ follows from (\[OmegaPL-f-higher\]) with (\[condConst-00\]) for large $\hat C$.
We now consider the following problem in the domain $\Omega^+({\phi})$: $$\begin{aligned}
&&{{\mathcal N}}(\psi){:=}A_{11}\psi_{{{\xi}}{{\xi}}}+
2A_{12}\psi_{{{\xi}}{{\eta}}}
+A_{22}\psi_{{{\eta}}{{\eta}}}=0 \qquad
\mbox{ in }\;\;\Omega^+({\phi}),
\label{iterationEquation} \\
&&{{\mathcal M}}(\psi){:=}\rho_2'(c_2^2-\hat{{{\xi}}}^2)\psi_{{\xi}}+
\big(\frac{\rho_2-\rho_1}{u_1}-\rho_2'\hat{{\xi}}\big)({{\eta}}\psi_{{\eta}}-\psi)
\label{iterationRH}\\
&&\qquad\qquad\,\, +E_1^{\phi}({{\xi}},{{\eta}})\cdot
D\psi+E_2^{\phi}({{\xi}},{{\eta}})\psi=0 \qquad\mbox{on }\;{\Gamma_{shock}}({\phi}),
\nonumber\\
&&\psi=0\qquad\,\,\mbox{on }\;{\Gamma_{sonic}},
\label{iterationCondOnSonicLine}
\\
&&\psi_\nu=0\qquad\,\, \mbox{on }\;{\Gamma_{wedge}},
\label{iterationCondOnWedge}
\\
&&\psi_{{\eta}}=-v_2\qquad\mbox{on }\;\partial \Omega^+({\phi})\cap
\{{{\eta}}=-v_2\}, \label{iterationCondOnSymmtryLine}\end{aligned}$$ where $A_{ij}=A_{ij}({{D}}\psi,{{\xi}},{{\eta}})$ will be defined below, and equation (\[iterationRH\]) is obtained from (\[RH-psi-2\]) by substituting ${\phi}$ into $E_i, i=1,2,$ i.e., $$E_i^{\phi}({{\xi}}, {{\eta}})=E_i({{D}}{\phi}({{\xi}}, {{\eta}}),{\phi}({{\xi}}, {{\eta}}), {{\eta}}).
\label{iteration-RH-error-term}$$ Note that, for ${\phi}\in{{\mathcal K}}$ and $({{\xi}}, {{\eta}})\in {{\mathcal D}}$, we have $({{D}}{\phi}({{\xi}}, {{\eta}}),{\phi}({{\xi}}, {{\eta}}), {{\eta}})\in
B_{\delta^*}(0)\times(-\delta^*, \delta^*)\times (-6\bar c_2/5,
6\bar c_2/5)$ by (\[condRewritingRH-0\])–(\[condRewritingRH\]). Thus, the right-hand side of (\[iteration-RH-error-term\]) is well-defined.
Also, we now fix $\alpha$ in the definition of ${{\mathcal K}}$. Note that the angles $\theta_{{{P_2}}}$ and $\theta_{{{P_3}}}$ at the corners ${{P_2}}$ and ${{P_3}}$ of $\Omega^+({\phi})$ satisfy (\[anglesCloseToPi2\]). Near these corners, equation (\[iterationEquation\]) is linear and its ellipticity constants near the corners are uniformly bounded in terms of the data. Moreover, the directions in the oblique derivative conditions on the arcs meeting at the corner ${{P_3}}$ (resp. ${{P_2}}$) are at the angles within the range $(7\pi/16, 9\pi/16)$, since (\[iterationRH\]) can be written in the form $\psi_{{\xi}}+e\psi_{{\eta}}-d\psi=0$, where $|e|\le C{\sigma}$ near ${{P_2}}$ from ${{\eta}}({{P_2}})=-v_2$, (\[u2-v2-bound\]), (\[RH-psi-2-error-term1\])–(\[RH-psi-2-error-term2\]), and (\[condConst-00\]). Then, by [@Lieberman88], there exists $\alpha_0\in(0, 1)$ such that, for any $\alpha\in (0,\alpha_0)$, the solution of (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]) is in $C^{1,\alpha}$ near and up to ${{P_2}}$ and ${{P_3}}$ if the arcs are in $C^{1,\alpha}$ and the coefficients of the equation and the boundary conditions are in the appropriate Hölder spaces with exponent $\alpha$. We use $\alpha=\alpha_0/2$ in the definition of ${{\mathcal K}}$ for $\alpha_0=\alpha_0(9\pi/16, 1/2)$, where $\alpha_0(\theta_0, \varepsilon)$ is defined in [@Lieberman88 Lemma 1.3]. Note that $\alpha\in(0, 1/2)$ since $\alpha_0\in(0, 1)$.
\[eqForIterationSection\] In this subsection, we fix ${\phi}\in{{\mathcal K}}$ and define equation (\[iterationEquation\]) such that
\(i) It is strictly elliptic inside the domain $\Omega^+({\phi})$ with elliptic degeneracy at the sonic circle ${\Gamma_{sonic}}=\partial
\Omega^+({\phi})\cap\partial B_{c_2}(0)$;
\(ii) For a fixed point $\psi={\phi}$ satisfying an appropriate smallness condition of $|{{D}}\psi|$, equation (\[iterationEquation\]) coincides with the original equation .
We define the coefficients $A_{ij}$ of equation (\[iterationEquation\]) in the larger domain ${{\mathcal D}}$. More precisely, we define the coefficients separately in the domains ${{\mathcal D}'}$ and ${{\mathcal D}''}$ and then combine them. In ${{\mathcal D}''}$, we define the coefficients of (\[iterationEquation\]) by substituting ${\phi}$ into the coefficients of (\[potent-flow-nondiv-psi-1\]), i.e., $$\label{iterationUniforDomEquation}
\begin{array}{ll}
A^1_{11}({{\xi}},{{\eta}})= c^2({{D}}{\phi}, {\phi}, \xi, \eta)
-({\phi}_{{\xi}}-{{\xi}})^2, \,\,
A^1_{22}({{\xi}},{{\eta}})=c^2({{D}}{\phi}, {\phi},
\xi, \eta) -({\phi}_{{\eta}}-{{\eta}})^2, \\
A^1_{12}({{\xi}},{{\eta}})=A^1_{21}({{\xi}},{{\eta}})=
-({\phi}_{{\xi}}-{{\xi}})({\phi}_{{\eta}}-{{\eta}}),
\end{array}$$ where ${\phi}, {\phi}_{{\xi}}$, and ${\phi}_{{\eta}}$ are evaluated at $({{\xi}},{{\eta}})$. Thus, (\[iterationEquation\]) in $\Omega^+({\phi})\cap{{\mathcal D}''}$ is a linear equation $$A^1_{11}\psi_{{{\xi}}{{\xi}}}+
2A^1_{12}\psi_{{{\xi}}{{\eta}}}
+A^1_{22}\psi_{{{\eta}}{{\eta}}}=0 \qquad
\mbox{ in }\;\;\Omega^+({\phi})\cap{{\mathcal D}''}.$$ From the definition of ${{\mathcal D}''}$, it follows that $\sqrt{{{\xi}}^2+{{\eta}}^2}\le c_2-\varepsilon$ in ${{\mathcal D}''}$. Then calculating explicitly the eigenvalues of matrix $(A^1_{ij})_{1\le
i,j\le 2}$ defined by (\[iterationUniforDomEquation\]) and using (\[condRewritingRH-0\]) yield that there exists $C=C(\gamma, \bar
c_2)$ such that, if $\varepsilon<\min(1, \bar c_2)/10$ and $\|{\phi}\|_{C^1}\le \varepsilon/C$, then $$\label{ellipticityInUniformDomain} {\varepsilon\bar c_2\over
8}|\mu|^2\le \sum_{i,j=1}^2A^1_{ij}({{\xi}},{{\eta}})\mu_i\mu_j\le 4\bar
c_2^2|\mu|^2\qquad \mbox{for any $({{\xi}}, {{\eta}})\in{{\mathcal D}''}$ and
$\mu\in{ {\bf R}}^2$.}$$ The required smallness of $\varepsilon$ and $\|{\phi}\|_{C^1}$ is achieved by choosing sufficiently large $\hat C$ in (\[condConst-00\]), since ${\phi}\in{{\mathcal K}}$.
In ${{\mathcal D}'}$, we use (\[equationForPsi-sonicStruct\]) and substitute ${\phi}$ into the terms $O_1,\dots,O_5$. However, it is essential that we do not substitute ${\phi}$ into the term $(\gamma+1)\psi_x$ of the coefficient of $\psi_{xx}$ in (\[equationForPsi-sonicStruct\]), since this nonlinearity allows us to obtain some crucial estimates (see Lemma \[quadraticGrowthPsi-Lemma\] and Proposition \[boundPsiXfromAbove-Prop\]). Thus, we make an elliptic cutoff of this term. In order to motivate our construction, we note that, if $$|O_k|\le \frac{x}{10\max(c_2,1)(\gamma+1)}, \qquad
\psi_x<\frac{4x}{3(\gamma+1)}\qquad\quad \mbox{ in }\, {{\mathcal D}'},$$ then equation (\[equationForPsi-sonicStruct\]) is strictly elliptic in ${{\mathcal D}'}$. Thus we want to replace the term $(\gamma+1)\psi_x$ in the coefficient of $\psi_{xx}$ in (\[equationForPsi-sonicStruct\]) by $\displaystyle
(\gamma+1)x\zeta_1\big(\frac{\psi_x}{x} \big)$, where $\zeta_1(\cdot)$ is a cutoff function. On the other hand, we also need to keep form (\[iterationEquation\]) for the modified equation in the $(\xi,\eta)$–coordinates, i.e., the form without lower-order terms. This form is used in Lemma \[negativeDerivPsiLemma\]. Thus we perform a cutoff in equation (\[potent-flow-nondiv-psi-1\]) in the $(\xi,\eta)$–coordinates such that the modified equation satisfies the following two properties:
\(i) Form (\[iterationEquation\]) is preserved;
\(ii) When written in the $(x,y)$–coordinates, the modified equation has the main terms as in (\[equationForPsi-sonicStruct\]) with the cutoff described above and corresponding modifications in the terms $O_1,\dots, O_5$ of (\[equationForPsi-sonicStruct\]).
Also, since the equations in ${{\mathcal D}'}$ and ${{\mathcal D}''}$ will be combined and the specific form of the equation is more important in ${{\mathcal D}'}$, we define our equation in a larger domain ${{\mathcal D}'}_{4\varepsilon}{:=}{{\mathcal D}}\cap \{c_2-r<4\varepsilon\}$.
We first rewrite equation (\[potent-flow-nondiv-psi-1\]) in the form $$I_1+I_2+I_3+I_4=0,$$ where $$\begin{aligned}
&&I_1{:=}\big(c^2(D\psi,\psi,\xi,\eta)-(\xi^2+\eta^2)\big)
\Delta\psi,\\
&&I_2{:=}{{\eta}}^2\psi_{{{\xi}}{{\xi}}}+{{\xi}}^2\psi_{{{\eta}}{{\eta}}}-2{{\xi}}{{\eta}}\psi_{{{\xi}}{{\eta}}},
\\
&& I_3{:=}2\big({{\xi}}\psi_{{\xi}}\psi_{{{\xi}}{{\xi}}}
+({{\xi}}\psi_{{\eta}}+{{\eta}}\psi_{{\xi}})\psi_{{{\xi}}{{\eta}}}+{{\eta}}\psi_{{\eta}}\psi_{{{\eta}}{{\eta}}}\big),\\
&&I_4{:=}-\frac{1}{2}\left(\psi_{{\xi}}(|{{D}}\psi|^2)_{{\xi}}+\psi_{{\eta}}(|{{D}}\psi|^2)_{{\eta}}\right).\end{aligned}$$ Note that, in the polar coordinates, $I_1,\dots,I_4$ have the following expressions: $$\begin{aligned}
&&I_1=
\big(c_2^2-r^2+(\gamma-1)(r\psi_r-\frac{1}{2}|{{D}}\psi|^2-\psi)\big)
\Delta\psi,\\
&&I_2= \psi_{\theta\theta}+r\psi_r,\\
&&I_3=r(|{{D}}\psi|^2)_r=2r\psi_r\psi_{rr}
+\frac{2}{r^2}\psi_\theta\psi_{r\theta}
-\frac{2}{r^2}\psi_{\theta}^2,\\
&& I_4=-\frac{1}{2}\big(\psi_r(|{{D}}\psi|^2)_r
+\frac{1}{r^2}\psi_\theta(|{{D}}\psi|^2)_\theta\big)\end{aligned}$$ with $|{{D}}\psi|^2=\psi_r^2+\frac{1}{r^2}\psi_\theta^2$ and $\Delta\psi=\psi_{rr}+\frac{1}{r^2}\psi_{\theta\theta}+\frac{1}{r}\psi_r$.
From this, by (\[coordNearSonic\]), we see that the dominating terms of (\[equationForPsi-sonicStruct\]) come only from $I_1,
I_2$, and the term $2r\psi_r\psi_{rr}$ of $I_3$, i.e., the remaining terms of $I_3$ and $I_4$ affect only the terms $O_1,\dots, O_5$ in (\[equationForPsi-sonicStruct\]). Moreover, the term $(\gamma+1)\psi_x$ in the coefficient of $\psi_{xx}$ in (\[equationForPsi-sonicStruct\]) is obtained as the leading term in the sum of the coefficient $(\gamma-1)r\psi_r$ of $\psi_{rr}$ in $I_1$ and the coefficient $2r\psi_r$ of $\psi_{rr}$ in $I_3$. Thus we modify the terms $I_1$ and $I_3$ by cutting off the $\psi_r$-component of first derivatives in the coefficients of second-order terms as follows. Let $\zeta_1\in C^\infty({ {\bf R}})$ satisfy $$\label{defZeta-1}
\zeta_1(s)=\left\{
\begin{array}{ll}
s,\quad&\displaystyle \mbox{if }\;|s|<4/\big(3(\gamma+1)\big),\\
\displaystyle 5\,\mbox{sign}(s)/[3(\gamma+1)], \quad&\displaystyle
\mbox{if }\; |s|>2/(\gamma+1),
\end{array}
\right.$$ so that $$\begin{aligned}
&&\zeta_1'(s)\ge 0,\quad
\zeta_1(-s)=-\zeta_1(s)\qquad \mbox{on }\;{ {\bf R}};\label{defZeta-1a} \\
&& \zeta_1''(s)\le 0\qquad \mbox{on }\;\{s\ge
0\}.\label{zeta-1-concave}\end{aligned}$$ Obviously, such a smooth function $\zeta_1\in C^\infty({ {\bf R}})$ exits. Property (\[zeta-1-concave\]) will be used only in Proposition \[boundPsiXfromAbove-Prop\]. Now we note that $\psi_{{\xi}}=\frac{{{\xi}}}{r}\psi_r-\frac{{{\eta}}}{r^2}\psi_\theta$ and $\psi_{{\eta}}=\frac{{{\eta}}}{r}\psi_r+\frac{{{\xi}}}{r^2}\psi_\theta$, and define $$\begin{aligned}
\hat I_1&{:=}& \Big(c_2^2-r^2+ (\gamma-1)r(c_2-r)
\zeta_1(\frac{{{\xi}}\psi_{{\xi}}+{{\eta}}\psi_{{\eta}}}{r(c_2-r)})
-(\gamma-1)(\frac{1}{2}|{{D}}\psi|^2+\psi)\Big)
\Delta\psi,\\
\hat I_3&{:=}& \;\; 2\Big( \frac{{{\xi}}}{r}
(c_2-r)\zeta_1(\frac{{{\xi}}\psi_{{\xi}}+{{\eta}}\psi_{{\eta}}}{r(c_2-r)})
-\frac{{{\eta}}}{r^2}({{\xi}}\psi_{{\eta}}-{{\eta}}\psi_{{\xi}})\Big)
({{\xi}}\psi_{{{\xi}}{{\xi}}}+{{\eta}}\psi_{{{\xi}}{{\eta}}})\\
&& + 2\Big( \frac{{{\eta}}}{r}
(c_2-r)\zeta_1(\frac{{{\xi}}\psi_{{\xi}}+{{\eta}}\psi_{{\eta}}}{r(c_2-r)})
+\frac{{{\xi}}}{r^2}({{\xi}}\psi_{{\eta}}-{{\eta}}\psi_{{\xi}})\Big)({{\xi}}\psi_{{{\xi}}{{\eta}}}
+{{\eta}}\psi_{{{\eta}}{{\eta}}}).\end{aligned}$$ The modified equation in the domain ${{\mathcal D}'}_{4\varepsilon}$ is $$\hat I_1+I_2+\hat I_3+I_4=0.
\label{iteration-equation-sonic-cartesian}$$ By (\[defZeta-1\]), the modified equation (\[iteration-equation-sonic-cartesian\]) coincides with the original equation (\[potent-flow-nondiv-psi-1\]) if $$\left|\frac{{{\xi}}}{r}\psi_{{\xi}}+\frac{{{\eta}}}{r}\psi_{{\eta}}\right|<\frac{4(c_2-r)}{3(\gamma+1)},$$ i.e., if $\displaystyle
\left|\psi_x\right|<4x/\big(3(\gamma+1)\big)$ in the $(x,y)$–coordinates. Also, equation (\[iteration-equation-sonic-cartesian\]) is of form (\[iterationEquation\]) in the $(\xi,\eta)$–coordinates.
Now we define (\[iterationEquation\]) in ${{\mathcal D}'}_{4\varepsilon}$ by substituting ${\phi}$ into the coefficients of (\[iteration-equation-sonic-cartesian\]) except for the terms involving $\displaystyle
\zeta_1(\frac{{{\xi}}\psi_{{\xi}}+{{\eta}}\psi_{{\eta}}}{r(c_2-r)})$. Thus, we obtain an equation of form (\[iterationEquation\]) with the coefficients: $$\label{iterationSonicDomEquation}
\begin{array}{ll}
A^2_{11}({{D}}\psi,{{\xi}},{{\eta}})=& c_2^2- (\gamma-1)\Big(r(c_2-r)
\zeta_1(\frac{{{\xi}}\psi_{{\xi}}+{{\eta}}\psi_{{\eta}}}{r(c_2-r)})
+\frac{1}{2}|{{D}}{\phi}|^2+{\phi}\Big)
\\
&-({\phi}_{{\xi}}^2+{{\xi}}^2)+2{{\xi}}\Big(\frac{{{\xi}}}{r}
(c_2-r)\zeta_1(\frac{{{\xi}}\psi_{{\xi}}+{{\eta}}\psi_{{\eta}}}{r(c_2-r)})-
\frac{{{\eta}}}{r^2}({{\xi}}{\phi}_{{\eta}}-{{\eta}}{\phi}_{{\xi}})\Big),
\\
A^2_{22}({{D}}\psi,{{\xi}},{{\eta}})=& c_2^2- (\gamma-1)\Big(r(c_2-r)
\zeta_1(\frac{{{\xi}}\psi_{{\xi}}+{{\eta}}\psi_{{\eta}}}{r(c_2-r)})
+\frac{1}{2}|{{D}}{\phi}|^2+{\phi}\Big)
\\
&-({\phi}_{{\eta}}^2+{{\eta}}^2)+2{{\eta}}\Big( \frac{{{\eta}}}{r}
(c_2-r)\zeta_1(\frac{{{\xi}}\psi_{{\xi}}+{{\eta}}\psi_{{\eta}}}{r(c_2-r)})+
\frac{{{\xi}}}{r^2}({{\xi}}{\phi}_{{\eta}}-{{\eta}}{\phi}_{{\xi}})\Big),
\\
A^2_{12}({{D}}\psi,{{\xi}},{{\eta}})=&
-({\phi}_{{\xi}}{\phi}_{{\eta}}+{{\xi}}{{\eta}})+2\Big( \frac{{{\xi}}{{\eta}}}{r}
(c_2-r)\zeta_1(\frac{{{\xi}}\psi_{{\xi}}+{{\eta}}\psi_{{\eta}}}{r(c_2-r)})+
\frac{{{\xi}}^2-{{\eta}}^2}{r^2}({{\xi}}{\phi}_{{\eta}}-{{\eta}}{\phi}_{{\xi}})\Big),
\\
A^2_{21}({{D}}\psi,{{\xi}},{{\eta}})=&A^2_{12}({{D}}\psi, {{\xi}},{{\eta}}),
\end{array}$$ where ${\phi}, {\phi}_{{\xi}}$, and ${\phi}_{{\eta}}$ are evaluated at $({{\xi}},{{\eta}})$.
Now we write (\[iteration-equation-sonic-cartesian\]) in the $(x,y)$–coordinates. By calculation, the terms $\hat I_1$ and $\hat
I_3$ in the polar coordinates are $$\begin{aligned}
&&\hat I_1=
\Big(c_2-r^2+(\gamma-1)\big(r(c_2-r)\zeta_1(\frac{\psi_r}{c_2-r})
-\frac{1}{2}|{{D}}\psi|^2-\psi\big)\Big)\Delta\psi,\\
&&\hat I_3=2r(c_2-r)\zeta_1(\frac{\psi_r}{c_2-r})\psi_{rr}
+\frac{2}{r^2}\psi_\theta\psi_{r\theta}
-\frac{2}{r^2}\psi_{\theta}^2.\end{aligned}$$ Thus, equation (\[iteration-equation-sonic-cartesian\]) in the $(x,y)$–coordinates in ${{\mathcal D}'}_{4\varepsilon}$ has the form $$\left(2x-(\gamma+1)x\zeta_1(\frac{\psi_x}{x})
+\tilde O_1 \right)\psi_{xx}
+\tilde O_2\psi_{xy}
+ \left({1\over c_2}+\tilde O_3\right)\psi_{yy}
-(1+\tilde O_4)\psi_{x} +\tilde O_5\psi_{y}=0
\label{cutOff-equation-sonicStruct}$$ with $$\begin{aligned}
&&\tilde O_i(p,z,x)= O_i(p,z,x)\qquad\mbox{ for }\; i=2, 5,
\nonumber
\\
&&\tilde O_1({{D}}\psi,\psi,x)= -\frac{x^2}{2c_2}+{\gamma+1\over
2c_2} \left(2x^2\zeta_1(\frac{\psi_x}{x})-\psi_x^2\right)
-(\gamma-1)\left(\psi+{1\over 2c_2(c_2-x)^2}\psi_y^2\right),
\nonumber
\\
&&\tilde O_3({{D}}\psi,\psi,x)={1\over c_2(c_2-x)^2}\bigg(
x(2c_2-x)-\frac{\gamma+1}{2(c_2-x)^2}\psi_y^2
\label{erTerms-xy-trunc} \\
&& \qquad\qquad\qquad\qquad\qquad\qquad\quad +(\gamma-1)\big(\psi+
(c_2-x)x\zeta_1(\frac{\psi_x}{x})+ \frac{\psi_x^2}{2} \big) \bigg),
\nonumber\\
&&\tilde O_4({{D}}\psi,\psi, x)=\frac{1}{c_2-x}\left(x -
{\gamma-1\over c_2}\big(\psi+(c_2-x)x\zeta_1(\frac{\psi_x}{x})
+\frac{\psi_x^2}{2} +\frac{\psi_y^2}{2 (c_2-x)^2}\big) \right),
\nonumber\end{aligned}$$ where $O_i(p,z,x), i=2,5,$ are given by (\[erTerms-xy-nontrunc\]). It follows that $\tilde O_1(p,z,x),\dots, \tilde O_5(p,z,x)$ satisfy estimates (\[estSmallterms\]). In the $(x,y)$–coordinates, this equation has the form $$\big( 2x-(\gamma+1)x\zeta_1(\frac{\psi_x}{x})
+O_1^{\phi}\big)\psi_{xx}
+O_2^{\phi}\psi_{xy}
+
\left({1\over c_2}+O_3^{\phi}\right)\psi_{yy}
-(1+O_4^{\phi})\psi_{x}
+O_5^{\phi}\psi_{y}=0,
\label{iteration-equation-sonicStruct}$$ with $\tilde O_k^{\phi}(p,x,y)$ defined by $$\begin{array}{ll}
&\tilde O_1^{\phi}(p,x, y)= -\frac{x^2}{c_2}+{\gamma+1\over 2c_2}
\big(2x^2\zeta_1(\frac{p_1}{x})-{\phi}_x^2\big) -{\gamma-1\over
c_2}\left({\phi}+{1\over 2(c_2-x)^2}{\phi}_y^2\right),
\\
&\tilde O_k^{\phi}(x,y)=\tilde O_k({{D}}{\phi}(x,y),{\phi}(x,y), x)
\qquad\mbox{ for }\; i= 2, 5,
\\
&\tilde O_3^{\phi}(p,x,y)={1\over
c_2(c_2-x)^2}\Big(x(2c_2-x)-\frac{\gamma+1}{2(c_2-x)^2}{\phi}_y^2\\
&\qquad\qquad\qquad\qquad\qquad \,\,-(\gamma-1)\big({\phi}+
(c_2-x)x\zeta_1\left(\frac{p_1}{x}\right)+\frac{1}{2}{\phi}_x^2\big)
\Big),
\\
&\tilde O_4^{\phi}(p,x, y)=\frac{1}{c_2-x}\left(x - {\gamma-1\over
c_2}\big({\phi}+(c_2-x)x\zeta_1\left(\frac{p_1}{x}\right)
+\frac{{\phi}_x^2}{2} +\frac{{\phi}_y^2}{2 (c_2-x)^2}\big) \right),
\qquad
\end{array}
\label{iteration-equation-sonicStruct-coef}$$ where $p=(p_1, p_2)$, and $({{D}}{\phi}, {\phi})$ are evaluated at $(x,y)$. The estimates in (\[estSmallterms\]), the definition of the cutoff function $\zeta_1$, and ${\phi}\in{{\mathcal K}}$ with (\[condConst-00\]) imply $$|\tilde O_1^{\phi}(p,x, y)|\le C|x|^{3/2}, \qquad |\tilde
O_k^{\phi}(x, y)|\le C|x|\qquad \mbox{for }\; k=2,\dots,5,
\label{estSmallterms-iter}$$ for all $p\in { {\bf R}}^2$ and $(x,y)\in {{\mathcal D}'}_{4\varepsilon}$. Indeed, using that ${\phi}\in{{\mathcal K}}$ implies $\|{\phi}\|_{2,\alpha,{{\mathcal D}'}}^{(par)} \leq M_1$, we find that, for all $p\in { {\bf R}}^2$ and $(x,y)\in {{\mathcal D}'}\equiv {{\mathcal D}'}_{2\varepsilon}$, $$\begin{aligned}
&&|\tilde O_1^{\phi}(p,x, y)|\le C (M_1^2+1)|x|^2\le C|x|^{3/2},
\nonumber
\\
&&|\tilde O_k^{\phi}(x, y)|\le C(1+M_1|x|)M_1|x|^{3/2}\le C|x|\qquad
\mbox{for }\; k=2,5, \label{estSmallterms-iter-S}
\\
&&|\tilde O_k^{\phi}(p,x,y)|\le C(|x|+M_1^2|x|^2)\le C|x|
\qquad \mbox{for }\; k=3,4.
\nonumber\end{aligned}$$ In order to obtain the corresponding estimates in the domain ${{\mathcal D}'}_{4\varepsilon}\setminus {{\mathcal D}'}_{2\varepsilon}$, we note that ${{\mathcal D}'}_{4\varepsilon}\setminus {{\mathcal D}'}_{2\varepsilon}\subset{{\mathcal D}''}$. Since $2\varepsilon\le x\le 4\varepsilon$ in ${{\mathcal D}'}_{4\varepsilon}\setminus {{\mathcal D}'}_{2\varepsilon}$ and ${\phi}\in{{\mathcal K}}$ implies $\|{\phi}\|_{2,\alpha,{{\mathcal D}''}}^{(-1-\alpha,
\Sigma_0)} \leq M_2{\sigma}$, we find that, for any $p\in { {\bf R}}^2$ and $(x,y)\in {{\mathcal D}'}_{4\varepsilon}\setminus {{\mathcal D}'}_{2\varepsilon}$, $$\begin{aligned}
&&\qquad\quad |\tilde O_1^{\phi}(p,x, y)|\le C
(1+M_2^2{\sigma}^2+M_2{\sigma})\varepsilon^2 \le C\varepsilon^2\le C|x|^2,
\nonumber
\\
&&\qquad\quad |\tilde O_k^{\phi}(x, y)|\le C(1+M_2{\sigma})M_2{\sigma}\le
C\varepsilon^2\le C|x|^2\qquad \mbox{for }\; k=2,5,
\label{estSmallterms-iter-U}
\\
&&\qquad\quad |\tilde O_k^{\phi}(p,x,y)|\le
C(\varepsilon+M_2^2{\sigma}^2+M_2{\sigma})\le C\varepsilon\le C|x| \qquad
\mbox{for }\; k=3,4. \nonumber\end{aligned}$$ Estimates (\[estSmallterms-iter-S\])–(\[estSmallterms-iter-U\]) imply (\[estSmallterms-iter\]).
The estimates in (\[estSmallterms-iter\]) imply that, if ${\phi}\in{{\mathcal K}}$ and $\varepsilon$ is sufficiently small depending only on the data (which is guaranteed by (\[condConst-00\]) with sufficiently large $\hat C$), equation (\[iteration-equation-sonicStruct\]) is nonuniformly elliptic in ${{\mathcal D}'}$. First, in the $(x,y)$–coordinates, writing (\[iteration-equation-sonicStruct\]) as $$a_{11}\psi_{xx}
+2a_{12}\psi_{xy}
+
a_{22}\psi_{yy}
+ a_1\psi_{x} +a_2\psi_{y}=0,$$ with $a_{ij}=a_{ij}({{D}}\psi, x,y)=a_{ji}$ and $a_{i}=a_{i}({{D}}\psi, x,y)$, and using (\[condRewritingRH-0\]), we have $$\frac{x}{6}|\mu|^2\le \sum_{i,j=1}^2a_{ij}(p,x,y)\mu_i\mu_j \le
\frac{2}{\bar c_2}|\mu|^2 \quad\mbox{for any }
(p,x,y)\in{ {\bf R}}^2\times{{\mathcal D}'}_{4\varepsilon} \,\mbox{and}
\,\mu\in{ {\bf R}}^2.$$ In order to show similar ellipticity in the $({{\xi}},{{\eta}})$–coordinates, we note that, by (\[condRewritingRH-0\]), the change of coordinates $({{\xi}},
{{\eta}})$ to $(x,y)$ in ${{\mathcal D}'}_{4\varepsilon}$ and its inverse have $C^1$ norms bounded by a constant depending only on the data if $\varepsilon<\bar c_2/10$. Then there exists $\tilde\lambda>0$ depending only on the data such that, for any $(p,{{\xi}},{{\eta}})\in{ {\bf R}}^2\times{{\mathcal D}'}_{4\varepsilon}$ and $\mu\in{ {\bf R}}^2$, $$\label{ellipticityInDegenerateDomain}
\tilde\lambda(c_2-r)|\mu|^2\le \sum_{i,j=1}^2A^2_{ij}(p,{{\xi}},{{\eta}})\mu_i\mu_j \le
\tilde{\lambda}^{-1}|\mu|^2,$$ where $A^2_{ij}(p,{{\xi}},{{\eta}}), i,j=1,2,$ are defined by (\[iterationSonicDomEquation\]), and $r=\sqrt{{{\xi}}^2+{{\eta}}^2}$.
Next, we combine the equations introduced above by defining the coefficients of (\[iterationEquation\]) in ${{\mathcal D}}$ as follows. Let $\zeta_2\in C^\infty({ {\bf R}})$ satisfy $$\zeta_2(s)=\left\{
\begin{array}{ll}
0,\quad&\displaystyle \mbox{if }\;s\le2\varepsilon,\\ \displaystyle
1,\quad&\displaystyle \mbox{if }\;s\ge 4\varepsilon,
\end{array}
\right.
\qquad
\mbox{and }\;\,\,
0\le\zeta_2'(s) \le 10/\varepsilon \quad \mbox{on }\;{ {\bf R}}.$$ Then we define that, for $p\in{ {\bf R}}^2$ and $({{\xi}},{{\eta}})\in{{\mathcal D}}$, $$\label{combineCoeffs}
A_{ij}(p,{{\xi}},{{\eta}})=\zeta_2(c_2-r)A_{ij}^1({{\xi}},{{\eta}})
+\big(1-\zeta_2(c_2-r)\big)A_{ij}^2(p,{{\xi}},{{\eta}}).$$ Then (\[iterationEquation\]) is strictly elliptic in ${{\mathcal D}}$ and uniformly elliptic in ${{\mathcal D}''}$ with ellipticity constant $\lambda>0$ depending only on the data and $\varepsilon$. We state this and other properties of $A_{ij}$ in the following lemma.
\[propertiesNonlinCoeffs\] There exist constants $\lambda>0$, $C$, and $\hat C$ depending only on the data such that, if $M_1,M_2, \varepsilon$, and ${\sigma}$ satisfy [(\[condConst-00\])]{}, then, for any ${\phi}\in{{\mathcal K}}$, the coefficients $A_{ij}(p,{{\xi}}, {{\eta}})$ defined by [(\[combineCoeffs\])]{}, $i,j=1,2$, satisfy
1. \[propertiesNonlinCoeffs-i1\] For any $({{\xi}}, {{\eta}})\in{{\mathcal D}}$ and $p,\mu\in{ {\bf R}}^2$, $$\displaystyle \lambda(c_2-r)|\mu|^2 \le
\sum_{i,j=1}^2A_{ij}(p,{{\xi}},{{\eta}})\mu_i\mu_j
\le \lambda^{-1}|\mu|^2 \qquad \mbox{with}\,\, r=\sqrt{{{\xi}}^2+{{\eta}}^2};$$
2. \[propertiesNonlinCoeffs-i2\] $A_{ij}(p,{{\xi}},{{\eta}})=A^1_{ij}({{\xi}}, {{\eta}})$ for any $({{\xi}}, {{\eta}})\in
{{\mathcal D}}\cap\{c_2-r>4\varepsilon\}$ and $p\in{ {\bf R}}^2$, where $A^1_{ij}({{\xi}}, {{\eta}})$ are defined by [(\[iterationUniforDomEquation\])]{}. Moreover, $$A^1_{ij}\in
C^{1,\alpha}(\overline{{{\mathcal D}}\cap\{c_2-r>4\varepsilon\}})$$ with $
\|A^1_{ij}\|_{{1,\alpha}
(\overline{{{\mathcal D}}\cap\{c_2-r>4\varepsilon\}})}\le C;
$
3. \[propertiesNonlinCoeffs-i3\] $|A_{ij}|+|D_{(p,{{\xi}},{{\eta}})}A_{ij}|
\le C$ for any $({{\xi}},{{\eta}})\in {{\mathcal D}}\cap \{0<c_2-r<12\varepsilon\}$ and $p\in{ {\bf R}}^2$.
Property (\[propertiesNonlinCoeffs-i1\]) follows from (\[ellipticityInUniformDomain\]) and (\[ellipticityInDegenerateDomain\])–(\[combineCoeffs\]). Properties (\[propertiesNonlinCoeffs-i2\])–(\[propertiesNonlinCoeffs-i3\]) follow from the explicit expressions (\[iterationUniforDomEquation\]) and (\[iterationSonicDomEquation\]) with ${\phi}\in{{\mathcal K}}$. In estimating these expressions in property (\[propertiesNonlinCoeffs-i3\]), we use that $|s\zeta_1'(s)|\le C$ which follows from the smoothness of $\zeta_1$ and (\[defZeta-1\]).
Also, equation (\[iterationEquation\]) coincides with equation (\[iteration-equation-sonicStruct\]) in the domain ${{\mathcal D}'}$. Assume that $\varepsilon<\kappa_0/24$, which can be achieved by choosing $\hat C$ large in (\[condConst-00\]). Then, in the larger domain ${{\mathcal D}}\cap\{c_2-r<12\varepsilon\}$, equation (\[iterationEquation\]) written in the $(x,y)$–coordinates has form (\[iteration-equation-sonicStruct\]) with the only difference that the term $x\zeta_1(\frac{\psi_x}{x})$ in the coefficient of $\psi_{xx}$ of (\[iteration-equation-sonicStruct\]) and in the terms $\tilde O_1^{\phi}$, $\tilde O_3^{\phi}$, and $\tilde
O_4^{\phi}$ given by (\[iteration-equation-sonicStruct-coef\]) is replaced by $$x\Big(\zeta_2(x)\zeta_1(\frac{{\phi}_x}{x}) +
(1-\zeta_2(x))\zeta_1(\frac{\psi_x}{x}) \Big).$$ From this, we have
\[propertiesNonlinCoeffs-xy\] There exist $C$ and $\hat C$ depending only on the data such that the following holds. Assume that $M_1,M_2, \varepsilon$, and ${\sigma}$ satisfy [(\[condConst-00\])]{}. Let ${\phi}\in{{\mathcal K}}$. Then equation [(\[iterationEquation\])]{} written in the $(x,y)$–coordinates in ${{\mathcal D}}\cap\{c_2-r<12\varepsilon\}$ has the form $$\label{nonlinIterEq-xy-lg}
\hat A_{11}\psi_{xx}+2\hat A_{12}\psi_{xy}+\hat A_{22}\psi_{yy}
+\hat A_{1}\psi_{x}+\hat A_{2}\psi_{y}=0,$$ where $\hat A_{ij}=\hat A_{ij}(\psi_x,x,y)$, $\hat A_{i}=\hat
A_{i}(\psi_x,x,y)$, and $\hat A_{21}=\hat A_{12}$. Moreover, the coefficients $\hat A_{ij}(p,x,y)$ and $\hat A_{i}(p,x,y)$ with $p=(p_1,p_2)\in { {\bf R}}^2$ satisfy
1. \[propertiesNonlinCoeffs-xy-i1\] For any $(x,y)\in{{\mathcal D}}\cap\{x<12\varepsilon\}$ and $p,\mu\in{ {\bf R}}^2$, $$\displaystyle {x\over 6}|\mu|^2\le \sum_{i,j=1}^2\hat
A_{ij}(p,x,y)\mu_i\mu_j
\le \frac{2}{\bar c_2}|\mu|^2;$$
2. \[propertiesNonlinCoeffs-xy-i3\] For any $(x,y)\in {{\mathcal D}}\cap \{x<12\varepsilon\}$ and $p\in{ {\bf R}}^2$, $$|(\hat A_{ij}, D_{(p,x,y)}\hat A_{ij})|+|(\hat A_{i},
D_{(p,x,y)}\hat A_{i})| \le C;$$
3. \[propertiesNonlinCoeffs-xy-i4-0\] $\hat A_{11}$, $\hat A_{22}$, and $\hat A_1$ are independent of $p_2$;
4. \[propertiesNonlinCoeffs-xy-i4\] $\hat A_{12}$, $\hat A_{21}$, and $\hat A_2$ are independent of $p$, and $$|(\hat A_{12},\hat A_{21},\hat A_{2})(x,y)|\le
C|x|, \quad |D(\hat A_{12},\hat A_{21},\hat A_{2})(x,y)| \le
C|x|^{1/2}.$$
The last inequality in Lemma \[propertiesNonlinCoeffs-xy\](\[propertiesNonlinCoeffs-xy-i4\]) is proved as follows. Note that $$(\hat A_{12}, \hat A_{2})(x,y)=(O_2, O_5)(D{\phi}(x,y), {\phi}(x,y),
x),$$ where $O_2$ and $O_5$ are given by (\[erTerms-xy-nontrunc\]). Then, by ${\phi}\in{{\mathcal K}}$ and (\[condConst-00\]), we find that, for $(x,y)\in {{\mathcal D}'}$, i.e., $x\in (0, 2\varepsilon)$, $$\begin{aligned}
|D(\hat A_{12},\hat A_{21},\hat A_{2})(x,y)|
&\le& C(1+M_1\varepsilon)|D{\phi}_y(x,y)|
+(1+M_1)|{\phi}_y(x,y)|\\
&\le& C(1+M_1\varepsilon)M_1x^{1/2}+C(1+M_1)M_1x^{3/2}\le Cx^{1/2};\end{aligned}$$ and, for $(x,y)\in {{\mathcal D}}\cap\{\varepsilon\le x\le
12\varepsilon\}\subset{{\mathcal D}''}$, we have ${ \mbox{dist}}(x,\Sigma_0)\ge
c_2/2\ge\bar c_2/4$ so that $$|D(\hat A_{12},\hat A_{21},\hat A_{2})(x,y)|
\le C(1+M_2{\sigma})M_2{\sigma}\le C\varepsilon\le C x.$$
The next lemma follows directly from both (\[defZeta-1\]) and the definition of $A_{ij}$.
\[cutOffEqIsOriginalEq\] Let $\Omega\subset{{\mathcal D}}$, $\psi\in C^2(\Omega)$, and $\psi$ satisfy equation [(\[iterationEquation\])]{} with ${\phi}=\psi$ in $\Omega$. Assume also that $\psi$, written in the $(x,y)$–coordinates, satisfies $|\psi_x|\le
4x/\big(3(\gamma+1)\big)$ in $\Omega'{:=}\Omega\cap\{c_2-r<4\varepsilon\}$. Then $\psi$ satisfies in $\Omega$.
\[overViewProcedureSubsection\] With the previous analysis, our iteration procedure will consist of the following ten steps, in which Steps 2–9 will be carried out in detail in §\[unifElliptApproxSection\]–§\[removeCutoffSection\] and the main theorem is completed in §\[proofSection\].
Fix ${\phi}\in{{\mathcal K}}$. This determines the domain $\Omega^+({\phi})$, equation (\[iterationEquation\]), and condition (\[iterationRH\]) on ${\Gamma_{shock}}({\phi})$, as described in §\[Constr-iter-section\]–§\[eqForIterationSection\] above.
In §\[unifElliptApproxSection\], using the vanishing viscosity approximation of equation (\[iterationEquation\]) via a uniformly elliptic equation $${{\mathcal N}}(\psi)+\delta\Delta\psi=0\,\,\qquad\mbox{for }\;\delta\in(0, 1)$$ and sending $\delta\to 0$, we establish the existence of a solution $\psi\in C(\overline{\Omega^+({\phi})})\cap
C^1(\overline{\Omega^+({\phi})}\setminus\overline{\Gamma_{sonic}}) \cap
C^2(\Omega^+({\phi}))$ to problem (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]). This solution satisfies $$0\le\psi\le C{\sigma}\qquad \mbox{in }\;\Omega^+({\phi}),
\label{L-ifty-BdIteratioOverview}$$ where $C$ depends only on the data.
For every $s\in(0, c_2/2)$, set ${\Omega''}_s:=
\Omega^+({\phi})\cap\{c_2-r>s\}$. By Lemma \[propertiesNonlinCoeffs\], if [(\[condConst-00\])]{} holds with sufficiently large $\hat C$ depending only on the data, then equation (\[iterationEquation\]) is uniformly elliptic in ${\Omega''}_s$ for every $s\in (0, c_2/2)$, the ellipticity constant depends only on the data and s, and the bounds of coefficients in the corresponding Hölder norms also depend only on the data and $s$. Furthermore, (\[iterationEquation\]) is linear on $\{c_2-r>4\varepsilon\}$, which implies that it is also linear near the corners ${{P_2}}$ and ${{P_3}}$. Then, by the standard elliptic estimates in the interior and near the smooth parts of $\partial\Omega^+({\phi})\cap \overline {{\Omega''}_s}$ and using Lieberman’s estimates [@Lieberman88] for linear equations with the oblique derivative conditions near the corners $(-u_2,-v_2)$ and ${\Gamma_{shock}}({\phi})\cap\{\eta=-v_2\}$, we have $$\label{ellipticEstimates-unif-halfDomain}
\|\psi\|^{(-1-\alpha,\Sigma_0)}_{2,\alpha,{\Omega''}_{s/2}}
\le C(s)(\|\psi\|_{L^\infty(\overline {{\Omega''}_s})}
+|v_2|),$$ if $\|\psi\|_{L^\infty(\overline {{\Omega''}_s})}+
|v_2|\le 1$, where the second term in the right-hand side comes from the boundary condition (\[iterationCondOnSymmtryLine\]), and the constant $C(s)$ depends only on the ellipticity constants, the angles at the corners ${{P_2}}={\Gamma_{shock}}({\phi})\cap\{\eta=-v_2\}$ and ${{P_3}}=(-u_2,-v_2)$, the norm of ${\Gamma_{shock}}({\phi})$ in $C^{1,\alpha}$, and $s$, which implies that $C(s)$ depends only on the data and $s$.
Now, using (\[L-ifty-BdIteratioOverview\]) and (\[u2-v2-bound\]), we obtain $\|\psi\|_{L^\infty(\overline{{\Omega''}_s})}+|v_2|\le 1$ if ${\sigma}$ is sufficiently small, which is achieved by choosing $\hat C$ in (\[condConst-00\]) sufficiently large. Then, from (\[ellipticEstimates-unif-halfDomain\]), we obtain $$\label{ellipticEstimates-unif-halfDomain-2}
\|\psi\|^{(-1-\alpha,\Sigma_0)}_{2,\alpha,{\Omega''}_{s/2}}\le
C(s){\sigma}$$ for every $s\in (0, c_2/2)$, where $C$ depends only on the data and $s$.
We work in the $(x, y)$–coordinates, and then equation (\[iterationEquation\]) is equation (\[iteration-equation-sonicStruct\]) in ${\Omega'}$.
Since ${\phi}\in{{\mathcal K}}$, the estimates in (\[estSmallterms-iter\]) hold for large $\hat C$ in (\[condConst-00\]) depending only on the data. We also rewrite the boundary condition (\[iterationRH\]) in the $(x, y)$–coordinates and obtain (\[RH-psi-3-xy\]) with $\hat E_i$ replaced by $\hat
E_i^{\phi}(x,y){:=}\hat E_i(D{\phi}(x,y),{\phi}(x,y),x,y)$. Using ${\phi}\in{{\mathcal K}}$, (\[RH-psi-2-error-term-xy-1\]), (\[RH-psi-2-error-term-xy-2\]), and (\[holder-hat-f-S\]) with $\hat f_{\phi}(0)=\hat f_0(0)=y_1$, we obtain $$|\hat E_i^{\phi}(x,y)|\le C(M_1\varepsilon+M_2{\sigma})\le C/\hat C,\qquad
i=1,2,
\label{RH-psi-2-error-term-iter}$$ for $(x,y)\in {\Gamma_{shock}}({\phi})\cap\{0<x<2\varepsilon\}$. Then, if $\hat
C$ in (\[condConst-00\]) is large, we find that the function $$w(x,y)=\frac{3x^2}{5(\gamma+1)}
$$ is a supersolution of equation (\[iteration-equation-sonicStruct\]) in ${\Omega'}({\phi})$ with the boundary condition (\[iterationRH\]) on ${\Gamma_{shock}}({\phi})\cap
\{0<x<2\varepsilon\}$. That is, the right-hand sides of (\[iterationRH\]) and (\[iteration-equation-sonicStruct\]) are negative on $w(x,y)$ in the domains given above. Also, $w(x,y)$ satisfies the boundary conditions (\[iterationCondOnSonicLine\])–(\[iterationCondOnWedge\]) within ${\Omega'}({\phi})$. Thus, $$0\le\psi(x,y)\le \frac{3x^2}{5(\gamma+1)}
\qquad\mbox{in }\;{\Omega'}({\phi}),
\label{L-ifty-BdIteratin-Sonic-Overview}$$ if $w\ge \psi$ on $x=\varepsilon$. By (\[L-ifty-BdIteratioOverview\]), $w\ge \psi$ on $x=\varepsilon$ if $$C{\sigma}\le \varepsilon^2,$$ where $C$ is a large constant depending only on the data, i.e., if (\[condConst-00\]) is satisfied with large $\hat C$. The details of the argument of Step 4.1 are in Lemma \[quadraticGrowthPsi-Lemma\].
. We use the parabolic rescaling in the rectangle $R_{z}$ defined by (\[parabRectangles\]) in which ${\Omega'}$ is replaced by ${\Omega'}({\phi})$. Note that $R_{z}\subset {\Omega'}$ for every $z=(x,y)\in{\hat\Omega'}({\phi})$. Thus, $\psi$ satisfies (\[iteration-equation-sonicStruct\]) in $R_{z}$. For every $z\in{\hat\Omega'}({\phi})$, we define the functions $\psi^{(z)}$ and ${\phi}^{(z)}$ by (\[parabRescaling\]) in the domain $Q_1^{(z)}$ defined by (\[rescaled-parabRectangles\]). Then equation (\[iteration-equation-sonicStruct\]) for $\psi$ yields the following equation for $\psi^{(z)}(S,T)$ in $Q_1^{(z)}$: $$\begin{aligned}
&&\quad\big((1+\frac{S}{4})\big(2-
(\gamma+1)\zeta_1(\frac{4\psi^{(z)}_S}{1+S/4})\big)
+xO_1^{({\phi},z)} \big)\psi^{(z)}_{SS}
+xO_2^{({\phi},z)}\psi^{(z)}_{ST}
\label{iteration-equation-sonicStruct-ParabRescaled}\\
&&\qquad\quad
+ \big({1\over c_2}+x O_3^{({\phi},z)}\big)\psi^{(z)}_{TT}
-({1\over 4}+xO_4^{({\phi},z)})\psi^{(z)}_{S}
+x^2O_5^{({\phi},z)}\psi^{(z)}_{T}=0,
\nonumber\end{aligned}$$ where the terms $O_k^{({\phi},z)}(S,T,p)$, $k=1,\dots, 5$, satisfy $$\label{rescaledErrorTermsEqEstimate}
\|O_k^{({\phi},z)}\|_{C^{1,\alpha}(\overline{Q_1^{(z)}}\times{ {\bf R}}^2)}\le C(1+M_1^2).$$ Estimate (\[rescaledErrorTermsEqEstimate\]) follows from the explicit expressions of $O_k^{({\phi},z)}$ obtained from both (\[iteration-equation-sonicStruct-coef\]) by rescaling and the fact that $$\|{\phi}^{(z)}\|_{C^{2,\alpha}(\overline{Q_1^{(z)}})}\le CM_1,$$ which is true since $\|{\phi}\|_{2,\alpha,{\Omega'}({\phi})}^{(par)}\le
M_1$. Now, since every term $O_k^{({\phi},z)}$ in (\[iteration-equation-sonicStruct-ParabRescaled\]) is multiplied by $x^{\beta_k}$ with $\beta_k\ge 1$ and $x\in (0, \varepsilon)$, condition (\[condConst-00\]) (possibly after increasing $\hat C$ depending only on the data) implies that equation (\[iteration-equation-sonicStruct-ParabRescaled\]) is uniformly elliptic in $Q_1^{(z)}$ and has the $C^{1,\alpha}$ bounds on the coefficients by a constant depending only on the data.
Now, if the rectangle $R_{z}$ does not intersect $\partial
\Omega^+({\phi})$, then $Q_1^{(z)}=Q_1$, where $Q_s=(-s, s)^2$ for $s>0$. Thus, the interior elliptic estimates in Theorem \[locEstElliptEq\] in Appendix imply $$\label{ellipticEstimates-interior}
\|\psi^{(z)}\|_{C^{2,\alpha}(\overline{Q_{1/2}})}\le C,$$ where $C$ depends only on the data and $\|\psi^{(z)}\|_{L^\infty(\overline{Q_{1}})}$. From (\[L-ifty-BdIteratin-Sonic-Overview\]), we have $$\|\psi^{(z)}\|_{L^\infty(\overline{Q_{1}})} \le 1/(\gamma+1).
$$ Therefore, we obtain (\[ellipticEstimates-interior\]) with $C$ depending only on the data.
Now consider the case when the rectangle $R_{z}$ intersects $\partial \Omega^+({\phi})$. From its definition, $R_{z}$ does not intersect ${\Gamma_{sonic}}$. Thus, $R_{z}$ intersects either ${\Gamma_{shock}}$ or the wedge boundary ${\Gamma_{wedge}}$. On these boundaries, we have the homogeneous oblique derivative conditions (\[iterationRH\]) and (\[iterationCondOnWedge\]). In the case when $R_{z}$ intersects ${\Gamma_{wedge}}$, the rescaled condition (\[iterationCondOnWedge\]) remains the same form, thus oblique, and we use the estimates for the oblique derivative problem in Theorem \[locEstElliptEq-Dirichlet\] to obtain $$\label{ellipticEstimates-bdry-rescaled}
\|\psi^{(z)}\|_{C^{2,\alpha}(\overline{Q^{(z)}_{1/2}})}\le C,$$ where $C$ depends only on the data, since the $L^\infty$ bound of $\psi^{(y)}$ in $Q^{(z)}_{1}$ follows from (\[L-ifty-BdIteratin-Sonic-Overview\]). In the case when $R_{z}$ intersects ${\Gamma_{shock}}$, the obliqueness in the rescaled condition (\[iterationRH\]) is of order $x^{1/2}$, which is small since $x\in (0, 2\varepsilon)$. Thus we use the estimates for the “almost tangential derivative" problem in Theorem \[locEstElliptEq-non-oblique\] to obtain (\[ellipticEstimates-bdry-rescaled\]).
Finally, rescaling back, we have $$\label{ellipticEstimates-par-halfDomain}
\|\psi\|_{2,\alpha,{\hat\Omega'}({\phi})}^{(par)}\le C.$$ The details of the argument of Step 4.2 are in Lemma \[EstParabolicHolder-Lemma\].
In Lemma \[extension-Lemma\], we extend $\psi$ from the domain $\Omega^+({\phi})$ to ${{\mathcal D}}$ working in the $(x,
y)$–coordinates (or, equivalently in the polar coordinates) near the sonic line and in the rest of the domain in the $({{\xi}},
{{\eta}})$–coordinates, by using the procedure of [@ChenFeldman1]. If $\hat C$ is sufficiently large, the extension of $\psi$ satisfies $$\begin{aligned}
\label{ellipticEstimates-par-Domain}
&& \|\psi\|_{2,\alpha,{{\mathcal D}'}}^{(par)}\le C,
\\
\label{ellipticEstimates-unif-Domain}
&&\|\psi\|^{(-1-\alpha,\Sigma_0)}_{2,\alpha,{{\mathcal D}''}}
\le C(\varepsilon){\sigma},\end{aligned}$$ with $C$ depending only on the data in (\[ellipticEstimates-par-Domain\]) and $C(\varepsilon)$ depending only on the data and $\varepsilon$ in (\[ellipticEstimates-unif-Domain\]). This is obtained by using (\[ellipticEstimates-par-halfDomain\]) and (\[ellipticEstimates-unif-halfDomain-2\]) with $s>0$ determined by the data and $\varepsilon$, and by using the estimates of the functions $f_{\phi}$ and $\hat f_{\phi}$ in (\[OmegaPL\]), (\[holder-hat-f\]), and (\[holder-hat-f-S\]).
We fix $\hat C$ in (\[condConst-00\]) large depending only on the data, so that Lemmas \[propertiesNonlinCoeffs\]–\[propertiesNonlinCoeffs-xy\] hold and the requirements on $\hat C$ stated in Steps 1–5 above are satisfied. Set $M_1=\max(2C,1)$ for the constant $C$ in (\[ellipticEstimates-par-Domain\]) and choose $$\varepsilon =\frac{1}{10\max((\hat CM_1)^4, \hat C)}.
\label{condConst-1-1}$$ This choice of $\varepsilon$ fixes $C$ in (\[ellipticEstimates-unif-Domain\]) depending only on the data and $\hat C$. Now set $M_2=\max(C,1)$ for $C$ from (\[ellipticEstimates-unif-Domain\]) and let $$0<{\sigma}\le {\sigma}_0:={({\hat
C}^{-1}-\varepsilon-\varepsilon^{1/4}M_1)\varepsilon^2 \over
2\left(\varepsilon^2\max(M_1, M_2)+M_2\right)},$$ where ${\sigma}_0>0$ since $\varepsilon$ is defined by (\[condConst-1-1\]). Then (\[condConst-00\]) holds with constant $\hat C$ fixed above.
Note that the constants ${\sigma}_0, \varepsilon, M_1$, and $M_2$ depend only on the data and $\hat C$.
With the constants ${\sigma}, \varepsilon, M_1$, and $M_2$ chosen in Step 6, estimates (\[ellipticEstimates-par-Domain\])–(\[ellipticEstimates-unif-Domain\]) imply $$\|{\phi}\|_{2,\alpha,{{\mathcal D}'}}^{(par)} \leq M_1, \qquad
\|\psi\|^{(-1-\alpha,\Sigma_0)}_{2,\alpha,{{\mathcal D}''}}
\le M_2{\sigma}.$$ Thus, $\psi\in{{\mathcal K}}({\sigma}, \varepsilon, M_1, M_2)$. Then the iteration map $J:{{\mathcal K}}\to{{\mathcal K}}$ is defined.
In Lemma \[extension-Lemma\] and Proposition \[existenceFixedPt\], by the argument similar to [@ChenFeldman1] and the fact that ${{\mathcal K}}$ is a compact and convex subset of $C^{1,\alpha/2}(\overline{{\mathcal D}})$, we show that the iteration map $J$ is continuous, by uniqueness of the solution $\psi\in C^{1,\alpha}(\overline{{\mathcal D}})\cap C^2({{\mathcal D}})$ of (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]). Then, by the Schauder Fixed Point Theorem, there exists a fixed point $\psi\in{{\mathcal K}}$. This is a solution of the free boundary problem.
Removal of the cutoff: By Lemma \[cutOffEqIsOriginalEq\], a fixed point $\psi={\phi}$ satisfies the original equation in $\Omega^+(\psi)$ if $|\psi_x|\le 4x/\big(3(\gamma+1)\big)$ in $\Omega^+(\psi)\cap \{c_2-r<4\varepsilon\}$. We prove this estimate in §\[removeCutoffSection\] by choosing $\hat C$ sufficiently large depending only on the data.
Since the fixed point $\psi\in {{\mathcal K}}$ of the iteration map $J$ is a solution of (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]) for ${\phi}=\psi$, we conclude
1. \[iterProcItem-first\] $\psi\in C^{1,\alpha}(\overline{\Omega^+(\psi)})\cap
C^{2,\alpha}(\Omega^+(\psi))$;
2. $\psi=0$ on ${\Gamma_{sonic}}$ by (\[iterationCondOnSonicLine\]), and $\psi$ satisfies the original equation in $\Omega^+(\psi)$ by Step 9;
3. $D\psi=0$ on ${\Gamma_{sonic}}$ since $\|{\phi}\|_{2,\alpha,{{\mathcal D}'}}^{(par)}\leq M_1$;
4. \[iterProcItem\] $\psi=\varphi_1-\varphi_2$ on ${\Gamma_{shock}}(\psi)$ by (\[shockPL\])–(\[shockIterDef\]) since ${\phi}=\psi$;
5. \[iterProcItem-last\] The Rankine-Hugoniot gradient jump condition (\[RH-psi\]) holds on ${\Gamma_{shock}}(\psi)$. Indeed, as we have showed in (\[iterProcItem\]) above, the function $\varphi=\psi+\varphi_2$ satisfies (\[cont-accross-shock-mod-phi\]) on ${\Gamma_{shock}}(\psi)$. Since $\psi\in{{\mathcal K}}$, it follows that $\psi$ satisfies (\[cont-accross-shock-psi-resolved\]). Also, $\psi$ on ${\Gamma_{shock}}(\psi)$ satisfies (\[iterationRH\]) with ${\phi}=\psi$, which is (\[RH-psi-2\]). Since $\psi\in{{\mathcal K}}$ satisfies (\[cont-accross-shock-psi-resolved\]) and (\[RH-psi-2\]), it has been shown in §\[equationForPsiSection\] that $\varphi$ satisfies (\[RH-mod-phi\]) on ${\Gamma_{shock}}(\psi)$, i.e., $\psi$ satisfies (\[RH-psi\]).
Extend the function $\varphi=\psi+\varphi_2$ from $\Omega:=\Omega^+(\psi)$ to the whole domain $\Lambda$ by using (\[phi-states-0-1-2\]) to define $\varphi$ in $\Lambda\setminus\Omega$. Denote $\Lambda_0:=\{\xi>\xi_0\}\cap
\Lambda$, $\Lambda_1$ the domain with $\xi<\xi_0$ and above the reflection shock $P_0{{P_1}}{{P_2}}$, and $\Lambda_2:=\Lambda\setminus(\overline \Lambda_0\cup \overline
\Lambda_1)$. Set $S_0:=\{\xi=\xi_0\}\cap\Lambda$ the incident shock and $S_1:=P_0{{P_1}}{{P_2}}\cap \Lambda$ the reflected shock. We show in §\[proofSection\] that $S_1$ is a $C^2$–curve. Then we conclude that the domains $\Lambda_0$, $\Lambda_1$, and $\Lambda_2$ are disjoint, $\partial \Lambda_0\cap \Lambda=S_0$, $\partial \Lambda_1\cap \Lambda=S_0\cup S_1$, and $\partial \Lambda_2\cap\Lambda=S_1$. Properties (\[iterProcItem-first\])–(\[iterProcItem-last\]) above and the fact that $\psi$ satisfies in $\Omega$ imply that $$\varphi\in W^{1,\infty}_{loc}(\Lambda), \quad \varphi\in
C^1(\overline{\Lambda_i}) \cap C^{1,1}(\Lambda_i)\quad\mbox{for
}i=0,1,2,$$ $\varphi$ satisfies equation a.e. in $\Lambda$ and the Rankine-Hugoniot condition (\[FBConditionSelfSim-0\]) on the $C^2$-curves $S_0$ and $S_1$, which intersect only at $P_0\in\partial \Lambda$ and are transversal at the intersection point. Using this, Definition \[def2.1\], and the remarks after Definition \[def2.1\], we conclude that $\varphi$ is a weak solution of Problem 2, thus of Problem 1. Note that the solution is obtained for every ${\sigma}\in(0,{\sigma}_0]$, i.e., for every $\theta_w\in [\pi/2-{\sigma}_0, \pi/2]$ by (\[angleCloseToPiOver2\]), and that ${\sigma}_0$ depends only on the data since $\hat C$ is fixed in Step 9.
Vanishing Viscosity Approximation and Existence of Solutions of\
Problem (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]) {#unifElliptApproxSection}
================================================================
In this section we perform Step 2 of the iteration procedure described in §\[overViewProcedureSubsection\]. Through this section, we keep ${\phi}\in{{\mathcal K}}$ fixed, denote by ${{\mathcal P}}:=\{P_1,
P_2, P_3, P_4\}$ the set of the corner points of $\Omega^+({\phi})$, and use $\alpha\in (0, 1/2)$ defined in §\[Constr-iter-section\].
We regularize equation (\[iterationEquation\]) by the vanishing viscosity approximation via the uniformly elliptic equations $${{\mathcal N}}(\psi)+\delta\Delta\psi=0\qquad\mbox{for }\;\delta\in(0, 1).$$ That is, we consider the equation $${{\mathcal N}}_\delta(\psi){:=}(A_{11}+\delta)\psi_{{{\xi}}{{\xi}}}+ 2A_{12}\psi_{{{\xi}}{{\eta}}}
+(A_{22}+\delta)\psi_{{{\eta}}{{\eta}}}=0 \qquad \mbox{ in
}\;\;\Omega^+({\phi}). \label{unif-ellipt-iterationEquation}$$ In the domain ${\Omega'}$ in the $(x,y)$–coordinates defined by (\[coordNearSonic\]), this equation has the form $$\begin{aligned}
\label{unif-ellipt-iteration-equation-sonicStruct}
&&\qquad\big(
\delta+2x-(\gamma+1)x\zeta_1\big(\frac{\psi_x}{x}\big)+O_1^{\phi}\big)\psi_{xx}
+O_2^{\phi}\, \psi_{xy}\\
&&\qquad\,\,+ \big({1\over c_2}+{\delta\over (c_2-x)^2}+O_3^{\phi}\big)\psi_{yy}
-(1-{\delta\over c_2-x}+O_4^{\phi})\psi_{x} +O_5^{\phi}\,\psi_{y}=0
\nonumber\end{aligned}$$ by using (\[iteration-equation-sonicStruct\]) and writing the Laplacian operator $\Delta$ in the $(x,y)$–coordinates, which is easily derived from the form of $\Delta$ in the polar coordinates. The terms $O_k^{\phi}$ in (\[unif-ellipt-iteration-equation-sonicStruct\]) are defined by (\[iteration-equation-sonicStruct-coef\]).
We now study equation (\[unif-ellipt-iterationEquation\]) in $\Omega^+({\phi})$ with the boundary conditions (\[iterationRH\])–(\[iterationCondOnSymmtryLine\]).
We first note some properties of the boundary condition (\[iterationRH\]). Using Lemma \[relatingNorms\] with $\alpha\in(0, 1/2)$ and (\[condConst-00\]), we find $\|{\phi}\|_{2,\alpha,{{\mathcal D}}}^{(-1-\alpha, \Sigma_0\cup{\Gamma_{sonic}})} \le C$, where $C$ depends only on the data. Then, writing (\[iterationRH\]) as $${{\mathcal M}}(\psi)({{\xi}},{{\eta}}):= b_1({{\xi}},{{\eta}})\psi_{{\xi}}+
b_2({{\xi}},{{\eta}})\psi_{{\eta}}+ b_3({{\xi}},{{\eta}})\psi=0 \qquad\mbox{on
}\;{\Gamma_{shock}}({\phi}) \label{iterationRH-lf}$$ and using (\[RH-psi-2-error-term1\])–(\[obliquenessRH\]), we obtain $$\label{estCoefsIterRH-0}
\|b_i\|^{(-\alpha,\{{{P_1}},{{P_2}}\})}_{1,\alpha, {\Gamma_{shock}}({\phi})}\le C
\qquad \mbox{for }\;i=1,2,3,$$ where $C$ depends only on the data.
Furthermore, ${\phi}\in{{\mathcal K}}$ with (\[condConst-00\]) implies that $$\|{\phi}\|_{C^1}\le M_1\varepsilon+M_2{\sigma}\le \varepsilon^{3/4}/\hat
C.$$ Then, using (\[RH-psi-2-error-term1\])–(\[obliquenessRH\]) and assuming that $\hat C$ in (\[condConst-00\]) is sufficiently large, we have $$\label{estCoefsIterRH}
\begin{array}{l}
(b_1({{\xi}}, {{\eta}}), b_2({{\xi}}, {{\eta}}))\cdot\nu({{\xi}}, {{\eta}})\ge
\frac{1}{4}\rho_2'(c_2^2-\hat{{{\xi}}}^2)>0\qquad\mbox{for any }\; ({{\xi}},
{{\eta}})\in{\Gamma_{shock}}({\phi}), \displaystyle
\\
b_1({{\xi}}, {{\eta}})\ge
\frac{1}{2}\rho_2'(c_2^2-\hat{{{\xi}}}^2)>0\qquad\mbox{for any }\; ({{\xi}},
{{\eta}})\in{\Gamma_{shock}}({\phi}), \displaystyle
\\
\left|b_2({{\xi}}, {{\eta}})-{{\eta}}\big(\frac{\rho_2-\rho_1}{u_1}-\rho_2'\hat{{\xi}}\big)\right|\le \varepsilon^{3/4}\qquad\mbox{for any }\; ({{\xi}},
{{\eta}})\in{\Gamma_{shock}}({\phi}), \displaystyle
\\
\left|b_3({{\xi}}, {{\eta}})+\big(\frac{\rho_2-\rho_1}{u_1}-\rho_2'\hat{{\xi}}\big)\right|\le\varepsilon^{3/4}\qquad\mbox{for any }\; ({{\xi}},
{{\eta}})\in{\Gamma_{shock}}({\phi}). \displaystyle
\end{array}$$
Now we write condition (\[iterationRH\]) in the $(x,y)$–coordinates on ${\Gamma_{shock}}({\phi})\cap\overline{{\mathcal D}'}$. Then we obtain the following condition of the form $${{\mathcal M}}(\psi)(x,y)=\hat b_1(x,y)\psi_x + \hat b_2(x,y)\psi_y + \hat
b_3(x,y)\psi=0 \qquad\mbox{on }\;{\Gamma_{shock}}({\phi})\cap\overline{{\mathcal D}'},
\label{iterationRH-lf-flattened}$$ where $\hat b_1(x,y)=b_1(\xi,\eta){\partial x\over\partial\xi}
+b_2(\xi,\eta){\partial x\over\partial\eta},
\hat b_2(x,y)=b_1(\xi,\eta){\partial y\over\partial\xi}
+b_2(\xi,\eta){\partial y\over\partial\eta},$ and $\hat b_3(x,y)=b_3(\xi,\eta)$. Condition (\[iterationRH\]) is oblique, by the first inequality in (\[estCoefsIterRH\]). Then, since transformation (\[coordNearSonic\]) is smooth on $\{0<c_2-r<2\varepsilon\}$ and has nonzero Jacobian, it follows that (\[iterationRH-lf-flattened\]) is oblique, that is, $$\label{obliqInxy-1}
(\hat b_1(x,y),\hat b_2(x,y))\cdot\nu_s(x,y)\ge C^{-1}>0 \qquad
\mbox{ on }\; {\Gamma_{shock}}({\phi})\cap\overline{{\mathcal D}'},$$ where $\hat\nu_s=\hat\nu_s(x,y)$ is the interior unit normal at $(x,y)\in{\Gamma_{shock}}({\phi})\cap {{\mathcal D}'}$ to ${\Omega}({\phi})$.
As we have showed in §\[sectionEqNearSonicLine\], writing the left-hand side of (\[RH-psi-2\]) in the $(x,y)$–coordinates, we obtain the left-hand side of (\[RH-psi-3-xy\]). Thus, (\[iterationRH-lf-flattened\]) is obtained from (\[RH-psi-3-xy\]) by substituting ${\phi}(x,y)$ into $\hat E_1$ and $\hat E_2$. Also, from (\[holder-hat-f-S\]) with $\hat
f_{\phi}(0)=\hat f_0(0)=y_1$, we estimate $|y-y_1|=|\hat
f_{\phi}(x)-\hat f_{\phi}(0)|\le CM_1\varepsilon$ on ${\Gamma_{shock}}\cap\{x<2\varepsilon\}$. Then, using (\[RH-psi-3-xy\])–(\[RH-psi-2-error-term-xy-2\]) and ${{\xi}}_1<0$, we find that, if $\hat C$ in (\[condConst-00\]) is sufficiently large depending only on the data, then $$\label{estCoefsIterRH-flattened}
\begin{array}{l}
\|\hat b_i\|^{(-1,\{{{P_1}}\})}_{1,\alpha, {\Gamma_{shock}}({\phi})\cap\overline{{\mathcal D}'}}\le
CM_1 \qquad \mbox{for }\;i=1,2,3, \displaystyle
\\
\hat b_1(x,y)\le -\frac{1}{2}\frac{\rho_2-\rho_1}{u_1}
\frac{{{\eta}}^2_1}{c_2}<0\qquad\mbox{for }\;
(x,y)\in{\Gamma_{shock}}({\phi})\cap\overline{{\mathcal D}'}, \displaystyle
\\
\hat b_2(x,y)\le -\frac{1}{2}{{\eta}}_1 \big(\rho_2'
+\frac{\rho_2-\rho_1}{u_1 c_2^2}|{{\xi}}_1| \big)<0\qquad\mbox{for }\;
(x,y)\in{\Gamma_{shock}}({\phi})\cap\overline{{\mathcal D}'}, \displaystyle
\\
\hat
b_3(x,y)\le-\frac{1}{2}\big(\rho_2'|{{\xi}}_1|+\frac{\rho_2-\rho_1}{u_1}
\big)<0\qquad\mbox{for }\; (x,y)\in{\Gamma_{shock}}({\phi})\cap\overline{{\mathcal D}'},
\displaystyle
\end{array}$$ where $C$ depends only on the data.
Now we state the main existence result for the regularized problem.
\[existSolUnifEllipt\] There exist $\hat C, C, \delta_0>0$ depending only on the data such that, if ${\sigma}, \varepsilon>0$ and $M_1, M_2\ge 1$ in [(\[defSetK\_R\])]{} satisfy [(\[condConst-00\])]{}, then, for every $\delta\in(0,\delta_0)$, there exists a unique solution $\psi\in C^{(-1-\alpha, {{\mathcal P}})}_{2,\alpha,\Omega^+({\phi})}$ of [(\[unif-ellipt-iterationEquation\])]{} and [(\[iterationRH\])]{}–[(\[iterationCondOnSymmtryLine\])]{}, and this solution satisfies $$\begin{aligned}
&&0\le\psi({{\xi}}, {{\eta}})\le C{\sigma}\qquad\mbox{for }\;({{\xi}},
{{\eta}})\in\Omega^+({\phi}), \label{L-infty-for-unif-ellipt}
\\
&&|\psi(x,y)|\le C{{\sigma}\over \varepsilon} x\qquad \mbox{for
}\;(x,y)\in{\Omega'}, \label{barier-for-unif-ellipt}\end{aligned}$$ where we have used coordinates [(\[coordNearSonic\])]{} in [(\[barier-for-unif-ellipt\])]{}. Moreover, for any $s\in (0,
c_2/4)$, there exists $C(s)>0$ depending only on the data and $s$, but independent of $\delta\in (0, \delta_0)$, such that $$\|\psi\|^{(-1-\alpha,
\{{{P_2}},{{P_3}}\})}_{2,\alpha,\Omega^+_s({\phi})}\le C(s){\sigma},
\label{Hoder-est-for-unif-ellipt}$$ where $\Omega^+_s({\phi}):=\Omega^+({\phi})\cap\{c_2-r>s\}$.
Note that equation (\[unif-ellipt-iterationEquation\]) is nonlinear and the boundary conditions (\[iterationRH\])–(\[iterationCondOnSymmtryLine\]) are linear. We find a solution of (\[iterationRH\])–(\[iterationCondOnSymmtryLine\]) and (\[unif-ellipt-iterationEquation\]) as a fixed point of the map $$\label{6.9a}
\hat J: C^{1,\alpha/2}(\overline{\Omega^+({\phi})}) \to
C^{1,\alpha/2}(\overline{\Omega^+({\phi})})$$ defined as follows: For $\hat \psi\in
C^{1,\alpha/2}(\overline{\Omega^+({\phi})})$, we consider the linear elliptic equation obtained by substituting $\hat\psi$ into the coefficients of equation (\[unif-ellipt-iterationEquation\]): $$a_{11}\psi_{{{\xi}}{{\xi}}}+ 2a_{12}\psi_{{{\xi}}{{\eta}}} +a_{22}\psi_{{{\eta}}{{\eta}}}=0
\qquad \mbox{ in }\;\;\Omega^+({\phi}),
\label{unif-ellipt-linear-iterationEquation}$$ where $$a_{ij}({{\xi}},{{\eta}})=A_{ij}(D\hat \psi({{\xi}},{{\eta}}),{{\xi}},{{\eta}}) +\delta\,
\delta_{ij}\qquad \mbox{ for }\;({{\xi}},{{\eta}})\in\Omega^+({\phi}),\,\,
i,j=1,2, \label{coef-linear-iterationEquation}$$ with $\delta_{ij}=1$ for $i=j$ and $0$ for $i\ne j$, $i,j=1,2$. We establish below the existence of a unique solution $\psi\in
C^{(-1-\alpha, {{\mathcal P}})}_{2,\alpha/2,\Omega^+({\phi})}$ to the linear elliptic equation (\[unif-ellipt-linear-iterationEquation\]) with the boundary conditions (\[iterationRH\])–(\[iterationCondOnSymmtryLine\]). Then we define $\hat J(\hat\psi)=\psi$.
We first state some properties of equation (\[unif-ellipt-linear-iterationEquation\]).
\[unifEllipticityOfLinearEq-Lemma\] There exists $\hat C>0$ depending only on the data such that, if ${\sigma}, \varepsilon>0$ and $M_1, M_2\ge 1$ in [(\[defSetK\_R\])]{} satisfy [(\[condConst-00\])]{}, and $\delta\in(0,1)$, then, for any $\hat \psi\in
C^{1,\alpha/2}(\overline{\Omega^+({\phi})})$, equation [(\[unif-ellipt-linear-iterationEquation\])]{} is uniformly elliptic in $\Omega^+({\phi})$: $$\label{ellipticityOfIterEq-Linear-0}
\delta|\mu|^2\le \sum_{i,j=1}^2 a_{ij}({{\xi}},{{\eta}})\mu_i\mu_j \le
2{\lambda}^{-1}|\mu|^2 \qquad\mbox{for }({{\xi}},
{{\eta}})\in\Omega^+({\phi}),\,
\mu\in{ {\bf R}}^2,$$ where $\lambda$ is from Lemma [\[propertiesNonlinCoeffs\]]{}. Moreover, for any $s\in (0, c_2/2)$, the ellipticity constants depend only on the data and are independent of $\delta$ in $\Omega^+_s({\phi})=\Omega^+({\phi})\cap\{c_2-r>s\}$: $$\label{ellipticityOfIterEq-Linear}
\lambda(c_2-s)|\mu|^2 \le\sum_{i,j=1}^2a_{ij}({{\xi}},{{\eta}})\mu_i\mu_j \le
2{\lambda}^{-1}|\mu|^2 \qquad\mbox{for
}z=({{\xi}},{{\eta}})\in\Omega^+_s({\phi}),\,\mu\in{ {\bf R}}^2.$$ Furthermore, $$\label{holder-coef}
a_{ij}\in C^{\alpha/2}(\overline{\Omega^+({\phi})}).$$
Facts (\[ellipticityOfIterEq-Linear-0\])–(\[ellipticityOfIterEq-Linear\]) directly follow from the definition of $a_{ij}$ and both the definition and properties of $A_{ij}$ in §\[eqForIterationSection\] and Lemma \[propertiesNonlinCoeffs\].
Since $A_{ij}(p,{{\xi}},{{\eta}})$ are independent of $p$ in $\Omega^+({\phi})\cap \{c_2-r>4\varepsilon\}$, it follows from (\[iterationUniforDomEquation\]), (\[iterationSonicDomEquation\]), and ${\phi}\in{{\mathcal K}}$ that $a_{ij}
\in C^{(-\alpha,\Sigma_0)}_{1,\alpha/2,\Omega^+({\phi})\cap {{\mathcal D}''}}
\subset C^\alpha(\overline{\Omega^+({\phi})\cap {{\mathcal D}''}})$.
To show $a_{ij}\in C^{\alpha/2}(\overline{\Omega^+({\phi})})$, it remains to prove that $a_{ij}\in
C^{\alpha/2}(\overline{\Omega({\phi})\cap {{\mathcal D}'}})$. To achieve this, we note that the nonlinear terms in the coefficients $A_{ij}(p,{{\xi}},{{\eta}})$ are only the terms $$(c_2-r)\zeta_1(\frac{{{\xi}}\psi_{{\xi}}+{{\eta}}\psi_{{\eta}}}{r(c_2-r)}).$$ Since $\zeta_1$ is a bounded and $C^\infty$-smooth function on ${ {\bf R}}$, and $\zeta'_1$ has compact support, then there exists $C>0$ such that, for any $s>0$, $q\in
{ {\bf R}}$, $$\label{propZeta1}
\left|s\zeta_1(\frac{q}{s})\right|\le
\big(\sup_{t\in{ {\bf R}}}|\zeta_1(t)|\big)s, \qquad
\left|D_{(q,s)}\big(s\zeta_1(\frac{q}{s})\big)\right| \le C.$$ Then it follows that the function $$F(p,{{\xi}}, {{\eta}})= (c_2-r)\zeta_1(\frac{{{\xi}}p_1+{{\eta}}p_2}{r(c_2-r)})$$ satisfies $|F(p,{{\xi}}, {{\eta}})|\le \|\zeta_1\|_{L^\infty({ {\bf R}})}(c_2-r)$ for any $(p,{{\xi}},{{\eta}})\in{ {\bf R}}^2\times{{\mathcal D}'}$, and $|{{D}}_{(p,{{\xi}},
{{\eta}})}F|$ is bounded on compact subsets of ${ {\bf R}}^2\times\overline{{\mathcal D}'}$. From this and $\hat\psi\in
C^{1,\alpha/2}(\overline{\Omega^+({\phi})})$, we have $a_{ij}\in
C^{\alpha/2}(\overline{\Omega^+({\phi})})$. Now we state some properties of equation (\[unif-ellipt-linear-iterationEquation\]) written in the $(x,y)$–coordinates.
\[unifEllipticityOfLinearEq-xy-Lemma\] There exist $\lambda>0$ and $C,\, \hat C>0$ depending only on the data such that, if ${\sigma}, \varepsilon>0$ and $M_1, M_2\ge 1$ in [(\[defSetK\_R\])]{} satisfy [(\[condConst-00\])]{}, and $\delta\in(0,1)$, then, for any $\hat \psi\in
C^{1,\alpha/2}(\overline{\Omega^+({\phi})})$, equation [(\[unif-ellipt-linear-iterationEquation\])]{} written in the $(x,y)$–coordinates has the structure $$\hat a_{11}\psi_{xx}+ 2\hat a_{12}\psi_{xy} +\hat a_{22} \psi_{yy}
+\hat a_1\psi_{x} +\hat a_2\psi_{y} =0 \quad\; \mbox{in
}\;\Omega^+({\phi})\cap{{\mathcal D}'}_{4\varepsilon},
\label{unif-ellipt-linear-xy-iterationEquation}$$ where $\hat a_{ij}=\hat a_{ij}(x,y)$ and $\hat a_i=\hat a_{i}(x,y)$ satisfy $$\label{holder-coef-xy}
\hat a_{ij}, \hat a_{i}\in
C^{\alpha/2}(\overline{\Omega^+({\phi})\cap{{\mathcal D}'}_{4\varepsilon}})
\qquad\mbox{for }\;i,j=1,2,$$ and the ellipticity condition $$\label{ellipticityOfIterEq-Linear-xy-0}
\begin{array}{l}
\displaystyle \delta\lambda|\mu|^2\le \sum_{i,j=1}^2 \hat
a_{ij}({{\xi}},{{\eta}})\mu_i\mu_j \le \lambda^{-1}|\mu|^2 \quad\mbox{ for
any }(x,y)\in\Omega^+({\phi})\cap{{\mathcal D}'}_{4\varepsilon},\, \mu\in{ {\bf R}}^2.
\end{array}$$ Moreover, $$\begin{aligned}
&& \qquad\qquad \delta\le \hat a_{11}(x,y)\le\delta+2x, \,\,\,
{1\over 2c_2} \le \hat a_{22}(x,y) \le {2\over c_2}, \,\,\, -2\le
\hat a_{1}(x,y)\le
-{1\over 2}, \nonumber\\
&&\qquad\qquad |(\hat a_{12}, \hat a_{21}, \hat a_{2})(x,y)|\le
C|x|,
\qquad |D(\hat a_{12}, \hat a_{21}, \hat a_{2})(x,y)| \le C|x|^{1/2},
\label{estSmallterms-iter-lin}
\\
&& \qquad \qquad|\hat a_{ii}(x,y)-\hat a_{ii}(0, \tilde y)|\le
C\,|(x,y)-(0,\tilde y)|^\alpha
\qquad\mbox{for }\;i=1,2,\nonumber\end{aligned}$$ for all $(x,y), (0,\tilde y)\in
\Omega^+({\phi})\cap{{\mathcal D}'}_{4\varepsilon}$.
By (\[condRewritingRH-0\]), if $\varepsilon\le \bar c_2/10$, then the change of variables from $({{\xi}}, {{\eta}})$ to $(x,y)$ in ${{\mathcal D}'}_{4\varepsilon}$ is smooth and smoothly invertible with Jacobian bounded away from zero, where the norms and lower bound of the Jacobian depend only on the data. Now (\[ellipticityOfIterEq-Linear-xy-0\]) follows from (\[ellipticityOfIterEq-Linear\]).
Equation (\[unif-ellipt-linear-iterationEquation\]) written in the $(x,y)$–coordinates can be obtained by substituting $\hat \psi$ into the term $\displaystyle x\zeta_1(\frac{\psi_x}{x})$ in the coefficients of equation (\[unif-ellipt-iteration-equation-sonicStruct\]). Using (\[propZeta1\]), the assertions in (\[holder-coef-xy\]) and (\[estSmallterms-iter-lin\]), except the last inequality, follow directly from (\[unif-ellipt-iteration-equation-sonicStruct\]) with (\[iteration-equation-sonicStruct-coef\]) and (\[erTerms-xy-nontrunc\]), ${\phi}\in{{\mathcal K}}$ with (\[condConst-00\]), and $\hat \psi\in
C^{1,\alpha/2}(\overline{\Omega^+({\phi})})$.
Then we prove the last inequality in (\[estSmallterms-iter-lin\]). We note that, from (\[unif-ellipt-iteration-equation-sonicStruct\]) and (\[iteration-equation-sonicStruct-coef\]), it follows that $\hat
a_{ii}(x,y)=F_{ii}({{D}}{\phi},{\phi},x,y)+G_{ii}(x)x\zeta_1(\frac{\hat
\psi_x}{x})$, where $F_{ii}$ and $G_{ii}$ are smooth functions, and ${\phi}$ and $\hat \psi$ are evaluated at $(x,y)$. In particular, since $\zeta_1(\cdot)$ is bounded, $\hat
a_{ii}(0,y)=F_{ii}({{D}}{\phi}(0,y),{\phi}(0,y),0,y)$. Thus, assuming $x>0$, we use the boundedness of $\zeta_1$ and $G_{ii}$, smoothness of $F_{ii}$, and ${\phi}\in{{\mathcal K}}$ with Lemma \[relatingNorms\] to obtain $$\begin{aligned}
&&|\hat a_{ii}(x,y)-\hat a_{ii}(0, \tilde y)|\\
&&\le |F_{ii}({{D}}{\phi}(x,y),{\phi}(x,y),x,y)-
F_{ii}({{D}}{\phi}(0,\tilde y),{\phi}(0,\tilde y),0,\tilde y)|\\
&&\quad + x|G_{ii}(x)\zeta_1(\frac{\hat \psi_x(x,y)}{x})|\\
&&\le Cx+C(M_1\epsilon^{1-\alpha}+M_2{\sigma})|(x,y)-(0,\tilde
y)|^\alpha \le C|(x,y)-(0,\tilde y)|^\alpha,\end{aligned}$$ where the last inequality holds since $\alpha\in(0, 1/2)$ and (\[condConst-00\]). If $x=0$, the only difference is that the first term is dropped in the estimates.
\[comparisonPrincipleOfUnifEllipt-Lemma\] There exists $\hat C>0$ depending only on the data such that, if ${\sigma}, \varepsilon>0$ and $M_1, M_2\ge 1$ in [(\[defSetK\_R\])]{} satisfy [(\[condConst-00\])]{}, and $\delta\in(0,1)$, the following comparison principle holds: Let $\psi\in C(\overline{\Omega^+({\phi})}) \cap
C^{1}(\overline{\Omega^+({\phi})}\setminus\overline{\Gamma_{sonic}}) \cap
C^2(\Omega^+({\phi}))$, let the left-hand sides of [(\[unif-ellipt-linear-iterationEquation\])]{}, [(\[iterationRH\])]{}, and [(\[iterationCondOnWedge\])]{}–[(\[iterationCondOnSymmtryLine\])]{} are nonpositive for $\psi$, and let $\psi\ge 0$ on ${\Gamma_{sonic}}$. Then $$\psi\ge 0 \qquad \mbox{in }\, \Omega^+({\phi}).$$
We assume that $\hat C$ is large so that (\[q2-u1\])–(\[OmegaPL\]) hold.
We first note that the boundary condition [(\[iterationRH\])]{} on ${\Gamma_{shock}}({\phi})$, written as (\[iterationRH-lf\]), satisfies $$(b_1, b_2)\cdot\nu>0, \qquad b_3<0 \qquad \mbox{on ${\Gamma_{shock}}({\phi})$,}$$ by (\[estCoefsIterRH\]) combined with $\hat{{\xi}}<0$ and $\rho_2>\rho_1$. Thus, if $\psi$ is not a constant in $\Omega^+({\phi})$, a negative minimum of $\psi$ over $\overline{\Omega^+({\phi})}$ cannot be achieved:
1. In the interior of $\Omega^+({\phi})$, by the strong maximum principle for linear elliptic equations;
2. In the relative interiors of ${\Gamma_{shock}}({\phi})$, ${\Gamma_{wedge}}$, and $\partial \Omega^+({\phi})\cap
\{{{\eta}}=-v_2\}$, by Hopf’s Lemma and the oblique derivative conditions [(\[iterationRH\])]{} and [(\[iterationCondOnWedge\])]{}–[(\[iterationCondOnSymmtryLine\])]{};
3. In the corners ${{P_2}}$ and ${{P_3}}$, by the result in Lieberman [@Lieberman87 Lemma 2.2], via a standard argument as in [@GilbargTrudinger Theorem 8.19]. Note that we have to flatten the curve ${\Gamma_{shock}}$ in order to apply [@Lieberman87 Lemma 2.2] near ${{P_2}}$, and this flattening can be done by using the $C^{1,\alpha}$ regularity of ${\Gamma_{shock}}$.
Using that $\psi\ge 0$ on ${\Gamma_{sonic}}$, we conclude the proof.
\[unifEstOfUnifEllipt-Lemma\] There exists $\hat C>0$ depending only on the data such that, if ${\sigma}, \varepsilon>0$ and $M_1, M_2\ge 1$ in [(\[defSetK\_R\])]{} satisfy [(\[condConst-00\])]{}, and $\delta\in(0,1)$, then any solution $\psi\in
C(\overline{\Omega^+({\phi})}) \cap
C^{1}(\overline{\Omega^+({\phi})}\setminus\overline{\Gamma_{sonic}}) \cap
C^2(\Omega^+({\phi}))$ of [(\[unif-ellipt-linear-iterationEquation\])]{} and [(\[iterationRH\])]{}–[(\[iterationCondOnSymmtryLine\])]{} satisfies [(\[L-infty-for-unif-ellipt\])]{}–[(\[barier-for-unif-ellipt\])]{} with the constant $C$ depending only on the data.
First we note that, since $\Omega^+({\phi})\subset\{{{\eta}}<c_2\}$, the function $$w({{\xi}},{{\eta}})=-v_2({{\eta}}-c_2)$$ is a nonnegative supersolution of (\[unif-ellipt-linear-iterationEquation\]) and (\[iterationRH\])–(\[iterationCondOnSymmtryLine\]): Indeed,
1. $w$ satisfies (\[unif-ellipt-linear-iterationEquation\]) and (\[iterationCondOnSymmtryLine\]);
2. $w$ is a supersolution of (\[iterationRH\]). This can be seen by using (\[iterationRH-lf\]), (\[estCoefsIterRH\]), $\rho_2>\rho_1$, $u_1>0$, $\rho'_2>0$ $\hat{{\xi}}<0$, and $|{{\eta}}|\le c_2$ to compute on ${\Gamma_{shock}}$: $${{\mathcal M}}(w)=-b_2v_2 -b_3v_2({{\eta}}-c_2)\le
-v_2\Big(\rho_2'|\hat{{\xi}}|
+\frac{\rho_2-\rho_1}{u_1}-\varepsilon^{3/4}(1+2c_2) \Big)<0$$ if $\varepsilon$ is small depending on the data, which is achieved by the choice of $\hat C$ in (\[condConst-00\]);
3. $w$ is a supersolution of (\[iterationCondOnWedge\]). This follows from ${{D}}w\cdot\nu=-c_2\cos\theta_w<0$ since the interior unit normal on ${\Gamma_{wedge}}$ is $\nu=(-\sin\theta_w, \cos\theta_w)$;
4. $w\ge 0$ on ${\Gamma_{sonic}}$.
Similarly, $\tilde w\equiv 0$ is a subsolution of (\[unif-ellipt-linear-iterationEquation\]) and (\[iterationRH\])–(\[iterationCondOnSymmtryLine\]). Thus, by the Comparison Principle (Lemma \[comparisonPrincipleOfUnifEllipt-Lemma\]), any solution $\psi\in
C(\overline{\Omega^+({\phi})}) \cap
C^{1}(\overline{\Omega^+({\phi})}\setminus\overline{\Gamma_{sonic}}) \cap
C^2(\Omega^+({\phi}))$ satisfies $$0\le\psi({{\xi}}, {{\eta}})\le w({{\xi}}, {{\eta}})\qquad\mbox{for any }\;({{\xi}},
{{\eta}})\in\Omega^+({\phi}).$$ Since $|v_2|\le C{\sigma}$, then (\[L-infty-for-unif-ellipt\]) follows.
To prove (\[barier-for-unif-ellipt\]), we work in the $(x,y)$–coordinates in ${{\mathcal D}'}\cap \Omega^+({\phi})$ and assume that $\hat C$ in (\[condConst-00\]) is sufficiently large so that the assertions of Lemma \[unifEllipticityOfLinearEq-xy-Lemma\] hold. Let $v(x,y)=L{\sigma}x$ for $L>0$. Then
\(i) $v$ is a supersolution of equation (\[unif-ellipt-linear-xy-iterationEquation\]) in ${\Omega'}\cap\{x<\varepsilon\}$: Indeed, the left-hand side of (\[unif-ellipt-linear-xy-iterationEquation\]) on $v(x,y)=L{\sigma}x$ is $\hat a_1(x,y) L{\sigma}$, which is negative in ${{\mathcal D}'}\cap
\Omega^+({\phi})$ by (\[estSmallterms-iter-lin\]);
\(ii) $v$ satisfies the boundary conditions (\[condOnSonicLine-Psi-xy\]) on $\partial\Omega^+({\phi})\cap
\{x=0\}$ and (\[condOnWedge-Psi-xy\]) on $\partial
\Omega^+({\phi})\cap \{y=0\}$;
\(iii) The left-hand side of (\[iterationRH-lf-flattened\]) is negative for $v$ on ${\Gamma_{shock}}\cap\{x<\varepsilon\}$: Indeed, ${{\mathcal M}}(v)(x,y)=L{\sigma}(\hat b_1 +\hat b_3x)<0$ by (\[estCoefsIterRH-flattened\]) and since $x\ge0$ in $\overline{\Omega'}$.
Now, choosing $L$ large so that $ L\varepsilon>C$ where $C$ is the constant in (\[L-infty-for-unif-ellipt\]), we have by (\[L-infty-for-unif-ellipt\]) that $v\ge\psi$ on $\{x=\varepsilon\}$. By the Comparison Principle, which holds since equation (\[unif-ellipt-linear-xy-iterationEquation\]) is elliptic and condition (\[iterationRH-lf-flattened\]) satisfies (\[obliqInxy-1\]) and $\hat b_3<0$ where the last inequality follows from (\[estCoefsIterRH-flattened\]), we obtain $v\ge\psi$ in $\Omega^+({\phi})\cap\{x<\varepsilon\}$. Similarly, $-\psi\ge -v$ in $\Omega^+({\phi})\cap\{x<\varepsilon\}$. Then (\[barier-for-unif-ellipt\]) follows.
\[C2alpha-Lin-UnifEllipt-Lemma\] There exists $\hat C>0$ depending only on the data such that, if ${\sigma}, \varepsilon>0$ and $M_1, M_2\ge 1$ in [(\[defSetK\_R\])]{} satisfy [(\[condConst-00\])]{}, and $\delta\in(0,1)$, any solution $\psi\in
C(\overline{\Omega^+({\phi})}) \cap
C^{1}(\overline{\Omega^+({\phi})}\setminus\overline{\Gamma_{sonic}}) \cap
C^2(\Omega^+({\phi}))$ of [(\[unif-ellipt-linear-iterationEquation\])]{} and [(\[iterationRH\])]{}–[(\[iterationCondOnSymmtryLine\])]{} satisfies $$\|\psi\|^{(-1-\alpha,
\{{{P_2}},{{P_3}}\})}_{2,\alpha/2,\Omega^+_s({\phi})} \le
C(s,\hat\psi){\sigma}\label{Hoder-est-for-lin-unif-ellipt}$$ for any $s\in (0, c_2/2)$, where the constant $C(s, \hat\psi)$ depends only on the data, $\|\hat\psi\|_{C^{1,\alpha/2}(\overline{\Omega^+({\phi})})}$, and $s$.
From (\[OmegaPL\]), (\[OmegaPL-f-higher\]), (\[estCoefsIterRH-0\])–(\[estCoefsIterRH\]), (\[ellipticityOfIterEq-Linear\])–(\[holder-coef\]), and the choice of $\alpha$ in §\[Constr-iter-section\], it follows by [@Lieberman88 Lemma 1.3] that $$\label{weakerLinearObliqueEstimate}
\|\psi\|^{(-1-\alpha, \Sigma_0\cup{\Gamma_{shock}}({\phi})\cup{\Gamma_{wedge}})}_{2,\alpha/2,\Omega^+_s({\phi})}\le
C(s,\hat\psi)(\|\psi\|_{C(\Omega^+({\phi}))}+|v_2|)\le C(s,\hat\psi){\sigma},$$ where we have used (\[u2-v2-bound\]) and Lemma \[unifEstOfUnifEllipt-Lemma\] in the second inequality.
In deriving (\[weakerLinearObliqueEstimate\]), we have used (\[OmegaPL-f-higher\]) and (\[estCoefsIterRH-0\]) only to infer that ${\Gamma_{shock}}({\phi})$ is a $C^{1,\alpha}$–curve and $b_i\in
C^\alpha(\overline{{\Gamma_{shock}}({\phi})})$. To improve (\[weakerLinearObliqueEstimate\]) to (\[Hoder-est-for-lin-unif-ellipt\]), we use the higher regularity of ${\Gamma_{shock}}({\phi})$ and $b_i$, given by (\[OmegaPL-f-higher\]) and (\[estCoefsIterRH-0\]) (and a similar regularity for the boundary conditions (\[iterationCondOnWedge\])–(\[iterationCondOnSymmtryLine\]), which are given on the flat segments and have constant coefficients), combined with rescaling from the balls $B_{d/2}(z)\cap \Omega^+({\phi})$ for any $z\in
\overline{\Omega^+_s({\phi})}\setminus\{{{P_2}}, {{P_3}}\}$ (with $d={ \mbox{dist}}(z, \{{{P_2}},{{P_3}}\}\cup\Sigma_0)$) into the unit ball and the standard estimates for the oblique derivative problems for linear elliptic equations.
Now we show that the solution $\psi$ is $C^{2,\alpha/2}$ near the corner ${{P_4}}={\Gamma_{sonic}}\cap{\Gamma_{wedge}}({\phi})$. We work in ${{\mathcal D}'}$ in the $(x,y)$–coordinates.
\[wedge-sonic-Lin-UnifEllipt-Lemma\] There exists $\hat C>0$ depending only on the data such that, if ${\sigma}, \varepsilon>0$ and $M_1, M_2\ge 1$ in [(\[defSetK\_R\])]{} satisfy [(\[condConst-00\])]{}, and $\delta\in(0,1)$, any solution $\psi\in
C(\overline{\Omega^+({\phi})}) \cap
C^{1}(\overline{\Omega^+({\phi})}\setminus\overline{\Gamma_{sonic}}) \cap
C^2(\Omega^+({\phi}))$ of [(\[unif-ellipt-linear-iterationEquation\])]{} and [(\[iterationRH\])]{}–[(\[iterationCondOnSymmtryLine\])]{} is in $C^{2,\alpha/2}(\overline{B_{\varrho}({{P_4}})\cap\Omega^+({\phi})})$ for sufficiently small ${\varrho}>0$.
In this proof, the constant $C$ depends only on the data, $\delta$, and $\|(\hat a_{ij}, \hat a_{i})\|_{
C^{\alpha/2}(\overline{\Omega^+({\phi})})}$ for $i, j=1,2$, i.e., $C$ is independent of ${\varrho}$.
[*Step 1.*]{} We work in the $(x,y)$–coordinates. Then ${{P_4}}=(0,0)$ and $\Omega^+({\phi})\cap B_{2{\varrho}}= \{x>0,y>0\})\cap
B_{2{\varrho}}$ for ${\varrho}\in (0, \varepsilon)$. Denote $$B_{\varrho}^{+}:=B_{\varrho}(0)\cap\{x>0\},\qquad B_{\varrho}^{++}:=B_{\varrho}(0)\cap\{x>0,
y>0\}.$$ Then $\psi$ satisfies equation (\[unif-ellipt-linear-xy-iterationEquation\]) in $B_{2{\varrho}}^{++}$ and $$\begin{aligned}
&&\psi=0 \qquad\mbox{on }\;{\Gamma_{sonic}}\cap B_{2{\varrho}} =B_{2{\varrho}}\cap \{x=0,
y>0\}, \label{condOnSonicLine-Psi-xy-loc}
\\
&&\psi_\nu\equiv\psi_y=0 \qquad\mbox{on }\;{\Gamma_{wedge}}\cap
B_{2{\varrho}}=B_{2{\varrho}}\cap \{y=0, x>0\}. \label{condOnWedge-Psi-xy-loc}\end{aligned}$$ Rescale $\psi$ by $$v(z)=\psi({\varrho}z)\qquad \mbox{ for } z=(x,y)\in B_2^{++}.$$ Then $v\in C(\overline{B_2^{++}}) \cap
C^{1}(\overline{B_2^{++}}\setminus\overline{\{x=0\}}) \cap
C^2(B_2^{++})$ satisfies $$\|v\|_{L^\infty(B_2^{++})}= \|\psi\|_{L^\infty(B_{2{\varrho}}^{++})},
\label{L-infty-for-unif-ellipt-v}$$ and $v$ is a solution of $$\begin{aligned}
&&\hat a_{11}^{({\varrho})}v_{xx}+ 2\hat a_{12}^{({\varrho})}v_{xy} +\hat
a_{22}^{({\varrho})}v_{yy} +\hat a_1^{({\varrho})}v_{x} +\hat a_2^{({\varrho})}v_{y}
=0 \qquad\mbox{in }\;B_2^{++}, \label{unif-ellipt-linear-xy-v}
\\
&&v=0\qquad\mbox{on }\;\partial B_2^{++}\cap \{x=0\},
\label{condOnSonicLine-v-xy-loc}
\\
&&v_\nu\equiv v_y=0\qquad\mbox{on }\;\partial B_2^{++}\cap \{y=0\},
\label{condOnWedge-v-xy-loc}\end{aligned}$$ where $$\label{def-rescaled-coef-v}
\hat a_{ij}^{({\varrho})}(x,y)=\hat a_{ij}({\varrho}x,{\varrho}y),\,\,\, \hat
a_{i}^{({\varrho})}(x,y)={\varrho}\,\hat a_{i}({\varrho}x,{\varrho}y)\quad\mbox{for }\;
(x,y)\in B_2^{++},\;i,j=1,2.$$ Thus, $\hat a_{ij}^{({\varrho})}$ satisfy (\[ellipticityOfIterEq-Linear-xy-0\]) with the unchanged constant $\lambda>0$ and, since ${\varrho}\le 1$, $$\label{holder-coef-xy-v}
\|(\hat a_{ij}^{({\varrho})}, \hat a_{i}^{({\varrho})})\|_{
C^{\alpha/2}(\overline{B_2^{++}})} \le \|(\hat a_{ij}, \hat
a_{i})\|_{ C^{\alpha/2}(\overline{\Omega^+({\phi})})} \qquad\mbox{for
}\;i,j=1,2.$$ Denote $Q:=\{z\in B_2^{++}\;:\;{ \mbox{dist}}(z, \partial B_2^{++})>1/50\}$. The interior estimates for the elliptic equation (\[unif-ellipt-linear-xy-v\]) imply $\|v\|_{C^{2,\alpha/2}(\overline{Q})}\le
C\|v\|_{L^\infty(B_2^{++})}$. The local estimates for the Dirichlet problem (\[unif-ellipt-linear-xy-v\])–(\[condOnSonicLine-v-xy-loc\]) imply $$\label{localEstNearBdry}
\|v\|_{C^{2,\alpha/2}(\overline{B_{1/10}(z)\cap B_2^{++}})} \le
C\|v\|_{L^\infty(B_2^{++})}$$ for every $z=(x,y)\in\{x=0, 1/2\le y\le 3/2\}$. The local estimates for the oblique derivative problem (\[unif-ellipt-linear-xy-v\]) and (\[condOnWedge-v-xy-loc\]) imply (\[localEstNearBdry\]) for every $z\in\{1/2\le x\le 3/2, y=0\}$. Then we have $$\label{Hoder-est-near-Br}
\|v\|_{C^{2,\alpha/2}(\overline{B_{3/2}^{++}\setminus
B_{1/2}^{++}})} \le C\|v\|_{L^\infty(B_2^{++})}.$$
We modify the domain $B_1^{++}$ by mollifying the corner at $(0,1)$ and denote the resulting domain by $D^{++}$. That is, $D^{++}$ denotes an open domain satisfying $$D^{++}\subset B_{1}^{++},\qquad D^{++}\setminus B_{1/10}(0,1)=
B_1^{++}\setminus B_{1/10}(0,1),$$ and $$\partial D^{++}\cap B_{1/5}(0,1)\qquad
\mbox{is a $C^{2,{\alpha/2}}$--curve}.$$ Then we prove the following fact: For any $g\in
C^{\alpha/2}(\overline{D^{++}})$, there exists a unique solution $w\in C^{2,\alpha/2}(\overline{D^{++}})$ of the problem: $$\begin{array}{ll}
&\hat a_{11}^{({\varrho})}w_{xx} +\hat a_{22}^{({\varrho})}w_{yy} +\hat
a_1^{({\varrho})}w_{x} =g \qquad\mbox{in }\;D^{++},
\\
&w=0\qquad\mbox{on }\;\partial D^{++}\cap \{x=0, y>0\},
\\
&w_\nu\equiv w_y=0\qquad\mbox{on }\;\partial D^{++}\cap \{x>0,
y=0\}, \\
&w=v\qquad\mbox{on }\;\partial D^{++}\cap \{x>0, y>0\},
\end{array}
\label{unif-ellipt-linear-xy-w}$$ with $$\label{Hoder-est-near-origin}
\|w\|_{C^{2,{\alpha/2}}(\overline{D^{++}})}\le
C(\|v\|_{L^\infty(B_{2}^{++})}+
\|g\|_{C^{\alpha/2}(\overline{D^{++}})}).$$
This can be seen as follows. Denote by $D^+$ the even extension of $D^{++}$ from $\{x,y >0\}$ into $\{x >0\}$, i.e., $$D^+:=D^{++}\cup\{(x,0)\; : \;x\in(0, 1)\}\cup D^{+-},$$ where $D^{+-}:=\{(x, y)\; : \;(x,-y)\in D^{++}\}$. Then $B^+_{7/8}\subset D^+\subset B^+_{1}$ and $\partial D^+$ is a $C^{2,{\alpha/2}}$–curve. Extend $F=(v,g,\hat a_{11}^{({\varrho})},\hat
a_{22}^{({\varrho})},\hat a_{1}^{({\varrho})})$ from $\overline{B_2^{++}}$ to $\overline{B_2^+}$ by setting $$F(x,-y)=F(x,y) \qquad\,\,\mbox{for }\;(x,y)\in \overline{B_2^{++}}.$$ Then it follows from (\[condOnSonicLine-v-xy-loc\])–(\[condOnWedge-v-xy-loc\]) and (\[Hoder-est-near-Br\]) that, denoting by $\hat v$ the restriction of (extended) $v$ to $\partial D^+$, we have $\hat v\in
C^{2,{\alpha/2}}(\partial D^+)$ with $$\label{Hoder-est-bdry-funct}
\|\hat v\|_{C^{2,{\alpha/2}}(\partial D^+)}\le
C\|v\|_{L^\infty(B_{2}^{++})}.$$ Also, the extended $g$ satisfies $g\in C^{\alpha/2}(\overline{D^+})$ with $\|g\|_{C^{\alpha/2}(\overline{D^+})}
=\|g\|_{C^{\alpha/2}(\overline{D^{++}})}$. The extended $(\hat
a_{11}^{({\varrho})}, \hat a_{22}^{({\varrho})}, \hat a_{1}^{({\varrho})})$ satisfy (\[ellipticityOfIterEq-Linear-xy-0\]) and $$\begin{aligned}
\|(\hat a_{11}^{({\varrho})}, \hat a_{22}^{({\varrho})}, \hat a_{1}^{({\varrho})})\|_{
C^{\alpha/2}(\overline{B_2^{+}})}&=&\|(\hat a_{11}^{({\varrho})}, \hat
a_{22}^{({\varrho})}, \hat a_{1}^{({\varrho})})\|_{
C^{\alpha/2}(\overline{B_2^{++}})}\\
&\le& \sum_{i,j=1}^2\|(\hat a_{ij}, \hat a_{i})\|_{
C^{\alpha/2}(\overline{\Omega^+({\phi})})}.\end{aligned}$$ Then, by [@GilbargTrudinger Theorem 6.8], there exists a unique solution $w\in C^{2,{\alpha/2}}(D^+)$ of the Dirichlet problem $$\begin{aligned}
&&\hat a_{11}^{({\varrho})}w_{xx} +\hat a_{22}^{({\varrho})}w_{yy} +\hat
a_1^{({\varrho})}w_{x} =g \qquad\mbox{in }\;D^{+},
\label{unif-ellipt-linear-xy-v-ext}
\\
&& w=\hat v\qquad\mbox{on }\;\partial D^{+}, \label{dirichlet-v-ext}\end{aligned}$$ and $w$ satisfies $$\label{Hoder-est-near-Br-w}
\|w\|_{C^{2,{\alpha/2}}(\overline{D^{+}})}\le C(\|\hat
v\|_{C^{2,{\alpha/2}}(\partial D^+)}+
\|g\|_{C^{\alpha/2}(\overline{D^+})}).$$ From the structure of equation (\[unif-ellipt-linear-xy-v-ext\]) and the symmetry of the domain and the coefficients and right-hand sides obtained by the even extension, it follows that $\hat w$, defined by $\hat w(x,y)=w(x,
-y)$ in $D^+$, is also a solution of (\[unif-ellipt-linear-xy-v-ext\])–(\[dirichlet-v-ext\]). By uniqueness for (\[unif-ellipt-linear-xy-v-ext\])–(\[dirichlet-v-ext\]), we find $$w(x,y)=w(x, -y) \qquad \mbox{in } D^+.$$ Thus, $w$ restricted to $D^{++}$ is a solution of (\[unif-ellipt-linear-xy-w\]), where we use (\[condOnSonicLine-v-xy-loc\]) to see that $w=0$ on $\partial
D^{++}\cap \{x=0, y>0\}$. Moreover, (\[Hoder-est-bdry-funct\]) and (\[Hoder-est-near-Br-w\]) imply (\[Hoder-est-near-origin\]). The uniqueness of the solution $w\in
C^{2,{\alpha/2}}(\overline{D^{++}})$ of (\[unif-ellipt-linear-xy-w\]) follows from the Comparison Principle (Lemma \[comparisonPrincipleOfUnifEllipt-Lemma\]).
Now we prove the existence of a solution $w\in
C^{2,{\alpha/2}}(\overline{D^{++}})$ of the problem: $$\begin{array}{ll}
&\hat a_{11}^{({\varrho})}w_{xx}+2\hat a_{12}^{({\varrho})}w_{xy} +\hat
a_{22}^{({\varrho})}w_{yy} +\hat a_1^{({\varrho})}w_{x} +\hat a_2^{({\varrho})}w_{y}
=0
\qquad\mbox{in }\;D^{++},\\
&w=0\qquad\mbox{on }\;\partial D^{++}\cap \{x=0, y>0\},
\\
&w_\nu\equiv w_y=0\qquad\mbox{on }\;\partial D^{++}\cap \{y=0,
x>0\}, \\
&w=v\qquad\mbox{on }\;\partial D^{++}\cap \{x>0, y>0\}.
\end{array}
\label{unif-ellipt-linear-xy-w-fullEq}$$ Moreover, we prove that $w$ satisfies $$\label{Hoder-est-near-Br-fullEq}
\|w\|_{C^{2,{\alpha/2}}(\overline{D^{++}})}\le
C\|v\|_{L^\infty(B_{2}^{++})}.$$
We obtain such $w$ as a fixed point of map $K:C^{2,{\alpha/2}}(\overline{D^{++}}) \to
C^{2,{\alpha/2}}(\overline{D^{++}})$ defined as follows. Let $W\in
C^{2,{\alpha/2}}(\overline{D^{++}})$. Define $$\label{defG-corner}
g=-2\hat a_{12}^{({\varrho})}W_{xy}-\hat a_2^{({\varrho})}W_{y}.$$ By (\[estSmallterms-iter-lin\]) and (\[def-rescaled-coef-v\]) with ${\varrho}\in(0,1)$, we find $$\label{estimateCoefR}
\|(a_{12}^{({\varrho})}, a_2^{({\varrho})})\|_{C^{\alpha/2}(\overline{D^{++}})}
\le C{\varrho}^{1/2},$$ which implies $$g\in C^{\alpha/2}(\overline{D^{++}}).$$ Then, by the results of Step 2, there exists a unique solution $w\in C^{2,{\alpha/2}}(\overline{D^{++}})$ of (\[unif-ellipt-linear-xy-w\]) with $g$ defined by (\[defG-corner\]). We set $K[W]=w$.
Now we prove that, if ${\varrho}>0$ is sufficiently small, the map $K$ is a contraction map. Let $W^{(i)}\in
C^{2,{\alpha/2}}(\overline{D^{++}})$ and $w^{(i)}:=K[W^{(i)}]$ for $i=1,2$. Then $w{:=}w^{(1)}-w^{(2)}$ is a solution of (\[unif-ellipt-linear-xy-w\]) with $$\begin{aligned}
&&g=-2\hat a_{12}^{({\varrho})}(W^{(1)}_{xy}-W^{(2)}_{xy})-
\hat a_2^{({\varrho})}(W^{(1)}_{y}-W^{(2)}_{y}), \\
&& v\equiv 0.\end{aligned}$$ Then $g\in C^{\alpha/2}(\overline{D^{++}})$ and, by (\[estimateCoefR\]), $$\|g\|_{C^{\alpha/2}(\overline{D^{++}})} \le
C{\varrho}^{1/2}\|W^{(1)}-W^{(2)}\|_{C^{2,{\alpha/2}}(\overline{D^{++}})}.$$ Since $v\equiv 0$ satisfies (\[condOnSonicLine-v-xy-loc\])–(\[condOnWedge-v-xy-loc\]), we can apply both (\[Hoder-est-near-origin\]) and the results of Step 2 to obtain $$\begin{aligned}
\|w^{(1)}-w^{(2)}\|_{C^{2,{\alpha/2}}(\overline{D^{++}})}&\le&
C{\varrho}^{1/2}\|W^{(1)}-W^{(2)}\|_{C^{2,{\alpha/2}}(\overline{D^{++}})}\\
&\le& {1\over
2}\|W^{(1)}-W^{(2)}\|_{C^{2,{\alpha/2}}(\overline{D^{++}})},\end{aligned}$$ where the last inequality holds if ${\varrho}>0$ is sufficiently small. We fix such ${\varrho}$. Then the map $K$ has a fixed point $w\in
C^{2,{\alpha/2}}(\overline{D^{++}})$ which is a solution of (\[unif-ellipt-linear-xy-w-fullEq\]).
Since $v$ satisfies (\[unif-ellipt-linear-xy-v\])–(\[condOnWedge-v-xy-loc\]), it follows from the uniqueness of solutions in $C(\overline{D^{++}})
\cap C^{1}(\overline{D^{++}}\setminus\overline{\{x=0\}}) \cap
C^2(D^{++})$ of problem (\[unif-ellipt-linear-xy-w-fullEq\]) that $w=v$ in $D^{++}$. Thus $v\in C^{2, {\alpha/2}}(\overline{D^{++}})$ so that $\psi\in C^{2,
{\alpha/2}}(\overline{B_{{\varrho}/2}({{P_4}})\cap\Omega^+({\phi})})$.
Now we prove that the solution $\psi$ is $C^{1,\alpha}$ near the corner ${{P_1}}={\Gamma_{sonic}}\cap{\Gamma_{shock}}({\phi})$ if $\delta$ is small.
\[C2alpha-near-P4-Lin-UnifEllipt-Lemma\] There exist $\hat C>0$ and $\delta_0\in (0,1)$ depending only on the data such that, if ${\sigma}, \varepsilon>0$ and $M_1, M_2\ge 1$ in [(\[defSetK\_R\])]{} satisfy [(\[condConst-00\])]{}, and $\delta\in (0,\delta_0)$, then any solution $\psi\in
C(\overline{\Omega^+({\phi})}) \cap
C^1(\overline{\Omega^+({\phi})}\setminus\overline{\Gamma_{sonic}}) \cap
C^2(\Omega^+({\phi}))$ of [(\[unif-ellipt-linear-iterationEquation\])]{} and [(\[iterationRH\])]{}–[(\[iterationCondOnSymmtryLine\])]{} is in $C^{1,{\alpha}}(\overline{B_{\varrho}({{P_1}})\cap\Omega^+({\phi})}) \cap
C^{2,{\alpha/2}}(B_{\varrho}({{P_1}})\cap\Omega^+({\phi}))$, for sufficiently small ${\varrho}>0$ depending only on the data and $\delta$, and satisfies $$\|\psi\|^{(-1-\alpha, \{{{P_1}}\})}_{2,\alpha/2,\Omega^+({\phi})}\le
C(\delta,\hat\psi){\sigma},
\label{Hoder-est-P4-for-lin-unif-ellipt}$$ where $C$ depends only on the data, $\delta$, and $\|\hat\psi\|_{C^{1,\alpha/2}(\overline{\Omega^+({\phi})})}$. Moreover, for $\delta$ as above, $$|\psi(x)|\le \tilde C(\delta)({ \mbox{dist}}(x, {{P_1}}))^{1+\alpha}
\qquad\mbox{for any }\; x\in \Omega^+({\phi}),
\label{growth-est-P4-for-lin-unif-ellipt}$$ where $\tilde C$ depends only on the data and $\delta$, and is independent of $\hat\psi$.
In Steps 1–3 of this proof below, the positive constants $C$ and $L_i, 1\le i\le 4,$ depend only on the data.
We work in the $(x,y)$–coordinates. Then the point ${{P_1}}$ has the coordinates $(0, y_{{{P_1}}})$ with $y_{{{P_1}}}
=\pi/2+\arctan{(|{{\xi}}_1|/{{\eta}}_1)}-\theta_w>0$. From (\[domain-in-rescaled-lemma\])–(\[holder-hat-f\]), we have $$\Omega^+({\phi})\cap B_\kappa({{P_1}}) =\{x>0,y<\hat
f_{\phi}(x)\}\cap B_\varepsilon({{P_1}}),$$ where $\hat f_{\phi}(0)=y_{{{P_1}}}$, $\hat f_{\phi}'(0)>0$, and $\hat
f_{\phi}>y_{{{P_1}}}$ on ${ {\bf R}}_+$ by (\[domain-in-xy-funct-0\]) and (\[holder-hat-f\]).
We change the variables in such a way that ${{P_1}}$ becomes the origin and the second-order part of equation (\[unif-ellipt-linear-iterationEquation\]) at ${{P_1}}$ becomes the Laplacian. Denote $${\mu}=\sqrt{\hat a_{11}({{P_1}})/\hat a_{22}({{P_1}})}.
\label{defLambdaDistortion}$$ Then, using (\[estSmallterms-iter-lin\]) and $x_{{{P_1}}}=0$, we have $$\sqrt{c_2\delta/2}\le {\mu}\le \sqrt{2c_2\delta}.
\label{estElliptDistortion}$$ Now we introduce the variables $$(X, Y):=(x/{\mu},y_{{{P_1}}}-y).$$ Then, for ${\varrho}=\varepsilon$, we have $$\label{domainResceled-distortion}
\Omega^+({\phi})\cap B_{\varrho}=\{X>0,\;Y> F(X)\}\cap B_{\varrho},$$ where $F(X)=y_{{{P_1}}}-\hat f_{\phi}({\mu}X)$. By (\[holder-hat-f\]), we have $0<\hat f_{\phi}'(X)\le C$ for all $X\in[0,2\varepsilon]$ if $\hat C$ is sufficiently large in (\[condConst-00\]) so that $2\varepsilon\le \kappa$. With this, we use $\hat f_{\phi}(0)=y_{{{P_1}}}$ and (\[estElliptDistortion\]) to obtain $$\begin{aligned}
F(0)=0, \qquad \quad-L_1\sqrt{\delta}\le F'(X)<0
\quad \mbox{ for }\;X\in [0,{\varrho}].
\label{properties-F-cap}\end{aligned}$$
We now write $\psi$ in the $(X,Y)$–coordinates. Introduce the function $$v(X,Y):=\psi(x,y)=\psi({\mu}X, y_{{{P_1}}}-Y).$$ Since $\psi$ satisfies equation (\[iterationRH-lf-flattened\]) and the boundary conditions (\[iterationCondOnWedge\]) and (\[unif-ellipt-linear-xy-iterationEquation\]), then $v$ satisfies $$\begin{aligned}
&&\qquad\,\, Av{:=}{1\over{\mu}^2}\tilde a_{11}v_{XX}-
{2\over{\mu}}\tilde a_{12}v_{XY} +\tilde a_{22} v_{YY}
+{1\over{\mu}}\tilde a_1v_{X} -\tilde a_2v_{Y}
=0\label{unif-ellipt-linear-XY-for-v}
\\
&&\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\quad \mbox{ in
}\;\;\{X>0,\;Y> F(X)\}\cap B_{\varrho},\nonumber
\\
&&\qquad\,\, Bv{:=}{1\over{\mu}}\tilde b_1 v_X - \tilde b_2 v_Y +
\tilde b_3 v=0 \qquad\mbox{on }\;\{X>0,\;Y=F(X)\}\cap B_{\varrho},
\label{RH-XY-for-v}
\\
&&\qquad\,\, v=0 \qquad\mbox{on }\;\{X=0,\;Y>0\}\cap B_{\varrho},
\label{cond-on-sonic-for-v}\end{aligned}$$ where $$\begin{aligned}
&&\tilde a_{ij}(X,Y)=\hat a_{ij}({\mu}X, y_{{{P_1}}}-Y), \,\, \tilde
a_{i}(X,Y)=\hat a_{i}({\mu}X, y_{{{P_1}}}-Y), \\
&&\tilde b_{i}(X,Y)=\hat b_{i}({\mu}X, y_{{{P_1}}}-Y).\end{aligned}$$ In particular, from (\[holder-coef-xy\]), (\[estSmallterms-iter-lin\]), and (\[defLambdaDistortion\]), we have $$\begin{aligned}
&&\qquad \tilde a_{ij}, \tilde a_{i}\in
C^{\alpha/2}(\overline{\{X>0,\;Y> F(X)\}\cap B_{\varrho}}),\\
&& \qquad \tilde a_{22}(0,0)={1\over{\mu}^2}\tilde a_{11}(0,0),\quad
\tilde a_{12}(0,0)=\tilde a_{2}(0,0)=0,
\label{estSmallterms-iter-lin-XY}
\\
&&\qquad |\tilde a_{ii}(X,Y)-\tilde a_{ii}(0, 0)|\le C|(X,
Y)|^\alpha \qquad\mbox{for }\;i=1,2,
\label{estSmallterms-iter-lin-contII}
\\
&&\qquad |\tilde a_{12}(X,Y)|+|\tilde a_{21}(X,Y)|+ |\tilde
a_{2}(X,Y)|\le C|X|^{1/2}, \quad |\tilde a_{1}(X,Y)|\le C.
\label{estSmallterms-iter-lin-contIJ}\end{aligned}$$ From (\[estCoefsIterRH-flattened\]), there exists $L_2>0$ such that $$\label{estCoefsIterRH-flattened-XY} -L_2^{-1}\le \tilde
b_i(X,Y)\le -L_2 \qquad\mbox{ for any }\;
(X,Y)\in\{X>0,\;Y=F(X)\}\cap B_{\varrho}.$$ Moreover, (\[obliqInxy-1\]) implies $$\label{obliqueness-blowup}
(\tilde b_1, \tilde b_2)\cdot\nu_F>0\qquad\mbox{ on
}\;\{X>0,\;Y=F(X)\}\cap B_{\varrho},$$ where $\nu_F=\nu_F(X,Y)$ is the interior unit normal at $(X,Y)\in\{X>0,\;Y=F(X)\}\cap B_{\varrho}$. Thus condition (\[RH-XY-for-v\]) is oblique.
We use the polar coordinates $(r, \theta)$ on the $(X,Y)$–plane, i.e., $$(X,Y)=(r\cos\theta, r\sin\theta).$$ From (\[properties-F-cap\]), we have $F, F'<0$ on $(0,{\varrho})$, which implies that $(X^2+F(X)^2)'>0$ on $(0,{\varrho})$. Then it follows from (\[properties-F-cap\]) that, if $\delta>0$ is a small constant depending only on the data and ${\varrho}$ is a small constant depending only on the data and $\delta$, there exist a function $\theta_F\in C^1({ {\bf R}}_+)$ and a constant $L_3>0$ such that $$\label{domain-XY-polar}
\{X>0,\;Y> F(X)\}\cap B_{\varrho}=\{0<r<{\varrho},\; \theta_F(r)<\theta<\pi/2\}$$ with $$\label{bdry-XY-polar}
-L_3\sqrt\delta\le\theta_F(r)\le 0.$$
Choosing sufficiently small $\delta_0>0$, we show that, for any $\delta\in(0,\delta_0)$, a function $$\label{supersolution-P4} w(r,\theta)=r^{1+\alpha}\cos
G(\theta),\qquad\mbox{with } G(\theta)={3+\alpha\over
2}(\theta-{\pi\over 4}),$$ is a positive supersolution of (\[unif-ellipt-linear-XY-for-v\])–(\[cond-on-sonic-for-v\]) in $\{X>0,\;Y> F(X)\}\cap B_{\varrho}$.
By (\[domainResceled-distortion\]) and (\[domain-XY-polar\])–(\[bdry-XY-polar\]), we find that, when $
0<\delta\le
\delta_0\le\big(\frac{(1-\alpha)\pi}{8(3+\alpha)L_3}\big)^2,
$ $$-{\pi\over 2}+{1-\alpha\over 16}\pi\le G(\theta)\le {\pi\over
2}-{1-\alpha\over 8}\pi\qquad \mbox{for all }(r,\theta)\in
\Omega^+({\phi})\cap B_{\varrho}.$$ In particular, $$\label{polarAngleBounds}
\cos(G(\theta))\ge \sin\big({1-\alpha\over 16}\pi\big)>0\qquad
\mbox{for all }(r,\theta)\in \overline{\Omega^+({\phi})\cap
B_{\varrho}}\setminus\{X=Y=0\},$$ which implies $$w> 0 \qquad\mbox{ in }\,\, \{X>0,\;Y> F(X)\}\cap B_{\varrho}.$$ By (\[domain-XY-polar\])–(\[bdry-XY-polar\]), we find that, for all $r\in(0, {\varrho})$ and $\delta\in (0,\delta_0)$ with small $\delta_0>0$, $$\cos(\theta_F(r))\ge 1-C\delta_0>0,\qquad |\sin(\theta_F(r))|\le
C\sqrt\delta_0.$$
Now, possibly further reducing $\delta_0$, we show that $w$ is a supersolution of (\[RH-XY-for-v\]). Using (\[estElliptDistortion\]), (\[RH-XY-for-v\]), (\[estCoefsIterRH-flattened-XY\]), the above estimates of $(\theta_F, G(\theta_F))$ derived above, and the fact that $\theta=\theta_F$ on $\{X>0,\;Y=F(X)\}\cap B_{\varrho}$, we have $$\begin{aligned}
Bw&\le&{\tilde b_1\over
{\mu}}r^\alpha\Big((\alpha+1)\cos(\theta_F)\cos(G(\theta_F))
+{3+\alpha\over 2}\sin(\theta_F)\sin(G(\theta_F)) \Big)\\
&& +Cr^\alpha|\tilde b_2|+Cr^{\alpha+1}|\tilde b_3|\\
&\le&-r^\alpha \Big((1-C\delta_0)({L_2\sin({1-\alpha\over
16}\pi)\over C\sqrt\delta_0}-{C\over L_2})
-C \Big)<0,\end{aligned}$$ if $\delta_0$ is sufficiently small. We now fix $\delta_0$ that satisfies all the smallness assumptions made above.
Finally, we show that $w$ is a supersolution of equation (\[unif-ellipt-linear-XY-for-v\]) in $(X,Y)\in \{X>0,\;Y>
F(X)\}\cap B_{\varrho}$ if ${\varrho}$ is small. Denote by $A_0$ the operator obtained by fixing the coefficients of $A$ in (\[unif-ellipt-linear-XY-for-v\]) at $(X,Y)=(0,0)$. Then $A_0=\tilde a_{22}(0,0)\Delta$ by (\[estSmallterms-iter-lin-XY\]). By (\[estSmallterms-iter-lin\]), we obtain $\tilde a_{22}(0,0)=\hat a_{22}(0, y_{{{P_1}}})\ge 1/(4\bar{c}_2)>0$. Now, by an explicit calculation and using (\[estElliptDistortion\]), (\[estSmallterms-iter-lin-XY\])–(\[estSmallterms-iter-lin-contIJ\]), (\[domain-XY-polar\]), and (\[polarAngleBounds\]), we find that, for $\delta\in(0,\delta_0)$ and $(X,Y)\in \{X>0,\;Y> F(X)\}\cap
B_{\varrho}$, $$\begin{aligned}
Aw(r,\theta) &=& a_2(0,0)\Delta w(r,\theta) +(A-A_0)w(r,\theta)
\\
&\le&\tilde a_{22}(0,0)r^{\alpha-1}\big((\alpha+1)^2
-({3+\alpha\over 2})^2\big)\cos(G(\theta))
\\
&&+Cr^{\alpha-1}\bigg( {1\over{\mu}^2}|\tilde a_{11}(X,Y)-\tilde
a_{11}(0,0)|
+|\tilde a_{22}(X,Y)-\tilde a_{22}(0,0)|
\bigg)\\
&&+{C\over {\mu}}r^{\alpha-1}|\tilde a_{12}(X,Y)| +
{C\over {\mu}}r^{\alpha}|\tilde a_{1}(X,Y)|+
Cr^{\alpha}|\tilde a_{2}(X,Y)|
\\
&\le& r^{\alpha-1}\left( - {(1-\alpha)(5+3\alpha)\over 8\bar{c}_2}
\sin\big({1-\alpha\over 16}\pi\big)+C{{\varrho}^{\alpha/ 2}\over\sqrt{\delta}} \right)<0\end{aligned}$$ for sufficiently small ${\varrho}>0$ depending only on the data and $\delta$.
