利用Z／P^αZ上矩阵构作多个结合类的结合方案与PBIB设计

本文利用有限局部环Ｚ／ｐαＺ上的１×ｎ矩阵集合构作具α个结合类的方案与ＰＢＩＢ设计，并计算了它们的参数．     

利用伪辛空间中2维全迷向子空间构作结合方案与PBIB设计

取有限域Ｆｑ上２×２＋２维伪辛空间中２维全迷向子空间作为处理，构作了具有４个结合类和２个结合类的结合方案与ＰＢＩＢ设计，并且计算了它们的参数     

利用有限域上伪辛几何中一类2维非迷向子空间构作PBIB设计

设Ｆｑ是特征为２的有限域，本文利用Ｆｑ上２ｖ＋２维伪辛几何中包含固定的１维非迷向子空间的一类的２维非迷向子空间作处理，构作了具有２（ｑ－１）个结合类的结合方法和ＰＢＩＢ设计，并计算了相应的参数。     

3倍角三角形的性质初探

在△ＡＢＣ中,若∠Ｃ＝ｎ∠Ｂ,∠Ｂ＝ｎ∠Ａ,ｎ∈Ｎ,则称△ＡＢＣ为。倍角三角形． 当ｎ＝１时,即为正三角形;当ｎ＝２时,则∠Ｃ＝２∠Ｂ,∠Ｂ＝２∠Ａ,此时 ∠Ａ：∠Ｂ：∠Ｃ＝２～０：２～１：２～２,我们称△ＡＢＣ为２倍角三角形． 关于２倍角三角形,文［１］已给出了若干有趣的性质． ２倍角三角形性质可以给出许多竞赛题以新解,简解,见文［２］． 当ｎ＝３时,∠Ｃ＝３∠Ｂ,∠Ｂ＝３∠Ａ,则∠Ａ：∠Ｂ：∠Ｃ＝３～０：３～１：３～２,称△ＡＢＣ为３倍角三角形,关于３倍角三角形,笔者初步得到如下性质： （１）当∠…     

虚二次域上不可分的正定Hermite型的构作

本文给出了构作虚二次域ＩＱ（－ｍ）上不可分的正定整Ｈｅｒｍｉｔｅ型的方法．对任意给定的自然数ｎ，ｄ和无平方因子的ｍ，当ｍ３（ｍｏｄ４），除了ｍ＝１时ｎ＝２，ｄ＝１；ｎ＝３，ｄ＝１，３；ｎ＝５，ｄ＝１和ｍ＝２时ｎ＝３，ｄ＝１这５个例外，证明了存在ＩＱ（－ｍ）上不可分的正定整Ｈｅｒｍｉｔｅ格，其秩为ｎ且判别式为ｄ，并给出它们的明确结构．在上述５个例外情形下，不存在具有上述性质的格．     

二维Landau-Lifshitz静态方程组的渐近性质

在本文中，我们讨论了ｕ∧（－Δｕ＋λ（ｕ，ｎ）ｎ）＝０，｜ｕ｜＝１，ｘ∈Ｂ１和ｕ＝（ｘ１，ｘ２，０）ｘ∈Ｂ１的轴对称解ｕλ的渐近行为，其中Ｂ１是Ｒ２中单位圆，ｕ＝（ｕ１，ｕ２，ｕ３）．我们证明了ｕλｕ∞在Ｈ２Ｂ１＼Ｂε，Ｒ３中．     

有限奇异酉几何中的计数定理和PBIB设计的构作

设Ｆ＾２ｑ是ｑ＾２个元素的有限域，这里ｑ是一个素数的方幂。本文计算了Ｆ＾２ｑ上奇异酉几何中包含在一个固定的（ｍ，ｒ，ｋ）型子空间里的（ｍ１，ｒ１，ｋ１）型子空间的个数，从而得到一个计数定理；然后分别利用（１，０，０）型子空间及（１，１，０）型子空间作处理构作某些结合方案和ＰＢＩＢ设计。     

增长曲线模型的参数估计及其性质

考虑增长曲线模型：Ｙｐ×ｎ＝ＡＢＣ＇＋εｐ×ｎ，Ｅε＝０．其中，ε＝（ε１，ε２，…，εｎ），ε１，ε２…，εｎ独立同分布，Ｅεｉεｉ＇＝Σｐ×ｐ＞０．该文利用协差阵的Σ的（一定意义下的）最小二乘估计Σ，分别给出了参数Ｂ，参数的线性函数ＡＢ，ｔｒ（Ｄ１’Ｂ）＋（Ｄ２Σ）（Ｄ２＝Ｄ‘２）（Ｄ２＝Ｄ２’）的估计Ｂｎ，Ｚｎ和ｔｒ（Ｄ１＇Ｂｎ）＋ｔｒ（Ｄ２Σｎ）．在ε１服从正态分布的情形下，给出了Ｚｎ，Σｎ的分布．并在ε１分布比较一般的情形和一定条件下给出了Ｚｎ，Ｂｎ，Σｎ和ｔｒ（Ｄ１’Ｂｎ）＋ｔｒ（Ｄ２Σｎ）的极限分布皆为正态分布（ｎ→∞）．而且Ｚｎ，和Σｎ，Ｂｎ，和Σｎ都是渐近独立的（ｎ→∞）．从而可构造参数Ｂ的置信区域和更好地进行判别分析，相关分析等．     

正2n边形的一个有趣性质

正２ｎ边形的一个有趣性质４１９６０５湖南沅陵六中周永国本文给出正２ｎ边形的一个性质，并由此得到了球的内接正Ｚｎ边形、棱锥以及等分圆周的有趣性质．引理设Ａ１Ａ２…Ａ２是正２ｎ边形，Ｐ为空间任一点，则证明作一正２ｎ边形Ｂ１Ｂ２…Ｂｎ，取边ＢｉＢｉ＋１的中...     

