nonlinear schrodinger equation with growing potential on infinite metric graphs abstract the paper deals with nonlinear schrodinger equations on infinite metric graphs we assume that the linear potential is infinitely growing we prove an existence and multiplicity result that covers both self focusing and defocusing cases furthermore under some additional assumptions we show that solutions obtained bifurcate from trivial ones we prove that these solutions are superexponentially localized our approach is variational and based on generalized nehari manifold

new exact analytical solutions for stokes first problem of maxwell fluid with fractional derivative approach the unsteady flow of an incompressible maxwell fluid with fractional derivative induced by a sudden moved plate has been studied using fourier sine and laplace transforms the obtained solutions for the velocity field and shear stress written in terms of generalized g functions are presented as sum of the similar newtonian solutions and the corresponding non newtonian contributions the non newtonian contributions as expected tend to zero for l 0 furthermore the solutions for ordinary maxwell fluid performing the same motion are obtained as limiting cases of general solutions and verified by comparison with previously known results finally the influence of the material and the fractional parameters on the fluid motion as well as a comparison among fractional maxwell ordinary maxwell and newtonian fluids is also analyzed by graphical illustrations

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

unsteady flow and heat transfer of a generalized maxwell fluid due to a hyperbolic sine accelerating plate this paper deals with the unsteady flow and heat transfer of a generalized maxwell fluid over a moving flat plate with variable temperature and hyperbolic sine velocity exact solutions are established for the velocity and temperature fields in terms of discrete fourier sine transform coupled with laplace transform for the fractional calculus graphs are sketched for values of parameters and associated dynamic characteristics are analyzed

reciprocity laws and galois representations recent breakthroughs suppose f x is a monic irreducible polynomial with integer coefficients if p is a prime number then reducing the coefficients of f x modulo p gives a new polynomial fp x which may be reducible we say that f x is split modulo p if fp x is the product of distinct linear factors example 1 1 1 the polynomial f x x2 1 is split modulo 5 since f5 x x 2 x 3 mod 5 but it is not split modulo 7 since f7 x is irreducible nor is it split modulo 2 since f2 x x 1 2 mod 2 has a repeated factor the first few p for which x2 1 is split modulo p are 5 13 17 29 37 41 53

translational flows of an oldroyd b fluid with fractional derivatives the objective of this paper is to study the unsteady flow of an oldroyd b fluid with fractional derivative model between two infinite coaxial circular cylinders using laplace and finite hankel transforms the motion of the fluid is produced by the inner cylinder that at time t 0 applies a time dependent longitudinal shear stress to the fluid velocity field and the adequate shear stress are presented in series form in terms of the generalized g and r functions the solutions that have been obtained satisfy all imposed initial and boundary conditions the corresponding solutions for ordinary oldroyd b fractional maxwell ordinary maxwell fractional second grade ordinary second grade and newtonian fluids performing the same motion are obtained as limiting cases of general solutions in particular the existing solutions for ordinary oldroyd b and second grade fluids are compared with the present solutions finally the influence of the pertinent parameters on the fluid motion as well as a comparison between models is underlined by graphical illustrations

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

the fontaine mazur conjecture for gl _2 introduction 641 1 breuil mezard conjecture and the p adic local langlands 644 1 1 the breuil mezard conjecture 644 1 2 review of colmez s functor 647 1 3 hilbert samuel multiplicities 650 1 4 representations and pseudo representations 653 1 5 gl2 qp representations mod p 657 1 6 local patching and multiplicities 661 1 7 from pseudo representations to representations 669 2 modularity via the breuil mezard conjecture 675 2 1 quaternionic forms 675 2 2 global patching and multiplicities 677 2 3 the breuil mezard conjecture 686 addendum to ki 1 687 references 689

