smoothing effect for some schrodinger equations abstract we study the cauchy problem for the following schrodinger equation in r n n n i t u 1 2 u v 1 u v 2 u 2 u t x r r n u 0 x r n where v 1 v 1 x 1 x 1 v 2 v 2 x k 2 3 k x k k r 1 k 3 0 1 min 2 n 2 0 2 3 min 2 n we prove the existence uniqueness and smoothing effect of global solutions of with not necessarily in the sobolev space h1 rn

combinatorial decompositions of a class of rings in this paper we introduce some combinatorial methods for the study of graded rings and use these methods to analyze a class of rings that are of importance in combinatorics our methods are based on a unique representation theorem which is shown section 2 to hold for any graded algebra and which in a sense is an extension of the direct sum decomposition for cohen macaulay algebras this unique representation is reminiscent of a similar result obtained by rees 20 both results yield a decomposition of the algebra however our result differs from that of rees in that it carries more information about the algebra our main results are concerned with a class of rings see section 3 for the definition associated to simplicial complexes there are several natural operations that can be performed on simplicial complexes to obtain new complexes which in turn have an interpretation for the corresponding rings we show sections 4 5 and 6 that our basic decompositions can be transferred during these operations from the original rings to the newly constructed ones the operations studied here are the chain transform or barycentric subdivision section 4 rank selection both for partially ordered sets and for simplicial complexes section s and localization section 6 as we develop these tools we give some applications in section 5 we give a new topological characterization of the cohen macaulay property for partially ordered sets an immediate consequence of which is the rankselection theorem baclawski 2 this characterization is closely related to those obtained by garsia in 14

domain and range of the modified wave operator for schrodinger equations with a critical nonlinearity we study the final problem for the nonlinear schrodinger equation i partial t u frac 1 2 delta u lambda u frac 2 n u quad t x in mathbf r times mathbf r n where lambda in bf r n 1 2 3 if the final data u in bf h 0 alpha left phi in bf l 2 left 1 left vert x right vert right alpha phi in bf l 2 right with frac n 2 alpha min left n 2 1 frac 2 n right and the norm vert widehat u vert bf l infty is sufficiently small then we prove the existence of the wave operator in l2 we also construct the modified scattering operator from h0 to h0 with frac n 2 delta alpha

alexander duality and stanley depth of multigraded modules abstract we apply miller s theory on multigraded modules over a polynomial ring to the study of the stanley depth of these modules several tools for stanley s conjecture are developed and a few partial answers are given for example we show that taking the alexander duality twice but with different centers is useful for this subject generalizing a result of apel we prove that stanley s conjecture holds for the quotient by a cogeneric monomial ideal

rapid computation of the discrete fourier transform algorithms for the rapid computation of the forward and inverse discrete fourier transform for points which are nonequispaced or whose number is unrestricted are presented the computational procedure is based on approximation using a local taylor series expansion and the fast fourier transform fft the forward transform for nonequispaced points is computed as the solution of a linear system involving the inverse fourier transform this latter system is solved using the iterative method gmres with preconditioning numerical results are given to confirm the efficiency of the algorithms

new tight frames of curvelets and optimal representations of objects with piecewise c2 singularities this paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise c 2 edges conceptually the curvelet transform is a multiscale pyramid with many directions and positions at each length scale and needle shaped elements at fine scales these elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets for instance curvelets obey a parabolic scaling relation which says that at scale 2 j each element has an envelope that is aligned along a ridge of length 2 j 2 and width 2 j we prove that curvelets provide an essentially optimal representation of typical objects f that are c 2 except for discontinuities along piecewise c 2 curves such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object for instance the n term partial reconstruction f c n obtained by selecting the n largest terms in the curvelet series obeys f f c n 2 l2 c n 2 log n 3 n this rate of convergence holds uniformly over a class of functions that are c 2 except for discontinuities along piecewise c 2 curves and is essentially optimal in comparison the squared error of n term wavelet approximations only converges as n 1 as n which is considerably worse than the optimal behavior

