Related papers: Maximum Likelihood Duality for Determinantal Varie…

We study the real rank of points with respect to a real variety $X$. This is a generalization of various tensor ranks, where $X$ is in a specific family of real varieties like Veronese or Segre varieties. The maximal real rank can be…

Probabilistic frames are a generalization of finite frames into the Wasserstein space of probability measures with finite second moment. We introduce new probabilistic definitions of duality, analysis, and synthesis and investigate their…

In this paper we announce a conjecture concerning enumeration of 2n x k n-times persymmetric matrices over F_2 by rank.

We introduce the concepts of an amazing hypercube decomposition and a double shortcut for it, and use these new ideas to formulate a conjecture implying the Combinatorial Invariance Conjecture of the Kazhdan--Lusztig polynomials for the…

In this paper, we compute the cohomology class of certain "special maximal-rank loci" originally defined by Aprodu and Farkas. By showing that such classes are nonzero, we are able to verify the non-emptiness portion of the Strong Maximal…

In this paper, we study the linear transformation model in the most general setup. This model includes many important and popular models in statistics and econometrics as special cases. Although it has been studied for many years, the…

We describe the fundamental constructions and properties of determinantal probability measures and point processes, giving streamlined proofs. We illustrate these with some important examples. We pose several general questions and…

Given an $m\times n$ binary matrix $M$ with $|M|=p\cdot mn$ (where $|M|$ denotes the number of 1 entries), define the discrepancy of $M$ as $\mbox{disc}(M)=\displaystyle\max_{X\subset [m], Y\subset [n]}\big||M[X\times Y]|-p|X|\cdot…

The paper contains an exposition of recent as well as old enough results on determinantal random point fields. We start with some general theorems including the proofs of the necessary and sufficient condition for the existence of the…

We characterize the existence of the maximum likelihood estimator for discrete exponential families. Our criterion is simple to apply as we show in various settings, most notably for exponential models of random graphs. As an application,…

We investigate the joint convergence of independent random Toeplitz matrices with complex input entries that have a pair-correlation structure, along with deterministic Toeplitz matrices and the backward identity permutation matrix.…

We consider double determinantal varieties, a special case of Nakajima quiver varieties. Li conjectured that double determinantal varieties are normal, irreducible, Cohen-Macaulay varieties whose defining ideals have a Gr\"obner basis given…

We consider partial symmetric Toeplitz matrices where a positive definite completion exists. We characterize those patterns where the maximum determinant completion is itself Toeplitz. We then extend these results with positive definite…

We develop $L$-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of cubic Hecke $L$-functions of prime moduli over the Eisenstein field using multiple Dirichlet series under the…

A mixed type dual to a nondifferentiable variational problem involving higher order derivative is formulated and duality results are proved under generalized invexity conditions. Special cases are generated from our results.

Let p_N be a random degree N polynomial in one complex variable whose zeros are chosen independently from a fixed probability measure mu on the Riemann sphere S^2. This article proves that if we condition p_N to have a zero at some fixed…

Buchweitz-Greuel-Schreyer conjectured in 1987 a lower bound on the ranks of matrix factorizations over certain local hypersurface rings. We study a graded version of this conjecture, and we show that it implies a novel conjecture concerning…

Let $\mu$ be a $p$-dimensional vector, and let $\Sigma_1$ and $\Sigma_2$ be $p \times p$ positive definite covariance matrices. On being given random samples of sizes $N_1$ and $N_2$ from independent multivariate normal populations…

Determinantal varieties -- the sets of bounded-rank matrices or tensors -- have attracted growing interest in low-rank optimization. The tangent cone to low-rank sets is widely studied and underpins a range of geometric methods. The…

Suppose $E$ is an elliptic curve over $\mathbb{Q}$ and $\chi$ is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that $L(E,\chi,1) = 0$. Via the Birch and Swinnerton-Dyer…