Square Roots of Hermitian Matrices and a Rational Algorithm for Checking Their Congruence

2019 ◽  
Vol 43 (3) ◽  
pp. 95-100
Author(s):  
Kh. D. Ikramov
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg

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, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.


2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
James Drummond ◽  
Jack Foster ◽  
Ömer Gürdoğan ◽  
Chrysostomos Kalousios

Abstract We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.


2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Andreas Blommaert ◽  
Thomas G. Mertens ◽  
Henri Verschelde

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.


2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.


1986 ◽  
Vol 93 (3) ◽  
pp. 213-214 ◽  
Author(s):  
William C. Waterhouse
Keyword(s):  

2011 ◽  
Vol 412 (39) ◽  
pp. 5458-5465
Author(s):  
Qi Cheng ◽  
Yu-Hsin Li
Keyword(s):  

The electrical resistivities of lithium -6 and lithium of natural isotopic composition have been studied between 4°K and room temperature. In addition, their absolute resistivities have been carefully compared at room temperature. These measurements show that the effect of ionic mass on electrical resistivity agrees with simple theoretical predictions, namely, that the properties of the conduction electrons in lithium do not depend on the mass of the ions, and that the characteristic lattice frequencies for the two pure isotopes are in the inverse ratio of the square roots of their ionic masses. A comparison with the specific heat results of Martin (1959, 1960), where the simple theory is found not to hold, indicates the possibility that anharmonic effects are present which affect the specific heat but not the electrical resistivity.


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