boundedness of automorphic l functions in vertical strips the purpose of this paper is to prove the boundedness in vertical strips of finite width for all the l functions that appear in constant terms of eisenstein series under a certain natural assumption on local normalized intertwining operators as a corollary we prove the boundedness for a number of important l functions among them the symmetic cube and triple product l functions attached to cusp forms on gl 2 as well as several rankin selberg product l functions where the local assumption was alredy proved to be valid in each case under the same assumption the finiteness of poles on all of c for all of these l functions is also proved a necessary step in the proof of boundedness in vertical strips

doubly lexical orderings of matrices every matrix has a doubly lexical ordering an ordering of the rows and columns so that the row vectors are lexically or lexicographically increasing and the column vectors are lexically increasing every graph has a lexical ordering a vertex ordering making the neighbourhood matrix doubly lexical an almost linear time doubly lexical ordering algorithm is given doubly lexical orderings unify the orderings characterizing certain classes of matrices and graphs including totally balanced matrices subtree matrices and chordal graphs

matrix sandwich problems abstract the matrix sandwich problem msp is introduced here as follows given a 0 1 valued matrix a where is interpreted as do not care does there exist a fill in of the asterisks with 0s and 1s such that the completed 0 1 valued matrix m satisfies property we study the computational complexity of this problem for several matrix properties including the ferrers property block decompositions and certain forbidden submatrices matrix sandwich problems are an important special case of matrix completion problems the latter being generally defined over the real numbers rather than simply 0 1

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

cuspidality of symmetric powers with applications the purpose of this paper is to prove that the symmetric fourth power of a cusp form on rm gl 2 whose existence was proved earlier by the first author is cuspidal unless the corresponding automorphic representation is of dihedral tetrahedral or octahedral type as a consequence we prove a number of results toward the ramanujan petersson and sato tate conjectures in particular we establish the bound q sp 1 9 sb v for unramified hecke eigenvalues of cusp forms on rm gl 2 over an arbitrary number field this is the best bound available at present

on moments of twisted l functions we study the average of the product of the central values of two l functions of modular forms f and g twisted by dirichlet characters to a large prime modulus q as our principal tools we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of q and we use the theory of deligne and katz to prove new bounds on bilinear forms in kloosterman sums with power savings when both variables are near the square root of q when at least one of the forms f and g is non cuspidal we obtain an asymptotic formula for the mixed second moment of twisted l functions with a power saving error term in particular when both are non cuspidal this gives a significant improvement on m young s asymptotic evaluation of the fourth moment of dirichlet l functions in the general case the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in kloosterman sums

regularity decay and best constants for dispersive equations we show that the strichartz l2w l2 estimates for solutions to the pseudo differential equations xu i tuand x 1u i tu are equivalent a necessary and sufficient condition for decay and regularity for solutions to the equation x u i tu is given

on the local langlands conejcture mathrm gl n the cyclic case let f be a locally compact nonarchimedean field and p its residue characteristic let f be a separable algebraic closure of f wf the weil group of f over f e a character of fx of order n and e the corresponding character of wf then there exists a canonical map from the set of equivalence classes a of irreducible degree n representations of wf such that e a a to the set of equivalence classes 7r of admissible irreducible supercuspidal representations of gl n f verifying odet x 7t this map is shown to be bijective and to preserve e factors for pairs

une preuve simple des conjectures de langlands pour gl n sur un corps p adique let f be a finite extension of p for each integer n 1 we construct a bijection from the set f 0 n of isomorphism classes of irreducible degree n representations of the absolute weil group of f onto the set f 0 n of isomorphism classes of smooth irreducible supercuspidal representations of gl n f those bijections preserve epsilon factors for pairs and hence we obtain a proof of the langlands conjectures for gl n over f which is more direct than harris and taylor s our approach is global and analogous to the derivation of local class field theory from global class field theory we start with a result of kottwitz and clozel on the good reduction of some shimura varieties and we use a trick of harris who constructs non galois automorphic induction in certain cases

