a rotation group whose subspace arrangement is not from a real reflection group abstract we exhibit a family of real rotation groups whose subspace arrangements are not contained in that of any real reflection group answering a question of martino and singh

descent equalities and the inductive mckay condition for types b and e abstract we establish the inductive mckay condition introduced by isaacs malle navarro 14 for finite simple groups of lie types b l l 2 e 6 e 6 2 and e 7 thus leaving open only the types d and d 2 we bring to the methods previously used by the authors for type c 8 some descent arguments using shintani s norm map this provides for types different from a d d 2 a uniform proof of the so called global requirement of the criterion given by the second author in 32 2 12 the local requirements from that criterion are verified through a detailed study of the normalizers of relevant levi subgroups and their characters

divisionally free arrangements of hyperplanes let mathcal a mathcal a mathcal a h be the triple of hyperplane arrangements we show that the freeness of mathcal a h and the division of chi mathcal a t by chi mathcal a h t imply the freeness of mathcal a this division theorem improves the famous addition deletion theorem and it has several applications which include a definition of divisionally free arrangements it is a strictly larger class of free arrangements than the classical important class of inductively free arrangements also whether an arrangement is divisionally free or not is determined by the combinatorics moreover we show that a lot of recursively free arrangements to which almost all known free arrangements are belonging are divisionally free

dispersive blowup of solutions of generalized korteweg de vries equations abstract the strong effect of dispersion on short wavelength disturbances featured by the korteweg de vries equation and some of its generalizations is exploited to provide solutions of these equations that correspond to infinitely smooth initial data which exhibit a specified loss of spatial smoothness at particular times the points in space time at which smoothness is lost may even comprise an arbitrary discrete subset of the upper half plane x t x r t 0 our results are related to recent work on smoothing of solutions of such equations some of which are sharpened here and they show that in certain aspects these earlier results are not far from being optimal the theory makes use of new results concerning well posedness of such equations in weighted sobolov spaces and some detailed analysis of the linear korteweg de vries equation

the module of logarithmic p forms of a locally free arrangement for an essential central hyperplane arrangement a v k n 1 we show that 1 a the module of logarithmic one forms with poles along a gives rise to a locally free sheaf on p n if and only if for all x la with rank x dim v the module 1 ax is free our main result says that in this case a t is essentially the chern polynomial the proof is based on a result of solomon terao 16 and a formula we give for the chern polynomial of a bundle e on p n in terms of the hilbert series of m2zh 0 p n i e m if 1 a has projective dimension one and is locally free we give a minimal

the spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices in this paper we determine graphs with the largest spectral radius among all the unicyclic and all the bicyclic graphs with n vertices and k pendant vertices respectively

a basis construction of the extended catalan and shi arrangements of the type a2 abstract in 10 terao proved the freeness of multi coxeter arrangements with constant multiplicities by giving an explicit construction of bases combining it with algebro geometric method yoshinaga proved the freeness of the extended catalan or shi arrangements in 12 however there have been no explicit constructions of the bases for the logarithmic derivation modules of the extended catalan and shi arrangements in this paper we give the first explicit construction of them when the root system is of the type a 2

remark on nonexistence of global solutions of the initial boundary value problem for the nonlinear klein gordon equation sufficient conditions are given so that the solutions of the initial boundary value problem for the nonlinear klein gordon equation do not exist for all t 0

the development version of the chevie package of gap3 abstract i describe the current state of the development version of the chevie package which allows to transform the theories of coxeter groups reductive algebraic groups complex reflection groups hecke algebras braid monoids etc into actual computations examples are given showing the code to check some results of lusztig

crossed s matrices and fourier matrices for coxeter groups with automorphism abstract we study crossed s matrices for braided g crossed categories and reduce their computation to a submatrix of the de equivariantization we study the more general case of a category containing the symmetric category rep a z with a a finite cyclic group and z a such that z 2 1 we give two examples of such categories which enable us to recover the fourier matrix associated with the big family of unipotent characters of the dihedral groups with automorphism as well as the fourier matrix of the big family of unipotent characters of the ree group of type f 4 2

on the regularity of solutions to a class of nonlinear dispersive equations we shall study special regularity properties of solutions to some nonlinear dispersive models the goal is to show how regularity on the initial data is transferred to the solutions this will depend on the spaces where regularity is measured

