the cohomology of the cotangent bundle of heisenberg groups abstract given a parabolic subalgebra g1 n of a semisimple lie algebra kostant ann math 1963 and griffiths acta math 1963 independently computed the g1 invariants in the cohomology group of n with exterior adjoint coefficients by a theorem of bott ann math 1957 this is the cohomology of the associated compact homogeneous space with coefficients in the sheaf of local holomorphic forms in this paper we determine explicitly the full module structure over the symplectic group of the cohomology group of the heisenberg lie algebra with exterior adjoint coefficients this is the cohomology of the cotangent bundle of the heisenberg group

on the negative squares of indefinite sturm liouville operators abstract the number of negative squares of all self adjoint extensions of a simple symmetric operator of defect one with finitely many negative squares in a krein space is characterized in terms of the behaviour of an abstract titchmarsh weyl function near 0 and these results are applied to a large class of symmetric and self adjoint indefinite sturm liouville operators with indefinite weight functions

bounding the equivariant betti numbers of symmetric semi algebraic sets abstract let r be a real closed field the problem of obtaining tight bounds on the betti numbers of semi algebraic subsets of r k in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to ole nik and petrovski thom and milnor these bounds are all exponential in the number of variables k motivated by several applications in real algebraic geometry as well as in theoretical computer science where such bounds have found applications we consider in this paper the problem of bounding the equivariant betti numbers of symmetric algebraic and semi algebraic subsets of r k we obtain several asymptotically tight upper bounds in particular we prove that if s r k is a semi algebraic subset defined by a finite set of s symmetric polynomials of degree at most d then the sum of the s k equivariant betti numbers of s with coefficients in q is bounded by s k d o d unlike the classical bounds on the ordinary betti numbers of real algebraic varieties and semi algebraic sets the above bound is polynomial in k when the degrees of the defining polynomials are bounded by a constant as an application we improve the best known bound on the ordinary betti numbers of the projection of a compact algebraic set improving for any fixed degree the best previously known bound for this problem due to gabrielov vorobjov and zell

long time large scale properties of the noisy driven diffusion equation we study the driven diffusion equation describing the dynamics of density fluc tuations x t in powders or traffic flows we have performed quite detailed numerical simulations of this equation in one dimension focusing in particular on the scaling behaviour of the correlation function t 0 0 one of our motivations was to assess the validity of various theoretical approaches such as renormalization group and different self consistent truncation schemes to these nonlinear dynamical equations although all of them are seen to predict correctly the scaling exponents only one of them where the non exponential nature of the relaxation is taken into account is able to reproduce satisfactorily the value of the numerical prefactors several other interesting issues such as the noise spectrum of the output current or the statistics of distance between jams showing a transition between a laminar regime for small noise to a jammed regime for higher noise are also investigated

two point two time closures applied to forced isotropic turbulence abstract the direct interaction approximation and the local energy transfer theory were used to calculate freely decaying turbulence for a range of initial taylor reynolds numbers 3 r t 0 129 and forced turbulence for evolved taylor reynolds numbers r t evolved 88 and r t evolved 232 in the case of the freely decaying turbulence the new feature of the work as compared to mccomb et al j fluid mech 245 1992 279 was the ability to make comparisons with a direct numerical simulation at the same initial conditions we were also able to obtain experimental results from the simulation with an estimate of the experimental error we present a small sample of results from an extensive investigation sufficient to confirm that both closures agree with the simulation to within the experimental error in broad agreement with our earlier investigations however the main motivation of the work was to study the application of the closures to stationary turbulence and here we faced new problems such as how to force the closures or the need to compute the time history integrals of the closures over the long transient period during which the numerical simulation settles down for this reason our results should be seen as rather preliminary and tentative in nature essentially we found that agreement between the closures and the simulation was not as good as for free decay for low wavenumber behaviour but was quite good for high wavenumber spectra and quantities determined by this region such as the taylor microscale we conclude that further work is needed to establish just what is a fair comparison between forced closure calculations and the analogous direct numerical simulation

subspace arrangements over finite fields cohomological and enumerative aspects abstract the enumeration of points on or off the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function we discuss the basic relations between these two points of view counting points is also related to the l adic cohomology of the arrangement as a variety we describe the eigenvalues of the frobenius map acting on this cohomology which corresponds to a finer decomposition of the zeta function the l adic cohomology groups and their decomposition into eigenspaces are shown to be fully determined by combinatorial data finally it is shown that the zeta function is determined by the topology of the corresponding complex variety in some important cases

a broad class of shellable lattices abstract we introduce a new class of lattices the modernistic lattices and their duals the comodernistic lattices we show that every modernistic or comodernistic lattice has shellable order complex we go on to exhibit a large number of examples of co modernistic lattices we show comodernism for two main families of lattices that were not previously known to be shellable the order congruence lattices of finite posets and a weighted generalization of the k equal partition lattices we also exhibit many examples of co modernistic lattices that were already known to be shellable to begin with the definition of modernistic is a common weakening of the definitions of semimodular and supersolvable we thus obtain a unified proof that lattices in these classes are shellable subgroup lattices of solvable groups form another family of comodernistic lattices that were already proven to be shellable we show not only that subgroup lattices of solvable groups are comodernistic but that solvability of a group is equivalent to the comodernistic property on its subgroup lattice indeed the definition of comodernistic exactly requires on every interval a lattice theoretic analogue of the composition series in a solvable group thus the relation between comodernistic lattices and solvable groups resembles in several respects that between supersolvable lattices and supersolvable groups

