a numerical continuation method based on pade approximants a continuation algorithm is presented with a new predictor which is based on a rational representation of the solution path this algorithm belongs to the class of asymptotic numerical methods that connect perturbation techniques with a discretization principle without the use of a correction process several examples from shell buckling and from contact mechanics are analyzed to assess the efficiency and the reliability of the method

perfect powers expressible as sums of two cubes abstract let n 3 this paper is concerned with the equation a 3 b 3 c n which we attack using a combination of the modular approach via frey curves and galois representations with obstructions to the solutions that are of brauer manin type we shall show that there are no solutions in coprime non zero integers a b c for a set of prime exponents n having dirichlet density 28 219 44 928 0 628 and for a set of exponents n having natural density 1

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

perceptual compensation for speaker differences and for spectral envelope distortion this study asks whether perceptual mechanisms that compensate for the spectral envelope distortion of transmission channels also contribute to compensation for speaker differences subjects identified test words that were played after a carrier sentence in some conditions the carriers were synthesized with f1 in low and high frequency ranges and in others they were distorted by filters whose frequency response is the spectral envelope of one vowel minus the spectral envelope of another the filter minus e and its inverse were used test words were drawn from an tch to etch continuum carriers filtered by minus e and its inverse give a phoneme boundary difference indicating compensation for spectral envelope distortion a phoneme boundary difference also occurs between carriers with f1 in low and high ranges indicating compensation for speaker differences neither of these effects is reduced by playing the carrier backwards even though a measurement of the perceived naturalness of carriers is sharply reduced by this manipulation analysis of carriers synthesized with low and high f1 showed that they have different long term spectra and subsequent experiments used time stationary filters to alter this characteristic the results showed that the long term spectra of the carriers govern their influence on the identity of subsequent test sounds however measurements of perceptual confusions among the carriers and of perceived talker differences between carriers revealed that other time varying factors are more important for voice identification

on the modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

automorphy of some residually dihedral galois representations during the period this research was conducted jack thorne served as a clay research fellow

numerical fluid dynamics by 1755 euler was explicitly envisaging analytical fluid dynamics as a mathematical science that of integrating the euler lagrange equations but by 1915 at least seven different analytical models were being used to explain the behavior of real fluids from 1945 on von neumann was trying to arithmetize the approximate solution of the equations associated with these models with the help of large scale high speed computers thus he was envisaging numerical fluid dynamics as a mathematical science after briefly reviewing the gradual disintegration of euler s concept of analytical fluid dynamics as a mathematical science this survey article tries to assess the present status and future prospects of numerical fluid dynamics through a series of case studies

on serre s conjecture for mod l galois representations over totally real fields in 1987 serre conjectured that any mod two dimensional irre ducible odd representation of the absolute galois group of the rationals came from a modular form in a precise way we present a generalisation of this conjecture to 2 dimensional representations of the absolute galois group of a totally real eld where is unramied the hard work is in formulating an analogue of the weight part of serre s conjecture serre furthermore asked whether his conjecture could be rephrased in terms of a mod langlands philosophy using ideas of emerton and vign eras we formulate a mod local global principle for the group d where d is a quaternion algebra over a totally real eld split above and at 0 or 1 innite places and show how it implies the conjecture

serre s modularity conjecture ii we provide proofs of theorems 4 1 and 5 1 of khare and wintenberger invent math doi 10 1007 s00222 009 0205 7 2009

