estimating isogenies on elliptic curves the object of this paper is to prove a new type of estimate for isogenies between elliptic curves this has several diophantine applications effective versions of serre s galois irreducibility theorem and shafarevich s theorem for example which are presented in another paper mw3 later articles will deal with the corresponding problems for abelian varieties of arbitrary dimension right at the beginning we emphasize that we are identifying elliptic curves e with weierstrass equations

prime power terms in elliptic divisibility sequences we study a problem on specializations of multiples of rational points on elliptic curves analogous to the mersenne problem we solve this problem when descent via isogeny is possible by giving explicit bounds on the indices of prime power terms in elliptic divisibility sequences associated to points in the image of a nontrivial isogeny we also discuss the uniformity of these bounds assuming the hall lang conjecture

existence of one signed periodic solutions of some second order differential equations via a krasnoselskii fixed point theorem this paper is devoted to study the existence of periodic solutions of the second order equation x 00 f t x where f is a caratheodory function by combining some new properties of green s function together with krasnoselskii fixed point theorem on compression and expansion of cones as applications we get new existence results for equations with jumping nonlinearities as well as equations with a repulsive or attractive singularity in this latter case our results cover equations with weak singularities and are compared with some recent results by i rachunkova m tvrdyand i vrkoc r 2002 elsevier science usa all rights reserved

multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces abstract the existence of multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces is discussed in this paper our nonlinearity may be singular in its dependent variable and our analysis relies on a nonlinear alternative of leray schauder type and on a fixed point theorem in cones

the difference between the weil height and the canonical height on elliptic curves estimates for the difference of the weil height and the canonical height of points on elliptic curves are used for many purposes both theoretical and computational in this note we give an explicit estimate for this difference in terms of the j invariant and discriminant of the elliptic curve the method of proof suggested by serge lang is to use the decomposition of the canonical height into a sum of local heights we illustrate one use for our estimate by computing generators for the mordell weil group in three examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 x ax b with a and b in the ring of integers of k the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 10 chapter viii 9 or 6 chapter vi the canonical height is determined by this property together with the fact that the difference

a parallel multi domain solution methodology applied to nonlinear thermal transport problems in nuclear fuel pins this paper describes an efficient and nonlinearly consistent parallel solution methodology for solving coupled nonlinear thermal transport problems that occur in nuclear reactor applications over hundreds of individual 3d physical subdomains efficiency is obtained by leveraging knowledge of the physical domains the physics on individual domains and the couplings between them for preconditioning within a jacobian free newton krylov method details of the computational infrastructure that enabled this work namely the open source advanced multi physics amp package developed by the authors is described details of verification and validation experiments and parallel performance analysis in weak and strong scaling studies demonstrating the achieved efficiency of the algorithm are presented furthermore numerical experiments demonstrate that the preconditioner developed is independent of the number of fuel subdomains in a fuel rod which is particularly important when simulating different types of fuel rods finally we demonstrate the power of the coupling methodology by considering problems with couplings between surface and volume physics and coupling of nonlinear thermal transport in fuel rods to an external radiation transport code

multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities abstract this paper is devoted to the study of nonlinear singular and non singular fourth order difference equations coupled with periodic boundary value conditions in the paper some existence and nonexistence results are given the results are deduced from krasnoselskii s fixed point theorems in cones an exhaustive study of the green s function related to a linear fourth order problem is done

on existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations the existence and multiplicity of positive solutions are established to periodic boundary value problems for singular nonlinear second order ordinary differential equations the arguments are based only upon the positivity of the green s functions and the krasnosel skii fixed point theorem as an example a periodic boundary value problem is also considered which comes from the theory of nonlinear elasticity

global bifurcation of positive solutions of a second order periodic boundary value problem with indefinite weight abstract in this paper we are concerned with the global structure and stability of positive solutions of periodic boundary value problem u t q t u t a t f u t 0 t 2 u 0 u 2 u 0 u 2 where q c r 0 is of periodic 2 and q t 0 t 0 2 a c r r is of periodic 2 and changes sign the proof of our main results are based on bifurcation techniques

hilbert s tenth problem and mazur s conjecture for large subrings of mathbb q we give the first examples of infinite sets of primes s such that hilbert s tenth problem over z s 1 has a negative answer in fact we can take s to be a density 1 set of primes we show also that for some such s there is a punctured elliptic curve e over z s 1 such that the topological closure of e z s 1 in e r has infinitely many connected components