infinitely ramified galois representations in this paper we show how to construct for most p 5 two types of surjective representations p gq gal q q gl2 zp that are ramified at an infinite number of primes the image of inertia at almost all of these primes will be torsion free the first construction is unconditional the catch is that we cannot say whether p igp gal qp q is crystalline or even potentially semistable the second construction assumes the generalized riemann hypothesis grh with this assumption we can further arrange that p igp is crystalline at p we remark that infinitely ramified reducible representations have been previously constructed by more elementary means

unsteady flow of a generalized maxwell fluid with fractional derivative due to a constantly accelerating plate the velocity field and the adequate shear stress corresponding to the unsteady flow of a generalized maxwell fluid are determined using fourier sine and laplace transforms they are presented as a sum of the newtonian solutions and the corresponding non newtonian contributions the similar solutions for maxwell and newtonian fluids performing the same motion are obtained as limiting cases of our general results graphical illustrations show that the velocity profiles corresponding to a generalized maxwell fluid are going to that for an ordinary maxwell fluid if a 1

hardy littlewood theory for semigroups soit t t t 0 un semi groupe symetrique sousmarkovien et soit n 2 alors on a les proprietes equivalentes a dim t t n b il existe c 2 0 s t t t f c 2 t n 2 f 1 f l 1 x t 0 en fait a b meme dans le cas n 2

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

flow heat and mass transfer of three dimensional fractional maxwell fluid over a bidirectional stretching plate with fractional fourier s law and fractional fick s law abstract this paper investigates the flow heat and mass transfer of fractional maxwell fluid over a bidirectional stretching sheet and research is carried out for the three dimensional case by analogy with the constitutive equation of fractional maxwell fluid fractional derivative is introduced into fourier s law and fick s law meanwhile the magnetic field and chemical reaction are considered furthermore the stretching speeds are not only power law dependent on time but also power law dependent on the distance of each space direction combining with l1 algorithm a newly finite difference method is developed to solve the governing equations and convergence of the method is verified by constructing a numerical example the influences of various physical parameters on velocity temperature and concentration are analyzed through three dimensional graphs the velocity fractional parameter presents an interesting effect on the velocity when the powers of each space direction coincide the smaller the velocity fractional parameter is the thinner the velocity boundary layer is on the contrary larger velocity fractional parameter results in decreasing velocity at first and then increasing for different powers of each space direction furthermore fractional fourier s law leads to more obvious heat transfer phenomena which is similar with the effect of fractional fick s law on the mass transfer

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

the dual cheeger constant and spectra of infinite graphs abstract in this article we study the top of the spectrum of the normalized laplace operator on infinite graphs we introduce the dual cheeger constant and show that it controls the top of the spectrum from above and below in a similar way as the cheeger constant controls the bottom of the spectrum moreover we show that the dual cheeger constant at infinity can be used to characterize that the essential spectrum of the normalized laplace operator shrinks to one point

flow of a generalized oldroyd b fluid due to a constantly accelerating plate the velocity field and the adequate shear stress corresponding to the unsteady flow of a generalized oldroyd b fluid due to a constantly accelerating plate have been established using fourier sine and laplace transforms in order to avoid lengthy calculations of residues and contour integrals the discrete inverse laplace transform method has been used the solutions that have been obtained presented under integral and series forms in terms of the generalized g and r functions satisfy all imposed initial and boundary conditions the similar solutions for generalized maxwell fluids as well as those for oldroyd b maxwell and newtonian fluids are obtained as limiting cases of our general solutions

unsteady mhd couette flow of a generalized oldroyd b fluid with fractional derivative this paper presents an analytical study for the magnetohydrodynamic mhd flow of a generalized oldroyd b fluid the fractional calculus approach is used to establish the constitutive relationship model of a viscoelastic fluid exact analytic solutions for the velocity field and shear stress in terms of fox h function are obtained by means of the laplace transform the influence of the relaxation and retardation times the orders of the time fractional derivative and the magnetic body force on the velocity and shear stress are analyzed it is shown that the ordinary oldroyd b fluid generalized second grade fluid and maxwell fluid are the limiting cases of the presented results

