the stein stromberg covering theorem in metric spaces abstract in 8 naor and tao extended to the metric setting the o d log d bounds given by stein and stromberg for lebesgue measure in r d deriving these bounds first from a localization result and second from a random vitali lemma here we show that the stein stromberg original argument can also be adapted to the metric setting giving a third proof we also weaken the hypotheses and additionally we sharpen the estimates for lebesgue measure

variation inequalities for the fejer and poisson kernels in this paper we show that the g th order variation operator for both the fejer and poisson kernels are bounded from l p to l p 1 2 counterexamples are given if g 2

on the littlewood paley stein g function we consider semigroups tt which are contractive on lp m for all p e q q and q e 1 2 we give an example on symmetric spaces of the noncompact type which shows that the littlewood paley stein g function associated to the infinitesimal generator of tt may be unbounded on lq m and on l m we prove that variants of the g function are bounded on these lebesgue spaces in this paper we present some new results concerning the lp boundedness of the littlewood paley stein g function suppose that tt is a co semigroup of operators on lp m where m is a a finite measure space and let a e n the littlewood paley stein function ga associated to the given semigroup is defined by the rule ga f 1 ta ttfi2 dt for all f for which the right hand side makes sense in classical cases where tt is the poisson semigroup acting on lp r or lp t the g function is an important tool in fourier analysis for instance it is frequently used to prove multiplier theorems and pointwise convergence results the reader is referred to 1 1 for a survey of its role in a general context the g function was considered by e m stein he proved that if tt is a symmetric diffusion semigroup then ga satisfies the estimate 1 aplifiip liga f iip bp fllp vf e lp m whenever p e 1 oo a proof of this result can be found in 10 a simpler and more general approach via transference techniques is presented in 2 a third proof is in 4 and 5 it is of an entirely different nature in that it is based on the fact that generators of symmetric diffusion semigroups and even more general semigroups 3 have an hx functional calculus one of the interesting applications of g functions is to functional calculus for infinitesimal generators of co semigroups stein proved a marcinkiewicz type result for those as a corollary of the theory he developed for the g function the relationship between g functions and functional calculus for the infinitesimal generator of tt was investigated in 4 and 5 our treatment here owes received by the editors march 7 1994 originally communiated to the proceedings of the ams by j marshall ash 1991 mathematics subject classification primary 42b25 47a60 secondary 43a85

random martingales and localization of maximal inequalities abstract let x d be a metric measure space for r 0 consider the hardy littlewood maximal operator m r f x def sup r r 1 b x r b x r f d we show that if there is an n 1 such that one has the microdoubling condition b x 1 1 n r b x r for all x x and r 0 then the weak 1 1 norm of m r has the following localization property m r l 1 x l 1 x sup r 0 m r r n r l 1 x l 1 x an immediate consequence is that if x d is ahlfors david n regular then the weak 1 1 norm of m r is n log n generalizing a result of stein and stromberg 1983 47 we show that this bound is sharp by constructing a metric measure space x d that is ahlfors david n regular for which the weak 1 1 norm of m 0 is n log n the localization property of m r is proved by assigning to each f l 1 x a distribution over random martingales for which the associated random doob maximal inequality controls the weak 1 1 inequality for m r

extending lipschitz functions via random metric partitions many classical problems in geometry and analysis involve the gluing together of local information to produce a coherent global picture inevitably the difficulty of such a procedure lies at the local boundary where overlapping views of the same locality must somehow be merged it is therefore desirable that the boundaries be smooth allowing a graceful transition from one viewpoint to the next for instance one may point to whitney s use of partitions of unity in studying what is now known as the whitney extension problem 36 37 in the present work we consider what is perhaps the most basic whitney type extension problem that of extending a lipschitz function so that it remains lipschitz often such a map is extended by first producing a cover of the new domain extending the mapping locally and then gluing together the individual pieces our main observation is that in many cases if one chooses a random cover from the right distribution the boundary can be made smooth on average even when the local maps are individually quite coarse this insight leads to the unification generalization and improvement of many known results as well as to new results for many interesting spaces

