on generalized cells in affine weyl groups abstract we determine the lowest generalized two sided cell for affine weyl groups we show that it consists of at most w 0 generalized left cells where w 0 denotes the corresponding finite weyl group for parameters coming from graph automorphisms we prove that this bound is exact for such parameters we also characterize all generalized left cells for finite and affine weyl groups

the principal series of p adic groups with disconnected center let g be a split connected reductive group over a local non archimedean field we classify all irreducible complex g representations in the principal series irrespective of the dis connectedness of the center of g this leads to a local langlands correspondence for principal series representations of g it satisfies all expected properties in particular it is functorial with respect to homomorphisms of reductive groups at the same time we show that every bernstein component s in the principal series has the structure of an extended quotient of bernstein s torus by bernstein s finite group both attached to s

parabolic projective functors in type a abstract we classify projective functors on the regular block of rocha caridi s parabolic version of the bgg category o in type a in fact we show that in type a the restriction of an indecomposable projective functor from o to the parabolic category is either indecomposable or zero as a consequence we obtain that projective functors on the parabolic category o in type a are completely determined up to isomorphism by the linear transformations they induce on the level of the grothendieck group which was conjectured by stroppel in 39

joint mean covariance models with applications to longitudinal data unconstrained parameterisation summary we provide unconstrained parameterisation for and model a covariance using covariates the cholesky decomposition of the inverse of a covariance matrix is used to associate a unique unit lower triangular and a unique diagonal matrix with each covariance matrix the entries of the lower triangular and the log of the diagonal matrix are unconstrained and have meaning as regression coefficients and prediction variances when regressing a measurement on its predecessors an extended generalised linear model is introduced for joint modelling of the vectors of predictors for the mean and covariance subsuming the joint modelling strategy for mean and variance heterogeneity gabriel s antedependence models dempster s covariance selection models and the class of graphical models the likelihood function and maximum likelihood estimators of the covariance and the mean parameters are studied when the observations are normally distributed applications to modelling nonstationary dependence structures and multivariate data are discussed and illustrated using real data a graphical method similar to that based on the correlogram in time series is developed and used to identify parametric models for nonstationary covariances

the distinguished involutions with a value n2 3n 3 in the weyl group of type dn let w s be a weyl group and h its associated hecke algebra let a z u u 1 be the laurent polynomial ring kazhdan and lusztig representation of coxeter groups and hecke algebras invent math 53 1979 165 184 introduced two a bases tw w w and cw w w for the hecke algebra h associated to w let yw y wul w l y ty then yw w w is also an a base for the hecke algebra in this paper we give an explicit expression for certain kazhdan lusztig basis elements cw as a linear combination of yx s in the hecke algebra of type dn in fact this gives also an explicit expression for certain kazhdan lusztig basis elements cw as a linear combination of tx s in the hecke algebra of type dn thus we describe also explicitly the kazhdan lusztig polynomials for certain elements of the weyl group we study also the joint relation among some elements in w and some distinguished involutions with a value n2 3n 3 in the weyl group of type dn

a conservative formulation of the multidimensional upwind residual distribution schemes for general nonlinear conservation laws in the present paper we consider the numerical solution of systems of general nonlinear hyperbolic conservation laws on unstructured grids by means of the residual distribution method we propose a new formulation of the first order linear optimal positive n scheme relying on a contour integration of the convective fluxes over the boundaries of an element full conservation is achieved for arbitrary flux functions while the robustness and the monotone shock capturing of the original n scheme is retained the new variant of the n scheme is combined with the conservative second order linear lda scheme to obtain a nonlinear second order monotone b scheme the performance of the new residual distribution schemes is evaluated on problems governed by the euler equations as an application to a more complex system of conservation laws lacking an exact conservative linearization we solve the ideal magnetohydrodynamics equations in two spatial dimensions

nonparametric modeling of longitudinal covariance structure in functional mapping of quantitative trait loci estimation of the covariance structure of longitudinal processes is a fundamental prerequisite for the practical deployment of functional mapping designed to study the genetic regulation and network of quantitative variation in dynamic complex traits we present a nonparametric approach for estimating the covariance structure of a quantitative trait measured repeatedly at a series of time points specifically we adopt huang et al s 2006 biometrika 93 85 98 approach of invoking the modified cholesky decomposition and converting the problem into modeling a sequence of regressions of responses a regularized covariance estimator is obtained using a normal penalized likelihood with an l 2 penalty this approach embedded within a mixture likelihood framework leads to enhanced accuracy precision and flexibility of functional mapping while preserving its biological relevance simulation studies are performed to reveal the statistical properties and advantages of the proposed method a real example from a mouse genome project is analyzed to illustrate the utilization of the methodology the new method will provide a useful tool for genome wide scanning for the existence and distribution of quantitative trait loci underlying a dynamic trait important to agriculture biology and health sciences

