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This book provides a comprehensive, in-depth overview of elementary mathematics as explored in Mathematical Olympiads around the world. It expands on topics usually encountered in high school and could even be used as preparation for a first-semester undergraduate course. This second volume covers Plane Geometry, Trigonometry, Space Geometry, Vectors in the Plane, Solids and much more. As part of a collection, the book differs from other publications in this field by not being a mere selection of questions or a set of tips and tricks that applies to specific problems. It starts from the most basic theoretical principles, without being either too general or too axiomatic. Examples and problems are discussed only if they are helpful as applications of the theory. Propositions are proved in detail and subsequently applied to Olympic problems or to other problems at the Olympic level. The book also explores some of the hardest problems presented at National and International Mathematics Olympiads, as well as many essential theorems related to the content. An extensive Appendix offering hints on or full solutions for all difficult problems rounds out the book.

Mathematics. --- Convex geometry. --- Discrete geometry. --- Polytopes. --- Projective geometry. --- Convex and Discrete Geometry. --- Projective Geometry. --- Discrete groups. --- Groups, Discrete --- Infinite groups --- Discrete mathematics --- Geometry, Modern --- Geometry, Plane. --- Plane. --- Plane geometry --- Convex geometry . --- Geometry --- Projective geometry --- Hyperspace --- Topology --- Combinatorial geometry

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This book consists of contributions from experts, presenting a fruitful interplay between different approaches to discrete geometry. Most of the chapters were collected at the conference “Geometry and Symmetry” in Veszprém, Hungary from 29 June to 3 July 2015. The conference was dedicated to Károly Bezdek and Egon Schulte on the occasion of their 60th birthdays, acknowledging their highly regarded contributions in these fields. While the classical problems of discrete geometry have a strong connection to geometric analysis, coding theory, symmetry groups, and number theory, their connection to combinatorics and optimization has become of particular importance. The last decades have seen a revival of interest in discrete geometric structures and their symmetry. The rapid development of abstract polytope theory has resulted in a rich theory featuring an attractive interplay of methods and tools from discrete geometry, group theory and geometry, combinatorial group theory, and hyperbolic geometry and topology. This book contains papers on new developments in these areas, including convex and abstract polytopes and their recent generalizations, tiling and packing, zonotopes, isoperimetric inequalities, and on the geometric and combinatorial aspects of linear optimization. The book is a valuable resource for researchers, both junior and senior, in the field of discrete geometry, combinatorics, or discrete optimization. Graduate students find state-of-the-art surveys and an open problem collection. .

Mathematics. --- Convex geometry. --- Discrete geometry. --- Polytopes. --- Mathematical optimization. --- Combinatorics. --- Convex and Discrete Geometry. --- Optimization. --- Geometry --- Combinatorial geometry --- Math --- Science --- Discrete groups. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Combinatorics --- Algebra --- Groups, Discrete --- Infinite groups --- Discrete mathematics --- Convex geometry . --- Hyperspace --- Topology

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This volume contains 17 surveys that cover many recent developments in Discrete Geometry and related fields. Besides presenting the state-of-the-art of classical research subjects like packing and covering, it also offers an introduction to new topological, algebraic and computational methods in this very active research field. The readers will find a variety of modern topics and many fascinating open problems that may serve as starting points for research.

Geometry. --- Mathematics --- Euclid's Elements --- Discrete groups. --- Combinatorics. --- Topology. --- Convex and Discrete Geometry. --- Polytopes. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Combinatorics --- Algebra --- Mathematical analysis --- Groups, Discrete --- Infinite groups --- Discrete mathematics --- Convex geometry . --- Discrete geometry. --- Hyperspace --- Topology --- Combinatorial geometry --- Convex geometry. --- Discrete mathematics. --- Discrete Mathematics. --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis

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This book offers a comprehensive and accessible exposition of Euclidean Distance Matrices (EDMs) and rigidity theory of bar-and-joint frameworks. It is based on the one-to-one correspondence between EDMs and projected Gram matrices. Accordingly the machinery of semidefinite programming is a common thread that runs throughout the book. As a result, two parallel approaches to rigidity theory are presented. The first is traditional and more intuitive approach that is based on a vector representation of point configuration. The second is based on a Gram matrix representation of point configuration. Euclidean Distance Matrices and Their Applications in Rigidity Theory begins by establishing the necessary background needed for the rest of the book. The focus of Chapter 1 is on pertinent results from matrix theory, graph theory and convexity theory, while Chapter 2 is devoted to positive semidefinite (PSD) matrices due to the key role these matrices play in our approach. Chapters 3 to 7 provide detailed studies of EDMs, and in particular their various characterizations, classes, eigenvalues and geometry. Chapter 8 serves as a transitional chapter between EDMs and rigidity theory. Chapters 9 and 10 cover local and universal rigidities of bar-and-joint frameworks. This book is self-contained and should be accessible to a wide audience including students and researchers in statistics, operations research, computational biochemistry, engineering, computer science and mathematics.

Rigidity (Geometry) --- Matrices. --- Geometric rigidity --- Rigidity theorem --- Discrete geometry --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Mathematical statistics. --- Discrete groups. --- Computational complexity. --- Statistical Theory and Methods. --- Convex and Discrete Geometry. --- Discrete Mathematics in Computer Science. --- Complexity, Computational --- Electronic data processing --- Machine theory --- Groups, Discrete --- Infinite groups --- Mathematics --- Statistical inference --- Statistics, Mathematical --- Statistics --- Probabilities --- Sampling (Statistics) --- Statistical methods --- Discrete mathematics --- Statistics . --- Convex geometry . --- Discrete geometry. --- Computer science—Mathematics. --- Geometry --- Combinatorial geometry --- Statistical analysis --- Statistical data --- Statistical science --- Econometrics

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This book presents the proceedings of the international conference Analytic Aspects in Convexity, which was held in Rome in October 2016. It offers a collection of selected articles, written by some of the world’s leading experts in the field of Convex Geometry, on recent developments in this area: theory of valuations; geometric inequalities; affine geometry; and curvature measures. The book will be of interest to a broad readership, from those involved in Convex Geometry, to those focusing on Functional Analysis, Harmonic Analysis, Differential Geometry, or PDEs. The book is a addressed to PhD students and researchers, interested in Convex Geometry and its links to analysis.

Convex domains. --- Mathematics. --- Algebra. --- Field theory (Physics). --- Group theory. --- Mathematical analysis. --- Analysis (Mathematics). --- Convex geometry. --- Discrete geometry. --- Convex and Discrete Geometry. --- Analysis. --- Field Theory and Polynomials. --- Group Theory and Generalizations. --- Geometry --- Combinatorial geometry --- 517.1 Mathematical analysis --- Mathematical analysis --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Mathematics --- Math --- Science --- Convex regions --- Convexity --- Calculus of variations --- Convex geometry --- Point set theory --- Discrete groups. --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Groups, Discrete --- Infinite groups --- Discrete mathematics --- Convex geometry . --- Algebraic fields. --- Polynomials. --- Algebraic number fields --- Algebraic numbers --- Fields, Algebraic --- Algebra, Abstract --- Algebraic number theory --- Rings (Algebra)