i simple pulsatile flow in an artery with a constriction abstract a smooth isolated axisymmetric occlusion in a straight vascular tube is a tractable problem for pulsatile flow calculations via finite difference approximations to the navier stokes equation steady flow depends on the reynolds number and two geometric parameters which describe the stenosis the mere addition of a simple harmonic to the mean flow adds two more parameters one is the reduced frequency or stokes number and the other e the ratio of unsteady to steady flux after describing steady stenosis flow examples the dynamic patterns of pulsatile flow are illustrated indicating the inadequacy of basing hypotheses of atherosclerosis on mean steady flow

directional maximal operators and lacunarity in higher dimensions we introduce a notion of lacunarity in higher dimensions for which we can bound the associated directional maximal operators in l p bbb r n with p1 in particular we are able to treat the classes previously considered by nagel stein wainger sj ogren sj olin and carbery closely related to this we find a characterization of the sets of directions which give rise to bounded maximal operators the bounds enable lebesgue type differentiation of integrals in l rm loc p bbb r n replacing balls by tubes which point in these directions

on a variable coefficient korteweg de vries model in fluid filled elastic tubes in this paper we investigate a variable coefficient korteweg de vries vckdv model in arterial mechanics a new lax pair of the vckdv model is constructed based on which the b cklund transformation solitonic and some other solutions of the vckdv model are obtained with symbolic computation the influence of the variable coefficients on solitonic propagation is investigated based on the solitonic solution which has the following three aspects i the coefficient of the nonlinear term affects the solitonic amplitude ii the coefficient of the dispersive term controls the solitonic velocity and propagation trace iii the coefficient of the dissipative term acts on both the solitonic amplitude and the velocity we also analyze the variations of velocity and pressure of blood flow along the scaled axial coordinate after the static deformation in the vicinity of stenosis

finite element simulation of pulsatile flow through arterial stenosis the problem of blood flow through a stenosis is solved using the incompressible navier stokes equations in a rigid circular tube presenting a partial occlusion calculations are based on a galerkin finite element method the time marching scheme employs a predictor corrector technique using a variable time step results are obtained for steady and physiological pulsatile flows computational experiments analyse the effect of varying the degree of stenosis the stricture length the reynolds number and womersley number the method gives results which agree well with previous computations for steady flows and experimental findings for steady and pulsatile flows

a contribution on the effects of inlet conditions when modelling stenoses using sudden expansions abstract using three diameter ratios and up to four inlet velocity profiles the laminar flow in axisymmetric sudden expansions is investigated numerically for inlet reynolds numbers of 250 or lower the vortex reattachment length is found to be a linear function of the reynolds number regardless of either the diameter ratio or inlet velocity profile however the inlet velocity profile plays an important role in contributing to a non linear relationship between the vortex length as a function of the expansion step height the wall shear stress distribution in the larger pipe is found to depend strongly upon the diameter ratio inlet velocity profile and reynolds number

pointwise theorems for amenable groups in this paper we prove the pointwise ergodic theorem for general locally compact amenable groups along folner sequences that obey some restrictions these restrictions are mild enough so that such sequences exist for all amenable groups we also prove a generalization of the shannon mcmillan breiman theorem to all discrete amenable groups

numerical investigation of the non newtonian blood flow in a bifurcation model with a non planar branch the pulsatile flow of non newtonian fluid in a bifurcation model with a non planar daughter branch is investigated numerically by using the carreau yasuda model to take into account the shear thinning behavior of the analog blood fluid the objective of this study is to deal with the influence of the non newtonian property of fluid and of out of plane curvature in the non planar daughter vessel on wall shear stress wss oscillatory shear index osi and flow phenomena during the pulse cycle the non newtonian property in the daughter vessels induces a flattened axial velocity profile due to its shear thinning behavior the non planarity deflects flow from the inner wall of the vessel to the outer wall and changes the distribution of wss along the vessel in particular in systole phase downstream of the bifurcation the velocity profiles are shifted toward the flow divider and low wss and high shear stress temporal oscillations characterized by osi occur on the outer wall region of the daughter vessels close to the bifurcation secondary motions become stronger with the addition of the out of plane curvature induced by the bending of the vessel and the secondary flow patterns swirl along the non planar daughter vessel a significant difference between the non newtonian and the newtonian pulsatile flow is revealed during the pulse cycle however reasonable agreement between the non newtonian and the rescaled newtonian flow is found calculated results for the pulsatile flow support the view that the non planarity of blood vessels and the non newtonian properties of blood are an important factor in hemodynamics and may play a significant role in vascular biology and pathophysiology