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

the based ring of the lowest two sided cell of an affine weyl group we show that lusztig s homomorphism from an affine hecke algebra to the direct summand of its asymptotic hecke algebra corresponding to the lowest two sided cell is related to the homomorphism constructed by chriss and ginzburg using equivariant k theory by a matrix over the representation ring of the associated algebraic group

second order blended multidimensional upwind residual distribution scheme for steady and unsteady computations multidimensional upwind residual distribution rd schemes have become an appealing alternative to more widespread finite volume and finite element methods fem for solving compressible fluid flows the rd approach allows to construct nonlinear second order and non oscillatory methods at the same time they are routinely used for steady state calculations of the complex flow problems e g 3d turbulent transonic industrial type simulations h deconinck k sermeus r abgrall status of multidimensional upwind residual distribution schemes and applications in aeronautics aaia paper 2000 2328 aiaa 2000 k sermeus h deconinck drag prediction validation of a multi dimensional upwind solver cfd based aircraft drag prediction and reduction vki lecture series 2003 02 von karman institute for fluid dynamics chausee do waterloo 72 b 1640 rhode saint genese belgium 2003 despite its maturity some problems are still present for the nonlinear schemes developed up to now namely a poor iterative convergence for the transonic problems and a decrease of accuracy in smooth parts of the flow caused by a weak l 2 instability m ricchiuto construction and analysis of compact residual discretizations for conservation laws on unstructured meshes ph d thesis universite libre de bruxelles von karman institute for fluid dynamics 2005 we have developed a new formulation of a blended scheme between the second order linear lda r abgrall m mezine residual distribution scheme for steady problems 33rd computational fluid dynamics course vki lecture series 2003 05 von karman institute for fluid dynamics chausee do waterloo 72 b 1640 rhode saint genese belgium 2003 scheme and the first order n scheme the blending coefficient is based on a simple shock capturing operator and it is properly scaled such that second order accuracy is preserved the approach is extended to unsteady flows problems using consistent formulation of the lda scheme with the mass matrix m mezine m ricchiuto r abgrall h deconinck monotone and stable residual distribution schemes on prismatic space time elements for unsteady conservation laws 33rd computational fluid dynamics course von karman institute for fluid dynamics 2003 r abgrall m mezine construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems j comput phys 188 2003 16 55 for the time integration a three point backward scheme is selected for its accuracy and robustness and the shock capturing operator is modified appropriately to handle moving shocks we present a numerical solution of several challenging test cases involving the solution of the euler equations from the subsonic to the hypersonic regime all the tests shows very good accuracy robustness and convergence properties

covariance matrix selection and estimation via penalised normal likelihood summary we propose a nonparametric method for identifying parsimony and for producing a statistically efficient estimator of a large covariance matrix we reparameterise a covariance matrix through the modified cholesky decomposition of its inverse or the one step ahead predictive representation of the vector of responses and reduce the nonintuitive task of modelling covariance matrices to the familiar task of model selection and estimation for a sequence of regression models the cholesky factor containing these regression coefficients is likely to have many off diagonal elements that are zero or close to zero penalised normal likelihoods in this situation with l1 and l2 penalities are shown to be closely related to tibshirani s 1996 lasso approach and to ridge regression adding either penalty to the likelihood helps to produce more stable estimators by introducing shrinkage to the elements in the cholesky factor while because of its singularity the l1 penalty will set some elements to zero and produce interpretable models an algorithm is developed for computing the estimator and selecting the tuning parameter the proposed maximum penalised likelihood estimator is illustrated using simulation and a real dataset involving estimation of a 102 x 102 covariance matrix

maximum likelihood estimation via the ecm algorithm a general framework two major reasons for the popularity of the em algorithm are that its maximum step involves only complete data maximum likelihood estimation which is often computationally simple and that its convergence is stable with each iteration increasing the likelihood when the associated complete data maximum likelihood estimation itself is complicated em is less attractive because the m step is computationally unattractive in many cases however complete data maximum likelihood estimation is relatively simple when conditional on some function of the parameters being estimated

on the category o for rational cherednik algebras we study the category o of representations of the rational cherednik algebra aw attached to a complex reflection group w w e construct an exact functor called knizhnik zamolodchikov functor o hw mod where hw is the finite iwahori hecke algebra associated to w we prove that the knizhnik zamolodchikov functor induces an equiv alence between o otor the quotient of o by the subcategory of aw modules supported on the discriminant and the category of finite dimensional hw modules the standard aw modules go under this equivalence to certain modules arising in kazhdan lusztig theory of cells provided w is a weyl group and the hecke algebra hw has equal parameters we prove that the category o is equivalent to the module category over a finite dimensional algebra a generalized q schur algebra associated to w

canonical left cells in affine weyl groups soit w s un groupe de weyl affine irreductible on montre que chaque cellule a droite et a gauche de w contient une cellule canonique a gauche

the joint relations and the set d1 in certain crystallographic groups abstract for any coxeter group w d kazhdan and g lusztig 1979 invent math 53 165 184 introduced some joint relations among the elements of w by which certain representations of w and its associated hecke algebra h are constructed recently in a letter to lusztig t a springer obtained some interesting results concerning the connection between some non zero structural coefficients of h and the joint relations in w when w is a certain crystallographic group he investigated some properties of a subset d 1 of w springer denoted d 1 by f in his letter which plays an important role for the joint relations in w in the present paper we shall develop some results of springer we shall deduce further joint relations in w from some non zero structural coefficients of h in particular we shall make some detailed discussion for the set d 1 this includes the approach to the proof of a conjecture of lusztig concerning the characterization of the set d 1 the connection between the finiteness of d 1 and some joint relations in w the explicit description for some elements of d 1 and the connectedness of the set d 0 d 2 1 see 3 2 and 5 2 for the notations

