Some Infinite Classes of Optimal (v, {3, 4}, 1, Q)-OOCs with {Q \in \{(\frac {1}{3}, \frac {2}{3}), (\frac {2}{3}, \frac{1}{3})\}}

Variable-weight optical orthogonal code (OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service requirements. It is proved that optimal (v, {3, 4}, 1, (1/2, 1/2))-OOCs exist for some complete congruence classes of v. In this paper, for ${Q \in \{(1/3, 2/3), (2/3, 1/3)\}}$ , by using skew starters, it is also proved that optimal (v, {3, 4}, 1, Q)-OOCs exist for some complete congruence classes of v.     

On a theory of elliptic functions based on the incomplete integral of the hypergeometric function {_{2}}F_{1}(\frac{1}{4},\frac{3}{4};\frac{1}{2};z)

Using the properties of conformal mappings and differential equations, we develop a class of elliptic functions associated with the hypergeometric function ${_{2}}F_{1}(\frac{1}{4},\frac{3}{4};1;z)$ . A detailed comparison is made with the classical Jacobi elliptic functions. Within the frame work of this theory, we provide a proof and new insight into a set of identities of Ramanujan associated with the above hypergeometric function.     

The rigidity of Clifford torusS^1 \left( {\sqrt {\frac{1}{n}} } \right) \times S^{n - 1} \left( {\sqrt {\frac{{n - 1}}{n}} } \right)

In this paper, we prove that ifM is ann-dimensional closed minimal hypersurface with two distinct principal curvatures of a unit sphereS ⁿ⁺¹ (1), thenS=n andM is a Clifford torus ifn≤S≤n+[2n ²(n+4)/3(n(n+4)+4)], whereS is the squared norm of the second fundamental form ofM.     

Interpolation spaces between(L_1^{W_0 } ,L_1^{W_1 } ) and (L1, L∞)

Let A₀, A₁ be a pair of normed spaces, having the property that the difference K(x, t; A₀, A₁) ?K(x, s; A₀, A₁) regarded as a function of xε A₀+ A₁ is a seminorm for t>s (here K is the Oklander-Peetre functional). All the pairs A, L of normed spaces, such that, if a linear operator is bounded from A₀ into L₁ and from A₁ into L_∞, then it is bounded from A into L, are characterized in the following article.     

Central limit theorem: Covergence in the norm\left\| u \right\|\left( {\int\limits_{ - \infty }^\infty {u^2 \left( x \right)e^{\frac{{x^2 }}{2}} dx} } \right)^{{\raise0.7ex\hbox{

Sufficient conditions for convergence in the central limit theorem (for identically distributed random variables) with respect to the topology specified in the title are given. These conditions hold for the uniform distribution although there exist distributions with smooth densities concentrated on a bounded interval for which the convergence result does not hold.     

Hybrid subconvexity bounds for twisted L-functions on GL(3)

Let q be a large prime, and χ the quadratic character modulo q. Let Φ be a self-dual Hecke-Maass cusp form for SL(3, Z), and uj a Hecke-Maass cusp form for Γ0(q) ? SL(2, Z) with spectral parameter tj. We prove, for the first time, some hybrid subconvexity bounds for the twisted L-functions on GL(3), such as L(1/2, Φ× uj ×χ) ?_(Φ,ε)(q(1 + |tj |))~(3/2-θ+ε) and L(1/2 + it, Φ×χ) ?_(Φ,ε)(q(1 + |t|))~(3/4-θ/2+ε) for any ε 0, where θ = 1/23 is admissible. The proofs depend on the first moment of a family of L-functions in short intervals. In order to bound this moment, we first use the approximate functional equations, the Kuznetsov formula, and the Voronoi formula to transform it to a complicated summation, and then we apply different methods to estimate it, which give us strong bounds in different aspects. We also use the stationary phase method and the large sieve inequalities.     

