1,074 results on '"Mathematics"'
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2. The Big Book of Real Analysis : From Numbers to Measures
- Author
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Syafiq Johar and Syafiq Johar
- Subjects
- Mathematics, Mathematical analysis, Sequences (Mathematics), Differential equations, Measure theory, Functions of real variables
- Abstract
This book provides an introduction to real analysis, a fundamental topic that is an essential requirement in the study of mathematics. It deals with the concepts of infinity and limits, which are the cornerstones in the development of calculus. Beginning with some basic proof techniques and the notions of sets and functions, the book rigorously constructs the real numbers and their related structures from the natural numbers. During this construction, the readers will encounter the notions of infinity, limits, real sequences, and real series. These concepts are then formalised and focused on as stand-alone objects. Finally, they are expanded to limits, sequences, and series of more general objects such as real-valued functions. Once the fundamental tools of the trade have been established, the readers are led into the classical study of calculus (continuity, differentiation, and Riemann integration) from first principles. The book concludes with an introduction to the studyof measures and how one can construct the Lebesgue integral as an extension of the Riemann integral. This textbook is aimed at undergraduate students in mathematics. As its title suggests, it covers a large amount of material, which can be taught in around three semesters. Many remarks and examples help to motivate and provide intuition for the abstract theoretical concepts discussed. In addition, more than 600 exercises are included in the book, some of which will lead the readers to more advanced topics and could be suitable for independent study projects. Since the book is fully self-contained, it is also ideal for self-study.
- Published
- 2024
3. PROJECT-BASED LEARNING (ABP) IN THE TEACHING OF MATHEMATICS IN THE CONTEXT OF COMPUTER PROGRAMMING.
- Author
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Inca Balseca, Cristian Luis, Inca Balseca, Evelyn Geovanna, Morocho Orellana, Julio Cesar, Morocho Caiza, Andrés Fernando, Coronel Maji, Franklin Marcelo, and Silva Godoy, Lizeth Fernanda
- Subjects
ACHIEVEMENT motivation ,COMPUTER programming ,PROJECT method in teaching ,ACADEMIC motivation ,DIFFERENTIAL equations ,MATHEMATICS - Abstract
Mathematics is one of the areas of knowledge where there are greater difficulties for its learning, due to its complexity, and the little usefulness that is perceived of them in real life, which is why pedagogical alternatives have been developed ( ABP). The usefulness of this tool in learning mathematics, which is sought by solving real problems through the use of mathematics. In this sense, the use of the PBL was evaluated in a group of 60 students of the Software career of the Polytechnic High School of Chimborazo (ESPOCH), 30 before the use of APB and 30 after its use comparing the changes in the academic performance of the students after the implementation of the PBL, as well as the perception of the students in relation to the interest, value, motivation and utilities of mathematics by solving problems using differential equations, The results found show that the students had a greater achievement when understanding and solving the equations adequately using derivation rules and analyzing and adequately interpret the information, which was related to a greater interest (50.85%), assessment (57.36%), satisfaction 57.62 (%) and utility 66.07%, despite the fact that the use of the PBL achieved a significant increase in grades, even the students state that they make a great effort to carry out projects in the area of mathematics, which was expressed by more than half of the students approached, the findings found lead to the conclusion that the use of this learning tool translates into a significant improvement in the acquisition of mathematical skills since a greater understanding and importance of the use of the itself, increases the interest and motivation of students for their study. [ABSTRACT FROM AUTHOR]
- Published
- 2023
4. New Trends in the Applications of Differential Equations in Sciences : NTADES 2022, Sozopol, Bulgaria, June 14–17
- Author
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Angela Slavova and Angela Slavova
- Subjects
- Differential equations, Mathematical analysis, Mathematical physics, Mathematics
- Abstract
This book convenes peer-reviewed, selected papers presented at the Ninth International Conference New Trends in the Applications of Differential Equations in Sciences (NTADES) held in Sozopol, Bulgaria, June 17–20, 2022. The works are devoted to many applications of differential equations in different fields of science. A number of phenomena in nature (physics, chemistry, biology) and in society (economics) result in problems leading to the study of linear and nonlinear differential equations, stochastic equations, statistics, analysis, numerical analysis, optimization, and more. The main topics are presented in the five parts of the book - applications in mathematical physics, mathematical biology, financial mathematics, neuroscience, and fractional analysis.In this volume, the reader will find a wide range of problems concerning recent achievements in both theoretical and applied mathematics. The main goal is to promote the exchange of new ideas and research between scientists, who develop and study differential equations, and researchers, who apply them for solving real-life problems. The book promotes basic research in mathematics leading to new methods and techniques useful for applications of differential equations.The NTADES 2022 conference was organized in cooperation with the Society of Industrial and Applied Mathematics (SIAM), the major international organization for Industrial and Applied Mathematics and for the promotion of interdisciplinary collaboration between applied mathematics and science, engineering, finance, and neuroscience.
- Published
- 2023
5. Calculus III : Practice Problems, Methods, and Solutions
- Author
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Mehdi Rahmani-Andebili and Mehdi Rahmani-Andebili
- Subjects
- Engineering mathematics, Mathematics, Mathematical analysis, Mathematical optimization, Calculus of variations, Differential equations
- Abstract
This study guide is designed for students taking a Calculus III course. The textbook includes examples, questions, and practice problems that will help students to review and sharpen their knowledge of the subject and enhance their performance in the classroom. The material covered in the book includes linear algebra and analytical geometry; lines, surfaces, and vector functions in three-dimensional coordinate systems; multiple-variable functions; multiple integrals and their applications; line integrals and their applications. Offering detailed solutions, multiple methods for solving problems, and clear explanations of concepts, this hands-on guide will improve students'problem-solving skills and foster a solid understanding of calculus, which will benefit them in all of their calculus-based courses.
- Published
- 2023
6. Analysis, Applications, and Computations : Proceedings of the 13th ISAAC Congress, Ghent, Belgium, 2021
- Author
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Uwe Kähler, Michael Reissig, Irene Sabadini, Jasson Vindas, Uwe Kähler, Michael Reissig, Irene Sabadini, and Jasson Vindas
- Subjects
- Mathematical analysis, Differential equations, Mathematics
- Abstract
This volume contains the contributions of the participants of the 13th International ISAAC Congress 2021, held in Ghent, Belgium.The papers, written by respected international experts, address recent results in mathematics, with a special focus on analysis. The volume provides to both specialists and non-specialists an excellent source of information on current research in mathematical analysis and its various interdisciplinary applications.
