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.     

L'equazioney\frac{{dy}}{{dx}} + y + \left| x \right|^v sgnx=0, (1≥v≥0)sgnx=0, (1≥v≥0)

Sunto Come viene indicato nell'introduzione l'A, studia il comportamento di tutti gli integrali dell'equazione oggetto della presente memoria, nel piano reale, per v ∈[0, 1], metendo in luce l'esistenza di due integrali fondamentali. Memoria dedicata a Masuo Hukuhara e Mitio Nagumo     

On the functional equation $$G\left( x,G\left( y,x\right) \right) = G\left( y,G\left( x,y\right) \right) $$Gx,Gy,x=Gy,Gx,y and means

Periodica Mathematica Hungarica - We consider the functional equation $$G\left( x,G\left( y,x\right) \right) =G\left( y,G\left( x,y\right) \right) $$, posed in Jarczyk and Jarczyk (Aequ Math...     

Twists and Singular Vectors in \widehat{sl}\left( {2\left| 1 \right.} \right) Representations

We propose new formulas for singular vectors in Verma modules over the affine Lie superalgebra . We analyze the coexistence of singular vectors of different types and identify the twisted modules arising as submodules and quotient modules of Verma modules. We show that with the twists (spectral flow transformations) properly taken into account, a resolution of irreducible representations can be constructed consisting of only the modules.     

On the regularity of the locally integrable solutions of the functional equations $$\sum\limits_{i = 1}^k {a_i \left( {x,t} \right)f\left( {x + \varphi _i \left( t \right)} \right) = b\left( {x,t} \right)} $$

Aequationes mathematicae -     

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.     

Factorization of the universalR-matrix forU_q \left( {\hat sl_2 } \right)^1

The factorization of the universal R-matrix corresponding to the so-called Drinfeld Hopf structure is described in the example of the quantum affine algebra . As a result of the factorization procedure, we deduce certain differential equations on the factors of the universal R-matrix that allow uniquely constructing these factors in the integral form. This article was written at the request of the Editorial Board Translated from Teoreticheskava i Matematicheskaya Fizika, Vol. 124, No. 2, pp. 179–214, August, 2000     

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.     

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| $ .     

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.