7,814 research outputs found
The role of funnels and punctures in the Gromov hyperbolicity of Riemann surfaces
No full text27 pages, no figures.-- MSC2000 codes: 30F20, 30F45.MR#: MR2243795 (2007e:30063)Zbl#: Zbl 1108.30031We prove results on geodesic metric spaces which guarantee that some spaces are not hyperbolic in the Gromov sense. We use these theorems in order to study the hyperbolicity of Riemann surfaces. We obtain a criterion on the genus of a surface which implies non-hyperbolicity. We also include a characterization of the hyperbolicity of a Riemann surface S* obtained by deleting a closed set from one original surface S. In the particular case when the closed set is a union of continua and isolated points, the results clarify the role of punctures and funnels (and other more general ends) in the hyperbolicity of Riemann surfaces.Research by all three authors partially supported by a grant from DGI (BFM 2003-04870), Spain. In addition, research by third author (Eva Tourís) was partially supported by a grant from DGI (BFM 2000-0022), Spain.Publicad
On Evaluation of Riemann Zeta function ζ(s)
No full textIn this paper, by using Fourier series theory, several summing formulae for Riemann Zeta function ζ(s) and Dirichlet series are deduced
Linear law for the logarithms of the Riemann periods at simple critical zeta zeros
No full textEach simple zero 1/2 + iγn of the Riemann zeta function on the critical line with γn > 0 is a center for the flow s˙ = ξ(s) of the Riemann xi function with an associated period Tn. It is shown that, as γn →∞, log Tn ≥ π/4 γn + O(log γn).
Numerical evaluation leads to the conjecture that this inequality can be replaced by an equality. Assuming the Riemann Hypothesis and a zeta zero separation conjecture γn+1 − γn≥ γn-θ for some exponent θ > 0, we obtain the upper bound log Tn ≤ γn2 + θ Assuming a weakened form of a conjecture of Gonek, giving a bound for the reciprocal of the derivative of zeta at each zero, we obtain the expected upper bound for the periods so, conditionally, log Tn = π/ 4 γn +O(log γn). Indeed, this linear relationship is equivalent to the given weakened conjecture, which implies the zero separation conjecture, provided the exponent is sufficiently large. The frequencies corresponding to the periods relate to natural eigenvalues for the Hilbert–Polya conjecture. They may provide a goal for those seeking a self-adjoint operator related to the Riemann hypothesis
The holomorphic flow of the Riemann zeta function
No full textThe flow of the Riemann zeta function, ś = ς(s), is considered, and phase portraits are presented. Attention is given to the characterization of the flow lines in the neighborhood of the first 500 zeros on the critical line. All of these zeros are foci. The majority are sources, but in a small proportion of exceptional cases the zero is a sink. To produce these portraits, the zeta function was evaluated numerically to 12 decimal places, in the region of interest, using the Chebyshev method and using Mathematica.
The phase diagrams suggest new analytic properties of zeta, of which some are proved and others are given in the form of conjectures
A Two Points Taylor's Formula for the Generalised Riemann Integral
No full textA two points Taylor’s formula for the generalised Riemann integral
and various bounds for the remainder are established. Moreover, particular
instances of interest are given
The Beesack-Darst-Pollard Inequalities and Approximation of the Riemann-Stieltjes Integral
No full textUtilising the Beesack version of the Darst-Pollard inequality, some
error bounds for approximating the Riemann-Stieltjes integral are given. Some
applications related to the trapezoid and mid-point quadrature rules are provided
Faithfulness of actions on Riemann-Roch spaces
No full textGiven a faithful action of a finite group G on an algebraic curve X of genus g > 1, we give explicit criteria for the induced action of G on the Riemann-Roch space H^0(X,O_X(D)) to be faithful, where D is a G-invariant divisor on X of degree at least 2g-2. This leads to a concise answer to the question when the action of G on the space H^0(X, \Omega_X^m) of global holomorphic polydifferentials of order m is faithful. If X is hyperelliptic, we furthermore provide an explicit basis of H^0(X, \Omega_X^m). Finally, we give applications in deformation theory and in coding theory and we discuss the analogous problem for the action of G on the first homology H_1(X, Z/mZ) if X is a Riemann surface
Estructura de superficies de Riemann, desigualdades isoperimétricas y medida p-armónica en grafos
No full textSe estructura el trabajo en torno a un capítulo introductorio y cinco capítulos temáticos que desarrollan teorías referidas a la descomposición de superficies, a la desigualdad isopemétrica en superficies de Riemann con punturas, a la desigualdad isoperimétrica en superficies de Denjoy, a la desigualdad isoperimétrica en superficies de Riemann generales y a la medida p-armónica en Arboles. En el capítulo introductorio se tratan conceptos importantes para poder exponer los resultados obtenidos y se describe, de forma breve cuáles son esos resultados
Accurate Approximations of the Riemann-Stieltjes Integral with (1,L) - Lipschitzian Integrators
No full textGrass-type inequalities for the Riemann-Stieltjes integral with (/,Z)-Lipschitzian integrators and applications for
the Cebysev functional are given. Sharp inequalities complementing results of Cebysev, Grass, Ostrowski and Lupas are given
Conditional Symmetries and Riemann Invariants for Hyperbolic Systems of PDEs
No full textThis paper contains an analysis of rank-k solutions in terms of Riemann invariants, obtained from interrelations between two concepts, that of the symmetry reduction method and of the generalized method of characteristics for first order quasilinear hyperbolic systems of PDEs in many dimensions. A variant of the conditional symmetry method for obtaining this type of solutions is proposed. A Lie module of vector fields, which are symmetries of an overdetermined system defined by the initial system of equations and certain first order differential constraints, is constructed. It is shown that this overdetermined system admits rank-k solutions expressible in terms of Riemann invariants. Finally, examples of applications of the proposed approach to the fluid dynamics equations in (k+1) dimensions are discussed in detail. Several new soliton-like solutions (among them kinks, bumps and multiple wave solutions) have been obtained
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