On asymptotics of the q-exponential and q-gamma functions

In this work we present a derivation for the complete asymptotic expansions of Euler?s q-exponential function and Jackson?s q-gamma function via Mellin transform. These formulas are valid everywhere, uniformly on any compact subset of the complex plane.     

Krull’s theory for the double gamma function

The Barnes’ G-function G(x) = 1/Γ₂, satisfies the functional equation logG(x + 1) − logG(x) = logΓ(x). We complement W. Krull’s work in Bemerkungen zur Differenzengleichung g(x + 1) − g(x) = φ(x), Math. Nachrichten 1 (1948), 365-376 with additional results that yield a different characterization of the function G, new expansions and sharp bounds for G on x > 0 in terms of Gamma and Digamma functions, a new expansion for the Gamma function and summation formulae with Polygamma functions.     

Omega theorems related to the general Euler totient function

We prove an omega estimate related to the general Euler totient function associated to a polynomial Euler product satisfying some natural analytic properties. For convenience, we work with a set of L-functions similar to the Selberg class, but in principle our results can be proved in a still more general setup. In a recent paper the authors treated a special case of Dirichlet L-functions with real characters. Greater generality of the present paper invites new technical difficulties. Effectiveness of the main theorem is illustrated by corollaries concerning Euler totient functions associated to the shifted Riemann zeta function, shifted Dirichlet L-functions and shifted L-functions of modular forms. Results are either of the same quality as the best known estimates or are entirely new.     

Inequalities for the double gamma function

We establish various new upper and lower bounds in terms of the classical gamma and digamma functions for the double gamma function (or Barnes G-function).     

Transference principles for semigroups and a theorem of Peller

A general approach to transference principles for discrete and continuous operator (semi)groups is described. This allows one to recover the classical transference results of Calderón, Coifman and Weiss and of Berkson, Gillespie and Muhly and the more recent one of the author. The method is applied to derive a new transference principle for (discrete and continuous) operator semigroups that need not be groups. As an application, functional calculus estimates for bounded operators with at most polynomially growing powers are derived, leading to a new proof of classical results by Peller from 1982. The method allows for a generalization of his results away from Hilbert spaces to Lp-spaces and—involving the concept of γ-boundedness—to general Banach spaces. Analogous results for strongly-continuous one-parameter (semi)groups are presented as well. Finally, an application is given to singular integrals for one-parameter semigroups.     

Weak convergence of a projection algorithm for variational inequalities in a Banach space

Let C be a nonempty, closed convex subset of a Banach space E. In this paper, motivated by Alber [Ya.I. Alber, Metric and generalized projection operators in Banach spaces: Properties and applications, in: A.G. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, in: Lecture Notes Pure Appl. Math., vol. 178, Dekker, New York, 1996, pp. 15-50], we introduce the following iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly-monotone operator A in a Banach space: x₁=x∈C and

xn+1=ΠCJ⁻¹(Jxn−λnAxn)     

Property (gb) and perturbations

An operator T acting on a Banach space X possesses property (gb) if , where σa(T) is the approximate point spectrum of T, is the essential semi-B-Fredholm spectrum of T and π(T) is the set of all poles of the resolvent of T. In this paper we study property (gb) in connection with Weyl type theorems, which is analogous to generalized Browder?s theorem. Several sufficient and necessary conditions for which property (gb) holds are given. We also study the stability of property (gb) for a polaroid operator T acting on a Banach space, under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic and Riesz operators commuting with T.     

Characterizing sequences for precompact group topologies

Motivated from [31], call a precompact group topology τ on an abelian group G ss-precompact (abbreviated from single sequence precompact  ) if there is a sequence u=(_un)$\mathbf{u}=(u_n)$ in G such that τ is the finest precompact group topology on G   making u=(_un)$\mathbf{u}=(u_n)$ converge to zero. It is proved that a metrizable precompact abelian group (G,τ)$(G,\tau )$ is ss-precompact iff it is countable. For every metrizable precompact group topology τ on a countably infinite abelian group G there exists a group topology η such that η is strictly finer than τ   and the groups (G,τ)$(G,\tau )$ and (G,η)$(G,\eta )$ have the same Pontryagin dual groups (in other words, (G,τ)$(G,\tau )$ is not a Mackey group in the class of maximally almost periodic groups).     