The estimate for mean values on prime numbers relative to \frac{4} {p} = \frac{1} {{n_1 }} + \frac{1} {{n_2 }} + \frac{1} {{n_3 }}

If n is a positive integer,let f （n） denote the number of positive integer solutions （n 1,n 2,n 3） of the Diophantine equation 4/n=1/n₁ + 1/n₂ + 1/n₃.For the prime number p,f （p） can be split into f 1 （p） + f 2 （p）,where f i （p） （i=1,2） counts those solutions with exactly i of denominators n 1,n 2,n 3 divisible by p.In this paper,we shall study the estimate for mean values ∑ p相似文献   

Optimal $\mathcal{L}^2$ error bounds for MITC3 type element

Summary. In this paper, we derive the optimal error bounds for the stabilized MITC3 element [3], the MIN3 type element [7] and the T3BL element [8]. In this way we have solved the problem proposed recently in [5] in a positive manner. Moreover, we estimate the difference between stabilized MITC3 and MIN3 and show it is of order uniform in the plate thickness. Received May 31, 2000 / Revised version received April 2, 2001 / Published online September 19, 2001     

Realization of 2D convolutional codes of rate \frac{1}{n} by separable Roesser models

In this paper, two-dimensional convolutional codes constituted by sequences in $(\mathbb F ^n)^{\mathbb{Z }^{2}}$ where $\mathbb F $ is a finite field, are considered. In particular, we restrict to codes with rate $\frac{1}{n}$ and we investigate the problem of minimal dimension for realizations of such codes by separable Roesser models. The encoders which allow to obtain such minimal realizations, called R-minimal encoders, are characterized.     

Congruences involving $$g_n(x)=\sum \limits _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k$$

Define \(g_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k\) for \(n=0,1,2,\ldots \). Those numbers \(g_n=g_n(1)\) are closely related to Apéry numbers and Franel numbers. In this paper we establish some fundamental congruences involving \(g_n(x)\). For example, for any prime \(p>5\) we have

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{g_k(-1)}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{g_k(-1)}{k^2}\equiv 0\pmod p. \end{aligned}$$

This is similar to Wolstenholme’s classical congruences

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{1}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{1}{k^2}\equiv 0\pmod p \end{aligned}$$

for any prime \(p>3\).

     

双圆盘加权Hardy空间上$T_{z_1^{k_1}z_2^{k_2}+\bar{z}_1^{l_1}\bar{z}_2^{l_2}}$的约化子空间

本文中, 我们主要刻画了Toeplitz算子$T=M_{z^k}+M^*_{z^l}$的约化子空间, 其中 $k_i, l_i$ ($i=1,2$) 均是正整数, $k=(k_1,k_2), l=(l_1,l_2)$ 且 $k\neq l$, $M_{z^k}$, $M_{z^l}$ 是双圆盘加权Hardy空间$\mathcal{H}_\omega^2(\mathbb{D}^2)$上的乘法算子. 对权系数 $\omega$ 适当限制, 我们证明了由 $z^m$ 生成的 $T$ 的约化子空间均是极小的. 特别地, Bergman 空间和加权 Dirichlet 空间 $\mathcal{D}_\delta(\mathbb{D}^2)(\delta>0)$ 均是满足该限制条件的加权Hardy空间. 作为应用, 我们刻画了 $\mathcal{D}_\delta(\mathbb{D}^2)(\delta>0)$ 上 Toeplitz 算子 $T_{z^k+\bar{z}^l}$ 的约化子空间, 该结论是对双圆盘Bergman 空间上相关结论的推广.     

Minimal generating and normally generating sets for the braid and mapping class groups of \mathbb{D }^{2}, \mathbb S ^{2} and \mathbb{R P^2}

We consider the (pure) braid groups $B_{n}(M)$ and $P_{n}(M)$ , where $M$ is the $2$ -sphere $\mathbb S ^{2}$ or the real projective plane $\mathbb R P^2$ . We determine the minimal cardinality of (normal) generating sets $X$ of these groups, first when there is no restriction on $X$ , and secondly when $X$ consists of elements of finite order. This improves on results of Berrick and Matthey in the case of $\mathbb S ^{2}$ , and extends them in the case of $\mathbb R P^2$ . We begin by recalling the situation for the Artin braid groups ( $M=\mathbb{D }^{2}$ ). As applications of our results, we answer the corresponding questions for the associated mapping class groups, and we show that for $M=\mathbb S ^{2}$ or $\mathbb R P^2$ , the induced action of $B_n(M)$ on $H_3(\widetilde{F_n(M)};\mathbb{Z })$ is trivial, $F_{n}(M)$ being the $n^\mathrm{th}$ configuration space of $M$ .     

Verallgemeinerte\mathfrak{L}^1 -Algebren

Ohne Zusammenfassung