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This book shows the importance of studying semilocal convergence in iterative methods through Newton's method and addresses the most important aspects of the Kantorovich's theory including implicated studies. Kantorovich's theory for Newton's method used techniques of functional analysis to prove the semilocal convergence of the method by means of the well-known majorant principle. To gain a deeper understanding of these techniques the authors return to the beginning and present a deep-detailed approach of Kantorovich's theory for Newton's method, where they include old results, for a historical perspective and for comparisons with new results, refine old results, and prove their most relevant results, where alternative approaches leading to new sufficient semilocal convergence criteria for Newton's method are given. The book contains many numerical examples involving nonlinear integral equations, two boundary value problems and systems of nonlinear equations related to numerous physical phenomena. The book is addressed to researchers in computational sciences, in general, and in approximation of solutions of nonlinear problems, in particular.

Newton-Raphson method. --- Method, Newton-Raphson --- Method of tangents --- Newton approximation method --- Newton iterative process --- Newton method --- Newton-Raphson algorithm --- Newton-Raphson formula --- Newton-Raphson process --- Newton's approximation method --- Newton's method --- Quadratically convergent Newton-Raphson process --- Raphson method, Newton --- -Second-order Newton-Raphson process --- Mathematics. --- Integral equations. --- Operator theory. --- Computer mathematics. --- Operator Theory. --- Computational Mathematics and Numerical Analysis. --- Integral Equations. --- Iterative methods (Mathematics) --- Computer science --- Equations, Integral --- Functional equations --- Functional analysis --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Mathematics

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In line with the emerging field of philosophy of mathematical practice, this book pushes the philosophy of mathematics away from questions about the reality and truth of mathematical entities and statements and toward a focus on what mathematicians actually do--and how that evolves and changes over time. How do new mathematical entities come to be? What internal, natural, cognitive, and social constraints shape mathematical cultures? How do mathematical signs form and reform their meanings? How can we model the cognitive processes at play in mathematical evolution? And how does mathematics tie together ideas, reality, and applications?Roi Wagner uniquely combines philosophical, historical, and cognitive studies to paint a fully rounded image of mathematics not as an absolute ideal but as a human endeavor that takes shape in specific social and institutional contexts. The book builds on ancient, medieval, and modern case studies to confront philosophical reconstructions and cutting-edge cognitive theories. It focuses on the contingent semiotic and interpretive dimensions of mathematical practice, rather than on mathematics' claim to universal or fundamental truths, in order to explore not only what mathematics is, but also what it could be. Along the way, Wagner challenges conventional views that mathematical signs represent fixed, ideal entities; that mathematical cognition is a rigid transfer of inferences between formal domains; and that mathematics' exceptional consensus is due to the subject's underlying reality. The result is a revisionist account of mathematical philosophy that will interest mathematicians, philosophers, and historians of science alike.

Mathematics --- Benedetto. --- Black-Scholes formula. --- Eugene Wigner. --- Friedrich W.J. Schelling. --- George Lakoff. --- Gilles Deleuze. --- Hermann Cohen. --- Hilary Putnam. --- Johann G. Fichte. --- Logic of Sensation. --- Mark Steiner. --- Rafael Nez. --- Stanislas Dehaene. --- Vincent Walsh. --- Water J. Freeman III. --- abbaco. --- algebra. --- arithmetic. --- authority. --- cognitive theory. --- combinatorics. --- conceptual freedom. --- constraints. --- economy. --- gender role stereotypes. --- generating functions. --- geometry. --- inferences. --- infinities. --- infinity. --- mathematical cognition. --- mathematical concepts. --- mathematical cultures. --- mathematical domains. --- mathematical entities. --- mathematical evolution. --- mathematical interpretation. --- mathematical language. --- mathematical metaphor. --- mathematical norms. --- mathematical objects. --- mathematical practice. --- mathematical signs. --- mathematical standards. --- mathematical statements. --- mathematics. --- natural order. --- natural sciences. --- nature. --- negative numbers. --- number sense. --- option pricing. --- philosophy of mathematics. --- reality. --- reason. --- relevance. --- semiosis. --- sexuality. --- stable marriage problem. --- Philosophy --- History.