Thus, all the estimates above hold for small $\delta_0>0$ and ${\varrho}>0$ depending only on the data.
Now, since $$\displaystyle\min_{\{X\ge 0,\;Y\ge F(X)\}\cap\partial
B_{\varrho}}w(X,Y)=L_4>0,$$ we use the Comparison Principle (Lemma \[comparisonPrincipleOfUnifEllipt-Lemma\]) (which holds since condition (\[RH-XY-for-v\]) satisfies (\[obliqueness-blowup\]) and $\tilde b_3<0$ by (\[estCoefsIterRH-flattened-XY\])) to obtain $${\|\psi\|_{L^\infty(\Omega^+({\phi}))}\over L_4} w \ge v\qquad\mbox{ in
}\;\{X>0,\;Y> F(X)\}\cap B_{\varrho}.$$ Similar estimate can be obtained for $-v$. Thus, using (\[L-infty-for-unif-ellipt\]), we obtain (\[growth-est-P4-for-lin-unif-ellipt\]) in $B_{\varrho}$. Since ${\varrho}$ depends only on the data and $\delta>0$, then we use (\[L-infty-for-unif-ellipt\]) to obtain the full estimate (\[growth-est-P4-for-lin-unif-ellipt\]).
Estimate (\[Hoder-est-P4-for-lin-unif-ellipt\]) can be obtained from (\[estCoefsIterRH-flattened\]), (\[holder-coef-xy\]), and (\[growth-est-P4-for-lin-unif-ellipt\]), combined with rescaling from the balls $B_{d_z/L}(z)\cap \Omega^+({\phi})$ for $z\in
\overline{\Omega^+_s({\phi})}\setminus\{{{P_1}}\}$ (with $d_z={ \mbox{dist}}(z,
{{P_1}})$ and $L$ sufficiently large depending only on the data) into the unit ball and the standard interior estimates for the linear elliptic equations and the local estimates for the linear Dirichlet and oblique derivative problems in smooth domains. Specifically, from the definition of sets ${{\mathcal K}}$ and $\Omega^+({\phi})$ and by (\[condConst-00\]), there exists $L\ge 1$ depending only on the data such that $$B_{d/L}(z)\cap (\partial \Omega^+({\phi})\setminus {\Gamma_{shock}})=\emptyset
\qquad\mbox{for any }\,\, z\in {\Gamma_{shock}}\cap\Omega_{\varrho},$$ and $$B_{d/L}(z)\cap (\partial \Omega^+({\phi})\setminus {\Gamma_{sonic}})=\emptyset
\qquad\mbox{for any } z\in {\Gamma_{sonic}}\cap\Omega_{\varrho}.$$ Then, for any $z\in \Omega^+({\phi})\cap B_{\varrho}({{P_1}})$, we have at least one of the following three cases:
1. $B_{d\over 10L}(z)\subset \Omega^+({\phi})$;
2. $z\in B_{d_{z_1}\over 2L}(z_1)$ and ${d_z\over d_{z_1}}\in({1\over 2}, 2)$ for some $z_1\in {\Gamma_{sonic}}$;
3. $z\in B_{d_{z_1}\over 2L}(z_1)$ and ${d_z\over d_{z_1}}\in({1\over 2}, 2)$ for some $z_1\in {\Gamma_{shock}}$.
Thus, it suffices to make the $C^{2,\alpha}$–estimates of $\psi$ in the following subdomains for $z_0=(x_0, y_0)$:
1. \[cases-HolderCorner-1\] $B_{d_{z_0}\over 20L}({z_0})$ when $B_{d_{z_0}\over 10L}({z_0})
\subset \Omega^+({\phi})$;
2. \[cases-HolderCorner-2\] $B_{d_{z_0}\over 2L}({z_0})\cap\Omega^+({\phi})$ for ${z_0}\in
{\Gamma_{sonic}}\cap B_{\varrho}({{P_1}})$;
3. \[cases-HolderCorner-3\] $B_{d_{z_0}\over 2L}({z_0})\cap\Omega^+({\phi})$ for ${z_0}\in
{\Gamma_{shock}}\cap B_{\varrho}({{P_1}})$.
We discuss only case (\[cases-HolderCorner-3\]), since the other cases are simpler and can be handled similarly.
Let ${z_0}\in {\Gamma_{shock}}\cap B_{\varrho}({{P_1}})$. Denote $\hat
d={d_{z_0}\over 2L}>0$. Without loss of generality, we can assume that $\hat d\le 1$.
We rescale $z=(x,y)$ near $z_0$: $$Z=(X, Y):={1\over \hat d}(x-x_0, y-y_0).$$ Since $B_{\hat d}(z_0)\cap (\partial \Omega^+({\phi})\setminus
{\Gamma_{shock}})=\emptyset$, then, for $\rho\in (0, 1)$, the domain obtained by rescaling $\Omega^+({\phi})\cap B_{\rho\hat d}(z_0)$ is $$\hat\Omega^{z_0}_\rho{:=}B_\rho\cap \big\{Y<\hat F(X){:=}{\hat
f_{\phi}(x_0+ \hat d X)-\hat f_{\phi}(x_0)\over \hat d}\big\},$$ where $\hat f_{\phi}$ is the function in (\[domain-in-rescaled-lemma\]). Note that $y_0=\hat f_{\phi}(x_0)$ since $(x_0, y_0)\in{\Gamma_{shock}}$. Since $L\ge 1$, we have $$\|\hat F\|_{C^{2,\alpha}([-1,1])}\le \|\hat f_{\phi}\|^{(-1-\alpha,
\{0\})}_{2,\alpha,{ {\bf R}}_+}$$ and $\|\hat f_{\phi}\|^{(-1-\alpha, \{0\})}_{2,\alpha,{ {\bf R}}_+}$ is estimated in terms of the data by (\[holder-hat-f\]).
Define $$\label{defV-mixed-corner}
v(Z)={1\over \hat d^{1+\alpha}}\psi(z_0+\hat dZ)\qquad \mbox{for }\;
Z\in \hat\Omega^{z_0}_1.$$ Then $$\label{V-mixed-corner-Linfty}
\|v\|_{L^\infty(\hat\Omega^{z_0}_1)}\le C$$ by (\[growth-est-P4-for-lin-unif-ellipt\]) with $C$ depending only on the data.
Since $\psi$ satisfies equation (\[unif-ellipt-linear-xy-iterationEquation\]) in $\Omega^+({\phi})\cap{{\mathcal D}'}_{4\varepsilon}$ and the oblique derivative condition (\[iterationRH-lf-flattened\]) on ${\Gamma_{shock}}\cap\overline{{{\mathcal D}'}_{4\varepsilon}}$, then $v$ satisfies an equation and an oblique derivative condition of the similar form in $\hat\Omega^{z_0}_1$ and on $\partial\hat\Omega^{z_0}_1\cap\{Y=\hat
F(X)\}$, respectively, whose coefficients satisfy properties (\[estCoefsIterRH-flattened\]) and (\[ellipticityOfIterEq-Linear-xy-0\]) with the same constants as for the original equations, where we have used $\hat d\le 1$ and the $C^{\alpha/2}$–estimates of the coefficients of the equation depending only on the data, $\delta$, and $\hat\psi$. Then, from the standard local estimates for linear oblique derivative problems, we have $$\|v\|_{C^{2,{\alpha/2}}(\overline{\hat{\Omega}^{z_0}_{1/2}})}\le C,$$ with $C$ depending only on the data, $\delta$, and $\hat\psi$.
We obtain similar estimates for cases (\[cases-HolderCorner-1\])–(\[cases-HolderCorner-2\]), by using the interior estimates for elliptic equations for case (\[cases-HolderCorner-1\]) and the local estimates for the Dirichlet problem for linear elliptic equations for case (\[cases-HolderCorner-2\]).
Writing the above estimates in terms of $\psi$ and using the fact that the whole domain $\Omega^+({\phi})\cap B_{\varrho}({{P_1}})$ is covered by the subdomains in (\[cases-HolderCorner-1\])–(\[cases-HolderCorner-3\]), we obtain (\[Hoder-est-P4-for-lin-unif-ellipt\]) by an argument similar to the proof of [@GilbargTrudinger Theorem 4.8] (see also the proof of Lemma \[partIntSeminorm-est-lemma\] below).
\[existence-Lin-UnifEllipt-Lemma\] There exist $\hat C>0$ and $\delta_0\in (0,1)$ depending only on the data such that, if ${\sigma}, \varepsilon>0$ and $M_1, M_2\ge 1$ in [(\[defSetK\_R\])]{} satisfy [(\[condConst-00\])]{}, and $\delta\in (0,\delta_0)$, there exists a unique solution $\psi\in
C^{(-1-\alpha, {{\mathcal P}})}_{2,{\alpha/2},\Omega^+({\phi})}$ of [(\[unif-ellipt-linear-iterationEquation\])]{} and [(\[iterationRH\])]{}–[(\[iterationCondOnSymmtryLine\])]{}. The solution $\psi$ satisfies [(\[L-infty-for-unif-ellipt\])]{}–[(\[barier-for-unif-ellipt\])]{}.
In this proof, for simplicity, we write $\Omega^+$ for $\Omega^+({\phi})$ and denote by $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, and $\Gamma_D$ the relative interiors of the curves ${\Gamma_{shock}}({\phi})$, $\Sigma_0({\phi})$, ${\Gamma_{wedge}}$, and ${\Gamma_{sonic}}$ respectively.
We first prove the existence of a solution for a general problem ${\mathcal P}$ of the form $$\sum_{i,j=1}^2 a_{ij}D^2_{ij} \psi=f\;\;\mbox{in }\Omega^+;\quad
\sum_{i=1}^2 b^{(k)}_{i}D_i \psi=g_i\;\;\mbox{on }\Gamma_k,\;
k=1,2,3;\quad \psi=0\;\;\mbox{on }\Gamma_D,$$ where the equation is uniformly elliptic in $\Omega^+$ and the boundary conditions on $\Gamma_k$, $k=1,2,3,$ are uniformly oblique, i.e., there exist constants $\lambda_1,\lambda_2, \lambda_3>0$ such that $$\begin{aligned}
&& \lambda_1|\mu|^2 \le
\sum_{i,j=1}^2
a_{ij}({{\xi}},{{\eta}})\mu_i\mu_j\le \lambda_1^{-1}|\mu|^2\qquad \mbox{for
all} \,\, ({{\xi}},{{\eta}})\in\Omega^+, \mu\in{ {\bf R}}^2,\\
&& \sum_{i=1}^2 b^{(k)}_{i}({{\xi}},{{\eta}})\nu_i({{\xi}},{{\eta}})\ge\lambda_2,\\
&&\displaystyle \left|{(b^{(k)}_1, b^{(k)}_2)\over |(b^{(k)}_1,
b^{(k)}_2)|} (P_k) -{(b^{(k-1)}_1, b^{(k-1)}_2)\over |(b^{(k-1)}_1,
b^{(k-1)}_2)|} (P_k)\right|\ge \lambda_3 \qquad\mbox{for }\;k=2,3,\end{aligned}$$ and $\|a_{ij}\|_{C^\alpha(\overline{\Omega^+})}+
\|b^{(k)}_{i}\|_{C^{1,\alpha}(\overline{\Gamma_k})}\le L$ for some $L>0$.
First we derive an apriori estimate of a solution of problem ${\mathcal P}$. For that, we define the following norm for $\psi\in
C^{k,\beta}(\Omega^+)$, $k=0,1,2,\dots$, and $\beta\in (0,1)$: $$\|\psi\|_{*,k,\beta}:=\sum_{i=2}^3 \|\psi\|^{-k+1-\beta,
\{P_i\}}_{k,\beta, B_{2{\varrho}}(P_i)\cap \Omega^+} + \sum_{i=1,4}
\|\psi\|^{-k+2-\beta, \{P_i\}}_{k,\beta, B_{2{\varrho}}(P_i)\cap
\Omega^+}+\|\psi\|_{C^{k,\beta}(\overline{
\Omega^+\setminus(\cup_{i=1}^4 B_{{\varrho}}(P_i))})},$$ where ${\varrho}>0$ is chosen small so that the balls $B_{2{\varrho}}(P_i)$ for $i=1,\dots,4$ are disjoint. Denote $C^{*,k,\beta}:=\{\psi\in
C^{*,k,\beta}\; : \;\|\psi\|_{*,k,\beta}<\infty\}$. Then $C^{*,k,\beta}$ with norm $\|\cdot\|_{*,k,\beta}$ is a Banach space. Similarly, define $$\|g_k\|_{*,\beta}:=\sum_{i=2}^3 \|g_k\|^{-\beta, \{P_i\}}_{k,\beta,
B_{2{\varrho}}(P_i)\cap \Gamma_k} + \sum_{i=1,4} \|g_k\|^{1-\beta,
\{P_i\}}_{k,\beta, B_{2{\varrho}}(P_i)\cap
\Gamma_k}+\|g_k\|_{C^{1,\beta}(\overline{
\Gamma_k\setminus(\cup_{i=1}^4 B_{{\varrho}}(P_i))})},$$ where the respective terms are zero if $B_{2{\varrho}}(P_i)\cap
\Gamma_k=\emptyset$. Using the regularity of boundary of $\Omega^+$, from the localized version of the estimates of [@Lieberman Theorem 2] applied in $B_{2r}(P_i)\cap \Omega^+$, $i=1,4$, and of the estimates of [@Lieberman88 Lemma 1.3] applied in $B_{2r}(P_i)\cap \Omega^+$, $i=2,3$, and the standard local estimates for the Dirichlet and oblique derivative problems of elliptic equations in smooth domains applied similarly to Step 4 in the proof of Lemma \[C2alpha-near-P4-Lin-UnifEllipt-Lemma\], we obtain that there exists $\beta=\beta(\Omega^+, \lambda_2,
\lambda_3)\in (0,1)$ such that any solution $\psi\in
C^{\beta}(\overline{\Omega^+}) \cap C^{1,\beta}(\overline{\Omega^+}
\setminus\overline\Gamma_D) \cap C^2(\Omega^+)$ of problem ${\mathcal P}$ satisfies $$\label{LiebermanEst-1}
\|\psi\|_{*,2,\beta}\le
C\big(\|f\|_{*,0,\beta}+\sum_{k=1}^3\|g_k\|_{*,\beta}
+\|\psi\|_{0,\Omega^+}\big)$$ for $C=C(\Omega^+, \lambda_1, \lambda_2, \lambda_3, L)$. Next, we show that $\psi$ satisfies $$\label{LiebermanEst-2}
\|\psi\|_{*,2,\beta}\le
C(\|f\|_{*,0,\beta}+\sum_{k=1}^3\|g_k\|_{*,\beta})$$ for $C=C(\Omega^+, \lambda_1, \lambda_2, \lambda_3, L)$. By (\[LiebermanEst-1\]), it suffices to estimate $\|\psi\|_{0,\Omega^+}$ by the right-hand side of (\[LiebermanEst-2\]). Suppose such an estimate is false. Then there exists a sequence of problems ${\mathcal P}^m$ for $m=1,2,\dots$ with coefficients $a_{ij}^m$ and $b_i^{(k),m}$, the right-hand sides $f^m$ and $g_k^m$, and solutions $\psi^m\in
C^{*,2,\beta}$, where the assumptions on $a_{ij}^m$ and $b_i^{(k),m}$ stated above are satisfied with uniform constants $\lambda_1, \lambda_2, \lambda_3$, and $L$, and $\|f^m\|_{*,0,\beta}+\sum_{k=1}^3\|g^m_k\|_{*,\beta}\to 0$ as $m\to\infty$, but $\|\psi^m\|_{0,\Omega^+}=1$ for $m=1,2,\dots$. Then, from (\[LiebermanEst-1\]), we obtain $\|\psi^m\|_{*,2,\beta}\le C$ with $C$ independent of $m$. Thus, passing to a subsequence (without change of notation), we find $a_{ij}^m\to a_{ij}^0$ in $C^{\beta/2}(\overline{\Omega^+})$, $b_i^{(k),m}\to b_i^{(k),0}$ in $C^{1,\beta/2}(\overline{\Gamma_k})$, and $\psi^m\to\psi^0$ in $C^{*,2,\beta/2}$, where $\|\psi^0\|_{0,\Omega^+}=1$, and $a_{ij}^0$ and $b_i^{(k),0}$ satisfy the same ellipticity, obliqueness, and regularity conditions as $a_{ij}^m$ and $b_i^{(k),m}$. Moreover, $\psi^0$ is a solution of the homogeneous Problem ${\mathcal P}$ with coefficients $a_{ij}^0$ and $b_i^{(k),0}$. Since $\|\psi^0\|_{0,\Omega^+}=1$, this contradicts the uniqueness of a solution in $C^{*,2,\beta}$ of problem ${\mathcal P}$ (the uniqueness for problem ${\mathcal P}$ follows by the same argument as in Lemma \[comparisonPrincipleOfUnifEllipt-Lemma\]). Thus (\[LiebermanEst-2\]) is proved.
Now we show the existence of a solution for problem ${\mathcal P}$ if $\hat C$ in (\[condConst-00\]) is sufficiently large. We first consider problem ${\mathcal P}_0$ defined as follows: $$\Delta \psi=f\;\;\mbox{in }\Omega^+;\quad D_\nu\psi=g_k\;\;\mbox{on
}\Gamma_k,\; k=1,2,3;\quad \psi=0\;\;\mbox{on }\Gamma_D.$$ Using the fact that $\Gamma_2$ and $\Gamma_3$ lie on $\eta=0$ and $\eta=\xi\tan\theta_w$ respectively, and using (\[angleCloseToPiOver2\]) and (\[OmegaPL-f-higher\]), it is easy to construct a diffemorphism $$F: \,\, \Omega^+\to Q:=\{(X, Y)\in(0,1)^2\}$$ satisfying $$\begin{aligned}
&&\|F\|_{C^{1,\alpha}(\overline\Omega^+)}\le C,\qquad
\|F^{-1}\|_{C^{1,\alpha}(\overline Q)}\le C,\\
&&F(\Gamma_D)=\Sigma_D{:=}\{X=1, Y\in(0,1)\},\end{aligned}$$ and $$\label{diffenorm}
\|DF^{-1}-Id\|_{C^\alpha(Q\cap\{X<\eta_1/2\})}\le
C\varepsilon^{1/4},$$ where $C$ depends only on the data, and $(\xi_1,\eta_1)$ are the coordinates of ${{P_1}}$ defined by (\[coord-P4\]) with $\eta_1>0$. The mapping $F$ transforms problem ${\mathcal P}_0$ into the following problem $\tilde{\mathcal P}_0$: $$\begin{aligned}
&&\sum_{i,j=1}^2 D_i(\tilde a_{ij}D_j u)=\tilde f\;\qquad\mbox{in
}Q;\\
&&\sum_{i,j=1}^2 \tilde a_{ij}D_ju\,\nu_i=\tilde g_k\;\qquad\mbox{on
}I_k,\; k=1,2,3;\\
&&u=0\;\qquad \mbox{on }\Sigma_D,\end{aligned}$$ where $I_k=F(G_k)$ are the respective sides of $\partial Q$, $\nu$ is the unit normal on $I_k$, $\|\tilde a_{ij}\|_{C^\alpha(\overline
Q)}\le C$, and $\tilde a_{ij}$ satisfy the uniform ellipticity in $\overline Q$ with elliptic constant $\tilde\lambda>0$. Using (\[diffenorm\]), we obtain $$\label{coeffsConorm}
\|\tilde a_{ij}-\delta_i^j\|_{C^\alpha(Q\cap\{X<\eta_1/2\})}\le
C\varepsilon^{1/4},$$ where $\delta_i^i=1$ and $\delta_i^j=0$ for $i\ne j$, and $C$ depends only on the data. If $\varepsilon>0$ is sufficiently small depending only on the data, then, by [@ChenFeldman3 Theorem 3.2, Proposition 3.3], there exists $\beta\in(0,1)$ such that, for any $\tilde f\in
C^\beta(\overline{Q})$ and $\tilde g_k\in C^{\beta}(\overline{I_k})$ with $k=1,2,3$, there exists a unique weak solution $u\in H^1(Q)$ of problem $\tilde {\mathcal P}_0$, and this solution satisfies $u\in
C^{\beta}(\overline Q) \cap C^{1,\beta}(\overline Q
\setminus\overline\Sigma_D)$. We note that, in [@ChenFeldman3 Theorem 3.2, Proposition 3.3], condition (\[coeffsConorm\]) is stated in the whole $Q$, but in fact this condition was used only in a neighborhood of $I_2=\{0\}\times(0,1)$, i.e., the results can be applied to the present case. We can assume that $\beta\le \alpha$. Then, mapping back to $\Omega^+$, we obtain the existence of a solution $\psi\in C^{\beta}(\overline{\Omega^+}) \cap
C^{1,\beta}(\overline{\Omega^+} \setminus\overline\Gamma_D) \cap
C^2(\Omega^+)$ of problem ${\mathcal P}_0$ for any $f\in
C^\beta(\overline{\Omega^+})$ and $g_k\in
C^{\beta}(\overline{\Gamma_k})$, $k=1,2,3$. Now, reducing $\beta$ if necessary and using (\[LiebermanEst-2\]), we conclude that, for any $(f, g_1, g_2, g_3)\in{\mathcal Y}^\beta{:=}\{(f, g_1, g_2,
g_3)\;: \;
\|f\|_{*,0,\beta}+\sum_{k=1}^3\|g_k\|_{*,\beta}<\infty\}$, there exists a unique solution $\psi\in C^{*,2,\beta}$ of problem ${\mathcal P}_0$, and $\psi$ satisfies (\[LiebermanEst-2\]).
Now the existence of a unique solution $\psi\in C^{*,2,\beta}$ of problem ${\mathcal P}$, for any $(f, g_1, g_2, g_3)\in{\mathcal Y}^\beta$ with sufficiently small $\beta\in (0,1)$, follows by the method of continuity, applied to the family of problems $t{\mathcal
P}+(1-t){\mathcal P}_0$ for $t\in[0,1]$. This proves the existence of a solution $\psi\in C^{*,2,\beta}$ of problem [(\[unif-ellipt-linear-iterationEquation\])]{} and [(\[iterationRH\])]{}–[(\[iterationCondOnSymmtryLine\])]{}.
Estimates (\[L-infty-for-unif-ellipt\])–(\[barier-for-unif-ellipt\]) then follow from Lemma \[unifEstOfUnifEllipt-Lemma\]. The higher regularity $\psi\in C^{(-1-\alpha,
{{\mathcal P}})}_{2,{\alpha/2},\Omega^+({\phi})}$ follows from Lemmas \[C2alpha-Lin-UnifEllipt-Lemma\]–\[C2alpha-near-P4-Lin-UnifEllipt-Lemma\] and the standard estimates for the Dirichlet problem near the flat boundary, applied in a neighborhood of ${\Gamma_{sonic}}\setminus(B_{{\varrho}/2}({{P_1}})\cup B_{{\varrho}/2}({{P_4}}))$ in the $(x,y)$–coordinates, where ${\varrho}>0$ may be smaller than the constant ${\varrho}$ in Lemmas \[wedge-sonic-Lin-UnifEllipt-Lemma\]–\[C2alpha-near-P4-Lin-UnifEllipt-Lemma\]. In fact, from Lemma \[wedge-sonic-Lin-UnifEllipt-Lemma\], we obtain even a higher regularity than that in the statement of Lemma \[existence-Lin-UnifEllipt-Lemma\]: $\psi\in
C^{(-1-\alpha,\{{{P_2}},{{P_3}},{{P_4}}\})}_{2,{\alpha/2},\Omega^+({\phi})}$. The uniqueness of solutions follows from the Comparison Principle (Lemma \[comparisonPrincipleOfUnifEllipt-Lemma\]). Lemma \[existence-Lin-UnifEllipt-Lemma\] justifies the definition of map $\hat{J}$ in defined by $\hat{J}(\hat\psi):=\psi$. In order to apply the Leray-Schauder Theorem, we make the following apriori estimates for solutions of the nonlinear equation.
\[estimates-nonlin-UnifEllipt\] There exist $\hat C>0$ and $\delta_0\in (0,1)$ depending only on the data such that the following holds. Let ${\sigma}, \varepsilon>0$ and $M_1, M_2\ge 1$ in [(\[defSetK\_R\])]{} satisfy [(\[condConst-00\])]{}. Let $\delta\in (0,\delta_0)$ and $\mu\in[0,1]$. Let $\psi\in C^{(-1-\alpha,
{{\mathcal P}})}_{2,{\alpha/2},\Omega^+({\phi})}$ be a solution of [(\[unif-ellipt-iterationEquation\])]{}, [(\[iterationRH\])]{}–[(\[iterationCondOnWedge\])]{}, and $$\psi_{{\eta}}=-\mu v_2\qquad\mbox{on }\;
\Sigma_0({\phi}){:=}\partial \Omega^+({\phi})\cap\{{{\eta}}=-v_2\}.
\label{iterationCondOnSymmtryLine-LS}$$ Then
1. \[estimates-nonlin-UnifEllipt-i1\] There exists $C>0$ independent of $\psi$ and $\mu$ such that $$\|\psi\|_{C^{1,\alpha}(\overline{\Omega^+({\phi})})}\le C;
$$
2. \[estimates-nonlin-UnifEllipt-i2\] $\psi$ satisfies [(\[L-infty-for-unif-ellipt\])]{}–[(\[barier-for-unif-ellipt\])]{} with constant $C$ depending only on the data;
3. \[estimates-nonlin-UnifEllipt-i2.1\] $\psi\in C^{(-1-\alpha, {{\mathcal P}})}_{2,\alpha,\Omega^+({\phi})}$. Moreover, for every $s\in (0, {c_2/2})$, estimate [(\[Hoder-est-for-unif-ellipt\])]{} holds with constant $C$ depending only on the data and $s$;
4. \[estimates-nonlin-UnifEllipt-i3\] Solutions of problem [(\[unif-ellipt-iterationEquation\])]{}, [(\[iterationRH\])]{}–[(\[iterationCondOnWedge\])]{}, and [(\[iterationCondOnSymmtryLine-LS\])]{} satisfy the following comparison principle: Denote by ${{\mathcal N}}_\delta(\psi)$, $B_1(\psi)$, $B_2(\psi)$, and $B_3(\psi)$ the left-hand sides of [(\[unif-ellipt-iterationEquation\])]{}, [(\[iterationRH\])]{}, [(\[iterationCondOnWedge\])]{}, and [(\[iterationCondOnSymmtryLine-LS\])]{} respectively. If $\psi_1,\psi_2\in
C^{(-1-\alpha,{{\mathcal P}})}_{2,\alpha,\Omega^+({\phi})}$ satisfy $$\begin{aligned}
&&{{\mathcal N}}_\delta(\psi_1)\le{{\mathcal N}}_\delta(\psi_2)
\qquad\mbox{ in } \Omega^+({\phi}),\\
&&B_k(\psi_1)\le B_k(\psi_2)
\qquad \mbox{on } {\Gamma_{shock}}({\phi}),\,
{\Gamma_{wedge}}, \,\mbox{and }\,\Sigma_0({\phi})
\mbox{ for } k=1,2,3,\\
&&\psi_1\ge \psi_2\qquad\mbox{ on } \,
{\Gamma_{sonic}},\end{aligned}$$ then $$\psi_1\ge \psi_2 \qquad\mbox{ in }\, \Omega^+({\phi}).$$ In particular, problem [(\[unif-ellipt-iterationEquation\])]{}, [(\[iterationRH\])]{}–[(\[iterationCondOnWedge\])]{}, and [(\[iterationCondOnSymmtryLine-LS\])]{} has at most one solution $\psi\in C^{(-1-\alpha, {{\mathcal P}})}_{2,\alpha,\Omega^+({\phi})}$.
The proof consists of six steps.
[*Step 1.*]{} Since a solution $\psi\in C^{(-1-\alpha,
{{\mathcal P}})}_{2,\alpha,\Omega^+({\phi})}$ of [(\[unif-ellipt-iterationEquation\])]{}, [(\[iterationRH\])]{}–[(\[iterationCondOnWedge\])]{}, and (\[iterationCondOnSymmtryLine-LS\]) with $\mu\in[0,1]$ is the solution of the linear problem for equation (\[unif-ellipt-linear-iterationEquation\]) with $\hat\psi{:=}\psi$ and boundary conditions (\[iterationRH\])–(\[iterationCondOnWedge\]) and (\[iterationCondOnSymmtryLine-LS\]). Thus, estimates (\[L-infty-for-unif-ellipt\])–(\[barier-for-unif-ellipt\]) with constant $C$ depending only on the data follow directly from Lemma \[unifEstOfUnifEllipt-Lemma\].
Now, from Lemma \[propertiesNonlinCoeffs\](\[propertiesNonlinCoeffs-i2\]), equation (\[unif-ellipt-iterationEquation\]) is linear in $\Omega^+({\phi})\cap \{c_2-r>4\varepsilon\}$, i.e., (\[unif-ellipt-iterationEquation\]) is (\[unif-ellipt-linear-iterationEquation\]) in $\Omega^+({\phi})\cap
\{c_2-r>4\varepsilon\}$, with coefficients $a_{ij}({{\xi}},{{\eta}})=A^1_{ij}({{\xi}},{{\eta}})+\delta\delta_{ij}$ for $A^1_{ij}$ defined by (\[iterationUniforDomEquation\]). Then, by Lemma \[propertiesNonlinCoeffs\](\[propertiesNonlinCoeffs-i2\]), $a_{ij}\in C^\alpha(\overline{\Omega^+({\phi})\cap
\{c_2-r>4\varepsilon\}})$ with the norm estimated in terms of the data. Also, ${\Gamma_{shock}}({\phi})$ and the coefficients $b_i$ of (\[iterationRH-lf\]) satisfy (\[OmegaPL-f-higher\]) and (\[estCoefsIterRH-0\])–(\[estCoefsIterRH\]). Then, repeating the proof of Lemma \[C2alpha-Lin-UnifEllipt-Lemma\] with the use of the $L^\infty$ estimates of $\psi$ obtained in Step 1 of the present proof, we conclude that $\psi\in C^{(-1-\alpha,
\{{{P_2}},{{P_3}}\})}_{2,\alpha,\Omega^+({\phi}) \cap
\{c_2-r>6\varepsilon\}}$ with $$\|\psi\|^{(-1-\alpha, \{{{P_2}},{{P_3}}\})}_{2,\alpha,\Omega^+({\phi})
\cap \{c_2-r>6\varepsilon\}}\le C{\sigma}\label{Hoder-est-for-nonlin-unif-ellipt}$$ for $C$ depending only on the data.
Now we prove (\[Hoder-est-for-unif-ellipt\]) for all $s\in (0, {c_2/2})$. If $s\ge 6\varepsilon$, then (\[Hoder-est-for-unif-ellipt\]) follows from (\[Hoder-est-for-nonlin-unif-ellipt\]). Thus, it suffices to consider the case $s\in (0, 6\varepsilon)$ and show that $$\|\psi\|_{C^{2,\alpha}(\overline{\Omega^+({\phi})\cap
\{{s/2}<c_2-r<6\varepsilon+{s/4}\}})}\le C(s){\sigma},
\label{Hoder-est-for-nonlin-unif-ellipt-s}$$ with $C$ depending only on the data and $s$. Indeed, (\[Hoder-est-for-nonlin-unif-ellipt\])–(\[Hoder-est-for-nonlin-unif-ellipt-s\]) imply (\[Hoder-est-for-unif-ellipt\]).
In order to prove (\[Hoder-est-for-nonlin-unif-ellipt-s\]), it suffices to prove the existence of $C(s)$ depending only on the data and $s$ such that $$\label{nonlin-est-unif-balls-inter}
\|\psi\|_{C^{2,\alpha}(\overline{B_{s/16}(z)})}\le
C(s)\|\psi\|_{L^\infty(B_{s/8}(z))}$$ for all $z:=({{\xi}},{{\eta}})\in \Omega^+({\phi})\cap
\{{s/2}<c_2-r<6\varepsilon+{s/4}\}$ with ${ \mbox{dist}}(z,
\partial\Omega^+({\phi}))>{s/8}$ and that $$\label{nonlin-est-unif-balls-oblique}
\|\psi\|_{C^{2,\alpha}(\overline{B_{s/8}(z)\cap\Omega^+({\phi})})}
\le C(s)\|\psi\|_{L^\infty(B_{s/4}(z)\cap\Omega^+({\phi}))}$$ for all $z\in({\Gamma_{shock}}({\phi})\cup{\Gamma_{wedge}})\cap\{{s/2}<c_2-r<6\varepsilon+{s/4}\}$. Note that all the domains in (\[nonlin-est-unif-balls-inter\]) and (\[nonlin-est-unif-balls-oblique\]) lie within $\Omega^+({\phi})\cap \{{s/4}<c_2-r<12\varepsilon\}$. We can assume that $\varepsilon < c_2/24$. Since equation (\[unif-ellipt-iterationEquation\]) is uniformly elliptic in $\Omega^+({\phi})\cap \{{s/4}<c_2-r<12\varepsilon\}$ by Lemma \[propertiesNonlinCoeffs\](\[propertiesNonlinCoeffs-i1\]), and the boundary conditions (\[iterationRH\]) and (\[iterationCondOnWedge\]) are linear and oblique with $C^{1,\alpha}$–coefficients estimated in terms of the data, then (\[nonlin-est-unif-balls-inter\]) follows from Theorem \[locEstElliptEq\] and (\[nonlin-est-unif-balls-oblique\]) follows from Theorem \[locEstElliptEq-oblique\] (in Appendix \[append-1-section\]). Since $\|\psi\|_{L^\infty(\Omega^+(\varphi))}\le 1$ by (\[L-infty-for-unif-ellipt\]), the constants in the local estimates depend only on the ellipticity, the constants in Lemma \[propertiesNonlinCoeffs\](\[propertiesNonlinCoeffs-i3\]), and, for the case of (\[nonlin-est-unif-balls-oblique\]), also on the $C^{2,\alpha}$–norms of the boundary curves and the obliqueness and $C^{1,\alpha}$–bounds of the coefficients in the boundary conditions (which, for condition (\[iterationRH\]), follow from (\[OmegaPL-f-higher\]) and (\[estCoefsIterRH-0\]) since our domain is away from the points ${{P_1}}$ and ${{P_2}}$). All these quantities depend only on the data and $s$. Thus, the constant $C(s)$ in (\[nonlin-est-unif-balls-inter\])–(\[nonlin-est-unif-balls-oblique\]) depends only on the data and $s$.
In this step, the universal constant $C$ depends only on the data and $\delta$, unless specified otherwise. We prove that $\psi\in C^{2,\alpha}(\overline{B_{\varrho}({{P_4}})\cap\Omega^+({\phi})})$ for sufficiently small ${\varrho}>0$, depending only on the data and $\delta$, and $$\|\psi\|_{C^{2,\alpha}(\overline{B_{\varrho}({{P_4}})\cap\Omega^+({\phi})})}\le
C. \label{Hoder-est-wedge-sonic-nonLin-unif-ellipt}$$
We follow the proof of Lemma \[wedge-sonic-Lin-UnifEllipt-Lemma\]. Since $B_{\varrho}({{P_4}})\cap\Omega^+({\phi})\subset {{\mathcal D}'}$ for small ${\varrho}$, we work in the $(x,y)$–coordinates. We use the notations $B^+_{\varrho}$ and $B^{++}_{\varrho}$, introduced in Step 1 of Lemma \[wedge-sonic-Lin-UnifEllipt-Lemma\], and consider the function $$v(x,y)={1\over {\varrho}}\psi({\varrho}x, {\varrho}y).$$ Then, by (\[barier-for-unif-ellipt\]), $v$ satisfies $$\label{L-infty-for-unif-ellipt-nonlin-v}
\|v\|_{L^\infty(B_2^{++})}\le 2C{{\sigma}\over \varepsilon}\le 1,$$ where the last inequality holds if $\hat C$ in (\[condConst-00\]) is sufficiently large. Moreover, $v$ is a solution of $$\begin{aligned}
&&\qquad\hat A_{11}^{({\varrho})}v_{xx}+ 2\hat A_{12}^{({\varrho})}v_{xy} +\hat
A_{22}^{({\varrho})}v_{yy} +\hat A_1^{({\varrho})}v_{x} +\hat A_2^{({\varrho})}v_{y}
=0 \qquad\mbox{in }\;B_2^{++}, \label{unif-ellipt-NONlinear-xy-v}
\\
&&\qquad v=0\qquad\mbox{on }\;B_{2}\cap \{x=0, y>0\},
\label{condOnSonicLine-v-xy-loc-NONlin}
\\
&&\qquad v_\nu\equiv v_y=0\qquad\mbox{on }\;B_{2}\cap \{y=0, x>0\},
\label{condOnWedge-v-xy-loc-NONlin}\end{aligned}$$ with $(A_{ij}^{({\varrho})}, A_{i}^{({\varrho})})=(A_{ij}^{({\varrho})},
A_{i}^{({\varrho})})(Dv, x, y)$, where we use (\[unif-ellipt-iteration-equation-sonicStruct\]) to find that, for $(x,y)\in B_2^{++}$, $p\in{ {\bf R}}^2$, $i,j=1,2,$ $$\label{def-rescaled-coef-v-NONlinear}
\begin{array}{l}
\displaystyle
\hat A_{11}^{({\varrho})}(p,x,y)=\hat A_{11}(p,{\varrho}x,{\varrho}y)+\delta, \\
\displaystyle
\hat A_{12}^{({\varrho})}(p,x,y)=\hat A_{21}^{({\varrho})}(p,x,y)=\hat A_{12}(p,{\varrho}x,{\varrho}y),\\
\displaystyle
\hat A_{22}^{({\varrho})}(p,x,y)=\hat A_{22}(p,{\varrho}x,{\varrho}y)+{\delta\over (c_2-{\varrho}x)^2},
\\
\displaystyle\hat A_{1}^{({\varrho})}(p, x,y)= {\varrho}\hat A_{1}(p, {\varrho}x,{\varrho}y)+{\delta\over (c_2-{\varrho}x)},\quad \hat A_{2}^{({\varrho})}(p, x,y)=
{\varrho}\hat A_{2}(p, {\varrho}x,{\varrho}y),
\end{array}$$ with $\hat A_{ij}$ and $\hat A_{i}$ as in Lemma (\[propertiesNonlinCoeffs-xy\]). Since ${\varrho}\le 1$, $\hat
A_{ij}^{({\varrho})}$ and $\hat A_{i}^{({\varrho})}$ satisfy the assertions of Lemma \[propertiesNonlinCoeffs-xy\](\[propertiesNonlinCoeffs-xy-i1\])–(\[propertiesNonlinCoeffs-xy-i3\]) with the unchanged constants. Moreover, $\hat A_{11}^{({\varrho})}$, $\hat
A_{22}^{({\varrho})}$, and $\hat A_{1}^{({\varrho})}$ satisfy the property in Lemma \[propertiesNonlinCoeffs-xy\](\[propertiesNonlinCoeffs-xy-i4-0\]). The property in Lemma \[propertiesNonlinCoeffs-xy\](\[propertiesNonlinCoeffs-xy-i4\]) is now improved to $$\label{estRescaledA-nonlin}
|(\hat A_{12}^{({\varrho})},\hat A_{21}^{({\varrho})},\hat
A_{2}^{({\varrho})})(x,y)|\le C{\varrho}|x|, \,\quad |D(\hat
A_{12}^{({\varrho})},\hat A_{21}^{({\varrho})}, \hat A_{2}^{({\varrho})})(x,y)|\le
C|{\varrho}x|^{1/2}.$$
Combining the estimates in Theorems \[locEstElliptEq\] and \[locEstElliptEq-Dirichlet\]–\[locEstElliptEq-oblique\] with the argument that has led to (\[Hoder-est-near-Br\]), we have $$\label{Hoder-est-near-Br-nonlin}
\|v\|_{C^{2,\alpha}(\overline{B_{3/2}^{++}\setminus B_{1/2}^{++}})}
\le C,$$ where $C$ depends only on the data and $\delta>0$ by (\[L-infty-for-unif-ellipt-nonlin-v\]), since $\hat
A_{ij}^{({\varrho})}$ and $\hat A_{i}^{({\varrho})}$ satisfy (\[locEstElliptEq-i1-0\])–(\[locEstElliptEq-i2-0\]) with the constants depending only on the data and $\delta$. In particular, $C$ in (\[Hoder-est-near-Br-nonlin\]) is independent of ${\varrho}$.
We now use the domain $D^{++}$ introduced in Step 2 of the proof of Lemma \[wedge-sonic-Lin-UnifEllipt-Lemma\]. We prove that, for any $g\in C^{\alpha}(\overline{D^{++}})$ with $
\|g\|_{C^{\alpha}(\overline{D^{++}})}\le 1,
$ there exists a unique solution $w\in
C^{2,\alpha}(\overline{D^{++}})$ of the problem: $$\begin{aligned}
&&\hat A_{11}^{({\varrho})}w_{xx} +\hat A_{22}^{({\varrho})}w_{yy} +\hat
A_1^{({\varrho})}w_{x} =g \qquad\mbox{in }\;D^{++},
\label{unif-ellipt-NONlinear-xy-w}
\\
&&w=0\qquad\mbox{on }\;\partial D^{++}\cap \{x=0, y>0\},
\label{condOnSonicLine-w-xy-loc-nonlin}
\\
&&w_\nu\equiv w_y=0\qquad\mbox{on }\;\partial D^{++}\cap \{x>0,
y=0\}, \label{condOnWedge-w-xy-loc-nonlin}
\\
&& w=v\qquad\mbox{on }\;\partial D^{++}\cap \{x>0, y>0\},
\label{dirichlet-w-nonlin}\end{aligned}$$ with $(A_{ii}^{({\varrho})}, A_{1}^{({\varrho})})=(A_{ii}^{({\varrho})},
A_{1}^{({\varrho})})(Dw, x, y)$. Moreover, we show $$\label{Hoder-est-near-origin-nonlin}
\|w\|_{C^{2,{\alpha}}(\overline{D^{++}})}\le C,$$ where $C$ depends only on the data and is independent of ${\varrho}$. For that, similar to Step 2 of the proof of Lemma \[wedge-sonic-Lin-UnifEllipt-Lemma\], we consider the even reflection $D^+$ of the set $D^{++}$, and the even reflection of $(v,g,\hat A_{11}^{({\varrho})},\hat A_{22}^{({\varrho})},\hat A_{1}^{({\varrho})})$ from $\overline{B_2^{++}}$ to $\overline{B_2^+}$, without change of notation, where the even reflection of $(\hat A_{11}^{({\varrho})},\hat
A_{22}^{({\varrho})}, \hat A_{1}^{({\varrho})})$, which depends on $(p,x,y)$, is defined by $$\hat A_{ii}^{({\varrho})}(p,x,-y)=\hat A_{ii}^{({\varrho})}(p,x,y), \,\,\, \hat
A_{1}^{({\varrho})}(p,x,-y)=\hat A_{1}^{({\varrho})}(p,x,y) \quad\,\,\mbox{for
}\;(x,y)\in \overline{B_2^{++}}.$$
Also, denote by $\hat v$ the restriction of (the extended) $v$ to $\partial D^+$. It follows from (\[condOnSonicLine-v-xy-loc-NONlin\])–(\[condOnWedge-v-xy-loc-NONlin\]) and (\[Hoder-est-near-Br-nonlin\]) that $\hat v\in
C^{2,{\alpha}}(\partial D^+)$ with $$\label{Hoder-est-bdry-funct-nonlin}
\|\hat v\|_{C^{2,\alpha}(\partial D^+)}\le C,$$ depending only on the data and $\delta$. Furthermore, the extended $g$ satisfies $g\in C^{\alpha}(\overline{D^+})$ with $\|g\|_{C^{\alpha}(\overline{D^+})}
=\|g\|_{C^{\alpha/2}(\overline{D^{++}})}\le 1$. The extended $\hat
A_{11}^{({\varrho})}, \hat A_{22}^{({\varrho})}$, and $\hat A_{1}^{({\varrho})}$ satisfy (\[locEstElliptEq-i1-0\])–(\[locEstElliptEq-i2-0\]) in $D^+$ with the same constants as the estimates satisfied by $A_{ii}$ and $A_i$ in $\Omega^+({\phi})$. We consider the Dirichlet problem $$\begin{aligned}
&&\hat A_{11}^{({\varrho})}w_{xx} +\hat A_{22}^{({\varrho})}w_{yy} +\hat
A_1^{({\varrho})}w_{x} =g \qquad\mbox{in }\;D^{+},
\label{unif-ellipt-NONlinear-xy-v-ext}
\\
&& w=\hat v\qquad\mbox{on }\;\partial D^{+},
\label{dirichlet-v-ext-nonlin}\end{aligned}$$ with $(A_{ii}^{({\varrho})}, A_{1}^{({\varrho})}):=(A_{ii}^{({\varrho})},
A_{1}^{({\varrho})})(Dw, x, y)$. By the Maximum Principle, $$\|w\|_{L^\infty(D^+)}\le \|\hat v\|_{L^\infty(D^+)}.$$ Thus, using (\[Hoder-est-bdry-funct-nonlin\]), we obtain an estimate of $\|w\|_{L^\infty(D^+)}$. Now, using Theorems \[locEstElliptEq\] and \[locEstElliptEq-Dirichlet\] and the estimates of $\|g\|_{C^{\alpha}(\overline{D^+})}$ and $\|\hat
v\|_{C^{2,\alpha}(\partial D^+)}$ discussed above, we obtain the a-priori estimate for the $C^{2,\alpha}$–solution $w$ of (\[unif-ellipt-NONlinear-xy-v-ext\])–(\[dirichlet-v-ext-nonlin\]): $$\label{Hoder-est-near-Br-w-nonlin}
\|w\|_{C^{2,\alpha}(\overline{D^{+}})}\le C,$$ where $C$ depends only on the data and $\delta$. Moreover, for every $\hat w\in C^{1,{\alpha}}(\overline{D^+})$, the existence of a unique solution $w\in C^{2,{\alpha}}(\overline{D^+})$ of the linear Dirichlet problem, obtained by substituting $\hat w$ into the coefficients of (\[unif-ellipt-NONlinear-xy-v-ext\]), follows from [@GilbargTrudinger Theorem 6.8]. Now, by a standard application of the Leray-Schauder Theorem, there exists a unique solution $w\in C^{2,{\alpha}}(\overline{D^+})$ of the Dirichlet problem (\[unif-ellipt-NONlinear-xy-v-ext\])–(\[dirichlet-v-ext-nonlin\]) which satisfies (\[Hoder-est-near-Br-w-nonlin\]).
From the structure of equation (\[unif-ellipt-NONlinear-xy-v-ext\]), especially the fact that $\hat A_{11}^{({\varrho})}$, $\hat A_{22}^{({\varrho})}$, and $\hat A_1^{({\varrho})}$ are independent of $p_2$ by Lemma \[propertiesNonlinCoeffs-xy\] (\[propertiesNonlinCoeffs-xy-i4-0\]), and from the symmetry of the domain and the coefficients and right-hand sides obtained by the even extension, it follows that $\hat w$, defined by $\hat
w(x,y)=w(x, -y)$, is also a solution of (\[unif-ellipt-NONlinear-xy-v-ext\])–(\[dirichlet-v-ext-nonlin\]). By uniqueness for problem (\[unif-ellipt-NONlinear-xy-v-ext\])–(\[dirichlet-v-ext-nonlin\]), we find $ w(x,y)=w(x, -y) $ in $D^+$. Thus, $w$ restricted to $D^{++}$ is a solution of (\[unif-ellipt-NONlinear-xy-w\])–(\[dirichlet-w-nonlin\]), where (\[condOnSonicLine-w-xy-loc-nonlin\]) follows from (\[condOnSonicLine-v-xy-loc-NONlin\]) and (\[dirichlet-v-ext-nonlin\]). Moreover, (\[Hoder-est-near-Br-w-nonlin\]) implies (\[Hoder-est-near-origin-nonlin\]).
The uniqueness of a solution $w\in
C^{2,{\alpha}}(\overline{D^{++}})$ of (\[unif-ellipt-NONlinear-xy-w\])–(\[dirichlet-w-nonlin\]) follows from the Comparison Principle (Lemma \[comparisonPrincipleOfUnifEllipt-Lemma\]).
Now we prove the existence of a solution $w\in
C^{2,{\alpha}}(\overline{D^{++}})$ of the problem: $$\begin{array}{ll}
&\hat A_{11}^{({\varrho})}w_{xx}+2\hat A_{12}^{({\varrho})}w_{xy} +\hat
A_{22}^{({\varrho})}w_{yy} +\hat A_1^{({\varrho})}w_{x} +\hat A_2^{({\varrho})}w_{y}
=0 \qquad\mbox{in }\;D^{++},
\\
&w=0\qquad\mbox{on }\;\partial D^{++}\cap \{x=0, y>0\},
\\
&w_\nu\equiv w_y=0\qquad\mbox{on }\;\partial D^{++}\cap \{y=0,
x>0\}, \\
& w=v\qquad\mbox{on }\;\partial D^{++}\cap \{x>0, y>0\},
\end{array}
\label{unif-ellipt-NONlinear-xy-w-fullEq}$$ where $(A_{ij}^{({\varrho})}, A_{i}^{({\varrho})}):=(A_{ij}^{({\varrho})},
A_{i}^{({\varrho})})(Dw, x, y)$. Moreover, we prove that $w$ satisfies $$\label{Hoder-est-near-Br-fullEq-nonlin}
\|w\|_{C^{2,{\alpha}}(\overline{D^{++}})}\le C$$ for $C>0$ depending only on the data and $\delta$.
Let $N$ be chosen below. Define $$\label{defSetM}
{{\mathcal S}(N)}:=\left\{W\in C^{2,\alpha}(\overline{D^{++}})\,\,:\,\,
\|W\|_{C^{2,\alpha}(\overline{D^{++}})}\le N\right\}.$$
We obtain such $w$ as a fixed point of the map $K:{{\mathcal S}(N)}\to {{\mathcal S}(N)}$ defined as follows (if $R$ is small and $N$ is large, as specified below). For $W\in {{\mathcal S}(N)}$, define $$\label{defG-corner-nonlin}
g=-2\hat A_{12}^{({\varrho})}(x,y)W_{xy}-\hat A_2^{({\varrho})}(x,y)W_{y}.$$ By (\[estRescaledA-nonlin\]), $$\|g\|_{C^{\alpha}(\overline{D^{++}})}\le CN \sqrt{{\varrho}}\le 1,$$ if ${\varrho}\le {\varrho}_0$ with ${\varrho}_0= {1\over CN^2}$, for $C$ depending only on the data and $\delta$. Then, as we have proved above, there exists a unique solution $w\in C^{2,\alpha}(\overline{D^{++}})$ of (\[unif-ellipt-NONlinear-xy-w\])–(\[dirichlet-w-nonlin\]) with $g$ defined by (\[defG-corner-nonlin\]). Moreover, $w$ satisfies (\[Hoder-est-near-origin-nonlin\]). Then, if we choose $N$ to be the constant $C$ in (\[Hoder-est-near-origin-nonlin\]), we get $w\in{{\mathcal S}(N)}$. Thus, $N$ is chosen depending only on the data and $\delta$. Now our choice of ${\varrho}_0= {1\over CN^2}$ and ${\varrho}\le
{\varrho}_0$ (and the other smallness conditions stated above) determines ${\varrho}$ in terms of the data and $\delta$. We define $K[W]:=w$ and thus obtain $K:{{\mathcal S}(N)}\to {{\mathcal S}(N)}$.
Now the existence of a fixed point of $K$ follows from the Schauder Fixed Point Theorem in the following setting: From its definition, ${{\mathcal S}(N)}$ is a compact and convex subset in $C^{2,{\alpha/2}}(\overline{D^{++}})$. The map $K:{{\mathcal S}(N)}\to {{\mathcal S}(N)}$ is continuous in $C^{2,{\alpha/2}}(\overline{D^{++}})$: Indeed, if $W_k\in {{\mathcal S}(N)}$ for $k=1,\dots$, and $W_k\to W$ in $C^{2,{\alpha/2}}(\overline{D^{++}})$, then it is easy to see that $W\in {{\mathcal S}(N)}$. Define $g_k$ and $g$ by (\[defG-corner-nonlin\]) for $W_k$ and $W$ respectively. Then $g_k\to g$ in $C^{{\alpha/2}}(\overline{D^{++}})$ since $(\hat A_{12}, \hat A_{2})
=(\hat A_{12}, \hat A_{2})(x,y)$ by Lemma \[propertiesNonlinCoeffs-xy\](\[propertiesNonlinCoeffs-xy-i4\]). Let $w_k=K[W_k]$. Then $w_k\in {{\mathcal S}(N)}$, and ${{\mathcal S}(N)}$ is bounded in $C^{2,\alpha}(\overline{D^{++}})$. Thus, for any subsequence $w_{k_l}$, there exists a further subsequence $ w_{k_{l_m}}$ converging in $C^{2,{\alpha/2}}(\overline{D^{++}})$. Then the limit $\tilde w$ is a solution of (\[unif-ellipt-NONlinear-xy-w\])–(\[dirichlet-w-nonlin\]) with the limiting function $g$ in the right-hand side of (\[unif-ellipt-NONlinear-xy-w\]). By uniqueness of solutions in ${{\mathcal S}(N)}$ to (\[unif-ellipt-NONlinear-xy-w\])–(\[dirichlet-w-nonlin\]), we have $\tilde w=K[W]$. Then it follows that the whole sequence $K[W_k]$ converges to $K[W]$. Thus $K:{{\mathcal S}(N)}\to {{\mathcal S}(N)}$ is continuous in $C^{2,{\alpha/2}}(\overline{D^{++}})$. Therefore, there exists $w\in {{\mathcal S}(N)}$ which is a fixed point of $K$. This function $w$ is a solution of (\[unif-ellipt-NONlinear-xy-w-fullEq\]).
Since $v$ satisfies (\[unif-ellipt-NONlinear-xy-v\])–(\[condOnWedge-v-xy-loc-NONlin\]), it follows from the uniqueness of solutions in $C(\overline{D^{++}})
\cap C^{1}(\overline{D^{++}}\setminus\overline{\{x=0\}}) \cap
C^2(D^{++})$ of problem (\[unif-ellipt-NONlinear-xy-w-fullEq\]) that $w=v$ in $D^{++}$. Thus, $v\in C^{2,\alpha}(\overline{D^{++}})$ and satisfies (\[Hoder-est-wedge-sonic-nonLin-unif-ellipt\]).
It remains to make the following estimate near the corner ${{P_1}}$: $$\|\psi\|^{(-1-\alpha, \{{{P_1}}\})}_{2,\alpha,\Omega^+({\phi})}\le
C,
\label{Hoder-est-P4-for-NONlin-unif-ellipt}$$ where $C$ depends only on the data, ${\sigma}$, and $\delta$.
Since $\psi$ is a solution of the linear equation [(\[unif-ellipt-linear-iterationEquation\])]{} for $\hat\psi=\psi$ and satisfies the boundary conditions [(\[iterationRH\])]{}–[(\[iterationCondOnSymmtryLine\])]{}, it follows from Lemma \[C2alpha-near-P4-Lin-UnifEllipt-Lemma\] that $\psi$ satisfies (\[growth-est-P4-for-lin-unif-ellipt\]) with constant $\hat C$ depending only on the data and $\delta$.
Now we follow the argument of Lemma \[C2alpha-near-P4-Lin-UnifEllipt-Lemma\] (Step 4): We consider cases (\[cases-HolderCorner-1\])–(\[cases-HolderCorner-3\]) and define the function $v(X,Y)$ by (\[defV-mixed-corner\]). Then $\psi$ is a solution of the nonlinear equation (\[unif-ellipt-iteration-equation-sonicStruct\]). We apply the estimates in Appendix \[append-1-section\]. From Lemma \[propertiesNonlinCoeffs-xy\] and the properties of the Laplacian in the polar coordinates, the coefficients of (\[unif-ellipt-iteration-equation-sonicStruct\]) satisfy (\[locEstElliptEq-i1-0\])–(\[locEstElliptEq-i2-0\]) with $\lambda$ depending only on the data and $\delta$. It is easy to see that $v$ defined by (\[defV-mixed-corner\]) satisfies an equation of the similar structure and properties (\[locEstElliptEq-i1-0\])–(\[locEstElliptEq-i2-0\]) with the same $\lambda$, where we use that $0\le \hat d\le 1$. Also, $v$ satisfies the same boundary conditions as in the proof of Lemma \[C2alpha-near-P4-Lin-UnifEllipt-Lemma\] (Step 4). Furthermore, since $\psi$ satisfies (\[growth-est-P4-for-lin-unif-ellipt\]), we obtain the $L^\infty$ estimates of $v$ in terms of the data and $\delta$, e.g., $v$ satisfies (\[V-mixed-corner-Linfty\]) in case (\[cases-HolderCorner-3\]). Now we obtain the $C^{2,\alpha}$–estimates of $v$ by using Theorem \[locEstElliptEq\] for case (\[cases-HolderCorner-1\]), Theorem \[locEstElliptEq-Dirichlet\] for case (\[cases-HolderCorner-2\]), and Theorem \[locEstElliptEq-oblique\] for case (\[cases-HolderCorner-3\]). Writing these estimates in terms of $\psi$, we obtain (\[Hoder-est-P4-for-NONlin-unif-ellipt\]), similar to the proof of Lemma \[C2alpha-near-P4-Lin-UnifEllipt-Lemma\] (Step 4).
Finally, we prove the comparison principle, assertion (\[estimates-nonlin-UnifEllipt-i3\]). The function $u=\psi_1-\psi_2$ is a solution of a linear problem of form (\[unif-ellipt-linear-iterationEquation\]), (\[iterationRH\]), (\[iterationCondOnWedge\]), and (\[iterationCondOnSymmtryLine\]) with right-hand sides ${{\mathcal N}}_\delta(\psi_1)-{{\mathcal N}}_\delta(\psi_2)$ and $B_k(\psi_1)- B_k(\psi_2)$ for $k=1,2,3$, respectively, and $u\ge 0$ on ${\Gamma_{sonic}}$. Now the comparison principle follows from Lemma \[comparisonPrincipleOfUnifEllipt-Lemma\]. Using Lemma \[existence-Lin-UnifEllipt-Lemma\] and the definition of map $\hat{J}$ in , and using Lemma \[estimates-nonlin-UnifEllipt\] and the Leray-Schauder Theorem, we conclude the proof of Proposition \[existSolUnifEllipt\].
Using Proposition \[existSolUnifEllipt\] and sending $\delta\to
0$, we establish the existence of a solution of problem (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]).
\[existSolDegenEllipt\] Let ${\sigma}, \varepsilon, M_1,$ and $M_2$ be as in Proposition [\[existSolUnifEllipt\]]{}. Then there exists a solution $\psi\in
C(\overline{\Omega^+({\phi})})\cap
C^1(\overline{\Omega^+({\phi})}\setminus\overline{\Gamma_{sonic}}) \cap
C^2(\Omega^+({\phi}))$ of problem [(\[iterationEquation\])]{}–[(\[iterationCondOnSymmtryLine\])]{} so that the solution $\psi$ satisfies [(\[L-infty-for-unif-ellipt\])]{}–[(\[Hoder-est-for-unif-ellipt\])]{}.
Let $\delta\in(0,\delta_0)$. Let $\psi_\delta$ be a solution of (\[unif-ellipt-iterationEquation\]) and (\[iterationRH\])–(\[iterationCondOnSymmtryLine\]) obtained in Proposition \[existSolUnifEllipt\]. Using (\[Hoder-est-for-unif-ellipt\]), we can find a sequence $\delta_j$ for $j=1,\dots$ and $\psi\in
C^1(\overline{\Omega^+({\phi})}\setminus\overline{\Gamma_{sonic}})\cap
C^2(\Omega^+({\phi}))$ such that, as $j\to\infty$, we have
1. $\delta_j \to 0$;
2. $\psi_{\delta_j}\to\psi$ in $C^1(\overline{\Omega^+_s({\phi})})$ for every $s\in (0,c_2/2)$, where $\Omega^+_s({\phi})=\Omega^+({\phi})\cap\{c_2-r>s\}$;
3. $\psi_{\delta_j}\to\psi$ in $C^2(K)$ for every compact $K\subset\Omega^+({\phi})$.
Then, since each $\psi_{\delta_j}$ satisfies (\[unif-ellipt-iterationEquation\]), (\[iterationRH\]), and (\[iterationCondOnWedge\])–(\[iterationCondOnSymmtryLine\]), it follows that $\psi$ satisfies (\[iterationEquation\])–(\[iterationRH\]) and (\[iterationCondOnWedge\])–(\[iterationCondOnSymmtryLine\]). Also, since each $\psi_{\delta_j}$ satisfies (\[L-infty-for-unif-ellipt\])–(\[Hoder-est-for-unif-ellipt\]), $\psi$ also satisfies these estimates. From (\[barier-for-unif-ellipt\]), we conclude that $\psi\in
C(\overline{\Omega^+({\phi})})$ and satisfies (\[iterationCondOnSonicLine\]).
Existence of the Iteration Map and Its Fixed Point {#fixedPtSection}
==================================================
In this section we perform Steps 4–8 of the procedure described in §\[overViewProcedureSubsection\]. In the proofs of this section, the universal constant $C$ depends only on the data.
We assume that ${\phi}\in{{\mathcal K}}$ and the coefficients in problem (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]) are determined by ${\phi}$. Then the existence of a solution $\psi\in
C(\overline{\Omega^+({\phi})})\cap
C^1(\overline{\Omega^+({\phi})}\setminus\overline{\Gamma_{sonic}}) \cap
C^2(\Omega^+({\phi}))$ of (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]) follows from Proposition \[existSolDegenEllipt\].
We first show that a comparison principle holds for (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]). We use the operators ${{\mathcal N}}$ and ${{\mathcal M}}$ introduced in (\[iterationEquation\]) and (\[iterationRH\]). Also, for $\mu>0$, we denote $$\begin{aligned}
&&\Omega^{+,\mu}({\phi}):=\Omega^+({\phi})\cap\{c_2-r<\mu\}, \quad
{\Gamma_{shock}}^\mu({\phi}):={\Gamma_{shock}}({\phi})\cap\{c_2-r<\mu\},
\\
&& {\Gamma_{wedge}}^\mu:={\Gamma_{wedge}}\cap\{c_2-r<\mu\}.\end{aligned}$$
\[comparisonPrincipleOfDegenEllipt-Lemma\] Let ${\sigma}, \varepsilon, M_1$, and $M_2$ be as in Proposition [\[existSolDegenEllipt\]]{}, and $\mu\in (0, \kappa)$, where $\kappa$ is defined in §[\[iteration-dom-subsect\]]{}. Then the following comparison principle holds: If $\psi_1, \psi_2\in
C(\overline{\Omega^{+,\mu}({\phi})}) \cap
C^{1}(\overline{\Omega^{+,\mu}({\phi})}\setminus\overline{\Gamma_{sonic}}) \cap
C^2(\Omega^{+,\mu}({\phi}))$ satisfy that $$\begin{aligned}
&&{{\mathcal N}}(\psi_1)\le{{\mathcal N}}(\psi_2) \qquad \text{in}\,\,
\Omega^{+,\mu}({\phi}),\\
&&{{\mathcal M}}(\psi_1)\le{{\mathcal M}}(\psi_2)\qquad \text{on}\,\,
{\Gamma_{shock}}^\mu({\phi}),\\
&&\partial_\nu \psi_1\le\partial_\nu \psi_2\qquad \mbox{on}\,\,
{\Gamma_{wedge}}^\mu, \\
&&\psi_1\ge\psi_2\qquad \mbox{on} \,\, {\Gamma_{sonic}}\,\mbox{and }\,\,
\Omega^+({\phi})\cap\{c_2-r=\mu\},\end{aligned}$$ then $$\psi_1\ge \psi_2 \qquad \mbox{in }\, \Omega^{+,\mu}.$$
Denote $\Sigma_\mu:=\Omega^+({\phi})\cap\{c_2-r=\mu\}$. If $\mu\in
(0, \kappa)$, then $\partial \Omega^{+,\mu}({\phi})=
{\Gamma_{shock}}^\mu({\phi})\cup{\Gamma_{wedge}}^\mu\cup\overline{\Gamma_{sonic}}\cup\overline{\Sigma_\mu}$.
From ${{\mathcal N}}(\psi_1)\le{{\mathcal N}}(\psi_2)$, the difference $\psi_1-\psi_2$ is a supersolution of a linear equation of form (\[unif-ellipt-linear-iterationEquation\]) in $\Omega^{+,\mu}({\phi})$ and, by Lemma \[propertiesNonlinCoeffs\] (\[propertiesNonlinCoeffs-i1\]), this equation is uniformly elliptic in $\Omega^{+,\mu}({\phi})\cap\{c_2-r>s\}$ for any $s\in (0,
\mu)$. Then the argument of Steps (i)–(ii) in the proof of Lemma \[comparisonPrincipleOfUnifEllipt-Lemma\] implies that $\psi_1-\psi_2$ cannot achieve a negative minimum in the interior of $\Omega^{+,\mu}({\phi})\cap\{c_2-r>s\}$ and in the relative interiors of ${\Gamma_{shock}}^\mu({\phi})\cap\{c_2-r>s\}$ and ${\Gamma_{wedge}}^\mu\cap\{c_2-r>s\}$. Sending $s\to 0+$, we conclude the proof.
\[uniquenessCor\] A solution $\psi\in C(\overline{\Omega^{+}({\phi})}) \cap
C^{1}(\overline{\Omega^{+}({\phi})}\setminus\overline{\Gamma_{sonic}}) \cap
C^2(\Omega^{+}({\phi}))$ of [(\[iterationEquation\])]{}–[(\[iterationCondOnSymmtryLine\])]{} is unique.
If $\psi_1$ and $\psi_2$ are two solutions, then we repeat the proof of Lemma \[comparisonPrincipleOfDegenEllipt-Lemma\] to show that $\psi_1-\psi_2$ cannot achieve a negative minimum in $\Omega^+({\phi})$ and in the relative interiors of ${\Gamma_{shock}}({\phi})$ and ${\Gamma_{wedge}}$. Now equation (\[iterationEquation\]) is linear, uniformly elliptic near $\Sigma_0$ (by Lemma \[propertiesNonlinCoeffs\]), and the function $\psi_1-\psi_2$ is $C^1$ up to the boundary in a neighborhood of $\Sigma_0$. Then the boundary condition [(\[iterationCondOnSymmtryLine\])]{} combined with Hopf’s Lemma yields that $\psi_1-\psi_2$ cannot achieve a minimum in the relative interior of $\Sigma_0$. By the argument of Step (iii) in the proof of Lemma \[comparisonPrincipleOfUnifEllipt-Lemma\], $\psi_1-\psi_2$ cannot achieve a negative minimum at the points ${{P_2}}$ and ${{P_3}}$. Thus, $\psi_1\ge \psi_2$ in $\Omega^+({\phi})$ and, by symmetry, the opposite is also true.
\[quadraticGrowthPsi-Lemma\] There exists $\hat C>0$ depending only on the data such that, if ${\sigma}, \varepsilon, M_1$, and $M_2$ satisfy [(\[condConst-00\])]{}, the solution $\psi\in
C(\overline{\Omega^+({\phi})})\cap
C^1(\overline{\Omega^+({\phi})}\setminus\overline{\Gamma_{sonic}}) \cap
C^2(\Omega^+({\phi}))$ of [(\[iterationEquation\])]{}–[(\[iterationCondOnSymmtryLine\])]{} satisfies $$0\le\psi(x,y)\le \frac{3}{5(\gamma+1)}x^2 \qquad\mbox{in
}\;{\Omega'}({\phi}):=\Omega^{+,2\varepsilon}({\phi}).
\label{L-ifty-BdIteratin-Sonic}$$
We first notice that $\psi\ge 0$ in $\Omega^+({\phi})$ by Proposition \[existSolDegenEllipt\]. Now we make estimate . Set $$w(x,y):=\frac{3}{5(\gamma+1)}x^2.$$
We first show that $w$ is a supersolution of equation (\[iterationEquation\]). Since (\[iterationEquation\]) rewritten in the $(x,y)$–coordinates in ${\Omega'}({\phi})$ has form (\[iteration-equation-sonicStruct\]), we write it as $${{\mathcal N}}_1(\psi)+{{\mathcal N}}_2(\psi)=0,$$ where $$\begin{aligned}
&&{{\mathcal N}}_1(\psi)= \big(
2x-(\gamma+1)x\zeta_1(\frac{\psi_x}{x})\big)\psi_{xx} +{1\over
c_2}\psi_{yy}
-\psi_{x},\\
&&{{\mathcal N}}_2(\psi)=O_1^{\phi}\psi_{xx}+O_2^{\phi}\psi_{xy}
+O_3^{\phi}\psi_{yy}-O_4^{\phi}\psi_{x}+O_5^{\phi}\psi_{y}.\end{aligned}$$ Now we substitute $w(x,y)$. By (\[defZeta-1\]), $$\zeta_1\big(\frac{w_x}{x}\big)=
\zeta_1\big(\frac{6}{5(\gamma+1)}\big)=\frac{6}{5(\gamma+1)},$$ thus $${{\mathcal N}}_1(w)=-\frac{6}{25(\gamma+1)}x.$$ Using (\[estSmallterms-iter\]), we have $$|{{\mathcal N}}_2(w)| = \Big|\frac{6}{5(\gamma+1)}O_1^{\phi}(Dw, x, y)
+\frac{6x}{5(\gamma+1)} O_4^{\phi}(Dw, x, y)\Big| \le Cx^{3/2}\le
C\varepsilon^{1/2}x,$$ where the last inequality holds since $x\in(0, 2\varepsilon)$ in ${\Omega'}({\phi})$. Thus, if $\varepsilon$ is small, we find $${{\mathcal N}}(w)<0\qquad\mbox{ in }\;{\Omega'}({\phi}).$$ The required smallness of $\varepsilon$ is achieved if (\[condConst-00\]) is satisfied with large $\hat C$.
Also, $w$ is a supersolution of (\[iterationRH\]): Indeed, since (\[iterationRH\]) rewritten in the $(x,y)$–coordinates has form (\[iterationRH-lf-flattened\]), estimates (\[estCoefsIterRH-flattened\]) hold, and $x>0$, we find $${{\mathcal M}}(w)=\hat b_1(x,y)\frac{6}{5(\gamma+1)}x+\hat
b_3(x,y)\frac{3}{5(\gamma+1)}x^2<0 \qquad \mbox{on }\, \,
{\Gamma_{shock}}({\phi})\cap\overline{{\mathcal D}'}.$$
Moreover, on ${\Gamma_{wedge}}$, $w_\nu\equiv w_y=0=\psi_\nu$. Furthermore, $w=0=\psi$ on ${\Gamma_{sonic}}$ and, by (\[L-infty-for-unif-ellipt\]), $\psi\le w$ on $\{x=2\varepsilon\}$ if $$C{\sigma}\le \varepsilon^2,$$ where $C$ is a large constant depending only on the data, i.e., if (\[condConst-00\]) is satisfied with large $\hat C$. Thus, $\psi\le w$ in ${\Omega'}({\phi})$ by Lemma \[comparisonPrincipleOfDegenEllipt-Lemma\]. We now estimate the norm $\|\psi\|_{2,\alpha,{\hat\Omega'}({\phi})}^{(par)}$ in the subdomain ${\hat\Omega'}({\phi}){:=}\Omega^+({\phi})\cap \{c_2-r<\varepsilon\}$ of ${\Omega'}({\phi}){:=}\Omega^+({\phi})\cap \{c_2-r<2\varepsilon\}$.
\[EstParabolicHolder-Lemma\] There exist $\hat C, C>0$ depending only on the data such that, if ${\sigma}, \varepsilon, M_1$, and $M_2$ satisfy [(\[condConst-00\])]{}, the solution $\psi\in
C(\overline{\Omega^+({\phi})})\cap
C^1(\overline{\Omega^+({\phi})}\setminus\overline{\Gamma_{sonic}}) \cap
C^2(\Omega^+({\phi}))$ of [(\[iterationEquation\])]{}–[(\[iterationCondOnSymmtryLine\])]{} satisfies $$\|\psi\|_{2,\alpha,{\hat\Omega'}({\phi})}^{(par)} \leq C.
\label{ParabolicHolder-BdIteratin-Sonic}$$
We assume $\hat C$ in (\[condConst-00\]) is sufficiently large so that ${\sigma}, \varepsilon, M_1$, and $M_2$ satisfy the conditions of Lemma [\[quadraticGrowthPsi-Lemma\]]{}.
We work in the $(x,y)$–coordinates and, in particular, we use (\[domain-in-rescaled-lemma\])–(\[holder-hat-f\]). We can assume $\varepsilon<\kappa/20$, which can be achieved by increasing $\hat C$ in (\[condConst-00\]).
For $z:=(x,y)\in {\hat\Omega'}({\phi})$ and $\rho\in(0,1)$, define $$\label{parabRectangles-1}
\quad \tilde R_{z,\rho}:=\left\{(s,t)\;\;:\;\;
|s-x|<\frac{\rho}{4}x, |t-y|<\frac{\rho}{4}\sqrt{x}\right\}, \quad
R_{z,\rho}:=\tilde R_{z,\rho} \cap \Omega^+({\phi}).$$ Since ${\Omega'}({\phi})=\Omega^+({\phi})\cap \{c_2-r<2\varepsilon\}$, then, for any $z\in {\hat\Omega'}({\phi})$ and $\rho\in(0,1)$, $$\label{localizeRectangle}
R_{z, \rho}\subset \Omega^+({\phi})\cap \{(s,t)\;\;:\;\;
\frac{3}{4}x<s<\frac{5}{4}x\}
\subset{\Omega'}({\phi}).
$$
For any $z\in {\hat\Omega'}({\phi})$, we have at least one of the following three cases:
1. $R_{z, 1/10}=\tilde R_{z, 1/10}$;
2. $z\in R_{z_w, 1/2}$ for $z_w=(x,0)\in {\Gamma_{wedge}}$;
3. $z\in R_{z_s, 1/2}$ for $z_s=(x,\hat f_{\phi}(x))\in {\Gamma_{shock}}({\phi})$.
Thus, it suffices to make the local estimates of $D\psi$ and $D^2\psi$ in the following rectangles with $z_0:=(x_0, y_0)$:
1. \[cases-ParabolicHolder-1\] $R_{z_0, {1/20}}$ for $z_0\in{\hat\Omega'}({\phi})$ and $R_{z_0,
1/10}=\tilde R_{z_0, 1/10}$;
2. \[cases-ParabolicHolder-2\] $R_{z_0, {1/2}}$ for ${z_0}\in {\Gamma_{wedge}}\cap \{x<\varepsilon\}$;
3. \[cases-ParabolicHolder-3\] $R_{z_0, {1/2}}$ for ${z_0}\in {\Gamma_{shock}}({\phi})\cap
\{x<\varepsilon\}$.
We first consider case (\[cases-ParabolicHolder-1\]) in Step 1. Then $$R_{z_0, {1/10}}=\big\{(x_0+\frac{x_0}{4}S,
y_0+\frac{\sqrt{x_0}}{4}T)\; : \;
(S,T)\in Q_{1/10}\big\},$$ where $Q_\rho{:=}(-\rho, \rho)^2$ for $\rho>0$.
Rescale $\psi$ in $R_{z_0, {1/10}}$ by defining $$\label{parabRescaling-1}
\psi^{(z_0)}(S, T):=\frac{1}{x_0^2}\psi(x_0+\frac{x_0}{4}S,
y_0+\frac{\sqrt{x_0}}{4}T) \qquad\mbox{for }\,\,(S, T)\in Q_{1/10}.$$ Then, by (\[L-ifty-BdIteratin-Sonic\]) and (\[localizeRectangle\]), $$\label{estInterLinfty-Rescaled-pf}
\|\psi^{(z_0)}\|_{C(\overline{Q_{1/10}})}\le 1/(\gamma+1).$$ Moreover, since $\psi$ satisfies equation (\[iteration-equation-sonicStruct\])–(\[iteration-equation-sonicStruct-coef\]) in $R_{z_0, {1/10}}$, then $\psi^{(z_0)}$ satisfies $$\begin{aligned}
\label{iteration-equation-sonicStruct-ParabRescaled-1}
&&\qquad \Big((1+\frac{1}{4}S)\big(2-
(\gamma+1)\zeta_1(\frac{4\psi^{(z_0)}_S}{1+S/4})\big) +x_0
O_1^{({\phi},z_0)} \Big)\psi^{(z_0)}_{SS}
+x_0 O_2^{({\phi},z_0)}\psi^{(z_0)}_{ST}
\\
&&\qquad\qquad
+ \big({1\over c_2}+x_0 O_3^{({\phi},z_0)}\big)\psi^{(z_0)}_{TT}
-({1\over 4}+x_0O_4^{({\phi},z_0)})\psi^{(z_0)}_{S} +x_0^2
O_5^{({\phi},z_0)}\psi^{(z_0)}_{T}=0 \nonumber\end{aligned}$$ in $Q_{1/10}$, where $$\label{iteration-equation-sonicStruct-coef-rescaled}
\begin{array}{ll}
\tilde O_1^{{\phi}, z_0}(p,S,T)=
-\frac{(1+{S/4})^2}{c_2}+{\gamma+1\over 2c_2}
\Big(2(1+{S/4})^2\zeta_1\big(\frac{4p_1}{1+{S/4}}\big)
-16|{\phi}^{(z_0)}_S|^2\Big)
\\
\quad \qquad\qquad\qquad-{\gamma-1\over c_2}\Big({\phi}^{(z_0)}+
{8x_0\over
(c_2-x_0(1+{S/4}))^2}|{\phi}^{(z_0)}_T|^2\Big), \\
\tilde O_2^{{\phi}, z_0}(p,S,T)=-{8\over c_2(c_2-x_0(1+{S/4}))^2}
\big(4x_0{\phi}^{(z_0)}_S+c_2-x_0(1+{S/4})\big){\phi}^{(z_0)}_T,
\\
\tilde O_3^{{\phi}, z_0}(p,S,T)\\
\quad ={1\over c_2(c_2-x_0(1+{S/4}))^2}
\bigg\{(1+{S/4})\big(2c_2-x_0(1+{S/4})\big)
\\
\qquad-(\gamma-1) \Big( x_0{\phi}^{(z_0)}+ (c_2-x_0(1+{S/4}))(1+{S/4})
\zeta_1\big(\frac{4p_1}{1+{S/4}}\big)+8x_0|{\phi}^{(z_0)}_S|^2 \Big)
\\
\qquad -\frac{8(\gamma+1)}{(c_2-x_0(1+{S/4}))^2}
x_0^2|{\phi}^{(z_0)}_T|^2\bigg\}, \\
\tilde O_4^{{\phi},z_0}(p,S,T) =
\frac{1}{c_2-x_0(1+{S/4})}\bigg\{1+{S/4} - {\gamma-1\over
c_2}\bigg(x_0{\phi}^{(z_0)} +8x_0|{\phi}^{(z_0)}_S|^2 \\
\qquad\qquad \qquad\quad +\big(c_2-x_0(1+{S/4})\big) (1+{S/4})
\zeta_1\big(\frac{4p_1}{1+{S/4}}\big) +\frac{
8|x_0{\phi}^{(z_0)}_T|^2} {(c_2-x_0(1+{S/4}))^2} \bigg)
\bigg\}, \\
\tilde O_5^{{\phi}, z_0}(p,S,T)={8\over c_2(c_2-x_0(1+{S/4}))^2}
\big(4x_0{\phi}^{(z_0)}_S+2c_2-2x_0(1+{S/4})\big) {\phi}^{(z_0)}_T,
\end{array}$$ where ${\phi}^{(z_0)}$ is the rescaled ${\phi}$ as in (\[parabRescaling-1\]). By (\[localizeRectangle\]) and ${\phi}\in{{\mathcal K}}$, we have $$\|{\phi}^{(z_0)}\|_{C^{2,\alpha}(\overline{Q_{1/10}})}\le CM_1,$$ and thus $$\label{errorTermsRescaled-pf}
\|\tilde O_k^{{\phi},
z_0}\|_{C^1(\overline{Q_{1/10}^{(z)}}\times{ {\bf R}}^2)} \le C(1+M_1^2),
\quad k=1,\dots, 5.$$ Now, since every term $O_k^{({\phi},z_0)}$ in (\[iteration-equation-sonicStruct-ParabRescaled-1\]) is multiplied by $x_0^{\beta_k}$ with $\beta_k\ge 1$ and $x_0\in (0,
\varepsilon)$, condition (\[condConst-00\]) (possibly after increasing $\hat C$) depending only on the data implies that equation (\[iteration-equation-sonicStruct-ParabRescaled-1\]) satisfies conditions (\[locEstElliptEq-i1-0\])–(\[locEstElliptEq-i2-0\]) in $Q_{1/10}$ with $\lambda>0$ depending only on $c_2$, i.e., on the data by (\[condRewritingRH-0\]). Then, using Theorem \[locEstElliptEq\] and (\[estInterLinfty-Rescaled-pf\]), we find $$\label{estInterHolder-Rescaled-1-pf}
\|\psi^{(z_0)}\|_{C^{2,\alpha}(\overline{Q_{1/20}})}\le C.$$
We then consider case (\[cases-ParabolicHolder-2\]) in Step 1. Let ${z_0}\in {\Gamma_{wedge}}\cap \{x<\varepsilon\}$. Using (\[domain-in-rescaled-lemma\]) and assuming that ${\sigma}$ and $\varepsilon$ are sufficiently small depending only on the data, we have $\overline{R_{z_0,
1}}\cap\partial\Omega^+({\phi})\subset{\Gamma_{wedge}}$ and thus, for any $\rho\in(0,1]$, $$R_{z_0, \rho}=\bigg\{(x_0+\frac{x_0}{4}S,
y_0+\frac{\sqrt{x_0}}{4}T)\; : \;
(S,T)\in Q_\rho\cap\{T>0\}\bigg\}.$$ The choice of parameters for that can be made as follows: First choose ${\sigma}$ small so that $|\bar{{\xi}}-{{\xi}}_1|\le |\bar{{\xi}}|/10$, where $\bar{{\xi}}$ is defined by (\[3.14\]), which is possible since ${{\xi}}_1\to \bar{{\xi}}$ as $\theta_w\to \pi/2$, and then choose $\varepsilon<(|\bar{{\xi}}|/10)^2$.
Define $\psi^{(z_0)}(S,T)$ by (\[parabRescaling-1\]) for $(S,T)\in
Q_1\cap\{T>0\}$. Then, by (\[L-ifty-BdIteratin-Sonic\]) and (\[localizeRectangle\]), $$\label{estInterLinfty-2-Rescaled-pf}
\|\psi^{(z_0)}\|_{C(\overline{Q_1}\cap\{T\ge 0\})}\le 1/(\gamma+1).$$ Moreover, similar to Step 2, $\psi^{(z_0)}$ satisfies equation (\[iteration-equation-sonicStruct-ParabRescaled-1\]) in $Q_1\cap\{T>0\}$, and the terms $\tilde O_k^{{\phi}, z_0}$ satisfy estimate (\[errorTermsRescaled-pf\]) in $Q_1\cap\{T>0\}$. Then, as in Step 2, we conclude that (\[iteration-equation-sonicStruct-ParabRescaled-1\]) satisfies conditions (\[locEstElliptEq-i1-0\])–(\[locEstElliptEq-i2-0\]) in $Q_1\cap\{T>0\}$ if (\[condConst-00\]) holds with sufficiently large $\hat C$. Moreover, since $\psi$ satisfies (\[iterationCondOnWedge\]), it follows that $$\partial_T\psi^{(z_0)}=0\qquad\mbox{on }
\{T=0\}\cap Q_{1/2}.$$ Then, from Theorem \[locEstElliptEq-oblique\], $$\label{estInterHolder-Rescaled-2-pf}
\|\psi^{(z_0)}\|_{C^{2,\alpha}(\overline{Q_{1/2}}\cap\{T\ge 0\})}\le
C.$$
We now consider case (\[cases-ParabolicHolder-3\]) in Step 1. Let ${z_0}\in {\Gamma_{shock}}({\phi})\cap \{x<\varepsilon\}$. Using (\[domain-in-rescaled-lemma\]) and the fact that $y_0=\hat
f_{\phi}(x_0)$ for $z_0\in{\Gamma_{shock}}({\phi})\cap\{x<\varepsilon\}$, and assuming that ${\sigma}$ and $\varepsilon$ are small as in Step 3, we have $\overline{R_{z_0,
1}}\cap\partial\Omega^+({\phi})\subset{\Gamma_{shock}}({\phi})$ and thus, for any $\rho\in(0,1]$, $$R_{z_0, \rho }=\bigg\{(x_0+\frac{x_0}{4} S,
y_0+\frac{\sqrt{x_0}}{4} T)\; : \;
(S,T)\in Q_\rho\cap\{T<\varepsilon^{1/4}F_{(z_0)}(S)\}\bigg\}$$ with $$F_{(z_0)}(S)=4\frac{\hat f_{\phi}(x_0+\frac{x_0}{4}S)- \hat
f_{\phi}(x_0)}{\varepsilon^{1/4}\sqrt{ x_0}}.$$ Then we use (\[holder-hat-f-S\]) and $x_0\in (0, 2\varepsilon)$ to obtain $$\begin{aligned}
&&F_{(z_0)}(0)=0,\\
&&\|F_{(z_0)}\|_{C^1([-{1/2}, {1/2}])} \le {\|\hat
f'_{\phi}\|_{L^\infty([0,2\varepsilon])}x_0\over\varepsilon^{1/4}
\sqrt{x_0}}\le C(1+M_1\varepsilon)\varepsilon^{1/4},\\
&& \|F_{(z_0)}''\|_{C^\alpha([-{1/2}, {1/2}])}
\le {\|\hat
f''_{\phi}\|_{L^\infty([0,2\varepsilon])}x_0^2
+[\hat f_{\phi}'']_{\alpha, (x_0/2, \varepsilon)}x_0^{2+\alpha}
\over 4\varepsilon^{1/4}\sqrt{x_0}}\le C(1+M_1)\varepsilon^{5/4},\end{aligned}$$ and thus, from (\[condConst-00\]), $$\label{norm-rescaled-bdry-func}
\|F_{(z_0)}\|_{C^{2,\alpha}([-{1/2},{1/2}])}\le C/\hat C\le 1$$ if $\hat C$ is large. Define $\psi^{(z_0)}(S, T)$ by (\[parabRescaling-1\]) for $(S,T)\in
Q_1\cap\{T<\varepsilon^{1/4}F_{(z_0)}(S)\}$. Then, by (\[L-ifty-BdIteratin-Sonic\]) and (\[localizeRectangle\]), $$\label{estInterLinfty-3-Rescaled-pf}
\|\psi^{(z_0)}\|_{C(\overline{Q_1}\cap\{T\le F_{(z_0)}(S)\})} \le
1/(\gamma+1).$$ Similar to Steps 2–3, $\psi^{(z_0)}$ satisfies equation (\[iteration-equation-sonicStruct-ParabRescaled-1\]) in $Q_1\cap\{T<\varepsilon^{1/4}F_{(z_0)}(S)\}$ and the terms $\tilde
O_k^{{\phi}, z_0}$ satisfy estimate (\[errorTermsRescaled-pf\]) in $Q_1\cap\{T<\varepsilon^{1/4}F_{(z_0)}(S)\}$. Then, as in Steps 2–3, we conclude that (\[iteration-equation-sonicStruct-ParabRescaled-1\]) satisfies conditions (\[locEstElliptEq-i1-0\])–(\[locEstElliptEq-i2-0\]) in $Q_1\cap\{T<\varepsilon^{1/4}F_{(z_0)}(S)\}$ if (\[condConst-00\]) holds with sufficiently large $\hat C$. Moreover, $\psi$ satisfies (\[iterationRH\]) on ${\Gamma_{shock}}({\phi})$, which can be written in form (\[iterationRH-lf-flattened\]) on ${\Gamma_{shock}}({\phi})\cap {{\mathcal D}'}$. This implies that $\psi^{(z_0)}$ satisfies $$\partial_S\psi^{(z_0)}
=\varepsilon^{1/4}\left(B_2\partial_T\psi^{(z_0)}+B_3\psi^{(z_0)}\right)
\qquad\mbox{on } \{T=\varepsilon^{1/4}F_{(z_0)}(S)\}\cap Q_{1/2},$$ where $$\begin{aligned}
&&B_2(S,T)=-{\sqrt{x_0}\over\varepsilon^{1/4}} {\hat b_2\over\hat
b_1}(x_0+\frac{x_0}{4}S,
y_0+\frac{\sqrt{x_0}}{4}T),\\
&&B_3(S,T)=-{x_0\over 4\varepsilon^{1/4}} {\hat b_3\over\hat
b_1}(x_0+\frac{x_0}{4}S,
y_0+\frac{\sqrt{x_0}}{4}T).\end{aligned}$$ From (\[estCoefsIterRH-flattened\]), $$\|(B_2, B_3)\|_{1,\alpha, \overline{Q_1}\cap
\{T\le\varepsilon^{1/4}F_{(z_0)}(S)\}}\le C\varepsilon^{1/4}M_1\le
C/\hat C\le 1.$$
Now, if $\varepsilon$ is sufficiently small, it follows from Theorem \[locEstElliptEq-non-oblique\] that $$\label{estInterHolder-Rescaled-3-pf}
\|\psi^{(z_0)}\|_{C^{2,\alpha}(\overline{Q_{1/2}}\cap
\{T\le\varepsilon^{1/4}F_{(z_0)}(S)\})}\le C.$$ The required smallness of $\varepsilon$ is achieved by choosing large $\hat C$ in (\[condConst-00\]).
Combining (\[estInterHolder-Rescaled-1-pf\]), (\[estInterHolder-Rescaled-2-pf\]), and (\[estInterHolder-Rescaled-3-pf\]) with an argument similar to the proof of [@GilbargTrudinger Theorem 4.8] (see also the proof of Lemma \[partIntSeminorm-est-lemma\] below), we obtain (\[ParabolicHolder-BdIteratin-Sonic\]). Now we define the extension of solution $\psi$ from the domain $\Omega^+({\phi})$ to the domain ${{\mathcal D}}$.
\[extension-Lemma\] There exist $\hat C, C_1>0$ depending only on the data such that, if ${\sigma}, \varepsilon, M_1$, and $M_2$ satisfy [(\[condConst-00\])]{}, there exists $C_2(\varepsilon)$ depending only on the data and $\varepsilon$ and, for any ${\phi}\in {{\mathcal K}}$, there exists an extension operator $${{\mathcal {P}}}_{\phi}: C^{1, \alpha}(\overline{\Omega^+({\phi})})\cap C^{2,
\alpha}(\overline{\Omega^+({\phi})}\setminus
\overline{{\Gamma_{sonic}}\cup\Sigma_0}) \to C^{1, \alpha}(\overline{{\mathcal D}})\cap
C^{2, \alpha}({{\mathcal D}})$$ satisfying the following two properties:
1. \[Uextended\_Est\] If $\psi\in C^{1, \alpha}(\overline{\Omega^+({\phi})})\cap
C^{2,\alpha}(\overline{\Omega^+({\phi})} \setminus
\overline{{\Gamma_{sonic}}\cup\Sigma_0})$ is a solution of problem [(\[iterationEquation\])]{}–[ (\[iterationCondOnSymmtryLine\])]{}, then $$\begin{aligned}
&&\|{{\mathcal {P}}}_{\phi}\psi\|_{2,\alpha, {{\mathcal D}'}}^{(par)}
\leq C_1,
\label{Hold-Ext-Sonic}\\
&&\|{{\mathcal {P}}}_{\phi}\psi\|^{(-1-\alpha, \Sigma_0)}_{2,\alpha,
{{\mathcal D}''}} \le C_2(\varepsilon){\sigma};
\label{Hold-Ext-Unif}\end{aligned}$$
2. \[Uextended\_Cont\] Let $\beta \in (0, \alpha)$. If a sequence ${\phi}_k \in {{\mathcal K}}$ converges to ${\phi}$ in $C^{1, \beta}(\overline{{{\mathcal D}}})$, then ${\phi}\in {{\mathcal K}}$. Furthermore, if $\psi_k\in C^{1,
\alpha}(\overline{\Omega^+({\phi}_k)})\cap C^{2,
\alpha}(\overline{\Omega^+({\phi}_k)}\setminus\overline{{\Gamma_{sonic}}\cup\Sigma_0})$ and $\psi \in C^{1, \alpha}(\overline{\Omega^+({\phi})})\cap C^{2,
\alpha}(\overline{\Omega^+({\phi})}\setminus
\overline{{\Gamma_{sonic}}\cup\Sigma_0})$ are the solutions of problems [(\[iterationEquation\])]{}–[(\[iterationCondOnSymmtryLine\])]{} for ${\phi}_k$ and ${\phi}$ respectively, then ${{\mathcal {P}}}_{{\phi}_k}\psi_k\rightarrow {{\mathcal {P}}}_{\phi}\psi$ in $C^{1,
\beta}(\overline{{{\mathcal D}}})$.
Let $\kappa>0$ be the constant in (\[domain-in-rescaled-lemma\]) and $\varepsilon<\kappa/20$. For any ${\phi}\in{{\mathcal K}}$, we first define the extension operator separately on the domains $\Omega_1{:=}\Omega^+({\phi})\cap\{c_2-r<\kappa\}$ and $\Omega_2{:=}\Omega^+({\phi})\cap\{c_2-r>\kappa/2\}$ and then combine them to obtain the operator ${{\mathcal {P}}}_{\phi}$ globally.
In the argument below, we will state various smallness requirements on ${\sigma}$ and $\varepsilon$, which will depend only on the data, and can be achieved by choosing $\hat C$ sufficiently large in (\[condConst-00\]). Also, the constant $C$ in this proof depends only on the data.
First we discuss some properties on the domains $\Omega^+({\phi})$ and ${{\mathcal D}}$ to be used below. Recall $\bar{{\xi}}<0$ defined by (\[3.14\]), and the coordinates $({{\xi}}_1,{{\eta}}_1)$ of the point ${{P_1}}$ defined by (\[coord-P4\]). We assume ${\sigma}$ small so that $|\bar{{\xi}}-{{\xi}}_1|\le |\bar{{\xi}}|/10$, which is possible since ${{\xi}}_1\to \bar{{\xi}}$ as $\theta_w\to \pi/2$. Then ${{\xi}}_1<0$. By (\[OmegaPL-f-higher\]) and ${{P_1}}\in {\Gamma_{shock}}({\phi})$, it follows that $$\label{domainExtension-0'}
{\Gamma_{shock}}({\phi})\subset \overline{{\mathcal D}}\cap \{{{\xi}}<{{\xi}}_1+\varepsilon^{1/4}\}.$$ Also, choosing $\varepsilon^{1/4}<|\bar{{\xi}}|/10$, we have $$\label{domainExtension-0}
{{\xi}}_1+\varepsilon^{1/4}<\bar{{\xi}}/2<0.$$ Furthermore, when ${\sigma}$ is sufficiently small, $$\label{domainExtension}
\mbox{\em if $({{\xi}}, {{\eta}})\in{{\mathcal D}}\cap\{{{\xi}}<{{\xi}}_1+\varepsilon^{1/4}\}$,
$({{\xi}}', {{\eta}})\in{{\mathcal D}}$, and ${{\xi}}'>{{\xi}}$, then
$|{{\xi}}'|<|{{\xi}}|$. }$$ Indeed, from the conditions in (\[domainExtension\]), we have $$-c_2<{{\xi}}<{{\xi}}_1+\varepsilon^{1/4}<\bar{{\xi}}/2<0.$$ Thus, $|{{\xi}}'|<|{{\xi}}|$ if ${{\xi}}'<0$. It remains to consider the case ${{\xi}}'>0$. Since ${{\mathcal D}}\subset
B_{c_2}(0)\cap\{{{\xi}}<{{\eta}}\cot\theta_w\}$, it follows that $|{{\xi}}'|\le c_2\cos \theta_w$. Thus $|{{\xi}}'|<|{{\xi}}|$ if $c_2\cos
\theta_w\le |\bar{{\xi}}|/2$. Using (\[condRewritingRH-0\]) and (\[angleCloseToPiOver2\]), we see that the last inequality holds if ${\sigma}>0$ is small depending only on the data. Then (\[domainExtension\]) is proved.
Now we define the extensions.
First, on $\Omega_1$, we work in the $(x,y)$–coordinates. Then $\Omega_1=\{0< x< \kappa,\, 0<y<\hat
f_{\phi}(x)\}$ by (\[domain-in-rescaled-lemma\]). Denote $Q_{(a,b)}{:=}(0,\kappa) \times (a,b)$. Define the mapping $\Phi:Q_{(-\infty,\infty)}\to Q_{(-\infty,\infty)}$ by $$\Phi(x,y)=(x,1-{y/\hat f_{\phi}(x)}).$$ The mapping $\Phi$ is invertible with the inverse $\Phi^{-1}(x,y)=(x, \hat f_{\phi}(x)(1-y))$. By definition of $\Phi$, $$\begin{aligned}
&&\Phi(\Omega_1)=Q_{(0,1)},\qquad
\Phi({\Gamma_{shock}}({\phi})\cap\{0<x<\kappa\})=(0,\kappa)\times\{0\},
\nonumber \\
&&\Phi({{\mathcal D}}\cap\{0<x<\kappa\})\subset Q_{(-1,1)},
\label{holder-hat-mapF}\end{aligned}$$ where the last property can be seen as follows: First we note that $\hat f_{\phi}(x)\ge {\hat f_{0,0}(0)\over 2}>0$ for $x\in(0,\kappa)$ by (\[fbFUnctionCloseToNormalPolar\]) and (\[holder-hat-f\]), then we use ${{\mathcal D}}\cap\{0<x<\kappa\}=\{0< x< \kappa,\, 0<y<\hat
f_0(x)\}$ and (\[holder-hat-f-S\]) to obtain ${y\over \hat
f_{\phi}(x)}>0$ on ${{\mathcal D}}\cap\{0<x<\kappa\}$ and $$\begin{aligned}
\sup_{(x,y)\in{{\mathcal D}}\cap\{0<x<\kappa\}}{y\over \hat
f_{\phi}(x)}&=&\sup_{x\in(0,\kappa)}{\hat f_0(x)\over \hat
f_{\phi}(x)} \le 1+{2\over \hat f_{0,0}(0)}\| \hat f_{\phi}-\hat
f_0\|_{C(0,\kappa)}\\
&<& 1+C(M_1\varepsilon+M_2{\sigma})<2,\end{aligned}$$ if $M_1\varepsilon$ and $M_2{\sigma}$ are small, which can be achieved by choosing $\hat C$ in (\[condConst-00\]) sufficiently large.
We first define the extension operator: $${{{\mathcal E}_2}}: C^{1,\beta}(\overline{Q_{(0,1)}})\cap
C^{2,\beta}(\overline{Q_{(0,1)}}\setminus\{x=0\}) \to
C^{1,\beta}(\overline{Q_{(-1,1)}})\cap
C^{2,\beta}(\overline{Q_{(-1,1)}}\setminus\{x=0\})$$ for any $\beta\in (0, 1]$. Let $v\in
C^{1,\beta}(\overline{Q_{(0,1)}})\cap
C^{2,\beta}(\overline{Q_{(0,1)}}\setminus\{x=0\})$. Define ${{{\mathcal E}_2}}v=v$ in $Q_{(0,1)}$. For $(x,y) \in Q_{(-1,0)}$, define $$\label{genExtension}
{{{\mathcal E}_2}}v(x,y)=
\sum_{i=1}^3 a_i v(x,-\frac{y}{i}),$$ where $a_1=6$, $a_2=-32$, and $a_3=27$, which are determined by $\sum_{i=1}^3 a_i\left(-\frac{1}{i} \right)^m=1$ for $ m=0, 1, 2$.
Now let $\psi\in C^{1, \alpha}(\overline{\Omega^+({\phi})})\cap
C^{2, \alpha}(\overline{\Omega^+({\phi})}\setminus
\overline{{\Gamma_{sonic}}\cup\Sigma_0})$. Let $$v=\psi|_{\Omega_1}\circ \Phi^{-1}.$$ Then $v\in C^{1,\beta}(\overline{Q_{(0,1)}})\cap
C^{2,\beta}(\overline{Q_{(0,1)}}\setminus\{x=0\})$. By (\[holder-hat-mapF\]), we have ${{\mathcal D}}\cap\{c_2-r<\kappa\}\subset\Phi^{-1}(Q_{(-1,1)})$. Thus, we define an extension operator on $\Omega_1$ by $${{\mathcal {P}}}_{\phi}^1 \psi=({{{\mathcal E}_2}}v)\circ\Phi \qquad \mbox{ on }\;
{{\mathcal D}}\cap\{c_2-r<\kappa\}.$$ Then ${{\mathcal {P}}}_{\phi}^1 \psi\in C^{1, \alpha}(\overline{{{\mathcal D}}_1})\cap
C^{2, \alpha}(\overline{{{\mathcal D}}_1}\setminus\overline{{\Gamma_{sonic}}})$ with ${{\mathcal D}}_1:={{\mathcal D}}\cap\{c_2-r<\kappa\}$.
Next we estimate ${{\mathcal {P}}}_{\phi}^1$ separately on the domains ${{\mathcal D}'}={{\mathcal D}}\cap\{c_2-r<2\varepsilon\}$ and ${{\mathcal D}}_1\cap\{c_2-r>\varepsilon/2\}$.
In order to estimate the Hölder norms of ${{\mathcal {P}}}_{\phi}^1$ on ${{\mathcal D}'}$, we note that $\Phi({\Omega'}({\phi}))=(0,2\varepsilon)\times(0,1)$ and ${{\mathcal D}'}\subset\Phi^{-1}((0,2\varepsilon)\times(-1,1))$ in the $(x,y)$–coordinates. We first show the following estimates, in which the sets are defined in the $(x,y)$–coordinates: $$\begin{aligned}
&&\qquad\quad \|\psi\circ \Phi^{-1}\|_{2,\alpha,
(0,2\varepsilon)\times(0, 1)}^{(par)} \le C\|\psi\|_{2,\alpha,
{\Omega'}({\phi})}^{(par)}\quad\mbox{for any } \psi\in C_{2,\alpha,
(0,2\varepsilon)\times(0, 1)}^{(par)}, \label{normOfSubstitution}
\\
&&\qquad\quad \|w\circ \Phi\|_{2,\alpha, {{\mathcal D}'}}^{(par)} \le
C\|w\|_{2,\alpha, (0,2\varepsilon)\times(-1,
1)}^{(par)}\quad\mbox{for any } w\in C_{2,\alpha,
(0,\varepsilon)\times(-1, 1)}^{(par)},\label{normOfSubstitution-bk}
\\
&&\qquad\quad \|{{{\mathcal E}_2}}v\|_{2,\alpha,(0,2\varepsilon)\times(-1,
1)}^{(par)} \le C\|v\|_{2,\alpha,(0,2\varepsilon)\times(0,
1)}^{(par)}\,\,\,\mbox{for any } v\in
C_{2,\alpha,(0,2\varepsilon)\times(-1, 1)}^{(par)}.
\label{simpleExtInHolderNorms}\end{aligned}$$ To show (\[normOfSubstitution\]), we denote $v=\psi\circ
\Phi^{-1}$ and estimate every term in definition (\[parabNormsApp\]) for $v$. Note that $v(x,y)=\psi(x, \hat
f_{\phi}(x)(1-y))$. In the calculations below, we denote $(v, Dv,
D^2v)=(v, Dv, D^2v)(x,y)$, $(\psi, D\psi, D^2\psi)=(\psi, D\psi,
D^2\psi)(x, \hat f_{\phi}(x)(1-y))$, and $(\hat f_{\phi}, \hat
f_{\phi}', \hat f_{\phi}'')= (\hat f_{\phi}, \hat f_{\phi}', \hat
f_{\phi}'')(x)$. We use that, for $x\in (0,2\varepsilon)$, $0<M_1x<2M_1\varepsilon<{2/\hat C}$ by (\[condConst-00\]). Then, for any $(x,y)\in (0,2\varepsilon)\times(0,1)$, we have $$\begin{aligned}
|v|&=&|\psi|\le \|\psi\|_{2,\alpha,
{\Omega'}({\phi})}^{(par)}x^2,\\
|v_x|&=&|\psi_x+(1-y)\psi_y \hat f_{\phi}'| \le \|\psi\|_{2,\alpha,
{\Omega'}({\phi})}^{(par)}\left(x+x^{3/2}(1+M_1x)\right)\le
C\|\psi\|_{2,\alpha,
{\Omega'}({\phi})}^{(par)}x,\\
|v_{xx}|&=&|\psi_{xx}+2
(1-y)\psi_{xy}\hat f_{\phi}'
+(1-y)^2\psi_{yy}(\hat f_{\phi}')^2
+(1-y)\psi_y\hat f_{\phi}''|\\
&\le& \|\psi\|_{2,\alpha,
{\Omega'}({\phi})}^{(par)}\left(1+x^{1/2}(1+M_1x)+x(1+M_1x)^2+M_1x^{3/2}\right)\\
&\le& C\|\psi\|_{2,\alpha, {\Omega'}({\phi})}^{(par)}.\end{aligned}$$ The estimates of the other terms in (\[parabNormsApp\]) for $v$ follow from similar straightforward (but lengthy) calculations. Thus, (\[normOfSubstitution\]) is proved. The proof of (\[normOfSubstitution-bk\]) is similar by using that $\hat
f_{\phi}(x)\ge {\hat f_{0,0}(0)/2}>0$ for $x\in(0,\kappa)$ from (\[fbFUnctionCloseToNormalPolar\]) and (\[holder-hat-f\]) and that $\hat f_{0,0}(0)$ depends only on the data. Finally, estimate (\[simpleExtInHolderNorms\]) follows readily from (\[genExtension\]).
Now, let $\psi\in C^{1, \alpha}(\overline{\Omega^+({\phi})})\cap
C^{2, \alpha}(\overline{\Omega^+({\phi})}\setminus
\overline{{\Gamma_{sonic}}\cup\Sigma_0})$ be a solution of (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]). Then $$\begin{aligned}
\|{{\mathcal {P}}}_{\phi}^1 \psi\|_{2,\alpha, {{\mathcal D}'}}^{(par)}
&=&\|{{{\mathcal E}_2}}(\psi|_{\Omega_1}\circ \Phi^{-1})\circ\Phi\|_{2,\alpha, {{\mathcal D}'}}^{(par)}
\;\le\; C\|{{{\mathcal E}_2}}(\psi|_{\Omega_1}\circ \Phi^{-1})\|_{2,\alpha,
(0,2\varepsilon)\times(-1, 1)}^{(par)}
\\
&\le& C\|\psi|_{\Omega_1}\circ \Phi^{-1}\|_{2,\alpha, (0,2\varepsilon)\times(0, 1)}^{(par)}
\; \le\; C\|\psi\|_{2,\alpha, {\Omega'}({\phi})}^{(par)}
\; \le\; C,\end{aligned}$$ where the first inequality is obtained from (\[normOfSubstitution-bk\]), the second inequality from (\[simpleExtInHolderNorms\]), the third inequality from (\[normOfSubstitution\]), and the last inequality from (\[ParabolicHolder-BdIteratin-Sonic\]). Thus, (\[Hold-Ext-Sonic\]) holds for ${{\mathcal {P}}}_{\phi}^1$.
Furthermore, using the second estimate in (\[holder-hat-f-S\]), noting that $M_2{\sigma}\le 1$ by (\[condConst-00\]), and using the definition of ${{\mathcal {P}}}_{\phi}^1$ and the fact that the change of coordinates $(x,y)\to({{\xi}}, {{\eta}})$ is smooth and invertible in ${{\mathcal D}}\cap\{{\varepsilon/2}<x<\kappa\}$, we find that, in the $({{\xi}},
{{\eta}})$–coordinates, $$\label{EstExt-1} \|{{\mathcal {P}}}_{\phi}^1
\psi\|_{C^{2,\alpha}(\overline{{\mathcal D}}\cap\{{\varepsilon/2}\le
c_2-r\le\kappa\})} \le
C\|\psi\|_{C^{2,\alpha}(\overline{{\Omega}^+({\phi})}\cap\{{\varepsilon/2}\le
c_2-r \le \kappa\})}.$$
Now we define an extension operator in the $({{\xi}},
{{\eta}})$–coordinates. Let $$\begin{aligned}
\tilde{{{\mathcal E}_2}}&\, :\, & C^1( [0,1]\times [-v_2, {{\eta}}_1])\cap
C^2([0,1]\times
(-v_2, {{\eta}}_1] )\\
&& \to C^1( [-1,1]\times [-v_2, {{\eta}}_1])\cap C^2( [-1,1]\times(-v_2,
{{\eta}}_1])\end{aligned}$$ be defined by $$\tilde{{{\mathcal E}_2}}v(X,Y):=
\sum_{i=1}^3 a_i v(-\frac{X}{i}, Y)\qquad
\mbox{for } \;(X,Y)\in(-1,0)\times (-v_2, {{\eta}}_1),$$ where $a_1, a_2$, and $a_3$ are the same as in (\[genExtension\]).
Let $\hat\Omega_2{:=}\Omega^+({\phi})\cap \{0\le{{\eta}}\le{{\eta}}_1\}$. Define the mapping $\Psi:\hat\Omega_2\to (0,1)\times (-v_2, {{\eta}}_1)$ by $$\Psi({{\xi}},{{\eta}}):=({{{\xi}}- f_{\phi}({{\eta}})\over
{{\eta}}\cot\theta_w-f_{\phi}({{\eta}})}, {{\eta}}),$$ where $f_{\phi}(\cdot)$ is the function from (\[shockPL\])–(\[OmegaPL\]). Then the inverse of $\Psi$ is $$\Psi^{-1}(X,Y)=(f_{\phi}(Y)+X(Y\cot\theta_w-f_{\phi}(Y)), Y),$$ and thus, from (\[OmegaPL-f-higher\]), $$\label{holder-hat-mapF-xx}
\|\Psi\|^{(-1-\alpha, [0, 1]\times\{-v_2,
{{\eta}}_1\})}_{2,\alpha, \hat\Omega_2}+
\|\Psi^{-1}\|^{(-1-\alpha, [0, 1]\times\{-v_2,
{{\eta}}_1\})}_{2,\alpha, (0,1)\times (-v_2, {{\eta}}_1)}\le C.$$ Moreover, by (\[OmegaPL-f-higher\]), for sufficiently small $\varepsilon$ and ${\sigma}$ (which are achieved by choosing large $\hat C$ in (\[condConst-00\])), we have ${{\mathcal D}}\cap\{-v_2<{{\eta}}<{{\eta}}_1\}\subset\Psi^{-1}([-1,1]\times [-v_2,
{{\eta}}_1])$. Define $${{\mathcal {P}}}_{\phi}^2 \psi:=\tilde{{{\mathcal E}_2}}(\psi\circ \Psi^{-1})\circ \Psi\qquad
\mbox{ on }\; {{\mathcal D}}\cap\{-v_2<{{\eta}}<{{\eta}}_1\}.$$ Then ${{\mathcal {P}}}_{\phi}^2 \psi\in C^{1, \alpha}(\overline{{{\mathcal D}}})\cap
C^{2,\alpha}(\overline{{{\mathcal D}}}
\setminus\overline{{\Gamma_{sonic}}\cup\Sigma_0})$ since ${{\mathcal D}}\setminus
\Omega^+({\phi})\subset {{\mathcal D}}\cap\{-v_2<{{\eta}}<{{\eta}}_1\}$. Furthermore, using (\[holder-hat-mapF-xx\]) and the definition of ${{\mathcal {P}}}_{\phi}^2$, we find that, for any $s\in(-v_2, {{\eta}}_1]$, $$\label{EstExt-2'}
\|{{\mathcal {P}}}_{\phi}^2 \psi\|^{(-1-\alpha,
\Sigma_0)}_{2,\alpha, {{\mathcal D}}\cap\{{{\eta}}\le s\}}
\le C({{\eta}}_1-s)\|\psi\|^{(-1-\alpha,
\{{{P_2}},{{P_3}}\})}_{2,\alpha, {\Omega}^+({\phi})\cap\{{{\eta}}\le s\}},$$ where $C({{\eta}}_1-s)$ depends only on the data and ${{\eta}}_1-s>0$.
Choosing $\hat C$ large in (\[condConst-00\]), we have $\varepsilon<\kappa/100$. Then (\[domain-in-rescaled-lemma\]) implies that there exists a unique point $P'={\Gamma_{shock}}({\phi})\cap\{c_2-r=\kappa/8\}$. Let $P'=({{\xi}}', {{\eta}}')$ in the $({{\xi}}, {{\eta}})$–coordinates. Then ${{\eta}}'>0$. Using (\[domainExtension-0’\]) and (\[domainExtension\]), we find $$\begin{aligned}
&&({{\mathcal D}}\setminus \Omega^+({\phi}))\cap\{c_2-r>\kappa/8\}\subset
{{\mathcal D}}\cap\{{{\eta}}\le {{\eta}}'\}, \\
&&\Omega^+({\phi})\cap\{{{\eta}}\le {{\eta}}'\}\subset
\Omega^+({\phi})\cap\{c_2-r>\kappa/8\}.\end{aligned}$$ Also, $\kappa/C\le {{\eta}}_1-{{\eta}}'\le C\kappa$ by (\[OmegaPL\]), (\[OmegaPL-f-higher\]), and (\[reflected-shock-s2\]). These facts and (\[EstExt-2’\]) with $s={{\eta}}'$ imply $$\label{EstExt-2} \|{{\mathcal {P}}}_{\phi}^2 \psi\|^{(-1-\alpha,
\Sigma_0)}_{2,\alpha, {{\mathcal D}}\cap\{c_2-r>\kappa/8\}} \le
C\|\psi\|^{(-1-\alpha, \{{{P_2}},{{P_3}}\})}_{2,\alpha,
{\Omega}^+({\phi})\cap\{c_2-r>\kappa/8\}}.$$
Finally, we choose a cutoff function $\zeta\in
C^\infty({ {\bf R}})$ satisfying $$\zeta\equiv 1 \mbox{ on }(-\infty, \kappa/4),\quad \zeta\equiv 0
\mbox{ on }(3\kappa/4, \infty),\quad \zeta'\le 0\,\, \mbox{ on }{ {\bf R}},$$ and define $${{\mathcal {P}}}_{\phi}\psi:=\zeta(c_2-r){{\mathcal {P}}}_{\phi}^1\psi+(1-\zeta(c_2-r)){{\mathcal {P}}}_{\phi}^2\psi
\qquad \mbox{in }{{\mathcal D}}.$$ Since ${{\mathcal {P}}}_{\phi}^k\psi=\psi$ on $\Omega^+({\phi})$ for $k=1,2$, so is ${{\mathcal {P}}}_{\phi}\psi$. Also, from the properties of ${{\mathcal {P}}}_{\phi}^k$ above, ${{\mathcal {P}}}_{\phi}\psi\in C^{1, \alpha}(\overline{{\mathcal D}})\cap C^{2,
\alpha}({{\mathcal D}})$ if $\psi\in C^{1,
\alpha}(\overline{\Omega^+({\phi})})\cap C^{2,
\alpha}(\overline{\Omega^+({\phi})}\setminus\overline{{\Gamma_{sonic}}\cup\Sigma_0})$. If such $\psi$ is a solution of (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]), then we prove (\[Hold-Ext-Sonic\])–(\[Hold-Ext-Unif\]): ${{\mathcal {P}}}_{\phi}\psi\equiv {{\mathcal {P}}}_{\phi}^1\psi$ on ${{\mathcal D}'}$ by the definition of $\zeta$ and by $\varepsilon<\kappa/100$. Thus, since (\[Hold-Ext-Sonic\]) has been proved in Step 2 for ${{\mathcal {P}}}_{\phi}^1\psi$, we obtain (\[Hold-Ext-Sonic\]) for ${{\mathcal {P}}}_{\phi}\psi$. Also, $\psi$ satisfies (\[Hoder-est-for-unif-ellipt\]) by Proposition \[existSolDegenEllipt\]. Using (\[Hoder-est-for-unif-ellipt\]) with $s={\varepsilon/2}$, (\[EstExt-1\]), and (\[EstExt-2\]), we obtain (\[Hold-Ext-Unif\]). Assertion (\[Uextended\_Est\]) is then proved.
Finally we prove assertion (\[Uextended\_Cont\]). Let ${\phi}_k \in {{\mathcal K}}$ converge to ${\phi}$ in $C^{1,
\beta}(\overline{{{\mathcal D}}})$. Then obviously ${\phi}\in {{\mathcal K}}$. By (\[nondegeneracy\])–(\[OmegaPL\]), it follows that $$\label{f-converges}
f_{{\phi}_k}\to f_{\phi}\qquad\mbox{in } C^{1,\beta}([-v_2, {{\eta}}_1]),$$ where $f_{{\phi}_k},f_{\phi}\in C^{(-1-\alpha, \{-v_2,
{{\eta}}_1\})}_{2,\alpha,(-v_2, {{\eta}}_1)}$ are the functions from (\[shockPL\]) corresponding to ${\phi}_k,{\phi}$, respectively. Let $\psi_k, \psi\in C^{1, \alpha}(\overline{\Omega^+({\phi}_k)})\cap
C^{2,
\alpha}(\overline{\Omega^+({\phi}_k)}\setminus\overline{{\Gamma_{sonic}}\cup\Sigma_0})$ be the solutions of problems (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]) for ${\phi}_k, {\phi}$. Let $\{\psi_{k_m}\}$ be any subsequence of $\{\psi_k\}$. By (\[Hold-Ext-Sonic\])–(\[Hold-Ext-Unif\]), it follows that there exist a further subsequence $\{{\phi}_{k_{m_n}}\}$ and a function $\bar\psi\in C^{1, \alpha}(\overline{{\mathcal D}})\cap C^{2,
\alpha}({{\mathcal D}})$ such that $${{\mathcal {P}}}_{{\phi}_{k_{m_n}}}\psi_{k_{m_n}}\to \bar\psi \qquad\mbox{in
$C^{2,{\alpha/2}}$ on compact subsets of ${{\mathcal D}}$
and in
$C^{1,{\alpha/2}}(\overline{{\mathcal D}})$}.$$ Then, using (\[f-converges\]) and the convergence ${\phi}_k \to
{\phi}$ in $C^{1, \beta}(\overline{{{\mathcal D}}})$, we prove (by the argument as in [@ChenFeldman1 page 479]) that $\bar\psi$ is a solution of problem (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]) for ${\phi}$. By uniqueness in Lemma \[uniquenessCor\], $\bar\psi=\psi$ in $\Omega^+({\phi})$. Now, using (\[f-converges\]) and the explicit definitions of extensions ${{\mathcal {P}}}_{\phi}^1$ and ${{\mathcal {P}}}_{\phi}^2$, it follows by the argument as in [@ChenFeldman1 pp. 477–478] that $$\begin{aligned}
&&\zeta{{\mathcal {P}}}_{{\phi}_{k_{m_n}}}^1(\psi_{k_{m_n}})\to
\zeta{{\mathcal {P}}}_{\phi}^1(\bar\psi_{|\Omega^+({\phi})})\qquad \mbox{ in
} C^{1,\beta}(\overline{{\mathcal D}}),\\
&&(1-\zeta){{\mathcal {P}}}_{{\phi}_{k_{m_n}}}^2(\psi_{k_{m_n}})\to
(1-\zeta){{\mathcal {P}}}_{\phi}^2(\bar\psi_{|\Omega^+({\phi})})\qquad \mbox{ in
} C^{1,\beta}(\overline{{\mathcal D}}).\end{aligned}$$ Therefore, $\bar\psi=\psi$ in ${{\mathcal D}}$. Since a convergent subsequence $\{\psi_{k_{m_n}}\}$ can be extracted from any subsequence $\{\psi_{k_{m}}\}$ of $\{\psi_k\}$ and the limit $\bar\psi=\psi$ is independent of the choice of subsequences $\{\psi_{k_{m}}\}$ and $\{\psi_{k_{m_n}}\}$, it follows that the whole sequence $\psi_k$ converges to $\psi$ in $C^{1,\beta}(\overline{{\mathcal D}})$. This completes the proof.
Now we denote by $\hat C_0$ the constant in (\[condConst-00\]) sufficiently large to satisfy the conditions of Proposition \[existSolDegenEllipt\] and Lemma \[extension-Lemma\]. Fix $\hat
C\ge \hat C_0$. Choose $M_1=\max(2C_1, 1)$ for the constant $C_1$ in (\[Hold-Ext-Sonic\]) and define $\varepsilon$ by (\[condConst-1-1\]). This choice of $\varepsilon$ fixes the constant $C_2(\varepsilon)$ in (\[Hold-Ext-Unif\]). Define $M_2=\max(C_2(\varepsilon),1)$. Finally, let $${\sigma}_0={{\hat C}^{-1}-\varepsilon-\varepsilon^{1/4}M_1 \over
2\left(M_2^2+ \varepsilon^2\max(M_1,M_2)\right)}\varepsilon^2.$$ Then ${\sigma}_0>0$, since $\varepsilon$ is defined by (\[condConst-1-1\]). Moreover, ${\sigma}_0$, $\varepsilon$, $M_1$, and $M_2$ depend only on the data and $\hat C$. Furthermore, for any ${\sigma}\in[0, {\sigma}_0]$, the constants ${\sigma}$, $\varepsilon$, $M_1$, and $M_2$ satisfy (\[condConst-00\]) with $\hat C$ fixed above. Also, $\psi\ge 0$ on $\Omega^+({\phi})$ by (\[L-infty-for-unif-ellipt\]) and thus $$\label{Ext-positive}
{{\mathcal {P}}}_{\phi}\psi\ge 0
\qquad\mbox{on }{{\mathcal D}}$$ by the explicit definitions of ${{\mathcal {P}}}_{\phi}^1, {{\mathcal {P}}}_{\phi}^2$, and ${{\mathcal {P}}}_{\phi}$. Now we define the iteration map $J$ by $J({\phi})={{\mathcal {P}}}_{\phi}\psi$. By (\[Hold-Ext-Sonic\])–(\[Hold-Ext-Unif\]) and (\[Ext-positive\]) and the choice of ${\sigma}$, $\varepsilon$, $M_1$, and $M_2$, we find that $J:{{\mathcal K}}\to{{\mathcal K}}$. Now, ${{\mathcal K}}$ is a compact and convex subset of $C^{1,{\alpha/2}}(\overline{{\mathcal D}})$. The map $J:{{\mathcal K}}\to{{\mathcal K}}$ is continuous in $C^{1,{\alpha/2}}(\overline{{\mathcal D}})$ by Lemma \[extension-Lemma\](\[Uextended\_Cont\]). Thus, by the Schauder Fixed Point Theorem, there exists a fixed point ${\phi}\in{{\mathcal K}}$ of the map $J$. By definition of $J$, such $\psi$ is a solution of (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]) with ${\phi}=\psi$. Therefore, we have
\[existenceFixedPt\] There exists $\hat C_0\ge 1$ depending only on the data such that, for any $\hat C\ge \hat C_0$, there exist ${\sigma}_0, \varepsilon>0$ and $M_1, M_2\ge 1$ satisfying [(\[condConst-00\])]{} so that, for any ${\sigma}\in (0, {\sigma}_0]$, there exists a solution $\psi\in{{\mathcal K}}({\sigma}, \varepsilon, M_1, M_2)$ of problem [(\[iterationEquation\])]{}–[(\[iterationCondOnSymmtryLine\])]{} with ${\phi}=\psi$ (i.e., $\psi$ is a “fixed point” solution). Moreover, $\psi$ satisfies [(\[Hoder-est-for-unif-ellipt\])]{} for all $s\in (0, c_2/2)$ with $C(s)$ depending only on the data and $s$.
Removal of the Ellipticity Cutoff {#removeCutoffSection}
=================================
In this section we assume that $\hat
C_0\ge 1$ is as in Proposition \[existenceFixedPt\] which depends only on the data, $\hat C\ge \hat C_0$, and assume that ${\sigma}_0,
\varepsilon>0$ and $M_1,M_2\ge 1$ are defined by $\hat C$ as in Proposition \[existenceFixedPt\] and ${\sigma}\in (0, {\sigma}_0]$. We fix a “fixed point” solution $\psi$ of problem (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]), that is, $\psi\in{{\mathcal K}}({\sigma}, \varepsilon, M_1, M_2)$ satisfying (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]) with ${\phi}=\psi$. Its existence is established in Proposition \[existenceFixedPt\]. To simplify notations, in this section we write $\Omega^+$, ${\Gamma_{shock}}$, and $\Sigma_0$ for $\Omega^+(\psi)$, ${\Gamma_{shock}}(\psi)$, and $\Sigma_0(\psi)$, respectively, and the universal constant $C$ depends only on the data.
We now prove that the “fixed point” solution $\psi$ satisfies $|\psi_x|\le 4x/\big(3(\gamma+1)\big)$ in $\Omega^+\cap
\{c_2-r<4\varepsilon\}$ for sufficiently large $\hat C$, depending only on the data, so that $\psi$ is a solution of the regular reflection problem; see Step 10 of §\[overViewProcedureSubsection\].
We also note the higher regularity of $\psi$ away from the corners and the sonic circle. Since equation (\[iterationEquation\]) is uniformly elliptic in every compact subset of $\Omega^+$ (by Lemma \[propertiesNonlinCoeffs\]) and the coefficients $A_{ij}(p,{{\xi}},{{\eta}})$ of (\[iterationEquation\]) are $C^{1,\alpha}$ functions of $(p,{{\xi}},{{\eta}})$ in every compact subset of ${ {\bf R}}^2\times\Omega^+$ (which follows from the explicit expressions of $A_{ij}(p,{{\xi}},{{\eta}})$ given by (\[iterationUniforDomEquation\]), (\[iterationSonicDomEquation\]), and (\[combineCoeffs\])), then substituting $p={{D}}\psi({{\xi}},{{\eta}})$ with $\psi\in{{\mathcal K}}$ into $A_{ij}(p,{{\xi}},{{\eta}})$, rewriting (\[iterationEquation\]) as a linear equation with coefficients being $C^{1,\alpha}$ in compact subsets of $\Omega^+$, and using the interior regularity results for linear, uniformly elliptic equations yield $$\label{higherRegOfPsi}
\psi\in C^{3,\alpha}\left( \Omega^+\right).$$
First we bound $\psi_x$ from above. We work in the $(x,y)$–coordinates in $\Omega^+\cap \{c_2-r<4\varepsilon\}$. By (\[domain-in-rescaled-lemma\]), $$\label{domain-in-rescaled-lemma-remCutoff}
\Omega^+({\phi})\cap\{c_2-r<4\varepsilon\} =\{0< x< \kappa,\,\,
0<y<\hat f_{\phi}(x)\},$$ where $\hat f_{\phi}$ satisfies (\[holder-hat-f\]).
\[boundPsiXfromAbove-Prop\] For sufficiently large $\hat C$ depending only on the data, $$\label{boundPsiXfromAbove}
\psi_x\le {4\over 3(\gamma+1)}x \qquad\mbox{ in }\;
\Omega^+\cap\{x\le 4\varepsilon\}.$$
To simplify notations, we denote $A=\frac{4}{3(\gamma+1)}$ and $$\Omega^+_s{:=}\Omega^+\cap\{x\le s\} \qquad\mbox{for}\,\, s>0.$$ Define a function $$\label{defV-remove-1}
v(x,y):=Ax-\psi_x(x,y) \qquad\mbox{on }\Omega^+_{4\varepsilon}.$$ From $\psi\in{{\mathcal K}}$ and (\[higherRegOfPsi\]), it follows that $$\label{v-regular-remCutoff}
v\in C^{0,1}\big(\overline{\Omega^+_{4\varepsilon}}\big) \cap
C^1\big(\overline{\Omega^+_{4\varepsilon}}\setminus\{x=0\}\big) \cap
C^2\left(\Omega^+_{4\varepsilon}\right).$$ Since $\psi\in{{\mathcal K}}$, we have $|\psi_x(x, y)|\le M_1x$ in $\Omega^+_{4\varepsilon}$. Thus $$\label{bdryCondV-sonic}
v=0\qquad\mbox{on }\, \partial\Omega^+_{4\varepsilon}\cap\{x=0\}.$$ We now use the fact that $\psi$ satisfies (\[iterationRH\]), which can be written as (\[iterationRH-lf-flattened\]) in the $(x,y)$–coordinates, and (\[estCoefsIterRH-flattened\]) holds. Since $\psi\in{{\mathcal K}}$ implies that $$|\psi(x,y)|\le M_1x^2, \qquad |\psi_y(x,y)|\le M_1x^{3/2},$$ it follows from (\[iterationRH-lf-flattened\]) and (\[estCoefsIterRH-flattened\]) that $$|\psi_x|\le C(|\psi_y|+|\psi|)\le CM_1x^{3/2}\qquad\mbox{ on } {\Gamma_{shock}}\cap\{x<2\varepsilon\},$$ and hence, by (\[condConst-00\]), if $\hat C$ is large depending only on the data, then $$|\psi_x|<A x\qquad\mbox{ on } {\Gamma_{shock}}\cap\{0<x<2\varepsilon\}.$$ Thus we have $$\label{bdryCondV-shock}
v\ge 0\qquad\mbox{on }{\Gamma_{shock}}\cap\{0<x<2\varepsilon\}.$$ Furthermore, condition (\[iterationCondOnWedge\]) on ${\Gamma_{wedge}}$ in the $(x,y)$–coordinates is $$\psi_y=0\qquad\mbox{ on }\; \{0<x<2\varepsilon, \, y=0\}.$$ Since $\psi\in{{\mathcal K}}$ implies that $\psi$ is $C^2$ up to ${\Gamma_{wedge}}$, then differentiating the condition on ${\Gamma_{wedge}}$ with respect to $x$, i.e., in the tangential direction to ${\Gamma_{wedge}}$, yields $\psi_{xy}=0$ on $\{0<x<2\varepsilon,\, y=0\}$, which implies $$\label{bdryCondV-wedge}
v_y= 0\qquad\mbox{on }{\Gamma_{wedge}}\cap\{0<x<2\varepsilon\}.$$ Furthermore, since $\psi\in{{\mathcal K}}$, $$\label{sigmaSmall}
|\psi_x|\le M_2{\sigma}\le A\varepsilon \qquad\mbox{on
}\; \Omega^+\cap\{\varepsilon/2\le x\le 4\varepsilon\},$$ where the second inequality holds by (\[condConst-00\]) if $\hat
C$ is large, depending only on the data. Thus, for such $\hat C$, $$\label{bdryCondV-upbdry}
v\ge 0\qquad\mbox{on
}\Omega^+_{4\varepsilon}\cap\{x=2\varepsilon\}.$$
Now we show that, for large $\hat C$, $v$ is a supersolution of a linear homogeneous elliptic equation on $\Omega^+_{2\varepsilon}$. Since $\psi$ satisfies equation (\[iteration-equation-sonicStruct\]) with (\[iteration-equation-sonicStruct-coef\]) in $\Omega^+_{4\varepsilon}$, we differentiate the equation with respect to $x$ and use the regularity of $\psi$ in (\[higherRegOfPsi\]) and the definition $v$ in (\[defV-remove-1\]) to obtain $$\label{eqnForV-diff-1}
\begin{array}{l}
\displaystyle a_{11}v_{xx}+a_{12}v_{xx}+ a_{22}v_{yy}\\
\quad +\left(A-v_x\right) \big(-1 +(\gamma+1) \big(\zeta_1(A-{v\over
x}) + \zeta_1'(A-{v\over x})({v\over x}-v_x) \big)\big)=E(x,y),
\end{array}$$ where $$\begin{aligned}
&&a_{11}=2x-(\gamma+1)x\zeta_1\big({\psi_x\over x}\big)+\hat O_1,
\quad a_{12}=\hat O_2,
\quad a_{22}={1\over c_2}+\hat O_3, \label{defA22-rem_cutoff-1}
\\
&& E(x,y)=\psi_{xx}\partial_x\hat O_1 +\psi_{xy}\partial_x\hat O_2
+\psi_{yy}\partial_x\hat O_3 -\psi_{xx}\hat O_4
-\psi_{x}\partial_x\hat O_4 \label{defE-rem_cutoff-1}
\\
&& \qquad\qquad +\psi_{xy}\hat O_5 +\psi_{y}\partial_x\hat O_5,
\nonumber\end{aligned}$$ with $$\label{defO-rem_cutoff} \hat O_k(x,y)=O_k^\psi(D\psi(x,y),x,y)\qquad
\mbox{ for }\;k=1,\dots, 5,$$ for $O_k^\psi$ defined by (\[iteration-equation-sonicStruct-coef\]) with ${\phi}=\psi$. From (\[defZeta-1\]), we have $$\zeta_1\left(A\right) = A.
$$ Thus we can rewrite (\[eqnForV-diff-1\]) in the form $$\label{eqnForV-diff-3}
a_{11}v_{xx}+a_{12}v_{xx}+ a_{22}v_{yy}+
bv_x+cv=-A\big((\gamma+1)A-1\big)+E(x,y),$$ with $$\begin{aligned}
&&b(x,y)=1 -(\gamma+1) \big(\zeta_1(A-{v\over x})+
\zeta_1'(A-{v\over x}) ({v\over x}-v_x-A) \big),
\label{def-b-rem_cutoff-1}
\\
&&c(x,y)=(\gamma+1){A\over x}\big(\zeta_1'(A-{v\over x})-
\int_0^1\zeta_1'(A-s{v\over x})ds \big), \label{def-c-rem_cutoff-1}\end{aligned}$$ where $v$ and $v_x$ are evaluated at the point $(x,y)$.