a discrete fourth order lidstone problem with parameters various existence multiplicity and nonexistence results for nontrivial solutions to a nonlinear discrete fourth order lidstone boundary value problem with dependence on two parameters are given using a symmetric green s function approach an existence result is also given for a related semipositone problem thus relaxing the usual assumption of nonnegativity on the nonlinear term

on the existence of positive solutions for nonlinear differential equations with periodic boundary conditions we prove the existence of positive solutions of second order nonlinear differential equations on a finite interval with periodic boundary conditions and give upper and lower bounds for these positive solutions obtained results yield positive periodic solutions of the equation on the whole real axis provided that the coefficients are periodic

eigenvalue problems for second order nonlinear dynamic equations on time scales abstract this paper is concerned with the existence and nonexistence of positive solutions of the second order nonlinear dynamic equation u t a t f u t 0 t 0 1 satisfying either the conjugate boundary conditions u 0 u 1 0 or the right focal boundary conditions u 0 u 1 0 where a and f are positive we show that there exists a 0 such that the above boundary value problem has at least two one and no positive solutions for 0 and respectively furthermore by using the semiorder method on cones of the banach space we establish an existence and uniqueness criterion for positive solution of the problem in particular such a positive solution u t of the problem depends continuously on the parameter i e u t is nondecreasing in lim 0 u 0 and lim u

new fixed point theorems for mixed monotone operators and local existence uniqueness of positive solutions for nonlinear boundary value problems abstract in this article we present a new fixed point theorem for a class of general mixed monotone operators which extends the existing corresponding results moreover we establish some pleasant properties of nonlinear eigenvalue problems for mixed monotone operators based on them the local existence uniqueness of positive solutions for nonlinear boundary value problems which include neumann boundary value problems three point boundary value problems and elliptic boundary value problems for lane emden fowler equations is proved the theorems for nonlinear boundary value problems obtained here are very general

computing heights on elliptic curves we describe how to compute the canonical height of points orn elliptic curves tate has given a rapidly converging series for archimedean local heights over r we describe a modified version of tate s series which also converges over c and give an efficient procedure for calculating local heights at non archimedean places in this way we can calculate heights over number fields having complex embeddings we also give explicit estimates for the tail of our series and present several examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 alxy a3y x3 a2x2 a4x a6 the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 11 viii section 9 or 6 chapter vi the canonical height is an extremely important theoretical tool in the arithmetic theory of elliptic curves being used for such diverse purposes as studying values of l functions 5 numbers of integral points 12 and transcendence theory 9 it is also important as a computational tool such as its use in zagier s algorithm for finding integral points up to large bounds 18 it is thus of interest to have an efficient method for calculating the canonical height of a point the usual definition of h as a limit h p limn 0 4 nh x 2np is not practical for computation instead one uses the fact that the canonical height can be written as a sum of local heights one term for each distinct absolute value on k 2 h p e na p vemk for example if k q then mk can be identified with the set of rational primes together with the usual absolute value on q the multiplicities nv are chosen so that the product formula holds and so that h is independent of the choice of the field k the local height corresponding to a non archimedean absolute value is given by intersection theory in a well known manner see e g 2 4 or 7 chapter 11 section 5 we will describe a quick way to compute non archimedean local heights in section 5 the local height for an archimedean absolute value is given by a transcendental function and so efficient computation is somewhat more difficult j tate 15 1988 american mathematical society 0025 5718 88 1 00 25 per page 339 received august 20 1987 revised october 21 1987 1980 mathematics subject classification 1985 revision primary 11g05 14k07 11d25 this work was partially supported by nsf grant dms 8612393 current address mathematics department brown university providence ri 02912 this content downloaded from 207 46 13 103 on thu 20 oct 2016 04 37 48 utc all use subject to http about jstor org terms 340 joseph h silverman has given an easily computed power series which works for real absolute values precisely for a given curve e and point p x y he gives a sequence of easily computed numbers co cl so that

existence and uniqueness of positive periodic solutions of functional differential equations abstract in this paper we consider the existence and uniqueness of positive periodic solution for the periodic equation y t a t y t h t f y t t by the eigenvalue problems of completely continuous operators and theory of concave or convex operators and its eigenvalue we establish some criteria for existence and uniqueness of positive periodic solution of above functional differential equations with parameter in particular the unique solution y t of the above equation depends continuously on the parameter finally as an application we obtain sufficient condition for the existence of positive periodic solutions of the nicholson blowflies model

doubly periodic solutions to a coupled telegraph system abstract while the nonlinearities are bounded below we establish the existence and multiplicity results of doubly periodic solutions for a coupled nonlinear telegraph system with a parameter the proof is based on a well known fixed theorem in a cone

