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General Fractional Derivatives: Theory, Methods and Applications
by Xiao-Jun YangGeneral Fractional Derivatives: Theory, Methods and Applications provides knowledge of the special functions with respect to another function, and the integro-differential operators where the integrals are of the convolution type and exist the singular, weakly singular and nonsingular kernels, which exhibit the fractional derivatives, fractional integrals, general fractional derivatives, and general fractional integrals of the constant and variable order without and with respect to another function due to the appearance of the power-law and complex herbivores to figure out the modern developments in theoretical and applied science. Features: Give some new results for fractional calculus of constant and variable orders. Discuss some new definitions for fractional calculus with respect to another function. Provide definitions for general fractional calculus of constant and variable orders. Report new results of general fractional calculus with respect to another function. Propose news special functions with respect to another function and their applications. Present new models for the anomalous relaxation and rheological behaviors. This book serves as a reference book and textbook for scientists and engineers in the fields of mathematics, physics, chemistry and engineering, senior undergraduate and graduate students. Dr. Xiao-Jun Yang is a full professor of Applied Mathematics and Mechanics, at China University of Mining and Technology, China. He is currently an editor of several scientific journals, such as Fractals, Applied Numerical Mathematics, Mathematical Modelling and Analysis, International Journal of Numerical Methods for Heat & Fluid Flow, and Thermal Science.
A General Framework for Reasoning On Inconsistency (SpringerBriefs in Computer Science)
by V. S. Subrahmanian Maria Vanina Martinez Leila Amgoud Cristian MolinaroThis SpringerBrief proposes a general framework for reasoning about inconsistency in a wide variety of logics, including inconsistency resolution methods that have not yet been studied. The proposed framework allows users to specify preferences on how to resolve inconsistency when there are multiple ways to do so. This empowers users to resolve inconsistency in data leveraging both their detailed knowledge of the data as well as their application needs. The brief shows that the framework is well-suited to handle inconsistency in several logics, and provides algorithms to compute preferred options. Finally, the brief shows that the framework not only captures several existing works, but also supports reasoning about inconsistency in several logics for which no such methods exist today.
A General Introduction to Data Analytics
by João Moreira André Carvalho Tomás HorvathA guide to the principles and methods of data analysis that does not require knowledge of statistics or programming A General Introduction to Data Analytics is an essential guide to understand and use data analytics. This book is written using easy-to-understand terms and does not require familiarity with statistics or programming. The authors—noted experts in the field—highlight an explanation of the intuition behind the basic data analytics techniques. The text also contains exercises and illustrative examples. Thought to be easily accessible to non-experts, the book provides motivation to the necessity of analyzing data. It explains how to visualize and summarize data, and how to find natural groups and frequent patterns in a dataset. The book also explores predictive tasks, be them classification or regression. Finally, the book discusses popular data analytic applications, like mining the web, information retrieval, social network analysis, working with text, and recommender systems. The learning resources offer: A guide to the reasoning behind data mining techniques A unique illustrative example that extends throughout all the chapters Exercises at the end of each chapter and larger projects at the end of each of the text’s two main parts Together with these learning resources, the book can be used in a 13-week course guide, one chapter per course topic. The book was written in a format that allows the understanding of the main data analytics concepts by non-mathematicians, non-statisticians and non-computer scientists interested in getting an introduction to data science. A General Introduction to Data Analytics is a basic guide to data analytics written in highly accessible terms.
General Investigations of Curved Surfaces: Edited with an Introduction and Notes by Peter Pesic
by Peter Pesic Adam Hiltebeitel James Morehead Karl Friedrich GaussGauss's theory of surfaces is among the purely mathematical achievements inspired by ideas that arose in connection with surveys of the surface of the earth. Long regarded as a masterpiece in content and form, this work features one of the author's most original contributions to mathematics--the discovery that Gauss termed the "Theorema Egregium." It consists of his penetrating definition of the concept of surface curvature and the theorem that the "Gauss curvature" is invariant under arbitrary isometric deformation of a curved surface. The profound effects of these concepts were soon generalized by Bernhard Riemann, and subsequent development included the important role of the Gauss-Riemann concept of curvature in modern relativity theory.This edition of Gauss's classic work features a new introduction, bibliography, and notes by science historian Peter Pesic. In addition, an informative appendix offers historical background.
