Mathematical Olympiad Series: Volume 23
Pages: 300
ISBN: 978-981-12-9410-5 (hardcover)
ISBN: 978-981-12-9475-4 (softcover)
For over two millennia, the complexities of elementary geometry have challenged learners, burdened by the intricacies of auxiliary graphics and cumbersome calculations. Inspired by Leibniz's query, this book introduces a groundbreaking method: point geometry. By operating directly on points, it integrates the strengths of coordinate, vector, and mass point methods, simplifying operations and problem-solving.
Central to this method is the identity approach, which streamlines complex problems into concise equations, unlocking multiple propositions with ease. Through meticulously crafted examples, readers are invited to explore the joy of mathematical thinking.
Beyond mathematics, point geometry holds promise for artificial intelligence, offering a simple yet rich knowledge representation and reasoning method. With most solutions generated by computer programs, the potential for simplifying reasoning methods is immense, paving the way for a brighter future in both education and AI advancement.
In this ambitious endeavor, the authors seek to simplify knowledge representation and reasoning, reduce the burden of learning, and accelerate the progress of artificial intelligence. This book is not just a guide to geometry; it's a catalyst for transformative thinking and discovery.
Overview of Point Geometry
Computational Methods
Identity-based Method 1: Analytical Approach
Identity-based Method 2: Undetermined Coefficients
Assorted Problems
High school math enthusiasts and mathematics competition contestants.
ISBN: 978-981-12-9284-2 (hardcover)
Geometry is often seen as one of the most beautiful aspects of mathematics. This beauty is probably a result of the fact that one can "see" this aspect of mathematics. Most people are exposed to the very basic elements of geometry throughout their schooling, concentrated in the secondary school curriculum. High schools in the United States offer one year of concentrated geometry teaching, allowing students to observe how a mathematician functions, since everything that is accepted beyond the basic axioms must be proved. However, as the course is only one year long, a great amount of geometry remains to be exposed to the general audience. That is the challenge of this book, wherein we will present a plethora of amazing geometric relationships.
We begin with the special relationship of the Golden Ratio, before considering unexpected concurrencies and collinearities. Next, we present some surprising results that arise when squares and similar triangles are placed on triangle sides, followed by a discussion of concyclic points and the relationship between circles and various linear figures. Moving on to more advanced aspects of linear geometry, we consider the geometric wonders of polygons. Finally, we address geometric surprises and fallacies, before concluding with a chapter on the useful concept of homothety, which is not included in the American year-long course in geometry.
Introduction
The Golden Ratio in Geometry
Unexpected Concurrencies
Unexpected Collinearities
Squares on Triangle Sides
Similar Triangles on Triangle Sides
Discovering Concyclic Points
Circle Wonders
Polygons and Polygrams
Geometric Surprises
Geometric Fallacies
Homothety/Similarity and Applications
This book is suitable for secondary/high school level students and teachers, as well as a general readership, particularly those interested in bridging the gaps in their knowledge of geometry.
Pages: 260
ISBN: 978-981-12-9431-0 (hardcover)
ISBN: 978-981-12-9477-8 (softcover)
In this book, potential theory is presented in an inclusive and accessible manner, with the emphasis reaching from classical to modern, from analytic to probabilistic, and from Newtonian to abstract or axiomatic potential theory (including Dirichlet spaces). The reader is guided through stochastic analysis featuring Brownian motion in its early chapters to potential theory in its latter sections. This path covers the following themes: martingales, diffusion processes, semigroups and potential operators, analysis of super harmonic functions, Dirichlet problems, balayage, boundaries, and Green functions.
The wide range of applications encompasses random walk models, especially reversible Markov processes, and statistical inference in machine learning models. However, the present volume considers the analysis from the point of view of function space theory, using Dirchlet energy as an inner product. This present volume is an expanded and revised version of an original set of lectures in the Aarhus University Mathematics Institute Lecture Note Series.