on diagonal lower estimate of heat kernels on graphs abstract we establish a new on diagonal lower estimate of continuous time heat kernels for large time on graphs to achieve the goal we first introduce an upper estimate of heat kernels in natural graph metric then use the upper estimate and the volume growth condition to show the validity of the on diagonal lower estimate

a class of descendant p groups of order p9 and higman s porc conjecture abstract we present a parametrised family of groups of order p 8 one for each prime p 3 and show that the number of descendants of order p 9 is not porc

periodic unidirectional flows of a viscoelastic fluid with the fractional maxwell model exact analytic solutions for the flows of viscoelastic fluids are constructed the flows are induced by general periodic oscillations of a plate based on the flow conditions described three flow situations are solved by fourier transform further the fractional calculus approach is taken into account in the constitutive relationship of viscoelastic fluid model finally some interesting flows caused by certain special oscillations are also established

analytical and numerical solutions of the unsteady 2d flow of mhd fractional maxwell fluid induced by variable pressure gradient abstract the nonlocal property of the fractional derivative can supply more precise mathematical models for depicting flow dynamics of complex fluid which cannot be modelled appropriately by normal integer order differential equations this paper studies the analytical and numerical methods of unsteady 2d flow of magnetohydrodynamic mhd fractional maxwell fluid in a rectangular pipe driven by variable pressure gradient the governing equation is formulated with caputo time dependent fractional derivatives whose orders are distributed in interval 0 2 a challenge is to firstly obtain the exact solution by combining modified separation of variables method with mikusinski type operational calculus meanwhile the numerical solution is also obtained by the implicit finite difference method whose validity has been confirmed by the comparison with the exact solution constructed different to the most classical works both the stability and convergence analysis of two dimensional multi term time fractional momentum equation are derived based on numerical analysis the results show that the velocity increases with the rise of the fractional parameter and relaxation time while an increase in the values of hartmann number leads to a slower velocity in the rectangular pipe

the time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems abstract in this paper a time fractional fourier law is obtained from fractional calculus according to the fractional fourier law a fractional heat conduction equation with a time fractional derivative in the general orthogonal curvilinear coordinate system is built the fractional heat conduction equations in other orthogonal coordinate systems are readily obtainable as special cases in addition we obtain the solution of the fractional heat conduction equation in the cylindrical coordinate system in terms of the generalized h function using integral transformation methods the fractional heat conduction equation in the case 0 1 interpolates the standard heat conduction equation 1 and the localized heat conduction equation 0 finally numerical results are presented graphically for various values of order of fractional derivative

on serre s conjecture for 2 dimensional mod p representations of gal q q we prove the existence in many cases of minimally ramified p adic lifts of 2 dimensional continuous odd absolutely irreducible mod p representations of the absolute galois group of q it is predicted by serre s conjecture that such representations arise from newforms of optimal level and weight

minorations pour les chaines de markov unidimensionnelles we prove a space time lower estimate for reversible random walks on graphs having linear growth this estimate complements a similar upper bound that follows from a work of hebisch and saloff coste

analytical solutions of multi term time fractional differential equations and application to unsteady flows of generalized viscoelastic fluid this paper derives analytical solutions for a class of new multi term fractional order partial differential equations which include the terms for spatial diffusion time fractional diffusion multi term and reaction these models can be used to describe the nonlinear relationship between the shear stress and shear rate of generalized viscoelastic oldroyd b fluid and burgers fluid by using a modified separation of variables method the governing fractional order partial differential equations are transformed into a set of fractional order ordinary differential equations mikusiski type operational calculus is then employed to obtain the exact solutions of the linear fractional ordinary differential equations with constant coefficients the solutions are expressed in terms of multivariate mittag leffler functions different situations for the unsteady flows of generalized oldroyd b fluid and burgers fluid due to a moving plate are considered via examples integral representations of the solutions are presented it is shown that the presented results reduce to the corresponding results for classical navierstokes oldroyd b maxwell and second grade fluids as special cases

