Non-homogeneous Polynomial Research Articles

Variety of solutions and dynamical behavior for YTSF equations

We construct non-homogeneous polynomial lump wave solutions of the Yu-Toda-Sasa-Fukuyama (YTSF) equation, based on a bilinear approach, enriching the formal diversity of lump waves. By studying the interaction between the lump solutions of the YTSF equation and the solitary wave solutions, we find a new aggregation effect and elastic collision effect. We obtain exact solutions, such as the solution of separated variables and periodic nonlinear wave solutions, by applying the Lie symmetry group method and the bilinear method.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/82/abstr.html

Multiple lump molecules and interaction solutions of the Kadomtsev–Petviashvili I equation

In this paper, a modified version of the solution in form of a Gramian formula is employed to investigate a new type of multiple lump molecule solution of the Kadomtsev–Petviashvili I equation. The high-order multiple lump molecules consisting of M N-lump molecules are constructed by means of the Mth-order determinant and the non-homogeneous polynomial in the degree of 2N. The interaction solutions describing P line solitons radiating P of the M N-lump molecules are constructed. The dynamic behaviors of some specific solutions are analyzed through numerical simulation. All the results will enrich our understanding of the multiple lump waves of the Kadomtsev–Petviashvili I equation.

Nonsingular zeros of polynomials defined over finite fields

The aim of this paper is to study the existence of nontrivial, nonsingular zeros of a nonhomogeneous polynomial defined over a finite field. To accomplish this, we determine conditions that guarantee the existence of a prescribed number of nonsingular zeros of a homogeneous form f over a finite field k that are not zeros of a homogeneous form h when f, h are relatively prime. The cases of quadratic and cubic polynomials are considered in detail. This extends previous results that have usually considered only the homogeneous case.

Invariant regions and existence of global solutions to a generalized m-component reaction-diffusion system with tridiagonal symmetric Toeplitz diffusion matrix

The aim of this paper is to construct invariant regions of a generalized m-component reaction-diffusion system with tridiagonal symmetric Toeplitz diffusion matrix and nonhomogeneous boundary conditions and polynomial growth for the nonlinear reaction terms. Using the eigenvalues and eigenvectors of the diffusion matrix and the parabolicity conditions. So we prove the global existence of solutions using Lyapunov functional.

On Bézout inequalities for non-homogeneous polynomial ideals

We introduce a “workable” notion of degree for non-homogeneous polynomial ideals and formulate and prove ideal theoretic Bézout inequalities for the sum of two ideals in terms of this notion of degree and the degree of generators.

Limit distributions of the Pearson statistics for nonhomogeneous polynomial scheme

Abstract For a nonhomogeneous polynomial scheme, conditions are found under which the Pearson statistic distributions converge to the distribution of nonnegative quadratic form of independent random variables with the standard normal distribution.

Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems

In this paper we research global dynamics and bifurcations of planar piecewise smooth quadratic quasi-homogeneous but non-homogeneous polynomial differential systems. We present sufficient and necessary conditions for the existence of a center in piecewise smooth quadratic quasi-homogeneous systems. Moreover, the center is global and non-isochronous, which cannot appear in smooth quadratic quasi-homogeneous systems. Then the global structures of piecewise smooth quadratic quasi-homogeneous but non-homogeneous systems are obtained. Finally we investigate limit cycle bifurcations of the piecewise quadratic quasi-homogeneous center and give the maximal number of limit cycles bifurcating from periodic orbits of the center by applying the Melnikov method for piecewise smooth near-Hamiltonian systems.

The connection between polynomial optimization, maximum cliques and Turán densities

In 1965, Motzkin–Straus established the connection between the maximum cliques and the Lagrangian of a graph, the maximum value of a quadratic function determined by a graph in the standard simplex. This connection gave a proof of the Turán’s classical result on Turán densities of complete graphs. In 1980’s, Sidorenko and Frankl–Füredi further developed this method for hypergraph Turán problems. However, the connection between the Lagrangian and the maximum cliques of a graph cannot be extended to hypergraphs. In 2009, S. Rota Bulò and M. Pelillo defined a homogeneous polynomial function of degree r determined by an r-uniform hypergraph and gave the connection between the minimum value of this polynomial function and the maximum cliques of an r-uniform hypergraph. In this paper, we provide a connection between the local (global) minimizers of non-homogeneous polynomial functions to the maximal (maximum) cliques of hypergraphs whose edges containing r−1 and r vertices. This connection can be applied to obtain an upper bound on the Turán densities of complete {r−1,r}-type hypergraphs.

Critère d’existence d’idempotent basé sur les algèbres de Rétrocroisement

ABSTRACTWe study the relationship of backcrossing algebras with mutation algebras and algebras satisfying ω-polynomial identities: we show that in a backcrossing algebra every element of weight 1 generates a mutation algebra and that for any polynomial identity f there is a backcrossing algebra satisfying f. We give a criterion for the existence of idempotent in the case of baric algebras satisfying a nonhomogeneous polynomial identity and containing a backcrossing subalgebra. We give numerous genetic interpretations of the algebraic results.

