This book discusses advances in maximal function methods related to Poincare and Sobolev inequalities, pointwise estimates and approximation for Sobolev functions, Hardy's inequalities, and partial differential equations. Capacities are needed for fine properties of Sobolev functions and characterization of Sobolev spaces with zero boundary values. The authors consider several uniform quantitative conditions that are self-improving, such as Hardy's inequalities, capacity density conditions, and reverse Hoelder inequalities. They also study Muckenhoupt weight properties of distance functions and combine these with weighted norm inequalities; notions of dimension are then used to characterize density conditions and to give sufficient and necessary conditions for Hardy's inequalities. At the end of the book, the theory of weak solutions to the p -Laplace equation and the use of maximal function techniques is this context are discussed. The book is directed to researchers and graduate students interested in applications of geometric and harmonic analysis in Sobolev spaces and partial differential equations.

This book presents what in our opinion constitutes the basis of the theory of the mu-calculus, considered as an algebraic system rather than a logic. We have wished to present the subject in a unified way, and in a form as general as possible. Therefore, our emphasis is on the generality of the fixed-point notation, and on the connections between mu-calculus, games, and automata, which we also explain in an algebraic way.
This book should be accessible for graduate or advanced undergraduate students both in mathematics and computer science. We have designed this book especially for researchers and students interested in logic in computer science, comuter aided verification, and general aspects of automata theory. We have aimed at gathering in a single place the fundamental results of the theory, that are currently very scattered in the literature, and often hardly accessible for interested readers.
The presentation is self-contained, except for the proof of the Mc-Naughton's Determinization Theorem (see, e.g., 97]. However, we suppose that the reader is already familiar with some basic automata theory and universal algebra. The references, credits, and suggestions for further reading are given at the end of each chapter.

This volume contains the proceedings of the Conference on Complex Analysis and Spectral Theory, in celebration of Thomas Ransford's 60th birthday, held from May 21-25, 2018, at Laval University, Quebec, Canada. Spectral theory is the branch of mathematics devoted to the study of matrices and their eigenvalues, as well as their infinite-dimensional counterparts, linear operators and their spectra. Spectral theory is ubiquitous in science and engineering because so many physical phenomena, being essentially linear in nature, can be modelled using linear operators. On the other hand, complex analysis is the calculus of functions of a complex variable. They are widely used in mathematics, physics, and in engineering. Both topics are related to numerous other domains in mathematics as well as other branches of science and engineering. The list includes, but is not restricted to, analytical mechanics, physics, astronomy (celestial mechanics), geology (weather modeling), chemistry (reaction rates), biology, population modeling, economics (stock trends, interest rates and the market equilibrium price changes). There are many other connections, and in recent years there has been a tremendous amount of work on reproducing kernel Hilbert spaces of analytic functions, on the operators acting on them, as well as on applications in physics and engineering, which arise from pure topics like interpolation and sampling. Many of these connections are discussed in articles included in this book.

The most useful tool for reviewing mathematical methods for economics classes-now with more content Schaum's Outline of Calculus for Business, Economics and Finance, Fourth Edition is the go-to study guide for help in economics courses, mirroring the courses in scope and sequence to help you understand basic concepts and get extra practice in topics like multivariable functions, exponential and logarithmic functions, and more. With an outline format that facilitates quick and easy review, Schaum's Outline of Calculus for Business, Economics and Finance, Fourth Edition supports the major bestselling textbooks in economics courses and is useful for a variety of classes, including Introduction to Economics, Economics, Econometrics, Microeconomics, Macroeconomics, Economics Theories, Mathematical Economics, Math for Economists and Math for Social Sciences. Chapters include Economic Applications of Graphs and Equations, The Derivative and the Rules of Differentiation, Calculus of Multivariable Functions, Exponential and Logarithmic Functions in Economics, Special Determinants and Matrices and Their Use in Economics, First-Order Differential Equations, and more. Features: NEW in this edition: Additional problems at the end of each chapter NEW in this edition: An additional chapter on sequences and series NEW in this edition: Two computer applications of Linear Programming in Excel 710 fully solved problems Outline format to provide a concise guide for study for standard college courses in mathematical economics Clear, concise explanations covers all course fundamentals Supplements the major bestselling textbooks in economics courses Appropriate for the following courses: Introduction to Economics, Economics, Econometrics, Microeconomics, Macroeconomics, Economics Theories, Mathematical Economics, Math for Economists, Math for Social Sciences

