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papers

Publications (29)

math.CA2017

On tau functions associated with linear systems

Gordon Blower, Samantha L. Newsham

Let $(- A, B, C)$ be a linear system in continuous time $t > 0$ with input and output space $C$ and state space $H$ . The function $I ¨ _{(x)} (t) = C e^{- (t + 2 x) A} B$ determines a Hankel…

math.PR2022

Transportation on spheres via an entropy formula

Gordon Blower

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…

math.FA2016

Operational calculus and integral transforms for groups with finite propagation speed

Gordon Blower, Ian Doust

Let $A$ be the generator of a strongly continuous cosine family $(cos (t A))_{t \in R}$ on a complex Banach space $E$ . The paper develops an operational calculus for integral…

math.FA2007

Hankel operators that commute with second-order differential operators

Gordon Blower

Suppose that $I^$ is a continuous and self-adjoint Hankel operator on $L^{2} (0, \infty)$ and that $L f = - (d / d x (a (x) df / d x)) + b (x) f (x)$ with $a (0) = 0$ . If $a$ and $b$ are both quadratic, h…

math.SP2024

Linear systems, spectral curves and determinants

Gordon Blower, Ian Doust

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}…

math.AP2019

Statistical Mechanics of the Periodic Benjamin-Ono Equation

Gordon Blower, Caroline Brett, Ian Doust

The periodic Benjamin-Ono equation is an autonomous Hamiltonian system with a Gibbs measure on $L^{2} (T)$ . The paper shows that the Gibbs measures on bounded balls of $L^{2}$ …

math.SP2014

Logarithmic Sobolev inequalities and spectral concentration for the cubic SchrÃ¶dinger equation

Gordon Blower, Caroline Brett, Ian Doust

The nonlinear SchrÃ¶dinger equation NLSE(p, Î²), -iu_t=-u_{xx}+Î²| u|^{p-2} u=0, arises from a Hamiltonian on infinite-dimensional phase space \Lp^2(\mT). For p\leq 6, Bourgain (Co…

math.FA2019

On determinant expansions for Hankel operators

Gordon Blower, Yang Chen

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 Christo…

math.AP2022

Hasimoto frames and the Gibbs measure of periodic nonlinear SchrÃ¶dinger Equation

Gordon Blower, Azadeh Khaleghi, Moe Kuchemann-Scales

The paper interprets the cubic nonlinear SchrÃ¶dinger equation as a Hamiltonian system with infinite dimensional phase space. There is a Gibbs measure which is invariant under the…

math.CA2010

On tau functions for orthogonal polynomials and matrix models

Gordon Blower

Let v be a real polynomial of even degree, and let Ïbe the equilibrium probability measure for v with support S; so that v(x)\geq 2\int \log |x-y| Ï(dy)+C_v for some constant C_v…

math.SP2014

Hill's Spectral Curves and the Invariant Measure of the Periodic KdV Equation

Gordon Blower, Caroline Brett, Ian Doust

This paper analyses the periodic spectrum of SchrÃ¶dinger's equation $- f^{''} + q f = I^» f$ when the potential is real, periodic, random and subject to the invariant measure $I^½_{N}^{2}$ of th…

math.PR2015

Concentration of the invariant measures for the periodic Zakharov, KdV, NLS and Gross--Piatevskii equations in 1D and 2D

Gordon Blower

This paper concerns Gibbs measures $I^½$ for some nonlinear PDE over the $D$ -torus $T^{D}$ . The Hamiltonian $H=\int_{{\bf T}^D} \Vert\nabla u\Vert^2 - \int_{{\bf T}^D} \vert u\v…

math.FA2015

Functional calculus on Venturi for Groups with Finite Propagation Speed

Gordon Blower, Ian Doust

Let $M$ be a complete Riemannian manifold with Ricci curvature bounded below and Laplace operator $I^$ . The paper develops a functional calculus for the cosine family $\cos…

nlin.SI2025

The noncommutative KP hierarchy and its solution via descent algebra

Gordon Blower, Simon J. A. Malham

We give the solution to the complete noncommutative Kadomtsev--Petviashvili (KP) hierarchy. We achieve this via direct linearisation which involves the Gelfand--Levitan--Marchenko…

math.CA2015

Kernels and point processes associated with Whittaker functions

Gordon Blower, Yang Chen

This article considers Whittaker's function $W_{I^º, I^¼}$ where $I^º$ is real and $I^¼$ is real or purely imaginary. Then $I ¨ (x) = x^{- I^¼ - 1/2} W_{I^º, I^¼} (x)$ arises as the scattering funct…

nlin.SI2023

The algebraic structure of the non-commutative nonlinear Schrodinger and modified Korteweg-de Vries hierarchy

Gordon Blower, Simon J. A. Malham

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 dispersio…

math.FA2009

On linear systems and tau functions associated with Lame's equation and Painleve's equation VI

Gordon Blower

Painleve's transcendental differential equation P_{VI} may be expressed as the consistency condition for a pair of linear differential equations with 2 by 2 matrix coefficients wit…

math.CA2018

Center of mass distribution of the Jacobi unitary ensembles: Painleve V, asymptotic expansions

