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

new candidates for arbitrage free stock price models via generalized conditional symmetry method abstract in this paper a new perspective upon generating arbitrage free stock price models is proposed the generalized conditional symmetry gcs method is applied to the governing second order 1 1 partial differential equation which does contain a rational parameter p drawn from the interval 1 2 1 we investigate the conditions that yield the concerned equation admitting a special class of second order gcss the determining system is solved in several special cases and from invariance surface condition associated to each of the gcs operator for all values of p some invariant solutions are pointed out new candidate models for arbitrage free stock price are derived

generalized multiple riccati equations rational expansion method with symbolic computation to construct exact complexiton solutions of nonlinear partial differential equations in this paper based on symbolic computation and the idea of rational expansion method a generalized multiple riccati equations rational expansion method gmrere is proposed to uniformly construct a series of exact complexiton solutions for nonlinear partial differential equations compared with most existing tanh function methods and rational expansion methods the proposed method not only recovers some known solutions but also finds some new and general complexiton solutions the efficiency of the method can be demonstrated on 2 1 dimensional burgers equation

multiplicity estimates on group varieties tions the field k will be the field q of algebraic numbers we remark at this point that the results remain true if we take instead of the field of complex numbers c its p adic analogue c for some fixed prime p and for k a corresponding subfield kp of cp we restrict ourselves to the complex case in order to avoid some minor complications appearing in the p adic domain these complications always arise when analytic functions come in our functions are defined globally in the case of complex numbers but only locally in the case when we deal with the p adic domain these difficulties can be avoided by a purely algebraic approach if we took this approach the only condition on the ground field would be that it should be algebraically closed and of characteristic zero but we prefer to avoid such an approach in order to keep the text understandable also for those who are mainly interested in the applications in transcendence

arithmetic of characteristic p special l values recently the second author has associated a finite fq t module h to the carlitz module over a finite extension of fq t this module is an analogue of the ideal class group of a number field in this paper we study the galois module structure of this module h for cyclotomic extensions of fq t we obtain function field analogues of some classical results on cyclotomic number fields such as the p adic class number formula and a theorem of mazur and wiles about the fitting ideal of ideal class groups we also relate the galois module h to anderson s module of circular units and give a negative answer to anderson s kummer vandiver type conjecture these results are based on a kind of equivariant class number formula which refines the second author s class number formula for the carlitz module

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

stochastic exact solutions of the wick type stochastic nls equation based upon the hermite transformation and the extended sub equation expansion method soliton like solutions of the stochastic schrodinger equation in the white noise environment are obtained via maple

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

un lemme matriciel effectif we show an almost optimal effective version of masser s matrix lemma lecture notes math 1290 109 148 1987 giving a lower bound of the height of an abelian variety in terms of its period lattices

some windows with very good sidelobe behavior some of the windows presented by harris 1 are not correct in terms of their reported peak sidelobes and optimal behavior we present corrected plots of harris windows and also derive additional windows with very good sidelobes and optimal behavior under several different constraints the temporal weightings are characterized as a sum of weighted cosines over a finite duration the plots enable the reader to select a window to suit his requirements in terms of bias due to nearby sidelobes and bias due to distant sidelobes

pade approximations and diophantine geometry abstract using methods of pade approximations we prove a converse to eisenstein s theorem on the boundedness of denominators of coefficients in the expansion of an algebraic function for classes of functions parametrized by meromorphic functions this result is applied to the tate conjecture on the effective description of isogenies for elliptic curves

positive line bundles on arithmetic varieties p 5 u integers is defined to count the volume of the lattice i l n of integral sections in the space j7 lon of real sections with supremum norm we want to prove that the leading term of this hilbert function as n f oo is given in terms of the height of x in as 1 this is known as a theorem of gillet and soule gs2 if x has a regular generic fiber beside this known result our proof uses hironaka s theorem on resolutions of singularities and minkowski s theorem on successive minima by hironaka s theorem we may construct