galois representations with conjectural connections to arithmetic cohomology in this paper we extend a conjecture of a ash and w sinnott relating niveau 1 galois representations to the mod p cohomology of congruence subgroups ofsln z to include galois representations of higher niveau we then present computational evidence for our conjecture in the case n 3 in the form of three dimensional galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture our examples include galois representations with nontrivial weight and level as well as irreducible three dimensional representations that are in no obvious way related to lower dimensional representations in addition we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture

mhd flow and heat transfer of a generalized burgers fluid due to an exponential accelerating plate with the effect of radiation this paper presents a research for the mhd flow and heat transfer of an incompressible generalized burgers fluid due to an exponential accelerating plate with the effect of radiation the fractional calculus approach is used to establish the constitutive relationship of the viscoelastic fluid exact analytic solutions are obtained for the velocity and temperature fields in integral and series form in terms of the g function by means of fourier sine transform and laplace transform moreover the figures are plotted to show the effects of different parameters on the velocity and temperature fields

on the critical set for fujita type blow up of solutions to the discrete laplacian parabolic equations with nonlinear source on networks abstract in this paper we are interested in long time behaviors of solutions to the discrete laplacian parabolic equations u t u f u with nonnegative and non trivial initial data in particular we assume that the function f is convex only on a short interval and f s f f s for 0 1 s 0 and we present a critical set depending on the function in the sense that if f is in the critical set then solutions blow up in finite time for any initial data and if not then solutions are global or blow up according to the size of initial data

chtoucas de drinfeld et correspondance de langlands on demontre la correspondance de langlands pour gl r sur les corps de fonctions la preuve generalise celle de drinfeld en rang 2 elle consiste a realiser la correspondance en rang r dans la cohomologie l adique des varietes modulaires de chtoucas de drinfeld de rang r

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

li yau inequality on graphs abstractwe prove the li yau gradient estimate for the heat kernel on graphs the onlyassumption is a variant of the curvature dimension inequality which is purely local and can be considered as a new notion of curvature for graphs we compute thiscurvature for lattices and trees and conclude that it behaves more naturally than thealready existing notions of curvature moreover we show that if a graph has non negative curvature then it has polynomial volume growth we also derive harnack inequalities and heat kernel bounds from the gradient esti mate and show how it can be used to strengthen the classical buser inequality relatingthe spectral gap and the cheeger constant of a graph 1 introduction and main ideas in their celebrated work 15 li and yau proved an upper bound on the gradient ofpositive solutions of the heat equation in its simplest form for an n dimensional com pact manifold with non negative ricci curvature the li yau gradient estimate statesthat a positive solution uof the heat equation

on nonexistence of global solutions for a semilinear heat equation on graphs abstract let g v e be a simple finite connected weighted graph satisfying curvature condition c d e n 0 and polynomial volume growth v x r c 0 r m be the normalized laplacian in this paper we prove that the semilinear heat equation u t u u 1 on g has no non negative global solutions for any bounded non negative and non trivial initial value in the case of m 2 the obtained result provides a significant complement to the work that was done recently by lin and wu 2017 concerning the existence and nonexistence of global solutions for the semilinear heat equation on graphs

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

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

analysis of the physical laplacian and the heat flow on a locally finite graph we study the physical laplacian and the corresponding heat flow on an infinite locally finite graph with possibly unbounded valence

oscillating flow of a viscoelastic fluid in a pipe with the fractional maxwell model the unidirectional oscillating flow of a viscoelastic fluid with the fractional maxwell model is studied the flow is produced by a periodic pressure gradient in an infinite straight pipe exact solutions are obtained in the time and frequency domains by using fourier transform the fractional maxwell model exhibits resonance phenomena similar to that of the maxwell model however the number of the resonance peaks the amplitudes of the velocity enhancement and the frequencies where resonance occurs are totally different the amplitudes of resonance peaks decay rapidly with frequency in addition a critical dimensionless radius is found for the fractional maxwell model below the critical radius the resonance behavior is weakened but above this value the enhancement is strengthened