Related papers: Pfaff tau-functions
We introduce and study a family of inhomogeneous symmetric functions which we call the Frobenius-Schur functions. These functions are indexed by partitions and differ from the conventional Schur functions in lower terms only. Our interest…
We establish identities of Pfaffian type for the theta function associated with twice or half the period matrix of a hyperelliptic curve. They are implied by the large size asymptotic analysis of exact Pfaffian identities for expectation…
We study the spectrum of the semiclassical Witten Laplacian $\Delta_{f}$ associated to a smooth function $f$ on ${\mathbb R}^d$. We assume that $f$ is a confining Morse--Bott function. Under this assumption we show that $\Delta_{f}$ admits…
The paper contains some new results and a review of recent achievements, concerning the multisupport solutions to matrix models. In the leading order of the 't Hooft expansion for matrix integral, these solutions are described by…
We consider integrals of tau functions of Zakharov-Shabat systems whose higher times are related to the eigenvalues of products of random matrices. Apart of random matrices there is the set of $n$ pairs of given matrices which play the role…
Like its precursor this paper is concerned with the Hopf algebra of noncommutative symmetric functions and its graded dual, the Hopf algebra of quasisymmetric functions. It complements and extends the previous paper but is also…
A half-tree is an edge configuration whose superimposition with a perfect matching is a tree. In this paper, we prove a half-tree theorem for the Pfaffian principal minors of a skew-symmetric matrix whose column sum is zero; introducing an…
For any complex number $\alpha$ and any even-size skew-symmetric matrix $B$, we define a generalization $\pfa{\alpha}(B)$ of the pfaffian $\pf(B)$ which we call the $\alpha$-pfaffian. The $\alpha$-pfaffian is a pfaffian analogue of the…
The main objective of this article is to present $\nu$-fractional derivative $\mu$-differentiable functions by considering 4-parameters extended Mittag-Leffler function (MLF). We investigate that the new $\nu$-fractional derivative…
In this paper, we prove the existence of tau functions of the discrete modified KP hierarchy and define the squared eigenfunction symmetry. Meanwhile, the Fay identity with its difference form, the squared eigenfunction potentials and the…
An element [\Phi] of the Grassmannian of n-dimensional subspaces of the Hardy space H^2, extended over the field C(x_1,..., x_n), may be associated to any polynomial basis {\phi} for C(x). The Pl\"ucker coordinates…
This is a survey article written for a workshop on L-functions and random matrix theory at the Newton Institute in July, 2004. The goal is to give some insight into how well-distributed sets of matrices in classical groups arise from…
In 2020, Hamaker, Pawlowski, and Sagan introduced the \emph{pattern quasisymmetric functions}, which are quasisymmetric functions associated with pattern-avoidance classes of permutations, and defined via expansions in fundamental…
By using the independence structure of points following a determinantal point process, we study the radii of the spherical ensemble, the truncation of the circular unitary ensemble and the product ensemble with parameter n and k. The…
A rapid transformation is derived between spherical harmonic expansions and their analogues in a bivariate Fourier series. The change of basis is described in two steps: firstly, expansions in normalized associated Legendre functions of all…
We give an asymptotic formula for correlations \[ \sum_{n\le x}f_1(P_1(n))f_2(P_2(n))\cdot \dots \cdot f_m(P_m(n))\] where $f\dots,f_m$ are bounded "pretentious" multiplicative functions, under certain natural hypotheses. We then deduce…
In this paper, we determine the complex-valued solutions of the functional equation $$ f(x\sigma(y))+f(\tau(y)x)=2f(x)f(y)$$ for all $x,y \in M$, where $M$ is a monoid, $\sigma$: $M\longrightarrow M$ is an involutive automorphism and…
A combinatorial construction proves an identity for the product of the Pfaffian of a skew-symmetric matrix by the Pfaffian of one of its submatrices. Several applications of this identity are followed by a brief history of Pfaffians.
Let $P$ be a prime polynomial in the variable $Y$ over a finite field and let $f$ be a quadratic irrational in the field of formal Laurant series in the variable $Y^{-1}$. We study the asymptotic properties of the degrees of the…
This paper is concerned with the study of random (Bernoulli and Markovian) product of matrices on a compact space of symbols. We establish the analyticity of the maximal Lyapunov exponent as a function of the transition probabilities, thus…