The General Kelvin-Helmholtz Stability Model: With Applications to Sand Wave and Water Wave Generation
by Hermann MoshagenThis book presents a generalized version of the classical Kelvin-Helmholtz instability, a useful tool which allows for new approaches when studying stability problems in fluid mechanics, as well as its important applications. It begins by providing an introduction to hydrodynamic stability and the Kelvin-Helmholtz (KH) instability. The author then develops the general KH stability model for a multi-layer flow system, which includes the conventional KH instability as a special case. This book also includes the detailed discussion of two important applications of this model: the generation of sand waves in alluvial channels and the generation of wind waves on water. Additionally, the effects of nonlinearities and the use of computational methods to study KH instability are included. This book serves as a concise and modern treatment of the KH stability model with specific attention paid to hydrodynamic stability analysis. It is ideal for graduate students interested in fluid dynamics as well as scientists and engineers in the fields of oceanography, geophysics, offshore engineering, and more.
General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions
by Qi Lü Xu ZhangThe classical Pontryagin maximum principle (addressed to deterministic finite dimensional control systems) is one of the three milestones in modern control theory. The corresponding theory is by now well-developed in the deterministic infinite dimensional setting and for the stochastic differential equations. However, very little is known about the same problem but for controlled stochastic (infinite dimensional) evolution equations when the diffusion term contains the control variables and the control domains are allowed to be non-convex. Indeed, it is one of the longstanding unsolved problems in stochastic control theory to establish the Pontryagin type maximum principle for this kind of general control systems: this book aims to give a solution to this problem. This book will be useful for both beginners and experts who are interested in optimal control theory for stochastic evolution equations.
General Quantum Numerical Analysis (ISSN)
by Svetlin G. Georgiev Khaled ZennirThis book is focused on the qualitative theory of general quantum calculus, the modern name for the investigation of calculus without limits. It centers on designing, analysing and applying computational techniques for general quantum differential equations. The quantum calculus or q-calculus began with F.H. Jackson in the early twentieth century, but this kind of calculus had already been worked out by Euler and Jacobi. Recently, it has aroused interest due to high demand of mathematics that models quantum computing and the connection between mathematics and physics.Quantum calculus has many applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hyper-geometric functions and other sciences such as quantum theory, mechanics and the theory of relativity.The authors summarize the most recent contributions in this area. General Quantum Numerical Analysis is intended for senior undergraduate students and beginning graduate students of engineering and science courses. The twelve chapters in this book are pedagogically organized, each concluding with a section of practical problems.
General Quantum Variational Calculus (Advances in Applied Mathematics)
by Svetlin G. Georgiev Khaled ZennirQuantum calculus is the modern name for the investigation of calculus without limits. Quantum calculus, or q-calculus, began with F.H. Jackson in the early twentieth century, but this kind of calculus had already been worked out by renowned mathematicians Euler and Jacobi.Lately, quantum calculus has aroused a great amount of interest due to the high demand of mathematics that model quantum computing. The q-calculus appeared as a connection between mathematics and physics. It has a lot of applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hypergeometric functions and other quantum theory sciences, mechanics, and the theory of relativity. Recently, the concept of general quantum difference operators that generalize quantum calculus has been defined. General Quantum Variational Calculus is specially designed for those who wish to understand this important mathematical concept, as the text encompasses recent developments of general quantum variational calculus. The material is presented in a highly readable, mathematically solid format. Many practical problems are illustrated, displaying a wide variety of solution techniques.This book is addressed to a wide audience of specialists such as mathematicians, physicists, engineers, and biologists. It can be used as a textbook at the graduate level and as a reference for several disciplines.
General Relativistic and Post-Newtonian Dynamics for Near-Earth Objects and Solar System Bodies (Springer Theses)
by Joseph O’LearyOwing to the increased accuracy requirements in fields such as astrometry and geodesy the general theory of relativity must be taken into account for any mission requiring highly accurate orbit information and for practically all observation and measurement techniques. This book highlights the confluence of Applied Mathematics, Physics and Space Science as seen from Einstein's general theory of relativity and aims to bridge the gap between theoretical and applied domains. The book investigates three distinct areas of general relativity: Exact solutions of the Einstein field equations of gravitation. Dynamics of near-Earth objects and solar system bodies. Relativistic orbitography. This book is an updated and expanded version of the author’s PhD thesis which was awarded the International Astronomical Union PhD prize in Division A: Fundamental Astronomy. Included is a new introduction aimed at graduate students of General Relativity and extended discussions and results on topics in post-Newtonian dynamics and general relativistic spacecraft propagation.
