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.     

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.     

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.     

A note on the congruence \left( {_{mp^k }^{np^k } } \right) \equiv \left( {_m^n } \right) (mod p r )

In the paper we discuss the following type congruences: $$\left( {_{mp^k }^{np^k } } \right) \equiv \left( {_m^n } \right)(\bmod p^r ),$$ where p is a prime, n, m, k and r are various positive integers with n ? m ? 1, k ? 1 and r ? 1. Given positive integers k and r, denote by W(k, r) the set of all primes p such that the above congruence holds for every pair of integers n ? m ? 1. Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets W(k, r) and inclusion relations between them for various values k and r. In particular, we prove that W(k + i, r) = W(k ? 1, r) for all k ? 2, i ? 0 and 3 ? r ? 3k, and W(k, r) = W(1, r) for all 3 ? r ? 6 and k ? 2. We also noticed that some of these properties may be used for computational purposes related to congruences given above.     

Certain imbedding theorems of the functional spacesW\frac{l}{p}_{, \overline a , } \left( G \right)

In the paper one investigates imbedding type theorems for the spaces \(W\frac{l}{p}_{, \overline a , x} \left( G \right)\) of real functions, defined in the domain G ? Eⁿ, which represent a further generalization of the generalized spaces of the Morrey type. One considers also the problem of the extremal choice of the parameters characterizing the indicated properties.     

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 \(p>3\).

     

Categories of bounded \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) - and \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) -modules

Let $ \mathfrak{g} $ be a reductive Lie algebra over $ \mathbb{C} $ and $ \mathfrak{k} \subset \mathfrak{g} $ be a reductive in $ \mathfrak{g} $ subalgebra. We call a $ \mathfrak{g} $ -module M a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module whenever M is a direct sum of finite-dimensional $ \mathfrak{k} $ -modules. We call a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module M bounded if there exists $ {C_M} \in {\mathbb{Z}_{{ \geqslant 0}}} $ such that for any simple finite-dimensional $ \mathfrak{k} $ -module E the dimension of the E-isotypic component is not greater than C _M dim E. Bounded $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -modules form a subcategory of the category of $ \mathfrak{g} $ -modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ - and $ \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ -modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V .     

Effect of a random real on\kappa \to \left( {\kappa ,{\text{ }}\left( {_{\omega _1 }^\alpha } \right)} \right)^2

It is consistent that $\kappa \to (\kappa ,{\text{ }}\left( {_{\omega _1 }^\alpha } \right))^2 $ holds in the random extension.     

On the continued fraction \frac{1 |}{| z } +\frac{1 |}{| 1 } + \frac{2 |}{| z } +\frac{3 |}{| 1 } + \frac{4 |}{| z} + \cdots

The article on hand deals with the continued fraction $$\frac{1 |}{| z } +\frac{1 |}{| 1 } + \frac{2 |}{| z } +\frac{3 |}{| 1 } + \frac{4 |}{| z} + \cdots.$$ The famous Indian mathematician Srinivasa Ramanujan has given a pre-presentation by a power series, but he however concealed a proof. Subsequently a proof has been established, but a direct verification is intricate. Here we give a quick and direct approach with comparitively little effort.     

On the Diophantine equation $$X^{2N}+2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5$$

We prove that for each prime p, positive integer \(\alpha \), and non-negative integers \(\beta \) and \(\gamma \), the Diophantine equation \(X^{2N} + 2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5\) has no solution with N, X, \(Z\in \mathbb {Z}^+\), \(N > 1\), and \(\gcd (X,Z) = 1\).     

Die Modulräume^3 M_{\mathbb{P}^3 }^{st} \left( { - 2, 3, c_3 } \right)

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