Codes defined by forms of degree 2 on non-degenerate Hermitian varieties in $${\mathbb{P}^{4}(\mathbb{F}_q)}$$

We study the functional codes of second order on a non-degenerate Hermitian variety as defined by G. Lachaud. We provide the best possible bounds for the number of points of quadratic sections of . We list the first five weights, describe the corresponding codewords and compute their number. The paper ends with two conjectures. The first is about minimum distance of the functional codes of order h on a non-singular Hermitian variety . The second is about distribution of the codewords of first five weights of the functional codes of second order on a non-singular Hermitian variety .      

Correct algebras over estimation algorithms in the set of regular recognition problems with nonoverlapping classes

Algebras over estimation algorithms in the set of regular problems with nonoverlapping classes are considered. A correctness criterion for the arbitrary degree algebraic closure of the model of estimation algorithms in the classification problems of this type is proposed; this criterion can be efficiently verified. An estimate of the minimal degree of the algebraic closure that is sufficient for constructing a correct classifier in an arbitrary regular problem with nonoverlapping classes is found.     

几何定理机器证明三十年

由于传统的兴趣和多种原因,几何定理的机器证明在自动推理的研究中占有重要的地位.自吴法发表至今30年,几何定理机器证明的研究和实践有了很大的进展.对无序几何命题而言,代数方法、数值方法均能有效地判定其真假,面积法(消点法)、搜索法更能生成其可读的证明.几何不等式机器证明的研究,由于多项式完全判别系统的建立,也有了突破.研究领域已由机器证明扩展为包括几何作图在内的一般几何问题的机器求解,并有了实际的应用.     

Structure of coordinate groups for algebraic sets in partially commutative nilpotent groups

The results obtained deal in algebraic geometry over partially commutative class two nilpotent ℚ-groups, where ℚ is a field of rationals. It is proved that two arbitrary non-Abelian partially commutative class two nilpotent ℚ-groups are geometrically equivalent. A necessary and sufficient condition of being universally geometrically equivalent is specified for two partially commutative class two nilpotent ℚ-groups. Algebraic sets for systems of equations in one variable, as well as for some special systems in several variables, are described. Dedicated to V. N. Remeslennikov on the occasion of his 70th birthday Translated from Algebra i Logika, Vol. 48, No. 2, pp. 378–399, May–June, 2009.     

Existence of global invariant jet differentials on projective hypersurfaces of high degree

Let be a smooth complex projective hypersurface. In this paper we show that, if the degree of X is large enough, then there exist global sections of the bundle of invariant jet differentials of order n on X, vanishing on an ample divisor. We also prove a logarithmic version, effective in low dimension, for the log-pair , where D is a smooth irreducible divisor of high degree. Moreover, these result are sharp, i.e. one cannot have such jet differentials of order less than n.     

The separability of the Gauss map versus the reflexivity

In projective algebraic geometry, various pathological phenomena in positive characteristic have been observed by several authors. Many of those phenomena concerning the behavior of embedded tangent spaces seem to be controlled by the separability of (the extension of function fields defined by) the Gauss map, or by the reflexivity with respect to the projective dual for a projective variety. The purpose of this paper is to survey the studies on the relationship between the separability of the Gauss map and the reflexivity for a projective variety: Is the separability of the Gauss map equivalent to the reflexivity for a projective variety?     

Bifurcation of Equilibria in a Class of Planar Piecewise Smooth Systems with 3 Parameters

The goal of this paper is to investigate the bifurcation properties of stationary points of a class of planar piecewise smooth systems with 3 parameters using the theory of differential inclusions. We especially study the existence of the stationary points on the line of discontinuity of this kind of planar piecewise smooth system.     

Co-fibered products of algebraic curves

We give examples of failure of the existence of co-fibered products in the category of algebraic curves.      

交换环上赋值的完全性

对有乘法单位元的交换环上的非精简显赋值定义一种"完全性"。首先就Von Neumann正则环成为完全赋值环给出一个充分必要条件(定理1);并对正则的完全赋值环证明在它的代数扩环上也能给出完全的拓展赋值(定理2)。其次,再对另一种特殊的环给出与定理1和定理2相同的结论(定理3,4)。更多还原