well posedness for the schrodinger korteweg de vries system we study well posedness of the cauchy problem associated to the schrodinger korteweg de vries system we obtain local well posedness for weak initial data where the best result obtained is for data in the sobolev space l 2 r x h 3 4 this result implies in particular the global well posedness in the energy space h 1 r x h 1 r both results considerably improve the previous ones by bekiranov ogawa and ponce 1997 guo and miao 1999 and tsutsumi 1993

modules of the 0 hecke algebra and quasisymmetric schur functions abstract we begin by deriving an action of the 0 hecke algebra on standard reverse composition tableaux and use it to discover 0 hecke modules whose quasisymmetric characteristics are the natural refinements of schur functions known as quasisymmetric schur functions furthermore we classify combinatorially which of these 0 hecke modules are indecomposable from here we establish that the natural equivalence relation arising from our 0 hecke action has equivalence classes that are isomorphic to subintervals of the weak bruhat order on the symmetric group focussing on the equivalence classes containing a canonical tableau we discover a new basis for the hopf algebra of quasisymmetric functions and use the cardinality of these equivalence classes to establish new enumerative results on truncated shifted reverse tableau studied by panova and adin king roichman generalizing our 0 hecke action to one on skew standard reverse composition tableaux we derive 0 hecke modules whose quasisymmetric characteristics are the skew quasisymmetric schur functions of bessenrodt et al this enables us to prove a restriction rule that reflects the coproduct formula for quasisymmetric schur functions which in turn yields a quasisymmetric branching rule analogous to the classical branching rule for schur functions

the decomposition of lie powers let g be a group f a field of prime characteristic p and v a finite dimensional fg module let l v denote the free lie algebra on v regarded as an fg submodule of the free associative algebra or tensor algebra t v for each positive integer r let l r v and t r v be the r th homogeneous components of l v and t v respectively here l r v is called the r th lie power of v our main result is that there are submodules b 1 b 2 of l v such that for all r b r is a direct summand of t r v and whenever m geqslant 0 and k is not divisible by p the module l p mk v is the direct sum of l p m b k l p m 1 b pk l 1 b p mk thus every lie power is a direct sum of lie powers of p power degree the approach builds on an analysis of t r v as a bimodule for g and the solomon descent algebra

twisted descent algebras and the solomon tits algebra abstract the notion of descent algebra of a bialgebra is lifted to the barratt joyal setting of twisted bialgebras the new twisted descent algebras share many properties with their classical counterparts for example there are twisted analogs of classical lie idempotents and of the peak algebra moreover the universal twisted descent algebra is equipped with two products and a coproduct and there is a fundamental rule linking all three this algebra is shown to be naturally related to the geometry of the coxeter complex of type a

spectral radius and hamiltonian graphs abstract in the paper we will present two sufficient conditions for a bipartite graph to be hamiltonian and a graph to be traceable respectively

global existence and blow up for the generalized sixth order boussinesq equation abstract in this paper we prove local well posedness in l 2 r and h 1 r for the generalized sixth order boussinesq equation u t t u x x u x x x x u x x x x x x u u x x our proof relies in the oscillatory integrals estimates introduced by kenig et al 1991 14 we also show that under suitable conditions a global solution for the initial value problem exists in addition we derive the sufficient conditions for the blow up of the solution to the problem

a numerical study of an ill posed boussinesq equation arising in water waves and nonlinear lattices filtering and regularization techniques1this research was supported by nsf grants no dms 9208061 and by the office of the vice president for research and associate provost for graduate studies at texas a m university 1 we consider an ill posed boussinesq equation which arises in shallow water waves and nonlinear lattices this equation has growing and decaying modes in the linear as well as nonlinear regimes and its linearized growth rate s for short waves of wavenumber k is given by s k 2 previous numerical studies have addressed numerical difficulties and construction of approximate solutions for ill posed problems with short wave instability up to s k e g kelvin helmholtz s k and rayleigh taylor s k instabilities these same issues are addressed and critically examined here for the present problem which has more severe short wave instability in order to develop numerical techniques for constructing good approximate solutions of this equation we use a finite difference scheme to investigate the effect of this short wave instability on the numerical accuracy of the exact solitary wave solution of this equation computational evidence is presented which indicates that numerical accuracy of the solutions is lost very quickly due to severe growth of numerical errors roundoff as well as truncation we use both filtering and regularization techniques to control growth of these errors and to provide better approximate solutions of this equation in the filtering technique numerical experiments with three types of spectral filters of increasing order of regularity are performed we examine the role of regularity of these filters on the accuracy of the numerical solutions numerical evidence is provided which indicates that the regularity of a filter plays an important role in improving the accuracy of the solutions in the regularization technique the ill posed equation is regularized by adding a higher order term to the equation two types of higher order terms are discussed i one that diminishes the growth rate of all modes below a cutoff wavenumber and sets the growth rate of all modes above it to zero and ii the other one diminishes the growth rate of all modes and the growth rate asymptotically approaches to zero as the wavenumber approaches infinity we have argued in favor of the first type of regularization and numerical results using a finite difference scheme are presented numerical evidence is provided which suggests that regularization in combination with the most regular c 2 here spectral filter for small values of the regularization parameter can provide good approximate solutions of the ill posed boussinesq equation for longer time than possible otherwise some of the ideas presented here can possibly be utilized for solving other ill posed problems with severe short wave instabilities and may have an important role to play in numerical studies of their solutions