- Published
- 2023
7. Advanced Topics in Fractional Differential Equations : A Fixed Point Approach
- Author
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Mouffak Benchohra, Erdal Karapinar, Jamal Eddine Lazreg, Abdelkrim Salim, Mouffak Benchohra, Erdal Karapinar, Jamal Eddine Lazreg, and Abdelkrim Salim
- Subjects
- Differential equations, Mathematics, Mathematical analysis, Integral equations, Difference equations, Functional equations, Functional analysis
- Abstract
This book explores fractional differential equations with a fixed point approach. The authors highlight the existence, uniqueness, and stability results for various classes of fractional differential equations. All of the problems in the book also deal with some form of of the well-known Hilfer fractional derivative, which unifies the Riemann-Liouville and Caputo fractional derivatives. Classical and new fixed point theorems, associated with the measure of noncompactness in Banach spaces as well as several generalizations of the Gronwall's lemma, are employed as tools. The book is based on many years of research in this area, and provides suggestions for further study as well. The authors have included illustrations in order to support the readers'understanding of the concepts presented. Includes illustrations in order to support readers understanding of the presented concepts · Approaches the topic of fractional differential equations while employing fixed point theorems as tools · Presents novel results, which build upon previous literature and many years of research by the authors
- Published
- 2023
8. Boundary Value Problems : Advanced Fractional Dynamic Equations on Time Scales
- Author
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Svetlin Georgiev and Svetlin Georgiev
- Subjects
- Mathematics, Mathematical analysis, Differential equations, Dynamical systems, Special functions
- Abstract
This book explores boundary value problems for fractional dynamic equations on arbitrary time scales, including Caputo fractional dynamic equations, impulsive Caputo fractional dynamic equations, and impulsive Riemann-Liouville fractional dynamic equations. The author provides an introduction to each fractional dynamic equation before delving into the problems. The book also covers initial value problems, boundary value problems, initial boundary value problems for each type of equation. The author provides integral representations of the solutions and proves the existence and uniqueness of the solutions.
- Published
- 2023
9. Boundary Value Problems : Essential Fractional Dynamic Equations on Time Scales
- Author
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Svetlin Georgiev and Svetlin Georgiev
- Subjects
- Mathematics, Mathematical analysis, Differential equations, Dynamical systems, Dynamics, Nonlinear theories, Special functions
- Abstract
This book explores boundary value problems for Riemann-Liouville fractional dynamic equations on arbitrary time scales as well as the shifting problem on the whole time scale. The author includes an introductory overview of fractional dynamic calculus on time scales. The book also introduces the Laplace transform on arbitrary time scales, including the bilateral Laplace transform, the Laplace transform of power series, and a deduction of an inverse formula. The author then discusses the generalized convolutions of functions on arbitrary time scales and the shifting problem for existence of solutions. The book moves on to cover boundary value problems and initial boundary value problems for some classes Riemann-Liouville fractional dynamic equations.
- Published
- 2023
10. Fractional Differential Equations : New Advancements for Generalized Fractional Derivatives
- Author
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Mouffak Benchohra, Erdal Karapınar, Jamal Eddine Lazreg, Abdelkrim Salim, Mouffak Benchohra, Erdal Karapınar, Jamal Eddine Lazreg, and Abdelkrim Salim
- Subjects
- Differential equations, Mathematics, Mathematical analysis, Integral equations, Difference equations, Functional equations, Functional analysis
- Abstract
This book covers problems involving a variety of fractional differential equations, as well as some involving the generalized Hilfer fractional derivative, which unifies the Riemann-Liouville and Caputo fractional derivatives. The authors highlight the existence, uniqueness, and stability results for various classes of fractional differential equations based on the most recent research in the area. The book discusses the classic and novel fixed point theorems related to the measure of noncompactness in Banach spaces and explains how to utilize them as tools. The authors build each chapter upon the previous one, helping readers to develop their understanding of the topic. The book includes illustrated results, analysis, and suggestions for further study.
- Published
- 2023
11. Analytical Methods for Solving Nonlinear Partial Differential Equations
- Author
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Daniel Arrigo and Daniel Arrigo
- Subjects
- Differential equations, Mathematics, Mathematical analysis
- Abstract
This textbook provides an introduction to methods for solving nonlinear partial differential equations (NLPDEs). After the introduction of several PDEs drawn from science and engineering, readers are introduced to techniques to obtain exact solutions of NLPDEs. The chapters include the following topics: Nonlinear PDEs are Everywhere; Differential Substitutions; Point and Contact Transformations; First Integrals; and Functional Separability. Readers are guided through these chapters and are provided with several detailed examples. Each chapter ends with a series of exercises illustrating the material presented in each chapter. This Second Edition includes a new method of generating contact transformations and focuses on a solution method (parametric Legendre transformations) to solve a particular class of two nonlinear PDEs.
- Published
- 2022
12. Multi-Valued Variational Inequalities and Inclusions
- Author
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Siegfried Carl, Vy Khoi Le, Siegfried Carl, and Vy Khoi Le
- Subjects
- Mathematical analysis, Differential equations, Operator theory, Mathematical optimization, Calculus of variations, Mathematics
- Abstract
This book focuses on a large class of multi-valued variational differential inequalities and inclusions of stationary and evolutionary types with constraints reflected by subdifferentials of convex functionals. Its main goal is to provide a systematic, unified, and relatively self-contained exposition of existence, comparison and enclosure principles, together with other qualitative properties of multi-valued variational inequalities and inclusions. The problems under consideration are studied in different function spaces such as Sobolev spaces, Orlicz-Sobolev spaces, Sobolev spaces with variable exponents, and Beppo-Levi spaces. A general and comprehensive sub-supersolution method (lattice method) is developed for both stationary and evolutionary multi-valued variational inequalities, which preserves the characteristic features of the commonly known sub-supersolution method for single-valued, quasilinear elliptic and parabolic problems. This method provides a powerful tool forstudying existence and enclosure properties of solutions when the coercivity of the problems under consideration fails. It can also be used to investigate qualitative properties such as the multiplicity and location of solutions or the existence of extremal solutions. This is the first in-depth treatise on the sub-supersolution (lattice) method for multi-valued variational inequalities without any variational structures, together with related topics. The choice of the included materials and their organization in the book also makes it useful and accessible to a large audience consisting of graduate students and researchers in various areas of Mathematical Analysis and Theoretical Physics.
