Format: Hardback, 173 pages, height x width: 235x155 mm, 7 Tables, color; 6 Illustrations, color;
66 Illustrations, black and white; XII, 173 p. 72 illus., 6 illus. in color., 1 Hardback
Series: Mathematics in Mind
Pub. Date: 26-Aug-2023
ISBN-13: 9783031315817
This book treats eighteenth-century Italian philosopher Giambattista Vico's theory of poetic logic for the first time as the originating force in mathematics, transforming instinctive counting and spatial perception into poetic (metaphorical) symbolism that dovetails with the origin of language. It looks at current work on mathematical cognition (from Lakoff and Nunez to Butterworth, Dehaene, and beyond), matching it against the poetic logic paradigm. In a sense, it continues from where Kasner and Newman left off, connecting contemporary research on the mathematical mind to the idea that the products of early mathematics were virtually identical to the first forms of poetic language. As such, this book informs the current research on mathematical cognition from a different angle, by looking back at a still relatively unknown philosopher within mathematics. The aim of this volume is to look broadly at what constitutes the mathematical mind through the Vichian lens of poetic logic. Vico was among the first to suggest that the essential nature of mind could be unraveled indirectly by reconstructing the sources of its "modifications" (his term for "creations"); that is, by examining the creation and function of symbols, words, and all the other uniquely human artifacts-including mathematics-the mind has allowed humans to establish "the world of civil society," Vico's term for culture and civilization. The book is of interest to cognitive scientists working on math cognition. It presents the theory of poetic logic as Vico articulated it in his book The New Science, examining its main premises and then applying it to an interpretation of the ongoing work in math cognition. It will also be of interest to the general public, since it presents a history of early mathematics through the lens of an idea that has borne fruit in understanding the origin of language and symbols more broadly.
List of FiguresPreface1. Imagination Prologue The Mathematical
Fantasia Poetic Wisdom Poetic Logic The Origins Question Psychological
Validity Epilogue2. Ingenuity Prologue The Mathematical
Ingegno Conjecture Proof and Decidability Epilogue3. Memory Prologue The
Mathematical Memoria Notation Diagrams Looping Structure Epilogue4.
Metaphor Prologue The Vichian Metafora Conceptual Metaphors in
Mathematics Image Schemas Conceptual Blending Mathematical
Infinity Epilogue5. Logic Prologue Logic and Mathematics Forms of Logic The
Bi-Part Soul The Origins Question Redux EpilogueGlossaryReferencesIndex
Format: Hardback, 340 pages, height x width: 235x155 mm, X, 340 p., 1 Hardback
Series: RSME Springer Series 9
Pub. Date: 30-Aug-2023
ISBN-13: 9783031364044
This monograph deals with matrix manifolds, i.e., manifolds for which there is a natural representation of their elements as matrix arrays. Classical matrix manifolds (Stiefel, Grassmann and flag manifolds) are studied in a more general setting. It provides tools to investigate matrix varieties over Pythagorean formally real fields. The presentation of the book is reasonably self-contained. It contains a number of nontrivial results on matrix manifolds useful for people working not only in differential geometry and Riemannian geometry but in other areas of mathematics as well. It is also designed to be readable by a graduate student who has taken introductory courses in algebraic and differential geometry.
Chapter 1 Algebraic preliminariesThe main goal of this chapter is to
introduce some notations and
terminologies. We assume that the reader is more or less familiar
with the basic concepts of algebraic geometry and linear algebra.
Chapter 2 Exceptional groups $G_2(K)$ and $F_4(K)$
This chapter takes up the systematic study of a generalization
of the exceptional compact Lie groups $G_2$ and $F_4$ on groups $G_2(K)$
and $F_4(K)$ provided $K$ is Pythagorean formally real field. The main result
stated in Theorem 2.48 says that any
Hermitian $3 \times 3$-matrix $A \in \mbox{Herm}_3(\mathbb{O}(K))$ can
be transformed to a diagonal form by some element of $F_4(K)$.
Chapter 3 Stiefel, Grassmann manifolds and generalizations
In this chapter we investigate and prove some properties of the
classical manifolds of Stiefel, Grassmann and flag manifolds.
All along this chapter $\mathcal{A}$ denotes the field of
reals, $\mathbb{R}$, the field of complex
numbers, $\mathbb{C}$, the skew field of
quaternions, $\mathbb{H}$ and, except if otherwise
said the octonion division algebra, $\mathbb{O}$.
Chapter 4 More classical matrix varieties
In this chapter we generalize Stiefel, Grassmann and flag
manifolds, defined in
Chapter 3, to what we call
here i-Stiefel, i-Grassmann and i-flag manifolds.
This "i" comes from idempotent. Those manifolds do not seem
to have being enough studied in the literature. In particular, they
do not have even a name. As in
Chapter 2,$ \mathcal{A}$ denotes the
field of reals, $\mathbb{R}$, the field of complex numbers, $\mathbb{C}$,
theskew field of quaternions, $\mathbb{H}$ and,
occasionally, the octonion division algebra $\mathbb{O}$.
