A note on the crossed product $$ \mathcal{R}(\mathcal{A},\alpha ) $$ associated with PSL2(ℝ)

Given the hyperbolic measure dxdy/y ² on the upper half plane ℍ, the rational actions of PSL₂(ℝ) on ℍ induces a continuous unitary representation α of this group on the Hilbert space L ²(ℍ, dxdy/y ²). Supposing that = {M _f : f ∈ L ^∞ (ℍ, dxdy/y ²)}, we show that the crossed product is of type I. In fact, the crossed product is *-isomorphic to the von Neumann algebra , where is the abelian group von Neumann algebra generated by the left regular representation of K. This work was supported by the Youth Foundation of Sichuan Education Department of China (Grant No. 2003B017)     

Itô’s formula for the L p -norm of stochastic {W^{1}_{p}} -valued processes

We prove Itô’s formula for the L _p -norm of a stochastic ${W^{1}_{p}}$ -valued processes appearing in the theory of SPDEs in divergence form.     

A note on the generalization of the summation formulas for Appell's and Kampé de Fériet's hypergeometric functions

In the present paper, a summation formula of a general triple hypergeometric series F⁽³⁾(x, y, z) introduced by Srivastava [10] is obtained. A particular case of this formula corresponds to a result of Shah [7] involving Kampé de Fériet's double hypergeometric function which can further be specialized to yield summation formulas of Srivastava [11] and Bhatt [2] for Appell's function F₂.     

Classical Limit of the Scaled Elliptic Algebra \mathcal{A} ħ,η (\widetilde{\mathfrak{s}\mathfrak{l}}_2)

The classical limit of the scaled elliptic algebra $\mathcal{A}$ ?,η ( $\widetilde{\mathfrak{s}\mathfrak{l}}_2$ ) is investigated. The limiting Lie algebra is described in two equivalent ways: as a central extension of the algebra of generalized automorphic sl2 valued functions on a strip and as an extended algebra of decreasing automorphic sl2 valued functions on the real line. A bialgebra structure and an infinite-dimensional representation in the Fock space are studied. The classical limit of elliptic algebra $\mathcal{A}$ q,p ( $\widetilde{\mathfrak{s}\mathfrak{l}}_2$ ) is also briefly presented.     

On the existence of minimum cubature formulas for Gaussian measure on ?2 of degree t supported by [\frac{t}{4}]+1 circles

In this paper we prove that there exists no minimum cubature formula of degree 4k and 4k+2 for Gaussian measure on ℝ² supported by k+1 circles for any positive integer k, except for two formulas of degree 4.     

A note on Linnik’s approach to the Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions

Abstract—Let χ (mod q), q > 1, be a primitive Dirichlet character. We first present a detailed account of Linnik’s deduction of the functional equation of L(s, χ) from the functional equation of ζ(s). Then we show that the opposite deduction can be obtained by a suitable modification of the method, involving finer arithmetic arguments.     

A note on pépin’s counter examples to the hasse principle for curves of genus 1

In a series of articles published in the C.R. Paris more than a century ago, T. PéPIN announced a list of “theorems” concerning the solvability of diophantine equations of the type ax⁴ +by ⁴ = z². In this article, we show how to prove these claims using the structure of 2-class groups of imaginary quadratic number fields. We will also look at a few related results from a modern point of view.     

A note on the diophantine equation (a n − 1)(b n − 1) = x 2

Let a, b be fixed positive integers such that a ≠ b, min(a, b) > 1, ν(a?1) and ν(b ? 1) have opposite parity, where ν(a ? 1) and ν(b ? 1) denote the highest powers of 2 dividing a ? 1 and b ? 1 respectively. In this paper, all positive integer solutions (x, n) of the equation (a ⁿ ? 1)(b ⁿ ? 1) = x ² are determined.     

Growth properties of the q-Dunkl transform in the space $$L^{p}_{q,\alpha }({\mathbb {R}}_{q},|x|^{2\alpha +1}d_{q}x)$$ L q , α p ( R q , | x | 2 α + 1 d q x )

The Ramanujan Journal - Our aim in this work is to prove an analogue of Titchmarsh’s theorem [19, Theorem 84] and Younis’s theorem [20, Theorem 3.3] on the image under the q-Dunkl...     

Bäcklund’s Theorem for n Dimensional Lorentzian Submanifold in the Minkowski Space {\mathbb{E}_1^2n-1}

In this paper, Bäcklund’s Theorem is introduced on the Lorentzian n-submanifold of the Minkowski space \({\mathbb{E}_{1}^{2n-1}}\) by using the method of moving frames. Also, we prove the Integrability Theorem for the Lorentzian n-submanifold of the Minkowski space \({\mathbb{E}_{1}^{2n-1}}\).     

A note on Green’s relations in the semigroups <Emphasis Type="Italic">T</Emphasis>(<Emphasis Type="Italic">X</Emphasis>,<Emphasis Type="Italic">ρ</Emphasis>)

Let X be a set and the full transformation semigroup on X. Let ρ be an equivalence relation on X and

Positive Solutions of the Equation (t) = −c(t)x(t − τ) in the Critical Case

The equation (t) − c(t)x(t − τ) is considered in the critical case. For it, the asymptotic behavior of dominant and subdominant solutions is studied. A generalization is made and connections with known results are discussed.