text fi _ mathcal w fi w modules and constraints on classical weyl group characters in this paper we study the characters of sequences of representations of any of the three families of classical weyl groups mathcal w n the symmetric groups the signed permutation groups hyperoctahedral groups or the even signed permutation groups our results extend work of church et al duke math j 2012 arxiv 1204 4533 geom topol 18 5 2951 2984 2014 on the symmetric groups we use the concept of an text fi mathcal w module an algebraic object that encodes the data of a sequence of mathcal w n representations with maps between them defined in the author s recent work j algebra 420 269 332 2014 we show that if a sequence v n of mathcal w n representations has the structure of a finitely generated text fi mathcal w module then there are substantial constraints on the growth of the sequence and the structure of the characters for n large the dimension of v n is equal to a polynomial in n and the characters of v n are given by a character polynomial in signed cycle counting class functions independent of n we determine bounds the degrees of these polynomials we continue to develop the theory of text fi mathcal w modules and we apply this theory to obtain new results about a number of sequences associated to the classical weyl groups the cohomology of complements of classical coxeter hyperplane arrangements and the cohomology of the pure string motion groups the groups of symmetric automorphisms of the free group

grobner methods for representations of combinatorial categories given a category c of a combinatorial nature we study the following fundamental question how does the combinatorial behavior of c affect the algebraic behavior of representations of c we prove two general results the first gives a combinatorial criterion for representations of c to admit a theory of grobner bases from this we obtain a criterion for noetherianity of representations the second gives a combinatorial criterion for a general rationality result for hilbert series of representations of c this criterion connects to the theory of formal languages and makes essential use of results on the generating functions of languages such as the transfer matrix method and the chomsky schutzenberger theorem our work is motivated by recent work in the literature on representations of various specific categories our general criteria recover many of the results on these categories that had been proved by ad hoc means and often yield cleaner proofs and stronger statements for example we give a new more robust proof that fi modules originally introduced by church ellenberg farb and a family of natural generalizations are noetherian we give an easy proof of a generalization of the lannes schwartz artinian conjecture from the study of generic representation theory of finite fields we significantly improve the theory of modules introduced by snowden in connection to syzygies of segre embeddings and we establish fundamental properties of twisted commutative algebras in positive characteristic

categories of fi type a unified approach to generalizing representation stability and character polynomials abstract representation stability is a theory describing a way in which a sequence of representations of different groups is related and essentially contains a finite amount of information starting with church ellenberg farb s theory of fi modules describing sequences of representations of the symmetric groups we now have good theories for describing representations of other collections of groups such as finite general linear groups classical weyl groups and wreath products s n g for a fixed finite group g this paper attempts to uncover the mechanism that makes the various examples work and offers an axiomatic approach that generates the essentials of such a theory character polynomials and free modules that exhibit stabilization we give sufficient conditions on a category c to admit such structure via the notion of categories of fi type this class of categories includes the examples listed above and extends further to new types of categories such as the categorical power fi m whose modules encode sequences of representations of m fold products of symmetric groups the theory is applied in 9 to give homological and arithmetic stability theorems for various moduli spaces e g the moduli space of degree n rational maps p 1 p m

on the invariants of the cohomology of complements of coxeter arrangements abstract we refine brieskorn s study of the cohomology of the complement of the reflection arrangement of a finite coxeter group w as a result we complete the verification of a conjecture by felder and veselov that gives an explicit basis of the space of w invariants in this cohomology ring

direct numerical simulation based measurement of the mean impulse response of homogeneous isotropic turbulence a technique for measuring the mean impulse response function of stationary homogeneous isotropic turbulence is proposed such a measurement is carried out here on the basis of direct numerical simulation dns a zero mean white noise volume forcing is used to probe the turbulent flow and the response function is obtained by accumulating the space time correlation between the white forcing and the velocity field this technique to measure the turbulent response in a dns numerical experiment is a research tool in that field of spectral closures where the linear response concept is invoked either by resorting to renormalized perturbations theories or by introducing the well known fluctuation dissipation relation fdr although the results obtained in the present work are limited to relatively low values of the reynolds number a preliminary analysis is possible both the characteristic form and the time scaling properties of the response function are investigated in the universal subrange of dissipative wave numbers a comparison with the response approximation given by the fdr is proposed through the independent dns measurement of the correlation function very good agreement is found between the measured response and kraichnan s description of random energy range advection effects

