fast reaction limit of a three component reaction diffusion system abstract we consider a three component reaction diffusion system with a reaction rate parameter and investigate its singular limit as the reaction rate tends to infinity the limit problem is given by a free boundary problem which possesses three regions separated by the free boundaries one component vanishes and the other two components remain positive in each region therefore the dynamics is governed by a system of two equations

on the unification of line processes outlier rejection and robust statistics with applications in early vision the modeling of spatial discontinuities for problems such as surface recovery segmentation image reconstruction and optical flow has been intensely studied in computer vision while line process models of discontinuities have received a great deal of attention there has been recent interest in the use of robust statistical techniques to account for discontinuities this paper unifies the two approaches to achieve this we generalize the notion of a line process to that of an analog outlier process and show how a problem formulated in terms of outlier processes can be viewed in terms of robust statistics we also characterize a class of robust statistical problems for which an equivalent outlier process formulation exists and give a straightforward method for converting a robust estimation problem into an outlier process formulation we show how prior assumptions about the spatial structure of outliers can be expressed as constraints on the recovered analog outlier processes and how traditional continuation methods can be extended to the explicit outlier process formulation these results indicate that the outlier process approach provides a general framework which subsumes the traditional line process approaches as well as a wide class of robust estimation problems examples in surface reconstruction image segmentation and optical flow are presented to illustrate the use of outlier processes and to show how the relationship between outlier processes and robust statistics can be exploited an appendix provides a catalog of common robust error norms and their equivalent outlier process formulations

on a theorem of benson and curtis iwahori conjectured 5 that the hecke algebra over q of a chevalley group g over a finite field f with respect to a bore1 subgroup is isomorphic to the group algebra over q of the weyl group the analogous statement with q replaced by its algebraic closure was proved by tits 3 ex 27 p 561 using a deformation argument in 11 benson and curtis proved using a case by case analysis that iwahori s conjecture is true over q whenever g is simple of type ze e with the help of springer they proved that when g is of type e or e there exists irreducible representations of the hecke algebra whose character involve 4 and hence iwahori s conjecture over q is false in general the main result of this paper is an explicit construction of an isomorphism of the hecke algebra over q q with the group algebra over q q of the weyl group the construction makes essential use of the polynomials p w of 8 at some point in the proof i have to appeal to the interpretation of p l in terms of multiplicities in verma modules 8 i 51 which has recently been established by brylinski and kashiwara 4 and independently by belinson and bernstein

a unification of knizhnik zamolodchikov and dunkl operators via affine hecke algebras some generalizations of the lusztig lascoux schutzenberger operators for affine hecke algebras are considered as corollaries we obtain lusztig s isomorphisms from affine hecke algebras to their degenerate versions a natural interpretation of the dunkl operators and a new class of differential difference operators generalizing dunkl s ones and the knizhnik zamolodchikov operators from the two dimensional conformal field theory

representations of rank two affine hecke algebras at roots of unity abstract in this paper we will fully describe the irreducible representations of the crystallographic rank two affine hecke algebras using algebraic and combinatorial methods for all possible values of q the focus is on the case when q is a root of unity of small order

guided wavelet shrinkage for edge aware smoothing edge aware smoothing has been extensively studied due to its wide range of applications in computer vision and graphics most published works have been focused on formulating the smoothing problem in the spatial domain in this paper we propose a new edge aware smoothing framework called guided wavelet shrinkage gws which is formulated in the wavelet domain as a maximum a posterior estimation problem we impose a number of desirable properties on the statistical models and the associated parameters in order to derive an effective and computationally efficient algorithm we compare the proposed gws with classical image denoising in the wavelet domain we have also compared different wavelet representations and found that the double density dual tree wavelet transform is the best choice we show that the gws can be configured as either self guidance or external guidance it can also be configured to operate in an iterative or non iterative way experimental results and comparison with many state of the algorithms demonstrate that the gws based algorithm can produce competitive results with mathcal o n computational complexity

bi exponential edge preserving smoother edge preserving smoothers need not be taxed by a severe computational cost we present in this paper a lean algorithm that is inspired by the bi exponential filter and preserves its structure a pair of one tap recursions by a careful but simple local adaptation of the filter weights to the data we are able to design an edge preserving smoother that has a very low memory and computational footprint while requiring a trivial coding effort we demonstrate that our filter a bi exponential edge preserving smoother or beeps has formal links with the traditional bilateral filter on a practical side we observe that the beeps also produces images that are similar to those that would result from the bilateral filter but at a much reduced computational cost the cost per pixel is constant and depends neither on the data nor on the filter parameters not even on the degree of smoothing