grobner bases of syzygies and stanley depth abstract let f be any free resolution of a finitely generated z n graded module over the polynomial ring k x 1 x n we show that for a suitable term order on f then for 0 p n the initial module of the p 1 th syzygy module z p f p is generated by terms m i e i where the m i are monomials in k x p 1 x n also for a large class of free resolutions f encompassing eliahou kervaire resolutions we show that a grobner basis for z p is given by the boundaries of generators of f p 1 we apply the above to give lower bounds for the stanley depth of the p 1 th syzygy module z p in particular showing it is greater or equal to p 1 we also show that if i is any squarefree ideal in k x 1 x n the stanley depth of i is at least of order 2 n

hilbert series and hilbert depth of squarefree veronese ideals abstract in this paper we obtain explicit formulas for the hilbert series and hilbert depth of squarefree veronese ideals in a standard graded polynomial ring

cohen macaulayness of semi invariants for tori in this paper we give a new method in terms of one parameter subgroups to study semi invariants for algebraic tori in some cases we obtain extensions to results in 7 in other cases we obtain different proofs

an algorithm for the numerical evaluation of the associated legendre functions that runs in time independent of degree and order abstract we describe a method for the numerical evaluation of normalized versions of the associated legendre functions p and q of degrees 0 1 000 000 and orders for arguments in the interval 1 1 our algorithm which runs in time independent of and is based on the fact that while the associated legendre functions themselves are extremely expensive to represent via polynomial expansions the logarithms of certain solutions of the differential equation defining them are not we exploit this by numerically precomputing the logarithms of carefully chosen solutions of the associated legendre differential equation and representing them via piecewise trivariate chebyshev expansions these precomputed expansions which allow for the rapid evaluation of the associated legendre functions over a large swath of parameter domain mentioned above are supplemented with asymptotic and series expansions in order to cover it entirely the results of numerical experiments demonstrating the efficacy of our approach are presented and our code for evaluating the associated legendre functions is publicly available

a non partitionable cohen macaulay simplicial complex abstract a long standing conjecture of stanley states that every cohen macaulay simplicial complex is partitionable we disprove the conjecture by constructing an explicit counterexample due to a result of herzog jahan and yassemi our construction also disproves the conjecture that the stanley depth of a monomial ideal is always at least its depth

saturation and castelnuovo mumford regularity abstract let i be a homogeneous ideal of the polynomial ring k x 0 x n where k is an arbitrary field avoiding the construction of a minimal graded free resolution of i we provide effective methods for computing the castelnuovo mumford regularity of i that also compute other cohomological invariants of k x 0 x n i

stanley depth of complete intersection monomial ideals and upper discrete partitions abstract let i be an m generated complete intersection monomial ideal in s k x 1 x n we show that the stanley depth of i is n m 2 we also study the upper discrete structure for monomial ideals and prove that if i is a squarefree monomial ideal minimally generated by 3 elements then the stanley depth of i is n 1

how to compute the stanley depth of a monomial ideal abstract let j i be monomial ideals we show that the stanley depth of i j can be computed in a finite number of steps we also introduce the fdepth of a monomial ideal which is defined in terms of prime filtrations and show that it can also be computed in a finite number of steps in both cases it is shown that these invariants can be determined by considering partitions of suitable finite posets into intervals

modified scattering operator for the hartree fock equation abstract we study the scattering problem for the hartree fock equation hrf i t u 1 2 u f u t x r r n n 2 where u u 1 u n t is a c n n 2 valued unknown function and f u f 1 u f n u t denotes a nonlinear term whose j th element is defined by f j u r n v x y k 1 n u k y 2 u j x u j y u k y u k x d y where v x x 1 r is called a coulomb potential we show that if 1 2 then the modified scattering operator for the system hrf is well defined from a neighborhood at the origin in the space h 0 to a neighborhood at the origin in the space h 0 where h 0 k l 2 1 x 2 k 2 l 2

