The Isometric Extension of an Into Mapping from the Unit Sphere S（t（2）^∞） to S（L^1（μ））

This is the first paper to consider the isometric extension problem of an into-mapping between the unit spheres of two different types of spaces. We prove that, under some conditions, an into-isometric mapping from the unit sphere S（t（2）^∞） to S（L^1（μ） can be （real） linearly isometrically extended.     

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 \(p>3\).

     

On the continuity of \text{ SLE }_{\kappa } in \kappa

We prove that for almost every Brownian motion sample, the corresponding SLE \(_\kappa \) curves parameterized by capacity exist and change continuously in the supremum norm when \(\kappa \) varies in the interval \([0,\kappa _0)\) , where \(\kappa _0=8(2-\sqrt{3})\approx 2.1\) . We estimate the \(\kappa \) -dependent modulus of continuity of the curves and also give an estimate on the modulus of continuity for the supremum norm change with \(\kappa \) .     

Some remarks on the moments of \left| {\varsigma \left( {\tfrac{1} {2} + it} \right)} \right| in short intervals

Some new results on power moments of the integral $$ J_k (t,G) = \frac{1} {{\sqrt {\pi G} }}\int_{ - \infty }^\infty { \left| {\varsigma \left( {\tfrac{1} {2} + it + iu} \right)} \right|^{2k} } e^{ - (u/G)^2 } du $$ (t ? T, T ^? ≦ G ? T, κ ∈ N) are obtained when κ = 1. These results can be used to derive bounds for moments of $ \left| {\varsigma \left( {\tfrac{1} {2} + it} \right)} \right| $ .     

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.     

On the diophantine equation F $$\left( {\left( {_n^x } \right)} \right) = b\left( {_m^y } \right)$$

As a generalization of the results of [3] and [22], we characterize those pairs (m,n) and those polynomials b{Z}[x] of prime degree for which equation (1) has only finitely many integer solutions.     

The exact order of\sum\nolimits_{k = 1}^{\text{n}} {|{\text{x - x}}_{\text{k}} } |^8 |{\text{l}}_{\text{k}} ({\text{x)|}}^{\text{t}}

In this paper we give the exact order of \(\sum\nolimits_{k = 1}^{\text{n}} {|{\text{x - x}}_{\text{k}} } |^5 .\) for any fixed nonnegative integers s and t, which is n^?s, n^?s lnn and n^1?t for s≤t?2, s=t?1 and s≥t, respectively.     

Application of Soliton Theory to the Construction of Isometric Immersions of M^{{n_{1} }} {\left( {c_{1} } \right)} \times M^{{n_{2} }} {\left( {c_{2} } \right)} into Constant Curvature Spaces M n (±1)

In this paper, we apply the soliton theory to the case of isometric immersion in differential geometry and obtain a family of isometric immersions from M ^{n 1}(c ₁) ×M ^{n 2}(c ₂) to space forms M ⁿ (c) by introducing 2-parameter loop algebra. Received July 14, 1999, Accepted June 15, 2000     

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 .     

Group-cograded Multiplier Hopf ${\left( { * {\text{ - }}} \right)}$algebras

Let G be a group and assume that (A _p ) _p∈G is a family of algebras with identity. We have a Hopf G-coalgebra (in the sense of Turaev) if, for each pair p,q ∈ G, there is given a unital homomorphism Δ _p,q : A _pq → A _p ⊗ A _q satisfying certain properties. Consider now the direct sum A of these algebras. It is an algebra, without identity, except when G is a finite group, but the product is non-degenerate. The maps Δ _p,q can be used to define a coproduct Δ on A and the conditions imposed on these maps give that (A,Δ) is a multiplier Hopf algebra. It is G-cograded as explained in this paper. We study these so-called group-cograded multiplier Hopf algebras. They are, as explained above, more general than the Hopf group-coalgebras as introduced by Turaev. Moreover, our point of view makes it possible to use results and techniques from the theory of multiplier Hopf algebras in the study of Hopf group-coalgebras (and generalizations). In a separate paper, we treat the quantum double in this context and we recover, in a simple and natural way (and generalize) results obtained by Zunino. In this paper, we study integrals, in general and in the case where the components are finite-dimensional. Using these ideas, we obtain most of the results of Virelizier on this subject and consider them in the framework of multiplier Hopf algebras. Presented by Ken Goodearl.     

Counting SL2（F2^s） Representations of Torus Knot Groups

In this paper,we count the number of SL2(F2^s)-representations of torus knot groups up to a conjugacy.For the finite field F2^s with character 2,the counting method is similar to that in out previous work[1].Explicit formulae of the effective counting are given in this paper.Twisted Alexander polynomials related to those reprsentations are discussed.     

Simulation of Weakly Self-Similar Stationary Increment $${\text{Sub}}_{\varphi } {\left( \Omega \right)}$$-Processes: A Series Expansion Approach

We consider simulation of -processes that are weakly selfsimilar with stationary increments in the sense that they have the covariance function

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.     

Some new estimates of the Fourier-Bessel transform in the space \mathbb{L}_2 \left( {\mathbb{R}_ + } \right)

The Fourier-Bessel integral transform $$g\left( x \right) = F\left[ f \right]\left( x \right) = \frac{1} {{2^p \Gamma \left( {p + 1} \right)}}\int\limits_0^{ + \infty } {t^{2p + 1} f\left( x \right)j_p \left( {xt} \right)dt}$$ is considered in the space $\mathbb{L}_2 \left( {\mathbb{R}_ + } \right)$ . Here, j _p (u) = ((2 ^p Γ(p+1))/(u ^p ))J _p (u) and J _p (u) is a Bessel function of the first kind. New estimates are proved for the integral $$\delta _N^2 \left( f \right) = \int\limits_N^{ + \infty } {x^{2p + 1} g^2 \left( x \right)dx, N > 0,}$$ in $\mathbb{L}_2 \left( {\mathbb{R}_ + } \right)$ for some classes of functions characterized by a generalized modulus of continuity.     

Boundedness of Singular Integrals on Multiparameter Weighted Hardy Spaces \text{{\textit{H}}}^\text{{\textit{p}}}_{\text{{\textit{w}}}}\ (\mathbb{R}^{\text{{\textit{n}}}}\times \mathbb{R}^{\text{{\textit{m}}}})

We apply the discrete version of Calderón??s identity and Littlewood?CPaley?CStein theory with weights to derive the $(H^p_w, H^p_w)$ and $(H^p_w, L^p_w) (0<p\le 1)$ boundedness for multiparameter singular integral operators. It turns out that even in the one-parameter case, our results substantially improve the known ones in the literature where w????A ₁ was needed. Our results in the multiparameter setting can be regarded as a natural extension of $L^p_w$ boundedness for p?>?1 for w????A _p to the case of weighted Hardy spaces $H^p_w$ for p????1, but under a weaker assumption that w belongs to the class of product A _???? weights with respect to rectangles in product spaces.     

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

Ohne Zusammenfassung