Volume 65, Issue 5

In the spirit of Euler sums we develop a set of identities for finite sums of products of harmonic numbers in higher order and reciprocal binomial coefficients. The new results complement some Euler sums of the type

A cross in Rn is a cluster of unit cubes comprising a central one and 2n arms. In their monograph Algebra and Tiling, Stein and Szabó suggested that tilings of ℝ n by crosses should be studied. The question of the existence of such a tiling has been answered by various authors for many special cases. In this paper we completely solve the problem for ℝ 2 . In fact we do not only characterize crosses for which there exists a tiling of ℝ 2 but for each cross we determine its maximum packing density.

Erdős and Niven proved that for any positive integers m and d, there are only finitely many positive integers n for which one or more of the elementary symmetric functions of 1/m, 1/(m + d), . . . , 1/(m + nd) are integers. In this paper, we show that if n ≥ 2, then none of the elementary symmetric functions of 1, 1/3, . . . , 1/(2n − 1) is an integer

Let R be a prime ring with center Z(R) and extended centroid C, H a non-zero generalized derivation of R and n ≥ 1 a fixed integer. In this paper we study the situations: (1) H(u 2 ) n −H(u) 2n ∈ C for all u ∈ L, where L is a non-central Lie ideal of R; (2) H(u 2 ) n − H(u) 2n = 0 for all u ∈ [I, I], where I is a nonzero right ideal of R.

Let R be a non-commutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, L a non-central Lie ideal of R, G a non-zero generalized derivation of R. If [G(u), u] n = [G(u), u], for all u ∈ L, with n > 1, then one of the following holds: (1) R satisfies the standard identity s 4 (x 1 , . . . , x 4 ) and there exist a ∈ U and α ∈ C such that G(x) = ax + xa + αx for all x ∈ R; (2) there exists γ ∈ C such that G(x) = γx for all x ∈ R.

We study the semigroup I cf λ of injective partial cofinite selfmaps of an infinite cardinal λ. We show that I cf λ is a bisimple inverse semigroup and each chain of idempotents in I cf λ is contained in a bicyclic subsemigroup of I cf λ , we describe the Green relations on I cf λ and we prove that every non-trivial congruence on I cf λ is a group congruence. Also, we describe the structure of the quotient semigroup I cf λ /σ, where σ is the least group congruence on I cf λ .

In this paper using some algebraic inequalities, Hölder inequality and a simple consequence of Keller’s chain rule we prove some new inequalities of Hardy type on a time scale T. These inequalities as special cases contain some integral and discrete inequalities when T = ℝ and T = ℕ.

The concept of the GAP-integral was introduced by the authors [GANGULY, D. K.-MUKHERJEE, R.: The generalized approximate Perron integral, Math. Slovaca 58 (2008), 31-42]. In this paper we prove the controlled convergence theorem for the GAP-integral and deduce other convergence theorems as corollaries.

In this paper we consider the following boundary value problem where f : [0, 1]×ℝ k ×ℝ k → ℝ k and the integral is meant in the sense of Riemann- Stieltjes. We give conditions for the existence of a solution for this boundary value problem using the properties of the Leray-Schauder topological degree. Our result extends some results in the references

In this paper, we consider the third order differential equations with functional arguments by using Riccati inequality

Sufficient conditions are obtained for the asymptotic behavior of solutions of the nonlinear neutral generalized pantograph equation with impulses

We consider Lagrangians for parametric variational problems defined on velocity manifolds and show that a Lagrangian is null precisely when its shadow, a family of vector forms, is closed. We also show that a null Lagrangian can be recovered (to within a constant) from its shadow, and therefore that such a Lagrangian is (again to within a constant) a sum of determinants of total derivatives.

We present in this paper some results for the existence of principal eigenvalues for equations or systems defined in R N involving Schrödinger operators with indefinite weight functions and with potentials which tend to infinity at infinity.

We prove that if a n = ∞ and (a n ) is non-decreasing, then a n = ∞ for any set A ⊂ ℕ of positive lower density. We introduce a Cauchy-like definition of I-convergence of series. We prove that the I-convergence of series coincides with the convergence on large set of indexes if and only if I is a P-ideal. It turns out that I-convergence of series a n implies I-convergence of (a n ) to zero. The converse implication does not hold for analytic P-ideals and it is independent of ZFC that there is I ideal of naturals for which I-convergence of (a n ) to zero implies I-convergence of series a n = ∞ for every sequence (a n ).

The present paper deals with a new type of Bernstein-Durrmeyer operators on mobile interval. First, we represent the operators in terms of hypergeometric series. We also establish local and global approximation results for these operators in terms of modulus of continuity. We obtain an asymptotic formula for these operators and in the last section we present better error estimation for the operators using King type approach

We show that extendible bilinear forms can be represented in an integral form. The representation requires the use of bimeasures. We then study some properties of these extendible bilinear forms and see how they are related to the Grothendieck inequality.

Let Δa,b denote an infinite lower triangular double-band matrix. In this paper, the spectrum, the point spectrum, the continuous spectrum and the residual spectrum of the matrix Δ a,b as a linear operator on the sequence space l p , where 1 ≤ p < ∞ are characterized completely

This paper is devoted to study some topological properties of the SG subgroup, π sg 1 (X, x), of the quasitopological fundamental group of a based space (X, x), π qtop 1 (X, x), its topological properties as a subgroup of the topological fundamental group π τ 1 (X, x) and its influence on the existence of universal covering of X. First, we introduce small generated spaces which have indiscrete topological fundamental groups and also small generated coverings which are universal coverings in the categorical sense. Second, we give a necessary and sufficient condition for the existence of the small generated coverings. Finally, by introducing the notion of semi-locally small generatedness we show that the quasitopological fundamental groups of semi-locally small generated spaces are topological groups.

Let X be a Banach function space over a nonatomic probability space Ω and let M- denote the collection of all uniformly integrable martingales on Ω. For f = (f n ) n∈Z+ ∈M u , let Mf denote the maximal function of f, and let f∞ denote the almost sure limit of f. We give some necessary and sufficient conditions for X to have the property that if f, g ∈M u and ‖Mg‖x ≤ ‖Mf‖x, then ‖g ∞ ‖X ≤ C‖f ∞ ‖x.

A linearization of nonlinear regression models is frequently used in practice. A recognition whether it can be done need some additional investigation. Some problems occur when constraints are involved in the model mainly when the model is singular. One possible solution is given in the paper.

Closedness of the solution map is investigated for a sequence of parametric inequality problems related to a “limit” problem governed by a pseudomonotone bifunction. The main result gives sufficient conditions for closedness of the solution map defined on the set of parameters.

The objective of this short note is to employ an equation error approach to identify a variable parameter in fourth-order partial differential equations. Existence and convergence results are given for the optimization problem emerging from the equation error formulation. Finite element based numerical experiments show the effectiveness of the proposed framework.