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This is part one of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory. .

Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Math --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic

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This is the second book of a two-volume textbook on real analysis. Both the volumes—Analysis I and Analysis II—are intended for honors undergraduates who have already been exposed to calculus. The emphasis is on rigor and foundations. The material starts at the very beginning—the construction of number systems and set theory (Analysis I, Chaps. 1–5), then on to the basics of analysis such as limits, series, continuity, differentiation, and Riemann integration (Analysis I, Chaps. 6–11 on Euclidean spaces, and Analysis II, Chaps. 1–3 on metric spaces), through power series, several variable calculus, and Fourier analysis (Analysis II, Chaps. 4–6), and finally to the Lebesgue integral (Analysis II, Chaps. 7–8). There are appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) is taught in two quarters of twenty-five to thirty lectures each.

Mathematical analysis. --- Analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Anàlisi matemàtica --- Matemàtica --- Àlgebra lineal --- Anàlisi combinatòria --- Anàlisi de Fourier --- Anàlisi estocàstica --- Anàlisi matemàtica no-estàndard --- Anàlisi numèrica --- Funcions --- Matemàtica per a enginyers --- Sèries infinites --- Teoria del potencial (Matemàtica) --- Teories no lineals --- Rutes aleatòries (Matemàtica) --- Àlgebra --- Càlcul

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MATRIX is Australia’s international, residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each lasting 1-4 weeks. This book is a scientific record of the five programs held at MATRIX in its first year, 2016: - Higher Structures in Geometry and Physics - Winter of Disconnectedness - Approximation and Optimisation - Refining C*-Algebraic Invariants for Dynamics using KK-theory - Interactions between Topological Recursion, Modularity, Quantum Invariants and Low- dimensional Topology The MATRIX Scientific Committee selected these programs based on their scientific excellence and the participation rate of high-profile international participants. Each program included ample unstructured time to encourage collaborative research; some of the longer programs also included an embedded conference or lecture series. The articles are grouped into peer-reviewed contributions and other contributions. The peer-reviewed articles present original results or reviews on selected topics related to the MATRIX program; the remaining contributions are predominantly lecture notes based on talks or activities at MATRIX.

Mathematics --- Research. --- Mathematics. --- Category theory (Mathematics). --- Homological algebra. --- Group theory. --- K-theory. --- Mathematical optimization. --- Topology. --- Category Theory, Homological Algebra. --- Group Theory and Generalizations. --- Optimization. --- K-Theory. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Algebraic topology --- Homology theory --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Homological algebra --- Algebra, Abstract --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Math --- Science --- Mathematical research --- Algebra.

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MATRIX is Australia’s international and residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each 1-4 weeks in duration. This book is a scientific record of the eight programs held at MATRIX in 2018: - Non-Equilibrium Systems and Special Functions - Algebraic Geometry, Approximation and Optimisation - On the Frontiers of High Dimensional Computation - Month of Mathematical Biology - Dynamics, Foliations, and Geometry In Dimension 3 - Recent Trends on Nonlinear PDEs of Elliptic and Parabolic Type - Functional Data Analysis and Beyond - Geometric and Categorical Representation Theory The articles are grouped into peer-reviewed contributions and other contributions. The peer-reviewed articles present original results or reviews on a topic related to the MATRIX program; the remaining contributions are predominantly lecture notes or short articles based on talks or activities at MATRIX.

Algebraic geometry. --- Dynamics. --- Ergodic theory. --- Partial differential equations. --- Category theory (Mathematics). --- Homological algebra. --- Bioinformatics. --- Computer science—Mathematics. --- Algebraic Geometry. --- Dynamical Systems and Ergodic Theory. --- Partial Differential Equations. --- Category Theory, Homological Algebra. --- Computational Biology/Bioinformatics. --- Mathematics of Computing. --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Partial differential equations --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Algebraic geometry --- Geometry --- Bio-informatics --- Biological informatics --- Biology --- Information science --- Computational biology --- Systems biology --- Homological algebra --- Algebra, Abstract --- Homology theory --- Data processing --- Mathematics.