global structure of positive solutions for superlinear second order periodic boundary value problems in this paper we consider the nonlinear eigenvalue problems gather u lambda h t f u 0 quad 0 0 for i 1 ldots m 2 with sum i 1 m 2 alpha i eta i 0 for t 0 in 0 1 f in c 0 infty 0 infty and f s 0 for s 0 and f 0 infty where f 0 lim s rightarrow 0 f s s we investigate the global structure of positive solutions by using the nonlinear krein rutman theorem

singular lidstone boundary value problem with given maximal values for solutions abstract the singular problem 1 nx 2n f t x x 2n 2 x 2j 0 x 2j t 0 0 j n 1 max x t 0 t t a depending on the parameter is considered here the positive caratheodory function f may be singular at the zero value of all its phase variables the paper presents conditions which guarantee that for any a 0 there exists a 0 such that the above problem with a has a solution x ac2n 1 0 t which is positive on 0 t the proofs are based on the regularization and sequential techniques and use the leray schauder degree and vitali s convergence theorem

positive solutions of even order system periodic boundary value problems we study a class of even order system boundary value problems with periodic boundary conditions a series of criteria are obtained for the existence of one two any arbitrary number and even a countably infinite number of positive solutions criteria for the nonexistence of positive solutions are also derived as for the second order case our results extend improve and supplement those in the literature for scalar and system boundary value problems several examples are given to demonstrate the applications moreover we obtain conditions for system periodic boundary value problems of a different form to have nontrivial solutions by transforming our main results to such problems

existence multiplicity and dependence on a parameter for a periodic boundary value problem the authors consider the boundary value problem y 2 y g t f y 0 0 t 2 y 0 y 2 y 0 y 2

divisors of mersenne numbers we add to the heuristic and empirical evidence for a conjecture of gillies about the distribution of the prime divisors of mersenne numbers we list some large prime divisors of mersenne numbers mp in the range 17000 p 105

on the number of positive solutions of nonlinear systems abstract we prove that appropriate combinations of superlinearity and sublinearity of f u with respect to at zero and infinity guarantee the existence multiplicity and nonexistence of positive solutions to boundary value problems for the n dimensional system u h t f u 0 0 is defined by u u 1 u n where u u 1 u n and covers the two important cases u u and u u p 2 u p 1 our methods employ fixed point theorems in a cone

periodic solutions to non autonomous second order systems abstract we establish the existence of periodic solution of a class of non autonomous second order systems x x v t x 0 where v t x v 1 t x v n t x if lim x v i t x x 0 i 1 n uniformly in t and v is bounded below or above for appropriate ranges of

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems

enhanced nonlinear iterative techniques applied to a nonequilibrium plasma flow we study the application of enhanced nonlinear iterative methods to the steady state solution of a system of two dimensional convection diffusion reaction partial differential equations that describe the partially ionized plasma flow in the boundary layer of a tokamak fusion reactor this system of equations is characterized by multiple time and spatial scales and contains highly anisotropic transport coefficients due to a strong imposed magnetic field we use newton s method to linearize the nonlinear system of equations resulting from an implicit finite volume discretization of the governing partial differential equations on a staggered cartesian mesh the resulting linear systems are neither symmetric nor positive definite and are poorly conditioned preconditioned krylov iterative techniques are employed to solve these linear systems we investigate both a modified and a matrix free newton krylov implementation with the goal of reducing cpu cost associated with the numerical formation of the jacobian a combination of a damped iteration mesh sequencing and a pseudotransient continuation technique is used to enhance global nonlinear convergence and cpu efficiency gmres is employed as the krylov method with incomplete lower upper ilu factorization preconditioning the goal is to construct a combination of nonlinear and linear iterative techniques for this complex physical problem that optimizes trade offs between robustness cpu time memory requirements and code complexity it is shown that a mesh sequencing implementation provides significant cpu savings for fine grid calculations performance comparisons of modified newton krylov and matrix free newton krylov algorithms will be presented

positive doubly periodic solutions of nonlinear telegraph equations abstract this paper deals with the existence of nontrivial doubly periodic solutions for the nonlinear telegraph equation u tt u xx cu t a t x u b t x f t x u t x r 2 where c 0 is a constant a b c r 2 r f c r 2 r r and a b f are 2 periodic in t and x we show the existence of positive doubly periodic weak solutions when 0 a t x c2 4 and f is either superlinear or sublinear on u by using the krasnoselskii s fixed point theorem in cones