Structure of Binary Transformations of Darboux Type and Their Application to Soliton Theory

On the basis of generalized Lagrange identity for pairs of formally adjoint multidimensional differential operators and a special differential geometric structure associated with this identity, we propose a general scheme of the construction of corresponding transformation operators that are described by nontrivial topological characteristics. We construct explicitly the corresponding integro-differential symbols of transformation operators, which are used in the construction of Lax-integrable nonlinear two-dimensional evolutionary equations and their Darboux–Bäcklund-type transformations.     

Multi-sequences with Almost Perfect Linear Complexity Profile and Function Fields over Finite Fields

Sequences with almost perfect linear complexity profile defined by Niederreiter (1997, Lecture Notes in Computer Science, Vol. 304, pp. 37–51, Springer-Verlag, Berlin/New York) are quite important for stream ciphers. In this paper, we investigate multi-sequences with almost perfect linear complexity profile and obtain a construction of such multi-sequences by using function fields over finite fields. Some interesting examples from this construction are presented to illustrate our construction.     

Lower Bounds of Complexity for Polarized Polynomials over Finite Fields

Siberian Mathematical Journal - We obtain an efficient lower bound of complexity for n-ary functions over a finite field of arbitrary order in the class of polarized polynomials. The complexity of...     

Veronese Varieties Over Finite Fields and Their Projections

Inthis paper Veronese varieties of degree d over aGalois field are studied. We also show that some of known capsembedded into classical varieties always are projections of Veronesevarieties.     

Galois Stability, Integrality and Realization Fields for Representations of Finite Abelian Groups

For a given field F of characteristic 0 we consider a normal extension E/F of finite degree d and finite Abelian subgroups GGL _n (E) of a given exponent t. We assume that G is stable under the natural action of the Galois group of E/F and consider the fields E=F(G) that are obtained via adjoining all matrix coefficients of all matrices gG to F. It is proved that under some reasonable restrictions for n, any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E=F(G) for prescribed integers n and t or prescribed n and d.     

完全二叉树理论的计算复杂度

完全二叉树的一阶理论已被证明具有量词消去的性质,进而计算了完全二叉树模型中元素的CB秩.本文利用有界Ehrenfeucht-Frassé博弈研究完全二叉树的一阶理论,证明了此理论的时间计算复杂度上界为22cn,空间计算复杂度上界为2dn(其中n为输入长度,c,d为合适的常数).     

Normal Bases and Their Dual-Bases over Finite Fields

In this paper, we prove the following results： 1） A normal basis N over a finite field is equivalent to its dual basis if and only if the multiplication table of N is symmetric; 2） The normal basis N is self-dual if and only if its multiplication table is symmetric and Tr（α^2） = 1, where α generates N; 3） An optimal normal basis N is self-dual if and only if N is a type-Ⅰ optimal normal basis with q = n = 2 or N is a type-Ⅱ optimal normal basis.     

Applications of Finite Fields to Combinatorics and Finite Geometries

We survey some applications of finite fields to finite geometries in part A and to combinatorics and error-correcting codes in parts B and C.     

有限域上的逻辑函数与其Chrestenson谱的关系

本文首先给出了有限域上逻辑函数的Chrestenson线性谱的新定义(不同于文献[1]所给出的),如同Chrestenson循环谱一样,重新定义的Chrestenson线性谱也是有限域Fq到复数域的映射,且证明了它们之间在实质意义下可以相互线性表出;最后我们还用重新定义的Chrestenson线性谱给出了有限域上逻辑函数的反演公式.     

Computational Complexity of Checking Identities in 0-Simple Semigroups and Matrix Semigroups over Finite Fields

In this paper we analyze the so-called word problem for (finite) combinatorial 0-simple semigroups and matrix semigroups from the viewpoint of computational complexity.     

RESEARCH ANNOUNCEMENTS——On Multiplication Tables of Normal Bases and Their Dual-bases Over Finite Fields

Let q be a power of a prime p and n be a positive integer,let K=Fq be the finite fiele with q elements and F=Fqn be the nte extension of K.N=｛αi|i=0,1,…,n-1｝is a normal basis of F over Fq,where αi=α^qi,i=0,1…,n-1.     

Restrictions of Algebraic Group Representations to Finite Subgroups

Suppose that H is a finite subgroup of a linear algebraic group,G. It was proved by Donkin that there exists a finite-dimensionalrational representation of G whose restriction to H is free.This paper gives a short proof of this in characteristic 0.The author also studies more closely which representations ofH can appear as a restriction of G.     

基于有限域中的二次特征生成的伪随机二进制格点

本文利用有限域的二次特征与乘法逆构造了大族的伪随机格点,并研究了其密码学性质:伪随机性、碰撞和雪崩效应.     

Majorana Representations of Finite Groups

In this paper we give a brief overview of the theory of Majorana representations of finite groups.     

Group-Like Algebras and Their Representations

The concept of “group-like algebras” was defined by the author as a special class of bF algebras. They generalize scheme rings (Bose–Mesner algebras) of noncommutative association schemes. We develop the representation theory for group-like algebras and symmetric bF algebras. We also study group-like algebras of association-scheme type with dimension 2 and 3.     

On Integral Representations and Boundary Properties of Spinor Fields

A new approach to the boundary value problem for the classic Dirac equation is proposed. This approach is based on a recent version of the metaharmonic quaternionic analysis developed in [14–16]. In particular, the following problem is studied: when and how a given function on a surface can be extended to a time-harmonic spinor field.     

有限复杂度的Koszul代数

首先给出了Koszul代数的张量积的复杂度,然后研究了Koszul遗传代数上的Koszul单列模,并证明了Koszul遗传代数上的Koszul模M的Koszul合成列在同构意义下是唯一的.