The concept of measures of noncompactness for operators has been introduced and thoroughly investigated by several authors (see, for instance, [2, 4, 8]). In this work, we extend this concept to the setting of multivalued operators. We employ an axiomatic framework to define measures of noncompactness for bounded linear relations, including those associated with upper semi-Fredholm perturbations. Leveraging these measures, we derive new results for certain classes of semi-Fredholm and Fredholm multivalued operators. Lastly, we establish localization results concerning some essential spectra of bounded linear relations on Banach spaces.
We consider the numerical solution of a ordinary differential equation with a discontinuity in its derivative. Specifically, we assume that the location of the discontinuity has been detected, but that a numerical error exists in this location. We study how this error propagates in the subsequent numerical solution beyond the discontinuity. We derive an analytical expression for this error, which exposes terms that can only be controlled, if possible, prior to the advent of the discontinuity. We also find a condition number that can amplify the location error significantly. An example illustrates our findings.
The concept of a trigonometrically convex function consists in replacing the linear weights inherent in the classical concept of convexity with trigonometric ones. It emerged in 2018, combined with a framework of integral inequalities. In this article, we innovate by establishing valuable bounds for the integral of the product of two functions, including at least one such trigonometrically convex function. To derive these bounds, we use a number of analytical tools, including sharp trigonometric inequalities and the Hermite–Hadamard, Chebyshev and Hölder integral inequalities. Our results are flexible; they can accommodate different convexity conditions. Detailed proofs are given for completeness.
A computational error was found in the above-mentioned paper. Although the error seemed minor at first and did not appear to alter the main conclusions, a more careful re-evaluation has uncovered additional insights into the structure and properties of rotatable 2 × 2 matrices.
This paper presents closed-form evaluations of two new Apéry-like series of weight 4 that involve harmonic numbers of the form H_2k. Several key results are derived and subsequently used to establish connections to the main series.
Various classes of convergent sequences, whose terms are characterized by linear combinations of harmonic number expressions with power function arguments, are formulated and used to obtain limit and integral expressions for Euler’s constant 𝛾. The sequences can be used to derive a known two parameter family of integral expressions for 𝛾, and their structure motivates an extensive generalization of these integrals, which are expressible in terms of parameter spaces of arbitrarily large dimension. Additionally, using inequalities derived from a digamma function identity, pairs of rational sequences are formed in which one member of the pair is an ascending lower bound for Euler’s Constant and the other a descending upper bound for 𝛾.
This paper presents explicit evaluations for a wide class of linear Euler-type sums containing four parameters, and related polylogarithmic integrals. These evaluations are given in terms of the Lerch transcendent function, which reduces to the Dirichlet beta function and Riemann zeta function. The identities that we derive in this work generalize many previously known results in the literature. Our derivations rely exclusively on real-analysis techniques.
The purpose of this article is to continue our studies on multiple 𝑞-hypergeometric functions by proving more 𝑞-integrals over the unit hypercube in dimension 𝑛 for 𝑞-Lauricella functions in the spirit of Koschmieder. Each time, the corresponding 𝑞-Appell function integral formulas after Feldheim are stated. We also introduce some new 𝑞-Humbert functions and prove integral representations for them. In an earlier paper we found 𝑞-integrals for all confluent double 𝑞-functions with the same parameters in the 𝑞- Beta function as in the original confluent function. This time confluent forms in another form are proved by a simple limiting process. In an earlier paper we found 𝑞-difference equations for all confluent double 𝑞-functions. In the end, we state these 𝑞-difference equations in an equivalent, more usual form, after Mellin. The corresponding partial differential equations are also given.
We characterize monic cubic CNS polynomials with only real roots in terms of relations between the other coefficients.
The Tribonacci-Lucas sequence {S_n}_n≥ 0 is defined by the linear recurrence relation S_n+3 = S_n+2 + S_n+1 + S_n, for n≥ 0, with the initial conditions S_0 =S_2= 3 and S_1 = 1. A palindromic number is a number that remains the same when its digits are reversed. This paper uses Baker's theory for nozero lower bounds for linear forms in logarithms of algebraic numbers, and reduction methods involving the theory of continued fraction to determine all Tribonacci-Lucas numbers that are palindromic concatenations of two distinct repdigits.