the majorization theorem of extremal pseudographs a pseudograph is a graph in which both loops and multiple edges are permitted suppose d1 d2 dn and d1 d2 dn are two positive non increasing degree sequences we write if and only if i 1ndi i 1ndi and i 1jdi i 1jdi for all j 1 2 n let be the class of connected undirected pseudographs with degree sequence let g and g be the spectral radius and signless laplacian spectral radius of g respectively in this paper the extremal pseudographs with the largest respectively signless laplacian spectral radii in are characterized furthermore we show that if g and g are the pseudographs with the largest respectively signless laplacian spectral radii in and respectively then g g and g g

blow up of smooth solutions of the korteweg de vries equation abstract the paper is devoted both to some initial boundary value problems and to the cauchy problem for the kdv equation blow up results are proved for these problems by using the nonlinear capacity approach 9

the spectral radius of trees on k pendant vertices abstract in this paper we consider the following problem of all trees of order n with k pendant vertices n k fixed which achieves the maximal spectral radius we show that the maximal spectral radius is obtained uniquely at t n k where t n k is a tree obtained from a star k 1 k and k paths of almost equal lengths by joining each pendant vertex of k 1 k to an end vertex of one path we also discuss the spectral radius of t n k and get some results

rationally smooth schubert varieties and inversion hyperplane arrangements abstract we show that an element w of a finite weyl group w is rationally smooth if and only if the hyperplane arrangement i associated to the inversion set of w is inductively free and the product d 1 1 d l 1 of the coexponents d 1 d l is equal to the size of the bruhat interval e w where e is the identity in w as part of the proof we describe exactly when a rationally smooth element in a finite weyl group has a chain billey postnikov decomposition for finite coxeter groups we show that chain billey postnikov decompositions are connected with certain modular coatoms of i

the majorization theorem of connected graphs abstract let d 1 d 2 d n and d 1 d 2 d n be two non increasing degree sequences we say is majorizated by denoted by if and only if i 1 n d i i 1 n d i and i 1 j d i i 1 j d i for all j 1 2 n if the degree of vertex v is resp not equal to 1 then we call v a pendant resp non pendant vertex of g we use c to denote the class of connected graphs with degree sequence suppose and are two non increasing c cyclic degree sequences let g and g be the graphs with greatest spectral radii in c and c respectively in this paper we shall prove that if g and g have the same number of pendant vertices and the degrees of all non pendant vertices of g are greater than c then g g

deletion theorem and combinatorics of hyperplane arrangements we show that the deletion theorem of a free arrangement is combinatorial i e whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice in fact we give a sufficient and necessary condition for the deletion theorem in terms of characteristic polynomials as a corollary we prove that whether a free arrangement has a free filtration is also combinatorial the proof is based on the result about a minimal set of generators of a logarithmic derivation module of a multiarrangement which satisfies the b 2 equality

reflection groups in algebraic geometry after a brief exposition of the theory of discrete reflection groups in spherical euclidean and hyperbolic geometry as well as their analogs in complex spaces we present a survey of appearances of these groups in various areas of algebraic geometry

restricting invariants of unitary reflection groups suppose that g is a finite unitary reflection group acting on a complex vector space v and x is the fixed point subspace of an element of g define n to be the setwise stabilizer of x in g z to be the pointwise stabilizer and c n z then restriction defines a homomorphism from the algebra of g invariant polynomial functions on v to the algebra of c invariant functions on x extending earlier work by douglass and rohrle for coxeter groups we characterize when the restriction mapping is surjective for arbitrary unitary reflection groups g in terms of the exponents of g and c and their reflection arrangements a consequence of our main result is that the variety of g orbits in the g saturation of x is smooth if and only if it is normal

dual braid monoids mikado braids and positivity in hecke algebras we study the rational permutation braids that is the elements of an artin tits group of spherical type which can be written x 1 y where x and y are prefixes of the garside element of the braid monoid we give a geometric characterization of these braids in type a n and b n and then show that in spherical types different from d n the simple elements of the dual braid monoid for arbitrary choice of coxeter element embedded in the braid group are rational permutation braids we conjecture this to hold also in type d n this property implies positivity properties of the polynomials arising in the linear expansion of their images in the iwahori hecke algebra when expressed in the kazhdan lusztig basis in type a n it implies positivity properties of their images in the temperley lieb algebra when expressed in the diagram basis

