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The study of Riemann problems has undergone a strong, steady growth in the last decade. The general direction of the research has headed toward understanding the wave structure of the solutions of more physically realistic systems. These systems fail either or both of the two main restrictions of the classical theory---that the system be strictly hyperbolic or genuinely nonlinear. The systems that have been studied tend to fall into the following broad classes: real gas dynamics (including combustion), visco-elastic materials, phase transitions, and multiphase flow in porous media. In addition to their usefulness in large-scale calculations, computational schemes have vastly improved the handling of discontinuity behavior.

This volume contains the proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Current Progress in Hyperbolic Systems: Riemann Problems and Computations, held at Bowdoin College in July 1988. The papers presented here provide a complete picture of recent research by some of the leaders in this field. Graduate students and beginning researchers will find this book a useful introduction to current work in this area. ... Read more

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At the turn of the twentieth century, the French mathematician Paul Painlevé and his students classified second order nonlinear ordinary differential equations with the property that the location of possible branch points and essential singularities of their solutions does not depend on initial conditions. It turned out that there are only six such equations (up to natural equivalence), which later became known as Painlevé I-VI. Although these equations were initially obtained answering a strictly mathematical question, they appeared later in an astonishing (and growing) range of applications, including, e.g., statistical physics, fluid mechanics, random matrices, and orthogonal polynomials. Actually, it is now becoming clear that the Painlevé transcendents (i.e., the solutions of the Painlevé equations) play the same role in nonlinear mathematical physics that the classical special functions, such as Airy and Bessel functions, play in linear physics. The explicit formulas relating the asymptotic behaviour of the classical special functions at different critical points play a crucial role in the applications of these functions. It is shown in this book that even though the six Painlevé equations are nonlinear, it is still possible, using a new technique called the Riemann-Hilbert formalism, to obtain analogous explicit formulas for the Painlevé transcendents. This striking fact, apparently unknown to Painlevé and his contemporaries, is the key ingredient for the remarkable applicability of these "nonlinear special functions". The book describes in detail the Riemann-Hilbert method and emphasizes its close connection to classical monodromy theory of linear equations as well as to modern theory of integrable systems. In addition, the book contains an ample collection of material concerning the asymptotics of the Painlevé functions and their various applications, which makes it a good reference source for everyone working in the theory and applications of Pa ... Read more

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In this volume, the one-dimensional and two-dimensional Riemann problems for the transportation equations in gas dynamics are solved constructively. In either the 1-D or 2-D case, there are only two kinds of solutions: one involves Dirac delta waves, and the other involves vacuums, which havea been merely discussed so far. The generalized Rankine-Hugoniot and entropy conditions for Dirac delta waves are clarified with viscous vanishing method. All of the existence, uniqueness and stability for viscous perturbations are proved analytically. ... Read more

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The primary goal of numerical simulation of compressible, inviscid time-dependent flow is to represent the time evolution of complex flow patterns. Developed by Matania Ben-Artzi and Joseph Falcovitz, the Generalized Riemann Problem (GRP) algorithm comprises some of the most commonly used numerical schemes of this process. This monograph presents the GRP methodology ranging from underlying mathematical principles through basic scheme analysis and scheme extensions. The book is intended for researchers and graduate students of applied mathematics, science and engineering. ... Read more

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This book examines qualitative properties of vector fields in the plane, in the spirit of Hilbert's Sixteenth Problem. Two principal topics explored are bifurcations of limit cycles of planar vector fields and desingularization of singular points for individual vector fields and for analytic families of such fields. In addition to presenting important new developments in this area, this book contains an introductory paper which outlines the general context and describes connections between the papers in the volume. The book will appeal to researchers and graduate students working in the qualitative theory of ordinary differential equations and dynamical systems. ... Read more

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The 16th Problem of Hilbert is one of the most famousremaining unsolved problems of mathematics. It concerns whether apolynomial vector field on the plane has a finite number of limitcycles. There is a strong connection with divergent solutions ofdifferential equations, where a central role is played by the StokesPhenomenon, the change in asymptotic behaviour of the solutions indifferent sectors of the complex plane.

The contributions to these proceedings survey both of these themes,including historical and modern theoretical points of view. Topicscovered include the Riemann-Hilbert problem, Painleve equations,nonlinear Stokes phenomena, and the inverse Galois problem. ... Read more

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Chapters: Feynman Diagram, Luminosity, Carrier Scattering, Wick's Theorem, Riemann-hilbert Problem, S-Matrix, Inverse Scattering Transform, Mhv Amplitudes, Scattering Length, Optical Theorem, Born Approximation, Dyson Series, Continuous Spectrum, Feshbach-fano Partitioning, Abeles Matrix Formalism, Soft Photons, Inverse Scattering Problem, Marchenko Equation, Diffraction Tomography. Source: Wikipedia. Pages: 112. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In theoretical quantum physics a physical quantity, like a cross section, is often calculated using the so called perturbation theory, where the quantity is represented as an infinite series, that is, a sum of ever decreasing terms. A Feynman diagram is an intuitive graphical representation of a term in this series. More precisely, a Feynman diagram is a graphical representation of a perturbative contribution to the transition amplitude or correlation function of a quantum mechanical or statistical field theory. Within the canonical formulation of quantum field theory a Feynman diagram represents term in the Wick's expansion of the perturbative S-matrix. Alternatively, the path integral formulation of quantum field theory represents the transition amplitude as a weighted sum of all possible histories of the system from the initial to the final state, in terms of either particles or fields. A Feynman diagram is then a contribution of a particular class of particle paths, which join and split as described by the diagram. The transition amplitude is then given as the matrix element of the S-matrix between the initial and the final states of the quantum system. Feynman diagrams were developed by Richard Feynman, and are named after him. There are many applications, primarily in quantum field theory, but also in other fields, e.g. in solid...More: http://booksllc.net/?id=11617 ... Read more

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This digital document is a journal article from European Journal of Operational Research, published by Elsevier in 2004. The article is delivered in HTML format and is available in your Amazon.com Media Library immediately after purchase. You can view it with any web browser.

