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Elliptic functions. --- Functions, Theta. --- Theta functions --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic
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Elliptic functions. --- 51 --- Elliptic functions --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic --- Mathematics --- 51 Mathematics
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In its first six chapters this 2006 text seeks to present the basic ideas and properties of the Jacobi elliptic functions as an historical essay, an attempt to answer the fascinating question: 'what would the treatment of elliptic functions have been like if Abel had developed the ideas, rather than Jacobi?' Accordingly, it is based on the idea of inverting integrals which arise in the theory of differential equations and, in particular, the differential equation that describes the motion of a simple pendulum. The later chapters present a more conventional approach to the Weierstrass functions and to elliptic integrals, and then the reader is introduced to the richly varied applications of the elliptic and related functions. Applications spanning arithmetic (solution of the general quintic, the functional equation of the Riemann zeta function), dynamics (orbits, Euler's equations, Green's functions), and also probability and statistics, are discussed.
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This book provides an introduction to the bifurcation theory approach to global solution curves and studies the exact multiplicity of solutions for semilinear Dirichlet problems, aiming to obtain a complete understanding of the solution set. This understanding opens the way to efficient computation of all solutions. Detailed results are obtained in case of circular domains, and some results for general domains are also presented. The author is one of the original contributors to the field of exact multiplicity results.
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A comprehensive treatment of elliptic functions is linked by these notes to a study of their application to elliptic curves. This approach provides geometers with the opportunity to acquaint themselves with aspects of their subject virtually ignored by other texts. The exposition is clear and logically carries themes from earlier through to later topics. This enthusiastic work of scholarship is made complete with the inclusion of some interesting historical details and a very comprehensive bibliography.
Elliptic functions. --- Curves, Elliptic. --- Elliptic curves --- Curves, Algebraic --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic --- Curves, Algebraic.
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This volume contains the proceedings of the conference on "C*-algebras and Elliptic Theory" held in Bedlewo, Poland, in February 2004. It consists of original research papers and expository articles focussing on index theory and topology of manifolds. The collection offers a cross-section of significant recent advances in several fields, the main subject being K-theory (of C*-algebras, equivariant K-theory). A number of papers is related to the index theory of pseudodifferential operators on singular manifolds (with boundaries, corners) or open manifolds. Further topics are Hopf cyclic cohomology, geometry of foliations, residue theory, Fredholm pairs and others. The wide spectrum of subjects reflects the diverse directions of research emanating from the Atiyah-Singer index theorem. Contributors: B. Bojarski, J. Brodzki, D. Burghelea, A. Connes, J. Eichhorn, T. Fack, S. Haller, Yu.A. Kordyukov, V. Manuilov, V. Nazaikinskii, G.A. Niblo, F. Nicola, I.M. Nikonov, V. Nistor, L. Rodino, A. Savin, V.V. Sharko, G.I. Sharygin, B. Sternin, K. Thomsen, E.V. Troitsky, E. Vasseli, A. Weber .
C*-algebras. --- Elliptic functions. --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic --- Algebras, C star --- Algebras, W star --- C star algebras --- W star algebras --- W*-algebras --- Banach algebras --- Functional analysis. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations
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This book consists of a collection of original, refereed research and expository articles on elliptic aspects of geometric analysis on manifolds, including singular, foliated and non-commutative spaces. The topics covered include the index of operators, torsion invariants, K-theory of operator algebras and L2-invariants. The results presented in this book, which is largely inspired and stimulated by the Atiyah-Singer index theorem, should be of interest to graduates and researchers in mathematical physics, differential topology and differential analysis.
C*-algebras --- Elliptic functions --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic --- Algebras, C star --- Algebras, W star --- C star algebras --- W star algebras --- W*-algebras --- Banach algebras --- Algebra. --- Functional analysis. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Mathematics --- Mathematical analysis
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Elliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics-the Birch and Swinnerton-Dyer Conjecture. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem.The key to the conjecture lies in elliptic curves, which may appear simple, but arise from some very deep-and often very mystifying-mathematical ideas. Using only basic algebra and calculus while presenting numerous eye-opening examples, Ash and Gross make these ideas accessible to general readers, and, in the process, venture to the very frontiers of modern mathematics.
Counting. --- Curves, Elliptic. --- Elliptic functions. --- Number theory. --- Elliptic functions --- Curves, Elliptic --- Number theory --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Fonctions elliptiques --- Courbes elliptiques --- Théorie des nombres --- Number study --- Numbers, Theory of --- Elliptic curves --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Algebra --- Curves, Algebraic --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic
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This book contains a detailed account of the result of the author's recent Annals paper and JAMS paper on arithmetic invariant, including μ-invariant, L-invariant, and similar topics. This book can be regarded as an introductory text to the author's previous book p-Adic Automorphic Forms on Shimura Varieties. Written as a down-to-earth introduction to Shimura varieties, this text includes many examples and applications of the theory that provide motivation for the reader. Since it is limited to modular curves and the corresponding Shimura varieties, this book is not only a great resource for experts in the field, but it is also accessible to advanced graduate students studying number theory. Key topics include non-triviality of arithmetic invariants and special values of L-functions; elliptic curves over complex and p-adic fields; Hecke algebras; scheme theory; elliptic and modular curves over rings; and Shimura curves.
Curves, Elliptic. --- Invariants. --- Number theory. --- Curves, Elliptic --- Invariants --- Number theory --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Elliptic functions. --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Number study --- Numbers, Theory of --- Mathematics. --- Algebraic geometry. --- Number Theory. --- Algebraic Geometry. --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic --- Geometry, algebraic. --- Algebraic geometry --- Geometry
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511.6 --- Elliptic functions --- Field extensions (Mathematics) --- Modular functions --- Rings of integers --- Integers, Rings of --- Algebraic number theory --- Rings (Algebra) --- Functions, Modular --- Group theory --- Number theory --- Extension fields (Mathematics) --- Algebraic fields --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic --- Algebraic number fields --- 511.6 Algebraic number fields
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