General Relativity: The Theoretical Minimum (The Theoretical Minimum)
by Leonard Susskind André CabannesThe latest volume in the New York Times–bestselling physics series explains Einstein&’s masterpiece: the general theory of relativity He taught us classical mechanics, quantum mechanics, and special relativity. Now, physicist Leonard Susskind, assisted by a new collaborator, André Cabannes, returns to tackle Einstein&’s general theory of relativity. Starting from the equivalence principle and covering the necessary mathematics of Riemannian spaces and tensor calculus, Susskind and Cabannes explain the link between gravity and geometry. They delve into black holes, establish Einstein field equations, and solve them to describe gravity waves. The authors provide vivid explanations that, to borrow a phrase from Einstein himself, are as simple as possible (but no simpler). An approachable yet rigorous introduction to one of the most important topics in physics, General Relativity is a must-read for anyone who wants a deeper knowledge of the universe&’s real structure.
General Relativity and Cosmology: A First Encounter (Graduate Texts in Physics)
by Ronald J. AdlerGravitational physics has now become a mainstream topic in physics and physics teaching. In particular cosmology and gravitational wave physics are at the focus of a great deal of current research. Thus it is important to introduce students to General Relativity as soon as reasonable. This textbook offers a brief but comprehensive treatment accessible to advanced undergraduate students, graduate students, and any physicist or mathematician interested in understanding the material in a short time. The author, an experienced teacher of the subject, has included numerous examples and exercises to help students consolidate the ideas they have learned.
General Relativity and its Applications: Black Holes, Compact Stars and Gravitational Waves
by Paolo Pani Valeria Ferrari Leonardo GualtieriContaining the latest, groundbreaking discoveries in the field, this text outlines the basics of Einstein’s theory of gravity with a focus on its most important astrophysical consequences, including stellar structures, black holes and the physics of gravitational waves. Blending advanced topics - usually not found in introductory textbooks - with examples, pedagogical boxes, mathematical tools and practical applications of the theory, this textbook maximises learning opportunities and is ideal for master and graduate students in Physics and Astronomy. Key features:• Provides a self-contained and consistent treatment of the subject that does not require advanced previous knowledge of the field.• Explores the subject with a new focus on gravitational waves and astrophysical relativity, unlike current introductory textbooks.• Fully up-to-date, containing the latest developments and discoveries in the field.
General Relativity Without Calculus
by Jose Natario"General Relativity Without Calculus" offers a compact but mathematically correct introduction to the general theory of relativity, assuming only a basic knowledge of high school mathematics and physics. Targeted at first year undergraduates (and advanced high school students) who wish to learn Einstein's theory beyond popular science accounts, it covers the basics of special relativity, Minkowski space-time, non-Euclidean geometry, Newtonian gravity, the Schwarzschild solution, black holes and cosmology. The quick-paced style is balanced by over 75 exercises (including full solutions), allowing readers to test and consolidate their understanding.
General Stochastic Measures: Integration, Path Properties and Equations
by Vadym M. RadchenkoThis book is devoted to the study of stochastic measures (SMs). An SM is a sigma-additive in probability random function, defined on a sigma-algebra of sets. SMs can be generated by the increments of random processes from many important classes such as square-integrable martingales and fractional Brownian motion, as well as alpha-stable processes. SMs include many well-known stochastic integrators as partial cases.General Stochastic Measures provides a comprehensive theoretical overview of SMs, including the basic properties of the integrals of real functions with respect to SMs. A number of results concerning the Besov regularity of SMs are presented, along with equations driven by SMs, types of solution approximation and the averaging principle. Integrals in the Hilbert space and symmetric integrals of random functions are also addressed.The results from this book are applicable to a wide range of stochastic processes, making it a useful reference text for researchers and postgraduate or postdoctoral students who specialize in stochastic analysis.
General Theory of Algebraic Equations
by Etienne BézoutThis book provides the first English translation of Bezout's masterpiece, the General Theory of Algebraic Equations. It follows, by almost two hundred years, the English translation of his famous mathematics textbooks. Here, Bézout presents his approach to solving systems of polynomial equations in several variables and in great detail. He introduces the revolutionary notion of the "polynomial multiplier," which greatly simplifies the problem of variable elimination by reducing it to a system of linear equations. The major result presented in this work, now known as "Bézout's theorem," is stated as follows: "The degree of the final equation resulting from an arbitrary number of complete equations containing the same number of unknowns and with arbitrary degrees is equal to the product of the exponents of the degrees of these equations." The book offers large numbers of results and insights about conditions for polynomials to share a common factor, or to share a common root. It also provides a state-of-the-art analysis of the theories of integration and differentiation of functions in the late eighteenth century, as well as one of the first uses of determinants to solve systems of linear equations. Polynomial multiplier methods have become, today, one of the most promising approaches to solving complex systems of polynomial equations or inequalities, and this translation offers a valuable historic perspective on this active research field.
