pointwise estimates and stability for dispersive diffusive shock waves we study the stability and pointwise behavior of perturbed viscous shock waves for a general scalar conservation law with constant diffusion and dispersion along with the usual lax shocks such equations are known to admit undercompressive shocks we unify the treatment of these two cases by introducing a new wave tracking method based on instantaneous projection giving improved estimates even in the lax case another important feature connected with the introduction of dispersion is the treatment of a non sectorial operator an immediate consequence of our pointwise estimates is a simple spectral criterion for stability in all l p norms p 1 for the lax case and p 1 for the undercompressive case our approach extends immediately to the case of certain scalar equations of higher order and would also appear suitable for extension to systems

organizing the arithmetic of elliptic curves abstract suppose that e is an elliptic curve defined over a number field k p is a rational prime and k is the maximal z p power extension of k in previous work b mazur k rubin elliptic curves and class field theory in ta tsien li ed proceedings of the international congress of mathematicians icm 2002 vol ii higher education press beijing 2002 pp 185 195 b mazur k rubin pairings in the arithmetic of elliptic curves in j cremona et al eds modular curves and abelian varieties progress in mathematics vol 224 2004 pp 151 163 we discussed the possibility that much of the arithmetic of e over k i e the mordell weil groups and their p adic height pairings the shafarevich tate groups and their cassels pairings over all finite extensions of k in k can be described efficiently in terms of a single skew hermitian matrix with entries drawn from the iwasawa algebra of k k in this paper using work of nekovar j nekovar selmer complexes preprint available at http www math jussieu fr nekovar pu we show that under not too stringent conditions such an organizing matrix does in fact exist we also work out an assortment of numerical instances in which we can describe the organizing matrix explicitly

on the iwasawa invariants of elliptic curves suppose that e is an elliptic curve defined over q and that p is a prime where e has good ordinary reduction under the assumption that e is modular one can define nonnegative integers e alg e anal e and anal e the algebraic iwasawa invariants alg e and e are defined in terms of the structure of the p primary subgroup sele q p of the selmer group for e over the cyclotomic zp extension q of q the definition of the analytic invariants e and anal e is in terms of the p adic l function for e constructed by mazur and swinnerton dyer msd74 we will recall these definitions below our purpose in this article is to prove in certain cases that e anal e 0 and that alg e anal e these equalities together with a deep theorem of kato imply the main conjecture for e over q in this introduction we will discuss the nature of our results and give an outline of the proofs for the case of modular elliptic curves we want to point out however that the theorems proven in the text apply to modular forms as well as to elliptic curves with multiplicative reduction at p we will not attempt to state the most general versions here if k is any algebraic extension of q then the selmer group for e over k is a certain subgroup of h gk e q tors where gk gal q k the selmer group fits into an exact sequence 0 e k q z sele k xe k

the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

asymptotic behavior near planar transition fronts for the cahn hilliard equation abstract we consider the asymptotic behavior of perturbations of planar wave solutions arising in the cahn hilliard equation in space dimensions d 2 such equations are well known to arise in the study of spinodal decomposition a phenomenon in which rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into regions in which one component or the other is dominant with these regions separated by steep transition layers a critical feature of the cahn hilliard equation in one space dimension is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability the analysis of planar waves in multiple space dimensions is further complicated by the fact that the leading eigenvalue for this linearized operator leading in the case of stability moves into the negative real half plane with cubic scaling 3 where r d 1 denotes a fourier variable associated with spatial components transverse to the planar wave under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from planar wave solutions establishing asymptotic stability for initial perturbations decaying at an appropriate algebraic rate in an l 1 norm of the transverse variables

computational studies for the stochastic landau lifshitz gilbert equation the stochastic landau lifshitz gilbert equation describes the thermally induced dynamics of magnetic moments in ferromagnetic materials solutions of this highly nonlinear stochastic pde are unit vector fields and satisfy an energy estimate these are crucial properties to construct a convergent discretization in space and time we propose a convergent finite element approximation of the problem based on the midpoint rule the numerical scheme preserves the underlying properties of the continuous problem further we construct a robust and efficient newton multigrid solver for the solution of the nonlinear systems associated with the discretized problems at each time level computational studies show the optimal convergence behavior of the scheme in the case of smooth solutions long time dynamics for finite ensembles of spins evidence the ergodicity of an invariant measure of the continuum model numerical experiments in two dimensions demonstrate pathwise finite time blow up behavior of the solution w

