on the eisenstein cohomology of arithmetic groups the cohomology h gamma e of an arithmetic subgroup gamma of a connected reductive algebraic group g defined over some algebraic number field f can be interpreted in terms of the automorphic spectrum of gamma with this framework in place there is a sum decomposition of the cohomology into the cuspidal cohomology i e classes represented by cuspidal automorphic forms for g and the so called eisenstein cohomology constructed as the span of appropriate residues or derivatives of eisenstein series attached to cuspidal automorphic forms on the levi components of proper parabolic f subgroups of g the main objective of this paper is to isolate a specific structural part in the eisenstein cohomology this pertains to regular eisenstein cohomology classes attached to cuspidal automorphic representations whose archimedean component is tempered it is shown that the cohomological degree of these classes is bounded from below by the constant q 0 g mathbb r 1 2 dim x g mathbb r rk g mathbb r rk k mathbb r where k mathbb r denotes a maximal compact subgroup of the real lie group g mathbb r where x g mathbb r is the associated symmetric space this investigation has various applications one of these is a vanishing result for the cohomology in the generic case i e where the representation determining the coefficient system e has regular highest weight in degrees below q 0 g mathbb r this is a sharp bound depending only on the underlying real lie group g mathbb r corollary 5 6 proposition 5 8 this result is supplemented by a qualitative structural result in the description of the cohomology in higher degrees by means of regular eisenstein cohomology classes

the cauchy problem for a generalized two component short pulse system with high order nonlinearities abstract this paper is concerned with the cauchy problem of a generalized short pulse system in the periodic setting local well posedness of this system is established in the besov space with s max 3 2 1 1 p it is then shown that the data to solution map is not uniformly continuous from b l r s b l r s to c 0 t b l r s c 0 t b l r s finally we provide a blow up result of the solution

propagation of ultra short optical pulses in cubic nonlinear media abstract we derive a partial differential equation that approximates solutions of maxwell s equations describing the propagation of ultra short optical pulses in nonlinear media and which extends the prior analysis of alterman and rauch phys lett a 264 2000 390 diffractive nonlinear geometric optics for short pulses preprint 2002 we discuss non rigorously conditions under which this approximation should be valid but the main contributions of this paper are 1 an emphasis on the fact that the model equation for short pulse propagation may depend on the details of the optical susceptibility in the wavelength regime under consideration 2 a numerical comparison of solutions of this model equation with solutions of the full nonlinear partial differential equation 3 a local well posedness result for the model equation and 4 a proof that in contrast to the nonlinear schrodinger equation which models slowly varying wavetrains this equation has no smooth pulse solutions which propagate with fixed shape and speed

the saito kurokawa lifting and functoriality certain nontempered liftings from pgl 2 pgl 2 to pgsp 4 are constructed using the theory of local and global theta lifts the resulting representations on pgsp 4 are the saito kurokawa representations the lifting is shown to be functorial under certain reasonable assumptions on the local langlands correspondence for pgsp 4

the cauchy problem for the fokas olver rosenau qiao equation abstract it is shown that the initial value problem for the fokas olver rosenau qiao equation forq is well posed in sobolev spaces h s s 5 2 in the sense of hadamard furthermore it is proved that the dependence on initial data is sharp i e the data to solution map is continuous but not uniformly continuous also peakon travelling wave solutions are derived on both the circle and the line and are used to prove that the solution map is not uniformly continuous in h s for s 3 2

the short pulse hierarchy we study a new hierarchy of equations containing the short pulse equation which describes the evolution of very short pulses in nonlinear media and the elastic beam equation which describes nonlinear transverse oscillations of elastic beams under tension we show that the hierarchy of equations is integrable we obtain the two compatible hamiltonian structures we construct an infinite series of both local and nonlocal conserved charges a lax description is presented for both systems for the elastic beam equations we also obtain a nonstandard lax representation

generalized deflated block elimination a stable algorithm is presented to solve a nonsingular bordered system of the form left begin array 20 c a b c t d end array right left begin array 20 c x y end array right left begin array 20 c f g end array right where b and c are n by m matrices and the n by n matrix a could be nearly singular with at most mu small singular values the algorithm needs only a solver for a and the solution to an m mu by m mu dense linear system it is thus well suited for problems for which a has easily exploitable structures and m mu ll n such as in continuation methods bifurcation problems and constrained optimization

the space change problem a space charge problem arising in modelling steady electric field in a neighborhood of an ionising conductor is considered mathematically the problem reduces to a system consisting of one elliptic equation and one nonlinear hyperbolic equation the existence of a weak solution is proved and its properties are studied in part rigorously and in part heuristically and numerically

periodic solutions of the short pulse model equation we develop a systematic procedure for constructing periodic solutions of the short pulse sp model equation that describes the propagation of ultrashort pulses in nonlinear media we first summarize a novel exact method of solution that consists of a hodograph transformation to convert the sp equation into the sine gordon sg equation we then exemplify some one and two phase periodic solutions of the sg equation for which the system of linear partial differential equations governing the inverse mapping can be integrated analytically to obtain periodic solutions of the sp equation in the form of the parametric representation we obtain a new class of two phase periodic solutions and investigate their properties in detail of particular interest is a nonsingular periodic solution that reduces to the breather solution in the long wave limit it may play an important role in studying the propagation of ultrashort pulses in a finite length optical system

an integrable coupled short pulse equation an integrable coupled short pulse csp equation is proposed for the propagation of ultra short pulses in optical fibers based on two sets of bilinear equations to a two dimensional toda lattice linked by a backlund transformation and an appropriate hodograph transformation the proposed csp equation is derived meanwhile its n soliton solutions are given by the casorati determinant in a parametric form the properties of one and two soliton solutions are investigated in detail same as the short pulse equation the two soliton solution turns out to be a breather type if the wave numbers are complex conjugate we also illustrate an example of soliton breather interaction

a new approach to the space charge problem in this paper we consider a model for the electric potential phi arising from the field due to a coronating electrode this results in the third order nonlinear partial differential equation nabla cdot delta phi nabla phi 0 which is of mixed type and we study here in two spatial dimensions we describe a new way of analysing this equation by constructing an orthogonal coordinate system phi psi and recasting the equation in terms of x y and delta phi as functions of phi psi it may be regarded as a synthesis of the method of characteristics for a hyperbolic equation and a hodograph method for laplace s equation this results in an effective algorithm for a numerical solution with finite differences

well posedness and continuity properties of the fornberg whitham equation in besov spaces abstract in this paper we prove well posedness of the fornberg whitham equation in besov spaces b 2 r s in both the periodic and non periodic cases this will imply the existence and uniqueness of solutions in the aforementioned spaces along with the continuity of the data to solution map provided that the initial data belongs to b 2 r s we also establish sharpness of continuity on the data to solution map by showing that it is not uniformly continuous from any bounded subset of b 2 r s to c t t b 2 r s furthermore we prove a cauchy kowalevski type theorem for this equation that establishes the existence and uniqueness of real analytic solutions and also provide blow up criterion for solutions

continuous properties of the data to solution map for a generalized μ camassa holm integrable equation this work studies a generalized type integrable equation with both quadratic and cubic nonlinearities the camassa holm and modified camassa holm equations are members of this family of equations it has been shown that the cauchy problem for this generalized camassa holm integrable equation is locally well posed for initial data u0 hs s 5 2 in this work we further investigate the continuity properties to this equation it is proved in this work that the data to solution map of the proposed equation is not uniformly continuous it is also found that the solution map is holder continuous in the hr topology when 0 r 5 2 in this work we further investigate the continuity properties to this equation it is proved in this work that the data to solution map of the proposed equation is not uniformly continuous it is also found that the solution map is holder continuous in the hr topology when 0 r s with holder exponent depending on both s and r

overconvergent hilbert modular forms we generalize the construction of the eigencurve by coleman mazur to the setting of totally real fields and show that a finite slope hilbert modular eigenform can be deformed into a one parameter family of finite slope eigenforms the key point is to show the overconvergence of the canonical subgroup and the complete continuity of the u p operator we deduce this form some general considerations in rigid analytic geometry

p adic banach spaces and families of modular forms let p be a prime cp the completion of an algebraic closure of the p adic numbers qp and k a nite extension of qp contained in cp let v be the valuation on cp such that v p 1 and let be the absolute value on cp such that x p v x for x cp suppose n is a positive integer prime to p let x1 np denote the modular curve over k which represents elliptic curves with 1 np structure and let up be the hecke operator on modular forms on x1 np which takes a form with q expansion n anq n to the modular form with q expansion n anpq n a modular form f is said to have slope q if there is a polynomial r t over cp such that r up f 0 and such that the newton polygon of r t has only one side and its slope is

eigenvarieties for reductive groups we develop the theory of overconvergent cohomology introduced by g stevens and we use it to give a construction of eigenvarieties associated to any reductive group g over q such that g r has discrete series we prove that the so called eigenvarieties are equidimensional and generically at over the weight space

integrable discretizations of the short pulse equation in this paper we propose integrable semi discrete and full discrete analogues of the short pulse sp equation the key construction is the bilinear form and determinant structure of solutions of the sp equation we also give the determinant formulas of n soliton solutions of the semi discrete and full discrete analogues of the sp equations from which the multi loop and multi breather solutions can be generated in the continuous limit the full discrete sp equation converges to the semi discrete sp equation and then to the continuous sp equation based on the semi discrete sp equation an integrable numerical scheme i e a self adaptive moving mesh scheme is proposed and used for the numerical computation of the short pulse equation

solitary wave solutions of the short pulse equation an exact nonsingular solitary wave solution of the schafer wayne short pulse equation is derived from the breather solution of the sine gordon equation by means of a transformation between these two integrable equations

category of nonlinear evolution equations algebraic structure and r matrix in this paper we deal with the category of nonlinear evolution equations nlees associated with the spectral problem and provide an approach for constructing their algebraic structure and r matrix first we introduce the category of nlees which is composed of various positive order and negative order hierarchies of nlees both integrable and nonintegrable the whole category of nlees possesses a generalized lax representation next we present two different lie algebraic structures of the lax operator one of them is universal in the category i e independent of the hierarchy while the other one is nonuniversal in the hierarchy i e dependent on the underlying hierarchy moreover we find that two kinds of adjoint maps are r matrices under the algebraic structures in particular the virasoro algebraic structures without a central extension of isospectral and nonisospectral lax operators can be viewed as reductions of our algebraic structure finally we give several concrete examples to illustrate

rigidity of p adic cohomology classes of congruence subgroups of gl n ℤ the main result of ash stevens 97 describes a framework for constructing padic analytic families of p ordinary arithmetic hecke eigenclasses in the cohomology of congruence subgroups of gl n q where the hecke eigenvalues vary p adic analytically as functions of the weight unanswered in that paper was the question of whether or not every p ordinary arithmetic eigenclass can be deformed in a positive dimensional family of arithmetic eigenclasses in this paper we make precise the notions of deformation and rigidity and investigate their properties rigidity corresponds to the non existence of positive dimensional deformations other than those coming from twisting by the powers of the determinant formal definitions are given in 3 when n 3 we will give a necessary and sufficient condition for a given p ordinary arithmetic eigenclass to be p adically rigid and we will use this criterion to give examples of p adically rigid eigenclasses on gl 3 more precisely we define a hecke eigenpacket to be a map from the hecke algebra to a coefficient ring which gives the hecke eigenvalues attached to a hecke eigenclass in the cohomology we then consider p adic analytic deformations of the hecke eigenpacket there are several advantages to this point of view as opposed to deforming the cohomology classes themselves for one thing a surjective map of coefficient modules need not induce a surjective map on the cohomology standard tools of algebra and geometry can be used to study the hecke algebra and the deformations of the eigenpackets moreover it is the hecke eigenvalues that carry the interesting arithmetic information attached to a cohomological eigenclass our main theoretical tool is the cohomology of congruence subgroups gl n z with coefficients in the space of bounded measures on the big cell of gl n see 2 in 4 we define the ring rord as a quotient of the hecke algebra acting on the p ordinary part of this cohomology in theorem 4 2 we prove that rord is universal in the sense that every p ordinary arithmetic hecke eigenpacket factors through rord we show that rord is a complete semilocal noetherian ring and study the geometry of its associated p adic rigid analytic space xord this leads

the l2 lefschetz numbers of hecke operators introduction 5 1 hecke operators and l2 cohomology 52 the spectral decomposition of cohomology 6 3 application of the trace formula 54 the functions pa t 5 5 proof of theorem 5 1 6 the main formula appendix bibliography