On the size of the Shafarevich--Tate group of elliptic curves over function fields

Let E be a nonconstant elliptic curve, over a global field K of positive, odd characterisitc. Assuming the finiteness of the Shafarevich-Tate group of E, we show that the order of theShafarevich-Tate group of E, is given by O ^{(N1/2+6 log(2)/ log(q))}, where N is the conductor of E,q isthe cardinality of the finite field of constants of K, and where the constant in the bound depends only on K. The method of proof is to workwith the geometric analog of the Birch-Swinnerton Dyer conjecture for thecorresponding elliptic surface over the finite field, as formulatedby Artin-Tate, and to examine the geometry of this elliptic surface.     

Toward a theorem of douady

A theorem of Douady says that the absolute Galois group of a rational function field F(x) in one variable over an algebraically closed field F of characteristic 0 is a free profinite group. A new method is proposed to extend Douady’s theorem from the case of the complex number field F = ℂ to the case of an arbitrary field.     

Note on g-Hilbertian fields and nonrational curves¶with the Hilbert property

In a recent paper, Fried and Jarden prove the existence, for all integers g, of non-Hilbertian fields K which cannot be covered by a finite number of sets of the form ϕ (X(K)), where X is a curve of genus ≤g and ϕ is a rational function on X of degree ≥ 2. (If no bound is given on the genus we recover the notion of Hilbertian field.) This generalizes the case g=0, obtained previously by Corvaja and Zannier with a more elementary method. By a suitable modification of that method, we give here a new proof of the result of Fried and Jarden which avoids the use of deep group theoretical results. By a somewhat related construction we give an example of a curve X/Q of any prescribed genus and a Hilbertian field K⊂ˉQ such that X/K has the Hilbert property, i.e. the set of rational points X(K) is not thin. Received: 10 March 1998 / Revised version: 20 April 1998     

Simple Components of the Center of FG/J(FG)

Let G be a finite group, F a field, FG the group ring of G over F, and J(FG) the Jacobson radical of FG. Using a result of Berman and Witt, we give a method to determine the structure of the center of FG/J(FG), provided that F satisfies a field theoretical condition.     

Application of the large sieve to the solution of additive problems of mixed type

A variant of the large-sieve: method, using a combination of results obtained by Lavrik, Montgomery, and Eombieri, is employed to derive asymptotic properties of the number of solutions of the equationNp+Na=n wherep is a prime ideal of some ideal class of a field K of degree n4, anda is a prime ideal of a class of an imaginary quadratic field.Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 73–81, July, 1971.The author wishes to thank A. I. Vinogradov for his help and advice in this work.     

Optimal gaussian approximation in the fluctuating field theory

We consider the problem of calculating the partition function given by the functional integral over an external field that fluctuates in space and in “time” τ ∈ [0, 1/T] (T is temperature). A method is presented for calculating such integrals with the help of the Gaussian approximation that takes into account dynamics and non-locality of the fluctuations. The method is based on the free energy minimum principle.     

Asymptotic Expansions for the Distribution of the Maximum of Gaussian Random Fields

Some asymptotic results are proved for the distribution of the maximum of a centered Gaussian random field with unit variance on a compact subset S of ^N . They are obtained by a Rice method and the evaluation of some moments of the number of local maxima of the Gaussian field above an high level inside S and on the border S. Depending on the geometry of the border we give up to N+1 terms of the expansion sometimes with exponentially small remainder. Application to waves maximum is shown.     

Estimate of the precision of determination of elastic parameters by the diffraction tomography method with the use of the finite-difference method (the SV-problem)

In the present paper, the finite-difference method is used for the solution of the direct problem, which enables us to take into account the diffraction phenomenon on local inhomogeneities of not weak contrast. Examples that allow us to estimate the precision of restoration of the parameters of such inhomogeneities, depending on the degree of their contrast, are given. The possibility of restoring the parameters of a local inhomogeneity in a medium containing interfaces with stepwise change of elastic properties for the case where source and receiver points are located on the free surface is demonstrated. An example of the numerical estimation of the precision of calculation of a scattered field in the Born approximation, approximating the wave field by the zero term of the ray method, is offered. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 136–152. Translated by Yu. V. Kiselev     

Witten deformation and the equivariant index

Let M be a compact Riemannian manifold endowed with an isometric action of a compact, connected Lie group. The method of the Witten deformation is used to compute the virtual representation-valued equivariant index of a transversally elliptic, first order differential operator on M. The multiplicities of irreducible representations in the index are expressed in terms of local quantities associated to the isolated singular points of an equivariant bundle map that is locally Clifford multiplication by a Killing vector field near these points.      