homological invariants in category for the general linear superalgebra we study three related homological properties of modules in the bgg category mathcal o for basic classical lie superalgebras with specific focus on the general linear superalgebra these are the projective dimension associated variety and complexity we demonstrate connections between projective dimension and singularity of modules and blocks similarly we investigate the connection between complexity and atypicality this creates concrete tools to describe singularity and atypicality as homological and hence categorical properties of a block however we also demonstrate how two integral blocks in category mathcal o with identical global categorical characteristics of singularity and atypicality will generally still be inequivalent this principle implies that category mathcal o for mathfrak gl m vert n can contain infinitely many non equivalent blocks which we work out explicitly for mathfrak gl 3 vert 1 all of this is in sharp contrast with category mathcal o for lie algebras but also with the category of finite dimensional modules for superalgebras furthermore we characterise modules with finite projective dimension to be those with trivial associated variety we also study the associated variety of verma modules to do this we also classify the orbits in the cone of self commuting odd elements under the action of an even borel subgroup

cells and the reflection representation of weyl groups and hecke algebras let f be the generic algebra of the finite crystallographic coxeter group w defined over the ring q u1 2 u 2 first the two sided cell corresponding to the reflection representation of f is shown to consist of the nonidentity elements of w having a unique reduced expression next the matrix entries of this representation are computed in terms of certain kazhdanlusztig polynomials finally the kazhdan lusztig polynomials just mentioned are described in case w is of type a1 lor b

parsimonious covariance matrix estimation for longitudinal data this article proposes a data driven method to identify parsimony in the covariance matrix of longitudinal data and to exploit any such parsimony to produce a statistically efficient estimator of the covariance matrix the approach parameterizes the covariance matrix through the cholesky decomposition of its inverse for longitudinal data this is a one step ahead predictive representation and the cholesky factor is likely to have off diagonal elements that are zero or close to zero a hierarchical bayesian model is used to identify any such zeros in the cholesky factor similar to approaches that have been successful in bayesian variable selection the model is estimated using a markov chain monte carlo sampling scheme that is computationally efficient and can be applied to covariance matrices of high dimension it is demonstrated through simulations that the proposed method compares favorably in terms of statistical efficiency with a highly regarded competing approach the estimator is applied to three

the lowest two sided cell of a coxeter group with complete graph abstract in this paper we give a description of the structure of the based ring of the lowest two sided cell for a weighted coxeter group with complete graph it is proved that this ring is generated in a simple way by some subrings which are isomorphic to the based rings of the lowest two sided cells for affine weyl groups of type a 1 or a 2

on certain distinguished involutions in the weyl group of type dn let w s be a weyl group let a z u u 1 be the laurent polynomial ring in an indeterminate u kazhdan and lusztig invent math 33 1979 165 184 introduced two a bases tw w w and cw w w for the hecke algebra h associated to w let yw y wul w l y ty then yw w w is also an a basis for the hecke algebra in this paper we assume w of type dn and we express certain kazhdan lusztig basis elements cw as a linear combination of yx s this in turn gives an explicit expression for certain kazhdan lusztig basis elements cw as a linear combination of tx s thus we describe explicitly the kazhdan lusztig polynomials for certain pairs of elements of w we also study the joint relation among some elements in w in particular we find certain distinguished involutions in the two sided cell t of w with a value 12 n2 n 4t2 2t for 1 2t n and n even 12 n2 n 4t2 2t for n odd where the two sided cell t does not contain the longest element w0 j in subgroup wj of w for any j s

left cells in the affine weyl group of typec4 we find a representative set of left cells of the affine weyl groupwaof typec4as well as its left cell graphs by applying an algorithm this algorithm was designed in my previous paper tohoku j math 46 1994 105 124 it is reformulated and improved in a more efficient form here these representatives of left cells are presented as the vertices of so called essential graphs so that the generalized invariants of left cells ofwaare actually described explicitly the latter almost characterize the left cells ofwa some comments and conjectures are proposed on cells of affine weyl groups mostly for the case of typecl l 2

left connectedness of some left cells in certain coxeter groups of simply laced type abstract let w be an irreducible finite or affine weyl group of simply laced type we show that any w w with a w 6 satisfies condition c w x w j y for some x y w and some j s with w j finite and l w j a w see 0 1 0 2 for the notation w j w j l w and a w we also show that if l is a left cell of w all of whose elements satisfy condition c then the distinguished involution d l of w in l satisfies d l z 1 z z 1 w j z for any z w j z e min l with j l z see 1 6 for the notation z 1 z and 0 3 for l z e min l and e l verifying a conjecture of mine in j y shi a survey on the cell theory of affine weyl groups adv sci china math 3 1990 79 98 conjecture 8 10 in our case if e l e min l then we show that the left cell l is left connected verifying a conjecture of lusztig in our case

simple transitive 2 representations of small quotients of soergel bimodules in all finite coxeter types but i2 12 i2 18 and i2 30 we classify simple transitive 2 representations for the quotient of the 2 category of soergel bimodules over the coinvariant algebra which

