and Applied Analysis 3 2. Preliminaries Let w α,β x 1 − x α 1 x β be a weight function in the usual sense for α, β > −1. The set of Jacobi polynomials {P α,β k x } k 0 forms a complete L 2 w α,β −1, 1 -orthogonal system, and ∥ ∥ ∥P α,β k ∥ ∥ ∥ 2 w α,β h α,β k 2 β 1Γ k α 1 Γ ( k β 1 ) ( 2k α β 1 ) Γ k 1 Γ ( k α β 1 ) . 2.1 Here, L2 w α,β −1, 1 is a weighted space defined by L2 w α,β −1, 1 {v : v...

The Moser-Trudinger embedding has been generalized in Adimurthi and Sandeep (2007) [2] to the following weighted version: if Ω⊂Rn is bounded, ωn−1 Hn−1 measure of unit sphere, then for α>0 β∈[0,n),supu∈B1⁡∫Ωeα|u|n/(n−1)|x|β≤C⇔ααn+βn≤1, where αn=nωn−11/(n−1) B1={u∈W01,n(Ω)|∫Ω|∇u|n≤1}. We prove that supremum attained on any domain Ω. paper also fills gaps proof Lin (1996) [21], which deals with c...

In this paper, by using new auxiliary functions, we study a class of contracting flows closed, star-shaped hypersurfaces in Rn+1 with speed rαβσk1β, where σk is the k-th elementary symmetric polynomial principal curvatures, α, β are positive constants and r distance from points on hypersurface to origin. We obtain convergence results under some assumptions k, β. When k≥2, 0<β≤1, α≥β+k, prove th...

Classical random matrix ensembles with orthogonal symmetry have the property that the joint distribution of every second eigenvalue is equal to that of a classical random matrix ensemble with symplectic symmetry. These results are shown to be the case r = 1 of a family of interrelations between eigenvalue probability density functions for generalizations of the classical random matrix ensembles...

Geodesically isotropic positive definite functions on compact two-point homogeneous spaces of dimension d have series representation as members weighted Lebesgue L1w([−1,1]), where the weight w(x)=wα,β(x)=(1−x)α(1+x)β is one related to Jacobi orthogonal polynomials P(α,β)(x) in [−1,1], and exponents α β are d. We derive some recurrence relations among coefficients representations under differen...

Abstract Despite the richness and biodiversity of invertebrates algae in Mediterranean Sea, these organisms are still poorly studied. The objective our research is discovery bioactive lead compounds from Sea sponge Hemimycale Collumella (HC). HC (189.0 g) was collected Mdiq costs on extracted with methanol to yield (10 which then subjected fractionation. A bio-guided protocol applied through ev...

Let f and g be holomorphic cusp forms for the modular group SL2(Z) of weight k1 k2 with Fourier coefficients λf(n) λg(n), respectively. For real α≠0 0<β≤1, consider a smooth resonance sum SX(f,g;α,β) λf(n)λg(n) against e(αnβ) over X≤n≤2X. Double square moments both are nontrivially bounded when their weights tend to infinity together. By allowing move, these double indeed associated automorphic...