Related papers: Inner Products on the Space of Complex Square Matr…

We describe explicit formulas for the product rule in $\kappa^*(\mathcal{M}_{g,n}^{ct})$.

On the set of mappings of the given set, we define the product of mappings. If A is associative algebra, then we consider the set of matrices, whose elements are linear mappings of algebra A. In algebra of matrices of linear mappings we…

Let $\textrm{Mat}_2(\mathbb{R})$ be the set of $2 \times 2$ matrices with real entries. For any $\varepsilon>0$ and any finitely--supported probability measure $\mu$ on $\textrm{Mat}_2(\mathbb{R})$, we prove that either \[ T(\mu) = \sum_{X,…

We give a new explicit construction of $n\times N$ matrices satisfying the Restricted Isometry Property (RIP). Namely, for some c>0, large N and any n satisfying N^{1-c} < n < N, we construct RIP matrices of order k^{1/2+c}. This overcomes…

We give canonical forms of selfadjoint and isometric operators on a complex vector space $U$ with scalar product given by a positive semidefinite Hermitian form, and of Hermitian forms on $U$. For an arbitrary system of semiunitary spaces…

The Bernstein Markov Property, shortly BMP, is an asymptotic quan- titative assumption on the growth of uniform norms of polynomials or rational functions on a compact set with respect to L {\mu} 2 -norms, where {\mu} is a positive finite…

We establish metrics of positive $2^\mathrm{nd}$-intermediate Ricci curvature, i.e. $\mathrm{Ric}_2>0$, on products of positively curved homogeneous spaces. Using these examples, we demonstrate that the Hopf conjectures, Petersen-Wilhelm…

We study a nonlinear decomposition of a positive definite matrix into two components: the inverse of another positive definite matrix and a symmetric matrix constrained to lie in a prescribed linear subspace. Equivalently, the inverse…

This article is a fundamental study in computable analysis. In the framework of Type-2 effectivity, TTE, we investigate computability aspects on finite and infinite products of effective topological spaces. For obtaining uniform results we…

We provide a manifestly positive expression for the volume of the moduli spaces of flat $\mathrm{U}(n)$-valued connections on punctured compact oriented surfaces. This volume is obtained by summing volumes of explicit polytopes describing…

In this note, we prove that for every two positive integers $m \geq n \geq 9$, there exist $n$ positive rational numbers whose product is 1 and sum is $m$.

Let H be any complex inner product space with inner product <, >. We say that f : C -->C is Hermitian positive definite on H if the matrix $$(f(<z^r,z^s>))_{r,s=1}^n \eqno(*)$$ is Hermitian positive definite for all choice of z^1,...,z^n in…

This paper defines Massey-type products for a homotopy inner product on an $A_\infty$ algebra, called Massey inner products. We include an explicit description of ordinary Massey products for $A_\infty$ algebras, and for $A_\infty$ modules,…

Let $\mathfrak{M}(\mathbb{D}, m, n, P)$ be the ring of all $m \times n$ matrices over a division ring $\mathbb{D}$, with the product given by $A \bullet B=A P B$, where $P$ is a fixed $n \times m$ matrix over $\mathbb{D}$. When $2\leq m, n…

In this paper, we introduce the concept of inner product on weak hypervector spaces and prove some results about them.

We develop the theory of subproduct systems over the monoid $\mathbb{N}\times \mathbb{N}$, and the non-self-adjoint operator algebras associated with them. These are double sequences of Hilbert spaces $\{X(m,n)\}_{m,n=0}^\infty$ equipped…

We prove anti-concentration bounds for the inner product of two independent random vectors, and use these bounds to prove lower bounds in communication complexity. We show that if $A,B$ are subsets of the cube $\{\pm 1\}^n$ with $|A| \cdot…

Let $N_{1,B}(n)$ denote the number of ones in the $B$-ary expansion of an integer $n$. Woods introduced the infinite product $P :=\prod_{n \geq 0} (\frac{2n+1}{2n+2})^{(-1)^{N_{1,2}(n)}}$ and Robbins proved that $P = 1/\sqrt{2}$. Related…

Several characterizations are given for a square matrix that can be written as the product of two positive (semidefinite) projections. Based on one of these characterizations, and the theory of alternating projections, a Matlab program is…

It is well known that the generalized (or quotient) singular values of a matrix pair $(A, C)$ can be obtained from the generalized eigenvalues of a matrix pencil consisting of two augmented matrices. The downside of this reformulation is…