on the potentialities of 3d 1d coupled models in hemodynamics simulations this work comprises a step towards the quantitative and qualitative analysis of coupled local and global hemodynamics phenomena in the arterial system the aim of this work is to present some numerical examples to put in evidence the importance of the use of 3d 1d coupled models in hemodynamics problems when carrying out simulations of rather complex situations accordingly some cases for which classical modeling cannot be applied are identified and solved the results obtained here allow us to assess some interrelations between local pointwise quantities defined at the level of the 3d model and global mean quantities defined at the level of the 1d model

the zero dispersion limit of the korteweg de vries equation with periodic initial data on etudie le probleme aux valeurs initiales pour l equation de korteweg de vries u t 6uu x e 2 u xxx 0 dans la limite e 0 on calcul la limite faible de la solution pour des donnees initiales periodiques quand e 0

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

affine stanley symmetric functions we define a new family inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of generating functions for w inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i which are affine analogues of stanley symmetric functions we establish basic properties of these functions including symmetry dominance and conjugation we conjecture certain positivity properties in terms of a subfamily of symmetric functions called affine schur functions as applications we show how affine stanley symmetric functions generalize the dual of the k schur functions of lapointe lascoux and morse as well as the cylindric schur functions of postnikov conjecturally affine stanley symmetric functions should be related to the cohomology of the affine flag variety

localization and dimension free estimates for maximal functions abstract in the recent paper a naor t tao random martingales and localization of maximal inequalities j funct anal 259 3 2010 731 779 naor and tao introduce a new class of measures with a so called micro doubling property and present via martingale theory and probability methods a localization theorem for the associated maximal functions as a consequence they obtain a weak type estimate in a general abstract setting for these maximal functions that is reminiscent of the n log n result of stein and stromberg in euclidean spaces the purpose of this work is twofold first we introduce a new localization principle that localizes not only in the time dilation parameter but also in space the proof uses standard covering lemmas and selection processes second we show that a uniform condition for micro doubling in the euclidean spaces provides dimension free estimates for their maximal functions in all l p with p 1 this is done introducing a new technique that allows to differentiate through dimensions

compression bounds for lipschitz maps from the heisenberg group to l 1 we prove a quantitative bi lipschitz non embedding theorem for the heisenberg group with its carnot caratheodory metric and apply it to give a lower bound on the integrality gap of the goemans linial semidefinite relaxation of the sparsest cut problem

regularity of gaussian processes on donne la validite de la conjecture de fernique 1974 sur l existence de mesures de majoration pouvant caracteriser la limitation des processus gaussiens

the effect of unsteadiness on the flow through stenoses and bifurcations abstract this paper is concerned with the influence of a stenosis or a bifurcation on the flow through a tube in particular the effect of unsteadiness is investigated using simple pulsatile and physiological type flows fig 1 the experimental investigations reported herein are concerned with velocity measurements and flow visualizations these measurements performed in a 60 bifurcation have permitted the reconstruction of the three dimensional velocity profiles the importance of the secondary flow in the branching is analyzed for various values of the flow parameters results of tests show a strong influence of unsteadiness on flow characteristics and then on hemodynamic factors one conclusion is the following if hemodynamic factors play an important role in the problems of atheroselerosis then for macrocirculation studies it is necessary to take into account unsteadiness and in particular the actual shape of the flow time forcing function

the tb theorem on non homogeneous spaces 0 introduction main objects and results 3 0 1 main results 4 0 2 an application of t1 heorem electric intensity capacity 7 0 3 how to interpret calderon zygmund operator t 9 0 3 1 bilinear form is defined on lipschitz functions 10 0 3 2 bilinear form is defined for smooth functions 11 0 3 3 apriori boundedness 12 0 4 plan of the paper 15

