Format: Hardback, 232 pages, height x width: 235x155 mm, 33 Illustrations,
black and white; VIII, 232 p. 33 illus., 1 Hardback
Series: Progress in Mathematics 345
Pub. Date: 09-Sep-2022
ISBN-13: 9783031087042
This monograph presents an approachable proof of Mirzakhani's curve counting theorem, both for simple and non-simple curves. Designed to welcome readers to the area, the presentation builds intuition with elementary examples before progressing to rigorous proofs. This approach illuminates new and established results alike, and produces versatile tools for studying the geometry of hyperbolic surfaces, Teichmuller theory, and mapping class groups. Beginning with the preliminaries of curves and arcs on surfaces, the authors go on to present the theory of geodesic currents in detail. Highlights include a treatment of cusped surfaces and surfaces with boundary, along with a comprehensive discussion of the action of the mapping class group on the space of geodesic currents. A user-friendly account of train tracks follows, providing the foundation for radallas, an immersed variation. From here, the authors apply these tools to great effect, offering simplified proofs of existing results and a new, more general proof of Mirzakhani's curve counting theorem. Further applications include counting square-tiled surfaces and mapping class group orbits, and investigating random geometric structures. Mirzakhani's Curve Counting and Geodesic Currents introduces readers to powerful counting techniques for the study of surfaces. Ideal for graduate students and researchers new to the area, the pedagogical approach, conversational style, and illuminating illustrations bring this exciting field to life. Exercises offer opportunities to engage with the material throughout. Basic familiarity with 2-dimensional topology and hyperbolic geometry, measured laminations, and the mapping class group is assumed
1. Introduction.-
2. Read Me.-
3. Geodesic Currents.-
4. Train Tracks.-
5. Radallas.-
6. Subconvergence of Measures.-
7. Approximating the Thurston Measure.-
8. The Main Theorem.-
9. Counting Curves.-
10. Counting Square Tiled Surfaces.-
11. Statistics of Simple Curves.-
12. Smoergasbord.- A. Radon Measures.- B. Computing Thurston Volumes.- References.- Index.
Format: Paperback / softback, 140 pages, height x width: 235x155 mm, X, 140 p., 1 Paperback / softback
Series: Lecture Notes in Mathematics 2307
Pub. Date: 24-Sep-2022
ISBN-13: 9783031082115
This book provides an organized exposition of the current state of the theory of commutative semigroup cohomology, a theory which was originated by the author and has matured in the past few years.
The work contains a fundamental scientific study of questions in the theory. The various approaches to commutative semigroup cohomology are compared. The problems arising from definitions in higher dimensions are addressed. Computational methods are reviewed. The main application is the computation of extensions of commutative semigroups and their classification.
Previously the components of the theory were scattered among a number of research articles. This work combines all parts conveniently in one volume. It will be a valuable resource for future students of and researchers in commutative semigroup cohomology and related areas.
1. The beginning.-
2. Beck cohomology.-
3. Symmetric cohomology.-
4. Calvo-Cegarra cohomology.-
5. The third cohomology group.-
6. The Overpath Method.-
7. Symmetric chains.-
8. Inheritance.-
9. Appendixes.
Format: Hardback, 187 pages, height x width: 235x155 mm, 11 Illustrations,
black and white; VIII, 187 p. 11 illus., 1 Hardback
Pub. Date: 27-Aug-2022
ISBN-13: 9789811925528
This book gives many helps for students of technical colleges who have had usual mathematical training. The material presented in this book exceeds the content of the spoken lessons, and so, it is also useful for other engineering specialities and even for students in mathematics.
The authors present in a small number of pages the basic notions and results of differential calculus concerning to: sequences and series of numbers, sequences and series of functions, power series, elements of topology in n-dimensional space, limits of functions, continuous functions, partial derivatives of functions of several variables, Taylor's formula, extrema of a function of several variables (free or with constrains), change of variables, dependent functions.
Chapter 1 - Sequences of Real Numbers.Chapter 2 - Real Number Series.Chapter 3 - Sequence of Functions (Functional Sequences).Chapter 4 - Series of Functions (Functional Series).Chapter 5 - Functions of Several Variables.Chapter 6 - Differential Calculus of Functions of Several Variables.
