scholarly journals DIFFERENTIAL STRUCTURE OF ABELIAN FUNCTIONS

2008 ◽  
Vol 19 (02) ◽  
pp. 145-171 ◽  
Author(s):  
KOJI CHO ◽  
ATSUSHI NAKAYASHIKI
Keyword(s):  
Abelian Variety ◽  
Differential Operators ◽  
Free Resolution ◽  
Holomorphic Differential ◽  
Differential Structure ◽  
Linear Basis ◽  
Higher Dimensional ◽  
Abelian Functions ◽  
Derivatives Of ◽  
Logarithmic Derivatives

The space of Abelian functions of a principally polarized abelian variety (J,Θ) is studied as a module over the ring [Formula: see text] of global holomorphic differential operators on J. We construct a [Formula: see text] free resolution in case Θ is non-singular. As an application, in the case of dimensions 2 and 3, we construct a new linear basis of the space of abelian functions which are singular only on Θ in terms of logarithmic derivatives of the higher-dimensional σ-function.

Higher order Wirtinger inequalities

1980 ◽  
Vol 85 (1-2) ◽  
pp. 87-110 ◽  
Author(s):  
Kurt Kreith ◽  
Charles A. Swanson
Keyword(s):  
Boundary Conditions ◽  
Quadratic Forms ◽  
Differential Operators ◽  
Conjugate Point ◽  
Wirtinger Inequality ◽  
Even Order ◽  
Linear Differential ◽  
Natural Boundary Conditions ◽  
Derivatives Of ◽  
Admissible Functions

SynopsisWirtinger-type inequalities of order n are inequalities between quadratic forms involving derivatives of order k ≦ n of admissible functions in an interval (a, b). Several methods for establishing these inequalities are investigated, leading to improvements of classical results as well as systematic generation of new ones. A Wirtinger inequality for Hamiltonian systems is obtained in which standard regularity hypotheses are weakened and singular intervals are permitted, and this is employed to generalize standard inequalities for linear differential operators of even order. In particular second order inequalities of Beesack's type are developed, in which the admissible functions satisfy only the null boundary conditions at the endpoints of [a, b] and b does not exceed the first systems conjugate point (a) of a. Another approach is presented involving the standard minimization theory of quadratic forms and the theory of “natural boundary conditions”. Finally, inequalities of order n + k are described in terms of (n, n)-disconjugacy of associated 2nth order differential operators.

On Logarithmic Derivatives of Functions in a Class of Starlike Mappings

1993 ◽  
Vol 36 (1) ◽  
pp. 38-44
Author(s):  
Alan D. Gluchoff
Keyword(s):  
Integral Means ◽  
Koebe Function ◽  
Starlike Mappings ◽  
Derivatives Of ◽  
Logarithmic Derivatives

AbstractThe purpose of this paper is to prove some facts about integral means of (d2/dz2)(log[f(z)/z])—or equivalently f″/f, for f in a class of starlike mappings of a "singular" nature. In particular it is noted that the Koebe function is not extremal for the Hardy means Mp(r,f″/f) for functions in this class.

scholarly journals Moments of logarithmic derivatives of L -functions

2018 ◽  
Vol 183 ◽  
pp. 40-61 ◽  
Author(s):  
Peter J. Cho ◽  
Henry H. Kim
Keyword(s):  
Derivatives Of ◽  
Logarithmic Derivatives

OMEGA THEOREMS FOR $\frac{L'}{L}(1, \chi_D)$

2013 ◽  
Vol 09 (03) ◽  
pp. 561-581 ◽  
Author(s):  
M. MOURTADA ◽  
V. KUMAR MURTY
Keyword(s):  
Classical Result ◽  
London Math ◽  
Quadratic Field ◽  
Analogous Result ◽  
Logarithmic Derivative ◽  
Dirichlet Character ◽  
Derivative Formula ◽  
Derivatives Of ◽  
Logarithmic Derivatives

A classical result of Chowla [Improvement of a theorem of Linnik and Walfisz, Proc. London Math. Soc. (2) 50 (1949) 423–429 and The Collected Papers of Sarvadaman Chowla, Vol. 2 (Centre de Recherches Mathematiques, 1999), pp. 696–702] states that for infinitely many fundamental discriminants D we have [Formula: see text] where χD is the quadratic Dirichlet character of conductor D. In this paper, we prove an analogous result for the logarithmic derivative [Formula: see text], and investigate the growth of the logarithmic derivatives of real Dirichlet L-functions. We show that there are infinitely many fundamental discriminants D (both positive and negative) such that [Formula: see text] and infinitely many fundamental discriminants 0 < D such that [Formula: see text] In particular, we show that the Euler–Kronecker constant γK of a quadratic field K satisfies γK = Ω( log log |dK|). We get sharper results assuming the GRH. Moreover, we evaluate the moments of [Formula: see text].