the number of solutions to linear diophantine equations and multivariate splines etude de la variation du nombre des solutions entieres non negatives de s equations diophantiennes lineaires en fonction des termes non homogenes

the inversion problem and applications of the generalized radon transform we prove that under certain conditions the inversion problem for the generalized radon transform reduces to solving a fredholm integral equation and we obtain the asymptotic expansion of the symbol of the integral operator in this equation we consider applications of the generalized radon transform to partial differential equations with variable coefficients and provide a solution to the inversion problem for the attenuated and exponential radon transforms

dualizing complexes of affine semigroup rings for an affine semigroup ring we construct the dualizing complex in terms of the semigroup and the homology of the face lattice of the polyhedral cone spanned by the semigroup as a consequence there are characterizations of locally cohen macaulay rings buchsbaum rings and cohen macaulay rings as well as serre s condition 9 introduction let s denote a finitely generated submonoid of the additive monoid n n being a positive integer by the affine semigroup ring i s of s over a field x let us denote the subring of x xl xn generated by all monomials xs s e s so one has up to isomorphisms a one to one correspondence between affine semigroup rings and affine varieties given parametrically by monomials now it is of some interest to characterize ring theoretic properties as cohenmacaulay gorenstein etc in terms of the semigroup s a first breakthrough in this direction was done by kempf et al kkms and hochster hol who showed that x s is a cohen macaulay ring provided it is normal moreover the normality of x s is described in terms of s see hol another particular case if s is a simplicial semigroup is treated by goto suzuki and watanabe gsw resp stanley st2 who showed that x s is a cohen macaulay ring if it satisfies the 22 condition of serre note that normality is equivalent to 5y2 and r1 moreover there is a description of the 5 condition in terms of s see theorem 6 3 in general the 52 condition is not sufficient for i s to be a cohen macaulay ring as follows by examples in hol resp th 1 2 on the other side grobner g posed the problem to classify cohen macaulay affine semigroup rings the solution of this problem was done by hoa and trung th related to structural properties of graded rings is a technical tool of the duality theory this is the dualizing complex introduced by grothendieck see ha in recent years it became a helpful tool in commutative algebra in particular its knowledge allows characterizations of cohen macaulay rings locally cohenmacaulay rings buchsbaum rings gorenstein rings and the 5 conditions of received by the editors december 18 1988 1980 mathematics subject classification 1985 revision primary 13h10 secondary 14m05 06a99 1990 american mathematical society 0002 9947 90 1 q0 25 per page

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

the alexander duality functors and local duality with monomial support abstract alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated n n graded module the functors associated with alexander duality provide a duality on the level of free and injective resolutions and numerous bass and betti number relations result as corollaries a minimal injective resolution of a module m is equivalent to the injective resolution of its alexander dual and contains all of the maps in the minimal free resolution of m over every z n graded localization results are obtained on the interaction of duality for resolutions with cellular resolutions and lcm lattices using injective resolutions theorems of eagon reiner and terai are generalized to all n n graded modules the projective dimension of m equals the support regularity of its alexander dual and m is cohen macaulay if and only if its alexander dual has a support linear free resolution alexander duality is applied in the context of the z n graded local cohomology functors hii for squarefree monomial ideals i in the polynomial ring s proving a duality directly generalizing local duality which is the case when i m is maximal in the process a new flat complex for calculating local cohomology at monomial ideals is introduced showing as a consequence that terai s formula for the hilbert series of hii s is equivalent to hochster s for hn i m s i

