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Combinatorial Identities for Stirling Numbers

Combinatorial Identities for Stirling Numbers Author Jocelyn Quaintance
ISBN-10 9789814725293
Release 2015-10-27
Pages 276
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' This book is a unique work which provides an in-depth exploration into the mathematical expertise, philosophy, and knowledge of H W Gould. It is written in a style that is accessible to the reader with basic mathematical knowledge, and yet contains material that will be of interest to the specialist in enumerative combinatorics. This book begins with exposition on the combinatorial and algebraic techniques that Professor Gould uses for proving binomial identities. These techniques are then applied to develop formulas which relate Stirling numbers of the second kind to Stirling numbers of the first kind. Professor Gould''s techniques also provide connections between both types of Stirling numbers and Bernoulli numbers. Professor Gould believes his research success comes from his intuition on how to discover combinatorial identities. This book will appeal to a wide audience and may be used either as lecture notes for a beginning graduate level combinatorics class, or as a research supplement for the specialist in enumerative combinatorics. Contents:Basic Properties of SeriesThe Binomial TheoremIterative SeriesTwo of Professor Gould''s Favorite Algebraic TechniquesVandermonde ConvolutionThe nth Difference Operator and Euler''s Finite Difference TheoremMelzak''s FormulaGeneralized Derivative FormulasStirling Numbers of the Second Kind S(n; k)Eulerian NumbersWorpitzky NumbersStirling Numbers of the First Kind s(n; k)Explicit Formulas for s(n; n — k)Number Theoretic Definitions of Stirling NumbersBernoulli NumbersAppendix A: Newton-Gregory ExpansionsAppendix B: Generalized Bernoulli and Euler Polynomials Readership: Undergraduates, graduates and researchers interested in combinatorial and algebraic techniques. Key Features:Professor Gould is an acknowledged expert in the field of Stirling number identitiesFor the first time in print, this book collects Professor''s Gould''s vast knowledge on this subject in one accessible locationThis book contains Professor Gould''s unique approaches to discovering and proving binomial identitiesThis book contains many fully-worked detailed proofs of the identities found in H W Gould''s "Combinatorial Identities: A Standardized Set of Tables Listing 500 Binomial Coefficient Summations"Keywords:Stirling Numbers of the First Kind;Stirling Numbers of the Second Kind;Bernoulli Numbers;Generalized Bernoulli Polynomials;Worpitzky Numbers;Eulerian Numbers;Binomial Theorem;Vandermonde Convolution;Euler''s Finite Difference Theorem;Melzak''s Formula "This book is a unique work that could appeal to a wide audience: from graduate students to specialists in enumerative combinatorics, to enthusiasts of Gould''s work." CERN Courier '

The Partition Method for a Power Series Expansion

The Partition Method for a Power Series Expansion Author Victor Kowalenko
ISBN-10 9780128045114
Release 2017-01-19
Pages 322
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The Partition Method for a Power Series Expansion: Theory and Applications explores how the method known as 'the partition method for a power series expansion', which was developed by the author, can be applied to a host of previously intractable problems in mathematics and physics. In particular, this book describes how the method can be used to determine the Bernoulli, cosecant, and reciprocal logarithm numbers, which appear as the coefficients of the resulting power series expansions, then also extending the method to more complicated situations where the coefficients become polynomials or mathematical functions. From these examples, a general theory for the method is presented, which enables a programming methodology to be established. Finally, the programming techniques of previous chapters are used to derive power series expansions for complex generating functions arising in the theory of partitions and in lattice models of statistical mechanics. Explains the partition method by presenting elementary applications involving the Bernoulli, cosecant, and reciprocal logarithm numbers Compares generating partitions via the BRCP algorithm with the standard lexicographic approaches Describes how to program the partition method for a power series expansion and the BRCP algorithm