interaction equations for short and long dispersive waves abstract we show the time local well posedness for a system of nonlinear dispersive equations for the water wave interaction formula it is shown that for any initial data u0 v0 hs r hs 1 2 r s 0 the solution for the above equation uniquely exists in a subset ofc t t hs c t t hs 1 2 and depends continuously on the data by virtue of a special structure of the nonlinear coupling the solution is stable under a singular limiting process

on the quiver presentation of the descent algebra of the symmetric group abstract we describe a presentation of the descent algebra of the symmetric group s n as a quiver with relations this presentation arises from a new construction of the descent algebra as a homomorphic image of an algebra of forests of binary trees which can be identified with a subspace of the free lie algebra in this setting we provide a short new proof of the known fact that the quiver of the descent algebra of s n is given by restricted partition refinement moreover we describe certain families of relations and conjecture that for fixed n n the finite set of relations from these families that are relevant for the descent algebra of s n generates the ideal of relations of an explicit quiver presentation of that algebra

shuffle compatible permutation statistics abstract since the early work of richard stanley it has been observed that several permutation statistics have a remarkable property with respect to shuffles of permutations we formalize this notion of a shuffle compatible permutation statistic and introduce the shuffle algebra of a shuffle compatible permutation statistic which encodes the distribution of the statistic over shuffles of permutations this paper develops a theory of shuffle compatibility for descent statistics statistics that depend only on the descent set and length which has close connections to the theory of p partitions quasisymmetric functions and noncommutative symmetric functions we use our framework to prove that many descent statistics are shuffle compatible and to give explicit descriptions of their shuffle algebras thus unifying past results of stanley gessel stembridge aguiar bergeron nyman and petersen

sufficient spectral conditions on hamiltonian and traceable graphs abstract in this paper we give sufficient conditions on the spectral radius for a bipartite graph being hamiltonian and traceable which expand the results of lu liu and tian 2012 10 furthermore we also provide tight sufficient conditions on the signless laplacian spectral radius for a graph to be hamiltonian and traceable which improve the results of yu and fan 2012 11

local well posedness for the sixth order boussinesq equation abstract in this work we study the local well posedness of the initial value problem for the nonlinear sixth order boussinesq equation u t t u x x u x x x x u x x x x x x u 2 x x where 1 we prove the local well posedness with initial data in non homogeneous sobolev spaces h s r for negative indices of s r

peak quasisymmetric functions and eulerian enumeration abstract via duality of hopf algebras there is a direct association between peak quasisymmetric functions and enumeration of chains in eulerian posets we study this association explicitly showing that the notion of cd index long studied in the context of convex polytopes and eulerian posets arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by stembridge thus eulerian posets having a nonnegative cd index for example face lattices of convex polytopes correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis we diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions and study the g polynomial for eulerian posets as an algebra homomorphism

lie idempotent algebras abstract besides the inner product inherited from each symmetric group algebra ksn on the direct sum ks nksn a second convolution product arises from the hopf algebra structure on ks for any subalgebra a of ks for both the inner and the convolution product it is shown that the lie idempotent algebra l a of a generated by all lie idempotents in a with respect to the convolution product is also closed under inner products its nth homogeneous component l n a is thus a subalgebra of ksn for all n provided that a contains a lie idempotent in ksk for all k n solomon s descent algebra d n is a subalgebra of l n a in this case a number of results on the structure of l n a is derived including a description of its jacobson radical its principal indecomposables and its cartan invariants furthermore solomon s epimorphism from d n onto the ring of class functions clk sn of sn extends to an epimorphism of algebras c n l n a cl k s n additional results are obtained concerning a bialgebra structure of l a in the particular case of a n d n this amounts to a new approach to several well known results on d n