Introduction
Martingales and Markov Processes
Brownian Motion and Ito-Calculus
Semigroups of Operators, Potentials, and Diffusion Equations
Harmonic Functions, Dynkin, and Transforms
Superharmonic Functions and Riesz Measures
Green Functions, Boundary Value Problems, and Kernels
Potential Theory, Capacity, Boundaries, Dirichlet Spaces, and Applications
Appendix: Kernels and More General Classes of Gaussian Processes
Advanced undergraduate and graduate students, researchers, and practitioners in mathematics, statistics, finance, and engineering, especially those focusing on probability theory, stochastic processes, and mathematical analysis. It is suitable for upper-level and graduate courses in mathematics, as well as physics, engineering, theoretical computer science, probability, and statistics, and is also perfect for self-study. Readers with interest in current developments in pure and applied mathematics, students and specialists alike.
Pages: 280
ISBN: 978-981-12-9413-6 (hardcover)
ISBN: 978-981-12-9476-1 (softcover)
Our physical world is embedded in a geometric environment. Plane geometry has many amazing wonders beyond those that are briefly touched on at school. The circle, one of the basic aspects of geometry, has a plethora of unexpected curiosities, which the authors present in an easily understandable fashion requiring nothing more than the very basics of school geometry to appreciate these curiosities and their justifications or proofs.
The book is intended to be widely appreciated by a general readership, whose love for geometry should be greatly enhanced through exploring these many unexpected relationships. Geometric Gems is also suitable for mathematics teachers, to enhance the education of their students with these highly motivating circle properties.
Circle Curiosities
Proofs of the Circle Curiosities
Introduction: The Geometry Toolbox
This book is suitable for a general readership, as well as secondary school teachers, high school students, and mathematics professors.
The comprehensive volume focuses on both research and survey papers presenting results in a broad spectrum of subjects in pure and applied mathematics, such as in approximation theory, optimization and their applications.
Topics within this book include Sobolev spaces, Banach spaces, locally convex spaces, integral operators, Szasz-Mirakyan operators, to name a few.
This useful reference text benefits professionals, academics, graduate students and advanced research scientists in theoretical computer science, computer mathematics and general applied mathematics. Effort was also made for the content to constitute a reference source for researchers in physics and engineering.
Extensions of the Cosine-Sine Functional Equation (O Ajebbar and E Elqorachi)
No Ulam Stability, but Modified Ulam Stability of First-order q-difference Equations with Periodic Coefficient (Douglas R Anderson)
Generalized Fractional Hermite-Hadamard-Noor Inequalities via N-polynomial Preinvex Functions (Muhammad Uzair Awan, Muhammad Ubaid Ullah, Michael Th Rassias, Sadia Talib Muhammad, Aslam Noor, Khalida Inayat Noor)
An approach of Locally Convex and Barrelled Spaces over an Arbitrary Field Equipped with a Valuation, without using "Convex", but Utilizing "Absolutely Convex"()
Random stability of a system of functional equations (Abasalt Bodaghi)
Multi-quintic mappings and their stability (Abasalt Bodaghi, Ali Bagheri Vakilabad and Themistocles M Rassias)
On Classical Orthogonal Polynomials On Lattices And Some Characterization Theorems (K Castillo, D Mbouna and J Petronilho)
A Closed-Loop Supply Chain Network Model for the Provision of 5G Services with UAVs and Trustworthiness Investments (G Colajanni, P Daniele and D Sciacca)
Kuratowski's Topology in b-metric Spaces (Stefan Czerwik)
'-Free Values of [nc tanθ (log n)] (S I Dimitrov)
Some Inequalities of Trapezoid Type for Double Integral Mean of Absolutely Continuous Functions (Silvestru Sever Dragomir)
Mathematical Models and Advancements in Cardiac Hemodynamics (Anastasios C Felias, Michael Th Rassias, Michail A Xenos, Minas E Paschopoulos)
Lyusternik chnirelmann Theory for Weighted Sobolev Spaces (Lucas Fresse and Viorica V Motreanu)