space time residual distribution on moving meshes abstract this article investigates the potential for an r adaptation algorithm to improve the efficiency of space time residual distribution schemes in the approximation of time dependent hyperbolic conservation laws e g scalar advection shallow water flows on unstructured triangular meshes in this adaptive framework the connectivity of the mesh and hence the number of degrees of freedom remain fixed but the mesh nodes are continually relocated as the flow evolves so that features of interest remain resolved as they move within the domain adaptive strategies of this type are well suited to the space time residual distribution framework because when the discrete representation is allowed to be discontinuous in time these algorithms can be designed to be positive and hence stable for any choice of time step even on the distorted space time prisms which arise from moving the nodes of an unstructured triangular mesh consequently a local increase in mesh resolution does not impose a more restrictive stability constraint on the time step which can instead be chosen according to accuracy requirements the order of accuracy of the fixed mesh scheme is retained on the moving mesh in the majority of applications tested space time schemes of this type are analogous to conservative ale formulations and automatically satisfy a discrete geometric conservation law so moving the mesh does not artificially change the flow volume for pure conservation laws for shallow water flows over variable bed topography the so called c property retention of hydrostatic balance between flux and source terms required to maintain the steady state of still flat water can also be satisfied by considering the mass balance equation in terms of free surface level instead of water depth even when the mesh is moved the r adaptation is applied within each time step by interleaving the iterations of the nonlinear solver with updates to mesh node positions the node movement is driven by a monitor function based on weighted approximations of the scaled gradient and laplacian of the local solution and regularised by a smoothing iteration numerical results are shown in two dimensions for both scalar advection and for shallow water flow over a variable bed which show that even for this simple implementation of the mesh movement reductions in cpu times of up to 60 can be attained without increasing the error

stabilized residual distribution for shallow water simulations we propose a stabilized residual distribution rd scheme for the simulation of shallow water flows the final discretization is obtained combining the stabilized rd approach proposed in abgrall j comp phys 214 2006 and ricchiuto and abgrall iccfd4 springer verlag 2006 with the conservative formulation already used in ricchiuto et al j comp phys 222 2007 to simulate shallow water flows the scheme proposed is a nonlinear variant of a lax friedrichs type discretization it is well balanced it actually yields second order of accuracy in smooth areas and it preserves the positivity of the height of the water in presence of dry areas this is made possible by the residual character of the discretization by properly adapting the stabilization operators proposed in abgrall j comp phys 214 2006 and ricchiuto and abgrall iccfd4 springer verlag 2006 and thanks to the positivity preserving character of the underlying lax friedrichs scheme we demonstrate the properties of the discretization proposed on a wide variety of tests

cells in coxeter groups i abstract the purpose of this article is to shed new light on the combinatorial structure of kazhdan lusztig cells in infinite coxeter groups w our main focus is the set d of distinguished involutions in w which was introduced by lusztig in one of his first papers on cells in affine weyl groups we conjecture that the set d has a simple recursive structure and can be enumerated algorithmically starting from the distinguished involutions of finite coxeter groups moreover to each element of d we assign an explicitly defined set of equivalence relations on w that altogether conjecturally determine the partition of w into left right cells we are able to prove these conjectures only in a special case but even from these partial results we can deduce some interesting corollaries

affine cellularity of affine yokonuma hecke algebras abstract we establish an explicit algebra isomorphism between the affine yokonuma hecke algebra y r n q and a direct sum of matrix algebras with coefficients in tensor products of affine hecke algebras of type a as an application of this result we show that y r n q is affine cellular in the sense of koenig and xi and further prove that it has finite global dimension when the parameter q is not a root of the poincare polynomial as another application we also recover the modular representation theory of y r n q previously obtained in 7

a sum rule for scale factors in the goldie rank polynomials this paper is a corollary to s and we adopt the same notation which is also that of s there we showed how two deep questions concerning the enveloping algebra of a semisimple lie algebra could be reduced to a purely combinatorial analysis of the kazhdan lusztig polynomials the first concerns the computation of sot l l wl l yl apart from its own intrinsic interest this question has deep interconnections with geometry via the component group a of the associated nilpotent orbit the second concerns the coefficients z which are ratios of goldie ranks these are central to the determination of the scale factors in the goldie rank polynomials which in turn relate to complete primeness and hence to unitarity see 7 1 4 16 7 121 it first seemed that the results in s did not give any easily extractable information however in 13 lusztig shows that for some canonical quotient a of a there exists a left cell of the weyl group w such that the character table of a can be read off in a well defined manner from data involving the hecke algebra module structure of the cell which we shall call a lusztig cell combined with s this gives several remarkable results first when rl z2 are involutions in a lusztig cell sot l l r a l 12 is determined by multiplying appropriate irreducible characters of a and identifying coefficients this implies that when a e s which occurs in type e it can happen that l l zl l r l is not multiplicity free providing thereby a counterexample to conjecture of 5 5 111 even in the diagonal case second if r is an involution in a lusztig cell then z is the degree of an irreducible character of a third we prove that z divides ial for every element w of a lusztig cell in partial answer to a conjecture in 9 3 231

an explicit residual based approach for shallow water flows we describe fully explicit residual based discretizations of the shallow water equations with friction on unstructured grids the schemes are obtained by properly adapting the explicit construction proposed in ricchiuto and abgrall 2010 57 in particular previous work on well balanced integration ricchiuto 2011 56 and preservation of the depth non negativity ricchiuto and bollermann 2009 60 is reformulated in the context of a genuinely explicit time stepping still based on a weighted residual approximation the paper discusses in depth how to achieve in this context an exact preservation of all the simple known steady equilibria and how to obtain a super consistent approximation for smooth non trivial moving equilibria the treatment of the wetting drying interface is also discussed giving formal conditions for the preservation of the non negativity of the depth for a particular case based on a nonlinear variant of a lax friedrichs type scheme the approach is analyzed and tested thoroughly the quality of the numerical results shows the interest in the proposed approach over previously proposed schemes in terms of accuracy and efficiency