global existence for 2d nonlinear schrodinger equations via high low frequency decomposition method abstract we study global existence of solutions for the cauchy problem of the nonlinear schrodinger equation i u t u u 2 m u in the 2 dimension case where m is a positive integer m 2 using the high low frequency decomposition method we prove that if 10 m 6 10 m 5 s 1 then for any initial value h s r 2 the cauchy problem has a global solution in c r h s r 2 and it can be split into u t e i t y t with y c r h 1 r 2 satisfying y t h 1 1 t 2 1 s 10 m 5 s 10 m 6 where is an arbitrary sufficiently small positive number

an improved local well posedness result for the one dimensional zakharov system abstract the 1d cauchy problem for the zakharov system is shown to be locally well posed for low regularity schrodinger data u 0 h k p and wave data n 0 n 1 h l p h l 1 p under certain assumptions on the parameters k l and 1 p 2 where u 0 h k p k u 0 l p generalizing the results for p 2 by ginibre tsutsumi and velo especially we are able to improve the results from the scaling point of view and also allow suitable k 0 l 1 2 i e data u 0 l 2 and n 0 n 1 h 1 2 h 3 2 which was excluded in the case p 2

enriched p partitions and peak algebras abstract we develop a more general view of stembridge s enriched p partitions and use this theory to outline the structure of peak algebras for the symmetric group and the hyperoctahedral group initially we focus on commutative peak algebras spanned by sums of permutations with the same number of peaks where we consider several variations on the definition of peak whereas stembridge s enriched p partitions are related to quasisymmetric functions the dual coalgebra of solomon s type a descent algebra our generalized enriched p partitions are related to type b quasisymmetric functions the dual coalgebra of solomon s type b descent algebra using these functions we move on to explore non commutative peak algebras spanned by sums of permutations with the same set of peaks while some of these algebras have been studied before our approach gives explicit structure constants with a combinatorial description

hyperoctahedral chen calculus for effective hamiltonians abstract we tackle the problem of unraveling the algebraic structure of computations of effective hamiltonians this is an important subject in view of applications to chemistry solid state physics or quantum field theory we show among other things that the correct framework for these computations is provided by the hyperoctahedral group algebras we define several structures on these algebras and give various applications for example we show that the adiabatic evolution operator in the time dependent interaction representation of an effective hamiltonian can be written naturally as a picard type series and has a natural exponential expansion

global rough solutions to the sixth order boussinesq equation abstract we show that the cauchy problem for the sixth order boussinesq equation with cubic nonlinearity is globally well posed in h s r provided 3 2 s 2

sharp local well posedness for the good boussinesq equation abstract in the present article we prove the sharp local well posedness and ill posedness results for the good boussinesq equation on 1d torus the initial value problem is locally well posed in h 1 2 t and ill posed in h s t for s 1 2 well posedness result is obtained from reduction of the problem into a quadratic nonlinear schrodinger equation and the contraction argument in suitably modified x s b spaces the proof of the crucial bilinear estimates in these spaces especially in the lowest regularity rely on some bilinear estimates for periodic functions in x s b spaces which are generalization of the bilinear refinement of the l 4 strichartz estimate on r our result improves the known local well posedness in h s t with s 3 8 given by oh stefanov 2012 similar ideas also establish the sharp local well posedness in h 1 2 r and ill posedness below h 1 2 for the nonperiodic case which improves the result of tsugawa and the author 2010 in h s r with s 1 2

skew quasisymmetric schur functions and noncommutative schur functions abstract recently a new basis for the hopf algebra of quasisymmetric functions qsym called quasisymmetric schur functions has been introduced by haglund luoto mason van willigenburg in this paper we extend the definition of quasisymmetric schur functions to introduce skew quasisymmetric schur functions these functions include both classical skew schur functions and quasisymmetric schur functions as examples and give rise to a new poset l c that is analogous to young s lattice we also introduce a new basis for the hopf algebra of noncommutative symmetric functions nsym this basis of nsym is dual to the basis of quasisymmetric schur functions and its elements are the pre image of the schur functions under the forgetful map nsym sym we prove that the multiplicative structure constants of the noncommutative schur functions equivalently the coefficients of the skew quasisymmetric schur functions when expanded in the quasisymmetric schur basis are nonnegative integers satisfying a littlewood richardson rule analogue that reduces to the classical littlewood richardson rule under as an application we show that the morphism of algebras from the algebra of poirier reutenauer to sym factors through nsym we also extend the definition of schur functions in noncommuting variables of rosas sagan in the algebra ncsym to define quasisymmetric schur functions in the algebra ncqsym we prove these latter functions refine the former and their properties and project onto quasisymmetric schur functions under the forgetful map lastly we show that by suitably labeling l c skew quasisymmetric schur functions arise in the theory of pieri operators on posets

low regularity solutions of non homogeneous boundary value problems of a higher order boussinesq equation in a quarter plane abstract in this article we study an initial boundary value problem of the sixth order boussinesq equation on a half line with nonhomogeneous boundary conditions u t t u x x u x x x x u x x x x x x u 2 x x 0 x 0 t 0 u x 0 x u t x 0 x x 0 u 0 t h 1 t u x x 0 t h 2 t u x x x x 0 t h 3 t t 0 where 1 it is shown that the problem is locally well posed in c r l o c h s r with initial condition h s r h s 1 r and boundary condition h 1 h 2 h 3 in the product space h s 1 3 r h s 1 3 r h s 3 3 r for 1 2 s 0

affine hecke algebras and generalized standard young tableaux abstract this paper introduces calibrated representations for affine hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations the primary technique is to provide indexing sets for controlling the weight space structure of finite dimensional modules for the affine hecke algebra using these indexing sets we show that 1 irreducible calibrated representations are indexed by skew local regions 2 the dimension of an irreducible calibrated representation is the number of chambers in the local region 3 each irreducible calibrated representation is constructed explicitly by formulas which describe the action of the generators of the affine hecke algebra on a specific basis in the representation space the indexing sets for weight spaces are generalizations of standard young tableaux and the construction of the irreducible calibrated affine hecke algebra modules is a generalization of a young s seminormal construction of the irreducible representations of the symmetric group in this sense young s construction has been generalized to arbitrary lie type

cyclotomic solomon algebras this paper introduces an analogue of the solomon descent algebra for the complex reflection groups of type g r 1 n as with the solomon descent algebra our algebra has a basis given by sums of distinguished coset representatives for certain reflection subgroups we explicitly describe the structure constants with respect to this basis and show that they are polynomials in r this allows us to define a deformation or q analogue of these algebras which depends on a parameter q we determine the irreducible representations of all of these algebras and give a basis for their radicals finally we show that the direct sum of cyclotomic solomon algebras is canonically isomorphic to a concatenation hopf algebra

well posedness for the cauchy problem associated to a periodic boussinesq equation abstract we consider the initial value problem associated to the nonlinear periodic sixth order boussinesq equation for given initial data in sobolev spaces h s t the local well posedness of this equation is established for s 1 2

a solomon descent theory for the wreath products _ we propose an analogue of solomon s descent theory for the case of a wreath product g g n where g is a finite abelian group our construction mixes a number of ingredients mantaci reutenauer algebras specht s theory for the representations of wreath products okada s extension to wreath products of the robinson schensted correspondence and poirier s quasisymmetric functions we insist on the functorial aspect of our definitions and explain the relation of our results with previous work concerning the hyperoctaedral group

well posedness for a quadratic derivative nonlinear schrodinger system at the critical regularity abstract we consider the cauchy problem for a quadratic derivative nonlinear schrodinger system in sobolev space h s in this paper under the null condition and the mass resonance relation we prove large data local well posedness if the space dimension d 3 and s s c d 2 and s s c or d 1 and s 0 where s c is the scaling critical regularity moreover under the same assumptions we also prove small data global well posedness and scattering in h s if d 3 and s s c our proof is based on a contraction argument using the u p and v p type function spaces

on the existence of a maximizer for the strichartz inequality it is shown that a maximizing function u l 2 does exist for the strichartz inequality e it x 2 u l 6 t l 6 x s u l 2 with s 0 being the sharp constant

random walks on quasisymmetric functions abstract conditions are provided under which an endomorphism on quasisymmetric functions gives rise to a left random walk on the descent algebra which is also a lumping of a left random walk on permutations spectral results are also obtained several important random walks are now realized this way stanley s qs distribution results from endomorphisms given by evaluation maps a shuffles result from the a th convolution power of the universal character and the tchebyshev operator of the second kind introduced recently by ehrenborg and readdy yields traditional riffle shuffles a conjecture of ehrenborg regarding the spectra for a family of random walks on ab words is proven a theorem of stembridge from the theory of enriched p partitions is also recovered as a special case

on adding a list of numbers and other one dependent determinantal processes adding a column of numbers produces carries along the way we show that random digits produce a pattern of carries with a neat probabilistic description the carries form a one dependent determinantal point process this makes it easy to answer natural questions how many carries are typical where are they located we show that many further examples from combinatorics algebra and group theory have essentially the same neat formulae and that any one dependent point process on the integers is determinantal the examples give a gentle introduction to the emerging fields of one dependent and determinantal point processes