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

formal subgroups of abelian varieties in this paper we generalize the result of 12 in the following sense let a be an abelian variety over a number field k let be the neron model of a over the ring of integers o k of k completing along its zero section defines a formal group widehat mathcal a over o k we prove that any formal subgroup of the generic fiber of widehat mathcal a whose closure in widehat mathcal a is smooth over an open subset of spec o k arises in fact from an abelian subvariety of a the proof is of a transcendental nature and uses the arakelovian formalism introduced by bost 3

zero estimates on group varieties ii the aim of this article is to give an account of a class of estimates that have in recent years been found useful in the theory of transcendental numbers as our space is limited and the estimates are quite technical in nature wc can provide no hints of proofs and we prefer to illustrate the results with examples rather than supply full statements we also mention some of the applications one basic method in transcendence theory can be described in general terms as follows suppose it is desired to establish a certain proposition usually it will be possible to associate with the problem a collection of meromorphic functions ft fnm such a way that assuming the falsity of the proposition imposes strong arithmetical conditions on the functions these enable us to construct a polynomial p p x1 xn such that the function f p f1 fn is not identically zero but has a large number of zeroes by means of analytic interpolation arguments it can then be deduced that takes many small values and by using once more the arithmetical conditions we conclude that has even more zeroes than initially arranged but often merely knowing upper bounds for the degree of p leads to upper bounds for the number of zeroes of and if we are lucky these yield the contradiction that proves the original proposition in this article we shall limit attention to the last stage of this argument as it is here that the most interesting recent progress has taken place a simple example will suffice to illustrate the sort of estimates involved to show that w and e cannot both be algebraic for w 0 suppose they are then we end up with a non zero polynomial p p x y of degree at most l 1 in x and of degree at most jf 1 in y such that the function

new applications of the singular manifold method to the 2 1 dimensional burgers equations new applications of the singular manifold method are presented to a 2 1 dimensional burgers equation and a higher order burgers equation by choosing different seed solutions auto backlund transformations cole hopf transformations and general functional separation solutions containing three arbitrary functions are obtained for the equations in question localized coherent structures are for the first time expressed by jacobi elliptic functions not infinite periodic limit and their interaction properties are numerically studied

lie transformations similarity reduction and solutions for the nonlinear madelung fluid equations with external potential the application of lie group methods to a system of coupled nonlinear partial differential equations representing what is usually called a madelung fluid is shown the generating operators of the transformation group that depends on five arbitrary group constants will be constructed and all subclasses of systems of ordinary differential equations derived by similarity reduction will be presented in tabular form two subclasses of physical interest are investigated in detail and the similarity solutions are compared with solutions found earlier by the application of inverse scattering transform techniques to the cubic nonlinear schrodinger equation similarity solutions for the madelung equations with linear external potential x f0x are presented

a new korteweg de vries equation based sub equation method and its application to the 2 1 dimensional korteweg de vries equation with the help of the symbolic computation system maple we present korteweg de vries equation based sub equation method being concise and straightforward it is applied to the 2 1 dimensional korteweg de vries equation it is shown that n soliton solution of the 2 1 dimensional korteweg de vries equation can be found by this new method the method can be applied to other nonlinear partial differential equations in mathematical physics

unlikely intersections with isogeny orbits in a product of elliptic schemes fix an elliptic curve e 0 without cm and a non isotrivial elliptic scheme over a smooth irreducible curve both defined over the algebraic numbers consider the union of all images of a fixed finite rank subgroup of arbitrary rank of e 0 g also defined over the algebraic numbers under all isogenies between e 0 g and some fiber of the g th fibered power mathcal a of the elliptic scheme where g is a fixed natural number as a special case of a slightly more general result we characterize the subvarieties of arbitrary dimension inside mathcal a that have potentially zariski dense intersection with this set in the proof we combine a generalized vojta remond inequality with the pila zannier strategy

special points on fibered powers of elliptic surfaces consider a fibered power of an elliptic surface we characterize its subvarieties that contain a zariski dense set of points that are torsion points in fibers with complex multiplication this result can be viewed as a mix of the manin mumford and andre oort conjecture and is related to a conjecture of pink the main technical tool is a new height inequality we also use it to give another proof of a case of gubler s result on the bogomolov conjecture over function fields