central auditory mechanisms of perceptual compensation for spectral envelope distortion the spectral envelope is a major determinant of the perceptual identity of many classes of sound including speech when sounds are transmitted from the source to the listener the spectral envelope is invariably and diversely distorted by factors such as room reverberation perceptual compensation for spectral envelope distortion was investigated here carrier sounds were distorted by spectral envelope difference filters whose frequency response is the spectral envelope of one vowel minus the spectral envelope of another the filter i minus q and its inverse were used subjects identified a test sound that followed the carrier the test sound was drawn from an an itch to qtch continuum perceptual compensation produces a phoneme boundary difference between i minus q and its inverse carriers were the phrase the next word is spoken by the same male speaker as the test sounds signal correlated noise derived from this phrase the same phrase spoken by a female speaker male and female versions played backwards and a repeated end point vowel the carrier and test were presented to the same ear to different ears and from different apparent directions by varying interaural time delay the results show that compensation is unlike peripheral phenomena such as adaptation and unlike phonetic perceptual phenomena the evidence favors a central auditory mechanism

serre s conjecture over f9 in this paper we show that an odd galois representation p i gal pq q gl2 f9 having nonsolvable image and satisfying certain local conditions at 3 and 5 is modular our main tools are ideas of taylor 21 and khare 10 which reduce the problem to that of exhibiting points on a hilbert modular surface which are defined over a solvable extension of q and which satisfy certain reduction properties as a corollary we show that hilbert blumenthal abelian surfaces with ordinary reduction at 3 and 5 are modular

the fermat type equations 2 or 3 solved through ℚ curves we solve the diophantine equations x 5 y 5 dz p with d 2 3 for a set of prime numbers of density 1 4 1 2 respectively the method consists in relating a possible solution to another diophantine equation and solving the later by using q curves and a generalized modular technique as in work of ellenberg and dieulefait jimenez along with some new techniques for eliminating newforms

hirzebruch zagier cycles and twisted triple product selmer groups let e be an elliptic curve over mathbb q and a another elliptic curve over a real quadratic number field we construct a mathbb q motive of rank 8 together with a distinguished class in the associated bloch kato selmer group using hirzebruch zagier cycles that is graphs of hirzebruch zagier morphisms we show that under certain assumptions on e and a the non vanishing of the central critical value of the twisted triple product l function attached to e a implies that the dimension of the associated bloch kato selmer group of the motive is 0 and the non vanishing of the distinguished class implies that the dimension of the associated bloch kato selmer group of the motive is 1 this can be viewed as the triple product version of kolyvagin s work on bounding selmer groups of a single elliptic curve using heegner points

on serre s uniformity conjecture for semistable elliptic curves over totally real fields let k be a totally real field and let s be a finite set of non archimedean places of k it follows from the work of merel momose and david that there is a constant b k s so that if e is an elliptic curve defined over k semistable outside s then for all p b k s the representation overline rho e p is irreducible we combine this with modularity and level lowering to show the existence of an effectively computable constant c k s and an effectively computable set of elliptic curves over k with cm e 1 cdots e n such that the following holds if e is an elliptic curve over k semistable outside s and p c k s is prime then either overline rho e p is surjective or overline rho e p sim overline rho e i p for some i 1 dots n

on icosahedral artin representations ii we prove that some new infinite families of odd two dimensional icosahedral representations of the absolute galois group of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i are modular and hence satsify the artin conjecture we also give an account of work of ramakrishna on lifting mod l galois representations to characteristic zero

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

1 n expansions in nonrelativistic quantum mechanics an extensive number of numerical computations of energy 1 n series using a recursive taylor series method are presented in this paper the series are computed to a high order of approximation and their behavior on increasing the order of approximation is examined

automorphic lifts of prescribed types we prove a variety of results on the existence of automorphic galois representations lifting a residual automorphic galois representation we prove a result on the structure of deformation rings of local galois representations and deduce from this and the method of khare and wintenberger a result on the existence of modular lifts of specified type for galois representations corresponding to hilbert modular forms of parallel weight 2 we discuss some conjectures on the weights of n dimensional mod p galois representations finally we use recent work of taylor to prove level raising and lowering results for n dimensional automorphic galois representations

the invariants of a genus one curve it was first pointed out by weil that we can use classical invariant theory to compute the jacobian of a genus one curve the invariants required for curves of degree n 2 3 4 were already known to the nineteenth century invariant theorists we have succeeded in extending these methods to curves of degree n 5 where although the invariants are too large to write down as explicit polynomials we have found a practical algorithm for evaluating them

