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Heavenly Mathematics traces the rich history of spherical trigonometry, revealing how the cultures of classical Greece, medieval Islam, and the modern West used this forgotten art to chart the heavens and the Earth. Once at the heart of astronomy and ocean-going navigation for two millennia, the discipline was also a mainstay of mathematics education for centuries and taught widely until the 1950's. Glen Van Brummelen explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. He conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation of its elegant proofs and often surprising conclusions. Heavenly Mathematics is illustrated throughout with stunning historical images and informative drawings and diagrams. This unique compendium also features easy-to-use appendixes as well as exercises that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

Spherical trigonometry. --- Trigonometry. --- Trig (Trigonometry) --- Geometry --- Mathematics --- Trigonometry, Spherical --- Trigonometry --- Abū 'l-Wafā. --- Abū Mahmūd al-Khujandī. --- Abū Nasr Mansūr ibn 'Alī ibn 'Irāq. --- Abū Sahl al-Kūhī. --- Albert Girard. --- B. M. Brown. --- Cesàro method. --- Christopher Columbus. --- Claudius Ptolemy. --- Earth. --- Elements. --- Georg Rheticus. --- Giuseppe Cesàro. --- Hipparchus of Rhodes. --- Islam. --- Islamic religious rituals. --- John Harrison. --- John Napier. --- Law of Cosines. --- Law of Sines. --- Leonhard Euler. --- Mathematical Collection. --- Mecca. --- Menelaus of Alexandria. --- Menelaus's Theorem. --- Moon. --- Napier's Rules. --- Opus palatinum. --- Planisphere. --- Ptolemy. --- Pythagorean Theorem. --- Rule of Four Quantities. --- Sphaerica. --- Sun. --- acute-angled triangle. --- angle. --- area. --- astrolabe. --- astronomical triangle. --- astronomy. --- cartography. --- celestial motion. --- celestial sphere. --- chronometer. --- classical Greece. --- dead reckoning. --- ecliptic. --- equatorial coordinates. --- geography. --- locality principle. --- logarithms. --- marteloio. --- mathematics. --- method of Saint Hilaire. --- navigation. --- oblique triangle. --- pentagramma mirificum. --- planar Law of Sines. --- plane trigonometry. --- planets. --- polygon. --- polyhedron. --- qibla. --- regular polyhedron. --- right-angled triangle. --- rising time. --- sphere. --- spherical Law of Sines. --- spherical astronomy. --- spherical geometry. --- spherical triangle. --- spherical trigonometry. --- star. --- stars. --- stereographic projection. --- table of sine. --- theorems. --- triangle. --- trigonometric table. --- trigonometry.

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Graph theory goes back several centuries and revolves around the study of graphs-mathematical structures showing relations between objects. With applications in biology, computer science, transportation science, and other areas, graph theory encompasses some of the most beautiful formulas in mathematics-and some of its most famous problems. The Fascinating World of Graph Theory explores the questions and puzzles that have been studied, and often solved, through graph theory. This book looks at graph theory's development and the vibrant individuals responsible for the field's growth. Introducing fundamental concepts, the authors explore a diverse plethora of classic problems such as the Lights Out Puzzle, and each chapter contains math exercises for readers to savor. An eye-opening journey into the world of graphs, The Fascinating World of Graph Theory offers exciting problem-solving possibilities for mathematics and beyond.