linear b spline finite element method for the improved boussinesq equation in this paper we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved boussinesq equation the finite element method with linear b spline basis functions is used to discretize the nonlinear partial differential equation in space and derive a second order system involving only ordinary derivatives it is shown that the coefficient matrix for the second order term in this system is invertible consequently for the first time the initial boundary value problem can be reduced to an explicit initial value problem to which many accurate numerical methods are readily applicable various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting wave interaction and blow up behavior

similarity solutions of the deformed maxwell bloch system the authors have found similarity reductions for the deformed maxwell bloch system to the fifth and second painleve equations asymptotics of the solutions of these equations which are relevant to two level atomic systems with pumping have also been derived

a method for the optimal pattern synthesis of linear arrays with prescribed nulls a new l sub spl infin optimal method for the synthesis of equispaced linear array functions with asymmetrical far field pattern functions is proposed this iterative method provides for the exact specification of the beam width while at the same time allowing for the specification of the relative levels of individual sidelobes by index or as a function of bearing e g angularly extended nulls as well as the realization of specified narrowband nulls the resulting array factors are optimal in the weighted l sub spl infin sense and in general have complex coefficients this new remez type method employs multiple objective functions to provide the degrees of freedom that are required for exact null placement examples which demonstrate the design flexibility offered by the method are included for various sum and difference patterns including superdirective and shaped beam arrays

generalized riccati equation rational expansion method and its application by directly extending the riccati equation rational expansion method a new method is developed for constructing a series of exact analytical solutions of an averaged dispersion managed fiber system equation adme including some new rational formal soliton like solutions triangle like solutions and rational solutions

mesures d independance lineaire de logarithmes dans un groupe algebrique commutatif this work falls within the theory of linear forms in logarithms over a connected and commutative algebraic group defined over the field of algebraic numbers overline mathbb q let g be such a group let w be a hyperplane of the tangent space at the origin of g defined over overline mathbb q and u be a complex point of this tangent space such that the image of u by the exponential map of the lie group g is an algebraic point then we obtain a lower bound for the distance between u and w which improves the results known before and which is in particular the best possible for the height of the hyperplane w the proof rests on baker s method and hirata s reduction as well as a new arithmetic argument chudnovsky s process of variable change which enables us to give a precise estimate of the ultrametric norms of some algebraic numbers built during the proof

hermitian vector bundles and extension groups on arithmetic schemes i geometry of numbers we define and investigate extension groups in the context of arakelov geometry the arithmetic extension groups ext xi f g we introduce are extensions by groups of analytic types of the usual extension groups extxi f g attached to ox modules f and g over an arithmetic scheme x in this paper we focus on the first arithmetic extension group ext x1 f g the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over x and we especially consider the case when x is an arithmetic curve namely the spectrum specok of the ring of integers in some number field k then the study of arithmetic extensions over x is related to old and new problems concerning lattices and the geometry of numbers namely for any two hermitian vector bundles f and g over x specok we attach a logarithmic sizesf g to any element of ext x1 f g and we give an upper bound on sf g in terms of slope invariants of f and g we further illustrate this notion by relating the sizes of restrictions to points in p1 z of the universal extension over pz1 to the geometry of psl2 z acting on poincare s upper half plane and by deducing some quantitative results in reduction theory from our previous upper bound on sizes finally we investigate the behaviour of size by base change i e under extension of the ground field k to a larger number field k when the base field k is q we establish that the size which cannot increase under base change is actually invariant when the field k is an abelian extension of k or when f g is a direct sum of root lattices and of lattices of voronoi s first kind the appendices contain results concerning extensions in categories of sheaves on ringed spaces and lattices of voronoi s first kind which might also be of independent interest

hodograph transformations of linearizable partial differential equations in this paper an algorithmic method is developed for transforming quasilinear partial differential equations of the form u t g u u nx f u u x cdots u n 1 x u mx equiv partial m u partial x m where dg du not equiv 0 into semilinear equations i e equations of the above form with g u 1 this crucially involves the use of hodograph transformations i e transformations involving the interchange of dependent and independent variables furthermore the most general quasilinear equation of the above form is found that can be mapped via a hodograph transformation to a semilinear form this algorithm provides a method for establishing whether a given quasilinear equation is linearizable i e is solvable in terms of either a linear partial differential equation or of a linear integral equation in particular this method is used to show how the painleve tests may be applied to quasilinear equations this appears to resolve the problem that solutions of lineariz