- Published
- 2021
13. Qualitative Behavior of Unbounded Solutions of Neutral Differential Equations of Third-Order.
- Author
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Kumar, M. Sathish, Elayaraja, R., Ganesan, V., Bazighifan, Omar, Al-Shaqsi, Khalifa, and Nonlaopon, Kamsing
- Subjects
- *
DIFFERENTIAL equations , *MATHEMATICS theorems , *MATHEMATICAL analysis , *BINOMIAL theorem , *MATHEMATICS - Abstract
New oscillatory properties for the oscillation of unbounded solutions to a class of thirdorder neutral differential equations with several deviating arguments are established. Several oscillation results are established by using generalized Riccati transformation and a integral average technique under the case of unbounded neutral coefficients. Examples are given to prove the significance of new theorems. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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14. Mathematical Analysis I
- Author
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Claudio Canuto, Anita Tabacco, Claudio Canuto, and Anita Tabacco
- Subjects
- Differential equations, Partial, Mathematical analysis, Mathematics, Differential equations
- Abstract
The purpose of the volume is to provide a support for a first course in Mathematics. The contents are organised to appeal especially to Engineering, Physics and Computer Science students, all areas in which mathematical tools play a crucial role. Basic notions and methods of differential and integral calculus for functions of one real variable are presented in a manner that elicits critical reading and prompts a hands-on approach to concrete applications. The layout has a specifically-designed modular nature, allowing the instructor to make flexible didactical choices when planning an introductory lecture course. The book may in fact be employed at three levels of depth. At the elementary level the student is supposed to grasp the very essential ideas and familiarise with the corresponding key techniques. Proofs to the main results befit the intermediate level, together with several remarks and complementary notes enhancing the treatise. The last, and farthest-reaching, level requiresthe additional study of the material contained in the appendices, which enable the strongly motivated reader to explore further into the subject. Definitions and properties are furnished with substantial examples to stimulate the learning process. Over 350 solved exercises complete the text, at least half of which guide the reader to the solution. This new edition features additional material with the aim of matching the widest range of educational choices for a first course of Mathematics.
- Published
- 2015
15. Local Minimization, Variational Evolution and Γ-Convergence
- Author
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Andrea Braides and Andrea Braides
- Subjects
- Mathematics, Differential equations, Mathematical optimization, Calculus of variations, Approximation theory, Mathematical analysis, Functional analysis
- Abstract
This book addresses new questions related to the asymptotic description of converging energies from the standpoint of local minimization and variational evolution. It explores the links between Gamma-limits, quasistatic evolution, gradient flows and stable points, raising new questions and proposing new techniques. These include the definition of effective energies that maintain the pattern of local minima, the introduction of notions of convergence of energies compatible with stable points, the computation of homogenized motions at critical time-scales through the definition of minimizing movement along a sequence of energies, the use of scaled energies to study long-term behavior or backward motion for variational evolutions. The notions explored in the book are linked to existing findings for gradient flows, energetic solutions and local minimizers, for which some generalizations are also proposed.
- Published
- 2014
16. Nonlinear Partial Differential Equations for Scientists and Engineers
- Author
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Lokenath Debnath and Lokenath Debnath
- Subjects
- Mathematical analysis, Differential equations, Mathematics, Engineering mathematics, Engineering—Data processing
- Abstract
'An exceptionally complete overview. There are numerous examples and the emphasis is on applications to almost all areas of science and engineering. There is truly something for everyone here. This reviewer feels that it is a very hard act to follow, and recommends it strongly. [This book] is a jewel.'- Applied Mechanics Review (Review of First Edition) This expanded and revised second edition is a comprehensive and systematic treatment of linear and nonlinear partial differential equations and their varied applications. Building upon the successful material of the first book, this edition contains updated modern examples and applications from areas of fluid dynamics, gas dynamics, plasma physics, nonlinear dynamics, quantum mechanics, nonlinear optics, acoustics, and wave propagation. Methods and properties of solutions are presented, along with their physical significance, making the book more useful for a diverse readership.
- Published
- 2013
17. Direct and Inverse Problems of Mathematical Physics
- Author
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R.P. Gilbert, Joji Kajiwara, Yongzhi S. Xu, R.P. Gilbert, Joji Kajiwara, and Yongzhi S. Xu
- Subjects
- Differential equations, Mathematical analysis, Physics, Astronomy, Functions of complex variables, Mathematics, Mathematical optimization
- Abstract
This volume consists of papers presented in the special sessions on'Wave Phenomena and Related Topics', and'Asymptotics and Homogenization'of the ISAAC'97 Congress held at the University of Delaware, during June 2-7, 1997. The ISAAC Congress coincided with a U.S.-Japan Seminar also held at the University of Delaware. The latter was supported by the National Science Foundation through Grant INT -9603029 and the Japan Society for the Promotion of Science through Grant MTCS-134. It was natural that the'participants of both meetings should interact and consequently several persons attending the Congress also presented papers in the Seminar. The success of the ISAAC Congress and the U.S.-Japan Seminar has led to the ISAAC'99 Congress being held in Fukuoka, Japan during August 1999. Many of the same participants will return to this Seminar. Indeed, it appears that the spirit of the U.S.-Japan Seminar will be continued every second year as part of the ISAAC Congresses. We decided to include with the papers presented in the ISAAC Congress and the U.S.-Japan Seminar several very good papers by colleagues from the former Soviet Union. These participants in the ISAAC Congress attended at their own expense. This volume has the title Direct and Inverse Problems of Mathematical Physics which consists of the papers on scattering theory, coefficient identification, uniqueness and existence theorems, boundary controllability, wave propagation in stratified media, viscous flows, nonlinear acoustics, Sobolev spaces, singularity theory, pseudo differential operators, and semigroup theory.
