Exploring Mathematical Analysis, Approximation Theory, and Optimization: 270 Years Since A.-M. Legendre’s Birth (Springer Optimization and Its Applications, 207)

Preface
Contents
On a Version of Jensen-Steffensen Inequality and a Note on Inequalities in Several Variables
	1 Introduction
	2 Variants of Jensen-Steffensen and Slater-Pečarić Inequalities
	3 Remarks on Convexity and Subquadracity in Several Variables
	References
A Class of Dynamic Unilateral Contact Problems with Sub-differential Friction Law
	1 Introduction
	2 The Contact Problem
	3 Proof of Theorem 1
	References
Square-Free Values of [nc tanθ(logn)]
	1 Notations
	2 Introduction and Statement of the Result
	3 Preliminary Lemmas
	4 Beginning of the Proof
	5 Estimation of S(1)c(x)
		5.1 Lower Bound for S(3)c(x)
		5.2 Upper Bound for S(4)c(x)
		5.3 Lower Bound for S(1)c(x)
	6 Upper Bound for S(2)c(x)
	7 The End of the Proof
	References
Ostrowski and Trapezoid Type Inequalities for Riemann-Liouville Fractional Integrals of Functions with Bounded Variation
	1 Introduction
	2 Some Identities
	3 Inequalities for Functions of Bounded Variation
	4 Inequalities for Lipschitzian Functions
	References
A Strong Maximum Principle for General Nonlinear Operators
	References
On the Application of Ergodic Theory to an Alternating Series Expansion for Real Numbers
	1 Introduction
	2 Preliminaries
	3 Auxiliary Results
		3.1 A General Alternating Series Algorithm (Knopfmachers' Algorithm)
			3.1.1 Alternating Series Expansions for Real Numbers
		3.2 Alternating SEL Series Expansion Algorithm
	4 On the Application of Ergodic Theory to an Alternating SEL Series Expansion
	References
Bounds for Similarity Condition Numbers of Unbounded Operators
	1 Introduction
	2 Operators with Hilbert-Schmidt Hermitian Components
	3 The Sylvester Equation
	4 The Finite Dimensional Case
	5 Proof of Theorem 1
	6  Operators with Schatten-von Neumann Hermitian Components
	7 A Generalization of Lemma 3
	8 Proof of Theorem 3
	9 Operators ``Close'' to Normal Ones
	10 Perturbations of Eigen-Projections
	11 Proof of Theorem 5
	12 Unboundedly Perturbed Selfadjoint Operators
	13 Applications of Condition Numbers
	14 Bibliographical Comments
	References
Legendre's Geometry and Trigonometry at the Evelpides School (Central Military School) During the Kapodristrian Period
	1 Introduction
	2 Mathematical Education in General
	3 Ioannis Karantinos
	4 Ecole Centrale Militaire – Central Military School (Kετ oε oεo)
		4.1 Geometry
		4.2 Trigonometry
	5 Conclusion
	References
		General National Archives of the Greek State (GAK)
The Overshadowing of Euclid's Geometry by Legendre's Géométrie in the Modern Greek Education
	1 Motivation
	2 The First Clues of the Greek Mathematics Context
	3 Change of Context, New Potential
	4 The Bavarian Reorganization of Greek Mathematics Culture
	5 Epilogue
	References
Finite Element Methods with Higher Order Polynomials
	1 Introduction
	2 Finite Element Method
		2.1 Basic Theory of the h-Version
		2.2 Shape Functions
		2.3 p-Version and Hierarchical Basis
		2.4 Spectral and hp Methods
		2.5 Convergence and Error Estimates
		2.6 Adaptive
	3 Applications in Fluid Mechanics
		3.1 The Poisson Equation
		3.2 The Stokes Equation
			3.2.1 Backward Facing Step
			3.2.2 Lid Driven Cavity
	4 Conclusions
	References
On Local Asymptotics for Orthonormal Polynomials
	1 Introduction: Compact Support in R
	2 The Unit Circle
