Volume 50, Issue 1

Let X be a set in ℝ n with positive Lebesgue measure. It is well known that the spectrum of the algebra L ∞ (X) of (equivalence classes) of essentially bounded, complex-valued, measurable functions on X is an extremely disconnected compact Hausdorff space.We show, by elementary methods, that the spectrum M of the algebra ℒ b (X, ℂ) of all bounded measurable functions on X is not extremely disconnected, though totally disconnected. Let ∆ = { δ x : x ∈ X} be the set of point evaluations and let g be the Gelfand topology on M. Then (∆, g) is homeomorphic to (X, Τ dis ),where T dis is the discrete topology. Moreover, ∆ is a dense subset of the spectrum M of ℒ b (X, ℂ). Finally, the hull h(I), (which is homeomorphic to M(L ∞ (X))), of the ideal of all functions in ℒ b (X, ℂ) vanishing almost everywhere on X is a nowhere dense and extremely disconnected subset of the Corona M \ ∆ of ℒ b (X, ℂ).

In this paper we investigate the asymptotic behaviour of the classical continuous and unbounded almost periodic function in the Lebesgue measure.Using diophantine approximations we show that this function can be estimated by functions of polynomial type and we give the best polynomial estimation.

In this paper,we study the properties of the multiplication operator acting on the bounded variation space BV[0, 1]. In particular,we show the existence of non-null compact multiplication operators on BV[0, 1] and non-invertible Fredholm multiplication operators on BV[0, 1].

In this note we point out errors in the proofs of first two theorems contained in Demonstratio Mathematica, Vol. XLVI, no 2, 2013, 405-413. We also rectify the erratic theorems by employing a proper setting.

The present paper deals with the rate of convergence of the general class of Durrmeyer operators, which are generalization of Ibragimov-Gadjiev operators. The special cases of the operators include somewell known operators as particular cases viz. Szász-Mirakyan-Durrmeyer operators, Baskakov-Durrmeyer operators. Herewe estimate the rate of convergence of Ibragimov-Gadjiev-Durrmeyer operators for functions having derivatives of bounded variation.

The purpose of this paper is to establish the rate of convergence in terms of the weighted modulus of continuity and Lipschitz type maximal function for the q-Szász-beta operators. We also study the rate of A-statistical convergence. Lastly, we modify these operators using King type approach to obtain better approximation.

In this paper, we study some local approximation properties of generalized Baskakov-Durrmeyer-Stancu operators. First, we establish a recurrence relation for the central moments of these operators, then we obtain a local direct result in terms of the second order modulus of smoothness. Further, we study the rate of convergence in Lipschitz type space and the weighted approximation properties in terms of the modulus of continuity, respectively. Finally, we investigate the statistical approximation property of the new operators with the aid of a Korovkin type statistical approximation theorem.

We deal with the approximation properties of a new class of positive linear Durrmeyer type operators which offer a reconstruction of integral type operators including well known Durrmeyer operators. This reconstruction allows us to investigate approximation properties of the Durrmeyer operators at the same time. It is first shown that these operators are a positive approximation process in Lp(ℝ + ) . While we are showing this property of the operators we consider the Ditzian-Totik modulus of smoothness and corresponding K-functional. Then, weighted norm convergence, whose proof is based on Korovkin type theorem on Lp(ℝ + ), is given. At the end of the paper we show several examples of classical sequences that can be obtained from the Ibragimov-Gadjiev-Durrmeyer operators.

In this paper, we introduce the q-analogue of the Jakimovski-Leviatan type modified operators introduced by Atakut with the help of the q-Appell polynomials.We obtain some approximation results via the well-known Korovkin’s theorem for these operators.We also study convergence properties by using the modulus of continuity and the rate of convergence of the operators for functions belonging to the Lipschitz class. Moreover,we study the rate of convergence in terms of modulus of continuity of these operators in aweighted space.

The purpose of this paper is to prove some fixed point results using JS-G-contraction on G-metric spaces, also to prove some fixed point results on G b -complete metric space for a new contraction. Our results extend and improve some results in the literature. Moreover, some examples are presented to illustrate the validity of our results.

Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued Fréchet differentiable functions to the unit operator, as well as other basic approximations including those under convexity. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators for which we study their approximation properties. We derive pointwise and uniform estimates, which imply the approximation of these operators to the unit assuming Fréchet differentiability of functions, and then we continue with basic approximations. At the end we study the special case where the approximated function fulfills a convexity condition resulting into sharp estimates. We give applications to Bernstein operators.