fermat type equations of signature 13 13 p via hilbert cuspforms in this paper we prove that for p 13649 equations of the form x 13 y 13 cz p have no non trivial primitive solutions a b c such that 13 not mid c for an infinite family of values for c our method consists on relating a solution a b c to the previous equation to a solution a b c 1 of another diophantine equation with coefficients in mathbb q sqrt 13 then we attach to a b c 1 a frey curve e a b defined over mathbb q sqrt 13 that is not a mathbb q curve we prove a modularity result of independent interest for certain elliptic curves over totally real abelian number fields satisfying some local conditions at 3 this theorem in particular implies modularity of e a b this enables us to use level lowering results and apply the modular approach via hilbert cuspforms over mathbb q sqrt 13 to prove the non existence of a b c 1 and consequently of a b c

an artificial small perturbation parameter and nonlinear plate vibrations abstract nonlinear natural in plane vibrations of a rectangular plate are studied using three small parameters firstly the nonlinearity is assumed to be small then a solution to a problem of the zeroth order linear is sought in the form of an asymptotic series with respect to the ratio of stiffness characteristics for internal resonance vibration modes are coupled via an infinite system of nonlinear algebraic equations and the artificial small parameter approach is proposed to solve the obtained system analytical formulas for the amplitude frequency characteristics are derived and the solutions are compared with numerical results

solving fermat type equations x 5 y 5 dz p in this paper we are interested in solving the fermat type equations x 5 y 5 dz p where d is a positive integer and p a prime number 7 we describe a new method based on modularity theorems which allows us to improve all earlier results for this equation we finally discuss the present limits of the method by looking at the case d 3

the mean matters effects of statistically defined nonspeech spectral distributions on speech categorization adjacent speech and even nonspeech contexts influence phonetic categorization four experiments investigated how preceding sequences of sine wave tones influence phonetic categorization this experimental paradigm provides a means of investigating the statistical regularities of acoustic events that influence online speech categorization and reciprocally reveals regularities of the sound environment tracked by auditory processing the tones comprising the sequences were drawn from distributions sampling different acoustic frequencies results indicate that whereas the mean of the distributions predicts contrastive shifts in speech categorization variability of the distributions has little effect moreover speech categorization is influenced by the global mean of the tone sequence without significant influence of local statistical regularities within the tone sequence further arguing that the effect is strongly related to the average spectrum of the sequence notched noise spectral complements of the tone sequences produce a complementary effect on speech categorization lastly these effects are modulated by the number of tones in the acoustic history and the overall duration of the sequence but not by the density with which the distribution defining the sequence is sampled results are discussed in light of stimulus specific adaptation to statistical regularity in the acoustic input and a speculative link to talker normalization is postulated

modularity of certain potentially barsotti tate galois representations where is the reduction of p these results were subject to hypotheses on the local behavior of p at x i e the restriction of p to a decomposition group at x and to irreducibility hypotheses on 1p in this paper we build on the methods of 461 451 and 121 and relax the hypotheses on local behavior in particular we treat certain adic representations which are not semistable at x but potentially semistable we do this using results of 61 generalizing a theorem of ramakrishna 321 see fontaine mazur 22 131 for a slightly different point of view the results in 61 show that certain potentially barsotti tate deformation problems are smooth allowing us to define certain universal deformations for with the necessary galoistheoretic properties to apply wiles method to carry out the proof that these deformations are indeed realized in the cohomology of modular curves i e that the universal deformation rings are hecke algebras we need to identify the corresponding cohomology groups and prove they have the modular theoretic properties needed to apply wiles method as in 151 and 121 the identification is made by matching local behavior of automorphic representations and galois representations via the local langlands correspondence together with fontaine s theory at the prime we work directly with cohomology of modular curves instead of