Weak semi-continuity of the duality product in Sobolev spaces

Given a weakly convergent sequence of positive functions in , we prove the equivalence between its convergence in the sense of obstacles and the lower semi-continuity of the term by term duality product associated to (the p-Laplacian of) weakly convergent sequences of p-superharmonic functions of . This result implicitly gives new characterizations for both the convergence in the sense of obstacles of a weakly convergent sequence of positive functions and for the weak l.s.c. of the duality product.     

On the generalized self-similar singularities for the Euler and the Navier-Stokes equations

In this paper we study blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the Euler and the Navier-Stokes equations, where the sense of convergence and self-similarity are considered in various generalized senses. We improve substantially, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities. Generalization of the self-similar transforms is also considered, and by appropriate choice of the parameterized transform we obtain new a priori estimates for the Euler and the Navier-Stokes equations depending on a free parameter.     

A double generalized Pareto distribution

Papastathopoulos and Tawn [Papastathopoulos, I., Tawn, J.A., 2013. A generalized Student’s t$t$-distribution. Statistics & Probability Letters 83, 70–77] proposed a generalization of Student’s t$t$ distribution to account for negative degrees of freedom. Here, an alternative distribution that has simpler mathematical properties is discussed. Several advantages are established for using the alternative distribution over Papastathopoulos and Tawn’s generalization.     

均质积分Brunn-Minkowski不等式的概率形式

本文运用两种不同的方法建立了一个L_p 空间中关于Firey 组合的强大数定理. 进一步, 本文获得了均质积分Brunn-Minkowski 不等式的概率形式.     

On a family of symmetric hypergeometric functions of several variables and their Euler type integral representation

This paper is devoted to the family {_Gn}$\{G_n\}$ of hypergeometric series of any finite number of variables, the coefficients being the square of the multinomial coefficients (_?1+?+_?n)!/(_?1!…_?n!)$({?}_1+?+{?}_n)!/({?}_1!\dots {?}_n!)$, where n∈_Z?1$n\in {\mathbb{Z}}_{?1}$. All these series belong to the family of the general Appell–Lauricella?s series. It is shown that each function _Gn$G_n$ can be expressed by an integral involving the previous one, _Gn−1$G_{n-1}$. Thus this family can be represented by a multidimensional Euler type integral, what suggests some explicit link with the Gelfand–Kapranov–Zelevinsky?s theory of A  -hypergeometric systems or with the Aomoto?s theory of hypergeometric functions. The quasi-invariance of each function _Gn$G_n$ with regard to the action of a finite number of involutions of C^?n${\mathbb{C}}^{?n}$ is also established. Finally, a particular attention is reserved to the study of the functions _G2$G_2$ and _G3$G_3$, each of which is proved to be algebraic or to be expressed by the Legendre?s elliptic function of the first kind.     

On a parabolic logarithmic Sobolev inequality

In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi [H. Kozono, Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys. 214 (2000) 191-200] have proved a logarithmic Sobolev inequality by means of isotropic (elliptic) BMO norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of anisotropic (parabolic) BMO norm. More precisely we give an upper bound for the L^∞ norm of a function in terms of its parabolic BMO norm, up to a logarithmic correction involving its norm in some Sobolev space. As an application, we also explain how to apply this inequality in order to establish a long-time existence result for a class of nonlinear parabolic problems.     

A parallel projection method for a system of nonlinear variational inequalities

In this paper, we introduce and consider a new system of nonlinear variational inequalities involving two different operators. Using the parallel projection technique, we suggest and analyze an iterative method for this system of variational inequalities. We establish a convergence result for the proposed method under certain conditions. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.