We discuss, extend, improve and enrich results on simulation functions established by several authors. Furthermore, by using Lemma 2.1 of Radenovic et al. [Bull. Iran. Math. Soc., 2012, 38, 625],we get much shorter and nicer proofs than the corresponding ones in the existing literature.

In this work we consider a class of contact manifolds (M, η) with an associated almost contact metric Structure (ϕ, ξ, η, g). This class contains, for example, nearly cosymplectic manifolds and the manifolds in the class C9 ⊕ C10 defined by Chinea and Gonzalez. All manifolds in the class considered turn out to have dimension 4n + 1. Under the assumption that the sectional curvature of the horizontal 2-planes is constant at one point, we obtain that these manifolds must have dimension 5.

We show that Müntz spaces, as subspaces of C[0, 1], contain asymptotically isometric copies of c 0 and that their dual spaces are octahedral.

The relation between derivatives of a polynomial of best approximation and the best approximation of the function is investigated in generalized Lebesgue spaces with variable exponent. In addition, the relationship between the fractional modulus of smoothness of the function and its de la Vallée-Poussin sums is studied.

This note exhibits a Banach-space operator such that neither the range nor the kernel is complemented both for the operator and its adjoint

Based on the existence of discrete Lp− subharmonic functions, a classification of infinite networks is carried out.

We consider a bounded linear operator A in a Hilbert space with a Hilbert-Schmidt Hermitian component (A − A*)/2i. A sharp norm estimate is established for functions of A nonregular on the convex hull of the spectrum. The logarithm, fractional powers and meromorphic functions of operators are examples of such functions. Our results are based on the existence of a sequence An (n = 1, 2, ...) of finite dimensional operators strongly converging to A, whose spectra belongs to the spectrum of A. Besides, it is shown that the resolvents and holomorphic functions of An strongly converge to the resolvent and corresponding function of A.

The aim of this paper is to introduce generalized condition (B) in a quasi-partial metric space acknowledging the notion of Künzi et al. [Künzi H.-P. A., Pajoohesh H., Schellekens M. P., Partial quasi-metrics, Theoret. Comput. Sci., 2006, 365, 237-246] and Karapinar et al. [Karapinar E., Erhan M.,Öztürk A., Fixed point theorems on quasi-partial metric spaces, Math. Comput.Modelling, 2013, 57, 2442-2448] and to establish coincidence and common fixed point theorems for twoweakly compatible pairs of self mappings. In the sequelwe also answer affirmatively two open problems posed by Abbas, Babu and Alemayehu [Abbas M., Babu G. V. R., Alemayehu G. N., On common fixed points of weakly compatible mappings satisfying generalized condition (B), Filomat, 2011, 25(2), 9-19]. Further in the setting of a quasi-partial metric space, the results obtained are utilized to establish the existence and uniqueness of a solution of the integral equation and the functional equation arising in dynamic programming. Our results are also justified by explanatory examples supported with pictographic validations to demonstrate the authenticity of the postulates.

A new property called P β -connectedness is introduced which is stronger than connectedness and equivalent to pre-connectedness. The properties of this notion are explored and its relationship with other forms of connectedness, for example hyperconnectedness etc. are discussed. Locally pre-indiscrete spaces are defined as the spaces in which pre-open sets are closed. In such spaces connectedness becomes equivalent to pre-connectedness and hence to P β -connectedness, and semi-connectedness becomes equivalent to P β - connectedness. The notion of locally P β -connected space is introduced. The behavior of P β -connectedness under several types of mappings is investigated. An intermediate value theorem is obtained.

In this paper, we shall establish sufficient conditions for the existence of solutions for second order semilinear functional evolutions equation with nonlocal conditions in Fréchet spaces. Our approach is based on the concepts of Hausdorff measure, noncompactness and Tikhonoff’s fixed point theorem. We give an example for illustration.

We study the existence of an almost periodic solution of discrete Volterra systems by means of fixed point theory. Using discrete variant of exponential dichotomy, we provide sufficient conditions for the existence of an almost periodic solution. Hence, we provide an alternative solution for the open problem proposed in the literature.

Given a Hermitian matrix M ∈ M 3 (ℂ) we describe explicitly the real diagonal matrices D M such that ║M + DM║ ≤ ║M + D║ for all real diagonal matrices D ∈ M 3 (ℂ), where ║ · ║ denotes the operator norm. Moreover, we generalize our techniques to some n × n cases.

The results concerninig pointwise approximation and product of summability methods corresponding to the theorems of Xh. Z. Krasniqi [Poincare J. Anal. Appl., 2014, 1, 1-8] and W. Łenski and B. Szal [Math. Slovaca, 2016, 66(6), 1-12] are generalized. Some special cases are also formulated as corollaries.