Format: Hardback, 175 pages, height x width: 254x178 mm, 28 Illustrations,
black and white; XXIV, 175 p. 28 illus., 1 Hardback
Series: Undergraduate Texts in Mathematics
Pub. Date: 09-Oct-2022
ISBN-13: 9783031130168
This textbook originates from a course taught by the late Ken Ireland in 1972. Designed to explore the theoretical underpinnings of undergraduate mathematics, the course focused on interrelationships and hands-on experience. Readers of this textbook will be taken on a modern rendering of Ireland’s path of discovery, consisting of excursions into number theory, algebra, and analysis. Replete with surprising connections, deep insights, and brilliantly curated invitations to try problems at just the right moment, this journey weaves a rich body of knowledge that is ideal for those going on to study or teach mathematics.
A pool of 200 ‘Dialing In’ problems opens the book, providing fuel for active enquiry throughout a course. The following chapters develop theory to illuminate the observations and roadblocks encountered in the problems, situating them in the broader mathematical landscape. Topics cover polygons and modular arithmetic; the fundamental theorems of arithmetic and algebra; irrational, algebraic and transcendental numbers; and Fourier series and Gauss sums. A lively accompaniment of examples, exercises, historical anecdotes, and asides adds motivation and context to the theory. Return trips to the Dialing In problems are encouraged, offering opportunities to put theory into practice and make lasting connections along the way.
Excursions in Number Theory, Algebra, and Analysis invites readers on a journey as important as the destination. Suitable for a senior capstone, professional development for practicing teachers, or independent reading, this textbook offers insights and skills valuable to math majors and high school teachers alike. A background in real analysis and abstract algebra is assumed, though the most important prerequisite is a willingness to put pen to paper and do some mathematics.
Preface.-
1. Dialing In Problems.-
2. Polygons and Modular Arithmetic.-
3. The Fundamental Theorem of Arithmetic.-
4. The Fundamental Theorem of Algebra.-
5. Irrational, Algebraic and Transcendental Numbers.-
6. Fourier Series and Gauss Sums.- Epilogue.- Notation.- Bibliography.- Index.
Format: Hardback, 511 pages, height x width: 235x155 mm, 161 Tables, color; 161 Illustrations,
color; 4 Illustrations, black and white; X, 511 p. 165 illus., 161 illus. in color., 1 Hardback
Series: Springer Actuarial
Pub. Date: 07-Oct-2022
ISBN-13: 9783031124082
This open access book discusses the statistical modeling of insurance problems, a process which comprises data collection, data analysis and statistical model building to forecast insured events that may happen in the future. It presents the mathematical foundations behind these fundamental statistical concepts and how they can be applied in daily actuarial practice.
Statistical modeling has a wide range of applications, and, depending on the application, the theoretical aspects may be weighted differently: here the main focus is on prediction rather than explanation. Starting with a presentation of state-of-the-art actuarial models, such as generalized linear models, the book then dives into modern machine learning tools such as neural networks and text recognition to improve predictive modeling with complex features.
Providing practitioners with detailed guidance on how to apply machine learning methods to real-world data sets, and how to interpret the results without losing sight of the mathematical assumptions on which these methods are based, the book can serve as a modern basis for an actuarial education syllabus.
1.Introduction and discussion of statistical modeling cycle, prediction versus explanation (Breiman 2001, Shmueli 2010).- 2.Exponential dispersion family (EDF), explicit derivation based on Barndorff-Nielsen (2014) and Jorgensen (1986, 1987, 1997), discussion of the properties of the EDF.- 3.Estimation theory based on Lehmann (1959, 1983). This includes maximum likelihood estimation (MLE), uniformly minimum variance unbiased (UMVU), sufficient statistics, Cramer-Rao information bound, consistency and asymptotic normality of MLEs.- 4.Prediction theory, this includes generalization losses, deviance losses, in- and out-of-sample performance, cross-validation, Akaike's information criterion, consistent loss functionals, proper scoring rules.- 5.Generalized linear models, this chapter is based on McCullagh-Nelder (1983) and Fahrmeir-Tutz (1994). Generalized linear models with insurance pricing application, balance property and unbiasedness, model validation, quasi-likelihoods, double generalized linear model.-
6. Bayesian methods, Markov-chain Monte Carlo (MCMC) methods, ridge and LASSO regularization, expectation-maximization (EM) algorithm, truncation and censoring. 7.- Deep learning. Feed-forward neural network, universality theorems, gradient descent algorithm and back-propagation, the balance property, bias regularization, embedding layers, auto-encoders, model-agnostic tools.- 8.Recurrent neural networks, gated-recurrent unit (GRU) networks, long short-term memory (LSTM) networks, natural language processing (NLP), word embedding.- 9.Convolutional neural networks and image recognition.- 10.Special topics such as mixture density networks (MDN), sieve estimators and asymptotic results for network predictions.
