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Let $(-A,B,C)$ be a continuous time linear system with state space a separable complex Hilbert space $H$, where $-A$ generates a strongly continuous contraction semigroup $(e^{-tA})_{t\geq 0}$ on $H$, and $\phi (t)=Ce^{-tA}B$ is the impulse response function. Associated to such a system is a Hankel integral operator $\Gamma_\phi$ acting on $L^2((0, \infty ); C)$ and a Schr{\"o}dinger operator whose potential is found via a Fredholm determinant by the Faddeev-Dyson formula. Fredholm determinants of products of Hankel operators also play an important role in the Tracy and Widom's theory of matrix models and asymptotic eigenvalue distributions of random matrices. This paper provide formulas for the Fredholm determinants which arise thus, and determines consequent properties of the associated differential operators. We prove a spectral theorem for self-adjoint linear systems that have scalar input and output: the entries of Kodaira's characteristic matrix are given explicitly with formulas involving the infinitesimal Darboux addition for $(-A,B,C)$. Under suitable conditions on $(-A,B,C)$ we give an explicit version of Burchnall-Chaundy's theorem, showing that the algebra generated by an associated family of differential operators is isomorphic to an algebra of functions on a particular hyperelliptic curve., Comment: 44 pages

We prove that each member of the non-commutative nonlinear Schrodinger and modified Korteweg--de Vries hierarchy is a Fredholm Grassmannian flow, and for the given linear dispersion relation and corresponding equivalencing group of Fredholm transformations, is unique in the class of odd-polynomial partial differential fields. Thus each member is linearisable and integrable in the sense that time-evolving solutions can be generated by solving a linear Fredholm Marchenko equation, with the scattering data solving the corresponding linear dispersion equation. At each order, each member matches the corresponding non-commutative Lax hierarchy field which thus represent odd-polynomial partial differential fields. We also show that the cubic form for the non-commutative sine--Gordon equation corresponds to the first negative order case in the hierarchy, and establish the rest of the negative order non-commutative hierarchy. To achieve this, we construct an abstract combinatorial algebra, the Poppe skew-algebra, that underlies the hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by compositions, endowed with the Poppe product -- the product rule for Hankel operators pioneered by Ch. Poppe for classical integrable systems. Establishing the hierarchy members at non-negative orders, involves proving the existence of a `Poppe polynomial' expansion for basic compositions in terms of `linear signature expansions' representing the derivatives of the underlying non-commutative field. The problem boils down to solving a linear algebraic equation for the polynomial expansion coefficients, at each order., Comment: 66 pages, 5 tables

Let $(-A,B,C)$ be a linear system in continuous time $t>0$ with input and output space ${\mathbb C}^2$ and state space $H$. The scattering functions $\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $\Gamma_{\phi_{(x)}}$; if $\Gamma_{\phi_{(x)}}$ is trace class, then the Fredholm determinant $\tau (x)=\det (I+\Gamma_{\phi_{(x)}})$ determines the tau function of $(-A,B,C)$. The paper establishes properties of algebras including $R_x=\int_x^\infty e^{-tA}BCe^{-tA}dt$ on $H$. Thus the paper obtains solutions of the sinh-Gordon PDE. The tau function for sinh-Gordon satisfies a particular Painl\'eve $\mathrm{III}'$ nonlinear ODE and describes a random matrix model, with asymptotic distribution found by the Coulomb fluid method to be the solution of an electrostatic variational problem on an interval.

The paper proves transportation inequalities for probability measures on spheres for the Wasserstein metrics with respect to cost functions that are powers of the geodesic distance. Let $\mu$ be a probability measure on the sphere ${\bf S}^n$ of the form $d\mu =e^{-U(x)}dx$ where $dx$ is the rotation invariant probability measure, and $(n-1)I+{\hbox{Hess}}\,U\geq {\kappa_U}I$, where $\kappa_U>0$. Then any probability measure $\nu$ of finite relative entropy with respect to $\mu$ satisfies ${\hbox{Ent}}(\nu\mid\mu) \geq (\kappa_U/2)W_2(\nu, \mu )^2$. The proof uses an explicit formula for the relative entropy which is also valid on connected and compact $C^\infty$ smooth Riemannian manifolds without boundary. A variation of this entropy formula gives the Lichn\'erowicz integral., Comment: 12 pages

The paper interprets the cubic nonlinear Schr\"odinger equation as a Hamiltonian system with infinite dimensional phase space. There is a Gibbs measure which is invariant under the flow associated with the canonical equations of motion. The logarithmic Sobolev and concentration of measure inequalities hold for the Gibbs measures, and here are extended to the $k$-point correlation function and distributions of related empirical measures. By Hasimoto's theorem, NLSE gives a Lax pair of coupled ODE for which the solutions give a system of moving frames. The paper studies the evolution of the measure induced on the moving frames by the Gibbs measure.

