the zak transform and sampling theorems for wavelet subspaces the zak transform is used for generalizing a sampling theorem of g waiter see ieee trans informat theory vol 38 p 881 884 1992 for wavelet subspaces cardinal series based on signal samples f a n n in z with a possibly unequal to 0 waiter s case are considered the condition number of the sampling operator and worst case aliasing errors are expressed in terms of zak transforms of scaling function and wavelet this shows that the stability of the resulting interpolation formula depends critically on a

on the homology of the hilbert scheme of points in the plane geir ellingsrud 1 and stein arild stromme 2 i matematisk institutt universitetet i oslo blindern n oslo 3 norway 2 matematisk institutt universitetet i bergen n 5014 bergen norway although several authors have been interested in the hilbert scheme hilba ip z parametrizing finite subschemes of length d in the projective plane i1 i2 f 1 f2 br among others not much is known about the topological properties of this space the picard group has been calculated i f2 and the homology groups of hilb3 ip 2 have been computed hi in this paper we give a precise description of the additive structure of the homology of hilba ip2 applying the results of birula bialynicki b1 b2 on the cellular decompositions de fined by a torus action to the natural action of a maximal torus of sl 3 on hilbe ipz a rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the chow groups and the homology groups are isomorphisms in particular there is no odd homology and the homology groups are all free the main objective of this work is to compute their ranks the betti numbers of hilba ipz as a byproduct of our method we get similar results on the homology of the punctual hilbert scheme and of the hilbert scheme of points in the affine plane it seems natural to generalize our results to any toric smooth surface however we give the results only for the rational ruled surfaces if with an indication of the necessary changes in the proofs for simplicity we work over the field of complex numbers but with an appropriate interpretation of the word homology our results remain valid over any base field

the douady space of a complex surface abstract we prove that a standard realization of the direct image complex via the so called douady barlet hilbert chow in the algebraic case morphism associated with a smooth complex analytic surface admits a natural decomposition in the form of an injective quasi isomorphism of complexes this is a more precise form of a special case of the decomposition theorems of beilinson bernstein deligne gabber and m saito of which the proof we present is independent in addition it is elementary and transparent and does not use either perverse sheaves the methods of positive characteristic nor saito s theory of mixed hodge modules the proof hinges on the special case of the bi disk in the complex affine plane where we make explicit use of a construction of nakajima s and of the corresponding representation theoretic interpretation foreseen by vafa and witten some consequences of the decomposition theorem the gottsche formula holds for complex surfaces interpretation of the rational cohomologies of douady spaces as a kind of fock space new proofs of results of briancon and ellingsrud and stromme on punctual hilbert schemes and computation of the mixed hodge structure of the douady spaces in the kahler case we also derive a natural connection with equivariant k theory for which in the case of algebraic surfaces bezrukavnikov and ginzburg have proposed a different approach

irrationality of motivic zeta functions let k0 varq 1 l denote the grothendieck ring of q varieties with the lefschetz class inverted we show that there exists a k3 surface x over q such that the motivic zeta function x t n symnx tn regarded as an element in k0 varq 1 l t is not a rational function in t thus disproving a conjecture of denef and loeser

parametrizations of ideals in k x y and k x y z abstract we parametrize the affine space of artinian affine ideals of k x y which have a given initial ideal with respect to the degree reverse lexicographic term order the fact that the term order is degree compatible allows us to extend the parametrization to the projective case namely zero dimensional subschemes of p 2 with some extra assumption and to determine the betti strata of these grobner cells this allows us to prove a formula due to a iarrobino for the codimension of the betti strata of codimension two punctual schemes in p 2

a theoretical study on six classifier fusion strategies we look at a single point in feature space two classes and l classifiers estimating the posterior probability for class spl omega sub 1 assuming that the estimates are independent and identically distributed normal or uniform we give formulas for the classification error for the following fusion methods average minimum maximum median majority vote and oracle

