scholarly journals Spherical Functions Approach to Sums of Random Hermitian Matrices

2017 ◽  
Vol 2019 (4) ◽  
pp. 1005-1029 ◽  
Author(s):  
Arno B J Kuijlaars ◽  
Pablo Román
Keyword(s):  
Spherical Functions ◽  
Hermitian Matrices

Olshanski spherical pairs of semigroups type

2019 ◽  
Vol 22 (03) ◽  
pp. 1950021
Author(s):  
Mohamed Bouali
Keyword(s):  
Integral Representation ◽  
Unitary Group ◽  
Inductive Limit ◽  
Complete Classification ◽  
Spherical Functions ◽  
Hermitian Matrices ◽  
Positive Type ◽  
Negative Type ◽  
Functions Of Positive Type

Let [Formula: see text] be the infinite semigroup, inductive limit of the increasing sequence of the semigroups [Formula: see text], where [Formula: see text] is the unitary group of matrices and [Formula: see text] is the semigroup of positive hermitian matrices. The main purpose of this work is twofold. First, we give a complete classification of spherical functions defined on [Formula: see text], by following a general approach introduced by Olshanski and Vershik.10 Second, we prove an integral representation for functions of positive-type analog to the Bochner–Godement theorem, and a Lévy–Khinchin formula for functions of negative type defined on [Formula: see text].

scholarly journals SPHERICAL FUNCTIONS ON THE SPACE OF p-ADIC UNITARY HERMITIAN MATRICES

2014 ◽  
Vol 10 (02) ◽  
pp. 513-558
Author(s):  
YUMIKO HIRONAKA ◽  
YASUSHI KOMORI
Keyword(s):  
Spherical Functions ◽  
Plancherel Formula ◽  
Symmetric Polynomials ◽  
Hermitian Matrices ◽  
Cartan Decomposition ◽  
Explicit Formulas ◽  
Spherical Fourier Transform ◽  
Rank 2 ◽  
Type C ◽  
Residual Characteristic

We investigate the space X of unitary hermitian matrices over 𝔭-adic fields through spherical functions. First we consider Cartan decomposition of X, and give precise representatives for fields with odd residual characteristic, i.e. 2 ∉ 𝔭. From Sec. 2.2 till the end of Sec. 4, we assume odd residual characteristic, and give explicit formulas of typical spherical functions on X, where Hall–Littlewood symmetric polynomials of type Cn appear as a main term, parametrization of all the spherical functions. By spherical Fourier transform, we show that the Schwartz space [Formula: see text] is a free Hecke algebra [Formula: see text]-module of rank 2n, where 2n is the size of matrices in X, and give the explicit Plancherel formula on [Formula: see text].

Spheric radial basis functions in Mahalanobis measuring for displaying local field features

2019 ◽  
Vol 952 (10) ◽  
pp. 2-9
Author(s):  
Yu.M. Neiman ◽  
L.S. Sugaipova ◽  
V.V. Popadyev
Keyword(s):  
Gravitational Field ◽  
Quadratic Form ◽  
Local Field ◽  
Euclidean Distance ◽  
Basis Functions ◽  
Spherical Functions ◽  
Local Models ◽  
Stationary Field ◽  
Modeling Process ◽  
The Earth

As we know the spherical functions are traditionally used in geodesy for modeling the gravitational field of the Earth. But the gravitational field is not stationary either in space or in time (but the latter is beyond the scope of this article) and can change quite strongly in various directions. By its nature, the spherical functions do not fully display the local features of the field. With this in mind it is advisable to use spatially localized basis functions. So it is convenient to divide the region under consideration into segments with a nearly stationary field. The complexity of the field in each segment can be characterized by means of an anisotropic matrix resulting from the covariance analysis of the field. If we approach the modeling in this way there can arise a problem of poor coherence of local models on segments’ borders. To solve the above mentioned problem it is proposed in this article to use new basis functions with Mahalanobis metric instead of the usual Euclidean distance. The Mahalanobis metric and the quadratic form generalizing this metric enables us to take into account the structure of the field when determining the distance between the points and to make the modeling process continuous.

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

scholarly journals Eigenbranes in Jackiw-Teitelboim gravity

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Andreas Blommaert ◽  
Thomas G. Mertens ◽  
Henri Verschelde
Keyword(s):  
Finite Volume ◽  
Chaotic System ◽  
Path Integral ◽  
Hermitian Matrices ◽  
The Matrix ◽  
Integral Contour

Abstract It was proven recently that JT gravity can be defined as an ensemble of L × L Hermitian matrices. We point out that the eigenvalues of the matrix correspond in JT gravity to FZZT-type boundaries on which spacetimes can end. We then investigate an ensemble of matrices with 1 ≪ N ≪ L eigenvalues held fixed. This corresponds to a version of JT gravity which includes N FZZT type boundaries in the path integral contour and which is found to emulate a discrete quantum chaotic system. In particular this version of JT gravity can capture the behavior of finite-volume holographic correlators at late times, including erratic oscillations.

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