blocks of affine quantum schur algebras abstract the affine quantum schur algebra is a certain important infinite dimensional algebra whose representation theory is closely related to that of quantum affine gl n finite dimensional irreducible modules for the affine quantum schur algebra s n r v were classified in 15 where v c is not a root of unity we will classify blocks of the affine quantum schur algebra s n r v in this paper

identification of simple representations for affine q schur algebras abstract we define an explicit action of the quantum loop algebra u q gl n on the tensor space and show that these actions are compatible with the natural embedding u q gl n u q gl n 1 as an application we identify simple representations of affine q schur algebras s n r c arising from simple polynomial representations of u q gl n frenkel and mukhin 2002 11 with those arising from simple modules of the affine hecke algebra h r c of type a rogawski 1985 26

regularization of wavelet approximations in this paper we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced the approach can apply readily to many other statistical contexts various new penalty functions are proposed the hard thresholding and soft thresholding estimators of donoho and johnstone are specific members of nonlinear regularized wavelet estimators they correspond to the lower and upper envelopes of a class of the penalized least squares estimators necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions the sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax to efficiently solve penalized least squares problems nonlinear regularized sobolev interpolators nrsi are proposed as initial estimators wh

the guided bilateral filter when the joint cross bilateral filter becomes robust the bilateral filter and its variants such as the joint cross bilateral filter are well known edge preserving image smoothing tools used in many applications the reason of this success is its simple definition and the possibility of many adaptations the bilateral filter is known to be related to robust estimation this link is lost by the ad hoc introduction of the guide image in the joint cross bilateral filter we here propose a new way to derive the joint cross bilateral filter as a particular case of a more generic filter which we name the guided bilateral filter this new filter is iterative generic inherits the robustness properties of the robust bilateral filter and uses a guide image the link with robust estimation allows us to relate the filter parameters with the statistics of input images a scheme based on graduated nonconvexity is proposed which allows converging to an interesting local minimum even when the cost function is nonconvex with this scheme the guided bilateral filter can handle non gaussian noise on the image to be filtered a complementary scheme is also proposed to handle non gaussian noise on the guide image even if both are strongly correlated this allows the guided bilateral filter to handle situations with more noise than the joint cross bilateral filter can work with and leads to high peak signal to noise ratio values as shown experimentally

matrix coefficients and iwahori hecke algebra modules abstract we establish a connection between certain unique models or equivalently unique functionals for representations of p adic groups and linear characters of their corresponding hecke algebras this allows us to give a uniform evaluation of the image of spherical and iwahori fixed vectors in the unramified principal series for this class of models we provide an explicit alternator expression for the image of spherical vectors under these functionals in terms of the representation theory of the dual group

a diffusive logistic equation with a free boundary and sign changing coefficient in time periodic environment abstract this paper concerns a diffusive logistic equation with a free boundary and sign changing intrinsic growth rate in heterogeneous time periodic environment in which the variable intrinsic growth rate may be very negative in a suitable large region see conditions h1 h2 4 3 such a model can be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front in the case of higher space dimensions with radial symmetry and when the intrinsic growth rate has a positive lower bound this problem has been studied by du guo peng 11 they established a spreading vanishing dichotomy the sharp criteria for spreading and vanishing and estimate of the asymptotic spreading speed in the present paper we show that the above results are retained for our problem

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

affine hecke algebras and generalized standard young tableaux abstract this paper introduces calibrated representations for affine hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations the primary technique is to provide indexing sets for controlling the weight space structure of finite dimensional modules for the affine hecke algebra using these indexing sets we show that 1 irreducible calibrated representations are indexed by skew local regions 2 the dimension of an irreducible calibrated representation is the number of chambers in the local region 3 each irreducible calibrated representation is constructed explicitly by formulas which describe the action of the generators of the affine hecke algebra on a specific basis in the representation space the indexing sets for weight spaces are generalizations of standard young tableaux and the construction of the irreducible calibrated affine hecke algebra modules is a generalization of a young s seminormal construction of the irreducible representations of the symmetric group in this sense young s construction has been generalized to arbitrary lie type

fundamental relationship between bilateral filtering adaptive smoothing and the nonlinear diffusion equation in this paper the relationship between bilateral filtering and anisotropic diffusion is examined the bilateral filtering approach represents a large class of nonlinear digital image filters we first explore the connection between anisotropic diffusion and adaptive smoothing and then the connection between adaptive smoothing and bilateral filtering previously adaptive smoothing was considered to be an inconsistent approximation to the nonlinear diffusion equation we extend adaptive smoothing to make it consistent thus enabling a unified viewpoint that relates nonlinear digital image filters and the nonlinear diffusion equation

