Format: Paperback / softback, 45 pages, height x width: 235x155 mm, 111 Illustrations,
black and white; X, 45 p. 111 illus.,
Pub. Date: 11-Nov-2022
ISBN-13: 9783658386399
This book deals with the geometry of visual space in all its aspects. As in any branch of mathematics, the aim is to trace the hidden to the obvious; the peculiarity of geometry is that the obvious is sometimes literally before one's eyes.
Starting from intuition, spatial concepts are embedded in the pre-existing mathematical framework of linear algebra and calculus. The path from visualization to mathematically exact language is itself the learning content of this book. This is intended to close an often lamented gap in understanding between descriptive preschool and school geometry and the abstract concepts of linear algebra and calculus. At the same time, descriptive geometric modes of argumentation are justified because their embedding in the strict mathematical language has been clarified.
The concepts of geometry are of a very different nature; they denote, so to speak, different layers of geometric thinking: some arguments use only concepts such as point, straight line, and incidence, others require angles and distances, still others symmetry considerations. Each of these conceptual fields determines a separate subfield of geometry and a separate chapter of this book, with the exception of the last-mentioned conceptual field "symmetry", which runs through all the others:
- Incidence: Projective geometry
- Parallelism: Affine geometry
- Angle: Conformal Geometry
- Distance: Metric Geometry
- Curvature: Differential Geometry
- Angle as distance measure: Spherical and Hyperbolic Geometry
- Symmetry: Mapping Geometry.
The mathematical experience acquired in the visual space can be easily transferred to much more abstract situations with the help of the vector space notion. The generalizations beyond the visual dimension point in two directions: Extension of the number concept and transcending the three illustrative dimensions.
This book is a translation of the original German 1st edition Geometrie ? Anschauung und Begriffe by Jost-Hinrich Eschenburg, published by Springer Fachmedien Wiesbaden GmbH, part of Springer Nature in 2020. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.
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What is geometry.- Parallelism: affine geometry.- From affine geometry to linear algebra.- Definition of affine space.- Parallelism and semiaffine mappings.- Parallel projections.- Affine coordinates and center of gravity.- Incidence: projective geometry.- Central perspective.- Far points and straight lines of projection.- Projective and affine space.-Semi-projective mappings and collineations.- Conic sections and quadrics; homogenization.- The theorems of Desargues and Brianchon.- Duality and polarity; Pascal's theorem.- The double ratio.- Distance: Euclidean geometry.- The Pythagorean theorem.- Isometries of Euclidean space.- Classification of isometries.- Platonic solids.- Symmetry groups of Platonic solids.- Finite rotation groups and crystal groups.- Metric properties of conic sections.- Curvature: differential geometry.- Smoothness.- Fundamental forms and curvatures.- Characterization of spheres and hyperplanes.- Orthogonal hyperface systems.- Angles: conformal geometry.- Conformal mappings.- Inversions.- Conformal and spherical mappings.- The stereographic projection.- The space of spheres.- Angular distance: spherical and hyperbolic geometry. The hyperbolic space. Distance on the sphere and in hyperbolic space. Models of hyperbolic geometry.- Exercises.- Solutions
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Format: Hardback, 584 pages, height x width: 235x155 mm, 3 Illustrations,
color; 97 Illustrations, black and white; XVI, 584 p. 100 illus., 3 illus. in color
Pub. Date: 15-Dec-2022
ISBN-13: 9783031118326
Now in its second edition, this introductory statistics textbook conveys the essential concepts and tools needed to develop and nurture statistical thinking. It presents descriptive, inductive and explorative statistical methods and guides the reader through the process of quantitative data analysis. This revised and extended edition features new chapters on logistic regression, simple random sampling, including bootstrapping, and causal inference.
The text is primarily intended for undergraduate students in disciplines such as business administration, the social sciences, medicine, politics, and macroeconomics. It features a wealth of examples, exercises and solutions with computer code in the statistical programming language R, as well as supplementary material that will enable the reader to quickly adapt the methods to their own applications.
Part I Descriptive Statistics: Introduction and Framework.- Frequency Measures and Graphical Representation of Data.- Measures of Central Tendency and Dispersion.- Association of Two Variables.- Part I Probability Calculus: Combinatorics.- Elements of Probability Theory.- Random Variables.- Probability Distributions.- Part III Inductive Statistics: Inference.- Hypothesis Testing.- Linear Regression.- Logistic Regression.- Part IV Additional Topics Simple Random Sampling and Bootstrapping.- Causality.- Part V Appendices: Introduction to R.- Solutions to Exercises.- Technical Appendix.- Visual Summaries.
Format: Paperback / softback, 283 pages, height x width: 235x155 mm,
5 Tables, color; 1 Illustrations, black and white; XI, 283 p. 1 illus.,
Series: Springer Undergraduate Mathematics Series
Pub. Date: 29-Oct-2022
ISBN-13: 9783031115301
This textbook offers an engaging account of the theory of ordinary differential equations intended for advanced undergraduate students of mathematics. Informed by the authorfs extensive teaching experience, the book presents a series of carefully selected topics that, taken together, cover an essential body of knowledge in the field. Each topic is treated rigorously and in depth.
The book begins with a thorough treatment of linear differential equations, including general boundary conditions and Greenfs functions. The next chapters cover separable equations and other problems solvable by quadratures, series solutions of linear equations and matrix exponentials, culminating in Sturm?Liouville theory, an indispensable tool for partial differential equations and mathematical physics. The theoretical underpinnings of the material, namely, the existence and uniqueness of solutions and dependence on initial values, are treated at length. A noteworthy feature of this book is the inclusion of project sections, which go beyond the main text by introducing important further topics, guiding the student by alternating exercises and explanations.
Designed to serve as the basis for a course for upper undergraduate students, the prerequisites for this book are a rigorous grounding in analysis (real and complex), multivariate calculus and linear algebra. Some familiarity with metric spaces is also helpful. The numerous exercises of the text provide ample opportunities for practice, and the aforementioned projects can be used for guided study. Some exercises have hints to help make the book suitable for independent study.
1 Linear Ordinary Differential Equations.- 2 Separation of Variables.- 3 Series Solutions of Linear Equations.- 4 Existence Theory.- 5 The Exponential of a Matrix.- 6 Continuation of Solutions.- 7 Sturm-Liouville Theory.
Format: Paperback / softback, 204 pages, height x width: 235x155 mm, 43 Illustrations,
color; 18 Illustrations, black and white; XII, 204 p. 61 illus., 43 illus. in color.
Pub. Date: 28-Nov-2022
ISBN-13: 9783031119996
The use of difference matrices and high-level MATLABR commands to implement finite difference algorithms is pedagogically novel. This unique and concise textbook gives the reader easy access and a general ability to use first and second difference matrices to set up and solve linear and nonlinear systems in MATLAB which approximate ordinary and partial differential equations. Prerequisites include a knowledge of basic calculus, linear algebra, and ordinary differential equations. Some knowledge of partial differential equations is a plus though the text may easily serve as a supplement for the student currently working through an introductory PDEs course. Familiarity with MATLAB is not required though a little prior experience with programming would be helpful.
In addition to its special focus on solving in MATLAB, the abundance of examples and exercises make this text versatile in use. It would serve well in a graduate course in introductory scientific computing for partial differential equations. With prerequisites mentioned above plus some elementary numerical analysis, most of the material can be covered and many of the exercises assigned in a single semester course. Some of the more challenging exercises make substantial projects and relate to topics from other typical graduate mathematics courses, e.g., linear algebra, differential equations, or topics in nonlinear functional analysis. A selection of the exercises may be assigned as projects throughout the semester. The student will develop the skills to run simulations corresponding to the primarily theoretical course material covered by the instructor. The book can serve as a supplement for the instructor teaching any course in differential equations. Many of the examples can be easily implemented and the resulting simulation demonstrated by the instructor. If the course has a numerical component, a few of the more difficult exercises may be assigned as student projects.
Established researchers in theoretical partial differential equations may find this book useful as well, particularly as an introductory guide for their research students. Those unfamiliar with MATLAB can use the material as a reference to quickly develop their own applications in that language. Practical assistance in implementing algorithms in MATLAB can be found in these pages. A mathematician who is new to the practical implementation of methods for scientific computation in general can learn how to implement and execute numerical simulations of differential equations in MATLAB with relative ease by working through a selection of exercises. Additionally, the book can serve as a practical guide in independent study, undergraduate or graduate research experiences, or for reference in simulating solutions to specific thesis or dissertation-related experiments.
1. Introduction.-
2. Review of elementary numerical methods and MATLAB(R).-
3. Ordinary Differential Equations.-
4. Partial Differential Equations.-
5. Advanced topics in semilinear elliptic BVP.- References.
Format: Hardback, 209 pages, height x width: 235x155 mm, 1 Illustrations, color;
56 Illustrations, black and white; XI, 209 p. 57 illus., 1 illus. in color
Pub. Date: 20-Nov-2022
ISBN-13: 9783031120992
This book offers an accessible introduction to random walk and diffusion models at a level consistent with the typical background of students in the life sciences. In recent decades these models have become widely used in areas far beyond their traditional origins in physics, for example, in studies of animal behavior, ecology, sociology, sports science, population genetics, public health applications, and human decision making. Developing the main formal concepts, the book provides detailed and intuitive step-by-step explanations, and moves smoothly from simple to more complex models. Finally, in the last chapter, some successful and original applications of random walk and diffusion models in the life and behavioral sciences are illustrated in detail. The treatment of basic techniques and models is consolidated and extended throughout by a set of carefully chosen exercises.
Introduction.- Discrete random Walks.- The Correlated Random Walk.- The Diffusion Limit.- The Wiener process.- More general Diffusion Processes.- Differential Equations for Probabilities.- Applications.
Format: Hardback, 252 pages, height x width: 235x155 mm, 14 Illustrations,
color; 1 Illustrations, black and white; X, 252 p. 15 illus., 14 illus. in color
Series: Springer Series in Computational Mathematics 59
Pub. Date: 20-Nov-2022
ISBN-13: 9783031132193
This book provides a comprehensive analysis of time domain boundary integral equations and their discretisation by convolution quadrature and the boundary element method.
Properties of convolution quadrature, based on both linear multistep and Runge?Kutta methods, are explained in detail, always with wave propagation problems in mind. Main algorithms for implementing the discrete schemes are described and illustrated by short Matlab codes; translation to other languages can be found on the accompanying GitHub page. The codes are used to present numerous numerical examples to give the reader a feeling for the qualitative behaviour of the discrete schemes in practice. Applications to acoustic and electromagnetic scattering are described with an emphasis on the acoustic case where the fully discrete schemes for sound-soft and sound-hard scattering are developed and analysed in detail. A strength of the book is that more advanced applications such as linear and non-linear impedance boundary conditions and FEM/BEM coupling are also covered. While the focus is on wave scattering, a chapter on parabolic problems is included which also covers the relevant fast and oblivious algorithms. Finally, a brief description of data sparse techniques and modified convolution quadrature methods completes the book.
Suitable for graduate students and above, this book is essentially self-contained, with background in mathematical analysis listed in the appendix along with other useful facts. Although not strictly necessary, some familiarity with boundary integral equations for steady state problems is desirable.
1 Some examples of causal convolutions.- 2 Convolution quadrature for hyperbolic symbols.- 3 Algorithms for CQ: linear multistep methods.- 4 Acoustic scattering in the time domain.- 5 Runge-Kutta CQ.- 6 Transient electromagnetism.- 7 Boundary-field formulations.- 8 Parabolic problems.- 9 Data sparse methods and other topics.