darmon points on elliptic curves over number fields of arbitrary signature we present new constructions of complex and pp adic darmon points on elliptic curves over base fields of arbitrary signature we conjecture that these points are global and present numerical evidence to support our conjecture

asymptotic approach for non linear periodical vibrations of continuous structures abstract an asymptotic approach for determining periodic solutions of non linear vibration problems of continuous structures such as rods beams plates etc is proposed starting with the well known perturbation technique the independent displacement and frequency is expanded in a power series of a natural small parameter it leads to infinite systems of interconnected non linear algebraic equations governing the relationships between modes amplitudes and frequencies a non trivial asymptotic technique based on the introduction of an artificial small parameter is used to solve the equations an advantage of the procedure is the possibility to take into account a number of vibration modes as examples free longitudinal vibrations of a rod and lateral vibrations of a beam under cubically non linear restoring force are considered resonance interactions between different modes are investigated and asymptotic formulae for corresponding backbone curves are derived

recipes to fermat type equations of the form x r y r cz p we describe a strategy to attack infinitely many fermat type equations of signature r r p where r ge 7 is a fixed prime and p is a prime allowed to vary indeed to a solution a b c of x r y r cz p we will attach several frey curves e e a b defined over totally real subfields of mathbb q zeta r we prove modularity of all the frey curves and the exsitence of a constant constant m r depending only on r such that for all p m r the representations bar rho e p are absolutely irreducible along the way we also prove modularity of certain elliptic curves that are semistable at all v mid 3 finally we illustrate our methods by proving arithmetic statements about equations of signature 7 7 p among which we emphasize that using a multi frey technique we show there is some constant m such that if p m then the equation x 7 y 7 3z p has no non trivial primitive solutions

ring theoretic properties of certain hecke algebras the purpose of this article is to provide a key ingredient of w2 by establishing that certain minimal hecke algebras considered there are complete intersections as is recorded in w2 a method going back to mazur m allows one to show that these algebras are gorenstein but for the complete intersection property a new approach is required the methods of this paper are related to those of chapter 3 of w2 the methods of section 3 of this paper are based on a previous approach of one of us a w we would like to thank henri darmon fred diamond and gerd faltings for carefully reading the first version of this article gerd faltings has also suggested a simplification of our argument as well as of the argument of chapter 3 of w2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper r t would like to thank a w for his invitation to collaborate and for sharing his many insights into the questions considered r t would also like to thank princeton university universite de paris 7 and harvard university for their hospitality during this collaboration a w was supported by an nsf grant

galois representations attached to q curves and the generalized fermat equation a 4 b 2 c p we prove that the equation a 4 b 2 c p has no solutions in coprime positive integers when p 211 the main step is to show that for all sufficiently large primes p every q curve over an imaginary quadratic field k with a prime of potentially multiplicative reduction greater than 6 has a surjective mod p galois representation the bound on p depends on k and the degree of the isogeny between e and its galois conjugate but is independent of the choice of e the proof of this theorem combines geometric arguments due to mazur momose darmon and merel with an analytic estimate of the average special values of certain l functions

on a conjecture of conrad diamond and taylor we prove a conjecture of conrad diamond and taylor on the size of certain deformation rings parametrizing potentially barsotti tate ga lois representations to achieve this we extend results of breuil and m ezard classifying galois lattices in semistable representations in terms of strongly divisible modules to the potentially crystalline case in hodge tate weights 0 1 we then use these strongly divisible modules to compute the desired deformation rings as a corollary we obtain new results on the modularity of potentially barsotti tate representations

moduli of finite flat group schemes and modularity we prove that under some mild conditions a two dimensional p adic galois representation which is residually modular and potentially barsotti tate at p is modular this provides a more conceptual way of establishing the shimura taniyama weil conjecture especially for elliptic curves which acquire good reduction over a wildly ramified extension of 3 the main ingredient is a new technique for analyzing flat deformation rings it involves resolving them by spaces which parametrize finite flat group scheme models of galois representations introduction