Since $\psi\in{{\mathcal K}}$ and $v$ is defined by (\[defV-remove-1\]), we have $$a_{ij}, b, c \in
C\big(\overline{\Omega^+_{4\varepsilon}}\setminus\{x=0\}\big).$$
Combining (\[defA22-rem\_cutoff-1\]) with (\[condConst-00\]), (\[defZeta-1\]), (\[estSmallterms-iter-S\]), and (\[defO-rem\_cutoff\]), we obtain that, for sufficiently large $\hat C$ depending only on the data, $$a_{11}\ge{1\over 6}x, \qquad a_{22}\ge{1\over 2c_2}, \qquad
|a_{12}|\le{1\over 3\sqrt{c_2}}x^{1/2} \qquad\mbox{on }\,
\Omega^+_{2\varepsilon}.$$ Thus, $4a_{11}a_{22}-(a_{12})^2\ge{2\over 9c_2}x$ on $\Omega^+_{2\varepsilon}$, which implies that equation (\[eqnForV-diff-3\]) is elliptic on $\Omega^+_{2\varepsilon}$ and uniformly elliptic on every compact subset of $\overline{\Omega^+_{2\varepsilon}}\setminus\{x=0\}$.
Furthermore, using (\[zeta-1-concave\]) and (\[def-c-rem\_cutoff-1\]) and noting $A>0$ and $x>0$, we have $$\label{cNegativeAtVneg} c(x,y)\le 0\qquad\mbox{for every
}\;(x,y)\in\Omega^+_{2\varepsilon}\;\mbox{ such that } v(x,y)\le 0.$$
Now we estimate $E(x,y)$. Using (\[defO-rem\_cutoff\]), (\[iteration-equation-sonicStruct-coef\]), (\[erTerms-xy-nontrunc\]), and $\psi\in{{\mathcal K}}$, we find that, on $\Omega^+_{2\varepsilon}$, $$\begin{aligned}
|\partial_x\hat O_1|&\le&
C\big(x+|\psi|+|D\psi|+x|\psi_{xx}|+|\psi_x\psi_{xx}|+|\psi_y\psi_{xy}|+|D\psi|^2\big)
\le CM_1^2x,
\\
|\partial_x\hat O_{2,5}|&\le&
C\big(|D\psi|+|D\psi|^2+|\psi_y\psi_{xx}|+(1+|\psi_x|)|\psi_{xy}|\big)
\le CM_1x^{1/2}(1+M_1x),
\\
|\partial_x\hat O_{3,4}|&\le& C\big(1+|\psi|+ \big|{\psi_x\over
x}\zeta_1'\big({\psi_x\over x}\big)\big| +(1+|D\psi|)|D^2\psi|
+|D\psi|^2\big) \\
&\le& CM_1(1+M_1x),\end{aligned}$$ where we have used the fact that $|s\zeta_1'(s)|\le C$ on ${ {\bf R}}$. Combining these estimates with (\[defE-rem\_cutoff-1\])–(\[defO-rem\_cutoff\]), (\[estSmallterms-iter\]), and $\psi\in{{\mathcal K}}$, we obtain from (\[defE-rem\_cutoff-1\]) that $$|E(x,y)|\le CM_1^2x(1+M_1x)\le {C/\hat C}\qquad \mbox{ on
}\Omega^+_{2\varepsilon}.$$ From this and $(\gamma+1)A>1$, we conclude that the right-hand side of (\[eqnForV-diff-3\]) is strictly negative in $\Omega^+_{2\varepsilon}$ if $\hat C$ is sufficiently large, depending only on the data.
We fix $\hat C$ satisfying all the requirements above (thus depending only on the data). Then we have $$\label{eqnForV-diff-4}
a_{11}v_{xx}+a_{12}v_{xx}+ a_{22}v_{yy}+ bv_x+cv<0 \qquad\mbox{on
}\Omega^+_{2\varepsilon},$$ the equation is elliptic in $\Omega^+_{2\varepsilon}$ and uniformly elliptic on compact subsets of $\overline{\Omega^+_{2\varepsilon}}\setminus \{x=0\}$, and (\[cNegativeAtVneg\]) holds. Moreover, $v$ satisfies (\[v-regular-remCutoff\]) and the boundary conditions (\[bdryCondV-sonic\])–(\[bdryCondV-wedge\]) and (\[bdryCondV-upbdry\]). Then it follows that $$v\ge 0\qquad \mbox{ on }\, \Omega^+_{2\varepsilon}.$$ Indeed, let $z_0:=(x_0, y_0)\in \overline{\Omega^+_{2\varepsilon}}$ be a minimum point of $v$ over $\overline{\Omega^+_{2\varepsilon}}$ and $v(z_0)<0$. Then, by (\[bdryCondV-sonic\])–(\[bdryCondV-shock\]) and (\[bdryCondV-upbdry\]), either $z_0$ is an interior point of $\Omega^+_{2\varepsilon}$ or $z_0\in
{\Gamma_{wedge}}\cap\{0<x<2\varepsilon\}$. If $z_0$ is an interior point of $\Omega^+_{2\varepsilon}$, then (\[eqnForV-diff-4\]) is violated since (\[eqnForV-diff-4\]) is elliptic, $v(z_0)<0$, and $c(z_0)\le 0$ by (\[cNegativeAtVneg\]). Thus, the only possibility is $z_0\in {\Gamma_{wedge}}\cap\{0<x<2\varepsilon\}$, i.e., $z_0=(x_0, 0)$ with $x_0>0$. Then, by (\[domain-in-rescaled-lemma-remCutoff\]), there exists $\rho>0$ such that $B_{\rho}(z_0)\cap \Omega^+_{2\varepsilon}=B_{\rho}(z_0)\cap\{y>0\}$. Equation (\[eqnForV-diff-4\]) is uniformly elliptic in $\overline{B_{\rho/2}(z_0)\cap\{y\ge0\}}$, with the coefficients $a_{ij}, b, c\in C(\overline{B_{\rho/2}(z_0)\cap\{y\ge0\}})$. Since $v(z_0)<0$ and $v$ satisfies (\[v-regular-remCutoff\]), then, reducing $\rho>0$ if necessary, we have $v<0$ in $B_{\rho}(z_0)\cap\{y>0\}$. Thus, $c\le 0$ on $B_{\rho}(z_0)\cap\{y>0\}$ by (\[cNegativeAtVneg\]). Moreover, $v(x,y)$ is not a constant in $\overline{B_{x_0/2}(x_0)\cap\{y\ge0\}}$ since its negative minimum is achieved at $(x_0,0)$ and cannot be achieved in any interior point, as we have showed above. Thus, $\partial_y v(z_0)>0$ by Hopf’s Lemma, which contradicts (\[bdryCondV-wedge\]). Therefore, $v\ge 0$ on $\Omega^+_{2\varepsilon}$ so that (\[boundPsiXfromAbove\]) holds on $\Omega^+_{2\varepsilon}$. Then, using (\[sigmaSmall\]), we obtain (\[boundPsiXfromAbove\]) on $\Omega^+_{4\varepsilon}$. Now we bound $\psi_x$ from below. We first prove the following lemma in the $({{\xi}}, {{\eta}})$–coordinates.
\[negativeDerivPsiLemma\] If $\hat C$ in [(\[condConst-00\])]{} is sufficiently large, depending only on the data, then $$\label{negativeDerivPsi}
\psi_{{{\eta}}}\le 0 \qquad\mbox{in } \Omega^+.$$
We divide the proof into six steps. [*Step 1*]{}. Set $w=\psi_{{{\eta}}}.$ From $\psi\in{{\mathcal K}}$ and (\[higherRegOfPsi\]), $$\label{w-regular-negativeDeriv}
w\in C^{0,\alpha}\big(\overline{\Omega^+}\big) \cap
C^1\big(\overline{\Omega^+}\setminus\overline{{\Gamma_{sonic}}\cup
\Sigma_0}\big) \cap C^2\left(\Omega^+\right).$$
In the next steps, we derive the equation and boundary conditions for $w$ in $\Omega^+$. To achieve this, we use the following facts:
\(i) If $\hat C$ in (\[condConst-00\]) is sufficiently large, then the coefficient $A_{11}$ of (\[iterationEquation\]) satisfies $$\label{A11-bdd-away-fromZero}
|A_{11}\left({{D}}\psi({{\xi}},{{\eta}}), {{\xi}},{{\eta}}\right)|\ge
{{\bar{c}_2}^2-\bar{{\xi}}^2\over 2}>0 \qquad\mbox{in } \Omega^+,$$ where $\bar{c}_2$ and $\bar\xi$ are defined in §\[section:4\]. Indeed, since $\bar{c}_2>|\bar\xi|$ by (\[sonic-intersect-shock-normal\]) and $(c_2,\tilde{{\xi}})\to (
\bar{c}_2,\bar{{\xi}})$ as $\theta_w\to \pi/2$ by §\[section:3.3\], we have $\displaystyle c_2^2-\tilde{{\xi}}^2\ge
{9({\bar{c}_2}^2-\bar{{\xi}}^2)/10}>0$ if ${\sigma}$ is small. Furthermore, for any $({{\xi}}, {{\eta}})\in{{\mathcal D}}$, we have $c_2\cos\theta_w\ge{{\xi}}\ge \tilde{{\xi}}$ and thus, assuming that $\sigma$ is small so that $|\tilde{{\xi}}|\le 2|\bar\xi|$ and $c_2\le
2\bar{c}_2$, we obtain $|{{\xi}}|\le C$. Now, since $\psi\in{{\mathcal K}}$, it follows that, if $\hat C$ in (\[condConst-00\]) is sufficiently large, then $A_{11}^1$ defined in (\[iterationUniforDomEquation\]) with ${\phi}=\psi$ implies $A_{11}^1\ge {({\bar{c}_2}^2-\bar{{\xi}}^2)/ 2}$ on ${{\mathcal D}}$, and $A_{11}^2$ in (\[iterationSonicDomEquation\]) with ${\phi}=\psi$ implies $A_{11}^2\ge {({\bar{c}_2}^2-\bar{{\xi}}^2)/ 2}$ on ${{\mathcal D}}\cap
\{c_2-r<4\varepsilon\}$. Then (\[A11-bdd-away-fromZero\]) follows from (\[combineCoeffs\]).
\(ii) Since $\psi$ satisfies equation (\[iterationEquation\]) in $\Omega^+$ with (\[A11-bdd-away-fromZero\]), we have $$\label{express2ndDeriv}
\psi_{{{\xi}}{{\xi}}}=-{2\hat A_{12}\psi_{{{\xi}}{{\eta}}}+\hat
A_{22}\psi_{{{\eta}}{{\eta}}}\over \hat A_{11}} \qquad\mbox{ in }\Omega^+,$$ where $\hat A_{ij}({{\xi}}, {{\eta}})=A_{ij}\left({{D}}\psi({{\xi}},{{\eta}}), {{\xi}},{{\eta}}\right)$ in $\Omega^+$.
We differentiate equation (\[iterationEquation\]) with respect to ${{\eta}}$ and substitute the right-hand side of (\[express2ndDeriv\]) for $\psi_{{{\xi}}{{\xi}}}$ to obtain the following equation for $w$: $$\label{negativeDerivPsi-eqnForW}
\hat A_{11}w_{{{\xi}}{{\xi}}}+2\hat A_{12}w_{{{\xi}}{{\eta}}}+\hat A_{22}w_{{{\eta}}{{\eta}}}
+ 2\big(\partial_{{\eta}}\hat A_{12}-{\partial_{{\eta}}\hat A_{11}\over \hat
A_{11}}
\hat A_{12}\big) w_{{\xi}}+\big(\partial_{{\eta}}\hat A_{22}-{\partial_{{\eta}}\hat A_{11}\over \hat
A_{11}} \hat A_{22}\big) w_{{\eta}}=0.
$$ By Lemma \[propertiesNonlinCoeffs\], (\[A11-bdd-away-fromZero\]), and $\psi\in{{\mathcal K}}$, the coefficients of (\[negativeDerivPsi-eqnForW\]) are continuous in $\overline{\Omega^+}\setminus \overline{{\Gamma_{sonic}}\cup \Sigma_0}$, and the equation is uniformly elliptic on compact subsets of $\overline{\Omega^+}\setminus \overline{{\Gamma_{sonic}}}$.
By (\[iterationCondOnSymmtryLine\]), we have $$\label{negativeDerivPsi-CondSymmetr-ForW}
w=-v_2\qquad\mbox{on }\;\Sigma_0:=\partial \Omega^+\cap
\{{{\eta}}=-v_2\}.$$
Since $\psi\in{{\mathcal K}}$, it follows that $|{{D}}\psi({{\xi}}, {{\eta}})|\le CM_1(c_2-r)$ for all $({{\xi}}, {{\eta}})\in\Omega^+\cap\{c_2-r\le 2\varepsilon\}$. Thus, $$\label{negativeDerivPsi-CondSonic-ForW}
w=0\qquad\mbox{on }\;{\Gamma_{sonic}}.$$
We derive the boundary condition for $\psi$ on ${\Gamma_{wedge}}$. Then $\psi$ satisfies (\[iterationCondOnWedge\]), which can be written as $$\label{iterationCondOnWedge-2}
-\sin\theta_w\,\psi_{{\xi}}+ \cos\theta_w\,\psi_{{\eta}}=0\qquad\,\, \mbox{on
}\;{\Gamma_{wedge}}.$$ Since $\psi\in{{\mathcal K}}$, we have $\psi\in C^2(\overline{\Omega^+}
\setminus\overline{{\Gamma_{sonic}}\cup\Sigma_0})$. Thus we can differentiate (\[iterationCondOnWedge-2\]) in the direction tangential to ${\Gamma_{wedge}}$, i.e., apply $\partial_\tau:=\cos\theta_w\,\partial_{{\xi}}+
\sin\theta_w\,\partial_{{\eta}}$ to (\[iterationCondOnWedge-2\]). Differentiating and substituting the right-hand side of (\[express2ndDeriv\]) for $\psi_{{{\xi}}{{\xi}}}$, we have $$\label{negativeDerivPsi-CondWedge-ForW}
\displaystyle \big(\cos(2\theta_w) +{\hat A_{12}\over \hat
A_{11}}\sin(2\theta_w)\big)w_{{\xi}}+\frac{1}{2}\sin(2\theta_w)\big(
1+{\hat A_{22}\over \hat A_{11}}\big)w_{{\eta}}=0 \qquad\mbox{on
}\;{\Gamma_{wedge}}.$$ This condition is oblique if ${\sigma}$ is small: Indeed, since the unit normal on ${\Gamma_{wedge}}$ is $(-\sin\theta_w, \cos\theta_w)$, we use (\[angleCloseToPiOver2\]) and (\[A11-bdd-away-fromZero\]) to find $$(\cos(2\theta_w) +{\hat A_{12}\over \hat A_{11}}\sin(2\theta_w),
\frac{1}{2}\sin(2\theta_w)(1+{\hat A_{22}\over \hat A_{11}}))\cdot (
-\sin\theta_w, \cos\theta_w)) \ge 1-C{\sigma}\ge {1\over 2}.$$
In this step, we derive the condition for $w$ on ${\Gamma_{shock}}$. Since $\psi$ is a solution of (\[iterationEquation\])–(\[iterationCondOnSymmtryLine\]) for ${\phi}=\psi$, the Rankine-Hugoniot conditions hold on ${\Gamma_{shock}}$: Indeed, the continuous matching of $\psi$ with $\varphi_1-\varphi_2$ across ${\Gamma_{shock}}$ holds by (\[shockPL\])–(\[shockIterDef\]) since ${\phi}=\psi$. Then (\[cont-accross-shock-psi-resolved\]) holds and the gradient jump condition (\[RH-psi\]) can be written in form (\[RH-psi-2\]). On the other hand, $\psi$ on ${\Gamma_{shock}}$ satisfies (\[iterationRH\]) with ${\phi}=\psi$, which is (\[RH-psi-2\]). Thus, $\psi$ satisfies (\[RH-psi\]).
Since $\psi\in{{\mathcal K}}$ which implies $\psi\in C^2(\overline\Omega^+
\setminus\overline{{\Gamma_{sonic}}\cup\Sigma_0})$, we can differentiate (\[RH-psi\]) in the direction tangential to ${\Gamma_{shock}}$. The unit normal $\nu_s$ on ${\Gamma_{shock}}$ is given by (\[norm-to-Shock\]). Then the vector $$\label{tangent-to-Shock-2}
\tau_s\equiv (\tau_s^1,\tau_s^2) :=({v_2+\psi_{{\eta}}\over
u_1-u_2},\; 1-{\psi_{{\xi}}\over u_1-u_2})$$ is tangential to ${\Gamma_{shock}}$. Note that $\tau_s\ne 0$ if $\hat C$ in (\[condConst-00\]) is sufficiently large, since $$\label{smallnessPsi-u-v}
|{{D}}\psi|\le C({\sigma}+\varepsilon)\quad \mbox{ in }\;
\overline{\Omega^+},\qquad |u_2|+|v_2|\le C{\sigma},$$ and $u_1>0$ from $\psi\in{{\mathcal K}}$ and §\[section:3.3\]. Thus, we can apply the differential operator $\partial_{\tau_s}=\tau_s^1\partial_{{\xi}}+ \tau_s^2\partial_{{\eta}}$ to (\[RH-psi\]).
In the calculation below, we use the notation in §\[equationForPsiSection\]. We have showed in §\[equationForPsiSection\] that condition (\[RH-psi\]) can be written in form (\[RH-psi-int1\]), where $F(p, z, u_2, v_2, {{\xi}}, {{\eta}})$ is defined by (\[RH-psi-func1\])–(\[RH-psi-func3\]) and satisfies (\[rewriteRH-reg-1\]). Also, we denote $$\label{tangent-to-Shock-3}
\hat\tau(p, u_2, v_2)\equiv (\hat\tau^1,\hat\tau^2)(p, u_2, v_2)
:=({v_2+p_2\over
u_1-u_2},\; 1-{p_1\over u_1-u_2}),$$ where $p=(p_1,p_2)\in{ {\bf R}}^2$ and $z\in{ {\bf R}}$. Then $\hat\tau\in
C^\infty(\overline{B_{\delta^*}(0)\times B_{u_1/50}(0)})$. Now, applying the differential operator $\partial_{\tau_s}$, we obtain that $\psi$ satisfies $$\label{diff-RH-1}
\Phi(D^2\psi, {{D}}\psi, \psi, u_2, v_2, {{\xi}}, {{\eta}})=0 \qquad\mbox{on}\;{\Gamma_{shock}},$$ where $$\label{Diff-RH-psi-func1}
\Phi(R, p, z, u_2, v_2, {{\xi}}, {{\eta}})= \sum_{i,j=1}^2\hat
\tau^iF_{p_j}R_{ij} +\sum_{i=1}^2\hat\tau^i(F_{z}p_i+F_{\xi_i})
\qquad \mbox{for}\,\, R=(R_{ij})_{i,j=1}^2,$$ and, in both (\[Diff-RH-psi-func1\]) and the calculation below, $D_{(\xi_1, \xi_2)}F$ denotes as $D_{(\xi, \eta)}F$, $\;(F_{p_j},
F_{z}, F_{\xi_i})$ as $(F_{p_j}, F_{z}, F_{\xi_i})(p, z, u_2, v_2,
{{\xi}}, {{\eta}})$, $\;(\hat\tau, \hat\nu)$ as $(\hat\tau, \hat\nu)(p,
u_2, v_2)$, and $\tilde\rho$ as $\tilde\rho(p, z, {{\xi}}, {{\eta}})$, with $\tilde\rho(\cdot)$ and $\hat\nu(\cdot)$ defined by (\[RH-psi-func2\]) and (\[RH-psi-func3\]), respectively. By explicit calculation, we apply (\[RH-psi-func1\])–(\[RH-psi-func3\]) and (\[tangent-to-Shock-3\]) to obtain that, for every $(p, z, u_2,
v_2, {{\xi}}, {{\eta}})$, $$\label{Diff-RH-lower-cancel}
\sum_{i=1}^2\hat\tau^i\, (F_{z}p_i+F_{\xi_i})
=(\rho_1-\tilde\rho)\,\hat\tau\cdot \hat\nu =0.$$
We note that (\[cont-accross-shock-psi-resolved\]) holds on ${\Gamma_{shock}}$. Using (\[diff-RH-1\]) and (\[Diff-RH-lower-cancel\]) and expressing $\xi$ from (\[cont-accross-shock-psi-resolved\]), we see that $\psi$ satisfies $$\label{diff-RH-2}
\tilde\Phi(D^2\psi, {{D}}\psi, \psi, u_2, v_2, {{\eta}})=0 \qquad\mbox{on}\;{\Gamma_{shock}},$$ where $$\label{Diff-RH-psi-func2}
\tilde\Phi(R, p, z, u_2, v_2, {{\eta}})= \sum_{i,j=1}^2\hat
\tau^i\Psi_{p_j}(p, z, u_2, v_2, {{\eta}})R_{ij},$$ $\Psi$ is defined by (\[RH-psi-func4\]) and satisfies $\Psi\in
C^\infty(\overline{\mathcal A})$ with $\|\Psi\|_{C^k(\overline{\mathcal A})}$ depending only on the data and $k\in{\bf N}$, and ${\mathcal A}=B_{\delta^*}(0)\times
(-\delta^*, \delta^*)\times B_{u_1/50}(0)\times (-6\bar c_2/5, 6\bar
c_2/5)$.
Now, from (\[RH-psi-func1\])–(\[RH-psi-func3\]), (\[RH-psi-func4\]), and (\[tangent-to-Shock-3\]), we find $$\hat\tau((0, 0),0,0)=(0,\,1),\quad D_{p}\Psi((0,0), 0, 0,0, {{\eta}})=
\big(\rho_2'(c_2^2-{\hat{{\xi}}}^2),\;
\big(\frac{\rho_2-\rho_1}{u_1}-\rho_2'\hat{{\xi}}\big){{\eta}}\big).$$ Thus, by (\[Diff-RH-psi-func2\]), we obtain that, on ${ {\bf R}}^{2\times
2}\times{\mathcal A}$, $$\begin{aligned}
\label{Diff-RH-psi-func-at-zero}
&&\qquad \tilde\Phi(R, p, z, u_2, v_2, {{\eta}})\\
&&\qquad =\rho_2'(c_2^2-{\hat{{\xi}}}^2) R_{21} +
\big(\frac{\rho_2-\rho_1}{u_1}-\rho_2'\hat{{\xi}}\big){{\eta}}R_{22}
+\sum_{i.j=1}^2 \hat E_{ij}(p, z, u_2, v_2, {{\eta}}) R_{ij},\nonumber\end{aligned}$$ where $\hat E_{ij}\in C^\infty(\overline{\mathcal A})$ and $$\begin{aligned}
|\hat E_{ij}(p,z, u_2, v_2, {{\eta}})| \le C(|p|+|z|+|u_2|+|v_2|)
\qquad\mbox{for any }(p, z, u_2, v_2, {{\eta}})\in {\mathcal A},\end{aligned}$$ with $C$ depending only on $\|D^2\Psi\|_{C^0(\overline{\mathcal
A})}$.
From now on, we fix $(u_2, v_2)$ to be equal to the velocity of state (2) obtained in §\[section:3.3\] and write $E_{ij}(p, z,
{{\eta}})$ for $\hat E_{ij}(p,z,u_2, v_2, {{\eta}})$. Then, from (\[diff-RH-2\]) and (\[Diff-RH-psi-func-at-zero\]), we conclude that $\psi$ satisfies $$\label{differentiateRH-3} \rho_2'(c_2^2-{\hat{{\xi}}}^2) \psi_{{{\xi}}{{\eta}}}+
\big(\frac{\rho_2-\rho_1}{u_1}-\rho_2'\hat{{\xi}}\big){{\eta}}\psi_{{{\eta}}{{\eta}}}
+\sum_{i,j=1}^2E_{ij}(D\psi, \psi, {{\eta}})D_{ij}\psi=0 \qquad\mbox{on
}\;{\Gamma_{shock}},$$ and $E_{ij}=E_{ij}(p, z, {{\eta}}), i,j=1,2,$ are smooth on $${\mathcal
B}{:=}\overline{B_{\delta^*}(0)\times(-\delta^*, \delta^*) \times
(-6\bar c_2/5, 6\bar c_2/5)}$$ and satisfy (\[RH-psi-2-error-term1\]) with $C$ depending only on the data. Note that $$({{D}}\psi({{\xi}}, {{\eta}}), \,\psi({{\xi}}, {{\eta}}),\,{{\eta}})\in {\mathcal B}
\qquad\mbox{on } {\Gamma_{shock}},$$ since $\psi\in{{\mathcal K}}$ and (\[condConst-00\]) holds with sufficiently large $\hat C$. Expressing $\psi_{{{\xi}}{{\xi}}}$ from (\[express2ndDeriv\]) and using (\[A11-bdd-away-fromZero\]), we can rewrite (\[differentiateRH-3\]) in the form $$\big(\rho_2'(c_2^2-{\hat{{\xi}}}^2)+ E_1(D\psi, \psi, {{\eta}})\big)
\psi_{{{\xi}}{{\eta}}}+ \big(\big(\frac{\rho_2-\rho_1}{u_1}-\rho_2'\hat{{\xi}}\big){{\eta}}+ E_2(D\psi, \psi, {{\eta}})\big)\psi_{{{\eta}}{{\eta}}}=0$$ on ${\Gamma_{shock}}$, where the functions $E_i=E_i(p, z, {{\eta}}), i=1,2,$ are smooth on ${\mathcal B}$ and satisfy (\[RH-psi-2-error-term1\]). Thus, $w$ satisfies $$\label{negativeDerivPsi-CondRH-ForW}
\big(\rho_2'(c_2^2-{\hat{{\xi}}}^2)+ E_1(D\psi, \psi, {{\eta}})\big)
w_{{{\xi}}}+ \big((\frac{\rho_2-\rho_1}{u_1}-\rho_2'\hat{{\xi}}){{\eta}}+
E_2(D\psi, \psi, {{\eta}})\big)w_{{{\eta}}}=0$$ on ${\Gamma_{shock}}$. Condition (\[negativeDerivPsi-CondRH-ForW\]) is oblique if $\hat C$ is sufficiently large in (\[condConst-00\]). Indeed, we have $c_2\ge {9\over 10}\bar{c}_2$, which implies $c_2^2-|\hat{{\xi}}|^2\ge \bar{c}_2\frac{\bar{c}_2-|\bar{\xi}|}{4}>0$ by using (\[inSonicRegion-in-shifed\]). Now, combining (\[norm-to-Shock\]) and (\[RH-psi-2-error-term1\]) with $\psi\in{{\mathcal K}}$ and (\[u2-v2-bound\]), we find that, on ${\Gamma_{shock}}$, $$\begin{aligned}
&&(\rho_2'(c_2^2-{\hat{{\xi}}}^2)+ E_1(D\psi, \psi, {{\eta}}),
\big(\frac{\rho_2-\rho_1}{u_1}-\rho_2'\hat{{\xi}}\big){{\eta}}+ E_2(D\psi, \psi, {{\eta}}) )\cdot\nu_s \\
&&\quad \ge\rho_2'\bar{c}_2\frac{\bar{c}_2-|\bar{\xi}|}{4}
-C(M_1\varepsilon+M_2{\sigma}) \ge
\rho_2'\bar{c}_2\frac{\bar{c}_2-|\bar{\xi}|}{8}>0.\end{aligned}$$ Also, the coefficients of (\[negativeDerivPsi-CondRH-ForW\]) are continuous with respect to $({{\xi}}, {{\eta}})\in{\Gamma_{shock}}$.
Both the regularity of $w$ in (\[w-regular-negativeDeriv\]) and the fact that $w$ satisfies equation (\[negativeDerivPsi-eqnForW\]) that is uniformly elliptic on compact subsets of $\overline{\Omega^+}\setminus
\overline{{\Gamma_{sonic}}}$ imply that the maximum of $w$ cannot be achieved in the interior of $\Omega^+$, unless $w$ is constant on $\Omega^+$, by the Strong Maximum Principle. Since $w$ satisfies the oblique derivative conditions (\[negativeDerivPsi-CondWedge-ForW\]) and (\[negativeDerivPsi-CondRH-ForW\]) on the straight segment ${\Gamma_{wedge}}$ and on the curve ${\Gamma_{shock}}$ that is $C^{2,\alpha}$ in its relative interior, and since equation (\[negativeDerivPsi-eqnForW\]) is uniformly elliptic in a neighborhood of any point from the relative interiors of ${\Gamma_{wedge}}$ and ${\Gamma_{shock}}$, it follows from Hopf’s Lemma that the maximum of $w$ cannot be achieved in the relative interiors of ${\Gamma_{wedge}}$ and ${\Gamma_{shock}}$, unless $w$ is constant on $\Omega^+$. Now conditions (\[negativeDerivPsi-CondSymmetr-ForW\])–(\[negativeDerivPsi-CondSonic-ForW\]) imply that $w\le 0$ on $\Omega^+$. This completes the proof. Using Lemma \[negativeDerivPsiLemma\] and working in the $(x,y)$–coordinates, we have
\[boundPsiXfromBelow-Prop\] If $\hat C$ in [(\[condConst-00\])]{} is sufficiently large, depending only on the data, then $$\label{boundPsiXfromBelow}
\psi_x\ge -{4\over 3(\gamma+1)}x \quad\mbox{ in }\;
\Omega^+\cap\{x\le 4\varepsilon\}.$$
By definition of the $(x,y)$–coordinates in (\[coordNearSonic\]), we have $$\label{psi-eta}
\psi_\eta=-\sin\theta\, \psi_x+{\cos\theta\over r}\psi_y,$$ where $(r,\theta)$ are the polar coordinates in the $({{\xi}},
{{\eta}})$–plane.
From (\[domainExtension\]), it follows that, for sufficiently small ${\sigma}$ and $\varepsilon$, depending only on the data, $${{\eta}}\ge{{\eta}}^*\qquad \mbox{for
all } \,\, ({{\xi}}, {{\eta}})\in{{\mathcal D}}\cap\{c_2-r<4\varepsilon\},$$ where $(l({{\eta}}^*), {{\eta}}^*)$ is the unique intersection point of the segment $\{(l({{\eta}}),{{\eta}})\ :\, {{\eta}}\in(0, {{\eta}}_1]\}$ with the circle $\partial B_{c_2-4\varepsilon}(0)$. Let $\bar{{\eta}}^*$ be the corresponding point for the case of normal reflection, i.e., $\bar{{\eta}}^*=\sqrt{(\bar c_2-4\varepsilon)^2-\bar{{\xi}}^2}$. By (\[sonic-intersect-shock-normal\]), $\bar{{\eta}}^*\ge \sqrt{\bar
c_2^2-\bar{{\xi}}^2}/2>0$ if $\varepsilon$ is sufficiently small. Also, from (\[reflected-shock-s2\])–(\[x1-in-shifed\]) and (\[u2-v2-bound\]), and using the convergence $(\theta_s, c_2,
\tilde\xi)\to(\pi/2, \bar c_2,\bar\xi)$ as $\theta_w\to \pi/2$, we obtain ${{\eta}}^*\ge \bar{{\eta}}^*/2$ and $c_2\le 2\bar c_2$ if ${\sigma}$ and $\varepsilon$ are sufficiently small. Thus, we conclude that, if $\hat C$ in (\[condConst-00\]) is sufficiently large depending only on the data, then, for every $({{\xi}},
{{\eta}})\in{{\mathcal D}}\cap\{c_2-r<4\varepsilon\}$, the polar angle $\theta$ satisfies $$\label{psi-eta-polar-angle-ineq}
\sin\theta=\frac{{{\eta}}}{\sqrt{{{\xi}}^2+{{\eta}}^2}}>0, \qquad |\cot\theta|=
\left|\frac{{{\xi}}}{{{\eta}}}\right|\le {8\bar c_2\over \sqrt{\bar
c_2^2-\bar{{\xi}}^2}}\le C.$$ From (\[psi-eta\])–(\[psi-eta-polar-angle-ineq\]) and Lemma \[negativeDerivPsiLemma\], we find that, on $\Omega^+\cap\{c_2-r<4\varepsilon\}$, $$\label{estFromBelowDer}
\psi_x=-{1\over \sin\theta}\psi_\eta+{\cot\theta\over r}\psi_y
\ge{\cot\theta\over r}\psi_y\ge -C|\psi_y|.$$ Note that $\psi\in{{\mathcal K}}$ implies $|\psi_y(x,y)|\le M_1x^{3/2}$ for all $(x,y)\in \Omega^+\cap\{c_2-r<2\varepsilon\}$. Then, using (\[estFromBelowDer\]) and (\[condConst-00\]) and choosing large $\hat C$, we have $$\psi_x\ge -{4\over 3(\gamma+1)}x \qquad\mbox{ in }\;
\Omega^+\cap\{x\le 2\varepsilon\}.$$ Also, $\psi\in{{\mathcal K}}$ implies $$|\psi_x|\le M_2{\sigma}\le
{4\over3(\gamma+1)}(2\varepsilon) \qquad\mbox{on }\;
\Omega^+\cap\{2\varepsilon\le x\le 4\varepsilon\},$$ where the second inequality holds by (\[condConst-00\]) if $\hat
C$ is sufficiently large depending only on the data. Thus, (\[boundPsiXfromBelow\]) holds on $\Omega^+_{4\varepsilon}$.
Proof of Main Theorem {#proofSection}
=====================
Let $\hat C$ be sufficiently large to satisfy the conditions in Propositions \[existenceFixedPt\] and \[boundPsiXfromAbove-Prop\]–\[boundPsiXfromBelow-Prop\]. Then, by Proposition \[existenceFixedPt\], there exist ${\sigma}_0,
\varepsilon>0$ and $M_1,M_2\ge 1$ such that, for any ${\sigma}\in (0,
{\sigma}_0]$, there exists a solution $\psi\in{{\mathcal K}}({\sigma},
\varepsilon, M_1, M_2)$ of problem [(\[iterationEquation\])]{}–[(\[iterationCondOnSymmtryLine\])]{} with ${\phi}=\psi$. Fix ${\sigma}\in (0, {\sigma}_0]$ and the corresponding “fixed point" solution $\psi$, which, by Propositions \[boundPsiXfromAbove-Prop\]–\[boundPsiXfromBelow-Prop\], satisfies $$|\psi_x|\le {4\over 3(\gamma+1)}x \qquad\mbox{ in }\;
\Omega^+\cap\{x\le 4\varepsilon\}.$$ Then, by Lemma \[cutOffEqIsOriginalEq\], $\psi$ satisfies equation (\[potent-flow-nondiv-psi-1\]) in $\Omega^+(\psi)$. Moreover, $\psi$ satisfies properties (\[iterProcItem-first\])–(\[iterProcItem-last\]) in Step 10 of §\[overViewProcedureSubsection\] by following the argument in Step 10 of §\[overViewProcedureSubsection\]. Then, extending the function $\varphi=\psi+\varphi_2$ from $\Omega:=\Omega^+(\psi)$ to the whole domain $\Lambda$ by using (\[phi-states-0-1-2\]) to define $\varphi$ in $\Lambda\setminus\Omega$, we obtain $$\varphi\in W^{1,\infty}_{loc}(\Lambda)\cap \left(\cup_{i=0}^2
C^1(\Lambda_i\cup S)\cap C^{1,1}(\Lambda_i)\right),$$ where the domains $\Lambda_i$, $i=0,1,2$, are defined in Step 10 of §\[overViewProcedureSubsection\]. From the argument in Step 10 of §\[overViewProcedureSubsection\], it follows that $\varphi$ is a weak solution of Problem 2, provided that the reflected shock $S_1=P_0{{P_1}}{{P_2}}\cap \Lambda$ is a $C^2$-curve.
Thus, it remains to show that $S_1=P_0{{P_1}}{{P_2}}\cap \Lambda$ is a $C^2$-curve. By definition of $\varphi$ and since $\psi\in{{\mathcal K}}({\sigma}, \varepsilon, M_1, M_2)$, the reflected shock $S_1=P_0{{P_1}}{{P_2}}\cap\Lambda$ is given by $S_1=\{{{\xi}}=f_{S_1}({{\eta}})
\;:\; {{\eta}}_{{{P_2}}}<{{\eta}}<{{\eta}}_{P_0}\}$, where ${{\eta}}_{{{P_2}}}= -v_2$, ${{\eta}}_{P_0}=|\hat{{\xi}}|{\sin\theta_s\sin\theta_w\over
\sin(\theta_w-\theta_s)}>0$, and $$\label{defFulReflShock}
f_{S_1}({{\eta}})= \left\{
\begin{array}{ll}
f_\psi({{\eta}})\quad & \mbox{if }{{\eta}}\in ({{\eta}}_{{{P_2}}}, {{\eta}}_{{{P_1}}}),\\
l({{\eta}})& \mbox{if }{{\eta}}\in ({{\eta}}_{{{P_1}}}, {{\eta}}_{P_0}),
\end{array}
\right.$$ where $l({{\eta}})$ is defined by (\[reflected-shock-s2\]), ${{\eta}}_{{{P_1}}}={{\eta}}_1>0$ is defined by (\[coord-P4\]), and ${{\eta}}_{P_0}>{{\eta}}_{{{P_1}}}$ if ${\sigma}$ is sufficiently small, which follows from the explicit expression of ${{\eta}}_{P_0}$ given above and the fact that $(\theta_s, c_2, \hat{{\xi}})\to(\pi/2,\bar{c}_2,
\bar{{\xi}})$ as $\theta_w\to\pi/2$. The function $f_\psi$ is defined by (\[shockPL\]) for ${\phi}=\psi$.
Thus we need to show that $f_{S_1}\in C^2([{{\eta}}_{{{P_2}}},
{{\eta}}_{P_0}])$. By (\[reflected-shock-s2\]) and (\[OmegaPL-f-higher\]), it suffices to show that $f_{S_1}$ is twice differentiable at the points ${{\eta}}_{{{P_1}}}$ and ${{\eta}}_{{{P_2}}}$.
First, we consider $f_{S_1}$ near ${{\eta}}_{{{P_1}}}$. We change the coordinates to the $(x,y)$–coordinates in (\[coordNearSonic\]). Then, for sufficiently small $\varepsilon_1>0$, the curve $\{{{\xi}}=f_{S_1}({{\eta}})\}\cap \{c_2-\varepsilon_1<r<c_2+\varepsilon_1\}$ has the form $\{y=\hat f_{S_1}(x)\; :
\;-\varepsilon_1<x<\varepsilon_1\}$, where $$\label{defFulReflShock-xy}
\hat f_{S_1}(x)= \left\{
\begin{array}{ll}
\hat f_\psi(x)\quad & \mbox{if }x\in(0, \varepsilon_1),\\
\hat f_0(x)& \mbox{if }x\in (-\varepsilon_1,0),
\end{array}
\right.$$ with $\hat f_0$ and $\hat f_\psi$ defined by (\[referenceFB-polar\]) and (\[domain-in-rescaled-lemma\]) for ${\phi}=\psi$. In order to show that $f_{S_1}$ is twice differentiable at ${{\eta}}_{{{P_1}}}$, it suffices to show that $\hat f_{S_1}$ is twice differentiable at $x=0$.
From (\[holder-hat-f\])–(\[holder-hat-f-S\]) and (\[referenceFB-polar\]), it follows that $\hat f_{S_1}\in
C^1((-\varepsilon_1, \varepsilon_1))$. Moreover, from (\[nondegenPolar-1\]), (\[domain-in-xy-0\]), (\[OmegaPL\]), and (\[holder-hat-f-S\]), we write $\varphi_1, \varphi_2,$ and $\psi$ in the $(x,y)$–coordinates to obtain that $$\label{defFulReflShock-deriv-xy}
\hat f_{S_1}'(x)= \left\{
\begin{array}{ll}
\displaystyle -{\partial_x(\varphi_1-\varphi_2-\psi)\over
\partial_y(\varphi_1-\varphi_2-\psi)} (x, \hat f_{S_1}(x))
\quad & \mbox{if }x\in(0, \varepsilon_1),\\
\displaystyle -{\partial_x(\varphi_1-\varphi_2)\over
\partial_y(\varphi_1-\varphi_2)} (x, \hat f_{S_1}(x)) & \mbox{if
}x\in (-\varepsilon_1,0],
\end{array}
\right.$$ and that $\hat f_0'(x)$ is given for $x\in(-\varepsilon_1,
\varepsilon_1)$ by the second line of the right-hand side of (\[defFulReflShock-deriv-xy\]). Using (\[nondegenPolar-1\]) and $\psi\in{{\mathcal K}}$ with (\[condConst-00\]) for sufficiently large $\hat C$, we have $$\label{defFulReflShock-deriv-xy-est}
|\hat f_{S_1}'(x)-\hat f_0'(x)|\le C|D_{(x,y)}\psi(x,\hat
f_{\psi}(x))| \qquad\mbox{for all }x\in(0, \varepsilon_1).$$ Since $\psi$ satisfies (\[iterationRH\]) with ${\phi}=\psi$, it follows that, in the $(x,y)$–coordinates, $\psi$ satisfies (\[iterationRH-lf-flattened\]) on $\{y=\hat f_{\psi}(x)\; :
\;x\in(0,\varepsilon_1)\}$, and (\[estCoefsIterRH-flattened\]) holds. Then it follows that $$|\psi_x(x,\hat f_{\psi}(x))|\le C(|\psi_y(x,\hat f_{\psi}(x))|
+|\psi(x,\hat f_{\psi}(x))|)\le Cx^{3/2},$$ where the last inequality follows from $\psi\in{{\mathcal K}}$. Combining this with (\[defFulReflShock-xy\]), (\[defFulReflShock-deriv-xy-est\]), and $\hat f_{S_1}, \hat f_0\in
C^1((-\varepsilon_1, \varepsilon_1))$ yields $$|\hat f_{S_1}'(x)-\hat f_0'(x)|\le Cx^{3/2}\qquad\mbox{for all }
x\in(-\varepsilon_1, \varepsilon_1).$$ Then it follows that $\hat f_{S_1}'(x)-\hat f_0'(x)$ is differentiable at $x=0$. Since $\hat f_0\in
C^\infty((-\varepsilon_1, \varepsilon_1))$, we conclude that $\hat
f_{S_1}$ is twice differentiable at $x=0$. Thus, $f_{S_1}$ is twice differentiable at ${{\eta}}_{{{P_1}}}$.
In order to prove the $C^2$–smoothness of $f_{S_1}$ up to ${{\eta}}_{{{P_2}}}=-v_2$, we extend the solution ${\phi}$ and the free boundary function $f_{S_1}$ into $\{{{\eta}}<-v_2\}$ by the even reflection about the line $\Sigma_0\subset\{{{\eta}}=-v_2\}$ so that ${{P_2}}$ becomes an interior point of the shock curve. Note that we continue to work in the shifted coordinates defined in §\[shiftCoordSection\], that is, for $({{\xi}}, {{\eta}})$ such that ${{\eta}}<-v_2$ and $({{\xi}}, -2v_2-{{\eta}})\in\overline{\Omega^+(\psi)}$, we define $(\varphi, \varphi_1)({{\xi}}, {{\eta}})=(\varphi, \varphi_1)({{\xi}},
-2v_2-{{\eta}})$ and $f_{S_1}({{\eta}})=-2v_2-{{\eta}}$ for $\varphi_1$ given by (\[phi-1-shifted\]). Denote $\Omega^+_{\varepsilon_1}({{P_2}}){:=}B_{\varepsilon_1}({{P_2}}) \cap \{{{\xi}}>f_{S_1}({{\eta}})\}$ for sufficiently small $\varepsilon_1 >0$. From $\varphi\in
C^{1,\alpha}(\overline{\Omega^+(\psi)})\cap
C^{2,\alpha}(\Omega^+(\psi))$ and (\[condOnSymmtryLinePhi\]), we have $$\varphi\in C^{1,\alpha}(\overline{\Omega^+_{\varepsilon_1}({{P_2}})})
\cap C^{2,\alpha}(\Omega^+_{\varepsilon_1}({{P_2}})).$$ Also, the extended function $\varphi_1$ is in fact given by (\[phi-1-shifted\]). Furthermore, from (\[nondegeneracy\]) and (\[OmegaPL\]), we can see that the same is true for the extended functions and hence $$\{{{\xi}}>f_{S_1}({{\eta}})\}\cap B_{\varepsilon_1}({{P_2}})
=\{\varphi<\varphi_1\}\cap B_{\varepsilon_1}({{P_2}}), \quad
f_{S_1}\in C^{1,\alpha}((-v_2-{\varepsilon_1\over 2},
-v_2+{\varepsilon_1\over 2})).$$ Furthermore, from (\[1.1.5\])–(\[1.1.6\]) and (\[condOnSymmtryLinePhi\]), it follows that the extended $\varphi$ satisfies equation (\[1.1.5\]) with (\[1.1.6\]) in $\Omega^+_{\varepsilon_1}({{P_2}})$, where we have used the form of equation, i.e., the fact that there is no explicit dependence on $({{\xi}}, {{\eta}})$ in the coefficients and that the dependence of $D\varphi$ is only through $|D\varphi|$. Finally, the boundary conditions (\[cont-accross-shock-mod-phi\]) and (\[RH-mod-phi\]) are satisfied on $\Gamma_{\varepsilon_1}({{P_2}})
{:=}\{{{\xi}}=f_{S_1}({{\eta}})\}\cap B_{\varepsilon_1}({{P_2}})$. Equation (\[1.1.5\]) is uniformly elliptic in $\Omega^+_{\varepsilon_1}({{P_2}})$ for $\varphi$, which follows from $\varphi=\varphi_2+\psi$ and Lemmas \[propertiesNonlinCoeffs\] and \[cutOffEqIsOriginalEq\]. Condition (\[RH-mod-phi\]) is uniformly oblique on $\Gamma_{\varepsilon_1}({{P_2}})$ for $\varphi$, which follows from §\[equationForPsiSection\].
Next, we rewrite equation (\[1.1.5\]) in $\Omega^+_{\varepsilon_1}({{P_2}})$ and the boundary conditions (\[cont-accross-shock-mod-phi\])–(\[RH-mod-phi\]) on $\Gamma_{\varepsilon_1}({{P_2}})$ in terms of $u{:=}\varphi_1-\varphi$. Substituting $u+\varphi_1$ for $\varphi$ into (\[1.1.5\]) and (\[RH-mod-phi\]), we obtain that $u$ satisfies $$F(D^2u, Du, u, {{\xi}}, {{\eta}})=0\quad\mbox{in
}\Omega^+_{\varepsilon_1}({{P_2}}),\qquad\; u=G(Du, u, {{\xi}}, {{\eta}})=0
\quad\mbox{on } \Gamma_{\varepsilon_1}({{P_2}}),$$ where the equation is quasilinear and uniformly elliptic, the second boundary condition is oblique, and the functions $F$ and $G$ are smooth. Also, from (\[nondegeneracy\]) which holds for the even extensions as well, we find that $\partial_{{\xi}}u>0$ on $\Gamma_{\varepsilon_1}({{P_2}})$. Then, applying the hodograph transform of [@KinderlehrerNirenberg §3], i.e., changing $({{\xi}}, {{\eta}})\to (X,Y)=(u({{\xi}}, {{\eta}}), {{\eta}})$, and denoting the inverse transform by $(X,Y)\to ({{\xi}}, {{\eta}})=(v(X,Y), Y)$, we obtain $$v\in C^{1,\alpha}(\overline{B^+_\delta((0, -v_2))}) \cap
C^{2,\alpha}(B^+_\delta((0, -v_2))),$$ where $B_\delta^+((0, -v_2)){:=}B_\delta((0, -v_2))\cap\{X>0\}$ for small $\delta>0$, $v(X,Y)$ satisfies a uniformly elliptic quasilinear equation $$\tilde F(D^2v, Dv, v, X, Y)=0 \qquad\mbox{in } B_\delta^+((0,
-v_2))$$ and the oblique derivative condition $$\tilde G(Dv,v,Y)=0 \qquad\mbox{on } \partial B_\delta^+((0, -v_2))\cap \{X=0\},$$ and the functions $\tilde F$ and $\tilde G$ are smooth. Then, from the local estimates near the boundary in the proof of [@Lieberman86 Theorem 2], $v\in C^{2,\alpha}(\overline{B^+_{\delta/2}((0, -v_2))})$. Since $f_{S_1}(\eta)=v(0, \eta)$, it follows that $f_{S_1}$ is $C^{2,\alpha}$ near ${{\eta}}_{{{P_2}}}=-v_2$.
It remains to prove the convergence of the solutions to the normal reflection solution as $\theta_w\to \pi/2$. Let $\theta_w^i\to
\pi/2$ as $i\to\infty$. Denote by $\varphi^i$ and $f^i$ the corresponding solution and the free-boundary function respectively, i.e., $P_0{{P_1}}{{P_2}}\cap \Lambda$ for each $i$ is given by $\{{{\xi}}=f^i({{\eta}}) \;:\;{{\eta}}\in({{\eta}}_{{{P_2}}}, {{\eta}}_{P_0})\}$. Denote by $\varphi^\infty$ and $f^\infty(\eta)=\bar{{\xi}}$ the solution and the reflected shock for the normal reflection respectively. For each $i$, we find that $\varphi^i-\varphi_2^i=\psi^i$ in the subsonic domain $\Omega^+_i$, where $\psi^i$ is the corresponding “fixed point solution” from Proposition \[existenceFixedPt\] and $\psi^i\in {{\mathcal K}}(\pi/2-\theta_w^i, \varepsilon^i, M_1^i, M_2^i)$ with (\[condConst-00\]). Moreover, $f^i$ satisfies (\[OmegaPL-f-higher\]). We also use the convergence of state (2) to the corresponding state of the normal reflection obtained in §\[section:3.3\]. Then we conclude that, for a subsequence, $f^i\to f^\infty$ in $C^1_{loc}$ and $\varphi^i\to\varphi^\infty$ in $C^1$ on compact subsets of $\{{{\xi}}>\bar{{\xi}}\}$ and $\{{{\xi}}<\bar{{\xi}}\}$. Also, we obtain $\|(D\varphi^i,
\varphi^i)\|_{L^\infty(K)}\le C(K)$ for every compact set K. Then $\varphi^i\to \varphi_\infty$ in $W^{1,1}_{loc}(\overline\Lambda)$ by the Dominated Convergence Theorem. Since such a converging subsequence can be extracted from every sequence $\theta_w^i\to
\pi/2$, it follows that $\varphi_{\theta_w}\to\varphi_\infty$ as $\theta_w\to \pi/2$.
Appendix: Estimates of Solutions to Elliptic Equations {#append-1-section}
======================================================
In this appendix, we make some careful estimates of solutions of boundary value problems for elliptic equations in ${ {\bf R}}^2$, which are applied in §\[unifElliptApproxSection\]–§\[fixedPtSection\]. Throughout the appendix, we denote by $(x,y)$ or $(X,Y)$ the coordinates in ${ {\bf R}}^2$, by ${ {\bf R}}^2_+:=\{y>0\}$, and, for $z=(x,0)$ and $r>0$, denote $B_r^+(z):=B_r(z)\cap{ {\bf R}}^2_+$ and ${ \Sigma}_r(z):=B_r(z)\cap\{y=0\}$. We also denote $B_r:=B_r(0)$, $B_r^+:=B_r^+(0)$, and ${ \Sigma}_r:={ \Sigma}_r(0)$.
We consider an elliptic equation of the form $$\label{nonlinEq-appdx}
A_{11}u_{xx}+2A_{12}u_{xy}+A_{22}u_{yy}+ A_{1}u_{x}+A_{2}u_{y}=f,$$ where $A_{ij}=A_{ij}(Du, x,y)$, $A_{i}=A_{i}(Du, x, y)$, and $f=f(x,y)$. We study the following three types of boundary conditions: (i) the Dirichlet condition, (ii) the oblique derivative condition, (iii) the “almost tangential derivative" condition.
One of the new ingredients in our estimates below is that we do not assume that the equation satisfies the “natural structure conditions", which are used in the earlier related results; see, e.g., [@GilbargTrudinger Chapter 15] for the interior estimates for the Dirichlet problem and [@LiebermanTrudinger] for the oblique derivative problem. For equation (\[nonlinEq-appdx\]), the natural structure conditions include the requirement that $|p||D_pA_{ij}|\le C$ for all $p\in{ {\bf R}}^2$. Note that equations (\[iteration-equation-sonicStruct\]) and (\[nonlinIterEq-xy-lg\]) do not satisfy this condition because of the term $x\zeta_1(\frac{\psi_x}{x})$ in the coefficient of $\psi_{xx}$. Thus we have to derive the estimates for the equations without the “natural structure conditions". We consider only the two-dimensional case here.
The main point at which the “natural structure conditions" are needed is the gradient estimates. The interior gradient estimates and global gradient estimates for the Dirichlet problem, without requiring the natural structure conditions, were obtained in the earlier results in the two-dimensional case; see Trudinger [@Trudinger85] and references therein. However, it is not clear how this approach can be extended to the oblique and “almost tangential" derivative problems. We also note a related result by Lieberman [@Lieberman87-1] for fully nonlinear equations and the boundary conditions without obliqueness assumption in the two-dimensional case, in which the Hölder estimates for the gradient of a solution depend on both the bounds of the solution and its gradient.
In this appendix, we present the $C^{2,\alpha}$–estimates of the solution only in terms of its $C$–norm. For simplicity, we restrict to the case of quasilinear equation (\[nonlinEq-appdx\]) and linear boundary conditions, which is the case for the applications in this paper. Below, we first present the interior estimate in the form that is used in the other parts of this paper. Then we give a proof of the $C^{2,\alpha}$–estimates for the “almost tangential" derivative problem. Since the proofs for the Dirichlet and oblique derivative problems are similar to that for the “almost tangential" derivative problem, we just sketch these proofs.
\[locEstElliptEq\] Let $u\in C^2(B_2)$ be a solution of equation [(\[nonlinEq-appdx\])]{} in $B_2$. Let $A_{ij}(p,x,y)$, $A_{i}(p,x,y)$, and $f(x,y)$ satisfy that there exist constants $\lambda>0$ and $\alpha\in(0,1)$ such that $$\begin{aligned}
\label{locEstElliptEq-i1-0} &&\qquad\lambda|\mu|^2 \le
\sum_{i,j=1}^nA_{ij}\mu_i\mu_j\le\lambda^{-1}|\mu|^2 \qquad\mbox{for
all } (x,y)\in B_2,\,\, p, \mu\in{ {\bf R}}^2,
\\
\label{locEstElliptEq-i2-0} &&\qquad \|
(A_{ij},\,A_{i})\|_{C^\alpha({ {\bf R}}^2\times\overline{B_2})}+\| D_p
(A_{ij},\,
A_{i})\|_{C({ {\bf R}}^2\times\overline{B_2})}+\|f\|_{C^\alpha(\overline{B_2})}
\le \lambda^{-1}.\end{aligned}$$ Assume that $\|u\|_{C(\overline{B_2})}\le M$. Then there exists $C>0$ depending only on $(\lambda, M)$ such that $$\label{localEst-appdx-2}
\|u\|_{C^{2,\alpha}(\overline{B_1})}\le C(\|u\|_{C(\overline{B_2})}
+\|f\|_{C^{\alpha}(\overline{B_2})}).$$
We use the standard interior Hölder seminorms and norms as defined in [@GilbargTrudinger Eqs. (4.17), (6.10)]. By [@GilbargTrudinger Theorem 12.4], there exists $\beta\in(0,1)$ depending only on $\lambda$ such that $$\begin{aligned}
[u]^*_{1,\beta, B_2}&\le& C(\lambda)(\|u\|_{0, B_2}
+\|f-A_1D_1u-A_2D_2u\|^{(2)}_{0, B_2})\\
&\le& C(\lambda, M)(1+\|f\|^{(2)}_{0, B_2}+\|Du\|^{(2)}_{0, B_2}).\end{aligned}$$ Then, applying the interpolation inequality [@GilbargTrudinger (6.82)] with the argument similar to that for the proof of [@GilbargTrudinger Theorem 12.4], we obtain $$\|u\|^*_{1,\beta, B_2}\le C(\lambda, M)(1+\|f\|^{(2)}_{0, B_2}).$$ Now we consider (\[nonlinEq-appdx\]) as a linear elliptic equation $$\sum_{i,j=1}^na_{ij}(x)u_{x_ix_j}+ \sum_{i=1}^n a_{i}(x)u_{x_i}=f(x)
\qquad\mbox{in }\,\, B_{3/2}$$ with coefficients $a_{ij}(x)=A_{ij}(Du(x),x)$ and $a_i=A_{i}(Du(x),x)$ in $C^\beta(\overline{B_{3/2}})$ satisfying $$\|(a_{ij}, a_i)\|_{C^\beta(\overline{B_{3/2}})}\le C(\lambda, M).$$ We can assume $\beta\le\alpha$. Then the local estimates for linear elliptic equations yield $$\|u\|_{C^{2,\beta}(\overline{B_{5/4}})}\le C(\lambda, M)
(\|u\|_{C(\overline{B_{3/2}})}
+\|f\|_{C^{\beta}(\overline{B_{3/2}})}).$$ With this estimate, we have $ \|(a_{ij},
a_i)\|_{C^\alpha(\overline{B_{5/4}})}\le C(\lambda, M). $ Then the local estimates for linear elliptic equations in $B_{5/4}$ yield (\[localEst-appdx-2\]). Now we make the estimates for the “almost tangential derivative" problem.
\[locEstElliptEq-non-oblique\] Let $\lambda>0$, $\alpha\in (0,1)$, and $\varepsilon\ge 0$. Let $\Phi\in C^{2,\alpha}({ {\bf R}})$ satisfy $$\label{bdryNormCond}
\|\Phi\|_{C^{2,\alpha}({ {\bf R}})}\le \lambda^{-1},$$ and denote $\Omega_R^+:=B_R\cap\{y>\varepsilon\Phi(x)\}$ for $R>0$. Let $u\in C^2(B_2^+)\cap C^1(\overline{B_2^+})$ satisfy [(\[nonlinEq-appdx\])]{} in $\Omega_2^+$ and $$\begin{aligned}
&&u_{x}=\varepsilon b(x,y)u_{y}+c(x,y) u\qquad \mbox{ on }\;
\Gamma_\Phi{:=}B_2\cap\{y=\varepsilon \Phi(x)\}.
\label{nonlinEq-appdx-nonObl-bc}\end{aligned}$$ Let $A_{ij}(p,x,y)$, $A_{i}(p,x,y)$, $a(x,y)$, $b(x,y)$, and $f(x,y)$ satisfy that there exist constants $\lambda>0$ and $\alpha\in(0,1)$ such that $$\begin{aligned}
\label{locEstElliptEq-i1} &&\qquad
\lambda|\mu|^2\le\sum_{i,j=1}^nA_{ij}\mu_i\mu_j\le\lambda^{-1}|\mu|^2
\qquad\mbox{for } (x,y)\in \Omega_2^+,\,\, p, \mu\in{ {\bf R}}^2,
\\
\label{locEstElliptEq-i2}
&&\qquad\|(A_{ij},\,A_{i})\|_{C^\alpha(\overline{\Omega_2^+}\times{ {\bf R}}^2)}+\|D_p
(A_{ij},\, A_{i})\|_{C(\overline{\Omega_2^+}\times{ {\bf R}}^2)}
+\|f\|_{C^{\alpha}(\overline{\Omega_2^+})} \le \lambda^{-1},
\\
\label{bdryCoefBds} &&\qquad
\|(b,c)\|_{C^{1,\alpha}(\overline{\Omega_2^+})}\le \lambda^{-1}.\end{aligned}$$ Assume that $\|u\|_{C(\overline{\Omega_2^+})}\le M$. Then there exist $\varepsilon_0(\lambda, M, \alpha)>0$ and $C(\lambda, M,
\alpha)>0$ such that, if $\varepsilon\in (0,\varepsilon_0)$, we have $$\label{localEst-appdx-nonOblique-est}
\|u\|_{C^{2,\alpha}(\overline{\Omega_1^+})}\le
C(\|u\|_{C(\overline{\Omega_2^+})}+\|f\|_{C^{\alpha}(\overline{\Omega_2^+})}).$$
To prove this theorem, we first flatten the boundary part $\Gamma_\Phi$ by defining the variables $(X,Y)=\Psi(x,y)$ with $(X,Y)=(x, y-\varepsilon\Phi(x))$. Then $(x,y)=\Psi^{-1}(X,Y)=(X,
Y+\varepsilon\Phi(X))$. From (\[bdryNormCond\]), we have $$\label{defFlatten-Norm}
\|\Psi-Id\|_{C^{2,\alpha}(\overline{\Omega_2^+})}+
\|\Psi^{-1}-Id\|_{C^{2,\alpha}(\overline{B_2^+})}\le
\varepsilon\lambda^{-1}.$$ Then, for sufficiently small $\varepsilon$ depending only on $\lambda$, the transformed domain ${{\mathcal D}}_2^+{:=}\Psi(\Omega_2^+)$ satisfies $$\label{rescaledDomain}
\begin{array}{l}
\displaystyle B_{2-2\varepsilon/\lambda}^+\subset{{\mathcal D}}_2^+\subset
B_{2+2\varepsilon/\lambda}^+, \quad
{{\mathcal D}}_2^+\subset{ {\bf R}}^2_+{:=}\{Y>0\}, \quad
\partial{{\mathcal D}}_2^+\cap\{Y=0\}=\Psi(\Gamma_\Phi);
\end{array}$$ the function $$v(X,Y)=u(x,y):=u(\Psi^{-1}(X,Y))$$ satisfies an equation of form (\[nonlinEq-appdx\]) in ${{\mathcal D}}_2^+$ with (\[locEstElliptEq-i1\])–(\[locEstElliptEq-i2\]) and the corresponding elliptic constants $\lambda/2$; and the boundary condition for $v$ by an explicit calculation is $$\label{nonlinEq-appdx-nonObl-bc-fltnd}
v_X=\varepsilon(b(\Psi^{-1}(X,0))+\Phi'(X))v_Y+c(\Psi^{-1}(X,0))v
\qquad \mbox{on }\;{{\mathcal D}}_2^+\cap\{Y=0\},$$ i.e., it is of form (\[nonlinEq-appdx-nonObl-bc\]) with (\[bdryCoefBds\]) satisfied on $\overline{{{\mathcal D}}_2^+}$ with elliptic constant $\lambda/4$. Moreover, by (\[defFlatten-Norm\])–(\[rescaledDomain\]), it suffices for this theorem to show the following estimate for $v(X,Y)$: $$\label{toProveThem-nontang}
\|v\|_{2,\alpha, B_{6/5}^+}\le C(\lambda,
M,\alpha)\big(\|v\|_{0,B_{2-2\varepsilon/\lambda}^+}+\|f\|_{\alpha,
B_{2-2\varepsilon/\lambda}^+}\big).$$ That is, we can consider the equation in $B_{2-2\varepsilon/\lambda}^+$ and condition (\[nonlinEq-appdx-nonObl-bc-fltnd\]) on ${ \Sigma}_{2-2\varepsilon/\lambda}$ or, by rescaling, we can simply consider our equation in $B_2^+$ and condition (\[nonlinEq-appdx-nonObl-bc-fltnd\]) on ${ \Sigma}_2{:=}B_2\cap\{Y=0\}$. In other words, without loss of generality, we can assume $\Phi\equiv 0$ in the original problem.
For simplicity, we use the original notation $(x,y, u(x,y))$ to replace the notation $(X, Y, v(X,Y))$. Then we assume that $\Phi\equiv 0$. Thus, equation (\[nonlinEq-appdx\]) is satisfied in the domain $B_2^+$, the boundary condition (\[nonlinEq-appdx-nonObl-bc\]) is prescribed on ${ \Sigma}_2=B_2\cap\{y=0\}$, and conditions (\[locEstElliptEq-i1\])–(\[bdryCoefBds\]) hold in $B_2^+$. Also, we use the partially interior norms [@GilbargTrudinger Eq. 4.29] in the domain $B_2^+\cup { \Sigma}_2$ with the related distance function $d_{z}={ \mbox{dist}}(z,\partial B_2^+\setminus { \Sigma}_2)$. The universal constant $C$ in the argument below depends only on $\lambda$ and $M$, unless otherwise specified.
As in [@GilbargTrudinger §13.2], we introduce the functions $w_i=D_iu$ for $i=1,2$. Then we conclude from equation (\[nonlinEq-appdx\]) that $w_1$ and $w_2$ are weak solutions of the following equations of divergence form: $$D_1\big({A_{11}\over A_{22}}D_1w_1+{2A_{12}\over
A_{22}}D_2w_1\big) +D_{22}w_1= D_1\big({f\over A_{22}}-{A_{1}\over A_{22}}D_1u
-{A_{2}\over A_{22}}D_2 u\big),
\label{eqnForDerivAppdx-1}$$ $$D_{11}w_2+ D_2\big({2A_{12}\over A_{11}}D_1w_2+{A_{22}\over
A_{11}}D_2w_2\big) = D_2\big({f\over A_{11}}-{A_{1}\over A_{11}}D_1
u-{A_{2}\over A_{11}}D_2 u\big). \label{eqnForDerivAppdx-2}$$
From (\[nonlinEq-appdx-nonObl-bc\]), we have $$\label{bc-w1}
w_1=g \qquad\mbox{ on } { \Sigma}_2,$$ where $$\label{bc-w1-defG} g:=\varepsilon
bw_2+cu\qquad\mbox {for }\;B_2^+.$$
We first obtain the following Hölder estimates of $D_1u$.
\[appDxnonObl-lemma\] There exist $\beta\in (0,\alpha]$ and $C>0$ depending only on $\lambda$ such that, for any $z_0\in B_2^+\cup { \Sigma}_2$, $$\label{scaledHolderAppdx-claim}
d_{z_0}^\beta[w_1]_{0,\beta, B_{d_{z_0}/16}({z_0})\cap B_2^+}\le
C(\|(Du, f)\|_{0,0,B_{d_{z_0}/2}({z_0})\cap B_2^+} +
d_{z_0}^\beta[g]_{0,\beta, B_{d_{z_0}/2}(z_0)\cap B_2^+}).$$
We first prove that, for $z_1\in { \Sigma}_2$ and $B_{2R}^+(z_1)\subset
B_2^+$, $$\label{scaledHolderAppdx-claim-1}
R^\beta[w_1]_{0,\beta, B_{R}^+({z_1})}\le
C(\|(Du, Rf)\|_{0,0,B_{2R}^+({z_1})}
+ R^\beta[g]_{0,\beta, B_{2R}^+(z_1)}).$$ We rescale $u$, $w_1$, and $f$ in $B_{2R}^+(z_1)$ by defining $$\label{rescaledU}
\hat u(Z)={1\over 2R}u(z_1+2RZ),\quad
\hat f(Z)={2R}f(z_1+2RZ)
\qquad\mbox{ for }\;
Z\in B_1^+,$$ and $\hat w_i=D_{Z_i}\hat u$. Then $\hat w_1$ satisfies an equation of form (\[eqnForDerivAppdx-1\]) in $B_1^+$ with $u$ replaced by $\hat u$ whose coefficients $\hat A_{ij}$ and $\hat A_i$ satisfy (\[locEstElliptEq-i1\])–(\[locEstElliptEq-i2\]) with unchanged constants (this holds for (\[locEstElliptEq-i2\]) since $R\le 1$). Then, by the elliptic version of [@LiebermanBook Theorem 6.33] stated in the parabolic setting (it can also be obtained by using [@LiebermanBook Lemma 4.6] instead of [@GilbargTrudinger Lemma 8.23] in the proofs of [@GilbargTrudinger Theorem 8.27, 8.29] to achieve $\alpha=\alpha_0$ in [@GilbargTrudinger Theorem 8.29]), we find constants $\tilde\beta(\lambda)\in (0,1)$ and $C(\lambda)$ such that $$[\hat w_1]_{0,\beta, B_{1/2}^+}\le C(\|(D\hat u, \hat f)\|_{0,0,B_1^+} +
[\hat w_1]_{0,\beta, B_1\cap\{y=0\}})$$ for $\beta=\min(\tilde\beta, \alpha)$. Rescaling back and using (\[bc-w1\]), we have (\[scaledHolderAppdx-claim-1\]).
If $z_1\in B_2^+$ and $B_{2R}(z_1)\subset B_2^+$, then an argument similar to the proof of (\[scaledHolderAppdx-claim-1\]) by using the interior estimates [@GilbargTrudinger Theorem 8.24] yields $$\label{scaledHolderAppdx-claim-2}
R^\beta[w_1]_{0,\beta, B_{R}({z_1})}\le
C\|(Du, Rf)\|_{0,0,B_{2R}({z_1})}.$$
Now let $z_0=(x_0,y_0)\in B_2^+\cup { \Sigma}_2$. When $y_0\le d_{z_0}/8$, then, denoting $z_0'=(x_0,0)$ and noting that $d_{z_0'}\ge d_{z_0}$, it is easy to check that $$\begin{aligned}
B_{d_{z_0}/16}(z_0)\cap B_2^+
\subset
B^+_{d_{z_0}/8}(z_0')
\subset
B_2^+,
\quad
B^+_{d_{z_0}/8}(z_0')
\subset
B_{d_{z_0}/2}(z_0)\cap B_2^+,\end{aligned}$$ and then applying (\[scaledHolderAppdx-claim-1\]) with $z_1=z_0'$ and $R={d_{z_0}/8}\le 1$ and using the inclusions stated above yield (\[scaledHolderAppdx-claim\]). When $y_0\ge d_{z_0}/8$, $B_{d_{z_0}/8}(z_0)\subset B_2^+$. Then applying (\[scaledHolderAppdx-claim-2\]) with $z_1=z_0$ and $R={d_{z_0}/16}\le 1$ yields (\[scaledHolderAppdx-claim\]). Next, we make the Hölder estimates for $Du$. We first note that, by (\[bdryCoefBds\]) and (\[bc-w1-defG\]), $g$ satisfies $$\begin{aligned}
\label{estimateG-appdx-1} &&\qquad |Dg|\le
C(\varepsilon|D^2u|+|Du|+|u|)\qquad \mbox{in }\;B_2^+,
\\
&&\qquad [g]_{0,\beta, B_{d_z/2}(z)\cap B_2^+} \le
C\left(\varepsilon [Du]_{0,\beta, B_{d_z/2}(z)\cap B_2^+} +
\|u\|_{1,0,B_{d_z/2}(z)\cap B_2^+} \right).\label{estimateG-appdx-2}\end{aligned}$$
\[appDxnonObl-DuHolder-lemma\] Let $\beta$ be as in Lemma [\[appDxnonObl-lemma\]]{}. Then there exist $\varepsilon_0(\lambda)>0$ and $C(\lambda)>0$ such that, if $0\le\varepsilon\le \varepsilon_0$, $$\begin{aligned}
d_{z_0}^\beta[Du]_{0,\beta, B_{d_{z_0}/ 32}({z_0})\cap B_2^+}
&\le&
C(\|u\|_{1,0,B_{d_{z_0}/ 2}({z_0})\cap B_2^+} +\varepsilon
d_{z_0}^\beta[Du]_{0,\beta, B_{d_{z_0}/ 2}(z_0)\cap B_2^+}
\nonumber
\\
&&\qquad+\|f\|_{0,0,B_{d_{z_0}/ 2}({z_0})\cap B_2^+} )
\label{scaledHolderAppdx-fullDu-claim}\end{aligned}$$ for any $z_0\in B_2^+\cup{ \Sigma}_2$.
The Hölder norm of $D_1u$ has been estimated in Lemma \[appDxnonObl-lemma\]. It remains to estimate $D_2u$. We follow the proof of [@GilbargTrudinger Theorem 13.1].
Fix $z_0\in B_2^+\cup{ \Sigma}_2$. In order to prove (\[scaledHolderAppdx-fullDu-claim\]), it suffices to show that, for every $\hat z\in B_{d_{z_0}/32}({z_0})\cap B_2^+$ and every $R>0$ such that $B_R(\hat z)\subset B_{d_{z_0}/16}({z_0})$, we have $$\label{MorreyEst}
\int_{B_R(\hat z)\cap B_2^+}|D^2u|^2dz\le {L^2\over
d_{z_0}^{2\beta}}R^{2\beta} ,$$ where $L$ is the right-hand side of (\[scaledHolderAppdx-fullDu-claim\]) (cf. [@GilbargTrudinger Theorem 7.19] and [@LiebermanBook Lemma 4.11]).
In order to prove (\[MorreyEst\]), we consider separately case (i) $B_{2R}(\hat z)\cap { \Sigma}_2\ne\emptyset$ and case (ii) $B_{2R}(\hat z)\cap { \Sigma}_2=\emptyset$.
We first consider case (i). Let $B_{2R}(\hat z)\cap { \Sigma}_2\ne\emptyset$. Since $B_R(\hat
z)\subset B_{d_{z_0}/32}({z_0})$, then $B_{2R}(\hat z)\subset
B_{d_{z_0}/16}({z_0})$ so that $$\label{R-le-d}
2R\le d_{z_0}.$$ Let $\eta\in C^1_0(B_{2R}(\hat z))$ and $\zeta=\eta^2(w_1-g)$. Note that $\zeta\in W^{1,2}_0(B_{2R}(\hat z)\cap B_2^+)$ by (\[bc-w1\]). We use $\zeta$ as a test function in the weak form of (\[eqnForDerivAppdx-1\]): $$\label{weakEqForDeriv}
\int_{B_2^+}{1\over A_{22}}\sum_{i,j=1}^2 A_{ij}D_iw_1D_j\zeta dz
=\int_{B_2^+}{1\over A_{22}}\big(\sum_{i=1}^2
A_{i}D_iu+f\big)D_1\zeta dz,$$ and apply (\[locEstElliptEq-i1\])–(\[locEstElliptEq-i2\]) and (\[estimateG-appdx-1\]) to obtain $$\begin{aligned}
\label{intermEstNonObliq}
\\
&&\int_{B_2^+}|Dw_1|^2\eta^2 dz \le
C\int_{B_2^+}\bigg(
\Big((\delta+\varepsilon)|Dw_1|^2+
\varepsilon|D^2u|^2\Big)\eta^2\nonumber
\\
&&\qquad\qquad\qquad\qquad
+({1\over\delta}+1)\left((|D\eta|^2+|f|\eta^2)(w_1-g)^2
+(|Du|^2+|u|^2)\eta^2\right)\bigg)dz, \nonumber\end{aligned}$$ where $C$ depends only on $\lambda$, and the sufficiently small constant $\delta>0$ will be chosen below. Since $$\label{intermEstNonObliq-1}
|Dw_1|^2=(D_{11}u)^2+(D_{12}u)^2,$$ it remains to estimate $|D_{22}u|^2$. Using the ellipticity property (\[locEstElliptEq-i1\]), we can express $D_{22}u$ from equation (\[nonlinEq-appdx\]) to obtain $$\int_{B_2^+}|D_{22}u|^2\eta^2 dz\le
C(\lambda)\int_{B_2^+}(|D_{11}u|^2+|D_{12}u|^2+|Du|^2)\eta^2 dz.$$ Combining this with (\[intermEstNonObliq\])–(\[intermEstNonObliq-1\]) and using (\[locEstElliptEq-i2\]) to estimate $|f|$ yield $$\begin{aligned}
\label{intermEstNonObliq-2}
\\
\int_{B_2^+}|D^2u|^2\eta^2 dz
&\le &
C\int_{B_2^+}\Big(
(\varepsilon+\delta)|D^2u|^2\eta^2\nonumber
\\
\nonumber && \quad\,
+({1\over\delta}+1)\left((|D\eta|^2+\eta^2)(w_1-g)^2
+(|Du|^2+|u|^2)\eta^2\right)\Big)dz.\end{aligned}$$ Choose $\varepsilon_0=\delta=(4C)^{-1}$. Then, when $\varepsilon\in
(0,\varepsilon_0)$, we have $$\label{intermEstNonObliq-3}
\int_{B_2^+}|D^2u|^2\eta^2 dz\le C\int_{B_2^+}\left(
(|D\eta|^2+\eta^2)(w_1-g)^2 +(|Du|^2+|u|^2)\eta^2\right)\,dz.$$
Now we make a more specific choice of $\eta$: In addition to $\eta\in C^1_0(B_{2R}(\hat z))$, we assume that $\eta\equiv 1$ on $B_{R}(\hat z)$, $0\le\eta\le 1$ on ${ {\bf R}}^2$, and $|D\eta|\le
{10/R}$. Also, since $B_{2R}(\hat z)\cap { \Sigma}_2\ne\emptyset$, then, for any fixed $z^*\in B_{2R}(\hat z)\cap { \Sigma}_2$, we have $|z-z^*|\le 2R$ for any $z\in B_{2R}(\hat z)$. Moreover, $(w_1-g)(z^*)=0$ by (\[bc-w1\]). Then, since $B_{2R}(\hat
z)\subset B_{d_{z_0}/16}({z_0})$, we find from (\[scaledHolderAppdx-claim\]), (\[estimateG-appdx-2\]), and (\[R-le-d\]) that, for any $z\in B_{2R}(\hat z)\cap B_2^+$, $$\begin{aligned}
|(w_1-g)(z)|
&=& |(w_1-g)(z) -(w_1-g)(z^*)|
\le
|w_1(z)-w_1(z^*)|+|g(z)-g(z^*)|\\
&\le&
{C\over d_{z_0}^\beta}\big(\|(Du,f)\|_{0,0,B_{d_{z_0}/2}({z_0})\cap B_2^+}
+d_{z_0}^\beta[g]_{0,\beta, B_{d_{z_0}/2}(z_0)\cap B_2^+}
\big)|z-z^*|^\beta
\\
&& +[g]_{0,\beta, B_{d_{z_0}/2}(z_0)\cap B_2^+}|z-z^*|^\beta
\\
&\le& C\big({1\over d_{z_0}^\beta}
\|(Du,f)\|_{0,0,B_{d_{z_0}/2}({z_0})\cap B_2^+} + \varepsilon
[Du]_{0,\beta, B_{d_{z_0}/2}(z_0)\cap B_2^+}
\\
&&\qquad +\|u\|_{0,0,B_{d_{z_0}/2}({z_0})\cap B_2^+} \big)R^\beta.\end{aligned}$$ Using this estimate and our choice of $\eta$, we obtain from (\[intermEstNonObliq-3\]) that $$\begin{aligned}
&&\int_{B_R(\hat z)\cap B_2^+}|D^2u|^2 dz\\
&&\le C\big({1\over d_{z_0}^{2\beta}}\|(Du,f)\|_{0,0,B_{d_{z_0}/
2}({z_0})\cap B_2^+}^2 +\varepsilon^2 [Du]_{0,\beta,
B_{d_{z_0}/2}(z_0)\cap B_2^+}^2\big)R^{2\beta}
\\
&&\quad +C\|u\|_{1,0,B_{d_{z_0}/2}({z_0})\cap
B_2^+}^2(R^{2\beta}+R^2),\end{aligned}$$ which implies (\[MorreyEst\]) for case (i).
Now we consider case (ii): $\hat z\in B_2^+$ and $R>0$ satisfy $B_R(\hat z)\subset B_{d_{z_0}/32}({z_0})$ and $B_{2R}(\hat z)\cap
{ \Sigma}_2=\emptyset$. Then $B_{2R}(\hat z)\subset
B_{d_{z_0}/16}({z_0})\cap B_2^+$. Let $\eta\in C^1_0(B_{2R}(\hat
z))$ and $\zeta=\eta^2(w_1-w_1(\hat z))$. Note that $\zeta\in
W^{1,2}_0(B_2^+)$ since $B_{2R}(\hat z)\subset B_2^+$. Thus we can use $\zeta$ as a test function in (\[weakEqForDeriv\]). Performing the estimates similar to those that have been done to obtain (\[intermEstNonObliq-3\]), we have $$\label{intermEstNonObliq-3-pr}
\int_{B_2^+}|D^2u|^2\eta^2 dz\le
C(\lambda)\int_{B_2^+}\big((|D\eta|^2+\eta^2)(w_1-w_1(\hat z))^2
+|Du|^2\eta^2\big)\,dz.$$ Choose $\eta\in C^1_0(B_{2R}(\hat z))$ so that $\eta\equiv 1$ on $B_{R}(\hat z)$, $0\le\eta\le 1$ on ${ {\bf R}}^2$, and $|D\eta|\le {10/
R}$. Note that, for any $z\in B_{2R}(\hat z)$, $$|w_1(z)-w_1(\hat z)| \le C\big({1\over
d_{z_0}^\beta}\|(Du,f)\|_{0,0,B_{d_{z_0}/2}({z_0})\cap B_2^+}
+\varepsilon [Du]_{0,\beta, B_{d_{z_0}/2}(z_0)\cap
B_2^+}\big)R^\beta$$ by (\[scaledHolderAppdx-claim\]) since $B_{2R}(\hat z)\subset
B_{d_{z_0}/16}({z_0})\cap B_2^+$. Now we obtain (\[MorreyEst\]) from (\[intermEstNonObliq-3-pr\]) similar to that for case (i). Then Lemma \[appDxnonObl-DuHolder-lemma\] is proved.
\[partIntSeminorm-est-lemma\] Let $\beta$ and $\varepsilon_0$ be as in Lemma [\[appDxnonObl-DuHolder-lemma\]]{}. Then, for $\varepsilon\in
(0,\varepsilon_0)$, there exists $C(\lambda)$ such that $$\label{partIntSeminorm-est}
[u]_{1,\beta, B_2^+\cup{ \Sigma}_2}^*\le
C(\|u\|_{1,0,B_2^+\cup{ \Sigma}_2}^*
+\varepsilon [u]_{1,\beta, B_2^+\cup{ \Sigma}_2}^*
+\|f\|_{0,0,B_2^+}),$$ where $[\cdot]^*$ and $\|\cdot\|^*$ denote the standard partially interior seminorms and norms [[@GilbargTrudinger Eq. 4.29]]{}.
Estimate (\[partIntSeminorm-est\]) follows directly from Lemma \[appDxnonObl-DuHolder-lemma\] and an argument similar to the proof of [@GilbargTrudinger Theorem 4.8]. Let $z_1, z_2\in
B_2^+$ with $d_{z_1}\le d_{z_2}$ (thus $d_{z_1, z_2}=d_{z_1}$) and let $|z_1-z_2|\le d_{z_1}/64$. Then $z_2\in
B_{d_{z_0}/32}({z_0})\cap B_2^+$ and, by Lemma \[appDxnonObl-DuHolder-lemma\] applied to $z_0=z_1$, we find $$\begin{aligned}
d_{z_1, z_2}^{1+\beta}{|Du(z_1)-Du(z_2)|\over |z_1-z_2|^\beta} &\le&
C(d_{z_1}\|u\|_{1,0,B_{d_{z_1}/2}({z_1})\cap B_2^+} +\varepsilon
d_{z_1}^{1+\beta}[Du]_{0,\beta, B_{d_{z_1}/2}(z_1)\cap B_2^+}\\
&&\qquad
+\|f\|_{0,0,B_{d_{z_1}/ 2}({z_1})\cap B_2^+})
\\
&\le& C(\|u\|_{1,0,B_2^+\cup{ \Sigma}_2}^*
+\varepsilon [u]_{1,\beta, B_2^+\cup{ \Sigma}_2}^*
+\|f\|_{0,0,B_2^+}),\end{aligned}$$ where the last inequality holds since $2d_z\ge d_{z_1}$ for all $z\in B_{d_{z_1}/2}(z_1)\cap B_2^+$. If $z_1, z_2\in B_2^+$ with $d_{z_1}\le d_{z_2}$ and $|z_1-z_2|\ge d_{z_1}/64$, then $$\begin{aligned}
d_{z_1, z_2}^{1+\beta}{|Du(z_1)-Du(z_2)|\over |z_1-z_2|^\beta} \le
64(d_{z_1}|Du(z_1)|+d_{z_2}|Du(z_2)|)\le
64\,\|u\|_{1,0,B_2^+\cup{ \Sigma}_2}^*.\end{aligned}$$ This completes the proof. Now we can complete the proof of Theorem \[locEstElliptEq-non-oblique\]. For sufficiently small $\varepsilon_0>0$ depending only on $\lambda$, when $\varepsilon\in
(0,\varepsilon_0)$, we use Lemma \[partIntSeminorm-est-lemma\] to obtain $$\label{partIntSeminorm-est-1}
[u]_{1,\beta, B_2^+\cup{ \Sigma}_2}^*\le
C(\lambda)(\|u\|_{1,0,B_2^+\cup{ \Sigma}_2}^* +\|f\|_{0,0,B_2^+}).$$ We use the interpolation inequality [@GilbargTrudinger Eq. (6.89)] to estimate $$\|u\|_{1,0,B_2^+\cup{ \Sigma}_2}^*\le
C(\beta,\delta)\|u\|_{0,B_2^+} +\delta[u]_{1,\beta,
B_2^+\cup{ \Sigma}_2}^*$$ for $\delta>0$. Since $\beta=\beta(\lambda)$, we choose sufficiently small $\delta(\lambda)>0$ to find $$\label{partIntSeminorm-est-2}
\|u\|_{1,\beta, B_2^+\cup{ \Sigma}_2}^*\le
C(\lambda)(\|u\|_{0,0,B_2^+} +\|f\|_{0,0,B_2^+})$$ from (\[partIntSeminorm-est-1\]). In particular, we obtain a global estimate in a smaller half-ball: $$\label{glob-norm-in-smaller-est-2}
\|u\|_{1,\beta, B_{9/5}^+}\le C(\lambda)(\|u\|_{0,0,B_2^+}
+\|f\|_{0,0,B_2^+}).$$
We can assume $\beta\le\alpha$. Now we consider (\[eqnForDerivAppdx-1\]) as a linear elliptic equation $$\label{linEq-InTangThProof}
\sum_{i,j=1}^2D_i(a_{ij}(x,y)D_jw_1)=D_1F \qquad\mbox{in }
B_{9/5}^+,$$ where $a_{ij}(x,y)=(A_{ij}/A_{22})(Du(x,y),x,y)$ for $i+j<4$, $a_{22}=1$, and $F(x,y)=\big({ A_{1}}D_1 u+{ A_{2}}D_2 u +f\big)/
A_{22}$ with $(A_{ij},A_i)=(A_{ij},A_i)(Du(x,y),x,y)$. Then (\[partIntSeminorm-est-2\]), combined with (\[locEstElliptEq-i2\]), implies $$\label{normsCoefs-InTangThProof}
\|a_{ij}\|_{0,\beta, B_{9/5}^+}\le C(\lambda, M).$$ From now on, $d_z$ denotes the distance related to the partially interior norms in $B_{9/5}^+\cup{ \Sigma}_{9/5}$, i.e., for $z\in
B_{9/5}^+$, $d_z:={ \mbox{dist}}(z,\partial B_{9/5}^+\setminus
{ \Sigma}_{9/5})$. Now, similar to the proof of Lemma \[appDxnonObl-lemma\], we rescale equation (\[linEq-InTangThProof\]) and the Dirichlet condition (\[bc-w1\]) from the balls $B^+_R(z'_1)\subset B_{9/5}^+$ and $B_R(z_1)\subset B_{9/5}^+$ with $R\le 1$ to $B=B_1^+$ or $B=B_1$, respectively, by defining $$(\hat w_1, \hat g, \hat a_{ij})(Z)=(w_1, g, a_{ij})(z_1+RZ),\quad
\hat F(Z)=RF(z_1+RZ)\qquad\mbox{for }Z\in B.$$ Then $ \sum_{i,j=1}^2D_i(\hat a_{ij}(x,y)D_j\hat w_1)=D_1\hat F $ in $B$, the ellipticity of this rescaled equation is the same as that for (\[linEq-InTangThProof\]), and $\|\hat a_{ij}\|_{0,\beta,
B}\le C$ for $C=C(\lambda, M)$ in (\[normsCoefs-InTangThProof\]), where we have used $R\le 1$. This allows us to apply the local $C^{1,\beta}$ interior and boundary estimates for the Dirichlet problem [@GilbargTrudinger Theorem 8.32, Corollary 8.36] to the rescaled problems in the balls $B^+_{3d_{z_0}/8}(z_0')$ and $B_{d_{z_0}/8}(z_0)$ as in Lemma \[appDxnonObl-lemma\]. Then, scaling back and multiplying by $d_{z_0}$, applying the covering argument as in Lemma \[appDxnonObl-lemma\], and recalling the definition of $F$, we obtain that, for any $z_0\in B_{9/5}^+\cup
{ \Sigma}_{9/5}$, $$\label{scaledHolderAppdx-claim-2nd}
\begin{array}{ll}
d_{z_0}^{2+\beta}[w_1]_{1,\beta, B_{d_{z_0}/16}({z_0})\cap
B_{9/5}^+} +d_{z_0}^2[w_1]_{1,0, B_{d_{z_0}/16}({z_0})\cap
B_{9/5}^+}\\
\displaystyle \quad \le C\Big(
d_{z_0}\|Du\|_{0,0,B_{d_{z_0}/2}({z_0})\cap B_{9/5}^+}
+d_{z_0}^{1+\beta}[u]_{1,\beta, B_{d_{z_0}/2}(z_0)\cap B_{9/5}^+}
+\|f\|_{0,\beta, B_{d_{z_0}/2}(z_0)\cap B_{9/5}^+}
\\
\displaystyle\qquad\,\,\,\,\,\,\,\,
+d_{z_0}^{2+\beta}[g]_{1,\beta,B_{d_{z_0}/2}(z_0)\cap B_{9/5}^+} +
\sum_{k=0,1} d_{z_0}^{k+1}[g]_{k,0,B_{d_{z_0}/2}(z_0)\cap B_{9/5}^+}
\Big),
\end{array}$$ where we have used $d_{z_0}<2$. Recall that $Dw_1=(D_{11}u, D_{12}u)$. Expressing $D_{22} u$ from equation (\[nonlinEq-appdx\]) by using (\[locEstElliptEq-i1\])–(\[locEstElliptEq-i2\]) and (\[partIntSeminorm-est-2\]) to estimate the Hölder norms of $D_{22} u$, in terms of the norms of $D_{11}u, D_{22}u$, and $Du$, and by using (\[bc-w1-defG\]) and (\[bdryCoefBds\]) to estimate the terms involving $g$ in (\[scaledHolderAppdx-claim-2nd\]), we obtain from (\[scaledHolderAppdx-claim-2nd\]) that, for every $z_0\in B_{9/5}^+\cup { \Sigma}_2$, $$\begin{array}{l}\displaystyle
d_{z_0}^{2+\beta}[D^2u]_{0,\beta, B_{d_{z_0}/16}({z_0})\cap B_{9/5}^+}
+d_{z_0}^2[D^2u]_{0,0, B_{d_{z_0}/16}({z_0})\cap B_{9/5}^+}
\\
\displaystyle \le C\Big(d_{z_0}\|Du\|_{C(B_{d_{z_0}/2}({z_0})\cap
B_{9/5}^+)} +d_{z_0}^{1+\beta}[u]_{1,\beta, B_{d_{z_0}/2}(z_0)\cap
B_{9/5}^+}\\
\qquad\,\, +d_{z_0}\|u\|_{1,0, B_{d_{z_0}/2}(z_0)\cap B_{9/5}^+}
+\|f\|_{0,\beta, B_{d_{z_0}/2}(z_0)\cap B_{9/5}^+} \\
\qquad\,\, +\varepsilon \big(d_{z_0}^{2+\beta}[D^2u]_{0,\beta,
B_{d_{z_0}/2}(z_0)\cap B_{9/5}^+} +d_{z_0}^{2}[D^2u]_{0,0,
B_{d_{z_0}/2}(z_0)\cap B_{9/5}^+}\big) \Big).
\end{array}$$ From this estimate, the argument of Lemma \[partIntSeminorm-est-lemma\] implies $$\label{scaled-C-1-alpha}
\|u\|_{2,\beta, B_{9/5}^+\cup{ \Sigma}_{9/5}}^*\le
C\big(\|u\|_{1,\beta,B_{9/5}^+\cup{ \Sigma}_{9/5}}^*
+\varepsilon\|u\|_{2,\beta, B_{9/5}^+\cup{ \Sigma}_{9/5}}^*
+\|f\|_{0,\beta,B_{9/5}^+}\big).$$ Thus, reducing $\varepsilon_0$ if necessary and using (\[glob-norm-in-smaller-est-2\]), we conclude $$\label{partIntSeminorm-est-4}
\|u\|_{2,\beta, B_{9/5}^+\cup{ \Sigma}_{9/5}}^* \le C(\lambda,
M)(\|u\|_{0,B_2^+}+\|f\|_{0,\beta,B_2^+}).$$ Estimate implies a global estimate in a smaller ball and, in particular, $ \|u\|_{1,\alpha, B_{8/5}^+} \le C(\lambda,
M)(\|u\|_{0,B_2^+}+\|f\|_{0,\beta,B_2^+}). $ Now we can repeat the argument, which leads from (\[glob-norm-in-smaller-est-2\]) to (\[partIntSeminorm-est-4\]) with $\beta$ replaced by $\alpha$, in $B_{8/5}^+$ (and, in particular, further reducing $\varepsilon_0$ depending only on $(\lambda, M,\alpha)$) to obtain $$\|u\|_{2,\alpha, B_{8/5}^+\cup{ \Sigma}_{8/5}}^* \le C(\lambda, M,
\alpha)(\|u\|_{0,B_2^+}+\|f\|_{0,\alpha,B_2^+}),$$ which implies (\[toProveThem-nontang\]) and hence (\[localEst-appdx-nonOblique-est\]) for the original problem. Theorem \[locEstElliptEq-non-oblique\] is proved. Now we show that the estimates also hold for the Dirichlet problem.
\[locEstElliptEq-Dirichlet\] Let $\lambda>0$ and $\alpha\in (0,1)$. Let $\Phi\in
C^{2,\alpha}({ {\bf R}})$ satisfy [(\[bdryNormCond\])]{} and $\Omega_R^+:=B_R\cap\{y>\Phi(x)\}$ for $R>0$. Let $u\in
C^2(\Omega_2^+)\cap C(\overline{\Omega_2^+})$ satisfy [(\[nonlinEq-appdx\])]{} in $\Omega_2^+$ and $$\begin{aligned}
&&u=g\qquad
\mbox{ on }\; \Gamma_\Phi{:=}B_2\cap\{y=\Phi(x)\},
\label{nonlinEq-appdx-Dirichlet-bc}\end{aligned}$$ where $A_{ij}=A_{ij}(Du, x,y)$ and $A_{i}=A_{i}(Du, x, y)$, $i,j=1,2$, and $f=f(x,y)$ satisfy [(\[locEstElliptEq-i1\])]{}–[(\[locEstElliptEq-i2\])]{}, and $g=g(x,y)$ satisfies $$\label{Rhs-Dirichlet}
\|g\|_{C^{2,\alpha}(\overline{\Omega_2^+})}\le \lambda^{-1},$$ with $(\lambda, \alpha)$ defined above. Assume that $\|u\|_{C(\Omega_2^+)}\le M$. Then $$\label{localEst-appdx-Dirichlet-est}
\|u\|_{C^{2,\alpha}(\overline{\Omega_1^+})}\le C(\lambda, M)
(\|u\|_{C(\overline{\Omega_2^+})}+\|f\|_{C^{\alpha}(\overline{\Omega_2^+})}
+\|g\|_{C^{2,\alpha}(\overline{\Omega_2^+})}).$$
By replacing $u$ with $u-g$, we can assume without loss of generality that $g\equiv 0$. Also, by flattening the boundary as in the proof of Theorem \[locEstElliptEq-non-oblique\], we can assume $\Phi\equiv 0$. That is, we have reduced to the case when (\[nonlinEq-appdx\]) holds in $B_2^+$ and $u=0$ on ${ \Sigma}_2$. Thus, $u_x=0$ on ${ \Sigma}_2$. Then estimate follows from Theorem \[locEstElliptEq-non-oblique\]. We now derive the estimates for the oblique derivative problem.
\[locEstElliptEq-oblique\] Let $\lambda>0$ and $\alpha\in (0,1)$. Let $\Phi\in
C^{2,\alpha}({ {\bf R}})$ satisfy [(\[bdryNormCond\])]{} and $\Omega_R^+:=B_R\cap\{y>\Phi(x)\}$ for $R>0$. Let $u\in
C^2(\Omega_2^+)\cap C^1(\overline{\Omega_2^+})$ satisfy $$\begin{aligned}
\label{nonlinEq-appdx-Obl-eq}
&&A_{11}u_{xx}+2A_{12}u_{xy}+A_{22}u_{yy}+
A_{1}u_{x}+A_{2}u_{y}=0 \qquad \mbox{ in }\;\Omega_2^+,\\
&&b_1u_{x}+b_2u_{y}+c u=0\qquad \mbox{ on }\; \Gamma_\Phi{:=}B_2\cap\{y=\Phi(x)\}, \label{nonlinEq-appdx-Obl-bc}\end{aligned}$$ where $A_{ij}=A_{ij}(Du, x,y)$ and $A_{i}=A_{i}(Du, x, y)$, $i,j=1,2$, satisfy [(\[locEstElliptEq-i1\])]{}–[(\[locEstElliptEq-i2\])]{}, and $b_i=b_i(x,y), i=1,2,$ and $c=c(x,y)$ satisfy the following obliqueness condition and $C^{1,\alpha}$–bounds: $$\begin{aligned}
\label{obliqueness-appdx}
&&b_2(x,y)\ge\lambda\qquad\mbox{for }\;(x,y)\in \Gamma_\Phi,
\\
\label{bdryCoefBds-obliq} &&\|(b_1, b_2,
c)\|_{C^{1,\alpha}(\overline{\Omega_2^+})}\le \lambda^{-1}.\end{aligned}$$ Assume that $\|u\|_{C(\overline{\Omega_2^+})}\le M$. Then there exists $C=C(\lambda, M, \alpha)>0$ such that $$\label{localEst-appdx-Oblique-est}
\|u\|_{C^{2,\alpha}(\overline{\Omega_1^+})}\le
C\|u\|_{C(\overline{\Omega_2^+})}.
$$
[*Step 1.*]{} First, we flatten the boundary $\Gamma_\Phi$ by the change of coordinates $(X,Y)=\Psi(x,y)=(x, y-\Phi(x))$. Then $(x,y)=\Psi^{-1}(X,Y)=(X, Y+\Phi(X))$. From (\[bdryNormCond\]), $
\|\Psi\|_{C^{2,\alpha}(\Omega_2^+)}+\|\Psi^{-1}\|_{C^{2,\alpha}({{\mathcal D}}_2^+)}\le
C(\lambda),$ where ${{\mathcal D}}_2^+{:=}\Psi(\Omega_2^+)$ satisfies $ {{\mathcal D}}_2^+\subset { {\bf R}}^2_+{:=}\{Y>0\}
$ and $
\Gamma_0{:=}\partial
{{\mathcal D}}_2^+\cap\{Y=0\}=\Psi(\Gamma_\Phi).
$ By a standard calculation, $v(X,Y)=u(x,y):=u(\Psi^{-1}(X,Y))$ satisfies the equation of form (\[nonlinEq-appdx-Obl-eq\]) in ${{\mathcal D}}_2^+$ and the oblique derivative condition of form (\[nonlinEq-appdx-Obl-bc\]) on $\Gamma_0$, where (\[locEstElliptEq-i1\])–(\[locEstElliptEq-i2\]) and (\[obliqueness-appdx\])–(\[bdryCoefBds-obliq\]) are satisfied with modified constant $\hat \lambda>0$ depending only on $\lambda$. Also, $\|v\|_{C({{\mathcal D}}_2^+)}\le M$. Thus, (\[localEst-appdx-Oblique-est\]) follows from $$\label{partIntSeminorm-est-obl}
\|v\|_{2,\alpha, {{\mathcal D}}_2^+\cup\Gamma_0}^*\le C(\lambda,
M,\alpha)\|v\|_{0,{{\mathcal D}}_2^+}.$$
Next we note that, in order to prove (\[partIntSeminorm-est-obl\]), it suffices to prove that there exist $K$ and $C$ depending only on $(\lambda, M, \alpha)$ such that, if $v$ satisfies (\[nonlinEq-appdx-Obl-eq\])–(\[nonlinEq-appdx-Obl-bc\]) in $B^+_1$ and ${ \Sigma}_1{:=}B_1\cap\{y=0\}$ respectively, (\[locEstElliptEq-i1\])–(\[locEstElliptEq-i2\]) and (\[obliqueness-appdx\])–(\[bdryCoefBds-obliq\]) hold in $B^+_1$, and $|v|\le M$ in $B_1^+$, then $$\label{appdx-Oblique-est-scaled-localized}
\|v\|_{C^{2,\alpha}(\overline{B_{1/K}^+})}\le C\|v\|_{C(B_1^+)}.$$ Indeed, if (\[appdx-Oblique-est-scaled-localized\]) is proved, then, using also the interior estimates (\[localEst-appdx-2\]) in Theorem \[locEstElliptEq\] and applying the scaling argument similar to the proof of Lemma \[appDxnonObl-lemma\], we obtain that, for any $z_0\in {{\mathcal D}}_2^+\cup { \Sigma}_2$, $$d_{z_0}^{2+\alpha}\|v\|_{C^{2,\alpha}(\overline{B_{d_{z_0}/(16K)}({z_0})\cap
{{\mathcal D}}_2^+})}\le C\|v\|_{C(B_{d_{z_0}/2}({z_0})\cap {{\mathcal D}}_2^+)}.$$ From this, we use the argument of the proof of Lemma \[partIntSeminorm-est-lemma\] to obtain (\[partIntSeminorm-est-obl\]).
Thus it remains to show (\[appdx-Oblique-est-scaled-localized\]). First we make a linear change of variables to normalize the problem so that $$\label{normalize-obliq-cond}
b_1(0)=0,\quad b_2(0)=1$$ for the modified problem. Let $$(X,Y)=\tilde\Psi(x,y):={1\over
b_2(0)}(b_2(0)x-b_1(0)y,y).$$ Then $$(x,y)=\tilde\Psi^{-1}(X,Y)=(X+b _1(0)Y, b_2(0)Y), \qquad
|D\tilde\Psi|+|D\tilde\Psi^{-1}|\le C(\lambda),$$ where the estimate follows from (\[obliqueness-appdx\])–(\[bdryCoefBds-obliq\]). Then the function $w(X,Y){:=}v(x,y)\equiv v(X+b _1(0)Y, b_2(0)Y)$ is a solution of the equation of form (\[nonlinEq-appdx-Obl-eq\]) in the domain $\tilde\Psi(B_1^+)$ and the boundary condition of form (\[nonlinEq-appdx-Obl-bc\]) on the boundary part $\tilde\Psi({ \Sigma}_1)$ such that (\[locEstElliptEq-i1\])–(\[locEstElliptEq-i2\]) and (\[obliqueness-appdx\])–(\[bdryCoefBds-obliq\]) are satisfied with constant $\hat \lambda>0$ depending only on $\lambda$, and (\[normalize-obliq-cond\]) holds, which can be verified by a straightforward calculation. Also, $\|w\|_{C(\tilde\Psi(B_1^+))}\le
M$.
Note that $\tilde\Psi(B_1^+)\subset { {\bf R}}^2_+:=\{Y>0\}$ and $\tilde\Psi({ \Sigma}_1)=\partial\tilde\Psi(B_1^+)\cap\{Y=0\}$. Moreover, since $|D\tilde\Psi|+|D\tilde\Psi^{-1}|\le C(\lambda)$, there exists $K_1=K_1(\lambda)>0$ such that, for any $r>0$, $
B_{r/K_1}\subset \tilde\Psi(B_r)\subset B_{K_1r}$. Thus it suffices to prove $$\|w\|_{C^{2,\alpha}(\overline{B_{r/2}^+})}\le C\|w\|_{C(B_r^+)}$$ for some $r\in (0, 1/K_1)$. This estimate implies (\[appdx-Oblique-est-scaled-localized\]) with $K=2K_1/r$.
As a result of the reduction performed in Step 1, it suffices to prove the following: There exist $\varepsilon\in (0, 1)$ and $C$ depending only on $(\lambda, \alpha, M)$ such that, if $u$ satisfies (\[nonlinEq-appdx-Obl-eq\]) and (\[nonlinEq-appdx-Obl-bc\]) in $B^+_{2\varepsilon}$ and on ${ \Sigma}_{2\varepsilon}$ respectively, if (\[locEstElliptEq-i1\])–(\[locEstElliptEq-i2\]) and (\[obliqueness-appdx\])–(\[bdryCoefBds-obliq\]) hold in $B^+_{2\varepsilon}$, and if (\[normalize-obliq-cond\]) holds and $\|u\|_{0,B_{2\varepsilon}^+}\le M$, then $$\|u\|_{2,\alpha,B_\varepsilon^+}\le C\|u\|_{0,B_{2\varepsilon}^+}.$$
We now prove this claim. For $\varepsilon>0$ to be chosen later, we rescale from $B_{2\varepsilon}^+$ into $B_2^+$ by defining $$\label{def-of-v-obliq}
v(x,y)={1\over \varepsilon}\big(u(\varepsilon x,\varepsilon
y)-u(0,0)\big)\qquad \mbox{for }\;(x,y)\in B_2^+.$$ Then $v$ satisfies $$\begin{aligned}
\label{nonlinEq-appdx-Obl-eq-rescaled}
&&\tilde A_{11}v_{xx}+2\tilde A_{12}v_{xy}+\tilde A_{22}v_{yy}+
\tilde A_{1}v_{x}+\tilde A_{2}v_{y}=\tilde f \qquad \mbox{ in }\;B_2^+,\\
&&v_{y}=\tilde b_1 v_{x}+ \tilde b_2 v_{y}+\tilde c v+\tilde c u(0,0)\qquad
\mbox{ on }\; { \Sigma}_2,
\label{nonlinEq-appdx-Obl-bc-rescaled}\end{aligned}$$ where $$\begin{aligned}
&&\tilde A_{ij}(p,x,y)=A_{ij}(p,\varepsilon x,\varepsilon y), \quad
\tilde A_{i}(p,x,y)=\varepsilon A_{i}(p,\varepsilon x,\varepsilon y),\\
&&\tilde b_1(x,y)=-b_1(\varepsilon x,\varepsilon y), \quad \tilde
b_2(x,y)=-b_2(\varepsilon x,\varepsilon y)+1, \quad \tilde
c(x,y)=-\varepsilon c(\varepsilon x,\varepsilon y).\end{aligned}$$ Then $\tilde A_{ij}$ and $\tilde A_i$ satisfy (\[locEstElliptEq-i1\])–(\[locEstElliptEq-i2\]) in $B_2^+$ and, using (\[bdryCoefBds-obliq\]), (\[normalize-obliq-cond\]), and $\varepsilon\le 1$, $$\label{bdryCoefBds-obliq-norm}
\|(\tilde b_1, \tilde b_2, \tilde c)\|_{1,\alpha, B_2^+}\le
C\varepsilon \qquad\text{for some } C=C(\lambda).$$
Now we follow the proof of Theorem \[locEstElliptEq-non-oblique\]. We use the partially interior norms [@GilbargTrudinger Eq. 4.29] in the domain $B_2^+\cup { \Sigma}_2$ whose distance function is $d_{z}={ \mbox{dist}}(z,\partial B_2^+\setminus { \Sigma}_2).$ We introduce the functions $w_i=D_iv$, $i=1,2$, to conclude from (\[nonlinEq-appdx-Obl-eq-rescaled\]) that $w_1$ and $w_2$ are weak solutions of equations $$\begin{aligned}
&&\qquad\quad D_1\big({\tilde A_{11}\over \tilde
A_{22}}D_1w_1+{2\tilde A_{12}\over \tilde A_{22}} D_2w_1\big)
+D_{22}w_1= -D_1\big({\tilde A_{1}\over \tilde A_{22}}D_1 v+{\tilde
A_{2}\over \tilde A_{22}}D_2 v\big), \label{eqnForDerivAppdx-1-obl}
\\
&&\qquad\quad D_{11}w_2+ D_2\big({2\tilde A_{12}\over \tilde
A_{11}}D_1w_2+{\tilde A_{22}\over \tilde A_{11}}D_2w_2\big) =
-D_2\big({\tilde A_{1}\over \tilde A_{11}}D_1 v+{\tilde A_{2}\over
\tilde A_{11}}D_2 v\big) \label{eqnForDerivAppdx-2-obl}\end{aligned}$$ in $B_2^+$, respectively. From (\[nonlinEq-appdx-Obl-bc-rescaled\]), we have $$\label{bc-w2}
w_2=\tilde g \qquad\mbox{ on } { \Sigma}_2,$$ where $\tilde g:=\tilde b_1 v_{x}+ \tilde b_2 v_{y}+\tilde c v+\tilde c
u(0,0)$ in $B_2^+$.
Using equation (\[eqnForDerivAppdx-2-obl\]) and the Dirichlet boundary condition (\[bc-w2\]) for $w_2$ and following the proof of Lemma \[appDxnonObl-lemma\], we can show the existence of $\beta\in (0,\alpha]$ and C depending only on $\lambda$ such that, for any $z_0\in B_2^+\cup { \Sigma}_2$, $$\label{scaledHolderAppdx-obl-w2}
d_{z_0}^\beta[w_2]_{0,\beta, B_{d_{z_0}/16}({z_0})\cap B_2^+}\le
C\big(\|Dv\|_{0,B_{d_{z_0}/2}({z_0})\cap B_2^+} +
d_{z_0}^\beta[\tilde g]_{0,\beta, B_{d_{z_0}/2}(z_0)\cap
B_2^+}\big).$$
Next we obtain the Hölder estimates of $Dv$ if $\varepsilon$ is sufficiently small. We first note that, by (\[bdryCoefBds-obliq-norm\]), $\tilde g$ satisfies $$\begin{aligned}
&&|D\tilde g|\le C\varepsilon(|D^2v|+|Dv|+
|v|+
\|u\|_{0,B_{2\varepsilon}^+})\qquad
\mbox{in }\;B_2^+,
\label{estimateG-obl-appdx-1}\\
&&[\tilde g]_{0,\beta, B_{d_z/2}(z)\cap{{\mathcal D}}_2^+}
\le C\varepsilon ( \|v\|_{1,\beta,B_{d_z/2}(z)\cap{{\mathcal D}}_2^+)} +
\|u\|_{0,B_{2\varepsilon}^+})
\qquad\quad\label{estimateG-obl-appdx-2}\end{aligned}$$ for $C=C(\lambda)$. The term $\varepsilon
\|u\|_{0,B_{2\varepsilon}^+}$ in (\[estimateG-obl-appdx-1\])–(\[estimateG-obl-appdx-2\]) comes from the term $\tilde c u(0,0)$ in the definition of $\tilde{g}$. We follow the proof of Lemma \[appDxnonObl-DuHolder-lemma\], but we now use the integral form of equation (\[eqnForDerivAppdx-2-obl\]) with test functions $\zeta=\eta^2(w_2-\tilde g)$ and $\zeta=\eta^2(w_2-w_2(\hat z))$ to get an integral estimate of $|Dw_2|$ and thus of $|D_{ij}v|$ for $i+j>2$, and then use (\[nonlinEq-appdx-Obl-eq-rescaled\]) to estimate the remaining derivative $D_{11}v$. In these estimates, we use (\[scaledHolderAppdx-obl-w2\])–(\[estimateG-obl-appdx-2\]). We obtain that, for sufficiently small $\varepsilon$ depending only on $\lambda$, $$\label{scaledHolderAppdx-fullDu-obliq}
\begin{array}{ll}
\displaystyle d_{z_0}^\beta[Dv]_{0,\beta,
B_{d_{z_0}/32}({z_0})\cap
B_2^+}\\
\le C\big(\|v\|_{C^1(B_{d_{z_0}/2}({z_0})\cap B_2^+)} +\varepsilon
d_{z_0}^\beta[Dv]_{0,\beta, B_{d_{z_0}/2}(z_0)\cap {{\mathcal D}}_2^+}
+\varepsilon d_{z_0}^\beta\|u\|_{0,B_{2\varepsilon}^+}\big)
\end{array}$$ for any $z_0\in B_2^+\cup { \Sigma}_2$, with $C=C(\lambda)$. Using (\[scaledHolderAppdx-fullDu-obliq\]), we follow the proof of Lemma \[partIntSeminorm-est-lemma\] to obtain $$\label{partIntSeminorm-est-obliq}
[v]_{1,\beta, B_2^+\cup { \Sigma}_2}^*\le C\big(\|v\|_{1,0,B_2^+\cup
{ \Sigma}_2}^* +\varepsilon [v]_{1,\beta, B_2^+\cup { \Sigma}_2}^*
+\varepsilon \|u\|_{0,B_{2\varepsilon}^+}\big).$$ Now we choose sufficiently small $\varepsilon>0$ depending only on $\lambda$ to have $$[v]_{1,\beta, B_2^+\cup { \Sigma}_2}^*\le
C(\lambda)(\|v\|_{1,0,B_2^+\cup { \Sigma}_2}^* +
\|u\|_{0,B_{2\varepsilon}^+}).$$ Then we use the interpolation inequality, similar to the proof of (\[partIntSeminorm-est-2\]), to have $$\label{partIntSeminorm-est-2-obl}
\|v\|_{1,\beta, B_2^+\cup { \Sigma}_2}^*\le
C(\lambda)(\|v\|_{0,B_2^+}+\|u\|_{0,B_{2\varepsilon}^+}).$$ By (\[def-of-v-obliq\]) with $\varepsilon=\varepsilon(\lambda)$ chosen above, (\[partIntSeminorm-est-2-obl\]) implies $$\label{partIntSeminorm-est-3-obl}
\|u\|_{1,\beta, B_{2\varepsilon}^+\cup B_{2\varepsilon}^0}^*\le
C(\lambda)\|u\|_{0,B_{2\varepsilon}^+}.$$ Then problem (\[nonlinEq-appdx-Obl-eq\])–(\[nonlinEq-appdx-Obl-bc\]) can be regarded as a linear oblique derivative problem in $B_{7\varepsilon/4}^+$ whose coefficients $a_{ij}(x,y):=A_{ij}(Du(x,y),x,y)$ and $a_{i}(x,y)$ $:=A_{i}(Du(x,y),x,y)$ have the estimate in $C^{0,\beta}(\overline{B_{7\varepsilon/4}^+})$ by a constant depending only on $(\lambda, M)$ from (\[partIntSeminorm-est-3-obl\]) and (\[locEstElliptEq-i2\]). Moreover, we can assume $\beta\le\alpha$ so that (\[bdryCoefBds-obliq\]) implies the estimates of $(b_i, c)$ in $C^{1,\beta}(\overline{B_{7\varepsilon/4}^+})$ with $\varepsilon=\varepsilon(\lambda)$. Then the standard estimates for linear oblique derivative problems [@GilbargTrudinger Lemma 6.29] imply $$\label{partIntSeminorm-est-4-obl}
\|u\|_{2,\beta,B_{3\varepsilon/2}^+}\le
C(\lambda,M)\|u\|_{0,B_{7\varepsilon/4}^+}.$$ In particular, the $C^{0,\alpha}(\overline{B_{3\varepsilon/2}^+})$–norms of the coefficients $(a_{ij}, a_i)$ of the linear equation (\[nonlinEq-appdx-Obl-eq\]) are bounded by a constant depending only on $(\lambda, M)$, which implies $$\|u\|_{2,\alpha,B_{\varepsilon}^+}\le C(\lambda,
M)\|u\|_{0,B_{3\varepsilon/2}^+},$$ by applying again [@GilbargTrudinger Lemma 6.29]. This implies the assertion of Step 2, thus Theorem \[locEstElliptEq-oblique\]. [**Acknowledgments**]{}. Gui-Qiang Chen’s research was supported in part by the National Science Foundation under Grants DMS-0505473, DMS-0244473, and an Alexander von Humboldt Foundation Fellowship. Mikhail Feldman’s research was supported in part by the National Science Foundation under Grants DMS-0500722 and DMS-0354729.
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Introduction
============
The photon dispersion relation in astrophysical plasmas is usually dominated by the electromagnetic interaction with the electrons of the medium. It was recently claimed [@MS96], however, that in a supernova (SN) core the photon interaction with the magnetic moments of the nucleons yields the dominant contribution to the refractive index $n_{\rm refr}$. Because the new contribution has the opposite sign of the usual plasma term, the photon four-momentum $K$ would actually become space-like, allowing for the Cherenkov processes $\gamma\nu\to\nu$ and $\nu\to\nu\gamma$ which could be of great importance for the neutrino opacities. Due to a numerical error in Ref. [@MS96] the overall magnitude of the nucleon magnetic-moment effect is in fact much smaller [@Raffelt97], but even after this correction it is not very much smaller than the electron plasma effect and thus deserves a closer look.
It is surprising, at first, that the nucleon magnetic moments contribute at all to the refractive index because the photon forward-scattering amplitude on fermions with a magnetic moment is identically zero. Most recently the photon polarization tensor for an ensemble of noninteracting spin-$\frac{1}{2}$ particles was calculated in great detail [@ON97] and it was found that indeed the magnetic moments alone do not produce any contribution to $n_{\rm refr}$. However, the underlying assumption of a collisionless system is far from justified in a SN core where the nucleon spin-spin interaction plays a dominant role. For photon frequencies below the nucleon spin relaxation rate $\Gamma_\sigma$ the hydrodynamic limit is the physically appropriate description (not the collisionless limit) justifying the use of the Pauli susceptibility in Ref. [@MS96]. In a SN core the spin relaxation rate is likely to be of the same order as the temperature $T$ [@JKRS96], typical photon frequencies are also of that order, so that for the entire spectrum of relevant photon frequencies neither the hydrodynamic nor the collisionless limits are truly justified. Therefore, an understanding of the photon refractive index and its frequency dependence requires a more general analysis than has been offered in either Ref. [@MS96] or [@ON97].
Perhaps the easiest way to appreciate this point is to consider photon absorption. In a collisionless system of neutral fermions (“neutrons”) with a magnetic moment $\mu_n$ the refractive index is $n_{\rm refr}=1$ up to order $\mu_n^2$ which implies that there is no Landau damping, i.e. no Cherenkov effect $\gamma n\leftrightarrow n$. The only photon damping occurs at order $\mu_n^4$ from magnetic Compton scattering $\gamma n\to n\gamma$. On the other hand, if our “neutrons” interact by a spin-dependent force (which for real neutrons is caused by pion exchange) we have the inverse-bremsstrahlung absorption process $\gamma n n\to nn$ so that we do have photon absorption to order $\mu_n^2$. Its rate far exceeds that of magnetic Compton scattering because in a SN core the $nn$ interaction rate is large. By virtue of the Kramers-Kronig relation one can then derive the associated refractive index $n_{\rm
refr}$ which does not vanish to order $\mu_n^2$. (Note that we always take $n_{\rm refr}$ to be a real quantity even though one sometimes describes absorption by an “imaginary part of the refractive index.”)
We proceed in Sec. \[sec:genrelations\] with the general photon dispersion relation in a pure neutron medium in terms of the dynamical spin-density structure function by virtue of the fluctuation-dissipation theorem and the Kramers-Kronig relation. In Sec. \[sec:semiheur\] we use a semi-heuristic expression for the dynamical spin-density structure function in the long-wavelength limit to obtain a quantitative estimate of the magnetic-moment refractive index in a SN core. In Sec. \[sec:summary\] we discuss and summarize our findings.
General Relations {#sec:genrelations}
=================
Photon Dispersion
-----------------
The main idea behind our treatment of the photon dispersion in a neutron medium is the observation that the photon absorption rate $\Gamma_{\rm abs}$ is dominated by the inverse-bremsstrahlung process $\gamma nn \to nn$ which is enabled by the tensor component of the pion-exchange force between neutrons. From the absorption rate we can determine the refractive index $n_{\rm refr}$ with the help of a Kramers-Kronig relation. The inverse-bremsstrahlung process itself can be calculated easily by the usual perturbative methods. However, a SN core is so dense and hot that these methods are not obviously justified [@JKRS96]. Therefore, it is more useful to begin with a discussion of photon absorption in the language of linear-response theory which allows us to identify more general properties of the photon refractive index than would be apparent on the perturbative level. In order to apply our general results to a SN core we will then, of course, have to take recourse to a semi-heuristic perturbative approach to $nn$ interactions (Sec. \[sec:semiheur\]).
Photon dispersion is caused by the medium’s response to applied electromagnetic fields. In the homogeneous and stationary case all relevant information is contained in the polarization tensor $\Pi^{\mu\nu}(K)$ where $K=(\omega,{\bf k})$ is the frequency and wavevector of the applied electromagnetic perturbation. If the medium is isotropic and parity conserving the polarization tensor is uniquely characterized by a pair of two response functions which are often chosen to be the longitudinal and transverse polarization functions $\pi_L=(1-\omega^2/k^2)\Pi^{00}$ and $\pi_T=\frac{1}{2}({\rm
Tr}\,\Pi-\pi_L)$ with $k=|{\bf k}|$ the wavenumber of the perturbation. Because of the assumed isotropy all quantities depend only on the wavenumber $k$, not on the direction of ${\bf k}$. The dispersion relation of propagating modes is determined by $\omega^2-k^2=\pi_{T,L}(\omega,k)$.
Another pair of medium response function is the dielectric permittivity $\epsilon$ and the magnetic permeability $\mu$ which give us the displacement ${\bf D}=\epsilon{\bf E}$ caused by an applied electric field and the magnetic field ${\bf H}=\mu^{-1}{\bf B}$ caused by an applied magnetic induction. However, the magnetic field ${\bf
H}$ and the transverse part of the displacement, characterized by ${\bf k}\cdot {\bf D}_T=0$, do not have independent meaning [@Kirzhnits87]. Therefore, among other possibilities one may equally well choose ${\bf H}={\bf B}$, ${\bf D}_T=\epsilon_T{\bf
E}_T$, and ${\bf D}_L=\epsilon_L{\bf E}_L$ with $\epsilon_L\equiv
\epsilon$ the longitudinal and $\epsilon_T\equiv\epsilon+(1-\mu^{-1})\,k^2/\omega^2$ the transverse dielectric permittivity. They are related to the polarization functions by $\epsilon_L=1-\pi_L/(\omega^2-k^2)$ and $\epsilon_T=1-\pi_T/\omega^2$ [@Weldon82]. In this language the dispersion relations take on their standard form $\epsilon_L(\omega,k)=0$ and $\omega^2\epsilon_T(\omega,k)=k^2$ [@Sitenko67].
We are presently only concerned with the dispersion relation of transverse modes (“photons”) because a medium consisting of magnetic dipoles is not expected to support longitudinal modes (longitudinal plasmons). From the above it follows immediately that the photon dispersion relation can be written in the form $$\label{eq:classicaldisp}
\frac{k^2}{\omega^2}=\epsilon(\omega,k)\,\mu(\omega,k).$$ With the usual definition of the photon refractive index $$\label{eq:defn}
n_{\rm refr} \equiv \frac{k}{\omega}$$ we arrive at the classical result $n^2_{\rm
refr}=\epsilon\mu$ [@Jackson]. It follows that the refractive index must be determined self-consistently as a solution of $$n_{\rm refr}^2(\omega,k)=\epsilon(\omega,k)\,\mu(\omega,k)$$ with $k=n_{\rm refr}\omega$ for any frequency $\omega$ of a propagating mode. Depending on the properties of the medium the long-wavelength approximation $\epsilon(\omega,k)\approx\epsilon(\omega,0)$ and $\mu(\omega,k)\approx\mu(\omega,0)$ may be justified, leading to the much simpler dispersion relation $n_{\rm
refr}^2(\omega)=\epsilon(\omega,0)\,\mu(\omega,0)$.
Sometimes it will be more useful to write the photon dispersion relation in the form $\omega^2-k^2=m_{\rm eff}^2$ in terms of a frequency-dependent “effective mass” $$\label{eq:effmass}
m_{\rm eff}^2(\omega)=(1-n_{\rm refr}^2)\,\omega^2,$$ where in fact $m_{\rm eff}^2<0$ is possible. For electric interactions and frequencies well above all resonances we obtain the well-known plasma effect dispersion relation which implies that $m_{\rm eff}^2>0$ and independent of frequency [@Jackson]. We will show that the same holds true for our magnetic case. Therefore, it is useful to define $$m_\gamma\equiv \lim_{\omega\to\infty} m_{\rm eff}(\omega)$$ as the (transverse) “photon mass” in the medium.
We will mostly be concerned with a medium of neutrons which interact with the electromagnetic field by their magnetic dipole moment. In the nonrelativistic limit they do not respond at all to an applied electric field so that we may use the approximation $\epsilon=1$. The magnetic permeability can be written as $\mu=1+\chi$ in terms of the magnetic susceptibility $\chi$. (We use rationalized units where $\alpha=e^2/4\pi \approx 1/137$ or else we would have to write $\mu=1+4\pi\chi$ [@Jackson].)
In general the magnetic susceptibility is a complex function of the real variables $\omega$ and $k$. Following common practice we write it in the form $$\chi(\omega,k) = \chi'(\omega,k) + i\,\chi''(\omega,k)$$ in terms of its real and imaginary parts. It is found that that $\chi''$ is an odd function of $\omega$ while $\chi'$ is even [@Forster]. Because we have defined the refractive index to be a real quantity the dispersion relation is $$\label{eq:nchi}
n_{\rm refr}^2(\omega,k) - 1 = \chi'(\omega,k)$$ with $k=n_{\rm refr}\omega$.
The imaginary part of the susceptibility describes dissipation: Usually one pictures a stationary beam of frequency $\omega$ along the $z$-direction which is characterized by a (real) wavenumber $k=n_{\rm
refr}\omega$ and a damping wavenumber $\kappa=\frac{1}{2}
\lambda^{-1}$ with $\lambda$ the photon mean free path. The amplitude of this beam varies as $e^{-i(\omega t - k z)-\kappa z}$, its intensity as $e^{-2\kappa z}=e^{-z/\lambda}$. The relativistic limit $|n_{\rm refr}-1| \ll 1$ implies $n_{\rm refr}^2-1=(n_{\rm
refr}+1)(n_{\rm refr}-1) \approx 2(n_{\rm refr}-1)$ or $n_{\rm
refr}-1\approx \frac{1}{2}\chi$. Therefore, one can picture $\frac{1}{2}\chi''$ to be an “imaginary part of the refractive index”, leading to the identification $\kappa=
\frac{1}{2}\chi''\omega$ or $\chi''=(\lambda\omega)^{-1}$.
