Related papers: Some counting questions for matrix products
This paper establishes that every positive-definite matrix can be written as a positive linear combination of outer products of integer-valued vectors whose entries are bounded by the geometric mean of the condition number and the dimension…
Results of Sierpinski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is "narrow" in a corresponding direction; that is, each line in that direction intersects the subset…
Let $D$ and $E$ be subspaces of the tensor product of the finite-dimensional Hilbert spaces $\mathbb{C}^m \otimes \mathbb{C}^n$. We show that the number of product vectors in $D$ with their partial conjugates in $E$ is uniformly bounded…
We address two variants of the classical necklace counting problem from enumerative combinatorics. In both cases, we fix a finite group $\mathcal{G}$ and a positive integer $n$. In the first variant, we count the ``identity-product…
I present a bijection on integer partitions that leads to recursive expressions, closed formulae and generating functions for the cardinality of certain sets of partitions of a positive integer $n$. The bijection leads also to a product on…
We investigate the product of $n$ complex non-Hermitian, independent random matrices, each of size $N\times N$ in the class of elliptic matrices, with independent identically distributed entries. The joint probability distribution of the…
In this paper we study the problem of finding explicit expressions for inner products on the space of complex square matrices $\Mn$. We show that, given an inner product $\lip \cdot, \cdot \rip$ on $\Mn$, with some conditions, there exist…
Let P be the direct product of countably many copies of the additive group Z of integers. We study, from a set-theoretic point of view, those subgroups of P for which all homomorphisms to Z annihilate all but finitely many of the standard…
It is known that a category of many-sorted algebras on pure sets of similarity type is "concretely equivalent" to a category of single-sorted algebras. In this paper, we characterize a single-sorted variety that corresponds to a many-sorted…
We consider products of $n$ random Hermitian matrices which generalize the one-matrix model and show its relation to Hurwitz numbers which count ramified coverings of certain type. Namely, these Hurwitz numbers count $2k$-fold ramified…
We prove upper and lower bounds for certain sums of products of fractional parts by using majoring and minorizing functions from Fourier analysis. In special cases the upper bounds are sharp if there exist counterexamples to the Littlewood…
We examine the relation of BSS-reducibility on subsets of the real numbers. The question was asked recently (and anonymously) whether it is possible for the halting problem H in BSS-computation to be BSS-reducible to a countable set.…
Consider the following parameterized counting variation of the classic subset sum problem, which arises notably in the context of higher homotopy groups of topological spaces: Let $\mathbf{v} \in \mathbb{Q}^d$ be a rational vector, $(T_{1},…
We consider a continuous analogue of Babai et al.'s and Cai et al.'s problem of solving multiplicative matrix equations. Given $k+1$ square matrices $A_{1}, \ldots, A_{k}, C$, all of the same dimension, whose entries are real algebraic, we…
We study the problem of constructing explicit families of matrices which cannot be expressed as a product of a few sparse matrices. In addition to being a natural mathematical question on its own, this problem appears in various…
Let $\A_0, \A_1, \ldots, \A_n$ be given square matrices of size $m$ with rational coefficients. The paper focuses on the exact computation of one point in each connected component of the real determinantal variety $\{\X \in\RR^n \: :\:…
In this paper we generalise the notion of linearity (in the sense of Lawvere) to a category C equipped with a compatible sum structure and product structure. In this context, any morphism f from an n-fold sum to an n-fold product has a…
A natural definition of the product of infinite matrices mimics the usual formulation of multiplication of finite matrices with the caveat (in the absence of any sense of convergence) that the intersection of the support of each row of the…
The multi-class prediction had gained popularity over recent years. Thus measuring fit goodness becomes a cardinal question that researchers often have to deal with. Several metrics are commonly used for this task. However, when one has to…
Positive-definite matrices materialize as state transition matrices of linear time-invariant gradient flows, and the composition of such materializes as the state transition after successive steps where the driving potential is suitably…