scholarly journals Abelian varieties isogenous to a power of an elliptic curve

2018 ◽  
Vol 154 (5) ◽  
pp. 934-959 ◽  
Author(s):  
Bruce W. Jordan ◽  
Allan G. Keeton ◽  
Bjorn Poonen ◽  
Eric M. Rains ◽  
Nicholas Shepherd-Barron ◽  
...  
Keyword(s):  
Elliptic Curve ◽  
Abelian Variety ◽  
Opposite Direction ◽  
Abelian Varieties ◽  
Sufficient Conditions ◽  
Higher Dimensional ◽  
Free Left ◽  
Necessary And Sufficient ◽  
Finitely Presented ◽  
Torsion Free

Let $E$ be an elliptic curve over a field $k$. Let $R:=\operatorname{End}E$. There is a functor $\mathscr{H}\!\mathit{om}_{R}(-,E)$ from the category of finitely presented torsion-free left $R$-modules to the category of abelian varieties isogenous to a power of $E$, and a functor $\operatorname{Hom}(-,E)$ in the opposite direction. We prove necessary and sufficient conditions on $E$ for these functors to be equivalences of categories. We also prove a partial generalization in which $E$ is replaced by a suitable higher-dimensional abelian variety over $\mathbb{F}_{p}$.

INTEGRATION BY PARTS AND SMOOTHNESS OF THE LAW FOR A CLASS OF STOCHASTIC EVOLUTION EQUATIONS

2004 ◽  
Vol 07 (01) ◽  
pp. 89-129 ◽  
Author(s):  
STEFANO BONACCORSI ◽  
MARCO FUHRMAN
Keyword(s):  
Additive Noise ◽  
Evolution Equations ◽  
Transition Probabilities ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Integration By Parts ◽  
Infinite Dimensional ◽  
Integration By Parts Formula ◽  
Derivatives Of ◽  
Logarithmic Derivatives

We consider a Markov process X in a Hilbert space H, solution of a semilinear stochastic evolution equation driven by an infinite-dimensional Wiener process, occurring in the equation as an additive noise. Using techniques of the Malliavin calculus, under suitable assumptions, we prove an integration by parts formula for the transition probabilities νt, t>0 (the laws of Xt). We deduce results on differentiability (i.e. existence of logarithmic derivatives) of νt along a set of directions h∈H which can be described in terms of the coefficients of the equation. The general results are then applied to various classes of non linear stochastic partial differential equations and systems.

scholarly journals A variational approach for an inverse dynamical problem for composite beams

2007 ◽  
Vol 18 (1) ◽  
pp. 21-55 ◽  
Author(s):  
ANTONIO MORASSI ◽  
GEN NAKAMURA ◽  
KENJI SHIROTA ◽  
MOURAD SINI
Keyword(s):  
Inverse Problem ◽  
Concrete Beam ◽  
Differential Operators ◽  
Reinforced Concrete Beam ◽  
Dynamical Problem ◽  
Mixed Data ◽  
Steel Beam ◽  
Projected Gradient Method ◽  
Boundary Measurements ◽  
Derivatives Of

This paper deals with a problem of nondestructive testing for a composite system formed by the connection of a steel beam and a reinforced concrete beam. The small vibrations of the composite beam are described by a differential system where a coupling takes place between longitudinal and bending motions. The motion is governed in space by two second order and two fourth order differential operators, which are coupled in the lower order terms by the shearing,k, and axial, μ, stiffness coefficients of the connection. The coefficientskand μ define the mechanical model of the connection between the steel beam and the concrete beam and contain direct information on the integrity of the system. In this paper we study the inverse problem of determiningkand μ by mixed data. The inverse problem is transformed to a variational problem for a cost function which includes boundary measurements of Neumann data and also some interior measurements. By computing the Gateaux derivatives of the functional, an algorithm based on the projected gradient method is proposed for identifying the unknown coefficients. The results of some numerical simulations on real steel-concrete beams are presented and discussed.

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