We stress that at finite temperature this simple interpretation is not complete because the medium can both absorb and spontaneously emit photons. The two processes are related by the usual detailed-balance factor $e^{-\omega/T}$. What is actually damped is not a mode $k$ of the electromagnetic field, but rather the deviation of its occupation number from a thermal distribution. It is easy to show that this damping occurs at a rate $1-e^{-\omega/T}$ times the “naive” absorption rate $\Gamma_{\rm abs}$ [@Weldon83]. Therefore, the appropriate interpretation of the imaginary part of the susceptibility is $$\chi''(\omega,k) = \frac{1}{\omega}\left(1-e^{-\omega/T}\right)\,
\Gamma_{\rm abs}(\omega)$$ with $k=n_{\rm refr}\omega$. In the limit $|n_{\rm refr}-1|\ll 1$ the “naive” absorption rate is $\Gamma_{\rm abs}=\lambda^{-1}$; it is given by the standard formula “absorption cross section times target density.”
Fluctuation-Dissipation Theorem
-------------------------------
To lowest order the neutrons can absorb photons only because they interact by a spin dependent force which enables the inverse-bremsstrahlung process $\gamma nn\to nn$. At the same time this spin-dependent force causes the neutron spins to fluctuate. The relationship between spin fluctuations and the absorptive part of the spin susceptibility is encapsuled in the fluctuation-dissipation theorem which will help us to understand some general properties of the photon refractive index.
In our case the most useful quantity to describe the neutron spin fluctuations is the dynamical spin-density structure function. Following the normalization convention of Ref. [@JKRS96] it is defined as $$S_{\sigma}(\omega,{\bf k}) = \frac{4}{3 n_n}\int_{-\infty}^{+\infty}
dt\, e^{i\omega t}\langle
\bbox{\sigma}(t,{\bf k})\cdot\bbox{\sigma}(0,-{\bf k}) \rangle,
\label{eq:defofS}$$ where $\bbox{\sigma}(t,{\bf k})$ is the spatial Fourier transform of the neutron spin-density operator $\bbox{\sigma}(t,{\bf r}) =
\frac{1}{2} \psi^{\dagger}(t,{\bf r}) \bbox{\tau} \psi(t,{\bf r})$. Here $\psi(t,{\bf r})$ is a Pauli two-spinor describing the nucleon field and $\bbox{\tau}$ is a vector of Pauli matrices. Further, $n_n$ is the neutron number density and $\langle\ldots\rangle$ denotes a thermal ensemble average. Of course, in an isotropic system the structure function depends only on $k=|{\bf k}|$. The normalization was chosen such that $$\label{eq:Snorm}
\int_{-\infty}^{+\infty} \frac{d\omega}{2\pi}\,S_{\sigma}(\omega,0)
= 1$$ for a case where there are no static spin-spin correlations between different neutrons which are taken to be nondegenerate. In the limit of vanishing spin-spin interactions we have $$\label{eq:Sdelta}
S_{\sigma}(\omega,0)\to2\pi\delta(\omega).$$ Moreover, it satisfies $$S_{\sigma}(-\omega,-{\bf k}) =
e^{-\omega/T}S_{\sigma}(\omega,{\bf k})
\label{eq:Sdetbalcond}$$ as required by the principle of detailed balance.
We next observe that the operator for the magnetization density for neutrons is ${\bf M}=2\mu_n\bbox{\sigma}$ where the factor 2 is the gyromagnetic ratio for a spin-$\frac{1}{2}$ particle and $\mu_n$ is the neutron magnetic moment, not to be confused with the magnetic permeability $\mu$ of the previous section. A relationship between the dissipative part of $\chi$ and spontaneous spin fluctuations can then be written in the form [@Forster] $$\chi''(\omega,{\bf k})=
\frac{1}{2}\int_{-\infty}^{+\infty}
dt\, e^{i\omega t}\biggl\langle
\frac{1}{3}\sum_{i=1}^3\bigl[M_i(t,{\bf k}),M_i(0,-{\bf k})\bigr]
\biggr\rangle,$$ where $[\,\cdot\,{,}\,\cdot\,]$ is the usual commutator. Comparing this with Eqs. (\[eq:defofS\]) and (\[eq:Sdetbalcond\]) reveals that this relationship is equivalent to $$\chi''(\omega,{\bf k}) = \frac{1}{2}\mu_n^2 n_n
\left(1 - e^{-\omega/T}\right)\,S_{\sigma}(\omega,{\bf k}).
\label{eq:flucdiss}$$ In this form it is known as the fluctuation-dissipation theorem [@Forster; @Pines].
Kramers-Kronig Relation
-----------------------
Once the imaginary part of the magnetic susceptibility is known the real part can be found by virtue of the well-known Kramers-Kronig relation $$\chi'(\omega,k) = {\rm P}\!\int_{-\infty}^{+\infty}
\frac{d\tilde\omega}{\pi}\,
\frac{\chi''(\tilde\omega,k)}{\tilde\omega - \omega}
\label{eq:KramersKronigchi}$$ where ${\rm P}$ denotes a Cauchy principal value integral. With the help of the fluctuation-dissipation theorem Eq. (\[eq:flucdiss\]) we find a direct relationship between the dispersive part of the magnetic susceptibility and the spin-density structure function $$\chi'(\omega,k) = 2 \mu_n^2 n_n\, {\rm P}\!
\int_{-\infty}^{+\infty}
\frac{d\tilde\omega}{2\pi}\,
\frac{\tilde\omega\,S_\sigma(\tilde\omega,k)}
{{\tilde\omega}^2 - \omega^2}.
\label{eq:chiSigma}$$ One may use detailed balance to write this in the form $$\chi'(\omega,k)=\chi_{\rm Pauli}\,{\rm P}\!
\int_{0}^{\infty}
\frac{d\tilde\omega}{\pi}\,
\frac{1-e^{\tilde\omega/T}}{\tilde\omega/T}\,
\frac{{\tilde\omega}^2\,S_\sigma(\tilde\omega,k)}
{{\tilde\omega}^2 - \omega^2},
\label{eq:chiSigmaB}$$ where $$\chi_{\rm Pauli}\equiv\frac{\mu_n^2 n_n}{T}$$ is the usual Pauli susceptibility for a system of collisionless spin-$\frac{1}{2}$ particles with a magnetic moment $\mu_n$.
Limiting Cases {#sec:limitingcases}
--------------
In order to understand the general behavior of the refractive index we begin with the static limit $\omega \rightarrow 0$. The static susceptibility $\chi_0(k)\equiv \chi'(0,k)$ has no imaginary part because $\chi''$ is an odd function of $\omega$. From Eq. (\[eq:chiSigmaB\]) we find for the real part $$\label{eq:staticsus}
\chi_0(k) = \chi_{\rm Pauli}\,
\int_{0}^{\infty}
\frac{d\tilde\omega}{\pi}\,
\frac{1-e^{-\tilde\omega/T}}{\tilde\omega/T}\,
S_{\sigma}(\tilde\omega,k).$$ In the collisionless limit the structure function becomes narrowly peaked around $\omega=0$. With the help of Eq. (\[eq:Sdelta\]) we thus recover the usual long-wavelength result result $\chi_0(0)=\chi_{\rm Pauli}$. When $S_\sigma(\omega,k)$ is not narrowly peaked on scales of the temperature, the static susceptibility decreases relative to the Pauli value—we show this effect explicitly in Fig. \[fig:static\] below in the framework of a heuristic toy model.
How large may the frequencies be that the static result is still approximately justified? The structure function in the long-wavelength limit $S_\sigma(\omega,0)$ has the interpretation of the autocorrelation function of a single neutron spin. Therefore, it is a broad, decreasing function of $\omega$ with a width representing something like the spin-relaxation or spin-fluctuation rate $\Gamma_\sigma$. If the external electromagnetic perturbation has a frequency much less than this, $\omega\ll\Gamma_\sigma$, we are in the hydrodynamic limit where the neutron spins may fully relax to a new thermodynamic equilibrium state on the time scale of a period of the perturbation. In this case we may use the static susceptibility to estimate the photon refractive index. Moreover, even though we just saw that the static susceptibility is not independent of the width of $S_\sigma(\omega)$, this dependence is weak so that in the hydrodynamic limit the Pauli susceptibility is a good estimate, justifying the approach of Ref. [@MS96] to photon dispersion in the limit $\omega\ll\Gamma_\sigma$.
The opposite limiting case is that of very large $\omega$. If $S_\sigma(\omega)$ falls off sufficiently fast beyond some frequency $\omega_0$ which is determined by the nature of the $nn$ interaction potential, then for $\omega\gg\omega_0$ the integral in Eq. (\[eq:chiSigma\]) is dominated by $|\tilde\omega|\alt\omega_0$, leading to $$\chi'(\omega,k)
=-2\,\frac{\mu_n^2 n_n}{\omega^2}
\int_{-\infty}^{+\infty} \frac{d\tilde\omega}{2\pi}\,
\tilde\omega\,S_\sigma(\tilde\omega,k).\label{eq:fsumintro}$$ The integral in this equation is the so-called $f$-sum of the structure function. Independently of the nature of the assumed $nn$ interaction the $f$-sum always exists and is given as a thermal expectation value of the tensor part of the $nn$ interaction potential [@Sigl]. Moreover, the $f$-sum is always positive because of the detailed-balance property Eq. (\[eq:Sdetbalcond\]).
For photon dispersion, this result corresponds to a positive value for the squared effective mass defined in Eq. (\[eq:effmass\]). With Eq. (\[eq:nchi\]) we find $$\label{eq:photonmass}
m_{\rm eff}^2 = 2 \mu_n^2 n_n
\int_{-\infty}^{+\infty} \frac{d\tilde\omega}{2\pi}\,
\tilde\omega\,S_\sigma(\tilde\omega,k).$$ If the momentum dependence of this expression is weak so that we may use the long-wavelength limit then the photon dispersion relation is that of a massive particle $\omega^2-k^2=m_\gamma^2$ with the transverse photon mass given by Eq. (\[eq:photonmass\]) with $k=0$ on the right-hand side. The appearance of this form has the same cause as in the usual plasma case, i.e. $n_{\rm refr}$ is given by the $f$-sum of the relevant dynamical structure functions.
The Pauli susceptibility is a positive number (the neutrons are a paramagnetic medium) so that in the hydrodynamic limit the photon dispersion relation is approximately characterized by $n_{\rm
refr}^2-1=\chi_{\rm Pauli}$ or $m_{\rm eff}^2=-\chi_{\rm
Pauli}\omega^2<0$. On the other hand in the large-frequency limit we have $m_{\rm eff}^2>0$ as given in Eq. (\[eq:photonmass\]). Moreover, on dimensional grounds the $f$-sum must take on the approximate value $\Gamma_\sigma$. Therefore, $$m_{\rm eff}^2\approx\chi_{\rm Pauli}
\times\cases{-\,\omega^2&for $\omega\ll\Gamma_\sigma$\cr
+\,T\Gamma_\sigma&for $\omega\gg\Gamma_\sigma$\cr}$$ gives us a rough picture of the behavior of the photon dispersion relation in a medium of neutron spins.
Semi-Heuristic Model {#sec:semiheur}
====================
In a SN core neither the collisionless nor the hydrodynamic limits are appropriate so that we need to come up with a concrete expression for the dynamical spin-density structure function in order to estimate the photon refractive index. In a dilute medium one may use the usual perturbative methods to compute the processes $\gamma
nn\leftrightarrow nn$. Because the relevant photon energies are small compared with the neutron mass the momentum transfer of the radiation to the neutron system may be neglected, an approximation which amounts to the long-wavelength limit which we shall henceforth adopt with the notation $S_\sigma(\omega)\equiv\lim_{k\to 0} S_\sigma(\omega,k)$. Next, one may extract $S^{(1)}_\sigma(\omega)$, where the superscript indicates that this is a lowest-order perturbative result. Independently of the details of the assumed $nn$ interaction potential one finds the generic representation [@JKRS96] $$S^{(1)}_\sigma(\omega)=\frac{\Gamma_\sigma}{\omega^2}\,s(\omega/T)$$ where $s(x)$ with $x=\omega/T$ is a slowly varying function of order unity. This factorization is somewhat arbitrary; we define what we call the neutron “spin-fluctuation rate” $\Gamma_\sigma$ such that for nondegenerate neutrons $s(0)=1$. Moreover, we have $$\label{eq:sdetbal}
s(-x)=s(x)\,e^{-x}$$ so that the detailed-balance relation for $S_{\sigma}(\omega)$ Eq. (\[eq:Sdetbalcond\]) is satisfied.
The lowest-order perturbative representation $S^{(1)}_\sigma(\omega)$ diverges at $\omega=0$ and thus violates the normalization rule Eq. (\[eq:Snorm\]). However, including multiple-scattering effects suggests the “resummed” representation [@JKRS96] $$\label{eq:resummedS}
S_\sigma(\omega)=\frac{\Gamma_\sigma}{\omega^2+\Gamma^2/4}
\,s(\omega/T).$$ In a very dilute medium this function is strongly peaked around $\omega=0$ so that it approaches $2\pi\delta(\omega)$. In this limit we have $\Gamma=\Gamma_\sigma$, i.e. we approach the classical limit of a Lorentzian correlation function $S_\sigma(\omega)=\Gamma_\sigma/(\omega^2+\Gamma_\sigma^2/4)$. We stress that the representation Eq. (\[eq:resummedS\]) is completely general if we interpret $\Gamma$ as a function of $\omega$ which in linear-response theory is related to the neutron spin’s “memory function” [@Forster]. In our heuristic description, however, we will use a constant value for $\Gamma$ which is fixed by the normalization requirement Eq. (\[eq:Snorm\]).
In order to calculate $\Gamma_\sigma$ and $s(x)$ in a dilute neutron medium we model the $nn$ interaction by one-pion exchange in Born approximation, an approach which has been common practice for SN and neutron-star physics since Friman and Maxwell’s seminal paper [@FM79] and which is further justified in Ref. [@HR97]. Further, we take the neutrons to be nondegenerate which is not a bad approximation during the early phases of SN core cooling. Finally, we neglect the mass in the pion propagator which is also a reasonable approximation for the large momentum transfers in typical $nn$ collisions in a SN core. All of these approximations go in the same direction of somewhat overestimating the $nn$ spin interaction rate. We also ignore static spin-spin correlations which could, in principle, both enhance or diminish our results.
to
Within this framework the spin-fluctuation rate is explicitly found to be [@HR97; @RS95] $$\label{eq:gamsig}
\Gamma_\sigma=4\sqrt{\pi}\,\alpha_\pi^2 n_n T^{1/2} m_N^{-5/2},$$ where $\alpha_\pi\equiv (f2m_N/m_\pi)^2/4\pi \approx15$ with $f\approx
1$ is the pion-nucleon “fine-structure constant,” $n_n$ is the neutron density, and $m_N$ the nucleon mass. Numerically we find $$\label{eq:numericalgamma}
\gamma_\sigma\equiv\Gamma_\sigma/T
=8.6\,\rho_{14}\,T_{10}^{-1/2},$$ where $\rho_{14}\equiv\rho/10^{14}\,{\rm g\,cm^{-3}}$ and $T_{10}\equiv T/10\,{\rm MeV}$. Moreover, one finds [@HR97; @RS95] $$\begin{aligned}
\label{eq:sx}
s(x)&=&\int_{\max(0,-x)}^\infty dv\,\,e^{-v}
\biggl[\sqrt{v(v+x)}\nonumber\\
&&\hskip3em
-\,\frac{x^2}{2(2v+x)}\,
\log\left(\frac{\sqrt{v+x}+\sqrt{v}}{\sqrt{v+x}-\sqrt{v}}\right)
\biggr],\end{aligned}$$ an expression which indeed fulfills the detailed balance requirement Eq. (\[eq:sdetbal\]) and which is smooth at $x=0$ with the derivative $s'(0)=1/2$ (Fig. \[fig:sx\]). We will use a simple analytic approximation to this integral [@HR97] $$s(x)\Big|_{x\geq0}
\approx \left(\frac{x}{4\pi}+\left[1+\left(12+\frac{3}{\pi}\right)x
\right]^{-1/12}\right)^{-1/2}\label{eq:sapp}$$ which reproduces the correct limiting behavior for $x\gg 1$ and for $x=0$ where it also has the correct derivative. It deviates from the true value by no more than 2.5% anywhere. For $x<0$ we use $s(x) = s(-x)\,e^{x}$ in accordance with detailed balance.
Our semiheuristic toy model is thus completely defined. In Fig. \[fig:norm\] we show the “downstairs $\Gamma$” of Eq. (\[eq:resummedS\]) as a function of $\Gamma_\sigma$ such that the normalization requirement Eq. (\[eq:Snorm\]) is obeyed. By construction we have $\Gamma=\Gamma_\sigma$ for $\Gamma_\sigma\to0$ with smaller values for a larger $\Gamma_\sigma$. This reduction is mostly due to the detailed-balance behavior which suppresses the classical structure function for negative $\omega$.
Further we consider the $f$-sum which is for our present model $$\label{eq:fsumdefinition}
\int_{-\infty}^{+\infty} \frac{d\omega}{2\pi}\,
\omega\,S_\sigma(\omega) = \Gamma_\sigma
\int_{-\infty}^{+\infty} \frac{dx}{2\pi}\,
\frac{x}{x^2 + \gamma^2/4}\, s(x),$$ where $\gamma\equiv\Gamma/T$. As claimed before it is equal to the spin fluctuation rate times a factor of order unity which is shown in Fig. \[fig:fsum\] as a function of $\Gamma_\sigma$.
Finally we show in Fig. \[fig:static\] the static long-wavelength susceptibility in units of the Pauli susceptibility for our toy model according to Eq. (\[eq:staticsus\]). It is a slowly decreasing function of the spin fluctuation rate.
The overall spin-density structure function $S_\sigma(\omega)$ in our toy model is shown in Fig. \[fig:heuristicS\] for several values of $\gamma_\sigma=\Gamma_\sigma/T$.
to
Next we study the photon dispersion relation implied by our model. In Fig. \[fig:chiplot\] we show the long-wavelength limit $\chi'(\omega)=n^2_{\rm refr}-1$ in units of the Pauli susceptibility as a function of $x=\omega/T$ for several values of $\gamma_\sigma$. The overall behavior is exactly as expected from our general discussion in Sec. \[sec:limitingcases\]. To see the large-$\omega$ behavior more clearly we show in Fig. \[fig:meffplot\] the equivalent quantity $m_{\rm eff}^2$ in units of $\chi_{\rm Pauli}
\Gamma_\sigma T$. As predicted, $m_{\rm eff}^2$ approaches an asymptotic value which is independent of frequency and which is of order $\chi_{\rm Pauli} \Gamma_\sigma T$. Of course, for small $\omega$ the squared “effective mass” $m_{\rm eff}^2$ begins at negative values. However, for all frequencies and all values of $\gamma_\sigma$ we find that $|m_{\rm eff}^2|<m_\gamma^2$ where the latter is the asymptotic value for $\omega\to\infty$.
to
to
In order to arrive at a numerical estimate for the magnetically induced photon transverse mass we write the nucleon magnetic moments in the usual form $\mu_N=\kappa_N e/2m_N$ with $m_N$ the nucleon mass and $\kappa_n=-1.91$ and $\kappa_p=2.79$. For simplicity we treat all nucleons as if they were neutrons which implies $$\chi_{\rm Pauli}=4.4\times10^{-3}\,\rho_{14}/T_{10},$$ where $\rho_{14}\equiv\rho/10^{14}\,{\rm g\,cm^{-3}}$ and $T_{10}\equiv T/10\,{\rm MeV}$. With the perturbative estimate Eq. (\[eq:numericalgamma\]) for the spin-fluctuation rate we then find $$m_\gamma\big|_{\rm magnetic}\approx
(\chi_{\rm Pauli} T\Gamma_\sigma)^{1/2}
\approx 1.9\,{\rm MeV}\,\rho_{14} T_{10}^{1/4}.$$ In a SN core we have densities of up to $10^{15}\,\rm g\,cm^{-3}$ and temperatures of up to 30–$60\,\rm MeV$, implying a perturbative spin-fluctuation rate far in excess of the temperature, i.e.$\gamma_\sigma=10$–100. It has been argued that in a SN core the true $\Gamma_\sigma$ cannot exceed a few times $T$ [@JKRS96; @Sigl]. Therefore, we have probably overestimated the transverse photon mass by at least a factor of a few.
We next consider the corresponding quantity caused by the interaction with electrons. One finds [@BS93] $m_\gamma^2=\frac{3}{2}\omega_P^2$ with $\omega_P$ the plasma frequency. For relativistic degenerate electrons it is $\omega_P^2=(4\alpha/3\pi)\,p_{F,e}^2$ with $p_{F,e}$ the electron Fermi momentum so that $$m_\gamma^2\big|_{\rm electrons}=\frac{2\alpha}{\pi}\,p_{F,e}^2.$$ Numerically, this corresponds to $$m_\gamma\big|_{\rm electrons}
=16.3~{\rm MeV}\,\,Y_e^{1/3}\,\rho_{14}^{1/3},$$ where $Y_e$ is the number of electrons per baryon. Evidently in the center of a SN core with $\rho_{14}\approx 8$, $T_{10}\approx4$ and initially $Y_e\approx 0.3$ the magnetic moment contribution could be almost as large as the electronic term. However, because we have probably overestimated the magnetic term by a factor of a few the electrons still dominate.
Discussion and Summary {#sec:summary}
======================
We have calculated the photon refractive index due to the interaction with the magnetic moments of the nucleons. For simplicity we have limited our discussion to nondegenerate neutrons. In the collisionless limit the forward-scattering amplitude vanishes identically so that the neutron magnetic moments alone do not cause any deviation of the photon dispersion relation from the vacuum behavior [@ON97]. However, because of strong neutron spin interactions the collisonless limit is far from justified in a SN core. On the basis of the fluctuation-dissipation theorem and the Kramers-Kronig relation we have derived a general expression for the photon refractive index in terms of the dynamical neutron spin-density structure function $S_\sigma(\omega,k)$. In an interacting medium it is a broad function of $\omega$, in contrast to the collisionless limit where it is proportional to $\delta(\omega)$.
We have found that for $\omega\ll\Gamma_\sigma$ (the neutron spin fluctuation rate) the “effective photon mass” $m_{\rm eff}^2$ begins with negative values $-\chi_{\rm Pauli}\omega^2$ in terms of the Pauli susceptibility of the neutron ensemble. However, as shown in Fig. \[fig:meffplot\] this function quickly turns around and then grows asymptotically to a positive value $m_\gamma^2\approx \chi_{\rm
Pauli} T \Gamma_\sigma$. In absolute terms this “transverse photon mass” is much larger than the maximum excursion of $m_{\rm eff}^2$ to negative values.
A numerical comparison for conditions relevant for a SN core reveals that the transverse photon mass caused by the neutron magnetic moment tends to be much smaller than that caused by the electron plasma effect, except for extreme densities and low electron fractions where the magnetic term may actually compete with the electronic one. A numerically accurate comparison is not possible because the neutron dynamical spin-density structure function is not known in any detail. We have only performed a relatively schematic estimate which involved many simplifying assumptions. However, it still appears safe to conclude that the negative magnetic $m_{\rm eff}^2$ at small frequencies cannot compete with the electronic plasma effect. This indicates that the total $m_{\rm eff}^2$ is always positive, i.e. the photon refractive index is always less than 1 and it is reasonably well estimated by the electronic plasma effect. This implies that the Cherenkov processes $\nu\leftrightarrow\nu\gamma$ remain forbidden.
A more quantitative analysis than has been presented here requires a better understanding of the dynamical nucleon spin-density structure function, or more precisely, of the dynamical spin and isospin susceptibilities of a hot and dense nuclear medium. We stress that the $\omega$ dependence is crucial for the photon dispersion relation as well as the neutrino opacities [@JKRS96], the static susceptibilities alone which have sometimes been studied in the literature are not enough.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank E. Braaten, P. Elmfors and G. Sigl for discussions or comments on the manuscript. This research was supported, in part, by the European Union under contract No. CHRX-CT93-0120 and by the Deutsche Forschungsgemeinschaft under grant No. SFB 375.
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abstract: 'We study an unconventional quantum Hall effect for the surface states of ultrathin Floquet topological insulators in a perpendicular magnetic field. The resulting band structure is modified by photon dressing and the topological property is governed by the low-energy dynamics of a single surface. An exchange of symmetric and antisymmetric surface states occurs by reversing the light’s polarization. We find a novel quantum Hall state in which the zeroth Landau level undergoes a phase transition from a trivial insulator state, with Hall conductivity $\sigma_{yx}=0$ at zero Fermi energy, to a Hall insulator state with $\sigma_{yx}=e^2/2h$. These findings open new possibilities for experimentally realizing nontrivial quantum states and unusual quantum Hall plateaux at $(\pm1/2,\pm3/2,\pm5/2,...)e^2/h$.'
author:
- 'M. Tahir$^{1,\ddagger}$ P. Vasilopoulos$^{1,\dag}$, and U. Schwingenschlögl$^{2,}$'
title: Unconventional quantum Hall effect in Floquet topological insulators
---
In the regime of the integral quantum Hall effect (IQHE) for conventional, two-dimensional (2D) systems, e.g., in a GaAs/AlGaAs heterostructure, the Hall conductivity takes the values $2(n+1) e^{2}/h=( 2, 4, 6,... )\ e^{2}/h$, where $h$ is the Planck constant, $e$ the electron charge, and $n$ an integer. In graphene though the IQHE plateaux appear at $4(n+1/2) e^{2}/h=(\pm 2, \pm 6, \pm 10,... )\ e^{2}/h$ [@KAS], and the “half integer” aspect is hidden under the 4-fold degeneracy associated with the spin and valley degrees of freedom [@VPS]. More recently, the IQHE has been assessed for silicene [@MTU] and MoS$_2$ [@XCG] in which the spin-orbit interaction rearranges, as in 2D systems [@XF], the Landau levels (LLs) in two groups and the plateaux appear at integer values of $ e^{2}/h$ due to a double degeneracy ($ \pm 0, \pm 1, \pm 2, \pm 4, \pm 6,...$). In topological insulators (TIs) electrons on both top and bottom surfaces contribute to the Hall conductivity and its plateaux, due to the surface degeneracy, have heights $2(n+1/2) \ e^{2}/h =( \pm 0, \pm 1, \pm 3, \pm 5,... )e^{2}/h$ [@DY; @HL; @CC]. Though the quest for a genuine “half-integer” QHE is long, a QHE like $(n+1/2) \ e^{2}/h =(\pm 0, \pm 1/2, \pm 3/2, \pm 5/2,...)\ e^{2}/h$ [*without any degeneracy prefactor*]{}, has not been observed. Then one wonders whether a “half-integer” QHE is possible in TIs by breaking their surface degeneracy.
TIs, well established theoretically [@MC] and experimentally [@MS], are a state of matter, that cannot appear in normal 2D systems with time-reversal symmetry [@BT]. They exhibit exotic properties such as disorder-protected conducting surface states, a single Dirac cone, quantum phase transitions [@HYY; @JGA], etc. These findings generated a strong interest in TIs that was further intensified by their potential applications in quantum computing [@AJ], optical devices [@OE], terahertz detectors [@XJ], etc.
More recently, the surface states of TIs driven by circularly polarized *off-resonant* light have become a subject of strong interest [@NG; @TT; @ANA; @HDT]. TIs driven by external time-periodic perturbations are known as Floquet TIs (FTIs). For such systems it is convenient to use the Floquet theory [@TT]. In the appropriate frequency regime the [*off-resonant*]{} light cannot generate real photon absorption or emission due to energy conservation. Accordingly, it does not directly excite electrons but instead modifies the electron band structure through second-order virtual-photon absorption processes. Averaged over time these processes result in an effective static alteration of the band structure. Illuminating, e.g., graphene or silicene with *off-resonant* light generates a Haldane-type gap [@ZMA].
Floquet bands were first realized in photonic crystals [@MJ] and have been verified by recent experiments on the surface states of FTIs [@YH; @HJJ]. These first studies of *off-resonant* light were limited to the band structure of FTIs and differ from many optical effects in TIs [@OE]. Also, no magnetic field was involved in these experiments.
In this work we identify a novel quantum Hall state of ultrathin FTIs in a magnetic field when their surface degeneracy is broken due to an *off-resonant* light. We evaluate their band structure and the longitudinal and Hall conductivities using linear response theory [@MK; @PV].
*Model formulation*. We consider surface states of ultrathin TIs in the ($x,y$) plane in the presence of circularly polarized *off-resonant* light [@TT] and hybridization [@HZ] between the top and bottom surface states. Extending the 2D Dirac-like Hamiltonian [@TT; @MTVP] by including an external perpendicular magnetic field $B$ gives $$H_{s}^{l}=v_{F}(\sigma _{x}\mathbf{\Pi }_{y}-\sigma _{y}\mathbf{\Pi }
_{x})+s\Delta _{h}\sigma _{z}+l\Delta _{\Omega }\sigma _{z}, \label{1}$$ where $s = +/-$ is for symmetric/antisymmetric surface states, $l=+/- $ for right-/left-handed circularly polarized *off-resonant* light, ($\sigma _{x}$, $\sigma _{y}$, $\sigma _{z}$) the Pauli matrices and $v_{F}$ the Fermi velocity. $\Delta _{h}$ is the hybridization energy between the top and bottom surface states that, depending on the thickness, varies from 20 meV to 120 meV [@YK]. $\Delta _{\Omega }=e^{2}v_{F}^{2}\hslash ^{2}A_{0}^{2}/\hslash
^{3}\Omega $ is the mass term induced by the *off-resonant* light with amplitude $E_{0}$, $\Omega $ the light’s frequency, and $A_0=E_{0}/\Omega $. It breaks the time-reversal symmetry and its value is about 50 meV [@YH; @HJJ]. $\mathbf{\Pi =p}+e\mathbf{A}$ is the 2D canonical momentum with vector potential $\mathbf{A}$. In the Landau gauge $\mathbf{A}=(0, Bx, 0)$, diagonalizing the Hamiltonian (1) gives the eigenvalues $$E_{n,s}^{\lambda ,l}=\lambda [\hslash ^{2}\omega _{c}^{2}n+\Delta
_{s,l}^{2}]^{1/2},\quad E_{0,s}^{0,l}=-\Delta _{s,l}, \label{2}$$ where $\lambda =\pm 1$ represents the electron/hole states, $\omega _{c}=v_{F}
\sqrt{2eB/\hslash }$, and $\Delta _{s,l}=l\Delta _{\Omega }+s\Delta _{h}$. The corresponding normalized eigenfunctions are $$\hspace*{-0.35cm}\Psi _{n,s}^{\lambda ,l}=\frac{e^{ik_{y}y}}{\sqrt{L_{y}}}\Big(
\begin{array}{c}
C_{n,s}^{\lambda ,l}\phi _{n-1} \\
D_{n,s}^{\lambda ,l}\phi _{n}
\end{array}
\Big),\,\,\,\Psi _{0,s}^{0,l}=\frac{e^{ik_{y}y}}{\sqrt{L_{y}}}\Big(
\begin{array}{c}
0 \\
\phi _{0}
\end{array}
\Big), \label{3}$$ where $C_{n,s}^{\lambda ,l}=[(E_{n,s}^{\lambda ,l}+\Delta
_{s,l})/2E_{n,s}^{\lambda ,l}]^{1/2}$, $D_{n,s}^{\lambda ,l}=
[(E_{n,s}^{\lambda ,l}-\Delta _{s,l})/2E_{n,s}^{\lambda ,l}]^{1/2}$; $\phi_{n}$ are the harmonic oscillator functions. Notice that Eq. (1) does not contain the Zeeman term $gs_zB$. We neglect it, because we consider only weak $B$ fields $\leq 1$ T, cf. Fig. 1. For a $g$ factor as large as 20 the Zeeman energy at $B=1$ T is $0.58$ meV and much smaller than all other energies.
A close inspection of Eq. (2) shows that we have a gapped Dirac spectrum, with gap $\Delta _{h}$, and electron-hole symmetry for zero *off-resonant* light, $\Delta _{\Omega }=0.$ For $\Delta _{\Omega }>\Delta _{h}>0$, the eigenvalues of Eq. (2) show a $\sqrt{B}$ dependence and the LLs split, see Fig. 1. We find an exchange of the symmetric (solid curves) and antisymmetric (dotted curves) surface states by changing the light’s polarization from right (red curves) to left (black curves). The parameters used are $v_{F}=0.5\times 10^{6}$ m/s, $\Delta _{h}=20$ meV, and $\Delta _{\Omega }=30$ meV ($ev_{F}A_{0}=0.48$ eV, $\hslash \Omega =7.5$ eV) [@TT]. The $n=0$ LL appears in the hole band for right-handed light (red curves, $l=1$) and in the electron band for left-handed light (black curves, $l=-1$), see Fig. 1. The exchange of surface states induced by the field $B$ and the *off-resonant* light in such FTIs is an entirely new phenomenon. The energies of the two surfaces are different for $B\rightarrow 0$, since the gap $l\Delta _{\Omega } \pm \Delta _{h}$ increases for one surface and decreases for the other.
We emphasize that the band gaps $l \Delta _{\Omega }+s \Delta _{h}$ at the two surfaces of ultrathin FTIs can be made different by, e.g., varying the light’s frequency or amplitude. This can create, e.g., for $l=1$, one surface with a small gap $ \Delta _{\Omega }- \Delta _{h}$ and the other one with a large gap $ \Delta _{\Omega }+ \Delta _{h}$; for $l=-1$ these gaps could be exchanged. Accordingly, only the antisymmetric or symmetric surface contributes to the transport properties depending on the light polarization ($ l=\pm 1$). Such a situation could be realized in experiments on FTIs, similar to those of Refs. [@YH; @HJJ], by varying the sample thickness [@YK] down to the limit of 6 nm below which different energies $\Delta_h$ have been reported [@YK]. To our knowledge this is a novel state of matter in FTIs like Bi$_{2}$Se$_{3}$, Bi$_{2}$Te$_{3}$, HgTe, and related materials.
{width="0.8\columnwidth" height="0.45\columnwidth"} {width="0.84\columnwidth" height="0.5\columnwidth"}
*Longitudinal conductivity*. For weak scattering potentials the current is due to hopping between orbit centres as a result of carrier collisions with, e.g., charged impurities [@MK]. In a normal magnetic field the diffusive contribution $\sigma_{xx}^{dif}$ to $\sigma_{xx}$ vanishes and only the collisional contribution $\sigma_{xx}^{col}\equiv\sigma_{xx}$ is important; it is given by [@MK; @PV] $$\sigma _{xx}=\frac{e^{2}\beta }{2S_{0}}\sum_{\zeta \neq \zeta ^{\prime
}}f(E_{\zeta })[1-f(E_{\zeta ^{\prime }})]W_{\zeta \zeta ^{\prime
}}(X_{\zeta }-X_{\zeta ^{\prime }})^{2}, \label{4}$$ where $f(E_{\zeta })=(\exp [\beta (E_{\zeta }-E_{F})]+1)^{-1}$ is the Fermi Dirac distribution function, $\beta=k_{B}T$, $T$ the temperature, $k_{B}$ the Boltzmann constant, and $E_{F}$ the Fermi energy. $W_{\zeta \zeta ^{\prime }}$ is the transition rate between the one-electron states $\left\vert \zeta \right\rangle $ and $\left\vert \zeta
^{\prime }\right\rangle $, and $e$ the charge of the electron. Here $f(E_{\zeta })=f(E_{\zeta ^{\prime }})$ for elastic scattering and $X_{\zeta }=\left\langle \zeta \right\vert x\left\vert \zeta \right\rangle$ with $x$ being the position operator.
The scattering rate is given by Fermi’s golden rule $$\hspace*{-0.25cm}W_{\zeta \zeta ^{\prime }}=F
\sum_{\zeta \neq
\zeta ^{\prime }}\left\vert U(\mathbf{q})\right\vert ^{2}\left\vert J_{\zeta
\zeta ^{\prime }}(u)\right\vert ^{2}\delta (E_{\zeta }-E_{\zeta ^{\prime
}})\delta _{k_{y},k_{y}^{\prime }+q_{y}}, \label{5}$$ with $F=2\pi N_i/S_{0}\hslash,\,q^2=q_{x}^{2}+q_{y}^{2}$, $u=l_{B}^{2}q^2/2$, and $N_i$ the impurity density. $J_{\zeta \zeta ^{\prime }}(u)=\left\langle \zeta \right\vert \exp
(i{\bf q\cdot r})\left\vert \zeta ^{\prime }\right\rangle $ are the form factors and $\left\vert \zeta \right\rangle \equiv \left\vert n,s,l,k_{y}\right\rangle $. $U(\mathbf{q})=U_{0}/(q^{2}+k_{s}^{2})^{1/2}$ with $U_{0}=e^{2}/(2\varepsilon
_{r}\varepsilon _{0})$. Further, $k_{s}$ is the screening wave vector, $\varepsilon
_{r}$ the relative permittivity, and $\varepsilon _{0}$ the permittivity of the vacuum. Furthermore, if the impurity potential is short-ranged (of the Dirac $\delta $-function type), one may use the approximation $k_{s}\gg q$ and obtain $U(\mathbf{q})\approx U_{0}/k_{s}$. Since the scattering is elastic and the eigenfunctions are degenerate in the quantum number $k_{x}$, cf. Eq. (3), only the $n\rightarrow n$ transitions are allowed. Further, we have $(X_{\zeta}-X_{\zeta ^{\prime }})^{2}=l_{B}^{4}q_{y}^{2}$, transform the sums over $k_y$ and $q$ into integrals, and evaluate them using cylindrical coordinates. The form factor $\left\vert J_{\zeta \zeta ^{\prime
}}(u)\right\vert ^{2}$ can be evaluated from the matrix element $\left\langle \zeta \right\vert \exp
(i{\bf q\cdot r})\left\vert \zeta ^{\prime }\right\rangle$. The result is $\left\vert J_{nn}(u)\right\vert ^{2}=\exp (-u)\big( \left\vert C_{n,s}^{\lambda ,l}
\right\vert ^{2}L_{n}(u)+\left\vert D_{n,s}^{\lambda ,l}
\right\vert ^{2}L_{n-1}(u)\big) ^{2}$ for $n=n^{\prime }$. With these details Eq. (4) takes the form $$\sigma _{xx}=\frac{e^{2}}{h}\frac{N_i\beta U_{0}^{2}}{4u_{sc}\hslash \omega _{c}}
\sum_{s,n }I_{n,s}^{\lambda ,l}\,f(E_{n,s}^{\lambda
,l})\,[1-f(E_{n,s}^{\lambda ,l})], \label{6}$$ where $f(E_{n,s}^{\lambda ,l})=(\exp [\beta (\lambda [\hslash
^{2}\omega _{c}^{2}n+(\Delta _{s,l})^{2}]^{1/2}-E_{F})]+1)^{-1}$ and $u_{sc}=l_{B}^{2}k_{s}^2/2$. The sum over $s$ is trivial since the two surfaces can be treated independently due to the different gaps. The factor $I_{n,s}^{\lambda ,l}$ in Eq. (6) is the integral $\int_{0}^{\infty }u\left\vert J_{nn}(u)\right\vert
^{2}du$ that can be evaluated analytically using the properties of the orthogonal polynomials $L_{n}(u)$. The result is $$I_{n,s}^{\lambda ,l}=(2n+1)\big\vert C_{n,s}^{\lambda ,l}\big\vert
^{4}-2n\big\vert C_{n,s}^{\lambda ,l}\big\vert ^{2}\big\vert
D_{n,s}^{\lambda ,l}\big\vert ^{2} +(2n-1)\big\vert D_{n,s}^{\lambda ,l}
\big\vert ^{4}. \label{8}$$ For $\Delta _{s,l}=0$, Eq. (7) reduces to $2n/4$, which means that the minima of $\sigma _{xx}$ occur at the odd factors $\nu =2n+1$ in accord with Ref. [@PV].
Since the band gap $l \Delta _{\Omega }+s \Delta _{h}$ becomes surface dependent, see Fig. 1, the longitudinal conductivity is dominated by one surface only, that of the symmetric or antisymmetric surface states. As usual, this conductivity, given by Eq. (6), exhibits Shubnikov-de Haas oscillations. For $\Delta _{h}=\Delta _{\Omega }=0$ we must consider both surfaces. The electron-hole spectrum is symmetric with a single peak (solid curve) at the Dirac point, as shown in Fig. 2(a), using the parameters [@DY; @HL; @CC]: $N_i=1\times 10^{13}$ m$^{-2}$, $\mu _{B}=5.788\times 10^{-5}$ eV/T, $T$ = 2 K, $B=$ 1 T, $k_{s}=10^{-7}$ m$^{-1}$, $v_F=5\times 10^{5}$ m/s, and $\epsilon _{r}=4$. We find a gap $\Delta _{h}$ at the Dirac point for $\Delta _{h}\neq 0$ and $\Delta _{\Omega}=0$ with symmetric electron-hole behaviour (dashed curve). This gives $\sigma _{xx}=0$ at the Dirac point and the peak at $E_F=0$ (solid curve) splits into two peaks, one in the electron ($s=-1$) and one in the hole band ($s=1$) in accord with Eq. (2).
For $\Delta _{\Omega}>\Delta_{h}>0$ the electron-hole spectrum is asymmetric and we consider only one surface depending on the light’s polarization. We consider only the symmetric surface states ($s=1$, black curves) for left-handed light $(l=-1)$ or the antisymmetric surface states ($s=-1$, red curves) for right-handed light $(l=1)$ and show $\sigma _{xx}$ in Fig. 2(b). As seen, the $n=0$ LL shifts into the hole or electron band. The shift can be understood with the help of the eigenvalues shown in Fig. 1: for right-(left-)handed light the $n=0$ LL moves into the hole (electron) band. This is a nontrivial state entirely new in FTIs. We notice in passing that were we to plot the current polarization $P=\big[\sigma_{xx}(l=1)-\sigma_{xx}(l=-1)\big]/\big[\sigma_{xx}(l=1)+\sigma_{xx}(l=-1)\big]$ we would have, on account of Fig. 2(b), only two peaks of height $P=1 (-1)$ centred at $E_F\approx -0.01 (0.01)$ eV. Also, had we considered the $s=-1$ surface with left-handed light $(l=-1)$ or the $s=1$ surface with right-handed light $(l=1)$, $\sigma_{xx}$ would be zero in the entire range of Fig. 2 since the corresponding surface states start at $\pm 0.05$ eV, cf. Fig. 1 for $B=1$ T. This is also corroborated by the fact that at very low temperatures the factor $\beta f(...)[1-f(...)]$ in Eq. (6) behaves as the function $\delta(E_{n,s}^{\lambda,l} -E_F)$.
{width="0.81\columnwidth" height="0.4\columnwidth"} {width="0.85\columnwidth" height="0.5\columnwidth"}
*Hall conductivity*. For linear responses to a weak source-to-drain electric field, the Hall conductivity is given by the Kubo-Greenwood formula [@MK; @PV] $$\sigma _{\mu \nu }
=\frac{i\hslash e^{2}}{S_{0}}\sum_{\zeta \neq \zeta
^{\prime }}\frac{(f_{\zeta }-f_{\zeta ^{\prime }})v_{\nu \zeta \zeta
^{\prime }}v_{\mu \zeta ^{\prime }\zeta }}{(E_{\zeta }-E_{\zeta ^{\prime
}})(E_{\zeta }-E_{\zeta ^{\prime }}+i\Gamma _{\zeta })}, \label{9}$$ where $v_{\nu \zeta \zeta ^{\prime }}$ and $v_{\mu \zeta ^{\prime }\zeta }$ are the nondiagonal matrix elements of the velocity operator with $\mu=x,y, \nu=x,y$. The sum runs over all quantum numbers of the states $\left\vert \zeta \right\rangle \equiv \left\vert
n,s,l,k_{y}\right\rangle $ and $\left\vert \zeta ^{\prime }\right\rangle
\equiv \left\vert n^{\prime },s^{\prime },l ^{\prime },k_{y}^{\prime
}\right\rangle $ provided $\zeta \neq \zeta ^{\prime }$. Assuming that the level broadening is approximately the same for all LLs, $\Gamma_{\zeta}=\Gamma $, one can show that the imaginary part of Eq. (8) vanishes. To obtain the most transparent results for the Hall conductivity $\sigma _{yx}$, we take $\Gamma=0$. The relevant velocity matrix elements are obtained from Eq. (1), for $\nu =x$ and $\mu =y$, and the evaluation follows the procedure detailed in Ref. [@PV]. The result for $\sigma _{yx}$ can be expressed as a sum of two terms, one (I) for $n\geq 1$ and the other (II) for $n=0$, i.e., $\sigma
_{yx}=\sigma _{yx}^{I}+\sigma _{yx}^{II}$, with $$\begin{aligned}
\hspace*{-0.1cm}\sigma _{yx}^{I} &=&\frac{e^2 }{h}\sum_{s,n=1}^{\infty
}\Big\{(n+1/2)[f_{n,s}^{+,l}-f_{n+1,s}^{+,l}+f_{n,s}^{-,l}-f_{n+1,s}^{-,l}]
\notag \\
&&-{\Delta
_{s,l}\over 2} \,[{f_{n,s}^{+,l}-f_{n,s}^{-,l}\over E_{n,s}^{+,l}}
-{f_{n+1,s}^{+,l}-f_{n+1,s}^{-,l}\over E_{n+1,s}^{+,l}}]\Big\}.
\label{13}\end{aligned}$$ The sum over $n$ starts at $n=1$, because the $n=0$ LL is treated separately. That over $s$ is trivial, as we consider only one surface at a time. For $n=0$ Eq. (8) gives $$\begin{aligned}
\sigma _{yx}^{II} &=&\frac{e^{2}}{h}\sum_{s}
\Big\{f_{0,s}^{+,l}+f_{0,s}^{-,l}-(f_{1,s}^{+,l}+f_{1,s}^{-,l})/2 \label{14} \\
&&+\Delta _{s,l}(f_{1,s}^{+,l}-f_{1,s}^{-,l})/2E_{1,s}^{+,l}\Big\}. \notag\end{aligned}$$
{width="0.8\columnwidth" height="0.4\columnwidth"} {width="0.85\columnwidth" height="0.5\columnwidth"}
At zero or very low temperature the sum over $s$ in Eqs. (9) and (10) has the value $1$ for $n=n_{F}$, since for a single surface the number of filled states is $1$. Here $n_F$ is the LL index at the Fermi energy. Notice also that for $\Delta _{s,l}=0$ Eqs. (9) and (10) take the form $\sigma _{yx}=2(e^{2}/h)(n+1/2)$ or in terms of the filling factor $\nu =2(n+1/2)$ as $\sigma _{yx}=\nu e^{2}/h$ for TIs [@DY; @HL; @CC; @JG]. An important aspect in the QHE in gapless graphene, TIs, silicene, MoS$_2$, etc. is the LL at zero energy or its modification in gapped systems.
Similar to the band structure and longitudinal conductivity, the unconventional QHE is dominated by one surface. That is, only the symmetric or antisymmetric surface contributes to it for left- or right-handed light, respectively. We show the Hall conductivity $\sigma _{yx}$ in Fig. 3, for $s=1$, as a function of $E_F$. In the limit $\Delta _{s,l}\to 0$ ((a), solid curve) these results reduce to an odd-integer QHE in TIs by multiplying with a factor 2 for surface degeneracy [@DY; @JG] (limit of zero strain in Ref. [@CC]), where the plateaus appear at ($\pm 1,\pm 3,\pm 5,...)e^2/h$, and both surfaces must be considered as in the case of Fig. 2(a). The results of Fig. 3(a), dashed curve, can be reduced to those of single-valley gapped graphene for $\Delta _{h}\neq 0$ [@PV], irrespective of a factor of 2 due to valley degeneracy, and to those of gapped TIs [@CJPC]. However, for $\Delta _{\Omega }>\Delta _{h}\neq 0$ only one surface must be considered, see Fig. 2(b) for $\sigma _{xx}$. The Hall plateaus occur at half-integer values in contrast to previous results for graphene and TIs, as shown in Fig. 3 (b). At the Dirac point we have $\sigma_{yx}=e^2/2h$ for right-handed light (dashed curve) and $\sigma_{yx}=-e^2/2h$ for left-handed light (solid curve), due to the occurrence of the $n=0$ LL in the hole and electron band, respectively. Again, had we considered the $s=-1$ surface with left-handed light or the $s=1$ surface with right-handed light, $\sigma_{yx}$ would vanish in the $E_F$ range of Fig. 3, since the corresponding states would start at $\pm 0.05$ eV and the occupation factors $f(...)$ would be zero.
The electron-hole symmetry is broken and the plateaus appear at $(\pm 1/2,\pm 3/2,\pm 5/2,...)e^2/h$. This shows a nontrivial transition at the Dirac point which could be experimentally tested. It occurs because the energy term $\Delta _{\Omega }$ due to the *off-resonant* light at the Dirac point can be externally tuned to higher values [@YH; @HJJ]. As for the influence of level broadening, i.e., finite $\Gamma$, on the results, on the basis of Ref. [@PV] we strongly expect they will not be altered qualitatively. After all, the QHE has already been realized on TIs [@CC]. These signatures of novel quantum phase transitions in FTIs are distinct from those in graphene or in TIs without light and relate to different values of the Hall conductivity. They could be tested in experiments similar to those performed on semiconducting silicon [@DD; @JK]. Moreover, radiation effects and light-dependent magnetotransport have been observed for Dirac fermions in the presence of [*on-resonant*]{} light [@PO; @JK2]. Accordingly, we believe our results can be tested in similar experiments using *off-resonant* light [@YH; @HJJ], a regime that is different from that of the optical absorption spectra [@OE; @WA].
*Summary*. We have identified an unconventional QHE in FTIs, in the presence of a perpendicular magnetic field, by evaluating their band structure and the Hall and longitudinal conductivities. The low-energy dynamics can be governed by a single-surface in a wide range of Fermi energies. This results in a nontrivial phase transition and unusual Hall plateaus at [*half-integer*]{} multiples of $e^{2}/h$ ($\pm 1/2,\pm 3/2,\pm 5/2, ...)$. In addition, reversing the light polarization leads to an exchange of surface states, in both the valence and conduction bands, and to a shift of the $n=0$ LL into the hole and electron bands, respectively. These findings suggest new directions in experimental research and device applications based on FTIs.\
[**Acknowledgments**]{}: This work was supported by the Canadian NSERC Grant No. OGP0121756 (MT, PV) and by funding from King Abdullah University of Science and Technology (KAUST) (US).\
Electronic addresses: $^{\ddagger}$m.tahir06@alumni.imperial.ac.uk,\
$^{\dag}$p.vasilopoulos@concordia.ca\
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---
abstract: |
The one-class classification problem is a well-known research endeavor in pattern recognition. The problem is also known under different names, such as outlier and novelty/anomaly detection. The core of the problem consists in modeling and recognizing patterns belonging only to a so-called target class. All other patterns are termed non-target, and therefore they should be recognized as such. In this paper, we propose a novel one-class classification system that is based on an interplay of different techniques. Primarily, we follow a dissimilarity representation based approach; we embed the input data into the dissimilarity space by means of an appropriate parametric dissimilarity measure. This step allows us to process virtually any type of data. The dissimilarity vectors are then represented through a weighted Euclidean graphs, which we use to (i) determine the entropy of the data distribution in the dissimilarity space, and at the same time (ii) derive effective decision regions that are modeled as clusters of vertices. Since the dissimilarity measure for the input data is parametric, we optimize its parameters by means of a global optimization scheme, which considers both mesoscopic and structural characteristics of the data represented through the graphs. The proposed one-class classifier is designed to provide both hard (Boolean) and soft decisions about the recognition of test patterns, allowing an accurate description of the classification process. We evaluate the performance of the system on different benchmarking datasets, containing either feature-based or structured patterns. Experimental results demonstrate the effectiveness of the proposed technique.\
[***Index terms—*** One-class classification; Entropic spanning graph; Modularity measure; Dissimilarity representation; Fuzzy set.]{}
author:
- 'Lorenzo Livi[^1][^2]'
- 'Alireza Sadeghian[^3]'
- 'Witold Pedrycz[^4]'
bibliography:
- '/home/lorenzo/University/Research/Publications/Bibliography.bib'
title: 'Entropic one-class classifiers'
---
Introduction
============
Pattern recognition problems involving the processing of patterns belonging to one class only are quite common [@Kemmler2013; @Tax19991191; @Juszczak20091859; @occ_sg_enricods__arxiv; @Tax:2002:UOG:944790.944809; @fuzzy_occ; @NIPS2002_2163; @wang2013position; @bodesheim1991divergence; @6619277; @6186735; @4049825; @pimentel2014review; @Ding2014313; @6722892; @bicego2009soft]. The interest in this type of problems is both methodological and of application-oriented character. In fact, one-class classification problems could be used to deal with tasks involving recognition of outliers in data. On the application side, instead, there are many real-world scenarios in which it is possible to obtain (or design) patterns only for the so-called “target” class. As an instance, we may cite the problem of determining whether a given machine/device is not working properly. Intuitively, patterns representing the correct functioning of the machine/device are “trivial” and “not informative”, in the sense that anything that is different from the observed faults is by definition an instance of correct functioning. As a consequence, in this case one would model only those patterns representing fault instances (see Fig. \[fig:occ\_example\]). However, modeling explicitly only one side of the decision boundary implies a more difficult setting with respect to (w.r.t.) well-established multi-class problems. In particular, the method for evaluating the performance of any one-class classifier (OCC) should take into account the implicit uncertainty rooted in the resulting decisions [@one-class_survey__2010; @Juszczak20091859; @fuzzy_occ; @wang2013position; @6722892].
 \[fig:occ\_example\]
The one-class classification setting has been adopted in many real-world applications (for a review, see [@one-class_survey__2010; @pimentel2014review]) such as the recognition of faults in smart electric grids [@occ_sg_enricods__arxiv], Raman spectroscopy [@Kemmler201329], events detection in videos [@piciarelli2008trajectory], document classification [@Manevitz01one-classsvms], medical imaging [@desir2012random], and oil spill detection [@oilspill__2010].
In this paper, we propose a novel OCC that is based on an interplay of different techniques. Our first objective is to make the proposed OCC applicable to any data type. To this end, we develop our approach on the basis of the dissimilarity space representation [@Duin2012826]. This choice, although increases the overall computational complexity, allows us to cover virtually any application context, regardless of the adopted representation for the input data (e.g., features, labeled graphs, etc.). Then, we represent the embedded data in the dissimilarity space (DS) by a complete Euclidean graphs, whose vertices denote input patterns and the edges their mutual (normalized) Euclidean distance in the DS. Such a graph allows us to (i) estimate the informativeness of the represented data and at the same time (ii) construct the decision regions (DRs) that we use to define the OCC model. Additionally, representing the data in the DS by means of the Euclidean graph provides us a way to bypass all problems related to the high-dimensionality of the DS. The informativeness of the embedded data is computed with the use of the $\alpha$-order Rényi entropy, estimated by means of the entropic spanning graph technique proposed by @gs:HeroEtAl2002. In particular, we use the minimum spanning tree, which is further analyzed with the aim of inducing a partition of the graphs into suitably compact and separated clusters of vertices. We designed a fast graph partitioning algorithm based on the concept of modularity [@4358966]. The derived DRs are then equipped with suitable membership functions [@Livi_ga_2013; @rizzi2002], which allow us to form both hard (Boolean) and soft decisions about the classification. To benchmark the proposed OCC, we test two types of data: features and labeled graphs (also termed attributed graphs) based patterns. We provide experiments and offer a comparative analysis for different datasets from well-known benchmarks, such as those coming from the UCI [@Bache+Lichman_2013] and IAM [@riesen+bunke2008] repositories.
The paper is structured as follows. Section \[sec:background\] offers an overview on the one-class classification context (Section \[sec:related\_works\]) by hence providing also a clear collocation for the proposed OCC system. Moreover, in Section \[sec:tech\_background\] we introduce the main technical background material used by the proposed OCC. Successively, in Section \[sec:eocc\] we present the details of the proposed OCC. In Section \[sec:exps\] we show and discuss the experimental evaluations. Finally, in Section \[sec:conclusions\] we draw our conclusions, providing also pointers to future directions.
Related Works and Background Material {#sec:background}
=====================================
A Review on One-Class Classification Methods {#sec:related_works}
--------------------------------------------
@pimentel2014review group the current one-class methods in five different categories: (i) probabilistic, (ii) distance-based, (iii) domain-based, (iv) reconstruction-based, and finally (v) information-theoretic techniques. Probabilistic methods are focused on the reconstruction of the generative probability density function (PDF) underlying the data at hand. Such methods are further subdivided into the usual parametric and non-parametric classes, where the former include methods based on the identification of the optimal parameters describing a pre-defined statistical model, while the latter include methods based on the reconstruction of the PDF directly from the data. Distance-based methods operate, essentially, by means of a suitable distance measure in the input space. Techniques of this category can be grouped into clustering-based and nearest neighbors based approaches. Reconstruction-based methods include classical data-driven approaches, such as neural networks and subspace-based methods. Domain-based methods revolve around the well-established Support Vector Data Description (SVDD) [@Tax19991191]. In this case, the objective is to model the target data via suitable decision regions/surfaces by optimizing a specific convex optimization problem. Finally, information-theoretic methods rely on information measures such as entropy, divergence, and mutual information. Intuitively, a non-target pattern is identified as one that alters significantly the information content of the data.
As stated before, an important class of OCCs revolves around the SVDD method proposed by @Tax19991191, which has been elaborated taking inspiration from the well-known support vector machine (SVM) [@Tax19991191; @SchWilSmoShaetal00; @wang2013position]. The classification model of SVDD is defined in terms of hyper-spheres, which cover the training set data through an SVM-like optimization problem (the minimization of the sphere radiuses is enforced). SVDD is particularly exploitable since it can be used jointly with positive definite kernel functions, which allow the generalization of the input domain. @SchWilSmoShaetal00 proposed an alternative approach to SVDD that employs a hyperplane, like in the conventional SVM case. The hyperplane is positioned to separated the region of the input space containing patterns form the region containing no data. Other more recent approaches include algorithms based on the minimum spanning tree (MST) [@Juszczak20091859], Gaussian processes [@Kemmler2013], and on Random forests [@Desir20133490]. The reader is referred to Ref. [@pimentel2014review] for a comprehensive survey portraying the state-of-the-art on one-class classification methods.
According to the aforementioned OCC systems categorization, the herein proposed OCC could be collocated in the intersection among probabilistic, distance-based, and information-theoretic based approaches. Notably, the proposed OCC exhibits some linkages with the system of @Juszczak20091859, in the sense that their solution relies on a MST. However, their approach is substantially different, since they do not use either the information-theoretic, fuzzy sets, or graph partitioning concepts that we exploit in this study. Moreover, our approach is dissimilarity-based, which opens a way to a multitude of applications in different areas. This last aspect recalls the OCC scheme by @NIPS2002_2163; however the authors use different techniques to design their system, which are based on linear programming and prototype selection. Graph-based, and in particular minimum spanning tree based, general clustering algorithms are popular in the literature [@10.1109/TPAMI.2012.226; @1432700; @xu2002clustering; @Galluccio201396; @Galluccio:2012:GBK:2184924.2185067], since, in fact, a graph provides a powerful data abstraction and a sound mathematical framework. However, to the best of our knowledge, the use of graph-based entropy estimation techniques for the design of the OCC model is missing in the OCC literature. The utilization of fuzzy sets to model the DRs establishes another connection with the so-called fuzzy one-class classifiers [@fuzzy_occ; @6613141; @Utkin:2012:FOC:2213741.2433967].
Technical Background Material {#sec:tech_background}
-----------------------------
In the following subsections, we introduce the main concepts used in the OCC proposed in this paper: dissimilarity representation, entropy estimation, and modularity of a graph partition. For more detailed discussions on dissimilarity representation we refer the reader to [@pkekalska+duin2005]; for graph-based entropy estimation to [@intrdim_shapes_hero; @gs:HeroEtAl2002]; finally, for modularity of a graph partition to [@fortunato2010; @4358966].
### Dissimilarity Representation {#sec:dr}
In the dissimilarity representation [@pkekalska+duin2005; @odse; @Duin2012826], the elements of an input dataset $\mathcal{S}\subset\mathcal{X}$ are characterized by considering their pairwise dissimilarity values. The key component is the definition of a nonnegative (bounded) dissimilarity measure $d: \mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}^{+}$, which is in charge of synthesizing all relevant commonalities among the input patterns of $\mathcal{S}$ into a single real-valued number. A set of prototypes, $\mathcal{R}$, called representation set (RS), is used to develop the dissimilarity matrix $\mathbf{D}$, which is given as $D_{ij}=d(x_i, r_j)$, for every $x_i\in\mathcal{S}$ and $r_j\in\mathcal{R}$. Of course, the determination of the most suitable $\mathcal{R}$, with usually $\mathcal{R}\subset\mathcal{S}$, is an important objective. Different techniques are discussed by @pkekalska+duin2005, which span from prototype selection strategies to criteria related to the embedded data. By using directly the rows of $\mathbf{D}$ as embedding vectors, we can obtain the so-called dissimilarity space representation (DSR). This is the fastest way to represent the input data as $\mathbb{R}^d$ vectors by starting from the dissimilarity values. In addition, any common algebraic structure can be defined on the dissimilarity space (DS), making this approach very flexible.
An important property of the DSR is that distances in the DS are just scaled by a factor equal to $\sqrt{|\mathcal{R}|}$ w.r.t. those in the input space (see [@pkekalska+duin2005 Sec. 4.4.1] for details).
### Graph-based Entropy Estimation {#sec:e_mst}
Let $X$ be a continuous random variable with PDF $p(\cdot)$. The $\alpha$-order Rényi entropy measure is defined as $$\label{eq:differential_entropy}
H_{\alpha}(X)=\frac{1}{1-\alpha}\log\left(\int p(x)^{\alpha}dx\right), \ \alpha\geq0, \alpha\neq1.$$
Let us assume to have a data sample $X_{n}$ of $n$ i.i.d. realizations of $X$, with $\underline{\mathbf{x}}_i\in X_{n}\subset\mathbb{R}^{d}, i=1, 2, ..., n$, and $d\geq 2$. Let $G$ be the complete Euclidean graph constructed over $X_{n}$. An edge $e_{ij}$ connecting $\underline{\mathbf{x}}_i$ and $\underline{\mathbf{x}}_j$ is weighted using a weight based on their distance, $|e_{ij}| = d_{2}(\underline{\mathbf{x}}_i, \underline{\mathbf{x}}_j)$. The $\alpha$-order Rényi entropy (\[eq:differential\_entropy\]) can be estimated according to a geometric interpretation of an entropic spanning graph of $G$. Examples of such graphs used in the literature are the MST, *k*-NN graph, Steiner tree, and TSP graph [@md_ent; @Bonev2013214; @pal_renyi_e_knn__2010; @odse2__arxiv; @4897236; @neemuchwala2005image; @oubel2005assessment; @intrdim_shapes_hero; @Hero_Asympt__1999]. In this paper, we will focus on the MST [@bonev__2008; @neemuchwala2005image]. Let $L_{\gamma}(G)$ be the *weighted length* of a MST, $T$, connecting the $n$ points in $X_{n}$, $$\label{eq:l_mst}
L_{\gamma}(G) = \displaystyle\sum_{e_{ij}\in T} |e_{ij}|^{\gamma} ,$$ where $\gamma\in(0, d)$ is a user-defined parameter. The Rényi entropy of order $\alpha\in(0, 1)$, elaborated using the MST length (\[eq:l\_mst\]), is defined as follows: $$\label{eq:rentropy_mst}
\hat{H}_{\alpha}(G) = \frac{d}{\gamma}\left[ \ln\left(\frac{L_{\gamma}(G)}{n^{\alpha}}\right) - \ln\left(\beta(L_{\gamma}(G), d)\right) \right],$$ where $\alpha=(d-\gamma)/d$. The $\beta(L_{\gamma}(G), d)$ term is a constant that is defined as $\beta(L_{\gamma}(G), d) \simeq \gamma/2\ln\left( d/2\pi e \right)$. As a consequence of $G$, the entropy estimator (\[eq:rentropy\_mst\]) is suitable for processing high-dimensional input data.
### Modularity of a Graph Partition {#sec:modularity}
A graph $G=(\mathcal{V}, \mathcal{E})$ is a pair of vertices and edges. An edge models a binary relation among two vertices. Normally, an edge either exists or does not exist. However, in the case of *weighted* graphs, every edge $e_{ij}\in\mathcal{E}$ is associated with a real-valued number called the weight, $w_{ij}=w(e_{ij})$, which determines the strength of the relation. The weighted adjacency matrix $\mathbf{A}$ of $G$ is defined as $A_{ij}=w_{ij}$. The degree of a vertex $v_i\in\mathcal{V}$ is defined as $\mathrm{deg}(v_i)=\sum_{j=1}^{|\mathcal{V}|} A_{ij}$. We will consider weighted graphs with weights in $[0, 1]$. If not diversely specified, when we refer to the “number” of edges we actually refer to the sum of their weights.
A partition [@Livi_ga_2013], $K(G)$, of order $k$ of a graph $G=(\mathcal{V}, \mathcal{E})$, is commonly intended as a partition of the vertex set $\mathcal{V}(G)$ into disjoint subsets (clusters, modules), $K(G)=\{\mathcal{C}_1, \mathcal{C}_2, ..., \mathcal{C}_k\}$. A well-established measure to determine the quality of $K(G)$ is the so-called modularity measure [@brandes+gaertler+wagner2003; @fortunato2010; @1367-2630-10-5-053039; @PhysRevE.71.046117; @4358966; @rosvall2007information], which basically quantifies how well $K(G)$ groups the vertices of $G$ into compact and separated clusters. Intuitively, in a graph a cluster of vertices is compact if the number of the intra-cluster edges is considerably greater than the one of the inter-cluster edges. The modularity measure $Q(\cdot, \cdot)$ is formally defined as follows, $$\begin{aligned}
\label{eq:modularity}
&Q(G, K(G)) = \frac{1}{2|\mathcal{E}(G)|}\sum_{i=1,j=1}^{k} \left( A_{ij}-\frac{\mathrm{deg}(v_i)\mathrm{deg}(v_j)}{2|\mathcal{E}(G)|} \right) \delta(\mathcal{C}_i, \mathcal{C}_j),\end{aligned}$$ where the graph $G$ is assumed to be partitioned according to a given $K(G)$. Eq. \[eq:modularity\] can be conveniently rewritten in terms of edges only ([@4358966; @1367-2630-10-5-053039]): $$\label{eq:modularity2}
Q(G, K(G)) = \sum_{l=1}^{k} \left[ \frac{|\mathcal{E}(\mathcal{C}_l)|}{|\mathcal{E}(G)|}-\left( \frac{\mathrm{deg}(\mathcal{C}_l)}{2|\mathcal{E}(G)|} \right)^2 \right].$$
In the above expression, $|\mathcal{E}(\mathcal{C}_l)|$ is the number of intra-cluster edges, and $\mathrm{deg}(\mathcal{C}_l)$ is the sum of degrees of the vertices in the l*th* cluster (considering all edges, i.e., also those with one end-point outside $\mathcal{C}_i$). Detailing the terms in (\[eq:modularity2\]), we have: $$\begin{aligned}
|\mathcal{E}(\mathcal{C}_l)| = \sum_{\{e_{ij}|\ v_i,v_j\in\mathcal{C}_l\}} w_{ij};\ \ \ \mathrm{deg}(\mathcal{C}_l) &= \sum_{v_i\in\mathcal{C}_l} \mathrm{deg}(v_i); \ \ \ |\mathcal{E}(G)| = \sum_{e_{ij}\in\mathcal{E}(G)} w_{ij}.\end{aligned}$$
The modularity of a graph $G$ is equal to the modularity of the partition of $G$ that maximizes (\[eq:modularity2\]). Finding such an optimal partition is NP-complete [@4358966], and therefore many heuristics has been proposed in the literature [@fortunato2010]. It is well-known that the modularity (\[eq:modularity2\]) assumes values in $[-1/2, 1]$ [@4358966] (the higher, the better). In this paper, we consider the normalized term $M(G, K(G))\in[0, 1]$, defined as: $$\label{eq:normalized_modularity}
M(G, K(G))=\log(3/2 + Q(G, K(G)))/\log(5/2).$$
The Proposed One-Class Classifier {#sec:eocc}
=================================
Given an input dataset $\mathcal{S}\subset\mathcal{X}$, we design a one-class classifier that is applicable to any input domain, $\mathcal{X}$. To fulfill such a requirement, we first embed the input data, $\mathcal{S}$, into an Euclidean space. Notably, we implement the embedding by constructing the DSR of $\mathcal{S}$ (see Sec. \[sec:dr\]). Let $d_{\mathrm{I}}: \mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}^+$ be a suitable dissimilarity measure. We assume here that $d_{\mathrm{I}}(\cdot, \cdot)$ depends on some (numerical) parameters, say $p$, which alter the resulting view of the input data, $\mathcal{S}$, in the DS. The derived DSR of $\mathcal{S}$ is accordingly denoted with $D(\mathcal{S}, \mathcal{R}, p)$, where $\mathcal{R}\subseteq\mathcal{S}$ is the RS, and $p$ is the specific parameters instance of $d_{\mathrm{I}}(\cdot, \cdot)$.
The embedded data are then represented by constructing a complete graph $G=(\mathcal{V}, \mathcal{E})$. Vertices $\mathcal{V}$ denote the patterns in $\mathcal{S}$, while edges denote their relations in terms of distance; each edge $e_{ij}\in\mathcal{E}$ has a weight given by $w_{ij}=d_2(v_i, v_j)$, where $d_2(\cdot, \cdot)$ is a suitable (normalized) Euclidean metric. Graph representations are popular for high-dimensional patterns [@gs:HeroEtAl2002; @oubel2005assessment]. Constructing $G$ allows us also to avoid the computational burden of determining the optimal RS, $\mathcal{R}$; the size of the RS corresponds to the dimensionality of the DS, and therefore usually a proper prototype selection method is used to consider the smallest but most informative subset of $\mathcal{R}$. Since $d_2(\cdot, \cdot)$ is Euclidean, $G$ can be used also to estimate the $\alpha$-order Rényi entropy [@intrdim_shapes_hero; @gs:HeroEtAl2002] of the underlying data distribution through the computation of the entropic MST (see Sec. \[sec:e\_mst\]). The entropy is a powerful mesoscopic data descriptor – it is also a measure of the *spread* of the data – that we use together with other terms to guide the synthesis of the OCC model. In the following, we denote with $G(p)$ the (complete) Euclidean graph constructed over $D(\mathcal{S}, \mathcal{R}, p)$, obtained by setting $d_{\mathrm{I}}(\cdot, \cdot)$ with the $p$ instances.
$G(p)$ provides an abstract framework in which we can develop the model, the related decision rules, and the synthesis of the OCC. Modules (i.e., vertex clusters) of $G(p)$, which we denote as a collection $K(G(p))=\{\mathcal{C}_1, \mathcal{C}_2, ..., \mathcal{C}_k\}$, are considered as suitable DRs, i.e., the OCC model; we derive such DRs by exploiting the concept of modularity (see Sec. \[sec:modularity\]). We maximize the modularity (\[eq:modularity2\]) of the graph by analyzing a partition derived directly on the entropic MST, which has to be computed in order to calculate the $\alpha$-order entropy. The point here is that the vertex degrees in the MST are a scaled approximation of those in $G(p)$ – highly central vertices in $G(p)$ remain well-connected also in the MST. Let $T(G(p))$ be a MST of $G(p)$. By using $T(G(p))$, we can induce a partition $K(G(p))=\{\mathcal{C}_1, \mathcal{C}_2, ..., \mathcal{C}_k\}$ whose quality can be evaluated by considering all edges of $G(p)$. Since the weights of $G(p)$ denote the pairwise normalized Euclidean distances in the DS, for the purpose of calculating the modularity (\[eq:modularity2\]) of $K(G(p))$, we consider instead the quantity $\overline{w}_{ij}=1-w_{ij}$ (the lower the distance, the higher the contribution in terms of modularity).
In the one-class classification setting we model the target class only. Therefore, we need to conceive an inference mechanism that takes into account the implicit uncertainty in the definition of the DR boundaries. To this aim, we equip each cluster $\mathcal{C}_i$ with a membership function [@pedrycz1990fuzzy; @Livi_ga_2013; @pedrycz1998introduction; @kuncheva2000fuzzy], which efficiently describes the uncertainty of the DR boundaries. In the following, we denote with $F(G(p))=\{\mathcal{F}_1, \mathcal{F}_2, ..., \mathcal{F}_k\}$ the fuzzy set-based DRs, which will be used during the test stage of the classifier to provide *soft* decisions.
The aim of the synthesis is to optimize the OCC model by searching for the best $p$ instances. We provide two objective functions (related optimization problems are always intended to realize maximization). The first one considers a linear convex combination of entropy and modularity, calculated on the training set $\mathcal{S}_{tr}\subset\mathcal{S}$ only; $\mathcal{S}_{tr}$ contains target patterns only. The two terms are clearly in conflict. In fact, the entropy favors the general spread–separation of the data in the DS, while the modularity constraints the data to group into compact clusters of $G(p)$. This combination yields solutions that help magnifying the structure of the DRs in $G(p)$. The second objective function is designed to train the OCC model by cross-validation, i.e., we effectively test the OCC model instances on a validation set, $\mathcal{S}_{vs}$, containing both target and non-target patterns. The first approach is considerably faster for what concerns the training stage, although the second one provides a more effective solution in terms of test set recognition performance.
Fig. \[fig:bloch\_scheme\] provides a block scheme describing the proposed OCC, while Fig. \[fig:functioning\] conveys the same information but using more intuitive illustrations.
![Block scheme of training and test stages of the OCC; first objective function is assumed. $\mathcal{S}_{tr}$ is used to synthesize the OCC model. The optimal parameters, $p^{*}$, are used to generate the fuzzy model, $F(G(p^{*}))$. $\mathcal{S}_{ts}$ is first embedded into the DS obtained by means of $p^{*}$, and successively it is tested. The OCC outputs both Boolean and membership grades to the target class. []{data-label="fig:bloch_scheme"}](./block_scheme)
OCC Model and Related Testing {#sec:occ_test}
-----------------------------
A test pattern is classified by considering the optimal parameters $p^*$, which yield the graph $G(p^*)$ constructed over $D(\mathcal{S}_{tr}, \mathcal{R}, p^*)$, and the derived hard $K(G(p^*))$ and fuzzy $F(G(p^*))$ partitions. Each fuzzy set $\mathcal{F}_i\in F(G(p^*)), i=1, 2, ..., k$, forms a fuzzy DR that is described by its membership function $\mu_{\mathcal{F}_{i}}(\cdot)$ and by a quantity $\tau_{i}>0$. The membership function $\mu_{\mathcal{F}_i}(\cdot)$ is parametrized by $\tau_{i}$, which is explicitly denoted as $\mu_{\mathcal{F}_{i}}(\cdot; \tau_i)$. The scalar $\tau_{i}$ is determined by considering a statistics of the $\mathcal{C}_i$ intra-cluster edge weights (e.g., average, standard deviation).
A test pattern $x\in\mathcal{S}_{ts}$ is first mapped to a dissimilarity vector $\underline{\mathbf{v}}$ by setting $d_{\mathrm{I}}(\cdot, \cdot)$ with $p^*$ and considering the dissimilarity w.r.t. $\mathcal{R}$. We define the soft decision function (SDF), which outputs a continuous value in $[0, 1]$ quantifying the membership degree of a test pattern to the target class, as: $$\begin{aligned}
\label{eq:soft_decision_func}
\mathrm{SDF}(\underline{\mathbf{v}}) = \displaystyle\bot_{i=1}^{k} \mu_{\mathcal{F}_{i}}(\underline{\mathbf{v}}; \tau_i).\end{aligned}$$
In the above expression, $\bot$ is a t-conorn (e.g., the maximum) and $\mu_{\mathcal{F}_{i}}(\cdot; \tau_i)$ is the membership function synthesized during the training – the membership is a function of the distance of $\underline{\mathbf{v}}$ w.r.t. the cluster representative, $R(\mathcal{C}_i)$. Please note that here we do not provide closed-form expressions for $\bot$ and $\mu_{\mathcal{F}_{i}}(\cdot; \tau_i)$, since those two factors are general and they can be implemented in different fashions. Later in the experiments section we will specify the setting that we adopted in this study.
It is important to provide also a hard (Boolean) decision function (HDF) about the classification. To this end, we exploit the cluster extent, $\tau_i$, for defining the HDF. Let $$\mathcal{F}_{\mathrm{max}} = {\operatornamewithlimits{arg\ max}}_{\mathcal{F}_i\in F(G(p^*))} \mu_{\mathcal{F}_{i}}(\underline{\mathbf{v}}; \tau_i)$$ be the fuzzy set in which $\underline{\mathbf{v}}$ achieves the maximum membership degree; $\mathcal{C}_{\mathrm{max}}$ is the corresponding non-fuzzy DR. Then, $$\label{eq:hard_decision_1}
\mathrm{HDF}(\underline{\mathbf{v}}) =
\begin{cases}
1 & \mathrm{if}\ d_{2}(\underline{\mathbf{v}}, R(\mathcal{C}_{\mathrm{max}}))\leq \tau_{\mathrm{max}}, \\
0 & \mathrm{otherwise.}
\end{cases}$$
Fig. \[fig:fuzzy\_model\_graph\] provides a schematic view of the test of an embedded pattern, $\underline{\mathbf{v}}_{\mathrm{test}}$. Testing of the OCC model is characterized by a computational complexity given by the embedding of $x$ and the computation of both SDF and HDF. The embedding can be performed in $O(|\mathcal{S}|+D |\mathcal{R}|)$, where $|\mathcal{S}|$ is the linear cost of deriving $\mathcal{R}$ from the input (training) data (that might be constant in the case $\mathcal{R}=\mathcal{S}$), and $D$ denotes the computational complexity of the dissimilarity measure for the input data. SDF can be computed by considering the $k$ different membership degrees; the same holds for HDF.
![Fuzzy model constructed over $G(p)$. A test pattern, $\underline{\mathbf{v}}_{\mathrm{test}}$, is classified as non-target by the HDF, while considering the SDF it gets a membership degree, $\mu_{\mathcal{F}_i}(\underline{\mathbf{v}}_{\mathrm{test}}; \tau_i)$, to the target class.[]{data-label="fig:fuzzy_model_graph"}](./FuzzyModel_Graph)
Synthesis of the OCC Model {#sec:fast_synthesis}
--------------------------
We cast the synthesis of the proposed OCC as the optimization of $G(p)$ w.r.t. the parameters $p\in\mathcal{P}$ of $d_{\mathrm{I}}(\cdot, \cdot)$. Note that in this paper, we make the fair assumption that $\mathcal{P}=[0, 1]^u$, where $u$ is the number of parameters characterizing $d_{\mathrm{I}}(\cdot, \cdot)$. The idea is to determine the best-performing parameters setting so that the following objective function is maximized: $$\label{eq:obj_func}
\max_{p\in\mathcal{P}} \eta \hat{H}_{\alpha}(G(p)) + (1-\eta) M(G(p), K(G(p))),\ \eta\in[0, 1].$$
We evaluate two conflicting quantities on $G(p)$: the entropy, $H(G(p))$, and the modularity, $M(G(p), K(G(p)))$. The entropy term favors the construction of a DS and related graph, $G(p)$, such that the overall distance among the vertices/patterns is maximized (the higher the entropy, the higher the spread of the data). On the other hand, evaluating the modularity of the derived partitioning, $K(G(p))$, constrains the optimization to search for solutions that magnify also the “community” structure of the graph – community is a term [@fortunato2010] that is used to denote a compact and populated cluster of vertices. This results in a graph $G(p)$ that is suitably optimized to derive the DRs in terms of compact and separated clusters of vertices.
However, to evaluate the modularity we first need to generate a partition, $K(G(p))$. In the next section, we describe the algorithm that we designed to derive a partition of $G(p)$, whose aim is to quickly find a reasonable approximation of the optimal modularity of $G(p)$.
### Greedy Edge Pruning Approach to Graph Partitioning
The following algorithm exploits the fact that we need to compute the MST, $T(G(p))$, of $G(p)$ to calculate the $\alpha$-order Rényi entropy (see Sec. \[sec:e\_mst\]). Once we have the MST, we form clusters on $T(G(p))$ iteratively by pruning (i.e., removing) those edges with higher Euclidean distance values. By construction of the MST, those edges are likely to be “bridges” among well-separated components of $G(p)$. A MST of a graph with $n$ vertices has $n-1$ edges, and hence the pruning loop is repeated at most $n-1$ times. Therefore, at iteration $i=0, 1, ..., n-1$, we partition the vertices of the MST into $k=i+1$ connected components. The connected components of the MST are used to derive a partition on $G(p)$, by considering exactly the same grouping of the vertices. However, in $G(p)$ we have full information of the edges, which we use to compute the modularity of the resulting partition (\[eq:modularity2\]). To terminate the procedure, we exploit the following greedy assumption. Since the MST is connected, by first removing edges with maximum weight, we will form the most interesting communities/clusters in terms of modularity. As a consequence, if at iteration $i+1$ we get a modularity value lower than the one obtained at iteration $i$, we stop the algorithm, returning the last computed partition. Algorithm \[alg:edge\_pruning\] delivers the pseudo-code of the herein described procedure.
The MST $T(G(p))$ and the graph $G(p)$ with $n$ vertices A partition $K(G(p))$ of $G(p)$ Set $i=0$ and best modularity $M_i=-1$ Let $L$ be a list with the $n-1$ edges of $T(G(p))$ in non-increasing order Remove $i$th edge from $T(G(p))$ and determine the resulting connected components, $K(T(G(p)))_i$ Derive the corresponding partitioning $K(G(p))_i$ of $G(p)$ by considering the same vertex grouping of $K(T(G(p)))_i$ Set $M_i$ according to the evaluation of (\[eq:modularity2\]) on $K(G(p))_i$ $K(G(p))_{i-1}$ $K(G(p))_{i}$
### Analysis of Computational Complexity {#sec:cca_eocc1}
The overall computational overhead of synthesizing the OCC using the herein explained approach is characterized by the sum of the following costs: (i) embedding, (ii) graph construction and entropy estimation, (iii) determination of graph partition, and finally (iv) the generation of the membership functions. The first three components must be considered into a suitable optimization loop, while the last one is performed only once at the end of the optimization cycle.
The dissimilarity representation of the training set $\mathcal{S}_{tr}, n=|\mathcal{S}_{tr}|$, costs $O(n|\mathcal{R}|D)$, where $D$ is the cost of the dissimilarity measure for the input data. The second cost can be summarized as follows: $$\begin{aligned}
\label{eq:rentropy_mst_complexity}
O\left( \frac{n(n-1)}{2}E + \frac{n(n-1)}{2}\times\log\left(\frac{n(n-1)}{2}\right) + (n-1) \right).\end{aligned}$$
The first term standing in (\[eq:rentropy\_mst\_complexity\]) accounts for the generation of $G(p)$, computing the respective Euclidean distances for the edge weights ($E$ is the cost). The second term quantifies the cost involved in the MST computation using the well-known Kruskal’s algorithm. The last term in (\[eq:rentropy\_mst\_complexity\]) concerns the computation of the MST length. The third cost is given by Algorithm \[alg:edge\_pruning\]. The main cycle is repeated a maximum of $n-1$ times. At each iteration, we derive the connected components on $T(G(p))$, which costs $O(n-1)$. To induce the partitioning on $G(p)$, we simply cycle through the vertices, grouping them according to the connected components. Eq. \[eq:modularity2\] can be computed in $O(n(n-1))$. Putting all together, we have for Algorithm \[alg:edge\_pruning\] $$O((n-1)\times( 2(n-1) + n(n-1) ),$$ which is dominated by a cubic computational complexity in the number of training patterns. The last cost (membership function elicitation) depends on the order of the derived best partition. In particular, for each cluster $\mathcal{C}_i$ we determine the membership function by first deriving $\tau_{\mathcal{C}_i}$. In the case average intra-cluster distances are considered, then the cost of this step is $O(2n)$.
The overall worst-case cost, considering as main parameter $n$, is given by the determination of the partition with best modularity. However, since we have designed the system to operate in the DS, the actual cost depends also on $D$, which may have a significant impact in case of complex input data types.
Synthesis By Cross-validation {#sec:synthesis_crossval}
-----------------------------
Eq. \[eq:obj\_func\] defines an objective that does not take into account explicitly the recognition capability of the synthesized model. A blind partitioning that derives $K(G(p))$ according to Algorithm \[alg:edge\_pruning\] might suffer from the problem of generating a too simple model, i.e., with too few DRs. In fact, Algorithm \[alg:edge\_pruning\] constraints the partition to be formed only by those clusters that induce an well-defined community structure in $G(p)$. However, since in the one-class setting we synthesize the model on the target class only, a well-formed cluster/community structure in $G(p)$ may not be easy to identify, especially in hard problems. By relaxing the imperative of finding the partition with maximum modularity, we can conceive another objective function that allows us to force the derivation of additional DRs. The alternative objective function to be considered reads as, $$\begin{aligned}
\label{eq:obj_func2}
\max_{p\in\mathcal{P}}\ &\beta P(\mathcal{S}_{tr}, \mathcal{S}_{vs}; F(G(p))) +& \\
\nonumber&(1-\beta) \left[\eta \hat{H}_{\alpha}(G(p)) + (1-\eta) M(G(p), K(G(p)))\right],\end{aligned}$$ where $\eta,\beta\in[0, 1]$. Eq. \[eq:obj\_func2\] takes explicitly into account a measure of recognition performance, $P(\mathcal{S}_{tr}, \mathcal{S}_{vs}; F(G(p)))$, achieved on a validation set, $\mathcal{S}_{vs}$. This term is combined (again with a linear convex combination) with (\[eq:obj\_func\]). Therefore, the final model does not necessarily imply the best possible modularity, $M(G(p), K(G(p)))$, and entropy, $\hat{H}_{\alpha}(G(p))$, of $G(p)$, focusing instead on the solutions that perform better also in terms of recognition. This choice, potentially, implies obtaining a more complex model (i.e., a partition characterized by more clusters, with lower overall modularity), although, as we will observe in the experiments, it usually provides also a more effective classification system on the test set.
Algorithm \[alg:crossval\] shows the pseudo-code of the herein described training scheme. DRs are derived by exploiting, basically, the same MST-based graph partitioning approach (Algorithm \[alg:edge\_pruning\]). In fact, DRs are derived incrementally, by first removing edges that are more likely to induce well-formed clusters (i.e., those edges with higher weights).
The training set $\mathcal{S}_{tr}$, the validation set $\mathcal{S}_{vs}$, and the parameters $p$ A fuzzy partition $F(G(p))$ Determine the DSR $D(\mathcal{S}_{tr}, \mathcal{R}, p)$ and $D(\mathcal{S}_{vs}, \mathcal{R}, p)$ Construct $G(p)$ over $D(\mathcal{S}_{tr}, \mathcal{R}, p)$ Estimate the entropy, $H(G(p))$, and get the MST, $T(G(p))$ Let $L$ be a list with the $n-1$ edges of $T(G(p))$ in non-increasing order Set $P_{\mathrm{max}}=0$ Remove $i$th edge from $T(G(p))$. Determine the resulting connected components, $K(T(G(p)))_i$ Derive $K(G(p))_i$ of $G(p)$ by considering the same vertex grouping of $K(T(G(p)))_i$ Set $M_i$ according to the evaluation of (\[eq:modularity2\]) on $K(G(p))_i$ Generate fuzzy model $F(G(p))_i$ from $K(G(p))_i$ Set $P_i$ as the evaluation of the objective function (\[eq:obj\_func2\]) on $\mathcal{S}_{vs}$ $P_{\mathrm{max}}=P_i$ $F(G(p))_{\mathrm{max}}$ corresponding to $P_{\mathrm{max}}$
### Analysis of Computational Complexity {#sec:cca_eocc2}
The training procedure described in Algorithm \[alg:crossval\] implies a substantial change in the OCC scheme illustrated in Fig. \[fig:bloch\_scheme\]. First, the modularity (\[eq:modularity2\]) is computed always exactly $n-1$ times. Additionally, the fuzzy model must be synthesized and then tested on $\mathcal{S}_{vs}$ inside the optimization cycle (see costs related to testing the model in Sec. \[sec:occ\_test\]). Those changes affect considerably the effective running time of the whole procedure, although the computational complexity, asymptotically, is not altered.
Experimental Evaluation {#sec:exps}
=======================
We start by describing the experimental setting and the considered performance measures (Sec. \[sec:pm\_expsettin\]). Then, in Sec. \[sec:exp\_synth\] we provide some preliminary and explanatory tests on synthetically generated problems. In Sec. \[sec:exp\_uci\] we perform experiments on different datasets of feature-based patterns taken from the UCI repository [@Bache+Lichman_2013]. Lastly, in Sec. \[sec:exp\_iam\], we discuss the results obtained over different IAM datasets [@riesen+bunke2008], containing patterns represented as labeled graphs.
Experimental Setting and Performance Measures {#sec:pm_expsettin}
---------------------------------------------
All non-synthetic datasets considered in this paper are originally conceived for multi-class classification problems. Accordingly, we convert them to fit the one-class setting by selecting one class as the target class, and considering all other classes as non-target. If not specified otherwise, we train the proposed OCC over the target patterns, $\mathcal{S}_{tr}$, while we test the model on all patterns, i.e., all targets not used during the training and all available non-target patterns. In the following, we shorten the proposed system as EOCC (standing for Entropic One-Class Classifier). The variant operating with the training scheme described in Sec. \[sec:fast\_synthesis\] is denoted as EOCC-1, while EOCC-2 is used to refer to the variant operating as described in Sec. \[sec:synthesis\_crossval\].
The global optimization is implemented by a genetic algorithm. It performs roulette wheel selection, two-point crossover, and random mutation on the parameters characterizing the dissimilarity measure. In addition, the genetic algorithm implements an elitism strategy which automatically imports the fittest individual into the next population; we set the population size to 30 individuals and the mutation rate to 0.3. Convergence criteria is determined by combining a maximum number of iterations/evolutions (here set to 100) and a check that evaluates if the best fitness is not changed over the last ten evolutions. Such settings are determined by a preliminary tuning of the system. Fuzzification of the vertex clusters is performed by generating Gaussian membership functions. The width/size of the Gaussian is set according to $\tau_{\mathcal{C}_i}$, which is computed as the intra-cluster average distance. Gaussian membership functions are symmetric and they can be described by a single parameter (i.e., the width/size). This fact is in agreement with the single-parameter description of the intra-cluster distances distribution. The RS, $\mathcal{R}$, is defined equal to $\mathcal{S}_{tr}$ (no selection is performed at all). Note that performing the DSR of the input data is not necessary when processing feature-based patterns. However, we do embed also this type of data to provide a uniform view of the experimental results. $\gamma$ in (\[eq:rentropy\_mst\]) is set to 0.02; $\eta,\beta$ in (\[eq:obj\_func\]) and (\[eq:obj\_func2\]) are both set to 0.5. Again, those settings are determined by a preliminary fine-tuning stage.
Software is implemented in standard C++ by means of the SPARE library [@spare_graph_2013]. Tests have been executed on a machine running a 64-bit Linux OS, equipped with an Intel(R) Core(TM) i7-3930K CPU 3.20GHz and 32 Gb of RAM.
When considering the SDF, we must rely on an appropriate test set performance measure that takes into account the “scorings” (in our case, the membership degrees to the target class) assigned to the test patterns. In particular, in this paper we consider the Area Under the ROC Curve (AUC); AUC is computed according to Ref. [@Fawcett:2006:IRA:1159473.1159475]. AUC is a robust statistics that gives the average probability that a target pattern is ranked higher than a non-target one. When using the HDF, instead, we evaluate the obtained confusion matrix by analyzing standard measures such as accuracy, recall, precision, F-measure and so on. The performance measure $P(\cdot, \cdot, \cdot)$ in (\[eq:obj\_func2\]) is defined as the accuracy on the validation set.
All results are reported in terms of averages of ten different runs executed with different random seeds; we report standard deviations and analysis of statistical significance with t-test.
Synthetic Data {#sec:exp_synth}
--------------
Fig. \[fig:ds1\_full\] illustrates an example in which the target class is distributed in three well-separated spherical clusters. EOCC is trained on the target instances (in red) and it is tested on the green and blue instances (the green instances actually belong to the target class, while those in blue are non-target instances) – see Fig \[fig:ds1\]. This is a very simple instance and in fact the EOCC solves the problem without errors (considering both training schemes). The obtained membership degrees to the target class are plotted in Fig. \[fig:ds1\_memberships\]. The fact that EOCC synthesizes three DRs corresponds to the best solution in terms of modularity, as demonstrated in Fig. \[fig:ds1\_modularity\], where the modularity value (\[eq:normalized\_modularity\]) of all possible partitions derivable from the MST are plotted – note that here we consider a specific instance of the parameters of the input data dissimilarity measure (weighted Euclidean metric). Finally, in Fig. \[fig:ds1\_entropy-modularity\] we show the entropy and modularity trends during the iterations of the optimization (\[eq:obj\_func\]). Since the problem is simple, the increments are numerically small, although the monotonic trend is clearly recognizable. It is worth noting the early convergence at the 30-th iteration, since in fact both entropy and modularity – and hence the fitness (\[eq:obj\_func\]) – are stuck at the same values.
Fig. \[fig:ds2\_full\] shows a more subtle problem. The target instances shown in red are the same as in the previous problem. Pattern instances used for testing the OCC are now centered at $x=0.5$, but with a very narrow variance of the y-axis. Target instances used for testing are in blue and violet, while those represented in green are non-target instances – see Fig. \[fig:ds2\]. Results achieved with EOCC-1 are good (for SDF, the AUC is 0.98), although it commits some errors considering the HDF (accuracy is 0.84, since eight test patterns of the target class are misclassified as non-target; represented in violet in figure). Three clusters are synthesized during the training. Fig. \[fig:ds2\_memberships\] reports the membership degrees obtained by EOCC-1. On the other hand, EOCC-2 performs better in terms of SDF (see Fig. \[fig:ds2\_memberships\_crossval\]; AUC is one, since EOCC-2 assigns always higher membership degrees to the target patterns. Moreover, also hard decisions are better, since zero errors are committed. Please note that, for clarity purpose, the validation data used for EOCC-2 is not shown in those examples. This test is exemplar for the different views that can be obtained by considering soft or binary decisions in the one-class classification setting. In fact, the SDF may still provide a consistent picture of the correct labeling of the tested patterns even in complex situations. Additionally, this test yields a first insight on the possible difference in terms of recognition performance among EOCC-1 and EOCC-2 – EOCC-2 is optimized explicitly towards those solutions that perform better.
Finally, Fig. \[fig:ds3\] shows two additional relevant properties of EOCC. In Fig. \[fig:ds3\_ob\] the training/test target data are distributed in two highly separated clusters denoting high variance on the y-axis. By assuming the weighted Euclidean metric as dissimilarity measure, $p$ of (\[eq:obj\_func\]) would consist in a real-valued vector in $[0, 1]^2$. EOCC synthesizes a model with two DRs by finding the best-performing solution, $p^*$, equal to $[0.995, 0.238]$ (average of five runs). This in fact corresponds to what we expected, since the x-coordinate plays the most important role in magnifying the separation (entropy), while the y-coordinate the compactness (modularity) of the partition. In Fig. \[fig:ds3\_isolated\] we show a situation in which the training data contains an isolated pattern (shown in the upper-left corner in green). In our interpretation of the one-class classification setting, such a pattern is not an outlier, since in fact outliers should be identified during the test stage. As a consequence, EOCC synthesizes four DRs.
Results on UCI Datasets {#sec:exp_uci}
-----------------------
Tab. \[tab:uci\_ds\] presents the details of the herein considered UCI datasets [@Bache+Lichman_2013]. @Juszczak20091859 provide results of experimentations completed for the one-class setting, which we use for comparison in our study; where possible, missing results have been retrieved from [@OCC_results]. Please note that to provide a consistent comparison with Ref. [@Juszczak20091859], we actually considered the versions of the UCI datasets downloaded from [@OCC_results]. We consider two versions of those data: (i) non-normalized (as downloaded from the reference) and (ii) normalized by ensuring zero-mean and unit variance for each component. The UCI datasets usually do not provide a validation set explicitly. Since our principal aim here is the comparison of the proposed EOCC with the results presented in Ref. [@Juszczak20091859], we need to consider the same training/test split scheme. As a consequence, in the case of EOCC-2 each validation set, $\mathcal{S}_{vs}$, is generated by applying a suitable zero-mean Gaussian noise to a randomly selected 10% of the target/non-target data in $\mathcal{S}_{ts}$. Nonetheless, as a demonstration of reliability of the method, we report also the results achieved defining a proper validation set by taking 10% of the target and non-target patterns from $\mathcal{S}_{ts}$ (in the following tables, this is indicated as “EOCC-2\_10%”). Since these datasets contain patterns described by features, the dissimilarity measure that we use is the weighted and normalized Euclidean metric, with weights $\underline{\mathbf{w}}\in[0, 1]^u$, where $u$ is the number of features characterizing the dataset at hand. For settings of the herein considered reference systems we refer the reader to Ref. [@Juszczak20091859].
Tab. \[tab:uci\_ds\_results\] and \[tab:uci\_ds\_results2\] show the average AUC results for EOCC-1, EOCC-2, and EOCC-2\_10%, respectively on the considered low- and high-dimensional UCI datasets. Results of Tab. \[tab:uci\_ds\_results\] show very competitive performances of all EOCC variants w.r.t. the others; in four out of seven seven datasets (i.e., BW, D, E, and I) we achieve the highest AUC, considering either the non-normalized and the normalized dataset instances. Data normalization, usually, does not affect the results, with the only exception for the L dataset, in which EOCC performances degrades significantly; it is worth noting that this is observed also for the competitors. Results of Tab. \[tab:uci\_ds\_results2\] still denote good EOCC performances, although we score the best AUC in two datasets only (S in the non-normalized case, and in SP normalized setting). Results on the other datasets are in general comparable, although we observe performance degradation for all three systems on C; however, results are never statistically worse than all competitors. Nonetheless, it is worth pointing out that, although EOCC relies on an entropy estimator (\[eq:rentropy\_mst\]) that in turn is (indirectly) based on the data PDF, we do not observe any severe performance breakdown when processing high-dimensional data, as it is usually observed with PDF-based methods (such as Parzen). This follows from the fact that we initially perform a DSR, whose dimensionality is given by the size of the representation set. Therefore, since we used $\mathcal{R}=\mathcal{S}_{tr}$ in the experiments, one should expect to see some performance degradation as the training data size grows (for instance, when considering AB, BW, D, BC-D, and CO). However, on the contrary we achieve good results on those datasets, demonstrating the effectiveness of the combined dissimilarity and graph based approach adopted in EOCC. Standard deviations are in general very low, denoting highly stable and reliable results.
[|c|c|c|c|c|c|c|]{} **UCI Dataset** & **Acronym** & **Target class** & **\# Target** & **\# Non-target** & **\# Params**\
\
Abalone & AB & 1 & 1407 & 2770 & 10\
Biomed & BI & normal & 127 & 67 & 5\
Breast Wisconsin & BW & benign & 458 & 241 & 9\
Diabetes (prima indians) & D & present & 500 & 268 & 8\
Ecoli & E & pp & 52 & 284 & 7\
Iris & I & Iris-setosa & 50 & 100 & 4\
Liver & L & healthy & 200 & 145 & 6\
\
Arrhythmia & AR & normal & 237 & 183 & 278\
Breast cancer wisconsin (diagnostic) & BC-D & B & 357 & 212 & 30\
Breast cancer wisconsin (prognostic) & BC-P & N & 151 & 47 & 33\
Colon & C & normal & 22 & 40 & 1908\
Concordia & CO & 2 & 400 & 3600 & 1024\
Sonar & S & M & 111 & 97 & 60\
Spectf & SP & 0 & 95 & 254 & 44\
[|c|c|c|c|c|c|c|c|]{} **System/Dataset** & **AB** & **BI** & **BW** & **D** & **E** & **I** & **L**\
\
EOCC-1 & 0.685(0.013) & 0.847(0.002) & **0.990(0.003)** & 0.607(0.046) & 0.953(0.002) & **1.000(0.000)** & 0.461(0.006)\
EOCC-2 & 0.831(0.001) & 0.864(0.003) & 0.989(0.001) & **0.717(0.005)** & **0.957(0.003)** & **1.000(0.000)** & 0.536(0.021)\
EOCC-2\_10% & 0.819(0.002) & 0.868(0.007) & 0.929(0.061) & 0.677(0.019) & 0.954(0.003) & **1.000(0.000)** & 0.493(0.021)\
Gauss & 0.861(0.002) & 0.900(0.004) & 0.823(0.002) & 0.705(0.003) & 0.929(0.003) & **1.000(0.000)** & 0.586(0.005)\
MoG & 0.853(0.005) & 0.912(0.009) & 0.785(1.003) & 0.674(0.003) & 0.920(0.004) & **1.000(0.000)** & 0.607(0.006)\
Naïve Parzen & 0.859(0.004) & **0.931(0.002)** & 0.965(0.004) & 0.679(0.003) & 0.930(0.008) & **1.000(0.000)** & **0.614(0.002)**\
Parzen & 0.863(0.001) & 0.900(0.011) & 0.723(0.005) & 0.676(0.004) & 0.922(0.004) & **1.000(0.000)** & 0.590(0.003)\
*k*-Means & 0.792(0.011) & 0.878(0.012) & 0.846(0.035) & 0.659(0.007) & 0.891(1.006) & **1.000(0.000)** & 0.578(1.000)\
1-NN & 0.865(0.001) & 0.891(0.008) & 0.694(0.006) & 0.667(0.007) & 0.902(0.009) & **1.000(0.000)** & 0.590(0.009)\
*k*-NN & 0.865(0.001) & 0.891(0.008) & 0.694(0.006) & 0.667(0.007) & 0.902(0.009) & **1.000(0.000)** & 0.590(0.009)\
Auto-encoder & 0.826(0.003) & 0.856(0.022) & 0.384(0.009) & 0.598(1.008) & 0.878(1.000) & **1.000(0.000)** & 0.564(0.009)\
PCA & 0.802(0.001) & 0.897(0.005) & 0.303(0.010) & 0.587(0.002) & 0.669(0.011) & 0.973(0.008) & 0.549(0.005)\
SOM & 0.814(0.003) & 0.887(0.008) & 0.790(0.023) & 0.692(0.007) & 0.890(0.011) & **1.000(0.000)** & 0.596(0.007)\
MST\_CD & **0.875(0.001)** & 0.898(0.010) & 0.765(0.018) & 0.669(0.007) & 0.897(0.009) & **1.000(0.000)** & 0.580(0.009)\
*k*-Centres & 0.760(0.008) & 0.878(0.024) & 0.715(0.124) & 0.606(0.016) & 0.863(0.012) & **1.000(0.000)** & 0.537(0.041)\
SVDD & 0.806(0.001) & 0.220(0.003) & 0.700(0.006) & 0.577(0.098) & 0.894(0.008) & **1.000(0.000)** & 0.470(0.014)\
MPM & 0.594(0.001) & 0.792(0.057) & 0.694(0.006) & 0.656(0.007) & 0.802(0.005) & **1.000(0.000)** & 0.587(0.009)\
LPDD & 0.697(0.001) & 0.865(0.026) & 0.800(0.005) & 0.668(0.007) & 0.896(0.005) & **1.000(0.000)** & 0.564(0.026)\
CHAMELEON & 0.706(0.004) & 0.727(0.019) & 0.669(0.008) & 0.651(0.010) & 0.758(0.016) & **1.000(0.000)** & 0.580(0.009)\
\
EOCC-1 & 0.693(0.005) & 0.867(0.005) & 0.853(0.020) & 0.670(0.024) & 0.928(0.011) & **1.000(0.000)** & 0.396(0.016)\
EOCC-2 & 0.831(0.001) & 0.878(0.006) & **0.995(0.001)** & **0.751(0.012)** & **0.957(0.004)** & **1.000(0.000)** & 0.460(0.026)\
EOCC-2\_10% & 0.841(0.002) & 0.862(0.016) & **0.995(0.002)** & 0.709(0.023) & 0.954(0.007) & **1.000(0.000)** & 0.452(0.026)\
Gauss & 0.862(0.000) & 0.899(0.005) & 0.985(0.001) & 0.721(0.003) & 0.929(0.003) & **1.000(0.000)** & 0.509(0.005)\
MoG & 0.860(0.003) & 0.911(0.008) & 0.984(0.002) & 0.738(0.003) & 0.929(0.003) & **1.000(0.000)** & 0.494(0.006)\
Naïve Parzen & 0.859(0.000) & **0.931(0.002)** & 0.987(0.001) & 0.678(0.003) & 0.930(0.008) & **1.000(0.000)** & 0.484(0.008)\
Parzen & **0.877(0.001)** & 0.915(0.009) & 0.991(0.001) & 0.756(0.002) & 0.929(0.005) & **1.000(0.000)** & 0.469(0.008)\
*k*-Means & 0.801(0.003) & 0.902(0.009) & 0.984(0.001) & 0.712(0.010) & 0.878(0.015) & **1.000(0.000)** & 0.469(0.014)\
1-NN & 0.862(0.001) & 0.914(0.012) & 0.991(0.001) & 0.721(0.002) & 0.906(0.008) & **1.000(0.000)** & 0.511(0.007)\
*k*-NN & 0.862(0.001) & 0.914(0.012) & 0.991(0.001) & 0.721(0.002) & 0.906(0.008) & **1.000(0.000)** & 0.511(0.007)\
Auto-encoder & 0.836(0.000) & 0.890(0.013) & 0.960(0.002) & 0.658(0.005) & 0.888(0.023) & **1.000(0.000)** & **0.608(0.008)**\
PCA & 0.826(0.001) & 0.776(0.031) & 0.920(0.004) & 0.640(0.006) & 0.655(0.013) & 0.920(0.008) & **0.608(0.008)**\
SOM & 0.838(0.003) & 0.908(0.006) & 0.990(0.002) & 0.709(0.009) & 0.898(0.004) & **1.000(0.000)** & 0.487(0.017)\
MST\_CD & - & 0.914(0.012) & 0.992(0.001) & 0.715(0.003) & 0.899(0.009) & **1.000(0.000)** & -\
*k*-Centres & 0.767(0.017) & 0.906(0.015) & 0.984(0.002) & 0.678(0.009) & 0.870(0.023) & **1.000(0.000)** & 0.483(0.006)\
SVDD & 0.791(0.002) & 0.915(0.009) & 0.988(0.001) & 0.732(0.005) & 0.922(0.010) & **1.000(0.000)** & 0.490(0.010)\
MPM & 0.735(0.002) & 0.909(0.010) & 0.991(0.001) & 0.729(0.003) & 0.922(0.007) & **1.000(0.000)** & 0.521(0.011)\
LPDD & 0.751(0.002) & 0.889(0.008) & 0.989(0.001) & 0.634(0.005) & 0.947(0.004) & **1.000(0.000)** & 0.506(0.005)\
CHAMELEON & - & - & - & - & - & - & -\
[|c|c|c|c|c|c|c|c|]{} **System/Dataset** & **AR** & **BC-D** & **BC-P** & **C** & **CO** & **S** & **SP**\
\
EOCC-1 & 0.683(0.007) & 0.933(0.000) & 0.554(0.002) & 0.631(0.008) & 0.550(0.016) & 0.439(0.016) & 0.727(0.014)\
EOCC-2 & 0.775(0.016) & 0.938(0.001) & 0.585(0.021) & 0.654(0.014) & 0.783(0.032) & **0.998(0.000)** & 0.917(0.000)\
EOCC-2\_10% & 0.707(0.019) & **0.946(0.005)** & 0.503(0.036) & 0.654(0.014) & 0.783(0.032) & 0.945(0.022) & 0.907(0.000)\
Gauss & 0.606(0.006) & - & 0.591(0.009) & 0.704(0.011) & 0.803(0.017) & 0.680(0.031) & 0.833(0.033)\
MoG & 0.577(0.166) & - & 0.511(0.017) & 0.500(0.000) & 0.500(0.011) & 0.704(0.035) & 0.776(0.031)\
Naïve Parzen & 0.774(0.007) & - & 0.535(0.015) & 0.700(0.015) & 0.846(0.007) & 0.532(0.039) & 0.902(0.037)\
Parzen & 0.577(0.166) & - & 0.586(0.029) & 0.364(0.224) & 0.502(0.022) & 0.805(0.031) & 0.879(0.027)\
*k*-Means & 0.766(0.006) & - & 0.536(0.021) & 0.668(0.031) & 0.862(0.025) & 0.698(0.037) & 0.923(0.017)\
1-NN & 0.760(0.008) & - & 0.595(0.025) & 0.713(0.033) & 0.901(0.008) & 0.763(0.043) & 0.926(0.029)\
*k*-NN & 0.760(0.008) & - & 0.595(0.025) & 0.713(0.033) & 0.901(0.009) & 0.696(0.048) & 0.923(0.015)\
Auto-encoder & 0.522(0.021) & - & 0.548(0.037) & 0.500(0.000) & 0.512(0.015) & 0.596(0.065) & 0.817(0.062)\
PCA & **0.807(0.010)** & - & 0.574(0.018) & 0.707(0.016) & 0.824(0.004) & 0.696(0.033) & 0.901(0.030)\
SOM & 0.772(0.007) & - & 0.523(0.030) & 0.682(0.026) & 0.887(0.020) & 0.801(0.034) & 0.975(0.021)\
MST\_CD & 0.796(0.006) & - & **0.611(0.026)** & **0.733(0.030)** & **0.911(0.001)** & 0.811(0.031) & **0.981(0.026)**\
*k*-Centres & 0.767(0.016) & - & 0.584(0.055) & 0.684(0.029) & 0.815(0.036) & 0.668(0.041) & 0.909(0.016)\
SVDD & 0.581(0.164) & - & 0.498(0.242) & 0.364(0.224) & 0.121(0.011) & 0.761(0.032) & **0.978(0.033)**\
MPM & 0.771(0.005) & - & 0.053(0.001) & 0.500(0.000) & 0.901(0.006) & 0.785(0.030) & **0.980(0.074)**\
LPDD & 0.577(0.166) & - & 0.539(0.183) & 0.418(0.200) & 0.864(0.004) & 0.636(0.027) & 0.934(0.033)\
CHAMELEON & 0.760(0.008) & - & - & 0.391(0.051) & 0.807(0.004) & 0.778(0.010) & 0.944(0.007)\
\
EOCC-1 & 0.657(0.035) & 0.843(0.026) & 0.492(0.021) & 0.452(0.010) & 0.550(0.016) & 0.291(0.026) & 0.664(0.014)\
EOCC-2 & 0.745(0.005) & **0.941(0.018)** & 0.490(0.040) & 0.506(0.009) & 0.783(0.032) & 0.349(0.019) & **0.959(0.002)**\
EOCC-2\_10% & 0.707(0.019) & 0.922(0.010) & 0.503(0.016) & 0.506(0.009) & 0.783(0.032) & 0.357(0.027) & 0.927(0.017)\
Gauss & 0.768(0.004) & - & 0.508(0.008) & 0.713(0.029) & 0.858(0.000) & 0.657(0.008) & 0.934(0.008)\
MoG & 0.761(0.004) & - & 0.526(0.016) & - & - & 0.643(0.015) & 0.948(0.008)\
Naïve Parzen & 0.774(0.007) & - & 0.538(0.022) & 0.700(0.015) & 0.846(0.000) & 0.569(0.012) & 0.892(0.009)\
Parzen & 0.773(0.005) & - & 0.522(0.017) & 0.364(0.224) & 0.000(0.000) & 0.695(0.008) & 0.958(0.011)\
*k*-Means & **0.787(0.006)** & - & 0.520(0.020) & 0.716(0.040) & 0.872(0.005) & 0.625(0.016) & 0.867(0.010)\
1-NN & 0.776(0.005) & - & 0.517(0.014) & **0.743(0.012)** & **0.888(0.000)** & 0.698(0.006) & **0.959(0.011)**\
*k*-NN & 0.776(0.005) & - & 0.517(0.014) & **0.743(0.012)** & **0.888(0.000)** & 0.698(0.006) & **0.959(0.011)**\
Auto-encoder & - & - & 0.520(0.011) & - & - & 0.594(0.014) & 0.850(0.001)\
PCA & 0.776(0.004) & - & **0.557(0.011)** & 0.707(0.019) & 0.853(0.000) & 0.608(0.009) & 0.807(0.020)\
SOM & **0.787(0.008)** & - & 0.511(0.021) & 0.729(0.019) & 0.878(0.001) & 0.711(0.012) & 0.860(0.003)\
MST\_CD & 0.778(0.005) & - & 0.527(0.018) & 0.735(0.022) & **0.888(0.000)** & **0.715(0.006)** & 0.957(0.011)\
*k*-Centres & 0.778(0.011) & - & 0.528(0.020) & 0.732(0.018) & 0.849(0.013) & 0.622(0.013) & 0.817(0.013)\
SVDD & 0.527(0.094) & - & 0.517(0.017) & 0.364(0.224) & 0.000(0.000) & 0.705(0.054) & 0.897(0.032)\
MPM & 0.771(0.005) & - & 0.518(0.018) & 0.000(0.000) & 0.324(0.000) & 0.696(0.008) & 0.901(0.028)\
LPDD & 0.783(0.006) & - & 0.531(0.017) & 0.368(0.224) & 0.741(0.013) & 0.644(0.005) & 0.956(0.009)\
CHAMELEON & - & - & - & - & - & - & -\
Results on IAM Graph Datasets {#sec:exp_iam}
-----------------------------
The IAM repository is a variegated database containing many different datasets of labeled graphs [@riesen+bunke2008]. We consider here six datasets, namely: AIDS, GREC, Letter-Low, Letter-High, Mutagenicity, and Protein. Tab. \[tab:iam\_ds\] shows the relevant information regarding the data. IAM datasets already contain a suitable validation set. Therefore, to adapt the considered dataset to our setting, we just move all non-target patterns of the training set into $\mathcal{S}_{ts}$. We process the input labeled graphs by means of the graph edit distance algorithm known as TWEC [@odse; @gm_survey]. TWEC is a fast (quadratic) heuristic solution to the graph edit distance problem [@gm_survey], which solves the assignment problem of the vertices by a greedy strategy. TWEC is characterized by three parameters ranging in $[0, 1]$, controlling the importance of insertion, deletion, and substitution edit operations.
In Tab. \[tab:details\_results\_iam\] we show the obtained results. To our knowledge, there are no results available for comparison considering the one-class classification setting over the IAM datasets. However, we report those results to demonstrate the wide and straightforward applicability of EOCC and for future experimental comparisons. The herein reported results confirm that EOCC-2 is more effective than EOCC-1, especially when considering harder datasets, such as M and P. Notably, the P dataset is known to be very hard (see results in [@odse; @odse2_ijcnn_2013] for the multi-class case); in Tab. \[tab:details\_results\_iam\] the obtained AUC denotes nearly a randomized classifier. However, the accuracy is fairly high ($\simeq 0.9$, with a precision of 1 and recall of $\simeq 0.21$). This proves that when considering the HDF, in this case the system correctly rejects all non-target patterns, while it rejects also some target instance. In general, the gap between EOCC-1 and EOCC-2 is reduced for both AUC and accuracy. The number of synthesized DRs is lower for EOCC-1. This aspect magnifies the potential of EOCC-1. In fact, in different experiments the results of EOCC-1 and EOCC-2 are comparable, although EOCC-1 solves the problem in a much lower computing time and with fewer DRs, i.e., with less resources. This is an important aspect to be evaluated on the basis of the specific application at hand. Standard deviations are acceptable, confirming the stability of the system.
**IAM Dataset** **Acronym** **Target class** **\# Target** **\# Non-target**
----------------- ------------- ------------------ --------------- ------------------- --
AIDS A a 200 1600
GREC G 1 50 1033
Letter-Low L-L A 150 1400
Letter-High L-H A 150 1400
Mutagenicity M mutagen 2401 1713
Protein P 1 99 332
: The considered IAM datasets. Full details in [@riesen+bunke2008].[]{data-label="tab:iam_ds"}
**Dataset** **AUC** **Accuracy** **\# DRs**
------------- -------------- -------------- ----------------
0.977(0.012) 0.942(0.002) 2.000(0.000)
0.974(0.014) 0.945(0.012) 3.400(2.607)
0.993(0.006) 0.969(0.008) 4.000(0.707)
1.000(0.000) 0.973(0.015) 8.000(0.000)
1.000(0.000) 0.988(0.002) 2.400(0.547)
1.000(0.000) 0.990(0.001) 3.800(0.447)
0.905(0.116) 0.840(0.144) 2.000(0.000)
0.967(0.004) 0.966(0.002) 29.400(2.302)
0.501(0.132) 0.493(0.223) 3.000(1.000)
0.682(0.212) 0.622(0.201) 123.000(1.050)
0.388(0.028) 0.158(0.036) 2.000(0.000)
0.554(0.025) 0.921(0.012) 26.000(1.732)
Conclusions and Future Directions {#sec:conclusions}
=================================
In this paper, we have proposed and evaluated a novel one-class classification system, called EOCC. The classifier has been designed by making use of an interplay of different techniques. The dissimilarity representation is exploited to make the system general-purpose. Graph-based techniques are employed for estimating information-theoretic quantities (i.e., the entropy) and for deriving the model of the classifier. The decision regions forming the model are obtained by exploiting the concept of modularity of a graph partition. The decision regions are hence defined as clusters of vertices, which are further equipped with suitable membership functions. This allows us to provide both hard (i.e., Boolean) and soft decisions about the recognition of test patterns. We have validated the system over two types of benchmarks: (i) different feature-based UCI datasets and (ii) six IAM datasets of labeled graphs. Overall, the comparisons made over the UCI datasets demonstrate the validity of the approach with respect to several state-of-the-art one-class classification systems taken from the literature. Results on the IAM datasets prove the versatility and the effectiveness of the system in processing labeled graphs (a less conventional pattern type).
The one-class classification setting is very useful in all those situations where only patterns of interest are known. Such patterns are termed target instances. Our solution is applicable to virtually any context, being based on the dissimilarity representation. This aspect is of particular interest, since it allows the user to model patterns according to their more suitable representation for the application at hand.
Future directions include usual improvements and variants of the herein discussed system (i.e., by changing suitable components but considering the same overall design). For instance, we used a complete Euclidean graph representation to estimate the entropy of the data mapped into the dissimilarity space. A more sparse representation could become handy when processing large volume of data. Therefore, in the future we will evaluate entropic spanning graphs based on the so-called *k*-NN graphs. Other global optimization techniques are of course of interest, as well as other membership function models for the decision regions (e.g., by exploiting non-symmetric membership functions). Particular attention will be devoted to the issue of *interpretability* of the system. Usually, researchers focus on improving performances of pattern recognition systems from the pure technical viewpoint, such as by paying attention on the generalization capability and the computing speed. However, when facing applications of true scientific interest, such systems should satisfy the requirement of producing results and using inference rules/functions that are easily understandable by humans. This fact would allow field experts to easily gather useful insights about the underlying problem. Our system is suitable for this mission, since it is conceived by exploiting fuzzy sets based techniques. Therefore, an important future goal is the evaluation of the system from this specific viewpoint.
[^1]: llivi@scs.ryerson.ca
[^2]: Corresponding author
[^3]: asadeghi@ryerson.ca
[^4]: wpedrycz@ualberta.ca
|
---
abstract: 'In this paper, we propose a new family of graphs, matrix graphs, whose vertex set $\mathbb{F}^{N\times n}_q$ is the set of all $N\times n$ matrices over a finite field $\mathbb{F}_q$ for any positive integers $N$ and $n$. And any two matrices share an edge if the rank of their difference is $1$. Next, we give some basic properties of such graphs and also consider two coloring problems on them. Let $\chi''_d(N\times n, q)$ (resp. $\chi_d(N\times n, q)$) denote the minimum number of colors necessary to color the above matrix graph so that no two vertices that are at a distance at most $d$ (resp. exactly $d$) get the same color. These two problems were proposed in the study of scalability of optical networks. In this paper, we determine the exact value of $\chi''_d(N\times n,q)$ and give some upper and lower bounds on $\chi_d(N\times n,q)$.'
author:
- |
[ [**Zhe Han**]{} [^1] [^2]]{}\
[Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China]{}
title: '**Two coloring problems on matrix graphs**'
---
16.3pt
[**Keywords:**]{} Coloring problem, coding theory, vertex coloring, Gabidulin codes.3cm
[**Mathematics Subject Classification**]{}: 05C15, 94B05, 94B65, 68R10
Introduction
============
Let $q=p^m$, where $p$ is a prime. For any two positive integers $N$ and $n$, let $\mathbb{F}^{N\times n}_q$ be the set of all $N\times n$ matrices over a finite field $\mathbb{F}_q$. In this paper we assume $n\leq N$ since the transpose of an $N\times n$ matrix becomes an $n\times N$ matrix. The rank, or rank weight, of a matrix $M\in \mathbb{F}^{N\times n}_q$, denoted by Rk$(M)$, is defined as the algebraic rank of this matrix over $\mathbb{F}_q$. The rank distance $d_{R}(M_1,M_2)$ between $M_1$ and $M_2$ is the rank of their difference $M_1-M_2$.
Let $G=(V ,E)$ be a simple undirected graph with the vertex set $V$ and the edge set $E$. For two vertices $v_i$ and $v_j$ ($i\not = j$) in $V$, the distance between $v_i$ and $v_j$, denoted by $d(v_i,v_j)$, is the number of edges in a shortest path joining $v_i$ and $v_j$ and the diameter of $G$ is the maximum distance between any two vertices of $G$.
A matrix graph, denoted by $\mathcal{M}_{N\times n}(q)$, is defined to be an undirected graph with the vertex set $\mathbb{F}^{N\times n}_q$ and the edge set $$E_{N\times n}(q)=\{(M_1,M_2)|M_1, M_2\in\mathbb{F}^{N\times n}_q, d_{R}(M_1,M_2)=1\}.$$
Recently, hypercubes were extensively studied due to their versatile and efficient topological structures of interconnection networks in [@Bhuyan; @Fu; @hanlu; @Jamison; @Kim; @Klotz; @Ngo; @NDG; @Ostergard; @Skupie; @Sullivan; @Wan]. The so-called $q$-ary $n$-cube is an undirected graph with the vertex set, $n$-dimensional vector space over $\mathbb{F}_q$, and the edge set, which contains all pairs such that their Hamming distance is $1$. The Hamming distance is the number of distinct coordinates between the pair.
Matrix graphs may be better than hypercubes and so it is worth studying their properties. We first give a basic lemma which will be used to prove the following Proposition 1.3. It is an old result with the first derivation of the formula in \[14, p.455\]; see also Lemma 2.1 in [@ling].
\[lem-3.1\] The number of $N \times n$ matrices over $\mathbb{F}_q$ with rank $k$ is $ \frac{\prod_{i=0}^{k-1}(q^N-q^i)(q^n-q^i)}{\prod_{i=0}^{k-1}(q^k-q^i)}$.
The matrix graph $\mathcal{M}_{N\times n}(q)$ is $\frac{(q^N-1)(q^n-1)}{q-1}$-regular connected graph of order $q^{Nn}$.
The distance of any two distinct matrices $M_1$ and $M_2$ in $\mathbb{F}^{N\times n}_q$ has $d(M_1,M_2)=d_R(M_1,M_2)=\mbox{Rk}(M_1-M_2)$.
The diameter of $\mathcal{M}_{N\times n}(q)$ is $n$.
Note that $\mathcal{M}_{N\times n}(q)$ is not a bipartite graph and $\mathcal{M}_{N\times 1}(q)$ is a complete graph.
The $\mathcal{M}_{N\times n}(q)$ is vertex transitive and then the edge connectivity is $\frac{(q^N-1)(q^n-1)}{q-1}$.
If $M$ is a fixed matrix in $\mathbb{F}^{N\times n}_q$, then the mapping $$\rho_M~:~A\mapsto A+M$$ is a permutation of the vertices of $\mathcal{M}_{N\times n}(q)$, where $+$ is the matrix addition. This mapping is an automorphism because for any two distinct matrices $M_1$ and $M_2$, $d_R( M_1,M_2)=1$ if and only if $d_R(M_1+M,M_2+M)=1$. There are $q^{Nn}$ such permutations and they form a subgroup of the automorphism group of $\mathcal{M}_{N\times n}(q)$. This subgroup acts transitively on $\mathbb{F}^{N\times n}_{q}$ because for any two vertices $M_1$ and $M_2$, the automorphism $\rho_{M_2-M_1}$ maps $M_1$ to $M_2$.
A coloring of $\mathcal{M}_{N\times n}(q)$ with $L$ colors is a mapping $\Gamma$ from the vertex set $\mathbb{F}^{N\times n}_q$ to ${\cal L}=\{1,2,\ldots,L\}$. A $d$-distance (resp. exactly $d$-distance) coloring of $\mathcal{M}_{N\times n}(q)$ is to color the vertices of $\mathcal{M}_{N\times n}(q)$ such that any two vertices with rank distance at most $d$ (resp. exactly $d$) have different colors. Note that for a coloring of $\mathcal{M}_{N\times n}(q)$ with $L$ colors $$\Gamma: \mathbb{F}^{N\times n}_q \longrightarrow \mathcal{L},$$ it is a $d$-distance coloring of $\mathcal{M}_{N\times n}(q)$ if and only if for any two distinct vertices $M_1,M_2\in \mathbb{F}^{N\times n}_q$, $$\Gamma(M_1)\neq \Gamma(M_2) \mbox{ if } d_R(M_1,M_2)\leq d$$ and it is an exactly $d$-distance coloring of $\mathbb{F}^{N\times n}_q$ if and only if for any two distinct vertices $M_1,M_2\in \mathbb{F}^{N\times n}_q$, $$\Gamma(M_1)\neq \Gamma(M_2) \mbox{ if } d_R(M_1,M_2)= d.$$ Denote $\chi'_d(N\times n,q)$ (resp. $\chi_d(N\times n,q)$) as the minimum number of colors needed for a $d$-distance (resp. an exactly $d$-distance) coloring of $\mathcal{M}_{N\times n}(q)$. Clearly, $\chi_{d}(N\times n,q)\leq \chi'_{d}(N\times n,q)$ for any $n$ and $N$.
These two coloring problems originally arose in the study of the scalability of optical networks [@Pavan]. In the rest of our paper, we determine exact value of $\chi'_{d}(N\times n,q)$ and give some bounds for $\chi_{d}(N\times n,q)$ for any $n$, $N$ and $d$.
The $d$-distance coloring and exactly $d$-distance coloring of $\mathcal{M}_{N\times n}(q)$ are equivalent to certain partitions of $\mathbb{F}^{N\times n}_q$, which are related to rank codes in coding theory. Therefore, we introduce rank codes in the next section.
Rank codes
===========
Algebraic coding theory can be considered as the theory of subsets of a finite-dimensional space over a finite field equipped with a norm function. The most known norm in coding theory is the Hamming weight of a vector. It turns out that the rank function of a matrix over a finite field can also be considered as a norm function. The interest in these codes is a consequence of their application in random network coding [@Koetter]. Explicitly, the concept of the rank metric was introduced by Loo-Keng Hua [@hua] as “Arithmetic distance”. Delsarte [@del] defined the rank distance on the set of bilinear forms (equivalently, on the set of rectangular matrices) and proposed the construction of optimal codes in bilinear form representation. Gabidulin [@gab] introduced the rank distance for the vector representation over extension fields and found connections between rank codes in the vector representation and in the matrix representation.
In the matrix representation, rank codes are defined as subsets of a normed space $\{\mathbb{F}^{N\times n}_q$, $Rk\}$ of $N\times n$ matrices over $\mathbb{F}_q$. The definitions of norm and distance of any two matrices are defined as above. The rank distance of a rank code $\mathcal{M}\subset \mathbb{F}^{N\times n}_q$ is defined as the minimal pairwise distance: $d(\mathcal{M})=d=\min\{Rk(M_i-M_j): M_i,M_j\in \mathcal{M}, i\neq j\}$.
The size $|\mathcal{M}|$ of related code with code distance $d$ satisfies the Singleton bound $|\mathcal{M}|\leq q^{N(n-d+1)}$. Codes reaching this bound are called maximum rank codes, or, MRD codes. A rank code $\mathcal{M}$ is called $\mathbb{F}_q$-linear if $\mathcal{M}$ is a subspace of $\mathbb{F}^{N\times n}_q$.
In [@del] the construction of optimal codes is proposed. Therefore for any $n\leq N$, a linear rank code $\mathcal{M}$ with code distance $d$ reaches the singleton bound $|\mathcal{M}|=q^{N(n-d+1)}$.
In the vector representation, rank codes are defined as subsets of a normed $n$-dimensional space $\{\mathbb{F}^{n}_{q^N}, Rk\}$ of $n$-vectors over an extension field $\mathbb{F}_{q^N}$, where the norm of a vector $\mathbf{v}\in \mathbb{F}^{n}_{q^N} $ is defined to be the column rank $Rk(\mathbf{v}|\mathbb{F}_q)$ of this vector over $\mathbb{F}_q$, that is, the maximal number of coordinates of $\mathbf{v}$ which are linearly independent over the base field $\mathbb{F}_q$. The rank distance between two vectors $\mathbf{v}_1, \mathbf{v}_2$ is the column rank of their difference $Rk(\mathbf{v}_1-\mathbf{v}_2|\mathbb{F}_q)$. The rank distance of a vector rank code $\mathcal{V}\subset \mathbb{F}^{n}_{q^N}$ is defined as the minimal pairwise distance: $d(\mathcal{V})=d=\min\{Rk(\mathbf{v}_i-\mathbf{v}_j|\mathbb{F}_q): \mathbf{v}_i,\mathbf{v}_j\in \mathcal{V},i\neq j\}$.