computing the stanley depth in the first chapter some definitions and necessary results from commutative algebra are given also some detail are given which leads us towards the stanley decompositions of multigraded s modules where s is a polynomi ring in n varables over a field f in the end of the chapter we give some important results about the stanledecompositions and stanley conjecture in chapter 2 for the given monomial primary ideals and of s we gave an upper bound for the stanley depth of s which is reached if are irreducible also we showed that stanley conjecture holds for i being some irreducible monomial ideals of s these results are published in our paper 23 for integers 1 n consider the ideal i n s in chapter 3 we gave an upper bound for the stanley depth of the ideal we gave similar upper bounds for the stanley depth of the ideal where in 2 is the square free veronese ideal of degree 2 in n variables these results are from our paper 11

interpolative butterfly factorization this paper introduces the interpolative butterfly factorization for nearly optimal implementation of several transforms in harmonic analysis when their explicit formulas satisfy certain analytic properties and the matrix representations of these transforms satisfy a complementary low rank property a preliminary interpolative butterfly factorization is constructed based on interpolative low rank approximations of the complementary low rank matrix a novel sweeping matrix compression technique further compresses the preliminary interpolative butterfly factorization via a sequence of structure preserving low rank approximations the sweeping procedure propagates the low rank property among neighboring matrix factors to compress dense submatrices in the preliminary butterfly factorization to obtain an optimal one in the butterfly scheme for an n times n matrix it takes o n log n operations and complexity to construct the factorization as a product of o log n sparse matrices each with o n nonz

stanley depth of multigraded modules abstract the stanley s conjecture on cohen macaulay multigraded modules is studied especially in dimension 2 in codimension 2 similar results were obtained by herzog soleyman jahan and yassemi as a consequence of our results stanley s conjecture holds in 5 variables

the regular fourier matrices and nonuniform fast fourier transforms for any triple of positive integers m n q the matrix f m n q called the m n q regular fourier matrix is defined the regular fourier matrices f m n q are then applied to set up new algorithms for nonuniform fast fourier transforms numerical results show that the accuracies obtained by our algorithms are much better than previously reported results with the same computation complexity the algorithms require o n cdot log 2n arithmetic operations where n is the number of data points

associated legendre functions on the cut abstract algorithms and a code are described for the computation of the associated legendre functions pvu cos pv u cos qvu cos and the normalized legendre polynomial p v u cos in the ranges 0 1 2 0 1 2 1 2 v the algorithms are based on power series expansions and recurrence relations they are executed in extended range arithmetic thereby admitting very extensive ranges of and v without causing overflow or underflow

depth and stanley depth of symbolic powers of cover ideals of graphs abstract let g be a graph with n vertices and let s k x 1 x n be the polynomial ring in n variables over a field k assume that j g is the cover ideal of g and j g k is its k th symbolic power we prove that the sequences sdepth s j g k k 1 and sdepth j g k k 1 are non increasing and hence convergent suppose that o g denotes the ordered matching number of g we show that for every integer k 2 o g 1 the modules j g k and s j g k satisfy the stanley s inequality we also provide an alternative proof for 9 theorem 3 4 which states that depth s j g k n o g 1 for every integer k 2 o g 1

evaluation of the spherical bessel transform of a whittaker function an application of a difference equation method a simple analytic expression for the spherical bessel transform of the zero range bound state wave function with the coulomb interaction present for the lth partial wave expressed as a whittaker function is obtained the result is given in terms of polynomials of degree l the exponential function and a simple hypergeometric function which is independent of l transformations of this latter function are derived in terms of more rapidly convergent series the method presented has much wider application since it relies essentially only on the existence of differential difference equations for the functions involved and the solution of the inhomogeneous difference and differential equations satisfied by the transform

accelerating the nonuniform fast fourier transform the nonequispaced fourier transform arises in a variety of application areas from medical imaging to radio astronomy to the numerical solution of partial differential equations in a typical problem one is given an irregular sampling of n data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain when the sampling is uniform the fast fourier transform fft allows this calculation to be computed in o n log n operations rather than o n 2 operations unfortunately when the sampling is nonuniform the fft does not apply over the last few years a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform ffts nuffts these rely on a mixture of interpolation and the judicious use of the fft on an oversampled grid a dutt and v rokhlin siam j sci comput 14 1993 pp 1368 1383 in this paper we observe that one of the standard interpolation or gridding schemes based on gaussians can be accelerated by a significant factor without precomputation and storage of the interpolation weights this is of particular value in two and three dimensional settings saving either 10 d n in storage in d dimensions or a factor of about 5 10 in cputime independent of dimension