from local kernel to nonlocal multiple model image denoising we review the evolution of the nonparametric regression modeling in imaging from the local nadaraya watson kernel estimate to the nonlocal means and further to transform domain filtering based on nonlocal block matching the considered methods are classified mainly according to two main features local nonlocal and pointwise multipoint here nonlocal is an alternative to local and multipoint is an alternative to pointwise these alternatives though obvious simplifications allow to impose a fruitful and transparent classification of the basic ideas in the advanced techniques within this framework we introduce a novel single and multiple model transform domain nonlocal approach the block matching and 3 d filtering bm3d algorithm which is currently one of the best performing denoising algorithms is treated as a special case of the latter approach

a diffusive logistic model with a free boundary in time periodic environment abstract we study the diffusive logistic equation with a free boundary in time periodic environment such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front for time independent environment in the cases of one space dimension and higher space dimensions with radial symmetry this free boundary problem has been studied in du and lin 2010 12 du and guo 2011 9 in both cases a spreading vanishing dichotomy was established and when spreading occurs the asymptotic spreading speed was determined in this paper we show that the spreading vanishing dichotomy is retained in time periodic environment and we also determine the spreading speed the former is achieved by further developing the earlier techniques and the latter is proved by introducing new ideas and methods

lagrangian construction for representations of hecke algebras 1 let k be a non archimedean local field with the residue field consisting of q elements according to langlands philosophy irreducible complex representations of a reductive k group g should roughly speaking correspond to homomorphisms of the weil deligne group into the socalled l group let zc g be the iwahori subgroup the category of admissible gkmodules generated by z fixed vectors is known to be equivalent to the category of finite dimensional representations of the hecke algebra h z g z langlands philosophy suggests via that equivalence the following classification of irreducible hecke algebra modules stated in purely complex terms and not involving any local held

bivariate shrinkage functions for wavelet based denoising exploiting interscale dependency most simple nonlinear thresholding rules for wavelet based denoising assume that the wavelet coefficients are independent however wavelet coefficients of natural images have significant dependencies we only consider the dependencies between the coefficients and their parents in detail for this purpose new non gaussian bivariate distributions are proposed and corresponding nonlinear threshold functions shrinkage functions are derived from the models using bayesian estimation theory the new shrinkage functions do not assume the independence of wavelet coefficients we show three image denoising examples in order to show the performance of these new bivariate shrinkage rules in the second example a simple subband dependent data driven image denoising system is described and compared with effective data driven techniques in the literature namely visushrink sureshrink bayesshrink and hidden markov models in the third example the same idea is applied to the dual tree complex wavelet coefficients

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

weighted guided image filtering it is known that local filtering based edge preserving smoothing techniques suffer from halo artifacts in this paper a weighted guided image filter wgif is introduced by incorporating an edge aware weighting into an existing guided image filter gif to address the problem the wgif inherits advantages of both global and local smoothing filters in the sense that 1 the complexity of the wgif is o n for an image with n pixels which is same as the gif and 2 the wgif can avoid halo artifacts like the existing global smoothing filters the wgif is applied for single image detail enhancement single image haze removal and fusion of differently exposed images experimental results show that the resultant algorithms produce images with better visual quality and at the same time halo artifacts can be reduced avoided from appearing in the final images with negligible increment on running times

asymptotic spreading in heterogeneous diffusive excitable media abstract we establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction diffusion equations of the type t u a t x u q t x u f t x u with compactly supported initial conditions at t 0 here a q f have a general dependence in t r and x r n we establish properties of families of propagation sets which are defined as families of subsets s t t 0 of r n such that lim inf t inf x s t u t x 0 the aim is to characterize such families as sharply as possible in particular we give some conditions under which 1 a given path t t 0 where t r n forms a family of propagation sets or 2 one can find such a family with s t x r n x r t and lim t r t this second property is called here complete spreading furthermore in the case q 0 and inf t x r r n f u t x 0 0 as well as under some more general assumptions we show that there is a positive spreading speed that is r t can be chosen so that lim inf t r t t 0 in the general case we also show the existence of an explicit upper bound c 0 such that lim sup t r t t c on the other hand we provide explicit examples of reaction diffusion equations such that for an arbitrary e 0 any family of propagation sets s t t 0 has to satisfy s t x r n x e t for large t in connection with spreading properties we derive some new uniqueness results for the entire solutions of this type of equations lastly in the case of space time periodic media we develop a new approach to characterize the largest propagation sets in terms of eigenvalues associated with the linearized equation in the neighborhood of zero