on sound propagation in a linear shear flow abstract it is shown that the acoustic wave equation in a linear shear flow always has a critical layer where the doppler shifted frequency vanishes since this is the only singularity of the wave equation apart from the point at infinity the power series solution about the critical layer has infinite radius of convergence two linearly independent solutions are even and odd functions of distance from the critical layer their plots show that acoustic oscillations are suppressed near the critical layer a linear combination of these solutions specifies the general acoustic field the constants of integration are determined from boundary conditions of which several examples are given the total acoustic field is illustrated for rigid and impedance wall conditions the cases illustrated include both real and complex wave fields these wave fields are small amplitude perturbations of the acoustic wave equation in a linear shear flow it is shown in the appendix that the perturbations of vorticity and dilatation are coupled and thus combine features of acoustic and hydrodynamic modes

on the optimal systems of subalgebras for the equations of hydrodynamic stability analysis of smooth shear flows and their group invariant solutions we present a unifying solution framework for the linearized compressible equations for two dimensional linearly sheared unbounded flows using the lie symmetry analysis the full set of symmetries that are admitted by the underlying system of equations is employed to systematically derive the one and two dimensional optimal systems of subalgebras whose connected group reductions lead to three distinct invariant ansatz functions for the governing sets of partial differential equations pdes the purpose of this analysis is threefold and explicitly we show that i there are three invariant solutions that stem from the optimal system these include a general ansatz function with two free parameters as well as the ansatz functions of the kelvin mode and the modal approach specifically the first approach unifies these well known ansatz functions by considering two limiting cases of the free parameters and related algebraic transformations the general ansatz function is reduced to either of them this f

castelnuovo mumford regularity of representations of certain product categories abstract we show in this paper that representations of a finite product of categories satisfying certain combinatorial conditions have finite castelnuovo mumford regularity if and only if they are presented in finite degrees and hence the category consisting of them is abelian these results apply to examples such as the categories fi m and fi g m

eulerian spectral closures for isotropic turbulence using a time ordered fluctuation dissipation relation in a previous paper 1 the kraichnan wyld perturbation expansion 2 3 was used to justify the introduction of a renormalized response function connecting two point covariances at different times the resulting relationship was specialized by suitable choice of initial conditions to the form of a fluctuation dissipation relation fdr this was further developed to reconcile the time symmetry of the covariance with the causality of the response function by the introduction of time ordering along with a counter term we pointed out that this formulation provides a solution to an old problem in turbulence theory that of representing the time dependence of the covariance and response by exponential forms 4 5 we showed that the derivative with respect to difference time of the covariance with this time ordering now vanishes at the origin this allows one to study the relationships between two time spectral closures and time independent theories such as the fokker planck theory of edwards 6 or the more recent renormalization group approaches we also showed that the renormalized response function is transitive with respect to intermediate times and reported a new langevin type equation for turbulence in this paper we interpret the second order response function as a mean field propagator and show that in addition to propagating two time covariances it also links single time covariances we then make use of its newly established properties to re derive the local energy transfer let response equation 7 and show that it now contains a counter term which removes the singularity of previous propagator equations at t t we also introduce a partial propagator representation and hence reformulate the let statistical equations furthermore we specialize the two time equations to time independent form by introducing exponential time dependences and

coincidences between homological densities predicted by arithmetic abstract motivated by analogies with basic density theorems in analytic number theory we introduce a notion and variations of the homological density of one space in another we use weil s number field function field analogy to predict coincidences for limiting homological densities of various sequences z n d 1 d m x of spaces of 0 cycles on manifolds x the main theorem in this paper is that these topological predictions which seem strange from a purely topological viewpoint are indeed true one obstacle to proving such a theorem is the combinatorial complexity of all possible collisions of points this problem does not arise in the simplest and classical case m n 1 2 of configuration spaces to overcome this obstacle we apply the bjorner wachs theory of lexicographic shellability from algebraic combinatorics

topology of order complexes of intervals in subgroup lattices abstract we conjecture that the order complex of an open interval in the subgroup lattice of a finite group has the homotopy type of a wedge of spheres and prove that if h g is a minimal counterexample to this conjecture then either g is almost simple or g hn where n is the unique minimal normal subgroup of g n is non abelian and h n 1

a polynomial linear search algorithm for the n dimensional knapsack problem presentation d un algorithme de recherche lineaire qui reconnait le probleme de sac a dos a n dimensions par etapes de 2n 4 logn o n 3

linear decision trees subspace arrangements and mobius functions let p be a set in rn given a point x we want to test if x e p our model of computation is a linear decision tree a rooted ternary tree t where with each interior node v is associated a linear function lv x l j ajxj b and the three edges connecting v to its descendants are labelled starting from the root we move down the tree at each internal node v we check whether iv x 0 and follow the appropriately labelled edge leaves are labelled yes and no and arriving at a leaf we read off the answer to the question is x e p linear decision trees are sometimes surprisingly powerful devices we mention here the result that some np complete problems like the knapsack problem have polynomial depth linear decision trees mh lower bound results on the size or depth of linear decision trees usually depend on counting the number of connected components of p or r n p this method was introduced by dobkin and lipton dl and further developed by steele and yao sy ben or bo and others in this paper we describe topological methods that can be applied to obtain lower bounds also if these sets are connected the basic idea of this paper was first explored in bly the euler characteristic of p or r n p was used to estimate the number of leaves in a linear decision tree for p here we improve and extend the topological methods of bly in particular we show that the euler characteristic can be replaced by the sum of the betti numbers of r n p in bl y the method was applied to the case when p is a subspace arrangement i e a union of a finite number of affine subspaces the euler characteristic of the union is essentially the mobius function of the intersection lattice of these subspaces in this paper using a formula of goresky and macpherson gm we work with the betti numbers of the complement of this union and thereby obtain better bounds the motivating problem in bly was the k equal problem given n real numbers xl xn decide if some k of them are equal it was shown that ev ry linear decision tree for the k equal problem has size at least 2njk 0 n and consequently depth at least q nlog 2njk which is best possible we