General Theory of Algebraic Equations
by Etienne ÉzoutThis book provides the first English translation of Bezout's masterpiece, the General Theory of Algebraic Equations. It follows, by almost two hundred years, the English translation of his famous mathematics textbooks. Here, Bézout presents his approach to solving systems of polynomial equations in several variables and in great detail. He introduces the revolutionary notion of the "polynomial multiplier," which greatly simplifies the problem of variable elimination by reducing it to a system of linear equations. The major result presented in this work, now known as "Bézout's theorem," is stated as follows: "The degree of the final equation resulting from an arbitrary number of complete equations containing the same number of unknowns and with arbitrary degrees is equal to the product of the exponents of the degrees of these equations." The book offers large numbers of results and insights about conditions for polynomials to share a common factor, or to share a common root. It also provides a state-of-the-art analysis of the theories of integration and differentiation of functions in the late eighteenth century, as well as one of the first uses of determinants to solve systems of linear equations. Polynomial multiplier methods have become, today, one of the most promising approaches to solving complex systems of polynomial equations or inequalities, and this translation offers a valuable historic perspective on this active research field.
The General Theory of Dirichlet's Series (Dover Books on Mathematics)
by G. H. Hardy Marcel RieszThis classic work explains the theory and formulas behind Dirichlet's series and offers the first systematic account of Riesz's theory of the summation of series by typical means. Its authors rank among the most distinguished mathematicians of the twentieth century: G. H. Hardy is famous for his achievements in number theory and mathematical analysis, and Marcel Riesz's interests ranged from functional analysis to partial differential equations, mathematical physics, number theory, and algebra.Following an introduction, the authors proceed to a discussion of the elementary theory of the convergence of Dirichlet's series, followed by a look at the formula for the sum of the coefficients of a Dirichlet's series in terms of the order of the function represented by the series. They continue with an examination of the summation of series by typical means and of general arithmetic theorems concerning typical means. After a survey of Abelian and Tauberian theorems and of further developments of the theory of functions represented by Dirichlet's series, the text concludes with an exploration of the multiplication of Dirichlet's series.
A General Theory of Entropy: Fuzzy Rational Foundations of Information-Knowledge Certainty (Studies in Fuzziness and Soft Computing #384)
by Kofi Kissi DompereThis book presents an epistemic framework for dealing with information-knowledge and certainty-uncertainty problems within the space of quality-quantity dualities. It bridges between theoretical concepts of entropy and entropy measurements, proposing the concept and measurement of fuzzy-stochastic entropy that is applicable to all areas of knowing under human cognitive limitations over the epistemological space. The book builds on two previous monographs by the same author concerning theories of info-statics and info-dynamics, to deal with identification and transformation problems respectively. The theoretical framework is developed by using the toolboxes such as those of the principle of opposites, systems of actual-potential polarities and negative-positive dualities, under different cost-benefit time-structures. The category theory and the fuzzy paradigm of thought, under methodological constructionism-reductionism duality, are used in the fuzzy-stochastic and cost-benefit spaces to point to directions of global application in knowing, knowledge and decision-choice actions. Thus, the book is concerned with a general theory of entropy, showing how the fuzzy paradigm of thought is developed to deal with the problems of qualitative-quantitative uncertainties over the fuzzy-stochastic space, which will be applicable to conditions of soft-hard data, fact, evidence and knowledge over the spaces of problem-solution dualities, decision-choice actions in sciences, non-sciences, engineering and planning sciences to abstract acceptable information-knowledge elements.
General Theory of Leibniz Algebras (Synthesis Lectures on Mathematics & Statistics)
by Leonid Kurdachenko Oleksandr Pypka Igor SubbotinThis book discusses many interesting results have been obtained in Leibniz algebras over the past two decades. The authors not only summarize recent results and methods successfully used in Leibniz algebras, but also show new prospective horizons. Any mathematical theories have a number of natural problems that arise in the process of its development, and these problems quite often have analogues in other areas such as differential geometry, homological algebra, classical algebraic topology, noncommutative geometry, etc. With this in mind the authors describe the general structure of Leibniz algebras that have already been discovered. This approach allows readers to see which parts of the theory should be developed further and also shows the significant differences of Leibniz algebras from Lie algebras. Recent results that constitute the naturally evolving general theory of the subject are then explored.