关于阶数为奇数的布尔矩阵行空间的基数

设Ｂｎ表示所有的ｎ阶布尔矩阵的集合，Ｒ（Ａ）表示Ａ∈Ｂｎ的行空间，｜Ｒ（Ａ）｜表示Ｒ（Ａ）的基数．设ｍ，ｎ为正整数，本文证明了（Ⅰ）ｍ∈［１，４６］，［１，７８］，分别存在Ａ∈Ｂ７，Ａ∈Ｂ８，使得｜Ｒ（Ａ）｜＝ｍ．（Ⅱ）当ｎ≥９为奇数时，则ｍ∈［１，２（ｎ＋３）／２＋２（ｎ＋１）／２＋…＋２３］，存在Ａ∈Ｂｎ，使得｜Ｒ（Ａ）｜＝ｍ．     

Locally symmetric minimal affine Lagrangian surfaces in C<Superscript>2</Superscript>

One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant curvature in C ² must be totally geodesic. In affine geometry the constancy of curvature corresponds to the local symmetry of a connection. In Opozda (Geom. Dedic. 121:155–166, 2006), we proposed an affine version of the theory of minimal Lagrangian submanifolds. In this paper we give a local classification of locally symmetric minimal affine Lagrangian surfaces in C ². Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian) geometry. The research supported by the KBN grant 1 PO3A 034 26.     

Affine normal curvature of hypersurfaces from the point of view of singularity theory

We consider smoothly embedded hypersurfaces under the action of the special affine group . We construct a differential invariant, called affine normal curvature, which assigns to a point and a tangent direction a number. We prove some of its nice properties which connect it with affine principal directions, affine umbilics, and affine mean curvature.      

On affine designs and Hadamard designs with line spreads

Rahilly [On the line structure of designs, Discrete Math. 92 (1991) 291-303] described a construction that relates any Hadamard design H on 4^m-1 points with a line spread to an affine design having the same parameters as the classical design of points and hyperplanes in AG(m,4). Here it is proved that the affine design is the classical design of points and hyperplanes in AG(m,4) if, and only if, H is the classical design of points and hyperplanes in PG(2m-1,2) and the line spread is of a special type. Computational results about line spreads in PG(5,2) are given. One of the affine designs obtained has the same 2-rank as the design of points and planes in AG(3,4), and provides a counter-example to a conjecture of Hamada [On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error-correcting codes, Hiroshima Math. J. 3 (1973) 153-226].     

Gheorghe Ţiţeica and the origins of affine differential geometry

We describe the historical and ideological context that brought to the fore the study of a centro-affine invariant that subsequently received much attention. The invariant was introduced by ?i?eica in 1907, and this discovery has been viewed by many as a consequence of Klein's Erlangen program. We thus present the starting point of affine differential geometry, as it was discovered by ?i?eica after his years in the Ph.D. program in Paris (1896–1899) under the guidance of Gaston Darboux.     

The homology of the space of affine flags containing a nilpotent element

We show that the homology of the space of Iwahori subalgebras containing a nilpotent element of a split semisimple Lie algebra over is isomorphic to the homology of the entire affine flag manifold.

     

The eight variable homogeneous degree three bent functions

We determine the affine equivalence classes of the eight variable degree three homogeneous bent functions using a new algorithm. Our algorithm applies to general bent functions and can systematically determine the automorphism groups. We provide a partial verification of the enumeration of eight variable degree three homogeneous bent functions obtained by Meng et al. We determine the affine equivalence classes of these functions.     

A group extensions approach to affine relative difference sets of even order

It is shown that a group extensions approach to central relative (k+1,k-1,k,1)-difference sets of even order leads naturally to the notion of an “affine” planar map; a notion analogous to the well-known planar map corresponding to a splitting relative (m,m,m,1)-difference set. Basic properties of affine planar maps are derived and applied to give some new results regarding abelian relative (k+1,k-1,k,1)-difference sets of even order and to give new proofs, in the even order case, for some known results. The paper concludes with computational non-existence results for 10,000<k?100,000.     

R4中关于Klingenberg确定的横截丛的仿射全脐曲面

如果对R4中属于仿射法平面的每个向量相应的形状算子是恒等算子的倍数 ,就称该曲面是全脐的 .本文关于Klingenberg确定的横截丛将仿射全脐曲面进行分类     

Volume distance to hypersurfaces: Asymptotic behavior of its hessian

The volume distance from a point p to a convex hypersurface $M?{\mathbb{R}}^{N+1}$ is defined as the minimum $(N+1)$-volume of a region bounded by M and a hyperplane H through the point. This function is differentiable in a neighborhood of M and if we restrict its hessian to the minimizing hyperplane $H(p)$ we obtain, after normalization, a symmetric bilinear form Q.In this paper, we prove that Q converges to the affine Blaschke metric when we approximate the hypersurface along a curve whose points are centroids of parallel sections. We also show that the rate of this convergence is given by a bilinear form associated with the shape operator of M. These convergence results provide a geometric interpretation of the Blaschke metric and the shape operator in terms of the volume distance.