‎Highest weight modules of the double affine Lie algebra $widehat{widehat{mathfrak{sl}}}_{2}$ are studied under a‎ ‎new triangular decomposition‎. ‎Singular vectors of Verma modules are‎ ‎determined using a similar condition with horizontal affine Lie‎ ‎subalgebras‎, ‎and highest weight modules are described under the‎ ‎condition $c_1>0$ and $c_2=0$.

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger–Wang, Chau–Weinkove and the author solved this global problem in W4,p under some restricti...

Students of the literature on entailment [7], [1] are familiar with the von Wright-Geach definition of entailment [6], [2]: p entails q if, and only if, by means of logic, it is possible to come to know the truth of •"/> D cp without coming to know the falsehood of p or the truth of q. A competitor of the Lewis account of entailment as strict implication, the von Wright-Geach definition has enj...

Given a finite set P ⊆ Rd, called a pattern, tP (n) denotes the maximum number of translated copies of P determined by n points in Rd. We give the exact value of tP (n) when P is a rational simplex, that is, the points of P are rationally affinely independent. In this case, we prove that tP (n) = n −mr (n), where r is the rational affine dimension of P , and mr (n) is the r-Kruskal-Macaulay fun...

Background: Surgery and accurate removal of the brain tumor in the operating room and after opening the scalp is one of the major challenges for neurosurgeons due to the removal of skull pressure and displacement and deformation of the brain tissue. This displacement of the brain changes the location of the tumor relative to the MR image taken preoperatively. Methods: This study, which is done...

Abstract The compounded Bell generalized class of distributions is proposed in this article as an alternative to the Poisson family distributions. Some properties and actuarial measures are presented. a special model named Weibull (BellW) obtained such linear representation density, r th moment, incomplete moment generating function using Wright hypergeometric Meijer’s G function, p order stati...

In this paper, we define a generalized class of starlike functions which are based upon some convolution operators on Hilbert space involving the Fox-Wright generalization of the classical hypergeometric pFq function (with p numerator and q denominator parameters). The various results presented in this paper include (for example) normed coefficient inequalities and estimates, distortion theorem...

As a follow-up of Haberl-Schuster’s “Asymmetric affine Lp Sobolev inequalities” and Cianchi-Lutwak-Yang-Zhang’s “Affine Moser-Trudinger and Morrey-Sobolev inequalities”, we establish sharp Moser-Trudinger and MorrySobolev inequalities induced by the positive part of a directional derivative on the unit Euclidean sphere. 1. Theorem In their 2009 JFA paper [1], Haberl-Schuster prove the following...

We establish a relation between the generating functions appearing in the S-duality conjecture of Vafa and Witten and geometric Eisenstein series for KacMoody groups. For a pair consisting of a surface and a curve on it, we consider a refined generating function (involving G-bundles with parabolic structures along the curve) which depends on the elliptic as well as modular variables and prove i...