Related papers: 2-parameter $\tau$-function for the first Painlev\…

We study the resurgent structure of the refined topological string partition function on a non-compact Calabi-Yau threefold, at large orders in the string coupling constant $g_s$ and fixed refinement parameter $\mathsf{b}$. For…

We construct a 2-parameter family of unitarily equivalent irreducible representations of the triply extended group $\g$ of translations of $\mathbb{R}^{4}$ associated with a family of its 4-dimensional coadjoint orbits and show how a…

The partition function for unitary two matrix models is known to be a double KP tau-function, as well as providing solutions to the two dimensional Toda hierarchy. It is shown how it may also be viewed as a Borel sum regularization of…

We investigate the structure of $\tau$-functions for the elliptic difference Painlev\'e equation of type $E_8$. Introducing the notion of ORG $\tau$-functions for the $E_8$ lattice, we construct some particular solutions which are expressed…

We find and study a two-parameter family of coupled Painlev\'e II systems in dimension four with affine Weyl group symmetry of several types. Moreover, we find a three-parameter family of polynomial Hamiltonian systems in two variables…

The \emph{Chow parameters} of a Boolean function $f: \{-1,1\}^n \to \{-1,1\}$ are its $n+1$ degree-0 and degree-1 Fourier coefficients. It has been known since 1961 (Chow, Tannenbaum) that the (exact values of the) Chow parameters of any…

We study factorizations of rational matrix functions with simple poles on the Riemann sphere. For the quadratic case (two poles) we show, using multiplicative representations of such matrix functions, that a good coordinate system on this…

The Painleve first equation can be represented as the equation of isomonodromic deformation of a Schrodinger equation with a cubic potential. We introduce a new algorithm for computing the direct monodromy problem for this Schrodinger…

In this paper we obtain large $z$ asymptotic expansions in the complex plane for the tau function corresponding to special function solutions of the Painlev\'e II differential equation. Using the fact that these tau functions can be written…

It is shown that all $\tau$-functions of BKP hierarchy can be written as Pfaffians of skew-symmetric matrices. $\tau$-functions of BKP hierarchy are parameterized by points in the universal orthogonal Grassmannian manifold (UOGM). The UOGM…

Motivated by the strong nearby Lagrangian conjecture, we constrain the parametrised Whitehead torsion of a family of closed exact Lagrangian submanifolds in a cotangent bundle. We prove the parametrised Whitehead torsion admits a…

We derive bilinear tau forms of the canonically quantized Painlev\'e equations, thereby relating them to those previously obtained from the $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations for the $\mathcal{N}=2$ supersymmetric gauge theory…

We construct a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painlev\'e equation (or higher-order analogues), and admitting a large family of monodromy-preserving…

For a pair of coupled Painlev\'e equations obtained as a similarity reduction of the Hirota-Satsuma systems we describe special parameter-families of solutions given in terms of mixtures of rational and Airy functions, and in terms of a…

We present a unified description of birational representation of Weyl groups associated with T-shaped Dynkin diagrams, by using a particular configuration of points in the projective plane. A geometric formulation of tau-functions is given…

The present paper is a continuation of our work [11], where we introduced a fractional operator calculus related to a fractional ${\psi}-$Fueter operator in the one-dimensional Riemann-Liouville derivative sense in each direction of the…

We present a systematic approach to the construction of Miura transformations for discrete Painlev\'e equations. Our method is based on the bilinear formalism and we start with the expression of the nonlinear discrete equation in terms of…

For a given spectral curve, we construct a family of symplectic dual spectral curves for which we prove an explicit formula expressing the $n$-point functions produced by the topological recursion on these curves via the $n$-point functions…

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

We introduce the tau-function of a rational d-connection and its isomonodromy transformations. We show that in a continuous limit our tau-function agrees with the Jimbo-Miwa-Ueno tau-function, compute the tau-function for the isomonodromy…