Related papers: A new solution to square matrix completion problem
In this paper we find a parametric solution to the hitherto unsolved problem of finding three positive integers such that their sum, the sum of their squares and the sum of their cubes are simultaneously perfect squares.
As two fundamental problems, graph cuts and graph matching have been investigated over decades, resulting in vast literature in these two topics respectively. However the way of jointly applying and solving graph cuts and matching receives…
We study the problem of exact completion for $m \times n$ sized matrix of rank $r$ with the adaptive sampling method. We introduce a relation of the exact completion problem with the sparsest vector of column and row spaces (which we call…
It is well-known that by adding integrality constraints to the semidefinite programming (SDP) relaxation of the max-cut problem, the resulting integer semidefinite program is an exact formulation of the problem. In this paper we show…
A positive definite completion problem pertains to determining whether the unspecified positions of a partial (or incomplete) matrix can be completed in a desired subclass of positive definite matrices. In this paper we study an important…
In this note, we present a new numerical method for solving backward stochastic differential equations. Our method can be viewed as an analogue of the classical finite element method solving deterministic partial differential equations.
We study several variations of line segment covering problem with axis-parallel unit squares in $I\!\!R^2$. A set $S$ of $n$ line segments is given. The objective is to find the minimum number of axis-parallel unit squares which cover at…
This is a contribution to the number theory of the dimer problem. The number of dimer coverings (i.e., perfect matchings) of a square lattice graph is discussed modulo powers of 2.
Symmetry is a common feature of many combinatorial problems. Unfortunately eliminating all symmetry from a problem is often computationally intractable. This paper argues that recent parameterized complexity results provide insight into…
A square trisection is a problem of assembling three identical squares from a larger square, using a minimal number of pieces. This paper presents an historical overview of the square trisection problem starting with its origins in the…
We use ideas from our previous work to obtain some theorems that will allow us to obtain the integer solution of a quadratic polynomial in two variables that represents a natural number
This paper addresses the numerical solution of the matrix square root problem. Two fixed point iterations are proposed by rearranging the nonlinear matrix equation $A - X^2 = 0$ and incorporating a positive scaling parameter. The proposals…
Ordinary approach to quantum algorithm is based on quantum Turing machine or quantum circuits. It is known that this approach is not powerful enough to solve NP-complete problems. In this paper we study a new approach to quantum algorithm…
Tensor completion is a natural higher-order generalization of matrix completion where the goal is to recover a low-rank tensor from sparse observations of its entries. Existing algorithms are either heuristic without provable guarantees,…
We show arithmetic triplets of Gaussian squares are in 3-to-1 correspondence with Pythagorean triples thereof. This correspondence would transform a solution to the Magic Square of Squares puzzle into a larger structure of perfect Gaussian…
In this study, a new form of quadratic spline is obtained, where the coefficients are determined explicitly by variational methods. Convergence is studied and parity conservation is demonstrated. Finally, the method is applied to solve…
In this paper, we propose a federated algorithm for solving large linear systems that is inspired by the classic randomized Kaczmarz algorithm. We provide convergence guarantees of the proposed method, and as a corollary of our analysis, we…
Traditional formulations of geometric problems from the Schubert calculus, either in Plucker coordinates or in local coordinates provided by Schubert cells, yield systems of polynomials that are typically far from complete intersections and…
Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…
In the total matching problem, one is given a graph $G$ with weights on the vertices and edges. The goal is to find a maximum weight set of vertices and edges that is the non-incident union of a stable set and a matching. We consider the…