Related papers: Optimal unit triangular factorization of symplecti…
We prove an asymptotic crystallization result in two dimensions for a class of nonlocal particle systems. To be precise, we consider the best approximation with respect to the 2-Wasserstein metric of a given absolutely continuous…
High-order methods gain more and more attention in computational fluid dynamics. However, the potential advantage of these methods depends critically on the availability of efficient elliptic solvers. With spectral-element methods, static…
This note contains a short proof of a classical result: any rational symplectic matrix can be put in diagonal form after right and left multiplication by integral symplectic matrices.
This manuscript explores many convolution (restricted summation) type sequences via certain types of matrix based factorizations that can be used to express their generating functions. The last primary (non-appendix) section of the thesis…
This thesis addresses the question of the maximal number of $d$-simplices for a simplicial complex which is embeddable into $\mathbb{R}^r$ for some $d \leq r \leq 2d$. A lower bound of $f_d(C_{r + 1}(n)) =…
An algorithm for matrix factorization of polynomials was proposed in \cite{fomatati2022tensor} and it was shown that this algorithm produces better results than the standard method for factoring polynomials on the class of summand-reducible…
A real symmetric matrix $M$ is completely positive semidefinite if it admits a Gram representation by (Hermitian) positive semidefinite matrices of any size $d$. The smallest such $d$ is called the (complex) completely positive semidefinite…
Conic optimization has recently emerged as a powerful tool for designing tractable and guaranteed algorithms for non-convex polynomial optimization problems. On the one hand, tractability is crucial for efficiently solving large-scale…
Let $\mathcal{A}=(A_{1},...,A_{n},...)$ be a finite or infinite sequence of $2\times2$ matrices with entries in an integral domain. We show that, except for a very special case, $\mathcal{A}$ is (simultaneously) triangularizable if and only…
We introduce a new infinite family of $d\times 2d$ equiangular tight frames. Many matrices in this family consist of two $d\times d$ circulant blocks. We conjecture that such equiangular tight frames exist for every $d$. We show that our…
Let $D$ be the ring of integers of a quadratic number field $\mathbb{Q}[\sqrt{d}]$. We study the factorizations of $2 \times 2$ matrices over $D$ into idempotent factors. When $d < 0$ there exist singular matrices that do not admit…
The factorization of skew-symmetric matrices is a critically understudied area of dense linear algebra, particularly in comparison to that of general and symmetric matrices. While some algorithms can be adapted from the symmetric case, the…
In this article, we focus on the communication costs of three symmetric matrix computations: i) multiplying a matrix with its transpose, known as a symmetric rank-k update (SYRK) ii) adding the result of the multiplication of a matrix with…
A matrix is called totally positive if every minor of it is positive. Such matrices are well studied and have numerous applications in Mathematics and Computer Science. We study how many times the value of a minor can repeat in a totally…
An algorithm is given to factor an integer with $N$ digits in $\ln^m N$ steps, with $m$ approximately 4 or 5. Textbook quadratic sieve methods are exponentially slower. An improvement with the aid of an a particular function would provide a…
We study categories of matrix factorizations. These categories are defined for any regular function on a suitable regular scheme. Our paper has two parts. In the first part we develop the foundations; for example we discuss derived direct…
It is proved that the discriminant of $n\times n$ real symmetric matrices can be written as a sum of squares, where the number of summands equals the dimension of the space of $n$-variable spherical harmonics of degree $n$. The…
We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix $X$ with gradient descent on a factorization of $X$. We conjecture and provide empirical and theoretical evidence that with small enough…
The capability of discretization of matrix elements in the problem of quadratic functional minimization with linear member built on matrix in N-dimensional configuration space with discrete coordinates is researched. It is shown, that…
We consider the problem of finding the best nonnegative rank-2 approximation of an arbitrary nonnegative matrix. We first revisit the theory, including an explicit parametrization of all possible nonnegative factorizations of a nonnegative…