On the Stochastic Properties of Birth-Death Circuit Chains for a Generalized Sample Path Case (Chrysoula Ganatsiou)
Recurrent Neural Networks for Solving Time-Varying Linear Optimization Problems (D Gerontitis and P Tzekis)
Solution Estimates and Stability Conditions for Linear Differential Equations with Unbounded Operators in a Banach Space (Michael Gil')
Numerical Solution of a Moving Boundary Problem Related to Phase Change Material (D Goeleven and R Oujja)
On Approximation of Certain Modified Szasz-Mirakyan Operators (Vijay Gupta, Gunjan Agrawal and Michael Th Rassias)
Certain Extension of Riemann Liouville Fractional Derivative Operator by Using Wiman's Function (Shilpi Jain, Rahul Goyal, RP Agarwal and Praveen Agarwal)
A Finite Element Scheme for an Initial Value Problem (Vassilios K Kalpakides)
Norm Inequalities for Generalized Fractional Integral Operators in Banach Spaces (Jichang Kuang)
Generalized Fractional Hilbert Type Integral Inequalities (Jichang Kuang and Michael Th Rassias)
Hyers-Ulam-Rassias Stability of Additive (ρE, ρE)-functional Inequalities and Hom-Derivations in Banach Algebras (Jung Rye Lee, Choonkil Park and Themistocles M Rassias)
An Additive (ρE, ρE)-Functional Inequality in Complex Banach Spaces (Jung Rye Lee, Choonkil Park and Michael Th Rassias)
The Effect of Two Exterior Masspoints on Bounds for Orthogonal Polynomials (D S Lubinsky)
The Stability of a Functional Equation in Two Variables (Laura Manolescu and Michael Th Rassias)
Static Nuel Games with Terminal Payoff (Symeon Mastrakoulis and Athanasios Kehagias)
Studying the Performance of LexOpt Algorithm for Unconstrained Optimization (Christina D Nikolakakou, Theodoula N Grapsa and George S Androulakis)
Strongly Biconvex Functions and Bivariational Inequalities (Muhammad Aslam Noor, Khalida Inayat Noor and Michael Th Rassias)
System of General Variational Inclusions (Muhammad Aslam Noor, Khalida Inayat Noor and Themistocles M Rassias)
Higher Order Bihemivariational Inequalities (Muhammad Aslam Noor, Khalida Inayat Noor, Themistocles M Rassias and Muhammad Uzair Awan)
On the Ranking of Players in Network Games with Local Average (Mauro Passacantando and Fabio Raciti)
A Two-Stage Game Theoretical Model of Social Interactions and Location Choice in City Areas (Mauro Passacantando, Fabio Raciti and Michael Th Rassias)
On a Theorem of Miron Nicolescu: Proofs, Generalizations and Applications to Zeros Separation of Some Special Functions (Adrian Petrusel and Ioan A Rus)
Minimization of the Free Energy with an Expectation Constraint (Iosif Pinelis)
An Extended Hardy-Hilbert's Integral Inequality with Internal Variables Involving Two Upper Limit Functions (M Th Rassias and B C Yang)
Implicit Sequential Contractions in Partially Ordered Metric Spaces (Mihai Turinici)
Equivalent Properties of Two Kinds of Hardy-Type Integral Inequalities with the General Nonhomogeneous Kernel in the Whole Plane (Bicheng Yang)
Researchers, professionals, academics and graduate students in theoretical computer science, computer maths, and general applied maths.
Pages: 848
ISBN: 978-981-12-9385-6 (hardcover)
This monograph is a testimony of the impact over Computational Analysis of some new trigonometric and hyperbolic types of Taylor's formulae with integral remainders producing a rich collection of approximations of a very wide spectrum.
This volume covers perturbed neural network approximations by themselves and with their connections to Brownian motion and stochastic processes, univariate and multivariate analytical inequalities (both ordinary and fractional), Korovkin theory, and approximations by singular integrals (both univariate and multivariate cases). These results are expected to find applications in the many areas of Pure and Applied Mathematics, Computer Science, Engineering, Artificial Intelligence, Machine Learning, Deep Learning, Analytical Inequalities, Approximation Theory, Statistics, Economics, amongst others. Thus, this treatise is suitable for researchers, graduate students, practitioners and seminars of related disciplines, and serves well as an invaluable resource for all Science and Engineering libraries.