- Published
- 2013
18. A Primer on PDEs : Models, Methods, Simulations
- Author
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Sandro Salsa, Federico Vegni, Anna Zaretti, Paolo Zunino, Sandro Salsa, Federico Vegni, Anna Zaretti, and Paolo Zunino
- Subjects
- Mathematics, Differential equations, Mathematical analysis
- Abstract
This book is designed as an advanced undergraduate or a first-year graduate course for students from various disciplines like applied mathematics, physics, engineering. It has evolved while teaching courses on partial differential equations during the last decade at the Politecnico of Milan. The main purpose of these courses was twofold: on the one hand, to train the students to appreciate the interplay between theory and modelling in problems arising in the applied sciences and on the other hand to give them a solid background for numerical methods, such as finite differences and finite elements.
- Published
- 2013
19. Fuzzy Partial Differential Equations and Relational Equations : Reservoir Characterization and Modeling
- Author
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Masoud Nikravesh, Lofti A. Zadeh, Victor Korotkikh, Masoud Nikravesh, Lofti A. Zadeh, and Victor Korotkikh
- Subjects
- Engineering mathematics, Engineering—Data processing, Mathematics, Mathematical analysis, Dynamical systems, Differential equations, Mathematics—Data processing
- Abstract
During last decade significant progress has been made in the oil indus try by using soft computing technology. Underlying this evolving technology there have, been ideas transforming the very language we use to describe problems with imprecision, uncertainty and partial truth. These developments offer exciting opportunities, but at the same time it is becoming clearer that further advancements are confronted by funda mental problems. The whole idea of how human process information lies at the core of the challenge. There are already new ways of thinking about the problems within theory of perception-based information. This theory aims to understand and harness the laws of human perceptions to dramatically im prove the processing of information. A matured theory of perception-based information is likely to be proper positioned to contribute to the solution of the problems and provide all the ingredients for a revolution in science, technology and business. In this context, Berkeley Initiative in Soft Computing (BISC), Univer sity of California, Berkeley from one side and Chevron-Texaco from another formed a Technical Committee to organize a Meeting entitled'State of the Art Assessment and New Directions for Research'to understand the signifi cance of the fields accomplishments, new developments and future directions. The Technical Committee selected and invited 15 scientists (and oil indus try experts as technical committee members) from the related disciplines to participate in the Meeting, which took place at the University of California, Berkeley, and March 15-17, 2002.
- Published
- 2013
20. The Implicit Function Theorem : History, Theory, and Applications
- Author
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Steven G. Krantz, Harold R. Parks, Steven G. Krantz, and Harold R. Parks
- Subjects
- Mathematical analysis, Differential equations, Geometry, Differential, Mathematics, History
- Abstract
The implicit function theorem is part of the bedrock of mathematical analysis and geometry. Finding its genesis in eighteenth century studies of real analytic functions and mechanics, the implicit and inverse function theorems have now blossomed into powerful tools in the theories of partial differential equations, differential geometry, and geometric analysis.There are many different forms of the implicit function theorem, including (i) the classical formulation for Ck functions, (ii) formulations in other function spaces, (iii) formulations for non-smooth function, and (iv) formulations for functions with degenerate Jacobian. Particularly powerful implicit function theorems, such as the Nash–Moser theorem, have been developed for specific applications (e.g., the imbedding of Riemannian manifolds). All of these topics, and many more, are treated in the present uncorrected reprint of this classic monograph. Originally published in 2002, The Implicit Function Theorem is an accessible and thorough treatment of implicit and inverse function theorems and their applications. It will be of interest to mathematicians, graduate/advanced undergraduate students, and to those who apply mathematics. The book unifies disparate ideas that have played an important role in modern mathematics. It serves to document and place in context a substantial body of mathematical ideas.
- Published
- 2013
21. The Implicit Function Theorem : History, Theory, and Applications
- Author
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Steven G. Krantz, Harold R. Parks, Steven G. Krantz, and Harold R. Parks
- Subjects
- Mathematical analysis, Differential equations, Geometry, Differential, Mathematics, History
- Abstract
The implicit function theorem is part of the bedrock of mathematical analysis and geometry. Finding its genesis in eighteenth century studies of real analytic functions and mechanics, the implicit and inverse function theorems have now blossomed into powerful tools in the theories of partial differential equations, differential geometry, and geometric analysis. There are many different forms of the implicit function theorem, including (i) the classical formulation for C^k functions, (ii) formulations in other function spaces, (iii) formulations for non- smooth functions, (iv) formulations for functions with degenerate Jacobian. Particularly powerful implicit function theorems, such as the Nash--Moser theorem, have been developed for specific applications (e.g., the imbedding of Riemannian manifolds). All of these topics, and many more, are treated in the present volume. The history of the implicit function theorem is a lively and complex story, and is intimately bound up with the development of fundamental ideas in analysis and geometry. This entire development, together with mathematical examples and proofs, is recounted for the first time here. It is an exciting tale, and it continues to evolve.'The Implicit Function Theorem'is an accessible and thorough treatment of implicit and inverse function theorems and their applications. It will be of interest to mathematicians, graduate/advanced undergraduate students, and to those who apply mathematics. The book unifies disparate ideas that have played an important role in modern mathematics. It serves to document and place in context a substantial body of mathematical ideas.
- Published
- 2012
22. Linear Integral Equations : Theory & Technique
- Author
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Ram P. Kanwal and Ram P. Kanwal
- Subjects
- Integral equations, Mathematics, Differential equations, Mathematical analysis, Mathematical physics
- Abstract
Many physical problems that are usually solved by differential equation methods can be solved more effectively by integral equation methods. Such problems abound in applied mathematics, theoretical mechanics, and mathematical physics. This uncorrected soft cover reprint of the second edition places the emphasis on applications and presents a variety of techniques with extensive examples.Originally published in 1971, Linear Integral Equations is ideal as a text for a beginning graduate level course. Its treatment of boundary value problems also makes the book useful to researchers in many applied fields.