approximation algorithms for the 0 extension problem in the 0 extension problem we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals our goal is to assign the rest of the nodes to terminals so as to minimize the sum over all edges of the product of the edge s weight and the distance between the terminals to which its endpoints are assigned this problem generalizes the multiway cut problem of dahlhaus et al siam j comput 23 1994 pp 864 894 and is closely related to the metric labeling problem introduced by kleinberg and tardos proceedings of the 40th ieee annual symposium on foundations of computer science new york 1999 pp 14 23 we present approximation algorithms for sc 0 extension in arbitrary graphs we present a o log k approximation algorithm k being the number of terminals we also give o 1 approximation guarantees for weighted planar graphs our results are based on a natural metric relaxation of the problem previously considered by karzanov european j combin 19 1998 pp 71 101 it is similar in flavor to the linear programming relaxation of garg vazirani and yannakakis siam j comput 25 1996 pp 235 251 for the multicut problem and similar to relaxations for other graph partitioning problems we prove that the integrality ratio of the metric relaxation is at least c sqrt lg k for a positive c for infinitely many k our results improve some of the results of kleinberg and tardos and they further our understanding on how to use metric relaxations

pointwise ergodic theorems for radial averages on the heisenberg group leth hn cn r denote the heisenberg group and let rdenote the normalized lebesgue measure on the sphere z 0 z r let x b m be a standard borel probability space on whichhacts measurably and ergodically by measure preserving transformations and let r denote the operator canonically associated with ronlp x we prove maximal and pointwise ergodic theorems inlp for radial averages ron the heisenberg grouphn n 1 the results are best possible for actions of the reduced heisenberg group the method of proof is to use the spectral theory of the banach algebra of radial measures on the group and decay estimates for its characters to establish maximal inequalities using spectral methods in particular littlewood paley stein square functions and analytic interpolation

best constants for the hardy littlewood maximal operator on finite graphs we study the behavior of averages for functions defined on finite graphs g in terms of the hardy littlewood maximal operator mg we explore the relationship between the geometry of a graph and its maximal operator and prove that mg completely determines g even though embedding properties for the graphs do not imply pointwise inequalities for the maximal operators optimal bounds for the p quasi norm of a general graph g in the range 0 p 1 are given and it is shown that the complete graph kn and the star graph sn are the extremal graphs attaining respectively the lower and upper estimates finally we study weak type estimates and some connections with the dilation and overlapping indices of a graph

subdyadic square functions and applications to weighted harmonic analysis abstract through the study of novel variants of the classical littlewood paley stein g functions we obtain pointwise estimates for broad classes of highly singular fourier multipliers on r d satisfying regularity hypotheses adapted to fine subdyadic scales in particular this allows us to efficiently bound such multipliers by geometrically defined maximal operators via general weighted l 2 inequalities in the spirit of a well known conjecture of stein our framework applies to solution operators for dispersive pde such as the time dependent free schrodinger equation and other highly oscillatory convolution operators that fall well beyond the scope of the calderon zygmund theory

characterization of sobolev spaces on the sphere abstract we introduce a new characterization of the sobolev spaces h on the unit sphere s d 1 where the smoothness index is any positive real number and d 2 this characterization is in terms of a square function and it avoids the notion of differentiability the result is fully based on zonal fourier multipliers special functions and the subtle use of classical analysis tools

three dimensional simulation of steady flow past a partial stenosis abstract the steady non uniform viscous flow past a partial stenosis in a circular vessel has been calculated numerically by a modified mac sola method the stenosis is represented by a half ring of nearly square cross section steady flows at reynolds numbers of 25 and 150 were generated by using the time dependent equations of motion and calculating until the solutions became convergent velocity field depictions indicate the occurrence of a small upstream and a large downstream separation zone at the stenosis pressure and shear distributions at various radial and angular positions show significant changes in these kinetic variables near the stenosis strong secondary flow development and decay are exemplified by changes in the vorticity component in the axial direction a change in the nature of this vorticity occurs as the stenosis is passed the size length of the separation zone near the vessel wall as determined by numerical calculation is confirmed by experimental observations at a reynolds number of 149

the zero dispersion limit of the korteweg de vries equation for initial potentials with non trivial reflection coefficient the inverse scattering method is used to determine the distribution limit as 0 of the solution u x t of the initial value problem ut 6uux 2uxxx 0 u x 0 v x where v x is a positive bump which decays sufficiently fast as x x the case v x 0 has been solved by peter d lax and c david levermore 8 9 10 the computation of the distribution limit of u x t as 0 is reduced to a quadratic maximization problem which is then solved

the best constant for the centered hardy littlewood maximal inequality we find the exact value of the best possible constant c for the weak type 1 1 inequality for the one dimensional centered hardy littlewood maximal operator we prove that c is the largest root of the quadratic equation 12c 2 22c 5 0 thus obtaining c 1 5675208 this is the first time the best constant for one of the fundamental inequalities satisfied by a centered maximal operator is precisely evaluated

