a parametric family of quintic thue equations for an integral parameter t z we investigate the family of thue equations f x y x 5 t 1 2 x 4 y 2t 3 4t 4 x 3 y 2 t 4 t 3 2t 2 4t 3 x 2 y 3 t 3 t 2 5t 3 xy 4 y 5 1 originating from emma lehmer s family of quintic fields and show that for t 3 28 10 15 the only solutions are the trivial ones with x 0 or y 0 our arguments contain some new ideas in comparison with the standard methods for thue families which gives this family a special interest

discontinuous displacement approximation for capturing plastic localization it is proposed to capture localized plastic deformation via the inclusion of regularized displacement discontinuities at element boundaries interfaces of the finite element subdivision the regularization is based on a kinematic assumption for an interface that resembles that which is pertinent to the classical shear band concept as a by product of the regularization an intrinsic band width is introduced as a constitutive property rather than a geometric feature of the finite element mesh in this way the spurious mesh sensitivity which is obtained when the displacement approximation is continuous can be avoided another consequence is that the interfacial relation between the elements is derived directly from the conventional constitutive properties of the continuously deforming material an interesting feature is that the acoustic tensor will not only play a role for diagnosing discontinuous bifurcation but will also serve as the tangent stiffness tensor of the interface up to within a scalar factor an analytical investigation of the behaviour of the interface is carried out and it is shown that dilatation may indeed accompany slip within a shear band for a general plasticity model the significance of proper mesh alignment is demonstrated for a simple problem in plane strain and plane stress it is shown that a unique structural post peak response in accordance with non linear fracture mechanics can be achieved when the plastic softening modulus is properly related to the bandwidth the paper concludes with a numerical simulation of the gradual development of a shear band in a soil slope

on the representation of unity by binary cubic forms if f x y is a binary cubic form with integer coefficients such that f x 1 has at least two distinct complex roots then the equation f x y 1 possesses at most ten solutions in integers x and y nine if f has a nontrivial automorphism group if further f x y is reducible over z x y then this equation has at most 2 solutions unless f x y is equivalent under gl2 z action to either x x2 xy y2 or x x2 2y2 the proofs of these results rely upon the method of thue siegel as refined by evertse together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational diophantine approximation along the way we completely solve all thue equations f x y 1 for f cubic and irreducible of positive discriminant df 106 as corollaries we obtain bounds for the number of solutions to more general cubic thue equations of the form f x y m and to mordell s equation y2 x3 k where m and k are nonzero integers

non algebraic integrability of the chew low reversible dynamical system of the cremona type and the relation with the 7th hilbert problem non resonant case abstract a smooth reversible dynamical system srds and a system of nonlinear functional equations defined by a certain rational quadratic cremona mapping and arising from the static model of the dispersion approach in the theory of strong interactions the chew low equations for p wave n scattering are considered this srds is splitted into 1 and 2 dimensional ones an explicit cremona transformation that completely determines the exact solution of the two dimensional system is found this solution depends on an odd function satisfying a nonlinear autonomous 3 point functional equation non algebraic integrability of srds under consideration is proved using the method of poincare normal forms and the siegel theorem on biholomorphic linearization of a mapping at a non resonant fixed point the proof is based on the classical feldman baker theorem on linear forms of logarithms of algebraic numbers which in turn relies upon solving the 7th hilbert problem by a i gel fond and t schneider and new powerful methods of a baker in the theory of transcendental numbers the general theorem following from the feldman baker theorem on applicability of the siegel theorem to the set of the eigenvalues c n of a mapping at a non resonant fixed point which belong to the algebraic number field a is formulated and proved the main results are presented in theorems 1 3 5 7 8 and remarks 3 7

on the linear independence measure of logarithms of rational numbers in this paper we give a general theorem on the linear independence measure of logarithms of rational numbers and in particular the linear independence measure of 1 log 2 log 3 log 5 and of 1 log 2 log 3 log 5 log 7 we also give a method to search for polynomials of smallest norm on a real interval a b which may be suitable for computing or improving the linear independence measure of logarithms of rational numbers

on the abc conjecture ii let x y and z be coprime positive integers with x y z in this paper we give upper bounds for z in terms of the greatest square free factor of xyz

solving resultant form equations over number fields we give an efficient algorithm for solving resultant form equations over number fields this is the first time that such equations are completely solved by reducing them to unit equations in two variables