stanley depth of squarefree monomial ideals in this paper we answer a question posed by y h shen we prove that if i is an m generated squarefree monomial ideal in the polynomial ring s k x1 xn with k a field then sdepthi n m 2 the proof is inductive and uses the correspondence between a stanley decomposition of a monomial ideal and a partition of a particular poset into intervals established by herzog vladoiu and zheng

imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized radon transform this paper treats the linearized inverse scattering problem for the case of variable background velocity and for an arbitrary configuration of sources and receivers the linearized inverse scattering problem is formulated in terms of an integral equation in a form which covers wave propagation in fluids with constant and variable densities and in elastic solids this integral equation is connected with the causal generalized radon transform grt and an asymptotic expansion of the solution of the integral equation is obtained using an inversion procedure for the grt the first term of this asymptotic expansion is interpreted as a migration algorithm as a result this paper contains a rigorous derivation of migration as a technique for imaging discontinuities of parameters describing a medium also a partial reconstruction operator is explicitly derived for a limited aperture when specialized to a constant background velocity and specific source receiver geometries our results are directly related to some known migration algorithms

local cohomology of modules of covariants abstract let g be a connected reductive algebraic group over an algebraically closed field of characteristic zero and let w u be two finite dimensional representations of g in this paper we compute the local cohomology of u sw g provided a certain relatively weak technical condition is true

a multilevel matrix decomposition algorithm for analyzing scattering from large structures a multilevel algorithm is presented for analyzing scattering from electrically large surfaces the algorithm accelerates the iterative solution of integral equations that arise in computational electromagnetics the algorithm permits a fast matrix vector multiplication by decomposing the traditional method of moment matrix into a large number of blocks with each describing the interaction between distant scatterers the multiplication of each block by a trial solution vector is executed using a multilevel scheme that resembles a fast fourier transform fft and that only relies on well known algebraic techniques the computational complexity and the memory requirements of the proposed algorithm are o n log sup 2 n

fast computation of fourier integral operators we introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation general hyperbolic equations and curvilinear tomography the problem is to numerically evaluate a so called fourier integral operator fio of the form int e 2 pi i phi x xi a x xi hat f xi d xi at points given on a cartesian grid here xi is a frequency variable hat f xi is the fourier transform of the input f a x xi is an amplitude and phi x xi is a phase function which is typically as large as xi hence the integral is highly oscillatory because a fio is a dense matrix a naive matrix vector product with an input given on a cartesian grid of size n by n would require o n 4 operations this paper develops a new numerical algorithm which requires o n 2 5 log n operations and as low as o sqrt n in storage space the constants in front of these estimates are small it operates by localizing the integral over polar wedges with small angular aperture in the frequency plane on each wedge the algorithm factorizes the kernel e 2 pi i phi x xi a x xi into two components 1 a diffeomorphism which is handled by means of a nonuniform fft and 2 a residual factor which is handled by numerical separation of the spatial and frequency variables the key to the complexity and accuracy estimates is the fact that the separation rank of the residual kernel is provably independent of the problem size several numerical examples demonstrate the numerical accuracy and low computational complexity of the proposed methodology we also discuss the potential of our ideas for various applications such as reflection seismology

