Related papers: Computing the permanent of (some) complex matrices

Given point sets $A$ and $B$ in $\mathbb{R}^d$ where $A$ and $B$ have equal size $n$ for some constant dimension $d$ and a parameter $\varepsilon>0$, we present the first deterministic algorithm that computes, in $n\cdot(\varepsilon^{-1}…

Consider the problem of estimating the median of N items to a precision epsilon, i.e., the estimate should be such that, with a high probability, the number of items, with values both smaller than and larger than this estimate, is less than…

In this paper we investigate the permanent of $(-1,1)$-matrices over fields of zero characteristics and our main goal is to provide a sharp upper bound for the value of the permanent of such matrices depending on matrix rank, solving Wang's…

Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…

Let $A$ be a set in a prime field $\mathbb{F}_p$. In this paper, we prove that $d\times d$ matrices with entries in $A$ determine almost $|A|^{3+\frac{1}{45}}$ distinct determinants and almost $|A|^{2-\frac{1}{6}}$ distinct permanents when…

We address the question of computing one selected term of an algebraic power series. In characteristic zero, the best algorithm currently known for computing the $N$th coefficient of an algebraic series uses differential equations and has…

Suppose we are given an oracle that claims to approximate the permanent for most matrices X, where X is chosen from the Gaussian ensemble (the matrix entries are i.i.d. univariate complex Gaussians). Can we test that the oracle satisfies…

We give a formula for the determinant of an $n\times n$ matrix with entries from a commutative ring with unit. The formula can be evaluated by a "straight-line program" performing only additions, subtractions and multiplications of ring…

Motivated by questions in robust control and switched linear dynamical systems, we consider the problem checking whether all convex combinations of k matrices in R^{n x n} are stable. In particular, we are interested whether there exist…

Let $f$ be a fixed (holomorphic or Maass) modular cusp form. Let $\cq$ be a Dirichlet character mod $q$. We describe a fast algorithm that computes the value $L(1/2,f\times\chi_q)$ up to any specified precision. In the case when $q$ is…

The process of alternately row scaling and column scaling a positive $n \times n$ matrix $A$ converges to a doubly stochastic positive $n \times n$ matrix $S(A)$, called the \emph{Sinkhorn limit} of $A$. Exact formulae for the Sinkhorn…

In this paper we describe an algorithm for the computation of canonical forms of finite subsets of $\mathbb{Z}^d$, up to affinities over $\mathbb{Z}$. For fixed dimension $d$, this algorithm has worst-case asymptotic complexity $O(n \log^2…

Base on some simple facts of Hadamard product, characterizations of positive definite preserving linear transformations on real symmetric matrix spaces with an additional assumption "$\ra T(E_{ii})=1, i=1,2,..., n$" or "$T(A)>0\to A> 0$",…

We introduce an elementary method for proving the absolute continuity of the time marginals of one-dimensional processes. It is based on a comparison between the Fourier transform of such time marginals with those of the one-step Euler…

We study the one-body reduced density matrix of a system of $N$ one-dimensional impenetrable anyons trapped by a harmonic potential. To this purpose we extend two methods developed to tackle related problems, namely the determinant approach…

The purpose of this paper is to compute the asymptotics of determinants of finite sections of operators that are trace class perturbations of Toeplitz operators. For example, we consider the asymptotics in the case where the matrices are of…

Hamiltonian matrices appear in a variety or problems in physics and engineering, mostly related to the time evolution of linear dynamical systems as for instance in ion beam optics. The time evolution is given by symplectic transfer…

We consider the product of determinantal point processes (DPPs), a point process whose probability mass is proportional to the product of principal minors of multiple matrices, as a natural, promising generalization of DPPs. We study the…

We study systems of equations of the form X1 = f1(X1, ..., Xn), ..., Xn = fn(X1, ..., Xn), where each fi is a polynomial with nonnegative coefficients that add up to 1. The least nonnegative solution, say mu, of such equation systems is…

This work studies the average complexity of solving structured polynomial systems that are characterized by a low evaluation cost, as opposed to the dense random model previously used. Firstly, we design a continuation algorithm that…