the first two obstructions to the freeness of arrangements abstract in his previous paper the author characterized free arrangements bythe vanishing of cohomology modules of a certain sheaf of graded modules overa polynomial ring thus the homogeneous components of these cohomologymodules can be viewed as obstructions to the freeness of an arrangement inthis paper the first two obstructions are studied in detail in particular thecomponent of degree zero of the first nontrivial cohomology module has a closerelation to formal arrangements and to the operation of truncation this enablesus to prove that in dimension greater than two every free arrangement is formal and not a proper truncation of an essential arrangement introductionin this paper we develop further the ideas from y where certain sheaf co homology was applied to the theory of arrangements let v be an dimensional 2 vector space over an arbitrary field kand sf an arrangement of n 2 hyperplanes in v for each h sfwe fix a linear functional ah v such that kera 77 let s d 0sdbe the graded symmetric algebra of v and der s der s s the gradeds module of derivations of s the following graded submodule of deri wasintroduced by h terao ti t3 and studied in many papers on arrangements d d s 6 der s 0 ah sah h sf if the module d is freethen the arrangement sf is called free let lo be the lattice of all intersections of hyperplanes from sf except

on the spectral radius of 0 1 matrices we determine the maximum spectral radius for 0 1 matrices with k2 andk2 1 1 s respectively and for symmetric 0 1 matrices with zero trace and e k21 s graphs with e edges in all cases equality is characterized

multiderivations of coxeter arrangements let v be an l dimensional euclidean space let g o v be a finite irreducible orthogonal reflection group let be the corresponding coxeter arrangement let s be the algebra of polynomial functions on v for h choose h v such that h ker h for each nonnegative integer m define the derivation module d m ders h s mh the module is known to be a free s module of rank l by k saito 1975 for m 1 and l solomon h terao 1998 for m 2 the main result of this paper is that this is the case for all m moreover we explicitly construct a basis for d m their degrees are all equal to mh 2 when m is even or are equal to m 1 h 2 mi 1 i l when m is odd here m1 ml are the exponents of g and h ml 1 is the coxeter number the construction heavily uses the primitive derivation d which plays a central role in the theory of flat generators by k saito or equivalently the frobenius manifold structure for the orbit space of g some new results concerning the primitive derivation d are obtained in the course of proof of the main result

partial symmetry reflection monoids and coxeter groups abstract this is the first of a series of papers in which we initiate and develop the theory of reflection monoids motivated by the theory of reflection groups the main results identify a number of important inverse semigroups as reflection monoids introduce new examples and determine their orders

kawahara solitons in boussinesq equations using a robust christov galerkin spectral method we develop a robust christov galerkin spectral technique for computing interacting localized wave solutions of and fourth and sixth order generalized wave equations to this end a special complete orthonormal system of functions in l 2 is used whose rate of convergence is shown to be exponential for the cases under consideration for the time stepping an implicit algorithm is chosen which makes use of the banded structure of the matrices representing the different spatial derivatives as featuring examples the head on collision of solitary waves is investigated for a sixth order generalized boussinesq equation and a fourth order boussinesq type equation with a linear term its solutions comprise monotone shapes sech es and damped oscillatory shapes kawahara solitons the numerical results are validated against published data in the literature using the method of variational imbedding

reflection subgroups of finite complex reflection groups abstract all simple extensions of the reflection subgroups of a finite complex reflection group g are determined up to conjugacy as a consequence it is proved that if the rank of g is n and if g can be generated by n reflections then for every set r of n reflections which generate g every subset of r generates a parabolic subgroup of g

on inductively free restrictions of reflection arrangements abstract let w be a finite complex reflection group acting on the complex vector space v and let a w a w v be the associated reflection arrangement in 6 we classified all inductively free reflection arrangements a w the aim of this note is to extend this work by determining all inductively free restrictions of reflection arrangements

simple root bases for shi arrangements abstract in his affirmative answer to the edelman reiner conjecture yoshinaga proved that the logarithmic derivation modules of the cones of the extended shi arrangements are free modules however we know very little about the bases themselves except their existence in this article we prove the unique existence of two distinguished bases which we call the simple root basis plus srb and the simple root basis minus srb they are characterized by nice divisibility properties relative to the simple roots