Nuclear Reactions

Nuclear Reactions Author Hans Paetz gen. Schieck
ISBN-10 9783642539862
Release 2014-02-11
Pages 365
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Nuclei and nuclear reactions offer a unique setting for investigating three (and in some cases even all four) of the fundamental forces in nature. Nuclei have been shown – mainly by performing scattering experiments with electrons, muons and neutrinos – to be extended objects with complex internal structures: constituent quarks; gluons, whose exchange binds the quarks together; sea-quarks, the ubiquitous virtual quark-antiquark pairs and last but not least, clouds of virtual mesons, surrounding an inner nuclear region, their exchange being the source of the nucleon-nucleon interaction. The interplay between the (mostly attractive) hadronic nucleon-nucleon interaction and the repulsive Coulomb force is responsible for the existence of nuclei; their degree of stability, expressed in the details and limits of the chart of nuclides; their rich structure and the variety of their interactions. Despite the impressive successes of the classical nuclear models and of ab-initio approaches, there is clearly no end in sight for either theoretical or experimental developments as shown e.g. by the recent need to introduce more sophisticated three-body interactions to account for an improved picture of nuclear structure and reactions. Yet, it turns out that the internal structure of the nucleons has comparatively little influence on the behavior of the nucleons in nuclei and nuclear physics – especially nuclear structure and reactions – is thus a field of science in its own right, without much recourse to subnuclear degrees of freedom. This book collects essential material that was presented in the form of lectures notes in nuclear physics courses for graduate students at the University of Cologne. It follows the course's approach, conveying the subject matter by combining experimental facts and experimental methods and tools with basic theoretical knowledge. Emphasis is placed on the importance of spin and orbital angular momentum (leading e.g. to applications in energy research, such as fusion with polarized nuclei) and on the operational definition of observables in nuclear physics. The end-of-chapter problems serve above all to elucidate and detail physical ideas that could not be presented in full detail in the main text. Readers are assumed to have a working knowledge of quantum mechanics and a basic grasp of both non-relativistic and relativistic kinematics; the latter in particular is a prerequisite for interpreting nuclear reactions and the connections to particle and high-energy physics.

Notes On The Binomial Transform Theory And Table With Appendix On Stirling Transform

Notes On The Binomial Transform  Theory And Table With Appendix On Stirling Transform Author Boyadzhiev Khristo N
ISBN-10 9789813234994
Release 2018-04-10
Pages 208
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The binomial transform is a discrete transformation of one sequence into another with many interesting applications in combinatorics and analysis. This volume is helpful to researchers interested in enumerative combinatorics, special numbers, and classical analysis. A valuable reference, it can also be used as lecture notes for a course in binomial identities, binomial transforms and Euler series transformations. The binomial transform leads to various combinatorial and analytical identities involving binomial coefficients. In particular, we present here new binomial identities for Bernoulli, Fibonacci, and harmonic numbers. Many interesting identities can be written as binomial transforms and vice versa. The volume consists of two parts. In the first part, we present the theory of the binomial transform for sequences with a sufficient prerequisite of classical numbers and polynomials. The first part provides theorems and tools which help to compute binomial transforms of different sequences and also to generate new binomial identities from the old. These theoretical tools (formulas and theorems) can also be used for summation of series and various numerical computations. In the second part, we have compiled a list of binomial transform formulas for easy reference. In the Appendix, we present the definition of the Stirling sequence transform and a short table of transformation formulas. Contents: Theory of the Binomial Transform: Introduction Prerequisite: Special Numbers and Polynomials Euler's Transformation for Series Melzak's Formula and Related Formulas Special Properties. Creating New Identities Binomial Transforms of Products Special Formulas and Power Series with Binomial Sums Table of Binomial Transforms: Assorted Binomial Formulas Identities Involving Harmonic Numbers Transforms of Binomial Coefficients Transforms of Special Numbers and Polynomials Transforms of Trigonometric and Hyperbolic Functions and Applications to Some Trigonometric Integrals Transforms of Some Special Functions Appendix: The Stirling Transform of Sequences Readership: Graduate and researchers in the areas of number theory, discrete mathematics, combinatorics, statistics working with applications using the binomial transform. Keywords: Binomial Coefficients;Binomial Identities;Binomial Sums;Binomial Transform;Euler's Series Transformation;Discrete Mathematics;Finite Differences;Stirling Numbers of the First Kind;Stirling Numbers of the Second Kind;Stirling Transform;Special Numbers and Polynomials;Harmonic Numbers;Bernoulli Numbers;Fibonacci Numbers;Melzak's Formula;Exponential Polynomials;Geometric Polynomials;Laguerre Polynomials;Trigonometric IntegralsReview: Key Features: This is the first, long-overdue book on the subject. (At present, there are no competing books) The book provides interesting new material for researchers in discrete mathematics and will serve as a valuable reference for binomial identities, binomial transform formulas, and Euler series transformations

Lecture Notes on Electron Correlation and Magnetism

Lecture Notes on Electron Correlation and Magnetism Author Patrik Fazekas
ISBN-10 9789810224745
Release 1999-01
Pages 777
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Readership: Graduate students and researchers in condensed matter physics.