This is part two of a two-volume introduction to real analysis and is intended for honours undergraduates who have already been exposed to calculus. The emphasis is on rigour and on foundations. The material starts at the very beginning--the construction of the number systems and set theory--then goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and finally to the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. There are also appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of twenty-five to thirty lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory. The fourth edition incorporates a large number of additional corrections reported since the release of the third edition, as well as some additional new exercises.

Get ahead in pre-calculus Pre-calculus courses have become increasingly popular with 35 percent of students in the U.S. taking the course in middle or high school. Often, completion of such a course is a prerequisite for calculus and other upper level mathematics courses. Pre-Calculus For Dummies is an invaluable resource for students enrolled in pre-calculus courses. By presenting the essential topics in a clear and concise manner, the book helps students improve their understanding of pre-calculus and become prepared for upper level math courses. Provides fundamental information in an approachable manner Includes fresh example problems Practical explanations mirror today's teaching methods Offers relevant cultural references Whether used as a classroom aid or as a refresher in preparation for an introductory calculus course, this book is one you'll want to have on hand to perform your very best.

We introduce a decoupling method on the Wiener space to define a wide class of anisotropic Besov spaces. The decoupling method is based on a general distributional approach and not restricted to the Wiener space. The class of Besov spaces we introduce contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our spaces) of forward diffusions and local times. It is shown that among our newly introduced Besov spaces there are spaces that characterize quantitative properties of directional derivatives in the Malliavin sense without computing or accessing these Malliavin derivatives explicitly. Regarding BSDEs, we deduce regularity properties of the solution processes from the Besov regularity of the initial data, in particular upper bounds for their Lp-variation, where the generator might be of quadratic type and where no structural assumptions, for example in terms of a forward diffusion, are assumed. As an example we treat sub-quadratic BSDEs with unbounded terminal conditions. Among other tools, we use methods from harmonic analysis. As a by-product, we improve the asymptotic behaviour of the multiplicative constant in a generalized Fefferman inequality and verify the optimality of the bound we established.

The author's goal for the book is that it's clearly written, could be read by a calculus student and would motivate them to engage in the material and learn more. Moreover, to create a text in which exposition, graphics, and layout would work together to enhance all facets of a student's calculus experience. They paid special attention to certain aspects of the text: 1. Clear, accessible exposition that anticipates and addresses student difficulties.2. Layout and figures that communicate the flow of ideas. 3. Highlighted features that emphasize concepts and mathematical reasoning including Conceptual Insight, Graphical Insight, Assumptions Matter, Reminder, and Historical Perspective.4. A rich collection of examples and exercises of graduated difficulty that teach basic skills as well as problem-solving techniques, reinforce conceptual understanding, and motivate calculus through interesting applications. Each section also contains exercises that develop additional insights and challenge students to further develop their skills. Achieve for Calculus redefines homework by offering guidance for every student and support for every instructor. Homework is designed to teach by correcting students' misconceptions through targeted feedback, meaningful hints, and full solutions, helping teach students conceptual understanding and critical thinking in real-world contexts.

