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It is known that a unitary matrix can be decomposed into a product of reflections, one for each dimension, and the Haar measure on the unitary group pushes forward to independent uniform measures on the reflections. We consider the sequence…

Yang-Baxter integrable dense $A_1^{(1)}$ and dilute $A_2^{(2)}$ loop models are considered on the torus in their simplest physical regimes. A combination of boundary conditions $(h,v)$ is applied in the horizontal and vertical directions…

We consider linear mappings on the $d$-dimensional torus, defined by $T(x) = Ax \pmod 1$, where $A$ is an invertible $d \times d$ integer matrix, with no eigenvalues on the unit circle. In the case $d = 2$ and $\det A = \pm 1$, we give a…

We study the error of the number of points of a unimodular lattice that fall in a strictly convex and analytic set having the origin and that is dilated by a factor $t$. The aim is to generalize the result of a previous article. We first…

We consider the eigenvectors of symmetric matrices with independent heavy tailed entries, such as matrices with entries in the domain of attraction of $\alpha$-stable laws, or adjacencymatrices of Erdos-Renyi graphs. We denote by…

A bounded Hilbert space operator $T$ for which the closure of the annulus \[ \mathbb{A}_r=\{z \ : \ r<|z|<1\} \subseteq \mathbb{C}, \qquad (0<r<1) \] is a spectral set is called an $\mathbb A_r$-contraction. A celebrated theorem due to…

Inspired by the bad scientist who keeps repeating an experiment 20 times to get a single outcome with $p < 0.05$, we consider matrices $A \in \mathbb{R}^{n \times n}$ whose rows are normalized in $\ell^2$ and for which $2^{-n}\sum_{x \in…

We reexamine the large N limit of matrix integrals over the orthogonal group O(N) and their relation with those pertaining to the unitary group U(N). We prove that lim_{N to infty} N^{-2} \int DO exp N tr JO is half the corresponding…

We obtain a uniform ergodic theorem for the sequence $\frac1{s(n)} \sum_{k=0}^n(\varDelta s)(n-k)\,T^k$, where $\varDelta$ is the inverse of the endomorphism on the vector space of scalar sequences which maps each sequence into the sequence…

The lattice definition of a two-dimensional topological field theory (TFT) is given generically, and the exact solution is obtained explicitly. In particular, the set of all lattice topological field theories is shown to be in one-to-one…

This paper defines the Iris function and provides two formulations of the matrix permanent. The first formulation, valid for arbitrary complex matrices, expresses the permanent of a complex matrix as a contour integral of a second order…

For any nonnegative self-adjoint operators A and B in a separable Hilbert space, we show that the Trotter-type formula $[(e^{i2tA/n}+e^{i2tB/n})/2]^n$ converges strongly in the closure of the intersection of the domains of A^{1/2} and…

Let $T\colon\mathbb{T}^d\to \mathbb{T}^d$, defined by $T x=Ax(\bmod 1)$, where $A$ is a $d\times d$ integer matrix with eigenvalues $1<|\lambda_1|\le|\lambda_2|\le\dots\le|\lambda_d|$. We investigate the Hausdorff dimension of the…

A new mathematical notation is proposed for the iteration of functions. It facilitates the application of the iteration of functions in mathematical and logical expressions, definitions of sets, and formulations of algorithms. Illustrations…

In the work [Bull, Austr. Math. Soc. 85 (2012), 315-234], S.R. Moghadasi has shown how the decomposition of the $N$-fold product of Lebesgue measure on $\mathbb R^n$ implied by matrix polar decomposition can be used to derive the…

An $n\times n$ complex matrix $A$ is called coninvolutory if $\bar AA=I_n$ and skew-coninvolutory if $\bar AA=-I_n$ (which implies that $n$ is even). We prove that each matrix of size $n\times n$ with $n>1$ is a sum of 5 coninvolutory…

We apply the renormalized perturbation theory (RPT) to the symmetric Anderson impurity model. Within the RPT framework exact results for physical observables such as the spin and charge susceptibility can be obtained in terms of the…

In this paper we investigate the algebraic structure of AdS/CFT in the strong-coupling limit. We propose an expression for the classical r-matrix with (deformed) u(2|2) symmetry, which leads to a quasi-triangular Lie bialgebra as the…

We show that for piecewise hereditary algebras, the periodicity of the Coxeter transformation implies the non-negativity of the Euler form. Contrary to previous assumptions, the condition of piecewise heredity cannot be omitted, even for…

Given an associative, not necessarily commutative, ring R with identity, a formal matrix calculus is introduced and developed for pairs of matrices over R. This calculus subsumes the theory of homogeneous systems of linear equations with…