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

the decomposition theorem perverse sheaves and the topology of algebraic maps we give a motivated introduction to the theory of perverse sheaves culminating in the decomposition theorem of beilinson bernstein deligne and gabber a goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties while most proofs are omitted we discuss several approaches to the decomposition theorem indicate some important applications and examples

pairs of commuting nilpotent matrices and hilbert function abstract let k be an infinite field there has been recent study of the family h n k of pairs of commuting nilpotent n n matrices relating this family to the fibre h n of the punctual hilbert scheme a n hilb n a 2 over the point np of the symmetric product sym n a 2 where p is a point of the affine plane a 2 v baranovsky the variety of pairs of commuting nilpotent matrices is irreducible transform groups 6 1 2001 3 8 r basili on the irreducibility of commuting varieties of nilpotent matrices j algebra 268 1 2003 56 80 a premet nilpotent commuting varieties of reductive lie algebras invent math 154 3 2003 653 683 in this study a pair of commuting nilpotent matrices a b is related to an artinian algebra k a b there has also been substantial study of the stratification of the local punctual hilbert scheme h n by the hilbert function as j briancon description de hilb n c x y invent math 41 1 1977 45 89 and others however these studies have been hitherto separate we first determine the stable partitions i e those for which p itself is the partition q p of a generic nilpotent element of the centralizer of the jordan nilpotent matrix j p we then explore the relation between h n k and its stratification by the hilbert function of k a b suppose that dim k k a b n and that k is algebraically closed of characteristic 0 or large enough p we show that a generic element of the pencil a b k has jordan partition the maximum partition p h whose diagonal lengths are the hilbert function h of k a b

irregular sampling for spline wavelet subspaces spline wavelets spl psi sub m t are important in time frequency localization due to i spl psi sub m can be arbitrarily close to the optimal case as m is sufficiently large ii spl psi sub m has compact support and simple analytic expression which lead to effective computation although the spline wavelet subspaces are so simple walter s well known sampling theorem does not hold if the order of spline m is even moreover when irregular sampling is considered in these spaces it is hard to determine the sampling density which is a serious problem in applications in this correspondence a general sampling theorem is obtained for m spl ges 3 in the sense of iterative construction and the sampling density spl delta sub m is estimated

sampling theorem and irregular sampling theorem for multiwavelet subspaces we generalize the multiwavelet sampling theorem by reproducing a kernel that is easy to use multiscaling functions that have orthogonality regularity short compact support symmetry and high approximation order satisfy the conditions in the sampling theorem which is not possible in the case of the scalar wavelet we consider the general cases of the uniform noninteger and irregular sampling a reconstruction from more general sets of points is necessary since the measurements may not be made at uniform points finally we establish a general irregular sampling theorem for multiwavelet subspaces and derive an estimate for the perturbations of uniform noninteger sampling in shift invariant spaces

on the estimation of independent binomial random variables using occurrence and sequential information we re visit the age old problem of estimating the parameters of a distribution from its observations traditionally scientists and statisticians have attempted to obtain strong estimates by extracting the information contained in the observations taken as a set however generally speaking the information contained in the sequence in which the observations have appeared has been ignored i e except to consider dependence information as in the case of markov models and n gram statistics in this paper we present results which to the best of our knowledge are the first reported results which consider how estimation can be enhanced by utilizing both the information in the observations and in their sequence of appearance the strategy known as sequence based estimation sbe works as follows we first quickly allude to the results pertaining to computing the maximum likelihood estimates mle of the data when the samples are taken individually we then derive the corresponding mle results when the samples are taken two at a time and then extend these for the cases when they are processed three at a time four at a time etc in each case we also experimentally demonstrate the convergence of the corresponding estimates we then suggest various avenues for future research including those by which these estimates can be fused to yield a superior overall cumulative estimate of the parameter of the distribution in pattern recognition pr and in other internet and compression applications we believe that our new estimates have great potential for practitioners especially when the cardinality of the observation set is small

the chow groups and the motive of the hilbert scheme of points on a surface abstract we compute the chow motive and the chow groups with rational coefficients of the hilbert scheme of points on a smooth algebraic surface