- Published
- 2012
23. Mathematical Analysis : Functions of One Variable
- Author
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Mariano Giaquinta, Giuseppe Modica, Mariano Giaquinta, and Giuseppe Modica
- Subjects
- Mathematical analysis, Functions of complex variables, Differential equations, Mathematics
- Abstract
For more than two thousand years some familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today the traditional place of mathematics in education is in grave danger. Unfortunately, professional representatives of mathematics share in the reponsibiIity. The teaching of mathematics has sometimes degen erated into empty drill in problem solving, which may develop formal ability but does not lead to real understanding or to greater intellectual indepen dence. Mathematical research has shown a tendency toward overspecialization and over-emphasis on abstraction. Applications and connections with other fields have been neglected... But... understanding of mathematics cannot be transmitted by painless entertainment any more than education in music can be brought by the most brilliant journalism to those who never have lis tened intensively. Actual contact with the content of living mathematics is necessary. Nevertheless technicalities and detours should be avoided, and the presentation of mathematics should be just as free from emphasis on routine as from forbidding dogmatism which refuses to disclose motive or goal and which is an unfair obstacle to honest effort. (From the preface to the first edition of What is Mathematics? by Richard Courant and Herbert Robbins, 1941.
- Published
- 2012
24. Dynamic Equations on Time Scales : An Introduction with Applications
- Author
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Martin Bohner, Allan Peterson, Martin Bohner, and Allan Peterson
- Subjects
- Mathematical analysis, Differential equations, Mathematics, System theory, Control theory, Mathematics—Data processing
- Abstract
On becoming familiar with difference equations and their close re lation to differential equations, I was in hopes that the theory of difference equations could be brought completely abreast with that for ordinary differential equations. [HUGH L. TURRITTIN, My Mathematical Expectations, Springer Lecture Notes 312 (page 10), 1973] A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both. [E. T. BELL, Men of Mathematics, Simon and Schuster, New York (page 13/14), 1937] The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his PhD thesis [159] in 1988 (supervised by Bernd Aulbach) in order to unify continuous and discrete analysis. This book is an intro duction to the study of dynamic equations on time scales. Many results concerning differential equations carryover quite easily to corresponding results for difference equations, while other results seem to be completely different in nature from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice, once for differential equa tions and once for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which is an arbitrary nonempty closed subset of the reals.
- Published
- 2012
25. Real Analysis
- Author
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Emmanuele DiBenedetto and Emmanuele DiBenedetto
- Subjects
- Mathematics, Mathematical analysis, Measure theory, Differential equations
- Abstract
This book is a self-contained introduction to real analysis assuming only basic notions on limits of sequences in ]RN, manipulations of series, their convergence criteria, advanced differential calculus, and basic algebra of sets. The passage from the setting in ]RN to abstract spaces and their topologies is gradual. Continuous reference is made to the ]RN setting, where most of the basic concepts originated. The first seven chapters contain material forming the backbone of a basic training in real analysis. The remaining two chapters are more topical, relating to maximal functions, functions of bounded mean oscillation, rearrangements, potential theory, and the theory of Sobolev functions. Even though the layout of the book is theoretical, the entire book and the last chapters in particular concern applications of mathematical analysis to models of physical phenomena through partial differential equations. The preliminaries contain a review of the notions of countable sets and related examples. We introduce some special sets, such as the Cantor set and its variants, and examine their structure. These sets will be a reference point for a number of examples and counterexamples in measure theory (Chapter II) and in the Lebesgue differentiability theory of absolute continuous functions (Chapter IV). This initial chapter also contains a brief collection of the various notions of ordering, the Hausdorff maximal principle, Zorn's lemma, the well-ordering principle, and their fundamental connections.
- Published
- 2012
26. Inverse Stefan Problems
- Author
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N.L. Gol'dman and N.L. Gol'dman
- Subjects
- Mathematical analysis, Differential equations, Mathematics—Data processing, Mathematics, Mathematical models, Materials—Analysis
- Abstract
In this monograph the theory and methods of solving inverse Stefan problems for quasilinear parabolic equations in regions with free boundaries are developed. The study of this new class of ill-posed problems is motivated by the needs of the mod eling and control of nonlinear processes with phase transitions in thermophysics and mechanics of continuous media. Inverse Stefan problems are important for the perfection of technologies both in high temperature processes (e.g., metallurgy, the aircraft industry, astronautics and power engineering) and in hydrology, exploitation of oil-gas fields, etc. The proposed book will complete a gap in these subjects in the preceding re searches of ill-posed problems. It contains the new theoretical and applied studies of a wide class of inverse Stefan problems. The statements of such problems on the determination of boundary functions and coefficients of the equation are considered for different types of additional information about their solution. The variational method of obtaining stable approximate solutions is proposed and established. It is implemented by an efficient computational scheme of descriptive regularization. This algorithm utilizes a priori knowledge of the qualitative structure of the sought solution and ensures a substantial saving in computational costs. It is tested on model and applied problems in nonlinear thermophysics. In particular, the results of calculations for important applications in continuous casting of ingots and in the melting of a plate with the help of laser technology are presented.