characteristic polynomials of subspace arrangements and finite fields let a be any subspace arrangement in rndefined over the integers and let fqdenote the finite field withqelements letqbe a large prime we prove that the characteristic polynomial a q of a counts the number of points in fnqthat do not lie in any of the subspaces ofa viewed as subsets of fnq this observation which generalizes a theorem of blass and sagan about subarrangements of the bnarrangement reduces the computation of a q to a counting problem and provides an explanation for the wealth of combinatorial results discovered in the theory of hyperplane arrangements in recent years the basic idea has its origins in the work of crapo and rota 1970 we find new classes of hyperplane arrangements whose characteristic polynomials have simple form and very often factor completely over the nonnegative integers

weak asymptotics method and the interaction of infinitely narrow δ solitons abstract we present a new method for studying the interaction of solitons for nonintegrable kdv type equations with small dispersion and test this method for the kdv equation

coxeter multiarrangements with quasi constant multiplicities we study structures of derivation modules of coxeter multiarrangements with quasi constant multiplicities by using the primitive derivation as an application we show that the characteristic polynomial of a coxeter multiarrangement with quasi constant multiplicity is combinatorially computable

hyperplane arrangements interval orders and trees abstract a hyperplane arrangement is a finite set of hyperplanes in a real affine space an especially important arrangement is the braid arrangement which is the set of all hyperplanes xi xj 1 1 i j n in rn some combinatorial properties of certain deformations of the braid arrangement are surveyed in particular there are unexpected connections with the theory of interval orders and with the enumeration of trees for instance the number of labeled interval orders that can be obtained from n intervals i1 in of generic lengths is counted there is also discussed an arrangement due to n linial whose number of regions is the number of alternating or intransitive trees as defined by gelfand graev and postnikov gelfand i m graev m i and postnikov a 1995 preprint finally a refinement is given related to counting labeled trees by number of inversions of a result of shi shi j y 1986 lecture notes in mathematics no 1179 springer verlag that a certain deformation of the braid arrangement has n 1 n 1 regions

arrangements defined by unitary reflection groups let v be a complex vector space of dimension i an arrangement in v is a finite set d of hyperplanes all containing the origin let l l d be the set of intersections of elements of r partially order l by reverse inclusion so that l has v as its minimal element and d as its set of atoms the poset l is a finite geometric lattice with rank function r x dim v x xe l without loss of generality we assume that n h h 0 is the maximal element of l and thus l has rank i the characteristic polynomial z l t of l is defined by

the moduli space of three qutrit states we study the invariant theory of trilinear forms over a three dimensional complex vector space and apply it to investigate the behavior of pure entangled three partite qutrit states and their normal forms under local filtering operations slocc we describe the orbit space of the slocc group sl 3 c 3 both in its affine and projective versions in terms of a very symmetric normal form parametrized by three complex numbers the parameters of the possible normal forms of a given state are roots of an algebraic equation which is proved to be solvable by radicals the structure of the sets of equivalent normal forms is related to the geometry of certain regular complex polytopes

derivations of an effective divisor on the complex projective line in this paper we consider an effective divisor on the complex projective line and associate with it the module d consisting of all the derivations 0 such that i i c i mi i for every i where i i is the ideal of p i the module d is graded and free of rank 2 the degrees of its homogeneous basis called the exponents form an important invariant of the divisor we prove that under certain conditions on m i the exponents do not depend on p i our main result asserts that if these conditions do not hold for m i then there exists a general position of n points for which the exponents do not change we give an explicit formula for them we also exhibit some examples of degeneration of the exponents in particular those where the degeneration is defined by the vanishing of certain schur functions as an application and motivation we show that our results imply terao s conjecture concerning the combinatorial nature of the freeness of hyperplane arrangements for certain new classes of arrangements of lines in the complex projective plane

quasi isolated blocks and brauer s height zero conjecture this paper has two main results firstly we complete the parametrisation of all p blocks of nite quasi simple groups by nding the so called quasi isolated blocks of exceptional groups of lie type for bad primes this relies on the explicit decomposition of lusztig induction from suitable levi subgroups our second major result is the proof of one direction of brauer s long standing height zero conjecture on blocks of nite groups using the