The paper considers the Euler system of PDE on a smooth compact Riemannian manifold of positive curvature without boundary, and the sphere ${\mathbb{S}}^2$ in particular. The paper interprets the Euler equations as a transport problem for the fluid density under dynamics governed by the gradient of the internal energy of the fluid. The paper develops the notion of transport cost in the tangent bundle, and compares its properties with the Wasserstein transportation cost on the manifold. There are applications to the discrete approximation to the Euler equations in the style of Gangbo and Wesdickenberg ({\sl Comm. Partial Diff. Equations} {\bf 34} (2009), 1041-1073), except that the analysis is heavily dependent upon the curvature of the underlying manifold. The internal energy is assumed to satisfy convexity conditions that allow analysis via $\Phi$-entropy entropy-production inequalities, and the results apply to the power law $\rho^\gamma$ where $1<\gamma<3/2$, which includes the case of a diatomic gas. The paper proves existence of weak solutions of the continuity equation, and gives a sufficient condition for existence of weak solutions to the acceleration equation.

The periodic Benjamin-Ono equation is an autonomous Hamiltonian system with a Gibbs measure on $L^2({\mathbb T})$. The paper shows that the Gibbs measures on bounded balls of $L^2$ satisfy some logarithmic Sobolev inequalities. The space of $n$-soliton solutions of the periodic Benjamin-Ono equation, as discovered by Case, is a Hamiltonian system with an invariant Gibbs measure. As $n\rightarrow\infty$, these Gibbs measures exhibit a concentration of measure phenomenon. Case introduced soliton solutions that are parameterised by atomic measures in the complex plane. The limiting distributions of these measures gives the density of a compressible gas that satisfies the isentropic Euler equations., Comment: 34 pages

Following the work of Conrey, Rubinstein and Snaith and Forrester and Witte we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group U(N) (also known as the CUE) and relate the moment to the solution of a Painleve differential equation. We also calculate a simple form for the asymptotic behaviour of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle.

Let $w$ be a semiclassical weight which is generic in Magnus's sense, and $(p_n)_{n=0}^\infty$ the corresponding sequence of orthogonal polynomials. The paper expresses the Christoffel--Darboux kernel as a sum of products of Hankel integral operators. For $\psi\in L^\infty (i{\mathbb R})$, let $W(\psi )$ be the Wiener-Hopf operator with symbol $\psi$. The paper gives sufficient conditions on $\psi$ such that $1/\det W(\psi )W(\psi^{-1})=\det (I-\Gamma_{\phi_1}\Gamma_{\phi_2})$ where $\Gamma_{\phi_1}$ and $\Gamma_{\phi_2}$ are Hankel operators that are Hilbert--Schmidt. For certain $\psi$, Barnes's integral leads to an expansion of this determinant in terms of the generalised hypergeometric ${}_nF_m$. These results extend those of Basor and Chen [2], who obtained ${}_4F_3$ likewise. The paper includes examples where the Wiener--Hopf factors are found explicitly., Comment: 30 pages

In this paper, we study the probability density function, $\mathbb{P}(c,\alpha,\beta, n)\,dc$, of the center of mass of the finite $n$ Jacobi unitary ensembles with parameters $\alpha\,>-1$ and $\beta >-1$; that is the probability that ${\rm tr}M_n\in(c, c+dc),$ where $M_n$ are $n\times n$ matrices drawn from the unitary Jacobi ensembles. We first compute the exponential moment generating function of the linear statistics $\sum_{j=1}^{n}\,f(x_j):=\sum_{j=1}^{n}x_j,$ denoted by $\mathcal{M}_f(\lambda,\alpha,\beta,n)$.