A rank code $\mathcal{V}$ in the vector representation is called $\mathbb{F}_{q^N}$-linear if $\mathcal{V}$ is a subspace of $\mathbb{F}_{q^N}^{n}$. Denote by $(N\times n,k,d)$ a code $\mathcal{V}$ over $\mathbb{F}_{q^N}$ of dimension $k\leq n$ and rank distance $d$. Such a code can be described in terms of a full rank generator matrix $G_k$ over the extension field $\mathbb{F}_{q^N}$ of size $k\times n$. Code vectors $\{\mathbf{v}\}$ are all linear combinations of rows of this matrix. Thus the size of a code is equal to $|\mathcal{V}|=q^{Nk}$.
Equivalently, a rank code $\mathcal{V}$ can be described in terms of a full rank parity-check matrix $H_{n-k}$ over $\mathbb{F}_{q^{N}}$ of size $(n-k)\times n$. It satisfies the condition $G_kH^{T}_{n-k}=O$, where $O$ is the all zero $k\times (n-k)$ matrix. Code vectors $(\mathbf{v})$ are all solutions of the linear system of equation $\mathbf{v}H^{T}_{n-k}=\mathbf{0}$. We give the similar property of linear rank codes to the linear hamming codes as follows.
\[lem:3.3\] Let $\mathcal{V}$ be a $(N\times n,k,d)$ linear rank code over $\mathbb{F}_{q^N}$ and $H$ be a parity-check matrix for $\mathcal{V}$. Then the following statements are equivalent:
\(i) $\mathcal{V}$ has rank distance $d$.
\(ii) any $d-1$ columns of $H$ are linearly independent over $\mathbb{F}_{q^N}$ and $H$ has a $d$ columns that are linearly dependent over $\mathbb{F}_{q^N}$.
Let $\mathbf{v}=(v_1,\ldots,v_n)\in \mathcal{V}$ be a codeword of rank weight $e$. Thus there are $n\times n$ invertible matrix $P$ such that $\mathbf{v}P=\mathbf{u}$, where $\mathbf{u}=(u_1,\ldots,u_e,0,\ldots,0)$ and where $u_i\neq 0$. So $\mathbf{v}=\mathbf{u}P^{-1}$. Then $\mathcal{V}$ contains a nonzero codeword $\mathbf{v}$ of rank weight $e$ if and only if $$\mathbf{0}=\mathbf{v}H^T=\mathbf{u}(H(P^{-1})^T)^T=u_1\mathbf{c}_1+\ldots+u_e\mathbf{c}_e,$$ which is true if and only if there are $e$ columns of $ H(P^{-1})^T$ (namely, $\mathbf{c}_{1}$,$\ldots$, $\mathbf{c}_{e}$) that are linear dependent over $\mathbb{F}_{q^N}$ if and only if there are $e$ columns of $H$ that are linear dependent over $\mathbb{F}_{q^N}$.
To say that the rank distance of $\mathcal{V}$ is $\geq d$ is equivalent to saying that $\mathcal{V}$ does not contain any nonzero word of $\leq d-1$, which is in turn equivalent to saying that any $d-1$ columns of $H$ are linearly independent over $\mathbb{F}_{q^N}$.
Similarly, to say that the rank distance of $\mathcal{V}$ is $\leq d$ is equivalent to saying that $\mathcal{V}$ contains a nonzero word of weight $\leq d$, which is in turn equivalent to saying that $H$ has $\leq d$ columns that are linear dependent over $\mathbb{F}_{q^N}$.
The above discussion easily leads to the desired results.
Likewise, general constructions of MRD codes in terms of parity-check matrices can be obtained [@gab].
\[prop:4\] Let $h_1,h_2,\ldots,h_n$ be a set of elements from the extension field $\mathbb{F}_{q^N}$ linearly independent over the base field $\mathbb{F}_q$. Let $s$ be a positive integer such that $\gcd(s,N)=1$. Then a parity matrix of the form $$H_{d-1}=\left(\begin{array}{cccc}
h_1& h_2& \ldots &h_n\\
h^{q^s}_{1} & h^{q^s}_{2} & \ldots & h^{q^s}_{n} \\
\ldots & \ldots & \ldots & \ldots \\
h^{q^{s(d-2)}}_{1} & h^{q^{s(d-2)}}_{2} & \ldots & h^{q^{s(d-2)}}_{n}
\end{array}\right )$$ defines an MRD $(N\times n,k,d)$ code with code length $n\leq N$, dimension $k=n-d+1$, and rank distance $d=n-k+1$.
Equidistant codes in rank norm are related to the exactly $d$-distance coloring problem. A code is said to be equidistant if the distance between any distinct codewords is the same (say $d$). A code is called a constant-weight code if each non-zero codeword is of the same weight. In [@Tuvi] the authors constructed an equidistant constant rank code as follows.
\[prop:5\] There exists an equidistant constant rank code over $\mathbb{F}_q$ with matrices of size ${n \choose 2}\times n$, rank $n-1$, rank distance $n-1$, and size $q^n-1$.
Exact value of $\chi'_{d}(N\times n,q)$
========================================
A code $C$ over $\mathbb{F}_{q^N}$ of length $n$ and minimum rank distance at least $d$ is called an $(N\times n,\ge d)_q$ code. Let $A_q(N\times n,d)$ (resp. $A_q(N\times n,\ge d)$) denote the maximum size of an $(N\times n, d)_q$ (resp. $(N\times n,\ge d)_q$) code. A code $C$ is called an $[N\times n, k]_q$ linear code if $C$ is a $k$-dimensional subspace of $\mathbb{F}^{n}_{q^N}$. An $[N\times n,k]_q$ linear code with minimum distance $d$ is denoted by an $[N\times n,k,d]_q$. For the fixed $N$, $n$, and minimum distance $d$, let $k(N\times n,d)_q$ denote the maximum dimension of an $[N\times n,k,d]_q$ code. A code $C$ of length $n$ is called an $(N\times n, \overline{\{d\}})_q$ forbidden distance code if $d_R(\mathbf{u},\mathbf{v})\not= d$ for any two distinct codewords $\mathbf{u},\mathbf{v}\in C$. Given $N, n$, and $d$, let $Q(N\times n, d)_q$ denote the maximum size of an $(N\times n, \overline{\{d\}})_q$ forbidden rank distance code.
An $(N\times n,L,d)_q$-partition (resp. $(N\times n, L,\overline{\{d\}})_q$-partition) of $\mathbb{F}^{N\times n}_q$ is a set of subsets $\{\mathcal{M}_i\}_{i=1}^{L}$ of $\mathbb{F}^{N\times n}_q$ satisfying (i) $\mathcal{M}_i\bigcap \mathcal{M}_j=\emptyset$ for $i\neq j$ and $\bigcup_{i=1}^{L}\mathcal{M}_i=\mathbb{F}^{N\times n}_q$, (ii) each $\mathcal{M}_i$ is an $(N\times n,\geq d)_q$ rank code (resp. $(N\times n,\overline{\{d\}})_q$ forbidden distance rank code). It is well known that a $d$-distance coloring (resp. an exactly $d$-distance) of $\mathbb{F}^{N\times n}_q$ with $L$ colors is equivalent to an $(N\times n,L,d+1)_q$-partition (resp. $(N\times n,L,\overline{\{d\}})_q$-partition) of $\mathbb{F}^{N\times n}_q$. Hence $\chi'_{d}(N\times n,q)$ (resp. $\chi_{d}(N\times n,q)$) is the minimum number $L$ of subsets in any $(N\times n, L, d+1)_q$ (resp. $(N\times n, L,\overline{\{d\}})_q$-partition) of $\mathbb{F}^{N\times n}_q$.
Since $q^{Nn}=|\mathbb{F}^{N\times n}_q|=\sum_{i=1}^{L}|\mathcal{M}_i|\le L A_{q}(N\times n,\geq d+1)\leq L A_{q}(N\times n,d+1) $ and $A_q(N\times n,d)$ is decreasing in $d$, we have $$\label{eq:basic}
\chi'_{d}(N\times n,q)\geq \frac{q^{Nn}}{A_{q}(N\times n,d+1)}.$$
Note that for an $[N\times n,k,d+1]_q$ linear rank code $\mathcal{M}$, the cosets of $\mathcal{M}$ form an $(N\times n,q^{n-k},d+1)_q$-partition of $\mathbb{F}^{N\times n}_q$. Hence, if there exists an $[N\times n,k,d+1]_q$ linear rank code, then $$\label{eq:code}
\chi'_d(N\times n,q)\leq q^{N(n-k)}.$$ In particular, $$\label{eq:max}
\chi'_d(N,n,q)\leq q^{N(n-k(n,d+1))}.$$ Furthermore, if $A_q(N\times n,d+1)=q^{N\times k(n,d+1)}$, i.e., $A_q(N\times n,d+1)$ is attained by an $[N\times n,k,d+1]_q$ linear rank code, then $$\label{eq:final}
\chi'_{d}(N\times n,q)=q^{N(n-k(n,d+1))}.$$
Summing the above discussion, we get our result on $\chi'_{d}(N\times n,q)$. It is clear that $\chi'_{d}(N\times n,q)=1$ if $d\geq n$.
For any positive integers $d\leq n$, we have $\chi'_{d}(N\times n,q)=q^{Nd}$.
By Proposition \[prop:4\], we have $k(n,d+1)=n-d$ and so the required result is obtained by Eq(\[eq:final\]).
Bounds of $\chi_{d}(N\times n,q)$
==================================
In this section, we consider bounds of $\chi_{d}(N\times n,q)$. Since the diameter of $M_{N\times n}(q)$ is $n$, $\chi_{d}(N\times n,q)= 1$ for any $d\ge n+1$. So we can assume $d\le n$.
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For $N\geq n$, we have $\chi_{1}(N\times n,q)= q^N$.
When $d=1$, it suffices to show that $\chi_{d}(N\times n,q)\geq q^N$ since $\chi_{d}(N\times n,q)\leq \chi'_{d}(N,n,q)$. Consider a set $S=\{A=(a_{ij})\in \mathbb{F}^{N\times n}_{q}: a_{ij}=0, \mbox{ for }1\leq i\leq N, 2\leq j\leq n \}$, in which any two of them have an edge, and so coloring them needs $$\left(\begin{array}{c}
N\\
0 \end{array} \right )+ \left(\begin{array}{c}
N\\
1 \end{array}\right )(q-1)+\ldots+\left(\begin{array}{c}
N\\
j \end{array}\right )(q-1)^j+\ldots+\left(\begin{array}{c}
N\\
N \end{array} \right )(q-1)^N=(1+q-1)^{N}.$$
We give some examples about equidistant rank codes with maximal size in each case.
When $N=n=2$, $q=2$ and $d=2$. The code $C_1=\left( \left(\begin{array}{cc}
1 & 0 \\
0 & 1\end{array} \right ),
\left(\begin{array}{cc}
0 & 0 \\
1 & 0\end{array} \right ),
\left(\begin{array}{cc}
0 & 1 \\
0 & 0\end{array} \right ), \right.$ $ \left. \left(\begin{array}{cc}
1 & 1 \\
1 & 1\end{array} \right )
\right )$ in matrix form.
Let $\alpha$ be a primitive element of the Galois field $\mathbb{F}_{2^3}$ such that $\alpha^3=\alpha+1$. We can construct an equidistant rank code $C_2$ of length $n=2$ over $\mathbb{F}_{2^3}$ and rank distance $2$, where $C_2=\{(1,\alpha), (\alpha,1+\alpha), (1+\alpha+\alpha^2,1), (0,1+\alpha^2),
(1+\alpha,\alpha^2), (1+\alpha^2, 1+\alpha+\alpha^2),(\alpha+\alpha^2, \alpha+\alpha^2), (\alpha^2,0)\}$.
We can also consider an example in [@sel]. Consider an equidistance rank code $C_3$ of length $n=3$ over $\mathbb{F}_{2^3}$ and rank distance $3$. The code $C_3$ is $\{(\alpha^2,0,0), (1,\alpha,\alpha^2), (0,1,\alpha), (1+\alpha^2, 1+\alpha,\alpha+\alpha^2), (\alpha+\alpha^2, \alpha+\alpha^2,1),
(1+\alpha,\alpha^2, 1+\alpha^2), (\alpha,1+\alpha+\alpha^2,1+\alpha), (1+\alpha+\alpha^2, 1+\alpha^2,1+\alpha+\alpha^2)\}$.
Therefore, we get the following proposition.
We have $$\label{eq:special}
\chi_{n}(N\times n,2)= 2^N, \mbox{ for pairs } (N,n)= (N,1),(2,2),(3,2),(3,3).$$
In addition, the size of an equidistant rank code over $\mathbb{F}_{2^N}$with length $n$ and rank distance $n$ should be a power of 2 and so we propose an open problem as follow: $$\label{eq: conj}
\chi_{n}(N\times n,2)= 2^N \mbox{ for all }N\geq n.$$
If this statement holds, it also means that there is a largest equidistant rank code over $\mathbb{F}_{2^N}$ with length $n$, distance $n$, size $2^N$.
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For $n\geq 3$, we have $\chi_{n-1}({n \choose 2}\times n ,q)\ge q^n-1$.
By Proposition \[prop:5\], it is forward to have $\chi_{n-1}({n \choose 2}\times n ,q)\ge q^n-1$.
0.2cm
In the following we discuss upper bounds of the exactly $d$-distance coloring of $\mathbb{F}^{N\times n}_q$. First we have $$\label{eq:nat}
\chi_{d}(N\times n,q)\leq \chi'_{d}(N\times n,q)=q^{Nd}.$$
Furthermore, we use the argument of linear forbidden rank distance codes. An $[N\times n,k]_q$ linear rank code $\mathcal{M}$ is called an $[N\times n,k,\overline{d}]_q$ linear forbidden rank distance code if $\mathcal{M}$ is also an $(N\times n, \overline{d})$ forbidden rank distance code, i.e., $Rk(M)\neq d$ for any nonzero codeword $M$ of $\mathcal{M}$. Given $N, n(n\leq N)$, and $d$, the maximum dimension of an $[N\times n,k,\overline{d}]_q$ linear forbidden distance code is denoted by $k(N\times n,\overline{d})$. Similar to $d$-distance coloring, we have
$$\chi_{d}(N\times n,q)\leq q^{N(n-k)}.$$
$$\label{eq:exactlc}
\chi_{d}(N\times n,q)\leq q^{N(n-k(N\times n,\overline{d}))}.$$
In the following lemma the lower bound on $k(N\times n,\overline{d})$ is given.
.2cm
\[lem:3.4\] We have $$k(N\times n,\overline{d})\geq nN-\left\lceil \log_{q}[2+{n-1\choose d-1}(q^N-1)^{d-1}]\right\rceil.$$
First if $$2+{n-1 \choose d-1}(q^N-1)^{d-1} \leq q^{mN},$$ then there exists a matrix $H$ over $\mathbb{F}_{q^N}$ of size $m\times n$ such that any column of $H$ is not equal to the $\mathbb{F}_{q^N}$-linear combination of any other $d-1$ columns of $H$ by Lemma \[lem:3.3\]. Hence there is a $[N\times n, k,\overline{d}]$ linear forbidden rank distance code, where the dimension $k\geq n-m$. Take $$Nm=\left\lceil \log_{q}[2+{n-1\choose d-1}(q^N-1)^{d-1}]\right\rceil.$$ Thus, $$\begin{aligned}
Nk(N, n,\overline{d}) &\geq & Ndim(\mathcal{V})\\
& \geq & Nn-Nm \\
& \geq & Nn-\left\lceil \log_{q}[2+{n-1\choose d-1}(q^N-1)^{d-1}]\right\rceil.\end{aligned}$$
This completes the proof.
.2cm
We have $$\chi_{d }(N\times n,q)\leq q^{\lceil \log_{q}[2+{n-1\choose d-1}(q^N-1)^{d-1}]\rceil}.$$
The theorem follows from Lemma \[lem:3.4\] and Eq (\[eq:exactlc\]).
In the following table we list some values of two bounds from Eq(12) and Eq(\[eq:nat\]).
$N$ $n$ $d$ $q$ Bound $(12)$ Bound (\[eq:nat\])
------ ----- ----- ----- -------------- --------------------
$6$ $4$ $2$ $2$ $2^8$ $2^{12}$
$6$ $4$ $3$ $2$ $2^{14}$ $2^{18}$
$6$ $4$ $2$ $3$ $3^7$ $3^{12}$
$6$ $4$ $3$ $3$ $3^{13}$ $3^{18}$
$5$ $3$ $2$ $2$ $2^{6}$ $2^{10}$
$5$ $3$ $3$ $3$ $3^{10}$ $3^{15}$
$10$ $7$ $4$ $2$ $2^{35}$ $2^{40}$
$10$ $7$ $4$ $3$ $2^{33}$ $2^{40}$
: \[figfcsr\] Bounds Comparison
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Acknowledgements {#acknowledgements .unnumbered}
================
This work is partially supported by National Natural Science Foundation of China (Nos. 61373019, 11171097 and 61170289).
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[^1]: email: hanzhe101@gmail.com.
[^2]: Correspondent author, email: mlu@math.tsinghua.edu.cn.
|
---
abstract: 'We study the 2D Kondo insulators in a uniform magnetic field using quantum Monte Carlo simulations of the particle-hole symmetric Kondo lattice model and a mean field analysis of the Periodic Anderson model. We find that the field induces a transition to an insulating, antiferromagnetically ordered phase with staggered moment in the plane perpendicular to the field. For fields in excess of the quasi-particle gap, corresponding to a metal in a simple band picture of the periodic Anderson model, we find that the metallic phase is unstable towards the spin density wave type ordering for any finite value of the interaction strength. This can be understood as a consequence of the perfect nesting of the particle and hole Fermi surfaces that emerge as the field closes the gap. We propose a phase diagram and investigate the quasi-particle and charge excitations in the magnetic field. We find good agreement between the mean-field and quantum Monte Carlo results.'
author:
- Igor Milat
- Fakher Assaad
- Manfred Sigrist
bibliography:
- 'references.bib'
title: Field induced magnetic ordering transition in Kondo insulators
---
Introduction {#sec:Introduction}
============
Kondo insulators, or heavy fermion semiconductors, are materials containing at least one atom per formula unit with a partially filled $f$ or $d$ shell and exhibiting properties similar to very narrow gap semiconductors. $CeRhAs$, $CeRhSb$, $YB_{12}$, $Ce_3Bi_4Pt_3$ and $SmB_6$ are the most thoroughly investigated examples [@riseborough.00]. In the canonical model, the formation of the gap in Kondo insulators is a consequence of the hybridization between the conduction band and the effective $f$-electron level which gives rise to quasi-particle and spin-gaps at low temperatures. Adopting a band picture one can close the gap by applying a high magnetic field, since the gap is on the meV scale. Although experiments on $YB_{12}$[@sugiyama.88], $SmB_6$[@cooley.99] and $Ce_3Bi_4Pt_3$[@jaime.00] seem to support this simple picture, the exact nature of the field induced insulator to metal transition as well as the role played by the strong correlations remains far from understood.
Magnetic instabilities of the periodic Anderson model at half-filling have been studied extensively by slave boson mean field approximations[@riseborough.92; @dorin.92; @doradzinski.98]. A phase diagram as a function of the interaction strength was established and some thermodynamic and transport properties have been calculated. In these studies, the only effect of the magnetic field is assumed to be the stabilization of the ferromagnetically ordered state with respect to other magnetic configurations.
Carruzzo and Yu[@carruzzo.1995] studied the one dimensional, half-filled Kondo lattice in magnetic field using DMRG and bosonisation techniques. They found that although the spin gap closed at the critical field, the charge gap remained due to umklapp scattering They conclude that the 1D half-filled Kondo lattice is insulating at all fields.
The effect of the field in disordered Kondo insulators was treated by CPA in Ref.. The authors find that, in the absence of magnetic ordering, the magnetic field induces the insulator to metal transition in the universality class of density driven metal-insulator transitions. Based on scaling arguments the field dependence of the quasi-particle gap as well as the critical field as a function of temperature and impurity concentration were derived.
In this paper we present a detailed study of the field induced quantum phase transition in 2D particle-hole symmetric models of Kondo insulators. We present a mean field calculation appropriate for the small-$U$ limit of the periodic Anderson model. We find that the magnetic field induces a phase transition from the paramagnetic insulator into a canted antiferromagnetic insulator which remains stable at all field strengths (until all the electrons in the system align with the field). While zero-energy spin modes exist, we find that the field does not close the quasi-particle gap, if the lattice is bipartite, so that the metallic ground state is never induced by the field. We investigate more carefully the large-field limit using two effective models and reach essentially the same conclusion - on the bipartite lattice, the interaction is a relevant perturbation and the ground state remains insulating at all fields. The approximate treatments are complemented by a quantum Monte Carlo study of the particle hole symmetric Kondo lattice model in 2D. We find good agreement between the results.
Recently, Beach and collaborators studied the effect of the magnetic field on the Kondo insulators using a large-$N$ type mean field analysis of the Kondo lattice model and quantum Monte Carlo simulations [@beach.03]. They find that a large enough magnetic field induces a phase transition to a metallic ground state from the insulating canted antiferromagnetic state. The phase transition into the metallic state occurs when the $f$ moments decouple from the conduction band, i.e. the hybridization mean field vanishes at a certain critical field. The question naturally arises whether this phase transition is real or possibly an artifact of the large-$N$ mean field approach. Here we will show results which lead to a different conclusion: the insulating state induced by the magnetic field remains stable up to the full polarization of all electrons in the system, if the system is completely particle-hole symmetric. Thus the particle-hole symmetric Kondo insulator is an insulator at all fields.
The paper is organized as follows: In the next section we introduce the models used to describe Kondo insulators. In section \[sec:meanfield\] the phase diagram in the presence of the magnetic field is obtained using a mean field approximation for the half-filled periodic Anderson model. In section \[sec:largeB\] we present a discussion of the Kondo lattice model in high magnetic fields. In section \[sec:qmc\] the results of the Quantum Monte Carlo simulations are presented and compared with the mean field calculations. We summarize our results in section \[sec:conclusion\] and briefly comment on their relevance for the experimental systems.
models {#sec:models}
======
The canonical model used to describe the physics of the Kondo insulators is the periodic Anderson model (PAM)[@riseborough.00]. The PAM Hamiltonian, including the uniform magnetic field in the $z$-direction is $$\begin{gathered}
\label{eq:Kpam}
H_{PAM} = -\sum_{<i,j>,\sigma} t_{ij} c^\dag_{i\sigma} c_{j\sigma}
%
+ \epsilon_f \sum_{i \sigma} f^\dag_{i\sigma} f_{i\sigma}
+ U \sum_{i} n^f_{i\uparrow}n^f_{i\downarrow} \\
+ \sum_{k,\sigma} (V f^\dag_{k\sigma} c_{k\sigma} + \text{H.c.}) -
g \mu_B \vec B \cdot \sum_i (\vec S^f_i + \vec S^c_i)\end{gathered}$$ Here all the symbols have their usual meaning. The PAM describes a two band system in which one band (conduction electron, $c$ band) is dispersive and uncorrelated and the other ($f$ band) dispersionless and strongly correlated. $t_{ij}$ is the hopping matrix element in the $c$ band and $U$ the local Coulomb interaction in the $f$ band. The two bands are mixed and the hybridization matrix element $V$ controls the mixing strength. In the particle-hole symmetric model that we consider in the following, $t_{ij} = t$ for nearest neighbor sites on the square lattice and zero otherwise and $\epsilon_f = -U/2$. The magnetic field is coupled to the $c$ and $f$ electron spins only. The $g$ factors of the $c$ and $f$ electrons are chosen to be the same, $g_c = g_f = 2$, for simplicity but choosing them differently would not change the qualitative aspects of our conclusions. In the following, the magnetic field is measured in the units of Zeeman energy.
In the non-interacting ($U=0$) case the ground state of the PAM is a paramagnetic band insulator with the quasi-particle gap $\Delta_{qp}^0
= \sqrt{(W/2)^2 + V^2} - W/2 \simeq \frac{V^2}{W}$, where $W$ is the conduction electron bandwidth. In the field, the Zeeman splitting reduces the quasi particle gap. For fields larger than $B_{c1} = \Delta_{qp}^0$, the gap vanishes and the ground state is metallic. In the fields beyond $B_{c2} = \sqrt{(W/2)^2 + V^2} +
W/2 \simeq W + 2V^2/W$, all the spins are aligned with the field. The fully polarized ground state consists of two completely filled bands and is a trivial band insulator.
Because of the particle-hole symmetry, the Fermi surfaces of the spin up electrons and the spin down holes in the metallic state at intermediate fields are perfectly nested with respect to $Q =
(\pi,\pi)$. The staggered susceptibility in the plane perpendicular to the field diverges logarithmically as $\omega \rightarrow 0$. This divergence makes the state unstable under perturbations coupling to the staggered magnetization. In particular, one expects that a staggered magnetization will be induced by any non-zero correlation on the $f$ sites. The ensuing ordered state is a canted antiferromagnet, characterized by both $m_z$ and $m_x$ different from zero.
When $U$ is large enough ($U/V \gg 1$) to suppress charge fluctuations on the $f$ sites, the low-energy physics of PAM is well described by the Kondo lattice model (KLM)[@tsunetsugu.1997; @sinjukow.02], $$\label{eq:Kklm}
H_{KLM} = -t \sum_{ \langle i,j \rangle \sigma} c^\dag_{i\sigma}
c_{j\sigma}
+ J \sum_i \vec S^c_i \cdot \vec S^f_i - 2 B_z \cdot \sum_i (S^{z,f}_i
+ S^{z,c}_i)$$ In the KLM, the charge fluctuations on the $f$ sites are completely suppressed, $f$ electrons are treated as spins and the hybridization is replaced by an antiferromagnetic exchange interaction between conduction electrons and $f$ spins. Formally PAM and KLM can be related by the Schrieffer-Wolff transformation[@sw.66; @sinjukow.02], yielding $J = 8V^2/U$.
The zero-temperature, zero-field phase diagram of the $2D$ particle-hole symmetric KLM has been well established by various numerical methods[@shi95; @capponi.2000; @zheng.03]. In the absence of the magnetic field the ground state of the KLM is a paramagnetic insulator at large $J/t$. There is a quantum critical point at $J/t
\simeq 1.4$ and for small $J/t$ the ground state is antiferromagnetically ordered.
The large $J/t$ paramagnetic state of the KLM is adiabatically connected to the $U = 0$ state of the PAM. In the particle-hole symmetric case, on finite lattices, this is guaranteed by theorems for the ground states of the two models [@tsunetsugu.1997].
Mean field analysis {#sec:meanfield}
===================
In this section we investigate the effect of the magnetic field on the small-$U$ PAM. In particular we want to investigate the spin density wave instability of the metallic state induced by the field in the non-interacting model. To this end, we perform the mean field decoupling of the interaction term in Eq.(\[eq:Kpam\]) by assuming the magnetization of the $f$ spins to have a uniform component along the field axis and a staggered component in the plane perpendicular to the field, $\langle \vec S^f_i \rangle = \vec m_i$ with $\vec m_i =
\left( (-)^i m_x, 0, m_z \right)$. This yields the mean field Hamiltonian (see the appendix \[sec:app-meanfield\] for details of the derivation),
$$\begin{gathered}
\label{eq:HMF}
H_{MF} = \sum_{k,\sigma} (\epsilon_k - p_\sigma B) c^\dag_{k\sigma}
c_{k\sigma} + \sum_{k,\sigma} (- p_\sigma )(B +U m_z)
f^\dag_{k\sigma} f_{k\sigma} + V \sum_{k,\sigma} ( c^\dag_{k\sigma}
f_{k\sigma} +
f^\dag_{k\sigma} c_{k\sigma} ) \\
- U m_x \sum_k ( f^\dag_{k+Q\uparrow} f_{k\downarrow} +
f^\dag_{k+Q\downarrow} f_{k\uparrow} ) + N U (m_x^2 +
m_z^2),\end{gathered}$$
with $p_\sigma = 1(\uparrow), -1(\downarrow)$ and $\epsilon_k = -\tfrac W 2 [\cos(k_x) + \cos(k_y)]$. The mean field Hamiltonian is quadratic in fermion operators and is easily diagonalized by a unitary transformation. In the presence of the staggered magnetization, the Brillouin zone is halved and one finds 8 quasi particle bands; the particle bands $$\begin{gathered}
E_{p,\pm}^{\sigma}(k) = \frac 1 {\sqrt 2}
\Bigl[ (B + U m_z)^2 + (U m_x)^2 + 2 V^2 + (\epsilon_k - p_\sigma B)^2 \pm \\
\pm \sqrt{ ((U m_x)^2 + (B + Um_z)^2 - (\epsilon_k - p_\sigma B)^2
)^2 + 4V^2[(\epsilon_k - 2 p_\sigma B - p_\sigma U m_z)^2 + (U
m_x)^2] } \Bigr]^{1/2}\end{gathered}$$ and the hole bands related by, $E_{h,s}^\sigma(k) =
-E_{p, s}^{\bar \sigma}(k)$. Note that the $k$ dependence of the quasi-particle bands originates only from the dispersion of the conduction electrons. On a bipartite lattice, with $\epsilon_{k+Q} =
-\epsilon_k$, the quasi particle bands satisfy, $E_{h,s}^\sigma(
\epsilon_k ) = -E_{p,\bar s}^{\bar \sigma} ( \epsilon_{k+Q} )$.
In the ground state, the particle bands are empty and the hole bands are completely filled. To obtain the ground state energy, the expression $$\label{eq:egs}
E_{gs} = \sum_{s,\sigma}{\sum_k}' E_{h,s}^\sigma(k)
+ \frac{(Um_x)^2 + (Um_z)^2}{U}$$ must be minimized with respect to $m_x$ and $m_z$, yielding the usual mean-field equations, $$\label{eq:mf}
\frac{\partial E_{gs}}{\partial(U m_{x,z})} = 0.$$ The prime on the summation sign in equation \[eq:egs\] indicates that the summation is to be taken over the magnetic Brillouin zone.
Mean field phase diagram {#sec:mean-field-phase}
------------------------
The minimization of the ground state energy for a range of $U$ and $B$ values was performed numerically and the obtained magnetization values are shown in Fig. \[fig:magnetizations\].
![Staggered (left) and parallel magnetizations vs. B and U for $W=1, V=1$ obtained by numerically minimizing the mean field equations. The grayed out plateau marks the paramagnetic phase.[]{data-label="fig:magnetizations"}](magnetizations.eps){width="\columnwidth"}
In zero field, the system is paramagnetic at small values of $U$ and antiferromagnetically ordered beyond $U_c \simeq 1.25 V$. The staggered magnetization grows as $(U-U_c)^{1/2}$ close to $U_c$ and tends to the fully saturated value $m_x = 1/2$ as $U \rightarrow
\infty$.
The magnetic field applied to the system reduces the value of $U$ at which the magnetic instability occurs. The phase boundary can be obtained by solving $$\label{eq:Uc}
\frac 2 {U_c(B)} =
-
\left. \frac{\partial^2 E_0}{\partial (U m_x)^2}
\right|_{m_x=0} ,$$ where $E_0 = \sum_{s=\pm,\sigma=\uparrow\downarrow} \sum_k
E_{h,s}^\sigma (k)$. At small fields, $B \ll B_{c1}$, the critical interaction strength falls off as the square root of the field, $U_c(B) - U_c(0) \propto -\sqrt{B}$, as expected in the mean field approach. At the phase boundary one finds the usual mean field critical exponents for the staggered magnetization, $m^x_s \propto (U -
U_c(B))^{1/2}$ and $m^x_s \propto (B-B_c(U))^{1/2}$. After the initial rise, $m_x$ goes through a maximum and falls of exponentially in large fields. On the phase boundary, the parallel susceptibility vanishes (it is zero in the paramagnetic phase, since the spin excitations are gaped). Close to the phase boundary, it behaves like $m_z \propto (B-B_c(U))^\alpha$, with $\alpha > 1$.
The right hand side of the Eq.(\[eq:Uc\]) is proportional to the static staggered susceptibility of the $U m_x = 0$ state, $\chi_0^{+-}(Q)$. This can be expressed using the familiar Lindhardt formula which in the case considered here reduces to $$\label{eq:chi0}
\chi_0^{+-}(Q) = - \left. {\sum_{k}}'
\frac{f(E_{h,-}^\downarrow(k)) - f(E_{p,-}^\uparrow(k))}
{E_{h,-}^\downarrow(k) - E_{p,-}^\uparrow(k)}\right|_{Um_x=0} .$$ In small fields, the quasi-particle gap provides a cut-off for the denominator in the sum on the right hand side and $\chi_0^{+-}(Q)$ is finite. When the field closes the gap, the denominator vanishes along the Fermi surface (Fermi lines), determined by the equation, $$\label{eq:epsilon0}
\epsilon_k = \pm \epsilon_0 = \pm \frac{V^2- B U m_z - B^2}{B+U m_z}.$$ Consequently, the staggered susceptibility diverges logarithmically in the field $B > \Delta_{qp}^0$ and there is no finite $U_c$. The system is ordered for any finite interaction strength. The divergence of $\chi_0^{+-}(Q)$ is a direct consequence of the perfect nesting, $E_{h,-}^\downarrow(k) = - E_{p,-}^\uparrow(k)$, and is found at all fields, if the conduction electron hopping is constrained to a bipartite lattice and the system is half filled.[^1]
The behavior of the staggered magnetization at small $U$ can be found by solving the mean field equations to leading logarithmic order in $U
m_x$. The details of the calculation are described in the appendix \[sec:app-meanfield\]. The resulting expression for the magnetization is $$\label{eq:mx}
m_x \propto \exp\left[ - \frac{(B+Um_z)^4+ V^2(B+Um_z)^2}
{V^4 \rho_0 U} \right],$$ where $\rho_0$ is taken to be the density of states on the $m_x = 0$ Fermi surface. It is interesting to note that the expression (\[eq:mx\]) is valid for the large field region and for the large $U$ region with $B \gg
8V^2/U$.
Quasi-particle spectrum {#sec:quasi-particle-gap}
-----------------------
The field dependence of the quasi-particle gap for a fixed value of the interaction is shown in the Fig. \[fig:qp-gap\].
![Quasi-particle gap (dots), staggered magnetization (dashed line) and total ($f$ + $c$) parallel magnetization (thin solid line) vs. magnetic field for $W = 4, V=1, U=1$. In the paramagnetic phase the gap decreases linearly with field. At large fields the gap follows the staggered magnetization.[]{data-label="fig:qp-gap"}](qpgap.eps){width="\columnwidth"}
In the paramagnetic phase, the gap decreases linearly with the field. In the ordered phase, the quasi-particle gap is proportional to the staggered moment and follows the same exponential dependence for large fields. It is important to realize that the quasi-particle gap always remains finite, so that the system is insulating.
The spectral functions for the electrons in the mean field model show infinitely sharp peaks at the quasi-particle band energies. The poles of the $\sigma=\uparrow$ electron spectral function in the ordered phase at various values of the field and interaction strengths are shown in Fig. \[fig:bands\].
![Poles of the $\sigma=\downarrow$ conduction electron spectral functions along the high symmetry lines of the Brillouin zone for field and interaction strengths indicated in the plots. The width of the lines indicates the weight in the pole. The plots on the right show the contours of constant quasi-particle gap magnitude with dashed lines indicating the position of the Fermi surfaces of the $m_x = 0$ state.[]{data-label="fig:bands"}](bands_gs.eps "fig:"){width="\columnwidth"}\
The width of the lines in the figure indicate the weights in the corresponding poles. Note that the gap at the Fermi surface is always finite, even though the exponentially small scale is not immediately apparent in the plots. The contour plots show the quasi-particle gap size in the Brillouin zone. The gap minima are indicated by the dashed lines in the contour plots. The location of the gap minima indicates also the position of the Fermi surfaces of the $m_x = 0$, metallic state.
It is interesting to observe the change in the character of the quasi-particles at the gap minima as the field and the interaction strength are varied. At small $U$ and $B \simeq \Delta_{qp}^0$, the low-energy quasi-particles are “heavy” and the minimum of the gap lies near the zone center. As the field is increased, the minimum moves towards the zone diagonal and the quasi-particles become more and more $c$ like.
When the gap minimum reaches the zone diagonal, the magnetization of the system along the field direction is exactly one half of the fully saturated value. The field strength at which this happens, $B_{1/2}$, depends on the interaction strength and can be obtained by setting $\epsilon_0 = 0$ with the limiting behavior: $$\label{eq:5}
B_{1/2} \rightarrow
\begin{cases}
V, & U \rightarrow 0 \\
\frac{2V^2}{U}, & U \rightarrow \infty
\end{cases}.$$ In the large $U$ limit, $B_{1/2}$ sets the energy scale at which the $f$ electrons align with the field.
Large B limit of the Kondo model {#sec:largeB}
================================
For large values of $U$, in the fields $B > B_{1/2}$, the $f$ electrons are almost completely aligned with the field. At the mean field level, the poles of the spectral function corresponding to the charge fluctuations on the $f$ sites move towards $\pm (B + U m_z)$, i.e. far from the Fermi level. In the large-$N$ mean field theory this eventually results in the complete decoupling of the $f$ electrons from the $c$ band and the decoupled metallic state obtains.[@beach.03]
In the particle hole symmetric case, the Fermi surfaces of the metallic state are perfectly nested. The perfect nesting makes the metallic state unstable at all fields in the small $U$ limit of the PAM. We will now demonstrate that also in the limit of large Coulomb interaction, i.e. for the KLM, the same instability arises.
Effective Hamiltonian approach {#sec:effH}
------------------------------
We consider the KLM Hamiltonian in the large magnetic field $B \gg J$. In the magnetic field, the $J=0$ ground state of the KLM is non degenerate and is given by $| \psi \rangle = \prod_{k<k_{F\uparrow}}
c^\dag_{k\uparrow}\prod_{k<k_{F\downarrow}} c^\dag_{k\downarrow}
\prod_i f^\dag_{i\uparrow} |\rangle$. Flipping the $f$ spin is an excitation with a gap given by the Zeeman energy. A canonical transformation approach can be used to generate an expansion in $(J/B)$ around the $J=0$ ground state. The effective Hamiltonian governing the low energy dynamics of the system is given by (the details of the derivation in the appendix \[sec:app-largeB\]), $$\label{eq:Heff}
\tilde H = \sum_{k\sigma} (\epsilon_k - p_\sigma \tilde B)
c^\dag_{k\sigma}c_{k\sigma} + \tilde U \sum_i
n_{i\uparrow}n_{i\downarrow}$$ where $\tilde U = \frac{J^2}{8 B}$, $\epsilon_k$ is the dispersion of the original conduction electron band and $\tilde B = (B - J/4 -
\tilde U/2 )$ is the effective magnetic field. In this effective model, the spin flip interaction between the conduction band and the fully polarized $f$-spin background of the KLM, has been replaced by a contact interaction between the $c$ electrons and the $f$ spins have decoupled from the dynamics.
If the conduction electron band is particle hole symmetric, so that $\epsilon_{k+Q} = - \epsilon_k$, the spin up hole and the spin down electron Fermi surfaces of the effective model are perfectly nested. Any non zero $\tilde U$ therefore induces magnetic ordering in the plane perpendicular to the applied field. A mean-field decoupling, with $\langle \vec s_i \rangle = ((-)^i m_x, 0, m_z)$, analogous to the one performed in section \[sec:meanfield\], yields the quasi particle bands $$E_{\sigma,\pm}(k) = \pm \sqrt{
(\epsilon_k - p_\sigma( \tilde B + \tilde U m_z) )^2 + (U m_x)^2
}$$ and the mean-field equation determining $m_x$, $$\frac 2 {\tilde U} = \int_{-W}^0 \rho(\epsilon)d\epsilon \left[
\frac 1 {E_{\uparrow,+}(\epsilon)} + \frac 1
{E_{\downarrow,+}(\epsilon)} \right].$$ In high magnetic fields, the up and down spin Fermi surfaces are well approximated by circles of radii $W - \tilde B$ centered at $(\pi,\pi)$ and $(0,0)$, respectively. We therefore can set $\rho(\epsilon) = \rho_0 = v_F^{-1}$ to obtain $$\tilde U m_x \propto 2 (W - \tilde B - \tilde U m_z)
\exp\left( -\frac 1 {\rho_0 \tilde U} \right).$$ The staggered magnetization and the quasi-particle gap are finite for any finite $\tilde U$, as long as $\tilde B + \tilde U m_z < W$. It is easy to see that, $\tilde B + \tilde U m_z = W$ is just the condition for system to fully polarize. This means that the staggered magnetization vanishes only in the completely polarized system. The completely polarized phase is a trivial insulator. The metallic state is, therefore, never obtained in the particle-hole symmetric case and is a bad starting point for the perturbation expansion.
Classical spins mean field {#sec:classical-spins-mean}
--------------------------
We have seen that a large magnetic field suppresses both charge and spin fluctuations on the $f$ sites. The physics of the high-field phase will, therefore, be well described by the KLM in which the $f$ spins are replaced by an array of statically arranged classical spins. Let the spin configuration be $$\vec S_i = \frac 1 2 \left(
\begin{array}[c]{c}
\sin \theta \cos Q r_i \\
-\sin \theta \sin Q r_i \\
\cos \theta
\end{array}
\right).$$ With $Q=(\pi,\pi)$, this corresponds to the same choice of $f$ magnetization as in section \[sec:meanfield\], with $1/2 \sin \theta
= m_x$ and $1/2 \cos \theta = m_z$, so that the system is fully polarized for $\theta = 0$. The problem is now reduced to one of the non-interacting conduction electrons in an external magnetic field, described by the Hamiltonian, $$\begin{gathered}
\label{eq:clH}
H = \sum_{k,\sigma} \left( \epsilon_k - p_\sigma \tilde B \right)
c^\dag_{k\sigma} c_{k\sigma} - N B \cos \theta \\
+ \frac{J \sin \theta}{4} \sum_k \left( c^\dag_{k-Q \uparrow} c_{k
\downarrow} + c^\dag_{k+Q \downarrow} c_{k \uparrow} \right),\end{gathered}$$ with $\tilde B = B - J/4 \cos \theta$. This is easily diagonalized to find the quasi-particle bands ($k$ in the magnetic Brillouin zone), $$\label{eq:clbands}
E_{\sigma,\pm}(k) =
\pm \sqrt{(\epsilon_k - p_\sigma \tilde B)^2 + (J/4)^2 \sin^2 \theta} .$$ In the ground state the “-” bands are completely filled. Minimizing the ground state energy and assuming the same circular Fermi surface approximation as in the previous subsection, the mean field equation determining the angle $\theta$ is obtained as, $$\begin{gathered}
B \tan \theta = \rho_0 \int_{-W}^0 \sum_\sigma \left[ \left(\frac J
4 \right)^2 \sin \theta
\right. \\
\left. -p_\sigma (\epsilon - p_\sigma \tilde B) \frac J 4 \tan
\theta \right] \frac 1 {E_{\sigma,+}}\end{gathered}$$ For $\sin \theta \ll 1$ ($f$ moments almost aligned with the field) we obtain, $$\label{eq:clmx}
J m_x \propto \sqrt{(W-\tilde B)(W + \tilde B)}
\exp\left[-\frac{8B}{\rho_0 J^2} + \frac{4 \rho_0 B +8}{\rho_0 J} \right].$$ The staggered magnetization vanishes only when $\tilde B = W$. It is easy to see that this is exactly the condition for the system to fully polarize. The full polarization field is equal to $W - J/4$ and agrees with the one obtained using the effective Hamiltonian. The dominant, small $J$ exponential dependence of the staggered magnetization $m_x
\propto \exp [ -8B/(\rho_0 J^2)]$ also agrees with the one obtained in the previous section. The subleading $1/(\rho_0 J)$ correction to the exponent arises because the Zeeman energy of the $f$ electrons has now been taken into account. As the quasi-particle gap is proportional to the staggered magnetization, the system stays insulating at all fields.
Quantum Monte Carlo {#sec:qmc}
===================
\
In this section we present QMC simulations of the Kondo lattice model in the magnetic field. As in the zero field case, the sign problem may be avoided only for particle-hole symmetric conduction bands. To compare with the mean-field results we adopt a projective approach in which the ground state, $ | \Psi_0 \rangle $, is filtered out from a trial wave function, $ | \Psi_T \rangle $, satisfying $ \langle \Psi_0
| \Psi_T \rangle \neq 0$. In this algorithm, the ground state expectation value of an observable $O$ is estimated via: $$\frac{ \langle \Psi_0 | O | \Psi_0 \rangle } { \langle \Psi_0 | \Psi_0 \rangle }
= \lim_{\Theta \rightarrow \infty}
\frac{ \langle \Psi_T | e^{- \Theta H /2 } O e^{ - \Theta H / 2} | \Psi_T \rangle }
{ \langle \Psi_T | e^{-\Theta H} | \Psi_T \rangle }.$$ In the QMC, we evaluate the right hand side of the above expression at finite values of $\Theta$ and then extrapolate to infinite values. The details of the algorithm – in particular the sign free formulation – has been described extensively in Ref. . Since we are working in the canonical ensemble, the total magnetization $$M_z = \frac{N^{\uparrow}_c + N^{\uparrow}_f - N^{\downarrow}_c - N^{\downarrow}_f }{N_u}$$ with $N_u$ the number of unit cells, is fixed during the simulations.
By measuring time displaced correlation functions, we can extract quasi-particles as well as spin gaps. Consider $$\begin{gathered}
\langle \Psi_0 | S^-(-q,\tau) S^+(q,0) | \Psi_0 \rangle = \\
\sum_n | \langle n | S^+(q) |\Psi_0 \rangle |^2
e^{-\tau(E_n(q,N,S_z+1) - E_0(N,S_z)) }\end{gathered}$$ where $E_n(q,N,S_z)$ are eigenstates of $H$ with momentum $\vec{q}$, particle number $N$ and total $z$-component of spin $S_z$. From the large $\tau$ behavior of the above correlation functions, we can extract the energy difference $E_0(q,N,S_z+1) - E_0(N,S_z)$ from which we can determine the spin gap: $$\Delta_{sp}(\vec{q}) = E_0(\vec{q},N,S_z+1) - E_0(N,S_z) - h$$ where $ h = \left[ E_0(N,S_z + 1) - E_0(N,S_z -1) \right] / 2 $. In the same manner, we compute the quasi-particle gap from the single-particle imaginary time displaced Green function. $$\Delta_{qp}(\vec{k}) = E_0 ( \vec{k}, N+1,S_z) - E_0(N,S_z) - \mu$$ with chemical potential: $\mu = [E_0(N+1,S_z) - E_0(N-1,S_z) ] / 2 $. In Fig. \[Gaps.fig\] we plot the gaps as a function of total magnetization at $J/t = 2.0$.
In the zero field case, the Kondo insulating state with finite quasi-particle and spin gaps is realized. At finite magnetizations and according to the mean-field approach, we expect a canted antiferromagnetic state and hence no spin gap. Fig. \[Gaps.fig\] confirms this point of view: the spin gap drops to zero at all finite values of the magnetization within the accuracy of the numerical simulation and the equal time spin-spin correlations in the plane perpendicular to the magnetic field show long range antiferromagnetic order. In the mean-field approach, magnetism stems from a Fermi surface instability and due to perfect nesting opens a gap on all the Fermi line. The QMC results of Fig. \[Gaps.fig\] are consistent with this prediction, since, as apparent, the quasi-particle gap survives at finite magnetizations.
{width="\textwidth"}
To further compare the mean-field approach to the QMC we have used the Maximum Entropy method to obtain the single particle spectral function $A(\vec{k},\omega)$. At zero field in the Kondo insulating phases, the dominant features of the spectral function are well described by hybridized bands (solid vertical lines in Fig. \[akom.fig\]a). From the single particle occupation number $ n_{\vec{k},\sigma} = \langle
c^{\dagger}_{\vec{k},\sigma} c_{\vec{k},\sigma} \rangle $ listed on the left hand side of Figs \[akom.fig\]a-c we can extract the total weight under the “photoemission” ($\omega < 0$) and the “inverse photoemission” ($\omega < 0$) spectra since $$\label{Sum_rule}
\int_{-\infty}^{0} {\rm d } \omega A^{\sigma}(\vec{k},\omega) = \pi n_{\vec{k},\sigma} \;\;
\int_{0}^{\infty} {\rm d } \omega A^{\sigma}(\vec{k},\omega) = \pi(1 - n_{\vec{k},\sigma} ).$$ In particular, one sees that the photoemission (inverse photo-emission) spectrum in the vicinity of $ \vec{k} = (\pi,\pi) $ $
(\vec{k} = (0,0)) $ has a small weight. Those heavy bands stem from the Kondo screening.
In the mean field approach, the effect of a magnetic field is to shift the spin down band up in energy until it ultimately crosses the Fermi surface, thus generating a metallic state. This metallic state is however unstable due to the underlying particle-hole symmetry. In Fig. \[akom.fig\]c we compare the finite field results with a rigid shift of the hybridized bands. The data are compatible with the interpretation that the down spin band has indeed crossed the Fermi surface but that at the crossing point the magnetic instability opens a gap. Furthermore on the photo-emission side around the $(\pi,\pi)$ point we see a very weak feature which we can identify as a shadow of the up band which has dropped below the Fermi energy in the vicinity of the $\vec{k} = (0,0) $ point.
Breaking of the spin symmetry by the magnetic field suppresses Kondo screening. We hence expect the weight of the features in the spectral function stemming from Kondo screening to be suppressed as a function of growing magnetization. For example, consider the inverse photoemission at $\vec{k} = (0,0) $ in Fig. \[akom.fig\]. As apparent the weight of this feature is reduced as a function of growing values of $M_z$ and will vanish when the $f$-spins become fully polarized. In our simulations, the $f$-spins are never completely aligned with the field as long as the system is not completely polarized, i.e. as long as $N_c^\uparrow + N_f^\uparrow <
N_u$. This supports the conclusion that the metallic state is never induced by the field.
Conclusion {#sec:conclusion}
==========
We studied the magnetic field induced quantum phase transition in the 2D particle-hole symmetric Kondo insulators using: i) a mean field approximation appropriate in the small $U$ limit of the periodic Anderson model, ii) two mean field approximations appropriate in the large field limit of the Kondo lattice model and iii) a quantum Monte Carlo simulation of the particle-hole symmetric 2D Kondo lattice model in the field.
We find a magnetic field induced quantum phase transition from a paramagnetic insulator into a canted antiferromagnetic insulator ground state. In the particle-hole symmetric case we studied, the antiferromagnetism can be understood as a spin density wave type instability of the perfectly nested quasi-particle Fermi surfaces that would arise in the field in the absence of interactions. Because of the perfect nesting, any finite interaction is a relevant perturbation and results in a finite quasi-particle gap. Consequently, the ground state of the interacting system remains insulating in all fields.
We conclude that the recently proposed insulator to metal transition induced by the field[@beach.03] is likely to be an artifact of the large-$N$ approximation to the Kondo lattice model in the particle-hole symmetric case. If, however, the particle-hole symmetry is violated a field-induced metal-insulator transition is possible in certain parameter ranges.
We find that the qualitative features of the phase diagram as well as of the quasi-particle excitations are well described by a simple mean field approximation to the periodic Anderson model. The magnetic field explicitly breaks the spin rotation symmetry and suppresses the charge fluctuations on $f$ electrons, essentially by fully polarizing the $f$ band.
The band structure of the Kondo insulators is non-trivial and deviations from particle-hole symmetry are to be expected in the real materials. In the absence of perfect nesting there would be a critical value of the field, controlled essentially by the nesting mismatch, at which the gap will close on some parts of the Fermi surface. Therefore one expects to eventually find a metallic state induced by the field. Finally we would like to mention that the conclusions we draw here are valid also for the three-dimensional systems, where the same kind of nesting features would appear for perfect particle-hole symmetry.
[**Acknowledgments**]{} We would like to thank P.A. Lee, T.M. Rice, K.D.S. Beach and S. Wehrli for helpful discussions. This work was financially supported by the Swiss Nationalfonds and, in particular, by the NCCR program MaNEP. F.F. Assaad thanks the DFG for financial support under the grant number of AS 120/1-1 as well as the hospitality of the ITP of the ETHZ where part of this work was carried out. The calculations were performed on the Cray-T3E as well as on the IBM-p690 in Jülich. We thank this institution for generous allocation of CPU time.
Mean field decoupling of the Hubbard term in the canted anti-ferromagnetic phase {#sec:app-meanfield}
================================================================================
We want to decouple the interaction term in the PAM, in the presence of a canted staggered magnetization. To this end, we select the spin quantization axis at each site to point in the direction of the local magnetization, $\vec m_i$. Using the operator identity $ f^\dag_{i\uparrow_i} f_{i\uparrow_i}
f^\dag_{i\downarrow_i} f_{i\downarrow_i} = 1/4 (f^\dag_{i\uparrow_i}
f_{i\uparrow_i} + f^\dag_{i\downarrow_i} f_{i\downarrow_i})^2 - 1/4
(f^\dag_{i\uparrow_i}f_{i\uparrow_i} - f^\dag_{i\downarrow_i}
f_{i\downarrow_i})^2$, where $\uparrow_i (\downarrow_i)$ denotes the spin with respect to the local quantization axis, the interaction term can be decoupled as $$\begin{gathered}
\label{eq:decoupling}
U \sum_i
f^\dag_{i\uparrow_i} f_{i\uparrow_i} f^\dag_{i\downarrow_i} f_{i\downarrow_i}
=
\frac {Un_f}2 \sum_i ( f^\dag_{i\uparrow_i} f_{i\uparrow_i}
+ f^\dag_{i\downarrow_i} f_{i\downarrow_i} ) \\
- N \frac{Un_f^2} 4
- U \sum_i |\vec m_i| ( f^\dag_{i\uparrow_i} f_{i\uparrow_i}
- f^\dag_{i\downarrow_i} f_{i\downarrow_i} ) + U |\vec M|^2 \\
+ \text{Fluct.} \end{gathered}$$ where $n_f = \langle f^\dag_{i\uparrow} f_{i\uparrow} +
f^\dag_{i\downarrow} f_{i\downarrow} \rangle$ is the average occupancy of the $f$ site and “Fluct.” denotes the terms neglected in the mean-field approximation. After the decoupling, a spin axis rotation to a common quantization axis (given by the direction of the external field) is performed using $\hat R = \exp \left( \sum_i -i/2 \vec \theta_i
\cdot \vec S^f_i \right)$ with $\vec \theta_i$ being the vector pointing along $\vec B \times \vec m_i$ and of magnitude equal to the angle between $\vec B$ and $\vec m_i$. Since, $$|\vec m_i| \hat R \tfrac 1 2 ( f^\dag_{i\uparrow_i} f_{i\uparrow_i}
- f^\dag_{i\downarrow_i} f_{i\downarrow_i} ) \hat R^\dag = \vec m
\cdot \vec S^f_i,$$ this yields the mean field Hamiltonian, Eq..\[eq:HMF\] of section \[sec:meanfield\].
To obtain the behavior of the staggered magnetization in fields $B >
\Delta_{qp}^0$ and at small $U$ we need to solve the mean field equations \[eq:mf\], in the limit when $U m_x \rightarrow 0$. As the only $k$ dependence of the quasi-particle bands comes through the $k$ dependence of the conduction electron energy, the summations over $k$ are readily transformed into integrals over the conduction electron energy, thus yielding ($\rho(\epsilon)$ is the conduction electron DOS), $$\frac{\partial}{\partial (Um_x)} \left[
\int_{-W}^0 \rho(\epsilon)
d \epsilon (E_{h,\pm}^\uparrow(\epsilon) +
E_{h,\pm}^\downarrow(\epsilon)) \right]
+ 2 \frac {Um_x}{U} = 0.$$ For $Um_x \rightarrow 0$, the dominant contribution to the integral comes from the band crossing the Fermi surface at $\epsilon_0$ (given by Eq. \[eq:epsilon0\]). For $\Delta_{qp}^0 < B < V$, this is $E_{h,+}^\downarrow$. For small $U$ and close to $\epsilon_0$ we can write, $$E_{h,+}^{\downarrow}(\epsilon) = \sqrt{\alpha^2 (Um_x)^2 + \beta^2
(\epsilon - \epsilon_0)^2},$$ with $$\begin{aligned}
\alpha &= \frac{V^2}{(B+Um_z)^2 + V^2} \\
\beta &= \frac{(B+Um_z)^2}{(B+Um_z)^2 + V^2}.\end{aligned}$$ The logarithmically divergent part of the mean field equation can now be written as (neglecting the non-divergent contributions), $$\frac 1 U = \frac{\alpha^2}{\beta} \int_0^{\epsilon_c}
\frac {d\epsilon' \rho(\epsilon_0+\epsilon')}
{\sqrt{\left(\frac{\alpha U m_x}{\beta}\right)^2 + \epsilon'^2}},$$ where we have introduced a cutoff $\epsilon_c$ which does not influence the exponential dependence. Eq. \[eq:mx\] in the text now follows by elementary integration.
Effective Hamiltonian in the large field {#sec:app-largeB}
========================================
The ground state of the $J=0$ KLM in the magnetic field is non-degenerate. All the $f$ spins are polarized in the direction of the field. The $f$ spin flip is an excitation with an energy gap of the size of the Zeeman energy. Let $P_n$ denote the portion of the Hilbert space with $n$ localized spins pointing opposite to the magnetic field and let $\mathcal P_n$ be the corresponding projector. In large magnetic fields, $B \gg J$, one expects the ground state and the low lying excitations to lie dominantly in $P_0$ and have only small components in the $P_n$, with $n>0$, subspaces. We split the $H_{KLM}$ into a $P_n$ diagonal part $$\begin{gathered}
\label{eq:H0}
H_0 = \sum_{k\sigma} (\epsilon_{k\sigma} - p_\sigma B )
c^\dag_{k\sigma} c_{k\sigma}
- 2 B \sum_\sigma p_\sigma S_i^z \\
+ \frac{J}{2} \sum_i S_i^z ( c^\dag_{i\uparrow}
c_{i\uparrow} - c^\dag_{i\downarrow} c_{i\downarrow})\end{gathered}$$ and the spin flip part $$V = \frac J 2 \sum_i \left( c^\dag_{i\uparrow} c_{i\downarrow}
S_i^- +
c^\dag_{i\downarrow} c_{i\uparrow} S_i^+ \right) .$$ Let $S$ be a Hermitian operator, such that $$\label{eq:S1}
[H_0, S] = - V .$$ The effective Hamiltonian, obtained by applying the canonical transformation $$% \label{eq:Heff}
\tilde H = e^{-S} H_{KLM} e^S$$ has no matrix elements between the states in $P_0$ and $P_{n>0}$ of order less then $\mathcal O ( J(J/B)^2 )$. We can therefore obtain the low-energy dynamics of the original problem correctly to order $J^2/B)$ by considering only the $P_0$ part of the Hilbert space and the Hamiltonian, $$\label{eq:HeffP0}
\tilde H^{P_0} = \mathcal P_0 e^S H_{KLM} e^{-S} P_0.$$ By expanding and rearranging the exponentials one obtains $$\label{eq:commutators}
\tilde H^{P_0} = \mathcal P_0 \left( H_0 + \frac 1 2 \left[V,
S\right] + \cdots \right) \mathcal P_0$$ Using Eq.(\[eq:S1\]) it is easy to obtain the matrix elements of $S$ between the eigenstates of $H_0$ from which the operator form easily follows, $$\label{eq:S}
S = \frac{J}{4B} \frac 1 {\sqrt N} \sum_{kq} \left(
S^+_q c^\dag_{k-q\downarrow} c_{k\uparrow}
- S^-_q c^\dag_{k-q\uparrow} c_{k\downarrow} \right),$$ thus yielding, $$[V,S] = \frac {J^2}{4B}
% \sum_q \left[ S_q^- s_{-q}^+,S_q^+s_{-q}^- \right]
\frac 1 N \sum_{qkk'} \left[ S_q^- c^\dag_{k\uparrow}c_{k-q\downarrow},
S_q^+ c^\dag_{k'\downarrow}c_{k'-q\uparrow} \right] .$$ Evaluating the commutator and projecting on the $P_0$ subspace, yields $$\mathcal P_0 [V,S] \mathcal P_0 =
- \frac {J^2}{4B} \frac 1 N \sum_{kk'q}
c^\dag_{k'\downarrow}c_{k'-q\uparrow} c^\dag_{k\uparrow}c_{k-q\downarrow}.$$ Substituting into Eq. (\[eq:commutators\]) and neglecting the terms corresponding to the dots, which are of order $\mathcal O(J^3/B^2)$ one obtains the effective Hamiltonian, $$\tilde H = H_0
+ \frac{J^2}{8B}\sum_i s^z_i
+ \frac{J^2}{8B} \sum_i n_{i\uparrow} n_{i\downarrow}
- \frac{J^2 N}{16 B}.$$ The last term in the above equation is a constant and can be dropped, the $s^z_i$ term is the contribution to the effective uniform magnetic field. This completes the derivation of the effective model described in section \[sec:largeB\].
[^1]: Note that, the shape of the Fermi lines is determined only by the conduction electron dispersion and the surface they enclose by the half-filling condition. Therefore, the perfect nesting in the half-filled system can not be removed by changing the ratio of $g$ factors or by changing $\epsilon_f$.
|
CERN-TH.6541/92\
KUL-TF-92/24\
hepth@xxx/9206097
[**The regularized BRST Jacobian of pure Yang-Mills theory**]{}\
5.mm [**F. De Jonghe$^1$, R. Siebelink$^2$, W. Troost$^3$, S. Vandoren$^4$**]{}\
Instituut voor theoretische fysica\
K. U. Leuven, B-3001 Leuven, Belgium\
[**P. van Nieuwenhuizen$^5$**]{} and [**A. Van Proeyen$^6$**]{}\
Theory Division, CERN\
CH-1211 Geneva 23, Switzerland
> The Jacobian for infinitesimal BRST transformations of path integrals for pure Yang-Mills theory, viewed as a matrix ${\mathord{\!\usebox{\uuunit}}}+\Delta J$ in the space of Yang-Mills fields and (anti)ghosts, contains off-diagonal terms. Naively, the trace of $\Delta J$ vanishes, being proportional to the trace of the structure constants. However, the consistent regulator ${{\cal R}}$, constructed from a general method, also contains off-diagonal terms. An explicit computation demonstrates that the regularized Jacobian $Tr\ \Delta J\exp -{{\cal R}}/M^2$ for $M^2\rightarrow \infty $ is the variation of a local counterterm, which we give. This is a direct proof at the level of path integrals that there is no BRST anomaly.
>
> ------------------------------------------------------------------------
>
> width 5.cm [$^1$ Aspirant, NFWO, Belgium; Bitnet FGBDA16 at BLEKUL11\
> $^2$ Aspirant, NFWO, Belgium; Bitnet FGBDA04 at BLEKUL11\
> $^3$ Bevoegdverklaard navorser, NFWO, Belgium; Bitnet FGBDA19 at BLEKUL11\
> $^4$ Bitnet FGBDA43 at BLEKUL11\
> $^5$ On leave from the Institute for Theoretical Physics, SUNY at Stony Brook, NY 11794, USA; bitnet VANNIEU at SUNYSBNP\
> $^6$ On leave from Instituut voor theoretische fysica, K.U. Leuven; Onderzoeksleider, N.F.W.O. Belgium; Bitnet FGBDA19 at BLEKUL11]{}
CERN-TH.6541/92\
KUL-TF-92/24\
June 1992
The rigid [^1] BRST symmetry [@BRST] of the quantum action for gauge theories has become an essential tool in the path-integral approach. One uses the BRST symmetry to obtain Ward identities [@Ward], which are then used to prove perturbative renormalizability [@renorm] and unitarity [@unit]. In principle, the Ward identities can contain an anomaly which appears in the path-integral approach as a deviation from unity of the Jacobian for BRST transformations. It is often argued that there is no anomaly in the BRST Ward identities because $\Delta J$ is proportional to the trace of the structure constants, which vanishes for semi-simple gauge groups. However, this argument, though widely repeated and presented in various textbooks, is patently false. For example, the same line of reasoning would conclude that there never are chiral anomalies since $tr\,\gamma
_5=0$. For chiral anomalies one knows that after regularization with a regulator ${{\cal R}}$, the regularized trace $tr\,\gamma _5 \exp -{{\cal R}}/M^2$ for $M\rightarrow \infty $ no longer vanishes in general and yields the chiral anomaly. For chiral symmetries, the choice of ${{\cal R}}$ is not very important, as “\[one\] can show that under broad assumptions the one loop anomalies depend only on the quantum numbers of the elementary fields, and not on the specific Lagrangians chosen" [@AGW]. This is due to the topological nature of the chiral anomaly. In general, the choice of ${{\cal R}}$ clearly matters. For example, the Jacobian for Weyl symmetry is field-independent, and if one were to choose a regulator which is also field-independent, one could never obtain anomalies proportional to the curvatures. Recently, a general method was developed for constructing a consistent regulator for the measure of any quantum field theory [@anomPV]. This regulator is equivalent to Pauli-Villars regularization of the Feynman diagrams of the effective action, and yields anomalies which satisfy the consistency conditions [@WZcc]. It is this regulator which we will use to compute the BRST anomaly. For BRST symmetry it is computationally not obvious that the regularized trace $Tr
\Delta J\exp -{{\cal R}}/M^2$ still vanishes in the limit $M\rightarrow \infty$, although one would expect so, given that indirect (cohomological) arguments say so [@coho]. In this article we will demonstrate by a direct calculation that the BRST anomaly in the Ward identities indeed vanishes [^2].
The derivation of the Ward identities is based on the Shakespeare theorem [^3] according to which one may rename the integration variables $\phi ^j$ everywhere (in the measure, in the action and in the coupling to external sources) by $\phi '^j=\phi ^j +
\delta \phi ^j$ $$\begin{aligned}
Z(J)&=&\int {\cal D}\phi ^j\,\exp {\textstyle{\frac{i}{\hbar }}}\left[
S_{qu}(\phi )+ \int J_j\phi ^j d^4x\right] \nonumber\\
&=&\int {\cal D}\phi'^j\,\exp {\textstyle{\frac{i}{\hbar }}}\left[
S_{qu}(\phi')+ \int J_j\phi'^j d^4x\right]\ .\end{aligned}$$ We take for $\delta \phi ^j$ an infinitesimal BRST transformation. The quantum action $S_{qu}(\phi )$ consists of an $\hbar =0$ part which is BRST invariant, and possibly local counterterms $\hbar M_1$, where $M_1$ is in general a power series in $\hbar $. This leads to the formal Ward identity $$\bigg< Tr\,\Delta J + i\delta M_1 +
{\textstyle{\frac{i}{\hbar }}}
\int J_j\delta \phi ^j d^4x \bigg>=0 \label{eq:Ward}$$ where $$\Delta J^j{}_k=\frac{{\raise.3ex\hbox{$\stackrel{\leftarrow}{\partial }$}}\delta\phi^j}{\partial \phi^k}$$ (with right derivatives) is the deviation of the Jacobian matrix from unity. We shall discuss the regularization of the trace $Tr\,\Delta J$ in pure Yang-Mills theory, with $\phi ^j$ equal to the fields $b$ (antighosts), $Q_\mu $ (quantum Yang-Mills fields) and $c$ (ghosts). However, for reasons to be explained, we shall use an action containing an extra, external, gauge field $B_\mu $, which interpolates between ordinary quantum field theory and the background field method. This action $S=S_{qu}(\hbar =0)$ reads $$S=Tr\int d^4x\left\{ -\textstyle{\frac{1}{4}}F_{\mu \nu }(Q)^2
-\textstyle{\frac{1}{2}}\left[
D^\mu (B)(Q_\mu -B_\mu )\right] ^2
- D_\mu (B) b\cdot D^\mu (Q) c
\right\} \label{SYMB}$$ where $D_\mu (X)Y=\partial _\mu Y +[X_\mu ,Y]$ and the trace $Tr$ is over gauge indices. For $B_\mu=0$, we obtain the Feynman gauge, while for $B_\mu \neq 0$ we recognize the action for the background field formalism (after shifting $Q_\mu $ to $Q_\mu +B_\mu $). This last action is invariant under two symmetries
1. local background symmetry, under which both the background field $B_\mu$ and the quantum field $Q_\mu $ transform as gauge fields, and $b$ and $c$ as vectors $$\begin{aligned}
\delta Q_{\mu} &=& D_\mu (Q) \lambda \nonumber\\
\delta B_\mu &=& D_\mu (B) \lambda \nonumber\\
\delta b&=& [b,\lambda ] \nonumber\\
\delta c&=& [c,\lambda ]\ .\label{bgrtr}\end{aligned}$$ with $\lambda $ the local commuting Lie-algebra valued Yang-Mills parameter.
2. rigid BRST symmetry, under which the background field is inert $$\begin{aligned}
\delta Q_\mu &=& D_\mu (Q) c\Lambda \nonumber\\
\delta B_\mu &=& 0 \nonumber\\
\delta b &=& - D^\mu (B) (Q^{\mu} - B^{\mu})\Lambda \nonumber\\
\delta c &=&{{\textstyle\frac{1}{2}}}[c,c] \Lambda \label{BRSTtr}\end{aligned}$$ with $\Lambda $ the constant, anticommuting BRST parameter.
In this article we shall discuss the Jacobians for these symmetries, as they appear in the path-integrals in [(\[eq:Ward\])]{}.
If one (erroneously) neglects regularization, one would conclude that the trace of $\Delta J$ in [(\[eq:Ward\])]{} vanishes for both symmetries in [(\[bgrtr\])]{} and [(\[BRSTtr\])]{}, as the diagonal entries are proportional to $f^a{}_{ab}$ in each case (where $f^a{}_{bc}$ are the structure constants of a Lie algebra, which are traceless for the semi-simple groups which we consider). As we explained above this conclusion is incorrect.
In view of the importance of BRST symmetry, we think that a direct, explicit calculation of the anomaly (the trace of $\Delta J
\exp -{{\cal R}}/M^2$) for both symmetries is useful. Of course we shall discuss how to determine ${{\cal R}}$. In addition, we shall determine by direct computation whether the anomaly can be written as the variation of a local counterterm (and thus removed) and when it actually vanishes.
We work in the generalized gauge with $B_\mu $ present in order to avoid an accidental vanishing of the anomaly. To draw a comparison with string theory, we prefer to work in a ‘general gauge’ (like $g_{\alpha\beta }=G_{\alpha\beta }$ where $G_{\alpha \beta }$ is an arbitrary background field) rather than a special gauge. If one were to choose the gauge $g_{\alpha\beta }=\eta _{\alpha \beta }$, the anomaly ${\cal A}= c\sqrt{g}R(g)=0$ would seem to vanish. Of course, this is not an allowed gauge since it cannot be reached using the gauge symmetries without anomalies. Similarly, in our case, we still have to prove that there are no Yang-Mills anomalies, so we should not use the Yang-Mills symmetry to choose a special gauge. In the Batalin-Vilkovisky field-antifield formalism [@BV] there is a natural way to work with general gauges [@bvsb]. Usually one eliminates antifields by putting $\phi ^*_j=\frac{\partial\Psi }{\partial\phi^j}$, where $\Psi $ is a ‘gauge fermion’. This is equivalent to first making a canonical transformation with a ‘generating function’ $\Psi $, and then projecting onto the hypersurface $\phi ^*_j=0$. However, if one does not project onto this hypersurface and keeps the antifield dependence, then these play the role of arbitrary gauge parameters. For the string e.g., the choice $\Psi =b^{\alpha \beta }(g_{\alpha
\beta} -\eta _{\alpha\beta})$ yields $g_{\alpha \beta }=\eta
_{\alpha \beta }-b^*_{\alpha \beta }$, where the last term is the antifield of the antighost. Clearly, whether the anomaly is expressed as a function of a general background metric $G_{\alpha \beta }$, or as a function of the $b^*_{\alpha \beta }$-antifield is the same thing. We have done the calculations also with antifields (for $B_\mu=0$, which is then sufficient), but for simplicity, we shall present here the calculations with [(\[SYMB\])]{} (and zero antifields), as this action is of practical interest and general enough for our purposes.
The most practical method for regularisation of Feynman diagrams is the dimensional regularisation method. It keeps gauge invariance at all stages when there are no $\gamma _5$ or similar dimension-dependent objects present. Therefore there are no anomalies in this regularization scheme in these cases. However, dimensional regularization can not be applied directly to the path integral measure. As shown by Fujikawa [@Fujikawa], in path integrals the anomalies come from the measure. However, in the original works, it was not known which regulators would give consistent [@WZcc] anomalies. This problem was solved in [@anomPV], where a general recipe was obtained which yields a consistent regulator for any quantum field theory, once the quantum action is given. Input for this method is a mass matrix $\phi ^iT_{ij}
\phi ^j$, which must be non-singular. Output is a consistent regulator ${{\cal R}}$, given by $${{\cal R}}^i\,_j = (T^{-1})^{ik}S_{kj} \ ;\qquad
S_{ij}=\frac{{\raise.3ex\hbox{$\stackrel{\rightarrow}{\partial}$}}}{\partial\phi^i}\frac{{\raise.3ex\hbox{$\stackrel{\leftarrow}{\partial }$}}}{\partial \phi^j}S\ .$$ The regularized trace (really a supertrace due to the presence of the (anti)ghosts, hence denoted by $str$) is then given by $$\begin{aligned}
{\cal A}&=&\lim_{M^2\rightarrow \infty }
str\ \Delta J \exp (T^{-1}S/M^2)\nonumber\\
&=&\lim_{M^2\rightarrow \infty }
str \Delta J_s e^{T^{-1}S/M^2}\ ,\label{anomJs}\end{aligned}$$ with $$\Delta J_s\equiv {{\textstyle\frac{1}{2}}}\left(
\Delta J + T^{-1} \Delta J^t T \right)\ .$$ The transposition on $\Delta J$ in this equation refers to a supertransposition [^4] of the $\Delta J$ matrix, a transposition of the derivative operators which amounts to an extra minus sign after partial integration, and a transposition of the Lie-algebra representation matrices which amounts to another minus sign (in the adjoint representation). This rewriting of ${\cal A}$ makes use of the super-symmetry of $S_{ij}$ and $T_{ij}$, and will simplify calculations later on.
It is conjectured (and checked in many examples) that\
- this method gives a consistent anomaly, i.e., the Wess-Zumino conditions are satisfied.\
- the expression for $\cal A$ is gauge dependent, but this dependence can be absorbed in a counterterm.\
- different choices of the mass term lead to different expressions of ${\cal A}$ which again differ by the variation of a local counterterm.\
This example will provide another check on the first two of these conjectures.
We choose a mass term which is invariant under rigid Yang-Mills transformations : $tr\ \int d^4x\left[ Q_\mu Q^\mu +2bc\right]$. We write the matrix-entries in order of decreasing ghost number, namely in the order $b$, $Q^\mu $, $c$ : this makes the triangular nature of the matrices to follow more manifest. Then $$T=\left( \begin{array}{ccc}
0 & 0 & 1 \\
0 & \eta _{\mu\nu }& 0 \\
-1 & 0 & 0 \end{array} \right)\ ;\qquad
T^{-1}=\left( \begin{array}{ccc}
0 & 0 &- 1 \\
0 & \eta ^{\mu\nu }& 0 \\
1 & 0 & 0 \end{array} \right)\ .$$
For the local background symmetry, $\Delta J$ is diagonal, with entries $f^a{}_{bc}\lambda ^c$, $f^a{}_{bc}\lambda ^c\delta ^\mu _\nu $ and $f^a{}_{bc}\lambda ^c$, respectively. One may verify that $\Delta J_s$ vanishes in this case. Hence, the background symmetry is preserved at the quantum level in an almost obvious way.
For the rigid BRST symmetry, to which we devote the rest of this article, we find that $\Delta J$ is off-diagonal and contains derivatives, $$\begin{aligned}
\Delta J=\left(\begin{array}{ccc}
0 & -D_{\nu}(B) & 0\\
0 & -c\delta ^\mu\, _\nu &-D^\mu (Q) \\
0 & 0 &-c
\end{array}\right)(x)\delta (x-y)\Lambda \ ,\end{aligned}$$ where entries such as $c$ stand for the matrices $f^a{}_{cb}c^c$. The symmetrized $\Delta J_s$ contains no derivatives but is purely algebraic: $$\Delta J_s = \frac{1}{2}\left(\begin{array}{ccc}
c & Q_\nu -B_\nu & 0\\
0 & 0 & -Q^\mu +B^\mu \\
0 & 0 & -c
\end{array}\right)(x)\delta (x-y)\Lambda \ .$$ This fact will greatly simplify the evaluation of the supertrace in [(\[anomJs\])]{}. Furthermore, the operator matrix $T^{-1}S$ is given by $$T^{-1}S=\left(\begin{array}{ccc}
D_\alpha (Q) D^\alpha (B) &-(D_\nu (B) b) & 0\\
c D^\mu (B) & R^\mu {}_\nu & (D^\mu (B) b)\\
0 & -D_\nu (B) c & D_\alpha (B) D^\alpha (Q)
\end{array}\right)(x)\delta (x-y)\ ,$$ where $$R^\mu {}_\nu =
D_\alpha(Q)D^\alpha(Q) \delta ^\mu_\nu - D^\mu (Q)D_\nu (Q)
+ D^\mu (B)D_\nu (B) + 2 F^\mu{}_\nu(Q)$$ and the covariant derivatives act as far as $\delta ^4(x-y)$, unless put within explicit brackets. This expression can be cast into the form $$T^{-1}S=
\left( \partial_{\alpha}{\mathord{\!\usebox{\uuunit}}}+{\cal Y_{\alpha}} \right)\eta ^{
\alpha \beta }
\left( \partial_{\beta}{\mathord{\!\usebox{\uuunit}}}+{\cal Y_{\beta}} \right) + E
\label{regulcast}$$ where ${\mathord{\!\usebox{\uuunit}}}$ and $\cal Y_\alpha $ are $6\times 6$ matrices with entries in the adjoint representation of the Yang-Mills Lie algebra, and $\alpha,\,\beta $ are ordinary Minkowski indices. The derivatives in (\[regulcast\]) are explicit, i.e., ${\cal
Y_{\alpha}}$ and $E$ do not contain free derivative operators any more. We find the following expression for $\cal Y_{\alpha}$ and $E$ in $d$ dimensions $$\begin{aligned}
{\cal Y_{\alpha}}&=& Q_\alpha {\mathord{\!\usebox{\uuunit}}}+ \frac{1}{2}
\left( \begin{array}{ccc}
-Q'_{\alpha} & 0 & 0 \\
c \delta ^\mu{}
_\alpha &-Q'^\mu\eta_{\nu\alpha}-Q'_\nu \delta ^\mu{}
_\alpha & 0 \\
0 & -c \eta_{\alpha\nu} & -Q'_\alpha
\end{array} \right)\nonumber\\
E&=& -\frac{1}{4}Q'^2 {\mathord{\!\usebox{\uuunit}}}+
\left( \begin{array}{ccc}
- {{\textstyle\frac{1}{2}}}D_\alpha (B)Q'^\alpha & -D_\nu(B) b & 0 \\
V^\mu &E^\mu{} _\nu & D^\mu(B) b \\
{\textstyle\frac{d}{4}}c^2 & -V_\nu ^T &{{\textstyle\frac{1}{2}}}D_\alpha (B)Q'^\alpha
\end{array} \right)\end{aligned}$$ where $Q'^{\nu}= Q^{\nu} - B^{\nu}$ and[^5] $$\begin{aligned}
V^\mu &=&-{{\textstyle\frac{1}{2}}}D^\mu(B) c+{{\textstyle\frac{1}{4}}}((d-1)Q'^\mu c-cQ'^\mu )\nonumber\\
-V_\nu ^T &=&
-{{\textstyle\frac{1}{2}}}D_\nu(B) c+{{\textstyle\frac{1}{4}}}( Q'_\nu c-(d-1)cQ'_\nu)\nonumber\\
E_{\mu \nu }&=&
2F_{\mu\nu}(B) +
3D_{[\mu }(B)Q'_{\nu ]} +({\textstyle \frac{3}{2}-\frac{d}{4}})
Q'_\mu Q'_\nu-Q'_\nu Q'_\mu\end{aligned}$$
The trace in [(\[anomJs\])]{} can be evaluated using the heat kernel. As long as $\Delta J_s$ is algebraic, i.e., contains no derivative operator (which is the case for all applications we consider in this paper), only the value of this kernel at coincident points is needed. In the limit of large $M^2$ it can be calculated by a variety of methods. We read off the result from [@Gilkey], generalized to the mixed bosonic and fermionic case. We are interested in the terms independent of $M^2$. In four dimensions they are usually denoted by $a_2$ and read $$a_2=
\frac{1}{(4\pi)^2} \left(
\frac{1}{12} W_{\alpha \beta} W^{\alpha \beta} +
\frac{1}{2} E^{2} +
\frac{1}{6}\Box E \right) \ ,$$ where $$\begin{aligned}
W_{\alpha \beta}&=&\partial _\alpha {\cal Y}
_\beta -\partial _\beta {\cal Y}_\alpha
+[{\cal Y}_\alpha,{\cal Y}_\beta]\nonumber\\
\Box E&=&\nabla _\alpha \nabla ^\alpha E \nonumber\\
\nabla_\alpha X&=&\partial _\alpha X + [{\cal Y}_\alpha ,X]\ .\end{aligned}$$ In two dimensions they are denoted by $a_1$ and read $$a_1= \frac{1}{4\pi } E\ .$$
As a check on these results we compute the trace anomalies. They are obtained by taking the trace of the Weyl Jacobian $J_W$ with the $a_n$ coefficients. For two dimensions the Weyl weights of $c$, $b$ and $Q_\mu$ are all equal, see [@BastPvN], hence the Weyl anomaly becomes proportional to the trace of $E$ ($d=2$). This vanishes, in agreement with the fact that the trace anomaly for spin 1 fields in $d=2$ is zero. For $d=4$ one has $J_W=\mbox{diag}
(0,{{\textstyle\frac{1}{2}}}\sigma (x),\sigma (x))$, see [@BastPvN], yielding the correct result. Another check on the correctness of $a_2$ (except the $\Box E$ term) is that after integration over space-time, they yield the one-loop counterterms, deduced by dimensional regularization and Feynman diagrams in [@GtHbf].
As we already mentioned, in the Yang-Mills case $\Delta J_s$ is also algebraic, due to the symmetrization in [(\[anomJs\])]{}. The anomaly is obtained by computing the supertrace $${\cal A} =
\frac{1}{(4\pi)^2}
str\ \Delta J_s
\left(
\frac{1}{12} W_{\alpha \beta} W^{\alpha \beta} +
\frac{1}{2} E^{2} +
\frac{1}{6}\Box E \right) \ . \label{AnGilk}$$ The problem of obtaining the BRST anomaly for pure Yang-Mills theory is thus reduced to the evaluation of the supertrace in [(\[AnGilk\])]{}. The result reads $$\begin{aligned}
{\cal A}=
\frac{1}{(4\pi )^2}\frac{1}{12}
tr\ [D^\nu(B) c]
\left[ 4 Q'_\mu Q'_\nu Q'^\mu -8 Q'^\mu D_{[\mu}(B)Q'_{\nu
]} -4Q'_\nu D_\mu(B)Q'^\mu \right.\nonumber\\
\left. +D_\mu(B) D^\mu(B) Q'_\nu -3
D_\nu(B) D_\mu(B) Q'^\mu \right]\ . \end{aligned}$$ If this is to be a consistent anomaly, its BRST variation should vanish. Indeed, it does. Expecting that there is no genuine BRST anomaly, this expression is expected to be the BRST variation of a local counterterm. Indeed, it is: ${\cal A}=\delta M_1$, with $$\begin{aligned}
M_1 =
\frac{1}{(4\pi )^2}\frac{1}{12} tr\ \left[
{\textstyle
\frac{3}{2}}(D_\mu(B)Q'^\mu)^2 - {{\textstyle\frac{1}{2}}}(D_\mu(B)Q'_\nu )(D^\mu(B) Q'^\nu ) \right.
\nonumber\\ \left.
-2 Q'^\mu(D_\mu (B)Q'_\nu )Q'^\nu +{\textstyle
\frac{3}{2}}Q'_\mu Q'_\nu Q'^\mu Q'^\nu - {{\textstyle\frac{1}{2}}}Q'^2 Q'^2\right]\ .\end{aligned}$$ For computations, a suitable alternative form is given by $$\begin{aligned}
M_1=
\frac{1}{(4\pi )^2}\frac{1}{12} tr\
\left[ \textstyle{\frac{3}{2}}(D_\mu(B) Q'^\mu
)^2-\textstyle{\frac{1}{2}}(D_\mu(B)Q'_\nu )(D^\nu(B) Q'^\mu )
\right.\nonumber\\ \left.
-(D_\mu(B)Q'_\nu )(D^\mu(B) Q'^\nu )
+Q'_\mu Q'_\nu Q'^\nu Q'^\mu +\textstyle{\frac{1}{4}}F'_{\mu \nu
}F'^{\mu \nu }\right) \label{M1c}\end{aligned}$$ where $$F'_{\mu \nu }= D_\mu(B)
Q'_\nu -D_\nu(B) Q'_\mu +[Q'_\mu , Q'_\nu ]=F_{\mu \nu
}(Q)-F_{\mu \nu }(B).$$ In the background field formalism $Q_\mu $ gets replaced by an external field $A_\mu $, and then one usually chooses the two external fields equal ($B_\mu =A_\mu $). In this case the anomaly vanishes without having to invoke a counterterm. In ordinary field theory one puts $B_\mu =0$; in this case one needs a nontrivial counterterm $M_1$.
In conclusion, we have seen that within the framework of path-integrals, one can give a direct and complete derivation of the BRST Ward identities, on which perturbative unitarity and renormalizability are based. The precise form of the local counterterm $M_1$ which must be added to the quantum action in order that the BRST anomaly cancels, depends of course on the regularization scheme used to compute the effective action. If one uses Pauli-Villars regularization with the given mass term, $M_1$ is given in [(\[M1c\])]{}. If one uses dimensional regularization, one has $M_1=0$. However, the precise form of $M_1$ is not needed in general.
[99]{} C. Becchi, A. Rouet and R. Stora, Commun. Math. Phys. [**42**]{} (1975) 127; Ann. Phys. [**98**]{} (1976) 287;\
I.V. Tyutin, Lebedev preprint FIAN No.39 (1975). G. ’t Hooft and M. Veltman, Nucl. Phys. [**B50**]{} (1972) 318. J. Zinn-Justin, in “[*Trends in Elementary Particle Theory*]{}”, Springer Lecture notes in physics [**37**]{}, eds. H. Rollnik and K. Dietz, Springer, Berlin, 1975. G. ’t Hooft, Nucl. Phys. [**B33**]{} (1971) 173. L. Alvarez-Gaumé and E. Witten, Nucl. Phys. [**B234**]{} (1984) 269. A. Diaz, W. Troost, P. van Nieuwenhuizen and A. Van Proeyen, Int. J. Mod. Phys. [**A4**]{} (1989) 3959;\
W. Troost, P. van Nieuwenhuizen and A. Van Proeyen, Nucl. Phys. [**B333**]{} (1990) 727;\
M. Hatsuda, W. Troost, P. van Nieuwenhuizen and A. Van Proeyen, Nucl. Phys. [**B335**]{} (1990) 166. J. Wess and B. Zumino, Phys. Lett. [**B37**]{} (1971) 95;\
W.A. Bardeen and B. Zumino, Nucl. Phys. [**B244**]{} (1984) 421. J.A. Dixon, preprint HUTMP 78/B64;\
F. Brandt, N. Dragon and M. Kreuzer, Phys. Lett. [**B231**]{} (1989) 263. W. Shakespeare, in “[*The most excellent and lamentable Tragedie, of Romeo and Iuliet*]{}“, act II scene 2, 1599. I.A. Batalin and G.A. Vilkovisky, Phys. Rev. [**D28**]{} (1983) 2567 (E:[**D30**]{} (1984) 508). A. Van Proeyen, preprint KUL-TF-91/35, to be published in ”Strings and Symmetries 1991“, proceedings of the conference in Stony Brook\
W. Troost and A. Van Proeyen, in ”Leuven Notes in Mathematical and Theoretical Physics", in preparation. K. Fujikawa, Phys. Rev. Lett. [**42**]{}(1979)1195; [**44**]{}(1980) 1733; Phys. Rev. [**D21**]{}(1980)2848 (E : [**D22**]{} (1980) 1499). P.B. Gilkey, J. Diff. Geom. [**10**]{} (1975) 601; [*Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem*]{} (Publish or Perish, Inc.) Math. lect. series no. 11, 1984. F. Bastianelli and P. van Nieuwenhuizen, preprint USITP-92-02; CERN-TH.6512/92. G. ’t Hooft, Nucl. Phys. [**B62**]{} (1973) 444.
[^1]: For local (gauged) BRST symmetry, see F.J. Ore and P. van Nieuwenhuizen, Nucl. Phys. [**B204**]{} (1982) 317.
[^2]: For convenience we will use the word ‘anomaly’ for this regulated trace, even when it is the variation of a local counterterm and can be removed.
[^3]: ... ô be some other name.\
> Whats in a name? that which we call a rose,\
> By any other word would smell as sweete,\
> So Romeo would were he not Romeo cald,\
> Retaine that deare perfection... [@Shakesp]
[^4]: All our matrices are supermatrices of bosonic type. Then the supertranspose for matrices with 2 lower indices, as $S$ and $T$, is defined by $(T^t)_{ij}= T_{ji} (-)^{i+j+ij}$ where $(-)^i$ is $+$ for entries related to $Q_\mu$ and $-$ for entries related to $b$ and $c$. This rule follows from the definition that $\phi ^i (T^t)_{ij}\varphi ^j = \varphi ^j T_{ji}\phi ^j$. For matrices with two upper indices, we have $(M^t)^{ij}=M^{ji}(-)^{ij}$, as can be derived for $T^{-1}$ if we impose $\left( T^{-1}\right) ^t T^t={\mathord{\!\usebox{\uuunit}}}$. Finally for matrices with mixed indices like $J^i{}_j$ the supertransposition rule follows from the product of matrices of the previous types, imposing $(MT)^t=T^tM^t$. We have $(J^t)_i{}^j=(-)^{i(j+1)} J^j{}_i$. For these matrices the supertrace is $str\ J= (-)^i J^i{}_i$ which has the property that $str\ J^t=str\ J$ and $str\ AB= str\ BA$. These properties, with the knowledge that $S$ and $T$ are super-symmetric by their definition, lead to the equality in [(\[anomJs\])]{}.
[^5]: Antisymmetrization $[\mu\nu]$ is done with weight 1, i.e., ${{\textstyle\frac{1}{2}}}(\mu \nu -\nu \mu )$.
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[**We study the reconstruction of visual stimuli from spike trains, recording simultaneously from the two H1 neurons located in the lobula plate of the fly [*Chrysomya megacephala*]{}. The fly views two types of stimuli, corresponding to rotational and translational displacements. If the reconstructed stimulus is to be represented by a Volterra series and correlations between spikes are to be taken into account, first order expansions are insufficient and we have to go to second order, at least. In this case higher order correlation functions have to be manipulated, whose size may become prohibitively large. We therefore develop a Gaussian-like representation for fourth order correlation functions, which works exceedingly well in the case of the fly. The reconstructions using this Gaussian-like representation are very similar to the reconstructions using the experimental correlation functions. The overall contribution to rotational stimulus reconstruction of the second order kernels - measured by a chi-squared averaged over the whole experiment - is only about 8% of the first order contribution. Yet if we introduce an instant-dependent chi-square to measure the contribution of second order kernels at special events, we observe an up to 100% improvement. As may be expected, for translational stimuli the reconstructions are rather poor. The Gaussian-like representation could be a valuable aid in population coding with large number of neurons.** ]{}
Introduction
=============
Living animals have to reconstruct a representation of the external world from the output of their sensory systems in order to correctly react to the demands of a rapidly varying environment. In many cases this sensory output is encoded into a sequence of identical action potentials, called spikes. If we represent the external world by a time-dependent stimulus function $s(t)$, the animal has to reconstruct $s(t)$ from a set of spikes. This decoding procedure generates an estimate $s_{e}(t)$ of the stimulus like a digital-to-analog converter.
Here we study this decoding procedure in a prominent example of spiking neurons: the two H1 neurons of the fly [*Chrysomya megacephala*]{}. The fly has two compound eyes with their associated neural processing systems [@hausen:1981; @hausen:1982b; @hausen:1984]. Motion detection starts at the photoreceptor cells, eight of them located in each one of the $\sim 5000$ ommatidia of each compound eye. They effect the transduction of photons into electrical signals, which are propagated via the lamina and medulla to the lobula plate. This neuropil is - inter alia - composed of horizontally and vertically directionally sensitive wide field neurons. The H1 neurons are horizontally sensitive and are excited by ipsilateral back to front motion and inhibited by oppositely moving stimuli. Each H1 neuron projects its axon to the contralateral lobula plate, exciting there two horizontal and two centrifugal cells. These cells mediate mutual inhibition between the two H1 neurons [@Haag1999; @Haag2001; @Farrow2003; @Haag2008; @Krapp2009][^1]. We subject the fly to rotational and translational stimuli - see Figure \[setup\]. If the fly rotates around a vertical axis, say clockwise when looking down the axis, the left neuron is inhibited and the right one is exited, so that the two neurons become an efficient rotational detector [@hausen:1984]. This can be seen in Figure \[raster\] (R1) & (R2). Even when recording only from the ipsilateral H1, one can simulate the response of the contralateral H1. In fact, since the two H1 cells have mirror symmetric directional sensitivities, the sign flipped stimulus induces a response in the ipsilateral H1 typical for the contralateral H1 cell [@Spikes]. The inset in (R2) shows this to be true to a very good approximation.
In forward translation none of H1 neurons is excited, corresponding to the low spike density regions in the raster-plots of Figure \[raster\] (T1) & (T2). In backward translation, both H1’s are excited and we expect a strong inhibition. Yet the spike rate is comparable to rotational excitation - compare Figure \[raster\] (R1) & (T1). Numerical computation confirms this visual impression. Nevertheless in translation the two H1’s fire mainly in sync, which leads to subtle differences with respect to rotation. As a consequence, our reconstructions will be much poorer for the translational case - see section \[sect:reconstr\].
If we want to take correlations between spikes into account, instead of treating them independently, we have to go at least to second order stimulus reconstructions. These require the computation of higher order spike-spike correlation functions and a subsequent matrix inversion. If one records from many neurons simultaneously, the size of these matrices may soon become prohibitively large. Here we present an efficient representation of these higher order correlation functions in terms of second order ones. The reconstruction now costs far less computationally, avoids large matrix inversions and gives excellent results. We test the quality of our reconstructions under both rotational and translational stimuli.
If this representation holds more generally, it may well make population coding computationally more tractable. We briefly discuss a perturbation scheme, which allows a stepwise inclusion of small effects.
Stimulus reconstruction from spike trains {#reconstr}
==========================================
Suppose we want to reconstruct the stimulus from the response of a single H1 neuron. We represent this response as a spike train $\rho(t)=\sum_{i=1}^{N_s} \delta(t-t_i)$, which is a sum of delta functions at the spike times $t_i$. $N_s$ is the total number of spikes generated by the neuron during the experiment.
The simplest reconstruction extracts the stimulus estimate via a linear transformation, see e.g. [@Spikes; @Bialek:1991],
$$\label{vol1}
s_e(t) = \int_{-\infty}^{\infty} k_1(\tau)\rho(t-\tau)d\tau,$$
with the kernel $k_1(t)$ to be determined.
For simplicity we effect an [*acausal*]{} reconstruction, i.e. we integrate from $-\infty$ to $+\infty$. Essentially the same results are obtained in a causal reconstruction. One way to implement causality proceeds to estimate the stimulus at time $t$, using as input the spike train up to time $t+t_0$. For the fly $t_0$ has to be $\gtrsim$ to 30 milliseconds. In this case equation \[vol1\] would read: $ s_e(t) = \int^{\infty}_{-t_0} k_1(\tau)\rho(t-\tau)d\tau.$
Equation \[vol1\] is the first term of a Volterra series [@volterra]: $$\label{vol2}
s_e(t) = \int_{-\infty}^{\infty} k_1(\tau)\rho(t-\tau)d\tau+
\int_{-\infty}^{\infty} k_2(\tau_1,\tau_2)\rho(t-\tau_1)\rho(t-\tau_2)d\tau_1 d\tau_2 +\ldots$$ There is no convergence proof for this expansion, but heuristically we may say that it should be a valid approximation, if the average number of spikes per correlation time $\tau_c$, $$\eta=\langle r \rangle \tau_c,$$ is small [@Spikes]. Here $ \langle r \rangle$ is the mean spike rate and $\tau_c$ a typical signal correlation time. For small $\eta$ each spike gives independent information about the stimulus. In our case $\eta\sim 0.6-0.8$, which is of the order of unity, so that higher order effects might be relevant.
The first order term, being proportional to $\sum_{i}^{N_s}k_1(t-i_i)$, independently adds contributions for each spike. Yet it is well established that pairs of spikes carry a significant amount of additional information beyond the single spike contributions [@brenner1]. This motivates the addition of the second order kernel $k_2(\tau_1,\tau_2)$, which includes correlations between up to two spikes.
In order to obtain the kernels $k_1$ and $k_2$ we choose to minimize the following functional - the $\chi^{(2)}$ error - $$\label{chi}
\chi^{(2)}(k_1,k_2) = \langle\int dt [s_e(t)-s(t)]^2 \rangle.$$ The brackets stand for an ensemble average with respect to the distribution of all possible stimuli in a given experiment. In a long experiment we average over $N_w\sim 10^5$ time windows of size $T_w$. Typically $T_w\sim 100$ milliseconds - see section \[MM\] for details. For ease of presentation, in the following our discussions will always refer to the rotational setup, unless explicitly stated otherwise as in section \[sect:reconstr\].
Since the functional \[chi\] is quadratic, the equations minimizing $\chi^{(2)}(k_1,k_2)$
$$\label{dchi}
\partial \chi^{(2)}/\partial k_j = 0, j=1,2$$
are linear in the unknowns $k_1,k_2$. E.g., if we keep only $k_1$, using therefore equation \[vol1\], we get:
$$\label{k11}
\tilde k_1(\omega) = \frac{ \langle\tilde s(\omega)^* \tilde\rho(\omega) \rangle}{ \langle\tilde\rho(\omega)^*\tilde\rho(\omega) \rangle},$$
where Fourier transforms are defined as $\tilde F(\omega) = \int dt F(t) e^{\imath\omega t}$.
We may include the second order term $k_2$, either as a correction to the first order reconstruction $s_1(t) = k_1\star\rho(t)$ [^2], or one may solve the coupled system \[dchi\].
If we record simultaneously from left and right H1, we obtain two spike trains $\rho_1(t)$ and $\rho_2(t)$. The expansion equation \[vol2\] generalizes to $$\nonumber
\label{vol22}
s_e(t) = K_1\star\rho_1(t)+ K_2\star\rho_2(t)+$$ $$K_{11}\star\rho_1\star\rho_1(t)+K_{12}\star\rho_1\star\rho_2(t)+K_{22}\star\rho_2\star\rho_2(t)+\ldots.$$ Here we have included the kernel $K_{12}$, which encodes effects correlating $\rho_1$ and $\rho_2$ [^3]. Notice that $K_{12}=K_{21}$.
To first order, keeping only $K_{1}$ and $K_{2}$ in the expansion \[vol22\], we get the following equations: $$\widetilde{S\!\!R}_a(\omega) = \sum_{b=1}^2 \tilde K_b(\omega) \tilde R_{ab}(\omega),\,\, a=1,2
\label{K1}$$ where $$\widetilde{S\!\!R}_a(\omega) = \int dt dt' \langle s(t')\rho_a(t'-t) \rangle e^{\imath\omega t}.$$ and $$R_{ab}(t_1,t_2) =\int dt \langle\rho_a(t-t_1)\rho_b(t-t_2) \rangle,\; a,b=1,2.$$ Due to time-translation invariance $R_{ab}(t_1,t_2)$ is only a function of the difference: $R_{ab}(t_1,t_2)=R_{ab}(t_1-t_2)$ and $ \tilde R_{ab}(\omega) = \int dt R_{ab}(t) e^{\imath\omega t}$. Analogous properties hold for all the following correlation functions involving only $\rho(t)$.
The solution of equations \[K1\] yields $${\tilde K}_a(\omega) = (L_a(\omega) R_{\hat{a}\hat{a}}-L_{\hat{a}}(\omega)R_{a \hat{a}}(\omega) )/\Delta,\; a=1,2$$ where $$L_a(\omega) = \langle s(\omega)\rho^*_a(\omega) \rangle,\\
\Delta = R_{11}R_{22}-R_{12}R_{21}$$ and $\hat{a}=3-a$. We obtain the first order reconstruction as $$s_1(t) = K_1\star\rho_1(t)+K_2\star\rho_2(t).$$
Since the second order contribution turns out to be small, we treat it as a perturbation to the first order reconstruction. We therefore expand $s_2(t) = s(t)-s_1(t)$ as: $$s_2(t) = K_{11}\star\rho_1\star\rho_1(t)+K_{12}\star\rho_1\star\rho_2(t)+K_{22}\star\rho_2\star\rho_2(t).$$
We now have to solve the following equations $$S\!\!R^{(2)}_{ab}(t_1,t_2)=
\int dt_3 dt_4 \sum_{c,d=1}^2 K_{cd}(t_1,t_2) R^{(4)}_{abcd}(t_1,t_2,t_3,t_4),
\label{K22}$$ where $$S\!\!R^{(2)}_{ab}(t_1,t_2) = \int dt \langle s_2(t)\rho_a(t-t_1)\rho_b(t-t_2) \rangle,$$ $$R^{(4)}_{abcd}(t_1,t_2,t_3,t_4) =\int dt \langle\rho_a(t-t_1)\rho_b(t-t_2)\rho_c(t-t_3)\rho_d(t-t_4) \rangle.$$
Although the system \[K22\] is linear, the matrices to be inverted may be very large. We have to invert the matrix $${\cal M}_{AT}^{BT'}\equiv R^{(4)}_{abcd}(t_1,t_2,t_3,t_4),
\label{4ptmat}$$ where $A,B$ are compound indices $A=[ab],B=[cd]$ labeling the neurons. $T=[t_1,t_2],T'=[t_3,t_4]$ are compound time indices of size $T_w^2$ each. If we compute the correlation functions using a time window of $T_w=128$ bins, with binsize $=2$ milliseconds, then the size of ${\cal M}_{AN}^{BN'} $ is $\sim 128^4\times2^4 \sim 5\times 10^9$. The matrices to be inverted may become prohibitively large, especially if we record from more than just two neurons [^4].
We therefore present below a Gaussian-like representation of $R^{(4)}_{abcd}$ with a small number of parameters and which requires no large matrix inversion.
Gaussian-like (Gl) representation for 4-point functions {#1G}
=======================================================
In this section we present a representation of the 4-point function ${ R}^{(4)}_{abcd}$ in terms of the 2-point function ${ R}^{(2)}_{ab}$, which is surprisingly good and which avoids the computation of the large matrices \[4ptmat\].
If our spike-generating process were Gaussian, we would have the following structure for $R^{(4)}$: $$\nonumber
R^{(4)}(1,2,3,4) = R(1,2)R(3,4)+R(1,3)R(2,4)+R(1,4)R(2,3)
\label{GR4}$$ $$-2 \langle\rho(t) \rangle^4,$$ where $\langle\rho(t) \rangle$ is just a constant, due to time-translation invariance[^5].
This suggests the following representation for $R^{(4)}$: $$\nonumber
R^{(4)}(1,2,3,4) = A \left[ R(1,2)R(3,4)+R(1,3)R(2,4)+\right.$$ $$\left. R(1,4)R(2,3) \right]-B,
\label{Fit1}$$ where $A$ and $B$ are constants to be adjusted[^6].
For two neurons we get the representation: $$R_{abcd}(1,2,3,4) = [R_{ab}(1,2) R_{cd}(3,4) +
R_{ac}(1,3) R_{bd}(2,4)+
\nonumber$$ $$R_{ad}(1,4) R_{bc}(2,3)] A_{abcd}+B_{abcd}
\label{2NRep}$$ with $a,b,c,d=1,2$ and $A_{abcd}$, $B_{abcd} $ constants to be determined.
The usefulness of our Gl-representation scheme depends on the quality of the 4-point functions obtained, which in turn hinges on the knowledge of the constants $A_{abcd}$ and $B_{abcd}$. There would be no point, if this required the computation of 4-point functions in large window sizes and a fitting procedure using these windows - exactly what we wanted to avoid. We therefore fit the constants $A_{abcd}$ and $B_{abcd}$ for a sequence of window sizes $T_w$, ranging from $10$ to $128$ bins, using $ R_{1111}(t_1,t_2=t_3=t_4=1)$ to fit to the experimental data. As can be seen in Figure \[AB\_dep\], at least in the fly’s case, the dependence of the parameters $A_{abcd},B_{abcd}$ on $T_w$ is only $0.05$% and therefore completely negligible. The constants $A_{abcd}$ and $B_{abcd}$ can therefore be computed very fast in small windows. In Figure \[Fig:4\_pt\_fits1\] we plot the fits to the first row $ R_{1111}(t_1,t_2=t_3=t_4=1)$ and its Gl approximation. As advertised we obtain a perfect fit.
In Figure \[Fig:4pt\_mat\] we show the Gl approximation for the $R_{1111}(t_1,t_2,t_3=t_4=1$ and its experimental version, which emphasizes the quality of the approximation. Using the same parameters for the other entries of $ R_{1111}$ and for $R_{2222}$ results in a fitting error about 20 % larger.
One of the utilities of this representation will become apparent, once we deal with the solution of equation \[K22\] in the next section.
A convenient set of functions to solve for second order kernels
===============================================================
At this point it is convenient to introduce a complete set of basis functions $f_\mu(t),\mu=1,2,..,n_f$ to expand our variables in. We thus trade continuous time-arguments for discreet Greek indices. We expand our second order kernels as: $$K_{ab}(t_1,t_2) = \sum_{\mu.\nu} \, f_\mu(t_1)f_\nu(t_2) {\cal D}^{ab}_{\mu\nu}.$$ We also expand our correlation functions: $$S\!\!R^{(2)}_{ab}(t_1,t_2) = \sum_{\mu.\nu} {\cal S}^{ab}_{\mu\nu}\, \, f_\mu(t_1)f_\nu(t_2)
\label{expSR}$$ and $$\nonumber
R^{(4)}_{abcd}(t_1,t_2,t_3,t_4)=$$ $$\sum_{\alpha\beta\mu.\nu} {\cal R}^{abcd}_{\alpha\beta\mu\nu}\,
f_\alpha(t_1)
f_\beta(t_2)f_\mu(t_3) f_\nu(t_4).
\label{expRRRR}$$
In order to efficiently compute our second order kernels it is crucial to select an adequate set for $f_\mu(t),\mu=1,2,..,n_f$. Depending on the case, it may be sufficient to use a small number $n_f$ of functions $f_\mu(t)$ to get a useful representation. If $n_f$ has only a slight dependence on window size $T_w$, this would allow one to increase $T_w$ without further computational costs.
Often a Fourier expansion is used, i.e. $f_\omega=e^{\imath\omega t}$. But we may exploit our liberty to choose the functions in a more profitable way. Since our 2-point function $R(t_1,t_2)$ is real, positive[^7] and symmetric in $t_1,t_2$, it posses a complete set of eigenfunctions $h_\mu(t)$: $$\int dt_2 R(t_1,t_2) h_\mu(t_2) = r_\mu h_\mu(t_1)$$ with eigenvalues $r_\mu, \mu=1,\ldots, N_w$. We now choose our functions as $f_\mu(t)=h_\mu(t)/\sqrt{r_\mu}$, which satisfy: $$\int dt_1 dt_2 f_\mu(t_1)R(t_1,t_2)f_\nu(t_2) = \delta_{\mu\nu}.$$ This choice will avoid large matrix inversions, if at least part of our higher order correlation functions can be built from $R(t_1,t_2)$.
Substituting the expansions \[expSR\] and \[expRRRR\] into equations \[K22\], we get a linear system to be solved for $ {\cal D}^{ab}_{\mu\nu}$: $$\label{Lin}
{\cal S}_{ab}^{\mu\nu} =
\sum_{cd,\alpha\beta} {\cal R}_{abcd}^{\mu\nu\alpha\beta}\, {\cal D}_{cd}^{\alpha\beta}$$
In order to avoid cluttering our expressions with indices, we introduce our representation first for one neuron only, suppressing thus the indices $a,b,..$, all set to $1$. We choose our functions $f_\mu(t)$ to diagonalize $ R^{11}(t_1,t_2)={ \langle \rho_1(t_1)\rho_2(t_2)\rangle}$: $$\int dt_1 dt_2 f_\mu(t_1)R^{11}(t_1,t_2)f_\nu(t_2) = \delta_{\mu\nu}.$$ The first of equations \[2NRep\] for ${\cal R}_{\mu\nu\alpha\beta}^{1111}$ becomes $${\cal R}_{\mu\nu\alpha\beta} = A (\delta_{\mu\nu}\delta_{\alpha\beta}+
2\delta_{\mu\alpha}\delta_{\nu\beta})-2B\,n_\alpha n_\beta n_\mu n_\nu ,$$ where $n_\mu = \int dt f_\mu(t) \langle\rho(t) \rangle$.
Using this expression and the shorthand ${\cal S}_{\mu\nu}\equiv{\cal S}_{\mu\nu}^{11}$ in equations \[Lin\], we get the following equations for the unknown coefficients ${\cal D}_{\mu\nu} \equiv{\cal D}_{\mu\nu}^{11}$ $${\cal S}_{\mu\nu} =A[ tr({\cal D})\delta_{\mu\nu}+2{\cal D}_{\mu\nu}]-2 B D_{nn} \,n_\nu n_\mu,
\label{Dequ1}$$ where $tr({\cal D})\equiv \sum_\mu{\cal D}_{\mu\mu} $ and $ D_{nn}\equiv \sum_{\alpha\beta}n_\alpha {\cal D}_{\alpha\beta}n_\beta$. The sums over $\mu,\alpha,\beta$ run from $1$ to $T_w$ bins.
This system can now easily be solved by:
1. taking the trace over $\mu\nu$ to compute $tr({\cal D})\equiv D$ and
2. multiplying by $n_\mu,n_\nu$ to compute $D_{nn}$.
We get
$${\cal D}_{\mu\nu} = [{\cal S}_{\mu\nu}/A-D\,\delta_{\mu\nu}+2Bn_\mu n_\nu\, D_{nn}]/2,
\label{EquD1}$$
with $$D = [2(1-n_4){\cal S}_{\mu\mu}+2n_2\,n_\mu{\cal S}_{\mu\nu} n_\nu]/\Delta,$$ $$D_{nn} = [(n+2)n_\mu {\cal S}_{\mu\nu} n_\nu - {\cal S}_{\mu\mu} n_2]/\Delta,$$ where $$\Delta = 2(T_w+2)(1-n_4)+2n^2, n_2\equiv \sum_\mu n_\mu n_\mu, n_4 \equiv (n_2)^2.$$
For two neurons we now have to decorate our formulas with the indices $a,b,\ldots$. To simplify our formulas, we assume symmetry between the two neurons: $R_{11}=R_{22}$, which in our case is very well satisfied - see Figure \[R11\_R22\].
The 4-point functions are now represented as $$\begin{array}{cc}
{\cal R}_{1111}^{\mu\nu\alpha\beta} = & [\delta_{\mu\nu}\delta_{\alpha\beta} + \delta_{\mu\alpha}\delta_{\nu\beta}+\delta_{\mu\beta}\delta_{\nu\alpha}] A_{1111}+B_{1111}^{\alpha\beta\mu\nu}
\label{4ptrep2}
\\
{\cal R}_{1112}^{\mu\nu\alpha\beta} =& [\delta_{\mu\nu}R^{\alpha\beta}_{12}+\delta_{\mu\alpha}R^{\nu\beta}_{12}+\delta_{\nu\alpha}R^{\mu\beta}_{12}]A_{1112}+ B_{1112}^{\alpha\beta\mu\nu}
\\
{\cal R}_{1122}^{\mu\nu\alpha\beta} =& [\delta_{\mu\nu}\delta_{\alpha\beta}+R^{\mu\alpha}_{12}R^{\nu\beta}_{12}+R^{\mu\beta}_{12}R^{\nu\alpha}_{12}]A_{1122}+ B_{1122}^{\alpha\beta\mu\nu}
\\
{\cal R}_{1222}^{\mu\nu\alpha\beta} =& [R^{\mu\nu}_{12}\delta_{\alpha\beta}+R^{\mu\alpha}_{12}\delta_{\nu\beta}+R^{\mu\beta}_{12}\delta_{\nu\alpha}]A_{1222}+ B_{1222}^{\alpha\beta\mu\nu}
\\
{\cal R}_{2222}^{\mu\nu\alpha\beta}=& [\delta_{\mu\nu}\delta_{\alpha\beta} + \delta_{\mu\alpha}\delta_{\nu\beta}+\delta_{\mu\beta}\delta_{\nu\alpha}]
A_{2222}+B_{2222}^{\alpha\beta\mu\nu}.
\end{array}$$
The intermediate steps 1 and 2 leading to equation \[Dequ1\] now increase, since we have to express several 4-point functions in terms of 2-point functions, not all of them being diagonal. In the particular case of the two H1 neurons though, we may further simplify this system, neglecting $R_{12}$. Its effect[^8] is very small indeed, since for rotational stimuli the action of the two neurons is complementary: an exciting stimulus for one neuron is inhibiting for the other - see Figure \[R11\_R22\]. Although for translational stimuli both neurons fire nearly synchronously, the dominant peak near $\tau=0$ in $R_{12}$ is absent, since synchrony is not exact. In the following we therefore neglect $K_{12}$. As can be seen in Figure \[K2\_1111\], $K_{12}$ is only $\sim K_{22}/5$. Since the contributions of $K_{11}$ and $K_{22}$ are already small, $K_{12}$’s $1$ % effect can be safely neglected for both types of stimuli.
Our equations now decouple and we get two sets identical to equations \[Lin\], one for each neuron.
Reconstructing the fly’s stimulus and measuring its quality {#sect:reconstr}
===========================================================
To test the quality of our reconstructions, we use the data with $\eta\sim 0.8, \tau = 10$ milliseconds and $ \langle r \rangle\sim 80$ spikes sec$^{-1}$.
We select a representative sample, one second long, of the experiment, in order to give a visual display of the reconstruction. In Figure \[Fig:recrot\] we show the first order reconstruction of the original stimulus using $K1$ and $K2$ and the second order reconstruction, where the effect of $K_{11}$ and $K_{22}$ is added - with and without the Gl-approximation. We conclude:
- Reconstructions using the experimental 4-point functions are very similar to their Gl-approximation.
- The reconstruction procedure is unable to reproduce the fast stimulus variations at the 2 milliseconds time scale. It is also clear that still higher order terms are not going to improve this deficiency. But the second order kernels always represent an improvement, since the black line in Figure \[Fig:recrot\] is always a better approximation to the stimulus than the blue one.
- We observe a stimulus-to-spike delay time of $t_{rot}\sim 20$ bins.
Although visual appraisement of the reconstruction quality is an indispensable guide to our intuition, numerical measures are less subjective. We naturally use the $\chi^{(2)} = \langle\int dt [s_e(t)-s(t)]^2 \rangle$ of Equation \[chi\], since its minimization was used to determine the kernels $k_i,K_j$. The reconstruction improvement due to second order kernel is reflected in $$\delta\chi^{(2)} \equiv \frac{ \chi^{(2)}_{1}-\chi^{(2)}_{12}}{\chi^{(2)}_1},$$ where $\chi^{(2)}_{1}$ takes only first order terms into account - $\chi^{(2)}_1 = \langle\int dt [s_1(t)-s(t)]^2 \rangle$, whereas second order terms are included in $\chi^{(2)}_{12} = \langle\int dt [s_1(t)+s_2(t)-s(t)]^2 \rangle$. $ \delta\chi^{(2)}$ is positive, but small of $\sim 8$%. The chi-squared difference between the experimental and Gl-reconstructions is only of $\sim 0.5$ %.
Although the $\chi^{(2)}$-improvement is small, second order terms are a important at specific stimulus-dependent instants. In order to assess the relevance of these, we measure [*local chi-squares*]{}, defined as: $$\chi_{1}^2(t,\Delta T) \equiv \int_{t-\Delta T}^{t+\Delta T} dt \langle (s_1(t)-s(t))^2 \rangle$$ and $$\chi_{12}^2(t)\equiv \int_{t-\Delta T}^{t+\Delta T} dt\langle (s_1(t)+s_2(t)-s(t))^2 ,\rangle$$ for $t=T_2$, where $T_2$ are instants when $\chi_{12}^2(t)$ is at least as important as $\chi_{1}^2(t)$. If $N_2$ is the number of such windows of size $\Delta T$ and $N_T$ the duration of the experiment in bins divided by the window-size in bins, we plot in Figure \[fig:lchi12\] the fraction of the stimulus-dependent instants vs. $\chi_{1}^2/\chi_{12}^2$. Although this fraction vanishes as we require the importance of second order terms to increase, they still make a sizable contribution. Unfortunately just looking at the mean stimulus around $T_2$ does not provide any insight and a more detailed analysis will be needed to reveal features, which might be relevant at these particular instants.
Here we only follow [@Spikes] and separate systematic from random errors, decomposing the estimate $\tilde s_e(\omega)$ into a frequency-dependent gain $g(\omega)$ and an effective noise $n_{eff}(\omega)$ referred to the input: $$\tilde s_e(\omega) = g(\omega)[\tilde s(\omega)+n_{eff}(\omega)].$$ Around $T_2$, we observe an overall improvement of 20% in $g(\omega)$. A further indication, that second order contributions, although drowned in averages over the whole experiment, may nevertheless have crucial importance in improving the code at specific moments.
Finally we discuss the reconstruction of translational stimuli. Although in real life there is a continuous intermingling of [*rotational*]{} and [*translational*]{} motion, for a start we have considered this artificial separation of stimuli. Thus we have computed all averages $\langle \cdot \rangle$ also for the translational setup. The kernels $K_a,K_{ab}$ are similar to the rotational ones, but there is a sign change. Whereas for rotational stimuli $K_1\sim -K_2, K_{11}\sim -K_{22}$, for the translational case we have $$\begin{array}{ccc}
{K_1}^{(trans)} & \sim {K_2}^{(trans)}& \sim {K_1}^{(rot) } ,\\
{K_{11}}^{(trans)}&\sim {K_{22}}^{(trans)}& \sim {K_{11}}^{(rot) } ,
\end{array}$$ The reconstructions shown in Figure \[Fig:rectrans\] are worse than the rotational ones. For positive stimuli, corresponding to unrealistic backward motion of the fly, both neurons fire vigorously, whereas in the opposite case none does. Interestingly, the delay-time is now $t_{trans}\sim 25$ bins, about $5$ bins larger than $t_{rot}$: inspite of their mutual inhibition, the neurons manage to fire, albeit a little bit retarded. The Gl-representation works equally well for this case. It would be interesting to subject the fly to a more realistic mixture of rotational and translational motion without separating the two and then compute correlation functions etc. We intend to come back to this issue in the future.
Gl-approximation in population coding: taming the matrix explosion {#Pop}
==================================================================
Although the spike generation process of the H1 neurons is not Gaussian, the parametrization \[Fit1\] is unexpectedly good. Actually we don’t know how to judge from the spike interval distribution, whether this surprise will happen or not. In fact, the interval distribution of the spike times looks more nearly Poisson, instead of Gaussian. We remark, that independent increment probability distributions, whether they are Poisson or not, never do justice to correlated spike trains. On the other hand, if the 2-point function $R(t)$ is to be a suitable building block to represent the 4-point function, then the parametrization, equation \[Fit1\], is uniquely selected to be the most general one respecting the symmetry of $R^{(4)}(1,2,3,4)$. Since first order computations treat each neuron independently and do not take their mutual correlations into account, in the future one certainly would want to perform second order reconstructions to study the fly’s visual system for more than two neurons. Our Gl-approximation makes these computations much more feasible. It should also work for correlation functions involving neurons not belonging to the fly’s lobula plate.
In order to apply our Gl-approximation, we imposed the requirement $R_{11} = R_{22}$ and we neglected $R_{12}$. This limitation may be relaxed in the following way[^9]. One could set $R_{12}=0$ and use a different set of functions for each neuron, diagonalizing thus all 2-point functions ${\cal R}_{aa}$ and compute the coefficients ${\cal D}_{ab}$. Then reexpand all variables in terms of one set of functions only and apply the procedure, which led to equation \[EquD1\] for $R_{12}\neq 0$. If this does not lead to a closed set of equations, small effects may always be taken into account by a perturbative scheme to arbitrary order. In fact, suppose we have solved equation \[Lin\] for some representation of ${\cal R}^{\mu\nu\alpha\beta}_{abcd}$ - e.g. as we did in section \[1G\]. Incorporating $R_{12}\neq 0$ and/or $R_{22}^{\mu\nu}\neq\delta_{\mu\nu}$ will change the ${\cal R}$-matrix into: $${\cal R'\,} = {\cal R}+
\delta{\cal R},$$ with $\delta{\cal R}$ supposedly [*small*]{}. The new equations to be solved are: $${\cal S}_{ab}^{\mu\nu} =
\sum_{cd,\alpha\beta} {\cal R'\,}_{abcd}^{\mu\nu\alpha\beta}\, {\cal D'\,}_{cd}^{\alpha\beta},
\label{Lin2}$$ where ${\cal D'}={\cal D}+\delta{\cal D}$ and ${\cal D}$ satisfies the unprimed equations \[Lin\]. Expanding both sides of equation \[Lin2\] to first order in the corrections, we get the equations $$-\sum_{cd,\alpha\beta}\delta {\cal R}_{abcd}^{\mu\nu\alpha\beta}\,{\cal D}_{cd}^{\alpha\beta} =
\sum_{cd,\alpha\beta} {\cal R}_{abcd}^{\mu\nu\alpha\beta}\,\delta {\cal D}_{cd}^{\alpha\beta}
\label{Lin3},$$ to be solved for the unknowns $ \delta {\cal D}$. $(-\delta {\cal R}\cdot{\cal D})$ replaces the left-hand-side of equation \[Lin\] and couples the neurons. The right-hand-sides of the above equation and equation \[Lin\] have the same form and can therefore be solved in the same manner.
The Gl-approximation could also be useful for other systems and this would be a considerable step forward in implementing coding involving a large population of neurons. One of the problems in second order reconstructions involving many neurons is the size-explosion of the 4-point function matrices alluded to at equation \[4ptmat\]. If, e.g. we record from four neurons using $128$ bin-sized windows, the length of the matrices to be inverted would be $\sim 128^8 \times 2^8 \sim 10^{19}$. With our approximation the size of the linear system to be solved grows only linearly with the number of neurons.
In order to use our approximation, one would have to check the windowsize independence of the parameters $A_{ab\ldots}$ and $B_{ab\ldots}$ for some subset of the complete matrix-indices, to convince oneself of the adequacy of the approximation. Since in our case the matrices were still manageable, we could compute the experimental 4-point functions to verify this point, but this will in general not be possible.
Materials and Methods {#MM}
======================
Flies, immobilized with wax, viewed two Tektronix 608 Monitors M1, M2, one for each eye, from a distance of $12cm$, as depicted in Figure \[setup\]. The monitors were horizontally centered, such that the mean spiking rates of the two neurons, averaged over several minutes, were equal. They were positioned, such that a straight line connecting the most sensitive spot of the compound eye to the monitor was perpendicular to the monitor’s screen. The light intensity corresponds roughly to that seen by a fly at dusk [@rob:1997]. The stimulus was a rigidly moving vertical bar pattern with horizontal velocity $v(t)$. We discretise time in bins of 2 milliseconds, which is roughly the refractory period of the H1 neurons. The fly therefore saw a new frame on the monitor every $\delta t = 2$ milliseconds, whose change in position $ \delta x$ was given by $\delta x(t) =
v(t) \delta t$.
The velocity $v(t)$ was generated by an Ornstein-Uhlenbeck process with correlation times $\tau_c = 0, 5$ and $10$ ms [^10], i.e. the stimulus was taken from a Gaussian distribution with correlation function $C(t) = e^{-t/\tau_c}$. Experimental runs for each $\tau_c$ lasted 45 minutes, consisting of 20 seconds long segments. In each segment, in the first 10 seconds the same stimulus was shown, whereas in the next 10 seconds the fly saw different stimuli.
Summary
========
The ability to reconstruct stimuli from the output of sensory neurons is a basic step in understanding how sensory systems operate. If intra- and inter-neuron correlations between the spikes emitted by neurons are to be taken into account, going beyond first order reconstructions is mandatory. In this case one has to face the size-explosion of higher order spike-spike correlation functions, the simplest being the 4-point correlation function necessary for a second order reconstruction. Our Gl-representation of the 4-point function in terms of 2-point functions tames this problem. If this representation holds more generally, the coding in large populations would become more feasible.
For our case of the two H1 neurons of the fly, correlations between them may be neglected, since they are only of $\sim 1$ %. We perform reconstructions using both the experimental and the Gl-approximation for the 4-point functions involved. Both are very similar, their chi-squared differing by $0.5$ %. To implement the Gl-program for the two neurons, we found it convenient to expand our variables in terms of eigenfunctions of 2-point matrices. We propose a perturbative scheme in order to take the neglected correlations into account.
We find that second order terms always improve the reconstruction, although measured by a chi-squared averaged over the whole experiment this improvement is only at the 8% level. Yet these terms can represent a $100$ % improvement at special instants as measured by an instant dependent chi-squared.\
[**Acknowledgments** ]{}\
\
We thank I. Zuccoloto for her help with the experiments. The laboratory was partially funded by FAPESP grant 0203565-4. NMF and BDLP were supported by FAPESP fellowships. We thank Altera Corporation for their University program and Scilab for its excellent software.
Bialek, W., Rieke, F., Steveninck, RR. van Warland, D.. . .
Brenner, N., Strong, S., K[ö]{}berle, R., Bialek, W. Steveninck, R. de Ruyter van.. . .
Farrow, K., Haag, J. Borst, A.. . .
Haag, J. Borst, A.. . .
Haag, J. Borst, A.. . .
Haag, J., Vermeulen, A. Borst, A.. . .
Hausen, K.. . .
Hausen, K.. . .
Hausen, K.. . M. All (), .
Krapp, H.. . .
Martin, S.. .
Rieke, F., Warland, D., Steveninck, R. de Ruyter van Bialek, W.. . .
Steveninck, RR. de Ruyter van, Lewen, G., Strong, S., K[ö]{}berle, R. Bialek, W.. . .
[^1]: Although experimental work has focussed on the vertical system, one expects analog results for the horizontal one.
[^2]: The symbol $\star$ stands for a convolution as in equation \[vol1\].
[^3]: Notice that we have not [*orthogonalized*]{} our expansion equation \[vol2\], so that there are $K_{11}(t_1,t_1)$ terms, which could have been absorbed in $K_1(t)$ and similarly for $K_2(t)$.
[^4]: We may solve the above system in Fourier space and select a subset of frequencies in order to reduce the size of the system.
[^5]: We write $(1,2,\ldots)$ instead of $(t_1,t_2,\ldots)$.
[^6]: Any structure built only from $R(t_1,t_2)$ could be used for our method to work.
[^7]: In case this is not true, we just add a convenient constant.
[^8]: The effect of $R^{12}$ may be included perturbatively- see section \[Pop\].
[^9]: Here we only provided an outline, leaving a detailed analysis for a future publication.
[^10]: Although we show results only for $\tau_c=10$ ms our conclusions are also valid for $\tau_c = 0, 5$ ms.
|
---
abstract: 'We describe the process of parametric amplification in a directional coupler of quadratically nonlinear and lossy waveguides, which belong to a class of optical systems with spatial parity-time (PT) symmetry in the linear regime. We identify a distinct spectral parity-time anti-symmetry associated with optical parametric interactions, and show that pump-controlled symmetry breaking can facilitate spectrally selective mode amplification in analogy with PT lasers. We also establish a connection between breaking of spectral and spatial mode symmetries, revealing the potential to implement unconventional regimes of spatial light switching through ultrafast control of PT breaking by pump pulses.'
author:
- 'Diana A. Antonosyan'
- 'Alexander S. Solntsev'
- 'Andrey A. Sukhorukov'
title: 'Parity-Time Anti-Symmetric Parametric Amplifier'
---
Light propagation in waveguiding structures with spatially distributed sections of loss and gain can be analogous to quantum wavepacket dynamics governed by a parity-time (PT) symmetric Hamiltonian [@Guo:2009-93902:PRL]. Below a certain gain/loss level, such systems support PT-symmetric optical modes, which then exhibit the same average loss or gain [@Ruschhaupt:2005-L171:JPA; @El-Ganainy:2007-2632:OL]. However when gain or loss is increased, the PT-symmetry of modes breaks, and a mode with the strongest gain (or smallest loss) dominates, as demonstrated experimentally [@Guo:2009-93902:PRL; @Ruter:2010-192:NPHYS]. The phase transition associated with such PT-symmetry breaking opens new possibilities for light manipulation, such as PT-symmetric lasers [@Feng:2014-972:SCI; @Hodaei:2014-975:SCI]. Such lasers can achieve single-mode operation, where small difference in medium gain leads to a dramatic difference in mode amplification below and above the PT breaking threshold.