blowup of solutions for improved boussinesq type equation the paper studies the existence and uniqueness of local solutions and the blowup of solutions to the initial boundary value problem for improved boussinesq type equation utt uxx uxxtt u xx by a galerkin approximation scheme combined with the continuation of solutions step by step and the fourier transform method it proves that under rather mild conditions on initial data the above mentioned problem admits a unique generalized solution u w2 0 t h2 0 1 as long as c2 r in particular when s asp where a 0 is a real number and p 1 is an integer specially a 0 if p is an odd number the solution blows up in finite time moreover two examples of blowup are obtained numerically

symmetries of differential equations from sophus lie to computer algebra the topic of this article is the symmetry analysis of differential equations and the applications of computer algebra to the extensive analytical calculations which are usually involved in it the whole area naturally decomposes into two parts depending on whether ordinary or partial differential equations are considered we show how a symmetry may be applied to lower the order of an ordinary differential equation or to obtain similarity solutions of partial differential equations the computer algebra packages sods and spde respectively which have been developed to perform almost all algebraic manipulations necessary to determine the symmetry group of a given differential equation are presented furthermore it is argued that the application of computer algebra systems has qualitatively changed this area of applied mathematics

a novel window for harmonic analysis this paper presents a new window design technique and discusses its effect on the detection of harmonic signals in the presence of nearby strong harmonic interference four design parameters are introduced to independently control the pattern falloff rate the overall sidelobe level the near sidelobe level and the depth of a steerable wide dip since the deep dip can be steered to any frequency the new window has effectively improved the detectability of a small tone without degrading resolvability contrary to the conventional approach of initially specifying the continuous weighting the new design technique starts with constructing the spectral window which meets the specifications and then employs the fast fourier transform to compute the discrete weighting no iterative sampling or perturbation procedure is required numerous examples are given to demonstrate the flexibility of the new window several examples of detecting a three tone signal have demonstrated the superiority of the new window in its detectability resolvability and accuracy of measuring the tone frequencies and amplitudes

explicit uniform estimation of rational points i estimation of heights by using the slope method in arakelov geometry we study the complexity of the singular locus of an arithmetic projective variety and explicit estimations of the arithmetic hilbert samuel function

isogroups and exact solutions for some klein gordon and liouville type equations in n dimensional euclidean space using the elementary symmetric functions of n euclidean coordinates xn as symmetry variables and the isovector approach the generalized forms of nonlinear klein gordon and liouville equations in n dimensional euclidean space are analyzed herein the construction of the components of the associated isovector field via transport property under lie derivative leads to orbital equations whose solutions yield invariant groups of transformations these invariant groups reduce the equations under consideration to nonlinear ordinary differential equations odes involving arbitrary functions of the new dependent and independent variables finally exact solutions of these resulting odes are tabulated for different forms of these arbitrary functions beside recovering the available results for particular forms of the said arbitrary functions involved in the ordinary differential equations some new exact solutions are reported

new classes of symmetries for partial differential equations new classes of symmetries for partial differential equations are introduced by writing a given partial differential equation s in a conserved form a related system t with potentials as additional dependent variables is obtained the lie group of point transformations admitted by t induces a symmetry group of s new symmetries may be obtained for s that are neither point nor lie backlund symmetries they are determined by a completely algorithmic procedure significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation

serre s uniformity problem in the split cartan case we prove that there exists an integer p 0 such that x split p q is made of cusps and cm points for any prime p p 0 equivalently for any non cm elliptic curve e over q and any prime p p 0 the image of the galois representation induced by the galois action on the p division points of e is not contained in the normalizer of a split cartan subgroup this gives a partial answer to an old question of serre