generalized q functions and dirichlet to neumann maps for elliptic differential operators the classical concept of q functions associated to symmetric and selfadjoint operators due to m g krein and h langer is extended in such a way that the dirichletto neumann map in the theory of elliptic differential equations can be interpreted as a generalized q function for couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit krein type formulas for the difference of the resolvents and trace formulas in an h 2 framework are obtained

why the characteristic polynomial factors we survey three methods for proving that the characteristic polynomial of a finite ranked lattice factors over the nonnegative integers and indicate how they have evolved recently the first technique uses geometric ideas and is is based on zaslavsky s theory of signed graphs the second approach is algebraic and employs results of saito and terao about free hyperplane arrangements finally we consider a purely combinatorial theorem of stanley about supersolvable lattices and its generalizations

computations for coxeter arrangements and solomon s descent algebra iii groups of rank seven and eight the authors would like to acknowledge support from the dfg priority program spp1489 algorithmic and experimental methods in algebra geometry and number theory

elliptic problems involving an indefinite weight we consider a selfadjoint elliptic eigenvalue problem which is derived formally from a variational problem of the form lu a o x u in q b1u 0 on f j 1 m where l is a linear elliptic operator of order 2m defined in a bounded open set q c rn n 2 with boundary f the bj are linear differential operators defined on f and w is a real valued function assuming both positive and negative values for our problem we prove the completeness of the eigenvectors and associated vectors in two function spaces which arise naturally in such an indefinite problem we also establish some results concerning the eigenvalues of the problem which complement the known results and investigate the structure of the principal subspaces

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

on finite rank perturbations of definitizable operators it was shown by p jonas and h langer that a selfadjoint definitizable operator a in a krein space remains definitizable after a finite rank perturbation in resolvent sense if the perturbed operator b is selfadjoint and the resolvent set b is nonempty it is the aim of this note to prove a more general variant of this perturbation result where the assumption on b is dropped as an application a class of singular ordinary differential operators with indefinite weight functions is studied

a formal derivation of the local energy transfer let theory of homogeneous turbulence a statistical closure of the navier stokes hierarchy which leads to equations for the two point two time covariance of the velocity field for stationary homogeneous isotropic turbulence is presented it is a generalisation of the self consistent field method due to edwards 1964 for the stationary single time velocity covariance the probability distribution functional p u t is obtained in the form of a series from the liouville equation by means of a perturbation expansion about a gaussian distribution which is chosen to give the exact two point two time covariance the triple moment is calculated in terms of an ensemble averaged infinitesimal velocity field propagator and shown to yield the edwards result as a special case the use of a gaussian zero order distribution has been found to justify the introduction of a fluctuation response relation which is in accord with modern dynamical theories in a sense this work completes the analogy drawn by edwards between turbulence and brownian motion originally edwards had shown that the noise input was determined by the correlation of the velocity field with the externally applied stirring forces but was unable to determine the system response now we find that the system response is determined by the correlation of the velocity field with internal quasi entropic forces this analysis is valid to all orders of perturbation theory and allows the recovery of the local energy transfer let theory which had previously been derived by more heuristical methods the let theory is known to be in good agreement with experimental results it is also unique among two point statistical closures in displaying an acceptable i e non markovian relationship between the transfer spectrum and the system response in accordance with experimental results as a result of the latter property it is compatible with the kolmogorov k41 spectral phenomenology

combinatorics of multigraded poincare series for monomial rings abstract backelin proved that the multigraded poincare series for resolving a residue field over a polynomial ring modulo a monomial ideal is a rational function the numerator is simple but until the recent work of berglund there was no combinatorial formula for the denominator berglund s formula gives the denominator in terms of ranks of reduced homology groups of lower intervals in a certain lattice we now express this lattice as the intersection lattice l a i of a subspace arrangement a i use crapo s closure lemma to drastically simplify the denominator in some cases such as monomial ideals generated in degree two and relate golodness to the cohen macaulay property for associated posets in addition we introduce a new class of finite lattices called complete lattices prove that all geometric lattices are complete and provide a simple criterion for golodness of monomial ideals whose lcm lattices are complete