Parametric amplifiers are commonly used as an integral part of optical setups enabling flexible wavelength conversion and tunable signal gain, extending the range of lasers where gain media are limited to particular wavelengths. Wave amplification is efficiently realized in the regime of difference-frequency generation in media with quadratic optical nonlinearity [@Boyd:2008:NonlinearOptics]. Importantly, the amplification rate is determined by the pump, enabling ultrafast all-optical tunability. In this work, we reveal the potential of PT-symmetric systems for optical parametric amplification, and identify a new regime of spectral PT anti-symmetry in such devices. Such devices can, on one hand, realize ultrafast spatial signal switching through pump-controlled breaking of PT symmetry, and on the other hand enable spectrally-selective mode amplification in analogy with PT lasers.
We consider a directional coupler composed of two waveguides in quadratically nonlinear medium, where modes exhibit different loss in each waveguide. It can be realized experimentally based on LiNbO$_3$ couplers where parametric gain was demonstrated [@Schiek:2005-11109:APL], as well as other platforms with $\chi^{(2)}$ nonlinearity. The loss can be introduced, for example, by depositing a thin layer of metal on top of th waveguide [@Guo:2009-93902:PRL]. An illustration of such structure with loss in one waveguide is presented in Fig. [\[fig:SchPTC\]]{}. In the linear regime, at low light intensities, such coupler realizes PT-symmetric optical system [@Guo:2009-93902:PRL]. However at higher intensities the effect of quadratic nonlinear interactions becomes important. It was predicted that in the regime of second-harmonic generation, when the signal and idler waves have identical spectra at half of the pump frequency, parametric interactions in PT couplers can support a rich variety of nonlinear modes [@Li:2013-53820:PRA]. Furthermore, the formation of quadratic solitons in spatially extended PT-symmetric structures was analyzed in detail [@Moreira:2012-53815:PRA; @Moreira:2013-13832:PRA]. However, the fundamentally important regime of parametric amplification in PT systems remained unexplored.
We analyze the process of optical parametric amplification based on nonlinear mixing between strong pump, signal and idler waves, as illustrated in Fig. [\[fig:SchPTC\]]{}. We model the wave propagation using coupled-mode equations [@Boyd:2008:NonlinearOptics] in the regime of narrowband and undepleted pump, $${\protect\label{eq:CMEA}}
\begin{split}
i \frac{\partial E_{s1}}{\partial z}
&=-\beta_{s1} E_{s1}-i\gamma_{s1} E_{s1} - C_{s} E_{s2}+i \chi_1 E_{p1} E^{\ast}_{i1},\\
i \frac{\partial E_{s2}}{\partial z}
&=-\beta_{s2} E_{s2}-i\gamma_{s2} E_{s2} - C_{s} E_{s1}+i \chi_2 E_{p2} E^{\ast}_{i2},\\
i \frac{\partial E_{i1}}{\partial z}
&=-\beta_{i1} E_{i1}-i\gamma_{i1} E_{i1} - C_{i} E_{i2}+i \chi_1 E_{p1} E^{\ast}_{s1},\\
i \frac{\partial E_{i2}}{\partial z}
&=-\beta_{i2} E_{i2}-i\gamma_{i2} E_{i2} - C_{i} E_{i1}+i \chi_2 E_{p2} E^{\ast}_{s2},\\
i \frac{\partial E_{p1}}{\partial z}
&=-\beta_{p1} E_{p1} - i\gamma_{p1} E_{p1} - C_{p} E_{p2},\\
i \frac{\partial E_{p2}}{\partial z}
&=-\beta_{p2} E_{p2} - i\gamma_{p2} E_{p2} - C_{p} E_{p1}.
\end{split}$$ Here the subscripts stand for signal (‘s’), idler (‘i’), and pump (‘p’) waves in two waveguides (‘1’ and ‘2’), $E$ are the mode amplitudes, $z$ is the propagation distance along the waveguides, $\beta$ are the propagation constants (for the pump mode, the propagation constant is adjusted to account for quasi-phase-matching through periodic poling of the ferroelectric domains [@Boyd:2008:NonlinearOptics]), $\gamma$ are the linear loss coefficients, $C$ are the mode coupling coefficients between the waveguides, and $\chi$ are effective quadratic nonlinear coefficients.
![\[fig:SchPTC\] Scheme of PT-symmetric nonlinear coupler with linear absorption in one waveguide.](fig01){width="0.6\columnwidth" height="0.7\textheight"}
We find that PT-symmetry in the regime of parametric amplification can be achieved in the near-degenerate case when the waveguide parameters are practically the same at signal and idler frequencies, i.e. $\gamma_{i1}=\gamma_{s1} = \gamma_1$, $\gamma_{i2}=\gamma_{s2} = \gamma_2$, and $C_{s}=C_{i}=C$. We assume that the waveguides are engineered such that $\beta_{s1} = \beta_{s2} = \beta_{s}$ and $\beta_{i1} = \beta_{i2} = \beta_{i}$, which is the regime required for linear PT-symmetry [@Guo:2009-93902:PRL; @Ruter:2010-192:NPHYS]. Under usual experimental conditions the mode at higher pump frequency is much stronger localized compared to signal an idler [@Solntsev:2014-31007:PRX], leading to suppressed coupling between the waveguides and also very small sensitivity to metal deposited on top of waveguides, meaning that $C_p \equiv 0$ and $\gamma_{p1} = \gamma_{p2} = 0$. We further consider a case of equal pump propagation constants in two waveguides, $\beta_{p1} = \beta_{p2} = \beta_p$. The latter condition does not need to be satisfied if pump is coupled to one waveguide, which already enables full range of mode switching and amplification control as we demonstrate below.
To reveal the PT-symmetry properties of Eqs. [([\[eq:CMEA\]]{})]{}, we represent the equations for signal and idler waves in the Hamiltonian form, $${\protect\label{eq:HEM}}
i\frac{\partial\mathbf{a}}{\partial z}=\mathcal{{H}}\mathbf{a},$$ where $$\mathbf{a}(z)=
\left( \begin{array}{cc}
a_{s1}(z) \\
a_{s2}(z) \\
a^{\ast}_{i1}(z) \\
a^{\ast}_{i2}(z)
\end{array} \right) =
\left( \begin{array}{cc}
E_{s1}(z) e^{ -i (\beta + \beta_s) z }\\
E_{s2}(z) e^{ -i (\beta + \beta_s) z } \\
E^{\ast}_{i1}(z) e^{ i (\beta + \beta_i) z } \\
E^{\ast}_{i2}(z) e^{ i (\beta + \beta_i) z }
\end{array} \right) ,$$ and $${\protect\label{eq:HF}}
\mathcal{H}=\left( \begin{array}{cccc}
\beta-i\gamma_{1} & -C & iA_{1} & 0 \\
-C & \beta-i\gamma_{2} & 0 & iA_{2} \\
iA^{\ast}_{1} & 0 & -\beta-i\gamma_{1} & C \\
0 & iA^{\ast}_{2} & C & -\beta-i\gamma_{2} \end{array} \right) ,$$ where $\beta = (\beta_p - \beta_s - \beta_i) / 2$ defines the phase mismatch of parametric wave mixing, and $A_{1,2}$ are the normalized input pump amplitudes, $A_1 = \chi_1 E_{p1}(z=0)$ and $A_2 = \chi_2 E_{p2}(z=0)$.
A key result of our analysis is that the Hamiltonian possesses a [*spectral anti-PT symmetry*]{}, corresponding to a negative sign on the right-hand side of the following equality, $${\protect\label{eq:AntiSC}}
\mathcal{P}_{1,+}\mathcal{P}_{2,+}\mathcal{T}\mathcal{H}
= -\mathcal{H}\mathcal{P}_{1,+}\mathcal{P}_{2,+}\mathcal{T}.$$ Here $\mathcal{T}$ is a time-reversal operator which changes $z \rightarrow -z$ and performs a complex conjugation. The [*parity operators operate in spectral domain*]{}, interchanging the signal and idler waves, $$\mathcal{P}_{1,\pm}=\left\{ a_{s1}\leftrightarrow \pm a^{\ast}_{i1} \right\},
\, \mathcal{P}_{2,\pm}=\left\{ a_{s2}\leftrightarrow \pm a^{\ast}_{i2} \right\}.$$ We define the parity operators with both symmetric (‘+’) and antisymmetric (‘-’) transformations, since the latter will be useful in the following analysis. We note that such unusual symmetry is fundamentally different from the previously studied antisymmetric PT-metamaterials with modulated dielectric and magnetic properties [@Ge:2013-53810:PRA].
Since the Hamiltonian is linear in the undepleted pump regime, the dynamics of signal and idler waves is defined by the eigenmode solutions, $${\protect\label{eq:emodes}}
\mathbf{a}(z) = \widetilde{\mathbf{a}}(\sigma) \exp( i \sigma z ),$$ where $\sigma$ is an eigenvalue. The real part, ${\rm Re}\left(\sigma\right)$, defines the phase velocity, whereas the imaginary part determines the modal gain, $\Gamma = - {\rm Im}(\sigma)$.
![\[fig:SigmaABN\] (a,b) The largest mode gain (white line marks a zero gain level) and (c,d) number of PT-symmetric mode pairs vs. the input pump in the first waveguide and the phase-mismatch. The linear losses are (a,c) $\gamma_{2s}=\gamma_{2i}=1$, (b,d) $\gamma_{2s}=\gamma_{2i}=3$. For all the plots $\gamma_{1s}=\gamma_{1i}=0$, $C=1$ and $A_2 = 0$. ](fig02){width="1\columnwidth" height="0.7\textheight"}
![\[fig:RIS\] Mode eigenvalues vs. the pump amplitude in the first waveguide ($A_{2}=0$): (a) negative imaginary part defining mode amplification, $\Gamma = - {\rm Im}(\sigma)$, and (b) real part defining propagation constant, ${\rm Re}(\sigma)$. Parameters are $\gamma_{1s}=\gamma_{1i}=0$, $\gamma_{2s}=\gamma_{2i}=1$, $C=1$, and $\beta=0$. ](fig03){width="1\columnwidth" height="0.7\textheight"}
We determine the effect of spectral anti-PT symmetry on the eigenmode properties by substituting Eq. [([\[eq:emodes\]]{})]{} into Eq. [([\[eq:HEM\]]{})]{} and applying PT operator. We obtain that if $\widetilde{\mathbf{a}}(\sigma)$ is an eigenmode, then $\widetilde{\mathbf{a}}(-\sigma^\ast) = \mathcal{P}_{1,+}\mathcal{P}_{2,+}\mathcal{T} \widetilde{\mathbf{a}}(\sigma)$ is also an eigenmode. There are two possibilities how these relations can be satisfied. First, the mode can be PT-symmetric, when PT transformation maps the mode profile to itself (up to an overall phase), which happens if $-\sigma^\ast = \sigma$, and accordingly ${\rm Re}(\sigma) = 0$. Such [*spectrally PT-symmetric modes would generally experience gain/loss different from other modes*]{}, since there are no specific relations for the value of ${\rm Im}(\sigma)$. Second, if the mode profile has broken PT-symmetry, the PT transformation relates two different modes with eigenvalues $\sigma_2 = - \sigma_1^\ast$. It follows that [*a pair of spectrally PT-broken modes experience the same loss or gain*]{}, ${\rm Im}(\sigma_1) = {\rm Im}(\sigma_2)$, but they have different phase velocities, ${\rm Re}(\sigma_1) = - {\rm Re}(\sigma_2)$. Remarkably, the established relations of mode symmetry and gain/loss are reversed in comparison to previously studied spatial PT-symmetry in directional couplers [@Guo:2009-93902:PRL; @Ruter:2010-192:NPHYS], due to the spectral [anti]{}-PT symmetry of parametric wave mixing.
We now demonstrate that the modal PT-breaking can be controlled by the pump beam. Due to the electronic nature of quadratic nonlinearity, such tuning can be ultrafast, directly following the pump profile in real time. Overall, there are four eigenmodes of the Hamiltonian. Accordingly, there can be three possible symmetry regimes: (i) there are two mode pairs with broken PT-symmetry, (ii) one pair of PT-broken modes and a pair of PT-symmetric modes, or (iii) two pairs of PT-symmetric modes. As an example, we present numerical analysis of the mode properties in a coupler with loss in the second waveguide, and pump coupled to the first waveguide. We show the largest mode gain $\Gamma={\rm Max}[-{\rm Im}(\sigma)]$ in Figs. [\[fig:SigmaABN\]]{}(a,b) and the number of PT-symmetric mode pairs in Figs. [\[fig:SigmaABN\]]{}(c,d), depending on the pump amplitude $A_1$ and the phase-mismatch, for different values of linear loss. We observe that in the regime when all modes have broken spectral PT symmetry \[blue shaded regions in Figs. [\[fig:SigmaABN\]]{}(c,d)\], the modes experience negative gain \[c.f. Figs. [\[fig:SigmaABN\]]{}(a,b)\]. This happens because pairs of eigenmodes exhibit the same amount of gain/loss, and effectively the amounts of gain and loss are averaged out between the eigenmodes. However upon transition to the region with spectrally PT-symmetric modes \[green and red shaded regions in Figs. [\[fig:SigmaABN\]]{}(c,d)\], there appears an unequal redistribution of gain and loss between the modes. One PT-symmetric eigenmode exhibits gain much larger then all other modes, while the latter experience stronger loss. Such sensitivity of amplification to PT-breaking threshold could be used to discriminate between multiple spectral modes, analogous to the concept of PT-lasers [@Feng:2014-972:SCI; @Hodaei:2014-975:SCI].
Next, we establish a connection between the spectral PT-symmetry identified above, and the spatial wave dynamics due to waveguide coupling. We first consider a special case which can be treated analytically, corresponding to perfect phase-matching, $\beta=0$, and additionally ${\rm Im}(A_{1}^\ast A_{2})=0$. The latter condition can be transformed to ${\rm Im}(A_{1}) = {\rm Im}(A_{2}) = 0$ under a substitution $\mathbf{a} \rightarrow \mathbf{a} \exp(i\varphi)$ with appropriately chosen constant phase $\varphi$. Then, we find that $$\begin{aligned}
\mathcal{P}_{1,+}\mathcal{P}_{2,-} \mathcal{H} & = &
\mathcal{H} \mathcal{P}_{1,+}\mathcal{P}_{2,-}, {\protect\label{eq:Ppm}} \\
\mathcal{P}_{1,-}\mathcal{P}_{2,+} \mathcal{H} & = &
\mathcal{H} \mathcal{P}_{1,-}\mathcal{P}_{2,+}. {\protect\label{eq:Pmp}}\end{aligned}$$ Accordingly, there appear two pairs of eigenmodes with $a^{\ast}_{i1}=\eta a_{s1}$ and $a^{\ast}_{i2}=-\eta a_{s2}$, where one pair with $\eta=+1$ has a profile symmetric with respect to $\mathcal{P}_{1,+}\mathcal{P}_{2,-}$ and second pair with $\eta=-1$ conforms to the symmetry $\mathcal{P}_{1,-}\mathcal{P}_{2,+}$. Remarkably, the signal dynamics of such modes is governed by equations resembling those for a linear PT-symmetric coupler [@Guo:2009-93902:PRL; @Ruter:2010-192:NPHYS], $${\protect\label{eq:CMsignal}} i \frac{\partial {\bf a}_{s}}{\partial z} = \mathcal{H}_r {\bf a}_{s},
\, \mathcal{H}_{r}=\left( \begin{array}{cccc}
i(\eta A_{1} -\gamma_{1}) & -C \\
-C & i(-\eta A_{2} -\gamma_{2}) \end{array} \right),$$ where ${\bf a}_{s} = (a_{s1}; a_{s2})$. We see that the linear loss is modified due to the parametric gain determined by the pump amplitudes. The coupler Eqs. [([\[eq:CMsignal\]]{})]{} are symmetric with respect to [*spatial PT-symmetry*]{}, up to a gauge transformation [@Guo:2009-93902:PRL] expressed through the identity matrix ($\mathcal{I}$) in the following relation, $${\protect\label{eq:CouplerHPT}}
\mathcal{P}_{12}\mathcal{T} (\mathcal{H}_r - \bar{\rho}\; \mathcal{I}) = (\mathcal{H}_r - \bar{\rho}\; \mathcal{I}) \mathcal{P}_{12}\mathcal{T} ,$$ where the [*spatial parity operator*]{} $\mathbf{P}_{12}$ swaps the mode amplitudes between the waveguides, $a_{s1} \leftrightarrow a_{s2}$ and $a_{i1} \leftrightarrow a_{i2}$. The coefficient $\bar{\rho} = (\eta A_{1} -\gamma_{1})/2 + (-\eta A_{2} -\gamma_{2})/2$ defines the average gain or loss between the two waveguides, which depends on the pump amplitudes and signal/idler mode symmetries corresponding to the different signs of ‘$\eta$’. The mode eigenvalues are $${\protect\label{eq:sigmaBeta0}}
\sigma = - i \bar{\rho} \pm
(1/2) \sqrt{C^2 -[\gamma_1-\gamma_2 - \eta (A_1 + A_2)]^2} .$$ We present characteristic dependencies of the eigenvalues on the pump amplitude in the first waveguide in Fig. [\[fig:RIS\]]{}. We find that both [*spatial and spectral PT-symmetry breaking occurs simultaneously*]{} at the threshold $|\gamma_1-\gamma_2 - \eta (A_1 + A_2)| = C$. However, the spatial and spectral symmetries are opposite: a mode pair is spatially PT-symmetric and has spectrally broken symmetry below threshold, whereas the situation is reversed above the threshold. We show the evolution of the signal intensity along the waveguides is shown in Fig. [\[fig:AZ\]]{}. At lower pump powers \[Fig. [\[fig:AZ\]]{}(a)\] signal periodically switched between the waveguides due to the beating between two modes exhibiting the same (negative) gain. However for stronger pump above the PT threshold \[Figs. [\[fig:AZ\]]{}(b)\], only one supermode with the strongest gain dominates and accordingly oscillations are suppressed.
![\[fig:AZ\] Evolution of the signal mode intensity along the waveguides: solid (green) lines — the first waveguide without loss, dashed (blue) line — the second waveguide with loss. The pump amplitude is (a) below PT-symmetry breaking threshold ($A_{1}=0.5$) and (b) above threshold ($A_{1}=1.5$). Parameters are $\gamma_{1s}=\gamma_{1i}=0$, $\gamma_{2s}=\gamma_{2i}=1$, $C=1$, and $\beta=0$. ](fig04){width="1\columnwidth" height="0.7\textheight"}
Finally, we investigate numerically a connection between spectral PT-breaking and spatial mode dynamics in a general case of non-zero phase mismatch. We present in Fig. [\[fig:A1B\]]{} the fraction of signal intensity in the first waveguide depending on the phase-mismatch and the pump amplitude for different propagating distances. We find that spatial dynamics strongly depends on the spatial PT-symmetry of modes in the linear regime (at low pump amplitudes $A_j \rightarrow 0$). If the linear modes are PT-symmetric, $|\gamma_1-\gamma_2| < C$, then increase of pump amplitude can control the period of mode coupling between the waveguides, while the oscillations get suppressed close to the spectral PT threshold, see Figs. [\[fig:A1B\]]{}(a,c) and Fig. [\[fig:SigmaABN\]]{}(c). However if the linear modes are PT-broken, $|\gamma_1-\gamma_2| > C$, then mode beating between the waveguides still occurs below the spectral PT threshold, but with very small modulation amplitude, see Figs. [\[fig:A1B\]]{}(b,d) and Fig. [\[fig:SigmaABN\]]{}(d). In all cases, the mode beating changes faster with pump amplitude at longer distances.
![\[fig:A1B\] Fraction of signal intensity in the first waveguide, $|a_{s1}|^{2}/(|a_{s1}|^{2}+|a_{s2}|^{2})$, vs. the input pump in the first waveguide and the phase-mismatch. Propagation distances are (a,b) $L=5C$ and (c,d) $L=10C$. Parameters correspond to (a,c) Figs. [\[fig:SigmaABN\]]{}(a,c) and (b,d) Figs. [\[fig:SigmaABN\]]{}(b,d). ](fig05){width="1\columnwidth" height="0.7\textheight"}
In conclusion, we have identified an anti-PT spectral symmetry of a parametric amplifier based on a quadratically nonlinear coupler with different losses in two waveguides. For pump powers below the threshold, the modes form pairs with broken PT symmetry and same gain/loss, whereas above the threshold one PT-symmetric mode experiences the largest gain. This can facilitate spectrally-selective mode amplification, and we expect that this effect will become even more pronounced in resonator structures in analogy with single-mode operation of PT microring lasers. We have further established an underlying connection between spectral anti-PT and conventional spatial PT mode symmetries, which reveals the possibility to control spatial light switching and amplification through parametric gain. Accordingly, the suggested platform can implement various unconventional regimes of light control previously suggested for PT-symmetric structures, such as unidirectional [@Bender:2013-234101:PRL] and nonreciprocal [@Peng:2014-394:NPHYS] operation, but with the advantage of ultrafast all-optical control of PT-symmetry regimes by pump pulses. We anticipate that, due to the universality of parametric amplification processes, these concepts will be extended to different physical mechanisms including Kerr-type optical nonlinearity, as well as a broad range of other systems including cold atomic Bose-Einstien condensates [@Vogels:2002-20401:PRL] in engineered potentials [@Graefe:2012-444015:JPA; @Cartarius:2012-13612:PRA] and exciton-polariton condensates [@Lien:2015-24511:PRB].
This work was supported by the Australian Research Council, including Discovery Project DP130100135 and Future Fellowship FT100100160.
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10.1103/PhysRevB.91.024511)
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---
abstract: 'We revisit the observed correlation between and velocities for Type II-P supernovae (SNe II-P) using 28 optical spectra of 13 SNe II-P and demonstrate that it is well modeled by a linear relation with a dispersion of about 300 kms$^{-1}$. Using this correlation, we reanalyze the publicly available sample of SNe II-P compiled by @dandrea10 and find a Hubble diagram with an intrinsic scatter of 11% in distance, which is nearly as tight as that measured before their sample is added to the existing set. The larger scatter reported in their work is found to be systematic, and most of it can be alleviated by measuring rather than velocities, due to the low signal-to-noise ratios and early epochs at which many of the optical spectra were obtained. Their sample, while supporting the mounting evidence that SNe II-P are good cosmic rulers, is biased toward intrinsically brighter objects and is not a suitable set to improve upon SN II-P correlation parameters. This will await a dedicated survey.'
author:
- 'Dovi Poznanski, Peter E. Nugent, and Alexei V. Filippenko'
title: 'Type II-P Supernovae as Standard Candles: The SDSS-II Sample Revisited'
---
Introduction
============
Type II supernovae (SNe) that undergo a long, bright, flat phase in photometric evolution are commonly referred to as plateau SNe (). These have been shown over the past decade to be good “standardizable candles,” and potential cosmological probes, in a manner similar to that of their more famous cousins, SNe Ia. @hamuy02 were the first to demonstrate this empirical “standardizable candle method” (SCM), a technique which was later streamlined and applied to larger samples by @nugent06 [hereafter N06] and @poznanski09a [hereafter P09]; see also @olivares10.
P09 used a sample of 34 to constrain the three parameters that define the correlation: a reference absolute magnitude in the $I$ band, a velocity term that is correlated with the luminosity during the plateau, and a color term that minimizes the contribution from intrinsic color inhomogeneity as well as from dust extinction. The resulting intrinsic scatter in the Hubble diagram was found to be 0.22mag, which is equivalent to about 10% in distance, similar to the result of N06 despite the larger sample. Since the samples used by N06 and P09 are dominated by nearby SNe that sport large distance uncertainties due to peculiar velocities, a robust derivation of the SCM parameters, together with a determination of its potential power, requires a set of SNe II-P in the Hubble flow.
Recently, @dandrea10 [hereafter D10] presented such a sample of 15 in the Hubble flow from the Sloan Digital Sky Survey II (SDSS-II) SN survey, albeit obtained during a project that was focused on SNe Ia. The SCM method relies on photospheric velocities measured from the $\lambda$5169 absorption line during the plateau phase, specifically about 50 days past explosion. As noted by D10, however, the optical spectra of their SNe II-P are usually of low signal-to-noise ratio (2n) and taken at an early epoch, when lines are not fully developed. This observation strategy, while complicating the application of the SCM method to , was dictated by the main purpose of their survey, which was to identify SNe Ia and measure their redshifts in order to use them as distance indicators [@kessler09]. D10 find that adding their sample to those analyzed previously increases the scatter significantly, to 15% in distance. This points to a failure of the data, method, or underlying model.
While D10 conscientiously explore many possible sources for this scatter, we show below that most of it is due to a systematic offset, and could be explained by two effects: the method used to determine the photospheric velocities, and an observational bias. In §\[s:vel\] we rederive photospheric velocities for their sample using absorption lines, as applied by N06 to their sample of higher redshift SNe. We calibrate the relation between velocities and the standard velocities in §\[s:hb\]. In §\[s:hd\] we update the Hubble diagram and discuss the specific selection effects that bias the SDSS-II sample.
![Velocities determined from the absorption minima of $\lambda$4861 and $\lambda$5169 using 28 spectra of 13 at phases of 5 to 40 days after explosion. The shaded area marks the 1$\sigma$ region for the derived correlation, $v_{\rm
Fe~II} = (0.84 \pm 0.05)\,v_{\rm H\beta} $. velocities can be well determined using , which is easier to detect in early-phase and low-2n spectra. Squares with dashed error bars mark velocities derived for objects with very high velocities using Eq. 2 of N06. The outlier at $v_{\rm Fe~II} \approx
8,400$kms$^{-1}$ is from a very early spectrum (2 days past explosion) of the low-luminosity, low-velocity SN2005cs.[]{data-label="f:hbeta"}](f1.pdf){width="3.0in"}
Photospheric Velocities {#s:vel}
=======================
In order to derive photospheric velocities, D10 apply to their spectra the algorithm developed by P09. Briefly, using the “SN Identification Code” (SNID; @blondin07), the spectra are cross-correlated with a library of high-2n templates for which velocities can be measured precisely. P09 successfully applied this method to 19 nearby SNe II-P that typically had a few high-2n spectra per object, many of them near day 50 after explosion. D10 correctly point out that this method may not be applicable to their sample, due to the early epoch at which they were obtained and their low 2n. The lines usually become fully developed after a few weeks past explosion. Even at later times those lines are often weak, making velocity derivation from low-2n spectra problematic.
We rederive the velocities for the SDSS-II sample using another method, and find values that are typically different from those of D10. These in turn improve the scatter in the Hubble diagram, as demonstrated in §\[s:hd\]. N06 showed that there is a correlation between the velocities measured from $\lambda$4861 and those derived from $\lambda$5169. In the next section we improve the determination of that correlation using many more objects and spectra.
We obtain the photospheric velocities for the SDSS-II sample as follows. From each spectrum we measure the velocity, and where feasible, the velocity. This is done by finding the minimum of the absorption line. When the 2n is too low we smooth the spectra with a Savitzky-Golay filter (which is similar to a running mean; @savitzky64). For most of the SDSS-II spectra, $\lambda$5169 is either undeveloped or buried in the noise. The velocities are translated to equivalent velocities using the relation found in §\[s:hb\] (with the uncertainty in that relationship folded in quadrature). For every spectrum that does have a direct velocity measurement, we calculate a weighted mean between the direct and indirect values. These velocities and uncertainties are then propagated to day 50 (as was done by N06, P09, and D10) and listed in Table \[t:vel\].
[lcc]{} SN 2007ld & 4110(570) & 3970(660)\
SN 2006iw & 4490(160) & 4660(270)\
SN 2007lb & 3950(340) & 4080(420)\
SN 2007lj & 4220(460) & 4110(450)\
SN 2006jl & 5560(310) & 4780(350)\
SN 2007lx & 3900(820) & 4170(910)\
SN 2007nw & 4290(750) & 4830(840)\
SN 2006kv & 3640(620) & 4050(950)\
SN 2007kw & 3930(460) & 4990(1060)\
SN 2006gq & 3400(150) & 3790(440)\
SN 2007ky & 3510(420) & 4060(480)\
SN 18321 & 5060(640) & 5820(700)\
SN 2006kn & 4220(440) & 5000(590)\
SN 2007kz & 4380(320) & 5490(370)\
SN 2007nr & 4430(350) & 4900(590)\
Correlation between and Velocities {#s:hb}
====================================
N06 propose an alternative velocity proxy to , using the absorption line that is both present early and significantly more prominent. We repeat the analysis of N06 using the much larger sample of P09 and derive a new relation between the two velocities.
We use 28 spectra of 13 that span ages of 5 to 40 days past explosion; both lines are readily identifiable in all of the spectra. We restrict ourselves to spectra obtained prior to day 40, since at later times $\lambda$5169 is at least as strong as , which also becomes blended with other lines that can bias the velocity measurement. We measure the velocities as described for the SDSS-II SNe in §\[s:vel\].
As seen in Figure \[f:hbeta\], we find a robust ($\chi^2/{\rm dof}
\approx 1$) linear correlation that can be represented as $v_{\rm
Fe~II} = (0.84 \pm 0.05)\,v_{\rm H\beta} $. The uncertainty is equivalent to an additional error of about 200–400kms$^{-1}$ over the relevant range of velocities. While this relation is somewhat different from the one derived by N06, in practice it gives similar results for the range of velocities probed by the SDSS-II sample.
Since there are very few objects in the P09 sample having velocities higher than 8,000kms$^{-1}$ in which both lines are well developed, we add to Figure \[f:hbeta\] nine early-epoch, high-velocity spectra where the lines are not detected. In order to approximate the velocity at those times, we use the velocity derived for these objects at day 50 in P09, and propagate them to the correct epoch using Equation 2 of N06 which models the time evolution of velocities. Most of the additional spectra seem to agree with the relationship derived above, with two notable exceptions. One is SN2005cs (2 days past explosion) with a derived velocity which is much too high. This object was a low-luminosity, low-velocity SN II-P, and is of little relevance to nonlocal samples. The second exception is that at velocities higher than 10,000kms$^{-1}$ the scatter increases significantly. This may be due to a combination of line blending at these high velocities, combined with very blue continua that make absorption minima harder to define and measure. All of the SNe in the D10 sample have a spectrum with velocity smaller than 10,000kms$^{-1}$. While these exceptions are not relevant to the reanalysis of the D10 sample, one should be cautious and remember that this correlation may break down for spectra with extremely high or low velocities.
{width="6.in"}
Hubble Diagram {#s:hd}
==============
When minimizing the “cost function” (Equation 4 of P09) in order to determine the best-fit parameters, P09 find an intrinsic scatter of 0.22 mag (D10 measure it to be 0.20 mag when minimizing a slightly different function), which is equivalent to $\sim$10% in distance, similar to the result of N06. D10 report an increase to 0.29 mag (which we measure to be 0.32 mag) when adding the SDSS-II sample, which is about 50% more scatter. An important clue to the source of this additional dispersion is provided by Figure 5 of D10, which clearly shows that the SDSS-II SNe are systematically offset from the Hubble law, with all but one of them falling below the line. In fact, the scatter is reduced significantly by artificially dimming the SDSS-II SNe by 0.4 mag, ending up even smaller than that reported by P09.
We repeat the analysis of D10 using all of their data, except for the velocities and their respective uncertainties, where we substitute those we find in §\[s:vel\]. As can be seen in Figure \[f:hd\], this eliminates a substantial fraction of the systematic offset, which in turn reduces the intrinsic scatter to 0.25 mag (11% in distance), only slightly higher than without this sample. Our best-fit parameters for the combined sample are nearly identical to those of P09, except for the reference absolute magnitude which is 0.1 mag brighter for the combined sample.
Deriving the parameters using only the SDSS-II sample, we still find preferred values for the correlation coefficients that are offset from those of P09. The SDSS-II SNe appear to be predominantly overluminous, and favor a small value for the velocity correlation coefficient. Following the suggestion made by D10, we show below that this is largely due to an observational bias arising from the SDSS-II follow-up criteria.
We have performed Monte Carlo simulations, starting from a SN population that mimics the properties of the P09 sample, to which we apply various observational cuts. For these simulated samples we then compute the best-fit correlation parameters. We find that a simple magnitude cut cannot produce such biased values. However, selecting objects that are *intrinsically brighter* skews the parameter derivation in the right direction. Folding in the larger velocity uncertainties allows us to reproduce the bias toward luminous SNe with little correlation. Indeed, when one selects only luminous SNe, it is not surprising that the best-fit reference absolute magnitude is brighter, and the correlation parameters smaller — thus pointing toward a weaker link between luminosity and velocity. The lack of statistical power of such a sample is reflected by the fact that the parameters derived from the combined set are almost identical to those recovered without including the SDSS-II sample. Because the SDSS-II sample does not cover a substantial region of parameter space, it cannot significantly constrain the SCM parameters.
D10 mention that the contrast between a SN and its host galaxy was a selection criterion for spectroscopic follow-up observations. This was obviously done to increase the efficiency of their spectroscopic observing runs and the number of SNe Ia they find. Unlike a regular that skews the distribution near the detection limit of the survey, such a criterion selects intrinsically bright objects, independent of redshift. This is evident from the very different luminosity functions of the two samples (Fig. 7 of D10). Figure \[f:contrast\] shows that nearly half of the SDSS-II SNe are brighter than their host galaxies up to a factor of a few, unlike the samples of P09 and N06 that are consistently fainter than their hosts. A better understanding of the SCM relation, based on larger, less biased samples, will allow a careful determination of the various potential biases, providing a crucial component for constraining cosmological parameters.
![Approximate $I$-band flux ratios (SN/host galaxy) for the Katzman Automatic Imaging Telescope (KAIT; P09) and Supernova Legacy Survey (SNLS; N06) samples (solid line) and for the SDSS-II sample from D10 (dotted line). About half of the SDSS-II SNe are brighter than their hosts galaxies, a selection effect that removes intrinsically faint objects and skews the estimation of correlation parameters.[]{data-label="f:contrast"}](f3.pdf){width="3.0in"}
Conclusions {#s:conc}
===========
We find that when measuring expansion velocities for the SDSS-II sample of by using , the SCM correlation gives a tight Hubble diagram, with a scatter of $\sim$11% in distance. This is comparable to the scatter obtained with previous samples of , despite a systematic offset among the SDSS-II objects, which is probably due to a strong bias favoring intrinsically bright SNe.
Our analysis shows that velocities correlate well with those derived from lines, with a scatter of about 300kms$^{-1}$, enabling the use of early-time spectra and data having low 2n.
The SCM parameters and the true tightness of the correlation remain to be tested with high-quality Hubble-flow data. We are currently compiling such a set of from the Palomar Transient Factory (@rau09 [@law09; @poznanski09b]), where cosmology with is one of the key projects and should lead to less biased samples.
We thank C. D’Andrea and his collaborators for making the SDSS-II data on SNe II-P available to the community, and J. S. Bloom for useful advice. D.P. is supported by an Einstein Fellowship. We acknowledge support from the US Department of Energy Scientific Discovery through Advanced Computing (SciDAC) program under contract DE-FG02-06ER06-04. A.V.F. is also grateful for funding from National Science Foundation grant AST-0908886 and the TABASGO Foundation.
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---
abstract: 'We discuss the use of a region of uniform and constant magnetic field in order to implement a two-state atomic polarizer for an H(2S) beam. We have observed that a device with such field configuration is capable of achieving an efficient polarization for a wide range of magnetic field intensities and atomic velocities. In addition, we establish a criterion that must be met to confirm a successful polarization. That is possible due to a specific beating pattern for the Lyman-$\alpha$ radiation expected for the outgoing two-state atomic beam.'
author:
- 'Amanda Alencar\*, I. Prazeres, C. R. de Carvalho, F. Impens, A. Medina, N. V. de Castro Faria, J. Robert\*\*, Ginette Jalbert'
bibliography:
- 'ReF.bib'
title: Quantum State Preparation of Hydrogen Atoms by Hyperfine Quenching
---
Introduction
============
Atomic structures are sensitive to the presence of external fields. For instance, light shifts may significantly alter the frequencies obtained in optical atomic clock [@katori2003ultrastable]. Spectral line widths are also sensitive to applied fields. Recently, it has been shown that the lifetime of hydrogen metastable state can be influenced not only by the field intensity, but also by the chosen geometry of the electromagnetic fields [@trappe2016geometric]. This paper focuses on the use of external fields to filter specific atomic states.
The atomic polarizer, which filters specific atomic states, is an essential component of matter-wave interferometers based on a multiple-state atomic source [@miniatura1992; @Kouchi2019entangled] such as hydrogen Stern-Gerlach interferometers [@miniaturaJPhysII91; @robertJPhysII92; @Impens17]. In the present work, we are interested in exploring the filtering of two specific metastable hyperfine structure states of the hydrogen atom. An implementation of this particular atomic polarizer can be obtained with of a region of static and uniform magnetic field. The role of the polarizer’s magnetic field is twofold: it is used to tune the energy levels of the metastable hydrogen and also to generate the commoving electric field responsible for the decay, as shown in the classical references [@Lamb50; @Lamb52a]. The couplings between $2S_{1/2}$ and $2P_{1/2}$ atomic states dressed by the magnetic field filter out two specific $2S_{1/2}$ hyperfine structure states. We analyse the Lyman-$\alpha$ radiation rate associated with these decays and show that a specific pattern is obtained. This pattern by itself can be seen as an indication of a successful polarization process. However, a more robust pattern can also be observed if, after the desired polarization, we add an external electric field and observe the resulting Lyman-$\alpha$ emission, this time coming from the states transmitted by the initial polarization stage.
The paper is organized as follows. Section \[Td\] reviews the influence of the magnetic field on the $2S_{1/2}$ and $2P_{1/2}$ hyperfine structure states of the hydrogen atom. We then derive the time-dependent probabilities to find the hydrogen atom in a specific $2S_{1/2}$ or $2P_{1/2}$ hyperfine structure state. These expressions are used in Section \[sec:fast\] to propose a polarizing device consisting in a region of uniform and constant magnetic field. Finally, we discuss a criterion of effectiveness for the polarizer based on the Lyman-$\alpha$ radiation patterns produced by the decay of the atomic states of interest.
Theoretical description {#Td}
=======================
Zeeman effect in hyperfine $H$ states
-------------------------------------
We review here the Zeeman effect in the hyperfine structure states $2S_{1/2}$ and $2P_{1/2}$ of the hydrogen atom. As usual, we use decomposition on a basis of orbital angular momentum, electron spin, and nucleus spin eigenstates (Appendix \[apendA\]) to analyse the Zeeman effect. For this purpose, we use a decomposition on the hyperfine eigenstates $\ket{2L_{J},F,M_F}$, where $\vec{F}=\vec{J}+\vec{I}=(\vec{L}+\vec{S})+\vec{I},$ with $\vec{L}$ standing for the orbital angular momentum, $\vec{S}$ the electron spin and $\vec{I}$ the spin of the nucleus. We consider the following hyperfine structure states: $$\begin{aligned}
&\ket{2S_{\frac{1}{2}},0,0},\ket{2S_{\frac{1}{2}},1,-1},\ket{2S_{\frac{1}{2}},1,0},\ket{2S_{\frac{1}{2}},1,1}, \nonumber \\
&\ket{2P_{\frac{1}{2}},0,0},\ket{2P_{\frac{1}{2}},1,-1},\ket{2P_{\frac{1}{2}},1,0},\ket{2P_{\frac{1}{2}},1,1}. \nonumber\end{aligned}$$
The $2P_{3/2}$ states were not considered in this study for being about ten times further from the $2S_{1/2}$ states, regarding energy intervals. For that reason, the influence of the $\ket{2P_{3/2},F,M_F}$ states in the $\ket{2S_{1/2},F,M_F}$ states lifetimes is much less significant than the influence of the $\ket{2P_{1/2},F,M_F}$ states.
We will consider the total Hamiltonian operator $
\hat{H}=\hat{H}_0+\hat{H}_B.
$ $\hat{H}_0$ is the non-perturbed Hamiltonian expressed in terms of hyperfine structure states [@Gasenzer12metastable]. $\hat{H}_B=(\hat{L}_z+g_e\hat{S}_{ez}-\tilde{g}_p\hat{S}_{pz})\mu_BB/\hbar$ describes the atom-magnetic field interaction, where $\mu_{B}$ is the Bohr magneton, $g_e$ is the gyromagnetic factor of the electron, $\tilde{g}_p= g_p \mu_N /\mu_B=g_p m_e / m_p$ is the gyromagnetic factor of the nucleus multiplied by the ratio between the electron mass and the proton mass.
The eigenvalues of the Hamiltonian $\hat{H}$ in the subspace spanned by the $\ket{2S_{1/2},F,M_F}$ states can be easily determined. The same can be done for the $\ket{2P_{1/2},F,M_F}$ states. The expressions for the mentioned Hamiltonian eigenvalues are displayed in Appendix \[apendB\] and Fig. \[Fig1\] shows them as a function of the magnetic field intensity. We take as reference the energy of the fine state $2P_{1/2}$ in the absence of magnetic field.
**![Zeeman splitting of the $2S_{1/2}$ and $2P_{1/2}$ hyperfine structure states. The energy eigenvalues represented are $h_1$ to $h_8$ for increasing energy at low magnetic field intensities.[]{data-label="Fig1"}](Fig01.jpg "fig:")
As expected, the energy eigenvalues associated to the angular momentum projections $M_F= \pm 1$, Eqs. (\[h2\],\[h4\],\[h6\],\[h8\]), behave linearly with the magnetic field intensity, while the energy eigenvalues associated to the angular momentum projections $M_F=0$, Eqs. (\[h1\],\[h3\],\[h5\],\[h7\]), behave quadratically with the magnetic field intensity when $(g_e+g_p)^{2} \mu_B ^2 B^2 \ll A_1 ^2$ and $(4-g_e+3g_p)^2\mu_B^2B^2 \ll 9A_2^2$, respectively, or $B \ll 63 $G. This behaviour is related to the mixture between the states $\ket{2S_{1/2},0,0}$ and $\ket{2S_{1/2},1,0}$ and between the states $\ket{2P_{1/2},0,0}$ and $\ket{2P_{1/2},1,0}$ caused by the external magnetic field. The new states can be written with respect to the $\ket{2S_{1/2},F,M_F}$ and $\ket{2P_{1/2},F,M_F}$ states as follows:
$$\label{1}
\ket{1}=m_1(B) \ket{2P_{\frac{1}{2}},0,0} +n_1(B) \ket{2P_{\frac{1}{2}},1,0}$$
$$\label{2}
\ket{2}=\ket{2P_{\frac{1}{2}},1,-1}$$
$$\label{3}
\ket{3}=m_3(B) \ket{2P_{\frac{1}{2}},0,0} +n_3(B) \ket{2P_{\frac{1}{2}},1,0}$$
$$\label{4}
\ket{4}=\ket{2P_{\frac{1}{2}},1,1}$$
$$\label{5}
\ket{5}=m_5(B) \ket{2S_{\frac{1}{2}},0,0} +n_5(B) \ket{2S_{\frac{1}{2}},1,0}$$
$$\label{6}
\ket{6}=\ket{2S_{\frac{1}{2}},1,-1}$$
$$\label{7}
\ket{7}=m_7(B) \ket{2S_{\frac{1}{2}},0,0} +n_7(B) \ket{2S_{\frac{1}{2}},1,0}$$
$$\label{8}
\ket{8}=\ket{2S_{\frac{1}{2}},1,1}$$
In the absence of an external magnetic field, we have $n_1(0)=m_3(0)=n_5(0)=m_7(0)=0$ while $m_1(0)=n_3(0)=m_5(0)=n_7(0)=1$. That means that all states are given in terms of only one state $\ket{2S_{1/2},F,M_F}$ or $\ket{2P_{1/2},F,M_F}$. For strong magnetic fields, on the other hand, we have $m_1(B)=n_1(B)=n_3(B)=m_5(B)=n_5(B)=n_7(B)\rightarrow 1/\sqrt{2}$, while $m_3(B)=m_7(B)\rightarrow -1/\sqrt{2}$. This means that, for high values of the magnetic field intensity, the states $\ket{5}$ and $\ket{7}$ are perfect mixtures of the states $\ket{2S_{1/2},0,0}$ and $\ket{2S_{1/2},1,0}$ and the states $\ket{1}$ and $\ket{3}$ are perfect mixtures of the states $\ket{2P_{1/2},0,0}$ and $\ket{2P_{1/2},1,0}$.
Figure \[Fig1\] reveals crossings between some of the $2S_{1/2}$ and $2P_{1/2}$ levels for well-defined values of the magnetic field. A strong coupling between certain states can be achieved by the motional electric field in the vicinity of the crossing regions. Selection rules impose $\Delta m_l=\pm 1$ with the motional electric field perpendicular to the quantization direction (See Appendix \[apendC\]).
Since the $2P_{1/2}$ states have short lifetime, of the order of $10^{-9}$s, it is possible to induce the decay of the $2S_{1/2}$ metastable states by creating a channel to the ground state through the unstable $2P_{1/2}$ states. Even a small external magnetic field may significantly enhance the decay probability of a moving atom in a $2S_{1/2}$ state. We study below how external fields affect the lifetimes of hyperfine states.
Time-dependent evolution of the atomic states
---------------------------------------------
The quenching process will be described using time dependent perturbation theory taking $\gamma_{2s}\approx 7 s^{-1}$ and $\gamma_{2p}\approx 6\times 10^{8} s^{-1}$ as the $2S_{1/2}$ and $2P_{1/2}$ decay constants, noting that $\gamma_{2s} \ll \gamma_{2p}$. The expressions below give the temporal variation of the probability amplitude of the atom to be observed in a certain state considering external perturbation and spontaneous decay $$\label{ddtcsdet}
\frac{d}{dt}c_j(t)=-\frac{1}{2}\gamma_{2s}c_j(t) + \sum_{k \in P} \frac{V_{jk}}{i\hbar}e^{i\omega_{jk} t}c_k(t)$$ $$\label{ddtcpdet}
\frac{d}{dt}c_k(t)=-\frac{1}{2}\gamma_{2p}c_k(t) + \sum_{j \in S} \frac{V_{kj}}{i\hbar}e^{i\omega_{kj} t}c_j(t)$$ where the sets $P = \{1,2,3,4 \} $ and $S = \{ 5,6,7,8 \} $ corresponds to the $2P_{1/2}$ and $2S_{1/2}$ states, respectively. The indices $j$ and $k$ denote $2S_{1/2}$ and $2P_{1/2}$ states, respectively. $c_j(t)$ and $c_k(t)$ are the corresponding time-dependent probability amplitudes. As discussed later, each state of the set $S$($P$) is coupled to two states of the set $P$($S$). However, as discussed in Appendix \[apendC\], only one of these two couplings will be relevant for the quantum evolution. The relevant coupling depends on the electric field induced in the atomic frame as well as on the magnetic field.
Since these states are separated by an energy $\Delta \varepsilon_{jk} = h_j-h_k = \hbar \omega_{jk} $, being those states coupled by an electric field $\vec{E}$, the matrix elements $V_{kj}$ are: $$V_{kj}=\bra{k}e\vec{E} \cdot \vec{r} \ket{j}=V_{jk}^*.$$ See Appendix \[apendC\].
The solutions $c_j(t)$ and $c_k(t)$ can be found by decoupling (\[ddtcsdet\]) and (\[ddtcpdet\]) through differentiation.
We take the atom initially in the $2S_{1/2}$ level so that $ c_j(0)=1 $ and $ c_k(0)=0 $. The solutions can then be written as: [$$\label{csdet}
c_j(t)=\Big( \frac{\lambda_{2}^{+}+\frac{1}{2}\gamma_{2s}}{\lambda_{2}^{+}-\lambda_{1}^{+}} \Big) e^{\lambda_{1}^{+} t}-\Big( \frac{\lambda_{1}^{+}+\frac{1}{2}\gamma_{2s}}{\lambda_{2}^{+}-\lambda_{1}^{+}} \Big)e^{\lambda_{2}^{+} t}$$ $$\label{cpdet}
c_k(t)=-\Big( \frac{(i\hbar)^{-1} V_{kj}}{\lambda_{2}^{-}-\lambda_{1}^{-}} \Big)e^{\lambda_{1}^{-} t}+\Big( \frac{(i\hbar)^{-1} V_{kj}}{\lambda_{2}^{-}-\lambda_{1}^{-}} \Big)e^{\lambda_{2}^{-} t}$$ ]{} with $\lambda_{1,2}^{+}= -\gamma_{+}/2 \pm \sqrt{\gamma_{+}^2/4-\omega_{0_+}^2}$ and $\lambda_{1,2}^{-}= -\gamma_{-}/2 \pm \sqrt{\gamma_{-}^2/4
-\omega_{0_-}^2}$ and where $\gamma_{\pm} = (\gamma_{2s}+\gamma_{2p})/2 \pm i\omega_{kj}$, $\omega_{0_+}^2= |V_{kj}|^2/\hbar^2+\gamma_{2s}\gamma_{2p}/4 + i\gamma_{2s}\omega_{kj}/2$ and $\omega_{0_-}^2= |V_{kj}|^2/\hbar^2+\gamma_{2s}\gamma_{2p}/4 - i\gamma_{2p}\omega_{kj}/2$. $\gamma_{\pm}$ and $\omega_{0_\pm}$ are dependent of the energy difference $\Delta \varepsilon_{jk}$, but that dependence is not explicitly expressed for the sake of simplicity.
The considered probabilities are then: $$\label{probtrabalhada}
\begin{aligned}
&|c_j(t)|^2 = \hspace{0.13cm} \frac{e^{-\Re(\gamma_+)t}}{\abs{2\beta_+}^2} \hspace{0.13cm} \Bigg\{ {} \bigg \{ \frac{\gamma_{2s}^2}{4}-\gamma_{2s} \Re\Big( \frac{\gamma_{+}}{2}+ \beta_+ \Big) \\
&+ \abs{ \frac{\gamma_{+}}{2}+ \beta_+ } ^2 \bigg \}\hspace{-0.03cm} e^{2\Re ( \beta_+ ) t} \hspace{-0.07cm} + \hspace{-0.08cm} \bigg \{\hspace{-0.07cm} \frac{\gamma_{2s}^2}{4}\hspace{-0.07cm} -\hspace{-0.05cm} \gamma_{2s} \hspace{-0.04cm} \Re\hspace{-0.03cm}\Big(\hspace{-0.03cm} \frac{\gamma_{+}}{2}\hspace{-0.07cm}-\hspace{-0.07cm} \beta_+ \hspace{-0.05cm} \Big) \\
& +\abs{ \frac{\gamma_{+}}{2}- \beta_+ } ^2 \bigg \} e^{-2\Re ( \beta_+ ) t} - \hspace{-0.02cm} \bigg \{ \frac{\gamma_{2s}^2}{4}-\gamma_{2s}\Re\Big( \frac{\gamma_{+}}{2} \Big) \\
& + \abs{\frac{\gamma_{+}}{2}}^2 -\abs{\beta_+}^2 \bigg \} \Big( e^{2i\Im ( \beta_+ ) t}+e^{-2i\Im ( \beta_+ ) t} \Big)\\
& - 2i {} \bigg \{ \Re\Big( \frac{\gamma_{+}}{2} \Big)\Im ( \beta_+ ) -\Im\Big( \frac{\gamma_{+}}{2} \Big)\Re\big( \beta_+ \big) \\
& -\frac{1}{2}\gamma_{2s}\Im\big( \beta_+ \big)\bigg \} \Big( e^{2i\Im ( \beta_+ ) t} - e^{-2i\Im ( \beta_+ ) t} \Big) \Bigg \},
\end{aligned}$$ and $$\label{modck2}
\begin{aligned}
|c_k(t)|^2 & = \frac{|V_{kj}|^2 e^{-\Re(\gamma_-)t}}{\hbar^2 \abs{2\beta_-}^2} \hspace{0.13cm} \Bigg\{ {} e^{2\Re ( \beta_- ) t} + \\
& + e^{-2\Re ( \beta_- ) t} -2 \cos \big( 2 \Im ( \beta_- ) t \big) \Bigg \}
\end{aligned}$$ where $\beta_{\pm}=\sqrt{\gamma^2_{\pm}/4-\omega_{0\pm
}^2}$.
Eq. (\[probtrabalhada\]), although seemingly complicated, falls for small values of the electric field, $|V_{kj}|^2 \ll \hbar^2 (\gamma_{2p}^2+4 \omega_{kj}^2)$, into the well-known expression [@Lamb50] $$\label{probapp1}
|c_j(t)|^2 \approx e^{-\gamma_{E}t},$$ where $\gamma_{E}=\gamma_{2p}|V_{kj}|^2/\big(\hbar^2 (\gamma_{2p}^2/4 + \omega_{kj}^2 )\big)$ is the resulting decay rate of the state $2S_{1/2}$ due to the action of an external electric field.
Decay for a resonant coupling
-----------------------------
When the energy difference between the two coupled states cancels, for a specific magnetic field intensity, a simpler interpretation of Eq. (\[probtrabalhada\]) is possible. With this condition, the parameters $\omega_{0+}$ and $\gamma_+$ turn real and can be interpreted as a frequency of oscillation and the damping constant, respectively, of a damped harmonic oscillator. Thus, $\gamma_{\pm}=\gamma=(\gamma_{2p}+\gamma_{2s})/2$ and $\omega_{0\pm}^2=\omega_0^2=|V_{kj}|^2/\hbar^2+(\gamma_{2s}\gamma_{2p})/4.$
Besides, Eq. (\[probtrabalhada\]) presents three different regimes depending on whether $\gamma ^2/4<\omega_0^2$, $\gamma ^2/4>\omega_0^2$ or $\gamma ^2/4=\omega_0^2$, corresponding to the under-damped, over-damped and critically damped regimes, respectively.
In the under-damped regime, we can write Eq. (\[probtrabalhada\]) as: $$\label{sub}
\begin{aligned}
|c_j(t)|^2= \bigg \{ & \frac{\bar{\gamma}^2}{4} + \omega^2 - \bigg ( \frac{\bar{\gamma}^2}{4} - \omega^2 \bigg )\cos(2\omega t)\\
& +\bar{\gamma} \omega \sin(2\omega t) \bigg \} \frac{e^{-\gamma t}}{2\omega^2},
\end{aligned}$$ where $\bar{\gamma}=(\gamma_{2p}-\gamma_{2s})/2$ and $\omega=\sqrt{\omega_{0
}^2-\gamma^2/4}$.
For a sufficiently strong damping, meeting the condition $\gamma ^2/4>\omega_0^2$, condition in which the decay happens without oscillations, we have that: $$\label{super}
\begin{aligned}
|c_j(t)|^2= \bigg \{ \Big( & \beta + \frac{\bar{\gamma}}{2}\Big )^2 e^{2\beta t} + \Big(\beta - \frac{\bar{\gamma}}{2}\Big )^2 e^{-2\beta t} \\
& + 2\beta^2 -\frac{1}{2} \bar{\gamma}^2 \bigg \} \frac{e^{-\gamma t}}{4\beta^2},
\end{aligned}$$ where $\beta=\sqrt{\gamma^2/4-\omega_{0
}^2}$.
In the critically damped regime, we can write: $$\label{critico}
|c_j(t)|^2= \bigg \{ 1 + \bar{\gamma} + \frac{1}{4}\bar{\gamma}^2 t^2 \bigg \} e^{-\gamma t}.$$
The three different regimes of decay can be achieved by varying the atom’s speed or with the addition of an external electric field.
Results and Discussion {#sec:fast}
======================
In this Section, we apply the previous analysis of the Zeeman energy levels to discuss the working principle of an atomic polarizer. We also analyse the Lyman radiation rate emitted by the selected atomic fragments. This rate exhibits a characteristic pattern.
The atomic polarizer
--------------------
We consider the decay rate of $2 S_{1/2}$ atoms (state $|6 \rangle =\ket{2S_{1/2},1,-1}$) propagating in a region of constant magnetic field with a velocity orthogonal to the field. In particular, we investigate the behaviour of this decay in the vicinity of the magnetic field $B_{3,6}=597 \: {\rm G}$ corresponding to the crossing between states $|3\rangle$ and $|6\rangle$. The probability of the atom being in the corresponding $2 S_{1/2}$ state as a function of time (given by Eq. (\[probtrabalhada\])) is presented on Figure \[Fig2\]. The relevant coupling in this region occurs between the states $|3\rangle$ and $|6\rangle$. Indeed, the couplings between $|6\rangle$ and $|4\rangle$ as well as $|6\rangle$ and $|2\rangle$ are forbidden by the selection rules. On the other hand, the coupling between $|6\rangle$ and $|1\rangle$ is strongly non-resonant and also suppressed by the asymptotic form of the eigenstate $|1 \rangle$ for strong magnetic field values (as detailed in Appendix \[apendC\]).
![a) Probability, $|c_{6}(t)|^2$, for a magnetic field intensity of $300 \: {\rm G}$ (dotted line), $400 \: {\rm G}$ (dashed line), $500 \: {\rm G}$ (dot-dashed line) and $B_{3,6}= 597 \: {\rm G}$ (solid line) for an atomic speed $v \: = \: 100 \: {\rm km/s}$. b) Probability $|c_{6}(t)|^2$ for the magnetic field $B_{3,6}$ and with the atomic velocities: 100 km/s (solid line) and 200 km/s (dashed line), corresponding to the underdamped regime; 17.7 km/s (dot-dashed), corresponding to the critically damped regime; and 8 km/s (dot-dot-dashed), corresponding to the overdamped regime. The dotted line shows the envelope $e^{-\frac{1}{2}(\gamma_{2s}+\gamma_{2p})t}$.[]{data-label="Fig2"}](Fig02.jpg)
Fig. \[Fig2\]a) shows that the decay depends strongly on the magnetic field intensity and is enhanced in the vicinity of the crossing value $B_{3,6}$. Fig. \[Fig2\]b) displays the time-dependent probability in the different regimes, Eqs.( \[sub\]-\[critico\]), at the crossing magnetic field $B_{3,6}$ for four atomic velocities, with $\vec{v} \perp \vec{B}$. In the underdamped regime, the probability of decay has an exponential envelope $e^{-\frac{1}{2}(\gamma_{2s}+\gamma_{2p})t}$. This can be interpreted as follows: the atom undergoes Rabi oscillations between states $\ket{6}$ and $\ket{3}$ and spends on average an equal time in both states. Since one of these states ($2S_{1/2}$) has a much smaller decay rate, the lifetime of the moving atom is approximately twice that of the $2P_{1/2}$ state. The maximum decay is obtained at the crossing magnetic field value $B_{3,6}$.
The discussion above can be extended to the state $\ket{5}$ (coupled to state $\ket{4}$). The remaining $2 S_{1/2}$ states $(\ket{7},\ket{8})$ do not exhibit energy crossings and are thus not resonantly coupled. Indeed, their lifetimes (less affected by the presence of the magnetic field) are orders of magnitude larger than that of the states $\ket{5}$ and $\ket{6}$. This fact enables one to build a device which quenches two of the four $2S_{1/2}$ Zeeman states, i.e., which filters out the states $\ket{5}$ and $\ket{6}$ and preserves the states $\ket{7}$ and $\ket{8}$, consisting of a region of uniform and constant magnetic field with appropriate length through which the atomic beam must travel, as further discussed in Appendix \[apendD\].
Detection signal with Lyman-$\alpha$ radiation and Criterion for testing a polarizing device
--------------------------------------------------------------------------------------------
Detection can be done by sensing the Lyman-$\alpha$ radiation emitted by the propagating fragments. Since the probability of emission of Lyman-$\alpha$ radiation is $10^8$ times higher when the atom is in a $2P_{1/2}$ state when compared to a $2S_{1/2}$ state, the radiation rate may be taken as proportional to the probability of the atom being in one of the $2P_{1/2}$ states: $$\label{Plyman2P}
R_{Lyman-\alpha} \propto \sum_{k=1}^{4} |c_k(t)|^2, $$ where the probabilities $|c_k(t)|^2$ are given by Eq.(\[modck2\]).
Let us consider a non-polarized H(2S) beam (with four equally populated $2S_{1/2}$ Zeeman states), propagating in the x-direction through a region with an orthogonal magnetic field yielding near-resonant energy levels $\ket{3},\ket{4},\ket{5},\ket{6}$.
![a) Joint Lyman-$\alpha$ emission originated by the quenching of the four $2S_{1/2}$ states when a H(2S) beam of velocity $\vec{v} = v \hat{x}$ with $v \: = \: 100 \: {\rm km/s}$ travels through a region with a magnetic field of $\vec{B}= B \hat{z}$ ($B= 565 G$). b) Joint Lyman-$\alpha$ emission originated by the quenching of states $\ket{7}$ and $\ket{8}$ in presence of an identical magnetic field and an additional external electric field $\vec{E}_{ext}=E_{ext} \hat{y}$ with $E_{ext}=600 \: \rm{V/cm}$.[]{data-label="Fig3"}](Fig03.jpg)
Figure \[Fig3\]a) shows the profile of a joint Lyman-$\alpha$ emission rate produced by the quenching of the four $2S_{1/2}$ Zeeman states. The dominant contribution to the Lyman-$\alpha$ radiation pattern comes from the coupling between the states $\ket{5}$ and $\ket{4}$, and $\ket{6}$ and $\ket{3}$, respectively, while the contribution from states $\ket{7}$ and $\ket{8}$ is negligible. For a magnetic field of 565 G we have an optimal value for a clear pattern, since the oscillations in the Lyman-$\alpha$ emission rate for the states that contribute the most get in phase, but significant variations in the magnetic field intensity still result in a similar pattern.
After roughly $t \simeq 20 \: {\rm ns}$ (corresponding to a path of a few mm at the considered velocity $v \: = \: 100 \: {\rm km/s}$), most atoms that were initially in the states $\ket{5}$ and $\ket{6}$ are in the ground state, and the resultant beam is almost completely composed by atoms in states $\ket{7}$ and $\ket{8}$, since the latter present a very low Lyman-$\alpha$ emission rate when compared to the former.
Although states $\ket{7}$ and $\ket{8}$ have a similar behaviour in the presence of external fields, regarding Lyman-$\alpha$ radiation emission, their simultaneous presence in the resulting beam drastically changes the radiation pattern in presence of an additional external electric field (fundamental to increase their decay rate). This can be seen in Figure \[Fig3\]b), which shows the Lyman radiation in presence of an additional external electric field for an atomic beam containing initially only equally populated $\ket{7},\ket{8}$ states. One notes an oscillatory pattern with a beat frequency $\Delta \omega =\big( \sqrt{\Delta \varepsilon_{81}^2+4|V_{81}|^2}-\sqrt{\Delta \varepsilon_{72}^2+4|V_{72}|^2} \big) / \hbar$, that can be calculated from $\vec{v}$, $\vec{B}$ and the external electric field $\vec{E}_{ext}$.
![a) Proposed setup for testing polarization with a fully working polarizer. A non-polarized H(2S) beam enters a region of constant magnetic field. In this regime, states $\ket{5}$ and $\ket{6}$ suffer quenching and the resulting Lyman-$\alpha$ radiation is detected by detector A. Upon leaving this first region, the beam is composed exclusively of states $\ket{7}$ and $\ket{8}$. It then travels to a second region with a constant magnetic field and an additional external electric field that quenches the remaining states. The Lyman-$\alpha$ radiation is then sensed by detector B. b) Testing setup with no polarizer. A non-polarized H(2S) beam travels through the apparatus until it gets to a region of constant electric and magnetic fields. Those fields cause intense quenching of the four hyperfine structure states and the resulting Lyman-$\alpha$ radiation is detected by detector B. The magnetic field is only present in two specific regions for illustrative purposes, it may remain constant throughout the setup.[]{data-label="Fig4"}](Fig04.jpg)
In theory, a polarizer consisting of a region of approximately constant magnetic field with adequate intensity and extension should be able to filter two of the four states of hyperfine structure, as discussed. However, inherent limitations in the apparatus construction, such as resulting inhomogeneities of the external magnetic field, can compromise the efficiency of the device. The study presented here leads to a simple and robust criterion for testing a polarizer after its construction and verify if the desired polarization has been achieved. This technique could be used prior to inserting the polarizing device into a more complex apparatus.
Figure \[Fig4\] shows a sketch of a setup for that verification. Fig. \[Fig4\]a) shows an initially non-polarized beam containing equal parts of the four hyperfine structure states. The beam then enters a region of constant magnetic field that causes quenching of two of the four states preferentially. The two surviving states then enter a region with an additional external electric field strong enough to cause a decay within a few millimetres of propagation, corresponding to a few nanoseconds. Detectors A and B should capture spectrums like the ones indicated in Fig. \[Fig4\]a). In the absence of a polarizer, the detection patterns would be as shown in Fig. \[Fig4\]b).
The pattern obtained by detector A in Fig. \[Fig4\]a) can be seen as an indication of a successful polarization process. However, the pattern from detector B is much more robust, as it is little affected by the atomic speed. See Appendix \[apendD\]. Indeed, the quenching of the two remaining states relies mostly on the external electric field, and not on the motional electric field as previously. Nevertheless, the speed still influences the detection time of the radiation, but this effect can be minimized by shortening the path and by using an atomic beam of narrow velocity distribution. The beating pattern does not depend on the specific polarizing process, as long as it delivers a beam with similar proportions of states $\ket{7}$ and $\ket{8}$.
Conclusion {#Conc}
==========
In this work, we have discussed how a tuning of the metastable hydrogen energy levels by a magnetic field can enable the realization of a two-state polarizer. This is apparent from the expressions of the time-dependent probability amplitudes for the $2S_{1/2}$ and $2P_{1/2}$ states. We have particularly investigated the effects of near-resonant coupling of the metastable states dressed by the magnetic field. We have also analysed the Lyman-$\alpha$ radiation rate, used in most detection schemes, for typical quenching conditions. The simultaneous presence of two $2S_{1/2}$ states after the polarization leads to a radiation pattern with beat notes that can be seen as an indication of successful polarization. We have established a simple and robust criterion for checking the effectiveness of a magnetic polarizer in atom-interferometers setup. Perspectives for this work include the design of specific electric and magnetic field geometries enabling the hydrogen polarization along a single atomic state instead of a two-state multiplicity.
acknowledgments
===============
This work was partially supported by the Brazilian agencies CNPq, CAPES, and FAPERJ. It is part of the INCT-IQ from CNPq.
Decomposition of $\ket{2L_J,F,M_F}$ in $\ket{L,M_L}\ket{M_e}\ket{M_p}$ {#apendA}
======================================================================
As usual, the decompositions of the states under study in terms of orbital angular momentum, electron spin, and nucleus spin are performed, for $L$ and $J$ fixed, as follows: $$\ket{F,M_F}=\sum_{M_L,M_e,M_p} C_{L,e,p} \ket{L,M_L}\ket{M_e}\ket{M_p},$$ where $M_e$ and $M_p$ are the projections of the electron and nucleus spin in the z axis, respectively, and can assume the values $\pm \hbar/2$, here represented as $\ket{\uparrow}$ and $\ket{\downarrow}$. The $C_{L,e,p}$ are Clebsch-Gordan coefficients. The results of the decompositions are the following: [$$\begin{aligned}
&\ket{2S_{\frac{1}{2}},0,0}=-\frac{1}{\sqrt{2}}\ket{0,0}_L\hspace{-0.04cm}\ket{\uparrow}_e\hspace{-0.03cm}\ket{\downarrow}_p\hspace{-0.07cm}+\hspace{-0.06cm}\frac{1}{\sqrt{2}}\ket{0,0}_L\hspace{-0.03cm}\ket{\downarrow}_e\hspace{-0.03cm}\ket{\uparrow}_p \\
&\ket{2S_{\frac{1}{2}},1,-1}= \ket{0,0}_L\ket{\downarrow}_e\ket{\downarrow}_p \\
&\ket{2S_{\frac{1}{2}},1,0}= \frac{1}{\sqrt{2}}\ket{0,0}_L\ket{\uparrow}_e\ket{\downarrow}_p+\frac{1}{\sqrt{2}}\ket{0,0}_L\ket{\downarrow}_e\ket{\uparrow}_p \\
&\ket{2S_{\frac{1}{2}},1,1}= \ket{0,0}_L\ket{\uparrow}_e\ket{\uparrow}_p \\
&\ket{2P_{\frac{1}{2}},0,0}=\frac{1}{\sqrt{3}}\ket{1,1}_L\ket{\downarrow}_e\ket{\downarrow}_p-\frac{1}{\sqrt{6}}\ket{1,0}_L\ket{\uparrow}_e\ket{\downarrow}_p \\
& \hspace{1.37cm} -\frac{1}{\sqrt{6}}\ket{1,0}_L\ket{\downarrow}_e\ket{\uparrow}_p+\frac{1}{\sqrt{3}}\ket{1,-1}_L\ket{\uparrow}_e\ket{\uparrow}_p \\
&\ket{2P_{\frac{1}{2}},1,-1}\hspace{-0.06cm}=\hspace{-0.07cm}\sqrt{\frac{2}{3}}\hspace{-0.05cm}\ket{1,-1}_L\hspace{-0.05cm}\ket{\uparrow}_e\hspace{-0.05cm}\ket{\downarrow}_p \hspace{-0.07cm}-\hspace{-0.05cm}\frac{1}{\sqrt{3}}\hspace{-0.05cm}\ket{1,0}_L\hspace{-0.05cm}\ket{\downarrow}_e\hspace{-0.05cm}\ket{\downarrow}_p\\
&\ket{2P_{\frac{1}{2}},1,0}\hspace{-0.05cm}=\hspace{-0.06cm}-\frac{1}{\sqrt{3}}\ket{1,1}_L\ket{\downarrow}_e\ket{\downarrow}_p\hspace{-0.06cm}+\hspace{-0.06cm}\frac{1}{\sqrt{6}}\ket{1,0}_L\ket{\uparrow}_e\ket{\downarrow}_p \\
& \hspace{1.37cm} -\frac{1}{\sqrt{6}}\ket{1,0}_L\ket{\downarrow}_e\ket{\uparrow}_p+\frac{1}{\sqrt{3}}\ket{1,-1}_L\ket{\uparrow}_e\ket{\uparrow}_p \\
&\ket{2P_{\frac{1}{2}},1,1}\hspace{-0.05cm}=\hspace{-0.06cm}-\sqrt{\frac{2}{3}}\ket{1,1}_L\hspace{-0.03cm}\ket{\downarrow}_e\ket{\uparrow}_p\hspace{-0.06cm}+\hspace{-0.06cm}\frac{1}{\sqrt{3}}\ket{1,0}_L\hspace{-0.03cm}\ket{\uparrow}_e\ket{\uparrow}_p
\end{aligned}$$]{}
Hamiltonian eigenvalues {#apendB}
=======================
The equations bellow express the energy eigenvalues of the $2S_{1/2}$ and $2P_{1/2}$ Zeeman states.
$$\label{h1}
h_1=-\frac{A_2}{4}-\frac{1}{6} \sqrt{ 9A_2^2 +\big( 4-g_e+3g_p \big)^2\mu_B^2B^2 }$$
$$\label{h2}
h_2=\frac{A_2}{4}-\frac{1}{6}(4-g_e-3g_p)\mu_BB$$
$$\label{h3}
h_3=-\frac{A_2}{4}+\frac{1}{6} \sqrt{ 9A_2^2 +\big( 4-g_e+3g_p \big)^2\mu_B^2B^2 }$$
$$\label{h4}
h_4=\frac{A_2}{4}+\frac{1}{6}(4-g_e-3g_p)\mu_BB$$
$$\label{h5}
h_5=-\frac{A_1}{4}+L-\frac{1}{2}\sqrt{A_1^2+(g_e+g_p)^2\mu_B^2B^2}$$
$$\label{h6}
h_6=\frac{A_1}{4}+L-\frac{1}{2}(g_e-g_p)\mu_BB$$
$$\label{h7}
h_7=-\frac{A_1}{4}+L+\frac{1}{2}\sqrt{A_1^2+(g_e+g_p)^2\mu_B^2B^2}$$
$$\label{h8}
h_8=\frac{A_1}{4}+L+\frac{1}{2}(g_e-g_p)\mu_BB$$
Coupling between the states $\ket{k}$ and $\ket{j}$ {#apendC}
===================================================
The states $\ket{j}$ and $\ket{k}$ can be written in terms of the states $\ket{2S_{1/2},F,M_F}$ and $\ket{2P_{1/2},F,M_F}$, respectively, as shown by Eqs. (\[1\]-\[8\]). Therefore, to determine the coupling between states $\ket{j}$ and $\ket{k}$ it is worth analysing the matrix elements $\bra{2P_{1/2},F,M_F}e \vec{E} \cdot \vec{r}\ket{2S_{1/2},F,M_F}$.
Considering the electric field in the $y$-direction in our particular case of study (the velocity of the atoms was considered to be in the $x$-direction and the magnetic field in the $z$-direction), the matrix elements of interest can be calculated. The non-zero matrix elements obtained were: $$\label{v16}
\bra{2P_{\frac{1}{2}},0,0}e \vec{E} \cdot \vec{r}\ket{2S_{\frac{1}{2}},1,-1}=-\frac{\sqrt{6}}{2}a_0ieE$$ $$\label{v36}
\bra{2P_{\frac{1}{2}},1,0}e \vec{E} \cdot \vec{r}\ket{2S_{\frac{1}{2}},1,-1}=\frac{\sqrt{6}}{2}a_0ieE$$ $$\label{v25}
\bra{2P_{\frac{1}{2}},1,-1}e \vec{E} \cdot \vec{r}\ket{2S_{\frac{1}{2}},0,0}=\frac{\sqrt{6}}{2}a_0ieE$$ $$\label{v27}
\bra{2P_{\frac{1}{2}},1,-1}e \vec{E} \cdot \vec{r}\ket{2S_{\frac{1}{2}},1,0}=-\frac{\sqrt{6}}{2}a_0ieE$$ $$\label{v45}
\bra{2P_{\frac{1}{2}},1,1}e \vec{E} \cdot \vec{r}\ket{2S_{\frac{1}{2}},0,0}=\frac{\sqrt{6}}{2}a_0ieE$$ $$\label{v47}
\bra{2P_{\frac{1}{2}},1,1}e \vec{E} \cdot \vec{r}\ket{2S_{\frac{1}{2}},1,0}=\frac{\sqrt{6}}{2}a_0ieE$$ $$\label{v18}
\bra{2P_{\frac{1}{2}},0,0}e \vec{E} \cdot \vec{r}\ket{2S_{\frac{1}{2}},1,1}=-\frac{\sqrt{6}}{2}a_0ieE$$ $$\label{v38}
\bra{2P_{\frac{1}{2}},1,0}e \vec{E} \cdot \vec{r}\ket{2S_{\frac{1}{2}},1,1}=-\frac{\sqrt{6}}{2}a_0ieE$$
From those results and from Eqs. (\[1\]-\[8\]) it is possible to see that the state $\ket{6}$ only couples with states $\ket{1}$ and $\ket{3}$. To calculate $\bra{1}e \vec{E} \cdot \vec{r}\ket{6}$ and $\bra{3}e \vec{E} \cdot \vec{r}\ket{6}$, for example, we do: [$$\begin{aligned}
\bra{1}e \vec{E} \cdot \vec{r}\ket{6} &= {} m_1(B)\bra{2P_{\frac{1}{2}},0,0}e \vec{E} \cdot \vec{r}\ket{2S_{\frac{1}{2}},1,-1} \\
&+ n_1(B)\bra{2P_{\frac{1}{2}},1,0}e \vec{E} \cdot \vec{r}\ket{2S_{\frac{1}{2}},1,-1} \\
&= (n_1-m_1)\frac{\sqrt{6}}{2}a_0ieE
\end{aligned}$$ ]{} [and $$\begin{aligned}
\bra{3}e \vec{E} \cdot \vec{r}\ket{6} &= {} m_3(B)\bra{2P_{\frac{1}{2}},0,0}e \vec{E} \cdot \vec{r}\ket{2S_{\frac{1}{2}},1,-1} \\
&+ n_3(B)\bra{2P_{\frac{1}{2}},1,0}e \vec{E} \cdot \vec{r}\ket{2S_{\frac{1}{2}},1,-1}\\
&= (n_3-m_3)\frac{\sqrt{6}}{2}a_0ieE
\end{aligned}$$ ]{}
Although the analysis of how the probability of the atom being in state $\ket{6}$ changes with time seems to depend on its interaction with states $\ket{1}$ and $\ket{3}$, the interaction of state $\ket{6}$ with state $\ket{1}$ has very little effect on its lifetime. That’s due to two main reasons. The first one is that state $\ket{3}$ is closer to state $\ket{6}$, concerning to energy, for magnetic field intensities lower than 800 G and, therefore, has stronger influence in its lifetime. But even more important is the fact that $|V_{16}|^2=(n_1-m_1)^23a_{0}^2e^2E^2/2$ is negligible when compared to $|V_{36}|^2=(n_3-m_3)^23a_{0}^2e^2E^2/2$ for most magnetic field intensities. That happens because $(n_1-m_1)^2$ goes fast to zero with the increase of the magnetic field intensity while $(n_3-m_3)^2$ goes to two, as shown in Figs. \[Fig5\] and \[Fig6\].
![The dashed line represents the coefficient $m_1(B)$, the dotted line represents the coefficient $n_1(B)$ and the solid line represents $|n_1(B)-m_1(B)|^2$.[]{data-label="Fig5"}](Fig05.jpg)
![The dashed line represents the coefficient $m_3(B)$, the dotted line represents the coefficient $n_3(B)$ and the solid line represents $|n_3(B)-m_3(B)|^2$.[]{data-label="Fig6"}](Fig06.jpg)
In order to plot $|c_6(t)|^2$, Figs. \[Fig2\]-\[Fig4\], we have calculated $\gamma_{+}=(\gamma_{2s}+\gamma_{2p})/2 \pm i\omega_{36}$ and $\omega_{0_+}^2= |V_{36}|^2/\hbar^2+\gamma_{2s}\gamma_{2p}/4 + i\gamma_{2s}\omega_{36}/2$, as mentioned previously.
Determining $\omega_{36}$ is even simpler than determining $|V_{36}|^2$, since we already know how the energies of the states of interest vary with the magnetic field.
$$\omega_{36}(B)=\frac{h_3(B)-h_6(B)}{\hbar}$$
Similarly to what was done for state $\ket{6}$, we can study the probability of the atom being in any of the Zeeman states by considering its coupling, by the electric field, with only one other state. Each state $\ket{j}$ couples with only two states $\ket{k}$ and one of these coupling is much stronger than the other.
The determination of $|V_{kj}|^2$ and $\omega_{kj}$ are all what is necessary, taking that we already know $\gamma_{2s}$ and $\gamma_{2p}$, to calculate $\gamma_{\pm}$ and $\omega_{0_{\pm}}^2$ and, therefore, to study the probability of the atom being in any of the states (\[1\]-\[8\]), given that it occupies the corresponding $2S_{1/2}$ Zeeman state at $t=0$.
Effects of magnetic field intensity and atomic velocity {#apendD}
=======================================================
As mentioned in Section \[sec:fast\], the magnetic field of the polarizer is capable of preserving states $\ket{7}$ and $\ket{8}$ while completely quenching states $\ket{5}$ and $\ket{6}$. This is true for a wide range of magnetic field intensities as can be seen in Fig. \[Fig7\]. We can see that the desired filtering is possible for all considered magnetic field intensities.
![a) Represents the Lyman-$\alpha$ radiation rate of an H(2S) beam travelling through a magnetic field region, with velocity 100 km/s perpendicular to the field, for five different values of magnetic field intensity: 460 G, 495 G, 530 G, 565 G, 600 G. The dotted and dashed lines represent the contribution from atoms originally in state $\ket{5}$ and $\ket{6}$, respectively. The solid line represents the Lyman-$\alpha$ radiation rate from all four states. Contributions from states $\ket{7}$ and $\ket{8}$ are not represented here due to their negligible influence on the total emission. b) Represents the populations of states $\ket{5}$, $\ket{6}$, $\ket{7}$ and $\ket{8}$ for the same atomic velocity and magnetic field intensities represented in a). The distinction between the populations of states $\ket{7}$ and $\ket{8}$ are not intended to be perceived from the image, it is only important to observe that they suffer little loss.[]{data-label="Fig7"}](Fig07.jpg)
Fig. \[Fig7\]a) allows us to see that, as mentioned, the Lyman-$\alpha$ emissions from states $\ket{5}$ and $\ket{6}$ get in phase for the magnetic field of 565 G allowing a clearer pattern. Fig. \[Fig7\]b) shows how the populations of the $\ket{j}$ states change with time for different magnetic field intensities, corresponding to the ones in Fig. \[Fig7\]a). We have chosen to represent the populations up to 40 ns (corresponding to 4 mm for the chosen atomic speed) because it better represents the selective filtering of the polarizer and helps distinguish between a low rate of Lyman-$\alpha$ emission and a low associated population. From Fig. \[Fig7\]b) we can notice that the decrease in the population happens in a significantly different rate for states $\ket{7}$ and $\ket{8}$ than it does for states $\ket{5}$ and $\ket{6}$. This difference is what allows the polarizer to work. The mentioned difference in decay rate happens for a wide range of magnetic field intensities. For a few centimetres long polarizer it is possible to use a much weaker magnetic field. However, as the field gets weaker, the polarization gets less efficient, meaning that less of states $\ket{7}$ and $\ket{8}$ remain when states $\ket{5}$ and $\ket{6}$ are completely extinct. It is important to adjust adequately the magnetic field intensity in the polarizer and its length so that we guarantee that states $\ket{5}$ and $\ket{6}$ are completely quenched. That adjustment must be done considering experimental convenience.
![a) Lyman-$\alpha$ radiation rate from the polarizer, with a perpendicular magnetic field of 565 G, for four atomic velocities: 10 km/s, 40 km/s, 70 km/s and 100 km/s. b) Lyman-$\alpha$ radiation rate from quenching of states $\ket{7}$ and $\ket{8}$ in the presence of a magnetic field of 565 G and an external electric field of 600 V/m for the velocities: 10 km/s, 40 km/s, 70 km/s and 100 km/s. Here, $\vec{v}$, $\vec{B}$ and $\vec{E}_{ext}$ are perpendicular to each other.[]{data-label="Fig8"}](Fig08.jpg)
The polarizer and its testing criterion are also robust to variations in the atomic velocity. As mentioned previously, the pattern from detector B in Fig. \[Fig4\]a) is much more robust to changes in atomic velocity than the patten from detector A. That difference can be noticed from Fig. \[Fig8\]. However, the polarizer’s operation is robust to wide variations in beam velocity, including a wide velocity distributions, despite the pattern from detector A being easily lost.
The effects of magnetic field intensity and velocity changes shown in Fig. \[Fig7\]a) and Fig. \[Fig8\]a) are somewhat related but not equivalent. Although they both influence the electric field in the atom frame, the magnetic field has an important effect on states energies that ultimately allows the filtering.
Fig. \[Fig8\]a) shows, for atomic velocities varying from 10 km/s to 100 km/s, the Lyman-$\alpha$ radiation rate emitted from a non-polarized H(2S) beam propagating perpendicular to a magnetic field of 565 G. The patter of emission changes with atomic velocity, but the desired polarization should occur if the beam propagates long enough through the field region. From Fig. \[Fig8\]b), we see that the proposed pattern for validating the polarizer’s effectiveness, for $\vec{E}_{ext}$ = 600 V/m, remains almost unchanged for atomic velocities varying from 10 km/s to 100 km/s. This means it can be used to validate the polarizer’s effectiveness for a wide range of atomic velocities. However, this pattern may not be observed, even for a successful polarization, if the beam either has a wide velocity distributions or has to travel a long way to the detection region, because those two factors affect the moment of detection.
From the understanding of how atomic velocities and field intensities affect the lifetimes of our states of interest, we can determine the relations between atomic velocity, magnetic field intensity and length of the field region required for a two-state polarizer as well as a criterion for validating its effectiveness.
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§[[**S**]{}]{}
PS. \#1[[**\#1**]{}]{} \#1\#2\#3[Phys. Lett. [**B**]{} ]{} \#1\#2\#3[Nucl. Phys. [**B**]{} ]{} \#1[Comm. Math. Phys. ]{} \#1[Phys. Lett. ]{}
\#1[Phys. Rev. ]{} \#1[Phys. Rev. Lett. ]{} \#1[Proc. Roc. Soc. ]{} \#1[Prog. Theo. Phys. ]{} \#1[Sov. J. Nucl. Phys. ]{} \#1[Theor. Math. Phys. ]{} \#1[Annals of Phys. ]{} \#1[Proc. Natl. Acad. Sci. USA ]{}
**[Branes at Orbifolded Conifold Singularities]{}**
**[and Supersymmetric Gauge Field Theories]{}**
Kyungho Oh$^{a}$ and Radu Tatar$^{b}$
$^a$ Dept. of Mathematics, University of Missouri-St. Louis, St. Louis, MO 63121, USA
[oh@math.umsl.edu]{}
$^b$ Dept. of Physics, Brown University, Providence, RI 02912, USA
tatar@het.brown.edu
Abstract
We consider D3 branes at orbifolded conifold singularities which are not quotient singularities. We use toric geometry and gauged linear sigma model to study the moduli space of the gauge theories on the D3 branes. We find that topologically distinct phases are related by a flop transition. It is also shown that an orbifold singularity can occur in some phases if we give expectation values to some of the chiral fields.
Introduction
============
Last years have witnessed great insights into understanding of supersymmetric gauge theory and supergravity theory. We now found that these are complementary descriptions of a single theory on solitonic brane solutions of M theory and string theory. Configurations containing NS fivebranes and D branes in string theory are tools for studying supersymmetric gauge field theory in various dimensions with different supersymmetries (see [@gku] for a complete set of references up to February 1998).
On the other hand, Maldacena’s conjecture proposes that M or string theory on the $AdS_{p} \times S^k$, with N units of supergravity $k$-form field through $S^{k}$ is dual to a $p - 1$ specific conformal field on the boundary of the $AdS_{p}$ space [@mal] (see [@mal1] for an extensive review and a complete set of references). The initial proposal gave conformal field theories with maximal supersymmetry, ${\cal N} = 4$ in four dimensions. This was obtained by studying D3-branes in flat space. An immediate generalization to D3-branes at orbifold singularities breaks more supersymmetry [@kac; @law].
Another important class is obtained by studying D3-branes at non-orbifold singularities like conifold singularity. The conifold singularity has been analyzed in [@kw] where an infrared theory on the worldvolume of D3 branes was proposed. Other results for the case of non-orbifold singularities and their connection to field theories in three and four dimensions have been obtained in [@ura; @dm; @afhs; @gr1; @mr; @ot; @bg; @lopez; @unge; @karch; @kw1; @ahn; @gk; @agata; @dia; @gns].
In [@ura; @dm], the authors have exploited the fact that the conifold singularity is dual to a system of perpendicular NS5 fivebranes intersecting over a 3+1 dimensional world-volume. Their result was generalized in [@unge; @karch] for more general conifolds. A duality between D3 branes on these general conifolds and configurations of NS and D4 branes was proposed together with relation between different resolutions of the singularity and displacements of NS branes. In [@karch], a mirror symmetry was proposed between orbifolded conifolds and generalized conifolds.
In the present paper we consider branes at orbifolded conifolds $\CC_{k l}$ which is an orbifold of the three dimensional conifold $xy-uv=0$ by a discrete group $\Z_k \times \Z_l$. We show that the Higgs branch of the moduli space of the gauge theory is the resolved or (partially) resolved conifold singularity, depending on the values of the FI parameters as holomorphic quotients. The moduli spaces for ${\cal N} = 2$ theories has been interpreted in terms of symplectic quotients in a linear sigma model approach in [@witten], and in terms of holomorphic quotients in the mathematical approach in [@agm]. In [@dgm], the latter approach has been extended to ${\cal N} = 1$ theories utilizing some ideas from [@agm; @dmo]. We also use toric geometry to study in detail the correspondence between D3 branes at orbifolded conifolds and brane configurations obtained after T-dualities (For details on toric geometry see [@agm]). In [@muto; @gre; @mra], D-branes on various other singularities have been studied in the lines of [@dgm]. The paper of [@unge] dealt with generalized conifolds $\CG_{k l}: x y = u^k v^l$.
The content of the paper is as follows: in section 2 we give a toric description of the quotient singularity of the three dimensional conifold. In section 3 we review relevant field theory and brane configurations. We argue why brane box model is more suitable for orbifolded conifolds. In section 4 we derive toric data for the simplest orbifolded conifold $\CC_{2,2}$. In section 5 we derive toric data for the orbifolded conifold $\CC_{2, 3}$. In section 6 we describe different phases of the vacuum moduli space.
Toric Geometry of Orbifolded Conifolds
======================================
In this section, we will briefly review toric singularity and its physical realization by the moduli space of the D-brane world-volume gauge theory on it via gauged linear sigma models to fix notations and terminologies. For detailed review, we refer to [@dgm]. A toric variety is a space which contains algebraic torus ${(\C^*)}^d$ as an open dense subset. For example, a projective space ${\bf P}^d = (\C^{d+1} -\{0\})/\C^*$ is a toric variety because it contains $(\C^*)^d \cong (\C^*)^{d+1}/\C^* \subset (\C^{d+1} -\{0\})/\C^*$. As in the case of the projective space, we will express our toric varieties as a quotient space (this can be thought of as a holomorphic quotient in the sense of the Geometric Invariant Theory [@mfk] or as a symplectic reduction as in gauged linear sigma model. In our cases, these two will be the same [@kir].): V\_= (\^q - F\_)//(\^\*)\^[q-d]{} where $q, F_\Delta$ and the action of $(\C^*)^{q-d}$ on $C^q$ are determined by a combinatorial data $\Delta$. Now we give a description of the combinatorial data $\Delta$ for Gorenstein canonical singularity (i.e. a singularity with a trivial canonical class, $K$). Consider vectors $v_1, \ldots , v_q$ in a lattice $\N = \Z^d \subset
\N_{\R}=\N \otimes \R =\R^d$ in general position. We introduce the corresponding homogeneous coordinates $x_i$ for of $\C^q- F_\Delta$ in the holomorphic quotients. In gauged linear sigma model, these correspond to matter multiplets. There will be $(q-d)$ independent relations \_[i=1]{}\^q Q\_i\^[(a)]{}v\_i = 0, a=1, …, q-d. Here $Q^{(a)}$’s correspond to the charges of the matter fields under $U(1)^{q-d}$ which is the maximal compact subgroup of $(\C^*)^{q-d}$. The D-term equations will be \_[i=1]{}\^[q]{} Q\^[(a)]{}\_i |x\_i|\^2 = r\_a, a =1, …, q-d. In the holomorphic quotient, the charge matrix whose column vectors consist of $Q^{(a)}$ determines the action of $(\C^*)^{q-d}$ on $\C^q$ i.e. the action of $(\lambda_1, \lambda_2, \ldots , \lambda_{q-d}) \in (\C^*)^{q-d}$ on $(x_1, \ldots , x_q) \in \C^q$ is given by (\_1\^[Q\_1\^[(1)]{}]{} \_2\^[Q\_1\^[(2)]{}]{} \_[q-d]{}\^[Q\_1\^[(q-d)]{}]{}x\_1, \_1\^[Q\_2\^[(1)]{}]{} \_2\^[Q\_2\^[(2)]{}]{} \_[q-d]{}\^[Q\_2\^[(q-d)]{}]{}x\_2, …, \_1\^[Q\_q\^[(1)]{}]{} \_2\^[Q\_q\^[(2)]{}]{} \_[q-d]{}\^[Q\_q\^[(q-d)]{}]{}x\_q) Here the action can be carried out as written or in two steps, an $(\R_+)^{q-d}$ action and a $U(1)^{q-d}$ action if Kähler. The quotient will depend on the gauge fixing determined by the $(\R_+)^{q-d}$ action i.e. the moment map. In the holomorphic approach, this corresponds to different spaces $F_\Delta$ which give rise to (partial) resolutions of the original space $V_\Delta$. In toric diagram, this corresponds to different triangulations of a convex cone in $\R^d$ determined by $\{ v_1, \ldots , v_q\}$. The collection of these combinatorial data is denoted by $\Delta$ called a fan. The quotient space $V_\Delta$ will have Gorenstein canonical singularity if there exists $u \in \Z^d$ such that $u \cdot v_i =1$ for all $i$ [@reid]. Thus $v_i$’s will lie on the hyperplane with normal $u$ at a distance $1/\|u \|$ from the origin in $\R^d$. This imposes the following condition on the charge vectors $Q^{(a)}$: \_[i=1]{}\^q Q\^[(a)]{}\_i = 0, a=1, …, q-d
To put our discussions in the language of the gauged linear sigma model, recall that $\C^q$ is a symplectic manifold with the standard symplectic form $\omega = i\sum_{i=1}^q dz^i \wedge d\bar{z}^{\bar{i}}$. The maximal compact subgroup $G:=U(1)^{q-d}$ of $(\C^*)^{q-d}$ acts covariantly on a symplectic manifold $(\C^q, \omega)$ by symplectomorphisms. The infinitesimal action will give rise to a moment map $\mu: \C^q \to
{\bf g}^*$ by Poisson brackets. In coordinates, the components of $\mu : \C^q \to \R^{q-d}$ are given by \[moment\] \_a = \_[i=1]{}\^q Q\^[(a)]{}\_i |x\_i|\^2 - r\_a where $r_a$ are undetermined additive constants. The symplectic reduction is then defined as V(r) \^[-1]{}(0)/G. The structure of $V(r)$ will depend on $r$. It follows from (\[moment\]) that every $(\C^*)^{q-d}$-orbit in $\C^q$ will contribute at most one point to $V(r)$. The value of $r$ will determine the contributing orbits. For a fixed $r$, the set of $(\C^*)^{q-d}$-orbits which do not contribute is precisely $F_\Delta$. The quotient space $V(r)$ carries a symplectic form $\omega_r$ by reducing $\omega$. The symplectic reduction carries a natural complex structure, in which the reduced symplectic form becomes a Kähler form.
Now we will consider quotient singularities of the conifold (i.e. orbifolded conifold). The conifold is a three dimensional hypersurface singularity in $\C^4$ defined by: : xy -uv = 0. The conifold can be realized as a holomorphic quotient of $\C^4$ by the $\C^*$ action given by [@witten; @kw] (A\_1, A\_2,B\_1, B\_2)(A\_1, A\_2,\^[-1]{} B\_1, \^[-1]{} B\_2)\^\*. Thus the charge matrix is the transpose of $Q^{'}
=(1,1,-1,-1)$ and $\Delta=\sigma$ will be a convex polyhedral cone in $\N^{'}_{\R}=\R^3$ generated by $v_1, v_2, v_3, v_4 \in \N^{'}=\Z^3$ where v\_1=(1,0,0), v\_2=(0,1,0),v\_3=(0,0,1),v\_4=(1,1,-1). The isomorphism between the conifold ${\cal C}$ and the holomorphic quotient is given by \[act\] x=A\_1B\_1, y=A\_2B\_2, u=A\_1B\_2, v=A\_2B\_1. We take a further quotient of the conifold ${\cal C}$ by a discrete group $\Z_k \times \Z_l$. Here $\Z_k$ acts on $A_i, B_j$ by \[zk\] (A\_1, A\_2, B\_1, B\_2) (e\^[-2i/k]{} A\_1, A\_2, e\^[2i/k]{}B\_1, B\_2), and $\Z_l$ acts by \[zl\] (A\_1, A\_2, B\_1, B\_2) (e\^[-2i/l]{} A\_1, A\_2, B\_1, e\^[2i/l]{}B\_2). Thus they will act on the conifold ${\cal C}$ by \[xy\] (x,y,u,v) (x,y,e\^[-2i/k]{}u, e\^[2i/k]{}v) and \[uv\] (x,y,u,v) (e\^[-2i/l]{}x,e\^[2i/l]{}y, u, v). Its quotient is called the hyper-quotient of the conifold or the orbifolded conifold and denoted by ${\cal C}_{kl}$. To put the actions (\[act\]), (\[zk\]) and (\[zl\]) on an equal footing, consider the over-lattice $\N$: = \^[’]{} + (v\_3-v\_1) + (v\_4 -v\_1). Now the lattice points $\sigma \cap \N$ of $\sigma$ in $\N$ is generated by $(k+1)(l+1)$ lattice points as a semigroup (These lattice points will be referred as a toric diagram.). The charge matrix $Q$ will be $(k+1)(l+1)$ by $(k+1)(l+1)-3$. The discrete group $\Z_k \times \Z_l \cong
\N / \N^{'}$ will act on the conifold $\C^4 // U(1)$ and its quotient will be the symplectic reduction $\C^{(k+1)(l+1)} // U(1)^{(k+1)(l+1)-3}$ with the moment map associated with the charge matrix $Q$. The new toric diagram for ${\cal C}_{kl}$ will also lie on the plane at a distance $1/\sqrt{3}$ from the origin with a normal vector $(1,1,1)$ and we draw a toric diagram on the plane for ${\cal C}_{57}$:
\#1\#2\#3\#4\#5[ @font ]{}
(1000,3705)(2000,-4897) (3601,-2161) (3601,-2761) (4201,-1561) (3601,-3361) (3601,-3961) (4801,-1561) (5401,-1561) (6001,-1561) (6601,-1561) (7201,-1561) (7201,-2161) (7201,-2761) (7201,-3361) (7201,-3961) (6601,-3961) (6001,-3961) (5401,-3961) (4801,-3961) (4201,-3961) (4201,-3361) (4801,-3361) (5401,-3361) (6001,-3361) (6601,-3361) (6601,-2761) (6601,-2161) (6001,-2161) (6001,-2761) (5401,-2161) (5401,-2761) (4801,-2161) (4801,-2761) (4201,-2161) (4201,-2761) (7801,-1561) (7801,-2161) (7801,-2761) (7801,-3361) (7801,-3961) (3601,-1561) (3601,-4561) (4201,-4561) (4801,-4561) (5401,-4561) (6001,-4561) (6601,-4561) (7201,-4561) (7801,-4561) (3451,-1336)[$v_1$]{} (7726,-1336)[$v_4$]{} (3451,-4861)[$v_3$]{} (7726,-4861)[$v_2$]{} (3601,-1561)[( 0,-1)[3000]{}]{} (3601,-4561)[( 1, 0)[4200]{}]{} (7801,-4561)[( 0, 1)[3000]{}]{} (7801,-1561)[(-1, 0)[4200]{}]{}
The action $(\ref{xy}), (\ref{uv})$ of $\Z_k \times \Z_l$ on the conifold ${\cal C}$ can be lifted to an action on $\C^4$ whose coordinates are $x,y, u, v$. The ring of invariants will be $\C[x^l, y^l, xy, u^k, v^k, uv]$ and the orbifolded conifold ${\cal C}_{kl}$ will be defined by the ideal $(xy-uv)\C[x^l, y^l, xy, u^k, v^k, uv]$. Thus after renaming variables, the defining equation for the orbifolded conifold will be \[con-eqn\] [C]{}\_[kl]{}: xy =z\^l, uv =z\^k.
Branes at the singularities and Gauge Theory
============================================
We now put branes to probe the geometric background space. Consider a system of $M$ D3 branes sitting at the orbifolded conifold $\CC_{kl}$ in the transversal direction. In the spirit of [@law], the corresponding gauge field theory on the world-volume of $D3$ has been obtained in [@ura] by investigating the action of the discrete group on the field theory of the conifold developed in [@kw].
The discrete group $\Z_k \times \Z_\l$ acts on the fields $A_i, B_i$ of the conifold theory as in (\[zk\]) and (\[zl\]). By starting with a conifold theory with a group $SU(klM) \times SU(klM)$, we obtain via the projection induced by the actions the following ${\cal N}=1$ supersymmetric generically chiral gauge theory for a specific choice for the Chan-Paton matrices: $$\prod_{i=1}^{k}\prod_{j = 1}^{l} SU(M)_{i,j} \times
\prod_{i=1}^{k}\prod_{j = 1}^{l} SU(M)_{i,j}^{'}
\label{group}$$ with matter content
----------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[**Field**]{} [**Repr.**]{}
$(A_1)_{i+1,j+1;i,j}$ $({\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}_{i+1,j+1},{\overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}}^{'}_{i,j})$
$(A_2)_{i,j;i,j}$ $({\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}_{i,j},{\overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}}^{'}_{i,j})$
$(B_1)_{i,j;i,j+1}$ $({\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}^{'}_{i,j},{\overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}}_{i, j+1})$
$(B_2)_{i,j;i+1,j}$ $({\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}^{'}_{i,j},{\overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}}}_{i+1,j})$
----------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
as explained in [@ura]. The superpotential is obtained by substituting the surviving fields into the conifold superpotential: W = \_[i,j]{} (A\_1)\_[i+1,j+1;i,j]{} (B\_1)\_[i,j;i,j+1]{} (A\_2)\_[i,j+1;i,j+1]{} (B\_2)\_[i,j+1;i+1,j+1]{}\
- \_[i,j]{} (A\_1)\_[i+1,j+1;i,j]{} (B\_1)\_[i,j;i+1,j]{} (A\_2)\_[i+1,j;i+1,j]{} (B\_2)\_[i+1,j;i+1,j+1]{} Moreover, by giving a vev to all the fields $(A_2)_{i,j;i-1,j-1}$, we obtain an $\prod_{i,j} SU(M)_{i,j}$ gauge theory with surviving chiral multiplets $(A_1)_{i,j;i-1,j-1}, (B_1)_{i,j;i,j+1}, (B_2)_{i,j;i,j+1}$. The superpotential for these fields will be W= \_[i,j]{}(A\_1)\_[i,j;i-1,j-1]{}(B\_1)\_[i-1,j-1;i-1,j]{} (B\_2)\_[i-1,j;i,j]{}\
-(A\_1)\_[i,j;i-1,j-1]{}(B\_1)\_[i-1,j-1;i,j-1]{} (B\_2)\_[i,j-1;i,j]{} This field theory is that appearing on D3 branes on an orbifold $\C^3/\Z_k \times \Z_l$.
We now discuss how to arrive from the configurations with D3 branes at conifold singularities to configurations with D4 or D5 branes together with both types of NS branes. >From (\[con-eqn\]), we see that the orbifolded conifold can be viewed as a $\C^{*}\times \C^*$ fibration over the $z$ plane. In other words, for generic values of $z$, the pairs of variables $(x, y)$ and $(u, v)$ describe $\C^{*}\times\C^{*} $. Because we have $\C^{*}\times\C^{*} $ fibration over the $z$ plane, we have two different kind $ U(1)$ orbits, one in each $\C^{*}$ fiber. So we can perform one T-duality or two T-dualities along each of these orbits. If we make one T-duality we obtain a configuration with $k$ NS branes on a circle and all the configuration is at a $Z_{l}$ singularity. As first explained in [@lykken] and developed in [@ura], this is a chiral theory. Because we still have a singularity which cannot be controlled by removing NS branes, it is more advantageous to do both T-dualities in order to use all the geometrical information. By making these, we arrive to brane box configurations with two compact direction, containing D5 branes together with both types of NS branes. So by using the geometry, we study the Kähler deformation of the orbifolded conifold with brane boxes. As explained in [@karch], in order to account the number of Kähler structure parameters necessary to completely solve the singularity, we need to modify the intersections of the NS branes by so-called diamonds. By closing the diamonds we turn off the B field and by rotating the diamonds on a plane perpendicular on the D5 brane we resolve the singularity to $\C^3/\Z_k \times \Z_l$.
The Orbifolded Conifold $\CC_{22}$
===================================
Consider a system of $M$ D3 branes sitting at the orbifolded conifold \_[2 2]{} : x y =uv =z\^2 As explained before, this is a chiral theory with the gauge group: $$\prod_{i,j = 1}^{2} SU(M)_{i,j} \times \prod_{i,j = 1}^{2} SU(M)^{'}_{i,j}.
\label{group1}$$ Because the T-dual theory contains NS branes which are perpendicular, the adjoint fields become massive and they are integrated out, leaving only quadratic terms in the superpotential. For simplicity we denote the 16 fields by:
[llll]{} A\_[11]{}= (A\_1)\_[22;11]{},& A\_[12]{} = (A\_1)\_[21;12]{},& A\_[13]{} = (A\_1)\_[12;21]{},& A\_[14]{} = (A\_1)\_[11;22]{},\
A\_[21]{} = (A\_2)\_[11;11]{},& A\_[22]{} = (A\_2)\_[12;12]{},&A\_[23]{} = (A\_2)\_[21;21]{},& A\_[24]{} = (A\_2)\_[22;22]{},\
B\_[11]{} = (B\_1)\_[11;12]{},& B\_[12]{} = (B\_1)\_[12;11]{},& B\_[13]{} = (B\_1)\_[21;22]{},& B\_[14]{} = (B\_1)\_[22;21]{},\
B\_[21]{} = (B\_2)\_[11;21]{},& B\_[22]{} = (B\_2)\_[12;22]{},& B\_[23]{} = (B\_2)\_[21;11]{},& B\_[24]{} = (B\_2)\_[22;12]{}.
The D term equations are: $$\begin{aligned}
\label{dz2}
|A_{14}|^2 + |A_{21}|^2 - |B_{12}|^2 - |B_{23}|^2 = \xi_1 \\ \nonumber
|A_{13}|^2 + |A_{22}|^2 - |B_{11}|^2 - |B_{24}|^2 = \xi_2 \\ \nonumber
|A_{12}|^2 + |A_{23}|^2 - |B_{14}|^2 - |B_{21}|^2 = \xi_3 \\ \nonumber
|A_{11}|^2 + |A_{24}|^2 - |B_{13}|^2 - |B_{22}|^2 = \xi_4 \\ \nonumber
|B_{21}|^2 + |B_{11}|^2 - |A_{11}|^2 - |A_{21}|^2 = \xi_5 \\ \nonumber
|B_{22}|^2 + |B_{12}|^2 - |A_{12}|^2 - |A_{22}|^2 = \xi_6 \\ \nonumber
|B_{23}|^2 + |B_{13}|^2 - |A_{13}|^2 - |A_{23}|^2 = \xi_7 \\ \nonumber
|B_{24}|^2 + |B_{14}|^2 - |A_{14}|^2 - |A_{24}|^2 = \xi_8 \\end{aligned}$$ where the FI parameters satisfy the constraint $$\sum_{i=1}^{8} \xi_i = 0$$ The superpotential is $$\begin{aligned}
\label{pot1}
W = A_{11} B_{11} A_{22} B_{22} + A_{12} B_{12} A_{21} B_{21} + A_{13}
B_{13} A_{24} B_{24} + A_{14}
B_{14} A_{23} B_{23} - \\ \nonumber
- A_{11} B_{21} A_{23} B_{13} - A_{12} B_{22} A_{24} B_{14} - A_{13}
B_{23} A_{21} B_{11}
- A_{14} B_{24} A_{22} B_{12} \\end{aligned}$$ There are 16 F-term constraints derived from this superpotential, not all of them independent. As opposed to other field theories considered previously in the literature, our case involves chiral fields so the F term equations will give equality between two products of three fields as for example the one obtained after taking the derivative with $A_{11}$ : $ B_{11} A_{22} B_{22} = B_{21} A_{23} B_{13}$ and the rest of 15 equations are similar. After solving the independent F-term equations, we arrive at 10 independent fields, the rest of 6 fields being expressed in terms of these. We chose $A_{24}, A_{13}, A_{14}, B_{12}, B_{13}, B_{14}, B_{21}, B_{22},
B_{23}, B_{24}$ as the independent variables. The solution for the F-term equations is:
[rrrrrrrrrrr]{} &A\_[24]{}& A\_[13]{}& A\_[14]{}& B\_[12]{}& B\_[13]{}& B\_[14]{}& B\_[21]{}& B\_[22]{}& B\_[23]{}& B\_[24]{}\
A\_[11]{}&0&0& 1& 0& -1& 1& -1& 0& 1& 0\
A\_[12]{}&0&1& 0& 0& 1& -1& 0& -1& 0& 1\
A\_[21]{}&1&0& 0& -1& 0& 1& -1& 1& 0& 0\
A\_[22]{}&1&1& -1& -1& 1& 0& 0& 0& 0& 0\
A\_[23]{}&1&1& -1& 0& 1& -1& 0& 0& -1& 1\
B\_[11]{}&0&0& 0& 1& 1& -1& 1& -1& -1& 1
We now proceed to obtain the vacuum moduli space in the usual way, i.e. by imposing the F-term constraints and the D-term constraints in the form of symplectic quotients as the gauged linear sigma model. If we impose only F-term constraints, we can identify the moduli, denoted by $\CM_F$, of the 16 fields as a cone $\M_+$ in $\M =\Z^{10}$ by expressing them in terms of 10 independent fields. To construct this as a symplectic quotient, we consider the dual cone $\N_+$ in $\N = \mbox{Hom } (\M, \Z)$. It turns out that the dual cone $\N_+$ is generated by $24$ lattice points. Thus we have a map T: \^[24]{} , which is shown in the Figure 2.
\[T\] [$$\left(
\begin{array}{rrrrrrrrrrrrrrrrrrrrrrrr}
0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 1& 0& 0& 0& 0& 0& 0& 1& 1& 1& 0& 0& 0\\
0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 1& 1& 1& 0& 0& 0& 0& 0& 1\\
0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 0& 1& 0& 0& 1& 0& 0& 0& 0& 0& 1& 0& 0& 1\\
0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 1& 0& 0& 1& 1& 0& 1& 0& 0& 1& 0\\
0& 0& 0& 0& 0& 0& 1& 1& 1& 0& 0& 0& 0& 1& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 0& 0& 1& 0& 1& 0& 0& 0& 0& 1& 1& 0& 1& 0& 1& 0& 0& 0& 0& 0& 0\\
0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 1& 0& 0& 1& 0& 1& 1& 0& 1\\
0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 1& 0& 0& 0& 0& 1& 1& 1\\
0& 0& 0& 1& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 0& 1& 1& 0\\
1& 0& 0& 1& 1& 1& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0
\end{array}
\right)$$ ]{}
The transpose of the kernel of $T$ is then a $14 \times 24$ charge matrix $Q$ which is shown Figure 3. \[Q\] [$$\left(
\begin{array}{rrrrrrrrrrrrrrrrrrrrrrrr}
2& -1& 0& -1& -1& 1& 1& 0& -1& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0&
1\\ 1& 0& 1& 0& -1& 0& -1& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0&
1& 0\\ 1& 0& 0& -1& -1& 1& 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0&
1& 0& 0\\ 1& 0& -1& -1& 0& 1& 1& 0& -1& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0&
0& 1& 0& 0& 0\\ 1& 0& 0& -1& 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0&
0& 1& 0& 0& 0& 0\\ 1& 0& -1& -1& 0& 1& 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0&
0& 1& 0& 0& 0& 0& 0\\ 1& -1& 1& 0& 0& -1& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0&
0& 1& 0& 0& 0& 0& 0& 0\\ 1& -1& 1& 0& -1& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0&
0& 1& 0& 0& 0& 0& 0& 0& 0\\ 1& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0&
0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 0& 0& 0& -1& 0& 0& 0& -1& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
0& 0& 1& 1& 0& -1& -1& 0& 0& 0& -1& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0&
0\\ 1& 0& -1& -1& 0& 0& 1& 0& -1& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0&
0& 0\\ 1& -1& 0& -1& 0& 0& 1& 0& -1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0&
0& 0& 0\\ 0& 0& 0& 1& 0& -1& -1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0&
0& 0& 0& 0
\end{array}
\right)$$]{}
Thus we have an exact sequence: 0 \^[14]{} \^[24]{} 0. >From this sequence, one can see that the moduli space $\CM_F$ can be expressed as a holomorphic quotient of $\C^{24}$ by $(\C^*)^{14}$ whose action is specified by $Q$ (or a symplectic quotient by $U(1)^{14}$.) via the map induced by $T$. To incooperate the D-term constraints, we need to see how the action of $(\C^*)^{10}$ on the toric variety $\CM_F$ is represented in these terms. Since the action of $(\C^*)^{10}$ on the open subset $(\C^*)^{10}\subset \CM_F$ must be the obvious multiplication, the action of $(\C^*)^{10}$ on $\C^{24}$ is specified the transpose of a $10\times 24$ matrix $U$ such that T\^[t]{}U = \_k. $U$ is shown in the Figure 4. \[U\] [$$\left(
\begin{array}{rrrrrrrrrrrrrrrrrrrrrrrr}
0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
-1& 0& 0& 1& 0& 0& -1& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 0& 0& -1& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
-1& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
-1& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
-1& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\
1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0
\end{array}
\right)$$]{}
The D-term equations are represented by a matrix $V$ in the Figure 5. \[V\] [$$\left(
\begin{array}{rrrrrrrrrr}
0& 0& 1& -1& 0& 0& 0& 0& -1& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0& -1\\
0& 0& 0& 0& 0& -1& -1& 0& 0& 0\\ 1& 0& 0& 0& -1& 0& 0& -1& 0& 0\\
0& 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 1& 0& 0\\
0& -1& 0& 0& 1& 0& 0& 0& 1& 0
\end{array}
\right)$$]{}
We ignored the charges on the dependent fields because they are already encoded in $Q$. Thus on $\C^{24}$, the D-term constraints are represented by the charge matrix $VU$. Finally the full set of charges is given by a $21\times 24$ charge matrix $\tilde{Q}$ (Figure 6.) by concatenating $Q$ and $VU$. The cokernel of its transpose gives toric data for our vacuum moduli space, denoted by $\CM$. After eliminating redundant variables, it is give in the form of a map $T_{\CM}: \Z^{9} \to \Z^3$: T\_=(
[rrrrrrrrr]{} 2& 1& 0& 1& 0& -1& 0& -1& -2\
0& 1& 2& 0& 1& 2& 0& 1& 2\
-1& -1& -1& 0& 0& 0& 1& 1& 1
) The lattice points given by $T_{\CM}$ lie on the plane with normal $(1,1,1)$ at a distance $1/\sqrt{3}$ from the origin. We depict these lattice points v\_[1]{} = (2, 0, -1),& v\_[2]{} = (1, 1, -1),& v\_[3]{} = (0, 2, -1),\
v\_[4]{} = (1, 0, 0),& v\_[5]{} = (0, 1, 0),& v\_[6]{} = (-1, 2, 0),\
v\_[7]{} = (0, 0, 1),& v\_[8]{} = (-1, 1, 1),& v\_[9]{} = (-2, 2, 1) in the planar diagram (Figure 7). This is exactly a toric diagram for the orbifolded conifold $\CC_{22}: xy=uv=z^2$. The corresponding charge matrix $Q_{\CM}$ for the toric data $T_{\CM}$ with the Fayet-Iliopoulos D-term parameters from (\[dz2\]) is as follows: \[QM\] Q\_=( 0& 0& 0& 2& -2& 0& -1& 0& 1& 2\_1 + \_2 + \_4 + \_5 - \_7\
0& 0& 0& 1& -1& 0& -1& 1& 0& \_1 - \_7\
0& 0& 0& 1& -2& 1& 0& 0& 0& \_1 - \_3 - \_6 - \_7\
1& 0& 0& -2& 0& 0& 1& 0& 0& -\_1 - \_2 - \_5 - \_6\
0& 0& 1& 0& -2& 0& 1& 0& 0& -\_2 - \_3 - \_6 - \_7\
0& 1& 0& -1& -1& 0& 1& 0& 0& -\_2 - \_6 ) For this choice of redundant variables, Fayet-Iliopoulos D-term parameters must satisfy \[fi-ieq\] \_1 > 0, \_4 > 0, -\_6 > 0, - \_7 > 0, -\_3 -\_6-\_7 > 0,\
-\_2 -\_6-\_7 > 0, -\_2-\_3 -\_6-\_7 > 0, -\_1-\_2 -\_5-\_6 > 0,\
-\_1-\_2-\_3 -\_5-\_6-\_7 > 0, \_1 +\_4+\_5 > 0, \_1+\_2 +\_4+\_5 > 0.
\#1\#2\#3\#4\#5[ @font ]{}
(3,5210)(451,-5488) (3601,-5161) (4801,-4561) (6001,-3961) (7201,-2161) (6001,-2761) (4801,-3361) (6001,-1561) (7201,-961) (8401,-361) (6001,-1561)[(-2,-3)[2400]{}]{} (3601,-5161)[( 2, 1)[2400]{}]{} (6001,-3961)[( 2, 3)[2400]{}]{} (8401,-361)[(-2,-1)[2400]{}]{}
(4726,-4861)[$v_2$]{} (3451,-5461)[$ v_1$]{} (5926,-4261)[$v_3$]{} (4726,-3661)[$v_4$]{} (5926,-3061)[$v_5$]{} (7126,-2461)[$v_6$]{} (5926,-1861)[$v_7$]{} (7126,-1261)[$v_8$]{} (8326,-661)[$v_9$]{}
The Orbifolded Conifold $x y = z^{2}, u v = z^3$
================================================
In this case we start with a system of D branes sitting at the orbifold conifold singularity \_[23]{}: x y = z\^[2]{}, u v = z\^3. By putting on the $M$ D3 branes on $\CC_{23}$, we obtain the field theory with the gauge group: $$\label{groupz3}
\prod_{i=1}^{2}\prod_{j=1}^{3} SU(M)_{i,j} \times
\prod_{i=1}^{2}\prod_{j=1}^{3}
SU(M)^{'}_{i,j}$$ The matter content for the theory with the gauge group (\[groupz3\]) is similar to the one encountered for the previous orbifolded conifold but we have 24 fields now instead of 16 as before. For simplicity we denote the 24 fields by:
[llll]{} A\_[11]{}= (A\_1)\_[22;11]{},& A\_[12]{} = (A\_1)\_[21;12]{},& A\_[13]{} = (A\_1)\_[32;21]{},& A\_[14]{} = (A\_1)\_[31;22]{}\
A\_[15]{} = (A\_1)\_[12;31]{},& A\_[16]{} = (A\_1)\_[11;32]{},& A\_[21]{} = (A\_2)\_[11;11]{},& A\_[22]{} = (A\_2)\_[12;12]{}\
A\_[23]{} = (A\_2)\_[21;21]{},& A\_[24]{} = (A\_2)\_[22;22]{},& A\_[25]{} = (A\_2)\_[31;31]{},& A\_[26]{} = (A\_2)\_[32;32]{}\
B\_[11]{} = (B\_1)\_[11;12]{},& B\_[12]{} = (B\_1)\_[12;11]{},& B\_[13]{} = (B\_1)\_[21;22]{},& B\_[14]{} = (B\_1)\_[22;21]{}\
B\_[15]{} = (B\_1)\_[31;32]{},& B\_[16]{} = (B\_1)\_[32:31]{},& B\_[21]{} = (B\_2)\_[11;21]{},& B\_[22]{} = (B\_2)\_[12;22]{}\
B\_[23]{} = (B\_2)\_[21;31]{},& B\_[24]{} = (B\_2)\_[22;32]{},& B\_[25]{} = (B\_2)\_[31;11]{},& B\_[26]{} = (B\_2)\_[32;12]{}
The superpotential is then: $$\begin{aligned}
\label{pot2}
W = A_{11} B_{11} A_{22} B_{22} + A_{12} B_{12} A_{21} B_{21} + A_{13}
B_{13} A_{24} B_{24} + A_{14}
B_{14} A_{23} B_{23} + \\ \nonumber
+ A_{15} B_{15} A_{26} B_{26}
+ A_{16} B_{16} A_{25} B_{25}
- A_{11} B_{21} A_{23} B_{13} - A_{12} B_{22} A_{24} B_{14} \\ \nonumber
- A_{13}
B_{23} A_{25} B_{15}
- A_{14} B_{24} A_{26} B_{16} -
- A_{15} B_{25} A_{21} B_{11} - A_{16} B_{26} A_{22} B_{12} \\end{aligned}$$ There are 24 F-term constraints derived from this superpotential, not all of them independent and by solving them we arrive at 14 independent fields, the rest of 10 fields being expressed in terms of these. We choose A\_[16]{}, A\_[26]{}, B\_[11]{}, B\_[12]{}, B\_[13]{}, B\_[14]{}, B\_[15]{}, B\_[16]{}, B\_[21]{}, B\_[22]{}, B\_[23]{}, B\_[24]{}, B\_[25]{}, B\_[26]{} as the independent fields.
The D term equations are: $$\begin{aligned}
\label{dz3}
|A_{16}|^2 + |A_{21}|^2 - |B_{12}|^2 - |B_{25}|^2 = \xi_1 \\ \nonumber
|A_{15}|^2 + |A_{22}|^2 - |B_{11}|^2 - |B_{26}|^2 = \xi_2 \\ \nonumber
|A_{12}|^2 + |A_{23}|^2 - |B_{14}|^2 - |B_{21}|^2 = \xi_3 \\ \nonumber
|A_{11}|^2 + |A_{24}|^2 - |B_{13}|^2 - |B_{22}|^2 = \xi_4 \\ \nonumber
|A_{14}|^2 + |A_{26}|^2 - |B_{16}|^2 - |B_{23}|^2 = \xi_5 \\ \nonumber
|A_{13}|^2 + |A_{25}|^2 - |B_{15}|^2 - |B_{24}|^2 = \xi_6 \\ \nonumber
|B_{21}|^2 + |B_{11}|^2 - |A_{11}|^2 - |A_{21}|^2 = \xi_7 \\ \nonumber
|B_{22}|^2 + |B_{12}|^2 - |A_{12}|^2 - |A_{22}|^2 = \xi_8 \\ \nonumber
|B_{23}|^2 + |B_{13}|^2 - |A_{13}|^2 - |A_{23}|^2 = \xi_9 \\ \nonumber
|B_{24}|^2 + |B_{14}|^2 - |A_{14}|^2 - |A_{24}|^2 = \xi_{10} \\ \nonumber
|B_{25}|^2 + |B_{15}|^2 - |A_{15}|^2 - |A_{25}|^2 = \xi_{11} \\ \nonumber
|B_{26}|^2 + |B_{16}|^2 - |A_{16}|^2 - |A_{26}|^2 = \xi_{12} \end{aligned}$$ where the FI parameters satisfy the constraint \_[i=1]{}\^[12]{} \_i = 0.
We want to implement the same procedure as in the previous $\Z_2 \times \Z_3$ orbifolded conifold $\CC_{22}$. As before, we can identify the moduli space $\CM_F$ of 24 fields under the F-term constraints as a cone $\M_+$ in $\M=\Z^{14}$. The dual cone $\N_+$ is generated by 80 lattice points represented by $T$. Thus $\CM_F$ can be expressed as a symplectic quotient $\C^{80}// U(1)^{66}$ whose action is specified by $Q$. This can be expressed as the following exact sequence: 0 \^[66]{} \^[80]{} 0. By further imposing $11$ D-term equations from (\[dz3\]), we obtain toric data for the vacuum moduli space $\CM$ as a three dimensional toric variety $\C^{80}// U(1)^{77}$. Because of huge sizes of the matrices involved, we only write the final toric data after eliminating redundant variables. It is given in the form of a map $T_{\CM}: \Z^{12} \to \Z^{3}$: T\_=(
[rrrrrrrrrrrr]{} 0& 1& 2& 3& -1& 0& 1& 2& -1&0&1&-2\
0& -1&-2&-3& 2& 1& 0& -1& 3&2&1&4\
1& 1& 1& 1& 0& 0& 0& 0& -1&-1&-1&-1
) The lattice points given by $T_{\CM}$ are as follows: v\_1= (0, 0, 1),& v\_2 =(1, -1, 1),& v\_3 =(2, -2, 1),& v\_4 =(3, -3, 1),\
v\_5=(-1, 2, 0),& v\_6=(0, 1, 0),& v\_7 =(1, 0, 0),& v\_8=(2, -1, 0),\
v\_9=(-1, 3, -1),& v\_[10]{}=(0, 2, -1),& v\_[11]{}=(1, 1, -1),& v\_[12]{}=(-2, 4, -1), which are drawn in Figure 8. This is exactly the toric data for the $\Z_3 \times \Z_2$ orbifolded conifold $\CC_{32}$. The corresponding charge matrix is given by Q\_=(
[cccccccccccc]{} 2& 0& 0& 0& 0& 0& 1& 0& -4& 1& 0& 0\
-3& 0& 0& 0& 0& 0& -1& 1& 3& 0& 0& 0\
1& 0& 0& 1& 0& 0& 1& 0& -3& 0& 0& 0\
-2& 1& 0& 0& 0& 0& -1& 0& 2& 0& 0& 0\
-2& 0& 0& 0& 0& 1& 0& 0& 1& 0& 0& 0\
1& 0& 0& 0& 0& 0& 0& 0& -2& 0& 1& 0\
0& 0& 0& 0& 0& 0& 1& 0& -2& 0& 0& 1\
-1& 0& 0& 0& 1& 0& -1& 0& 1& 0& 0& 0\
-1& 0& 1& 0& 0& 0& 1& 0& -1& 0& 0& 0
). The Fayet-Iliopoulos D-term parameters corresponding to each row is given by (
[c]{} -3\_1 + \_3 - 2\_4 - \_5 + 2\_6 - 3\_7 + \_9 - 2\_[10]{} - \_[11]{}\
3\_1 - \_3 + 2\_4 + \_5 - 2\_6 + 2\_7 - \_8 - 2\_9 + \_[10]{}\
-3\_1 - \_2 - 2\_4 - \_5 + \_6 - 3\_7 - \_8 - 2\_[10]{} - \_[11]{}\
2\_1 - \_3 + \_4 - 2\_6 + \_7 - \_8 - 2\_9\
\_1 - \_2 - \_3 - 2\_6 - \_8 - 2\_9 - \_[11]{}\
-2\_1 - \_4 - \_5 + \_6 - 2\_7 - \_[10]{} - \_[11]{}\
-\_1 - \_4 - \_5 - \_7 - \_[10]{} - \_[11]{}\
\_1 + \_4 + \_5 + \_7 + \_[10]{}\
-\_1 - \_2 - \_3 - \_4 - \_5 - \_6 - \_7 - \_8 - \_9 - \_[10]{} - \_[11]{}
)
\#1\#2\#3\#4\#5[ @font ]{}
(433,2219)(451,-4897) (4801,-2761) (4201,-3961) (5401,-3361) (6601,-2761) (3001,-4561) (3601,-3361) (2401,-3961) (1201,-4561) (8401,-2761) (7201,-3361) (6001,-3961) (4801,-4561) (4801,-2761)[(-2,-1)[3600]{}]{} (1201,-4561)[( 1, 0)[3600]{}]{} (4801,-4561)[( 2, 1)[3600]{}]{} (8401,-2761)[(-1, 0)[3600]{}]{}
(1051,-4861)[$v_1$]{} (2251,-4261)[$v_2$]{} (3451,-3661)[$v_3$]{} (4651,-3061)[$v_4$]{} (2926,-4861)[$v_5$]{} (4051,-4261)[$v_6$]{} (5251,-3661)[$v_7$]{} (6451,-3061)[$v_8$]{} (4651,-4861)[$v_9$]{} (5926,-4261)[$v_{10}$]{} (7051,-3661)[$v_{11}$]{} (8326,-3061)[$v_{12}$]{}
Partial Resolutions
===================
In order to see (partial) resolutions of the singularities in the formalism used above, we need to turn on the Fayet-Iliopoulos terms. This will correspond to triangulations of the convex cone in toric geometry and moving the center of the moment map in symplectic reduction.
Before starting the actual discussion, we make some observations about the general cases. In [@unge] it was considered the case of generalized conifolds of type $x y = u^{k} v^{k}$ and their resolutions. Their partial resolutions are conifold singularities, pinch point singularities and orbifold singularities and are obtained for different values of the FI parameters. In the T dual picture, D3 branes at $x y = u^{k} v^{k}$ singularities transform into $k$ NS branes, $k$ NS$^{'}$ branes on circle together with D4 branes having the circle as one of the worldvolume coordinates. Partial resolutions of the singularity are obtained in the T-dual picture by moving one NS brane in the $x^7$ direction (in field theory this means to give expectation values to one field thus breaking the product of two gauge groups to a diagonal one). This smoothen the singularity to $x y = u^{k-1} v^{k}$. By removing a NS$^{'}$ brane, the singularity is smoothen to $x y = u^{k} v^{k-1}$. In [@unge], the starting point was D3 at $x y = u^2 v^2$ singularity whose T dual contains 2 NS and 2 NS$^{'}$ branes. By removing the two NS branes one arrives at the conifold singularity, by removing one NS and one NS$^{'}$ one arrives at the conifold and by removing either one NS or one NS$^{'}$ the pinch point singularity is obtained. This of course means that we resolve the initial “worse” singularity to a “smoother” one. By removing NS branes we have complete control on the spacetime singularity.
In the case of orbifolded conifolds we need to use brane box models obtained by making two T-dualities. In this case, the resolutions are obtained either by moving NS and NS$^{'}$ branes with respect to each other or by opening diamonds at the intersections of the NS and NS$^{'}$ branes.