time invariant orthonormal wavelet representations a simple construction of an orthonormal basis starting with a so called mother wavelet together with an efficient implementation gained the wavelet decomposition easy acceptance and generated a great research interest in its applications an orthonormal basis may not however always be a suitable representation of a signal particularly when time or space invariance is a required property the conventional way around this problem is to use a redundant decomposition we address the time invariance problem for orthonormal wavelet transforms and propose an extension to wavelet packet decompositions we show that it is possible to achieve time invariance and preserve the orthonormality we subsequently propose an efficient approach to obtain such a decomposition we demonstrate the importance of our method by considering some application examples in signal reconstruction and time delay estimation

modules simples en caracteristique p de l algebre de hecke du pro p iwahori de gl3 f resume pour f un corps p adique et n n n 2 on ne connait pas de facon generale les representations lisses irreductibles modulo p de gl n f le foncteur des invariants par le pro p iwahori i 1 invite a les apprehender par l etude des modules simples a droite sur l algebre de hecke h f p gl n f i 1 dans le cas n 2 on connait les h f p gl n f i 1 modules simples ayant un caractere central parmi eux on distingue les modules que l on appelle supersinguliers ils ne correspondent pas via le foncteur des invariants par le pro p iwahori a des sous quotients d induites paraboliques de gl 2 f m f vigneras representations modulo p of the p adic group gl 2 f compos math 140 2004 333 358 nous considerons le cas n 3 il s agit de donner la classification des h f p gl 3 f i 1 modules simples ayant un caractere central et de decrire conjecturalement les modules supersinguliers l ensemble de ces modules supersinguliers est en bijection non unique avec l ensemble des f p representations irreductibles de dimension 3 du groupe de weil de f

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

traveling wave solutions of a time periodic competitive system with a free boundary abstract we consider a time periodic competition system with a free boundary which can be used to describe two species that are struggling on the free boundary to obtain their own habitats and the final competition result is the regional partition of two species in 15 we have studied the classical traveling wave solutions of the temporal homogeneous problems in this paper we consider the time periodic case and study periodic traveling wave solutions of the problem more precisely we will prove the existence and uniqueness of the periodic traveling wave solution and show that the speed of the periodic traveling wave is uniquely determined by and monotonically dependent on the time periodic terms in the system moreover we will present some sufficient conditions to guarantee that the wave travels with a positive speed

decomposition numbers for generic iwahori hecke algebras of noncrystallographic type we compute the e modular decomposition matrices for the generic iwahori hecke algebras of typei2 m form n m 2 h3 andh4 for alle n leading to nontrivial decomposition maps the results are obtained by a combined use of different ideas from computational representation theory and by application of the computer algebra systemsgap chevie vectorenumeratorandmeataxe

the double density dual tree dwt this paper introduces the double density dual tree discrete wavelet transform dwt which is a dwt that combines the double density dwt and the dual tree dwt each of which has its own characteristics and advantages the transform corresponds to a new family of dyadic wavelet tight frames based on two scaling functions and four distinct wavelets one pair of the four wavelets are designed to be offset from the other pair of wavelets so that the integer translates of one wavelet pair fall midway between the integer translates of the other pair simultaneously one pair of wavelets are designed to be approximate hilbert transforms of the other pair of wavelets so that two complex approximately analytic wavelets can be formed therefore they can be used to implement complex and directional wavelet transforms the paper develops a design procedure to obtain finite impulse response fir filters that satisfy the numerous constraints imposed this design procedure employs a fractional delay allpass filter spectral factorization and filterbank completion the solutions have vanishing moments compact support a high degree of smoothness and are nearly shift invariant

on the classification of finite dimensional irreducible modules for affine bmw algebras in this paper we classify the finite dimensional irreducible modules for affine bmw algebra over an algebraically closed field with arbitrary characteristic

kazhdan lusztig representations and whittaker space of some genuine representations we prove a formula for the dimension of whittaker functionals of irreducible constituents of a regular unramified genuine principal series for covering groups the formula explicitly relates such dimension to the kazhdan lusztig representations associated with certain right cells of the weyl group we also state a refined version of the formula which is proved under some natural assumption the refined formula is also verified unconditionally in several important cases