bilinear estimates and applications to 2d nls the three bilinearities uv uv uv for functions u v r2 0 t 7 c are sharply estimated in function spaces xs b associated to the schrodinger operator i t these bilinear estimates imply local wellposedness results for schrodinger equations with quadratic nonlinearity improved bounds on the growth of spatial sobolev norms of finite energy global in time and blow up solutions of the cubic nonlinear schrodinger equation and certain generalizations are also obtained

a decomposition of the descent algebra of the hyperoctahedral group elements of the hyperoctahedral group bn can be represented as lists of integers 1 2 n where the list of absolute value of the k s is a permutation of the integers from 1 to n the descent set of such a is des k 0 n 1 k k 1 where k 0 we set 0 0 a descent class as of bn is the formal sum of all elements of bn with a given descent set s 0 1 n 1 in this paper we shall give a combinatorial proof of a result of solomon 6 stating that the product in the group algebra q bn of two descent classes is a linear combination of descent classes we refer to the linear span of these descent classes it is a subalgebra bn of q bn as solomon s hyperoctahedral descent algebra we derive from our combinatorial multiplication rule for bn that there is a family of ideals jv n of bn such that j0 n j1 n jn n with bn v 1 bnjv n in 4 garsia and reutenauer relate a similar decomposition of the multiplicative structure of the descent algebra of the symmetric group sn to the action of the symmetric group on the free lie algebra it develops that one can introduce a bn analog of the free lie algebra and give similar results for bn

spectral radius and hamiltonicity of graphs abstract let g be a graph of order n and g be the largest eigenvalue of its adjacency matrix let g be the complement of g write k n 1 v for the complete graph on n 1 vertices together with an isolated vertex and k n 1 e for the complete graph on n 1 vertices with a pendent edge we show that if g n 2 then g contains a hamiltonian path unless g k n 1 v if strict inequality holds then g contains a hamiltonian cycle unless g k n 1 e if g n 1 then g contains a hamiltonian path unless g k n 1 v if g n 2 then g contains a hamiltonian cycle unless g k n 1 e

a multiplication rule for the descent algebra of typed here we give an interpretation of solomon s rule for multiplication in the descent algebra of coxeter groups of typed dn we describe an ideal i such that dn i is isomorphic to the descent algebra of the hyperoctahedral group bn 2

low regularity global well posedness for the zakharov and klein gordon schrodinger systems we prove low regularity global well posedness for the id zakharov system and the 3d klein gordon schrodinger system which are systems in two variables u r d x x r t c and n r d x x r t r the zakharov system is known to be locally well posed in u n l 2 x h and the klein gordon schrodinger system is known to be locally well posed in u n l 2 x l 2 here we show that the zakharov and klein gordon schrodinger systems are globally well posed in these spaces respectively by using an available conservation law for the l 2 norm of u and controlling the growth of n via the estimates in the local theory

the peak algebra and the descent algebras of types b and d we show the existence of a unital subalgebra n of the symmetric group algebra linearly spanned by sums of permutations with a common peak set which we call the peak algebra we show that n is the image of the descent algebra of type b under the map to the descent algebra of type a which forgets the signs and also the image of the descent algebra of type d the algebra n contains a two sided ideal n which is defined in terms of interior peaks this object was introduced in previous work by nyman 2003 we find that it is the image of certain ideals of the descent algebras of types b and d we derive an exact sequence of the form 0 n n n 2 0 we obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs in particular we construct two new commutative semisimple subalgebras of the descent algebra of dimensions n and n 2 1 by grouping permutations according to their number of peaks or interior peaks we discuss the hopf algebraic structures that exist on the direct sums of the spaces n and n over n 0 and explain the connection with previous work of stembridge 1997 we also obtain new properties of his descents to peaks map and construct a type b analog

a decomposition of the group algebraof a hyperoctahedral group the descent algebra of a finite coxeter group w is a subalgebra of the group algebra defined by solomon descent algebras of symmetric groups have properties that are not shared by other coxeter groups for instance the natural map from the descent algebra of a symmetric group to its character ring is a surjection with kernel equal the jacobson radical thus the descent algebra implicitly encodes information about the representations of the symmetric group and a complete set of primitive idempotents in the character ring leads to a decomposition of the group algebra into a sum of right ideals indexed by partitions stanley asked whether this decomposition of the regular representation of a symmetric group could be realized as a sum of representations induced from linear characters of centralizers this question was answered positively by bergeron bergeron and garsia using a connection with the free lie algebra on n letters and independently by douglass pfeiffer and rohrle who connected the decomposition with the configuration space of n tuples of distinct complex numbers the mantaci reutenauer algebra of a hyperoctahedral group is a subalgebra of the group algebra that contains the descent algebra bonnafe and hohlweg showed that the natural map from the mantaci reutenauer algebra to the character ring is a surjection with kernel equal the jacobson radical in 2008 bonnafe asked whether the analog to stanley s question about the decomposition of the group algebra into a sum of induced linear characters holds in this paper we give a positive answer to bonnafe s question by explicitly constructing the required linear characters