a power law model of blood flow through a tapered overlapping stenosed artery a mathematical model of non newtonian blood flow through a tapered stenotic artery is considered the non newtonian model chosen is characterized by the generalized power law model incorporating the effect of tapering due to the pulsatile nature of blood flow the flow is assumed to be unsteady laminar two dimensional and axisymmetric the governing equations of motion in terms of the viscous shear stress in the cylindrical coordinate system are solved numerically using a finite difference scheme numerical results obtained for the positive taper angle show that the blood flow characteristics such as the axial velocity profiles flow rate and wall shear stress have lower values while the resistive impedances have higher values than the newtonian model

a k tableau characterization of k schur functions we study k schur functions characterized by k tableaux proving combinatorial properties such as a k pieri rule and a k conjugation this new approach relies on developing the theory of k tableaux and includes the introduction of a weight permuting involution on these tableaux that generalizes the bender knuth involution this work lays the groundwork needed to prove that the set of k schur littlewood richardson coefficients contains the 3 point gromov witten invariants structure constants for the quantum cohomology ring

quantum cohomology of g p and homology of affine grassmannian let g be a simple and simply connected complex algebraic group p g a parabolic subgroup we prove an unpublished result of d peterson which states that the quantum cohomology qh g p of a flag variety is up to localization a quotient of the homology h gr g of the affine grassmannian gr g of g as a consequence all three point genus zero gromov witten invariants of g p are identified with homology schubert structure constants of h gr g establishing the equivalence of the quantum and homology affine schubert calculi

long time behavior of the continuum limit of the toda lattice and the generation of infinitely many gaps from c initial data we analyze a continuum limit of the finite non periodic toda lattice through an associated constrained maximization problem over spectral density functions the maximization problem was derived by deift and mclaughlin using the lax levermore approach initially developed for the zero dispersion limit of the kdv equation it encodes the evolution of the continuum limit for all times including evolution through shocks the formation of gaps in the support of the maximizer is indicative of oscillations in the toda lattice and the lack of strong convergence of the continuum limit for large times the maximizer tends to have zero gaps which is the continuum analogue of the sorting property of the finite lattice using methods from logarithmic potential theory we show that this behavior depends crucially on the initial data we exhibit initial data for which the zero gap ansatz holds uniformly in the spatial parameter at large times and other initial data for which this uniformity fails at all times we then construct an example of c smooth initial data generating at a later time infinitely many gaps in the support of the maximizer while for even larger times all gaps have closed

the best constant in the davis inequality for the expectation of the martingale square function a method is introduced for the simultaneous study of the square function and the maximal function of a martingale that can yield sharp norm inequalities between the two one application is that the expectation of the square function of a martingale is not greater than 3 times the expectation of the maximal function this gives the best constant for one side of the davis two sided inequality the martingale may take its values in any real or complex hilbert space the elementary discrete time case leads quickly to the analogous results for local martingales m indexed by 0 some earlier inequalities are also improved and closely related the levy martingale is embedded in a large family of submartingales

gromov witten invariants for and pontryagin product for ω we give an explicit formula for t equivariant 3 pointed genus zero gromov witten invariants for g b we derive it by finding an explicit formula for the pontryagin product on the equivariant homology of the based loop group k

modulated waves in a semiclassical continuum limit of an integrable nls chain a one dimensional integrable lattice system of odes for complex functions qn that exhibits dispersive phenomena in the phase is studied we consider wave solutions of the local form qn qexp i kn c in which q k and modulate on long time and long space scales t and x n such solutions arise from initial data of the form qn 0 q n exp i n the phase derivative giving the local value of the phase difference k formal asymptotic analysis as 0 yields a first order system of pdes for q and as functions of x and t a certain finite subchain of the discrete system is solvable by an inverse spectral transform we propose formulae for the asymptotic spectral data and use them to study the limiting behavior of t he solution in the case of initial data qn 1 which yield hyperbolic pdes in the formal limit we show that the hyperbolic case is amenable to lax levermore theory the associated maximization problem in the spectral domain is solved by means of a scalar riemann hilbert problem for a special class of data for all times before breaking of the formal pdes under certain assumptions on asymptotic behaviors the phase and amplitude modulation of the discrete systems is shown to be governed by the formal pdes modulation equations after breaking time are not studied full details of the wkb theory and numerical results are left to a future exposition c 1999 john wiley sons inc

