Abstract We consider ℓ p -direct sums (1 ≤ p < ∞ ) and c 0 -direct sums of countably many normed spaces and find the dual of these spaces. We characterize the support functionals of arbitrary elements in these spaces to characterize smoothness and approximate smoothness, both locally and globally. These results let us answer the Chmieliński, Khurana, and Sain question raised in [4] on the existence of a non-approximately smooth normed space whose every element is smooth. We also characterize Birkhoff-James orthogonality and its pointwise symmetry in these spaces.
The main aim of this paper is to use the k-Hilfer fractional derivative to derive certain P & oacute;lya-Szeg & ouml; fractional integral inequalities. Further fractional integral inequalities are obtained. The results presented here extend and generalize various existing inequalities associated with the Riemann-Liouville, Caputo, Saigo, and Hilfer fractional operators. The findings contribute to the growing theory of fractional calculus and offer potential tools for the analysis of fractional differential and integral equations.
In this manuscript, we accelerate the local convergence of a third-order biparametric iterative method in & Ropf; or & Copf; by assuming that the first-order Fr & eacute;chet derivative satisfies the Generalized continuity condition. We extend this analysis by using the H & ouml;lder continuity condition, which allows us to solve more numerical problems. Our study also shows the sizes of the convergence balls, the smallest error bounds that can be computed, and the fact that the answer is unique. Several math tests show that this third-order method gives better results than the midpoint method established by I.K. Argyros and S. George [4]. This method solves problems that earlier studies have not been able to solve.
The present paper investigates the asymptotic behavior of positive generalized self-similar solutions for a nonlinear hybrid problem involving nth-order derivative porous medium equations. We provide sufficient conditions for the existence and uniqueness of weak solutions that have compact support and dynamic characteristics. Furthermore, we establish the behavior of these solutions by examining a specific set of variables and their signs, which must meet certain conditions to determine whether the solutions exist globally or locally in time.
This article presents a unified framework that extends the scope of two existing theorems on multivariate Hardy-Hilbert-type integral inequalities. Key to this extension is the use of two additional adjustable parameters that increase flexibility and generality. The framework also has the originality of including the incomplete lower gamma function in the integral definitions governed by a parameter. Detailed proofs are given, mainly based on the Laplace transform, the generalized Young inequality, the generalized H & ouml;lder integral inequality and changes of variables. This article thus provides a new comprehensive foundation for future research in generalized multivariate integral inequalities.
These notes were made for the Preparatory School on Lefschetz Properties, held from 6th to 10th May 2024, in Krak & oacute;w, ahead of the conference "Lefschetz properties in algebra, geometry, topology, and combinatorics". They are a short introduction to Jordan type of an Artinian algebra, and collect basic results known so far, with examples to illustrate them along the way.
These notes represent background and supplementary material for our course in the Preparatory School for the conference “Lefschetz Properties in Algebra, Geometry and Combinatorics.” The school was held in Kraków in May, 2024 and the conference took place in June, 2024.
These lecture notes were prepared for the Lefschetz Preparatory School, a graduate summer course held in Krakow, May 6-10, 2024. They present the story of the algebraic Lefschetz properties from their origin in algebraic geometry to some recent developments in commutative algebra. The common thread of the notes is a bias towards topics surrounding the algebraic Lefschetz properties that have a topological flavor. These range from the Hard Lefschetz Theorem for cohomology rings to commutative algebraic analogues of these rings, namely artinian Gorenstein rings, and topologically motivated operations among such rings.
Let function $f$ be normalized, analytic and univalent in the unit disk ${\mathbb D}=\{z:|z|<1\}$ and $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$. Using a method based on Grusky coefficients we study several problems over that class of univalent functions: upper bound of the third logarithmic coefficient, upper bound of the coefficient difference $|a_4|-|a_3|$, the special case of the generalized Zalcman conjecture $|a_2a_3-a_4|$ and upper bounds of the second and the third Hankel determinant. Obtained results improve the previous ones.
We consider $\ell_p$-direct sums ($1\leq p<\infty$) and $c_0$-direct sums of countably many normed spaces and find the duals of these spaces. We characterize the support functionals of arbitrary elements in these spaces to characterize smoothness and approximate smoothness, both locally and globally. These results let us obtain examples of spaces that are not approximately smooth but where every non-zero element is approximately smooth. We also characterize Birkhoff-James orthogonality and its pointwise symmetry in these spaces.
Josephson junctions find applications for construction of various devices in measuring, transmitting, receiving, and amplifying systems as well as classical and quantum computing devices. Some of the existing and future applications are related to shape engineering, which allows for adjustment of the parameters of the junction to the needs of various applications. The article includes review of both a description of the applications and theoretical considerations devoted to the Josephson junctions and their properties.
The paper aims to investigate lambda-statistical convergence using modulus function and a generalized difference operator for double sequences of functions for order gamma is an element of (0, 1]. Further, we prove that the statistical convergence in the newly formed sequence spaces is not well defined for gamma > 1. Finally, we examine relevant inclusion relations concerning lambda-statistical convergence and strongly lambda-summable in the environment of the newly defined classes of double sequences of functions. Some interesting examples related to the examined results are also discussed in this paper.
In this paper we introduce the notion of delta(mu)-connectedness on a mu -proximity space. It has been proved that delta(mu)-connectedness can be characterized by delta(mu)-continuous functions. We initiate the idea of delta(mu)-chain and establish some results related to this. The concepts of delta(mu)-component and delta(mu)-quasi component have been introduced and their interrelation has been studied.
In this note we provide a complete classification of weak combinatorics of the so-called maximizing line arrangements in the complex projective plane.
By rewriting the differential entropy in a form of a differ-integral function's limit, and deforming the ordinary derivative to a fractional-order one, we derive in this paper a novel generalized fractional-order differential entropy along with its related information measures. When the order of fractional differentiation alpha -> 1, the ordinary Shannon's differential entropy is recovered, which corresponds to the results from first-order ordinary differentiation.
The aim of this article is to introduce the concept of centrally-extended Jordan endomorphisms and proving that if $R$ is a non-commutative prime ring of characteristic not two, and $G$ is a CE- Jordan epimorphism such that $[G(x), x] \in Z(R)$ ($[G(x), x^*] \in Z(R)$) for all $x \in R$, then $R$ is an order in a central simple algebra of dimension at most $4$ over its center or there is an element $\lambda$ in the extended of $R$ such that $G(x) = \lambda x$ ($G(x) = \lambda^* x^*$) for all $x \in R$.
The homotopy type of the complement manifold of a complexified toric arrangement has been investigated by d'Antonio and Delucchi in a paper that shows the minimality of such topological space. In this work we associate to a given toric arrangement a matrix that represents the arrangement over the integers. Then, we consider the family of toric arrangements for which this matrix has maximal rank. Our goal is to prove, by means of basic linear algebra arguments, that the complement manifold of the toric arrangements that belong to this family is diffeomorphic to that of centered toric arrangements and thus it is a minimal topological space, too.
Abstract In this article, we introduce the notion of centrally-extended generalized Jordan derivations and characterize the structure of a prime ring (resp. *-prime ring) R that admits a centrally-extended generalized Jordan derivation F satisfying [F(x), x] ∈ Z(R) (resp. [F(x), x*] ∈ Z(R)) for all x ∈ R.
Abstract In the present paper, a theorem on θ −|T; δ|k summability method of an infinite series is proved, and also by using this method, a result on summability of a trigonometric Fourier series is obtained.