on the stanley depth of powers of edge ideals abstract let k be a field and s k x 1 x n be the polynomial ring in n variables over k let g be a graph with n vertices assume that i i g is the edge ideal of g and p is the number of its bipartite connected components we prove that for every positive integer k the inequalities sdepth i k i k 1 p and sdepth s i k p hold as a consequence we conclude that s i k satisfies stanley s inequality for every integer k n 1 also it follows that i k i k 1 satisfies stanley s inequality for every integer k 0 furthermore we prove that if i g is a non bipartite graph or ii at least one of the connected components of g is a tree with at least one edge then i k satisfies stanley s inequality for every integer k n 1 moreover we verify a conjecture of the author in special cases

computation of spherical harmonic expansion coefficients via fft s abstract a method for numerically expanding an arbitrary function on the sphere in a series of spherical harmonics which makes use of the speed of a fast fourier transform is described discussions of the operation count storage requirements accuracy and an algebraic and a numerical example are included a comparison with straightforward integration is made throughout also a new method for evaluating the spherical harmonics is discussed

long range scattering for non linear schrodinger and hartree equations in space dimension n 2 we consider the scattering problem for the non linear schrodinger nls equation with a power interaction with critical powerp 1 2 n in space dimensionsn 2 and 3 and for the hartree equation with potential x 1 in space dimensionn 2 we prove the existence of modified wave operators in thel2 sense on a dense set of small and sufficiently regular asymptotic states

stanley depths of certain stanley reisner rings abstract let be a simplicial complex on n s k x 1 x n the polynomial ring in n variables over a field k and k s i the stanley reisner ring of with respect to k it is proved that the stanley conjecture holds for k i e sdepth s k depth s k when i has four associated prime ideals it is also obtained that sdepth s k size s i

prime filtrations of monomial ideals and polarizations abstract we show that an arbitrary monomial ideal i is pretty clean if and only if its polarization i p is clean this yields a new characterization of pretty clean monomial ideals in terms of the arithmetic degree and it also implies that a multicomplex is shellable if and only the simplicial complex corresponding to its polarization is non pure shellable we also discuss stanley decompositions in relation to prime filtrations

non commutative resolutions of toric varieties abstract let r be the coordinate ring of an affine toric variety we prove using direct elementary methods that the endomorphism ring end r a where a is the finite direct sum of all isomorphism classes of conic r modules has finite global dimension equal to the dimension of r this gives a precise version and an elementary proof of a theorem of spenko and van den bergh implying that end r a has finite global dimension furthermore we show that end r a is a non commutative crepant resolution if and only if the toric variety is simplicial for toric varieties over a perfect field k of prime characteristic we show that the ring of differential operators d k r has finite global dimension

the ubiquity of smooth hilbert schemes we investigate the geography of hilbert schemes parametrizing closed subschemes of projective space with specified hilbert polynomials we classify hilbert schemes with unique borel fixed points via combinatorial expressions for their hilbert polynomials these expressions naturally lead to an arrangement of nonempty hilbert schemes as the vertices of an infinite full binary tree here we discover regularities in the geometry of hilbert schemes specifically under natural probability distributions on the tree we prove that hilbert schemes are irreducible and nonsingular with probability greater than 0 5

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

the action of the special orthogonal group on planar vectors integrity bases via a generalization of the symbolic interpretation of molien functions the present article completes the mathematical description initiated in the paper by dhont and zhilinskii 2013 the action of the orthogonal group on planar vectors invariants covariants and syzygies j phys a math theor 46 455202 of the algebraic structures that emerge from the symmetry adapted polynomials in the coordinates of n planar vectors under the action of the so 2 group the set of covariant polynomials contains all the polynomials that transform according to the weight of so 2 and is a free module for but a non free module for the sum of the rational functions of the molien function for covariants describes the decomposition of the ring of invariants or the module of covariants as a direct sum of submodules a method for extracting the generating function for covariants from the comprehensive generating function for all polynomials is introduced the approach allows the direct construction of the integrity basis for the module of covariants decomposed as a direct sum of submodules and gives insight into the expressions for the molien functions found in our earlier paper in particular a generalized symbolic interpretation in terms of the integrity basis of a rational function is discussed where the requirement of associating the different terms in the numerator of one rational function with the same subring of invariants is relaxed

