Building Symmetric Designs With Building Sets

We introduce a uniform technique for constructing a family of symmetric designs with parameters (v(q ^m+1-1)/(q-1), kq ^m ,q ^m), where m is any positive integer, (v, k, ) are parameters of an abelian difference set, and q = k ²/(k - ) is a prime power. We utilize the Davis and Jedwab approach to constructing difference sets to show that our construction works whenever (v, k, ) are parameters of a McFarland difference set or its complement, a Spence difference set or its complement, a Davis–Jedwab difference set or its complement, or a Hadamard difference set of order 9 · 4 ^d , thus obtaining seven infinite families of symmetric designs.     

Hausdorff Dimension of Symmetric Perfect Sets

We consider nowhere dense perfect subsets of [0, 1] that are symmetric but have no additional nice properties. We prove that if E = E_n is a symmetric perfect set and the length of the basic intervals in E_n is denoted by l_n then the Hausdorff dimension of E is

Sets Computing the Symmetric Tensor Rank

Let ${\nu_{d} : \mathbb{P}^{r} \rightarrow \mathbb{P}^{N}, N := \left( \begin{array}{ll} r + d \\ \,\,\,\,\,\, r \end{array} \right)- 1,}$ denote the degree d Veronese embedding of ${\mathbb{P}^{r}}$ . For any ${P\, \in \, \mathbb{P}^{N}}$ , the symmetric tensor rank sr(P) is the minimal cardinality of a set ${\mathcal{S} \subset \nu_{d}(\mathbb{P}^{r})}$ spanning P. Let ${\mathcal{S}(P)}$ be the set of all ${A \subset \mathbb{P}^{r}}$ such that ${\nu_{d}(A)}$ computes sr(P). Here we classify all ${P \,\in\, \mathbb{P}^{n}}$ such that sr(P) <  3d/2 and sr(P) is computed by at least two subsets of ${\nu_{d}(\mathbb{P}^{r})}$ . For such tensors ${P\, \in\, \mathbb{P}^{N}}$ , we prove that ${\mathcal{S}(P)}$ has no isolated points.     

Low Correlation Noise Stability of Symmetric Sets

Journal of Theoretical Probability - We study the Gaussian noise stability of subsets A of Euclidean space satisfying $$A=-A$$ . It is shown that an interval centered at the origin, or its...     

基于代数等价路径的一致P-函数非线性互补问题的可行内点算法

对一致P-函数非线性互补问题，提出了一种新的基于代数等价路径的可行内点算法，并讨论了计算复杂性．该算法可以在任一内部可行点启动，并且全局收敛；当初始点靠近中心路径时，此算法便成为中心路径跟踪算法，特别对于单调线性互补问题，总迭代次数为O（√nL），其中L是问题的输入长度。     

自相似集之间的等变映射

定义了复射动力系统以及它们之间等变映射的概念 ,在此基础上又引出了自相似集之间等变映射的概念并讨论了在此类映射的映射下自相似集象的 Hausdorff维数的状态 .     

对称几乎可约矩阵的两个指数集

本文完全确定出ｎ（＞２）阶对称非本原几乎可约布尔矩阵的幂敛指数集和最大密度指数集．     

Equivalent Sets of Consequences of Linear Inequality Systems

Different types of linear inequality systems have different consequence inequalities. Investigating several types of linear inequality systems, the present paper gives explicitly those consequences of the given system of linear inequalities that are all consistent if and only if the original system is consistent. Our results generalize the well-known Kuhn-Fourier theorem, and present important particular cases.     

对称n筛计算式

本文引进一种筛法,由此给出从1,2,…,2a中经过n次两两互素的筛数筛减后的乘余数点个数的计算公式。     

Difference Sets Corresponding to a Class of Symmetric Designs

We study difference sets with parameters(v, k, ) = (p ^s(r ^2m - 1)/(r - 1), p ^s-1 r ^2m-2 r - 1)r ^{2m -2}, where r = r ^s - 1)/(p - 1) and p is a prime. Examples for such difference sets are known from a construction of McFarland which works for m = 1 and all p,s. We will prove a structural theorem on difference sets with the above parameters; it will include the result, that under the self-conjugacy assumption McFarland's construction yields all difference sets in the underlying groups. We also show that no abelian .160; 54; 18/-difference set exists. Finally, we give a new nonexistence prove of (189, 48, 12)-difference sets in Z ₃ × Z ₉ × Z ₇.     

Symmetric ‐Designs and ‐Difference Sets

In this paper, we introduce a method to construct ‐designs, which are also known as partial geometric designs, by using subsets of certain finite groups. We introduce the concept of ‐difference sets and investigate the existence and nonexistence of these structures. We also provide some nonexistence results on ‐designs based on the fact that ‐designs yield directed strongly regular graphs.     

Measurable Sets With Excluded Distances

For a set of distances D = {d ₁,..., d _k } a set A is called D-avoiding if no pair of points of A is at distance d _i for some i. We show that the density of A is exponentially small in k provided the ratios d ₁/d ₂, d ₂/d ₃, …, d _k-1/d _k are all small enough. This resolves a question of Székely, and generalizes a theorem of Furstenberg–Katznelson–Weiss, Falconer–Marstrand, and Bourgain. Several more results on D-avoiding sets are presented. Received: January 2007, Revision: February 2008, Accepted: February 2008     

Some Equivalent Norms in Sobolev-Besov Spaces on Symmetric Riemannian Manifolds

In this paper we introduce local means on Riemannian symmetricmanifolds of the noncompact type corresponding to the Laplace-Beltramioperator, and investigate equivalent norms in the Sobolev andBesov spaces defined via these means.     

具有对称的集合和完备度量空间上两个自映射的共同不动点

本文给出了具有对称的集合上两个自映射有唯一共同不动点的充要条件.同时,还给出了完备度量空间上两个自映射有共同唯一不动点的条件.     

Explicit Constructions of Centrally Symmetric k-Neighborly Polytopes and Large Strictly Antipodal Sets

We present explicit constructions of centrally symmetric $2$ -neighborly $d$ -dimensional polytopes with about $3^{d/2}\approx (1.73)^d$ vertices and of centrally symmetric $k$ -neighborly $d$ -polytopes with about $2^{{3d}/{20k^2 2^k}}$ vertices. Using this result, we construct for a fixed $k\ge 2$ and arbitrarily large $d$ and $N$ , a centrally symmetric $d$ -polytope with $N$ vertices that has at least $\left( 1-k^2\cdot (\gamma _k)^d\right) \genfrac(){0.0pt}{}{N}{k}$ faces of dimension $k-1$ , where $\gamma _2=1/\sqrt{3}\approx 0.58$ and $\gamma _k = 2^{-3/{20k^2 2^k}}$ for $k\ge 3$ . Another application is a construction of a set of $3^{\lfloor d/2 -1\rfloor }-1$ points in $\mathbb R ^d$ every two of which are strictly antipodal as well as a construction of an $n$ -point set (for an arbitrarily large $n$ ) in $\mathbb R ^d$ with many pairs of strictly antipodal points. The two latter results significantly improve the previous bounds by Talata, and Makai and Martini, respectively.