Every financial professional wants and needs an advantage. A firm foundation in advanced mathematics can translate into dramatic advantages to professionals willing to obtain it. Many are not—and that is the advantage these books offer the astute reader. Published under the collective title of Foundations of Quantitative Finance, this set of ten books presents the advanced mathematics finance professionals need to advance their careers. These books develop the theory most do not learn in Graduate Finance programs, or in most Financial Mathematics undergraduate and graduate courses. As a high-level industry executive and authoritative instructor, Robert R. Reitano presents the mathematical theories he encountered and used in nearly three decades in the financial industry and two decades in education where he taught in highly respected graduate programs. Readers should be quantitatively literate and familiar with the developments in the first books in the set. The set offers a linear progression through these topics, though each title can be studied independently since the collection is extensively self-referenced. Book III: The Integrals of Lebesgue and (Riemann-) Stieltjes, develops several approaches to an integration theory. The first two approaches were introduced in the Chapter 1 of Book I to motivate measure theory. The general theory of integration on measure spaces will be developed in Book V, and stochastic integrals then studies on Book VIII. Book III Features: Extensively referenced to utilize materials from earlier books. Presents the theory needed to better understand applications. Supplements previous training in mathematics, with more detailed developments. Built from the author's five decades of experience in industry, research, and teaching. Published and forthcoming titles in the Robert Reitano Quantitative Finance Series: Book I: Measure Spaces and Measurable Functions. Book II: Probability Spaces and Random Variables, Book III: The Integrals of Lebesgue and (Riemann-) Stieltjes Book IV: Distribution Functions and Expectations Book V: General Measure and Integration Theory Book VI: Densities, Transformed Distributions, and Limit Theorems Book VII: Brownian Motion and Other Stochastic Processes Book VIII: Itô Integration and Stochastic Calculus 1 Book IX: Stochastic Calculus 2 and Stochastic Differential Equations Book 10: Applications and Classic Models

Every financial professional wants and needs an advantage. A firm foundation in advanced mathematics can translate into dramatic advantages to professionals willing to obtain it. Many are not—and that is the advantage these books offer the astute reader. Published under the collective title of Foundations of Quantitative Finance, this set of ten books presents the advanced mathematics finance professionals need to advance their careers. These books develop the theory most do not learn in Graduate Finance programs, or in most Financial Mathematics undergraduate and graduate courses. As a high-level industry executive and authoritative instructor, Robert R. Reitano presents the mathematical theories he encountered and used in nearly three decades in the financial industry and two decades in education where he taught in highly respected graduate programs. Readers should be quantitatively literate and familiar with the developments in the first books in the set. The set offers a linear progression through these topics, though each title can be studied independently since the collection is extensively self-referenced. Book III: The Integrals of Lebesgue and (Riemann-) Stieltjes, develops several approaches to an integration theory. The first two approaches were introduced in the Chapter 1 of Book I to motivate measure theory. The general theory of integration on measure spaces will be developed in Book V, and stochastic integrals then studies on Book VIII. Book III Features: Extensively referenced to utilize materials from earlier books. Presents the theory needed to better understand applications. Supplements previous training in mathematics, with more detailed developments. Built from the author's five decades of experience in industry, research, and teaching. Published and forthcoming titles in the Robert Reitano Quantitative Finance Series: Book I: Measure Spaces and Measurable Functions. Book II: Probability Spaces and Random Variables, Book III: The Integrals of Lebesgue and (Riemann-) Stieltjes Book IV: Distribution Functions and Expectations Book V: General Measure and Integration Theory Book VI: Densities, Transformed Distributions, and Limit Theorems Book VII: Brownian Motion and Other Stochastic Processes Book VIII: Itô Integration and Stochastic Calculus 1 Book IX: Stochastic Calculus 2 and Stochastic Differential Equations Book 10: Applications and Classic Models

The goal of the book is to summarize those methods for evaluating Feynman integrals that have been developed over a span of more than fifty years. The book characterizes the most powerful methods and illustrates them with numerous examples starting from very simple ones and progressing to nontrivial examples. The book demonstrates how to choose adequate methods and combine evaluation methods in a non-trivial way. The most powerful methods are characterized and then illustrated through numerous examples. This is an updated textbook version of the previous book (Evaluating Feynman integrals, STMP 211) of the author.