on use of the em algorithm for penalized likelihood estimation summary the em algorithm is a popular approach to maximum likelihood estimation but has not been much used for penalized likelihood or maximum a posteriori estimation this paper discusses properties of the em algorithm in such contexts concentrating on rates of conver gence and presents an alternative that is usually more practical and converges at least as quickly the em algorithm is a general approach to maximum likelihood estimation rather than a specific algorithm dempster et al 1977 discussed the method and derived basic properties demonstrating that a variety of procedures previously developed rather informally could be unified the common strand to problems where the approach is applicable is a notion of incomplete data this includes the conventional sense of missing data but is much broader than that the em algorithm demon strates its strength in situations where some hypothetical experiment yields data from which estimation is particularly convenient and economical the incomplete data actually at hand are regarded as observable functions of these complete data the resulting algorithms while usually slow to converge are often extremely simple and remain practical in large problems where no other approaches may be feasible dempster et al 1977 briefly refer to the use of the same approach to the problem of finding the posterior mode maximum a posteriori estimate in a bayesian estima

hochschild homology of affine hecke algebras abstract let h h r q be an affine hecke algebra with complex possibly unequal parameters q which are not roots of unity we compute the hochschild and the cyclic homology of h it turns out that these are independent of q and that they admit an easy description in terms of the extended quotient of a torus by a weyl group both of which are canonically associated to the root datum r for positive q we also prove that the representations of the family of algebras h r q c come in families which depend analytically on analogous results are obtained for graded hecke algebras and for schwartz completions of affine hecke algebras

some homological properties of category o iii we prove that thick category o associated to a semi simple complex finite dimensional lie algebra is extension full in the category of all modules we also prove the weak alexandru conjecture both for regular blocks of thick category o and the associated categories of harish chandra bimodules but disprove it for singular blocks

lusztigʼs a function for coxeter groups of rank 3 abstract we show that lusztig s a function of a coxeter group is bounded if the rank of the coxeter group is 3

application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes we consider the numerical solution of the shallow water equations on unstructured grids we focus on flows over wet areas the extension to the case of dry bed will be reported elsewhere the shallow water equations fall into the category of systems of conservation laws which can be symmetrized thanks to the existence of a mathematical entropy coinciding in this case with the total energy our aim is to show the application of a particular class of conservative residual distribution rd schemes to the discretization of the shallow water equations and to analyze their discrete accuracy and stability properties we give a review of conservative rd schemes showing relations between different approaches previously published and recall l stability and accuracy criteria characterizing the schemes in particular the accuracy of the rd method in presence of source terms is analyzed and conditions to construct rth order discretizations on irregular triangular grids are proved it is shown that the rd approach gives a natural way of obtaining high order discretizations which moreover preserves exactly the steady lake at rest solution independently on mesh topology nature of the variation of the bottom and polynomial order of interpolation used for the unknowns we also consider more general analytical solutions which are less investigated from the numerical view point on irregular grids linearity preserving rd schemes yield a truly second order approximation we also sketch a strategy to achieve discretizations which preserve exactly some of these solutions numerical results on steady and time dependent problems involving smooth and non smooth variations of the bottom topology show very promising features of the approach

representations of affine hecke algebras and based rings of affine weyl groups it is known that an interesting part of the study of the representation theory of p adic groups can be reduced to the study of the representation theory of affine hecke algebras b v let w s be an extended affine weyl group and hk qo the corresponding hecke algebra over a field k with a nonzero parameter qo k when k is the complex numbers field and qo is not a root of unity a classification of simple representations of hk qo was established in kl2 deligne langlands lusztig classification for affine type a a classification of simple representations of hk qo was obtained in am for any qo and arbitrary sufficiently large k when k is algebraically closed and has positive characteristic the representations of hk qo were studied by vign ras as part of her study of modular representations of p adic groups v in this paper we shall verify a conjecture of lusztig l6 7 a by means of the based ring of an extended affine weyl group theorem 3 3 the conjecture says that if the parameter qo is not a root of the corresponding poincar polynomial then the classification established in kl2 remains valid the restriction is necessary for the classification see remark 3 4 a

cells of harish chandra modules for real classical groups we compute the w module structure of a large number of cells of harish chandra modules for a classical group g where w is the complex weyl group

affine cellular algebras abstract graham and lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras many classes of finite dimensional algebras including various hecke algebras and diagram algebras have been shown to be cellular and the theory due to graham and lehrer successfully has been applied to these algebras we will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field affine hecke algebras of type a and infinite dimensional diagram algebras like the affine temperley lieb algebras are shown to be examples of our definition the isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties in this way representation theory of non commutative algebras is linked with commutative algebra moreover conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification these conditions are shown to be satisfied for the affine hecke algebra of type a if the quantum parameter is not a root of the poincare polynomial