- Published
- 2012
27. Scientific Computing with Mathematica® : Mathematical Problems for Ordinary Differential Equations
- Author
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Addolorata Marasco, Antonio Romano, Addolorata Marasco, and Antonio Romano
- Subjects
- Mathematical models, Mathematical analysis, Computer software, Differential equations, Mathematics—Data processing, Mathematics
- Abstract
Many interesting behaviors of real physical, biological, economical, and chemical systems can be described by ordinary differential equations (ODEs). Scientific Computing with Mathematica for Ordinary Differential Equations provides a general framework useful for the applications, on the conceptual aspects of the theory of ODEs, as well as a sophisticated use of Mathematica software for the solutions of problems related to ODEs. In particular, a chapter is devoted to the use ODEs and Mathematica in the Dynamics of rigid bodies. Mathematical methods and scientific computation are dealt with jointly to supply a unified presentation. The main problems of ordinary differential equations such as, phase portrait, approximate solutions, periodic orbits, stability, bifurcation, and boundary problems are covered in an integrated fashion with numerous worked examples and computer program demonstrations using Mathematica. Topics and Features:•Explains how to use the Mathematica package ODE.m to support qualitative and quantitative problem solving •End-of- chapter exercise sets incorporating the use of Mathematica programs •Detailed description and explanation of the mathematical procedures underlying the programs written in Mathematica •Appendix describing the use of ten notebooks to guide the reader through all the exercises. This book is an essential text/reference for students, graduates and practitioners in applied mathematics and engineering interested in ODE's problems in both the qualitative and quantitative description of solutions with the Mathematica program. It is also suitable as a self-
- Published
- 2012
28. Fields, Flows and Waves : An Introduction to Continuum Models
- Author
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David F. Parker and David F. Parker
- Subjects
- Mathematical analysis, Mathematics, Physics, Astronomy, Differential equations, Mechanics
- Abstract
Many phenomena in the physical and biological sciences involve the collective behaviour of (very large) numbers of individual objects. For example, the be haviour of gases ultimately concerns the interacting motions of uncountably many atoms and molecules, but to understand flow in nozzles, around aircraft and in meteorology it is best to treat velocity and density as continuous func tions of position and time and then to analyse the associated flows. Although modern electronics involves ever smaller components, even the semiconduc tor devices used widely in electronic communications and in digital processing involve collective phenomena, such as electric currents and fields, which are continuously varying functions of position and time. Diffusion and reaction between various chemical constituents, the growth and spread of biological or ganisms and the flow of traffic on major highways are all phenomena which may be described and analysed in terms of fields and flows, while sound, light and various other electromagnetic phenomena involve both fields and waves. Treating these using a continuum model, which does not attempt to trace the motion and evolution of individual objects, often gives good predictions. The mathematical concepts and techniques which underlie such treatments are the subject of this book. This book is designed as a first introduction to the use of mathematical techniques, within continuum theories.
- Published
- 2012
29. A NEW METHOD TO PROVE THE NONUNIFORM DICHOTOMY SPECTRUM THEOREM IN Rn.
- Author
-
YONGHUI XIA, YUZHEN BAI, and O'REGAN, DONAL
- Subjects
- *
MATHEMATICAL analysis , *SHIFT systems , *DIFFERENTIAL equations , *MATHEMATICS , *CONTRADICTION , *SIMILARITY (Geometry) - Abstract
This paper presents a new method to prove the nonuniform dichotomy spectrum theorem. Chu et al. [Bull. Sci. Math. 139 (2015), pp. 538-557] and Zhang [J. Funct. Anal. 267 (2014), pp. 1889-1916] generalized the dichotomy spectrum in Siegmund [J. Dynam. Differential Equations 14 (2002), pp. 243-258] to the nonuniform dichotomy spectrum and the authors in these works employed linear integral manifolds (stable and unstable) to establish the spectral theorem. They then used the spectrum theorem to study reducibility. We prove the nonuniform dichotomy spectrum by way of contradiction. In particular, we employ the nonuniform kinematically similarity (nonuniform reducibility) to reduce the shift system into two blocks and then we get a contradiction based on a technique in mathematical analysis. The method in the proof is completely different from previous works. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
30. A NEW METHOD TO PROVE THE NONUNIFORM DICHOTOMY SPECTRUM THEOREM IN Rn.
- Author
-
YONGHUI XIA, YUZHEN BAI, and O'REGAN, DONAL
- Subjects
MATHEMATICAL analysis ,SHIFT systems ,DIFFERENTIAL equations ,MATHEMATICS ,CONTRADICTION ,SIMILARITY (Geometry) - Abstract
This paper presents a new method to prove the nonuniform dichotomy spectrum theorem. Chu et al. [Bull. Sci. Math. 139 (2015), pp. 538-557] and Zhang [J. Funct. Anal. 267 (2014), pp. 1889-1916] generalized the dichotomy spectrum in Siegmund [J. Dynam. Differential Equations 14 (2002), pp. 243-258] to the nonuniform dichotomy spectrum and the authors in these works employed linear integral manifolds (stable and unstable) to establish the spectral theorem. They then used the spectrum theorem to study reducibility. We prove the nonuniform dichotomy spectrum by way of contradiction. In particular, we employ the nonuniform kinematically similarity (nonuniform reducibility) to reduce the shift system into two blocks and then we get a contradiction based on a technique in mathematical analysis. The method in the proof is completely different from previous works. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
31. SUBORDINATION RESULTS FOR CERTAIN CLASS OF ANALYTIC FUNCTIONS ASSOCIATED WITH MITTAG-LEFFLER FUNCTION.
- Author
-
YASSEN, MANSOUR F.
- Subjects
- *
ANALYTIC functions , *MATHEMATICAL functions , *GEOMETRIC function theory , *DIFFERENTIAL equations , *MATHEMATICAL analysis , *MATHEMATICS - Abstract
In this paper, we introduce a new class of analytic functions associated with Mittag-Leffler fuction in the open unit disk. Several properties of functions belonging to this class are derived. [ABSTRACT FROM AUTHOR]
- Published
- 2019
32. DIFFERENTIAL SUBORDINATION FOR ANALYTIC FUNCTIONS ASSOCIATED WITH LEAF-LIKE DOMAINS.
- Author
-
Sivasubramanian, S., Govindaraj, M., Murugusundaramoorthy, G., and Cho, N. E.
- Subjects
- *
MATHEMATICAL functions , *DIFFERENTIAL equations , *MATHEMATICAL analysis , *MATHEMATICS , *MATHEMATICAL domains - Abstract
In our present investigation, we obtain several differential subordination results involving leaf-like domains. Moreover, certain sharp coefficient estimates are investigated when the class of functions lies in leaf-like domains. [ABSTRACT FROM AUTHOR]
- Published
- 2019
33. Some identities involving generalized degenerate tangent polynomials arising from differential equations.
- Author
-
Ryoo, C. S.