on the solutions of a family of quartic thue equations in this paper we solve a certain family of diophantine equations associated with a family of cyclic quartic number fields in fact we prove that for n 5 10 6 and n n 1 191 x 10 19 with n n 2 n 2 4 square free the thue equation n x y x 4 n 2 x 3 y n 3 2n 2 4n 2 x 2 y 2 n 2 xy 3 y 4 1 has no integral solution except the trivial ones 1 0 1 0 0 1 0 1

accessory parameters for punctured spheres this paper contains some qualitative results about the accessory parameters for punctured spheres with signature we show that the fuchsian uniformizing connection and hence also the accessory parameters for the surface depends real analytically on moduli we also show that the important invariants of a uniformization of a punctured sphere such as the accessory parameters fuchsian groups poincare metrics and covering maps vary continuously under degenerations such as coalescing of punctures the branched holomorphic universal covering group of a riemann surface s of finite analytic type g n with 2g 2 n 0 can be described as the monodromy group of the schwarzian differential equation where o is a meromorphic connection on s with singularities of very simple type and only at the punctures of s the connection o depends on 3g 3 n unknown constants called accessory parameters despite numerous attempts see hejhal he2 for some historical remarks these parameters remain elusive they can be determined only in the cases where the covering group is elementary or the surface has no moduli see 2 2 nevertheless the study of the parameters has led to some surprising results hejhal he2 has used variations of accessory parameters to study poincare series and in the work of zograf and takhtadzhyan zt1 zt2 zt3 these accessory parameters appear in calculations of the potential for the weil petersson metric on teichmuller space in this paper we study the simplest case that is n punctured spheres g 0 in this case the connection o is a rational function we show that o and hence also the accessory parameters depends real analytically on moduli and that various objects attached to a uniformization such as the uniformizing received by the editors november 30 1987 1980mathematicssubject classification 1985 revision primary 30f10 30f35 32g15 14h15 research partially supported by national science foundation grants 8401280 and 8120790 1989 american mathematical society 0002 9947 89 1 00 25 per page

there are only finitely many diophantine quintuples a set of m positive integers is called a diophantine m tuple if the product of its any two distinct elements increased by 1 is a perfect square diophantus found a set of four positive rationals with the above property the first diophantine quadruple was found by fermat the set 1 3 8 120 baker and davenport proved that this particular quadruple cannot be extended to a diophantine quintuple in this paper we prove that there does not exist a diophantine sextuple and that there are only finitely many diophantine quintuples

on the diophantine equation ax n by n if a b and n are positive integers with b a and n 3 then the equation of the title possesses at most one solution in positive integers x and y with the possible exceptions of a b n satisfying b a 1 2 a min 0 3n 83 and 17 n 347 the proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometric functions the theory of linear forms in logarithms and recent computational methods related to lattice basis reduction additionally we compare and contrast a number of these last mentioned techniques

on divisors of lucas and lehmer numbers let un be the nth term of a lucas sequence or a lehmer sequence in this article we shall establish an estimate from below for the greatest prime factor of un which is of the form n exp log n 104 log log n in doing so we are able to resolve a question of schinzel from 1962 and a conjecture of erd s from 1965 in addition we are able to give the first general improvement on results of bang from 1886 and carmichael from 1912

the mixed littlewood conjecture for pseudo absolute values in this paper we study the mixed littlewood conjecture with pseudo absolute values we show that if p is a prime and mathcal d is a pseudo absolute value sequence satisfying mild conditions then begin aligned inf n in mathbb n n n p n mathcal d vert n alpha vert 0 quad text for text all alpha in mathbb r end aligned our proof relies on a measure rigidity theorem due to lindenstrauss and lower bounds for linear forms in logarithms due to baker and wustholz we also deduce the answer to the related metric question of how fast the infimum above tends to zero for almost every alpha

primitive divisors of lucas and lehmer sequences stewart reduced the problem of determining all lucas and lehmer sequences whose nth element does not have a primitive divisor to solving certain thue equations using the method of tzanakis and de weger for solving thue equations we determine such sequences for n 30 further computations lead us to conjecture that for n 30 the nth element of such sequences always has a primitive divisor