Maximal growth rate at the Rosensweig instability: theory,experiment, and numerics

We investigate the growth of a pattern of liquid crests emerging in a layer of magnetic liquid when subjected to a magnetic field oriented normally to the fluid surface. After a step like increase of the magnetic field, the temporal evolution of the pattern amplitude is measured by means of a Hall-sensor array. The extracted growth rate is compared with predictions from linear stability analysis by taking into account the nonlinear magnetization curve M (H). The remaining discrepancy can be resolved by numerical calculations via the finite element method. By starting with a finite surface perturbation it can reproduce the temporal evolution of the pattern amplitude and the growth rate. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

High speed computation of group characters

This paper describes a practical procedure for computing the ordinary irreducible characters of finite groups (of orders up to 1000 or so). The novelty of the method consists of transposing the problem from the field of complex numbers into the field of integers modulop for a suitable primep. It is much easier to compute the modular characters in the latter field, and from these characters we can calculate the ordinary irreducible characters in algebraic form.     

Maximum Modulus Estimates for the Linear Steady Stokes System

A maximum modulus estimate for the Stokes system in bounded domains of ℝⁿ (n ≥ 2) is established via methods of hydrodynamical potential theory. The method is based on the unique solvability of the boundary integral equations' system resulting from the double layer potential ansatz together with a projection onto the normal field on the boundary. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Criterion for Designs in ℤ4-codes on the Symmetrized Weight Enumerator

The Assmus–Mattson theorem is known as a method to find designs in linear codes over a finite field. It is an interesting problem to find an analog of the theorem for Z ₄-codes. In a previous paper, the author gave a candidate of the theorem. The purpose of this paper is to give an improvement of the theorem. It is known that the lifted Golay code over Z ₄ contains 5-designs on Lee compositions. The improved method can find some of those without computational difficulty and without the help of a computer.     

On the study of the decomposition of matrix polynomials into factors

We present a survey of studies of questions on the decomposition of matrix polynomials into factors. The main attention is devoted to results obtained by P. S. Kazimirs 'kii on a method of solving the problem of determining the regular factors of a matrix polynomial over an algebraically closed field of characteristic zero. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Multiobjective Optimization of the Flow Around a Cylinder Using Model Order Reduction

In this article an efficient numerical method to solve multiobjective optimization problems for fluid flow governed by the Navier Stokes equations is presented. In order to decrease the computational effort, a reduced order model is introduced using Proper Orthogonal Decomposition and a corresponding Galerkin Projection. A global, derivative free multiobjective optimization algorithm is applied to compute the Pareto set (i.e. the set of optimal compromises) for the concurrent objectives minimization of flow field fluctuations and control cost. The method is illustrated for a 2D flow around a cylinder at Re = 100. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Axisymmetric deformation of bimetallic shells of revolution in the supercritical region

Geometrically nonlinear relationships of the theory of thin layered shells are applied to analyze axisymmetric strain of bimetallic shells of revolution in a temperature field. One-dimensional nonlinear boundary-value problems are solved by a combination of the linearization method and the discrete orthogonalization method. A numerical approach is proposed to solve the boundary-value problems in the supercritical strain region.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 52–56, 1990.     

Propagation of Creeping Waves Along the Curvilinear Surface of an Anisotropic Elastic Body

An analytic expression for the wave field of a creeping wave propagating along the curvilinear surface of an anisotropic elastic body is constructed by the boundary layer method. Bibliography: 14 titles.Dedicated to L. A. Molotkov on the occasion of his seventieth birthday__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 9–29.     

Analysis of the immersed boundary method for a finite element Stokes problem

Convergence results are presented for the immersed boundary (IB) method applied to a model Stokes problem. As a discretization method, we use the finite element method. First, the immersed force field is approximated using a regularized delta function. Its error in the W^{?1, p} norm is examined for 1 ≤ p < n/(n ? 1), with n representing the space dimension. Subsequently, we consider IB discretization of the Stokes problem and examine the regularization and discretization errors separately. Consequently, error estimate of order h^{1 ? α} in the W^{1, 1} × L¹ norm for the velocity and pressure is derived, where α is an arbitrary small positive number. The validity of those theoretical results is confirmed from numerical examples.     

Boundary integral equations and maximum modulus estimates for the Stokes system

A maximum modulus estimate for the Stokes system in bounded domains of ℝⁿ (n ≥ 2) is established via methods of hydrodynamical potential theory. The method is based on the unique solvability of the boundary integral equations' system resulting from the double layer potential ansatz together with a projection onto the normal field on the boundary. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

We compute a local linearization for the nonlinear, inverse problem of identifying the stored energy function of a hyperelastic material from the full knowledge of the displacement field. The displacement field is described as a solution of the nonlinear, dynamic, elastic wave equation, where the first Piola–Kirchhoff stress tensor is given as the gradient of the stored energy function. We assume that we have a dictionary at hand such that the energy function is given as a conic combination of the dictionary's elements. In that sense, the mathematical model of the direct problem is the nonlinear operator that maps the vector of expansion coefficients to the solution of the hyperelastic wave equation. In this article, we summarize some continuity results for this operator and deduce its Fréchet derivative as well as the adjoint of this derivative. Because the stored energy function encodes mechanical properties of the underlying, hyperelastic material, the considered inverse problem is of highest interest for structural health monitoring systems where defects are detected from boundary measurements of the displacement field. For solving the inverse problem iteratively by the Landweber method or Newton‐type methods, the knowledge of the Fréchet derivative and its adjoint is of utmost importance. Copyright © 2016 John Wiley & Sons, Ltd.