Homogeneous and Non-homogeneous Polynomial Based Eigenspaces to Extract the Features on Facial Images

Homogeneous and Non-homogeneous Polynomial Based Eigenspaces to Extract the Features on Facial Images

Connection between a class of polynomial optimization problems and maximum cliques of non-uniform hypergraphs

In 1965, Motzkin and Straus provided a connection between the order of a maximum clique in a graph $$G$$G and a global solution of a quadratic optimization problem determined by $$G$$G which is called the Lagrangian function of $$G$$G. This connection and its extensions have been useful in both combinatorics and optimization. In 2009, Rota Bulo and Pelillo extended the Motzkin---Straus result to $$r$$r-uniform hypergraphs by establishing a one-to-one correspondence between local (global) minimizers of a family of homogeneous polynomial functions of degree $$r$$r (different from Lagrangian function) and the maximal (maximum) cliques of an $$r$$r-uniform hypergraph. In this paper, we study similar optimization problems related to non-uniform hypergraphs and obtain some extensions of their results to non-uniform hypergraphs. In particular, we provide a one-to-one correspondence between local (global) minimizers of a family of non-homogeneous polynomial functions and the maximal (maximum) cliques of $$\{1, r\}$${1,r}-hypergraphs. An application of a main result gives an upper bound on the Turan density of complete $$\{1, r\}$${1,r}-hypergraphs. The approach applied in the proof follows from the approach in Rota Bulo and Pelillo (2009).

Action du groupe des opérateurs de gamétisation sur les identités ω-polynomiales

The gametization process reduces the study of non-commutative and non-associative algebras satisfying non-homogeneous polynomial identities with variables in X={x1,…,xn} to algebras verifying simpler identities. However after a gametization, certain identities remain invariant and other identities, said universal invariant, are invariant for every gametization. Now in the case n=1, for all algebras satisfying a universal invariant polynomial identity studied until now, we know that the existence of an idempotent is not certain. Using an action of the gametization operators group on the non-commutative and non-associative algebra of polynomials K〈X〉, we give all identities which are invariant and universal invariant by gametization.

A new proof for the correctness of the F5 algorithm

In 2002, Faugere presented the famous F5 algorithm for computing Grobner basis where two cri- teria, syzygy criterion and rewritten criterion, were proposed to avoid redundant computations. He proved the correctness of the syzygy criterion, but the proof for the correctness of the rewritten criterion was left. Since then, F5 has been studied extensively. Some proofs for the correctness of F5 were proposed, but these proofs are valid only under some extra assumptions. In this paper, we give a proof for the correctness of F5B, an equivalent version of F5 in Buchberger's style. The proof is valid for both homogeneous and non-homogeneous polynomial systems. Since this proof does not depend on the computing order of the S-pairs, any strategy of selecting S-pairs could be used in F5B or F5. Furthermore, we propose a natural and non-incremental variant of F5 where two revised criteria can be used to remove almost all redundant S-pairs.

On the polynomial vector fields on

Let X be a polynomial vector field of degree n on M, M = ℝm. The dynamics and the algebraic-geometric properties of the vector fields X have been studied intensively, mainly for the case when M = ℝm, and especially when n = 2. Several papers have been dedicated to the study of the homogeneous polynomial vector field of degree n on $\mathbb{S}^2$, mainly for the case where n = 2 and M = $\mathbb{S}^2$. But there are very few results on the non-homogeneous polynomial vector fields of degree n on $\mathbb{S}^2$. This paper attempts to rectify this slightly.

Generalization of the polarization formula for nonhomogeneous polynomials and analytic mappings on Banach spaces

We propose an analogue of the classical polarization formula for any nonhomogeneous polynomial and analytic mapping on complex Banach spaces.

Simple method to obtain symmetry harmonics of point groups

We propose a simple, self-consistent method to obtain basis functions of irreducible representations of a finite point group. Our method is based on eigenproblem formulation of a projection operator represented as a nonhomogeneous polynomial of angular momentum L. The method is shown to be more efficient than the usual numerical methods when applied to the analysis of high-order symmetry harmonics in cubic and icosahedral groups. For low-order symmetry harmonics the method provides rational coefficients of expansion in the Y(L,M) basis.

Nonintegrability of nonhomogeneous nonlinear lattices

We study the integrability of nonlinear lattices with nonhomogeneous polynomial potentials. We prove a nonintegrability theorem for these dynamical systems.

Multivariate chi-square distribution for non-homogeneous polynomial scheme

Multivariate chi-square distribution for non-homogeneous polynomial scheme

Short Communication:On a Poisson Approximation for the Distribution of the Number of Empty Cells in a Nonhomogeneous Allocation Scheme

A nonhomogeneous polynomial scheme of allocation of particles into cells is considered in which different allocation laws of particles are allowed. Explicit estimates are given for the accuracy of the Poisson approximation of the distribution of the number of empty cells in an arbitrary subset of cells. The proof of the main theorem uses the combination of the Chen--Stein method with the common probability space method described in the monograph [A.D. Barbour, L. Holst, and S. Janson, Poisson Approximation, Oxford University Press, Oxford, 1992].

Non-integrability of the truncated Toda lattice Hamiltonian at any order

Two sufficient conditions for the non-existence of an additional analytic integral are given for Hamiltonian systems with non-homogeneous polynomial potential of an arbitrary degree. An application is made to the truncated three-particle Toda lattice, which is proved to be non-integrable at any order.