classical and modular approaches to exponential diophantine equations i fibonacci and lucas perfect powers this is the first in a series of papers whereby we combine the classical approach to exponential diophantine equations linear forms in logarithms thue equations etc with a modular approach based on some of the ideas of the proof of fermat s last theorem in this paper we give new improved bounds for linear forms in three logarithms we also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the fibonacci sequence are 0 1 8 and 144 and the only perfect powers in the lucas sequence are 1 and 4

multi potential based discontinuous bifurcation model for jointed rock masses and its application based on the localization mechanism of jointed rock masses damage localization in jointed rock masses can be treated in terms of discontinuous bifurcation the discontinuous bifurcation model for jointed rock masses in the framework of multi potential theory is presented in this paper a numerical method is used to obtain the solution of the eigenvalue problem resulting from the bifurcation model consideration of elements embedded with discontinuous deformation mode is made and corresponding codes are incorporated into standard fem programs thus enabling the applicability of the programs for localization problems with the aid of the enhanced fem program localization bands i e dominant cracking bands in specimens under tension or compression are simulated numerically by using the proposed model numerical results show different localization bands in distinct load cases to demonstrate the potential application of this program analysis of xiluodu arch dam which is one of the largest arch dams in southwest china serves as an example of engineering cases the results of xiluodu arch dam from numerical analysis and physical modeling test are compared and coincided very well with each other

on the resolution of relative thue equations an efficient algorithm is given for the resolution of relative thue equations the essential improvement is the application of an appropriate version of wildanger s enumeration procedure based on the ellipsoid method of fincke and pohst recently relative thue equations have gained an important application e g in computing power integral bases in algebraic number fields the presented methods can surely be used to speed up those algorithms the method is illustrated by numerical examples

on a conjecture of whittaker concerning uniformization of hyperelliptic curves this article concerns an old conjecture due to e t whittaker aiming to describe the group uniformizing an arbitrary hyperelliptic riemann surface y 2 2g 2 i 1 x a i as an index two subgroup of the monodromy group of an explicit second order linear differential equation with singularities at the values a i whittaker and collaborators in the thirties and r rankin some twenty years later were able to prove the conjecture for several families of hyperelliptic surfaces characterized by the fact that they admit a large group of symmetries however general results of the analytic theory of moduli of riemann surfaces developed later imply that whittaker s conjecture cannot be true in its full generality recently numerical computations have shown that whittaker s prediction is incorrect for random surfaces and in fact it has been conjectured that it only holds for the known cases of surfaces with a large group of automorphisms the main goal of this paper is to prove that having many automorphisms is not a necessary condition for a surface to satisfy whittaker s conjecture

p adic logarithmic forms and a problem of erdős for any natural number m 1 let p m denote the greatest prime divisor of m by the problem of erd s in the title of the present paper we mean the general version of his problem that is his conjecture from 1965 that frac p 2 n 1 n to infty quad rm as n to infty see erd s 10 and its far reaching generalization to lucas and lehmer numbers in the present paper the author delivers three refinements upon yu 40 required by c l stewart for solving completely the problem of erd s see stewart 25 the author gives also some remarks on the solution of this problem aiming to be more streamlined with respect to the p adic theory of logarithmic forms

finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions a new finite element formulation aimed at the solution of problems involving strain localization is presented the proposed formulation incorporates displacement interpolated embedded localization lines results are shown to converge to an exact solution when the mesh is refined and also to be quite insensitive to mesh distortions

a toric extension of faltings diophantine approximation on abelian varieties for divisors on abelian varieties faltings established an optimal bound on the proximity of rational points to the same we extend this both to the quasiprojective category where the role of abelian varieties is played by their toroidal extensions and to holomorphic maps from the line proving along the way some wholly general dynamic intersection estimates in value distribution theory of independent interest

a thue equation with quadratic integers as variables in an earlier paper we determined all the solutions in z of a cubic thue equation with coefficients in a quadratic number field it is now shown that the method used there can be used to solve the more general problem of determining all the solutions of the thue equation in a ring of quadratic integers

on the complexity of the orbit problem we consider higher dimensional versions of kannan and lipton s orbit problem determining whether a target vector space may be reached from a starting point x under repeated applications of a linear transformation a answering two questions posed by kannan and lipton in the 1980s we show that when has dimension one this problem is solvable in polynomial time and when has dimension two or three the problem is in nprp

a note on divisible points of curves let c be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least 3 we partially answer a question posed by levin on points on c for which a non trivial power lies again on c our results have connections to zilber s conjecture on intersections with tori and yield to methods arising in transcendence theory and the theory of o minimal structures