This paper extends the investigation into the uniqueness problems concerning meromorphic functions and their differences or difference polynomials and explores the conditions under which two transcendental meromorphic functions 𝑓 (𝑧) and 𝑔(𝑧), with hyper-order less than one, they share certain values or small functions. For instance, 𝑓 (𝑧) 𝑛 and share common values together with 𝑔(𝑧) 𝑛 and share common values. Moreover, we address scenarios where 𝑓 (𝑧) 𝑛 and 𝑔 (𝑘) (𝑧 + 𝑐) share values 𝐼𝑀 together with 𝑔(𝑧) 𝑛 and 𝑓 (𝑘) (𝑧 + 𝑐) share values 𝐼𝑀, leading to conclusions about the relationship between the functions.
We study the property of Kelley and the property of Kelley weakly on Hausdorff continua. We extend results known for metric continua to the class of Hausdorff continua. We also present new results about these properties.
In this paper, we study controlled fusion frame in tensor product of Hilbert spaces and discuss some of its properties. We describe the resolution of the identity operator on a tensor product of Hilbert spaces using the theory of controlled fusion frame. Finally, we discuss alternative dual with the help of controlled fusion frame in tensor product of Hilbert spaces.
In the present paper, we study the asymptotic properties of an exponential-type operator which was recently constructed. It is connected with 𝑝(𝑥) = 𝑥 4/3 . The main result is a pointwise complete asymptotic expansion valid for locally smooth functions. All coefficients are derived and explicitly given.
Alfréd Rényi, the founding director of the Mathematical Institute of the Hungarian Academy of Sciences was the first mathematician who proved a density theorem for the zeros of Dirichlet’s 𝐿-functions with variable moduli. This was based on a refinement of the large sieve of Linnik, developed by Rényi himself. He used this to show a weaker form of the binary Goldbach conjecture. His density theorem was the first forerunner of the famous Bombieri–Vinogradov theorem. We give a simple alternative proof of a weaker form of the Bombieri–Vinogradov theorem, based only on classical facts about 𝐿-functions (including Siegel’s theorem) and a simple but ingenious idea of Halász, but without using any form of the large sieve.
Following previous observations on 𝑞-Appell and 𝑞-Lauricella functions, the purpose of this article is to find canonical 𝑞-difference equations for the four intermediate 𝑞-Lauricella functions . The convergence regions for the above functions have already been considered in previous papers/studies. To save space, these 𝑞-difference equations are written in vector form. Furthermore, many more solutions of these 𝑞-difference equations for the two first functions are proved and the proofs are almost identical to another 𝑞-Lauricella function article. The reason is that the order of the four functions above is by order of symmetry; like in physics, the molecules (our parameters) strive to obtain maximum symmetry. Furthermore, a 𝑞-Laplace integral expressions for the first function in the form 𝑞-confluent functions is used to find more solutions.
Recent results have provided important functional generalizations, extensions and improvements of the Hardy and Levinson integral inequalities. However, they require some assumptions on the main functions, such as monotonicity or convexity assumptions, which remain somewhat restrictive. In this article, we propose two new ideas of functional generalizations, one based on a series expansion approach and the other on an integral approach. Both achieve the goal of offering adaptable generalizations and extensions of the Hardy and Levinson integral inequalities. They are formulated in two different general theorems, which are proved in detail. Several examples of new integral inequalities are derived.
The aim of this paper is to study the interrelationship between various forms of ( F , G )-shadowing property and represent it through the diagram. We show that asymptotic shadowing is equivalent to (ℕ 0 , F 𝑐𝑓 )-shadowing property and that (ℕ 0 , F 𝑐𝑓 )-shadowing implies ( F 𝑐𝑓 , F 𝑐𝑓 )-shadowing. Necessary examples are discussed to support the diagram. We also give characterization for maps to have the ( F , G )-shadowing property through the shift map on the inverse limit space. Further, we relate the ( F , G )-shadowing property to the positively F 𝑠 -expansive map. Also, we obtain the necessary and sufficient condition for the identity map to have (ℕ 0 , F 𝑡 )-shadowing property.
We prove that, when 𝑛 goes to infinity, Kostant’s problem has negative answer for almost all simple highest weight modules in the principal block of the BGG category O for the Lie algebra sl 𝑛 (ℂ).