on the ramanujan conjecture and finiteness of poles for certain l functions in this paper we prove a number of general results about automorphic l functions which appear in the constant terms of eisenstein series for an arbitrary quasi split connected reductive algebraic group over a number field these l functions were first introduced by langlands for chevalley groups 20 and through them he was led to make some of his deep conjectures 21 one significance of these l functions is that all the automorphic l functions studied so far are among them there are three general results first we obtain a uniform line of absolute convergence for all of these l functions theorem 5 1 a uniform estimate for the hecke eigenvalues of generic cusp forms on many absolutely simple quasi split connected reductive algebraic groups over number fields corollary 5 4 and an improvement on the best available estimate for the fourier coefficients of maass wave forms corollary 5 5 also follow next we establish a meromorphic continuation and functional equation for each of these l functions theorem 6 1 finally in theorem 6 2 we prove finiteness of poles on the whole complex plane for an important class of these l functions corollaries 6 6 through 6 10 more precisely let g be a quasi split connected reductive algebraic group over a number field f set g g a f where a f is the ring of adeles of f fix a borel subgroup b of g over f and let u be its unipotent radical let p be a maximal f parabolic subgroup of g with p d b in the context of the problems studied here nothing new will be obtained if one drops the maximality condition on p write p mn a levi decomposition n c u and let b u p m and n be the corresponding groups of adelic points for every place v of f let gv g fj similarly we have bv us pv me and nv let x xv be a generic character of u f u cf section 3 then each xv is generic let r ov 7tv be a cusp form on m we shall say 7t is

algebraic trace functions over the primes we study sums over primes of trace functions of l adic sheaves using an extension of our earlier results on algebraic twists of modular forms to the case of eisenstein series and bounds for type ii sums based on similar applications of the riemann hypothesis over finite fields we prove general estimates with power saving for such sums we then derive various concrete applications

smoothness and non smoothness of the fundamental solution of time dependent schrodinger equations the fundamental solutione t s x y of time dependent schrodinger equationsi u t 1 2 u v t x u is studied it is shown that e t s x y is smooth and bounded fort s if the potential is sub quadratic in the sense thatv t x o x 2 at infinity in one dimension ifv t x v x is time independent and super quadratic in the sense thatv x c 1 x 2 e at infinity c 0 and e 0 thene t s x y is nowherec1

a proof of langland s conjecture on plancherel measures complementary series of p adic groups analysis of p adic reductive groups our first result theorem 7 9 proves a conjecture of langlands on normalization of intertwining operators by means of local langlands root numbers and l functions at least when the group is quasi split and the inducing representation is generic assuming two natural conjectures in harmonic analysis of p adic groups we also prove the validity of the conjecture in general theorem 9 5 as our second result we obtain all the complementary series and special representations of quasi split p adic groups coming from rank one parabolic subgroups and generic supercuspidal represen

simple linear time algorithms to test chordality of graphs test acyclicity of hypergraphs and selectively reduce acyclic hypergraphs chordal graphs arise naturally in the study of gaussian elimination on sparse symmetric matrices acyclic hypergraphs arise in the study of relational data bases rose tarjan and lueker siam j c

unitary equivalence and scattering theory for stark like hamiltonians let h0 v0 x1 h h0 v x be self adjoint in l2 rn v0 depends only on one coordinate and tends monotonically to as x1 v is a real h0 compact potential short range with respect to v0 in particular the cases v0 x1 sgn x1 x1 0 2 and v x c x 1 are included 1 being the stark effect it is shown that a h is spectrally absolutely continuous over the entire real axis apart from a possible discrete sequence of eigenvalues of finite multiplicity and rapidly decaying eigenfunctions h0 has no eigenvalues and b the wave operators w h h0 exist and are complete