spectral analysis of planar transition fronts for the cahn hilliard equation abstract we consider the spectrum of the linear operator that arises upon linearization of the cahn hilliard equation in dimensions d 2 about a planar transition front a solution that depends on only one distinguished space variable and that has different values at in previous work the author has established conditions on this spectrum under which such planar transition fronts are asymptotically stable and we verify here that those conditions hold for all such waves arising in a general form of the cahn hilliard equation

diagonal cycles and euler systems ii the birch and swinnerton dyer conjecture for hasse weil artin l functions this article establishes new cases of the birch and swinnerton dyer conjecture in analytic rank 0 for elliptic curves over mathbb q viewed over the fields cut out by certain self dual artin representations of dimension at most 4 when the associated l function vanishes to even order ge 2 at its central point two canonical classes in the corresponding selmer group are constructed and shown to be linearly independent assuming the non vanishing of a garrett hida p adic l function at a point lying outside its range of classical interpolation the key tool for both results is the study of certain p adic families of global galois cohomology classes arising from gross kudla schoen diagonal cycles in a tower of triple products of modular curves

full discretization of semilinear stochastic wave equations driven by multiplicative noise a fully discrete approximation of the semilinear stochastic wave equation driven by multiplicative noise is presented a standard linear finite element approximation is used in space and a stochastic trigonometric method is used for the temporal approximation this explicit time integrator allows for mean square error bounds independent of the space discretization and thus does not suffer from a step size restriction as in the often used stormer verlet leapfrog scheme furthermore it satisfies an almost trace formula i e a linear drift of the expected value of the energy of the problem numerical experiments are presented and confirm the theoretical results

asymptotic behavior near transition fronts for equations of generalized cahn hilliard form we consider the asymptotic behavior of perturbations of standing wave solutions arising in evolutionary pde of generalized cahn hilliard form in one space dimension such equations are well known to arise in the study of spinodal decomposition a phenomenon in which the rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into its two components with their concentrations separated by sharp transition layers motivated by work of bricmont kupiainen and taskinen 5 we regard the study of standing waves as an interesting step toward understanding the dynamics of these transitions a critical feature of the cahn hilliard equation is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from standing wave solutions establishing phase asymptotic orbital stability for initial perturbations decaying with appropriate algebraic rate

convergence of finite element solutions of stochastic partial integro differential equations driven by white noise numerical approximation of a stochastic partial integro differential equation driven by a space time white noise is studied by truncating a series representation of the noise with finite element method for spatial discretization and convolution quadrature for time discretization sharp order convergence of the numerical solutions is proved up to a logarithmic factor numerical examples are provided to support the theoretical analysis

p adic l functions and p adic periods of modular forms let e be an elliptic curve which is defined over q and has stable reduction modulo a given prime p assuming that e is modular one can associate to e a p adic l function lp e s see mz swd a v vi mz t t for its construction in various cases this function is defined by a certain interpolation property and is analytic for sezp in this paper we will assume that e has split multiplicative reduction at p under this assumption the interpolation property implies that lp e 1 0 we will prove a formula for ep e 1 which was discovered experimentally by mazur tate and teitelbaum mz t t by tate s p adic urfiformization theory there is a p adic integer q epzp which we refer to as the tate period for e and a p adic analytic isomorphism

two fully discrete schemes for fractional diffusion and diffusion wave equations with nonsmooth data we consider initial boundary value problems for the subdiffusion and diffusion wave equations involving a caputo fractional derivative in time we develop two fully discrete schemes based on the piecewise linear galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward euler method second order backward difference method and establish error estimates optimal with respect to the regularity of problem data these two schemes are first and second order accurate in time for both smooth and nonsmooth data extensive numerical experiments for two dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity

variation of iwasawa invariants in hida families let gq gl2 k be an absolutely irreducible modular galois representation over a finite field k of characteristic p assume further that is p ordinary and p distinguished in the sense that the restriction of to a decomposition group at p is reducible and non scalar the hida family h of is the set of all p ordinary p stabilized newforms f with mod p galois representation isomorphic to if is unramified at p then one must also fix an unramified line in and require that the ordinary line of f reduces to this fixed line these newforms are a dense set of points in a certain p adic analytic space of overconvergent eigenforms consisting of an intersecting system of branches i e irreducible components t a indexed by the minimal primes a of a certain hecke algebra to each modular form f h one may associate the iwasawa invariants an f an f alg f and alg f the analytic resp algebraic invariants are the number of zeroes of the p adic l function resp of the characteristic power series of the dual of the selmer group of f while the invariants are the exponents of the powers of p dividing the same objects in this paper we prove the following results on the behavior of these iwasawa invariants as f varies over h

module de congruences pour gl 2 d un corps imaginaire quadratique et theorie d iwasawa d un corps cm biquadratique