eigenspace arrangements of reflection groups the lattice of intersections of reflecting hyperplanes of a complex reflection group w may be considered as the poset of 1 eigenspaces of the elements of w in this paper we replace 1 with an arbitrary eigenvalue and study the topology and homology representation of the resulting poset after posing the main question of whether this poset is shellable we show that all its upper intervals are geometric lattices and then answer the question in the affirmative for the infinite family g m p n of complex reflection groups and the first 31 of the 34 exceptional groups by constructing cl shellings in addition we completely determine when these eigenspaces of w form a k 1 resp free arrangement for the symmetric group we also extend the combinatorial model available for its intersection lattice to all other eigenvalues by introducing balanced partition posets presented as particular upper order ideals of dowling lattices study the representation afforded by the top co homology group and give a simple map to the posets of pointed d divisible partitions 1 main question and results let v denote an n dimensional c vector space a reflection in v is any nonidentity element r in gl v of finite order that fixes some hyperplane hr and a finite subgroup w of gl v is called a reflection group if it is generated by reflections example the action of the symmetric group sn on n 1 2 n gives rise to a faithful action on c via ei e i where e1 e2 en denote the standard basis vectors since the transpositions act as reflections and generate the group this representation realizes sn as a reflection group in gl c we shall refer to it as the defining representation for an element g w and root of unity let v g denote the eigenspace of g in v define e w to be the w poset partially ordered set of all such eigenspaces v g g w ordered by reverse inclusion with w action given by h v g v hgh the eigenvalue 1 gives the lattice lw of intersections of reflecting hyperplanes for w see 29 lemma 4 4 the minimal elements of e w i e the inclusion maximal eigenspaces forw are the focus of springer s theory of regular elements 35 and each has dimension equal to the number a d of degrees d1 d2 dn of w that are divisible by d the order of see proposition 3 7 below when w is crystallographic the poset e w itself appears in broue malle and michel s sylow theory 8 2010 mathematics subject classification primary 20f55 05e10 05e18 55p20 14n20

shellable nonpure complexes and posets ii this is a direct continuation of shellable nonpure complexes and posets i which appeared in transactions of the american mathematical society 348 1996 1299 1327 8 interval generated lattices and dominance order in this section and the following one we will continue exemplifying the applicability of lexicographic shellability to nonpure posets let f i1 i2 in be a family of intervals of integers by which is meant sets of the form a b a a 1 b a b we assume that there are no containments among these intervals and that they are ordered so that their left and right endpoints are increasing let l f be the lattice of all sets that are unions of subfamilies of f ordered by inclusion such interval generated lattices l f were introduced and studied by greene g define an edge labeling of l f as follows if a b is a covering and a max b a then a b a if a 1 a and a is the left endpoint of some i f

solution of functional equations and reduction of dimension in the local energy transfer theory of incompressible three dimensional turbulence it is shown that the set of integrodifferential and algebraic functional equations of the local energy transfer theory may be considerably reduced in dimension for the case of isotropic turbulence this is achieved without restricting the solution space the basis for this is a complete analytical solution to the functional equations q k t t h k t t q k t t and h k t s h k s t h k t t the solution is proved to depend only on a single function straight phi k t solely determining q and h hence the dimension of both the dependent and the independent variables is reduced by one from the latter the corresponding two integrodifferential equations are lowered to a single integrodifferential equation for straight phi k t extended by an integral side condition on the k dependence of straight phi k t in the limit nu 0 a partial solution to the reduced set of equations is presented in the appendix

small rational model of subspace complement this paper concerns the rational cohomology ring of the complement m of a complex subspace arrangement we start with the de concini procesi differential graded algebra that is a rational model for m inside it we find a much smaller subalgebra d quasi isomorphic to the whole algebra d is described by defining a natural multiplication on a chain complex whose homology is the local homology of the intersection lattice l whence connecting the de concini procesi model with the goresky macpherson formula for the additive structure of h m the algebra d has a natural integral version that is a good candidate for an integral model of m if the rational local homology of l can be computed explicitly we obtain an explicit presentation of the ring h m q for example this is done for the cases where l is a geometric lattice and where m is a k equal manifold

a path integral for turbulence in incompressible fluids in this paper a classical path integral is formulated for incompressible fluids that evolve according to the navier stokes equation the path integral propagates probability distributions deterministically on the space gvol of solenoidal velocity fields we construct a set of isp 2 charges associated with the geometry of gvol and its poisson structure and a pair of supersymmetry charges connected with the hamiltonian these charges generate exact symmetries of the classical path integral when the viscosity is set equal to zero when the effect of dissipation is included the charges associated with the poisson structure and the hamiltonian are no longer conserved charges that generate kolmogorov scaling and galilean transformations are also constructed the classical path integral is formulated in terms of vorticity as well

group actions on arrangements of linear subspaces and applications to configuration spaces for an arrangement of linear subspaces in ri that is invariant under a finite subgroup of the general linear group gin ir we develop a formula for the g module structure of the cohomology of the complement m a our formula specializes to the well known goresky macpherson theorem in case g 1 but for g 1 the formula shows that the g module structure of the complement is not a combinatorial invariant as an application we are able to describe the free part of the cohomology of the quotient space ma4ig our motivating examples are arrangements in cn that are invariant under the action of sn by permuting coordinates a particular case is the k equal arrangement first studied by bj6rner lovasz and yao motivated by questions in complexity theory in these cases ma4 and ma sn are spaces of ordered and unordered point configurations in cn many of whose properties are reduced by our formulas to combinatorial questions in partition lattices more generally we treat point configurations in rd and provide explicit results for the k equal and the k divisible cases

extension of the momentum transfer model to time dependent pipe turbulence we analyze a possible extension of gioia and chakraborty s momentum transfer model of friction in steady turbulent pipe flows phys rev lett 96 044502 2006 to the case of time and or space dependent turbulent flows the end result is an expression for the stress at the wall as the sum of a steady and a dynamic component the steady part is obtained by using the instantaneous velocity in the expression for the stress at the wall of a stationary flow the unsteady part is a weighted average over the history of the flow acceleration with a weighting function similar to that proposed by vardy and brown j sound vibr 259 1011 2003 j sound vibr 270 233 2004 but naturally including the effect of spatial derivatives of the mean flow as in the brunone model brunone et al j water res plan manage 126 236 2000

