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

boundary integral equations for the scattering of electromagnetic waves by a homogeneous chiral obstacle the time harmonic scattering of electromagnetic waves by a homogeneous chiral media are considered herein the scattering problem is reduced using bohren s decomposition to an equivalent boundary integral equation which is shown to be uniquely solvable except for a discrete set of values of electromagnetic parameters of the scatterer it should be noted that the boundary integral operators appearing in this equation are the same as for a dielectric scatterer and hence their mapping properties are well known

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

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

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

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

two dimensional blasius viscous flow of a power law fluid over a semi infinite flat plane analytic results are obtained for the similarity equation governing the two dimensional blasius viscous flow of a power law fluid over a semi infinite flat plane via taylor series for small values of the independent similarity variable then an analytic perturbative procedure is used to determine an approximate solution that exhibits the correct asymptotic behavior this perturbation method allows for the computation of the shear stress at the wall something which is impossible with a taylor series approach it is found that the perturbation solutions converge sufficiently rapidly indeed a first order approximation gives qualitatively accurate results furthermore we employ the perturbation method to deduce the influence of the power law index n on the obtained similarity solutions

magnetohydrodynamic boundary layer on a flat plate further analytic results further analytic results are deduced with the magnetohydrodynamic boundary layer equations for a flat plate the asymptotic behavior of the solutions is deduced using the scaling group method then an analytic perturbative procedure is used to determine an approximate solution that exhibits this asymptotic behavior

a priori estimates for a singular limit approximation of the constitutive laws for chiral media in the time domain abstract the equations for the evolution of electromagnetic fields in chiral media in the time domain are nonlocal in time in this work we study the validity of a singular limit local in time approximation for these nonlocal in time equations by estimating the size of the difference of the fields as predicted by both models in particular we establish an a priori estimate for this difference depending on the time horizon properties of the domain spatial properties of the initial data and the source terms and the chirality measure of the approximating model

strongly elliptic boundary integral equations for electromagnetic transmission problems we study a boundary integral equation method for transmission problems for strongly elliptic differential operators which yields a strongly elliptic system of pseudodifferential operators and which therefore can be used for numerical computations with galerkin s procedure the method is shown to work for the vector helmholtz equation in 3 with electromagnetic transmission conditions we propose a slightly modified system of boundary values in order for the corresponding bilinear form to be coercive over h 1 we analyse the boundary integral equations using the calculus of pseudodifferential operators here the concept of the principal symbol is used to derive existence and regularity results for the solution

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

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

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

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

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

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

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

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

electromagnetic scattering by a homogeneous chiral obstacle boundary integral equations and low chirality approximations time harmonic electromagnetic waves are scattered by a homogeneous chiral obstacle the corresponding transmission problem is reduced to a pair of coupled integral equations over s where s is the interface between the obstacle and the surrounding medium this is done using a generalization of the stratton chu representation that is valid for chiral media the integral equations obtained are a generalization of those obtained by muller for a homogeneous dielectric obstacle finally we develop approximations for low chirality obstacles these approximations can be computed using simple modifications to existing codes for solving muller s equations

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

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

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

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

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

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

an exterior boundary value problem for the maxwell equations with boundary data in a sobolev space we treat the time harmonic maxwell equations in an exterior domain with prescribed boundary data n e in the sobolev space of square integrable tangential fields with square integrable surface divergence by using boundary integral equation methods existence and uniqueness results are established furthermore we investigate the completeness of electric and magnetic dipoles distributed on an inner surface in this sobolev space

a challenging nonlinear problem for numerical techniques in this paper we show that a nonlinear boundary value problem describing blasius viscous flow of a kind of non newtonian fluid has an infinite number of explicit analytic solutions these solutions are rather sensitive to the second order derivative at the boundary and the difference of the second derivatives of two obviously different solutions might be less than 10 1 0 0 0 therefore it seems impossible to find out all of these solutions by means of current numerical methods thus this nonlinear problem might become a challenge to current numerical techniques

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

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

boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle time harmonic electromagnetic waves are scattered by a homogeneous dielectric obstacle the corresponding electromagnetic transmission problem is reduced to a single integral equation over s for a single unknown tangential vector field where s is the interface between the obstacle and the surrounding medium in fact several different integral equations are derived and analysed including two previously known equations due to e marx and j r mautz and two new singular integral equations mautz s equation is shown to be uniquely solvable at all frequencies a new uniquely solvable singular integral equation is also found the paper also includes a review of methods using pairs of coupled integral equations over s it is these methods that are usually used in practice although single integral equations seem to offer some computational advantages

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

a new perturbative approach to nonlinear problems a recently proposed perturbative technique for quantum field theory consists of replacing nonlinear terms in the lagrangian such as 4 by 2 1 and then treating as a small parameter it is shown here that the same approach gives excellent results when applied to difficult nonlinear differential equations such as the lane emden thomas fermi blasius and duffing equations

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

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