a numerical method for a partial integro differential equation a method is considered for the integration in time of a partial integro differential equation the discretization technique employed is patterned after an idea of ch lubich error bounds are derived for both smooth and nonsmooth initial data

dirichlet and neumann eigenvalue problems on domains in euclidean spaces abstract we obtain here some inequalities for the eigenvalues of dirichlet and neumann value problems for general classes of operators or system of operators acting in l 2 or l 2 c m r d d 1

pointwise green s function bounds for multidimensional scalar viscous shock fronts we determine detailed pointwise bounds for the green s function of the linearized operator about a multidimensional scalar viscous shock front these extend the pointwise semigroup methods introduced by howard and zumbrun in the one dimensional case to multidimensions sharpening lp estimates obtained by goodman and miller using a weighted norm approach moreover our results apply to shocks of arbitrary strength as previous results did not as described in a companion paper the bounds we obtain are sufficient to give a straightforward treatment of the nonlinear lp asymptotic behavior of the front under small perturbation the analysis of the multidimensional case involves several new features not found in the one dimensional case concerned with the geometry of propagating signals

pointwise green s function estimates toward stability for multidimensional fourth order viscous shock fronts abstract for the case of multidimensional viscous conservation laws with fourth order smoothing only we develop detailed pointwise estimates on the green s function for the linear fourth order convection equation that arises upon linearization of the conservation law about a viscous planar wave solution as in previous analyses in the case of second order smoothing our estimates are sufficient to establish that spectral stability implies nonlinear stability though the full development of this result will be considered in a companion paper

some remarks on the two variable main conjecture of iwasawa theory for elliptic curves without complex multiplication we establish several results towards the two variable main conjecture of iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields namely i the existence of an appropriate p adic l function building on works of hida and perrin riou ii the basic structure theory of the dual selmer group following works of coates hachimori venjakob et al and iii the implications of dihedral or anticyclotomic main conjectures with basechange the result of i is deduced from the construction of hida and perrin riou which in particular is seen to give a bounded distribution the result of ii allows us to deduce a corank formula for the p primary part of the tate shafarevich group of an elliptic curve in the z p 2 extension of an imaginary quadratic field finally iii allows us to deduce a criterion for one divisibility of the two variable main conjecture in terms of specializations to cyclotomic characters following a suggestion of greenberg as well as a refinement via basechange c 2011 elsevier inc all rights reserved

spectral analysis for transition front solutions in multidimensional cahn hilliard systems abstract we consider the spectrum associated with the linear operator obtained when a cahn hilliard system on r n is linearized about a planar transition front solution in the case of single cahn hilliard equations on r n it s known that under general physical conditions the leading eigenvalue moves into the negative real half plane at a rate 3 where is the fourier transform variable corresponding with components transverse to the wave moreover it has recently been verified that for single equations this spectral behavior implies nonlinear stability in the current analysis we establish that the same cubic rate law holds for a broad range of multidimensional cahn hilliard systems the analysis of nonlinear stability will be carried out separately

two variable p adic l functions attached to eigenfamilies of positive slope consider a classical cusp eigenform f n 1 a n f q n of weight k 2 for 0 n with a dirichlet character mod n and let l f s n 1 n a n f n s denote the l function of f twisted with an arbitrary dirichlet character for a prime number p 5 consider a family of cusp eigenforms f k of weight k k f k n 1 a n f k q n containing f f k such that the fourier coefficients a n f k are given by certain p adic analytic functions k a n f k the purpose of this paper is to construct a two variable p adic l function attached to coleman s family f k of cusp eigenforms of a fixed positive slope v p p 0 where p p k is an eigenvalue which depends on k of the atkin operator u u p our p adic l function interpolates the special values l f k s at points s k with s 1 2 k 1 we give a construction using the rankin selberg method and the theory of p adic integration on a profinite group y with values in an affinoid k algebra a where k is a fixed finite extension of q p our p adic l functions are p adic mellin transforms of certain a valued measures in their turn such measures come from eisenstein distributions with values in certain banach a modules m m n a of families of overconvergent forms over a

galerkin finite element methods for stochastic parabolic partial differential equations we study the finite element method for stochastic parabolic partial differential equations driven by nuclear or space time white noise in the multidimensional case the discretization with respect to space is done by piecewise linear finite elements and in time we apply the backward euler method the noise is approximated by using the generalized l2 projection operator optimal strong convergence error estimates in the l2 and dot h 1 norms with respect to the spatial variable are obtained the proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem the computational analysis and numerical example are given