- Subjects
- *
DIFFERENTIAL equations , *POLYNOMIALS , *MATHEMATICAL functions , *MATHEMATICAL analysis , *MATHEMATICS - Abstract
In this paper, we study differential equations arising from the generating functions of generalized degenerate tangent polynomials. We give explicit identities for the generalized degenerate tangent polynomials arising from differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2019
34. Existence of positive solution for fully third-order boundary value problems.
- Author
-
Yongxiang Li and Ibrahim, Elyasa
- Subjects
- *
BOUNDARY value problems , *MATHEMATICAL inequalities , *MATHEMATICAL analysis , *MATHEMATICS , *DIFFERENTIAL equations - Abstract
In this paper, we are concerned with the existence of positive solutions of the fully third-order boundary value problem... is continuous. Some inequality conditions on f to guarantee the existence of positive solution are presented. These inequality conditions allow that ƒ(t; x; y; z) may be superlinear or sublinear growth on x, y and z as |(x; y; z)| → 0 and |(x; y; z)| → ∞. [ABSTRACT FROM AUTHOR]
- Published
- 2019
35. Differential identities, 2 × 2 upper triangular matrices and varieties of almost polynomial growth.
- Author
-
Giambruno, Antonio and Rizzo, Carla
- Subjects
- *
DIFFERENTIAL equations , *MATRICES (Mathematics) , *ALGEBRA , *POLYNOMIALS , *MATHEMATICAL analysis , *MATHEMATICS - Abstract
Abstract We study the differential identities of the algebra U T 2 of 2 × 2 upper triangular matrices over a field of characteristic zero. We let the Lie algebra L = Der (U T 2) of derivations of U T 2 (and its universal enveloping algebra) act on it. We study the space of multilinear differential identities in n variables as a module for the symmetric group S n and we determine the decomposition of the corresponding character into irreducibles. If V is the variety of differential algebras generated by U T 2 , we prove that unlike the other cases (ordinary identities, group graded identities) V does not have almost polynomial growth. Nevertheless we exhibit a subvariety U of V having almost polynomial growth. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
36. Geometrical characteristics of the stability domain in the restricted problem of eight bodies.
- Author
-
CEBOTARU, ELENA
- Subjects
- *
GEOMETRICAL constructions , *MATHEMATICAL analysis , *NUMERICAL calculations , *DIFFERENTIAL equations , *MATHEMATICS - Abstract
The eight-body Newtonian problem is studied. Applying the symbolic calculation system Mathematica the stationary solutions, their stability in numerical form and the geometric characteristics of the stability domain are studied. [ABSTRACT FROM AUTHOR]
- Published
- 2019
37. A NOTE ON BOUNDED SOLUTIONS OF AN ITERATIVE EQUATION.
- Author
-
HOU YU ZHAO and JIA LIU
- Subjects
- *
ITERATIVE methods (Mathematics) , *DIFFERENTIAL equations , *MATHEMATICAL analysis , *FIXED point theory , *FUNCTIONAL analysis , *EQUATIONS , *MATHEMATICS - Abstract
In this paper, we use Schauder and Banach fixed point theorems to study the existence, uniqueness and stability of bounded nonhomogeneous iterative functional differential equations of the form x' (t) = λ1x(t) + λ2x[2](t)+...+λnx[n](t)+f(t). [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
38. INFINITELY MANY WEAK SOLUTIONS FOR A FOURTH-ORDER EQUATION WITH NONLINEAR BOUNDARY CONDITIONS.
- Author
-
TAVANI, MOHAMMAD REZA HEIDARI and NAZARI, ABDOLLAH
- Subjects
- *
EQUATIONS , *CRITICAL point theory , *CALCULUS of variations , *BOUNDARY value problems , *DIFFERENTIAL equations , *MATHEMATICAL analysis , *CALCULUS , *MATHEMATICS - Abstract
Existence results of infinitely many solutions for a fourth-order differential equation are established. This equation depends on two real parameters. The approach is based on an infinitely many critical points theorem. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
39. Mathematical Analysis II
- Author
-
Claudio Canuto, Anita Tabacco, Claudio Canuto, and Anita Tabacco
- Subjects
- Mathematics, Mathematical analysis, Functional analysis, Differential equations
- Abstract
The purpose of this textbook is to present an array of topics in Calculus, and conceptually follow our previous effort Mathematical Analysis I.The present material is partly found, in fact, in the syllabus of the typical second lecture course in Calculus as offered in most Italian universities. While the subject matter known as `Calculus 1'is more or less standard, and concerns real functions of real variables, the topics of a course on `Calculus 2'can vary a lot, resulting in a bigger flexibility. For these reasons the Authors tried to cover a wide range of subjects, not forgetting that the number of credits the current programme specifications confers to a second Calculus course is not comparable to the amount of content gathered here. The reminders disseminated in the text make the chapters more independent from one another, allowing the reader to jump back and forth, and thus enhancing the versatility of the book. On the website: http://calvino.polito.it/canuto-tabacco/analisi 2, the interested reader may find the rigorous explanation of the results that are merely stated without proof in the book, together with useful additional material. The Authors have completely omitted the proofs whose technical aspects prevail over the fundamental notions and ideas. The large number of exercises gathered according to the main topics at the end of each chapter should help the student put his improvements to the test. The solution to all exercises is provided, and very often the procedure for solving is outlined.
- Published
- 2011
40. The Legacy of Niels Henrik Abel : The Abel Bicentennial, Oslo, 2002
- Author
-
Olav Arnfinn Laudal, Ragni Piene, Olav Arnfinn Laudal, and Ragni Piene
- Subjects
- Algebraic geometry, Mathematical analysis, Functional analysis, Mathematics, History, Differential equations
- Abstract
This book contains a series of research papers on subjects related to the work of Niels Henrik Abel, written by some of the foremost specialists in their fields. Some of the authors have been specifically invited to present papers, discussing the influence of Abel in a mathematical-historical context. Others have submitted papers presented at the Abel Bicentennial Conference, Oslo June 3-8, 2002. The idea behind the book has been to produce a text covering a substantial part of the legacy of Abel, as perceived at the beginning of the 21st century. It is accompanied by a CD-ROM with a large amount of information related to Niels Henrik Abel, such as on the Abel Centennial in 1902 and the Abel Bicentennial Conference in 2002, the launching of the Abel Prize, Abel monuments, and stamps, banknotes, coins etc. issued in honour of Niels Henrik Abel.