In this paper we study a particular Painlev\'e V (denoted ${\rm P_{V}}$) that arises from Multi-Input-Multi-Output (MIMO) wireless communication systems. Such a $P_V$ appears through its intimate relation with the Hankel determinant that describes the moment generating function (MGF) of the Shannon capacity. This originates through the multiplication of the Laguerre weight or the Gamma density $x^{\alpha} {\rm e}^{-x},\;x> 0,$ for $\alpha>-1$ by $(1+x/t)^{\lambda}$ with $t>0$ a scaling parameter. Here the $\lambda$ parameter "generates" the Shannon capacity, see Yang Chen and Matthew McKay, IEEE Trans. IT, 58 (2012) 4594--4634. It was found that the MGF has an integral representation as a functional of $y(t)$ and $y'(t)$, where $y(t)$ satisfies the "classical form" of $P_V$. In this paper, we consider the situation where $n,$ the number of transmit antennas, (or the size of the random matrix), tends to infinity, and the signal-to-noise ratio (SNR) $P$ tends to infinity, such that $s={4n^{2}}/{P}$ is finite. Under such double scaling the MGF, effectively an infinite determinant, has an integral representation in terms of a "lesser" $P_{III}$. We also consider the situations where $\alpha=k+1/2,\;\;k\in \mathbb{N},$ and $\alpha\in\{0,1,2,\dots\}$ $\lambda\in\{1,2,\dots\},$ linking the relevant quantity to a solution of the two dimensional sine-Gordon equation in radial coordinates and a certain discrete Painlev\'e-II. From the large $n$ asymptotic of the orthogonal polynomials, that appears naturally, we obtain the double scaled MGF for small and large $s$, together with the constant term in the large $s$ expansion. With the aid of these, we derive a number of cumulants and find that the capacity distribution function is non-Gaussian., Comment: 30 pages

This article considers Whittaker's function $W_{\kappa ,\mu }$ where $\kappa$ is real and $\mu$ is real or purely imaginary. Then $\varphi (x)=x^{-\mu -1/2}W_{\kappa ,\mu }(x)$ arises as the scattering function of a continuous time linear system with state space $L^2(1/2, \infty )$ and input and output spaces ${\bf C}$. The Hankel operator $\Gamma_\varphi$ on $L^2(0, \infty )$ is expressed as a matrix with respect to the Laguerre basis and gives the Hankel matrix of moments of a Jacobi weight $w$. The operation of translating $\varphi$ is equivalent to multiplying $w$ by an exponential factor to give $w_\varepsilon$. The determinant of the Hankel matrix of moments of $w_\varepsilon$ satisfies the $\sigma$ form of Painlev\'e's transcendental differential equation $PV$. It is shown that $\Gamma_\varphi$ gives rise to the Whittaker kernel from random matrix theory, as studied by Borodin and Olshanski (Comm. Math. Phys. 211 (2000), 335--358).\par, Comment: 19 pages

Let $A$ be the generator of a strongly continuous cosine family $(\cos (tA))_{t\in {\bf R}}$ on a complex Banach space $E$. The paper develops an operational calculus for integral transforms and functions of $A$ using the generalized harmonic analysis associated to certain hypergroups. It is shown that characters of hypergroups which have Laplace representations give rise to bounded operators on $E$. Examples include the Mellin transform and the Mehler--Fock transform. The paper uses functional calculus for the cosine family $\cos( t\sqrt {\Delta})$ which is associated with waves that travel at unit speed. The main results include an operational calculus theorem for Sturm--Liouville hypergroups with Laplace representation as well as analogues to the Kunze--Stein phenomenon in the hypergroup convolution setting., Comment: arXiv admin note: substantial text overlap with arXiv:1304.5868. Substantial revision to version 1

This paper concerns Gibbs measures $\nu$ for some nonlinear PDE over the $D$-torus ${\bf T}^D$. The Hamiltonian $H=\int_{{\bf T}^D} \Vert\nabla u\Vert^2 - \int_{{\bf T}^D} \vert u\vert^p$ has canonical equations with solutions in $\Omega_N=\{ u\in L^2({\bf T}^D) :\int \vert u\vert^2\leq N\}$. For $D=1$ and $2\leq p<6$, $\Omega_N$ supports the Gibbs measure $\nu (du)=Z^{-1}e^{-H(u)}\prod_{x\in {\bf T}} du(x)$ which is normalized and formally invariant under the flow generated by the PDE. The paper proves that $(\Omega_N, \Vert\cdot\Vert_{L^2}, \nu )$ is a metric probability space of finite diameter that satisfies the logarithmic Sobolev inequalities for the periodic $KdV$, the focussing cubic nonlinear Schr\"odinger equation and the periodic Zakharov system. For suitable subset of $\Omega_N$, a logarithmic Sobolev inequality also holds in the critical case $p=6$. For $D=2$, the Gross--Piatevskii equation has $H=\int_{{\bf T}^2} \Vert\nabla u\Vert^2-\int_{{\bf T}^2} (V\ast \vert u\vert^2 ) \vert u\vert^2$, for a suitable bounded interaction potential $V$ and the Gibbs measure $\nu$ lies on a metric probability space $(\Omega , \Vert\cdot\Vert_{H^{-s}}, \nu )$ which satisfies $LSI$. In the above cases, $(\Omega, d, \nu )$ is the limit in $L^2$ transportation distance of finite-dimensional $(\Omega_n, \Vert \cdot \Vert,\nu_n)$ given by Fourier sums., Comment: 28 pages