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A form p on R^n (homogeneous n-variate polynomial) is called positive semidefinite (p.s.d.) if it is nonnegative on R^n. In other words, the zero vector is a global minimizer of p in this case. The famous 17th conjecture of Hilbert [Bull. Amer. Math. Soc. (N.S.), 37 (4) (2000) 407] (later proven by Artin [The Collected Papers of Emil Artin, Addison-Wesley Publishing Co., Inc., Reading, MA, London, 1965]) is that a form p is p.s.d. if and only if it can be decomposed into a sum of squares of rational functions. In this paper we give an algorithm to compute such a decomposition for ternary forms (n=3). This algorithm involves the solution of a series of systems of linear matrix inequalities (LMI's). In particular, for a given p.s.d. ternary form p of degree 2m, we show that the abovementioned decomposition can be computed by solving at most m/4 systems of LMI's of dimensions polynomial in m. The underlying methodology is largely inspired by the original proof of Hilbert, who had been able to prove his conjecture for the case of ternary forms. ... Read more

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Chapters: Goldbach's Conjecture, Continuum Hypothesis, Consistency, Diophantine Set, Hilbert's Third Problem, Hilbert's Tenth Problem, Hilbert's Second Problem, Hilbert's Fifth Problem, Hilbert's First Problem, Riemann Hypothesis, Kepler Conjecture, Hilbert's Program, Hilbert's Sixteenth Problem, Hilbert's Twenty-First Problem, Hilbert's Twelfth Problem, Hilbert's Fourteenth Problem, Hilbert's Seventeenth Problem, Hilbert's Sixth Problem, Hilbert's Thirteenth Problem, Hilbert's Eighteenth Problem, Hilbert Number, Hilbert's Seventh Problem, Hilbert's Ninth Problem, Hilbert's Nineteenth Problem, Zetagrid, Hilbert's Eleventh Problem, Hilbert's Fourth Problem, Hilbert's Eighth Problem, Hilbert's Fifteenth Problem, Hilbert's Twentieth Problem, Hilbert's Twenty-Second Problem, Hilbert's Twenty-Third Problem. Source: Wikipedia. Pages: 155. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture about the distribution of the zeros of the Riemann zeta-function which states that all non-trivial zeros of the Riemann zeta function have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann hypothesis implies results about the distribution of prime numbers that are in some ways as good as possible. Along with suitable generalizations, it is considered by some mathematicians to be the most important unresolved problem in pure mathematics (Bombieri 2000). The Riemann zeta-function (s) is defined for all complex numbers s 1. It has zeros at the negative even integers (i.e. at s = 2, 4, 6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that: The real part of any...More: http://booksllc.net/?id=19344125 ... Read more

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Chapters: History of Geometry, Charles Sanders Peirce, Hilbert's Problems, Foundations of Mathematics, Charles Sanders Peirce Bibliography, History of Elementary Algebra, Timeline of Category Theory and Related Mathematics, Cantor's First Uncountability Proof, Numerical Approximations of Π, Timeline of Mathematics, History of Grandi's Series, Classical Hamiltonian Quaternions, History of Calculus, History of Trigonometry, Kerala School of Astronomy and Mathematics, Weil Conjectures, Brouwer-hilbert Controversy, Antiquarian Science Book, Madhava Series, Egyptian Mathematics, New Algebra, Babylonian Mathematics, Criticism of Non-Standard Analysis, History of Group Theory, Hobbes-wallis Controversy, Controversy Over Cantor's Theory, Leibniz and Newton Calculus Controversy, History of Mathematical Notation, History of Computer Science, Contributions of Leonhard Euler to Mathematics, History of Computing, Séminaire de Géométrie Algébrique Du Bois Marie, the Story of Maths, the Value of Science, Chronology of Computation of Π, Hindu Units of Measurement, Whitehead's Point-Free Geometry, Unifying Theories in Mathematics, History of Combinatorics, History of Statistics, History of Manifolds and Varieties, Existential Graph, Background and Genesis of Topos Theory, the Compendious Book on Calculation by Completion and Balancing, Special Functions, Timeline of Abelian Varieties, History of Probability, Eudemus of Rhodes, the Quadrature of the Parabola, Algebraic Logic, Jyā, Koti-Jyā and Utkrama-Jyā, Preintuitionism, Ishango Bone, Quaternion Society, Sand Table, Haridatta, Galley Division, Italian School of Algebraic Geometry, Regiomontanus' Angle Maximization Problem, Pierre Hérigone, Method of Exhaustion, Jyotirmimamsa, Salvatore Pincherle, Utpala, C. M. Whish, Elementary Calculus: an Infinitesimal Approach, de Vetula, Hekat, Pons Asinorum, Isis, Mathematics Genealogy Project, Analytical Society, Ramanujan's Lost Notebook, Entitative...More: http://booksllc.net/?id=3967710 ... Read more

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Bolibrukh presents the negative solution of Hilbert's twenty-first problem for linear Fuchsian systems of differential equations. Methods developed by Bolibrukh in solving this problem are then applied to the study of scalar Fuchsian equations and systems with regular singular points on the Riemann sphere. ... Read more