A Comprehensive Treatment of q Calculus

A Comprehensive Treatment of q Calculus Author Thomas Ernst
ISBN-10 9783034804301
Release 2012-09-08
Pages 492
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To date, the theoretical development of q-calculus has rested on a non-uniform basis. Generally, the bulky Gasper-Rahman notation was used, but the published works on q-calculus looked different depending on where and by whom they were written. This confusion of tongues not only complicated the theoretical development but also contributed to q-calculus remaining a neglected mathematical field. This book overcomes these problems by introducing a new and interesting notation for q-calculus based on logarithms.For instance, q-hypergeometric functions are now visually clear and easy to trace back to their hypergeometric parents. With this new notation it is also easy to see the connection between q-hypergeometric functions and the q-gamma function, something that until now has been overlooked. The book covers many topics on q-calculus, including special functions, combinatorics, and q-difference equations. Apart from a thorough review of the historical development of q-calculus, this book also presents the domains of modern physics for which q-calculus is applicable, such as particle physics and supersymmetry, to name just a few.​

The Legacy of Alladi Ramakrishnan in the Mathematical Sciences

The Legacy of Alladi Ramakrishnan in the Mathematical Sciences Author Krishnaswami Alladi
ISBN-10 1441962638
Release 2010-08-26
Pages 575
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In the spirit of Alladi Ramakrishnan’s profound interest and contributions to three fields of science — Mathematics, Statistics, and Physics — this volume contains invited surveys and research articles from prominent members of these communities who also knew Ramakrishnan personally and greatly respected his influence in these areas of science. Historical photos, telegrams, and biographical narratives of Alladi Ramakrishnan’s illustrious career of special interest are included as well.

Like a Splinter in Your Mind

Like a Splinter in Your Mind Author Matt Lawrence
ISBN-10 1405125241
Release 2004-07-16
Pages 232
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Like a Splinter in Your Mind leads readers through the myriad of philosophical themes within the Matrix trilogy, helping them to gain a better understanding of the films and of philosophy itself. Offers a way into philosophy through the Matrix films. Covers thirteen of the biggest philosophical questions in thirteen self-sufficient chapters suitable for course use. Demonstrates how each of these questions is illustrated through the events and characters of the films. Considers whether sentient machines are possible, and whether we should expect them to face the same existentialist issues that we do. Familiarises readers with key issues in metaphysics, epistemology, ethics, philosophy of mind, race and gender, existentialism, Taoism and mysticism. Includes a chapter that explains some of the technical elements of the films and confusing aspects of the plot. Also includes a Matrix glossary, and a cast of characters and their related symbolism.

Fighting Infection in the 21st Century

Fighting Infection in the 21st Century Author P. W. Andrew
ISBN-10 063205817X
Release 2000-11-28
Pages 288
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The 20th century has seen improvements in both public health and disease prevention which, in turn, have had a dramatic impact on our lives. Success in preventing infection by vaccination and treating infection with antibiotics led some to believe that infectious disease was a thing of the past. However, the adaptability of pathogens and the emergence of new diseases has presented microbiologists with a fresh set of challenges as we enter the new millennium. While celebrating past successes and highlighting developing problems, this volume aims to address some of the issues facing microbiologists in the future. Covering a wide range of topics, it will provide an invaluable resource for microbiologists and an excellent reference for advanced students.