Suitable for graduate students and professional researchers in operator theory and/or analysis Numerous applications in related scientific fields and areas.

Certain constants occupy precise balancing points in the cosmos of number, like habitable planets sprinkled throughout our galaxy at just the right distances from their suns. This book introduces and connects four of these constants (phi, pi, e and i), each of which has recently been the individual subject of historical and mathematical expositions. But here we discuss their properties, as a group, at a level appropriate for an audience armed only with the tools of elementary calculus. This material offers an excellent excuse to display the power of calculus to reveal elegant truths that are not often seen in college classes. These truths are described here via the work of such luminaries as Nilakantha, Liu Hui, Hemachandra, Khayyam, Newton, Wallis, and Euler.

General Fractional Derivatives with Applications in Viscoelasticity introduces the newly established fractional-order calculus operators involving singular and non-singular kernels with applications to fractional-order viscoelastic models from the calculus operator viewpoint. Fractional calculus and its applications have gained considerable popularity and importance because of their applicability to many seemingly diverse and widespread fields in science and engineering. Many operations in physics and engineering can be defined accurately by using fractional derivatives to model complex phenomena. Viscoelasticity is chief among them, as the general fractional calculus approach to viscoelasticity has evolved as an empirical method of describing the properties of viscoelastic materials. General Fractional Derivatives with Applications in Viscoelasticity makes a concise presentation of general fractional calculus.

An engaging, accessible introduction into how numbers work and why we shouldn't be afraid of them, from maths expert Rachel Riley. Do you know your fractions from your percentages? Your adjacent to your hypotenuse? And who really knows how to do long division, anyway? Puzzled already? Don't blame you... But fret not! You won't be At Sixes and Sevens for long. In this brilliant, well-rounded guide, Countdown's Rachel Riley will take you back to the very basics, allow you to revisit what you learnt at school (and may have promptly forgotten, *ahem*), build your understanding of maths from the get-go and provide you with the essential toolkit to gain confidence in your numerical abilities. Discover how to divide and conquer, make your decimal debut, become a pythagoras professional and so much more with these easy-to-learn tips and tricks. Packed full of working examples, fool-proof methods, quirky trivia and brainteasers to try from puzzle-pro Dr Gareth Moore, this book is an absolute must-read for anyone and everyone who ever thought maths was 'above' them. Because the truth is: you can do it. What's more, it can be pretty fun too!

Since the publication of the first edition of this book, the area of mathematical finance has grown rapidly, with financial analysts using more sophisticated mathematical concepts, such as stochastic integration, to describe the behavior of markets and to derive computing methods. Maintaining the lucid style of its popular predecessor, Introduction to Stochastic Calculus Applied to Finance, Second Edition incorporates some of these new techniques and concepts to provide an accessible, up-to-date initiation to the field. New to the Second Edition Complements on discrete models, including Rogers' approach to the fundamental theorem of asset pricing and super-replication in incomplete markets Discussions on local volatility, Dupire's formula, the change of numeraire techniques, forward measures, and the forward Libor model A new chapter on credit risk modeling An extension of the chapter on simulation with numerical experiments that illustrate variance reduction techniques and hedging strategies Additional exercises and problems Providing all of the necessary stochastic calculus theory, the authors cover many key finance topics, including martingales, arbitrage, option pricing, American and European options, the Black-Scholes model, optimal hedging, and the computer simulation of financial models. They succeed in producing a solid introduction to stochastic approaches used in the financial world.

This monograph (in two volumes) deals with non scalar variational problems arising in geometry, as harmonic mappings between Riemannian manifolds and minimal graphs, and in physics, as stable equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and accessible to non specialists. Topics are treated as far as possible in an elementary way, illustrating results with simple examples; in principle, chapters and even sections are readable independently of the general context, so that parts can be easily used for graduate courses. Open questions are often mentioned and the final section of each chapter discusses references to the literature and sometimes supplementary results. Finally, a detailed Table of Contents and an extensive Index are of help to consult this monograph