General Theory of Light Propagation and Imaging Through the Atmosphere (Progress in Optical Science and Photonics #20)
by T. Stewart McKechnieThis 2nd edition lays out an updated version of the general theory of light propagation and imaging through Earth’s turbulent atmosphere initially developed in the late ‘70s and ‘80s, with additional applications in the areas of laser communications and high-energy laser beam propagation. New material includes a chapter providing a comprehensive mathematical tool set for precisely characterizing image formation with the anticipated Extremely Large Telescopes (ELTS), enabling a staggering range of star image shapes and sizes; existing chapters rewritten or modified so as to supplement the mathematics with clearer physical insight through written and graphical means; a history of the development of present-day understanding of light propagation and imaging through the atmosphere as represented by the general theory described. Beginning with the rudimentary, geometrical-optics based understanding of a century ago, it describes advances made in the 1960s, including the development of the ‘Kolmogorov theory,’ the deficiencies of which undermined its credibility, but not before it had done enormous damage, such as construction of a generation of underperforming ‘light bucket’ telescopes. The general theory requires no a priori turbulence assumptions. Instead, it provides means for calculating the turbulence properties directly from readily-measurable properties of star images.
The General Theory of Relativity
by Andrew Debenedictis Anadijiban DasThe General Theory of Relativity: A Mathematical Exposition will serve readers as a modern mathematical introduction to the general theory of relativity. Throughout the book, examples, worked-out problems, and exercises (with hints and solutions) are furnished. Topics in this book include, but are not limited to: tensor analysis the special theory of relativity the general theory of relativity and Einstein's field equations spherically symmetric solutions and experimental confirmations static and stationary space-time domains black holes cosmological models algebraic classifications and the Newman-Penrose equations the coupled Einstein-Maxwell-Klein-Gordon equations appendices covering mathematical supplements and special topics Mathematical rigor, yet very clear presentation of the topics make this book a unique text for both university students and research scholars. Anadijiban Das has taught courses on Relativity Theory at The University College of Dublin, Ireland, Jadavpur University, India, Carnegie-Mellon University, USA, and Simon Fraser University, Canada. His major areas of research include, among diverse topics, the mathematical aspects of general relativity theory. Andrew DeBenedictis has taught courses in Theoretical Physics at Simon Fraser University, Canada, and is also a member of The Pacific Institute for the Mathematical Sciences. His research interests include quantum gravity, classical gravity, and semi-classical gravity.
General Topology (Dover Books on Mathematics)
by John L. Kelley"The clarity of the author's thought and the carefulness of his exposition make reading this book a pleasure," noted the Bulletin of the American Mathematical Society upon the 1955 publication of John L. Kelley's General Topology. This comprehensive treatment for beginning graduate-level students immediately found a significant audience, and it remains a highly worthwhile and relevant book for students of topology and for professionals in many areas. A systematic exposition of the part of general topology that has proven useful in several branches of mathematics, this volume is especially intended as background for modern analysis. An extensive preliminary chapter presents mathematical foundations for the main text. Subsequent chapters explore topological spaces, the Moore-Smith convergence, product and quotient spaces, embedding and metrization, and compact, uniform, and function spaces. Each chapter concludes with an abundance of problems, which form integral parts of the discussion as well as reinforcements and counter examples that mark the boundaries of possible theorems. The book concludes with an extensive index that provides supplementary material on elementary set theory.
General Topology
by Stephen WillardAmong the best available reference introductions to general topology, this volume is appropriate for advanced undergraduate and beginning graduate students. Its treatment encompasses two broad areas of topology: "continuous topology," represented by sections on convergence, compactness, metrization and complete metric spaces, uniform spaces, and function spaces; and "geometric topology," covered by nine sections on connectivity properties, topological characterization theorems, and homotopy theory. Many standard spaces are introduced in the related problems that accompany each section (340 exercises in all). The text's value as a reference work is enhanced by a collection of historical notes, a bibliography, and index. 1970 edition. 27 figures.
General Topology and Applications: Fifth Northeast Conference (Annals Of The New York Academy Of Sciences Ser. #Vol. 659)
by Susan J. AndimaThis book is based on the proceedings of the Fifth Northeast Conference on General Topology and Applications, held at The College of Staten Island – The City University of New York. It provides insight into the relationship between general topology and other areas of mathematics.
Generalized Additive Models
by T.J. HastieThis book describes an array of power tools for data analysis that are based on nonparametric regression and smoothing techniques. These methods relax the linear assumption of many standard models and allow analysts to uncover structure in the data that might otherwise have been missed. While McCullagh and Nelder's Generalized Linear Models shows how to extend the usual linear methodology to cover analysis of a range of data types, Generalized Additive Models enhances this methodology even further by incorporating the flexibility of nonparametric regression. Clear prose, exercises in each chapter, and case studies enhance this popular text.