The Craft of Probabilistic Modelling

The Craft of Probabilistic Modelling Author J. Gani
ISBN-10 9781461386315
Release 2012-12-06
Pages 313
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This book brings together the personal accounts and reflections of nineteen mathematical model-builders, whose specialty is probabilistic modelling. The reader may well wonder why, apart from personal interest, one should commission and edit such a collection of articles. There are, of course, many reasons, but perhaps the three most relevant are: (i) a philosophicaJ interest in conceptual models; this is an interest shared by everyone who has ever puzzled over the relationship between thought and reality; (ii) a conviction, not unsupported by empirical evidence, that probabilistic modelling has an important contribution to make to scientific research; and finally (iii) a curiosity, historical in its nature, about the complex interplay between personal events and the development of a field of mathematical research, namely applied probability. Let me discuss each of these in turn. Philosophical Abstraction, the formation of concepts, and the construction of conceptual models present us with complex philosophical problems which date back to Democritus, Plato and Aristotle. We have all, at one time or another, wondered just how we think; are our thoughts, concepts and models of reality approxim&tions to the truth, or are they simply functional constructs helping us to master our environment? Nowhere are these problems more apparent than in mathematical model ling, where idealized concepts and constructions replace the imperfect realities for which they stand.

Engineering Optimization

Engineering Optimization Author Xin-She Yang
ISBN-10 0470640413
Release 2010-07-20
Pages 384
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An accessible introduction to metaheuristics and optimization, featuring powerful and modern algorithms for application across engineering and the sciences From engineering and computer science to economics and management science, optimization is a core component for problem solving. Highlighting the latest developments that have evolved in recent years, Engineering Optimization: An Introduction with Metaheuristic Applications outlines popular metaheuristic algorithms and equips readers with the skills needed to apply these techniques to their own optimization problems. With insightful examples from various fields of study, the author highlights key concepts and techniques for the successful application of commonly-used metaheuristc algorithms, including simulated annealing, particle swarm optimization, harmony search, and genetic algorithms. The author introduces all major metaheuristic algorithms and their applications in optimization through a presentation that is organized into three succinct parts: Foundations of Optimization and Algorithms provides a brief introduction to the underlying nature of optimization and the common approaches to optimization problems, random number generation, the Monte Carlo method, and the Markov chain Monte Carlo method Metaheuristic Algorithms presents common metaheuristic algorithms in detail, including genetic algorithms, simulated annealing, ant algorithms, bee algorithms, particle swarm optimization, firefly algorithms, and harmony search Applications outlines a wide range of applications that use metaheuristic algorithms to solve challenging optimization problems with detailed implementation while also introducing various modifications used for multi-objective optimization Throughout the book, the author presents worked-out examples and real-world applications that illustrate the modern relevance of the topic. A detailed appendix features important and popular algorithms using MATLAB® and Octave software packages, and a related FTP site houses MATLAB code and programs for easy implementation of the discussed techniques. In addition, references to the current literature enable readers to investigate individual algorithms and methods in greater detail. Engineering Optimization: An Introduction with Metaheuristic Applications is an excellent book for courses on optimization and computer simulation at the upper-undergraduate and graduate levels. It is also a valuable reference for researchers and practitioners working in the fields of mathematics, engineering, computer science, operations research, and management science who use metaheuristic algorithms to solve problems in their everyday work.

Concrete Mathematics

Concrete Mathematics Author Ronald L. Graham
ISBN-10 0201558025
Release 1994-01-01
Pages 657
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This book, updated and improved, introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills--the skills needed to solve complex problems, to evaluate horrendous-looking sums, to solve complex recurrence relations, and to discover subtle patterns in data. It is an indispensable text and reference, not only for computer scientists but for all technical professionals in virtually every discipline.

Discrete Approximation Theory

Discrete Approximation Theory Author George A Anastassiou
ISBN-10 9789813145856
Release 2016-09-29
Pages 348
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In this monograph, we present the authors' recent work of the last seven years in Approximation Theory. Chapters are self-contained and can be read independently and advanced courses can be taught out of this book. Here our generalized discrete singular operators are of the following types: Picard, Gauss–Weierstrass and Poisson–Cauchy operators. We treat both the unitary and non-unitary, univariate and multivariate cases of these operators, which are not necessarily positive operators. The book's results are expected to find applications in many areas of pure and applied mathematics, and statistics. As such, it is suitable for researchers, graduate students, and seminars of related subjects, and serves well as an invaluable resource for all science libraries.