integral operators and integral cohomology classes of hilbert schemes the methods of integral operators on the cohomology of hilbert schemes of points on surfaces are developed they are used to establish integral bases for the cohomology groups of hilbert schemes of points on a class of surfaces and conjecturally for all simply connected surfaces

reconstruction algorithms in irregular sampling a constructive solution of the irregular sampling problem for band limited functions is given we show how a band limited function can be com pletely reconstructed from any random sampling set whose density is higher than the nyquist rate and give precise estimates for the speed of convergence of this iteration method variations of this algorithm allow for irregular sampling with derivatives reconstruction of band limited functions from local averages and irregular sampling of multivariate band limited functions in the irregular sampling problem one is asked whether and how a band limited function can be completely reconstructed from its irregularly sam pled values f xi this has many applications in signal and image processing seismology meteorology medical imaging etc finding constructive solutions of this problem has received considerable attention among mathematicians and engineers the mathematical literature provides several uniqueness results 1 2 17 18 19 it is now part of the folklore that for stable sampling the sampling rate must be at least the nyquist rate 18 these results as deep as they are have had little impact for the applied sciences because they were not constructive if the sampling set is just a perturbation of the regular oversampling then a reconstruction method has been obtained in a seminal paper by duffin and schaeffer 6 see also 29 if for some l 0 a 0 and o 0 the sampling points xk k e z satisfy a xk ok a k i then the norm equivalence a ir f x 2dx with w n o this norm equivalence implies that it is possible to reconstruct through an iterative procedure the so called frame method most of the later work on constructive methods consists of variations of this method 3 21 22 26 the above conditions on the sampling set exclude random irregular sampling sets e g sets with regions of higher sampling density a partial but undesirable remedy to handle highly irregular sampling sets would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration

wavelets of multiplicity r a multiresolution approximation km m z of l2 r is of multi plicity r 0 if there are r functions x r whose translates form a riesz basis for v in the general theory we derive necessary and sufficient conditions for the translates of x r y x y r to form a riesz basis for v the resulting reconstruction and decomposition sequences lead to the construction of dual bases for v0 and its orthogonal complement w0 in vx the general theory is applied in the construction of spline wavelets with mul tiple knots algorithms for the construction of these wavelets for some special cases are given

irreducible components of the equivariant punctual hilbert schemes abstract let h ab be the equivariant hilbert scheme parameterizing the zero dimensional subschemes of the affine plane invariant under the natural action of the one dimensional torus t ab t b t a t k we compute the irreducible components of h ab they are in one to one correspondence with the set of possible hilbert functions as a by product of the proof we give new proofs of results by ellingsrud and stromme namely the main lemma of the computation of the betti numbers of the hilbert scheme h l parametrizing the zero dimensional subschemes of the affine plane of length l invent math 87 1988 343 and a description of bialynicki birula cells on h l by means of explicit flat families invent math 91 1988 365 in particular we give precise conditions when this last description applies

weighted sampling and reconstruction in weighted reproducing kernel spaces in this paper we discuss sampling and reconstruction of signals in the weighted reproducing kernel space associated with an idempotent integral operator we show that any signal in that space can be stably reconstructed from its weighted samples taken on a relatively separated set with sufficiently small gap we also develop an iterative reconstruction algorithm for the reconstruction of a signal from its weighted samples taken on a relatively separated set with sufficiently small gap

decision templates for multiple classifier fusion an experimental comparison abstract multiple classifier fusion may generate more accurate classification than each of the constituent classifiers fusion is often based on fixed combination rules like the product and average only under strict probabilistic conditions can these rules be justified we present here a simple rule for adapting the class combiner to the application c decision templates one per class are estimated with the same training set that is used for the set of classifiers these templates are then matched to the decision profile of new incoming objects by some similarity measure we compare 11 versions of our model with 14 other techniques for classifier fusion on the satimage and phoneme datasets from the database elena our results show that decision templates based on integral type measures of similarity are superior to the other schemes on both data sets