Format: Hardback, 430 pages, height x width: 235x155 mm, 55 Illustrations,
black and white; XX, 430 p. 55 illus., 1 Hardback
Series: Graduate Texts in Mathematics 294
Pub. Date: 10-Oct-2022
ISBN-13: 9783031133787
This textbook introduces the study of partial differential equations using both analytical and numerical methods. By intertwining the two complementary approaches, the authors create an ideal foundation for further study. Motivating examples from the physical sciences, engineering, and economics complete this integrated approach.
A showcase of models begins the book, demonstrating how PDEs arise in practical problems that involve heat, vibration, fluid flow, and financial markets. Several important characterizing properties are used to classify mathematical similarities, then elementary methods are used to solve examples of hyperbolic, elliptic, and parabolic equations. From here, an accessible introduction to Hilbert spaces and the spectral theorem lay the foundation for advanced methods. Sobolev spaces are presented first in dimension one, before being extended to arbitrary dimension for the study of elliptic equations. An extensive chapter on numerical methods focuses on finite difference and finite element methods. Computer-aided calculation with Maple? completes the book. Throughout, three fundamental examples are studied with different tools: Poisson’s equation, the heat equation, and the wave equation on Euclidean domains. The Black?Scholes equation from mathematical finance is one of several opportunities for extension.
Partial Differential Equations offers an innovative introduction for students new to the area. Analytical and numerical tools combine with modeling to form a versatile toolbox for further study in pure or applied mathematics. Illuminating illustrations and engaging exercises accompany the text throughout. Courses in real analysis and linear algebra at the upper-undergraduate level are assumed.
1 Modeling, or where do differential equations come from.- 2 Classification and characteristics.- 3 Elementary methods.- 4 Hilbert spaces.- 5 Sobolev spaces and boundary value problems in dimension one.- 6 Hilbert space methods for elliptic equations.- 7 Neumann and Robin boundary conditions.- 8 Spectral decomposition and evolution equations.- 9 Numerical methods.- 10 Maple (R), or why computers can sometimes help.- Appendix.
Format: Paperback / softback, 157 pages, height x width: 235x155 mm, VIII, 157 p.
Series: Lecture Notes in Mathematics 2309
Pub. Date: 06-Oct-2022
ISBN-13: 9783031117985
This monograph uncovers the full capabilities of the Riemann integral. Setting aside all notions from Lebesgue’s theory, the author embarks on an exploration rooted in Riemann’s original viewpoint. On this journey, we encounter new results, numerous historical vignettes, and discover a particular handiness for computations and applications.
This approach rests on three basic observations. First, a Riemann integrability criterion in terms of oscillations, which is a quantitative formulation of the fact that Riemann integrable functions are continuous a.e. with respect to the Lebesgue measure. Second, the introduction of the concepts of admissible families of partitions and modified Riemann sums. Finally, the fact that most numerical quadrature rules make use of carefully chosen Riemann sums, which makes the Riemann integral, be it proper or improper, most appropriate for this endeavor.
A Modern View of the Riemann Integral is intended for enthusiasts keen to explore the potential of Riemann's original notion of integral. The only formal prerequisite is a proof-based familiarity with the Riemann integral, though readers will also need to draw upon mathematical maturity and a scholarly outlook.
Preface.
Chapter
1. Introduction.
Chapter
2. The -Riemann Integral.
Chapter
3. A Convergence Theorem.
Chapter
4. The Modified -Riemann Sums.
Chapter
5. The Pattern and Uniform Integrals.
Chapter
6. The Improper and Dominated Integrals.
Chapter
7. Coda.- Appendix I.- Appendix II.- References.- Index.