Irresistible Integrals

Irresistible Integrals Author George Boros
ISBN-10 0521796369
Release 2004-06-21
Pages 306
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The problem of evaluating integrals is well known to every student who has had a year of calculus. It was an especially important subject in nineteenth century analysis and it has now been revived with the appearance of symbolic languages. The authors use the problem of exact evaluation of definite integrals as a starting point for exploring many areas of mathematics. The questions discussed are as old as calculus itself. In presenting the combination of methods required for the evaluation of most integrals, the authors take the most interesting-rather than the shortest-path to the results. They illuminate connections with many subjects, including analysis, number theory, algebra and combinatorics. This is a guided tour of exciting discovery for undergraduates and their teachers in mathematics, computer science, physics, and engineering.

Quantum Calculus

Quantum Calculus Author Bashir Ahmad
ISBN-10 9789813141544
Release 2016-06-07
Pages 288
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The main objective of this book is to extend the scope of the q-calculus based on the definition of q-derivative [Jackson (1910)] to make it applicable to dense domains. As a matter of fact, Jackson's definition of q-derivative fails to work for impulse points while this situation does not arise for impulsive equations on q-time scales as the domains consist of isolated points covering the case of consecutive points. In precise terms, we study quantum calculus on finite intervals. In the first part, we discuss the concepts of qk-derivative and qk-integral, and establish their basic properties. As applications, we study initial and boundary value problems of impulsive qk-difference equations and inclusions equipped with different kinds of boundary conditions. We also transform some classical integral inequalities and develop some new integral inequalities for convex functions in the context of qk-calculus. In the second part, we develop fractional quantum calculus in relation to a new qk-shifting operator and establish some existence and qk uniqueness results for initial and boundary value problems of impulsive fractional qk-difference equations. Contents:PreliminariesQuantum Calculus on Finite IntervalsInitial Value Problems for Impulsive qk-Difference Equations and InclusionsBoundary Value Problems for First-Order Impulsive qk-Integro-Difference Equations and InclusionsImpulsive qk-Difference Equations with Different Kinds of Boundary ConditionsNonlinear Second-Order Impulsive qk-Difference Langevin Equation with Boundary ConditionsQuantum Integral Inequalities on Finite IntervalsImpulsive Quantum Difference Systems with Boundary ConditionsNew Concepts of Fractional Quantum Calculus and Applications to Impulsive Fractional qk-Difference EquationsIntegral Inequalities via Fractional Quantum CalculusNonlocal Boundary Value Problems for Impulsive Fractional qk-Difference EquationsExistence Results for Impulsive Fractional qk-Difference Equations with Anti-periodic Boundary ConditionsImpulsive Fractional qk-Integro-Difference Equations with Boundary ConditionsImpulsive Hybrid Fractional Quantum Difference Equations Readership: Mathematics and physics researchers.

Encyclopedia of Mathematics and Society

Encyclopedia of Mathematics and Society Author Sarah J. Greenwald
ISBN-10 1587658461
Release 2012
Pages 1191
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Presents articles showing the math behind our daily lives. Explains how and why math works, and allows readers to better understand how disciplines such as algebra, geometry, calculus, and others affect what we do every day.

Lecture Notes on Generalized Heegaard Splittings

Lecture Notes on Generalized Heegaard Splittings Author Martin Scharlemann
ISBN-10 9789813109131
Release 2016-04-25
Pages 140
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This book is part of the series of three books arise from lectures organized by Hitoshi Murakami at RIMS, Kyoto University in the summer of 2001. The lecture series was aimed at a broad audience that included many graduate students. Its purpose lay in familiarizing the audience with the basics of 3-manifold theory and introducing some topics of current research. The first portion of the lecture series was devoted to standard topics in the theory of 3-manifolds. The middle portion was devoted to a brief study of Heegaard splittings and generalized Heegaard splittings. In the standard schematic diagram for generalized Heegaard splittings, Heegaard splittings are stacked on top of each other in a linear fashion. This can cause confusion in those cases in which generalized Heegaard splittings possess interesting connectivity properties. Fork complexes were invented in an effort to illuminate some of the more subtle issues arising in the study of generalized Heegaard splittings. Contents:PreliminariesDefinition and Examples of Heegaard SplittingsProperties of Heegaard SplittingsTwo Theorems on Heegaard SplittingsGeneralized Heegaard Splittings Readership: Graduate students and researchers in topology.