freeness of multi reflection arrangements via primitive vector fields abstract in 2002 terao showed that every reflection multi arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations we first generalize terao s result to multi arrangements stemming from well generated unitary reflection groups where the multiplicity of a hyperplane depends on the order of its stabilizer here the exponents depend on the exponents of the dual reflection representation we then extend our results further to all imprimitive irreducible unitary reflection groups in this case the exponents turn out to depend on the exponents of a certain galois twist of the dual reflection representation that comes from a beynon lusztig type semi palindromicity of the fake degrees

free arrangements of hyperplanes and supersolvable lattices a finite family of hyperplanes through the origin in c is called an iarrangement of hyperplanes or simply an l arrangement an l arrangement is said to be free if the module of logarithmic vector fields holomorphic vector fields tangent to all hyperplanes in the arrangement is a free module the accurate definition will be given in 2 1 let a be a free i arrangement then a set do d of i 1 nonnegative integers can be defined 2 3 these d d are called the exponents of a on the other hand one can associate a geometric lattice l a with any arrangement a 3 1 the characteristic polynomial xlca t e z t is defined by using the mobius function 3 4 the second author 7 proved that

coxeter and crystallographic arrangements are inductively free abstract using the classification of finite weyl groupoids we prove that crystallographic arrangements a large subclass of the class of simplicial arrangements which was recently defined are hereditarily inductively free in particular all crystallographic reflection arrangements are hereditarily inductively free among them the arrangement of type e 8 with little extra work we prove that also all coxeter arrangements are inductively free

characterization of a free arrangement and conjecture of edelman and reiner we consider a hyperplane arrangement in a vector space of dimension four or higher in this case the freeness of the arrangement is characterized by properties around a fixed hyperplane as an application we prove the freeness of cones over certain truncated affine weyl arrangements which was conjectured by edelman and reiner

recursively free reflection arrangements abstract let a a w be the reflection arrangement of the finite complex reflection group w by terao s famous theorem the arrangement a is free in this paper we classify all reflection arrangements which belong to the smaller class of recursively free arrangements moreover for the case that w admits an irreducible factor isomorphic to g 31 we obtain a new computer free proof for the non inductive freeness of a w since our classification implies the non recursive freeness of the reflection arrangement a g 31 we can prove a conjecture by abe about the new class of divisionally free arrangements which he recently introduced

arrangements of ideal type abstract in 2006 sommers and tymoczko defined so called arrangements of ideal type a i stemming from ideals i in the set of positive roots of a reduced root system they showed in a case by case argument that a i is free if the root system is of classical type or g 2 and conjectured that this is also the case for all types this was established only very recently in a uniform manner by abe barakat cuntz hoge and terao the set of non zero exponents of the free arrangement a i is given by the dual of the height partition of the roots in the complement of i in the set of positive roots generalizing the shapiro steinberg kostant theorem which asserts that the dual of the height partition of the set of positive roots gives the exponents of the associated weyl group our first aim in this paper is to investigate a stronger freeness property of the a i we show that all a i are inductively free with the possible exception of some cases in type e 8 in the same paper from 2006 sommers and tymoczko define a poincare polynomial i t associated with each ideal i which generalizes the poincare polynomial w t for the underlying weyl group w solomon showed that w t satisfies a product decomposition depending on the exponents of w for any coxeter group w sommers and tymoczko showed in a case by case analysis in types a n b n and c n and some small rank exceptional types that a similar factorization property holds for the poincare polynomials i t generalizing the formula of solomon for w t they conjectured that their multiplicative formula for i t holds in all types in our second aim to investigate this conjecture further the same inductive tools we develop to obtain inductive freeness of the a i are also employed here we also show that this conjecture holds inductively in almost all instances with only a small number of possible exceptions

