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In this paper we deduce some tight Turán type inequalities for Tricomi confluent hypergeometric functions of the second kind, which in some cases improve the existing results in the literature. We also give alternative proofs for some already established Turán type inequalities. Moreover, by using these Turán type inequalities, we deduce some new inequalities for Tricomi confluent hypergeometric functions of the second kind. The key tool in the proof of the Turán type inequalities is an integral representation for a quotient of Tricomi confluent hypergeometric functions, which arises in the study of the infinite divisibility of the Fisher-Snedecor F distribution.

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A semigroup is called eventually regular if each of its elements has a regular power. In this paper we study certain fundamental congruences on an eventually regular semigroup. We generalize some results of Howie and Lallement (1966) and LaTorre (1983). In particular, we give a full description of the semilattice of group congruences (together with the least such a congruence) on an arbitrary eventually regular (orthodox) semigroup. Moreover, we investigate UBG-congruences on an eventually regular semigroup. Finally, we study the eventually regular subdirect products of an E-unitary semigroup and a Clifford semigroup.

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The set of Cohen-Macaulay monomial ideals with a given radical contains the so-called Cohen-Macaulay modifications. Not all Cohen-Macaulay squarefree monomial ideals admit nontrivial Cohen-Macaulay modifications. We present classes of Cohen-Macaulay squarefree monomial ideals with infinitely many nontrivial Cohen-Macaulay modifications.

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The Bn (k) poly-Bernoulli numbers — a natural generalization of classical Bernoulli numbers (B n = Bn (1)) — were introduced by Kaneko in 1997. When the parameter k is negative then Bn (k) is a nonnegative number. Brewbaker was the first to give combinatorial interpretation of these numbers. He proved that Bn (−k) counts the so called lonesum 0–1 matrices of size n × k. Several other interpretations were pointed out. We survey these and give new ones. Our new interpretation, for example, gives a transparent, combinatorial explanation of Kaneko’s recursive formula for poly-Bernoulli numbers.

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Over the (1, 1)-dimensional real supercircle, we consider the K(1)-modules D λ,μ k of linear differential operators of order k acting on the superspaces of weighted densities, where K(1) is the Lie superalgebra of contact vector fields. We give, in contrast to the classical setting, a classification of these modules. This work is the simplest superization of a result by Gargoubi and Ovsienko.

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A ring R is called right SSP (SIP) if the sum (intersection) of any two direct summands of R R is also a direct summand. Left SSP (SIP) rings are defined similarly. There are several interesting results on rings with SSP. For example, R is right SSP if and only if R is left SSP, and R is a von Neumann regular ring if and only if Mn(R) is SSP for some n > 1. It is shown that R is a semisimple ring if and only if the column finite matrix ring ℂFM(R) is SSP, where ℕ is the set of natural numbers. Some known results are proved in an easy way through idempotents of rings. Moreover, some new results on SSP rings are given.

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Let G be a finite group. A subgroup H of G is said to be s-permutable in G if H permutes with all Sylow subgroups of G. Let H be a subgroup of G and let H sG be the subgroup of H generated by all those subgroups of H which are s-permutable in G. A subgroup H of G is called n-embedded in G if G has a normal subgroup T such that H G = HT and HTH sG, where H G is the normal closure of H in G. We investigate the influence of n-embedded subgroups of the p-nilpotency and p-supersolvability of G.

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In 1944, Santaló asked about the average number of normals through a point of a given convex body. Since then, numerous results appeared in the literature about this problem. The aim of this paper is to add to this list some new, recent developments. We point out connections of the problem to static equilibria of rigid bodies as well as to geometric partial differential equations of surface evolution.

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S. Banach in [1] proved that for any function fL 2(0, 1), f ≁ 0, there exists an ONS (orthonormal system) such that the Fourier series of this function is not summable a.e. by the method (C, α), α > 0.

D. Menshov found the conditions which should be satisfied by the Fourier coefficients of the function for the summability a.e. of its Fourier series by the method (C, α), α > 0.

In this paper the necessary and sufficient conditions are found which should be satisfied by the ONS functions (φ n(x)) so that the Fourier coefficients (by this system) of functions from class Lip 1 or A (absolutely continuous) satisfy the conditions of D. Menshov.

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Motivated by the well known Kadec-Pełczynski disjointification theorem, we undertake an analysis of the supports of non-zero functions in strongly embedded subspaces of Banach functions spaces. The main aim is to isolate those properties that bring additional information on strongly embedded subspaces. This is the case of the support localization property, which is a necessary condition fulfilled by all strongly embedded subspaces. Several examples that involve Rademacher functions, the Volterra operator, Lorentz spaces or Orlicz spaces are provided.

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