In this paper we prove the existence and uniqueness of coincident (fixed) points for nonlinear mappings of any number of arguments under a (ψ, θ, φ)-weak contraction condition without O-compatibility. The obtained results extend, improve and generalize some well-known results in the literature to be discussed below. Moreover, we present an example to show the efficiency of our results.

The aim of this paper is to explain the contributions of G. Lemaître to Spinor Theory. At the end of the paper, we edited also, for the first time two short manuscripts: Spineurs et Quanta and Les spineurs et la physique quantique, written by Lemaître in December 1955 and in January 1956. This edition is a way of honouring Professor Michael Heller because he was the first, with Professor Odon Godart, who discovered, classified and published unedited manuscripts of G. Lemaître.

Inspired by the work of Heller and Sasin [1], we construct in this paper Weil homomorphism in a locally free sheaf W of ϕ-fields [2] over a structured space. We introduce the notion of G-consistent, linear connection on this sheaf, what allows us to clearly define Chern, Pontrjagin and Euler characteristic classes. We also show proper equalities between those classes.

We study the representations of transitive transformation groupoids with the aim of generalizing the Mackey theory. Using the Mackey theory and a bijective correspondence between the imprimitivity systems and the representations of a transformation groupoid we derive the irreducibility theory. Then we derive the direct sum decomposition for representations of a groupoid together with the formula for the multiplicity of subrepresentations. We discuss a physical interpretation of this formula. Finally, we prove the claim analogous to the Peter-Weyl theorem for a noncompact transformation groupoid. We show that the representation theory of a transitive transformation groupoids is closely related to the representation theory of a compact groups.

We prove the asymptotic global stability of the de Sitter solution in the Friedmann-Robertson-Walker conservative and dissipative cosmology. In the proof we construct a Lyapunov function in an exact form and establish its relationship with the first integral of dynamical system determining evolution of the flat Universe. Our result is that de-Sitter solution is asymptotically stable solution for general form of equation of state p = (ρ, H), where dependence on the Hubble function H means that the effect of dissipation are included.

The famous „twin paradox” of special relativity is of purely geometric nature and formulated in curved spacetimes of general relativity motivates investigations of the timelike geodesic structure of these manifolds. Except for the maximally symmetric spacetimes the search for the longest timelike curves is hard, complicated and requires both advanced methods of global Lorentzian geometry and solving the intricate geodesic deviation equation. This article is a theoretical introduction to the problem. First we describe the procedure of determining the locally longest curves; it is algorithmic in the sense of consisting of a small number of definite steps and is effective if the geodesic deviation equation may be solved. Then we discuss the problem of globally maximal timelike curves; due to its nonlocal nature there is no prescription of how to solve it in finite number of steps. In the case of sufficiently high symmetry of the manifold also the globally longest curves may be found. Finally we briefly present some results recently found.

We investigate examples of Gauss-Bonnet theorem and the scalar curvature for the two-dimensional commutative sphere with quasi-spectral triples obtained by modifying the order-one condition.

Recent renewed interest in Sasakian manifolds is due mainly to the fact that they can provide examples of generalized Einstein manifolds, manifolds which are of great interest in mathematical models of various aspects of physical phenomena. Sasakian manifolds are odd dimensional counterparts of Kählerian manifolds to which they are closely related. The paper presents a foliated approach to Sasakian manifolds on which the author gave several lectures. The paper concentrates on cohomological properties of Sasakian manifolds and of transversely holomorphic and Kählerian foliations. These properties permit to formulate obstructions to the existence of Sasakian structures on compact manifolds.

Noncommutative or ‘quantum’ differential geometry has emerged in recent years as a process for quantizing not only a classical space into a noncommutative algebra (as familiar in quantum mechanics) but also differential forms, bundles and Riemannian structures at this level. The data for the algebra quantisation is a classical Poisson bracket while the data for quantum differential forms is a Poisson-compatible connection. We give an introduction to our recent result whereby further classical data such as classical bundles, metrics etc. all become quantised in a canonical ‘functorial’ way at least to 1st order in deformation theory. The theory imposes compatibility conditions between the classical Riemannian and Poisson structures as well as new physics such as typical nonassociativity of the differential structure at 2nd order. We develop in detail the case of ℂℙ n where the commutation relations have the canonical form [w i , w̄ j ] = iλδ ij similar to the proposal of Penrose for quantum twistor space. Our work provides a canonical but ultimately nonassociative differential calculus on this algebra and quantises the metric and Levi-Civita connection at lowest order in λ.