analysis on the blow up of solutions to a class of integrable peakon equations abstract we investigate the blow up mechanism of solutions to a class of quasilinear integrable equations which could possess peakons the dynamics of the blow up quantity along the characteristics is established by the riccati type differential inequality which involves the interaction among three parts a local nonlinearity a nonlocal term and a term stemming from the weak linear dispersion to analyse the interplay among these quantities we provide two different approaches the first one is designed for the case when the equations do not exhibit a weak linear dispersion and hence focuses on the interplay between the first two parts the method is based on a refined analysis on either evolution of the solution u and its gradient u x that is c u u x or the growth rate of the relative ratio u x u the second one handles the general situation when all of three parts are present the idea is to extract the truly blow up component from the riccati type differential inequality and utilizes the morawetz type identity or higher order conservation laws to show that such a component blows up in finite time before the other component degenerates

the spectral radius of bicyclic graphs with prescribed degree sequences restricted to the bicyclic graphs with prescribed degree sequences we determine the unique graph with the largest spectral radius with respect to the adjacency matrix

smoothing properties for the higher order nonlinear schrodinger equation with constant coefficients abstract we study local and global existence and smoothing properties for the initial value problem associated to a higher order nonlinear schrodinger equation with constant coefficients which appears as a model for propagation of pulse in an optical fiber

inductive freeness of ziegler s canonical multiderivations for reflection arrangements abstract let a be a free hyperplane arrangement in 1989 ziegler showed that the restriction a of a to any hyperplane endowed with the natural multiplicity is then a free multiarrangement we initiate a study of the stronger freeness property of inductive freeness for these canonical free multiarrangements and investigate them for the underlying class of reflection arrangements more precisely let a a w be the reflection arrangement of a complex reflection group w by work of terao each such reflection arrangement is free thus so is ziegler s canonical multiplicity on the restriction a of a to a hyperplane we show that the latter is inductively free as a multiarrangement if and only if a itself is inductively free

free and non free multiplicities on the a3 arrangement abstract we give a complete classification of free and non free multiplicities on the a 3 braid arrangement namely we show that all free multiplicities on a 3 fall into two families that have been identified by abe terao wakefield 2007 and abe nuida numata 2009 the main tool is a new homological obstruction to freeness derived via a connection to multivariate spline theory

the characteristic polynomial of a multiarrangement abstract given a multiarrangement of hyperplanes we define a series by sums of the hilbert series of the derivation modules of the multiarrangement this series turns out to be a polynomial using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arrangement the characteristic polynomial of an arrangement is a combinatorial invariant but this generalized characteristic polynomial is not however when the multiarrangement is free we are able to prove the factorization theorem for the characteristic polynomial the main result is a formula that relates global data to local data of a multiarrangement given by the coefficients of the respective characteristic polynomials this result gives a new necessary condition for a multiarrangement to be free consequently it provides a simple method to show that a given multiarrangement is not free

universal deformations of the finite quotients of the braid group on 3 strands abstract we prove that the quotients of the group algebra of the braid group on 3 strands by a generic quartic and quintic relation respectively have finite rank this is a special case of a conjecture by broue malle and rouquier for the generic hecke algebra of an arbitrary complex reflection group exploring the consequences of this case we prove that we can determine completely the irreducible representations of this braid group of dimension at most 5 thus recovering a classification of tuba and wenzl in a more general framework

free arrangements and coefficients of characteristic polynomials ziegler showed that the multirestriction of a free arrangement is also free after ziegler s work several results concerning the reverse direction i e characterizing freeness of an arrangement via that of its multirestriction have appeared in this paper we prove a new characterization of freeness in which the second betti number of the arrangement plays a crucial role

new method and new results on the order of spectral radius this paper presents a simple method to order trees unicyclic graphs and bicyclic graphs according to their spectral radii which is based on the application of the majorization theorem of tree unicyclic and bicyclic graphs respectively moreover we obtain some new results on the order of spectral radii of unicyclic and bicyclic graphs by employing this new method

generalized exponents of a free arrangement of hyperplanes and shepherd todd brieskorn formula we define an n arrangement as a finite family of hyperplanes through the origin in c 1 in 11 and 12 we studied the free arrangement and defined its structure sequence their definitions will be given again in sect 2 in this article we say the generalized exponents instead of the structure sequence let d o d 1 d be the generalized exponents of a free n arrangement x let xj u h our main result is hex

computational proof of the mackey formula for q 2 let g be a connected reductive group defined over a finite field with q elements we prove that the mackey formula for the lusztig induction and restriction holds in g whenever q 2 or g does not have a component of type e