Chapter 5 Algebraic generalizations of matrix varieties
We use
Chapters 1 and 2 to define and extend results of
Chapters 3 and 4 to matrix varieties over more general division
algebras.
That includes extending the classical definitions of Riemannian,
Hermitian and symplectic manifolds.
All along this chapter $K$ is a formally real
Pythagorean field and $\mathcal{A}$ denotes either $K$, the
complex $K$-algebra $\mathbb{C}(K)$, the quaternion $K$-algebra
$\mathbb{H}(K)$ or the
octonion $K$-algebra $\mathbb{O}(K)$.
Chapter 6 Curvature, geodesics and distance on matrix varieties
In this chapter we study more closely the Riemannian structure
of classical matrix manifolds introduced in
Chapters 3 and
4.
Here, $\mathcal{A} = \mathbb{R},\, \mathbb{C},\, \mathbb{H}$ and
occasionally $\mathbb{O}$.
We also extend, whenever it is possible, definitions and
results to the general case treated in
Chapter 5,
where $\mathcal{A} = K,\, \mathbb{C}(K),\, \mathbb{H}(K),\, \mathbb{O}(K)$
for $K$ a Pythagorean formally real field.
Format: Hardback, 412 pages, height x width: 235x155 mm, 15 Tables, color; 17 Illustrations,
color; 100 Illustrations, black and white; VIII, 412 p. 117 illus., 17 illus. in color., 1 Hardback
Series: Advances in Mechanics and Mathematics 49
Pub. Date: 08-Sep-2023
ISBN-13: 9783031356544
This monograph presents the geometric foundations of continuum mechanics. An emphasis is placed on increasing the generality and elegance of the theory by scrutinizing the relationship between the physical aspects and the mathematical notions used in its formulation. The theory of uniform fluxes in affine spaces is covered first, followed by the smooth theory on differentiable manifolds, and ends with the non-smooth global theory. Because continuum mechanics provides the theoretical foundations for disciplines like fluid dynamics and stress analysis, the author's extension of the theory will enable researchers to better describe the mechanics of modern materials and biological tissues. The global approach to continuum mechanics also enables the formulation and solutions of practical optimization problems. Foundations of Geometric Continuum Mechanics will be an invaluable resource for researchers in the area, particularly mathematicians, physicists, and engineers interested in the foundational notions of continuum mechanics.
1. Introduction.-
2. Prelude: Finite Dimensional Systems.- Part I
Algebraic Theory: Uniform Fluxes.-
3. Simplices in Affine Spaces and Their
Boundaries.-
4. Uniform Fluxes in Affine Spaces.-
5. From Uniform Fluxes to
Exterior Algebra.- Part II: Smooth Theory.-
6. Smooth Analysis on Manifolds:
A Short Review.-
7. Interlude: Smooth Distributions of Defects.-
8. Smooth
Fluxes.-
9. Frames, Body Points, and Spacetime Structure.-
10. Stresses.-
11.
Smooth Electromagnetism on Manifolds.-
12. The Elasticity Problem.-
13.
Symmetry and Dynamics.- Part III Non-Smooth, Global Theories.-
14. Banachable
Space of Sections of Vector Bundles over Compact Manifolds.-
15. Manifolds of
Sections and Embeddings.-
16. The General Framework for Global Analytic
Stress Theory.-
17. Dual Spaces Corresponding to Spaces of Differentiable
Sections of a Vector Bundle: Localization of Sections and Functionals.-
18.
de Rham Currents.-
19. Interlude: Singular Distributions of Defects in
Bodies.-
20. Vector-Valued Currents.-
21. The Representation of Forces by
Stresses and Hyperstresses.-
22. Simple Forces and Stresses.-
23. Whitney's
Geometric Integration Theory and Non-Smooth Bodies.-
24. Optimal Fields and
Load Capacity of Bodies.- Index.
