176 pp., 5 x 8 in, 25 b&w illus., 3 figures
Paperback
9780262544269
Published: September 27, 2022
Why mathematics is not merely formulaic: an argument that to write a mathematical proof is tantamount to inventing a story.
In The Meaning of Proofs, mathematician Gabriele Lolli argues that to write a mathematical proof is tantamount to inventing a story. Lolli offers not instructions for how to write mathematical proofs, but a philosophical and poetic reflection on mathematical proofs as narrative. Mathematics, imprisoned within its symbols and images, Lolli writes, says nothing if its meaning is not narrated in a story. The minute mathematicians open their mouths to explain something?the meaning of x, how to find y?they are framing a narrative.
Every proof is the story of an adventure, writes Lolli, a journey into an unknown land to open a new, connected route; once the road is open, we correct it, expand it. Just as fairy tales offer a narrative structure in which new characters can be inserted into recurring forms of the genre in original ways, in mathematics, each new abstract concept is the protagonist of a different theory supported by the general techniques of mathematical reasoning. In ancient Greece, there was more than an analogy between literature and mathematics, there was direct influence. Euclid's proofs have roots in poetry and rhetoric. Mathematics, Lolli asserts, is not the mere manipulation of formulas.
Paperback
454 pp., 7 x 10 in, 2 color illus., 75 b&w illus.
9780262542159
Published: October 25, 2022
An approachable introduction to elementary sheaf theory and its applications beyond pure math.
Sheaves are mathematical constructions concerned with passages from local properties to global ones. They have played a fundamental role in the development of many areas of modern mathematics, yet the broad conceptual power of sheaf theory and its wide applicability to areas beyond pure math have only recently begun to be appreciated. Taking an applied category theory perspective, Sheaf Theory through Examples provides an approachable introduction to elementary sheaf theory and examines applications including n-colorings of graphs, satellite data, chess problems, Bayesian networks, self-similar groups, musical performance, complexes, and much more.
With an emphasis on developing the theory via a wealth of well-motivated and vividly illustrated examples, Sheaf Theory through Examples supplements the formal development of concepts with philosophical reflections on topology, category theory, and sheaf theory, alongside a selection of advanced topics and examples that illustrate ideas like cellular sheaf cohomology, toposes, and geometric morphisms.
Sheaf Theory through Examples seeks to bridge the powerful results of sheaf theory as used by mathematicians and real-world applications, while also supplementing the technical matters with a unique philosophical perspective attuned to the broader development of ideas.
The open access edition of this book was made possible by generous funding from Arcadia ? a charitable fund of Lisbet Rausing and Peter Baldwin.
Paperback
248 pp., 7 x 10 in, 28 b&w illus.
9780262544344
Published: October 25, 2022
The first collection of Leibniz's key writings on the binary system, newly translated, with many previously unpublished in any language.
The polymath Gottfried Wilhelm Leibniz (1646?1716) is known for his independent invention of the calculus in 1675. Another major?although less studied?mathematical contribution by Leibniz is his invention of binary arithmetic, the representational basis for today's digital computing. This book offers the first collection of Leibniz's most important writings on the binary system, all newly translated by the authors with many previously unpublished in any language. Taken together, these thirty-two texts tell the story of binary as Leibniz conceived it, from his first youthful writings on the subject to the mature development and publication of the binary system.
As befits a scholarly edition, Strickland and Lewis have not only returned to Leibniz's original manuscripts in preparing their translations, but also provided full critical apparatus. In addition to extensive annotations, each text is accompanied by a detailed introductory gheadnoteh that explains the context and content. Additional mathematical commentaries offer readers deep dives into Leibniz's mathematical thinking. The texts are prefaced by a lengthy and detailed introductory essay, in which Strickland and Lewis trace Leibniz's development of binary, place it in its historical context, and chart its posthumous influence, most notably on shaping our own computer age.
Format: Paperback / softback, 560 pages, height x width: 235x155 mm, weight:
884 g, 1 Illustrations,
color; 14 Illustrations, black and white; XXI, 560 p. 15 illus., 1 illus. in color.,
Series: Modeling and Simulation in Science, Engineering and Technology
Pub. Date: 20-Jun-2022
ISBN-13: 9783030696559
This textbook, now in its fourth edition, offers a rigorous and self-contained introduction to the theory of continuous-time stochastic processes, stochastic integrals, and stochastic differential equations. Expertly balancing theory and applications, it features concrete examples of modeling real-world problems from biology, medicine, finance, and insurance using stochastic methods. No previous knowledge of stochastic processes is required. Unlike other books on stochastic methods that specialize in a specific field of applications, this volume examines the ways in which similar stochastic methods can be applied across different fields. Beginning with the fundamentals of probability, the authors go on to introduce the theory of stochastic processes, the Ito Integral, and stochastic differential equations. The following chapters then explore stability, stationarity, and ergodicity. The second half of the book is dedicated to applications to a variety of fields, including finance, biology, and medicine. Some highlights of this fourth edition include a more rigorous introduction to Gaussian white noise, additional material on the stability of stochastic semigroups used in models of population dynamics and epidemic systems, and the expansion of methods of analysis of one-dimensional stochastic differential equations. An Introduction to Continuous-Time Stochastic Processes, Fourth Edition is intended for graduate students taking an introductory course on stochastic processes, applied probability, stochastic calculus, mathematical finance, or mathematical biology. Prerequisites include knowledge of calculus and some analysis; exposure to probability would be helpful but not required since the necessary fundamentals of measure and integration are provided. Researchers and practitioners in mathematical finance, biomathematics, biotechnology, and engineering will also find this volume to be of interest, particularly the applications explored in the second half of the book.
