On the riesz charge of the lower envelope of δ-subharmonic functions

By potential theoretic methods involving the Cartan fine topology a recent result by two of the authors is extended as follows: The Riesz charge of the lower envelope of a family of 3 or more -subharmonic functions (no longer supposed continuous) in the plane equals the infimum of the charges of the lower envelopes of all pairs of functions from the family. As a key to this it is shown in two different ways that the (fine) harmonic measures of any 3 pairwise disjoint finely open planar sets have Borel supports with empty intersection. One proof of this uses the Jordan curve theorem and the fact that the set of inaccessible points of the fine boundary of a fine domain is Borel and has zero harmonic measure; the other involves Carleman-Tsuji type estimates together with a fine topology version of a recent result of P. Jones and T. Wolff on harmonic measure and Hausdorff dimension.     

Some properties of the psi function and evaluations of γ

In this paper, the authors show some monotonicity and concavity of the classical psi function, by which several known results are improved and some new asymptotically sharp estimates are obtained for this function. In addition, applying the new results to the psi function,the authors improve the well-known lower and upper bounds for the approximate evaluation of Euler's constant γ.     

On the differential properties of the support function of the∈-subdifferential of a convex function

We study differential properties of the support function of the-subdifferential of a convex function; applications in algorithmics are also given.     

Some discrete Fourier transform pairs associated with the Lipschitz–Lerch Zeta function

It is shown that there exists a companion formula to Srivastava’s formula for the Lipschitz–Lerch Zeta function [see H.M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000) 77–84] and that together these two results form a discrete Fourier transform pair. This Fourier transform pair makes it possible for other (known or new) results involving the values of various Zeta functions at rational arguments to be easily recovered or deduced in a more general context and in a remarkably unified manner.     

Some properties of polynomials orthogonal in a space with “intermediate” topology. Kernel function and extremal properties

A space of holomorphic functions is considered. A topology in this space is intermediate between the topology of uniform convergence and the topology of uniform convergence on compact sets. The properties of systems of orthonormal polynomials are studied in Hilbert spaces with this topology.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 915–923, July, 1993.     

Lower types of δ-subharmonic functions of fractional order

It is proved that the lower types of functions T(r, u) and N(r, u)=N(r, u₁)+N(z, u₂) relative to the proximate order (r) of a function u=U₁–u₂ of fractional order -subharmonic in ^m, m>- 2, coincide, that is, are simultaneously minimal or mean. In the case of an arbitrary proximate order (r), the assertion is, in general, false.Translated from Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1280–1284, September, 1992.     

On the Möbius function of the locally finite poset associated with a numerical semigroup

Let S be a numerical semigroup and let (?,≤ _S ) be the (locally finite) poset induced by S on the set of integers ? defined by x≤ _S y if and only if y?x∈S for all integers x and y. In this paper, we investigate the Möbius function associated to (?,≤ _S ) when S is an arithmetic semigroup.     

Explicit calculation of the partition function of a one-dimensional δ-function bose gas

We present a method to calculate explicitly the partition function of a one-dimensional δ-function bose gas. The cluster integrals are obtained for up to the four particles and are shown to be the same as those by the thermal Bethe ansatz method (TBA).     

Some properties of solutions of a hyperbolic modification of the Bürgers equation

The results of dispersion analysis of the equation and relevant computer-assisted experiments are presented. The existence of solutions with sharpenings (collapses) and solutions of oscillatory type is discovered. Bibliography: 6 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 13–18.     

Integral means of δ-subharmonic functions and classes of entirely regular growth

Classes of functions of finite -type -subharmonic in m are investigated. A membership criterion that defines when a function belongs to the class expressed in terms of its integral q-means is established. Classes of entirely regular growth of functions that are -subharmonic in ^m are introduced and the behavior of these functions at infinity and the distribution of Riesz-associated measures are studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 408–416, March, 1992.     

Some properties of the Schröder numbers

In the paper, the authors present some properties, including convexity, complete monotonicity, product inequalities, and determinantal inequalities, of the large Schröder numbers and find three relations between the Schröder numbers and central Delannoy numbers. Moreover, the authors sketch generalizing the Schröder numbers and central Delannoy numbers and their generating functions.     

Some properties of the family Γ of modular Lie superalgebras

In this paper, we continue to investigate some properties of the family Γ of finite-dimensional simple modular Lie superalgebras which were constructed by X. N. Xu, Y. Z. Zhang, L. Y. Chen (2010). For each algebra in the family, a filtration is defined and proved to be invariant under the automorphism group. Then an intrinsic property is proved by the invariance of the filtration; that is, the integer parameters in the definition of Lie superalgebras Γ are intrinsic. Thereby, we classify these Lie superalgebras in the sense of isomorphism. Finally, we study the associative forms and Killing forms of these Lie superalgebras and determine which superalgebras in the family are restrictable.     

Analytic properties of the Green’s function for the equation of internal gravitational waves in a stratified medium with shear flows

Theoretical and Mathematical Physics - We discuss the problem of constructing the Green’s function for the equation for linear internal gravity waves in a finite layer of a stratified medium...     

Some properties of extended remainder of binet’s first formula for logarithm of gamma function

In the paper, we extend Binet’s first formula for the logarithm of the gamma function and investigate some properties, including inequalities, star-shaped and sub-additive properties and the complete monotonicity, of the extended remainder of Binet’s first formula for the logarithm of the gamma function and related functions.