	3 Local Limits for Varying Exponential Weights
	4 Fixed Exponential Weights
	References
New Trends in Geometric Function Theory
	1 Introduction
	2 Preliminaries and Basic Concepts
		2.1 Subordination
		2.2 Convolution (Hadamard Product)
		2.3 Caratheodory Functions and Related Classes
		2.4 Some Basic Properties of the Class P
		2.5 Some Extensions of Class P
	3 Some Linear Operators
	4 Comprehensive Subclasses of Analytic Functions
	5 Coefficient Results
	6 Necessary Conditions
	7 Some Inclusion and Radius Problems
	8 q-Calculus Approach to Geometric Function Theory
		8.1 Sufficient Conditions for the Functions in Cq
		8.2 Fekete-Szego Inequalities for Functions in Kq and Cq
	References
A Unified Approach to Extended General Quasi Variational Inclusions
	1 Introduction
	2 Basic Definitions and Results
	Special Cases
	3 Resolvent Techniques
	4 Dynamical Systems Technique
	Conclusion
	References
On a Reverse Hilbert-Type Inequality in the Whole Plane with Multi-Parameters
	1 Introduction
	2 Some Lemmas
	3 Main Results and Some Particular Inequalities
	References
Generating Functions for the Fubini Type Polynomials and Their Applications
	1 Introduction
	2 Generating Function for Certain Families of Special Numbers and Polynomials
		2.1 Generating Functions for Combinatorial Numbers
		2.2 Generating Functions for Apostol Type Numbers and Polynomials, Members of Apell Polynomials
	3 Generating Functions for Fubini Type Numbers and Polynomials with Their Properties
	4 Derivative Formulas and Identities Including Fubini Type Numbers and Polynomials
	5 Relations Among the Lah Numbers, Stirling and Fubini Type Numbers and Polynomials
	6 Computational Algorithms and Drawing Plots and Surfaces for the Polynomials am( k) ( x) with Their Generating Function
	7 Generating Functions for the Fubini Type Numbers and Polynomials of Negative Higher Order
	8 Generating Functions for Families of Two Variable Fubini Type Polynomials
	9 Asymptotic Representation for Families of Fubini Type Numbers
	10 Generating Functions for Two Parametric Kinds of Fubini-Type Polynomials with Their Generating Functions
	11 Generating Functions for Families of Unified Form Fubini Type Polynomials
	12 Generating Functions for Unified and Modified Presentation Fubini Numbers and Polynomials
		12.1 Interpolation Functions for the Numbers am( v) ( μ;,b)  and the Polynomials am( v) ( x,μ;,b,c)
		12.2 Relations Among the Function Zv( s;x,μ;,b,c) , the Hurwitz-Lerch Zeta Function, the Lipschitz Function, the Hurwitz Zeta Function, Polylogarithm Function, and the Riemann Zeta Function
	References
Kleene Fixed Point Theorems and Applications
	1 Introduction
	2 Dependent Choice Principle
	3 Conv-Cauchy Structures
	4 Increasing Maps
	5 Progressive Case
	6 Increasing Maps in Metric Ordered Spaces
	7 Metrical Contractions (Progressive Methods)
	References
On Ergodic States, Spontaneous Symmetry Breaking and Quasi-Averages
	1 Introduction and Summary
	2 Setup: Continuous SSB, ODLRO and Examples
	3 Selection of Pure States by the Bogoliubov Quasi-Averages: Spin Systems
	4 Continuous Boson Systems: Quasi-Averages, Condensates, and Pure States
	5 Concluding Remarks
	Appendix
	References
Improvement of the Hardy Inequality and Legendre Polynomials
	1 Introduction
	2 Proof of Theorem 1
		2.1 Space Setting
		2.2 The Eigenvalue Problem
		2.3 (Improved) Hardy-Poincaré Inequality
		2.4 The Special Case g=(1+r2)-2
	References