Foreword.- Preface to the Fourth Edition.- Preface to the Third Edition.- Preface to the Second Edition.- Preface.- Part I: Theory of Stochastic Processes.- Fundamentals of Probability.- Stochastic Processes.- The Ito Integral.- Stochastic Differential Equations.- Stability, Stationary, Ergodicity.- Part II: Applications of Stochastic Processes.- Applications to Finance and Insurance.- Applications to Biology and Medicine.- Measure and Integration.- Convergence of Probability Measures on Metric Spaces.- Diffusion Approximation of a Langevin System.- Elliptic and Parabolic Equations.- Semigroups of Linear Operators.- Stability of Ordinary Differential Equations.- References.- Nomenclature.- Index.
Format: Paperback / softback, 919 pages, height x width: 235x155 mm, weight: 1430 g, 89 Illustrations, color; 202 Illustrations,
black and white; XXXII, 919 p. 291 illus., 89 illus. in color.,
Series: Interdisciplinary Applied Mathematics 53
Pub. Date: 03-Jun-2022
ISBN-13: 9783030754549
This book covers methods of Mathematical Morphology to model and simulate random sets and functions (scalar and multivariate). The introduced models concern many physical situations in heterogeneous media, where a probabilistic approach is required, like fracture statistics of materials, scaling up of permeability in porous media, electron microscopy images (including multispectral images), rough surfaces, multi-component composites, biological tissues, textures for image coding and synthesis. The common feature of these random structures is their domain of definition in n dimensions, requiring more general models than standard Stochastic Processes.The main topics of the book cover an introduction to the theory of random sets, random space tessellations, Boolean random sets and functions, space-time random sets and functions (Dead Leaves, Sequential Alternate models, Reaction-Diffusion), prediction of effective properties of random media, and probabilistic fracture theories.
1. Introduction.- Part I Tools for Random Structures.- 2 Introduction to Random Closed Sets and to Semi-continuous Random Functions.- 3 Quantitative Analysis of Random Structures.- Part II Models of Random Structures.- 4 Excursion sets of Gaussian RF.- 5 Stochastic Point Processes and Random Trees.- 6 Boolean Random Sets.- 7 Random Tessellations.- 8 The Mosaic Model.- 9 Boolean Random Functions.- 10 Random Tessellations and Boolean Random Functions.- 11 Dead Leaves Models: from Space Tessellations to Random Functions.- 12 Sequential Cox Boolean and Conditional Dead Leaves Models.- 13 Sequential Alternate Random Functions.- 14 Primary Grains and Primary Functions.- 15 Dilution Random Functions.- 16 Reaction-Diffusion and Lattice Gas Models.- 17 Texture Segmentation by Morphological Probabilistic Hierarchies.- Part III Random Structures and Change of Scale.- 18 Change of Scale in Physics of Random Media.- 19 Digital Materials.- 20 Probabilistic Models for Fracture Statistics.- 21 Crack Paths in Random Media.
Format: Paperback / softback, 287 pages, height x width: 235x155 mm, weight: 480 g, 30 Illustrations, color;
22 Illustrations, black and white; XXI, 287 p. 52 illus., 30 illus. in color.
Pub. Date: 01-Jun-2022
ISBN-13: 9789811604492
This book offers a unique account on the life and works of Srinivasa Ramanujan?often hailed as the greatest gnaturalh mathematical genius. Sharing valuable insights into the many stages of Ramanujanfs life, this book provides glimpses into his prolific research on highly composite numbers, partitions, continued fractions, mock theta functions, arithmetic, and hypergeometric functions which led the author to discover a new summation theorem. It also includes the list of Ramanujanfs collected papers, letters and other material present at the Wren Library, Trinity College in Cambridge, UK. This book is a valuable resource for all readers interested in Ramanujanfs life, work and indelible contributions to mathematics.
1. Life of Srinivasa Ramanujan.-
2. Ramanujan at Cambridge.-
3. Ramanujan's Mathematics: Glimpses.-
4. Hardy on Ramanujan.-
5. Ramanujan and Hypergeometric Series.-
6. Chandrasekhar (Chandra) and Ramanujan.-
7. Books on Ramanujan and Busts of Ramanujan.-
8. Ramanujan Birth Anniversaries and Documentaries on Ramanujan.-
9. Relevance of Ramanujan Today.