semiclassical eigenvalue distribution of the zakharov shabat eigenvalue problem abstract in this paper we consider the semi classical limit of the non self adjoint zakharov shabat eigenvalue problem we conduct a series of careful numerical experiments which provide strong evidence that the number of eigenvalues scales like 1 just as in the self adjoint case and that the eigenvalues appear to approach a limiting curve one general choice of potential functions produces a y shaped spectrum we give an asymptotic argument which predicts a critical value for the phase for which the straight line spectra bifurcates to produce the y shaped spectra this asymptotic prediction agrees quite well with numerical experiments the asymptotic argument also predicts a symmetry breaking in the eigenfunctions which we are able to observe numerically we also show that the number of eigenvalues living away from the real axis for a restricted class of potentials is bounded by c 1 where c is an explicit constant a complete theory of the shape of the eigenvalue curve and a general bound on the number of eigenvalues is still lacking

the weak type 1 1 bounds for the maximal function associated to cubes grow to infinity with the dimension let md be the centered hardy littlewood maximal function associated to cubes in r d with lebesgue measure and let cd denote the lowest constant appearing in the weak type 1 1 inequality satised by md we show that cd 1 as d 1 thus answering for the case of cubes a longstanding

pulsatile flow of non newtonian fluids through arterial stenoses the problem of blood flow through stenoses is solved using the incompressible generalized newtonian model the herschel bulkley bingham and power law fluids are incorporated the geometry corresponds to a rigid circular tube with a partial occlusion calculations are performed by a galerkin finite element method for the pulsatile case a predictor corrector time marching scheme is used with an adaptive time step results are obtained for steady and pulsatile physiological flows computations show that the memory effects taken into account in the model affect deeply the flow compared with newtonian reference case the disturbances are stronger by their vorticity intensity and persist after the geometrical obstacle this is especially true for severe stenoses

schubert polynomials for the affine grassmannian let g be a complex simply connected simple group and k a maximal compact subgroup let f c t denote the field of formal laurent series and o c t the ring of formal power series the quotient qr g f g 0 is known as the affine grassmannian of g it is known that qr is homotopy equivalent to the space of based loops k into the compact group k see 9 30 the multiplication on qk endows the homology h qr and cohomology h qr with structures of dual hopf algebras these hopf algebras were first identified by bott 4 another descrip tion is given by ginzburg 10 which we shall not use the bruhat decomposition of g f induces a stratification of qr by schubert cells ilw indexed by grassman nian elements w of the affine weyl group waffwe denote by aw g h qr and aw g h qr the schubert classes in homology and cohomology when g sl n c bott identifies h qr and h qr with a subring an and a quotient an of the ring of symmetric functions our main theorem theorem 7 1 identifies the schubert classes aw g qr and aw g h qr as explicit symmet ric functions under this isomorphism these symmetric functions are combinatori ally defined in homology the aw g h qr are represented by lascoux lapointe morse s fc schur functions sw x g an and in cohomology the aw g h qr are represented by the dual fc schur functions or affine schur functions fw x g an see 21 24 17 our theorem was originally conjectured by mark shimozono the conjecture was made explicit in the cohomology case by jennifer morse kostant and kumar 14 studied the topology of homogeneous spaces for arbi trary kac moody groups and in particular calculated the structure constants of h qr in the schubert basis using the algebraic construction of the nilhecke ring our connection between the topology and the combinatorics proceeds via the study of three subalgebras of the nilhecke ring the first subalgebra is the nilcoxeter al gebra ao which is the algebra generated by the divided difference operators the second algebra is a certain centralizer algebra aaff s which we call the peterson subalgebra peterson 28 constructs an isomorphism j from the t equivariant ho mology hr qr of the affine grassmannian to z ff s the third algebra b is a combinatorially defined subalgebra of a0 which we call the affine fomin stanley