Format: Hardback, 690 pages, height x width: 235x155 mm, 45 Tables, color; 45 Illustrations,
color; 19 Illustrations, black and white; X, 690 p. 64 illus., 45 illus. in color., 1 Hardback
Series: Statistics for Biology and Health
Pub. Date: 06-Sep-2023
ISBN-13: 9783031358500
This book provides an introduction to computer-based methods for the analysis of genomic data. Breakthroughs in molecular and computational biology have contributed to the emergence of vast data sets, where millions of genetic markers for each individual are coupled with medical records, generating an unparalleled resource for linking human genetic variation to human biology and disease. Similar developments have taken place in animal and plant breeding, where genetic marker information is combined with production traits. An important task for the statistical geneticist is to adapt, construct and implement models that can extract information from these large-scale data. An initial step is to understand the methodology that underlies the probability models and to learn the modern computer-intensive methods required for fitting these models. The objective of this book, suitable for readers who wish to develop analytic skills to perform genomic research, is to provide guidance to take this first step. This book is addressed to numerate biologists who typically lack the formal mathematical background of the professional statistician. For this reason, considerably more detail in explanations and derivations is offered. It is written in a concise style and examples are used profusely. A large proportion of the examples involve programming with the open-source package R. The R code needed to solve the exercises is provided. The MarkDown interface allows the students to implement the code on their own computer, contributing to a better understanding of the underlying theory. Part I presents methods of inference based on likelihood and Bayesian methods, including computational techniques for fitting likelihood and Bayesian models. Part II discusses prediction for continuous and binary data using both frequentist and Bayesian approaches. Some of the models used for prediction are also used for gene discovery. The challenge is to find promising genes without incurring a large proportion of false positive results. Therefore, Part II includes a detour on False Discovery Rate assuming frequentist and Bayesian perspectives. The last chapter of Part II provides an overview of a selected number of non-parametric methods. Part III consists of exercises and their solutions. Daniel Sorensen holds PhD and DSc degrees from the University of Edinburgh and is an elected Fellow of the American Statistical Association. He was professor of Statistical Genetics at Aarhus University where, at present, he is professor emeritus.
Overview.- Part I: Fitting Likelihood and Bayesian Models.- Likelihood.-
Computing the Likelihood.- Bayesian Methods.- McMC in Practice.- Part II:
Prediction.- Fundamentals of Prediction.- Shrinkage Methods.- Digression on
Multiple Testing: False Discovery Rates.- Binary Data.- Bayesian Prediction
and Model Checking.- Nonparametric Methods: A Selected Overview.- Part III:
Exercises and Solutions.- Exercises.- Solution to Exercises.
Format: Paperback / softback, 193 pages, height x width: 235x155 mm, 30 Tables, color;
30 Illustrations, color; 9 Illustrations, black and white; X, 193 p. 39 illus., 30 illus. in color., 1 Paperback / softback
Series: Surveys and Tutorials in the Applied Mathematical Sciences 12
Pub. Date: 09-Aug-2023
ISBN-13: 9783031345142
This primer on averaging theorems provides a practical toolbox for applied mathematicians, physicists, and engineers seeking to apply the well-known mathematical theory to real-world problems. With a focus on practical applications, the book introduces new approaches to dissipative and Hamiltonian resonances and approximations on timescales longer than 1/ . Accessible and clearly written, the book includes numerous examples ranging from elementary to complex, making it an excellent basic reference for anyone interested in the subject. The prerequisites have been kept to a minimum, requiring only a working knowledge of calculus and ordinary and partial differential equations (ODEs and PDEs). In addition to serving as a valuable reference for practitioners, the book could also be used as a reading guide for a mathematics seminar on averaging methods. Whether you're an engineer, scientist, or mathematician, this book offers a wealth of practical tools and theoretical insights to help you tackle a range of mathematical problems.
1. Introduction.-
2. First order periodic averaging.-
3. Periodic
solutions.-
4. Second order periodic averaging.-
5. First order general
averaging.-
6. Approximations on timescales longer than 1/ .-
7. Averaging
over spatial variables.-
8. Hamiltonian reasonances.-
9. The emergence of
tori.-
10. Averaging for partial differential equations.
Format: Hardback, 602 pages, height x width: 235x155 mm,
41 Illustrations, black and white; VIII, 602 p. 41 illus., 1 Hardback
Series: Algorithms and Computation in Mathematics 30
Pub. Date: 16-Sep-2023
ISBN-13: 9783031346514
Defined as solutions of linear differential or difference equations with polynomial coefficients, D-finite functions play an important role in various areas of mathematics. This book is a comprehensive introduction to the theory of these functions with a special emphasis on computer algebra algorithms for computing with them: algorithms for detecting relations from given data, for evaluating D-finite functions, for executing closure properties, for obtaining various kinds of "explicit" expressions, for factoring operators, and for definite and indefinite symbolic summation and integration are explained in detail.
The book comes "with batteries included" in the sense that it requires no background in computer algebra as the relevant facts from this area are summarized in the beginning. This makes the book accessible to a wide range of readers, from mathematics students who plan to work themselves on D-finite functions to researchers who want to apply the theory to their own work. Hundreds of exercises invite the reader to apply the techniques in the book and explore further aspects of the theory on their own. Solutions to all exercises are given in the appendix.
When algorithms for D-finite functions came up in the early 1990s, computer proofs were met with a certain skepticism. Fortunately, these times are over and computer algebra has become a standard tool for many mathematicians. Yet, this powerful machinery is still not as widely known as it deserves. This book helps to spread the word that certain tasks can be safely delegated to a computer algebra system, and also what the limitations of these techniques are.
Chapter.
1. Background and fundamental conceptsChapter.
2. The
recurrence case in one variableChapter.
3. The differential case in one
variableChapter.
4. OperatorsChapter.
5. Summation and integrationAnswers to
ExercisesSoftwareNotationsReferencesIndex