on modular representations of bar q q arising from modular forms where g is the galois group gai i q and f is a finite field of characteristic i 3 suppose that p is modular of level n i e that it arises from a weight 2 newform of level dividing n and trivial nebentypus character then p is an odd representation the matrix p c where c is a complex conjugation in g has eigenvalues 1 1 since 1 and 1 are distinct in f p is absolutely irreducible and has a model over every subfield of f containing the set trace p we assume that f has been chosen so that it is generated by this set assume that p is a prime which exactly divides n we write p i n and restrict p to a decomposition group of g for the prime p view the restriction as a representation p p of gal t p qp we say that p is finite at p 34 or 35 w if there is a finite flat f vector space scheme h over zp for which the action of gal t p qp on the f vector space h 1 p gives pp i f p 9 is finite at p if and only if pp is unramified it is clear that p is finite at p whenever p is already modular of level n p in w of 34 cf 35 serre conjectured the converse i e that if p is finite then p is modular of level n p soon thereafter mazur 1 20 proved this conjecture in the case

solving thue equations without the full unit group the main problem when solving a thue equation is the computation of the unit group of a certain number field it is shown that the knowledge of a subgroup of finite index is in fact sufficient two examples linked with the primitive divisor problem for lucas and lehmer sequences are given la principale difficulte lors de la resolution d une equation de thue reside dans le calcul du groupe des unites du corps de nombres associe on montre qu il suffit en fait de connaitre un sous groupe d indice fini de ce groupe on donne deux exemples li

properties of discontinuous bifurcation solutions in elasto plasticity explicit expressions for the spectral properties of the bifurcation problem involving discontinuities for general elastic plastic materials are presented it then follows that the classical value of the critical hardening modulus derived by rice 1976 proc 14th iutam congr delft the netherlands pp 207 220 north holland amsterdam is in fact the only possible one furthermore from the spectral analysis it follows in a very straightforward fashion that bifurcation displaying elastic unloading on one side of the singular surface can never precede bifurcation with plastic loading on both sides of this surface explicit analytical results for the critical bifurcation directions and the corresponding hardening modulus are derived for non associated volumetric flow rules while the devialoric portion is associated the considered yield and potential functions may depend on all three stress invariants and may involve mixed isotropic and kinematic hardening the result obtained by rudnicki and rice 1975 j mech phys solids 23 371 394 for a drucker pruger material appear as a special case other criteria that are investigated arc those of coulomb and rankine less

the solution of triangularly connected decomposable form equations an algorithm is given to solve the equations of the title it generalizes an earlier algorithm to solve discriminant form equations an application is given to finding curves of genus 2 with good reduction outside a given finite set of primes and weierstrass points in given number fields

distribution of integral division points on the algebraic torus let k be a number field with algebraic closure overline k and let s be a finite set of places of k containing all the infinite ones let it gamma 0 be a finitely generated subgroup of mathbb g textup m overline k and let it gamma subset mathbb g textup m overline k be the division group attached to it gamma 0 here is an illustration of what we will prove in this article fix a proper closed subinterval i of 0 infty and a nonzero effective divisor d on mathbb g textup m which is not the translate of any torsion divisor on the algebraic torus mathbb g textup m by any point of it gamma with height belonging to i then we prove a statement which easily implies that the set of integral division points on mathbb g textup m with height near i i e the set of points of it gamma with standard absolute logarithmic weil height in j which are s integral on mathbb g textup m relative to d is finite for some fixed subinterval j of 0 infty properly containing i we propose a conjecture on the nondensity of integral division points on semi abelian varieties with prescribed height values which generalizes some previously known conjectures as well as this finiteness result for mathbb g textup m finally we also propose an analogous version for a dynamical system on mathbb p 1

on the structure of the whittaker sublocus of the moduli space of algebraic curves we prove the compactness of whittaker sublocus of moduli space of riemann surfaces complex algebraic curves this is the subset of points representing hyperelliptic curves which satisfy whittaker s conjecture on the uniformization of hyperelliptic curves via monodromy of fuchsian differential equations in the last part of the article we drive our attention to the statement made by r a rankin more than forty years ago to the effect that the conjecture has not been proved for any algebraic equation containing irremovable arbitrary constants we combine our compactness result with other facts coming from teichmuller theory to show that in the most natural interpretations of this sentence we can think of this is in fact impossible