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S. Khrushchev 《Journal of Computational and Applied Mathematics》2010,233(6):1548-1553
A proof of Markoff’s Great Theorem on the Lagrange spectrum using continued fractions is sketched. Markoff’s periods and Jean Bernoulli sequence 1 are used to obtain a simple algorithm for the computation of the Lagrange spectrum below 3. 相似文献
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Anitha Srinivasan 《Journal of Number Theory》2011,131(8):1420-1428
Using the Markoff theorem on indefinite binary quadratic forms it is shown that every element of the class group of any real quadratic order of discriminant d has an ideal of norm less than or equal to . This bound is best possible as there are quadratic orders of discriminant d which have ideal classes where the least norm is . These discriminants are explicitly given and their connection to the Markoff conjecture is highlighted. 相似文献
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Markoff triples and quasifuchsian groups 总被引:1,自引:0,他引:1
We study the global behaviour of trees of Markoff triples overthe complex numbers. We relate this to the space of type-preservingrepresentations of the punctured torus group into SL(2,C). Inparticular, we explore which Markoff triples correspond to quasifuchsianrepresentations. We derive a variation of McShane's identityfor quasifuchsian groups. In the case of non-discrete representations,we attempt to relate the asymptotic behaviour of Markoff triplesto the realisability of laminations in hyperbolic 3-space. Wealso consider how some of these issues might be related formore general surfaces. 1991 Mathematics Subject Classification:57M50. 相似文献
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We study the (relative) SL(2,C) character varieties of the one-holed torus and the action of the mapping class group on the (relative) character variety. We show that the subset of characters satisfying two simple conditions called the Bowditch Q-conditions is open in the relative character variety and that the mapping class group acts properly discontinuously on this subset. Furthermore, this is the largest open subset for which this holds. We also show that a generalization of McShane's identity holds for all characters satisfying the Bowditch Q-conditions. Finally, we show that further variations of the McShane-Bowditch identity hold for characters which are fixed by an Anosov element of the mapping class group and which satisfy a relative version of the Bowditch Q-conditions, with applications to identities for incomplete hyperbolic structures on punctured torus bundles over the circle, and also for closed hyperbolic 3-manifolds which are obtained by hyperbolic Dehn surgery on such manifolds. 相似文献
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François Guéritaud 《Geometriae Dedicata》2008,134(1):203-216
This note defines a family of Laurent polynomials indexed in which generalize the Markoff numbers and relate to the character variety of the one-cusped torus. We describe which monomials
appear in each polynomial and prove all the coefficients are positive integers. We also conjecture a generalization of that
positivity result.
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Rodney T. Hansen 《International Journal of Mathematical Education in Science & Technology》2013,44(4):535-536
The multinomial theorem and a basic property of multinomial coefficients yield a way of counting the number of chains of coloured beads of a fixed length where each colour can be used repeatedly. This coupled with a simple counting argument forms a concise proof of Fermat's theorem. 相似文献
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We exhibit a counterexample to a fiber theorem stated by F. Fumagalli in [Francesco Fumagalli, On the homotopy type of the Quillen complex of finite soluble groups, J. Algebra 283 (2) (2005) 639–654] and show how it affects the rest of Fumagalli's paper. As a consequence, whether the poset Ap(G) is homotopy equivalent to a wedge of spheres for any finite solvable group G seems to remain an open question. 相似文献
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Archiv der Mathematik - 相似文献
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We give here a full account of Markoff's celebrated result on badly approximable numbers. The proofs rely exclusively on the classical theory of simple continued fractions, together with Harvey Cohn's method using words in the free group with two generators for the determination of the structure of periods of the continued fractions of Markov irrationals. Appendix A gives a short self-contained presentation of the results on continued fractions used here and Appendix B gives short proofs of some results on the still open uniqueness problem for Markoff numbers. 相似文献
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Bernhard Dietz 《Monatshefte für Mathematik》1983,96(4):265-267
Letf be a reduced real indefinite binary quadratic form, having an arithmetic minimum that is the inverse of a left endpoint of a gap of the Markoff spectrum. Then the roots off are eventually periodic continued fractions, provided that $$\min \{ |f(p,q)|:p^2 + q^2 > 0,p,q \operatorname{int} egers\} $$ exists. 相似文献
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The Uniqueness of the Prime Markoff Numbers 总被引:1,自引:0,他引:1
Given the Diophantine equation a2+b2+c2=3abc, a solution tripleof natural numbers (a, b, c) can be arranged in ascending orderso that abc. Then, given the largest element c, one can askwhether this uniquely determines the triple. This is referredto as the Markoff conjecture. The paper proves that, if c isprime, then there is indeed only one triple that solves theequation with c as the largest element. The proof uses onlystandard algebraic number theory, but it was prompted by geometricconsiderations. 相似文献
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T. W. Cusick 《Monatshefte für Mathematik》1987,103(2):85-91
If the interval (a, b) is a gap in the Markoff spectrum, then the numbera can be written as the sum of two quadratic irrationals. A similar result is obtained for one type of right endpointb. 相似文献
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Lajos Takcs 《Journal of Combinatorial Theory, Series A》1981,30(3):345-348
In 1954, M. Kac discovered a probabilistic interpretation of a theorem of G. Szegö of Toeplitz matrices and demonstrated that this theorem can be provedin an elementary way by using a combinatorial identify of G. A. Hunt. In this paper Hunt's combinatorial identity is derived from a more general combinatorial result. 相似文献
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Chandler Davis 《Linear algebra and its applications》1977,18(1):33-43
Problem: Given operators Aj ? O on Hilbert space , with ΣAj = 1, to find commuting projectors Ej on a Hilbert space ? such that (for all j) x1Ajy = x1Ejy for, x, y ∈ . This paper gives an explicit construction, quite different from the familiar solution. 相似文献
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