Longjun Zhan, Gordon Blower, Yang Chen +1

In this paper, we study the probability density function, $P (c, I^\pm, I^^{2}, n) d c$ , of the center of mass of the finite $n$ Jacobi unitary ensembles with parameters $I^\pm > - 1$ a…

math.FA2006

Operators associated with the soft and hard spectral edges of unitary ensembles

Gordon Blower

Using Hankel operators and shift-invariant subspaces on Hilbert space, this paper develops the theory of the operators associated with soft and hard edges of eigenvalue distributio…

math.FA2022

Linear systems, Hankel products and the sinh-Gordon equation

Gordon Blower, Ian Doust

Let $(- A, B, C)$ be a linear system in continuous time $t > 0$ with input and output space $C^{2}$ and state space $H$ . The scattering functions $I ¨ _{(x)} (t) = C e^{- (t + 2 x) A} B$ de…

math.PR2005

Concentration inequalities on product spaces with applications to Markov processes

Gordon Blower, FranÃ§ois Bolley

For a stochastic process with state space some Polish space, this paper gives sufficient conditions on the initial and conditional distributions for the joint law to satisfy Gaussi…

math.AP2025

Linear systems, determinants and solutions of the Kadomtsev-Petviashvili equation

Gordon Blower, Simon J. Malham

Let $(- A, B, C)$ be a linear system in continuous time $t > 0$ with input and output space $C$ and state space $H$ . The scattering (or impulse response) functions $Ï_{(x)}(t…

nlin.SI2024

Direct linearisation of the non-commutative Kadomtsev-Petviashvili equations

Gordon Blower, Simon J. A. Malham

We prove that the non-commutative Kadomtsev-Petviashvili (KP) equation and a `lifted' modified Kadomtsev-Petviashvili (mKP) equation are directly linearisable, and thus integrable…

math-ph2017

Hongmei Chen, Min Chen, Gordon Blower +1

In this paper we study a particular PainlevÃ© V (denoted $P_{V}$ ) that arises from Multi-Input-Multi-Output (MIMO) wireless communication systems. Such a $P_{V}$ appears throug…

math.FA2008

Linear systems and determinantal random point fields

Gordon Blower

Tracy and Widom showed that fundamentally important kernels in random matrix theory arise from differential equations with rational coefficients. More generally, this paper conside…

math.FA2007

Integrable operators and the squares of Hankel operators

Gordon Blower

Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distributions of large self-adjoint random matrices from the generalized unitary e…

math-ph2019

Mixed moments of characteristic polynomials of random unitary matrices

Emma C. Bailey, Sandro Bettin, Gordon Blower +4

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 unit…

math.AP2020

Convexity and transport for the isentropic Euler equations on the sphere

Gordon Blower

The paper considers the Euler system of PDE on a smooth compact Riemannian manifold of positive curvature without boundary, and the sphere $S^{2}$ in particular. The paper…

math.AP2009

A logarithmic Sobolev inequality for the invariant measure of the periodic Korteweg--de Vries equation

Gordon Blower

The periodic KdV equation u_t=u_{xxx}+Î²uu_x arises from a Hamiltonian system with infinite-dimensional phase space L^2(T). Bourgain has shown that there exists a Gibbs measure Î½o…

Publications (29)

On tau functions associated with linear systems

Transportation on spheres via an entropy formula

Operational calculus and integral transforms for groups with finite propagation speed

Hankel operators that commute with second-order differential operators

Linear systems, spectral curves and determinants

Statistical Mechanics of the Periodic Benjamin-Ono Equation

Logarithmic Sobolev inequalities and spectral concentration for the cubic SchrÃ¶dinger equation

On determinant expansions for Hankel operators

Hasimoto frames and the Gibbs measure of periodic nonlinear SchrÃ¶dinger Equation

On tau functions for orthogonal polynomials and matrix models

Hill's Spectral Curves and the Invariant Measure of the Periodic KdV Equation

Concentration of the invariant measures for the periodic Zakharov, KdV, NLS and Gross--Piatevskii equations in 1D and 2D

Functional calculus on Venturi for Groups with Finite Propagation Speed

The noncommutative KP hierarchy and its solution via descent algebra

Kernels and point processes associated with Whittaker functions

The algebraic structure of the non-commutative nonlinear Schrodinger and modified Korteweg-de Vries hierarchy

On linear systems and tau functions associated with Lame's equation and Painleve's equation VI

Center of mass distribution of the Jacobi unitary ensembles: Painleve V, asymptotic expansions

Operators associated with the soft and hard spectral edges of unitary ensembles

Linear systems, Hankel products and the sinh-Gordon equation

Concentration inequalities on product spaces with applications to Markov processes

Linear systems, determinants and solutions of the Kadomtsev-Petviashvili equation

Direct linearisation of the non-commutative Kadomtsev-Petviashvili equations

Single-use MIMO system, PainlevÃ© transcendents and double scaling

Linear systems and determinantal random point fields

Integrable operators and the squares of Hankel operators

Mixed moments of characteristic polynomials of random unitary matrices

Convexity and transport for the isentropic Euler equations on the sphere

A logarithmic Sobolev inequality for the invariant measure of the periodic Korteweg--de Vries equation