windows with rapidly decaying sidelobes and steerable sidelobe dips a window in the form of a sum of cosine terms is considered and the temporal weights are evaluated analytically so that it can have rapidly decaying sidelobes as well as steerable sidelobe dips an analytic expression is derived so that the window designer can exploit the trade off between a high sidelobe falloff rate slfor and many steerable sidelobe dips ssld the treatment is based on using the l degrees of freedom inherent in the l temporal weights of the l 1 term window in partially achieving the goals of a high slfor and many ssld in the extreme case where all degrees of freedom are used in achieving the maximum slfor a simple closed form expression is derived for the temporal weights thus formalizing and generalizing the results of nuttall 1981 the other extreme case where all degrees of freedom are used in creating many ssld has been recently treated by zhong et al see ibid vol 40 no 6 p 1452 1459 1992 combining both desired features in one unified mathematical framework is the main objective of this paper simulation results demonstrate the superior performance of a window having both features to that of a window having only one feature in extracting a weak sinusoidal signal dominated by a leakage from another stronger signal

design of windows with steerable sidelobe dips based on the frequency sampling method a novel and simple window design technique was established upon three theorems the technique produces a set of windows which have simple expressions and the steerable sidelobe dip ssld property by steering the dips in the sidelobes it is easy to reduce the edge of the mainlobe or to minimize the height of sidelobes or to create a hole in the sidelobes if its desired to reduce spectral leakage at a particular frequency it is found that windows constructed as a weighted sum of cosines over a finite support and with a 6 db oct sidelobe falloff rate can all be simulated by appropriate ssld distributions in this sense many good windows such as hamming blackman blackman harris and minimum sidelobe behaviors can be created with a minor constraint this renders design flexibility for various applications finally the new technique can be easily incorporated with other techniques to produce windows with hybrid merits

on a unified approach to transformations and elementary solutions of painleve equations an algorithmic method is developed for investigating the transformation properties of second order equations of painleve type this method which utilizes the singularity structure of these equations yields explicit transformations which relate solutions of the painleve equations ii vi with different parameters these transformations easily generate rational and other elementary solutions of the equations the relationship between painleve equations and certain new equations quadratic in the second derivative of painleve type is also discussed

application of the extended homotopy perturbation method to a kind of nonlinear evolution equations based on the homotopy perturbation method we propose a new extended homotopy perturbation method we apply the method to solve a kind of nonlinear evolution equations with the help of the symbolic computation system maple the success of the paper lies on the fact that we do not conform to the traditional homotopy mode but construct a new homotopy of the given equations the results reveal that we can obtain the approximation solution of a kind of nonlinear evolution equations

the anisotropic heisenberg spin chain and the nonlinear schrodinger equation abstract it is shown that the solutions of the continuous anisotropic heisenberg spin chain ahsc can be obtained from the linear integral equation which was proposed in a previous paper for the solutions of the isotropic heisenberg spin chain ihsc and the nonlinear schrodinger equation nls an explicit expression is obtained for the miura transformation which maps the solutions of the ahsc on solutions of the nls in the second part of the paper we investigate the similarity solutions of these partial differential equations which leads to ordinary differential equations of painleve type as an application we discuss some new solutions of painleve iv

nonclassical symmetries of a porous medium equation with absorption in this paper new symmetry reductions and exact solutions are presented for the porous medium equation with absorption those spatial forms for which the equation can be reduced to an ordinary differential equation are studied the symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by bluman and cole and are unobtainable by lie classical method

applications of some recent techniques for the exact solutions of the small disturbance potential flow equation of nonequilibrium transonic gas dynamics abstract using the isovector approach as developed in 1 and the direct method of clarkson and kruskal 2 we have obtained similarity and other reductions to ordinary differential equations and exact solutions of the small disturbance potential flow equation in two dimension nonequilibrium transonic gas dynamics namely k c x a xxx kc 2 xx d x xx k xyy b xx yy 0 further the lie algebra of the transformation groups yielded by the isovector approach has enabled us to remark on its solvability and nilpotency

lie transformations nonlinear evolution equations and painleve forms we present the results of a systematic investigation of invariance properties of a large class of nonlinear evolution equations under a one parameter continuous lie group of transformations it is shown that in general the corresponding invariant variables the subclass of which is the usual similarity variables lead to ordinary differential equations of painleve type in the case of inverse scattering transform solvable equations as conjectured by ablowitz ramani and segur this is found to be also true for certain higher spatial dimensional versions such as the kadomtsev petviashivilli two dimensional sine gordon and ernst equations for the nonsolvable equations considered here this invariance study leads to ordinary differential equations with movable critical points

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant