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The sixth Painlev\'e equation is hiding extremely rich geometric structures behind its outward appearance. This article tries to give as a total picture as possible of its dynamical natures, based on the Riemann-Hilbert approach recently…
The Riemann-Hilbert method is employed to carry out an asymptotic analysis of a family of $\sigma$-Painlev\'e V functions associated with Hankel determinants involving the confluent hypergeometric function of the second kind. In the…
We give the solution to the complete noncommutative Kadomtsev--Petviashvili (KP) hierarchy. We achieve this via direct linearisation which involves the Gelfand--Levitan--Marchenko (GLM) equation. This is a linear integral equation in which…
We further study the orthogonal polynomials with respect to the generalized Airy weight based on the work of Clarkson and Jordaan [{\em J. Phys. A: Math. Theor.} {\bf 54} ({2021}) {185202}]. We prove the ladder operator equations and…
Let $\Omega \subset \RR^d$, $d \geqslant 1$, be a bounded domain with piecewise smooth boundary $\partial \Omega $ and let $U$ be an open subset of a Banach space $Y$. Motivated by questions in "Uncertainty Quantification," we consider a…
This paper considers a class of distributed resource allocation problems where each agent privately holds a smooth, potentially non-convex local objective, subject to a globally coupled equality constraint. Built upon the existing method,…
The goal of this article is to prove that the determinantal formulas of the Painlev'e 2 system identify with the correlation functions computed from the topological recursion on their spectral curve for an arbitrary non-zero monodromy…
We apply the Painleve test for integrability of partial differential equations to a system of two coupled Burgers-type equations found by Foursov, which was recently shown by Sergyeyev to possess infinitely many commuting local generalized…
We probe the application of the calculus of conormal distributions, in particular the Pull-Back and Push-Forward Theorems, to the method of layer potentials to solve the Dirichlet and Neumann problems on half-spaces. We obtain full…
The Painlev\'e--Kovalevskaya test is applied to find three matrix versions of the Painlev\'e II equation. All these equations are interpreted as group-invariant reductions of integrable matrix evolution equations, which makes it possible to…
We investigate the Lax equation in the context of infinite-dimensional Lie algebras. Explicit solutions are discussed in the sequentially complete asymptotic estimate context, and an integral expansion (sums of iterated Riemann integrals…
Chatterjee, Diaconis and Sly (2011) recently established the consistency of the maximum likelihood estimate in the $\beta$-model when the number of vertices goes to infinity. By approximating the inverse of the Fisher information matrix, we…
We obtain the optimal proxy variance for the sub-Gaussianity of Beta distribution, thus proving upper bounds recently conjectured by Elder (2016). We provide different proof techniques for the symmetrical (around its mean) case and the…
We consider the extended discrete KP hierarchy and show that similarity reduction of its subhierarchies lead to purely discrete equations with dependence on some number of parameters together with equations governing deformations with…
Perfectly incompressible materials do not exist in nature but are a useful approximation of several media which can be deformed in non-isothermal processes but undergo very small volume variation. In this paper the linear analysis of the…
We directly evaluate the probability amplitudes in Jackiw-Teitelboim (JT) gravity using the Lorentzian path integral formulation. By imposing boundary conditions on the scale factor and the dilaton field, the Lorentzian path integral…
We give an extension of the two-component KP hierarchy by considering additional time variables. We obtain the linear $2\times 2$ system by taking into consideration the hierarchy through a reduction procedure. The Lax pair of the…
In this paper, we develop a two-stage distributed algorithm that enables nodes in a graph to cooperatively estimate the spectrum of a matrix $W$ associated with the graph, which includes the adjacency and Laplacian matrices as special…
We consider T-systems and Y-systems arising from cluster mutations applied to quivers that have the property of being periodic under a sequence of mutations. The corresponding nonlinear recurrences for cluster variables (coefficient-free…
A Riemannian gradient descent algorithm and a truncated variant are presented to solve systems of phaseless equations $|Ax|^2=y$. The algorithms are developed by exploiting the inherent low rank structure of the problem based on the…