functional mapping of quantitative trait loci underlying growth trajectories using a transform both sides logistic model the incorporation of developmental control mechanisms of growth has proven to be a powerful tool in mapping quantitative trait loci qtl underlying growth trajectories a theoretical framework for implementing a qtl mapping strategy with growth laws has been established this framework can be generalized to an arbitrary number of time points where growth is measured and becomes computationally more tractable when the assumption of variance stationarity is made in practice however this assumption is likely to be violated for age specific growth traits due to a scale effect in this article we present a new statistical model for mapping growth qtl which also addresses the problem of variance stationarity by using a transform both sides tbs model advocated by carroll and ruppert 1984 journal of the american statistical association 79 321 328 the tbs based model for mapping growth qtl cannot only maintain the original biological properties of a growth model but also can increase the accuracy and precision of parameter estimation and the power to detect a qtl responsible for growth differentiation using the tbs based model we successfully map a qtl governing growth trajectories to a linkage group in an example of forest trees the statistical and biological properties of the estimates of this growth qtl position and effect are investigated using monte carlo simulation studies the implications of our model for understanding the genetic architecture of growth are discussed

a proof of lusztig s conjectures for affine type g2 with arbitrary parameters we prove lusztig s conjectures bf p1 bf p15 for the affine weyl group of type tilde g 2 for all choices of parameters our approach to compute lusztig s mathbf a function is based on the notion of a balanced system of cell representations for the hecke algebra we show that for arbitrary coxeter type the existence of balanced system of cell representations is sufficient to compute the mathbf a function and we explicitly construct such a system in type tilde g 2 for arbitrary parameters we then investigate the connection between kazhdan lusztig cells and the plancherel theorem in type tilde g 2 allowing us to prove bf p1 and determine the set of duflo involutions from there the proof of the remaining conjectures follows very naturally essentially from the combinatorics of weyl characters of types g 2 and a 1 along with some explicit computations for the finite cells

the second lowest two sided cell in an affine weyl group abstract let w a be an irreducible affine weyl group with w 0 the associated weyl group the present paper is to study the second lowest two sided cell qr of w a let n qr be the number of left cells of w a in qr we conjecture that the equality n qr 1 2 w 0 should always hold when w a is either a n 1 n 2 or of rank 4 this equality can be verified by the existing data see 0 3 then the main result of the paper is to prove the inequality n qr 1 2 w 0 in all cases

motives with exceptional galois groups and the inverse galois problem we construct motivic l adic representations of textup gal overline mathbb q mathbb q into exceptional groups of type e 7 e 8 and g 2 whose image is zariski dense this answers a question of serre the construction is uniform for these groups and is inspired by the langlands correspondence for function fields as an application we solve new cases of the inverse galois problem the finite simple groups e 8 mathbb f ell are galois groups over mathbb q for large enough primes l

maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix the positive definiteness constraint is the most awkward stumbling block in modelling the covariance matrix pourahmadi s 1999 unconstrained parameterisation models covariance using covariates in a similar manner to mean modelling in generalised linear models the new covariance parameters have statistical interpretation as the regression coefficients and logarithms of prediction error variances corresponding to regressing a response on its predecessors in this paper the maximum likelihood estimators of the parameters of a generalised linear model for the covariance matrix their consistency and their asymptotic normality are studied when the observations are normally distributed these results along with the likelihood ratio test and penalised likelihood criteria such as bic for model and variable selection are illustrated using a real dataset

a unipotent support for irreducible representations 1 1 let g f be the group of fq rational points of a connected reductive algebraic group g defined over a finite field f throughout this paper we shall assume that the characteristic of the ground field is large the following is part of problem ii in l9 let p be an irreducible complex character of g f show that there is a unique umpotent class c in g which has the property that cga c cfqi p g 0 and has maximal dimension among classes with this property one of the results of this paper is a solution of this problem we also prove the following refinement if g e gf is such that tr g p 0 then the unipotent part of g lies in c or in a conjugacy class of dimension dim c we call c the unipotent support of p another map from the set of irreducible representations of gf to unipotent classes in g has been defined in l8 13 43 this was the composition of two maps one from irreducible representations of gf to a set of parameters in the langlands dual of g and one from this set of parameters to unipotent classes in g another conjectural definition of this map which did not pass through the langlands dual has been proposed by kawanaka kl k2 k3 in terms of generalized gelfand graev representations in analogy with real lie groups he uses the name wave front set for the unipotent class attached to a representation one of the results of this paper is a proof see section 11 of kawanaka s conjecture that his wave front set is well defined and that it coincides with the one defined in l8 this has been known earlier from the work of kawanaka in certain special cases we also show that the unipotent support of an irreducible representation coincides with the wave front set of the dual representation

the leading coefficient of certain kazhdan lusztig polynomials of the permutation group sn abstract in this paper we show that the leading coefficient y w of certain kazhdan lusztig polynomials p y w of the permutation group s n of 1 2 n are not greater than 1 more precisely we show that the leading coefficients y w are not greater than 1 whenever a y a w where a s n n is the function defined in g lusztig cells in affine weyl groups in algebraic groups and related topics in adv stud pure math vol 6 kinokunia north holland 1985 pp 255 287 see 1 5 for a simple interpretation of the function a

