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Quasicrystals are non-periodic solids that were discovered in 1982 by Dan Shechtman, Nobel Prize Laureate in Chemistry 2011. The underlying mathematics, known as the theory of aperiodic order, is the subject of this comprehensive multi-volume series. This first volume provides a graduate-level introduction to the many facets of this relatively new area of mathematics. Special attention is given to methods from algebra, discrete geometry and harmonic analysis, while the main focus is on topics motivated by physics and crystallography. In particular, the authors provide a systematic exposition of the mathematical theory of kinematic diffraction. Numerous illustrations and worked-out examples help the reader to bridge the gap between theory and application. The authors also point to more advanced topics to show how the theory interacts with other areas of pure and applied mathematics.

Aperiodicity --- Crystallography, Mathematical --- Aperiodic tilings --- Quasicrystals --- Pavage (mathématiques) --- Quasicristaux --- Mathematics --- Mathématiques --- Mathématiques. --- Aperiodic tilings. --- Quasi-crystals --- Condensed matter --- Crystals --- Aperiodic point sets --- Sets, Aperiodic point --- Discrete geometry --- Point set theory --- Tiling (Mathematics) --- Mathematics.

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The aim of the book is to give a broad introduction of topology to undergraduate students. It covers the most important and useful parts of the point-set as well as the combinatorial topology. The development of the material is from simple to complex, concrete to abstract, and appeals to the intuition of readers. Attention is also paid to how topology is actually used in the other fields of mathematics. Over 150 illustrations, 160 examples and 600 exercises will help readers to practice and fully understand the subject. Contents: Set and Map Metric Space Graph Topology Topological Concepts Complex Topological Properties Surface Topics in Point Set Topology Index

Topology --- Point set theory --- Algebraic topology --- Assemblage of points --- Groups of points --- Point sets --- Points, Assemblage of --- Points, Groups of --- Set theory --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Algebras, Linear --- Mathematics --- Physical Sciences & Mathematics

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Category theory. Homological algebra --- Homotopy theory. --- Categories (Mathematics) --- Point set theory. --- Homotopie. --- Catégories (mathématiques) --- Ensembles de points (mathématiques) --- Homotopy theory --- Point set theory --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Assemblage of points --- Groups of points --- Point sets --- Points, Assemblage of --- Points, Groups of --- Set theory --- Deformations, Continuous

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One of the greatest revolutions in mathematics occurred when Georg Cantor (1845-1918) promulgated his theory of transfinite sets. This revolution is the subject of Joseph Dauben's important studythe most thorough yet writtenof the philosopher and mathematician who was once called a "corrupter of youth" for an innovation that is now a vital component of elementary school curricula. Set theory has been widely adopted in mathematics and philosophy, but the controversy surrounding it at the turn of the century remains of great interest. Cantor's own faith in his theory was partly theological. His religious beliefs led him to expect paradoxes in any concept of the infinite, and he always retained his belief in the utter veracity of transfinite set theory. Later in his life, he was troubled by recurring attacks of severe depression. Dauben shows that these played an integral part in his understanding and defense of set theory.

Cantor, Georg --- Aristotle. --- Arithmetic. --- Berlin, Germany. --- Borchardt, C. W. --- Catholic church. --- Consistency. --- Continuity. --- Derived Set(s). --- Diagonal method. --- Epistemology. --- Equivalence. --- Finitism. --- Fourier series. --- Goldscheider, F. --- Grattan-Guinness, I. --- Hermite, C. --- Jeiler, I. --- Joseph of Arimathea. --- Kerry, B. --- Kronecker, L. --- Limit points. --- Logic. --- Mathematics. --- Neo-Thomism. --- Order types. --- Paradoxes. --- Point sets. --- Power(s). --- Quaternions. --- Schoenflies, A. --- Set theory. --- Subsets. --- Theologians. --- Transfinite numbers. --- Trigonemetric series.

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What is a number? What is infinity? What is continuity? What is order? Answers to these fundamental questions obtained by late nineteenth-century mathematicians such as Dedekind and Cantor gave birth to set theory. This textbook presents classical set theory in an intuitive but concrete manner. To allow flexibility of topic selection in courses, the book is organized into four relatively independent parts with distinct mathematical flavors. Part I begins with the Dedekind–Peano axioms and ends with the construction of the real numbers. The core Cantor–Dedekind theory of cardinals, orders, and ordinals appears in Part II. Part III focuses on the real continuum. Finally, foundational issues and formal axioms are introduced in Part IV. Each part ends with a postscript chapter discussing topics beyond the scope of the main text, ranging from philosophical remarks to glimpses into landmark results of modern set theory such as the resolution of Lusin's problems on projective sets using determinacy of infinite games and large cardinals. Separating the metamathematical issues into an optional fourth part at the end makes this textbook suitable for students interested in any field of mathematics, not just for those planning to specialize in logic or foundations. There is enough material in the text for a year-long course at the upper-undergraduate level. For shorter one-semester or one-quarter courses, a variety of arrangements of topics are possible. The book will be a useful resource for both experts working in a relevant or adjacent area and beginners wanting to learn set theory via self-study.

Logic. --- Computer science --- Topology. --- Algebra. --- Mathematics. --- Global analysis (Mathematics) --- Logic, Symbolic and mathematical. --- Mathematical Logic and Foundations. --- Analysis. --- Set theory --- Point set theory --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Theory --- Aggregates --- Classes (Mathematics) --- Ensembles (Mathematics) --- Mathematical sets --- Sets (Mathematics) --- Theory of sets --- Assemblage of points --- Groups of points --- Point sets --- Points, Assemblage of --- Points, Groups of --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Argumentation --- Deduction (Logic) --- Deductive logic --- Dialectic (Logic) --- Logic, Deductive --- Math --- Analysis, Global (Mathematics) --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Mathematical analysis. --- Analysis (Mathematics). --- Mathematical logic. --- Discrete mathematics. --- Discrete Mathematics. --- Geometry --- Polyhedra --- Algebras, Linear --- Mathematical analysis --- Intellect --- Philosophy --- Psychology --- Science --- Reasoning --- Thought and thinking --- Algebra, Abstract --- Metamathematics --- Syllogism --- 517.1 Mathematical analysis --- Methodology --- Global analysis (Mathematics). --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Set theory. --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis