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Subresultants of two univariate polynomials are one of the most classic and ubiquitous objects in computational algebra and algebraic geometry. In 1948, Habicht discovered and proved interesting relationships among subresultants. Those…

We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral) and Jacobi $\beta$-ensembles of $N\times N$ random matrices. More specifically, we calculate scaling limits of the expectation value of products of…

Motivated by Tweedie's formula for the Compound Decision problem, we examine the theoretical foundations of empirical Bayes estimators that directly model the marginal density $m(y)$. Our main result shows that polynomial log-marginals of…

We construct a generalize Ishida complex to compute the local cohomology with monomial support of modules over quotients of polynomial rings by cellular binomial ideals. As a consequence, we obtain a combinatorial criterion to determine…

We consider min-max optimization problems for polynomial functions, where a multivariate polynomial is maximized with respect to a subset of variables, and the resulting maximal value is minimized with respect to the remaining variables.…

Consideration of a question of E. R. Berlekamp led Carlitz, Roselle, and Scoville to give a combinatorial interpretation of the entries of certain matrices of determinant~1 in terms of lattice paths. Here we generalize this result by…

For a finite loop $Q$, let $P (Q)$ be the set of elements that can be represented as a product containing each element of $Q$ precisely once. Motivated by the recent proof of the Hall-Paige conjecture, we prove several universal…

We give a proof of the Universality Conjecture for orthogonal (beta=1) and symplectic (beta=4) random matrix ensembles of Laguerre-type in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated…

We establish sparsity and summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions of countably-parametric solutions of linear elliptic and parabolic divergence-form partial differential equations with…

We prove that the Buchweitz-Greuel-Schreyer Conjecture on the minimal rank of a matrix factorization holds for a generic polynomial of given degree and strength. The proof introduces a notion of the secondary strength of a polynomial, and…

We derive analytical expression of matrix factorization/completion solution by variational Bayes method, under the assumption that observed matrix is originally the product of low-rank dense and sparse matrices with additive noise. We…

In this paper we deal with the estimation of population variance of the study variable y using auxiliary information on variable x. A family of ratio and product-type estimators are proposed using suitable transformation on both random…

We assume the direct sum <A> o <B> for the signal subspace. As a result of post- measurement, a number of operational contexts presuppose the a priori knowledge of the LB -dimensional "interfering" subspace <B> and the goal is to estimate…

In high-dimensional statistical inference, sparsity regularizations have shown advantages in consistency and convergence rates for coefficient estimation. We consider a generalized version of Sparse-Group Lasso which captures both…

The Hanson-Wright inequality establishes exponential concentration for quadratic forms $X^T M X$, where $X$ is a vector with independent sub-Gaussian entries and with parameters depending on the Frobenius and operator norms of $M$. The most…

The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the…

We propose a new pivot selection technique for symmetric indefinite factorization of sparse matrices. Such factorization should maintain both sparsity and numerical stability of the factors, both of which depend solely on the choices of the…

The concept of a composed product for univariate polynomials has been explored extensively by Brawley, Brown, Carlitz, Gao, Mills, et al. Starting with these fundamental ideas and utilizing fractional power series representation (in…

Let $j\geq 3$ be any fixed integer and $f$ be a primitive holomorphic cusp form of even integral weight $\kappa\geq 2$ for the full modular group $SL(2,\mathbb{Z})$. We write $\lambda_{{\rm{sym}^j }f}(n)$ for the $n^\text{th}$ normalized…

We study the Darcy boundary value problem with log-normal permeability field. We adopt a perturbation approach, expanding the solution in Taylor series around the nominal value of the coefficient, and approximating the expected value of the…