higher order derivative nonlinear schrodinger equation in the critical case we consider the cauchy problem for the higher order derivative nonlinear schrodinger equation our purpose in this paper is to prove the large time asymptotic behavior of solutions under the non zero mass condition u0xdx 0

large time asymptotics of solutions to the generalized korteweg de vries equation we study the asymptotic behavior for large time of solutions to the cauchy problem for the generalized korteweg de vries gkdv equationut u 1 u x 13uxxx 0 wherex t rwhen the initial data are small enough if the power of the nonlinearity is greater than 3 then the solution of the cauchy problem has a quasilinear asymptotic behavior for large time more precisely we show that the solutionu t satisfies the decay estimate u t l c 1 t 1 3 1 1 for 4 uux t l ct 2 3 1 t 1 3and using these estimates we prove the existence of the scattering stateu l2such that u t u t u l2 ct 3 3for any small initial data belonging to the weighted sobolev spaceh1 1 f l2 1 x 2 1 2 1 2x 1 2 f l2 whereu t is the airy free evolution group

normalisers of parabolic subgroups in finite unitary reflection groups abstract it is well known that the normaliser of a parabolic subgroup of a finite coxeter group is the semidirect product of the parabolic subgroup by the stabiliser of a set of simple roots we show that a similar result holds for all finite unitary reflection groups namely the normaliser of a parabolic subgroup is a semidirect product and in many but not all cases the complement can be obtained as the stabiliser of a set of roots in addition we record when the complement acts as a reflection group on the space of fixed points of the parabolic subgroup

integrals involving products of airy functions their derivatives and bessel functions abstract a new integral representation of the hankel transform type is deduced for the function f n x z z n 1 a i x z a i x z with x r z 0 and n n this formula involves the product of airy functions their derivatives and bessel functions the presence of the latter allows one to perform various transformations with respect to z and obtain new integral formulae of the type of the mellin transform k transform laplace and fourier transform some integrals containing airy functions their derivatives and chebyshev polynomials of the first and second kind are computed explicitly a new representation is given for the function a i z 2 with z c

milnor fiber complexes for shephard groups abstract the symmetry group of a regular polytope p is a finite coxeter group w the intersection of the unit sphere with the reflecting hyperplanes of w induces a simplicial triangulation of the sphere called the coxeter complex w in case p is convex radial projection maps the barycentric subdivision of p homeomorphically onto w the symmetry group of a regular complex polytope p is a shephard group g in this paper we construct a simplicial complex g in the milnor fiber of a g invariant polynomial of minimal positive degree which shares many of the properties of the coxeter complex in case p is non starry we construct a linear complex b p p which is homeomorphic to g thus the relation of b p to g is the same as the relation of the barycentric subdivision of a convex polytope to the coxeter complex

a simplicial complex of nichols algebras we translate the concept of restriction of an arrangement in terms of hopf algebras in consequence every nichols algebra gives rise to a simplicial complex decorated by nichols algebras with restricted root systems as applications some of these nichols algebras provide weyl groupoids which do not arise for diagonal nichols algebras and in fact we realize all root systems of finite weyl groupoids of rank greater than three further our result explains the root systems of the folded nichols algebras over nonabelian groups and of generalized satake diagrams

root system chip firing i interval firing jim propp recently introduced a variant of chip firing on a line where the chips are given distinct integer labels hopkins mcconville and propp showed that this process is confluent from some but not all initial configurations of chips we recast their set up in terms of root systems labeled chip firing can be seen as a root firing process which allows the moves for alpha in phi whenever langle lambda alpha vee rangle 0 where phi is the set of positive roots of a root system of type a and lambda is a weight of this root system we are thus motivated to study the exact same root firing process for an arbitrary root system actually this central root firing process is the subject of a sequel to this paper in the present paper we instead study the interval root firing processes determined by for alpha in phi whenever langle lambda alpha vee rangle in k 1 k 1 or langle lambda alpha vee rangle in k k 1 for any k ge 0 we prove that these interval firing processes are always confluent from any initial weight we also show that there is a natural way to consistently label the stable points of these interval firing processes across all values of k so that the number of weights with given stabilization is a polynomial in k we conjecture that these ehrhart like polynomials have nonnegative integer coefficients