a decomposition of solomon s descent algebra abstract a descent class in the symmetric group sn is the collection of permutations with a given descent set it was shown by l solomon j algebra41 1976 255 264 that the product in the group algebra q sn of two descent classes is a linear combination of descent classes thus descent classes generate a subalgebra of q sn we refer to it here as solomon s descent algebra and denote it by n this algebra is not semisimple but it has a faithul representation in terms of upper triangular matrices the main goal of this paper is a decomposition of its multiplicative structure it develops that n acts in a natural way on the so called lie monomials this action has a purely combinatorial description and is a crucial tool in the construction of a complete set of indecomposable representations of n in particular we obtain a natural basis of irreducible orthogonal idempotents indexed by partitions of n for the quotient n n natural bases of nilpotents and idempotents for the subspaces e ne for two arbitrary partitions and are also constructed and the dimensions of these spaces are given a combinatorial interpretation in terms of the so called decreasing factorization of an arbitary word into a product of lyndon words

modular lie powers and the solomon descent algebra let v be an r dimensional vector space over an infinite field f of prime characteristic p and let l n v denote the nth homogeneous component of the free lie algebra on v we study the structure of l n v as a module for the general linear group gl r f when n pk and k is not divisible by p and where r n our main result is an explicit 1 1 correspondence multiplicity preserving between the indecomposable direct summands of l k v and the indecomposable direct summands of l n v which are not isomorphic to direct summands of v n our approach uses idempotents of the solomon descent algebras and in addition a correspondence theorem for permutation modules of symmetric groups

spectral analogues of moon moser s theorem on hamilton paths in bipartite graphs abstract in 1962 erd s proved a theorem on the existence of hamilton cycles in graphs with given minimum degree and number of edges significantly strengthening in case of balanced bipartite graphs moon and moser proved a corresponding theorem in 1963 in this paper we establish several spectral analogues of moon and moser s theorem on hamilton paths in balanced bipartite graphs and nearly balanced bipartite graphs one main ingredient of our proofs is a structural result of its own interest involving hamilton paths in balanced bipartite graphs with given minimum degree and number of edges

the shifted poirier reutenauer algebra based on the shifted schensted correspondence and the shifted knuth equivalence a shifted analog of the poirier reutenauer algebra is introduced it is a right coideal subalgebra of the poirier reutenauer algebra and turns out to be a higher lift of schur s p functions its close relations with the peak subalgebra and the stembridge algebra of peak functions are also uncovered

on the quiver of the descent algebra abstract we study the quiver of the descent algebra of a finite coxeter group w the results include a derivation of the quiver of the descent algebra of types a and b our approach is to study the descent algebra as an algebra constructed from the reflection arrangement associated to w

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

skew schubert functions and the pieri formula for flag manifolds we show the equivalence of the pieri formula for flag manifolds with certain identities among the structure constants for the schubert basis of the polynomial ring this gives new proofs of both the pieri formula and of these identities a key step is the association of a symmetric function to a finite poset with labeled hasse diagram satisfying a symmetry condition this gives a unified definition of skew schur functions stanley symmetric functions and skew schubert functions defined here we also use algebraic geometry to show the coefficient of a monomial in a schubert polynomial counts certain chains in the bruhat order obtainng a combinatorial chain construction of schubert polynomials

duality between quasi symmetrical functions and the solomon descent algebra abstract the ring qsym of quasi symmetric functions is naturally the dual of the solomon descent algebra the product and the two coproducts of the first extending those of the symmetric functions correspond to a coproduct and two products of the second which are defined by restriction from the symmetric group algebra a consequence is that qsym is a free commutative algebra

