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The mathematics of Frobenius in context a journey through 18th to 20th century mathematics / Thomas Hawkins.
| Author/creator | Hawkins, Thomas |
| Format | Electronic |
| Publication Info | New York ; Heidelberg : Springer, [2013] |
| Description | xiii, 699 pages : illustrations ; 25 cm. |
| Supplemental Content | Full text available from Springer Nature - Springer Mathematics and Statistics eBooks 2013 English International |
| Supplemental Content | Full text available from Springer Books |
| Subjects |
| Series | Sources and studies in the history of mathematics and physical sciences Sources and studies in the history of mathematics and physical sciences. ^A515093 |
| Contents | Overview of Frobenius' career and mathematics -- A Berlin education -- Student years : 1867-1870 -- Postdoctoral years : 1870-1874 -- Professor at the Zürich polytechnic : 1874-1892 -- Berlin professor : 1892-1917 -- Berlin-style linear algebra -- The paradigm : Weierstrass' memoir of 1858 -- Generic reasoning -- Stability of solutions to Bÿ + Ay = 0 -- Lagrange -- Laplace -- Sturm -- Cauchy's theory of determinants -- Cauchy and the principal axes theorem -- The three-dimensional principal axes theorem -- The n-dimensional principal axes theorem (1829) -- Cauchy's proof of his reality theorem -- A very remarkable property -- Jacobi's generic formula -- Cauchy's method of integration -- Weierstrass' memoir of 1858 -- Further development of the paradigm : 1858-1874 -- Weierstrass' unpublished theory -- Christoffel and Hermitian symmetry -- Kronecker on complex multiplication and bilinear forms -- Weierstrass' theory of elementary divisors -- The canonical form of Camille Jordan -- Singular families and disciplinary ideals : Kronecker's memoirs -- Singular families of quadratic forms 1868-1874 -- The first disciplinary ideal -- The second disciplinary ideal -- Bilinear families xt (uA + vAt)y revisited -- Generalization of Weierstrass' theory -- The mathematics of Frobenius -- The problem of Pfaff -- Mathematical preliminaries and caveats -- The problem of Pfaff -- The contributions of Clebsch -- Frobenius' solution to the problem of Pfaff -- The algebraic classification theorem -- The analytic classification theorem -- The integrability theorem -- Initial reactions -- Cartan's calculus of differential forms -- Paradigmatic aspects of Frobenius' paper -- The Cayley-Hermite problem and matrix algebra -- The disquisitiones arithmeticae of Gauss -- Eisenstein and Hermite -- The Cayley-Hermite problem -- Cayley's memoir of 1858 -- Frobenius' memoir on matrix algebra -- The minimal polynomial -- Fusion with Weierstrass-Kronecker theory -- The problem of Rosanes -- The Cayley-Hermite problem -- Orthogonal transformations -- A theorem on division algebras -- Arithmetic investigations : linear algebra -- Two Gaussian problems for bilinear forms -- Solution to problem (I) : invariant factors -- Frobenius' proof of lemma 8.5 -- Applications -- Linear systems of equations and congruences -- Alternating forms -- Modules -- Solution to problem (II) : the containment theorem -- Outline of Frobenius' proof of theorem 8.16 -- The work of H.J.S. Smith -- A rational theory of elementary divisors -- The rationality paradox -- Frobenius' approach and its scope -- A rational canonical form -- Arithmetic investigations : groups -- Origins of the fundamental theorem of finite abelian groups -- Gauss -- Kummer -- Schering -- Kronecker -- Dedekind -- The Frobenius-Stickelberger paper -- Schering's theorem via the Smith-Frobenius normal form -- Cyclic factorization of (Z/MZ)x -- Summary -- Analytic densities and Galois groups -- A challenging paper by Kronecker -- First density theorem and conjecture -- Correspondence with dedekind -- Counterexample to the first conjecture -- An outline of Frobenius' proof of theorem 9.14 -- Second density theorem and conjecture -- Group lists and group equations -- An abstract proof of Sylow's first theorem -- Double cosets -- Double cosets and Sylow's three theorems -- Abelian functions : problems of Hermite and Kronecker -- Abelian functions and the Jacobi inversion problem -- Hermite's Abelian matrix problem -- Abelian matrices -- Hermite's problem -- Kronecker and Weber on Hermite's problem -- Frobenius' solution to Hermite's problem -- Kronecker's complex multiplication problem -- Elliptic functions with complex multiplication -- Kronecker's problem -- Frobenius' solution to Kronecker's problem -- Geometric applications of Frobenius' results -- Hurwitz -- Humbert -- Scorza -- Lefschetz -- Postscript : A. A. Albert -- Frobenius' generalized theory of theta functions -- Weierstrass' lectures on theta functions -- Weierstrass on general Abelian functions -- Frobenius' theory of Jacobian functions -- A fundamental existence theorem -- Connection with the Riemann-Weierstrass conditions (I)-(II) on a period malrix -- A formula for the number of independent Jacobian functions of a given type -- An application of theorem 11.10 -- Assimilation into the mainstream -- Developments in France -- The contributions of Wirtinger -- New foundations for the theories of abelian functions and varieties -- The group determinant problem -- The fountain of youth -- Dedekind characters and group determinants -- Frobenius learns about group determinants -- Theta functions with integral characteristics -- Group characters and representations 1896-1897 -- Frobenius' letter of 12 April -- Frobenius' letter of 17 April -- Frobenius' paper "On group characters" -- The missing link -- Matrix representations -- Alternative routes to representation theory -- Hypercomplex numbers and lie groups -- T. Molien and É. Cartan -- W. Burnside -- H. Maschke -- Characters and representations after 1897 -- Frobenius' theory of induced characters -- Characteristic units and young tableaux -- Hypercomplex number systems à la Frobenius -- Applications to finite groups by Burnside and Frobenius -- I. Schur -- Polynomial representations of GL(...) -- Projective representations and factor sets -- Schur's lemma and representations of SO (...) -- Index theory -- R. Brauer -- Generalized index theory -- Modular representations -- Artin L-functions and induction theorems -- Loose Ends -- Congruence problems and matrix square roots -- A gap in Weierstrass' theory -- Two matrix congruence problems -- Frobenius' solution -- Cayley, Sylvester, and matrix square roots -- Frobenius' proof of his square-root theorem -- The spread of Frobenius-style matrix algebra -- Assimilation of Frobenius' rational elementary divisor theory -- The module-theoretic approach to elementary divisors -- Loewy on differential equations and matrix complexes -- Krull's theory of generalized abelian groups -- Van der Waerden's moderne algebra -- Nonnegative matrices -- The work of Perron -- Stolz's theorem revisited -- Generalized continued fraction algorithms -- Perron's lemma -- Perron's theorem -- Frobenius' theory of nonnegative matrices -- Frobenius' papers of 1908 and 1909 -- Frobenius' 1912 paper on nonnegative matrices -- Outline of Frobenius' proof of theorem 17.19 -- Markov chains 1908-1936 -- Markov's paper of 1908 -- Frobenius' theory and Markov chains -- The mathematics of Frobenius in retrospect -- References -- Index. |
| Bibliography note | Includes bibliographical references and index. |
| Access restriction | Available only to authorized users. |
| Technical details | Mode of access: World Wide Web |
| Issued in other form | 9781461463337 |
| Genre/form | Electronic books. |
| LCCN | 2013941038 |
| ISBN | 9781461463320 (alk. paper) |
| ISBN | 1461463327 (alk. paper) |
Availability
| Library | Location | Call Number | Status | Item Actions |
|---|---|---|---|---|
| Electronic Resources | ✔ Available |