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2023 | Buch

On a Version of Jensen-Steffensen Inequality and a Note on Inequalities in Several Variables Kapitel lesen Erstes Kapitel lesen

Exploring Mathematical Analysis, Approximation Theory, and Optimization

270 Years Since A.-M. Legendre’s Birth

herausgegeben von: Nicholas J. Daras, Michael Th. Rassias, Nikolaos B. Zographopoulos

Verlag: Springer International Publishing

Buchreihe : Springer Optimization and Its Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book compiles research and surveys devoted to the areas of mathematical analysis, approximation theory, and optimization. Being dedicated to A.-M. Legendre's work, contributions to this volume are devoted to those branches of mathematics and its applications that have been influenced, directly or indirectly, by the mathematician. Additional contributions provide a historical background as it relates to Legendre's work and its association to the foundation of Greece's higher education.

Topics covered in this book include the investigation of the Jensen-Steffensen inequality, Ostrowski and trapezoid type inequalities, a Hilbert-Type Inequality, Hardy’s inequality, dynamic unilateral contact problems, square-free values of a category of integers, a maximum principle for general nonlinear operators, the application of Ergodic Theory to an alternating series expansion for real numbers, bounds for similarity condition numbers of unbounded operators, finite element methods with higher order polynomials, generating functions for the Fubini type polynomials, local asymptotics for orthonormal polynomials, trends in geometric function theory, quasi variational inclusions, Kleene fixed point theorems, ergodic states, spontaneous symmetry breaking and quasi-averages.

It is hoped that this book will be of interest to a wide spectrum of readers from several areas of pure and applied sciences, and will be useful to undergraduate students, graduate level students, and researchers who want to be kept up to date on the results and theories in the subjects covered in this volume.