two exponential type integrators for the good boussinesq equation we introduce two exponential type integrators for the good boussinesq equation they are of orders one and two respectively and they require lower spatial regularity of the solution compared to classical exponential integrators for the first integrator we prove first order convergence in h r for solutions in h r 1 with r 1 2 this new integrator even converges with lower order in h r for solutions in h r i e without any additional smoothness assumptions for the second integrator we prove second order convergence in h r for solutions in h r 3 with r 1 2 and convergence in l 2 for solutions in h 3 numerical results are reported to illustrate the established error estimates the experiments clearly demonstrate that the new exponential type integrators are favorable over classical exponential integrators for initial data with low regularity

spectral conditions and hamiltonicity of a balanced bipartite graph with large minimum degree abstract in this paper we establish some spectral conditions for a balanced bipartite graph with large minimum degree to be traceable or hamiltonian

a generating set of solomon s descent algebra abstract solomon s descent algebra is generated by sums of descent classes corresponding to certain hook shapes this particularly implies that the ring of class functions of any finite symmetric group s n is generated by the irreducible characters corresponding to certain hook partitions of n as another consequence a second generating set of solomon s descent algebra and of the ring of class functions of s n is obtained related to the major index of permutations

homomorphisms between solomon s descent algebras in a previous paper see a garsia and c reutenauer adv in math 77 1989 189 262 we have studied algebraic properties of the descent algebras n and shown how these are related to the canonical decomposition of the free lie algebra corresponding to a version of the poincare birkhoff witt theorem in the present paper we study homomorphisms between these algebras n the existence of these homomorphisms was suggested by properties of some directed graphs that we constructed in the previous paper reference above describing the structure of the descent algebras more precisely examination of the graphs suggested the existence of homomorphisms n n s and n n s we were then able to construct for any s 0 s a surjective homomorphism s n n s and an embedding s n s n which reflects these observations the homomorphisms s may also be defined as derivations of the free associative algebra q t 1 t 2 which sends t i on t i s if one identifies the basis element d s of n with some word coding s on the alphabet t t 1 t 2 we show that this mapping is indeed a homomorphism using the combinatorial description of the multiplication table of n given in the previous paper reference above

a quiver presentation for solomon s descent algebra the descent algebra w is a subalgebra of the group algebra qw of a finite coxeter group w which supports a homomorphism with nilpotent kernel and commutative image in the character ring of w thus w is a basic algebra and as such it has a presentation as a quiver with relations here we construct w as a quotient of a subalgebra of the path algebra of the hasse diagram of the boolean lattice of all subsets of s the set of simple reflections in w from this construction we obtain some general information about the quiver of w and an algorithm for the construction of a quiver presentation for the descent algebra w of any given finite coxeter group w

the peak algebra of the symmetric group revisited abstract the linear span p n of the sums of all permutations in the symmetric group sn with a given set of peaks is a sub algebra of the symmetric group algebra due to nyman this peak algebra is a left ideal of the descent algebra d n and the direct sum p of all p n is a hopf sub algebra of the direct sum d of all d n dual to the stembridge algebra of peak functions in our self contained approach peak counterparts of several results on the descent algebra are established including a simple combinatorial characterization of the algebra p n an algebraic characterization of p n based on the action on the poincare birkhoff witt basis of the free associative algebra the display of peak variants of the classical lie idempotents an eulerian type sub algebra of p n a description of the jacobson radical of p n and its nil potency index of the principal indecomposable and irreducible p n modules and of the cartan matrix of p n furthermore it is shown that the primitive lie algebra of p is free and that p is its enveloping algebra

tree expansion in time dependent perturbation theory the computational complexity of time dependent perturbation theory is well known to be largely combinatorial whatever the chosen expansion method and family of parameters combinatorial sequences goldstone and other feynman type diagrams etc we show that a very efficient perturbative expansion both for theoretical and numerical purposes can be obtained through an original parametrization by trees and generalized iterated integrals we emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics these tools are applied to the adiabatic approximation and the effective hamiltonian we prove perturbatively and nonperturbatively the convergence of morita s generalization of the gell mann and low wave function we show that summing all the terms associated with the same tree leads to an utter simplification where the sum is simpler than any of its terms finally we recover the rayleigh schrodinger time independent equation for the wave operator and we give an explicit nonrecursive expression for the term corresponding to an arbitrary tree

a symmetry of the descent algebra of a finite coxeter group abstract the descent algebra d w of a finite coxeter group w discovered by solomon in 1976 is a subalgebra of the group algebra of w due to solomon it is intimately linked to the representation theory of w by means of a homomorphism of algebras mapping d w into the algebra of class functions of w for w of type a jollenbeck and reutenauer derived the identity x y y x for all x y d w where class functions of w have been extended to the group algebra of w linearly they conjectured that this symmetry property of d w holds for arbitrary finite coxeter groups w this conjecture actually a combinatorial refinement is proven here as a consequence several properties of the characters of w afforded by the primitive idempotents of d w may be derived at once including a symmetry of the corresponding character table and a combinatorial description of their intertwining numbers with the descent characters of w this recovers and extends results of gessel reutenauer and scharf thibon on the symmetric group and of poirier on the hyperoctahedral group

