THE HOMOGENEOUS POLYNOMIAL SOLUTIONS FOR THE GRUSHIN OPERATOR

In this paper, the author computes the dimension of space of homogeneous Grushin-harmonic functions, and give an orthogonal basis of them. Moreover, the author describes the nodal curves of these homogenous Grushin-harmonic basis. As an application of the orthogonal basis, the author proves a Liouville-type theorem for the Grushin operator, that is the Grushin-harmonic functions are homogeneous polynomials provided that the frequency of such a function is equal to some constant.

ON A SET OF h-HARMONIC HOMOGENEOUS POLYNOMIALS

In this paper, we study the homogeneous polynomials orthogonal with the weight function h(x (d))=x 2k 1 1…x 2k d d on S d-1. We obtain the explicit formula on a basis of the orthogonal homogeneous polynomials of degree n. It is simpler than the formula in [2], and can be regarded as an extension of [1] under the weighted case.

Novel stability criteria for fuzzy Hopfield neural networks based on an improved homogeneous matrix polynomials technique

The global stability problem of Takagi-Sugeno（T-S） fuzzy Hopfield neural networks（FHNNs） with time delays is investigated.Novel LMI-based stability criteria are obtained by using Lyapunov functional theory to guarantee the asymptotic stability of the FHNNs with less conservatism.Firstly,using both Finsler＇s lemma and an improved homogeneous matrix polynomial technique,and applying an affine parameter-dependent Lyapunov-Krasovskii functional,we obtain the convergent LMI-based stability criteria.Algebraic properties of the fuzzy membership functions in the unit simplex are considered in the process of stability analysis via the homogeneous matrix polynomials technique.Secondly,to further reduce the conservatism,a new right-hand-side slack variables introducing technique is also proposed in terms of LMIs,which is suitable to the homogeneous matrix polynomials setting.Finally,two illustrative examples are given to show the efficiency of the proposed approaches.

Extensions of 2-Homogeneous Polynomials on c_0

We prove that every 2-homogeneous polynomial on the complex co has a unique normpreserving extension to its bidual l∞.

Finite Indeterminacy of Homogenous Polynomial Germs under Some Subgroups R_(I_r) of R

Using the finite determinacy relation with the regular sequence in the Ring Theory and the complete intersection in Analytic Geometry, the finite indeterminacy of homogeneous polynomial germs under some subgroups R1（r） of R in both real and complex case is proven by the homogeneity of the polynomial germs. It results in the finite determinacy of homogeneous polynomial germs needn＇t be discussed respectively.

V-BILIPSCHITZ DETERMINACY OF WEIGHTED HOMOGENEOUS ANALYTIC FUNCTION-GERMS ON WEIGHTED HOMOGENEOUS REAL ANALYTIC VARIETIES

In this article, we provide estimates for the degree of V bilipschitz determinacy of weighted homogeneous function germs defined on weighted homogeneous analytic variety V satisfying a convenient Lojasiewicz condition.The result gives an explicit order such that the geometrical structure of a weighted homogeneous polynomial function germs is preserved after higher order perturbations.

On a generalized homogeneous Hahn polynomial

We investigate a generalized homogeneous Hahn polynomial in some detail. This polynomial includes as special cases the homogeneous Hahn polynomial and the homogeneous Rogers-Szeg o polynomial.A generating function, which contains a known generating function as a special case, is given. We also give a finite series generating function. Some results on the asymptotic expansion for this polynomial are derived.Certain results on zeros are also obtained. We deduce several results on zeros of certain entire functions involving this generalized Hahn polynomial. As results, one of Zhang(2017)’s results as well as others is obtained. Finally,we derive several general results on q-congruences of the generalized q-Apéry polynomials, from which two qcongruences involving the generalized homogeneous Hahn polynomial are deduced.

MODIFIED ROPER-SUFFRIDGE OPERATOR FOR SOME SUBCLASSES OF STARLIKE MAPPINGS ON REINHARDT DOMAINS

In this note, the author introduces some new subcIasses of starlike mappings S^＊Ωn1p2,…,pn（β,A,B）={f∈H（Ω）：｜itanβ＋（1-itanβ）2/p（z）аp/аz（z）Jf^-1（z）f（z）-1-AB/1-B^2｜〈B-A/1-B^2},on Reinhardt domains Ωn1p2,…,pn=z∈C^n：｜z1｜^2＋n∑j=2｜zj｜^pj〈1}where - 1≤A〈B〈1,q=min{p2,…,pn}≥1,l=max{p2,…,pn}≥2 and β ∈（-π/2,π/2）.Some different conditions for P are established such that these classes are preserved under the following modified Roper-Suffridge operator F（z）=（f（z1）＋f＇（z1）Pm（z0）,（f＇（z1））^1/mz0）＇where f is a normalized biholomorphic function on the unit disc D, z = （z1,z0） ∈Ωn1p2,…,pn,z0=（z2,…,zn）∈ C^n-1.Another condition for P is also obtained such that the above generalized Roper-Suffridge operator preserves an almost spirallike function of type/3 and order β These results generalize the modified Roper-Suffridge extension oper-ator from the unit ball to Reinhardt domains. Notice that when p2 = p3 …=pn = 2,our results reduce to the recent results of Feng and Yu.