Inhaltsverzeichnis

Frontmatter
On a Version of Jensen-Steffensen Inequality and a Note on Inequalities in Several Variables
Abstract
We deal with special versions of Jensen-Steffensen inequality for convex functions, and we note on convex and subquadratic functions in one and several variables.
Shoshana Abramovich
A Class of Dynamic Unilateral Contact Problems with Sub-differential Friction Law
Abstract
We study a class of dynamic unilateral contact problems with sub-differential friction law, and thermal effects, for time depending long memory visco-elastic materials, with or without the clamped condition. We describe the mechanical problem, derive its variational formulation, and after specifying the assumptions on the data and operators, we prove an existence and uniqueness of weak solution on displacement and temperature fields.
Oanh Chau, Adrien Petrov, Arnaud Heibig
Square-Free Values of
S. I. Dimitrov
Ostrowski and Trapezoid Type Inequalities for Riemann-Liouville Fractional Integrals of Functions with Bounded Variation
Abstract
In this chapter we establish some Ostrowski and generalized trapezoid type inequalities for the Riemann-Liouville fractional integrals of functions of bounded variation and of Lipschitzian functions. Applications for mid-point and trapezoid inequalities are provided as well. They generalize the know results holding for the classical Riemann integral.
Silvestru Sever Dragomir
A Strong Maximum Principle for General Nonlinear Operators
Lucas Fresse, Viorica V. Motreanu
On the Application of Ergodic Theory to an Alternating Series Expansion for Real Numbers
Abstract
By considering a general alternating series algorithm introduced by A. and J. Knopfmacher according to which every real number may be expressed by alternating series representations in terms of rationals we present some important results arising from the application of ergodic theory to an alternating series expansion for real numbers in terms of rationals generating from another algorithm called alternating Sylvester-Engel-Lüroth series expansion (alternating SEL series expansion) which gives generalized versions of the corresponding three alternating series expansions constructing from the general alternating series algorithm of A. and J. Knopfmacher.
Chryssoula Ganatsiou, Ilias K. Savvas
Bounds for Similarity Condition Numbers of Unbounded Operators
Michael Gil’
Legendre’s Geometry and Trigonometry at the Evelpides School (Central Military School) During the Kapodristrian Period
Abstract
The Military Academy (Central Military School) was the first institution of higher education in liberated Greece. The model used was that of the French École Polytechnique. This effect was due to the fact that the professor of this course, Ioannis Karantinos, studied in 1820 at the Ecole Polytechnique. He translated France mathematical books in modern Greek language. Legendre’s geometry and trigonometry were the basis for the education of the Evelpides.
Andreas Kastanis
The Overshadowing of Euclid’s Geometry by Legendre’s Géométrie in the Modern Greek Education
Abstract
Euclid’s Elements is an important heritage of the Neo-Hellenic civilization. Despite this fact Euclid is very difficult to be found in the Greek mathematics education as it was developing after the liberation from the Ottoman Empire.
Nikos Kastanis
Finite Element Methods with Higher Order Polynomials
Abstract
The Finite Element Method (FEM) has recently been implemented in the fluid mechanics field to solve the instabilities that arise as a result of the equations’ non-linearities. For this reason, novel formulations of FEM were introduced, including the use of orthogonal polynomials and high-order polynomials. In this review, the focus rests on studying and analysing the aforementioned formulations and describing their improvements over the classical method. Initially, a theoretical background of FEM is introduced, with an emphasis on evaluating the basis of the function space. Additionally, the p-version of FEM is analysed, using Legendre polynomials. A comparison of the classical h-version and the p-version in terms of convergence. Moreover, other formulations that yield, using higher-order polynomials, such as hp-FEM and Spectral Element Method, are briefly reviewed. Finally, applications on FEM are presented, revealing the effects of the increase in the degree of the polynomials when solving a fluid mechanics problem.
Konstantina C. Kyriakoudi, Michail A. Xenos
On Local Asymptotics for Orthonormal Polynomials
Eli Levin, D. S. Lubinsky
New Trends in Geometric Function Theory
Abstract
This chapter is a comprehensive survey about the study and recent developments in the area of geometric function theory. The definitions of certain classes of analytic functions are given in a unified and generalized form. Basic properties, necessary conditions, distortion and coefficient results together with the Fekete-Szego and Hankel determinant problems are highlighted. Certain linear operators and the use of q-calculus in this area are also included.
Khalida Inayat Noor, Mohsan Raza
A Unified Approach to Extended General Quasi Variational Inclusions
Abstract
In this article, we introduce and consider some new classes of extended general quasi variational inclusions, which provide us with a unified, natural and simple framework to consider a wide class of unrelated problems arising in pure and applied sciences. We establish the equivalence between the general quasi variational inclusions and the fixed point problems. This alternative equivalence formulation is applied to discuss the existence of a solution as well as to propose some iterative methods. Convergence analysis is investigated under certain mild conditions. We introduce a new class of dynamical systems associated with extended general quasi variational inclusions. We have used the dynamical systems to suggest and analyzed some implicit iterative methods for solving the extended general quasi variational inclusions. Since the extended general quasi variational inclusions include quasi variational inclusions, absolute vale equations, complementarity problems, variational inequalities, and related optimization problems as special cases, our results continue to hold for these problems. It is an interesting problem to compare these methods with other technique for solving quasi variational inclusions for further research activities.
Muhammad Aslam Noor, Khalida Inayat Noor, Michael Th. Rassias
On a Reverse Hilbert-Type Inequality in the Whole Plane with Multi-Parameters
Abstract
In the present paper we make use of weight coefficients and methods from real and complex analysis, in order to establish a reverse Hilbert-type inequality in the whole plane with multi-parameters. The corresponding constant factor of the inequality is proved to be the best possible. Moreover, we also consider equivalent forms and a few particular inequalities.
Michael Th. Rassias, Bicheng Yang, Andrei Raigorodskii
Generating Functions for the Fubini Type Polynomials and Their Applications
Abstract
One of the aims of this chapter is to give Fubini type numbers and polynomials discovered with the help of generating functions or defined by combinatorial methods and also their general properties with known methods or techniques that we have found. The second purpose of this chapter is to give formulas and relations that we have just found, besides the known ones, using generating functions and their functional equations. The third purpose of this chapter is to give the relations between Fubini-type numbers and polynomials and other special numbers and polynomials. The fourth of the purposes of this chapter will be to give tables with Fubini-type numbers and polynomials, as well as other special numbers and special polynomials. In addition, by using Wolfram Mathematica version 12.0, graphs of Fubini type polynomials and their generating functions, surface graphs and mathematical codes will be given. The fifth purpose of this chapter, some known applications in the theory of approximation with Fubini-type numbers and polynomials are summarized. The sixth of the purposes of this chapter is to give zeta-type functions that interpolate Fubini-type numbers and polynomials at negative integers. Moreover, throughout this chapter, we are tried diligently to present the results obtained in comparison with other known results and their reductions, taking into account the relevant sources.
Yilmaz Simsek, Neslihan Kilar
Kleene Fixed Point Theorems and Applications
Abstract
A couple of Kleene fixed point principles is discussed on (partially) ordered spaces. Then, some applications of these—including the Caristi-Kirk principles and Banach-Matkowski results—are given.
Mihai Turinici
On Ergodic States, Spontaneous Symmetry Breaking and Quasi-Averages
Abstract
It is shown that the Bogoliubov quasi-averages select the pure or ergodic states in the ergodic decomposition of the thermal (Gibbs) state. Our examples include quantum spin systems and many-body boson systems. As a consequence, we elucidate the problem of equivalence between Bose-Einstein condensation and the quasi-average spontaneous symmetry breaking of the gauge invariance recently discussed for continuous boson systems. The multi-mode extended van den Berg-Lewis-Pulé condensation of type III demonstrates that the only physically reliable quantities are those that defined by Bogoliubov quasi-averages.
Walter F. Wreszinski, Valentin A. Zagrebnov
Improvement of the Hardy Inequality and Legendre Polynomials
Abstract
We discuss the improvement of the Hardy inequality on the whole space and we give its connection with the Legendre polynomials.
Nikolaos B. Zographopoulos
Metadaten
Titel
Exploring Mathematical Analysis, Approximation Theory, and Optimization
herausgegeben von
Nicholas J. Daras
Michael Th. Rassias
Nikolaos B. Zographopoulos
Copyright-Jahr
2023
Electronic ISBN
978-3-031-46487-4
Print ISBN
978-3-031-46486-7
DOI
https://doi.org/10.1007/978-3-031-46487-4

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