steerable pyramids and tight wavelet frames in l_ 2 bbr d we present a functional framework for the design of tight steerable wavelet frames in any number of dimensions the 2 d version of the method can be viewed as a generalization of simoncelli s steerable pyramid that gives access to a larger palette of steerable wavelets via a suitable parametrization the backbone of our construction is a primal isotropic wavelet frame that provides the multiresolution decomposition of the signal the steerable wavelets are obtained by applying a one to many mapping n th order generalized riesz transform to the primal ones the shaping of the steerable wavelets is controlled by an m m unitary matrix where m is the number of wavelet channels that can be selected arbitrarily this allows for a much wider range of solutions than the traditional equiangular configuration steerable pyramid we give a complete functional description of these generalized wavelet transforms and derive their steering equations we describe some concrete examples of transforms including some built around a mallat type multiresolution analysis of l2 rd and provide a fast fourier transform based decomposition algorithm we also propose a principal component based method for signal adapted wavelet design finally we present some illustrative examples together with a comparison of the denoising performance of various brands of steerable transforms the results are in favor of an optimized wavelet design equalized principal component analysis which consistently performs best

guided image filtering in this paper we propose a novel explicit image filter called guided filter derived from a local linear model the guided filter computes the filtering output by considering the content of a guidance image which can be the input image itself or another different image the guided filter can be used as an edge preserving smoothing operator like the popular bilateral filter 1 but it has better behaviors near edges the guided filter is also a more generic concept beyond smoothing it can transfer the structures of the guidance image to the filtering output enabling new filtering applications like dehazing and guided feathering moreover the guided filter naturally has a fast and nonapproximate linear time algorithm regardless of the kernel size and the intensity range currently it is one of the fastest edge preserving filters experiments show that the guided filter is both effective and efficient in a great variety of computer vision and computer graphics applications including edge aware smoothing detail enhancement hdr compression image matting feathering dehazing joint upsampling etc

rational and polynomial representations of yangians abstract we define natural classes of rational and polynomial representations of the yangian of the general linear lie algebra we also present the classification and explicit realizations of all irreducible rational representations of the yangian

drinfeld functor and finite dimensional representations of yangian we extend the results of drinfeld on the drinfeld functor to the case l n we present the character of finite dimensional representations of the yangian y mathfrak s mathfrak l n in terms of the kazhdan lusztig polynomials as a consequence

some examples of square integrable representations of semisimple p adic groups we construct irreducible representations of the hecke algebra of an affine weyl group analogous to kilmoyer s reflection representation corresponding to finite weyl groups and we show that in many cases they correspond to a square integrable representation of a simple p adic group

fast global image smoothing based on weighted least squares this paper presents an efficient technique for performing a spatially inhomogeneous edge preserving image smoothing called fast global smoother focusing on sparse laplacian matrices consisting of a data term and a prior term typically defined using four or eight neighbors for 2d image our approach efficiently solves such global objective functions in particular we approximate the solution of the memory and computation intensive large linear system defined over a d dimensional spatial domain by solving a sequence of 1d subsystems our separable implementation enables applying a linear time tridiagonal matrix algorithm to solve d three point laplacian matrices iteratively our approach combines the best of two paradigms i e efficient edge preserving filters and optimization based smoothing our method has a comparable runtime to the fast edge preserving filters but its global optimization formulation overcomes many limitations of the local filtering approaches our method also achieves high quality results as the state of the art optimization based techniques but runs 10 30 times faster besides considering the flexibility in defining an objective function we further propose generalized fast algorithms that perform l norm smoothing 0 2 and support an aggregated robust data term for handling imprecise data constraints we demonstrate the effectiveness and efficiency of our techniques in a range of image processing and computer graphics applications

the minimal basis for the centre of an iwahori hecke algebra abstract this paper arose out of an attempt to generalize the q q q 1 basis for the centre of an iwahori hecke algebra h q found by jones to z q q 1 and to other types considering the iwahori hecke algebra h over a subring z of z q1 2 q 1 2 where q1 2 q 1 2 we use a new and natural definition of positivity on h to describe the minimal z basis for z h in terms of a partial order on the positive part of h the main result is to show that this minimal basis is the set of primitive minimal elements of the positive part of the centre for any weyl group in addition the primitive minimal positive central elements can be characterized as exactly those positive central elements which specialize on setting 0 the equivalent of setting q 1 in z q1 2 q 1 2 to the sum of elements in a conjugacy class and which apart from the shortest elements from that conjugacy class sum contain no other terms corresponding to shortest elements of any conjugacy class a constructive algorithm is provided for obtaining the minimal basis we use the results of jones to achieve the result in type an without the need for character theory and give the result for all weyl groups using the character theoretical results of geck and rouquier finally we discuss the non crystallographic cases and give some explicit examples