The discussion is similar for both types of ${\bf Z}_k \times {\bf Z}_l$ orbifolded conifolds discussed in this paper. Let us consider the $\Z_2\times \Z_2$ orbifolded conifold case. From (\[QM\]), we have a moment map $\mu_{\CM}: \C^9 \to \R^{6}$: \_=( 2|p\_3|\^2 -2|p\_4|\^2 -|p\_6|\^2 +|p\_8|\^2 -2\_1 - \_2 - \_4 - \_5 +\_7\
|p\_3|\^2-|p\_4|\^2 -|p\_6|\^2 +|p\_7|\^2 - \_1 + \_7\
|p\_3|\^2-2|p\_4|\^2 +|p\_5|\^2 -\_1 + \_3 + \_6 +\_7\
|p\_0|\^2 -2|p\_3|\^2 +|p\_6|\^2+\_1 + \_2 + \_5 + \_6\
|p\_2|\^2 -2|p\_4|\^2 +|p\_6|\^2+\_2 + \_3 + \_6 + \_7\
|p\_1|\^2 -|p\_3|\^2-|p\_4|\^2 +|p\_6|\^2 +\_2 + \_6 ) where $p_i$ are homogeneous coordinates of $\C^9$. Then the $\CM$ is the symplectic reduction $\mu_{\CM}^{-1}(0)/U(1)^6$. >From the conditions of (\[fi-ieq\]), Fayet-Iliopoulos parameters of the resulting $U(1)^6$ gauged linear sigma model satisfy inequalities -2\_1 - \_2 - \_4 - \_5 +\_7<0,\
- \_1 + \_7 <0,\
-\_1 + \_3 + \_6 +\_7<0,\
\_1 + \_2 + \_5 + \_6 <0,\
\_2 + \_3 + \_6 + \_7<0. But the condition (\[fi-ieq\]) does not determine the sign of the last coordinate $\xi_2 + \xi_6$ of the center of the moment map $\mu_{\CM}$. Notice that the last coordinate of the moment map $\mu_{\CM}$ which is flopped as the sign of $\xi_2 +\xi_6$ changes. When $ \xi_2 +\xi_6>0$, it is parameterized by the homogeneous coordinates $p_3$ and $p_4$. When $ \xi_2 +\xi_6<0$, it is parameterized by the homogeneous coordinates $p_1$ and $p_6$. These two phases are topologically different. Thus the D-brane vacuum moduli space $\CM$ does have topologically distinct phases which are related by a flop transition. This phenomenon has been observed for orbifold singularities [@muto; @gre]. We can see this flop in the toric diagram which is shown in Figure 9 . \#1\#2\#3\#4\#5[ @font ]{}
(434,5218)(-2051,-5197) (7202,-4262) (8402,-3662) (9602,-1862) (8402,-2462) (7202,-3062) (8402,-1262) (9602,-662) (10802,-62) (8401,-1261)[(-2,-3)[1200]{}]{} (7201,-3061)[( 0,-1)[1200]{}]{} (7201,-4261)[( 2, 3)[1200]{}]{} (8401,-2461)[( 0, 1)[1200]{}]{} (7127,-4562)[$v_2$]{} (8327,-3962)[$v_3$]{} (9527,-2162)[$v_6$]{} (9527,-962)[$v_8$]{} (10727,-362)[$v_9$]{} (6676,-3211)[$v_4$]{} (8551,-1561)[$v_7$]{} (8551,-2761)[$v_5$]{} (1803,-4262) (3003,-3662) (4203,-1862) (3003,-2462) (1803,-3062) (3003,-1262) (4203,-662) (5403,-62) (3002,-1261)[(-2,-3)[1200]{}]{} (1802,-3061)[( 0,-1)[1200]{}]{} (1802,-4261)[( 2, 3)[1200]{}]{} (3002,-2461)[( 0, 1)[1200]{}]{} (1728,-4562)[$v_2$]{} (2928,-3962)[$v_3$]{} (4128,-2162)[$v_6$]{} (4128,-962)[$v_8$]{} (5328,-362)[$v_9$]{} (1277,-3211)[$v_4$]{} (3152,-1561)[$v_7$]{} (3152,-2761)[$v_5$]{} (6001,-4861) (601,-4861) (1801,-3061)[( 2, 1)[1200]{}]{} (8401,-1261)[(-2,-5)[1200]{}]{} (5101,-2461)[( 1, 0)[1500]{}]{} (6601,-2461)[(6.6667,10.0000)[.]{}]{} (6601,-2461)[(6.6667,10.0000)[.]{}]{} (6526,-2461)[( 1, 0)[ 75]{}]{} (6526,-2461)[(-1, 0)[1350]{}]{} (7201,-3061)[(6.6667,10.0000)[.]{}]{} (1801,-3061)(-7.50000,-11.25000)[161]{}[(6.6667,10.0000)[.]{}]{} (601,-4861)(12.06030,6.03015)[200]{}[(6.6667,10.0000)[.]{}]{} (3001,-3661)(7.50000,11.25000)[321]{}[(6.6667,10.0000)[.]{}]{} (5401,-61)(-12.06030,-6.03015)[200]{}[(6.6667,10.0000)[.]{}]{} (3001,-1261)(12.12121,-6.06061)[100]{}[(6.6667,10.0000)[.]{}]{} (4201,-661)(0.00000,-13.48315)[90]{}[(6.6667,10.0000)[.]{}]{} (4201,-1861)(-12.12121,-6.06061)[100]{}[(6.6667,10.0000)[.]{}]{} (3001,-2461)(0.00000,-13.55422)[84]{}[(6.6667,10.0000)[.]{}]{} (3001,-3586)(0.00000,-12.50000)[7]{}[(6.6667,10.0000)[.]{}]{} (7201,-3061)(-7.50000,-11.25000)[161]{}[(6.6667,10.0000)[.]{}]{} (6001,-4861)(12.12121,6.06061)[100]{}[(6.6667,10.0000)[.]{}]{} (7201,-4261)(12.12121,6.06061)[100]{}[(6.6667,10.0000)[.]{}]{} (8401,-3661)(7.50000,11.25000)[321]{}[(6.6667,10.0000)[.]{}]{} (10801,-61)(-12.06030,-6.03015)[200]{}[(6.6667,10.0000)[.]{}]{} (8401,-1261)(12.12121,-6.06061)[100]{}[(6.6667,10.0000)[.]{}]{} (9601,-1861)(0.00000,13.48315)[90]{}[(6.6667,10.0000)[.]{}]{} (8401,-2461)(12.12121,6.06061)[100]{}[(6.6667,10.0000)[.]{}]{} (8401,-2536)(0.00000,-13.55422)[84]{}[(6.6667,10.0000)[.]{}]{} (451,-5161)[$v_1$]{} (5851,-5161)[$v_1$]{} (5626,-2311)
For special values of $\xi_i$, there are several singularity types. Of course, we get the orbifolded conifold $\CC_{22}$ when all $\xi$ are zero. But the singularity becomes partially resolved, when fields get expectation values in terms of the FI parameters. One of the most interesting case is when we give expectation values to the fields $A_{2i}, i= 1 ,\cdots, 4$. This region corresponds to $\xi_5 + \xi_1 =\xi_6 + \xi_2=\xi_7 + \xi_3
=\xi_8 + \xi_4$. Hence the last three coordinates of the center of the moment map $\mu_{\CM}$ are zeros. Thus one can see that the lower left half triangle of the toric diagram will not be triangulated. So we will have an orbifold singularity $\C^3/\Z_2 \times
\Z_3$ for generic values of $\xi_i$ under these circumstances (Figure 10). \#1\#2\#3\#4\#5[ @font ]{}
(33,5210)(1,-5488) (3601,-5161) (4801,-4561) (6001,-3961) (7201,-2161) (6001,-2761) (4801,-3361) (6001,-1561) (7201,-961) (8401,-361) (6001,-1561)[(-2,-3)[2400]{}]{} (3601,-5161)[( 2, 1)[2400]{}]{} (6001,-3961)[( 0, 1)[2400]{}]{} (6076,-3886)(4.99426,7.49139)[470]{}[(6.6667,10.0000)[.]{}]{} (6001,-1561)(8.05369,4.02685)[299]{}[(6.6667,10.0000)[.]{}]{} (7126,-961)(9.37500,0.00000)[9]{}[(6.6667,10.0000)[.]{}]{} (7201,-961)[(6.6667,10.0000)[.]{}]{} (7201,-961)(0.00000,-9.02256)[134]{}[(6.6667,10.0000)[.]{}]{} (7201,-2161)(-8.05369,-4.02685)[150]{}[(6.6667,10.0000)[.]{}]{} (6001,-1561)(8.05369,-4.02685)[150]{}[(6.6667,10.0000)[.]{}]{} (4726,-4861)[$v_2$]{} (3451,-5461)[$v_1$]{} (5926,-4261)[$v_3$]{} (4726,-3661)[$v_4$]{} (5926,-3061)[$v_5$]{} (7126,-2461)[$v_6$]{} (5926,-1861)[$v_7$]{} (7126,-1261)[$v_8$]{} (8326,-661)[$v_9$]{}
The configuration of D3 branes at this singularity is T-dual to a $2 \times 2$ brane box with trivial identification of the unit cell. In the language of [@karch], giving expectation values to the fields $A_{2i}$, i.e. going to a baryonic branch, means to rotate the diamonds which lie at the intersections of the NS and NS’ branes. One can have similar discussions for the ${\bf Z_2} \times {\bf Z_3}$ orbifolded conifold.
Conclusions
===========
In this paper, we have used the toric geometry and Witten’s gauged linear sigma model to identify the Higgs moduli space of the field theory on the world volume of branes at the orbifolded conifold singularity of type $\CC_{kl}$ which is a $\Z_k \times \Z_l$ quotient of the conifold $xy -uv = 0$. We have shown that the Higgs moduli space does have phases related by a flop transition and topology change can occur. It is also observed that the orbifold singularity can be obtained as one of the phases of the Higgs moduli space. In field theory, this corresponds to giving expectation value to some hypermultiplets.
Moreover, we have studied a correspondence between brane configurations and brane at singularities for the case of orbifolded conifolds of type $\CC_{kl}$.
Acknowledgments
===============
We would like to thank Dieter Lust and Andreas Karch for very important discussions. The work of K. Oh is supported in part by NSF grant PHY-9970664. R. Tatar would like to thank the Department of Mathematics at U. Missouri-St. Louis for hospitality.
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abstract: 'We have produced ultrathin epitaxial graphite films which show remarkable 2D electron gas (2DEG) behavior. The films, composed of typically 3 graphene sheets, were grown by thermal decomposition on the (0001) surface of 6H-SiC, and characterized by surface-science techniques. The low-temperature conductance spans a range of localization regimes according to the structural state (square resistance to at , with positive magnetoconductance). Low resistance samples show characteristics of weak-localization in two dimensions, from which we estimate elastic and inelastic mean free paths. At low field, the Hall resistance is linear up to , which is well-explained by $n$-type carriers of density per graphene sheet. The most highly-ordered sample exhibits Shubnikov - de Haas oscillations which correspond to nonlinearities observed in the Hall resistance, indicating a potential new quantum Hall system. We show that the high-mobility films can be patterned via conventional lithographic techniques, and we demonstrate modulation of the film conductance using a top-gate electrode. These key elements suggest electronic device applications based on nano-patterned epitaxial graphene (NPEG), with the potential for large-scale integration.'
author:
- Claire Berger
- Zhimin Song
- Tianbo Li
- Xuebin Li
- 'Asmerom Y. Ogbazghi'
- Rui Feng
- Zhenting Dai
- 'Alexei N. Marchenkov'
- 'Edward H. Conrad'
- 'Phillip N. First'
- 'Walt A.'
date: 'October 7, 2004'
title: 'Ultrathin epitaxial graphite: 2D electron gas properties and a route toward graphene-based nanoelectronics.'
---
The exceptional electronic transport properties of low-dimensional graphitic structures have been amply demonstrated in carbon nanotubes and nanotube-based transistors. Ballistic transport has been observed up to room temperature [@Frank98; @Poncharal02; @Liang01], and quantum interference effects at cryogenic temperatures [@Tans97--0; @Bachtold99; @Schonenberger99]. Simple nanotube transistors [@Tans98--0; @Martel98], and interconnected logic gates [@Bachtold01] have been demonstrated, which rely on the ability to control the nanotube conductance via an electrostatic gate. The basic transport parameters of these devices are so compelling that nanotubes are considered to be a candidate material system to eventually supplant silicon in many electronic devices.
An under-appreciated fact is that most electronic properties of carbon nanotubes are shared by other low-dimensional graphitic structures. For example, planar nanoscopic graphene ribbons (i.e. ribbons of a single sheet of graphite) have been studied theoretically [@Wakabayashi01--0; @Nakada96--1], and they exhibit properties that are similar to nanotubes. Graphene ribbons with either metallic or semiconducting electronic structure are possible, depending on the crystallographic direction of the ribbon axis [@Wakabayashi01--0]. Thus, if suitable methods were developed to support and align graphene sheets, it would be possible to combine the advantages of nanotube-like electronic properties with high-resolution planar lithography to achieve large-scale integration of ballistic devices. An essential difference between nanotubes and planar graphene ribbons is the presence of dangling bonds at the edges. Normally these would be hydrogen-terminated, with little influence on the valence electronic properties. However, edge atoms could be passivated with donor or acceptor molecules, thus tuning the electronic properties without affecting the graphitic backbone of the device.
This Letter presents recent results [@Berger04] that show the two-dimensional nature of electrical transport in ultrathin graphite (multilayered graphene) grown epitaxially on SiC(0001). 6H-SiC is a large bandgap () semiconductor, which provides an insulating substrate at temperatures below for the $n$-type (nitrogen) doping employed here. We use magnetoconductance measurements and the physics of weak-localization to determine transport parameters of the graphite 2D electron gas (2DEG), and we show that the character of the magnetotransport/localization spans a wide range of behaviors, depending on the amount of disorder in the film or substrate. Quantum oscillations in the magnetoconductance and in the Hall resistance are found for the most ordered sample. The character of these features suggests that the quantum Hall effect could be observed at lower temperatures, higher fields, or in ultrathin graphite films of only slightly higher mobility. To our knowledge, these are the first transport measurements on oriented and patterned graphite films of only a few monolayers thickness (hence “graphene” films), although related transport experiments have been done on thicker (65–100 graphene layers) free-standing graphite microdisks, which were nano-patterned by focused-ion-beam (FIB) lithography [@Dujardin01].
Given the large mean free paths measured in high-quality graphites [@Kaburagi96], the unusual electronic dispersion of graphene, and the fact that the carriers lie near an air-exposed surface, this unique 2DEG system holds great scientific potential. Furthermore, with sufficiently high-quality material, ballistic and coherent devices analogous to nanotube designs [@Javey03] would be possible. This goal requires that the epitaxial graphene can survive the processing necessary for creation of submicron ribbons [@Wakabayashi01--0; @Nakada96--1], and that the 2DEG can be gated electrostatically. Below we also demonstrate these critical elements for the realization of electronic devices based on nano-patterned epitaxial graphene (NPEG).
Ultrathin epitaxial graphite films were produced on the Si-terminated (0001) face of single-crystal 6H-SiC by thermal desorption of Si [@Bommel75; @Charrier02; @Forbeaux98--0; @SampleNote]. After surface preparation by oxidation [@Cho03] or H$_{2}$ etching [@Ramachandran98], samples were heated by electron bombardment in ultrahigh vacuum (UHV; base pressure ) to in order to remove the oxide (some samples were oxidized/de-oxidized several times to improve the surface quality). Scanning force microscopy images showed that the best initial surface quality was obtained with H$_2$ etching (sample A). After verifying by Auger electron spectroscopy (AES) that the oxide was removed, samples were heated to temperatures ranging from to for 1–20 min. Under these conditions, thin graphite layers are formed [@Bommel75; @Charrier02; @Forbeaux98--0], with the layer thickness determined predominantly by the temperature. Multilayered-graphene film thicknesses were estimated by modeling the ratio of measured intensities in the Si 92- and C 271- Auger peaks ( incident energy) [@Li04; @Tanuma91--0; @Tanuma91--1].
![\[fig:surface\] (a)-(d) LEED patterns from Graphite/SiC(0001). The sample was heated several times to successively higher temperatures. (a) for 10 min. Immediately after oxide removal, showing SiC $1 \times 1$ pattern at . AES C:Si ratio 1:2. (b) , 3 min. The $\sqrt{3}\times\sqrt{3}$ reconstruction is seen at . AES ratio 1:1.9. (c) , 20 min. pattern showing diffracted beams from the $6\sqrt{3} \times 6\sqrt{3}$ unit cell. Examples of first-order SiC and graphite spots are marked. Note the surrounding hexagons of “$6 \times 6$” spots. AES C:Si ratio 2:1 ( graphite). (d) , 8 min. LEED pattern. AES ratio 7.5:1 ( graphite).(e) STM image of a surface region of the sample described in Fig. \[fig:surface\]d. Inset: Atomically-resolved region (different sample, similar preparation). (f): $dI/dV$ spectra (log scale) acquired from the regions marked with corresponding line types in the image at top. The solid line is an average of 396 spectra at different positions, the dashed line an average of 105. With a few “glitchy” exceptions, individual spectra in each region showed negligible variation from the average $dI/dV$ shown. ](\figsurface){width="\columnwidth"}
Figures \[fig:surface\] (a)–(d) show low energy electron diffraction patterns (LEED) at different stages during the growth of a 2.5-monolayer (ML) graphite film grown in-situ. Figure \[fig:surface\]e displays an STM image from the sample obtained after stage (d). The image reveals a distinct $6 \times 6$ corrugation of the overlayer [@Tsai92] and a raised region along a step on the surface. This modulation has been previously attributed to variations of the interlayer interaction arising from Moiré coincidences between the graphite and SiC lattices within a fundamental $6\sqrt{3} \times 6\sqrt{3}$ surface unit cell [@Bommel75; @Tsai92; @Starke98--1; @Owman96--0]. LEED confirms that the graphene sheets register epitaxially with the underlying SiC, as shown in Figs. \[fig:surface\]c and \[fig:surface\]d. The mean height difference between terraces (), indicates that the step in Fig. \[fig:surface\]e is a bilayer step in the SiC substrate. Terrace sizes (corresponding to a single $6 \times 6$ domain) are found by STM to be up to several hundred nanometers in extent. Preliminary high-resolution LEED studies indicate that the graphene layers are strained in-plane by 0.3–0.5%, with a mean structural coherence length of greater than .
Also shown in Fig. \[fig:surface\]f are derivative tunneling spectra ($dI/dV$ vs. $V$) acquired within the respective boxed regions of the image. The $dI/dV$ spectrum obtained from the lower terrace (solid line) is consistent with that of a zero-gap semiconductor, as found typically for bulk graphite. On the upper terrace, the $6 \times 6$ domain images somewhat differently, and the $dI/dV$ spectrum (dashed line) displays a region of constant, finite conductance around the Fermi energy (zero bias). Spectral shapes are very uniform within each $6 \times 6$ domain. The $dI/dV$ curves show that the electronic properties of the film are not entirely homogeneous. This may relate to differing lateral registry (i.e. not orientational) of the graphite on the SiC substrate, or electron confinement within $6 \times 6$ domains. $dI/dV$ spectra acquired over the buckled region at the step edge are nearly identical to those found on the upper terrace, suggesting that the graphite layer remains continuous over the step.
DC and low-frequency AC conductance measurements were made for temperatures $T=\textrm{0.3--50}]{K}$, and for magnetic fields $H$ from on graphite films with thicknesses of typically 3 graphene sheets (see Table \[tab:samples\]). For Hall-effect measurements, samples were defined using standard optical lithography (photoresist coating, plasma etching, photoresist removal via solvents). Four contacts were painted with silver paste directly on the surface or on evaporated Pd-Au pads on mm-size samples. For samples B, C and E, the voltage probe distance $d_V$ is . For the Hall bar samples A and D $d_V = 600$ and respectively (see photo inset, Fig. \[fig:conductance\]a). Reported values below are the square conductance $G$.
Sample
-------- ----------- ------ ---- ----- ----------- ------ -------------
A 10 3 ML 1.5 k$\Omega$ 1100 cm$^{2}$/Vs
B $\infty$ $>5$ 2.2
C 9 3 22
D 10 3 33 15
E 9 3 225
F 7 2.5
: \[tab:samples\] Sample properties. Ratio of intensities in the C() and Si() AES peaks, calculated thickness in graphene monolayers, square resistance at , and mobility (where measured).
The 2D nature of electrical transport in the film is vividly demonstrated in Fig. \[fig:conductance\]a by the large anisotropy in the magnetoconductance \[$\textrm{MC}=G(H)$\]: For $\mathbf{H}$ perpendicular to the graphene plane the differential magnetoconductance ($\textrm{dMC}=dG/dH$) is large and positive ( at ), whereas there is essentially no response when $\mathbf{H}$ lies in the plane (in-plane MC was measured with $\mathbf{H}$ transverse to the current direction). This anisotropy is found in all of our samples, and indicates that the motion of charge carriers is confined to the graphene planes. The observed positive dMC is in contrast to bulk graphite [@Soule64], which shows negative dMC for well-ordered single-crystals, and also a large anisotropy. However, carbon foils fabricated from exfoliated graphite [@Schaijk98] and partially graphitic carbons [@Bayot90] have positive MCs initially, which become negative at large fields. This behavior is a consequence of disorder-induced localization in the sample.
Figure \[fig:conductance\]a shows systematic changes in the perpendicular MC for samples of successively larger zero-field conductance (i.e. decreasing disorder). For sample C the initial slope is larger, at , and the MC attains an approximately temperature-independent maximum of at $H \approx \unit[3.5]{T}$. Sample A shows even more structure. Following a large initial dMC ( at ), the MC maximizes near , then decreases, followed by a series of (Shubnikov-de Haas) oscillations. A temperature-independent “fixed point” at $G=\unit[680]{\mu S}$ and $H=\unit[7.4]{T}$ is also observed.
As a function of temperature, the conductance increases proportional to $\ln T$ at low $T$, as shown in Fig. \[fig:conductance\]b for samples A–C. This is quite characteristic of a 2D electron gas in the weak localization regime [@Abrahams79; @Bergmann84]. The least conductive samples (D and E; $G < e^{2}/h=\unit[38.8]{\mu S}$) deviate slightly from the $\ln T$ dependence, which is indicative of a transition to strong localization \[see for instance Ref. \].
In a 2D system, carriers will localize at low temperature due to constructive quantum interference of time-reversed paths for carriers scattered elastically from static disorder [@Abrahams79]. The interference is reduced (conductance increased) by breaking time-reversal symmetry through the application of a magnetic field, or by increasing temperature. These coherent effects are manifest when the elastic mean free path $\ell_e$ is smaller than the inelastic mean free path $\ell_i(T)$. For $k_F\ell_e \gg 1$ ($k_F$ is the Fermi wave vector), the system is in the weak localization regime. Strong localization occurs for smaller $k_F\ell_e$ [@Minkov02; @StrongLocal].
Shown in the center-panel inset of Fig. \[fig:conductance\]a is the perpendicular MC of sample B for five different temperatures (circles), and fits to the data according to 2D weak-localization theory (lines) [@Bergmann84]. The entire family of MC curves is fit by a single temperature-dependent parameter,$\ell_i(T)$. Note that for transport in two dimensions, the mean free paths are obtained from the modeling without knowledge of either the Fermi velocity or the carrier effective mass. For sample B, we find $\ell_e =
\unit[15]{nm}$ and $\ell_i = \unit[100]{nm}$ at $T = \unit[4]{K}$. Weak localization effects are observed over a much smaller range of magnetic field for sample A, but a similar estimate gives $\ell_e =
\unit[\textrm{20--30}]{nm}$, and a much larger inelastic mean free path $\ell_i(4K)\sim\unit[300]{nm}$. From the carrier density $n=\unit[10^{12}]{cm^{-2}}$ per graphene sheet (see below), we find for sample A $k_F\ell_{e} \sim 5$, in agreement with the weak localization regime.
In two cases we have observed a reversal of the dMC. At the maximum MC for sample C (Fig. \[fig:conductance\]a), the conductance per graphene sheet (Table \[tab:samples\]) is $1.5 e^{2}/h$, i.e. comparable to the conductance quantum. This behavior, and the large change in MC of this sample (550%) are reminiscent of disordered 2DEGs, which have been explained in terms of a transition from an Anderson insulator to a quantum Hall liquid [@Jiang93]. The second case is that of sample A, which underwent an improved substrate preparation. For this sample, we also observe an initial maximum in the MC, but at much lower field (). Apparently weak-localization dominates the MC behavior of sample A in the low-field region, but the longer scattering paths are dephased by a smaller magnetic field. The subsequent appearance of Shubnikov-de Haas oscillations indicates quantization of the electron energy spectrum, and wave function coherence on a scale comparable to the cyclotron radius ($\sim
\unit[30]{nm}$ at ), which is consistent with the elastic scattering lengths estimated above. The significance of the quasi-temperature-independent fixed point at and a field of is not yet fully understood. Note that $1/H$ at the crossing point is equal to the mobility obtained from Hall measurements (see below). This correspondence would be an expected consequence of electron-electron interactions under weak localization [@Minkov02; @Altshuler85], but the observed quantum oscillations show that at high fields the system is beyond this regime.
The Hall resistance $R_{xy}$ was measured for samples D and A in the Hall bar configuration (photo inset, Fig. \[fig:conductance\]a) at a bias current of . For sample D at , $R_{xy}$ versus $H$ is linear from 0 to , with the slope corresponding to a density $n = \unit[10^{13}]{cm^{-2 }}$ $n$-type charge carriers, and a mobility of . The Hall voltage is also linear up to in sample A, from which we determine $n = \unit[3.6\times 10^{12}]{cm^{-2 }}$ ($n$-type), and an enhanced mobility of . The observation of a linear Hall effect is particularly remarkable, since single-crystal graphite samples display a substantial quadratic component at small fields [@Schaijk98; @Berlincourt55; @Du04; @Tokumoto04], due to three sub-bands (one electron, two hole). Apparently, our samples cannot be thus described. The carrier densities found here are comparable to those of other 2DEG systems, although the density per graphene sheet ($\sim \unit[10^{12}]{cm^{-2}}$) is higher than that found in high-quality graphites [@Soule64; @Du04; @Tokumoto04]. It remains to be determined what effect substrate doping has on the carrier density and mobility in the graphite film.
For the case of sample A, a carrier density can be obtained from the period (in $1/H$) of the Shubnikov-de Haas oscillations. Assuming a circular 2D Fermi surface, we estimate $n =
\unit[10^{12}]{cm^{-2}}$ ($k_F = \unit[2.5 \times 10^{6}]{cm^{-1}}$). The carrier density determined by the Hall effect is very nearly a factor 3 larger, which is the number of graphene layers measured via AES. Clearly, the simplest explanation would be that each graphene sheet supports a 2D electron gas which remains confined within the sheet. This would be consistent with the large anisotropy in conductivity for bulk graphite [@Soule64]. The temperature dependences of samples A–C also support this interpretation: $dG/d(\ln T)$ falls between $2.5
e^{2}/\pi h$ and $3.5 e^{2}/\pi h$, about a factor 3 larger than the predicted weak-localization contribution to the conductance, $3.5 (e^{2}/\pi h)\ln T$ [@Bergmann84].
Above in sample A, we observe nonlinearities in $R_{xy}$ vs. $H$, as shown in Fig. \[fig:conductance\]a (lower-panel inset) for $T=\unit[4]{K}$. These coincide with the Shubnikov-de Haas oscillations in the magnetoconductance, showing that they have the same origin: Either broadened quantum Hall plateaus, or bulk magneto-quantum oscillations in a metallic system. If the Hall conductance at the location of the local maximum ($\approx\unit[5.5]{T}$) is normalized with respect to the number of sheets, one obtains a conductance $\approx 4 e^{2}/h$ which suggests a quantum Hall effect (see also predictions in Ref. ). Experiments at lower temperatures and higher fields will be necessary to verify this conjecture.
At , we also observe a pronounced zero-bias anomaly in the highly resistive sample D. The conductance is found to increase by about a factor 10 as the bias voltage is increased from 0 to [@Thermal]. For weak electron-electron interactions it can be understood in terms of enhanced scattering of carriers near the Fermi energy: the wavelengths of these carriers are commensurate with the Friedel oscillations surrounding impurities, thus they scatter strongly. The coherence is lost at higher bias (higher kinetic energies). Zero-bias anomalies have also been observed in carbon nanotubes [@Bockrath99; @Tarkiainen01; @Yi03].
It should be noted that the samples are remarkably stable over time. For instance, measurements in Figs. \[fig:conductance\]a and \[fig:conductance\]b were made 4 months earlier than those in the lower-panel inset of Fig. \[fig:conductance\]a, with no particular storage precautions. The features observed are essentially the same, except for a slight decrease in conductance and carrier density. The results presented above for sample A also show that the multi-sheet epitaxial graphene film survives conventional lithographic processing extraordinarily well.
Finally, as a preliminary demonstration of the device potential of this new 2DEG system, a large-area gated graphite-channel field-effect transistor (FET) structure was assembled. A schematic of the “device” is shown in Fig. \[fig:gating\], as well as the measured source-drain resistance as a function of gate voltage. The top-gate structure consisted of a conductive coating on a 100-nm-thick insulating aluminum oxide layer. The gate covered only a portion of the graphite film between the source and drain electrodes, leaving large ungated leakage paths (see inset, Fig. \[fig:gating\]). Consequently, the resistance modulation is rather small (2%), but these results show clearly that multi-sheet epitaxial graphene films can be gated, in distinct contrast to thicker samples [@Kempa03--1]. Thus we anticipate that FET-type devices will be possible, particularly when the channel electronic structure is controlled by patterning the graphite into a narrow strip [@Wakabayashi01--0; @Dujardin01; @Cancado04].
![\[fig:gating\] Conductance as a function of gate voltage for sample F at . Inset, sketch of the sample showing the contacts and top-gate geometry (S, D, G = source, drain, gate). The top-gate is only partially effective due to the open geometry. Nevertheless, a 2% change in conductance is observed.](\figgating){width="0.6\columnwidth"}
The experimental results presented here demonstrate the rich scientific promise of ultrathin epitaxial graphite (“graphene”) films. Several points should be appreciated: First, the production method allows graphitic films to be grown epitaxially, as evidenced by LEED and STM measurements. From Auger spectroscopy we further conclude that the layers involve only a few graphene sheets. Remarkably, the films are electrically continuous over several mm. Magnetoconductance measurements clearly reveal 2D electron gas properties, including large anisotropy, high mobility, and 2D localization, in samples patterned by conventional lithography. Quantum oscillations observed in both the magnetoconductance and the Hall resistance indicate a potential new quantum Hall system. Finally, control of the 2D electron gas carrier density via electrostatic gating was also demonstrated.
Considered with prior research in graphitic systems, these results provide ample evidence that the graphite/SiC system could provide a platform for a new breed of seamlessly-integrated ballistic-carrier devices based on nano-patterned epitaxial graphene. Such an architecture could have many advantages for nanoelectronics, including potentially coherent devices, energy efficiency, and facile integration with molecular devices.\
This work was funded by Intel Research and by the Department of Energy (DE-FG02-02ER45956). Support from Georgia Tech, CNRS-France, and the NSF (ECS-0404084), is also gratefully acknowledged. We thank Drs. Thierry Klein, Jacques Marcus and Frédéric Gay, CNRS-LEPES, Grenoble-France, for their generosity in allowing us to use their cryostats, and Dr. P. G. Neudeck, NASA Glenn Research Center, for supplying the sample used in Fig. \[fig:surface\]. We are especially grateful to Dr. Thierry Grenet for providing invaluable assistance in obtaining the data in Fig. \[fig:gating\].
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abstract: 'We calculate single-particle properties of short-range ordered stripe states, using Monte-Carlo simulations of collective charge-density wave (CDW) order parameters coupled to fermions on a 2d square lattice. For superconducting bond-centered stripes with a $d$-wave form factor, we find a valence-bond “glass” which coexists with low-energy quasiparticles featuring interference phenomena, in agreement with recent scanning-tunneling-microscopy (STM) measurements on underdoped [Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$]{} and [Ca$_{2-x}$Na$_x$CuO$_2$Cl$_2$]{}. Together with earlier work, our calculations provide a link between CDW signatures seen in STM and those in magnetic neutron scattering.'
author:
- Matthias Vojta
date: 'Sep 17, 2008'
title: ' Electronic properties of disordered valence-bond stripes in cuprate superconductors '
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Introduction
============
Charge-density wave (CDW) phenomena have been detected in a number of superconducting cuprates. Most prominent are the uni-directional spin and charge modulations, termed “stripes”, in [La$_{2-x}$Ba$_x$CuO$_4$]{} and [La$_{2-x}$Sr$_x$CuO$_4$]{} (with Nd or Eu co-doping), being strongest near 1/8th doping.[@jt95; @jt97; @yamada98; @pnas; @abbamonte] In other cuprates, notably [Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$]{} and [Ca$_{2-x}$Na$_x$CuO$_2$Cl$_2$]{}, scanning tunneling microscopy (STM) measurements have found signatures of short-range charge order.[@hana; @ali; @mcelroy; @kohsaka].
Remarkably, the existence of stripe states was postulated in early theory work on the Hubbard model,[@zaanen; @schulz; @machida] far before experimental indications for such phases were found. Later on, ideas of frustrated phase separation as driving force of stripe formation were worked out in detail,[@ek94] and CDW quantum criticality was proposed as source of both non-Fermi liquid behavior and superconductivity.[@castellani] For the vast number of subsequent theoretical activities we refer to the review articles Ref. .
While the role of charge order for the overall properties of cuprates is under debate, a plausible hypothesis is that tendencies toward charge ordering are common to underdoped cuprates.[@ek94; @castellani; @jan; @stevek; @ahcn04; @doug] Even compounds not displaying long-range order are influenced by the proximity to a charge-ordered state. In particular, impurities will act as random-field pinning centers for the collective charge modes, leading to static short-range order (as observed in STM).[@stevek; @maestro; @robertson] Moreover, charge order will influence the magnetic excitations, believed to be the pairing glue:[@natphys] It was recently shown [@vvk] that short-range-ordered stripes give rise to an “hour-glass” magnetic spectrum, very similar to that observed in neutron scattering experiments both on [La$_{2-x}$Ba$_x$CuO$_4$]{}[@jt04] and [YBa$_2$Cu$_3$O$_{6+\delta}$]{}.[@hayden]
While neutron and X-ray scattering were used to detect superstructure modulations from long-range charge order,[@jt95; @jt97; @abbamonte] there is relatively little information on the electronic structure of stripe states. Both STM and photoemission indicate the presence of coherent, gapless nodal quasiparticles (QP) in (1,1) direction, whereas antinodal QP in (1,0) direction are rather incoherent and likely dominated by charge ordering.[@mcelroy; @shen; @hana07] For the compound [La$_{2-x}$Ba$_x$CuO$_4$]{}, a $d$-wave-like gap was recently reported,[@valla] which may be attributed to static stripes or to fluctuating superconductivity.[@li]
In this paper, we present a detailed study of local electronic properties of disordered stripe states in cuprates, using a CDW order-parameter approach plus a mean-field theory for the single-particle dynamics. A central ingredient is the $d$-wave-like form factor of the charge order,[@MVOR] which causes the modulations to be located primarily on Cu-O-Cu bonds instead of on Cu sites. Our results reproduce central features of the STM data of Refs. . As we employ the [*same*]{} model for the collective CDW modes as in Ref. , used there to calculate spin excitations in the presence of disordered stripes, our results provide a link between different probes of stripe physics.
The remainder of the paper is organized as follows: In Sec. \[sec:model\] we describe the employed model together with the approximations and their physical background. Sec. \[sec:res\] presents the main numerical results, with focus on describing the STM data of Refs. . A discussion and conclusion closes the paper.
Phenomenological modelling {#sec:model}
==========================
Our phenomenological model consists of coupled CDW fluctuations and electrons, with the action ${\cal S} = {\cal S}_\psi + {\cal S}_c + {\cal S}_{c\psi}$. To account for the strong commensuration effects observed experimentally, all fields will be defined for discrete lattice coordinates.[@zachar98]
Lattice CDW order-parameter theory
----------------------------------
The CDW part ${\cal S}_\psi$ captures the tendency toward stripe ordering and is identical to that of Ref. : Two complex fields $\psi_{x,y} ({\vec r}, \tau)$ represent the amplitude of horizontal and vertical stripe order at wavevectors ${\vec K}_{x,y}$, such that the real field $
Q_x ({\vec r} ) = {\rm Re}\,\psi_x ({\vec r}) e^{i {\vec K}_x \cdot {\vec r}}
$ (similarly for $Q_y$) measures the modulation of both the charge density and bond order (i.e., kinetic energy or pairing amplitude), for $\vec r$ on sites and bonds, respectively. Then, $\delta\rho({\vec r}_j) = Q_x + Q_y$ is the deviation of the local hole density from its spatial average. We restrict our attention to ${\vec K}_{x}\!=\!(\pi/2,0)$ and ${\vec K}_y\!=\!(0,\pi/2)$, i.e., a charge modulation period of 4 lattice spacings.[@ali; @hana] The complex phase of $\psi_{x,y}$ represents the sliding degree of freedom of the density wave.
Fluctuations of the charge order are described by a $\psi^4$-type theory ${\cal S}_\psi$ for the O(4) field $\psi = (\psi_x,\psi_y)$. The precise form of ${\cal S}_\psi$ will determine the character of the fluctuations (amplitude vs. phase). The STM data of Ref. , with modulations present everywhere in real space, point toward small amplitude fluctuations; in addition, the calculated spin-fluctuation spectra of Ref. were only compatible with experiment under the assumption of dominant phase fluctuations. Hence, we employ $$\begin{aligned}
&&\mathcal{S}_{\psi} = \int \!d\tau \! \sum_i \Bigl[
\left| \partial_\tau \psi_{ix} \right|^2 +
\left| \partial_\tau \psi_{iy} \right|^2 +
s_x |\psi_{ix}|^2 + s_y |\psi_{iy}|^2 \nonumber\\
&&+
c_{1x}^2 | \psi_{ix}-\psi_{i+x,x} |^2 +
c_{2x}^2 | \psi_{ix}-\psi_{i+y,x} |^2 \nonumber\\
&&+
c_{1y}^2 | \psi_{iy}-\psi_{i+x,y} |^2 +
c_{2y}^2 | \psi_{iy}-\psi_{i+x,y} |^2
\nonumber \\
&&+
u_1 \psi_i^4 + u_2 \psi_i^6
+ v |\psi_{ix}|^2 |\psi_{iy}|^2 \nonumber\\
&&+ w \left( \psi_{ix}^4 \!+\! \psi_{ix}^{\ast 4}
\!+\! \psi_{iy}^4 \!+\! \psi_{iy}^{\ast 4} \right) \Bigr]
\label{spsi}\end{aligned}$$ with $\psi_{ix}\equiv \psi_x(\vec{r}_i)$ and $\psi_i^2\!\equiv\!|\psi_{ix}|^2\!+\!|\psi_{iy}|^2$. A combination of $u_1\!<\!0$ and $u_2\!>\!0$ suppresses amplitude fluctuations of $\psi$. The quartic $v |\psi_x|^2 |\psi_y|^2$ term regulates the repulsion or attraction between horizontal and vertical stripes; we shall mainly employ $v\!>\!0$ leading to stripe-like order (whereas $v\!<\!0$ results in checkerboard structures). The phase-sensitive $w$ term provides commensurate pinning and selects bond-centered (instead of site-centered) stripes[@kohsaka; @vvk] for $w\!>\!0$.
Fermions
--------
To calculate electronic properties in the presence of collective charge modes, we start from a BCS model of fermions on the square lattice of Cu atoms:[@notation] $$\begin{aligned}
{\cal S}_c &=& \int \! d\tau \! \sum_{{\vec k}} \left[
\bar{c}_{{\vec k}\sigma} (\partial_\tau\!+\!\epsilon_{\vec k}\!-\!\mu) c_{{\vec k}\sigma} +
\Delta_{\vec k} (c_{{\vec k}\uparrow} c_{{-\vec k}\downarrow} + c.c.)
\right]
\nonumber\\
\label{sc}\end{aligned}$$ where summation over spin indices $\sigma$ is implied. The single-particle dispersion consists of hopping to first, second, and third neighbors, with $t\!=\!-0.15$ eV, $t'\!=\!-t/4$, $t''\!=\!t/12$. The chemical potential is $\mu\!=\!-0.12$ eV, leading to a hole doping of $\approx 11\%$. The pairing is of $d$-wave type, $\Delta_{\vec k} = \Delta_0 (\cos k_x\!-\!\cos k_y)$ with $\Delta_0\!=\!24$ meV.
The coupling to the collective CDW fields $Q_{x,y}$ reads $$\begin{aligned}
{\cal S}_{c\psi} &=& \int d\tau \sum_i \Big[
\kappa_1 Q_x({\vec r}_i) \bar{c}_{i\sigma} c_{i\sigma} \nonumber\\
+&&\!\!\!\!\!\!\!\!\!\!
\big(
\kappa_2 Q_x({\vec r}_{i+x/2}) \bar{c}_{i\sigma} c_{i+x,\sigma} \,+\,
\kappa_3 Q_x({\vec r}_{i+y/2}) \bar{c}_{i\sigma} c_{i+y,\sigma} \nonumber\\
+&&\!\!\!\!\!\!\!\!\!\!
\kappa_4 Q_x({\vec r}_{i+x/2}) c_{i\uparrow} c_{i+x\downarrow} +
\kappa_5 Q_x({\vec r}_{i+y/2}) c_{i\uparrow} c_{i+y\downarrow} \!+\! c.c.
\big) \nonumber\\
+&&\!\!\!\!\!\!\!\!\!\!
[x \leftrightarrow y] \Big]
\label{scpsi}
$$ with $Q_x({\vec r}_{i+x/2})=[Q_x({\vec r}_{i})+Q_x({\vec r}_{i+x})]/2$. The coupling constants $\kappa_{1\ldots5}$ decide about the electronic struture of the CDW state, by implementing modulations of charge densities and bond kinetic and pairing energies. In the simplest picture, stripes correspond to modulations in the on-site charge densities. Those are induced by $\kappa_1$ and lead to a nearly $\vec k$-independent ($s$-wave) CDW form factor $\phi_2({\vec k}) = \langle c_{{\vec k+\vec K},\sigma}^\dagger c_{{\vec k}\sigma}
\rangle$.[@MVOR; @notation] However, local ordering can instead be dominated by physics on Cu-O-Cu [*bonds*]{}: Stripe formation is driven by the competition between kinetic and magnetic energies, both living on bonds.[@ssrmp; @sr; @vs] We have recently argued[@MVOR] that such a bond-dominated stripe state will have modulations in $\langle c_{i\sigma}^\dagger c_{i+\Delta,\sigma}\rangle$ with locally different signs on horizontal and vertical bonds, implying a strong $d$-wave component of $\phi_2({\bf k})$, see Fig. 1 of Ref. . Modulations on bonds are induced by $\kappa_{2\ldots5}$, with the $d$-wave character encoded, e.g., in $\kappa_2=-\kappa_3$.
A few remarks are in order: In the advocated model, Eqs. (\[spsi\],\[sc\],\[scpsi\]), correlation effects are included via $\epsilon_{\vec k}$ being a renormalized quasiparticle dispersion and via $Q_{x,y}$ representing collective CDW tendencies, while genuine Mott physics is absent. Dispersion renormalizations are standard in mean-field theories of correlated electrons; here we refrain from a self-consistent calculation of the dispersion and instead use plausible hopping parameters extracted from photoemission experiments. The separation of degrees of freedom into quasiparticles and collective CDW fields, both with full spatial or momentum dependence, is phenomenological and cannot be rigorously justified. However, e.g., in the context of electrons interacting with antiferromagnetic fluctuations, this has proven to be a fruitful route of investigation.[@chub]
Observables
-----------
STM experiments determine the spatially resolved local density of states (LDOS), $\rho(\vec{r},E)$, up to an $\vec{r}$-dependent tunnel matrix element (which depends on the set-point conditions.[@kohsaka; @hana07]) To separate physical modulations from set-point effects, the LDOS ratios $$\begin{aligned}
Z(\vec{r},E) &=& \frac{\rho(\vec{r},E)}{\rho(\vec{r},-E)}, \nonumber\\
R(\vec{r},E) &=& \frac{\int_0^E d\omega\rho(\vec{r},\omega)}{\int_{-E}^0 d\omega \rho(\vec{r},\omega)}
$$ have been used. In a weakly doped Mott insulator, both $Z$ and $R$ (measuring spectral particle–hole asymmetry) can be shown to be proportional to the hole density.[@pwa_as; @mohit_as] In Ref. , spatial modulations were observed in $R(\vec{r},E)$. In the following, we shall assume that these reflect modulations in the hole density.[@mohit_as; @ZR_foot]
Perfect CDW order
-----------------
Perfectly ordered CDW states are described by ${\cal S}_c+{\cal S}_{c\psi}$, with $\psi_{x,y}$ taken to be constant. From the diagonalized fermionic Hamiltonian all electronic properties can be obtained.
Sample results for the real-space densities of different types of CDW are displayed in Fig. \[fig:ord\]. Here, $\psi_x=(1+i)/\sqrt{2}$, $\psi_y=0$ for the bond-centered stripes in panels a) and b), while $\psi_x=\psi_y=(1+i)/\sqrt{2}$ for the checkerboards in panels c) and d). The couplings $\kappa$ were taken to induce $s$-wave-like \[panels b) and d)\] or $d$-wave-like \[panels a) and c)\] modulations, and the overall $\kappa$ amplitude was chosen such that the resulting modulation of fermionic densities is about 30-40%. To facilitate comparison with STM data,[@kohsaka] which show a strong modulation on the [*bonds*]{} of the CuO$_2$ plane,[@beyond] we included the bond charge densities $\langle c_{i\sigma}^\dagger c_{i+\Delta,\sigma} + h.c.\rangle$ (i.e. kinetic energies) in Fig. \[fig:ord\] – those are shown in between the square-lattice sites.[@bondfoot] Clearly, Fig. \[fig:ord\]a with $d$-wave stripes is most compatible with experiment.[@kohsaka]
=3.4in
Pinning and adiabatic approximation
-----------------------------------
The treatment of $\mathcal{S}_c\!+\!\mathcal{S}_{\psi}\!+\!\mathcal{S}_{c\psi}$ requires additional input. Pinning is important especially in the disordered phase of $\mathcal{S}_{\psi}$: Quenched disorder (e.g. from dopant impurities) acts as a random field and renders [*static*]{} a short-range ordered stripe configuration. In such a situation, the electronic properties can be approximately calculated by diagonalizing ${\cal S}_c+{\cal S}_{c\psi}$ for fixed static configurations of $\psi_{x,y}$. We generate these from classical lattice Monte Carlo (MC) simulations of $\mathcal{S}_{\psi}$, using a standard Metropolis algorithm at a finite effective temperature ($T=1$) in a regime where the stripe correlation length is of order $\xi \approx 10$.[@pinfoot] The numerical procedure parallels that of the adiabatic approximation of Ref. , with the difference that the ingredient of pinning is crucial to obtain a static signal in STM. (The presence or absence of pinning was irrelevant to the finite-frequency spin fluctuations described in Ref. .) In contrast to earlier work dealing with fermionic properties in the presence of disordered stripes,[@salkola] our modelling implements the $d$-wave bond character, and it properly describes short-range order via $S_\psi$ , i.e., stripe segments coexist with checkerboard domain walls.[@vvk; @maestro; @robertson]
Choice of parameters and validity of approximation
--------------------------------------------------
The parameters of $\mathcal{S}_c\!+\!\mathcal{S}_{c\psi}$, Eqs. (\[sc\],\[scpsi\]), used in our simulations are taken as in the static-stripe calculation above, i.e., for the fermionic sector we use values for $t$, $t'$, $t''$, and $\Delta$, which are standard in the BCS mean-field description of cuprates, and the couplings $\kappa$ are taken as in Fig. \[fig:ord\]a.
The CDW part of the action, $\mathcal{S}_{\psi}$ , is designed to capture the complicated non-universal physics of the strongly correlated CDW formation on the lattice scale. The combination of $s_{x,y}$, $u_1$ and $u_2$ decides about the importance of amplitude vs. phase fluctuations of the CDW,[@vvk] we have used $s_x\!=\!s_y\!=\!-4\ldots-3$, $u_1\!=\!-1.15$, $u_2\!=\!0.1$. Choosing $v\!=\!0.2$ prefers stripes over checkerboards, but allows for some checkerboard structure between stripe domains.[@vvk] Finally, $w\!=\!0.05$ is taken for a moderate commensurate lattice pinning toward bond-centered stripes. The precise values of the mass $s$ and the gradient $c$ were used to tune the CDW correlation length $\xi$, which was between 10 and 30 in our simulations. Note that an overall scale factor in $\mathcal{S}_{\psi}$ is free and determines the typical amplitude of $\psi$ which we have normalized to unity. The $\mathcal{S}_{\psi}$ parameters here are identical to those used in Figs. 1a, 2a of Ref. for the description of the spin excitations of fluctuating stripes. Moreover, the charge configurations generated from the MC simulations visually match the STM results in the sense that short and medium stripe segments coexist with checkerboard-like domain walls. This property is robust with respect to parameter changes of 20% and more, provided that the correlation length $\xi$ is kept fixed. As we employ classical MC simulations for $\mathcal{S}_{\psi}$, with time gradients absent, the parameters cannot be translated directly into physical energies or velocities.
Our approach assumes that a mean-field picture of both superconductivity and charge order is a reasonable starting point for the description of cuprates. The adiabatic approximation neglects inelastic processes and stripe dynamics, which can be justified if the latter is slow (compared to the observed fermions), as happens in the proximity to a CDW ordering transition. Thus, the approximation is invalid for energies below a typical stripe fluctuation frequency; for strong impurity pinning this scale is small or zero.[@pinfoot] The quasiparticle picture in cuprates may break down at elevated energies; as far as this happens due to inelastic physics, it is not captured by our approach (while some elastic disorder physics is captured).
Further, we assume that dimerization and bond order are the driving forces behind stripe ordering,[@vs; @vvk] whereas magnetic long-range order is less important. For simplicity, we therefore neglect both order and fluctuations in the triplet channel. Note that this does not mean that we ignore local-moment physics entirely, but instead we assume that those moments form singlet valence bonds, which is accounted for by modulated hoppings ($\kappa_{2,3}$) in $\mathcal{S}_{c\psi}$ . We note that the coupling to magnetic fluctuations will contribute to the broadening primarily of antinodal quasiparticles,[@soc] but a calculation including spatial disorder and inelastic processes is beyond the scope of the present paper.
Numerical results {#sec:res}
=================
We now turn to the numerical results obtained from $\mathcal{S}_c\!+\!\mathcal{S}_{\psi}\!+\!\mathcal{S}_{c\psi}$ for short-range ordered stripes.
=3.1in
Local densities
---------------
Fig. \[fig:dis1\] displays the order parameter field $(Q_x\!+\!Q_y)$ together with the resulting fermionic charge density, the LDOS $\rho(\vec{r},E)$, and $Z(\vec{r},E)$ at a high energy of $0.3$ eV, for one fixed $\psi_{x,y}$ configuration[@pinfoot] for a $d$-wave coupling in ${\cal S}_{c\psi}$. The stripe modulation, being prominent on the bonds, leads to a large contrast in Fig. \[fig:dis1\]b, while the contrast in both the site-LDOS $\rho$ and $Z$ (Figs. \[fig:dis1\]c,d) is weaker. (Both $\rho$ and $Z$ show strong modulations around the gap energy, Fig. \[fig:spectra\] below.) The result in Fig. \[fig:dis1\]b has a striking similarity to the “glassy” structures in Figs. 3,4 of Ref. . In particular, the modulation locally breaks the C$_4$ rotation symmetry down to C$_2$ and is primarily located on the Cu-O-Cu bonds. The latter fact – which originates in the $d$-wave form factor – can be nicely seen in the Fourier-transformed density, Fig. \[fig:ftdens\]. Stripe order is manifest in peaks at $(\pi/2,0)$, $(0,\pi/2)$ and $(3\pi/2,0)$, $(0,3\pi/2)$, with the signal at $(3\pi/2,0)$ being much stronger compared to $(\pi/2,0)$ (whereas for $s$-wave stripes the peaks are roughly equal in intensity). Again, this is in agreement with STM data, Fig. 6 of Ref. .
We note that the present comparison between theory and experiment does not easily allow to deduce the amplitude of the actual modulations: The only observables free of set-point effects are $Z$ and $R$. However, a reliable calculation of these has to has to cope with Mott physics not included in our model.[@ZR_foot] A rough estimate, however, relates the experimentally observed $R$ contrast of $\approx\pm30\%$ to a bond modulation of similar magnitude.
=3.1in
Nodal quasiparticles and quasiparticle interference
---------------------------------------------------
One outstanding feature of the STM results on underdoped [Ca$_{2-x}$Na$_x$CuO$_2$Cl$_2$]{}is the presence of quasiparticle interference (QPI) features in the low-energy spectra, in a situation where the high-energy spectra are dominated by period-4 modulations.[@hana07]
=3.5in
Our calculations qualitatively reproduce this physics. For QPI to occur, we have to add realistic disorder as source of QP scattering: Following Ref. , we use a combination of 2% extended potential scatterers (strength 40 meV, size 1.2), 5% extended pairing scatterers (strength $\Delta_0$, size 1.5), and 0.2% pointlike unitary scatterers (strength 2.5 eV). The real-space results of such a calculation are displayed in Fig. \[fig:qpi\]. Both the LDOS and the $Z$ map at higher energies are clearly dominated by stripe segments, whereas the signal below $\approx 25$ meV shows the typical QPI modulations (compare e.g. Fig. 3c of Ref. ). Extracting the scattering wavevectors from the Fourier transform of our data (not shown) is difficult due to the small system size; the only unambigous peak is at the so-called $q_7$ wavevector (corresponding to the diagonal modulations in $Z$ at low $E$, Fig. \[fig:qpi\]).
We point out two features of our results. (i) The $Z$ map is more sensitive to QPI than the LDOS, because, to leading order, QPI modulations at positive and negative $E$ are anti-phase, while stripe modulations are in-phase. Nevertheless, the strong period-4 modulations seen in the experimental low-energy LDOS [@hana07] are likely due to set-point effects. (ii) Real-space localization of antinodal QP [*cannot*]{} be made responsible for the loss of QPI at higher $E$. We have calculated the inverse participation ratio (not shown) as an indicator of localization, and have observed no localization signatures on scales up to several $\xi$ (while these length scales are sufficient to observe QPI).
More generally, the compatibility of stripes with long-lived nodal QP has been pointed out in the past.[@vs; @MVOR; @granath01; @wavev; @granath08] For small stripe amplitude, this already follows from the fact that the ordering wavevector $\bf Q$ does [*not*]{} connect the nodal points.[@vs; @wavev] In our case, nodal QP survive even for [*large*]{} stripe amplitude due to the $d$-wave character of the charge order (provided that the $s$-wave component remains small).[@MVOR]
Within our simulations, the survival of coherent nodal QP in the presence of disordered stripes is also seen in the LDOS spectra in Fig. \[fig:spectra\], taken along two different line cuts indicated in Fig. \[fig:dis1\]. While strong inhomogeneities occur at elevated energies, in particular near the gap energy (note the period-4 modulation in Fig. \[fig:spectra\]b around $-50$ meV), the low-energy part of the LDOS is essentially homogeneous,[@granath08] again in striking similarity to STM data.[@mcelroy; @kohsaka] (A detailed comparison of our spectra with experiment reveals several differences, which we believe to be related to Mott physics not captured here.)
Conclusions
===========
We have determined electronic properties of short-range ordered stripe states, coexisting with superconductivity. Agreement with salient features of STM experiments, in particular stripy LDOS modulations at elevated energies coexisting with QP interference at low energies, is found for valence-bond stripes with $d$-wave-like form factor, singling out a specific mean-field plus stripe disorder model.
As the [*same*]{} collective-mode description was used earlier to model magnetic excitations in the presence of fluctuating or disordered stripes, our calculations give a unified account of stripe signatures seen in STM and in neutron scattering, and strongly indicate that similar physics underlies the modulated states observed in different underdoped cuprates.
Very recent STM experiments[@extinct] indicate the quasiparticle interference disappears not only at high energies, but at a very specific location in momentum space, approximately at the boundary of the antiferromagnetic Brillouin zone. Such a feature is not part of the present theory, and likely requires to take into account either antiferromagnetic fluctuations or other precursors of strong Mott physics.
=3.2in
We thank J. C. Davis, H. Takagi, and A. Yazdani for discussions, and R. K. Kaul, S. Sachdev, T. Vojta, and A. Wollny for collaborations on related work. This research was supported by the DFG through SFB 608 (Köln) and the Research Unit FG 538.
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|
---
abstract: |
Recent active galactic nucleus (AGN) and quasar surveys have revealed a population showing rapid AGN luminosity variability by a factor of $\sim10$. Here we present the most drastic AGN luminosity decline by a factor of $\gtrsim 10^{3}$ constrained by a ${\textit{NuSTAR}}$ X-ray observation of the nearby galaxy Arp 187, which is a promising “dead” quasar whose current activity seems quiet but whose past activity of ${L_{\rm bol}}\sim
10^{46}$ erg s$^{-1}$ is still observable at a large scale by its light echo. The obtained upper bound of the X-ray luminosity is $\log (L_{\rm 2-10~keV}/{\rm erg}~{\rm s}^{-1}) < 41.2$, corresponding to $\log (L_\mathrm{bol}/{\rm erg}~{\rm s}^{-1}) < 42.5$, indicating an inactive central engine. Even if a putative torus model with ${N_{\rm H}}\sim 1.5 \times 10^{24}$ cm$^{-2}$ is assumed, the strong upper-bound still holds with $\log (L_{\rm 2-10~keV}/{\rm erg}~{\rm s}^{-1}) < 41.8$ or $\log (L_\mathrm{bol}/{\rm erg}~{\rm s}^{-1}) < 43.1$. Given the expected size of the narrow line region, this luminosity decrease by a factor of $\gtrsim 10^3$ must have occurred within $\lesssim 10^4$ yr. This extremely rapid luminosity/accretion shutdown is puzzling and it requires one burst-like accretion mechanism producing a clear outer boundary for an accretion disk. We raise two possible scenarios realizing such an accretion mechanism: a mass accretion 1) by the tidal disruption of a molecular cloud and/or 2) by the gas depletion as a result of vigorous nuclear starformation after rapid mass inflow to the central engine.
author:
- Kohei Ichikawa
- Taiki Kawamuro
- Megumi Shidatsu
- Claudio Ricci
- 'Hyun-Jin Bae'
- Kenta Matsuoka
- Jaejin Shin
- Yoshiki Toba
- Junko Ueda
- Yoshihiro Ueda
bibliography:
- 'arp187NuSTAR.bib'
title: '[*NuSTAR*]{} Discovery of Dead Quasar Engine in Arp 187'
---
Introduction {#sec:intro}
============
One of the fundamental questions on supermassive black holes (SMBHs) is how they stop growing their mass. The recent and ongoing quasar surveys have revealed massive SMBHs with masses of ${M_{\rm BH}}\gtrsim 10^9\ {M_{\odot}}$ at $z>7$ [e.g., @mor11], and interestingly, there seems to be a redshift-independent maximum mass limit at ${M_{\rm BH}}\sim 10^{10.5} {M_{\odot}}$ [e.g., @net03; @kor13]. This suggests that there is a fundamental quenching mechanism of the SMBH growth independently from the cosmic evolution, and possible mechanisms have been discussed theoretically by several authors [e.g., @nat09; @kin16; @ina16].
However, it is still observationally difficult to find quasars in the final growing/dying phase. The Soltan argument requires the total AGN lifetime is the order of $10^{7-9}$ yr [@sol82; @mar04], and even a single episode of AGN activity should be longer than $10^5$ yr [@sch15], and possibly $10^{6-7}$ yr [e.g., @mar04; @hop06]. This long lifetime implies that it is extremely difficult to witness the “newly-born” or “dying” phase of each AGN within the human timescale of $\lesssim 100$ yr.
One solution for this issue is using the difference in the physical size among AGN indicators, some of which would give us the quasar time variability longer than the human timescale. AGN have multiple indicators with different physical scales from $10$–$100$ $R_{\rm g}$ [X-ray corona and UV-optically bright accretion disk; @dai10; @mor10], $\sim 0.1$–$10$ pc [AGN tori; @bur13; @ich15], to $\sim 1-10$ kpc [narrow-line region or AGN jet; @ben02; @ode98], and the luminosities of the AGN indicators are tightly correlated with each other [@ich12; @ich17a; @ich19a; @tob14; @asm15; @ued15]. Recent observations have revealed an interesting AGN population that shows strong AGN activity at large scales with $\sim$1 kpc but much weaker one at small scales ($<$ 10 pc), suggesting a fading activity of the central engine. They are called fading AGN and currently $\sim20$ such sources have been reported [e.g., @sch13; @ich16; @ich19b; @kee17; @kaw17; @vil18; @wyl18; @sar18b].
Out of those $\sim20$ sources, Arp 187, a merger remnant infrared galaxy located at $z=0.04$ ($D_L = 178$ Mpc), is the most promising “dying” or “dead” quasar candidate, which completely lack current AGN signatures on small scales ($<10$ pc), but previous AGN activity estimated by the large scale AGN indicators ($\gtrsim 1$ kpc) must have reached quasar level luminosity. Previous VLA and ALMA 5–100 GHz radio observations have revealed the bimodal jet lobes with $\sim5$ kpc size, whose kinematic jet age of $8\times10^4$ yr. On the other hand, the central radio-core is absent, suggesting that the central engine is already faint or even quenched. The optical spectrum indicates that Arp 187 has narrow line region with the estimated size of $\sim1$ kpc, and the expected AGN luminosity reaches $L_\mathrm{bol}=
1.5 \times 10^{46}$ erg s$^{-1}$ [@ich19b]. On the nuclear AGN indicators, $\sim10$ pc scale AGN torus emission was not detected in the *Spitzer*/IRS mid-infrared spectrum, whose emission is dominated by the host galaxy, suggesting the absence of the current AGN torus activity with the upper-bound of $L_\mathrm{bol} < 6 \times 10^{43}$ erg s$^{-1}$ [@ich16].
However, we still lack a strong constraint on the current activity. In this letter, we report the first [*NuSTAR*]{} hard X-ray observation for this target. Thanks to its strong penetration power against absorption, [*NuSTAR*]{} puts the strongest constraint on the current AGN luminosity even in the case of heavy obscuration, allowing us to conclude that Arp 187 has an inactive central engine.
[*NuSTAR*]{} Observations and Results {#sec:obs}
=====================================
{width="100.00000%"}
The [*NuSTAR*]{} data were obtained with an on-source exposure of 82 ksec (GO cycle-4 Program 04037, PI: K. Ichikawa). Following the “[*NuSTAR*]{} Analysis Quickstart Quide” [^1], we reprocessed the data from [*NuSTAR*]{} detector modules of FPMA and FPMB with the standard [*NuSTAR*]{} script of `nupipeline`, which has two options to remove times with high background (i.e., `saamode` and `tentable`). From the telemetry report on count rates over the focal plane, we found slightly higher rates in orbits around the standard SAA area ($\sim $ 2 counts s$^{-1}$) than typical values ($\lesssim$ 1 count s$^{-1}$). Thus, `saamode=optimized` was adopted. Even if a more strict option of `saamode=strict` is used, our conclusion is unchanged. By contrast, such increase cannot be clearly seen in the so-called tentacle region [@for14] near the SAA, but by following recommendation of the [*NuSTAR*]{} team, we adopted `tentacle=yes`. Alternative option of `tentable=no` indeed provides a similar result, thus having little impact on our conclusion. The left panel of Figure \[fig:nustar\] shows an exposure-corrected 8–24 keV image, created by combining the FPMA and FPMB data and smoothed by a Gaussian function with $\sigma = 2$ pixels.
As indicated in the X-ray image, we defined a source region as a circle with a 30-arcsec radius centered at the optical position of the galaxy, and the background region was selected from the same chip as an off-source area with a 90 arcsec radius. The larger size was set to avoid local statistical fluctuations of the background level. We confirm insignificant change of our conclusion, even if a background spectrum is taken from a 30-arcsec circle near the source region. Note that, in the field-of-view, an X-ray source was serendipitously detected in (R.A, Decl.)$\sim (05:04:49.325,-10:16:40.17)$ with $\approx$ 8.8$\sigma$ significance at 8–24 keV, and its counterpart is likely to be GALEXASC J050449.00–101633.6 at $(05:04:49.0, -10:16:33.7)$. Its 2–10 keV flux estimated by a power-law model fit is $\sim 7\times10^{-14}$ erg cm$^{-2}$ s$^{-1}$. Given its location far from our target Arp 187 with an angular separation of $\approx$ 2 arcmin, which is at least six times larger than the positional uncertainty of [*NuSTAR*]{} [up to $\simeq 20$ arcsec, e.g., @lan17], we conclude that the emission does not originate from Arp 187 and hereafter we will not discuss this source.
The right panel of Figure \[fig:nustar\] shows obtained spectra of Arp 187 at 3–50 keV from the two regions in the left panel. The source spectrum shows no significant excess (2.9$\sigma$ and 1.5$\sigma$ in the 3.0–8.0 keV and 8.0–24 keV bands, respectively) to the background one. By considering an un-absorbed cut-off power-law component with the photon index of 1.7 and cut-off energy of 360 keV [@kaw16b][^2], the 3$\sigma$ upper limits of the 8–24 keV flux and luminosity are estimated to be $3.8\times10^{-14}$ erg cm$^{-2}$ s$^{-1}$ and $1.4\times10^{41}$ erg s$^{-1}$, equivalent to the 2–10 keV luminosity of $1.6\times10^{41}$ erg s$^{-1}$, corresponding to $L_\mathrm{bol} < 3.2 \times 10^{42}$ erg s$^{-1}$ with a bolometric correction factor of 20 [@Vas09]. Hereafter, all upper-limits on X-ray fluxes are at $3\sigma$ level. This estimate is not so sensitive to absorption in the sight-line up to $\log(N_{\rm H}/{\rm cm}^{-2}) \sim 23$. To consider more heavily obscured cases, we adopt a putative torus model as follows:
TBabs*cabs*zpowerlw*zhighect
+zpowerlw*zhighect
*mtable{e-torus_20161121_2500M.fits}
+atable{refl_fe_torus.fits},
represented in XSPEC terminology[^3]. This takes account of an absorbed and Compton scattered power-law component, a reflected continuum and an accompanying fluorescent iron-K$\alpha$ line. The photon index of the power-law, inclination and opening angles of the torus are set to 1.7, 70$^\circ$, and 60$^\circ$, respectively. Even under a Compton-thick absorption of ${N_{\rm H}}= 1.5 \times 10^{24}$ cm$^{-2}$ in the torus equatorial plane, the upper bound of the intrinsic luminosity is still very low with $\log ({L_{2-10}}/{\rm erg}~{\rm s}^{-1}) = 41.75$, or the bolometric luminosity of $\log
({L_{\rm bol}}/{\rm erg}~{\rm s}^{-1}) = 43.05$. Note that other well-known torus models, such as MYTorus and Borus [@Yaq12; @Bal18], also gives similar luminosity upper-bounds with the difference by a factor of 1.2. Lastly, we mention that the X-ray luminosity expected from the star-formation in the infrared [@ued14] is consistent with the 0.5–8 keV upper bound ($\sim 2\times10^{41}$ erg s$^{-1}$) from the extrapolation based on the 3–8 keV band, where a canonical power-law model seen in star-forming galaxies with $\Gamma=2.0$ and $3\times10^{21}$ cm$^{-2}$ [@min12] is utilized.
Discussion {#sec:discussion}
==========
{width="80.00000%"}
Very Faint AGN Even If It Is Highly Obscured
--------------------------------------------
Our [*NuSTAR*]{} result shows the strongest current luminosity constraints with $\log ({L_{\rm bol}}/\mathrm{erg~s}^{-1})< 42.5$ for $\log({N_{\rm H}}/{\rm cm}^{-2}) \lesssim 23$, and $\log ({L_{\rm bol}}/\mathrm{erg~s}^{-1})< 43.1$ for $\log({N_{\rm H}}/{\rm cm}^{-2}) \simeq 24.2$. This indicates that the central engine of Arp 187 is currently very faint even if it is highly obscured by gas. This is consistent with the absence of the AGN torus emission in the *Spitzer*/IRS spectra, which gives us the 3$\sigma$ upper-bound luminosity of $\log ({L_{\rm bol}}/\mathrm{erg}~\mathrm{s}^{-1}) < 43.8$ [@ich16].
One would expect that Arp 187 might be obscured by thicker absorption of $N_\mathrm{H}=10^{25}$ cm$^{-2}$. In this case, the expected upper-bound reaches to $\log ({L_{2-10}}/{\rm erg}~{\rm s}^{-1}) = 42.92$, or $\log ({L_{\rm bol}}/{\rm erg}~{\rm s}^{-1}) = 44.22$, exceeding the upper-bound obtained from the *Spitzer*/IRS spectra. However, this situation is unlikely because the reprocessed infrared emission should be observed even in such highly obscured situation, contributing to the *Spitzer*/IRS spectra [e.g., @yan18]. Thus, we conclude that the central engine of Arp 187 is likely to be dead, even if we consider the Compton-thick level obscuration, but the extreme absorption reaching $N_\mathrm{H}=10^{25}$ cm$^{-2}$ is also unlikely.
The Drastic Luminosity Decline
------------------------------
One important goal of our study is to constrain how rapidly the AGN in Arp 187 has dropped its luminosity. As already described in Section \[sec:intro\], the mutli-wavelength observations indicate that Arp 187 has experienced a luminosity decline in the past $\sim10^4$ yr. Figure \[fig:example\] summarizes the long-term decline together with the X-ray upper-bound we have obtained. The luminosity and the look-back time are obtained by combining the observational results of several AGN indicators with different physical scales [@ich16; @ich19b].
Figure \[fig:example\] shows that, thanks to its sensitivity in the hard X-ray band, [*NuSTAR*]{} (blue point) gives us a nearly two orders of magnitude fainter luminosity constraint than a previous estimate in the *Swift*/BAT 105 month catalog [purple; @oh18]. In addition, the [*NuSTAR*]{} observation gives the constraint on the current luminosity better than the MIR observations. Compared to the luminosity of $\log ({L_{\rm bol}}/{\rm erg}~{\rm s}^{-1})=46.15$ (see the black point) obtained from the NLR tracing the AGN activity $10^{3-4}$ yr ago, Arp 187 has experienced the luminosity decline at least by a factor of $>10^3$.
Naively, this drastic luminosity experience indicates that the accretion rate in Arp 187 should have drastically dropped over $>10^3$ times within $10^4$ yr. This seemingly short timescale itself is consistent with the viscous timescale of the UV emitting region [see the discussion of @ich19b]. There however remains another question of how such drastic decline of accretion was achieved. A gradual decrease of a external gas supply to the accretion disk cannot produce such a drastic luminosity decline. One suggestion is thus that the accretion disk has a clear outer disk boundary out of which the accretion rate drastically drops over $>10^3$ times. Therefore, one burst-like accretion event is preferable for realizing such a drastic accretion rate change.
Tidal Disruption Event in Arp 187?
----------------------------------
One might argue that a tidal disruption event (TDE) of a star could reproduce such a drastic accretion change. However, there are three difficulties in the case of Arp 187. First, the estimated BH mass of Arp 187 is $6.7\times10^{8}
{M_{\odot}}$, which thus requires a massive star above the main sequence, such as the red giant, to be tidally disrupted by the tidal field of the SMBH [e.g., @ree88]. The second is the luminosity problem: even if a red super giant, whose total mass is typically $\lesssim 50 {M_{\odot}}$, is tidally disrupted, it would be hard for the large BH ($\sim 7\times10^8 {M_{\odot}}$) to reach the expected Eddington ratio of Arp 187 ($\lambda_{\rm Edd}\sim 0.1$), or an accretion rate of $\sim 2.5 {M_{\odot}}$/yr [e.g., see Figure 5 of @mac13]. Third, the expected time scale: considering the rapid luminosity decline of TDEs which decays roughly as $L \propto t^{-5/3}$, the maximum observable timescale as AGN or quasar would be maximum $\lesssim 10$ yr. If a TDE is assumed to have happened at the time of jet-launch, or $10^{4}$–$10^{5}$ yr ago (see Figure \[fig:example\] or Section \[sec:intro\]), the estimated NLR size should be expanded only up to $\sim 10$ pc scale, and the \[\] would cool on timescales of $\sim100$ yr and thus such feature is no longer observable at the current stage. This is in clear disagreement with the observations, which leads us to exclude a TDE of a star as the origin of the accretion episode currently observed in Arp 187.
The other possibility is the TDE of a giant molecular cloud (GMC). Arp 187 is a good environment to produce such an event because of the starforming galaxy with plenty gas mass of $\sim 2\times10^9 {M_{\odot}}$ in the central $\sim900$ pc [@ued14]. The tidal radius of a GMC cloud is big enough as $R_{\rm TDE} = 200\times(R_{\rm GMC}/20~{\rm pc}) \times (M_{\rm BH}/10^8 M_{\odot})^{1/3} \times (M_{\rm GMC}/10^5 M_{\odot})^{-1/3}$ pc, where a canonical range of GMC radii is $R_{\rm GMC} = 10$–$50$ pc and that of GMC masses is $M_{\rm GMC} = 10^4$–$10^6$ $M_{\rm sun}$ in local galaxies [e.g., @bol08]. Although this idea is exclusively applied to Sgr A$\star$ [e.g., @bon08] and further theoretical studies are required to examine the case of much bigger SMBHs with ${M_{\rm BH}}> 10^8 {M_{\odot}}$, a GMC with mass of $\sim10^6 {M_{\odot}}$ can feed the SMBH of Arp 187 with the sub-Eddington level for $M_{\rm GMC}/(2.5 {M_{\odot}}~{\rm yr}^{-1}) \sim 4\times10^5$ yr. This would be long enough to produce the expected-size NLR by keeping the estimated past luminosity of $\log (L_{\rm bol}/{\rm erg}~{\rm s}^{-1}) \simeq 46.15$.
Accretion Disk Outer Boundary After Nuclear Starburst
-----------------------------------------------------
Our observation indicates the rapid luminosity decline in the final phase of quasar activity in Arp 187. One question raised from this result is whether this drastic luminosity decline is unique event only for Arp 187 or a rather common behaviour in the final phase of quasars.
Once the accretion rate somehow exceeds a certain value, it may naturally produce the drastic accretion rate gap, resulting in the drastic luminosity decline in the final phase of a quasar. By utilizing the nuclear starburst disk model by [@tho05], [@bal08] and [@ina16] discussed such a possibility that once the rapid accretion rate of $>10 {M_{\odot}}$ yr$^{-1}$ is achieved, at around $\sim1-10$ pc, vigorous star formation starts to deplete most of the gas and the accretion rate rapidly decreases by a factor of $\sim10^{2-3}$ times at some point, making a strong accretion rate gap. This is in good agreement with our expectation of the clear outer accretion disk boundary.
Considering that Arp 187 is a merger remnant, such a rapid accretion flow with $>10 {M_{\odot}}$ yr$^{-1}$ could be achieved by a previous major merger [e.g., @hop10]. The expected lifetime of such accretion disk is $t_\mathrm{life} \sim t_\mathrm{vis} (r=1~\mathrm{pc}) \sim 5\times10^7$ yr, which is long enough to produce the NLR and actually consistent with the typical quasar lifetime [e.g., @mar04b]. Based on those indirect observational signatures, quasars who experienced a drastic accretion inflow might follow the same luminosity decline in their future after consuming most of the gas in the accretion disk. On the other hand, a smooth accretion which have never exceeded the critical accretion rate of $\sim10 {M_{\odot}}$ yr$^{-1}$ will show more slower luminosity decline longer than $\sim10^4$ yr.
We acknowledge the anonymous referee for helpful suggestions that strengthened the paper. We thank Mitsuru Kokubo, Ryo Tazaki, and Takuma Izumi for fruitful discussion. This work is supported by Program for Establishing a Consortium for the Development of Human Resources in Science and Technology, Japan Science and Technology Agency (JST) and is partially supported by Japan Society for the Promotion of Science (JSPS) KAKENHI (18K13584; KI, 18J01050 and 19K14759; YT, 17K05384; YU). T.K. was financially supported by the Grant-in-Aid for JSPS Fellows for young researchers (PD). CR acknowledges support from the CONICYT+PAI Convocatoria Nacional subvencion a instalacion en la academia convocatoria año 2017 PAI77170080.
[^1]: <https://heasarc.gsfc.nasa.gov/docs/nustar/analysis/nustar_quickstart_guide.pdf>
[^2]: Even if we adopt another plausible parameter set of $\Gamma = 1.8$ and cut-off energy of 200 keV, found for a large hard X-ray selected AGN sample by [@ric17], the upper limit of 2–10 keV luminosity increases only by $\approx$10%, thus having little impact on our conclusion.
[^3]: The fits files of e-torus models were originally created by [@Ike09]. The first one is publicly available from <https://heasarc.gsfc.nasa.gov/xanadu/xspec/models/etorus.html> and the second one was privately obtained from [@Ike09]
|
---
abstract: 'The coupling constant of $ \rho\rar \eta\gamma $ decay is calculated in the framework of light cone QCD sum rules. A comparison of our prediction on the coupling constant with the result obtained from analysis of the experimental data is performed.'
author:
- |
\
[T. M. Aliev$^a$ [^1], [. I]{} Kan[i]{}k$^a$ [^2], A. Özpineci$^b$ [^3],]{}\
[a Physics Department, Middle East Technical University, 06531 Ankara, Turkey]{}\
[b The Abdus Salam International Center for Theoretical Physics, I-34100, Trieste, Italy]{}
title: ' [ ]{} '
---
16.3 true cm 23.0 true cm -0.8 true in 0.00 true in
0[[\^[\*0]{}]{}]{} 5[\_5]{} ß[S\_s\^[??]{}]{} Ø[[O]{}]{} *[[\^0 ]{}]{}*
PACS numbers: 11.55.Hx, 13.40.Hq.
Introduction
============
Radiative transition between vector(V) and pseudoscalar(P) mesons represent an important source of information on low energy hadron physics. These transitions are governed by magnetic dipole (M1) radiation of the photon and had played one of the central roles for checking the predictions of quark model and SU(3) symmetry, as well as it was very useful in the determination of the magnetic dipole moment of $N^*(1535)$ in $\gamma N \rightarrow \eta N$ process [@R20]. Recently the theoretical activity on the VP$\gamma$ magnetic dipole transition have increased(see [@R1] and references therein), in particular, due to the fact that the analysis of the radiative $V \rar P\gamma
$ decay with $\eta$ and $\eta'$ mesons in final state can provide insights to the long standing issue of the $\eta$ and $\eta'$ mixing (for review see [@R2] and references therein). The usual parametrization of $\eta - \eta'$ mixing in the octet–singlet basis, which will be used in this work, is as follows: the current-particle matrix elements are defined as 0 | J\_[5]{}\^| P(p) = -i f\_P\^p\_(=8;0;P=,’ \[eq1\]) where $ J_{5\mu}^8$, the $SU(3)_F$ octet axial vector current, is given by J\_[5]{}\^8=(\_\_5 u+\_\_5 d-2\_\_5 s) \[eq2\] and $ J_{5\mu}^0$, the $SU(3)_F$ singlet current is given by: J\_[5]{}\^0=(\_\_5 u+\_\_5 d+\_\_5 s) \[eq3\]
Two mixing angles $\theta_8$ and $\theta_0$ are required in order to consistently describe mixing [@R3]. Accordingly the couplings in Eq.(\[eq1\]) can be defined as follows f\_\^8 = f\_8 cos\_8 &,& f\_\^0 = - f\_0 sin\_0\
f\_[’]{}\^8 = f\_8 sin\_8 &,& f\_[’]{}\^0 = f\_0 cos\_0 \[eq4\]
Alternatively, two independent axial vector currents with distinct flavors can be considered J\_[5]{}\^q &=& (\_\_5 u+\_\_5 d)\
J\_[5]{}\^s &=& \_\_5s \[eq5\] and the couplings of these currents with $\eta$ and $\eta'$ mesons are defined similarly to Eq.(\[eq1\]) : f\_\^q = f\_q cos\_q &,& f\_[’]{}\^q = f\_q sin\_q\
f\_\^s = -f\_s sin\_s &,& f\_[’]{}\^s =f\_s cos\_s \[eq6\]
As we see that in each basis there are two angles and in [@R2] it was shown that, to a very good accuracy, the mixing can described in terms of single angle $\varphi$ since $ |\varphi_s -
\varphi_q|/|\varphi_s + \varphi_q| << 1$ which is also confirmed by a QCD sum rules calculation [@R4].
In present work we will follow to the first approach, neglecting the mixing angles $\theta_0$ and $\theta_8$ due to their smallness and calculate the coupling constant of $\rho\rightarrow\eta\gamma$ decay in framework of light cone QCD sum rules (more about of light cone QCD sum rules and its applications can be found in [@R5] and [@R6]).
The paper is organized as follows: In section 2 we derive light cone QCD sum rules for $\rho\rightarrow\eta\gamma$ decay constant, section 3 is devoted to the numerical analysis of the sum rules and contain our conclusions.
Light Cone QCD Sum Rules for $\rho\rightarrow\eta\gamma$ coupling constant
==========================================================================
In this section we calculate the coupling constant of $\rho\rightarrow\eta\gamma$ decay using light cone QCD sum rules method. In order to calculate this coupling constant we consider the following two point correlation function \_ = i d\^4 x e\^[i p\_2 x]{} 0 | T{ J\_\^(x) J\_\^(0)} | 0 \_\[eq7\] where $\gamma$ denotes the external electromagnetic field, and $J_\nu^\eta$ and $J_\mu^\rho$ are the interpolating currents with $\eta$ and $\rho$ meson quantum numbers. Here we would make the following remark. As we already noted that both of the mixing angles in Eq. (\[eq4\]) are small, in the following discussion we will neglect the mixing, i.e we take $J_\mu^\eta \equiv
J_{5\mu}^8$.
At the phenomenological level the Eq.(\[eq7\]) can be expressed as: \_ = \[eq8\] where $p_1 = p_2 + q $ and q is the photon momentum. The matrix elements entering Eq.(\[eq8\]) are defined as 0 | J\_\^| &=& m\_f\_\_\^\[eq9\]\
0 |J\_[5 ]{}\^8| (p\_2)&=& -i f\_\^8 p\_[2]{} \[eq10\] The remaining matrix element $\la \eta(p_2)| \rho(p_1)\ra_\gamma$ which describes the M1 transition, can be parameterized as, (p\_2)| (p\_1)\_= e \_\_\^p\_[1]{} \_\^q\_F(q\^2) \[eq11\] where $\varepsilon^\gamma$ is the photon polarization vector. Since the photon is real, we need the value of $F(q^2)$ only at the point $q^2=0$. We can use an alternative parametrization for the $\rho\eta\gamma$ vertex: L\_[int]{} = - g\_\_ (\_\_-\_\_)(\_A\_- \_A\_) \[eq12\]
Comparing Eqns. (\[eq11\]) and (\[eq12\]) we see that F(q\^2=0) \[eq13\]
Using Eqns. (\[eq8\]) – (\[eq12\]), for the physical part of the sum rules we get \_\^[ph]{} = \[eq14\]
Our next task is the calculation of correlator Eq.(\[eq1\]) from the QCD side. The correlator receives both perturbative and non-perturbative contributions. In calculation of the non-perturbative contributions by the OPE on the light cone one needs to know the matrix elements of nonlocal operators between vacuum and the photon states;i.e. $\la \gamma(q)|\bar q \Gamma_i q |0 \ra$ where $\Gamma_i$ is an arbitrary Dirac matrix. These matrix elements can be expressed in terms of photon wave functions with definite twist. In calculations we neglect twist 3 three particle photon wave functions since their contributions are small. Twist two and twist four photon wave functions are defined as [@R7; @R8; @R9]: (q) | |q \_q |0&=& i e\_q |q q\_0\^1 du e\^[iuqx]{} {(\_q\_- \_q\_)\[(u)+x\^2 \] +\
&&\[qx(\_x\_- \_x\_)+ x (x\_q\_- q\_x\_)\]g\_2 (u)} \[eq16\]\
(q) | |q \_\_5q |0&=& e\_q \_ \^q\^x\^\_0\^1 du e\^[iuqx]{}(u) \[eq17\] where for simplicity we use $\overline{q} \Gamma q$ to denote $J_{5\mu}^8=\frac{1}{\sqrt{6}}(\overline{u}\Gamma u+\overline{d} \Gamma d-2
\overline{s}\Gamma s)$. In Eqs.(\[eq16\]) and (\[eq17\]) $e_q$ is the corresponding quark charge, $\chi$ is the magnetic susceptibility, $\varphi(u)$, and $\psi(u)$ are the leading twist two and $g_1(u)$, and $g_2(u)$ are the twist four photon wave functions.
After standard calculations we get the following expression for correlator from QCD side in the coordinate representation: \_ = && (e\_u - e\_d) d\^4 x e\^[i p\_2 x]{} {\[6(x)\_ q\_x\_- 6(qx)\_\_x\_+ 6x\^2\_\_q\_\
&& -12 x\_\_ \_q\_x\_\]+f\^2 x\^2 \[x\_\_\_q\_x\_+\
&& x\_\_\_q\_x\_\]\_0\^1 du e\^[iuqx]{}(u)}/\^4 x\^6 \[eq18\]
The sum rules for $g_{\rho\eta\gamma}$ are obtained by equating the phenomenological and theoretical parts (in Eq.(\[eq18\]) it is necessary to perform Fourier transformation first) of the correlator.
Performing double Borel transformation on variables $p_2^2 = p^2$ and $p_1^2 = (p+q)^2$ on both sides of the correlation function in order to suppress the contributions of the continuum and higher states (procedure of subtraction in light cone sum rules one can find in [@R10; @R11]), and also to remove the subtraction terms in the dispersion relation, we obtain the following sum rules for $g_{\rho\eta\gamma}$ coupling constant g\_= (1- u\_0) { M\^2 E\_0(s\_0/M\^2) + f (u\_0) } \[eq19\] where $E_0(s_0/M^2)=1-e^{-s_0/M^2}$ is the function used to subtract continuum, $s_0$ is the continuum and u\_0= , M\^2= \[eq15\] where $M_1^2$ and $M_2^2$ are the Borel mass parameters in $\rho$ and $\eta$ channels. Note that in Eq.(\[eq15\]) we take into account $F_\eta^u=F_\eta^d$. The masses of $\eta$ meson is close to the $\rho$ meson mass. For this reason, it is natural to set $M_1^2=M_2^2=2M^2$ from which it follows that $u_0 = 1/2$.
Numerical Analysis
==================
In this section we present our numerical result on $g_{\rho\eta\gamma}$ coupling constant. From sum rules Eq.(\[eq19\]) we see that for estimating $g_{\rho\eta\gamma}$ coupling constant first of all one needs to know the photon wave function $\psi(u)$. It was shown in [@R8; @R9] that the photon wave function do not deviate remarkably from its asymptotic form which is given by $\psi(u)=1$ [@R8; @R7]. The values of the other constants appearing in the sum rules are: $m_\rho =
0.77~GeV$, $m_\eta=0.55~GeV$, $f= 0.028 GeV^2$. Leptonic decay constant of $\rho$ meson $f_\rho = 0.15~GeV$ follows from experimental result of the $\rho \rightarrow e^+ e^- $ decay , $\Gamma(\rho \rightarrow e^+ e^-)= (6,85 \pm 0,11)KeV $[@R13]. More recent analysis shows that the coupling of $\eta$ meson with the octet axial vector current is $f_\eta^8 = 0.159~GeV$ [@R2] and this result we will use in our analysis.
In Fig. 1 we present the dependence of the coupling constant $g_{\rho\eta\gamma}$ on the Borel parameter $ M^2 $ at three different values of the continuum threshold: $s_0= 1.4~GeV^2,~
1.6~ GeV^2,~ 1.8~GeV^2 $. Since the Borel mass $ M^2 $ is an auxiliary parameter and the physical quantities should not depend on it, we must look for the region where $g_{\rho\eta\gamma}$ is practically independent of $M^2$. We obtain that this condition is satisfied when $1~GeV^2 \leq M^2 \leq 1.4~GeV^2$. From this figure we also obtained that the variation of $s_0$ from $s_0= 1.4~GeV^2$ to $s_0= 1.8~GeV^2$ causes a change on the result on $g_{\rho\eta\gamma}$ of about $10\%$. Therefore one can say that the result $g_{\rho\eta\gamma}$ is insensitive to $s_0$ and $M^2$. Our final prediction on the coupling constant is g\_=(1.4 0.2) \[eq20\] where the error is attributed to the variation of $s_0$, $M^2$ and neglected twist three photon wave functions. At the end we would like to compare our prediction on $g_{\rho\eta\gamma}$ with experimental result. The decay width of the $\rho\rightarrow\eta\gamma$ decay is given by ()=m\_(1-m\_\^2/m\_\^2)\^3 \[eq21\] The experimental value is $\Gamma(\rho^0\rightarrow\eta\gamma) = (57 \pm 10)~
KeV$[@R13]. Using this value of $\Gamma(\rho^0\rightarrow\eta\gamma)$, the $g_{\rho\eta\gamma}$ coupling constant is obtained from Eq.(\[eq21\]) as: g\_ = (1.42 0.12) \[eq22\] which is very close to the sum rule prediction. Finally we note that the coupling constant $g_{\omega\eta\gamma}$ can be obtained from $g_{\rho\eta\gamma}$ with the help of the relation $g_{\rho\eta\gamma}= 3 g_{\omega\eta\gamma}$
$\left. \right.$
[99]{}
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[^1]: e-mail: taliev@metu.edu.tr
[^2]: e-mail: e114288@metu.edu.tr
[^3]: e-mail: ozpineci@ictp.trieste.it
|
---
abstract: 'Confocal microscopy in combination with real-space particle tracking has proven to be a powerful tool in scientific fields such as soft matter physics, materials science and cell biology. However, 3D tracking of anisotropic particles in concentrated phases remains not as optimized compared to algorithms for spherical particles. To address this problem, we developed a new particle-fitting algorithm that can extract the positions and orientations of fluorescent rod-like particles from three dimensional confocal microscopy data stacks, even when the fluorescent signals of the particles overlap considerably. We demonstrate that our algorithm correctly identifies all five coordinates of uniaxial particles in both a concentrated disordered phase and a liquid-crystalline smectic-B phase. Apart from confocal microscopy images, we also demonstrate that the algorithm can be used to identify nanorods in 3D electron tomography reconstructions. Lastly, we determined the accuracy of the algorithm using both simulated and experimental confocal microscopy data-stacks of diffusing silica rods in a dilute suspension. This novel particle-fitting algorithm allows for the study of structure and dynamics in both dilute and dense liquid-crystalline phases (such as nematic, smectic and crystalline phases) as well as the study of the glass transition of rod-like particles in three dimensions on the single particle level.'
author:
- 'T. H. Besseling'
- 'M. Hermes'
- 'A. Kuijk'
- 'B. de Nijs'
- 'T.-S. Deng'
- 'M. Dijkstra'
- 'A. Imhof'
- 'A. van Blaaderen'
bibliography:
- './library\_rod\_fitting\_3D.bib'
title: 'Determination of the positions and orientations of concentrated rod-like colloids from 3D microscopy data'
---
[^1]
[^2]
Introduction
============
Colloidal particles are applied throughout industry, for example in paints, personal care products, food, ceramics and pharmaceutics [@Croll2002; @Letchford2007; @Sozer2009]. Additionally, they are also applied in recent commercially available products such as the electronic ink in e-readers [@Sucher1998]. As a result, the characterisation of the structure and dynamics of colloidal suspensions is important for many industrial applications. Furthermore, hard-sphere colloidal suspensions have proven to serve as a model system to investigate phenomena such as crystallization, the glass transition and flow induced behaviour on the single particle level [@VanBlaaderen1995; @Weeks2000; @Dassanayake2000; @Gasser2001; @Dinsmore2001; @Cohen2006; @Yethiraj2003; @Besseling2012a]. In many of these studies, an image processing technique was applied based on the algorithm described by Crocker and Grier [@Crocker1996]. In their algorithm, spherical particles are located in 2D digital microscopy images using a local brightness maxima criterion. The position is refined by calculating the brightness-weighted centroid of a cluster of pixels. This method was extended to 3D either slice-by-slice [@VanBlaaderen1995; @Dassanayake2000] or by considering the full 3D image [@Dinsmore2001]. Crocker and Grier also reported a method to obtain the trajectories of the individual particles in time, known as particle tracking [@Crocker1996]. Since then, there have been numerous algorithms that locate or track spherical particles with increased accuracy or performance [@Lu2007; @Jenkins2008; @Gao2009; @Besseling2009a; @Vissers2011; @Kurita2012a; @Leocmach2013]. These extensions and alternatives are all based on processing images of *spherical* particles. However, due to recent progress in particle synthesis, well-defined (shape) anisotropic colloids are becoming widely available, see e.g. Refs. . These particles can often be observed directly with a (confocal) microscope and therefore enable quantitative measurement of not only their positional but also their rotational degrees of freedom. Therefore, a rapid increase in the number of algorithms that extract coordinates of anisotropic particles from microscopy images has taken place [@Mohraz2005; @Anthony2006; @Han2006; @Hunter2011; @Zhao2011; @Chakrabarty2013]. Most of these algorithms are based on processing of 2D (bright-field) images of quasi-2D systems. Mohraz and Solomon, however, were one of the first to determine the 3D position and orientation of uniaxial ellipsoidal particles, i.e. all five coordinates, using 3D confocal microscopy and a novel anisotropic feature-finding algorithm [@Mohraz2005]. Their algorithm identifies the points that are located on the central axis (or backbone) of a rod. These points are then grouped together by cluster analysis as individual rod-backbones, from which the centroid location and orientation are determined. This algorithm enabled the quantitative determination of the 3D translational and rotational motion of a dilute suspension of ellipsoids [@Mukhija2007].\
Quantitative 3D real-space study of concentrated phases of anisotropic particles is, however, much less progressed compared to studies on spherical colloids. Progress has been made for suspensions of ellipsoids, where nematic order was found using a centrifugal field [@Mukhija2011] and local crystalline order with an external electric field [@Shah2012]. In contrast with the system of ellipsoidal particles, it was recently shown by some of us that a system of fluorescent silica rod-like particles forms both nematic and smectic phases in equilibrium [@Kuijk2011; @Kuijk2012a]. However, determination of all the 3D positions and orientations of the particles in these dense phases was not possible due to the significant overlap of fluorescent signals. In this paper we demonstrate a novel 3D image processing algorithm that is capable of quantifying fluorescent silica rods in concentrated (liquid-crystalline) phases. The algorithm is tailored to work even when the fluorescent signals of the particles overlap considerably and a threshold method and subsequent clusters analysis alone does not suffice. The algorithm in principle also works for other uniaxial particles such as ellipsoids or dumbbells.
This paper is organized as follows. First, we describe the basics of particle-locating algorithms. Second, we describe our algorithm in detail. Third, we demonstrate the performance of the algorithm with 3D image stacks of concentrated fluorescent silica rods. Then, we illustrate that our algorithm can also be applied to 3D electron tomography data of gold nanorods. Next, we evaluate the accuracy of the algorithm by measuring the translational and rotational motion of non-overlapping rod-like particles. Finally, we compare our results with recent progress in the field and give an outlook on further studies that the algorithm enables.
Methods
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Locating particles in confocal microscopy data sets
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The aim is to identify and locate (rod-shaped) particles in a set of real-space images (or snapshots) and to obtain the full configuration of the system. A specific configuration of a system of particles is given by a set of parameters, one for each degree of freedom of every particle. In the case of rods, these degrees of freedom for particle $i$ are centre position $\mathbf{r}_i$, orientation $\mathbf{\hat{u}}_i$ and possibly length $l_i$, diameter $d_i$ and brightness $b_i$. If the length and diameter are known in advance they can be fixed, but if the particles vary in size they can also be left as free parameters. If the particles vary in brightness this can be added as an additional degree of freedom. Variations in brightness can be caused by the synthesis method, scattering or shading in the sample, but also by photo bleaching. In the case of fully symmetric, homogeneously dyed rods it is not possible to distinguish between the two ends of the rods. However, we also synthesised rods with a gradient in brightness, with one bright and one much darker end [@Kuijk2014a], of which the orientation could be fully determined. To keep the notation short we introduce ${\mathbf p}_i=\{ {\mathbf{\hat u}}_i,l_i,d_i,b_i\}$ which contains all the degrees of freedom except the position.
To obtain the configuration (${\mathbf r}_i$ and ${\mathbf p}_i$) we need to elaborate on what is measured. In case of fluorescent confocal laser scanning microscopy we can assume that the imaging system is linear so that we can add intensities. The measured image intensity $M(\mathbf{r})$ at position $\mathbf{r}$ can be written as the sum of the ideal (noiseless or averaged) images of the single particles, $$M(\mathbf{r}) = \sum_{i=1}^N \mathrm{RSP}(\mathbf{r}-\mathbf{r}_i,\mathbf{p}_i),$$ and $\mathrm{RSP}(\mathbf{r},\mathbf{p}_i)$ is the image of a single particle placed in the origin, or rod spread function (RSP). The image of a single particle at the origin depends on all the internal degrees of freedom of the particle such as orientation, length, diameter and brightness, but also on the point spread function (PSF) of the imaging system. It is given by $$\begin{aligned}
\mathrm{RSP}(\mathbf{r},\mathbf{p}_i) &=&
\int \mathrm{d}\mathbf{r}' \rho_\mathrm{dye}(\mathbf{r}',\mathbf{p}_i) \mathrm{PSF}(\mathbf{r}-\mathbf{r}')\nonumber\\
&=&(\rho_\mathrm{dye}(\mathbf{p}_i) \ast \mathrm{PSF})(\mathbf{r}),\end{aligned}$$ which is a convolution ($\ast$) of the dye distribution $\rho_\mathrm{dye}(\mathbf{r},\mathbf{p}_i)$ of particle $i$ placed in the origin and the PSF. In a dilute sample this $\mathrm{RSP}(\mathbf{r},\mathbf{p}_i)$ can be measured directly but it can also be calculated when the dye distribution is simple and the parameters of the optical systems are known.
Different approaches to obtain the particle coordinates are possible. If all the parameters such as the PSF and RSP are known, the locating problem becomes in principle a deconvolution. However, the RSP and the PSF can be time consuming to determine accurately, and deconvolutions are sensitive to small changes in the kernel function [@Cannell2006]. This is unfortunate since e.g. polydispersity will introduce changes in the RSP which would make the deconvolution difficult. If the RSP is not known, there exist several other possible options. The first option is to assume that the overlap between the RSPs is not too severe and to determine centre-of-mass and orientation with methods that are insensitive to the details of the optical system. This is the method used by centroiding algorithms and is also the method used in this article.
Another option is to use a Bayesian method [@Besag1991]. This method searches for the configuration that has the largest probability of having resulted in the observed image. This method has proven to work well for two-dimensional data sets [@Al-Awadhi2004]. It is, however, slow and complex and therefore not practical for large three-dimensional data sets.
Generation of test images
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To test our algorithm we generated 8-bit confocal-like images from sets of computer-generated particle trajectories. Using the centres-of-mass ${\mathbf r}_i$ and particle orientations ${\mathbf{\hat u}}_i$, we generated 3D stacks of $xy$-images of spherocylinders with aspect ratio $l$/$d$ = 5, where $l$ is the end-to-end length of the particle and $d$ the diameter. This was done by calculating the closest distances $D$ to a line segment, representing the backbone of a particle. The distance from a point in the origin to a line segment from $\mathbf{x}_1$ to $\mathbf{x}_2$ with length $l= |\mathbf{x}_1-\mathbf{x}_2|$ is given by $$D(\mathbf{x}_1,\mathbf{x}_2)=\left.
\begin{array}{rcl}
|\mathbf{x}_1| & \text{if} &\alpha<0, \\
\sqrt{|\mathbf{x}_1|^2-\alpha^2 } &\text{if} &0<\alpha<l, \\
|\mathbf{x}_2| & \text{if} &\alpha>l, \\
\end{array} \right.$$ where $\alpha=(\mathbf{\hat{u}} \cdot \mathbf{x}_1)$ and $\mathbf{\hat{u}}= (\mathbf{x}_1-\mathbf{x}_2)/l$ the unit vector along the length of the line segment. If this distance was less than the diameter of the particle, the pixel was given a value of 0.95. This was then repeated for all particles. We approximated the effect of the PSF in our test images by convolving them with a Gaussian kernel with fixed standard deviation $\sigma_x/d = \sigma_y/d = 0.3$ and $\sigma_z/d = 0.3,0.6$ and $0.9$ with $d$ the diameter of the particle. The full-width-at-half-maximum (FWHM) of the Gaussian function, given by $2\sqrt{2 \ln 2}\,\sigma_i$, is a direct measure of the resolution of the images. Besides variation of resolution, we also varied the amount of noise in the images. Although noise from modern detectors is essentially photon-limited, suggesting a Poisson distribution [@Art2006], we added noise to each pixel in our images with a simple Gaussian distribution with standard deviation $\sigma_n = 0.10 - 0.30$. Because the amount of noise is known a priori, it is still straightforward to calculate the signal to noise ratio (SNR), which we define as SNR = $(\sigma_g^2 / \sigma_n^2 - 1)^{1/2}$, with $\sigma_g^2$ the variance of the constructed image and $\sigma_n^2$ the variance of the noise [@Jenkins2008]. Finally, we converted all our data, with pixel-values between 0 and 1, to 8-bit grayscale tiff images.
Our algorithm
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To demonstrate the three-dimensional rod tracking algorithm we will first illustrate all steps of the algorithm with an artificially created set of images of a single rod, shown in Fig. \[fig:single\]. This will allow us to demonstrate clearly what is going on on a single pixel/voxel level. Later we will demonstrate how the algorithm fares with real colloidal suspensions. The following description is for three dimensions but most of the steps are straightforward to modify for two dimensions.
#### Reading
In Figs. \[fig:single\]a-c we show three orthogonal slices through a generated 3D image that acts as the source image. The particle shown in the image has a diameter $d = 13.0$ pixels, and is blurred with a Gaussian kernel $\sigma_x/d = \sigma_y/d = 0.3$ and $\sigma_z/d$ = $0.9$. Gaussian pixel noise of $\sigma_n = 0.1$ was added to the image. The first step is to read in these source images. To avoid accumulating rounding errors and to allow the use of images of arbitrary bit depth we perform all image manipulations on floating point numbers between zero and one. Next, the image is rescaled to make sure the voxels are cubic, which is often not the case for confocal microscopy image stacks. The rescaling avoids having to account for different $x$, $y$ and $z$ scales in all following routines. To make sure no information is lost, this is done by enlarging the image using a bicubic interpolation. Care should be taken not to use overexposed images since this will result in a loss of information and an increase of positional error. See Ref. for a more detailed description and the optimal shape of the intensity histogram. We generally choose the magnification such that the particles are approximately 10 pixels in diameter. Larger magnification results in a large file size without any additional benefit.
#### Filter
The aim of the first filter step is to reduce image noise. We apply a Gaussian blur to the image, i.e. a convolution with a Gaussian kernel, that acts as a low pass filter. The optimal width of the function depends on the noise level in the images; a value between 1.5 and 3 pixels was found to give the best results for the images obtained in the present paper. A value that is too large will result in the loss of resolution and in missing particles, a value that is too small will result in additional, incorrectly identified, particles. To ensure a black background for the particles, a background value is subtracted from every pixel. This background value is assumed to be mostly the result of photon noise, but it can also originate from other sources such as fluorescence from the solvent or immersion fluid. Pixels that have a negative value after the background value has been subtracted, are set to zero. In most cases a background value between 0.01 and 0.1 is used. This value should be chosen such that approximately half the empty pixels (not containing a particle) of the image are zero. We also save a copy of the image that has not been filtered. This allows us to perform the final fitting step on the original image. An example of a computer-generated image that has been filtered is shown Fig. \[fig:single\]d.
![ \[fig:single\] The different stages of identification of the position and orientation of a single rod-shaped particle. (a,b,c) Orthogonal slices through a computer generated 3D image stack. The particle has a diameter of 13.0 pixels and is blurred with a Gaussian kernel with width $\sigma_x/d = \sigma_y/d = 0.3$, and $ \sigma_z/d = 0.9$. Pixel noise has been added by adding Gaussian noise with $\sigma_n$ = 0.1. (d) The same image after the filter-step. (e) After a threshold step, the pixels above the threshold are marked in yellow. (f) After a backbone step, the pixels identified as backbone pixels are marked in yellow. (g,h,i) The rod as it is located, viewed from the $xy$, $yz$ and $xz$ plane. (j) The histogram of the average intensity along the rod length after smoothing and background removal. The dashed vertical lines mark the fitted end-points of the rod. ](fig1_v4.pdf){width="45.00000%"}
#### Well separated particles
When the intensity distributions of the individual particles do not overlap significantly we apply what we call a threshold method. This threshold method works as follows. A typical value for the threshold is between 0.4 and 0.7 and can be determined by plotting a histogram or by a quick test on a single image in a program like Photoshop, Gimp or ImageJ. The next step is to group all connected pixels above the threshold value into sets, as described in the next section. This methods works when these sets of pixels belong each to a single particle and each particle only corresponds to a single set of pixels. In Fig. \[fig:single\]e an example is shown of the threshold method applied to a single particle. All pixels above the threshold are marked in yellow. The particle coordinates can be obtained by applying a fit to these sets of pixels, as described later in this section. When this threshold method works, it is preferred over more complex methods since it is both robust and accurate.
#### General case
When a threshold does not successfully separate the image into regions belonging to single particles another method has to be used. The first step of this method is similar to the Crocker and Grier algorithm and is aimed at providing the final fitting steps with a good initial starting point.
In this step, we roughly locate the line segment starting from one end of the rod and ending at the other end, called the backbone of the particle. To locate the backbone, we look at all voxels brighter than a predetermined cut-off value. A good value for this is in general between 0.1 and 0.5 depending on the intensity fluctuations between the rods. For these bright pixels we then check whether they are part of a backbone. To do this we first note that all local maxima should be part of the backbone. To check if the brightness of the pixel is a local maximum we compare its intensity to that of all pixels within a distance $r_\mathrm{bb}$. If none of these pixels are brighter the pixel is a local maximum. To find the parts of the backbone that are not on a local maximum, we look at the distribution of brighter pixels around the pixel in question. If the pixel is part of the backbone they should be on a ridge. Backbone pixels can have brighter pixels to one side or two sides but all these brighter pixels should be more or less on a line through the pixel in question. So to check if the pixel is part of a backbone we need to check if the pixels brighter than the pixel in question are on a straight line. To do this we fit a line to these bright pixels and sum the squared residuals $\chi$, the squared distance between the brighter pixels and the line. If these bright pixels are part of the backbone of a rod this number will be low since the pixels will form an almost perfect line while on other places they will not form a line and the residuals will be much higher. We found that $r_\mathrm{bb}=3$ pixels and a maximum value $\chi_\mathrm{max}=80$ work well for all our data. This step depends on the initial filtering and on the thickness of the rod in pixels. Fig. \[fig:single\]f shows the pixels that have been identified as backbone pixels in yellow.
After having identified the backbone pixels, we group them into connected clusters. Due to noise there can be small gaps between the backbone pixels of a rod, so we use the same search range $r_\mathrm{bb}$ as before to identify neighbouring pixels. This should work as long as the diameter of a rod is larger than $r_\mathrm{bb}$.
We now have groups of pixels most likely belonging to a single rod. To continue, we fit (least square) a straight line to these pixels using a singular value decomposition and the algorithm described in Ref. . The coordinates resulting from this fit are accurate, but still have a strong pixel bias since they only fit to a few backbone pixels. To eliminate this bias and to obtain more accurate results, we use these coordinates, lengths and orientations as a starting point to fit the real image again.
#### Fitting
The fitting steps work best when applied to the unfiltered image. The Gaussian blur filter will result in an additional overlap of the RSPs which can result in a decreased accuracy. The fitting is done in three steps; first the centre of mass of each group of pixels is computed, then the orientation is fitted and finally the length is fitted. The position is taken from the centre of mass, weighted with the pixel intensity, of the pixels within half a diameter from the previous fit. The orientation is obtained by fitting a straight line to these pixels where the fit is weighted with the intensity of the pixels using the same least square fitting algorithm as for the backbones. The length is obtained by calculating the average intensity of pixels along the rod length, see Fig. \[fig:single\]j. The histogram that is obtained from this is smoothed with a Gaussian kernel to avoid noise. The end points are then obtained by determining where the histogram value drops below $I_\mathrm{end}I_\mathrm{max}$, where $I_\mathrm{max}$ is the maximum intensity value in the smoothed histogram and $I_\mathrm{end}$ is a parameter that can be set manually. Usually a value of $I_\mathrm{end}$ = 0.6 - 0.8 was found to give good results, see the (blue) dashed lines in Fig. \[fig:single\]j. To obtain sub-pixel accuracy we fit a straight line to the 2 pixels above and 2 pixels below the point where the histogram crosses this value. To determine which pixels to take into account in the generation of the histogram and the other fits, we use the pixels within one radius of the central line segment of the previous fit. Therefore, the result of the fit might improve when the step is repeated. The fitting algorithm normally converges in one or two steps. If this is not the case there is something wrong with the data or one of the parameters. Figs. \[fig:single\]g-i show the same orthogonal sections as Figs. \[fig:single\]a-c with the backbone of the rod highlighted in yellow and the outline of the rod (resulting from the fit) highlighted in magenta.
#### Filtering
The final step is to filter out particles that are found more than once, particles that do not contain enough intensity or sometimes particles that are not long enough. Ideally not much filtering is required.
3D particle tracking
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To study particle dynamics, we applied our algorithm to time-series of 3D image-stacks. We first identified the positions and orientations of the rods in each 3D stack separately. Then, we obtained the particle trajectories using standard IDL-based routines [@Crocker1996]. To uniquely track the tip of the (up-down indistinguishable) rods, it is required that the angular displacements between successive frames $[{\mathbf{\hat u}}(t+1) - {\mathbf{\hat u}}(t)]^2 < 2$. Therefore, care was taken that displacements with $[{\mathbf{\hat u}}(t+1) - {\mathbf{\hat u}}(t)]^2 > 2$ were negligible. We then calculated the mean squared displacement (MSD) and the mean squared angular displacement (MSAD). We fitted the MSD to the expression $$\label{eq:msd}
\langle \Delta {\bf r}^2 (t) \rangle =6\,D_t\,t + 6 \, \epsilon_t^2, $$ with $D_t$ the rotationally averaged translational diffusion coefficient and $\epsilon_t$ the error in measurement of each of the coordinates of the particle [@Savin2005]. For the MSAD we used the expression [@Dhont1996; @Cheong2010] $$\label{eq:msad}
\langle \Delta {\mathbf{\hat u}}^2 (t) \rangle = 2[1-(1-\epsilon_r^2)\exp({-2 D_r t})], $$ with $D_r$ the average rotational diffusion coefficient and $\epsilon_r$ the measurement error in the determination of ${\mathbf{\hat u}}(t)$. For short times, equation reduces to $$\langle \Delta {\mathbf{\hat u}}^2 (t) \rangle = 4D_r t + 2\epsilon_r^2\,.$$\[eq:msad\_short\] To estimate the sedimentation velocity at infinite dilution, assuming complete decoupling of rotations, translations and sedimentation [@Homogeneous1979], we use the Svedberg equation [@Svedberg1940] $$\label{eq:sed}
v_{sed} = \frac{v_p\,D_t\,g\,(\rho_p-\rho_s)}{k_B\,T}, $$ with $v_p$ the volume of the particle, $g$ the gravitational acceleration, $\rho_p$ the mass density of the particle and $\rho_s$ the mass density of the solvent.
Expressions for the diffusion coefficients
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To test the validity of our experimental measurements of the diffusion coefficients, we compared them to analytical expressions for hard cylinders at infinite dilution, as proposed by Tirado, Martinez and de la Torre [@Tirado1984], $$\begin{aligned}
D_\perp &= \frac{k_B T}{4 \pi \eta \,l}(\log p + \delta_\perp), \\
D_\parallel &= \frac{k_B T}{2 \pi \eta \,l}(\log p + \delta_\parallel), \\
D_t &= \frac{2}{3}D_\perp + \frac{1}{3}D_\parallel \label{eq:dtrans}, \\
D_r &= \frac{3 k_B T}{\pi \eta \,l^3}(\log p + \delta_r) \label{eq:Dr},\end{aligned}$$ with $\eta$ the solvent viscosity, $p = l/d$ the aspect ratio of the particle and $\delta_i$ a correction term for the finite aspect ratio of the cylinders, given by [@Tirado1984] $$\begin{aligned}
\delta_\perp &= 0.839 + 0.185/p + 0.233/p^2, \\
\delta_\parallel &= - 0.207 + 0.980/p - 0.133/p^2, \\
\delta_r &= - 0.662 + 0.917/p - 0.050/p^2.\end{aligned}$$
Experimental methods
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### Dense sediments of silica rods
For the preparation of dense samples of silica rods, two different batches of particles were used. The first batch consisted of rods with length $l = 2.37$ $\mu$m ($\delta$ = 10%) and diameter $d$ = 640 nm ($\delta$ = 7.5%), with $\delta$ the polydispersity (standard deviation over the mean) [@Kuijk2011]. A transmission electron microscopy (TEM) image of these particles is shown in Fig. \[fig:particles\]a. The particles contained a non-fluorescent core, a 30 nm fluorescein isothiocyanate (FITC) labelled shell, and a 190 nm non-fluorescent outer shell. For the second batch of silica rods, with length $l = 2.6$ $\mu$m (8.5%) and diameter $d$ = 630 nm (6.3%), rhodamine isothiocyanate (RITC) dye was added during synthesis, which resulted in an intensity gradient of dye molecules along the major axis of the particle [@Kuijk2014a]. The particles were coated with a 175 nm non-fluorescent outer shell. Particle suspensions were prepared by dispersing the rods in an index-matching mixture ($n_D^{21} = 1.45$) of either dimethylsulfoxide (DMSO) and ultrapure water (Millipore system) or glycerol and ultrapure water. The particles were first dispersed in DMSO or glycerol, after which water was added until the suspension was index-matched by eye. This resulted in mixtures of 91 wt% DMSO in water and 85 wt% glycerol in water.
Next, sample cells were constructed with standard microscopy slides and No.1.0-1.5 glass coverslips (Menzel-Gläzer). After the cells were filled with the suspension, they were sealed with UV-glue (Norland No. 68). The suspensions were imaged with a confocal microscope (Leica SP2 or Leica SP8) using a 63x/1.4 or 100x/1.4 oil-immersion confocal objective (Leica). We corrected the 3D images for distortion of the axial (*z*) distances due to the refractive index mismatch between sample ($n_D^{21}$ = 1.45) and immersion oil ($n_D^{21}$ = 1.51), which resulted in an increase of axial distances of 5% [@Besseling2014]. Fig. \[fig:particles\]b shows a 3D confocal microscopy image of a single rod suspended in 85 wt% glycerol in water. In Figs. \[fig:particles\]c-e, three orthogonal slices through this 3D volume are shown. The larger width of the PSF in the axial ($z$) direction is clearly visible. Notice that the pixel size in *x,y* (50 nm) is smaller than in *z* (78 nm). Figs. \[fig:particles\]f-h show the same rod after rescaling to cubic pixels, filtering and particle-fitting. In Fig. \[fig:particles\]i, we show the intensity histograms of two rods that were oriented parallel to the *xy* image plane of the confocal microscope. The continuous (red) line shows the intensity histogram of a single uniformly dyed rod and the dashed (green) line that of a gradient-dyed rod [@Kuijk2014a].
![ \[fig:particles\] Particle fitting in 3D. (a) Transmission electron microscopy (TEM) micrograph of fluorescently labelled silica rods with length $l = 2.37$ $\mu$m ($\delta$ = 10%) and diameter $d$ = 640 nm ($\delta$ = 7.5%). (b) 3D confocal microscopy image of a single rod suspended in 85wt% glycerol in water. (c-e) Three orthogonal slices through the 3D confocal image shown in (b). The scale bar is 800 nm. (f-h) The particle after rescaling, filtering and fitting. The magenta outline indicates the final fit from which the position and orientation is computed. (i) The (normalized) intensity histograms along the major axis of two differently dyed particles that are oriented in the $xy$ plane, obtained from confocal microscopy images. The (red) solid line is from a uniformly dyed silica rod. The (green) dashed line is from a silica rod with a gradient in dye distribution.](intensity_profiles_v9.pdf){width="45.00000%"}
### Freely diffusing silica rods
For the experimental measurements on a dilute suspension of silica rods we used particles with length $l$ = 3.3 $\mu$m ($\delta$ = 10%) and diameter $d$ = 550 nm ($\delta$ = 11%), as measured with TEM. The particles were fluorescently labelled with a 30 nm (FITC) shell. The particles were dispersed in an index matching mixture of 85 wt% glycerol in water. The density of the solvent mixture was $\rho = 1.222$ g/ml [@Segur1951] and the viscosity $\eta = 92$ cP (22$^\circ$C), as measured with an SV10 viscometer (A&D Company). This mixture not only matches the refractive index of the particles ($n_D^{25}=1.45$), the high viscosity slows down the particle dynamics enough to measure their short-time self-diffusion in 3D. Because the density of this mixture is significantly lower than the density of the particles $\rho = 1.9$ g/ml [@Kuijk2014a], sedimentation cannot be avoided. We assume, however, complete decoupling between translational motion, rotational motion and sedimentation [@Homogeneous1979]. A fused quartz capillary (Vitrocom) was filled with a dilute suspension (volume fraction $\phi < 1\%$) of the fluorescent silica rods. The suspension was imaged with a confocal microscope (Leica SP8) equipped with a fast 12 kHz resonant scanner and hybrid detector. Images with 8-bit pixel-depth were acquired using a white light laser with a selected wavelength of 488 nm. A confocal glycerol immersion objective 63x/1.3 (Leica) was used, which is optimized for refractive index $n_D = 1.45$. If we assume a Poisson distribution of the noise, we can easily estimate the signal to noise ratio (SNR) of a single image because of the photon counting mode of the hybrid detector. We use the definition SNR = $\sqrt{n_p}$ with $n_p$ the number of detected photons in the brightest part of the image [@Sheppard2006]. To avoid hydrodynamic interactions with the wall, particles were imaged 20 $\mu$m deep into the sample. We recorded 800 repeats of 3D image stacks consisting of 512 x 261 x 66 pixels with voxel size 144 x 144 x 331 nm. The time to record a single 3D volume was $\tau = 1.80$ s. During this time, the particles are expected to translate on average $\sqrt{2\, D_t \, \tau} = 110$ nm in each direction and rotate only $\sqrt{4\, D_r \, \tau} = 0.1$ rad.
### AuNRs@SiO$_2$ & 3D electron tomography
For the fabrication of a spherical cluster of nanorods, we first synthesized gold nanorods following the method described in Ref. . Next, the gold rods were coated with a layer of mesoporous silica (AuNRs@SiO$_2$) [@Gorelikov2008], which resulted in particles with length $l = 119$ nm and diameter $d = 68$ nm, as measured with TEM. Afterwards, clusters were fabricated via an emulsification process [@Peng2013; @deNijs2014]. Brightfield TEM tilt series of an 11-particle NR-cluster were acquired by tilting the sample over a range of -65$^\circ$ to 65$^\circ$ and recording images every 2$^\circ$. Images were taken on a Tecnai 20 (FEI) transmission electron microscope, operating at 200 kV with an LaB$_6$ electron source, in bright field mode. Tomographic reconstructions of the images were made with the iMOD software package using the simultaneous iterative reconstruction technique (SIRT) [@Kremer1996; @Mastronarde1997]. After reconstruction, the data stack was filtered using a low frequency Fourier filter (iMOD) and inverted to ensure light particles on a dark background to enable individual particle identification.
Results
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Determination of 3D particle positions & orientations in dense suspensions
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To test our 3D particle-fitting algorithm we identified the fluorescent particles in a concentrated suspension of silica rods, as shown in Fig. \[fig:dense\_B31\]. The particles were uniformly dyed, had a length $l$ = 2.37 $\mu$m (10%), diameter $d$ = 640 nm (7.5 %) aspect ratio $l/d$ = 3.7 and were dispersed in a 85 wt% glycerol in water mixture. Small regions of hexagonally stacked particles existed in the sample (Fig. \[fig:dense\_B31\]c), however there was no long-ranged order in the sample and particles seemed jammed or arrested in different orientations. Fig. \[fig:dense\_B31\]a shows that 5.2 $\mu$m deep in the sample, the fluorescent signals of the particles did not overlap significantly in *xy*, despite the high particle concentration. This is due to the 190 nm non-fluorescent outer shell of the particles which was deliberately grown around the particles during synthesis to resolve them individually, even when they were lying side-by-side. However, the orthogonal slices in Figs. \[fig:dense\_B31\]b-c show that particle signals did overlap in the *z*-direction, even after noise filtering. Nevertheless, by visual inspection of the (magenta) particle outlines in Figs. \[fig:dense\_B31\]d-f we conclude that the algorithm correctly identified the orientations and positions of the particles, despite the high particle concentration.
![Local order in a dense sediment of rods with length $l$ = 2.37 $\mu$m (10%), diameter $d$ = 640 nm (7.5 %) and aspect ratio $l/d$ = 3.7, dispersed in a glycerol/water mixture. The dimensions of the image volume were 512 x 201 x 79 pixels with voxel sizes 60 x 60 x 83 nm in $x$,$y$ and $z$. The time to record the complete stack was 3.37 s. (a-c) Close-ups of orthogonal slices through the 3D image, after filtering and (d-f) after particle identification. The scale bars are 3 $\mu$m. (g) Computer rendered 3D reconstruction of the sample with the RGB value of the colour indicating the particle orientations. \[fig:dense\_B31\]](tracking_3D_2.pdf){width="45.00000%"}
Fig. \[fig:dense\_B31\]g shows a computer generated reconstruction of the sample, with colours indicating the 3D orientation of the particles.\
Fig. \[fig:dense\_gradient\] shows a second example of the performance of our fitting-algorithm in a concentrated suspension. The rods in this sample had length $l$ = 2.6 $\mu$m (8.5 %), diameter $d$ = 630 nm (6.3 %) and were dispersed in an index-matching mixture of DMSO/water. After the particles had been left to sediment for several days, they ordered into smectic layers, more or less parallel to the *xy*-plane, as can be seen from Fig. \[fig:dense\_gradient\]a (12.5 $\mu$m deep in the sample). It can also be seen that the particles had an intensity gradient along their major axis and that there was significant overlap of the fluorescent signals in the *xy*-image (Fig. \[fig:dense\_gradient\]a). As expected, it was even more difficult to resolve individual particles in the $z$-direction (Figs. \[fig:dense\_gradient\]b-c), however, it is clear from the hexagonal pattern in Fig. \[fig:dense\_gradient\]b that the particles formed a smectic-B phase. The magenta outlines in Fig. \[fig:dense\_gradient\]d-f show the result of the particle fitting. By visual inspection of the outlines in the complete image-stack (containing 1699 particles), we conclude that $>$ 98% of the particles had been correctly identified by the algorithm. In Fig. \[fig:dense\_gradient\]g we show a 3D reconstruction of a part of the image-stack, which clearly shows 3D orientational order, smectic layering, and transverse (red and blue) particles.
![Smectic-B phase of rods with length $l$ = 2.6 $\mu$m (8.5 %), diameter $d$ = 630 nm (6.3 %) and aspect ratio $l/d$ = 4.1, dispersed in a DMSO/water mixture. The dimensions of the image volume were 256 x 256 x 151 pixels with voxel size 58 x 58 x 104 nm in $x$,$y$ and $z$. The time to record the image stack was 73.3 s. (a-c) Orthogonal sections after filtering. (a) 12.5 $\mu$m deep in the sample, particles were ordered in smectic-like layers. (d-f) Identified particles are outlined in magenta. All scale bars are 3 $\mu$m. (g) Computer rendered 3D reconstruction of the sample with the RGB value of the colour indicating the particle orientations. \[fig:dense\_gradient\]](tracking_3D_3.pdf){width="45.00000%"}
Determination of 3D positions and orientations of gold nanorods
---------------------------------------------------------------
Although our algorithm was written for analysis of confocal microscopy images, it is also applicable to other 3D image-stacks of uniaxial symmetric particles. As an example, we show results of the identification of gold nanorods (AuNRs) from a 3D transmission electron microscopy (TEM) tomographic reconstruction in Fig. \[fig:gold\_rods\]. The TEM micrograph in Fig. \[fig:gold\_rods\]a shows the gold nanorods (in black), that were coated with a layer of mesoporous silica (dark grey).
![ \[fig:gold\_rods\] Identification of the positions and orientations of 11 gold nanorods coated with mesoporous silica (AuNRs@SiO$_2$), confined in a small spherical cluster. (a) A single TEM image that was part of the tilt-series used for the tomographic reconstruction. (b) An *xy* and (c) *zy*-view of the 3D electron tomogram. The images were inverted for particle identification. (d) Corresponding *xy* and (e) *zy*-views of the filtered images with the identified particles outlined in red. (f) 3D reconstruction of the nanorod cluster. Colours indicate their 3D orientations. ](gold_rods_cluster_v5.pdf){width="45.00000%"}
Fig. \[fig:gold\_rods\]b-c show two orthogonal sections through the 3D reconstruction of the cluster. The images were inverted to enable particle identification with our algorithm. Fig. \[fig:gold\_rods\]d-e show the same orthogonal sections after filtering, with identified particles outlined in red. Finally, Fig. \[fig:gold\_rods\]f shows the 3D reconstruction, with color-coding of the 3D orientation of the rods. The algorithm had identified all 11 particles and the reconstruction clearly shows that there was some degree of orientational ordering inside the cluster.\
We are aware that a substantial amount of information on the 3D structure of the nanoparticles can be measured directly (and manually) from the 3D tomogram itself. Our image-processing algorithm however can determine unambiguously the 3D positions and orientations of the particles and can therefore be useful for the quantification of (larger) nanoparticle assemblies and should in principle also work on other types of samples, e.g. self-assembled clusters of nano-dumbbells [@Grzelczak2012].
Testing the accuracy of the algorithm for non-overlapping particle signals
--------------------------------------------------------------------------
In this section, we assess the accuracy of our algorithm in more detail. We focus on the fitting accuracy of the algorithm when applied to images containing particle signals that are well separated. Although this situation is much less demanding compared to partially overlapping signals, care has to be taken when fitting this type of data as well. The main reason is that the (fluorescent) diameter of typical rod-like particles used in our experiments ($d_{fl} \sim 300$ nm) is comparable to the resolution of a typical confocal microscope ($200 - 300$ nm in the lateral and $500 - 700$ nm in the axial direction [@Wilson1990; @Besseling2014]). Additionally, the PSF itself is anisotropic, which can result in a (strongly) distorted particle shape. Things become progressively worse when there is a refractive-index mismatch between the sample and immersion fluid, which deteriorates the PSF, introduces an intensity fall-off with height and distorts axial distances [@Hell1993; @Besseling2014].
We therefore determined the accuracy of our algorithm using two approaches. In the first approach we investigated both the effect of the theoretically approximated PSF and the effect of noise on particle tracking accuracy using computer generated data. The second approach consisted of an experimental measurement of the translational and rotational diffusion of a dilute suspension of silica rods.\
Details on the construction of the test-images and variation of the theoretically approximated PSF and noise can be found in the Supplementary Information. The results are summarized in Table \[tab:errors\], which shows that for our worst-case scenario of a $z$-resolution of 636 nm and signal to noise ratio of 1.7, we obtain for the error in the determination of the main-axis of the rod $\epsilon_r = 0.07$ rad, which corresponds to a small measurement error of $4.1^\circ$, see also Fig. S1. Additionally, we did not find any significant pixel-bias in either the position or the orientation (see Fig. S2). For the error in the positional measurement, we found $\epsilon_t/d$ $\sim$ 0.05, which indicates sub-pixel accuracy.
\[h\]
-------------- ----------- ------------ ------ --------------------
$\sigma_z/d$ FWHM (nm) $\sigma_n$ SNR $\epsilon_r$ (rad)
0.3 212 0.09 13.5 0.025
0.6 424 0.09 11.4 0.026
0.9 636 0.09 11.2 0.036
0.9 636 0.18 3.8 0.048
0.9 636 0.27 1.7 0.071
-------------- ----------- ------------ ------ --------------------
: Static measurement error $\epsilon_r$ in the determination of the main axis of the rod, assuming $d$ = 300 nm. The error increases with both $\sigma_z$ and $\sigma_n$. For the worst case scenario of $\sigma_z/d = 0.9$ and $\sigma_n = 0.27$, the value for $\epsilon_r$ remains rather small.[]{data-label="tab:errors"}
Finally, we measured the diffusive motion in a dilute suspension of fluorescent silica rods experimentally, which provides a real-life test of the accuracy of our algorithm. The rods that were used had length $l$ = 3.3 $\mu$m ($\delta$ = 10%), diameter $d$ = 550 nm ($\delta$ = 11%) and aspect ratio $l$/$d$ = 6.0. From the number of photons in the brightest part of the image we estimated the signal to noise ratio to be SNR $\approx 3$, which is in the range stated in Table \[tab:errors\]. The tracking results, averaged over 8 particles, are shown in Fig. \[fig:track\_exp\]. A typical translational trajectory of 12 min is shown in Fig. \[fig:track\_exp\]a. From a fit to the average linear displacements (of all 8 particles) in the z-direction, we estimate the sedimentation speed to be $v_{sed} = 0.331\,\pm\,$0.005 $\mu$m/min. This value is slightly higher but comparable to the value of $v_{sed} = 0.28$ $\mu$m/min that we obtained from equation . For further analysis we subtracted the average linear displacements from the trajectories. Fig. \[fig:track\_exp\]b shows a rotational trajectory of 12 min for a single particle. In Fig. \[fig:track\_exp\]c we show the probability distribution of the norm of the displacement $|\Delta {\mathbf r}|$, for three different time-steps $\Delta$t. In Fig. \[fig:track\_exp\]d we show the same distribution for the norm of the displacements of the unit vector $|\Delta {\mathbf{\hat u}}|$. The solid black lines in Figs. \[fig:track\_exp\]c,d are fits proportional to $|\alpha|^2 \exp(-|\alpha|^2)$ with $\alpha = {\Delta\mathbf r},{\Delta\mathbf{\hat u}}$ respectively.
\[ht!\] {width="100.00000%"}
To extract the translational diffusion coefficient, we calculated the rotationally averaged mean squared displacement $\langle \Delta \mathbf{r}^2 \rangle$, as can be seen in Fig. \[fig:track\_exp\]e. For $\Delta t > 10$ s we found that $\langle \Delta \mathbf{r}^2 \rangle \sim t^{0.97}$ indicating diffusive behaviour. The statistical error in the individual measurement points is smaller than the symbol size. Fitting the data with equation , we obtain the short-time rotationally averaged translation diffusion coefficient $D_t = (3.06\,\pm\,0.01)\,\times\,10^{-3}$ $\mu$m$^2$/s and static error $\epsilon_t = 45, 46$ and $59$ nm in the $x$, $y$ and $z$-direction respectively, which confirms that we can locate the particles with sub-pixel accuracy. The value for $D_t$ is in strong agreement with the theoretical value obtained from equation which is $D_t$ = 3.2 $\times\,10^{-3}$ $\mu$m$^2$/s. Finally, we calculated the mean squared angular displacement $\langle \Delta \mathbf{\hat {u}}^2 \rangle$, as shown in Fig. \[fig:track\_exp\]f. This time we obtained $\langle \Delta \mathbf{\hat {u}}^2 \rangle \sim t^{0.92}$ and for the short-time rotational diffusion coefficient $D_r$ = (1.32$\,\pm\,0.02)\,\times\,10^{-3}$ rad$^2$/s. This is in good agreement with the theoretical value $D_r$ = 1.5 $\times\,10^{-3}$ rad$^2$/s, obtained from equation . For the corresponding rotational relaxation time we found $\tau_r = 1/(2 \, D_r)$ = 3.8 $\times\,10^2$ s, which confirms that we measured in the short-time diffusion regime. From the fit we also obtained $\epsilon_r$ = 0.07 rad, which corresponds to a small angular uncertainty of $\sim$ 4$^\circ$.