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

a fluctuation relaxation relation for homogeneous isotropic turbulence a liouville type equation is solved perturbatively for the exact pdf of fluctuating velocities by expansion about an invariant gaussian measure which is an approximation to the symmetric part of the pdf and is constrained to give the correct covariance the antisymmetric part of the pdf is calculated to the lowest nontrivial order and leads to closed equations for the covariance in terms of a response function which arises from the liouville operator averaged against the ground state distribution as the latter is a stationary gaussian pdf we obtain an expression for the system response which is linear in the covariance

a krein space approach to symmetric ordinary differential operators with an indefinite weight function on considere les proprietes spectrales du probleme differentiel l f 1 n p 0 f n n 1 n 1 p 1 f n 1 n 1 p n f rf sur un intervalle fini ou infini a b avec des coefficients reels localement sommables 1 p 0 p 1 p n r sous l hypothese que p 0 0 et que la fonction poids r change de signe sur a b

mobius functions of lattices abstract we introduce the concept of a bounded below set in a lattice this can be used to give a generalization of rota s broken circuit theorem to any finite lattice we then show how this result can be used to compute and combinatorially explain the mobius function in various examples including non crossing set partitions shuffle posets and integer partitions in dominance order next we present a generalization of stanley s theorem that the characteristic polynomial of a semimodular supersolvable lattice factors over the integers we also give some applications of this second main theorem including the tamari lattices

on the eigenvalues of nonselfadjoint problems involving indefinite weights on etudie la distribution asymptotique des valeurs propres de problemes aux valeurs limites elliptiques comprenant des fonctions poids

boundary value problems for elliptic partial differential operators on bounded domains for a symmetric operator or relation a with infinite deficiency indices in a hilbert space we develop an abstract framework for the description of symmetric and self adjoint extensions a of a as restrictions of an operator or relation t which is a core of the adjoint a this concept is applied to second order elliptic partial differential operators on smooth bounded domains and a class of elliptic problems with eigenvalue dependent boundary conditions is investigated

representation stability for families of linear subspace arrangements abstract church ellenberg farb 8 used the language of fi cha to identify certain sequences of hyperplane arrangements with s n actions that satisfy cohomological representation stability here we vastly extend their results and define when a collection of arrangements is finitely generated using this notion we get stability results to general linear subspace arrangements not necessarily of hyperplanes a wide class of group actions replacing fi by a general category c we show that the cohomology of such collections of arrangements satisfies a strong form of representation stability with many concrete applications for example this implies that their betti numbers are always given by certain polynomials for this purpose we use the theory of representation stability for quite general classes of groups developed in a 17 we apply this theory to get classical cohomological stability of quotients of linear subspace arrangements with coefficients in certain constructible sheaves

bounds on the non real spectrum of differential operators with indefinite weights ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non negative operators in krein spaces under the assumption that 0 and infty are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non real spectrum is contained in a compact set and with definite type real spectrum outside this set the main results are quantitative estimates for this set which are applied to sturm liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains

rational points and cohomology of discriminant varieties abstract let g be a finite unitary reflection group acting in a complex vector space v c n the discriminant varietyxg of g is defined as the space of regular orbits of g on v classical examples include the varieties of complex polynomials of degree n with distinct resp non zero distinct roots the normaliser g of g in gl v acts on xg in this work we determine the action of g g on the cohomology of xg in the classical cases this amounts to computing the cohomology of xg with certain local coefficient systems our methods are to compute equivariant weight polynomials by means of explicit counting of the rational points of certain varieties over finite fields and then to exploit the weight purity of the relevant varieties we obtain some power series identities as a byproduct

representation theory and homological stability abstract we introduce the idea of representation stability and several variations for a sequence of representations v n of groups g n a central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability this makes it possible to extend classical theorems of homological stability to a much broader variety of examples representation stability also provides a framework in which to find and to predict patterns from classical representation theory littlewood richardson and murnaghan rules stability of schur functors to cohomology of groups pure braid torelli and congruence groups to lie algebras and their homology to the equivariant cohomology of flag and schubert varieties to combinatorics the n 1 n 1 conjecture the majority of this paper is devoted to exposing this phenomenon through examples in doing this we obtain applications theorems and conjectures beyond the discovery of new phenomena the viewpoint of representation stability can be useful in solving problems outside the theory in addition to the applications given in this paper it is applied by church ellenberg farb in preparation 20 to counting problems in number theory and finite group theory representation stability is also used by church 2012 19 to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds

homotopy types of subspace arrangements via diagrams of spaces we prove combinatorial formulas for the homotopy type of the union of the subspaces in an affine compactified affine spherical or projective subspace arrangement from these formulas we derive results of goresky macpherson on the homology of the arrangement and the cohomology of its complement the union of an arrangement can be interpreted as the direct limit of a diagram of spaces over the intersection poset a closely related space is obtained by taking the homotopy direct limit of this diagram our method consists in constructing a combinatorial model diagram over the same poset whose homotopy limit can be compared to the original one by usual homotopy comparison results for diagrams of spaces

eulerian field theoretic closure formalisms for fluid turbulence the formalisms of wyld ann phys 14 143 1961 and martin siggia and rose msr phys rev a 8 423 1973 address the closure problem of a statistical treatment of homogeneous isotropic turbulence hit based on techniques primarily developed for quantum field theory in the wyld formalism there is a well known double counting problem for which an ad hoc solution was suggested by lee ann phys 32 292 1965 we show how to implement this correction in a more natural way from the basic equations of the formalism this leads to what we call the improved wyld lee renormalized perturbation theory msr had noted that their formalism had more vertex functions than wyld s formalism and based on this felt wyld s formalism was incorrect however a careful comparison of both formalisms here shows that the wyld formalism follows a different procedure to that of the msr formalism and so the treatment of vertex corrections appears in different ways in the two formalisms taking that into account along with clarifications made to both formalisms we find that they are equivalent and we demonstrate this up to fourth order

k parabolic subspace arrangements in this paper we study k parabolic arrangements a generalization of the k equal arrangement for any finite real reflection group when k 2 these arrangements correspond to the well studied coxeter arrangements brieskorn 1971 showed that the fundamental group of the complement of the type w coxeter arrangement over mathbb c is isomorphic to the pure artin group of type w khovanov 1996 gave an algebraic description for the fundamental group of the complement of the 3 equal arrangement over mathbb r we generalize khovanov s result to obtain an algebraic description of the fundamental group of the complement of the 3 parabolic arrangement for arbitrary finite reflection group our description is a real analogue to brieskorn s description

on new stability modes of plane canonical shear flows using symmetry classification in the work of nold and oberlack phys fluids 25 104101 2013 it was shown that three different instability modes of the linear stability analysis perturbing a linear shear flow can be derived in the common framework of lie symmetry methods these modes include the normal mode the kelvin mode and a new mode not reported before as this was limited to linear shear we now present a full symmetry classification for the linearised navier stokes equations which are employed to study the stability of an arbitrary plane shear flow if viscous effects for the perturbations are neglected then we obtain additional symmetries and new ansatz functions for a linear an algebraic an exponential and a logarithmic base shear flow if viscous effects are included in the formulation then the linear and a quotient type base flow allow for additional symmetries the symmetry invariant solutions derived from the new and classical generic symmetries for all different flow types naturally lead to algebraic growth an

reviving the method of particular solutions fox henrici and moler made famous a method of particular solutions for computing eigenvalues and eigenmodes of the laplacian in planar regions such as polygons we explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties the crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant similar methods may be used to improve other mesh free algorithms for a variety of computational problems

action principle in nonequilibrium statistical dynamics we introduce a variational method for approximating distribution functions of dynamics with a liouville operator l in terms of a nonequilibriumaction functional for two independent left and right trial states the method is valid for deterministic or stochastic markov dynamics and for stationary or time dependent distributions a practical rayleigh ritz procedure is advanced whose inputs are a finitely parametrized ansatz for the trial states leading to a parametric action for their evolution the euler lagrange equations of the action principle are hamiltonian in form generally noncanonical this permits a simple identification of fixed points as critical points of the parametric hamiltonian we also establish a variational principle for low order statistics such as mean values and correlation functions by means of the least effective action the latter is a functional of the given variable which is positive and convex as a consequence of h older realizability inequalities its value measures the cost for a fluctuation from the average to occur and in a weak noise limit it reduces to the onsager machlup action in general the effective action is shown to arise from the nonequilibrium action functional by a constrained variation this result provides a rayleigh ritz scheme for calculating just the desired low order statistics with internal consistency checks less demanding than for the full distribution textcopyright 1996 the american physical society

functional and graphical methods for classical statistical dynamics i a formulation of the martin siggia rose method a formulation of the martin siggia rose msr method for describing the statistical dynamics of classical systems is presented the present formulation is very similar in structure to the original msr operator formalism and very different from the alternative functional integral formalism of janssen de dominicis peliti and others the need for imposing certain boundary conditions in the msr formalism as pointed out by deker is clarified the basic results of this paper include a construction of the msr formalism in a way that demonstrates its internal consistency a definition of a functional whose functional derivatives give all the correlation functions and response functions of an ensemble of mechanical systems a graphical expression for the correlation functions and response functions a graphical expression for the legendre transform of the functional and of the resulting vertex functions and a graphical derivation of the appropriate dyson equation the present formulation is applicable to systems with highly non gaussian statistics including systems of classical particles described in terms of the particle density in single particle phase space in this paper we consider only the case of ensembles of systems whose coordinates are continuous and whose time evolution is described by deterministic first order differential equations that are local in time the method is easily extended to systems whose dynamics is governed by stochastic differential equations and to spin systems