BIFURCATION ANALYSIS AND FEEDBACK CONTROL OF A 3D CHAOTIC SYSTEM

In this paper, we analyze a three-dimensional differential system derived from the Chen system based on the first Lyapunov coefficient, and apply it to investigate the local bifurcation. And we present some insights on bifurcation and stability, also obtain some conditions for subcfitical and supercritical. Finally, we give some numerical simulation studies of system in order to verify analytic results.

On the Complexity of Finding Tensor Ranks

The purpose of this note is to give a linear algebra algorithm to find out if a rank of a given tensor over a field F is at most k over the algebraic closure of F,where K is a given positive integer.We estimate the arithmetic complexity of our algorithm.

ON FUNCTIONAL DECOMPOSITION OF MULTIVARIATE POLYNOMIALS WITH DIFFERENTIATION AND HOMOGENIZATION

This paper gives a theoretical analysis for the algorithms to compute functional decomposition for multivariate polynomials based on differentiation and homogenization which were proposed by Ye, Dai, and Lam （1999） and were developed by Faugere, Perret （2006, 2008, 2009）. The authors show that a degree proper functional decomposition for a set of randomly decomposable quartic homoge- nous polynomials can be computed using the algorithm with high probability. This solves a conjecture proposed by Ye, Dal, and Lam （1999）. The authors also propose a conjecture which asserts that the decomposition for a set of polynomials can be computed from that of its homogenization and show that the conjecture is valid with high probability for quartic polynomials. Finally, the authors prove that the right decomposition factors for a set of polynomials can be computed from its right decomposition factor space.

Rotational invariants constructed by the products of three spherical harmonic polynomials

The rotational invariants constructed by the products of three spherical harmonic polynomials are expressed generally as homogeneous polynomials with respect to the three coordinate vectors in the compact form, where the coefficients are calculated explicitly in this paper.

A Hybrid Second-Order Method for Homogenous Polynomial Optimization over Unit Sphere

In this paper, we propose a hybrid second-order method for homogenouspolynomial optimization over the unit sphere in which the new iterate is generated byemploying the second-order information of the objective function. To guarantee theconvergence, we recall the shifted power method when the second-order method doesnot make an improvement to the objective function. As the Hessian of the objectivefunction can easily be computed and no line search is involved in the second-orderiterative step, the method is not time-consuming. Further, the new iterate is generatedin a relatively larger region and thus the global maximum can be likely obtained. Thegiven numerical experiments show the efficiency of the proposed method.

A Curious Identity on Multiple Sums over Fields with Applications

Let F be a field,and let e,k be integers such that 1≤e≤|F\{0}|and k≥0.We show that for any subset{a1,……,ae}■F\{0},the curious identity∑(i1+……ie)∈Z^(e)≥0,i1+……+ie=k a_(1)^(i1)…a_(e)^(ie)=∑i=1 e a_(i)^(k+e-1)/∏i≠j=1 e(a_(i)-a_(j))holds with Z≥0 being the set of nonnegative integers.As an application,we prove that for any subset{a_(1)…,a_(e)}■F_(q)\{0}with F_(q)being the finite field of q elements and e,l being integers such that 2≤e≤q-1 and 0≤l≤e-2,∑(i_(1),…,i_(e))∈Z^(e)≥0,i_(1)+…i_(e)=q-e+l a_(1)^(i1)…a_(e)^(ie)=0 Using this identity and providing an extension of the principle of cross-classification that slightly generalizes the one obtained by Hong in 1996,we show that if r is an integer with 1≤r≤q-2,then for any subset{a_(1),…a_(r)}■F_(q)^(*)we have x^(q-1)-1/∏i=1 r(x-a_(i))-∑i=1 q-1-r(∑i_(1)+…+i_(r)=q-1-r-i^(a_(1)^(i1)…a_(r)^(ir)))x^(i).This implies#{x∈Fq*|∑i=0 q-1-4(∑_(i1)+…+ir=q-1-r-i^(a_(1)^(i1)…a_(r)^(ir)))x^(i)=0}=q-1-r.

A Kinematic Formula for Integral Invariant of Degree 4 in Real Space Form

In this paper,kinematic formulae in a real space form are investigated.A kinematic formula for homogeneous polynomial of degree 4 on the second fundamental forms in a real space form is obtained.The formula obtained is a concrete form of the result of Howard and the analogue of the known formula of Chen and Zhou.