the module structure of the solomon tits algebra of the symmetric group abstract let w s be a finite coxeter system tits defined an associative product on the set of simplices of the associated coxeter complex the corresponding semigroup algebra is the solomon tits algebra of w it contains the solomon algebra of w as the algebra of invariants with respect to the natural action of w on for the symmetric group s n there is a 1 1 correspondence between and the set of all set compositions or ordered set partitions of 1 n the product on has a simple combinatorial description in terms of set compositions we study in detail the representation theory of the solomon tits algebra of s n over an arbitrary field and show how our results relate to the corresponding results on the solomon algebra of s n this includes the construction of irreducible and principal indecomposable modules a description of the cartan invariants of the ext quiver and of the descending loewy series our approach builds on a twisted hopf algebra structure on the direct sum of all solomon tits algebras

cohomology of coxeter arrangements and solomon s descent algebra the authors would like to acknowledge support from the dfg priority programme spp1489 algorithmic and experimental methods in algebra geometry and number theory part of the research for this paper was carried out while the authors were staying at the mathematical research institute oberwolfach supported by the research in pairs programme the second author wishes to acknowledge support from science foundation ireland

quadratic dispersive generalized benjamin ono equation abstract this paper is concerned with quadratic dispersive generalized benjamin ono equation 1 1 1 2 we prove sharp local well posedness in h s s 1 4 with a solution map that is analytic from h s r to c 0 t h s r and the ill posed result in h s r as soon as s s in the sense that the flow map u 0 u t cannot be continuous

schubert polynomials for the classical groups the task of a theory of schubert polynomials is to produce explicit representatives for schubert classes in the cohomology ring of a flag variety and to do so in a manner that is as natural as possible from a combinatorial point of view to explain more fully let us review a special case the schubert calculus for grassmannians where one asks for the number of linear spaces of given dimension satisfying certain geometric conditions a typical problem is to find the number of lines meeting four given lines in general position in 3 space answer below for each of the four given lines the set of lines meeting it is a schubert variety in the grassmannian and we want the number of intersection points of these four subvarieties in the modem solution of this problem the schubert varieties induce canonical elements of the cohomology ring of the grassmannian called schubert classes the product of these schubert classes is the class of a point times the number of intersection points counted with appropriate multiplicities this reformulation of the problem though one of the great achievements of algebraic geometry is only part of a solution it remains to give a concrete model for the cohomology ring that makes explicit computation with schubert classes possible as it happens the cohomology rings of grassmannians can be identified with quotients of a polynomial ring so that schubert classes correspond to schur functions intersection numbers such as we are considering then turn out to be littlewood richardson coefficients for example the answer to our four lines 4 problem is the coefficient of the schur function s 2 2 in the product s 1 x or 2 for an extended treatment and history of the subject see 10 11 18 the identification of schur functions as schubert polynomials for grassmannians is a consequence of a more general and now highly developed theory of schubert polynomials for the flag varieties of the special linear groups sl n c the starting point for this more general theory is a construction of

signless laplacian spectral radius and hamiltonicity we give tight conditions on the signless laplacian spectral radius of a graph for the existence of hamiltonian paths and cycles

quadratic forms for the 1 d semilinear schrodinger equation this paper is concerned with 1 d quadratic semilinear schr6dinger equations we study local well posedness in classical sobolev space hs of the associated initial value problem and periodic boundary value problem our main interest is to obtain the lowest value of s which guarantees the desired local well posedness result we prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered

global existence of smooth solutions and stability of solitary waves for a generalized boussinesq equation certain generalizations of one of the classical boussinesq type equations u tt u xx u 2 u xx xx are considered it is shown that the initial value problem for this type of equation is always locally well posed it is also determined that the special solitary wave solutions of these equations are nonlinearly stable for a range of their phase speeds these two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations this contrasts with the results of blow up obtained recently by kalantarov and ladyzhenskaya for the same type of equation and casts additional light upon the results for the original version of this class of equations obtained via inverse scattering theory by deift tomei and trubowitz

a proof of solomon s rule abstract we put forward a proof of solomon s rule in terms of matrices for multiplication in the descent algebra of the symmetric group our proof exploits the graphs that we can obtain from all the subsets of the set of transpositions i i 1 n 1i 1

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation