Paperback
This book is a complete introduction to vector analysis, especially within the context of computer graphics. The author shows why vectors are useful and how it is possible to develop analytical skills in manipulating vector algebra. Even though vector analysis is a relatively recent development in the history of mathematics, it has become a powerful and central tool in describing and solving a wide range of geometric problems. The book is divided into eleven chapters covering the mathematical foundations of vector algebra and its application to, among others, lines, planes, intersections, rotating vectors, and vector differentiation.
"Each chapter presents some topic from vector analysis and contains a well-developed derivation and mathematical demonstration that makes following the topic easier. ... The book is written in a very accessible fashion. The author gives many examples presenting the notations and problems considered, making study easier. The book is suitable for undergraduate students of computer science, mathematics, and engineering, and is an ideal reference for researchers and professionals in computer graphics." (Krzysztof Gdawiec, zbMATH 1478.68008, 2022)
Preface.- History of Vector Analysis.- Linear Equations.- Vector Algebra.- Products of Vectors.- Differentiating Vector-Valued Functions.- Vector Differential Operators.- Tangent and Normal Vectors.- Straight Lines.- The Plane.- Intersections.- Rotating Vectors.- Index.
Format: Hardback, 488 pages, height x width: 235x155 mm, weight: 922 g, 1 Illustrations,
black and white; XIX, 488 p. 1 illus.
Series: Theory and Applications of Computability
Pub. Date: 26-Jul-2022
ISBN-13: 9783031113666
Reverse mathematics studies the complexity of proving mathematical theorems and solving mathematical problems. Typical questions include: Can we prove this result without first proving that one? Can a computer solve this problem? A highly active part of mathematical logic and computability theory, the subject offers beautiful results as well as significant foundational insights.
This text provides a modern treatment of reverse mathematics that combines computability theoretic reductions and proofs in formal arithmetic to measure the complexity of theorems and problems from all areas of mathematics. It includes detailed introductions to techniques from computable mathematics, Weihrauch style analysis, and other parts of computability that have become integral to research in the field.
Provides a complete introduction to reverse mathematics, including necessary background from computability theory, second order arithmetic, forcing, induction, and model construction
Offers a comprehensive treatment of the reverse mathematics of combinatorics, including Ramsey's theorem, Hindman's theorem, and many other results
Provides central results and methods from the past two decades, appearing in book form for the first time and including preservation techniques and applications of probabilistic arguments
Includes a large number of exercises of varying levels of difficulty, supplementing each chapter
The text will be accessible to students with a standard first year course in mathematical logic. It will also be a useful reference for researchers in reverse mathematics, computability theory, proof theory, and related areas.
1 introduction.- Part I Computable mathematics: 2 Computability theory.- 3 Instance-solution problems.- 4 Problem reducibilities.- Part II Formalization and syntax: 5 Second order arithmetic.- 6 Induction and bounding.- 7 Forcing.- Part III Combinatorics: 8 Ramsey's theorem.- 9 Other combinatorial principles.- Part IV Other areas: 10 Analysis and topology.- 11 Algebra.- 12 Set theory and beyond.
Format: Paperback / softback, 283 pages, height x width: 235x155 mm, 307 Illustrations,
color; 6 Illustrations, black and white; XI, 283 p. 313 illus., 307 illus. in color.
Series: Mathematics Study Resources 2
Pub. Date: 26-Oct-2022
ISBN-13: 9783658388096
The aim of the book is to study symmetries and tesselation, which have long interested artists and mathematicians. Famous examples are the works created by the Arabs in the Alhambra and the paintings of the Dutch painter Maurits Escher. Mathematicians did not take up the subject intensively until the 19th century. In the process, the visualisation of mathematical relationships leads to very appealing images. Three approaches are described in this book. In Part I, it is shown that there are 17 principally different possibilities of tesselation of the plane, the so-called "plane crystal groups". Complementary to this, ideas of Harald Heesch are described, who showed how these theoretical results can be put into practice: He gave a catalogue of 28 procedures that one can use creatively oneself - following in the footsteps of Escher, so to speak - to create artistically sophisticated tesselation. In the corresponding investigations for the complex plane in Part II, movements are replaced by bijective holomorphic mappings. This leads into the theory of groups of Moebius transformations: Kleinian groups, Schottky groups, etc. There are also interesting connections to hyperbolic geometry. Finally, in Part III, a third aspect of the subject is treated, the Penrose tesselation. This concerns results from the seventies, when easily describable and provably non-periodic parquetisations of the plane were given for the first time.
Part I: Escher seen over his shoulders.- Part II: Furniture transformations.- Part III: Penrose tesselation.
Format: Paperback / softback, 226 pages, height x width: 235x155 mm, 28 Tables, color;
23 Illustrations, color; 147 Illustrations, black and white; XVI, 226 p. 170 illus., 23 illus. in color
Pub. Date: 18-Oct-2022
ISBN-13: 9783031135651
This is open access book provides plenty of pleasant mathematical surprises. There are many fascinating results that do not appear in textbooks although they are accessible with a good knowledge of secondary-school mathematics. This book presents a selection of these topics including the mathematical formalization of origami, construction with straightedge and compass (and other instruments), the five- and six-color theorems, a taste of Ramsey theory and little-known theorems proved by induction.
Among the most surprising theorems are the Mohr-Mascheroni theorem that a compass alone can perform all the classical constructions with straightedge and compass, and Steiner's theorem that a straightedge alone is sufficient provided that a single circle is given. The highlight of the book is a detailed presentation of Gauss's purely algebraic proof that a regular heptadecagon (a regular polygon with seventeen sides) can be constructed with straightedge and compass.