Graph theory. --- Graph theory --- Graphs, Theory of --- Theory of graphs --- Combinatorial analysis --- Topology --- Extremal problems --- 1-Factorization Conjecture. --- 1-factorable graph. --- 2-factorable graph. --- Alfred Bray Kempe. --- Alspach's Conjecture. --- Around the World Problem. --- Art Gallery Problem. --- Arthur Cayley. --- Brick-Factory Problem. --- Cayley's Tree Formula. --- Chinese Postman Problem. --- Christian Goldbach. --- Erdős number. --- Euler Identity. --- Euler Polyhedron Formula. --- Eulerian graph. --- First Theorem of Graph Theory. --- Five Color Theorem. --- Five Queens Problem. --- Four Color Conjecture. --- Four Color Problem. --- Gottfried Leibniz. --- Graceful Tree Conjecture. --- Hall's Theorem. --- Hamiltonian graph. --- Herbert Ellis Robbins. --- Icosian Game. --- Instant Insanity. --- Internet. --- Job-Hunters Problem. --- King Chicken Theorem. --- Kirkman's Schoolgirl Problem. --- Knight's Tour Puzzle. --- Kruskal's Algorithm. --- Kuratowski's Theorem. --- Königsberg Bridge Problem. --- Leonhard Euler. --- Lights Out Puzzle. --- Marriage Theorem. --- Minimum Spanning Tree Problem. --- Paul Erdős. --- Peter Guthrie Tait. --- Petersen graph. --- Petersen's Theorem. --- Pierre Fermat. --- Polyhedron Problem. --- Problem of the Five Princes. --- Prüfer code. --- Ramsey number. --- Reconstruction Problem. --- Road Coloring Theorem. --- Robbins's Theorem. --- Sir William Rowan Hamilton. --- Steiner triple system. --- Thomas Penyngton Kirkman. --- Three Friends or Three Strangers Problem. --- Three Houses and Three Utilities Problem. --- Traveling Salesman Problem. --- Traveller's Dodecahedron. --- Tutte's Theorem. --- Vizing's Theorem. --- Voyage Round the World. --- Wagner's Conjecture. --- What Is Mathematics?. --- William Tutte. --- bipartite graph. --- bridge. --- chromatic index. --- coloring. --- complete graph. --- complex numbers. --- connected graph. --- crossing number. --- cyclic decomposition. --- decision tree. --- distance. --- dominating set. --- edge coloring. --- geometry of position. --- graceful graph. --- graph theory. --- graph. --- icosian calculus. --- irregular graph. --- irregular multigraph. --- isomorphic graph. --- leaf. --- mathematicians. --- mathematics. --- orientation. --- oriented graph. --- planar graph. --- problem solving. --- regular graph. --- round robin tournament. --- subgraph. --- theorem. --- tree. --- vertex coloring. --- voting. --- weighted graph.

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Outer billiards is a basic dynamical system defined relative to a convex shape in the plane. B. H. Neumann introduced this system in the 1950's, and J. Moser popularized it as a toy model for celestial mechanics. All along, the so-called Moser-Neumann question has been one of the central problems in the field. This question asks whether or not one can have an outer billiards system with an unbounded orbit. The Moser-Neumann question is an idealized version of the question of whether, because of small disturbances in its orbit, the Earth can break out of its orbit and fly away from the Sun. In Outer Billiards on Kites, Richard Schwartz presents his affirmative solution to the Moser-Neumann problem. He shows that an outer billiards system can have an unbounded orbit when defined relative to any irrational kite. A kite is a quadrilateral having a diagonal that is a line of bilateral symmetry. The kite is irrational if the other diagonal divides the quadrilateral into two triangles whose areas are not rationally related. In addition to solving the basic problem, Schwartz relates outer billiards on kites to such topics as Diophantine approximation, the modular group, self-similar sets, polytope exchange maps, profinite completions of the integers, and solenoids--connections that together allow for a fairly complete analysis of the dynamical system.

Hyperbolic spaces. --- Singularities (Mathematics) --- Transformations (Mathematics) --- Geometry, Plane. --- Plane geometry --- Algorithms --- Differential invariants --- Geometry, Differential --- Geometry, Algebraic --- Hyperbolic complex manifolds --- Manifolds, Hyperbolic complex --- Spaces, Hyperbolic --- Geometry, Non-Euclidean --- Abelian group. --- Automorphism. --- Big O notation. --- Bijection. --- Binary number. --- Bisection. --- Borel set. --- C0. --- Calculation. --- Cantor set. --- Cartesian coordinate system. --- Combination. --- Compass-and-straightedge construction. --- Congruence subgroup. --- Conjecture. --- Conjugacy class. --- Continuity equation. --- Convex lattice polytope. --- Convex polytope. --- Coprime integers. --- Counterexample. --- Cyclic group. --- Diameter. --- Diophantine approximation. --- Diophantine equation. --- Disjoint sets. --- Disjoint union. --- Division by zero. --- Embedding. --- Equation. --- Equivalence class. --- Ergodic theory. --- Ergodicity. --- Factorial. --- Fiber bundle. --- Fibonacci number. --- Fundamental domain. --- Gauss map. --- Geometry. --- Half-integer. --- Homeomorphism. --- Hyperbolic geometry. --- Hyperplane. --- Ideal triangle. --- Intersection (set theory). --- Interval exchange transformation. --- Inverse function. --- Inverse limit. --- Isometry group. --- Lattice (group). --- Limit set. --- Line segment. --- Linear algebra. --- Linear function. --- Line–line intersection. --- Main diagonal. --- Modular group. --- Monotonic function. --- Multiple (mathematics). --- Orthant. --- Outer billiard. --- Parallelogram. --- Parameter. --- Partial derivative. --- Penrose tiling. --- Permutation. --- Piecewise. --- Polygon. --- Polyhedron. --- Polytope. --- Product topology. --- Projective geometry. --- Rectangle. --- Renormalization. --- Rhombus. --- Right angle. --- Rotational symmetry. --- Sanity check. --- Scientific notation. --- Semicircle. --- Sign (mathematics). --- Special case. --- Square root of 2. --- Subsequence. --- Summation. --- Symbolic dynamics. --- Symmetry group. --- Tangent. --- Tetrahedron. --- Theorem. --- Toy model. --- Translational symmetry. --- Trapezoid. --- Triangle group. --- Triangle inequality. --- Two-dimensional space. --- Upper and lower bounds. --- Upper half-plane. --- Without loss of generality. --- Yair Minsky.

