Related papers: Short Solutions To Homogenous Linear congruences

The existence and multiplicity of positive periodic solutions for first non-autonomous singular systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. The proof of our…

We prove that a class of superlinear indefinite problems with homogeneous Neumann boundary conditions admits an arbitrarily high number of positive solutions, provided that the parameters of the problem are adequately chosen. The…

Suppose that M is countable, binary, primitive, homogeneous, and simple, and hence 1-based. We prove that the SU-rank of the complete theory of M is~1. It follows that M is a random structure. The conclusion that M is a random structure…

Let $G$ be the simple algebraic group $\mathrm{SL}_2$ defined over an algebraically closed field $k$ of characteristic $p > 0$. Using results of A. Parker, we develop a method which gives, for any $q \in \mathbb{N}$, a closed form…

In this note, we establish a new closed formula for the solution of homogeneous second-order linear difference equations with constant coefficients by using matrix theory. This, in turn, gives new closed formulas concerning all sequences of…

We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific congruences for Hecke eigenvalues. In particular, we show that Ramanujan-type congruences are…

Linear systems are the bedrock of virtually all numerical computation. Machine learning poses specific challenges for the solution of such systems due to their scale, characteristic structure, stochasticity and the central role of…

In this note, simple proofs of certain well-known results involving the positive square root of positive matrices are given.

We describe a solution of the word problem in free fields (coming from non-commutative polynomials over a commutative field) using elementary linear algebra, provided that the elements are given by minimal linear representations. It relies…

For a sequence of nonnegative random variables, we provide simple necessary and sufficient conditions to ensure that each sequence of its forward convex combinations converges in probability to the same limit. These conditions correspond to…

Given a flat vector bundle over a compact Riemannian manifold, Corlette and Donaldson proved that it admits harmonic metrics if and only if it is semi-simple. In this paper, we extend this equivalence to arbitrary vector bundles without any…

Moraga and Yeong conjectured that for a smooth complex projective variety $X$ of dimension $n$, an ample line bundle $A$ on $X$ and an integer $m \ge 3 n + 1$, very general elements of the adjoint linear system $|\omega_{X} \otimes…

This paper considers the problem of testing whether there exists a non-negative solution to a possibly under-determined system of linear equations with known coefficients. This hypothesis testing problem arises naturally in a number of…

Solving sparse linear systems from discretized PDEs is challenging. Direct solvers have in many cases quadratic complexity (depending on geometry), while iterative solvers require problem dependent preconditioners to be robust and…

Let $V$ be a finite relational vocabulary in which no symbol has arity greater than 2. Let $M$ be countable $V$-structure which is homogeneous, simple and 1-based. The first main result says that if $M$ is, in addition, primitive, then it…

We provide a countable series of bisimple $\mathcal{H}$-trivial finitely presented congruence-free monoids.

This paper proves the existence of weak solutions to the spatially homogeneous Boltzmann equation for Maxwellian molecules, when the initial data are chosen from the space of all Borel probability measures on R^3 with finite second moments…

Algebras defined over fields of characteristic zero and positive characteristic usually do not behave the same way. However, for certain algebras, for example the group algebras, they behave the same way as the characteristic zero case at…

Recently there has been quite a bit of study carried out related to arithmetic properties of overpartitions into non-multiples of two co-prime integers. The paper [19] by Nadji et al. looked into congruences modulo $3$ and powers of $2$ for…

We prove that suitably generic pairs of linear equations on an even number of variables are uncommon. This verifies a conjecture of Kam\v{c}ev, Morrison and the second author. Moreover, we prove that any large system containing such a…