the relation lr on some elements of the affine weyl group c n abstract let w s be the affine weyl group of type c n with s its coxeter generator set let 2 n 1 be the set of all partitions 1 r of 2 n 1 such that j 1 2 k 1 j is odd for any k n with 2 k 1 r for any j s let w j be the longest element in the parabolic subgroup of w generated by j we define a map w j j s 2 n 1 and study the preorder lr on the set w j j s and its relation with the partial order on the set w j j s where iterating star operations and primitive pairs play an important role

the steinberg variety and representations of reductive groups abstract we give an overview of some of the main results in geometric representation theory that have been proved by means of the steinberg variety steinberg s insight was to use such a variety of triples in order to prove a conjectured formula by grothendieck the steinberg variety was later used to give an alternative approach to springer s representations and played a central role in the proof of the deligne langlands conjecture for hecke algebras by kazhdan and lusztig

generalized likelihood ratio statistics and wilks phenomenon likelihood ratio theory has had tremendous success in parametric inference due to the fundamental theory of wilks yet there is no general applicable approach for nonparametric inferences based on function estimation maximum likelihood ratio test statistics in general may not exist in nonparametric function estimation setting even if they exist they are hard to find and can not be optimal as shown in this paper we introduce the generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics a new s wilks phenomenon is unveiled we demonstrate that a class of the generalized likelihood statistics based on some appropriate nonparametric estimators are asymptotically distribution free and follow chi 2 distributions under null hypotheses for a number of useful hypotheses and a variety of useful models including gaussian white noise models nonparametric regression models varying coefficient models and generalized varying coefficient models we further demonstrate that generalized likelihood ratio statistics are asymptotically optimal in the sense that they achieve optimal rates of convergence given by ingster they can even be adaptively optimal in the sense of spokoiny by using a simple choice of adaptive smoothing parameter our work indicates that the generalized likelihood ratio statistics are indeed general and powerful for nonparametric testing problems based on function estimation

isomorphisms between affine hecke algebras of type a2 abstract we prove that two affine hecke algebras h p and h q of type a 2 with nonzero parameters p q c are not isomorphic whenever p q and p q 1 theorem 3 1 however they are isomorphic locally more precisely we prove that the quotient algebras h s q and h t p are isomorphic provided that n s q n t p for type a 2 theorem 2 4

the cells in the weighted coxeter group c n l m abstract the affine weyl group c n s can be realized as the fixed point set of the affine weyl group a m s m m 2 n 1 2 n 2 n 1 under a certain group automorphism m n let l m be the length function of a m the aim of the present paper is to give a combinatorial description for all the left cells of a m which have non empty intersection with c n then we use this description to deduce some formulae for the number of left cells of the weighted coxeter group c n l m in the set e associated to any partition of m 1

the based ring of the lowest generalized two sided cell of an extended affine weyl group abstract let c 0 be the lowest generalized two sided cell of an extended affine weyl group w with unequal parameters we first prove that certain conjectures of lusztig called p1 p15 hold for c 0 which implies the existence of the based ring of c 0 and then we determine the structure of the based ring as an application we use the structure of the based ring to study certain simple modules of the hecke algebra with unequal parameters associated to w namely those attached to c 0 further we give a set of prime ideals p of the center z of the affine hecke algebra h such that the reduced algebra k p h is simple over k p where k p frac z p is the residue field of z at p in particular we show that the algebra h z frac z is split simple over the field frac z

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

well balanced residual distribution for the ale spherical shallow water equations on moving adaptive meshes we consider the numerical approximation of the shallow water equations swes in spherical geometry for oceanographic applications to provide enhanced resolution of moving fronts present in the flow we consider adaptive discrete approximations on moving triangulations of the sphere to this end we restate all arbitrary lagrangian eulerian ale transport formulas as well as the volume transformation laws for a 2d manifold using these results we write the set of ale swes on the sphere we then propose a residual distribution discrete approximation of the governing equations classical properties as the dgcl and the c property well balancedness are reformulated in this more general context an adaptive mesh movement strategy is proposed the discrete framework obtained is thoroughly tested on standard benchmarks in large scale oceanography to prove their potential as well as the advantage brought by the adaptive mesh movement

sparse covariance matrix estimation by dca based algorithms this letter proposes a novel approach using the formula see text norm regularization for the sparse covariance matrix estimation scme problem the objective function of scme problem is composed of a nonconvex part and the formula see text term which is discontinuous and difficult to tackle appropriate dc difference of convex functions approximations of formula see text norm are used that result in approximation scme problems that are still nonconvex dc programming and dca dc algorithm powerful tools in nonconvex programming framework are investigated two dc formulations are proposed and corresponding dca schemes developed two applications of the scme problem that are considered are classification via sparse quadratic discriminant analysis and portfolio optimization a careful empirical experiment is performed through simulated and real data sets to study the performance of the proposed algorithms numerical results showed their efficiency and their superiority compared with seven state of the art methods

tabular algebras and their asymptotic versions we introduce tabular algebras which are simultaneous generalizations of cellular algebras in the sense of graham lehrer and table algebras in the sense of arad blau we show that if a tabular algebra is equipped with a certain kind of trace map then the algebra has a corresponding asymptotic version whose structure can be explicitly determined we also study various natural examples of tabular algebras

comparison ofa functions abstract we show agreement of lusztig sa function with thea function previously given by c k fan j amer math soc 10 no 1 1997 139 167

kazhdan lusztig cells and the frobenius schur indicator abstract let w be a finite coxeter group it is well known that the number of involutions in w is equal to the sum of the degrees of the irreducible characters of w following a suggestion of lusztig we show that this equality is compatible with the decomposition of w into kazhdan lusztig cells the proof uses a generalisation of the frobenius schur indicator to symmetric algebras which may be of independent interest