- Published
- 2011
41. Quotients of internally quasicontinuous functions*.
- Author
-
Szyszkowska, Paulina
- Subjects
- *
MATHEMATICAL analysis , *MATHEMATICS theorems , *MATHEMATICS , *MATHEMATICAL functions , *DIFFERENTIAL equations - Abstract
In this paper, we characterize the family of quotients of internally quasicontinuous functions. Moreover, we study cardinal invariants related to quotients in the case of internally quasicontinuous functions and the complement of this family. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
42. Quotients of internally quasicontinuous functions*.
- Author
-
Szyszkowska, Paulina
- Subjects
MATHEMATICAL analysis ,MATHEMATICS theorems ,MATHEMATICS ,MATHEMATICAL functions ,DIFFERENTIAL equations - Abstract
In this paper, we characterize the family of quotients of internally quasicontinuous functions. Moreover, we study cardinal invariants related to quotients in the case of internally quasicontinuous functions and the complement of this family. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
43. Rapid computation of L-functions attached to Maass forms.
- Author
-
Booker, Andrew R. and Then, Holger
- Subjects
- *
MATHEMATICAL functions , *DIFFERENTIAL equations , *MATHEMATICS , *SET theory , *MATHEMATICAL analysis - Abstract
Let L be a degree-2L-function associated to a Maass cusp form. We explore an algorithm that evaluates t values of L on the critical line in time O(t1+𝜖). We use this algorithm to rigorously compute an abundance of consecutive zeros and investigate their distribution. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
44. On the oscillation of impulsive vector partial differential equations with distributed deviating arguments.
- Author
-
Chatzarakis, George E., Sadhasivam, Vadivel, and Raja, Thangaraj
- Subjects
- *
NUMERICAL analysis , *MATHEMATICAL analysis , *DIFFERENTIAL equations , *ADJOINT differential equations , *MATHEMATICS - Abstract
In this paper, we consider a class of nonlinear impulsive neutral partial functional differential equations with continuous distributed deviating arguments. For this class, we establish sufficient conditions for the
H -oscillation of the solutions, using impulsive differential inequalities and an averaging technique with two different boundary conditions. We provide an example to illustrate the main result. [ABSTRACT FROM AUTHOR]- Published
- 2018
- Full Text
- View/download PDF
45. HOMOGENEOUS FUNCTIONALLY ALEXANDROFF SPACES.
- Author
-
LAZAAR, SAMI, RICHMOND, TOM, and SABRI, HOUSSEM
- Subjects
- *
SET theory , *MATHEMATICS , *MATHEMATICAL functions , *DIFFERENTIAL equations , *MATHEMATICAL analysis - Abstract
A function $f:X\rightarrow X$ determines a topology $P(f)$ on $X$ by taking the closed sets to be those sets $A\subseteq X$ with $f(A)\subseteq A$. The topological space $(X,P(f))$ is called a functionally Alexandroff space. We completely characterise the homogeneous functionally Alexandroff spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
46. ON A PROPERTY OF DIFFERENTIAL EQUATIONS INTEGRABLE USING MEROMORPHIC DOUBLE-PERIODIC FUNCTIONS.
- Author
-
Petrovitch, Michel
- Subjects
DIFFERENTIAL equations ,NUMERICAL analysis ,MEROMORPHIC functions ,MATHEMATICS ,MATHEMATICAL analysis - Abstract
Copyright of Theoretical & Applied Mechanics is the property of Theoretical & Applied Mechanics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
- Published
- 2018
- Full Text
- View/download PDF
47. On the moment generating function for random vectors via inverse survival function.
- Author
-
Song, Pingfan, Tan, Changchun, and Wang, Shaochen
- Subjects
- *
VECTORS (Calculus) , *MATHEMATICAL functions , *DIFFERENTIAL equations , *MATHEMATICAL analysis , *MATHEMATICS - Abstract
Abstract In this paper, we propose one new alternative formula for moment generating function of random vectors via the inverse survival function. A recursive formula for moment generating function of random vector is obtained and as application, we derive the corresponding alternative formula for mixed moment. Several examples are also presented along with the theory. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
48. Applications of Differential Equations in General Problem Solving.
- Author
-
Klopfenstein, R. W.
- Subjects
- *
DIFFERENTIAL equations , *NUMERICAL analysis , *CALCULUS , *BESSEL functions , *MATHEMATICAL analysis , *MATHEMATICS - Abstract
A large class of problems leading to digital computer processing can be formulated in terms of the numerical solution of systems of ordinary differential equations. Powerful methods are in existence for the solution of such systems. A good general purpose routine for the solution of such systems furnishes a powerful tool for processing many problems. This is true from the point of view of ease of programming, ease of debugging, and minimization of computer time. A number of examples are discussed in detail. [ABSTRACT FROM AUTHOR]
- Published
- 1965
- Full Text
- View/download PDF
49. Algorithms.
- Author
-
Herriot, J. G.
- Subjects
- *
LEGENDRE'S functions , *ALGORITHMS , *MATHEMATICS , *MATHEMATICAL functions , *ALGEBRA , *DIFFERENTIAL equations , *COMPUTER systems , *ARITHMETIC , *MATHEMATICAL analysis - Abstract
The article presents mathematical procedures using Legendre functions for cases involving arguments larger than one. The integer Legendre 1 procedure is said to generate the associated Legendre functions of the first kind. Integer Legendre 2 generates associated Legendre functions of the second kind to d significant digits. The integer Legendre 3 generates the Legendre functions of the second kind to d significant digits and stores in the array Q. Legendre 1 procedure evaluates the Legendre functions to d significant digits. Other procedures include Legendre 2, conical and toroidal.
- Published
- 1965
50. STABILITY OF NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS.
- Author
-
GUILING CHEN, DINGSHI LI, VAN GAANS, ONNO, and LUNEL, SJOERD VERDUYN
- Subjects
- *
DIFFERENTIAL equations , *MATHEMATICAL functions , *MATHEMATICAL physics , *MATHEMATICAL analysis , *MATHEMATICS - Abstract
We present new criteria for asymptotic stability of two classes of nonlinear neutral delay differential equations. By using two auxiliary functions on a contraction condition, we extend the results in [12]. Also we give two examples that illustrate our results. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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