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Research on interior-point methods (IPMs) has dominated the field of mathematical programming for the last two decades. Two contrasting approaches in the analysis and implementation of IPMs are the so-called small-update and large-update methods, although, until now, there has been a notorious gap between the theory and practical performance of these two strategies. This book comes close to bridging that gap, presenting a new framework for the theory of primal-dual IPMs based on the notion of the self-regularity of a function. The authors deal with linear optimization, nonlinear complementarity problems, semidefinite optimization, and second-order conic optimization problems. The framework also covers large classes of linear complementarity problems and convex optimization. The algorithm considered can be interpreted as a path-following method or a potential reduction method. Starting from a primal-dual strictly feasible point, the algorithm chooses a search direction defined by some Newton-type system derived from the self-regular proximity. The iterate is then updated, with the iterates staying in a certain neighborhood of the central path until an approximate solution to the problem is found. By extensively exploring some intriguing properties of self-regular functions, the authors establish that the complexity of large-update IPMs can come arbitrarily close to the best known iteration bounds of IPMs. Researchers and postgraduate students in all areas of linear and nonlinear optimization will find this book an important and invaluable aid to their work.

Interior-point methods. --- Mathematical optimization. --- Programming (Mathematics). --- Mathematical optimization --- Interior-point methods --- Programming (Mathematics) --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Mathematical programming --- Goal programming --- Algorithms --- Functional equations --- Operations research --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Simulation methods --- System analysis --- 519.85 --- 681.3*G16 --- 681.3*G16 Optimization: constrained optimization; gradient methods; integer programming; least squares methods; linear programming; nonlinear programming (Numericalanalysis) --- Optimization: constrained optimization; gradient methods; integer programming; least squares methods; linear programming; nonlinear programming (Numericalanalysis) --- 519.85 Mathematical programming --- Accuracy and precision. --- Algorithm. --- Analysis of algorithms. --- Analytic function. --- Associative property. --- Barrier function. --- Binary number. --- Block matrix. --- Combination. --- Combinatorial optimization. --- Combinatorics. --- Complexity. --- Conic optimization. --- Continuous optimization. --- Control theory. --- Convex optimization. --- Delft University of Technology. --- Derivative. --- Differentiable function. --- Directional derivative. --- Division by zero. --- Dual space. --- Duality (mathematics). --- Duality gap. --- Eigenvalues and eigenvectors. --- Embedding. --- Equation. --- Estimation. --- Existential quantification. --- Explanation. --- Feasible region. --- Filter design. --- Function (mathematics). --- Implementation. --- Instance (computer science). --- Invertible matrix. --- Iteration. --- Jacobian matrix and determinant. --- Jordan algebra. --- Karmarkar's algorithm. --- Karush–Kuhn–Tucker conditions. --- Line search. --- Linear complementarity problem. --- Linear function. --- Linear programming. --- Lipschitz continuity. --- Local convergence. --- Loss function. --- Mathematician. --- Mathematics. --- Matrix function. --- McMaster University. --- Monograph. --- Multiplication operator. --- Newton's method. --- Nonlinear programming. --- Nonlinear system. --- Notation. --- Operations research. --- Optimal control. --- Optimization problem. --- Parameter (computer programming). --- Parameter. --- Pattern recognition. --- Polyhedron. --- Polynomial. --- Positive semidefinite. --- Positive-definite matrix. --- Quadratic function. --- Requirement. --- Result. --- Scientific notation. --- Second derivative. --- Self-concordant function. --- Sensitivity analysis. --- Sign (mathematics). --- Signal processing. --- Simplex algorithm. --- Simultaneous equations. --- Singular value. --- Smoothness. --- Solution set. --- Solver. --- Special case. --- Subset. --- Suggestion. --- Technical report. --- Theorem. --- Theory. --- Time complexity. --- Two-dimensional space. --- Upper and lower bounds. --- Variable (computer science). --- Variable (mathematics). --- Variational inequality. --- Variational principle. --- Without loss of generality. --- Worst-case complexity. --- Yurii Nesterov. --- Mathematical Optimization --- Mathematics --- Programming (mathematics)