sur le changement de signe des sommes de kloosterman combining sieve methods with automorphic form theory and techniques from l adic cohomology we prove that the sign of kloosterman sums kl 1 1 n changes infinitely often as n ranges over the squarefree integers having all their prime factors larger than n 1 23 9

on symmetric power l invariants of iwahori level hilbert modular forms we compute the arithmetic cal l invariants of greenberg benois of twists of symmetric powers of p adic galois representations attached to iwahori level hilbert modular forms under some technical conditions our method uses the automorphy of symmetric powers and the study of analytic galois representations on p adic families of automorphic forms over symplectic and unitary groups combining these families with some explicit plethysm in the representation theory of rm gl 2 we construct global galois cohomology classes with coefficients in the symmetric powers and provide formulae for the cal l invariants in terms of logarithmic derivatives of hecke eigenvalues

local coefficients as artin factors for real groups introduction the purpose of this paper is to prove the equality of certain local coefficients of arithmetic significance which were attached to representations of quasi split real reductive algebraic groups in 27 with their corresponding artin factors attached by local class field theory 21 as a consequence we establish an identity satisfied by certain normalized intertwining operators it seems to be useful in applications of the trace formula 1 29 more precisely let g be the group of real points of a quasi split reductive algebraic group over r let a be the set of simple roots defined by a fixed minimal parabolic subgroup po m0a0u of g fix 6 c a and let p p9 be the corresponding standard parabolic subgroup of g and write p man for its langlands decomposition fix a nondegenerate character x f u i a h o be an irreducible admissible x gic banach in particular x generic unitary representation of m cf section 1 given v g a the complex dual of the lie algebra of a let i v o 0 be the continuously quasi unitarily if o is unitary induced representation indp ga e v 9 and let v v o 0 be its space section 0 then f a vfoo o now let w be the weyl group of ao in g choose w g w such that w 0 c a let n be the unipotent group opposite to n define n u c wn w where w is a representative of w in g for g f a 00 define

global estimates for the schrodinger equation abstract let u u x t be a solution to the ivp for the schrodinger equation iu 1 v x u hu u x 0 u 0 x p ac l 2 r n pac is the projection on the absolutely continuous subspace of h assume that for some e 0 the multiplication operator 1 x 1 i v x h 1 e r n l 2 r n is bounded then u x t u1 x t u2 x t where for every s 1 2 r r n 1 x 2 s 1 h 1 4 u 1 x t 2 dxdt c u 0 l 2 r n 2 and for every integer j sup i h j u 2 t l 2 r n c j u 0 l 2 r n t r

minorations de sommes d exponentielles keywords complete exponential sums almost prime modulus lower bound positive density multidimensional kloosterman sum reference tan article 1998 003doi 10 1215 s0012 7094 98 09507 2 record created on 2008 11 14 modified on 2017 05 12

autour de la conjecture de sato tate pour les sommes de kloosterman i keywords almost primes sieve identity of buchstab type kloosterman sum sato tate conjecture distribution function sato tate distribution estimates for multiple exponential sums reference tan article 1995 001doi 10 1007 bf01884290 record created on 2008 11 14 modified on 2017 05 12

invariant distributions and time averages for horocycle flows there are infinitely many obstructions to existence of smooth solutions of the cohomological equation uu f where u is the vector field generating the horocycle flow on the unit tangent bundle sm of a riemann m surface of finite area and f is a given function on sm we study the sobolev regularity of these obstructions construct smooth solutions of the cohomological equation and derive asymptotics for the ergodic averages of horocycle flows

parametrices for a class of schrodinger equations the article describes the construction of a parametrix global in space in the initial value problem for the schrodinger equation i u t u q x u v x u with q a real quadratic form and v a real potential in the hypothesis on v is that v x const x and v x const x 1 as x by conjugating the equation with the unitary group exp ith h q x the problem is reduced to a hyperbolic pseudodifferential operator to which a global version of the geometric optics method applies the symbolic calculus used by the construction is based on weights that reflect the role of the propagation set

regularity of solutions to the schrodinger equation soit f dans l espace de schwartz s r n et soit s t f x u x t rn exp ix exp it 2 f d x r n t r f est la transformee de fourier de f on considere u solution de l equation de schrodinger u i u t on pose s f x sup 0 t 1 s t f x x r n on introduit des espaces de sobolev h s en posant h s f s f hd s s r ou f hd s rn 1 2 s f 2 d 1 2 on etudie pour quelles valeurs de s on a b s f x 2 dx 1 2 c b f hd s f s r n pour toutes les boucles b de r n

regularity and decay of solutions to the stark evolution equation long time behavior and regularity are studied for solutions of the stark equationut i x1 v x u u 0 x l2 rn it is shown that for a class of short range potentialsv x the gain of local smoothness and the decay as t are close to those of the corresponding schrodinger equationut i v x u

local smoothing and convergence properties of schrodinger type equations abstract the initial value problem for the schrodinger type equation u t ip d u u 0 x u 0 x x r n is considered here p is a principal type polynomial of order m i e p c 1 m 1 r or a certain type of other real symbol such as p m m 1 the following results are proved a if u 0 l 2 r n then for any fixed r 0 write u 0 u 1 u 2 where u 1 and u 2 are square integrable and the fourier transform of u 1 vanishes for r and that of u 2 vanishes for r if u 1 t x e itp d u 1 x u 2 t x e itp d u 2 x then u 1 is real analytic in r r n and satisfies for every multi index sup t x d u 1 t x c u 0 l 2 while 1 x 2 1 2 u 2 t x l 2 r t h m 1 2 r x n b if u 0 l 2 s r n s 0 then for every t 0 u t h loc m 1 s r n c if u 0 h s r n s 1 2 then for a e x r n u t x u 0 x as t 0

modularity of the rankin selberg l series and multiplicity one for sl 2 let f g be primitive cusp forms holomorphic or otherwise on the upper half plane h of levels n m respectively with unitarily normalized l functions l s f equation and l s g equation when p does not divide n resp m the inverse roots p p resp p p are nonzero with sum ap resp bp for every p prime to nm set lp s f g 1 p pp s 1 p pp s 1 p pp s 1 p pp s 1 let l s f g denote the incomplete euler product of lp s f g over all p not dividing nm this is closely related to the convolution l series sum over n 1 a sub n b sub n n s whose miraculous properties were first studied by rankin and selberg

when kloosterman sums meet hecke eigenvalues by elaborating a two dimensional selberg sieve with asymptotics and equidistributions of kloosterman sums from ell adic cohomology as well as a bombieri vinogradov type mean value theorem for kloosterman sums in arithmetic progressions it is proved that for any given primitive hecke maass cusp form of trivial nebentypus the eigenvalue of the n th hecke operator does not coincide with the kloosterman sum mathrm kl 1 n for infinitely many squarefree n with at most 100 prime factors this provides a partial negative answer to a problem of katz on modular structures of kloosterman sums

on the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs golumbic kaplan and shamir in their paper m c golumbic h kaplan r shamir graph sandwich problems j algorithms 19 1995 449 473 on graph sandwich problems published in 1995 left the status of sandwich problems for strongly chordal graphs and chordal bipartite graphs open we prove that the sandwich problem for strongly chordal graphs is np complete we also give some comments on the computational complexity of the sandwich problem for chordal bipartite graphs

on the arithmetic of shalika models and the critical values of l functions for gl2n let pi be a cohomological cuspidal automorphic representation of rm gl 2n bbb a over a totally real number field f suppose that pi has a shalika model we define a rational structure on the shalika model of pi f comparing it with a rational structure on a realization of pi f in cuspidal cohomology in top degree we define certain periods omega epsilon pi f we describe the behavior of such top degree periods upon twisting pi by algebraic hecke characters chi of f then we prove an algebraicity result for all the critical values of the standard l functions l s pi otimes chi here we use the recent work of b sun on the non vanishing of a certain quantity attached to pi infty as applications we obtain algebraicity results in the following cases firstly for the symmetric cube l functions attached to holomorphic hilbert modular cusp forms we also discuss the situation for higher symmetric powers secondly for certain self dual of symplectic type rankin selberg l functions for rm gl 3 times rm gl 2 assuming langlands functoriality this generalizes to certain rankin selberg l functions of rm gl n times rm gl n 1 thirdly for the degree four l functions attached to siegel modular forms of genus 2 and full level moreover we compare our top degree periods with periods defined by other authors we also show that our main theorem is compatible with conjectures of deligne and gross

unique perfect phylogeny is intractable a phylogeny is a tree capturing evolution and ancestral relationships of a set of taxa e g species reconstructing phylogenies from molecular data plays an important role in many areas of contemporary biological research a phylogeny is perfect if in rough terms it correctly captures all input data determining if a perfect phylogeny exists was shown to be intractable in 1992 by mike steel steel 1992 32 and independently by bodlaender et al bodlaender et al 1992 4 in light of this a related problem was proposed by steel 1992 32 given a perfect phylogeny determine if it is the unique perfect phylogeny for the given dataset where the dataset is provided as a set of quartet 4 leaf trees it was suggested that this problem may be more tractable steel 1992 32 and determining its complexity became known as the quartet challenge steel 2012 33 in this paper we resolve this question by showing that the problem is conp complete we prove this by relating perfect phylogenies to satisfying assignments of boolean formulae to this end we cast the question as a chordal sandwich problem as a particular consequence of our method we show that the unique minimal chordal sandwich problem is conp complete and counting minimal chordal sandwiches is p complete

pathwidth bandwidth and completion problems to proper interval graphs with small cliques we study two related problems motivated by molecular biology given a graph g and a constant k does there exist a supergraph g of g that is a unit interval graph and has clique size at most k given a graph g and a proper k coloring c of g does there exist a supergraph g of g that is properly colored by c and is a unit interval graph we show that those problems are polynomial for fixed k on the other hand we prove that the first problem is equivalent to deciding if the bandwidth of g is at most k 1 hence it is np hard and w t hard for all t we also show that the second problem is w 1 hard this implies that for fixed k both of the problems are unlikely to have an o n alpha algorithm where alpha is a constant independent of k a central tool in our study is a new graph theoretic parameter closely related to pathwidth an unexpected useful consequence is the equivalence of this parameter to the bandwidth of the graph

real zeros and size of rankin selberg l functions in the level aspect in this paper some asymptotic formulas are proved for the harmonic mollified second moment of a family of rankin selberg l functions one of the main new input is a substantial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average a first consequence is a new subconvexity bound for rankin selberg l functions in the level aspect moreover infinitely many rankin selberg l functions having at most eight non trivial real zeros are produced and some new non trivial estimates for the analytic rank of the family studied are obtained

twisted endoscopy and reducibility of induced representations for p adic groups 1 introduction this is the first in a series of papers in which we study the reducibility of representations induced from discrete series representations of the levi factors ofmaximal parabolic subgroups of p adic groups on the unitary axis the problem is equivalent to determining certain local langlands l functions which are of arithmetic significance by themselves one of the main aims of this paper is to interpret our results in the direction of the parametrization problem by means of the theory of twisted endoscopy for which our method seems to be very suitable when the representation is supercuspidal we also study the reducibility of the induced representations off the unitary axis let f be a p adic field of characteristic zero fix a positive integer n 1 and let g be either of the groups sp2n s02n or s02n l in all three cases there is a conjugacy class of maximal parabolic subgroups which has gl as its levi factor let p mn be the standard parabolic subgroup ofg in this conjugacy class then m gl let tr be a discrete series representation of m gln f and given s c let i s tr be the representation of g g f induced from tr r idet ls let i tr i 0 tr