Although the mathematics used in the book is elementary (Euclidean and analytic geometry, algebra, trigonometry), students in secondary schools and colleges, teachers, and other interested readers will relish the opportunity to confront the challenge of understanding these surprising theorems.
1. The Collapsing Compass. -
2. Trisection of an Angle. -
3. Squaring the Circle. -
4. The Five-Color Theorem. -
5. How to Guard a Museum. -
6. Induction. -
7. Solving Quadratic Equations. -
8. Ramsey Theory. -
9. Langford's Problem. -
10. The Axioms of Origami. -
11. Lill's Method and the Beloch Fold. -
12. Geometric Constructions Using Origami. -
13. A Compass Is Sufficient. -
14. A Straightedge and One Circle is Sufficient. -
15. Are Triangles with Equal Areas and Perimeters Congruent?. -
16. Construction of a Regular Heptadecagon.
Format: Hardback, 495 pages, height x width: 235x155 mm, 2 Tables, color; 55 Illustrations,
color; 47 Illustrations, black and white; XXII, 495 p. 102 illus., 55 illus. in color.
Series: Springer Biographies
Pub. Date: 18-Oct-2022
ISBN-13: 9783031113086
During his lifetime, Kurt Goedel was not well known outside the professional world of mathematicians, philosophers and theoretical physicists. Early in his career, for his doctoral thesis and then for his Habilitation (Dr.Sci.), he wrote earthshaking articles on the completeness and provability of mathematical-logical systems, upsetting the hypotheses of the most famous mathematicians/philosophers of the time. He later delved into theoretical physics, finding a unique solution to Einstein's equations for gravity, the 'Goedel Universe', and made contributions to philosophy, the guiding theme of his life. This book includes more details about the context of Goedel's life than are found in earlier biographies, while avoiding an elaborate treatment of his mathematical/scientific/philosophical works, which have been described in great detail in other books. In this way, it makes him and his times more accessible to general readers, and will allow them to appreciate the lasting effects of Goedel's contributions (the latter in a more up-to-date context than in previous biographies, many of which were written 15-25 years ago). His work spans or is relevant to a wide spectrum of intellectual endeavor, and this is emphasized in the book, with recent examples. This biography also examines possible sources of his unusual personality, which combined mathematical genius with an almost childlike naivete concerning everyday life, and striking scientific innovations with timidity and hesitancy in practical matters. How he nevertheless had a long and successful career, inspiring many younger scholars along the way, with the help of his loyal wife Adele and some of his friends, is a fascinating story in human nature.
Preface.- Prologue.- La Belle Epoque in Brunn. Beginnings.- School Days. A New Nation.- Moving to the Capital. Student Life in 1920's Vienna.- The Wiener Kreis and the Mathematical Colloquium. Graduate work.- Private Life in Vienna.- Goedel's Doctoral Thesis. The Incompleteness Theorems.- The Mathematician in Vienna. Habilitation.- Matters of Health.- A Sojourn Abroad: 1933/34 - Princeton.- Back to Vienna. The First Breakdown.- 'Commuting' between Vienna and Princeton - The Late 1930's. Marriage.- The Continuum Hypothesis.- Professional Uncertainty. A Long Journey Eastwards.- Princeton and the IAS - Philosophy, Einstein and von Neumann. A Bizarre Birthday Present: Goedel's Universe.- Reception, Recognition, Honors. Einstein's Loss. The Professor at Princeton.- The 1960's: Fame and Seclusion.- Later Years. Philosophy, Cosmology, Logic, Computability.- Goedel's Legacy - The Lessons of an Unusual Life.- Epilogue.
Format: Hardback, 186 pages, height x width: 240x168 mm, 35 Tables, color; 4 Illustrations,
color; 1 Illustrations, black and white; XII, 186 p. 5 illus., 4 illus. in color.
Series: Synthesis Lectures on Mathematics & Statistics
Pub. Date: 18-Oct-2022
ISBN-13: 9783031112355
This book systematically establishes the almost periodic theory of dynamic equations and presents applications on time scales in fuzzy mathematics and uncertainty theory. The authors introduce a new division of fuzzy vectors depending on a determinant algorithm and develop a theory of almost periodic fuzzy multidimensional dynamic systems on time scales. Several applications are studied; in particular, a new type of fuzzy dynamic systems called fuzzy q-dynamic systems (i.e. fuzzy quantum dynamic systems) is presented. The results are not only effective on classical fuzzy dynamic systems, including their continuous and discrete situations, but are also valid for other fuzzy multidimensional dynamic systems on various hybrid domains. In an effort to achieve more accurate analysis in real world applications, the authors propose a number of uncertain factors in the theory. As such, fuzzy dynamical models, interval-valued functions, differential equations, fuzzy-valued differential equations, and their applications to dynamic equations on time scales are considered.
Generalized Hukuhara Difference and Division for Interval and Fuzzy Arithmetic.- An Embedding Theorem and the Multiplication of Fuzzy Vectors.- Calculus of Fuzzy Vector-valued Functions on Time Scales.- Shift Almost Periodic Fuzzy Vector-valued Functions.- Division of Fuzzy Vector-valued Functions Depending on a Determinant Algorithm.- Almost Periodic Generalized Fuzzy Multidimensional Dynamic Equations and Applications.