Sums of Squares of Integers

We find a formula for the sum of the first n squares:Sn = 12 22 32 42 … n2.We haveS1 = 1, S2 = 12 22 = 5, S3 = l2 22 32 = 14,S4 = 30, S5 = 55, S6 = 91.and so on. To make things a little simpler, we will also use the sum of the squares of the first zero terms , S0 = 02 = 0. Arranging these values in a table,we find thatSince the third differenes △3Sn are constant,these data values must follow a cubic pattern;that is, the formula for the sum of the squares of the first n integers is a cubic function,     

Vertices of π-Irreducible Characters of Groups of Odd Order

If G is a group of odd order and χ ∈ Irr(G) lifts an irreducible Brauer character ?, then we associate to χ a canonical pair (Q, δ) up to G-conjugacy, where Q is a vertex of ? and δ ∈ Irr(Q) is a linear character of Q. We show that (Q, δ) is a Navarro vertex for χ. We also discuss examples.     

Stability of Solution of a Class of Quasilinear Elliptic Equations

In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equationsin space W(θ,p)(Ω), which is bigger than W1,p(Ω). Next, using revise reverse Holder inequality we prove that if ωc is uniformly p-think, then there exists a neighborhood U of p, such that for all t ∈U, the weak solutions of equation corresponding t are bounded uniformly. Finally, we get the stability of weak solutions on exponent p.     

On the Growth of Components of Meromorphic Solutions of Systems of Complex Differential Equations

This paper investigates the problem of the growth of the components of meromorphic solutions of a class of a system of complex algebraic differential equations, and generalized some of N. Toda＇s results concerning the growth of differential equations to the case of systems of differential equations. The paper considers the existence of admissible solutions of the system of differential equations.     

Correlation of zeros of automorphic L-functions

We compute the n-level correlation of normalized nontrivial zeros of a product of L-functions:L(s,π1)···L(s,πk), where πj, j=1,...,k, are automorphic cuspidal representations of GLmj(QA). Here the sizes of the groups GLmj(QA) are not necessarily the same. When these L(s,πj) are distinct, we prove that their nontrivial zeros are uncorrelated, as predicted by random matrix theory and verified numerically. When L(s,πj) are not necessarily distinct, our results will lead to a proof that the n-level correlation of normalized nontrivial zeros of the product L-function follows the superposition of Gaussian Unitary Ensemble (GUE) models of individual L-functions and products of lower rank GUEs. The results are unconditional when m1,...,mk 4,but are under Hypothesis H in other cases.     

Extension of a Theorem of Carleson-Duren

1. Introduction A theorem of Carleson,as generalized by Duren characterizes those positive measure μ on the unit disc U={z∈C:|z|<1} for which the H~p norm domiates the L~q(μ) norm of elements of H~p. Later on, Hasting proved an analogous results with H~p replaced by A~p, the Bergman space of fuctions f     

Estimates of groups of deviations of Faber sums and strong summability of faber series on classes of ψ-integrals

We establish upper bounds for a group of *-deviations of Faber sums on the classes of -integrals in a complex plane introduced by Stepanets.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 451–461, April, 2004.     

The Degree of Unsolvability of Models

The purpose of this paper is to describe the complexity of models by theirdegrees of unsolvability,J.Richter defined the degree of a structure to be deg ()=sup{deg(),deg(R_i),i=1,…,n},Where is a model for afinite language L={R_i,i=1,…,n}and the universe of is a subset of ω. Shepointed out that, according to her definition, there can be models which areisomorphic but their degrees are different.Also,her discussions are restricted tofinite languages and models whose universes are subsets of ω.     

Lie Bialgebras of a Family of Lie Algebras of Block Type

Lie bialgebra structures on a family of Lie algebras of Block type are shown to be triangular coboundary.     

Superderivation Algebras of Modular Lie Superalgebras of (O)-Type

The authors consider a family of finite-dimensional Lie superalgebras of O-type over an algebraically closed field of characteristic p > 3. It is proved that the Lie superalgebras of O-type are simple and the spanning sets are determined. Then the spanning sets are employed to characterize the superderivation algebras of these Lie superalgebras. Finally, the associative forms are discussed and a comparison is made between these Lie superalgebras and other simple Lie superalgebras of Cartan type.     

Ad-Nilpotent Elements,Isomorphisms, and the Weisfeiler Filtration of Infinite-Dimensional Modular Odd Contact Superalgebras

The Weisfeiler filtrations of infinite-dimensional odd contact superalgebras over fields of positive characteristic are proved to be invariant under automorphisms by characterizing ad-nilpotent elements. Thereby, we are able to obtain an intrinsic characterization of odd contact superalgebras and a property of automorphisms of these Lie superalgebras.     

奇Hamiltonian型滤过李超代数HO

本文应用象空间极小维数的方法研究了Cartan型模李超代数HO的非平凡可迁滤过(HO,R)的不变性.应用滤过的不变性得到模李超代数HO的某些内蕴性质,从而得到此类李超代数在同构意义下的分类.     

有限维单Cartan型模李超代数HO

本文构造了一族有限维Cartan型模李超代数-奇Hamilton模李超代数HO, 并证明了其单性.通过建立Cartan型模李超代数W,H,K,S和HO的维数公式,讨 论了奇Hamilton模李超代数HO与Cartan型模李超代数W,S,H,K的同构关系.     

模李超代数研究的若干进展

近几年，模李超代数（即素特征域上的李超代数）的研究取得了一些进展。本文分析以下6个方面介绍近几年在模李超代数研究中取得的一些新结果。1，限制李超代数，2，Cartan型模李超代数；3，深度1的Z－阶化李超代数；4，深度1的滤过李超代数。5；Cartan型模李超代数的某些内蕴性质；6，公开的问题。     

具有非退化Killing型的广义李超代数

设F是特征数为零的域.本文证明了F上的有限维广义李超代数的Killing型是相容的、不变的与上对称的.进而证明了具有非退化Killing型的有限维广义李超代数的 若干性质，最后得出这种广义李超代数必为有限个典型的广义李超代数的直积.     

泛阶化李超代数

对于给定的负阶化李超代数K-,本文定义了K-型泛阶化李超代数并证明了它的存在性．进而引出阶化Cartan型李超代数,并且证得阶化Cartan型李超代数 W(m,n),K(m,n,ωA),S(m,n)和H(m,n)分别可以用某种泛阶化李超代数来刻画．     

Left supersymmetric Structures on Lie Superalgebras

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Filtration, automorphisms, and classification of infinite-dimensional odd contact superalgebras

The principal filtration of the infinite-dimensional odd contact Lie superalgebra over a field of characteristic p>2 is proved to be invariant under the automorphism group by investigating ad-nilpotent elements and determining certain invariants such as subalgebras generated by some ad-nilpotent elements. Then, it is proved that two automorphisms coincide if and only if they coincide on the -1 component with respect to the principal grading. Finally, all the odd contact superalgebras are classified up to isomorphisms.     

Infinite-dimensional modular special odd contact superalgebras

The infinite-dimensional special odd contact Lie superalgebras SKO(n,n+1;c) over a field of positive characteristic are studied. In particular, we prove that Lie superalgebras SKO(n,n+1;c) are simple. Besides, the Weisfeiler filtration of SKO(n,n+1;c) is proved to be invariant under automorphisms by characterizing -nilpotent elements. Thereby, we obtain some properties of these Lie superalgebras.