spectral analysis of singular ordinary differential operators with indefinite weights abstract in this paper we develop a perturbation approach to investigate spectral problems for singular ordinary differential operators with indefinite weight functions we prove a general perturbation result on the local spectral properties of selfadjoint operators in krein spaces which differ only by finitely many dimensions from the orthogonal sum of a fundamentally reducible operator and an operator with finitely many negative squares this result is applied to singular indefinite sturm liouville operators and higher order singular ordinary differential operators with indefinite weight functions

partitions into even and odd block size and some unusual characters of the symmetric groups etude des partitions en blocs de taille paire et impaire et des caracteres insolites des groupes symetriques demonstration d une conjecture de r p stanley

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

lagrangian statistical mechanics applied to non linear stochastic field equations abstract we consider non linear stochastic field equations such as the kpz equation for deposition and the noise driven navier stokes equation for hydrodynamics we focus on the fourier transform of the time dependent two point field correlation k t we employ a lagrangian method aimed at obtaining the distribution function of the possible histories of the system in a way that fits naturally with our previous work on the static distribution our main result is a non linear integro differential equation for k t which is derived from a peierls boltzmann type transport equation for its fourier transform in time k that transport equation is a natural extension of the steady state transport equation we previously derived for k 0 we find a new and remarkable result which applies to all the non linear systems studied here the long time decay of k t is described by k t exp a k t where a is a constant and is system dependent

homology representations arising from the half cube abstract we construct a cw decomposition c n of the n dimensional half cube in a manner compatible with its structure as a polytope for each 3 k n the complex c n has a subcomplex c n k which coincides with the clique complex of the half cube graph if k 4 the homology of c n k is concentrated in degree k 1 and furthermore the k 1 st betti number of c n k is equal to the k 2 nd betti number of the complement of the k equal real hyperplane arrangement these betti numbers which also appear in theoretical computer science numerical analysis and engineering are the coefficients of a certain pascal like triangle sloane s sequence a119258 the coxeter groups of type d n act naturally on the complexes c n k and thus on the associated homology groups

the homology representations of the k equal partition lattice we determine the character of the action of the symmetric group on the homology of the induced subposet of the lattice of partitions of the set 1 2 nr obtained by restricting block sizes to the set 1 k k 1 a plethystic formula for the generating function of the frobenius characteristic of the representation is given we combine techniques from the theory of nonpure shellability recently developed by bj6rner and wachs with symmetric function techniques developed by sundaram for determining representations on the homology of subposets of the partition lattice

fi modules and stability for representations of symmetric groups in this paper we introduce and develop the theory of fi modules we apply this theory to obtain new theorems about the cohomology of the configuration space of n distinct ordered points on an arbitrary connected oriented manifold the diagonal coinvariant algebra on r sets of n variables the cohomology and tautological ring of the moduli space of n pointed curves the space of polynomials on rank varieties of n n matrices the subalgebra of the cohomology of the genus n torelli group generated by h1 and more the symmetric group sn acts on each of these vector spaces in most cases almost nothing is known about the characters of these representations or even their dimensions we prove that in each fixed degree the character is given for n large enough by a polynomial in the cycle counting functions that is independent of n in particular the dimension is eventually a polynomial in n in this framework representation stability in the sense of church farb for a sequence of sn representations is converted to a finite generation property for a single fi module

homological stability for configuration spaces of manifolds let cn m be the configuration space of n distinct ordered points in m we prove that if m is any connected orientable manifold closed or open the homology groups hi cn m are representation stable in the sense of church and farb arxiv 1008 1368 applying this to the trivial representation we obtain as a corollary that the unordered configuration space bn m satisfies classical homological stability for each i hi bn m hi bn 1 m for n i this improves on results of mcduff segal and others for open manifolds applied to closed manifolds this provides natural examples where rational homological stability holds even though integral homological stability fails

spectral theory of elliptic differential operators with indefinite weights the spectral properties of a class of non self adjoint second order elliptic operators with indefinite weight functions on unbounded domains are investigated it is shown under an abstract regularity assumption that the non real spectrum of the associated elliptic operators in l 2 is bounded in the special case where r n decomposes into subdomains and with smooth compact boundaries and the weight function is positive on and negative on it turns out that the non real spectrum consists only of normal eigenvalues that can be characterized with a dirichlet to neumann map

group invariant solutions and optimal systems for multidimensional hydrodynamics the group properties of the three dimensional 3 d one temperature hydrodynamic equations including nonlinear conduction and a thermal source are presented a subgroup corresponding to axisymmetric geometry is chosen and the details of the construction of the one and two dimensional optimal systems are shown the two dimensional optimal system is used to generate 23 intrinsically different reductions of the 2 d partial differential equations to ordinary differential equations these ordinary differential equations can be solved to provide analytic solutions to the original partial differential equations two example analytic solutions are presented a 2 d axisymmetric flow with a p2 asymmetry and a 3 d spiraling flow