fine selmer groups of elliptic curves over p adic lie extensions the aim of this paper is to discuss variants for elliptic curves of some deep conjectures of classical cyclotomic iwasawa theory let f be a finite extension of q p an odd prime number and f cyc the cyclotomic zp extension of q let k f cyc denote the maximal unramified abelian p extension of f cyc in which every prime of f cyc above p splits completely and define y f cyc to be the galois group of k f cyc over f cyc put g f cyc f and write for the iwasawa algebra of iwasawa proved that y f cyc is always a finitely generated torsion module and he conjectured 12 that in fact y f cyc is always a finitely generated zp module at present this conjecture has only been proven when f is abelian over q see 5 26 perhaps surprisingly it does not seem to have been pointed out in the literature that there is a precise analogue of iwasawa s conjecture for elliptic curves over f cyc let e be an elliptic curve over f and let

nonlinear stability for multidimensional fourth order shock fronts we consider the question of stability for planar wave solutions that arise in multidimensional conservation laws with only fourth order regularization such equations arise for example in the study of thin films for which planar waves correspond to fluid coating a pre wetted surface an interesting feature of these equations is that both compressive and undercompressive planar waves arise as solutions compressive or undercompressive with respect to asymptotic behavior relative to the un regularized hyperbolic system and numerical investigation by bertozzi munch and shearer indicates that undercompressive waves can be nonlinearly stable proceeding with pointwise estimates on the green s function for the linear fourth order convection regularization equation that arises upon linearization of the conservation law about the planar wave solution we establish that under general spectral conditions such as appear to hold for shock fronts arising in our motivating thin films equations compressive waves are stable for all dimensions d 2 and undercompressive waves are stable for dimensions d 3 in the special case d 1 compressive waves are stable under a very general spectral condition we also consider an alternative spectral criterion valid for example in the case of constant coefficient regularization for which we can establish nonlinear stability for compressive waves in dimensions d 3 and undercompressive waves in dimensions d 5 the case of stability for undercompressive waves in the thin films equations for the critical dimensions d 1 and d 2 remains an interesting open problem

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

correction of high order bdf convolution quadrature for fractional evolution equations we develop proper correction formulas at the starting k 1 steps to restore the desired k th order convergence rate of the k step bdf convolution quadrature for discretizing evolution equations

on the hida deformations of fine selmer groups abstract in this paper we study the fine selmer group of p adic galois representations and their deformations we show that for an infinite family of elliptic cuspforms if the invariant of the dual fine selmer group is zero for one member of the family then the same holds for all the other members further the invariants are equal for all but finitely many members in the family

on certain zeta functions attached to two hilbert modular forms i the case of hecke characters to make our exposition smooth let us first introduce some notational conventions for an algebraic number field k of finite degree we denote by jk the set of all embeddings of k into c and by ik the free z module generated by the elements of jk we then put rik ik0 r and cik ik0 c if p eapau e ik with a e jk and po e z we put xp fl xa pfor 0 x e k this is meaningful for p e cik if xa are all real and positive throughout the paper we denote by d the complex upper half plane and by f a totally real algebraic number field of degree n now the zeta function to be studied in this paper when suitably specialized has the form

numerical approximation of some linear stochastic partial differential equations driven by special additive noises this paper is concerned with the numerical approximation of some linear stochastic partial differential equations with additive noises a special representation of the noise is considered and it is compared with general representations of noises in the infinite dimensional setting convergence analysis and error estimates are presented for the numerical solution based on the standard finite difference and finite element methods the effects of the noises on the accuracy of the approximations are illustrated results of the numerical experiments are provided

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

strong order of convergence of a fully discrete approximation of a linear stochastic volterra type evolution equation in this paper we investigate a discrete approximation in time and in space of a hilbert space valued stochastic process u t t 0 t satisfying a stochastic linear evolution equation with a positive type memory term driven by an additive gaussian noise the equation can be written in an abstract form as du t 0 b t s au s ds dt dw t 0 t u 0 u0 h where w is a q wiener process on h l d and where the main example of b we consider is given by b t t 1 0 0 such that a has finite trace and that q is bounded from h into d a for some real with 1 1 the discretization is achieved via an implicit euler scheme and a laplace transform convolution quadrature in time parameter t t n and a standard continuous finite element method in space parameter h let un h be the discrete solution at t n t we show that e un h u t 1 2 o h t for any 1 1 2 and 1 1 m kovacs department of mathematics and statistics university of otago po box 56 dunedin 9054 new zealand e mail mkovacs maths otago ac nz j printems laboratoire d analyse et de mathematiques appliquees cnrs umr 8050 61 avenue du general de gaulle universite paris est 94010 creteil france e mail printems u pec fr 2 mihaly kovacs jacques printems