Discussion
==========
In this paper we demonstrated a new image-processing algorithm that is capable of extracting the positions and orientations of fluorescent rod-like particles in both dilute and concentrated suspensions. Although the algorithm was written for three dimensions, most steps are straightforward to modify for two dimensions [@Besseling]. Mohraz and Solomon [@Mohraz2005] were the first, as far as we know, to describe an algorithm that can detect the position and orientation of ellipsoidal particles in 3D confocal microscopy images and this work follows a similar approach. The algorithm of Mohraz and Solomon groups clusters of pixels together to form backbones but does not use additional fitting steps, which we found necessary to correctly identify particles when there is significant overlap of particle signals. The difference in particle geometry (ellipsoids versus rods) combined with the small (fluorescent) particle diameter in our study might be the reason why we find that using only a maximum threshold and cluster analysis is not sufficient to identify rods in concentrated suspensions, even when the rods have a large ($>
150$ nm) non-fluorescent shell and a considerable electric double layer ($\sim
100$ nm) [@Kuijk2011; @Kuijk2012a]. The rod-like particles used in this study have a repulsive interaction potential and therefore form dense smectic-like phases, which we now can identify on the single-particle level in the bulk. The algorithm also enables the study of glassy phases of anisotropic particles in three dimensions, which is promising since all current real-space glass-transition studies of anisotropic particles so far are either 2D [@Yunker2011a; @Zheng2011; @Mishra2013] or tracer-host [@Edmond2012]. Finally, we would like to mention that the algorithm is also applicable to study the dynamics of (concentrated) ‘active colloids’ (e.g. self-propelled particles and bacteria), a field that is rapidly emerging [@Ball2013].\
Since the typical fluorescent diameter of the rod-like particles is around 300 nm, deconvolution of the image-stacks before particle fitting can be useful when particles are difficult to resolve individually. The necessary higher (Nyquist) sampling rate, however, is not always practical or even not possible for faster moving particles. Additionally, deconvolutions are sensitive to small changes in the rod-spread-function (RSP), introduced by e.g. polydispersity. A clear improvement of the algorithm, therefore, is to fit the particles with the RSP, analogous to the fitting of the sphere-spread-function (SSF) reported by Jenkins *et al.* [@Jenkins2008], which is work currently ongoing. With this type of extension of the current algorithm it should also be possible to accurately measure in-situ particle polydispersity, which is known to have a large effect on e.g. the liquid-crystalline phase behaviour [@Bates1998].\
By measurement of freely diffusing rods, we acquired additional information on the accuracy of our algorithm. Although the motion is analysed in the lab-frame and therefore translational and rotational motion should be coupled [@Han2006], we did not observe such behaviour since the friction anisotropy in our 3D measurement is small $D_\parallel$/$D_\perp$ = 1.3 and because we averaged over an ensemble of particles and over many initial orientations. We found that the error in locating the rods ($\epsilon_t = 45, 46$ and $59$ nm in the $x$, $y$ and $z$-direction respectively) confirms sub-pixel accuracy and agrees roughly with the criterion for spherical particles that $\epsilon_t \sim M/N$, with $M$ the pixel-size and $N$ the diameter of the particle in pixels [@Crocker1996]. The value for the (short-time) rotational diffusion coefficient $D_r$ = (1.32$\,\pm\,0.02)\,\times\,10^{-3}$ rad$^2$/s, is one order of magnitude larger than previously accessible with 3D confocal microscopy [@Mukhija2007] which is, however, due to the equipment rather than the image-processing. The error in the determination of the orientation of the rod ($\epsilon_r$ = 0.07 rad) is in the range of the values obtained via simulated test-images, shown in Table \[tab:errors\]. The rule-of-thumb that $\epsilon_r \sim 1/P_a$ with $P_a$ the half-length of the rods in pixels [@Mukhija2007] seems to hold quite well in our case, since $1/P_a$ = 0.08 in our measurements.
Conclusion
==========
We developed an algorithm that extracts the positions and orientations of rod-like particles from 3D confocal microscopy images. The algorithm is tailored to a system of fluorescently labelled silica rods and can identify these particles even in the bulk of 3D concentrated phases where the fluorescent signals of the particles overlap considerably. This allowed us to determine the 3D positions and orientations of particles in a concentrated disordered phase and in a liquid-crystalline smectic-B phase. The algorithm also works on electron tomography reconstructions of gold nanorods, which enables the 3D quantification of (large) nano-particle assemblies. It is also expected to work on other uniaxial particles such as ellipsoids or dumbbells. We determined the accuracy of the algorithm for varying $z$-resolution and noise levels from generated 3D test-images. Despite the (anisotropic) distortion of the theoretically approximated point spread function (PSF) and the low signal to noise ratio (SNR), the error in the determination of the orientation of the particles remained small. These results confirmed that we can accurately track rod-like particles with (fluorescent) diameters down to 300 nm. With our algorithm and a fast confocal microscope we determined the translational and rotational motion of a dilute suspension of sedimenting silica rods. We demonstrated that the measured diffusive motion was in good agreement with theory (neglecting sedimentation) and that we can track the particles with sub-pixel resolution.\
This novel algorithm therefore allows for studies of structure and dynamics on the particle level of dense liquid-crystalline phase behaviour (such as nematic, smectic and crystalline phases), but also allows for studies of the glass transition of anisotropic rod-like particles in three dimensions. Of course, the algorithm will also be applicable to dilute suspensions or in cases where rod-like particles are used as tracers, such as in biophysical or micro-rheology studies.
Acknowledgement
---------------
The authors thank E. Weeks and A. Gantapara for useful discussion, H. Bakker, C. Kennedy and B. Liu for useful feedback on the algorithm and H. Meeldijk for help with the electron microscopy data. This research was carried out partially (THB) under project number M62.7.08SDMP25 in the framework of the Industrial Partnership Program on Size Dependent Material Properties of the Materials innovation institute (M2i) and the Foundation of Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO). Part of the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ ERC Grant Agreement no. \[291667\].
Author Contributions
--------------------
AvB initiated the research. MD, AI and AvB supervised the research. MH and THB contributed equally to this work. MH and THB wrote and tested the particle fitting algorithm and analysed the data. AK performed synthesis of the fluorescent silica rods. TSD performed synthesis of the gold nanorods, fabrication of the nanorod-clusters and acquired the tilt-series of TEM images. BdN performed the electron tomography reconstruction. THB and AK performed the confocal microscopy experiments. THB, MH and AvB co-wrote the manuscript.
[^1]: Both authors contributed equally
[^2]: Both authors contributed equally
|
---
abstract: 'The properties of charmonium states are or will be intensively studied by the B-factories Belle II and BESIII, the LHCb and PANDA experiments and at a future Super-$c$-$\tau$ Factory. Precise lattice calculations provide valuable input and several results have been obtained by simulating up, down and strange quarks in the sea. We investigate the impact of a charm quark in the sea on the charmonium spectrum, the renormalization group invariant charm-quark mass $\Mc$ and the scalar charm-quark content of charmonium. The latter is obtained by the direct computation of the mass-derivatives of the charmonium masses. We do this investigation in a model, QCD with two degenerate charm quarks. The absence of light quarks allows us to reach very small lattice spacings down to $0.023~$fm. By comparing to pure gauge theory we find that charm quarks in the sea affect the hyperfine splitting at a level below 2%. The most significant effects are 5% in $M_c$ and 3% in the value of the charm quark content of the $\eta_c$ meson. Given that we simulate two charm quarks these estimates are upper bounds for the contribution of a single charm quark. We show that lattice spacings $<0.06~$fm are needed for safe continuum extrapolations of the charmonium spectrum with O($a$) improved Wilson quarks. A useful relation for the projection to the desired parity of operators in two-point functions computed with twisted mass fermions is proven.'
address:
- 'Department of Physics, University of Cyprus, P.O. Box 20537,1678 Nicosia, Cyprus'
- 'Department of Physics, Bergische Universit[ä]{}t Wuppertal, Gaussstr. 20, 42119 Wuppertal, Germany'
author:
- Salvatore Calì
- Francesco Knechtli
- Tomasz Korzec
title: 'How much do charm sea quarks affect the charmonium spectrum?'
---
|
---
abstract: 'This paper reports on polarimetric radiation properties based on the switching modes of normal PSR B2020+28 by analysing the data acquired from the Nanshan 25-m radio telescope at 1556 MHz. With nearly 8 hours quasi-continuous observation, the data presented some striking and updated phenomena. The change of relative intensity between the leading and trailing components is the predominant feature of mode switching. The intensity ratio between the leading and trailing components are measured for the individual profiles averaged over 30 seconds. It is found that there is an excess of high ratios over the normal distribution, which indicates that two modes exist in the pulsar. The distribution of abnormal mode has a narrower width indicating that the abnormal mode is more stable than the normal mode. A total of 76 mode switching events are detected in our data. It spends 89% in the normal mode and 11% in the abnormal mode. The intrinsic distributions of mode timescales are constrained with power-law distributions. The significant difference in the index of the duration distribution between normal and abnormal modes possibly indicates that the timescale for the abnormal mode to get stable is shorter than that for the normal mode. The frequent switching between both modes may indicate that the oscillations between different magnetospheric states are rapid.'
author:
- 'Z. G. Wen'
- 'N. Wang'
- 'W. M. Yan'
- 'J. P. Yuan'
- 'Z. Y. Liu'
- 'M. Z. Chen'
- 'J. L. Chen'
title: The mode switching of PSR B2020+28
---
Introduction
============
Pulsars are rapidly rotating, extremely dense neutron stars that emit radio frequency electromagnetic radiation from regions above their magnetic polar caps. As a pulsar rotates, the radiation originating from these regions sweeps through space much like a beam of a lighthouse. It sweeps across our line of sight once per rotation, hence we can receive the pulse signal. The pulsar presents a regularly spaced pulses with a repetition respectively stable period (hereafter $P$). In this case, the rotation frequency $\nu=1/P$. The individual pulses are very weak and vary with time significantly. In order to produce a stable profile, it requires the coherent addition of many hundreds or even thousands of pulses together. However, some pulsars show two or more patterns of averaged pulse profiles, which is called mode switching (or mode changing) phenomenon. Since the discovery of mode switching in PSR B1237+25 [@Backer+1970], this phenomenon has been seen in a few dozens of pulsars. However, the integrated profiles of these pulsars are quite complex, e.g. PSR B1237+25 has 5 components [@Backer+1970; @Rankin+1986], PSR B0329+54 has triple components [@Rankin+1986; @Chen+etal+2011]. The recent observations of PSRs B0723+26 [@Sobey+etal+2015] and B0943+10 [@Bilous+etal+2014] with LOFAR show two relatively stable states (bright and quiet modes). The forced Markov process was recently analysed by @Cordes+2013 to model state changing pulsars.
The radio emission from pulsars is highly polarized [@Lyne+Smith+1968]. The mean pulse profile and the polarization measurements can yield a wealth of information, not only about the emission process itself, including the pulse emission mechanism, the beaming of pulsar radiation and the geometry of the system, but also about the medium through which it propagates. The integrated pulse profiles can be described with core/double-cone geometric model [@Rankin+1983]. The observed polarization position angle (hereafter $PA$) variations can be approximately described by the Rotating Vector Model (RVM) [@Radhakrishnan+Cooke+1969]. This model results from the idea that the radiation is polarized in the plane of curvature of field lines emanating from a magnetic pole on the star [@Komesaroff+1970]. For a simple dipole field, the observed PA variation is then determined by the projected direction of the magnetic axis as the star rotates. The rapid swings often observed near the profile midpoint imply that magnetic axis is nearly aligned with the observer’s line of sight at that profile phase. The observed PA swings are various. @Yan+etal+2011 presented polarization profiles of 20 pulsars with smooth, continuous and discontinuous PA variation. Discontinuities of approximately $90^\circ$ are often observed [@Manchester+etal+1975; @Backer+Rankin+1980; @Stinebring+etal+1984; @Han+etal+2009], and these are interpreted as resulting from overlapping emission from orthogonally polarized emission modes [@McKinnon+Stinebring+1998; @Gangadhara+1997]. Such orthogonal polarization modes may have resulted from the radiation emitted by positrons and electrons while moving along the curved magnetic field lines [@Gangadhara+1997], or generated as the wave propagates through the pulsar magnetosphere [@Petrova+2001]. The circular polarization is usually relatively weaker than the linear polarization. It is most often associated with the central or core component of the profile, often with a sense reversal near the profile mid-point [@Rankin+1983].
PSR B2020+28 is a relatively strong pulsar with flux density of 38 mJy at 1400 MHz. The profile of this pulsar is relatively simple, having only two resolved components separated by 0.027 in pulse phase. The period P is 0.3434 seconds, and its first derivative is $\rm{1.89\times10^{-15}\, s\,s^{-1}}$. Correspondingly, its characteristic age ($\rm{\tau=2.87\times10^6\, years}$) is no more than the pulsar median. It was discovered with the east-west arm of the Bologna Cross telescope at 408 MHz [@Bonsignori-Facondi+etal+1973]. Polarimetric observations were obtained with the 76-m Lovell telescope at Jodrell Bank at radio frequencies centred around 230, 400, 600, 920, 1400, 1600 MHz, which gives the $W_{10}$, $W_{50}$, $W_e$, $L$, $|V|$, $V$ and polarization profiles respectively [@Gould+Lyne+1998]. There exist two orthogonal modes of polarization that have position angles separated by $90^\circ$ and opposite senses of circular polarization [@Cordes+etal+1978; @Stinebring+etal+1984]. This pulsar also shows significant pattern change and is complicated by interstellar scintillation [@Wang+etal+2001]. However, the profiles of normal and abnormal modes have not been published so far. The radiation of various properties from pulsars are explained most naturally by a simple picture in which the observed radiation is produced by the acceleration of charged particles streaming outward along open filed lines above the poles of an essentially dipolar magnetic field.
In this paper we show a detailed investigation of the emission behavior of PSR B2020+28. In section 2, we describes the observing system and our observations. Section 3 presents data analysis and results. The implications of the results and conclusions are discussed in section 4.
Observations {#sect:Obs}
============
The observations of PSR B2020+28 were carried out using the Nanshan 25-m radio telescope on 2012 June 12. A dual-channel cryogenic receiver was used to receive orthogonal circular polarizations at a centre radio frequency of 1556 MHz, and with total bandwidth of 512 MHz. The system temperature, $T_{sys} = T_{rec} + T_{sky} + T_{spi}$ (in which $T_{rec}$, $T_{spi}$ and $T_{sky}$ are the receiver, spillover and the sky noise temperature, respectively), is approximately 32 K. An ortho-mode transducer (OMT) was used to resolve the electromagnetic wave into left-handed circular ($L$) and right-handed circular ($R$) basis modes. To determine the relative gain of the two polarization channels and the phase between them, a calibration signal is injected at an angle of $45^\circ$ to the feed probes. The two independent polarizations were then amplified and down-converted to an intermediate frequency in the range of 0$-$512 MHz with a local oscillator at 1300 MHz. The band-limited signals were fed to an updated back-end signal processing system, the third-generation Digital Filterbank System (DFB3), since 2010. After conversions from analogue voltages to digital signals at the Nyquist rate with 9-bit sampling, the DFB3 uses field-programmable gate array (FPGA) processors to produce a maximum of 8192 polyphase filterbank frequency channels and averaged pulse profiles. In order to obtain enough signal-to-noise ratio (S/N), about 90 pulse periods ( 30 s) were averaged and a time resolution of 512 bins per pulse period was used in our observations. The data lasting nearly 8 hours from 1024 frequency channels were then recorded for off-line processing, which contain 75400 single pulses.
The data were calibrated carefully. The averaged pulses were obtained by de-dispersing the data at a dispersion measure (DM) of $24.6\ \rm{pc\ cm^{-3}}$. Four Stokes parameters were recorded and have been corrected for dispersion, interstellar Faraday rotation and various instrumental polarization effects following the method described by [@Yan+etal+2011]. The data are analysed offline using the PSRCHIVE package [@Hotan+etal+2004] and corrected for parallactic angle and the orientation of the feed. The four Stokes parameters are accessible in function (1), $$\centering
\left[\begin{array}{c}I \\
Q \\
U \\
V \\
\end{array}\right]=\left[\begin{array}{c}|L|^2+|R|^2 \\
2Re(L*R) \\
2Im(L*m) \\
|R|^2-|L|^2 \\
\end{array}\right]
\label{eq:LebsequeI}$$ where \* indicates a complex conjugate [@Lorimer+2005].
Data reduction and analysis
===========================
The structure and brightness of individual pulses are observed to vary significantly, but the average of many hundreds of individual pulses is usually stable, leading to a characteristic profile that is often unique to pulsar. However, some pulsars exhibits two or more discrete and well-defined pulse profile morphologies, and they switches abruptly, which is known as mode-switching. The change of relative intensity between the different components is the predominant feature of mode switching [@Chen+etal+2011].
At 1.5 GHz, the average pulse profile of PSR B2020+28 is relatively simple, having double well resolved main emission components (leading and trailing components), which is common to many pulsars. They are connected by a pronounced bridge emission. The outer boundaries of main components are confined at 10% of pulse peak of the average profile, while the inner boundary is defined as the minimal intensity between them. The examples of average pulse profile (solid line) and phase boundaries (vertical dotted lines) are shown in Fig. \[waveform\]. The polarization characteristics of the mean pulse profile provide a framework for understanding the emission processes in pulsars. The averages of linear polarization (dashed line) is a maximum in the leading component at 30% compared with 22% in the trailing component. Two components are depolarized relative to low-frequency observations, but the effect is most noticeable in the leading component of the pulse [@Cordes+etal+1978; @Stinebring+etal+1984]. The averages circular waveform (dash-dotted line) is almost negligible in the leading component and a sense reversal slightly trails the leading nulls in the linear polarization. The extrema of the circular polarization occur under the linear maximum in the trailing component. The position angle rotation (lower panel) in the saddle region shows a S-shaped sweep with total swing only about 30 degrees. Whereas, two peaks in the leading and trailing regions show up.
![Polarization waveform of PSR B2020+28. The uniformly weighted values of the Stokes parameters are averaged over nearly 8 hours, with total intensity $I$, circular polarization $V$, and the linear polarization $L = (Q^2 + U^2)^{1/2}$ (where $Q$ and $U$ are the linear Stokes parameters), are displayed in solid, dot-dashed and dashed lines respectively. The boundaries are plotted with dotted lines to distinguish leading and trailing components. The position angle as a function of pulse phase, $\chi =1/2\tan^{-1}(U/Q)$, is given in the lower panel.[]{data-label="waveform"}](waveform.eps){width="7.8cm"}
The dynamic spectrum is shown in Fig. \[Dynamic\_spectrum\]. It seems like that the intensity fluctuations with time and frequency are very similar to the case of the observations by @Wang+etal+2001. The scintles (the red patches) have a characteristic timescale (the scintillation timescale) of three thousand seconds. The horizontal stripes in the dynamic spectrum are because the bad frequency channels are affected by Radio Frequency Interference (RFI). At the edges of the observing frequency, there is no useful data as the bandpass rolls off. The scintillation bandwidth (the characteristic frequency scale of the scintiles) is clearly less than the total bandwidth of the observation. The effects of such large fluctuations can be addressed by subtracting a running mean of length one fifth of the scintillation timescale.
![The dynamic spectrum of scintillations for PSR B2020+28. Here the measured signal to noise of the pulsar signal is plotted as a function of both time and frequency. The horizontal and vertical stripes in the dynamic spectrum are where data were affected by RFI.[]{data-label="Dynamic_spectrum"}](Dynamic_Spectrum.eps){width="8.2cm"}
The whole 8-hour observations with 912 scintillation-corrected sub-integrations, each is averaged over 30 seconds, are shown in Fig. \[seq\]. The variation of relative peak intensity ($R_I$) between the leading and trailing components are shown in the right panels. It varies frequently in a wide range from 0.5 to 1.6. No strong regularity is found easily by visual inspection. Discrete Fourier Transform (FFT) and auto-correlation were done on the whole 8-hour time sequence of $R_I$ values, and no presence of periodic signals was detected in either methods.
{width="8.0cm"} {width="8.0cm"} {width="8.0cm"} {width="8.0cm"}
The distribution of $R_I$ over nearly 8 hours is shown in Fig. \[histogram\], which extends over an extremely wide range. The histogram consists of a broad Gaussian component and a long tail component. As we can see, the best-fit solution gives us a combination of two Gaussian components corresponding to two emission modes. The pulses with $R_I$ in the range of $0.53 \sim 1.13$ are classified as normal mode, the remainder with $R_I$ in the range of $1.13 \sim 1.52$ are classified as abnormal mode. The distribution of the abnormal mode is considerably narrower than that of the normal mode, which may suggest that it is more stable than the normal mode.
![$\rm{R_I}$ distribution versus integration time of 30 seconds by using the data of nearly 8 hours observation. The solid line represents the fitting based on the combination of two Gaussian components (dashed lines). The lower panel shows the fit residuals.[]{data-label="histogram"}](hist.eps){width="8.0cm"}
Representative example of a sequence of 15 sub-integrated pulse profiles, arbitrarily normalized, is shown in the left panel of Fig. \[demo\]. The mode switching phenomenon is presented apparently as $R_I$ varies with time, which is shown in the right panel. The integrated pulse profiles from 445 to 447 switch from normal mode to abnormal mode. The averaged pulse profiles of both modes are shown in the insets. Through a careful inspection on the pulse profiles, it is noted that an additional component appears on the leading edge of the first main component for the abnormal pulse profiles.
{width="12.0cm"}
The integrated polarization properties of the normal (left panel) and abnormal (right panel) modes are given in Fig. \[Modes\_Profiles\]. The flux density in the abnormal mode is 1.3 times stronger than that of the normal mode. Note that the polarization profiles are slightly different as well. For the abnormal mode, the linear polarization intensity in the leading component increases by a factor of almost 45%, the circular polarization intensity decreases by a factor of almost 11%, compared with those of the normal mode. There are no significant variations in PA between both modes.
{width="8.0cm"} {width="8.0cm"}
To further investigate the relative properties of normal and abnormal modes, the durations in each mode are obtained. The histograms for the timescales of the normal (left) and abnormal (right) modes are shown in Fig. \[duration\]. A curve of best fit was calculated assuming a single power law (dashed line). The exponent, $
\alpha$, derived from the power-law best fits for the normal and abnormal modes are -1.0(1), -2.7(1) respectively. There is a significant difference in the power-law index of the duration distribution between normal and abnormal modes.
{width="8.0cm"} {width="8.0cm"}
In order to further to analysis the pulse-to-pulse fluctuation properties. The sub-integrated pulse profiles which are in the same range of $R_I$ are superposed in a group. The details of the division are shown in Table 1. The time proportions of seven groups are all more than $6$ per cent of the total observation time, which allow high S/N to analysis the polarization properties. The intervals of $R_I$ between two adjacent groups are nearly 0.1. The averaged polarization waveforms of seven groups are shown in Fig. \[waveforms\]. It is found that the seven groups all have the same pulse widths. The intermediate “saddle” regions in total and linear polarization profiles are relatively broad compared with the widths of the lobes, especially the linear polarization profiles. With the increase of $R_I$, the fractional circular polarization is decreased by a factor of almost 50%, and a factor of 26% in the linear polarization is increased. There is no much difference between these seven groups in PA variation.
[clclccccc]{} Group No. & $R_I$ Range & percentage & $R_I$ & $\sigma$ & $R_L$ & $\sigma$ & $R_L/R_I$ & $\sigma$\
(a) & $<0.7$ & 6.95% & 0.6897 & 0.0058 & 0.9558 & 0.0213 & 1.3859 &0.0330\
(b) & $0.7-0.8$ & 19.12% & 0.7864 & 0.0029 & 1.1990 & 0.0186 & 1.5246 &0.0244\
(c) & $0.8-0.9$ & 25.62% & 0.8768 & 0.0027 & 1.2908 & 0.0192 & 1.4722 &0.0223\
(d) & $0.9-1.0$ & 22.48% & 0.9729 & 0.0025 & 1.1649 & 0.0157 & 1.1974 &0.0165\
(e) & $1.0-1.1$ & 11.92% & 1.0711 & 0.0036 & 1.7747 & 0.0297 & 1.6569 &0.0283\
(f) & $1.1-1.2$ & 7.19 % & 1.1710 & 0.0045 & 1.7284 & 0.0393 & 1.476 &0.0340\
(g) & $>1.2$ & 6.72% & 1.2998 & 0.0041 & 2.1518 & 0.0041 & 1.6555 &0.0263\
{width="17.0cm"}
Following to the calculation of $R_I$, the linear polarization intensity ratio ($R_L$) is derived between two peaks in the linear polarization profile. The errors of $R_I$ and $R_L$ are derived from the standard deviation for each group. By means of the error transfer formula, the errors of $R_L/R_I$ are calculated. The intensity ratio ($R_L$) is also changing with that of averaged pulse profiles. And $R_I$ and $R_L$ are positive correlating after fitting with a straight line (see Fig. \[Fit\]), which indicates that the fractional linear polarization is constant. If this correlation is confirmed in other pulsars, $R_L$ may be also used as an indicator to identify the mode switching phenomenon and to figure out emission mechanism more than $R_I$.
![The positive correlation between $R_I$ and $R_L$ after fitting with a straight line. Corresponding error bars denote $3\sigma$ errors.[]{data-label="Fit"}](fit.eps){width="8.0cm"}
Discussion and conclusions
==========================
We report new results on the emission properties of pulsar B2020+28 based on detailed analysis of 8-hour observations at a center frequency 1556 MHz using the Nanshan 25-m radio telescope. A total of 76 mode switching events are detected. It spends 89% in the normal mode and 11% in the abnormal mode.
The major difference between the normal and abnormal modes is the intensity ratio between the leading and trailing components and the length of moding timescales. The variation of $R_I$ has no strong regularity, and the mode switching phenomenon seems a random process. The distribution of $R_I$ for the abnormal mode is narrower than that of the normal mode by a factor of 65%, which may indicate that the abnormal mode is more stable than the normal mode. The intrinsic timescale distributions, constrained for this pulsar for the first time, provide valuable information to understand the physics of mode switching phenomenon. The durations of abnormal mode are extremely short, which are less than 250 pulse periods. They may indicate that the timescale for the abnormal mode to get stable is shorter than that for the normal mode. The short durations of both modes are very common and long durations are much less, which may be represented by an elementary Markov process [@Cordes+2013].
The strong linear polarization is an outstanding characteristics of pulsars, which is usually associated with cone emission around magnetic field lines [@Gedalin+Dzigan+2005]. The stronger linear polarization in the leading and trailing components than that in the intermediate “saddle” region reveals the double-lobed structure in the total intensity is from the conal emissions. The circular polarization often accompanies core emission and is generally the strongest in the central or ’core’ regions of a profile. The sense reversal often occurs near the middle of the profile [@Rankin+1983]. But for PSR B2020+28, the circular polarization changes senses at the leading component and reaches peak value at the trailing component.
The mechanisms of mode switching phenomenon are proposed by many authors. Pulsar switches between different magnetospheric states are likely to be caused by changes of particle current flow in the pulsar magnetosphere [@Lyne+etal+1971; @Bartel+etal+1982]. In some cases, the pulse profile changes are also correlated with large changes in spin-down rates. [@Lyne+etal+2010]. Changes between two distinct emission states in PSR J0742$-$2822 is correlated with the changes in the derivative of the pulse frequency [@Keith+etal+2013]. The alteration of the temperature of pulsar surface could trigger different sparking modes in the inner vacuum gap, thus results in mode switching phenomenon [@Zhang+etal+1997]. The switches in the magnetosphere geometry or/and redistribution of the currents flowing in the magnetosphere change the pulsar emission beam and its orientation with respect to the line of sight and hence lead to the mode switching phenomenon [@Timokhin+2010]. For PSR B2020+28, the frequent switching between normal and abnormal modes suggests that the oscillations between two different magnetospheric states are rapid.
Normally, the pulse-to-pulse variability are hard to identify in sources, because summing over many pulses is required to achieve sufficient S/N. To distinguish mode switch and pulse-to-pulse fluctuations in pulse shapes, fitting to the distribution of $R_I$ with two Gaussian components has smaller residuals than that with one. Furthermore, the whole data set are examined carefully, approximately third of pulse profiles show an additional leading component in the intervals of abnormal mode. And this additional component is also detected in several normal pulse profiles. As shown in Fig. \[duration\], the duration of abnormal mode is relatively short, the detected abnormal pulse profiles may be contaminated by some normal pulses, so the additional weak leading emission becomes blurred. Therefore, the association between the additional component with intervals of abnormal mode is expected to exist in PSR B2020+28.
The relative pulse-to-pulse total intensity fluctuation is identified in variable linear and circular polarization. The evolution of fractional polarization with the increase of $R_I$ was also identified. The stable PA variations of different groups imply that the geometry of the pulsar emission beam remains constant, while the emission strength varies in different emission regions, as a result of the intensity variations of total and linear polarization. It is noted that most points deviate from the straight line shown in Fig. \[Fit\]. Such deviation maybe due to the error in the systematic polarization calibration, or perhaps caused by the intrinsic variation.
Mode switching is expected to occur over a broad-range of wavelengths [@Sobey+etal+2015], even up to X-rays. Therefore, we would require multi-frequency simultaneous single pulse observations to better understand these emission characteristics, which may lead to a better understanding of the magnetospheric emission mechanism.
We are grateful to the referee for valuable suggestions. This work was supported by National Basic Research Program of China grants 973 Programs 2015CB857100 and 2012CB82180, the Pilot-B project grant XDB09010203, and the West Light Foundation of Chinese Academy of Sciences (WLFC) No.XBBS201422. WMY is supported by NSFC (11203063, 11273051) and\
WLFC (XBBS201123). JPY is supported by NSFC 2012CB821801 and 11173041. We thank members of the Pulsar Group at Xinjiang Astronomical Observatory for helpful discussions.
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abstract: 'We measure the mass function for a sample of 840 young star clusters with ages between 10–300 Myr observed by the Panchromatic Hubble Andromeda Treasury (PHAT) survey in M31. The data show clear evidence of a high-mass truncation: only 15 clusters more massive than $>10^4$ are observed, compared to $\sim$100 expected for a canonical $M^{-2}$ pure power-law mass function with the same total number of clusters above the catalog completeness limit. Adopting a Schechter function parameterization, we fit a characteristic truncation mass of $M_c = 8.5^{+2.8}_{-1.8} \times 10^3$ . While previous studies have measured cluster mass function truncations, the characteristic truncation mass we measure is the lowest ever reported. Combining this M31 measurement with previous results, we find that the cluster mass function truncation correlates strongly with the characteristic star formation rate surface density of the host galaxy, where $M_c \propto$ [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{}$^{\sim1.1}$. We also find evidence that suggests the observed $M_c$–[$\Sigma_{\mathrm{SFR}}$]{} relation also applies to globular clusters, linking the two populations via a common formation pathway. If so, globular cluster mass functions could be useful tools for constraining the star formation properties of their progenitor host galaxies in the early Universe.'
author:
- 'L. Clifton Johnson, Anil C. Seth, Julianne J. Dalcanton, Lori C. Beerman, Morgan Fouesneau, Daniel R. Weisz, Timothy A. Bell, Andrew E. Dolphin, Karin Sandstrom, Benjamin F. Williams'
title: 'Panchromatic Hubble Andromeda Treasury XVIII. The High-mass Truncation of the Star Cluster Mass Function'
---
Introduction {#intro}
============
Star cluster populations are observational tracers of star formation activity in galaxies out to $\sim$100 Mpc distances. By comparing the properties of star cluster populations to the properties of overall star formation activity, studies of nearby galaxies have established that there is a correlation between the star formation rate (SFR) surface density, [$\Sigma_{\mathrm{SFR}}$]{}, and the fraction of stars that form in long-lived star clusters [e.g., @Adamo15; @Johnson16_gamma]. This correlation demonstrates a close connection between star clusters and their formation environment, where the rate of cluster formation is linked to the total SFR, but also to local galactic properties such as gas surface density and interstellar pressure [@Kruijssen12]. One implication of this result is that star clusters can reveal the characteristics of past star formation episodes long after they have ended. While cluster destruction through evaporation due to two-body relaxation, tidal shocks, and other processes will erode low-mass star cluster populations over time, globular clusters and other massive clusters provide long-lived records of star formation activity.
The mass function of star clusters is another observable property that we can exploit to study episodes of past star formation. Numerous studies have characterized the mass function of young star clusters using a power-law distribution ($dN/dM \propto M^{\alpha}$) with an index of $\alpha$=$-2.0\pm0.3$ that holds over a wide range of cluster mass [e.g., @Zhang99; @Gieles06a; @PZ10; @Fall12]. A power-law mass function slope of $-2$ has the notable property that total cluster mass is distributed equally among logarithmic intervals of cluster mass. This behavior is consistent with predictions for cluster formation via random sampling from a hierarchical gas distribution, and predictions for clump mass distributions from turbulent fractal clouds [see @Elmegreen08conf and references therein]. The observed similarity in shape of the young cluster mass function across a wide range of star-forming environments is often cited as evidence in favor of universal (or “quasi-universal”) descriptions of cluster formation behavior [e.g., @Fall12].
There is on-going debate as to whether the high-mass ($>$10$^4$ ) portion of the cluster mass function also follows a power-law distribution, or instead turns over and truncates at some maximum cluster mass. A pure power-law form is often assumed due to the lack of obvious features in smoothly declining cluster mass distributions and limitations imposed by low number statistics at the high-mass end [e.g., @Chandar10-LMC; @Whitmore10]. However, multiple studies have presented evidence in support of an exponential high-mass truncation through direct mass function fitting [e.g., @Gieles09; @Adamo15], through indirect modeling of the most massive and most luminous clusters [e.g., @Bastian08; @Bastian12_MF], or both [@Larsen09]. Modeling the truncated mass distribution using a @Schechter76 function ($dN/dM \propto M^{\alpha} \exp(-M/M_c)$, where $M_c$ is the characteristic truncation mass), these investigations report mass function truncations with $M_c$ of $\sim$10$^5$ in normal star forming galaxies, and larger values ($\sim$10$^6$ ) for the interacting, starburst Antennae galaxies.
A definitive consensus on the behavior of the high-mass end of the cluster mass function has not yet emerged. Small sample sizes of massive clusters, relatively small differences between predictions for a pure power-law and a Schechter function, and the indirect nature of some analyses all contribute to the lingering uncertainty. Nonetheless, measured truncation masses appear to increase systematically with star formation intensity, as observed on galaxy-wide scales [@Larsen09], as well as within individual galaxies [@Adamo15].
We present results from an unparalleled study of the star cluster mass function in the neighboring Local Group galaxy M31, based on data from the *Hubble* Space Telescope (HST) obtained by the Panchromatic *Hubble* Andromeda Treasury survey [PHAT; @Dalcanton12]. High spatial resolution imaging from HST resolves individual member stars in M31’s star clusters, and we use these observations to measure cluster ages and masses through color-magnitude diagram (CMD) fitting of the cluster’s resolved stars. This approach provides stronger constraints on young cluster properties than those obtained through multi-band SED fitting, and avoids large uncertainties caused by stochastic variations in the integrated light of low-mass clusters [see e.g., @Fouesneau10; @Krumholz15].
We measure the cluster mass function for a well-characterized sample of 1249 young star clusters drawn from the PHAT cluster catalog [@Johnson15_AP]. Robust cluster identifications and catalog completeness determinations combine to yield a sample of clusters that is well-suited for a mass function investigation. The catalog’s $\sim$10$^3$ 50% completeness limit for young clusters, combined with our well-characterized completeness function, provides an unprecedented range of masses available for mass function fitting.
The cluster population in M31 allows us to analyze the properties of the cluster mass function in a galaxy that falls at the low-intensity end of the galactic [$\Sigma_{\mathrm{SFR}}$]{} spectrum, providing valuable leverage for evaluating possible systematic variations of the high-mass truncation of the cluster mass function. Previous observations have focused on galaxies with moderate star formation activity [e.g., M51 and M83; @Gieles09; @Adamo15], as well as high intensity starburst galaxy mergers [e.g., the Antennae; @Zhang99; @Whitmore10]. Our study of M31 extends the range of star formation environments analyzed by an order of magnitude in [$\Sigma_{\mathrm{SFR}}$]{}, providing significant leverage on measuring environmentally-dependent variations of cluster mass function truncations.
We structure the paper as follows. We begin by introducing the PHAT cluster sample and CMD fitting in Section \[data\]. Next, we introduce a probabilistic cluster mass function fitting technique in Section \[analysis\], and present results in Section \[results\]. We compare our M31 results to Schechter mass function measurements from other young cluster systems and discuss the systematic variation of high-mass truncation masses with [$\Sigma_{\mathrm{SFR}}$]{} in Section \[discuss\_ymc\]. In Section \[discuss\_gc\], we consider the implications that a $M_c$–[$\Sigma_{\mathrm{SFR}}$]{} relation may have on the interpretation of old globular cluster systems. We summarize our results in Section \[summary\].
Data
====
  
We draw our cluster sample from the Andromeda Project (AP) cluster catalog [@Johnson15_AP]. This catalog identifies 2753 star clusters that span a wide age and mass range. The AP catalog was constructed from visual cluster identifications in optical (F475W, F814W; equivalent to $g$ and $I$) images from the PHAT survey data by volunteer citizen scientists, facilitated through a website hosted by the Zooniverse organization. The final sample of clusters was selected according to a candidate’s frequency of identification, where each image was examined by $>$80 AP volunteers. We adopt a cluster identification threshold that maximizes completeness and minimizes contamination with respect to the expert-derived PHAT Year 1 cluster catalog [@Johnson12] and its initial 25% survey coverage.
The completeness of the cluster catalog was measured using a suite of 3000 synthetic clusters. Each synthetic cluster was injected into an AP search image and subsequently identified and analyzed in the same way as the genuine clusters; see Section 2.2 in @Johnson15_AP for detailed properties of the artificial cluster sample. We compute survey-averaged 50% completeness limits as a function of cluster mass in two bins in cluster age, 10–100 Myr and 100–300 Myr, following a strategy similar to that used in @Johnson16_gamma to account for the spatial variation of completeness and star formation across the survey. First, we bin the synthetic cluster results according to local red giant branch stellar surface density (roughly equivalent to bins of galactocentric radius) to account for the variation of completeness as a function of background stellar density. Second, we weight the synthetic results in each bin based on local [$\Sigma_{\mathrm{SFR}}$]{} to account for the difference in spatial distribution of synthetic clusters versus that of young clusters and star formation. Third, we calculated a weighted average across the bins of synthetic cluster results, using weights assigned by integrated SFR. Finally, we fit the weighted and combined completeness results using a logistic function parameterization and find 50% completeness in mass at 740 for the 10–100 Myr age bin and 1080 for the 100–300 Myr age bin.
Photometric measurements of individual cluster stars were drawn from the catalog of 117 million resolved stars measured as part of the PHAT survey. The completeness limits of this stellar catalog allow the detection of main sequence stars down to $\sim$3 . Please refer to @Dalcanton12 and @Williams14 for full details on the survey’s crowded field stellar photometry analysis. We extract optical (F475W, F814W) CMDs for each cluster, and obtain constraints on cluster parameters through CMD fitting. We use the MATCH software package to perform maximum-likelihood CMD analysis following techniques described in @Dolphin02. For cluster fitting, we adopt a M31 distance modulus of 24.47 [785 kpc; @McConnachie05], a binary fraction of 0.35 with uniform mass ratio distribution, a @Kroupa01 IMF for masses from 0.15 to 120 , and stellar models from the Padova group [@Marigo08] that include updated low-mass asymptotic giant branch tracks [@Girardi10]. We employ a restrictive prior on \[M/H\] (from $-0.2$ to $0.1$) to constrain solutions to $\sim$$Z_{\sun}$ in an effort to match gas phase metallicity observations within M31 [e.g., @Zurita12]. Cluster masses from MATCH reflect initial masses, unaffected by mass loss from stellar evolution. Cluster ages and masses for the PHAT young cluster sample were published as an appendix in @Johnson16_gamma; we publish a full catalog of cluster parameters, demonstrate the reliability of these results using synthetic cluster tests, and compare CMD-based fits to those derived from integrated light SED fitting in A. Seth et al. (in preparation).
We select a sample of young clusters with ages between 10–300 Myr for mass function analysis. We adopt a 10 Myr lower limit due to the uncertain and subjective nature of cluster identification at younger ages. @Gieles11 demonstrate that differentiating between long-lived clusters and rapidly expanding, unbound associations becomes well-defined for ages $>$10 Myr, so we adopt this lower age bound at little expense in terms of integrated star formation and number of clusters. The upper age bound of 300 Myr is based on the limit where CMD fitting becomes dramatically less precise when the MS turnoff drops below the completeness limit of the stellar photometry. CMD fitting yields 1249 clusters with best fit ages between 10–300 Myr and masses between 300–20,000 , where the median age uncertainty is 0.2 dex and the median mass uncertainty is 0.04 dex. We plot the derived mass distribution for the young cluster sample in the left panel of Figure \[fig\_data\].
Before we proceed with analysis of the cluster mass function, we note that the age distribution of the PHAT young cluster sample is consistent with a near-constant formation history and little or no cluster destruction [@Fouesneau14; @Johnson16_gamma]. The absence of significant cluster mass loss and destruction over the age and mass range we analyze has an important implication: it is safe to assume that the present day mass function we observe can be interpreted as the initial cluster mass function. In other words, we expect little or no evolution in the shape of the mass function with age due to cluster destruction. The center and right panels of Figure \[fig\_data\] show that the completeness-corrected mass distributions for age-based subsamples appear qualitatively similar to one another, in agreement with the assumption of no evolution. When the mass functions for the age subsamples are duration-normalized to account for different bin widths, the two samples also show close agreement in their normalization. This indicates similar cluster formation rates during these two epochs. Nonetheless, we will test the assumption of negligible cluster dissolution quantitatively and investigate possible age-dependencies of our results by separately analyzing 10–100 Myr and 100–300 Myr subsamples in addition to the full cluster sample.
Analysis
========
We derive mass function constraints using probabilistic modeling, following an approach similar to that used by @Weisz13 for initial stellar mass function fitting. The likelihood function of an observed cluster with mass $M$ is given as $$p_{\rm cluster}(M | \vec{\theta}, \tau) \equiv \frac{1}{Z}\ p_{\rm MF}(M | \vec{\theta})\ p_{\rm obs}(M | \tau),$$ where $p_{\rm MF}(M | \vec{\theta})$ is the cluster mass distribution function as defined by the set of parameters $\vec{\theta}$, and $p_{\rm obs}(M | \tau)$ is the observational completeness function, which depends on cluster age, $\tau$. Finally, $Z$ is the normalization required for $p_{\rm cluster}(M | \vec{\theta}, \tau)$ to properly integrate to 1, given as $$Z = \int p_{\rm MF}(M | \vec{\theta})\ p_{\rm obs}(M | \tau)\ dM.$$
We adopt a @Schechter76 functional form for the cluster mass distribution, whose shape is controlled by two parameters, $\vec{\theta} = \{\alpha, M_c\}$; $\alpha$ is the low-mass power-law index and $M_c$ is the characteristic mass that defines the exponential high-mass truncation. This distribution follows the form $$\label{eq_s}
p_{\rm MF}(M | \alpha, M_c) \propto M^{\alpha} \exp(-M/M_c).$$ Note that the Schechter function simplifies to a simple power-law function ($dN/dM \propto M^{\alpha}$) in the limit that $M_c \to \infty$. We model the age-dependent cluster completeness function using a logistic function, parameterized by the 50% mass completeness limit, $M_{\rm lim}$, and maximum slope, $a_{\rm lim}$. The values of the completeness function parameters depend on cluster age, such that ($M_{\rm lim}$, $a_{\rm lim}$)=(740 , 5.0) for 10–100 Myr old clusters, and ($M_{\rm lim}$, $a_{\rm lim}$)=(1080 , 5.0) for 100–300 Myr old clusters. To ensure that we are not too sensitive to the completeness corrections, we restrict the model and data to masses greater than the 50% completeness limit, such that $$p_{\rm obs}(M | \tau) =
\begin{cases}
\left(1+\exp\left[\frac{-a_{\mathrm{lim}}(\tau)(M-M_{\mathrm{lim}}(\tau))}{M_{\sun}} \right] \right)^{-1}, \\
\hspace{100pt} M > M_{\mathrm{lim}}(\tau)\\
0, \\
\hspace{100pt} \mathrm{otherwise}.
\end{cases}$$
We use Bayes’ theorem to derive the posterior probability distribution function of the Schechter function parameters, given as $$\label{eq_bayes}
p(\vec{\theta} | \{M_i\}, \tau) \propto p_{\rm cluster}( \{M_i\} | \vec{\theta}, \tau)\ p(\vec{\theta}),$$ where $\{M_i\}$ is the set of $N$ cluster masses, $p_{\rm cluster}( \{M_i\} | \vec{\theta}, \tau)$ is the likelihood function for a set of cluster masses, and $p(\vec{\theta})$ is the prior probability for the Schechter function parameters. The likelihood function for a set of cluster masses is defined as the product of the individual cluster mass probabilities: $$\begin{gathered}
\label{eq_s_like}
p_{\rm cluster}( \{M_i\} | \alpha, M_c, \tau) = \\
\prod_{i=1}^{N} \frac{1}{Z}\ M_i^{\alpha} \exp(-M_i/M_c)\ p_{\rm obs}(M_i | \tau),\end{gathered}$$ where the normalization term becomes $$\label{eq_s_norm}
Z = \int_{M_{\mathrm{lim}}}^{\infty} M^{\alpha} \exp(-M/M_c)\ p_{\rm obs}(M | \tau)\ dM.$$ We adopt uniform top-hat prior probability distributions that generously cover the range of possible parameter values: $-3 \le \alpha \le -1$ and $3 \le \log (M_c$/$M_\sun$) $\le 8$. These uninformative priors on $\alpha$ and $M_c$ are sufficiently broad to enclose all points in parameter space with non-trivial likelihoods, such that the fitting results are not sensitive to their specific limits. Finally, we integrate the normalization term numerically during the course of fitting.
The probabilistic framework we use here for cluster mass function fitting assumes negligible uncertainties on individual cluster masses. @Weisz13 demonstrate that this simplifying assumption does not significantly bias fitting results in the limit of small (0.1) fractional mass uncertainties. As the fractional error on the masses increases to 0.5 and beyond, fitting results become more and more affected. The PHAT CMD-based cluster masses have a median uncertainty of 0.04 dex, and these mass uncertainties are smallest at the high-mass end of the cluster sample where individual masses have the greatest leverage over $M_c$ results. Therefore, we are confident that the assumption of negligible mass errors does not significantly impact the results presented here.
Power-law Functional Form {#powerlaw}
-------------------------
We also adapt this probabilistic framework to fit a non-truncated, power-law functional form of the cluster mass distribution. For this purpose, we adopt $$\label{eq_pl}
p_{\rm MF}(M | \vec{\theta}) \propto M^{\alpha},$$ and power-law equivalents of the likelihood function for the set of cluster masses and its normalization (Eqs. \[eq\_s\_like\] and \[eq\_s\_norm\]) are given as $$\label{eq_pl_like}
p_{\rm cluster}( \{M_i\} | \alpha, \tau) = \prod_{i=1}^{N} \frac{1}{Z}\ M_i^{\alpha}\ p_{\rm obs}(M_i | \tau)$$ and $$\label{eq_pl_norm}
Z = \int_{M_{\mathrm{lim}}}^{\infty} M^{\alpha}\ p_{\rm obs}(M | \tau)\ dM.$$
Sampling the Posterior Probability Distributions {#powerlaw}
------------------------------------------------
We use a Markov Chain Monte Carlo (MCMC) technique to sample the posterior probability distributions of the Schechter and power-law mass function parameters. In particular, we use the `emcee`[^1] Python package [@ForemanMackey13] and its implementation of an affine invariant ensemble sampler from @GoodmanWeare10. For the MCMC calculation, we use 500 walkers, each producing 600 step chains, of which we discard the first 100 burn-in steps. We report the median value of the marginalized posterior probability distribution function (PDF) for each of the Schechter function parameters, $p(M_c | \{M_{i}\}, \tau)$ and $p(\alpha | \{M_{i}\}, \tau)$, accompanied by a 1$\sigma$ confidence interval defined by the 16th to 84th percentile range of the marginalized posterior. For the power-law function, we report the median and 1$\sigma$ confidence interval for the single parameter, $\alpha$.
Results
=======
Schechter Fitting Results {#results_sch}
-------------------------
 
Schechter function fitting results for the 10–300 Myr PHAT young cluster sample are shown in Figure \[fig\_fit\_sch\], derived for 840 clusters whose masses are greater than the 50% mass completeness limit of the cluster’s age bin. In the left panel, we compare the observed, completeness-corrected cluster mass distribution to Schechter function fits. We draw pairs of $M_c$ and $\alpha$ parameter values from the posterior PDF and normalize these functions to match the completeness-corrected number of clusters above the most restrictive completeness limit (from the 100–300 Myr age bin) at 1080 . We stress that the binned mass distribution shown here is only used for visualization purposes, and that our results are based on probabilistic fitting of individual, unbinned cluster masses.
We find that the PHAT young cluster sample is well-described by a Schechter function with $M_c$ = $8.5^{+2.8}_{-1.8} \times 10^3$ ($\log M_c$/= $3.93^{+0.13}_{-0.10}$) and $\alpha$ = $-1.99 \pm 0.12$. These results are based on the one-dimensional marginalized posterior PDFs, which we present in the right panel of Figure \[fig\_fit\_sch\] along with the two-dimensional posterior PDF that shows the covariance between the Schechter function parameters. The characteristic truncation mass reported here is the lowest value ever obtained for a star cluster population, which is more than an order of magnitude below the $2 \times 10^5$ value derived for a sample of star forming galaxies by @Larsen09. The index of the low-mass slope agrees perfectly with the canonical value of $-2$, supporting the notion that the mass function for the M31 PHAT cluster sample is otherwise rather typical at lower cluster mass.
### Testing for Age Dependence {#results_age}
![Two-dimensional posterior constraints on $\alpha$ and $M_c$ for the 10–100 Myr (blue) and 100–300 Myr (red) cluster samples, overlaid on sample-wide (10–300 Myr) constraints (grayscale). Contours represent 1, 2, and 3$\sigma$ confidence intervals.[]{data-label="fig_agecomp"}](results-v2_contour_all.pdf)
A notable signature of mass dependent cluster dissolution is a flattening of the low-mass slope of the cluster mass function with increasing age [@Gieles09]. We test for age-dependence in our Schechter mass function fits by dividing the sample into two age bins: 10–100 Myr and 100–300 Myr. A comparison of the two-dimensional posterior PDFs for all three cases of age binning is presented in Figure \[fig\_agecomp\]. Note that the $M_c$ constraint from the younger age bin alone is significantly weaker due to the reduced number of clusters; only 324 clusters in the 10–100 Myr age range lie above the bin’s 50% mass completeness limit. This demonstrates that our large sample of clusters, obtained by integrating over a wide age range and down to low cluster mass, was key to obtaining a robust result. Nevertheless, we obtain very similar results for the two separate age bins as we did for the total 10–300 Myr sample, and find no significant age dependence of the mass function shape.
The $\alpha$ constraints for the two age bins show a marginal trend of a flatter slope for older ages, but both bins are also consistent with a single $-2$ power-law slope at $\sim$1.5$\sigma$ confidence. Therefore, the Schechter function fitting results show no significant or definitive signature of cluster dissolution on $\sim$100 Myr timescales, in agreement with previous PHAT analysis of age and mass distributions [@Fouesneau14]. Together, these results suggest that characteristic cluster dissolution timescales longer than the age range examined here ($>$300 Myr). We will pursue constraints on the timescales and mass dependence of cluster dissolution in future work (M. Fouesneau et al., in preparation).
### Comparison to Previous Work {#results_prev}
Previous studies of the young cluster mass function in M31 did not detect a truncation mass of $\sim$10$^4$ . For example, @Vansevicius09 compare their ground-based M31 cluster sample [and the sample from @Caldwell09] with a Schechter function distribution and argue that their results are consistent with the @Larsen09 spiral galaxy sample average $M_c$ value of $2\times10^5$ , although they did not perform any fitting.
There are a number of points to consider when comparing our PHAT results to the work of @Vansevicius09 and @Caldwell09. First, these two studies were both significantly limited by the low-mass completeness cutoffs of their catalogs. @Vansevicius09 and @Caldwell09 have 50% completeness limits at $\log(M/M_{\sun})$ of 3.7 and 4.0, respectively, which is comparable to the $M_c$ value we measured for PHAT. Without a full accounting of the cluster population at masses below the knee of the distribution, it is difficult to properly constrain the characteristic truncation mass. Second, the @Vansevicius09 sample only contains a single $10^5$ cluster at masses greater than $5\times10^4$ , revealing extremely sparse sampling near their preferred $M_c$ value of $2\times10^5$ . Third, both of these works analyze clusters from a broader age range, including clusters with ages between 1–3 Gyr. We prefer to restrict our analysis to an age regime where we can obtain robust cluster fits from CMD fitting. Fourth, both @Vansevicius09 and @Caldwell09 derive masses using conversions based on fully-sampled mass functions. As mentioned in the introduction, this strategy can lead to significant mass discrepancies due to the stochastic contribution of luminous evolved members.
Finally, the potential exists that the cluster population surveyed by these previous works, which include clusters that lie on the southwest side of the M31 disk opposite that of the PHAT survey region, might truly represent a different star formation environment with higher intrinsic values of $M_c$. The southwest portion of M31 hosts the star forming complex NGC206 [@Hunter96] and vigorous star formation near the split in the 10 kpc star forming ring [@Gordon06], and is known to host a number of notable massive ($10^4$–$10^5$ ) young clusters [e.g., VdB01; @Perina09]. Indeed, @Elmegreen97 point out that the southwestern portion of the M31 disk hosts a spiral arm segment [S4; also OB79–82 in the parlance of @vdB64] with particularly high intensity star formation, highlighting this same region of interest. With this in mind, we note that our results apply only to the PHAT survey region covering the NE quadrant of M31, and that variations across the disk of M31 are possible. Further study of the active southwest portion of the M31 disk could provide an interesting counterpoint to the more moderate star formation surveyed by PHAT.
### Fitting of Radially-selected Cluster Subsamples {#results_radial}
@Adamo15 present Schechter function fitting results for M83 that show a radial trend in truncation mass, such that $M_c$ decreases with increasing galactocentric radius. These results motivate us to ask: beyond the survey-wide results presented, can radial trends in $M_c$ be detected in M31? Adopting region definitions from @Johnson16_gamma, we assemble inner disk, 10 kpc ring, and outer disk spatial subsamples. Unfortunately, the present M31 cluster dataset from the PHAT survey does not provide strong constraints on radial trends in $M_c$ due to low number statistics in regions outside the 10 kpc star-forming region, which dominates the PHAT cluster sample ($>$60% of the total). There are only 144 and 82 clusters with masses greater than the 50% completeness limit in the inner disk and outer disk regions, respectively. These cluster counts are far smaller than the 324 young cluster sample that yielded weak constraints on $M_c$. Preliminary analysis yields weak constraints for the outer disk region ($>$0.5 dex uncertainty on $M_c$), and only a lower limit for the inner disk region. Further analysis is required to confirm the robustness of these fitting results in the low number statistics regime.
Power-law Fitting Results and Comparison to Schechter Function {#results_pl}
--------------------------------------------------------------
 
While we find that the observed cluster mass distribution is well-described by a Schechter function, we also fit a power-law functional form for comparison. We present power-law fitting results for the PHAT young cluster sample in Figure \[fig\_fit\_pl\]. Similar to Figure \[fig\_fit\_sch\], we compare realizations of the power-law function to the observed mass distribution in the left panel, where we draw $\alpha$ values from the posterior PDF and normalize the functions to match the completeness-corrected number of clusters above a limiting mass of 1080 . We find that the PHAT young cluster sample is best described by a power-law function with $\alpha$ = $-2.49 \pm 0.05$, and we plot the posterior PDF for $\alpha$ in the right panel of Figure \[fig\_fit\_pl\]. This fitted power-law index is much steeper than the canonical $-2$ value, and tends to over-predict the number of clusters at masses greater than 10$^4$ .
{width=".34\textwidth"} {width=".34\textwidth"} {width=".3\textwidth"}
The Schechter and power-law functional forms fitted to the observed PHAT cluster mass function yield similar predictions for low-mass clusters, but diverge significantly for high-mass clusters. We compare the fitting results for the two functional forms in Figure \[fig\_fitcomp\] using differential and cumulative curves in the left and center panels, respectively. We observe that the fitted power-law function systematically over-predicts the number of massive clusters, whereas the exponential truncation of the Schechter function allows a significantly better fit to the observed distribution.
As a quantitative illustration of the difference between the fitted Schechter and power-law functions at high cluster mass, we compare the number of clusters with masses $>$10$^4$ observed by PHAT to predictions from the fitted functions. We transform the posterior PDFs derived for the Schechter and power-law function parameters into PDFs of $N_{\rm cluster}$ with mass greater than $>$10$^4$ , assuming a normalization set to match the total completeness-corrected number of clusters above a limiting mass of 1080 . We plot the resulting PDFs for the Schechter and power-law fits in the right panel of Figure \[fig\_fitcomp\], and compare these predictions to the observed value of fifteen $>$10$^4$ clusters.
This comparison shows that the 15 observed $>$10$^4$ clusters is incompatible with the $33^{+5}_{-3}$ prediction for the power-law function fit at high significance ($>$4$\sigma$), while well-matched to the $17^{+4}_{-3}$ prediction for the Schechter function. While this illustration uses an arbitrary threshold cluster mass of 10$^4$ , we find that the fitted power-law mass function is discrepant at $>$3$\sigma$ significance for any threshold mass greater than 8$\times$10$^3$ .
We also note that the discrepancy in the observed number of $>$10$^4$ clusters would be even worse for a shallower power-law mass function. A prediction of 101 $>$10$^4$ clusters, calculated for a canonical $-2$ power-law slope similarly normalized to the total number of clusters above a limiting mass of 1080 , is clearly discrepant with the observed population of PHAT clusters.
In addition to the specific comparison of Schechter and power law fits at the high mass end, we also compute Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) test statistics and probabilities to assess the overall goodness-of-fit for each functional form to the observed data. We acknowledge that the Schechter function does not perfectly capture the observed distribution, as shown in the center cumulative distribution panel of Figure \[fig\_fitcomp\]. A sharper truncation would improve the fit, but the two-parameter Schechter function provides a satisfactory fit to the data. We derive KS and AD probabilities of 0.376 and 0.430, respectively, demonstrating that the observed data are consistent with our most likely Schechter function. In contrast, we find KS and AD probabilities for the most likely power-law function of 0.007 and 0.019, respectively, allowing us to discard the null hypothesis that our data were drawn from the most likely power law distribution with high confidence. Please note that KS and AD probabilities for both functional forms were computed via simulation to properly assess the significance of the test results.
The systematic over-prediction of the massive cluster population by the power-law mass function model argues strongly for the existence of a high-mass truncation of the cluster mass function, and rules out the notion of a universal, pure power-law cluster mass function where the maximum cluster mass is driven only by sampling statistics. The exponentially-truncated Schechter function serves as a useful description of the high-mass end of the cluster mass distribution, allowing us to compare the M31 mass function to those in other galaxies.
Discussion {#discuss}
==========
Mass Function Truncations for Young Cluster Systems: Correlation with [$\Sigma_{\mathrm{SFR}}$]{} {#discuss_ymc}
-------------------------------------------------------------------------------------------------
In this section we examine whether the physical conditions of star formation in the PHAT survey region of M31 can explain the low value of $M_c$ measured here relative to previous studies. We combine our M31 mass function result with those from the literature and find a clear correlation between the mass function truncation, $M_c$, and the SFR surface density, [$\Sigma_{\mathrm{SFR}}$]{}.
We complement the PHAT $M_c$ result with young cluster mass function measurements from the literature. We use the value of $\log (M_c/M_{\odot})$=$6.3^{+0.7}_{-0.3}$ for the Antennae, as calculated by @Jordan07 using the 2.5–6.3 Myr cluster mass distribution data from @Zhang99. We also use results from @Gieles09 for M51, and survey-wide results from @Adamo15 for M83. The M51 and M83 measurements are consistent with $M_c \sim 2 \times 10^5$ , which is the value reported by @Larsen09 for a combined analysis of $\sim$20 nearby spiral galaxies (of which M51 and M83 were members). We observe that $M_c$ values among the four galaxies vary by $>$2 orders of magnitude. While the current sample of galaxies with robust Schechter function fits in the literature is relatively small, we benefit greatly from the large dynamic range spanned in characteristic truncation mass and star formation activity.
The four galaxies studied here span a wide range of star formation intensity, from relatively quiescent activity in M31, to merger-induced starburst activity in the Antennae. M31’s low SFR is characteristic of a “green valley” galaxy [@Mutch11], and its star formation activity is primarily contained within a 10 kpc star-forming ring, which may be associated with the outer Lindblad resonance of a central bar [@Athanassoula06; @Blana17]. The Antennae serve as the prototype for a galaxy merger, providing one of the youngest and closest laboratories for studying high-intensity star formation and massive cluster formation [e.g., @Whitmore10; @Johnson15_Antennae]. In between, M51 and M83 both show signs of recent or on-going galaxy interactions that produce strong present-day star formation, high-amplitude spiral arms, and bar-driven gas flows.
We quantify $M_c$ variations as a function of [$\Sigma_{\mathrm{SFR}}$]{}, an observable metric of star formation intensity. Unlike an unnormalized galaxy-integrated SFR that scales strongly with global galaxy mass, [$\Sigma_{\mathrm{SFR}}$]{} tends to better differentiate galaxies according to differences in local star formation properties. Furthermore, we calculate galaxy-wide [$\Sigma_{\mathrm{SFR}}$]{} values using a SFR-weighted average of kpc-scale [$\Sigma_{\mathrm{SFR}}$]{} observations, represented hereafter as [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{}. This weighted average provides a characteristic, global metric that accurately represents the properties of the local environments in which stars are forming.
We derive new [$\Sigma_{\mathrm{SFR}}$]{} measurements for each of the four galaxies in our sample, yielding a homogeneous set of observations that is well-suited for $M_c$-[$\Sigma_{\mathrm{SFR}}$]{} correlation analysis. For each galaxy, we construct a map of [$\Sigma_{\mathrm{SFR}}$]{} using a kpc-scale spatial kernel, and obtain a global [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} measurement by computing a SFR-weighted average over the set of local measurements represented in the map. In addition to the weighted-average, we also report the narrowest percentile range containing 68% ($\pm1\sigma$) of the SFR-weighted local [$\Sigma_{\mathrm{SFR}}$]{} measurements for each galaxy. This interpercentile range serves as a reminder that the global [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} values we calculate represent an underlying distribution of local star formation environments.
We present a detailed description of the [$\Sigma_{\mathrm{SFR}}$]{} calculations in Appendix \[appendix\_sigsfr\], including a discussion and justification regarding our use of a SFR-weighted [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{}. Briefly, we use spatially resolved star formation history maps from @Lewis15 to compute SFR averaged over 10-100 Myr and produce [$\Sigma_{\mathrm{SFR}}$]{} maps of the PHAT survey region in M31, following the same methodology used by @Johnson16_gamma. For M51, M83, and the Antennae, we use GALEX FUV and Spitzer 24$\mu$m imaging to produce [$\Sigma_{\mathrm{SFR}}$]{} maps, following the SFR calibration and methodology used by @Leroy08. We present $M_c$ and [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} measurements for the galaxy sample in Table \[tbl\_mcdata\] and plot these results in Figure \[fig\_mc\_ymc\].
[lcccc]{} M31 & PHAT & $3.93^{+0.13}_{-0.10}$ & $-2.68^{+0.26}_{-0.38}$ & This Work\
M51 & & $5.27^{+0.11}_{-0.14}$ & $-1.44^{+0.40}_{-0.46}$ & @Gieles09\
M83 & 0.45–4.5 kpc & $5.20^{+0.08}_{-0.09}$ & $-1.52^{+0.34}_{-0.28}$ & @Adamo15\
Antennae & & $6.3^{+0.7}_{-0.3}$ & $-0.53^{+0.46}_{-0.49}$ & @Jordan07\
Normal Galaxies & & 5.32 $\pm$ 0.10 & & @Larsen09
![Comparison of $M_c$ fits for young cluster samples as a function of [$\Sigma_{\mathrm{SFR}}$]{} for M31, M83 [@Adamo15], M51 [@Gieles09], and Antennae [@Zhang99; @Jordan07]. Solid vertical bars denote fitting uncertainties for $M_c$, and dotted horizontal bars denote the narrowest 68% interpercentile range of local [$\Sigma_{\mathrm{SFR}}$]{} measurements within each galaxy. We perform a linear fit to the data, and find that $M_c \propto$ [$\Sigma_{\mathrm{SFR}}$]{}$^{1.1}$.[]{data-label="fig_mc_ymc"}](result_mc-sigsfr_final.pdf)
Figure \[fig\_mc\_ymc\] shows a strong correlation between $M_c$ and [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{}, spanning $>$2 orders of magnitude in each quantity. The observed trend suggests a strong dependence of the cluster mass function truncation on the characteristics of the galactic star forming environment. We quantify the observed relationship between $M_c$ and [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} by fitting a linear relation to the observed data in log $M_c$–log [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} parameter space: $$\label{eq_mc_sigsfr}
\log M_c = (1.07 \pm 0.10) \times \log \langle\Sigma_{\mathrm{SFR}}\rangle + (6.82 \pm 0.20).$$ The fitting suggests a near-linear proportionality between the mass function truncation and [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{}, such that $M_c \propto$ [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{}$^{1.1}$. The quoted uncertainties on the fitted slope account for $M_c$ uncertainties only; uncertainties on the slope increase to $\pm0.2$ if 0.2 dex uncertainties on [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} measurements were included, or they would increase to $\pm0.3$ if the 68% interpercentile range is used to define the [$\Sigma_{\mathrm{SFR}}$]{} confidence interval.
The $M_c$-[$\Sigma_{\mathrm{SFR}}$]{} relation we identify here is defined at galaxy-integrated scales. This choice of averaging scale provides the large star cluster number statistics required to place strong constraints on cluster mass function shape and the presence of a high-mass truncation. However, galaxy-wide averaging obscures the complexity of physical dependencies related to star cluster formation, beyond the [$\Sigma_{\mathrm{SFR}}$]{} dependence we characterize here. Therefore, we stress caution when extrapolating cluster formation behavior at smaller scales using Equation \[eq\_mc\_sigsfr\]. Spatially-resolved observational follow-up work examining the physical drivers of massive cluster formation has the potential to further our understanding of star formation in high gas density, high star formation efficiency environments, but obtaining statistical robust constraints in the regime of small cluster number statistics will be a significant challenge (as discussed in Section \[results\_radial\]).
We note that $M_c$ and cluster formation efficiency ($\Gamma$ = $M_{\rm cluster}$/$M_{\rm total}$) have both been shown to vary systematically with [$\Sigma_{\mathrm{SFR}}$]{}, and seek to clarify the interconnected yet distinct nature of these variations. Assuming a fixed normalization for the low-mass end of a Schechter mass function, decreasing $M_c$ tends to decrease the integrated stellar mass of a cluster population. As a result, the observed variation in $M_c$ leads to correlated declines in both $\Gamma$ and [$\Sigma_{\mathrm{SFR}}$]{}. However, only a small fraction of the total $\Gamma$ variation observed can be explained by the variation in $M_c$ alone. For example, decreasing $M_c$ from $10^6$ to $10^4$ only produces a factor of $\sim$2 change in $\Gamma$, while observations and theoretical predictions show evidence for more than an order of magnitude change over the same range of [$\Sigma_{\mathrm{SFR}}$]{} [@Kruijssen12; @Johnson16_gamma]. Therefore, variations in $\Gamma$ do not stem solely from differences in high mass cluster formation as a function of [$\Sigma_{\mathrm{SFR}}$]{}, but reflect broad differences in cluster formation over a wide range of masses.
### Physical Drivers of $M_c$-[$\Sigma_{\mathrm{SFR}}$]{} Correlation: Pressure {#discuss_pmp}
We explore the role that interstellar pressure may play in driving the observed $M_c$-[$\Sigma_{\mathrm{SFR}}$]{} correlation. Large stellar densities observed in massive clusters and globular clusters suggest extremely high gas densities in progenitor molecular clouds at the time of formation [@Elmegreen97]. Maintaining such high densities is likely to require large external pressures to keep the natal gas confined, which motivates our specific interest in pressure over other environmental parameters. While the coupling between external and internal pressures for host molecular clouds is currently debated, observational evidence favoring the influence of galactic environment and external pressure on molecular cloud properties has begun to emerge [@Hughes13; @Colombo14]. These confining pressures may be set by the equilibrium conditions of star-forming disks, or may be produced transiently over large spatial scales in galaxy mergers [@Renaud15_Antennae] or over small scales in molecular cloud collisions [@Fukui14].
For the simple case of an equilibrium star-forming disk, we can examine whether observed variations in $M_c$ are consistent with the predicted scaling behavior of pressure as a function of [$\Sigma_{\mathrm{SFR}}$]{}. We approximate the dependence between mid-plane pressure ($P_{\rm{mp}}$) and [$\Sigma_{\mathrm{SFR}}$]{} for the case of a stable star-forming galaxy disk following the logic presented in @Elmegreen09. We combine the expectation that $P_{\rm{mp}}$ scales as [$\Sigma_{\mathrm{gas}}$]{}$^2$ with the empirical Kennicutt-Schmidt relation [@Kennicutt98] where [$\Sigma_{\mathrm{SFR}}$]{} $\propto$ [$\Sigma_{\mathrm{gas}}$]{}$^{\rm{1.4}}$, and we predict that $P_{\rm{mp}} \propto$ [$\Sigma_{\mathrm{SFR}}$]{}$^{\rm{1.4}}$. This predicted dependence is steeper than the observed trend, where $M_c$ $\propto$ [$\Sigma_{\mathrm{SFR}}$]{}$^{1.1}$, suggesting that transient enhancements of interstellar pressure or other environmental characteristics drive the behavior of high-mass cluster formation.
Establishing that pressure, or another physical driver, is responsible for the mass function truncation variations will require additional study. Rather than relying on indirect scaling arguments, obtaining observational estimates of interstellar pressure and other environmental variables and directly analyzing their correlation with $M_c$ observations could help identify key galactic properties that influence massive cluster formation behavior.
Another avenue of study involves the comparison of the star cluster mass function with the giant molecular cloud (GMC) mass function. As clusters are formed out of molecular gas, and the GMC mass function is known to vary with galactic environment [e.g., @Colombo14], understanding the connection between the behavior of these two mass distributions may shed light on the underlying physics involved. To this point, @Kruijssen14 suggested that the maximum mass scale of both star clusters and GMCs might have a common origin, tied to the Toomre mass [@Toomre64]. The CARMA survey of M31 GMCs (A. Schruba et al., in preparation) and other extragalactic GMC surveys with ALMA and other facilities will provide many opportunities to study the connection between cluster and molecular cloud mass functions in detail, and to test theoretical explanations for observed behavior.
Mass Function Truncations for Globular Cluster Systems: Similarity to Young Clusters? {#discuss_gc}
-------------------------------------------------------------------------------------
Old globular cluster systems have a dramatically different mass function shape compared to the young cluster systems discussed in the previous section. The globular cluster mass function (GCMF) is commonly parameterized using a Gaussian or log-normal form, and shows a clear peak at a near-constant mass of $2\times10^5$ [e.g., @Jordan07; @Villegas10].
Early theoretical work proposed that globular clusters formed in a way that differs from young clusters forming today, following a mass distribution which peaks at a characteristic mass scale [e.g., @Peebles68; @Fall85]. In contrast, more recent work has argued that globular clusters form with an initial power-law (or Schechter function) mass distribution that evolves to a peaked distribution due to dynamical evolution and destruction processes [e.g., @Gnedin97; @Fall01; @Kruijssen15]. The use of an initial power-law mass function is motivated by cluster formation behavior observed at low redshift, and assumes cluster formation proceeds similarly at all redshifts. In this case, globular cluster populations today are the surviving relics of a population that formed in the same way that young massive clusters do in the present day. The small number of young massive clusters presently formed at low redshift, relative to the large number of old massive globular clusters, results from an overall decline of the cosmic star formation history since $z\sim2$ [@Madau14], leading to a corresponding decline in local [$\Sigma_{\mathrm{SFR}}$]{} and massive cluster formation.
Globular cluster systems in early-type galaxies show systematic variations in their luminosity function shapes. The width of the peaked luminosity functions are observed to increase with host galaxy mass, as observed for Virgo cluster members [@Jordan06; @Jordan07], Fornax cluster members [@Villegas10], and seven brightest cluster galaxies in other massive galaxy clusters [@Harris14]. @Jordan07 demonstrate that this increase in width of the globular cluster luminosity function, and subsequently the GCMF, can be interpreted either as an increase in the dispersion ($\sigma_{\rm LN}$) of a traditional log-normal functional form, or as an increase in $M_c$ for an evolved Schechter function — a functional form inspired by @Fall01 that accounts for cluster mass loss. This behavior is broadly similar to the mass function variations observed for young cluster systems. We therefore compare these two sets of $M_c$ measurements and investigate a possible connection between globular cluster and young cluster formation. If the two cluster populations follow the same $M_c$–[$\Sigma_{\mathrm{SFR}}$]{} correlation, this could signal they form through a common formation pathway.
### Globular Cluster $M_c$ Measurements {#discuss_gc_data}
We compare the young cluster $M_c$ measurements from Section \[discuss\_ymc\] to globular cluster $M_c$ measurements from the ACS Virgo Cluster Survey [VCS; @Cote04] published by @Jordan07. These authors fit the data using an evolved Schechter function, allowing a direct comparison between the two sets of results[^2].
We plot globular cluster $M_c$ values as a function of present-day host galaxy mass in the left panel of Figure \[fig\_mc\_gc\], reproducing the data and result from @Jordan07. These data points reflect binned results based on $z$-band luminosity function fitting, where globular cluster systems for small subsets of galaxies (1–9; see their Table 3) are stacked to boost cluster number statistics. We obtain $M_c$ masses by converting luminosity function fitting results into cluster mass parameter space via $z$-band mass-to-light ratios ($\Upsilon_z$) derived from SSP models, then transform from present-day to initial stellar mass by accounting for stellar evolution-based mass loss and death. We use the Flexible Stellar Population Synthesis code [FSPS; @Conroy09; @Conroy10] to calculate the conversions, where $\Upsilon_z$ are based on a nominal 13 Gyr cluster age and a @Kroupa01 stellar IMF, and are relatively insensitive to metallicity in agreement with @Jordan07. The conversion from present-day to initial cluster mass accounts for the $\sim$45% of SSP mass returned to the ISM over the nominal 13 Gyr cluster lifetime. We derive galaxy stellar masses ($M_{*\rm{, galaxy}}$) using $z$-band luminosities from @Ferrarese06, ($g-z$) colors and distances from @Blakeslee09, and color-based stellar mass-to-light ratios from @Into13.
We fit the following linear relation for $M_c$ as a function of $M_{*\rm{, galaxy}}$: $$\begin{gathered}
\label{eq_mc_mstar}
\log (M_c/M_{\sun}) = (0.35 \pm 0.07) \times \log (M_{* \mathrm{, galaxy}}/M_{\sun}) \\
+ (2.6 \pm 0.8).\end{gathered}$$ We emphasize that the observed correlation is unlikely to be directly linked to stellar mass. Instead, we expect that the stronger underlying correlation is that the intensity of star formation is higher in progenitor galaxies that go on to form more massive galaxies as compared to progenitor galaxies that merge to form less massive galaxies. In this scenario, present day host galaxy mass serves as a proxy for galaxy assembly history.
We also compare mass function constraints for the Milky Way and M31 globular cluster systems to demonstrate that the GCMF behavior shown here is not a special feature of early-type host galaxies in galaxy cluster environments. We use the Milky Way $M_c$ reported in @Jordan07, corrected using the $\Upsilon_z$ values derived above for the VCS measurements, and pair it with the Galactic stellar mass determination from @McMillan11. We performed our own globular cluster luminosity function fit for M31, as described in Appendix \[appendix\_m31gcmf\], and pair this $M_c$ measurement with the M31 stellar mass from @Tamm12. These data points are not included in the $M_c$–$M_{*\rm{, galaxy}}$ fit, but appear to follow a similar trend as found for the VCS galaxies. The deviation of the late-type spiral galaxies toward higher $M_{*\rm{, galaxy}}$ with respect to the relation for early-type hosts is plausibly explained by differences in galaxy evolution. Progenitor galaxies with similar properties, and $M_c$ values, at the epoch of globular cluster formation will diverge in terms of present day stellar mass if their star formation histories differ significantly, as explained by @Mistani16 for the case of field versus galaxy cluster dwarf galaxies.
### $M_c$ Comparison and [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} Predictions {#discuss_gc_results}
The VCS globular cluster systems span a $\sim$1 dex range in $M_c$, from $\sim$10$^{6}$–10$^{7}$ , which overlaps with the upper range of $M_c$ values observed in young cluster systems (right panel of Figure \[fig\_mc\_gc\]). Given the comparable $M_c$ values and existing models which assume young massive cluster formation and globular cluster formation are governed by the same physical processes, we hypothesize that the same $M_c$–[$\Sigma_{\mathrm{SFR}}$]{} relation observed for young cluster systems also holds for globular cluster systems. If true, then globular clusters in these early-type galaxies must have formed in star forming environments with [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} values between 0.1–1.0 kpc$^{-2}$ — within $\pm$0.5 dex of the Antennae’s [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{}. We highlight this portion of the $M_c$–[$\Sigma_{\mathrm{SFR}}$]{} correlation in the right panel of Figure \[fig\_mc\_gc\] with a gray box. Furthermore, we combine the $M_c$–$M_{*\rm{, galaxy}}$ and $M_c$–[$\Sigma_{\mathrm{SFR}}$]{} correlations and infer properties of globular cluster formation environments as a function of present day host galaxy mass, presented as a upper x-axis in the left panel of Figure \[fig\_mc\_gc\].
The $M_c$ values ascribed to the globular cluster systems, and hence the [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} values assigned according to the young cluster $M_c$–[$\Sigma_{\mathrm{SFR}}$]{} relation presented in Section \[discuss\_ymc\], depend on assumptions made about globular cluster mass loss. We explicitly account for stellar evolution-based mass loss in this study through $\Upsilon_z$, and the constant mass loss term, $\Delta$, is included as part of the evolved Schechter function parameterization[^3] to account for additional sources of mass loss. The @Jordan07 fitting results for $\Delta$ call for negligible mass function evolution at the high-mass end, in agreement with predictions for most forms of globular cluster mass loss. However, globular cluster formation models developed to explain multiple population phenomena call for large amounts of mass loss ($>$90% of initial mass) in order to explain the observed ratio of enriched to unenriched populations and obtain the necessary dilution of enriching material [e.g., @DErcole08; @Conroy12]. Inferred $M_c$ and [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} values would increase in the case of large, cluster mass-independent mass loss. Recent observational studies disfavor formation models with large fractional mass loss [@Larsen12; @Larsen14; @Bastian15; @Schiavon17_Nrich], but the matter is far from settled.
The 0.1–1.0 kpc$^{-2}$ [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} values for globular cluster formation inferred here are lower than the most extreme values observed in intense starbursts and luminous infrared galaxies: 10–100 kpc$^{-2}$ [e.g., @Kennicutt12]. The lower [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} predictions are partially explained by their galaxy-averaged spatial scales, where the underlying distribution includes more extreme [$\Sigma_{\mathrm{SFR}}$]{} values in smaller, localized regions. However, the predicted 0.1–1.0 kpc$^{-2}$ range is not unreasonable when considering that a significant fraction of globular clusters (especially metal-poor systems) form in lower-mass progenitors before merging and accreting onto more massive halos, and these progenitors are not expected to host 10–100 kpc$^{-2}$ starburst conditions.
To place these [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} predictions into context, we highlight a number of numerical simulation studies that make related predictions about the properties of globular cluster formation. In a study by @Peng08 investigating the specific frequency ($S_N$) of relatively low-mass ($M_z > -19$; $\log (M_{*}/M_{\sun}) < 9.6$) Virgo cluster galaxies, the authors examined theoretical predictions for star and cluster formation histories from the Millennium simulation [@Springel05]. We select a comparable sample of low-mass ($\log (M_{*}/M_{\sun}) < 9.6$) VCS galaxies and find an average truncation mass of log ($M_c$/) $\sim 5.9$. Paired with simulation-based predictions of log ([$\Sigma_{\mathrm{SFR}}$]{}/ kpc$^{-2}$) $\sim -1.5$ at the $z\sim4.5$ peak globular cluster formation epoch, the resulting prediction lies to the left of the observed $M_c$–[$\Sigma_{\mathrm{SFR}}$]{} relation. However, this result relies on a large number of assumptions (e.g., semi-analytic star formation prescriptions, approximate galaxy size estimates) that may bias the [$\Sigma_{\mathrm{SFR}}$]{} prediction. The latest generation of cosmological simulations that include full baryonic physics [e.g., the Illustris simulation; @Vogelsberger14] and high-resolution zoom-in galaxy simulations [e.g., the FIRE simulations; @Hopkins14; @Hopkins17] motivate a new look at [$\Sigma_{\mathrm{SFR}}$]{} predictions during the epoch of globular cluster formation, building on studies that already explore kpc-scale [$\Sigma_{\mathrm{SFR}}$]{} at high redshift [@Orr17] and the impact of galaxy environment on globular cluster formation [@Mistani16].
Another recent numerical simulation study by @Li17 is also closely related to our exploration of globular cluster formation, $M_c$ values, and influence of star-forming environment. The authors implement star cluster-based star formation in a cosmological galaxy formation simulation and find that the resulting cluster initial mass function is well-described by a Schechter function. @Li17 find a positive correlation between the characteristic truncation mass and SFR for high redshift ($z > 3$) cluster formation, such that $M_c \propto \rm{SFR}^{1.6}$. While we encourage future comparisons based on [$\Sigma_{\mathrm{SFR}}$]{} rather than an unnormalized SFR (see discussion in Section \[discuss\_ymc\]), we find a similar correlation between $M_c$ and integrated SFR for our four galaxy sample, such that $M_c \propto \rm{SFR}^{1.5}$. This result supports our hypothesis that globular cluster formation at high redshift follows similar relations as young cluster formation at the present day. Further examination of the cluster mass function behavior in the context of theoretical globular cluster formation models [e.g., @Kruijssen15; @Renaud17] is clearly desirable.
A universal correlation between cluster formation and star formation environment has important implications. If this hypothesis is true, measurements of the upper end of the GCMF could allow observers to infer important details about the hierarchical build-up of galaxies and the physical conditions of star formation in the early universe through studies of globular cluster systems. There are many aspects of the globular cluster formation we are yet to fully understand (e.g., their specific frequencies, metallicity distributions, destruction and mass loss mechanisms, the origin of He and light-element abundance variations within individual clusters), but $M_c$ measurements could serve as an important tool for studying star formation in the early Universe.
Summary
=======
We find evidence for a high-mass truncation of the star cluster mass function within the PHAT survey region in M31. Parameterized using a Schechter function, this exponential truncation has a characteristic mass of $M_c$ = $8.5^{+2.8}_{-1.8} \times 10^3$ . This truncation mass is the lowest value ever observed for a star cluster population, and provides strong evidence of an upper mass limit for the PHAT cluster sample that rules out a universal power-law cluster mass distribution where the maximum cluster mass is set by sampling statistics.
When we combine the M31 mass function fit and previous $M_c$ results for young cluster systems from the literature, we identify a strong systematic correlation between the truncation mass of the star cluster mass function and star formation environment, as characterized by [$\Sigma_{\mathrm{SFR}}$]{}. The characteristic truncation mass increases with increasing star formation intensity, such that $M_c \propto$ [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{}$^{\sim1.1}$. This scaling relation might suggest an underlying physical dependence driven by interstellar pressure, but further study is required to confirm the relationship and its physical underpinnings.
Finally, we highlight that globular cluster systems also show systematic variations in the high-mass truncation of their mass distributions. We hypothesize that these mass function variations are the result of the same environmentally-dependent truncation relation that we observe for young cluster systems in nearby galaxies. This proposed commonality between ancient globular clusters and young clusters forming today could represent a long-sought link demonstrating that, while star formation in the early Universe was generally more active and intense, star cluster formation follows the same universal trends across all of cosmic time. Furthermore, it could enable the use of a galaxyÕs globular clusters systems to make quantitative statements about its early formation environment.
Calculating a Characteristic Galaxy-averaged [$\Sigma_{\mathrm{SFR}}$]{} {#appendix_sigsfr}
========================================================================
Motivation
----------
Defining a robust, galaxy-averaged [$\Sigma_{\mathrm{SFR}}$]{} is important when exploring the link between cluster mass function truncation measurements and [$\Sigma_{\mathrm{SFR}}$]{} on galaxy-wide scales. [$\Sigma_{\mathrm{SFR}}$]{} is known to vary by more than an order of magnitude within galaxies as measured on 0.1–1.0 kpc scales [e.g., @Leroy08]. Galaxy-wide [$\Sigma_{\mathrm{SFR}}$]{} measurements are often calculated simply by dividing a global SFR by an estimate of total galaxy area [e.g., using $R_{25}$; @Larsen00]. These estimates make an implicit assumption that star formation is distributed uniformly across the galaxy, leading to [$\Sigma_{\mathrm{SFR}}$]{} estimates that are biased toward small values due to the centrally-concentrated and clumpy spatial distribution of star formation within galaxies.
In this work, we calculate SFR-weighted [$\Sigma_{\mathrm{SFR}}$]{} values, denoted here as [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{}, to provide characteristic, galaxy-integrated measurements that are useful for comparing star formation behavior across our sample of galaxies. This weighted average accurately summarizes the distribution of local, kpc-scale properties of star formation within a galaxy, accounting for the fact that a large fraction of stellar mass forms in a small fraction of the galaxy area — in regions that lie in the upper tail of the [$\Sigma_{\mathrm{SFR}}$]{} distribution. We highlight that this averaging technique is conceptually similar to the analysis techniques employed by @Leroy16 in their analysis of gas surface density and ISM properties.
The [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} measurement is defined by two scales: a 2–3 kpc$^2$ measurement scale, and a full-galaxy averaging scale. The choice of a uniform, sample-wide measurement scale minimizes scale-dependent differences in [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} across the sample, and is set by the available spatial resolution of the SFR tracer observations for the sample’s distant galaxies. In addition, our adoption of a $\gtrsim$1 kpc measurement scale minimizes potential biases on SFR estimates due to the effects of discreteness and non-constant star formation histories when sampling smaller spatial scales and integrated SFRs [see e.g., @Schruba10]. The choice of galaxy-integrated averaging is driven by the need for large samples of star clusters in order to obtain statistically-significant constraints on the shape of the cluster mass function at its high mass end.
The [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} values we derive represent an underlying distribution of local [$\Sigma_{\mathrm{SFR}}$]{} values and star-forming conditions. This fact becomes particularly important when considering how galaxy-scale correlations based on galaxy-averaged kpc-scale [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} presented in this paper translate to cloud-scale cluster formation behavior. We stress caution when extrapolating behavior across dissimilar spatial scales.
[$\Sigma_{\mathrm{SFR}}$]{} Calculations
----------------------------------------
We characterize local [$\Sigma_{\mathrm{SFR}}$]{} distributions and global average values following steps described in Section \[discuss\_ymc\]. We begin by integrating SFRs over a common spatial scale (2–3 kpc$^2$) within each of the four galaxies in our sample to map [$\Sigma_{\mathrm{SFR}}$]{} locally. Next, we derive a SFR-weighted average and accompanying interpercentile range to represent the distribution of local [$\Sigma_{\mathrm{SFR}}$]{} measurements. We use the narrowest percentile range that contains 68% of the weighted [$\Sigma_{\mathrm{SFR}}$]{} measurements to characterize the dispersion of the distribution.
Galaxy-specific observational data and SFR estimation techniques fall into two groups: nearby galaxies (M31) and distant galaxies (M83, M51, Antennae). For M31, we measure [$\Sigma_{\mathrm{SFR}}$]{} using the same procedure used in @Johnson16_gamma: maps of [$\Sigma_{\mathrm{SFR}}$]{} were created by averaging the 10-100 Myr star formation history derived from CMD fitting [@Lewis15] using a deprojected circular 2 kpc$^2$ tophat spatial kernel. The use of a larger spatial kernel in this work versus the results published in @Johnson16_gamma has little effect on the derived distribution nor the weighted average of [$\Sigma_{\mathrm{SFR}}$]{}. The use of a 2 kpc$^2$ as opposed to 0.5 kpc$^2$ kernel ($r \sim 0.8$ kpc versus 0.4 kpc) results in a 0.05 dex reduction in characteristic [$\Sigma_{\mathrm{SFR}}$]{}.
For M83, M51, and the Antennae, we process GALEX and Spitzer imaging to produce [$\Sigma_{\mathrm{SFR}}$]{} maps based on the FUV+24$\mu$m SFR calibration from @Leroy08. The images were downloaded from the IRSA and MAST data archives, and include data products produced by the Local Volume Legacy survey [@Dale09] and MIPS Local Galaxy Survey [@Bendo12]. We convolve the FUV and 24$\mu$m images to a common 11 arcsec resolution and combine their flux densities according to Equation D11 from @Leroy08 to produce [$\Sigma_{\mathrm{SFR}}$]{} maps. We use a common 3 kpc$^2$ tophat spatial kernel ($r \sim 1$ kpc) to compute matched-resolution local [$\Sigma_{\mathrm{SFR}}$]{} measurements and a SFR-weighted average [$\Sigma_{\mathrm{SFR}}$]{} for each of the three galaxies. We note that this newly derived [$\langle \Sigma_{\mathrm{SFR}} \rangle$]{} measurements differ from previous values presented in the literature. These differences occur for a variety of reasons, primarily driven by our use of SFR-weighted averaging, and differences in choice of SFR tracer and calibration.
$M_c$ for M31 Globular Cluster System {#appendix_m31gcmf}
=====================================
We performed an evolved Schechter function fit to the M31 globular cluster system to provide a $M_c$ measurement that can be compared to the VCS results presented by @Jordan07. The evolved Schechter function takes the form $$p_{\textrm{MF}}(M|M_c, \Delta) \propto \frac{1}{(M+\Delta)^2}\ \exp \left ( - \frac{M+\Delta}{M_c} \right ),$$ where $M$ is cluster mass, $M_c$ is the characteristic truncation mass, and $\Delta$ is the cumulative mass loss. We fit the luminosity-based version of this function $$p_{\textrm{MF}}(m|m_c, \delta) \propto \frac{10^{-0.4(m-m_c)}}{(10^{-0.4(m-m_c)}+10^{-0.4(\delta-m_c)})^2}\ \exp ( -10^{-0.4(m-m_c)} ),$$ where $m \equiv C - 2.5 \log M$, $\delta \equiv C - 2.5 \log \Delta$, and $m_c \equiv C - 2.5 \log M_c$ represent magnitude-based versions of the mass-based variables, and C is the conversion factor defined by the solar absolute magnitude and the adopted cluster mass-to-light ratio.
We use the photometry catalog of old globular clusters from @Peacock10 and fit the $z$-band luminosity function using the same probabilistic framework described in Section \[analysis\]. We assume $\Upsilon_z$ of 3.0 $M_{\sun}$/$L_{\sun}$ to transform our fitting results back into mass space under the same assumptions used for the VCS $M_c$ data points (i.e., calculated using FSPS with 13 Gyr nominal age, @Kroupa01 IMF, and correction for stellar evolution mass losses to initial mass). As a result, we find $$\log (M_c / M_{\sun}) = 6.27^{+0.11}_{-0.10};\ \log (\Delta / M_{\sun}) = 6.05^{+0.14}_{-0.11} .$$ We pair this $M_c$ measurement with a M31 stellar mass determination from @Tamm12 of $\log(M_*/M_{\sun})=11.1 \pm 0.09$ and use these observations to place M31 on the left panel of Figure \[fig\_mc\_gc\].
[^1]: <http://dan.iel.fm/emcee/>
[^2]: Results from @Villegas10 and @Harris14 are not included because only Gaussian function fitting results are published. While these additional fits would boost the sample’s number statistics, the @Jordan07 results are representative of the larger, combined dataset.
[^3]: Please note that the fitted $M_c$ values in the evolved Schechter function parameterization represent initial values and do not need to be corrected for $\Delta$-parameterized mass loss.
|
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abstract: 'We discuss the formation of globules in planetary nebulae, typified by those observed in the Helix Nebula. We show that the properties of the globules, their number, mass, separation, and overall geometry strongly support a scenario in which globules are formed by the fragmentation of a swept-up shell as opposed to models in which the knots form in the AGB wind. We show that the RT or other instabilities which lead to the break-up of shells formed in the nebulae by fast winds or ionization fronts can produce arrays of globules with the overall geometry and within the mass range observed. We also show that the presence of a magnetic field in the circumstellar gas may play an important role in controlling the fragmentation process. Using field strengths measured in the precursor AGB envelopes, we find that close to the central star where the fields are relatively strong, the wavelengths of unstable MRT modes are larger than the shell dimensions, and the fragmentation of the shell is suppressed. The wavelength of the most unstable MRT mode decreases with increasing distance from the star, and when it becomes comparable to the shell thickness, it can lead to the sudden, rapid break-up of an accelerating shell. For typical nebula parameters, the model results in numerous fragments with a mass scale and a separation scale similar to those observed. Our results provide a link between global models of PN shaping in which shells form via winds and ionization fronts, and the formation of small scale structures in the nebulae.'
date: '?? and in revised form ??'
---
Introduction
============
Globules are the most striking small-scale structures seen in planetary nebulae (PNe). They consist of dense molecular condensations embedded in and around the periphery of the ionized gas (e.g., [@hu02 Huggins 2002]). In optical images their photo-ionized surfaces are seen in H$\alpha$ and other lines, illuminated by the radiation of the central star. They often have cometary tails extending away from the star in the radial direction. Because of their small size, globules are only resolved at high resolution in nearby PNe, e.g., in NGC 7293 (the Helix Nebula) and NGC 6720 (the Ring Nebula), but they are expected to be a common feature of evolved PNe with a significant component of molecular gas.
The large number and the similarity of the globules in a PN like NGC 7293 point to an underlying formation mechanism with rather specific characteristics. In this paper we review the properties of globules, we ask whether simple models can explain some of their general characteristics, and we explore the possible role of magnetic fields in the globule formation process.
Properties of the globules
==========================
Table 1 lists the measured properties of globules in NGC 7293 that are likely relevant to the formation process. The evolution of mature globules is dominated by photo-ionization processes, but we are interested here in the mechanisms that determine quantities such as the number and mass-scale of fragments from which the mature globules form.
Most of the properties listed in Table 1 are self-explanatory. The typical angular spacing of the globules (relative to the central star) is an especially useful quantity because it does not vary with expansion or the evolution of the globules: it is estimated here from the surface density in the main ring given by [@mei05]. The general distribution of the molecular gas seen in spatio-kinematic CO maps consists of partial shells; the ratio $\Delta r/r$ in the table is taken from high velocity resolution observations made at the systemic velocity. The spatio-kinematics of the low excitation ionized gas (e.g., [@me05 Meaburn 2005]) support the shell picture.
NGC 6720 shares some of the characteristics of NGC 7293 but is a factor $\sim 5$ younger. Less of the neutral envelope is in globules, and they are at an earlier stage of development. Their morphology and relation to the nebula shell structure are of special interest, and are illustrated in Fig. 1.
-------------------- -------------- ------------------------- ------------
globule mass $m_g$ $10^{-5}$ $M_\odot$ [@hu02]
shell mass $M_s$ 0.2 $M_\odot$ [@yo99]
number of globules $N$ 20,000 [@mei05]
distance from star $r$ 6–15$\times 10^{17}$ cm
angular spacing $\theta$ 0.02 rd This paper
shell width/radius $\Delta r/r$ 1/20 [@fo91]
-------------------- -------------- ------------------------- ------------
: Properties of the Globules in NGC 7293[]{data-label="tab:kd"}
Shell models
============
General characteristics
-----------------------
The thin, shell-like distribution of the densest gas in NGC 7293 (and NGC 6720) provides strong support for a model of globule formation based on the fragmentation of a swept-up shell. For the break-up of a shell of radius $R_s$ into fragments of size $\Delta R_s$, equal to the shell thickness, we expect: $M_s \sim Nm_g$, $N\sim
4\pi/\theta^2$, and $\theta \sim \Delta R_s/R_s$. These relations are approximately satisfied by the independently measured quantities given in Table 1.
In order to construct a physical model, we consider for simplicity the case of a shell driven by a constant, momentum-conserving wind ([@ka90 Kahn & Breitschwerdt 1990]) which sweeps up the precursor AGB envelope with an $r^{-2}$ density distribution. This simple case leads to a shell model that travels with a constant velocity. In reality, fragmentation of the shell will require modest accelerations. The difference between the two cases, in terms of shell conditions, will likely be small. The constant velocity shell is completely specified by the AGB wind velocity ($U$) and mass-loss rate ($\dot{M}$), the shell velocity ($V_s$), and the sound speed in the shell ($c_s$). Fig. 1 shows the properties of this shell as a function of the shell radius for $U=15$ kms$^{-1}$, $\dot{M}= 10^{-4}$ $M_\odot$, $V =
23.5$ kms$^{-1}$, and $c_s = 1.5$ kms$^{-1}$ (we assume the gas is in a PDR). The left hand panels show the shell mass, and thickness. Note that $\Delta R_s/R_s \sim 1/100$ is close to that observed. Note also that $M_s$ does not reach a few tenths of a solar mass until the shell is large $\sim 10^{17} $ cm. The top right panel shows the mass of fragments if the shell breaks up at radius $R_s$ on a size scale $\Delta R_s$. Note that the mass of a fragment is small at small $R_s$, and only reaches $>10^{-6}$ $M_\odot$ at $R_s >
10^{17}$ cm.
Fragmentation
-------------
Several different processes have been suggested for the actual break-up of PN shells including the NTSI, the TSI and the related ISFI, and the RT instability (e.g., [@dw98 Dwarkadas & Balick 1998], [@ga99 Garcia-Segura 1999]). In simulations, the NTSI may develop at an early phase but it may not lead to fragmentation, and from the discussion above it is doubtful that it could generate the ensemble of observed globules at that time. The RT instability is well-studied and it occurs when the shell is accelerated. The onset of ionization is one of several means of shell acceleration and the RT instability may couple to the ISFI at that stage. Note also that propagation of the shell down a steeper than $\rho \sim r^{-2}$ gradient will also produce an acceleration. For a nominal acceleration of 10 kms$^{-1}$/1000 yr, the RT growth time for the length scale equal to $\Delta R_s$ is shown in the bottom left panel. It is significantly less than the expansion time of the shell for all values of $R_s$. Thus the shell is fragile. If it accelerates near this nominal level before it reaches $10^{17}$ cm, it will break-up into low-mass fragments with a low total mass.
Effect of a magnetic field
==========================
The role of magnetic fields in PN formation is an area of ongoing debate. Significant fields are measured in the envelopes of AGB stars in the SiO, H$_2$O, and OH maser lines, and it can be expected that the fields will be swept up into PN shells. We explore here how this may affect the fragmentation process.
The presence of a tangential magnetic field at an interface (the situation expected in a swept-up shell) typically has a stabilizing effect. The theory is well studied for the RT instability, and simulations for the magnetic case have been reported by [@ju95]. The field has two effects. First, it suppresses all RT modes at short wavelengths with a cut-off given by $\lambda_c =
B^2/a\rho$, where $a$ is the acceleration and $\rho$ is the density of the shell (assumed to be much denser than the driving wind). Second, the wavelength of the fastest growing mode is $2\lambda_c$, with a growth rate similar to the non-magnetic case.
For the shell model discussed earlier, the curves with subscript M in Fig 2. show the effects of a magnetic field in the AGB wind. The field is assumed to have the form $B=(r/10^{16})^2$ mG, based on an ensemble of circumstellar envelopes ([@vl02 Vlemmings 2002]). At early times, the field contributes to the pressure when it is swept into the shell, with the result that the shell thickness increases, and the potential break-up mass for small $R_s$ is $10^{-6}$–$10^{-5}$ $M_\odot$, close to that observed. The break-up can not, however, occur at these scales because at small $R_s$ the magnetic-RT critical wavelength is much larger than the shell thickness. The break-up of the shell is suppressed in the early phases.
At larger $R_s$, where the fields become weaker, the critical wavelength decreases. and when it becomes comparable to the shell thickness, the growth time for the instability drops rapidly to a low value. The effect is like a switch. If the system is accelerating, it leads to the sudden, rapid break-up of the shell. At these scales the mass of the fragments and the total mass are in the observed ranges.
Conclusions {#sec:concl}
===========
The properties of globules in PNe support a scenario in which globules are formed by fragmentation of a swept-up shell. Instabilities in simple shell models can produce arrays of globules with the overall geometry and within the mass range observed. The magnetic field in the AGB wind may play a key role in controlling the fragmentation process. Our results provide a link between global models of PN shaping and globule formation.
This work is supported in part by NSF grants AST 03-07277 and AST 05-07519.
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|
---
abstract: 'We present followup spectroscopic observations of quasar candidates in the Small Magellanic Cloud selected by Eyer from the OGLE database. Of twelve observed objects identified as “QSO Candidate”, five are confirmed quasars, with the emission redshifts ranging from 0.28 to 2.16. Two of those quasars were also recently identified independently in the MACHO database by Geha et al. We discuss the prospects of using variability-based selection technique for quasar searches behind other dense stellar fields. An additional criterion utilizing the color-color diagram should reduce the number of stars in the candidate lists.'
author:
- 'A. Dobrzycki, L. M. Macri, K. Z. Stanek, and P. J. Groot'
title: 'Variability-selected quasars behind the Small Magellanic Cloud'
---
Introduction
============
Searches for quasars in dense stellar fields — such as the Magellanic Clouds — were in the past hampered by difficulties in selecting candidates. Optical followup on X-ray selected objects had to deal with large number of candidates in X-ray source error boxes, while variability studies required monitoring of vast number of objects. At the same time, such quasars are of great astrophysical interest, for example as reference points for analysis of proper motions and as background sources for absorption studies. Until recently, only a handful of confirmed quasars behind the LMC and SMC were known (Blanco & Heathcote 1986; Crampton et al. 1997; Tinney et al. 1997; Anguita, Loyola, & Pedreros 2000; see also Kahabka et al.1999, Haberl et al. 2000 and Kahabka, de Boer, & Brüns 2001). Almost all of those quasars were located behind the outer, sparse parts of the Clouds.
Recent developments — such as the launch of the [*Chandra X-ray Observatory*]{} with its superb spatial resolution in X-rays, and availability of large photometry databases (OGLE, MACHO) — are now making systematic searches for quasars in dense stellar fields possible. Recently, Dobrzycki et al. (2002) found four X-ray quasars among serendipitous sources in four [*Chandra*]{} observations of objects in the LMC coinciding with the OGLE fields.
A characteristic that was often used in quasar surveys was their irregular variability. Several such surveys were performed or are on-going (Hawkins 1983; Meusinger & Brunzendorf 2001; Rengstorf et al. 2001). In the Magellanic Clouds, this technique has been used by Geha et al. (2002), who published a list of forty seven quasars behind the LMC and SMC, selected from the MACHO database.
Between 1997 and 2001, large parts of the Magellanic Clouds were monitored for microlensing events by the Optical Gravitational Lensing Experiment (OGLE-II: Udalski, Kubiak & Szymański 1997). Udalski et al. (1998) released $BVI$ photometry and astrometry of 2.2 million objects from the central parts of the SMC[^1]. In addition, a large catalog of 68,000 variable objects observed by OGLE-II in both the LMC and the SMC was prepared by Żebruń et al. (2001)[^2], based on a version of the image subtraction software (Alard & Lupton 1998) developed by Woźniak (2000).
Eyer (2002) presented an algorithm for selecting quasar candidates from objects in the OGLE database. He searched the database for slowly and irregularly varying blue objects and identified QSO candidates (“QCs”) towards the Magellanic Clouds: 118 QCs towards the LMC and 15 towards the SMC. Eyer also identified several “Unclassified” objects, which had similar light curve characteristics.
In 2002 September, we performed followup observations of twelve brightest “QSO Candidates” from the SMC from the Eyer’s list with the Magellan 6.5-meter Baade telescope. Five of them turned out to be previously unknown quasars; an excellent success rate. We also observed all four of Eyer’s “Unclassified” objects in the direction of the SMC and none of them turned out to be a QSO.
Coincidentally, less then a week after our observations had been completed, the paper by Geha et al. (2002) was posted on astro-ph. In this paper, the authors performed an analysis of the MACHO variable star database. They selected 360 quasar candidates behind the Magellanic Clouds, and completed followup observations of 259 of them. In that way, they identified forty seven quasars: thirty eight behind the LMC and nine behind the SMC. Three of their SMC objects were on the list of Eyer’s candidates: two quasars and a Be star.
In this paper, we present the identifications of the new quasars behind the SMC, and we discuss the prospects for application of Eyer’s method for quasar searches behind other dense stellar fields.
Observations and identifications
================================
The optical spectra were obtained on 2002 September 16-18 with the Magellan Baade 6-5 meter telescope. We used the LDSS-2 imaging spectrograph, with a 2048$\times$2048 SITe\#1 CCD with a scale of 0.38 arcsec/pixel, a gain of 1 $e^-/$ADU, and a readout noise of $7e^-$. The slit width was 1.03 arcsec and the grism setting was 300 l/mm, yielding a nominal resolution of 13.3 Å. Exposure times ranged from 120 to 600 seconds. All observations were carried out with the slit oriented in the east-west direction. Additionally, we observed two spectrophotometric standards, LTT 1788 and LTT 7379 (Hamuy et al. 1992). Following each observation, a He-Ne arc lamp spectrum was acquired for wavelength calibration purposes. Spectra were reduced in the standard way using IRAF.
Figure \[fig:spectra\] shows the spectra of twelve of Eyer’s QC objects. There are five confirmed quasars among them, and we show their spectra on the left panel, while on the right panel one can find spectra of objects that turned out to be stars. We summarize the object properties in Table \[tab:identifications\]. We also observed all four of Eyer’s “Unclassified” objects, which all turned out to be stars; for completeness, we include this information in Table \[tab:identifications\].
[cccccc]{} 003850.79–731053.1 & 17.683 & 17.870 & 16.822 & S1 & QSO, ${\mbox{$z_{\rm em}$}}=0.28$\
004743.68–731630.1 & 17.532 & 17.666 & 17.211 & S3 & F star\
004818.25–731242.8 & 17.446 & 17.493 & 17.193 & S4 & A star\
004905.87–730257.5 & 17.973 & 18.002 & 17.833 & S5 & B star\
005136.59–732016.5 & 18.036 & 18.349 & 17.241 & S6 & Be star\
005316.80–724219.9 & 19.263 & 19.432 & 18.860 & S7 & G star\
005418.96–723737.7 & 18.137 & 18.194 & 17.893 & S8 & Be star\
005448.97–722544.6 & 19.001 & 19.180 & 18.263 & S9 & QSO, ${\mbox{$z_{\rm em}$}}=1.79$\
010127.64–722422.6 & 19.058 & 19.191 & 18.673 & S13 & F star\
010234.69–725424.1 & 18.346 & 18.689 & 17.640 & S12 & QSO, ${\mbox{$z_{\rm em}$}}=2.12$\
010244.89–721521.7 & 18.892 & 19.385 & 18.412 & S11 & QSO, ${\mbox{$z_{\rm em}$}}=1.06$\
010721.61–724845.5 & 18.970 & 19.218 & 18.279 & S15 & QSO, ${\mbox{$z_{\rm em}$}}=2.16$\
004504.34–724449.9 & 17.906 & 18.245 & 17.356 & S22 & F star\
004702.90–730800.7 & 18.036 & 18.506 & 17.346 & S23 & F star\
005039.12–724154.3 & 18.871 & 19.138 & 18.305 & S24 & F star\
005137.19–731429.2 & 17.221 & 17.240 & 16.973 & S25 & Be star\
The spectra of all five quasars show at least two emission lines, enabling unambiguous determination of emission redshifts. As mentioned earlier, two of those quasars (QSO J010234.69–725424.1 and QSO J010721.61–724845.5) were independently identified by Geha et al.(2002).
Discussion
==========
We have identified five variability-selected quasars among twelve brightest Quasar Candidates identified by Eyer (2002) based on the characteristics of their light curves in the OGLE database. The method is very efficient. As expected, the contaminants in the list of candidates were predominantly early type stars in the SMC. Qualitatively, we do not see any obvious trends or differences between the light curves of quasars and stars. In Figure \[fig:lightcurves\] we show the OGLE light curves of the QC objects, arranged similarly to Fig. \[fig:spectra\].
It appears, however, that the source colors will help in improving the candidate lists. In Figure \[fig:colcol\] we show the color-color diagram for the twelve sources presented in this paper, plus twenty four Eyer’s QCs from the LMC region which were identified by Geha et al. (2002) and Huchra (2002, private communication). One can clearly see that variability-selected stars and quasars occupy different regions in the color-color space. We note here that the Faint Sky Variability Survey (Groot et al. 2002) data seem to show a similar effect.
In Fig. \[fig:colcol\] we also show the known X-ray selected quasars in both the LMC and the SMC for which OGLE photometry is available. Those quasars also separate well from the stars. We note that one of Eyer’s selection criteria for quasar candidates was a color cutoff, $V-I<0.9$, but this criterion was not applied to the X-ray-selected quasars before they were identified. They are typically redder than Eyer’s objects.
As mentioned earlier, until very recently only a handful of confirmed quasars were known in the general direction of the SMC (Crampton et al. 1997; Tinney et al. 1997). All those quasars are located in fairly sparse stellar fields away from the center of the SMC, which limits their applicability to studies of SMC proper motion or investigations of absorption properties of the SMC. Quasars presented in the present paper and objects from Geha et al. (2002) lie behind the dense parts of the SMC. Note that those quasars — extremely interesting and useful objects in their own right — are in reality byproducts of monitoring surveys, unrelated to the original scientific goals of the surveys. It is an excellent example that such projects can lead to unexpected, yet very valuable results.
Geha et al. (2002) independently identified two of our quasars. We note that those two objects are the largest redshift quasars among our five, but this most likely is just a coincidence, not a result of the difference in the methods applied by Geha et al. and Eyer. Geha et al.’s quasar redshifts span a large redshift range. We note, however, that two methods, which, after all, are based on a similar concept, apparently have a rather small intersection in the final candidate lists: there are only ten objects in common in both the SMC and the LMC. This fact and the fact that X-ray-selected quasars are redder than the variability-selected objects indicate that both techniques are conservative and both will miss some quasars. The two methods should therefore be considered as complementing one another, rather than as competing.
We note here that the same two quasars lie relatively close to [*ROSAT*]{} X-ray sources listed in Kahabka et al. (1999) and Haberl et al. (2000), although neither one of the X-ray sources was classified as a probable QSO.
One of the quasars identified in this paper, QSO J003850.79–731053.1 (${\mbox{$z_{\rm em}$}}=0.28, V=17.7$) is a very promising candidate for studies of absorption in the SMC. Its brightness should enable good spectroscopy with the Cosmic Origin Spectrograph aboard the Hubble Space Telescope. The other four quasars and quasars from Geha et al. (2002) are very well positioned to be reference points for SMC proper motion studies. We add here that we also identified six other X-ray selected quasars behind the dense parts of the SMC; we will present them in the forthcoming paper (Dobrzycki et al. 2003, in preparation).
We were somewhat surprised to find a G-type star among Eyer’s QC objects, especially since he did utilize color selection in constructing the list of candidates. However, it was also the faintest of the observed objects, for which the photometry is likely rather uncertain.
Excellent efficiency of the variability-based method bodes well for searches of quasars behind other dense stellar fields for which monitoring photometry databases are available. A first successful search has already been published by Geha et al. (2002), who found 38 QSOs behind the LMC in the MACHO database.
The paper by Eyer (2002) contains a list of 118 QSO candidates behind the LMC. At present, identifications are known for $\sim$25 of them (Geha et al. 2002; Huchra 2002, private communication). As noted earlier, the color-color diagram for identified Eyer’s QSO candidates indicates that quasars are typically redder in $V-I$ than QCs that turned out to be stars (Fig. \[fig:colcol\]). Remaining objects from Eyer’s LMC candidate list split roughly evenly between the quasar and stellar regions in the color-color diagram, indicating that the spectroscopic followup should yield a large number of quasars.
Geha et al. (2002) noted that their quasars tend to be bluer when they brighten. OGLE time coverage in $B$ and $V$ filters is much sparser than in $I$, which, combined with relatively small size of our sample, does not allow us to make quantitative statements of that nature. We do note, however, that this effect is present in the brightest of our quasars, QSO J003850.79–731053.1: the well defined features in the light curve of this object (top left panel on Fig. \[fig:lightcurves\]) have corresponding changes in the $V-I$ color consistent with the effect seen in MACHO quasars.
Another extremely interesting region to search for quasars where the method could be applied is the Galactic Bulge. To our knowledge, no quasars have been identified so far in the vicinity of the Galactic center. There are, however, several regions where interstellar extinction is quite low ($A_V<3$) and where one could, in principle, see quasars. The best known such area is Baade’s Window (e.g. Stanek 1996), but there are several other.
Identifying quasars behind the Galactic Bulge would be very valuable. Recently, Sumi, Eyer, & Woźniak (2002) have shown that there is a statistically significant difference between proper motions of faint versus bright red clump stars in one of the OGLE bulge fields. Finding fixed reference points for this study, which quasars could provide, would be an extremely interesting result.
On one hand, the search toward the Galactic bulge will be made easier by the fact that there will be few early type stars, which in the Magellanic Clouds are the primary contaminants in the candidate lists. On the other hand, the quasars will be considerably reddened even in the low extinction windows, making them less conspicuous as blue objects. Also, the surface density of objects that need to be analyzed will be very large, and as a result there will likely be a sizeable number of artifacts, etc., which will contaminate the candidate lists.
We would like to thank L. Eyer, B. Paczyński and J. Stocke for helpful discussions, J. Huchra for observing the LMC candidates, and the referee, M. Geha, for helpful comments and, especially, for her suggestion to explore the information contained in the color-color diagram. AD acknowledges support from NASA Contract No. NAS8-39073 (CXC). LMM was supported by the Hubble Fellowship grant HF-01153.01-A from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.
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[^1]: Data available from ftp://bulge.princeton.edu/ogle/ogle2/maps/smc/
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---
abstract: 'The spin density wave and its temperature dependence in oxypnictide are studied in a three-band model. The spin susceptibilities with various interactions are calculated in the random phase approximation(PPA). It is found that the spin susceptibility peaks around the M point show a spin density wave(SDW) with momentum (0, $\pi$) and a clear stripe-like spin configuration. The intra-band Coulomb repulsion enhances remarkably the SDW but the Hund’s coupling weakens it. It is shown that a new resonance appears at higher temperatures at the $\Gamma$ point indicating the formation of a paramagnetic phase. There is a clear transition from the SDW phase to the paramagnetic phase.'
author:
- 'Mingsheng Long, Liangbin Hu, W. LiMing'
title: 'Spin density wave in oxypnictide superconductors in a three-band model'
---
INTRODUCTION
============
The high temperature superconductivity in the newly discovered oxypnictides, LnFeAsO (Ln = La,Pr,Ce,Sm ), has attracted great attention aiming to identify the mechanism of superconductivity in these materials[@Kamihara]. Recently the transition temperature $T_c$ is dramatically raised from $26K$ to $43K$[@Chen]. In addition to the high $T_c$, these materials display many other interesting properties. The most interesting phenomena is the presence of competition between the magnetically ordered ground states[@dong] of spin density wave(SDW) and superconductivity, but a controversy model explains the superconductivity by means of antiferromagnetic spin fluctuation in LaFeAsO[@Mazin]. Pure oxypnictide is not superconducting but shows an anomaly at about $150 K$ in both resistivity and dc magnetic susceptibility[@Kamihara; @Ma]. Both experimental and theoretical evidences show that the anomaly is caused by the SDW instability[@dong; @Chen1; @McGuire].
It is shown by the first principle calculations that the band structure of LnFeAsO near the Fermi surface (FS) is formed by a hole-like pocket centered around the $\Gamma$ point and an electron-like pocket around the M point in the extended Brillouin zone(BZ)(one Fe atom per unit cell)[@Singh; @Boeri; @Haule]. A strong FS nesting effect exists between the hole and electron pockets with commensurate wave vectors, $(\pi, 0)$ and its symmetric ones. This leads to a strong SDW instability, and is believed to cause the the anomaly at $150 K$.
Raghu [*et al*]{} calculated the SDW in the iron oxypnictides within a minimal two-band model in the phase random approximation(RPA). They found that the SDW is enhanced significantly by the intra-band Coulomb repulsion. The influence of the inter-band Coulomb interaction and the Hund’s coupling, however, have not yet been fully studied in literatures. Different researchers select different groups of interaction parameters, but the relation between them has not been revealed. In addition, the temperature dependence of the SDW has hardly been theoretically considered. As pointed out by Lee and Wen[@Lee], a three-band model reproduces more accurately the band structure near the Fermi surface of oxypnictides. In this work we study the SDW in a three-band model in the RPA and the temperature dependence of the spin susceptibility. The magnetic instability is studied for a wide range of interaction parameters. We found a new resonance of the spin susceptibility around the $\Gamma$ point at higher temperatures, indicating the formation of a paramagnetic phase. There appears a transition from the SDW phase to the paramagnetic phase when temperature increases.
\[hop\] {width="6cm" height="5cm"}
Model Hamiltonian
=================
The FeAs layer in LaFeAsO forms a square lattice, where Fe ions locate on the lattice sites and an As ion sits at the center of each square. Various band structure calculations showed that the main contribution to the density of states near the FS comes from $d_{xz},d_{yz}$ and $d_{xy}$ of the five $3d$ orbitals of Fe atoms[@Lee]. The left two orbitals are far apart from the FS.
In the three band model, the hopping terms between $d_{xz}$, $d_{yz}$ and $d_{xy}$ as illustrated in Fig.1 are included in the Hamiltonian. They are written as[@4; @Lee] $$H_0=\sum_{{\bf k}\sigma}\Psi_{k\sigma}^\dagger M_{\bf
k}\Psi_{k\sigma},$$ where the three-component field $\Psi_{k\sigma}$ is defined as $\Psi_{k\sigma}=(
d_{xz\sigma}({\bf k}), d_{yz \sigma}({\bf k}),d_{xy\sigma}({\bf k}))^T$ and the Matrix $M_{\bf k}$ is given by $$\begin{aligned}
\label{Mk}
M_{\bf
k}=\left(
\begin{array}{ccc}
\varepsilon_{11}(k)&\varepsilon_{12}(k)&\varepsilon_{13}(k)\\
\varepsilon_{21}(k) &\varepsilon_{22}(k)&\varepsilon_{23}(k)\\
\varepsilon_{31}(k)& \varepsilon_{32}(k)& \varepsilon_{33}(k)\\
\end{array}\right),\end{aligned}$$ with elements $$\begin{aligned}
\nonumber&&\varepsilon_{11}=-2t_1\cos k_x-2t_2\cos k_y-4t_3\cos k_x\cos k_y\\&&\nonumber
\varepsilon_{22}=-2t_2\cos k_x-2t_1\cos k_y-4t_3\cos k_x\cos k_y\\&&\nonumber
\varepsilon_{33}=-2t_7\cos k_x-2t_7\cos k_y-4t_8\cos k_x\cos k_y\\&&\nonumber
\varepsilon_{12}=\varepsilon_{21}=-4t_4\sin k_x\sin k_y\\&&\nonumber
\varepsilon_{13}=-\varepsilon_{31}=-2it_5\sin k_x\\&&\nonumber
\varepsilon_{23}=-\varepsilon_{32}=-2it_6\sin k_y.\nonumber\end{aligned}$$ The hopping parameters are set to $t_1 = -1.0(\approx0.4 eV), t_2 = 0.7, t_3 = -0.80,
t_4 = 0.6, t_5=t_6=-0.35, t_7=-0.3, t_8=0.2, \mu = 1.15$, in units of $|t_1|$ [@4].
\[energy\] {width="8cm" height="7cm"}
The energy dispersion is plotted in Fig. 2(a) in the extended BZ and 2(b) along the $\Gamma\rightarrow X \rightarrow M \rightarrow\Gamma$ in the folded BZ(two Fe atoms per unit cell). Fig. 2(d) plots the density of states of the band structure. It is seen that there are two dominating Van Hove singularities near the Fermi level, close to the hole and electron pockets. The density of states is significantly different from the two-band model, of which the Van Hove singularities are more distant from the Fermi level. A stronger nesting effect is expected in this three-band model. The third band is mixed with the two conventional bands and thus should have great contribution to the spin susceptibility.
The spin susceptibility
=======================
The static spin susceptibility in the non-interacting case is given by $\chi^{(0)}({\bf
q}) = \sum_{ll'}\chi^0_{ll'}({\bf q})$ with $$\begin{aligned}
\label{chill} \chi^0_{ll'}({\bf q})&={1\over 2N}\sum_{k}{f(\varepsilon_{{\bf
k}l})-f(\varepsilon_{{\bf k+q}l'})\over \varepsilon_{{\bf k+q}l'}-\varepsilon_{{\bf
k}l}}|<{k+q,l'}|{k,l}>|^2\end{aligned}$$ where $l,l'=1,2,3$ are band indexes, $f(\varepsilon)=1/(e^{\beta(\varepsilon-\mu)}+1)$ is the Fermi distribution function, $\beta=1/kT$, and $\varepsilon_{{\bf k}l}$ and $|{\bf
k},l> $ are the eigenvalues and eigenvectors respectively of the Hamiltonian matrix (\[Mk\]).
We fix $kT = 0.02(\sim 93K)$ for a finite lattice with 64 $\times$ 64 ${\bf k}$ meshes in the extended BZ to calculate the static spin susceptibility, which is shown in Fig.3(upper). The static spin susceptibility shows great peaks at ${(0,\pm\pi)}$ and ${(\pm\pi,0)}$. This indicates that a collinearly-striped antiferromagnetic (AFM) order phase, a spin density wave (SDW), exhibits in oxypnictide. This feature is in agreement with the neutron scattering measurements[@Ma; @McGuire; @T]. The SDW comes from the strong nesting between the hole pocket at the $\Gamma$ point and the electron pocket at the M point in the extended BZ. It is observed experimentally that this SDW peak is significantly suppressed by F doping[@dong]. This is reasonable because a down-shift of the Fermi level tends to reduce the size of electron pocket and enlarge the hole pocket thus to suppress the nesting between them.
\[chi\] {width="5cm" height="4cm"}
{width="5cm" height="4cm"}
Now we consider the interactions: the intra-band Coulomb repulsion $U$, the inter-band coulomb interaction $U'$, the Hund’s coupling $J$. The interaction Hamiltonian is written as $$\begin{aligned}
\nonumber H_{int}&=&U\sum_{il}n_{il\uparrow}n_{il\downarrow}+U'\sum_{i,l\neq
l'}n_{il}n_{il'}\\&+&J\sum_{i,l\neq l'}{\bf S}_{il}\cdot{\bf S}_{il'}
%+J'\sum_{il\neq
%l'}d_{il\uparrow}^\dagger d_{il\downarrow}^\dagger d_{il'\downarrow}d_{il\uparrow}\end{aligned}$$
In the RPA[@2], the spin susceptibility, $\chi^s({\bf q})$, and the charge-orbital susceptibility, $\chi^c({\bf q})$, with interactions are given by $$\begin{aligned}
\label{chis}
\chi^s({\bf q})&&=[\hat{I}-U^s\chi^0({\bf q})]^{-1}\chi^0({\bf q}),\\
\chi^c({\bf q})&&=[\hat{I}+U^c\chi^0({\bf q})]^{-1}\chi^0({\bf q}),\label{chic}\end{aligned}$$ where $\chi^0({\bf q})$ is a $3\times 3$ matrix with elements defined in (\[chill\]) and $U^{s(c)}$ are the interaction matrices $$\begin{aligned}
\nonumber U^s=\left[\begin{array}{ccc}U&-J&-J\\
-J&U&-J\\
-J&-J&U\end{array}\right],U^c=\left[\begin{array}{ccc}U&2U'&2U'\\
2U'&U&2U'\\
2U'&2U'&U\end{array}\right]\end{aligned}$$
It is seen from (\[chis\]) and (\[chic\]) there will appears a magnetic instability when the following relations are satisfied: $$\begin{aligned}
det[\hat{I}\mp U^{s(c)}\chi^0({\bf q})]=0\label{det}\end{aligned}$$ To show the magnetic instability at the M point the determinants (\[det\]) at the SDW momentum, $(0, \pi)$, are calculated for different interaction parameters and are plotted in Fig.4. The parameters on the contour lines labeled by “$-0-$” in these two diagrams give zero determinants thus lead to magnetic instabilities. For a fixed $U$ the spin susceptibility decreases with increasing Hund’s coupling $J$ thus reduces the SDW peak at the M point. This result is quite different from that made by Raghu [*et al*]{}[@Raghu], who found stronger spin fluctuations with increasing $J$. The lower diagram in Fig.3 shows the spin susceptibility close to the magnetic instability with $U
= 3.0, J = 0.0$, where the SDW peak is much enhanced due to the intraband Coulomb repulsion relative to that of the bare spin susceptibility. When $J$ increases, however, the SDW peak drops significantly until the SDW phase fully disappears. It is interesting to notice that, when $U<5$, a ferromagnetic coupling ($J<0$) is beneficial to the formation of the SDW on the M point. On the other hand, the charge-orbital instability appears in the region $-4.2<U'<0$. That is, when a Coulomb attraction exists between different bands a charge-orbital instability may occur. Apart from this region the charge-orbital susceptibility also drops rapidly. It is worthy to notice that the charge-orbital susceptibility depends weakly on the value of $U$, the intra-band repulsion, except for the case of very strong inter-band repulsion.
\[detJU\] {width="5cm" height="4cm"} {width="5cm" height="4cm"}
The SDW phase has striking temperature dependence. It is found that at higher temperatures a new resonance appears at the $\Gamma$ point, which starts to increase rapidly at a temperature $kT \sim 0.08$, as shown in Fig.5. This resonance corresponds to the formation of a paramagnetic phase in the material. The SDW peak at the M point drops to a nearly stable value at this temperature. This result is qualitatively in agreement with the experimental observation for the stripe-like AFM phase in LaOFeAs, which forms under a temperature 134K[@Clarina]. At $kT_c=0.12$ the intensity of paramagnetic phase starts to surpass that of the striped AFM phase. It is found that temperature changes hardly the bare spin susceptibility but smooths its distribution in the BZ. Reducing the intra-band coupling parameter $U$, the relative intensity between the paramagnetic phase and the SDW phase reduces significantly. Hence it is confirmed that this new resonance comes from the intra-band coupling. Therefore, this temperature dependence of the spin susceptibility provides detailed information for the band structure of the oxypnictide material. More experimental evidences for this temperature dependence are expected. At very low temperatures the spin susceptibility showed more abundant structures but requires further stringent calculations.
\[tem\] {width="5cm" height="4cm"} {width="5cm" height="4cm"}
We tried to solve the superconducting gap equation by including a pairing coupling term into the interaction Hamiltonian but failed to find a stable gap function with some symmetries, such as $s, s^-, d_{x^2-y^2}, d_{xy}$, etc. This failure also exists in some literatures, e.g., Yao [*et al*]{} found only eigenvalues 0.1-0.4 with the $d_{xy}-$wave symmetry[@Z]. But Yanagi [*et al*]{} claimed that some groups of interaction parameters give eigenvalue unity for this gap symmetry in the same material[@YANAGI]. Therefore, the gap symmetry remains controversy and requires further studies.
In conclusion, a three-band model is set up to reproduce the band structure near the fermi surfaces of oxypnictide. It shows a hole pocket around the $\Gamma$ point and a electron pocket around M point in the extended BZ. The spin susceptibility with various interactions are calculated in the random phase approximation. It is found that the spin susceptibility peaks around the M point, showing a SDW with momentum (0, $\pi$) and a clear stripe-like spin arrangement. The intra-band coupling enhances remarkably the SDW but the Hund’s coupling weakens it. The interaction parameters are determined for the magnetic instability in oxypnictide. Finally the temperature dependence of the spin susceptibility is studied. It is found that a new resonance appears at higher temperatures at the $\Gamma$ point indicating the formation of a paramagnetic phase. There is a transition between the SDW phase and the paramagnetic phase.
This work was supported by the National Natural Science Foundation of China (Grant No. 10874049), the State Key Program for Basic Research of China (No. 2007CB925204) and the Natural Science Foundation of Guangdong province ( No. 07005834 ).
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