decomposable compositions symmetric quasisymmetric functions and equality of ribbon schur functions abstract we define an equivalence relation on integer compositions and show that two ribbon schur functions are identical if and only if their defining compositions are equivalent in this sense this equivalence is completely determined by means of a factorization for compositions equivalent compositions have factorizations that differ only by reversing some of the terms as an application we can derive identities on certain littlewood richardson coefficients finally we consider the cone of symmetric functions having a nonnnegative representation in terms of the fundamental quasisymmetric basis we show the schur functions are among the extremes of this cone and conjecture its facets are in bijection with the equivalence classes of compositions

the spectral transform in the semiclassical limit of a finite discrete nls chain the direct eigenvalue problem associated with an inverse scattering method for a finite nonlinear schrodinger chain is studied in the semiclassical limit in the case that the initial data for the chain are less than unity in modulus the eigenvalue problem is unitary and the associated norming constants are real valued formal asymptotic wkb analysis is performed and formulas for the asymptotic spectral density and norming constants are obtained they are supported by known facts about the discrete transform rigorous asymptotics and numerical calculations

mathematical modelling of pulsatile flow of casson s fluid in arterial stenosis the effects of non newtonian nature of blood and pulsatility on flow through a stenosed artery have been investigated it is of interest to note that the thickness of the viscous flow region changes with axial distance an important result is that the mean and steady flow rates decrease as the yield stress increases the critical value of the yield stress at which the flow rate behaviour changes from one type to another has been determined another important result of pulsatility is the mean resistance to flow is greater than its steady flow value whereas the mean value of the wall shear for pulsatile blood flow is equal to steady wall shear stress the velocity profiles and associated physiological characteristics involved in the analysis have been determined many standard results regarding casson and newtonian fluids flow uniform and steady flow in an artery can be obtained in the present analysis as the special cases

a graph theoretic game and its application to the k server problem this paper investigates a zero sum game played on a weighted connected graph g between two players the tree player and the edge player at each play the tree player chooses a spanning tree t and the edge player chooses an edge e the payoff to the edge player is cost t e defined as follows if e lies in the tree t then cost t e 0 if e does not lie in the tree then cost t e cycle t e w e where w e is the weight of edge e and cycle t e is the weight of the unique cycle formed when edge e is added to the tree t the main result is that the value of the game on any n vertex graph is bounded above by exp o sqrt log n log log n it is conjectured that the value of the game is o log n the game arises in connection with the k server problem on a road network i e a metric space that can be represented as a multigraph g in which each edge e represents a road of length w e it is shown that if the value of the game on g is val g w then there is a randomized strategy that achieves a competitive ratio of k 1 val g w against any oblivious adversary thus on any n vertex road network there is a randomized algorithm for the k server problem that is k cdot exp o sqrt log n log log n competitive against oblivious adversaries at the heart of the analysis of the game is an algorithm that provides an approximate solution for the simple network design problem specifically for any n vertex weighted connected multigraph the algorithm constructs a spanning tree t such that the average over all edges e of cost t e is less than or equal to exp o sqrt log n log log n this result has potential application to the design of communication networks it also improves substantially known estimates concerning the existence of a sparse basis for the cycle space of a graph

quantum cohomology and the k schur basis we prove that structure constants related to hecke algebras at roots of unity are special cases of k littlewood richardson coefficients associated to a product of k schur functions as a consequence both the 3 point gromov witten invariants appearing in the quantum cohomology of the grassmannian and the fusion coefficients for the wzw conformal field theories associated to su l are shown to be k littlewood richardson coefficients from this mark shimozono conjectured that the k schur functions form the schubert basis for the homology of the loop grassmannian whereas k schur coproducts correspond to the integral cohomology of the loop grassmannian we introduce dual k schur functions defined on weights of k tableaux that given shimozono s conjecture form the schubert basis for the cohomology of the loop grassmannian we derive several properties of these functions that extend those of skew schur functions

lp compression traveling salesmen and stable walks we show that if h is a group of polynomial growth whose growth rate is at least quadratic then the lp compression of the wreath product z oh equals max 1 p 1 2 we also show that the lp compression of z oz equals max p 2p 1 2 3 and the lp compression of z oz 0 the zero section of z oz equipped with the metric induced from z o z equals max p 1 2p 3 4 the fact that the hilbert compression exponent of z o z equals 3 while the hilbert compression exponent of z o z 0 equals 3 4 is used to show that there exists a lipschitz function f z o z 0 l2 which cannot be extended to a lipschitz function defined on all of z o z