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

expression of certain kazhdan lusztig basis elements cw over the hecke algebra of type an abstract in this paper we determine expression of kazhdan lusztig basis elements c w over the hecke algebra of type a n where w a j i is an element in weyl group of type a n we also find joined relation of some elements in w

left cells in the weyl group of type e8 abstract the aim of the present paper is to give an explicit description for all the left cells of the weyl group w of type e 8 we first find a representative set for the left cells of w and then use it to describe these cells our results show that most of the left cells in w can be characterized by their generalized invariants alone then the remaining ones which are called exceptional left cells are characterized by the generalized invariants together with the a values

fully commutative elements in the weyl and affine weyl groups abstract let w be a weyl or an affine weyl group and let w c be the set of fully commutative elements in w we associate each w w c to a digraph g w by using g w we give a graph theoretic description for lusztig s a function on w c and describe explicitly all the distinguished involutions of w c the results verify two conjectures in our case one was proposed by myself in adv sci china math 3 1990 79 98 conjecture 8 10 and the other by lusztig in t asai et al open problems in algebraic groups in r hotta ed problems from the conference on algebraic groups and representations katata august 29 september 3 1983

on the determination of kazhdan lusztig cells for affine weyl groups with unequal parameters abstract let w be a coxeter group and l be a weight function on w following lusztig we have a corresponding decomposition of w into left cells which have important applications in representation theory we study the case where w is an affine weyl group of type g 2 using explicit computation with coxeter and chevie we show that 1 there are only finitely many possible decompositions into left cells and 2 the number of left cells is finite in each case thus confirming some of lusztig s conjectures in this case a key ingredient of the proof is a general result which shows that the kazhdan lusztig polynomials of affine weyl group are invariant under large enough translations

nonparametric estimation of large covariance matrices of longitudinal data summary estimation of an unstructured covariance matrix is difficult because of its positivedefiniteness constraint this obstacle is removed by regressing each variable on its predecessors so that estimation of a covariance matrix is shown to be equivalent to that of estimating a sequence of varying coefficient and varying order regression models our framework is similar to the use of increasing order autoregressive models in approximating the covariance matrix or the spectrum of a stationary time series as an illustration we adopt fan zhang s 2000 two step estimation of functional linear models and propose nonparametric estimators of covariance matrices which are guaranteed to be positive definite for parsimony a suitable order for the sequence of auto regression models is found using penalised likelihood criteria like aic and bic some asymptotic results for the local polynomial estimators of components of a covariance matrix are established two longitudinal datasets are analysed to illustrate the methodology a simulation study reveals the advantage of the nonparametric covariance estimator over the sample covariance matrix for large covariance matrices

computing kazhdan lusztig cells for unequal parameters abstract following lusztig we consider a coxeter group w together with a weight function l this gives rise to the pre order relation l and the corresponding partition of w into left cells we introduce an equivalence relation on weight functions such that in particular l is constant on equivalent classes we shall work this out explicitly for w of type f 4 and check that several of lusztig s conjectures concerning left cells with unequal parameters hold in this case even for those parameters which do not admit a geometric interpretation the proofs involve some explicit computations using chevie

joint relations on elements of the symmetric group abstract we introduce two methods which reduce the checking of the joint relations on element pairs x y i e checking if x y 0 see 1 2 inside a two sided cell of the symmetric group s n to some special cases up to star operations one involves the involutions in s n and the other involves the set for n see 3 1 and 5 1 we give a detailed investigation for the properties of the set in terms of standard tableaux also we study the sets u s and u s and give an affirmative answer to a question of n h xi see proposition 2 9

high order residual distribution scheme for the time dependent euler equations of fluid dynamics abstract in the present work a high order finite element type residual distribution scheme is designed in the framework of multidimensional compressible euler equations of gas dynamics the strengths of the proposed approximation rely on the generic spatial discretization of the model equations using a continuous finite element type approximation technique while avoiding the solution of a large linear system with a sparse mass matrix which would come along with any standard ode solver in a classical finite element approach to advance the solution in time in this work we propose a new residual distribution rd scheme which provides an arbitrary explicit high order approximation of the smooth solutions of the euler equations both in space and time the design of the scheme via the coupling of the rd formulation ricchiuto and abgrall 2010 abgrall 2006 with a deferred correction dec type method liu et al 2008 minion 2003 allows to have the matrix associated to the update in time which needs to be inverted to be diagonal the use of bernstein polynomials as shape functions guarantees that this diagonal matrix is invertible and ensures strict positivity of the resulting diagonal matrix coefficients this work is the extension of abgrall et al 2016 and abgrall 2017 to multidimensional systems we have assessed our method on several challenging benchmark problems for one and two dimensional euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions

a counter example to a conjecture of lusztig abstract in lusztig 2003 7 lusztig defined two functions a and a on a coxeter group w and conjectured that they are the same in the present paper we give a counter example to lusztig s conjecture then we propose a new conjecture for the description of two sided cells of w which can be regarded as an improvement of lusztig s conjecture some examples are given to support our conjecture