Trigonometric and Hyperbolic Taylor Formulae and Opial and Ostrowski Inequalities
Perturbed A-Generalized Logistic Function Activated Complex Valued Trigonometric
Perturbed A-Generalized Logistic Function Activated Complex Valued Multivariate
Perturbed Hyperbolic Tangent Function Activated Complex Valued Trigonometric and Hyperbolic
Perturbed Hyperbolic Tangent Function Activated Complex Valued Multivariate Trigonometric
General Sigmoid Function Activated Complex Valued Trigonometric and Hyperbolic
General Multiple Sigmoid Functions Activated Complex Valued Multivariate Trigonometric
Approximation of Multiple Time Separating Random Functions by Neural Networks
Brownian Motion Approximation by Perturbed Neural Networks
Approximation of Brownian Motion Over Simple Graphs
Multivariate Fuzzy-Random with Perturbed Neural Network Approximation
Trigonometric and Hyperbolic Korovkin Properties
Integral Inequalities Using the New Conformable Derivatives
About Proportional Fractional Calculus and Inequalities
New Radial Ostrowski Inequalities on the Ball
New Ostrowski Inequalities on a Spherical Shell
Trigonometric and Hyperbolic Polya Inequalities
New Opial and Polya Inequalities Over a Spherical Shell
Trigonometric and Hyperbolic Poincare, Sobolev and Hilbert achpatte Inequalities
Trigonometric Induced Degree of Approximation by Smooth Picard Singular Integral Operators
Trigonometric Derived Lp Degree of Approximation by Smooth Picard Singular Integral Operators
Parametrized and Trigonometric Induced Quantitative Convergence of Smooth Picard
Parametrized and Trigonometric Lp Approximation with Rates of Smooth Picard
Update on Uniform Approximation by Smooth Picard Multivariate Singular Integral Operators
Trigonometric Derived Multivariate Smooth Picard Singular Integrals
Trigonometric Produced Rate of Approximation of Various Smooth Singular Integral Operators
Trigonometric Derived Lp Quantitative Approximation by Various Smooth Singular Integral Operators
Parametrized and Trigonometric Produced Uniform Quantitative Approximation
Parametrized Trigonometric Implied Lp Degree of Quantitative Approximation
Trigonometric Implied Multivariate Smooth Gausseierstrass Singular Integrals
Trigonometric Based Multivariate Smooth Poissonauchy Singular Integrals Quantitative
Trigonometric Implied Multivariate Smooth Trigonometric Singular Integrals Quantitative
Graduate students and researchers interested in approximation theory, from the fields of Pure Mathematics, Statistics, Engineering and Computer Science.
Pages: 230
ISBN: 978-981-12-9640-6 (hardcover)
The process of authoring this book is inspired by the recent increased activity of research on dynamic equations on time scales and other closely related areas. This monograph is the first published book that attempts to provide a comprehensive view of the theory and applications of set dynamic equations on time scales. The main focus of the book is the qualitative theory of set dynamic equations and their applications to fuzzy dynamic equations. The key topics include the solvability of set dynamic equations, stability of set dynamic equations, and applications to certain types of fuzzy dynamic equations.
There are five chapters in the book, through which the authors examine a wide scope of the concept of set dynamic equations and their applications. Each chapter focuses on theory and proofs to enrich the reader's understanding of the topic.
This book will be particularly useful to those experts who work in applied analysis, in general. It will also be a good reference for computer scientists since it covers fuzzy dynamic equations. Researchers and graduate students at various levels interested in learning about set dynamic equations and related fields will find this text a valuable resource of both introductory and advanced material.
Essentials of Time Scales Theory
Set Functional Calculus
Solvability of Set Dynamic Equations
Stability of Set Dynamic Equations
Applications to Fuzzy Dynamic Equations
The monograph can be used as a reference book for advanced undergraduate and graduate students, professors, and researchers in the field of time scales theory and related areas. It can also be used as a textbook for a "Special Topics" graduate course.