alexander duality for stanley reisner rings and squarefree nn graded modules abstract let s k x1 xn be a polynomial ring and let s be its canonical module first we will define squarefreeness for n n graded s modules a stanley reisner ring k s i its syzygy module syzi k and extis k s are always squarefree this notion will simplify some standard arguments in the stanley reisner ring theory next we will prove that the i linear strand of the minimal free resolution of a stanley reisner ideal i s has the same information as the module structure of extis k s where is the alexander dual of in particular if k has a linear resolution we can describe its minimal free resolution using the module structure of the canonical module of k which is cohen macaulay in this case we can also give a new interpretation of a result of herzog and co workers which states that k is sequentially cohen macaulay if and only if i is componentwise linear

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

stanley conjecture in small embedding dimension we show that stanley s conjecture holds for a polynomial ring over a field in four variables in the case of polynomial ring in five variables we prove that the monomial ideals with all associated primes of height two are stanley ideals

simplicity of rings of differential operators in prime characteristic let w be a finite dimensional representation of a linearly reductive group g over a field k motivated by their work on classical rings of invariants levasseur and stafford asked whether the ring of invariants under g of the symmetric algebra of w has a simple ring of differential operators in this paper we show that this is true in prime characteristic indeed if r is a graded subring of a polynomial ring over a perfect field of characteristic p 0 and if the inclusion r hookrightarrow s splits then d k r is a simple ring in the last section of the paper we discuss how one might try to deduce the characteristic zero case from this result as yet however this is a subtle problem and the answer to the question of levasseur and stafford remains open in characteristic zero http www luc ac be research algebra 1991 mathematics subject classification 16s32 16g60 13a35

fast monte carlo algorithms for matrices ii computing a low rank approximation to a matrix in many applications the data consist of or may be naturally formulated as an m times n matrix a it is often of interest to find a low rank approximation to a i e an approximation d to the matrix a of rank not greater than a specified rank k where k is much smaller than m and n methods such as the singular value decomposition svd may be used to find an approximation to a which is the best in a well defined sense these methods require memory and time which are superlinear in m and n for many applications in which the data sets are very large this is prohibitive two simple and intuitive algorithms are presented which when given an m times n matrix a compute a description of a low rank approximation d to a and which are qualitatively faster than the svd both algorithms have provable bounds for the error matrix a d for any matrix x let x f and x 2 denote its frobenius norm and its spectral norm respectively in the first algorithm c columns of a are randomly chosen if the m times c matrix c consists of those c columns of a after appropriate rescaling then it is shown that from c tc approximations to the top singular values and corresponding singular vectors may be computed from the computed singular vectors a description d of the matrix a may be computed such that mathrm rank d le k and such that left a d right xi 2 le min d mathrm rank d le k left a d right xi 2 poly k 1 c left a right 2 f holds with high probability for both xi 2 f this algorithm may be implemented without storing the matrix a in random access memory ram provided it can make two passes over the matrix stored in external memory and use o cm c 2 additional ram the second algorithm is similar except that it further approximates the matrix c by randomly sampling r rows of c to form a r times c matrix w thus it has additional error but it can be implemented in three passes over the matrix using only constant additional ram to achieve an additional error beyond the best rank k approximation that is at most epsilon a 2 f both algorithms take time which is polynomial in k 1 epsilon and log 1 delta where delta 0 is a failure probability the first takes time linear in mbox max m n and the second takes time independent of m and n our bounds improve previously published results with respect to the rank parameter k for both the frobenius and spectral norms in addition the proofs for the error bounds use a novel method that makes important use of matrix perturbation theory the probability distribution over columns of a and the rescaling are crucial features of the algorithms which must be chosen judiciously

local cohomology with support in ideals of maximal minors and sub maximal pfaffians abstract we compute the gl equivariant description of the local cohomology modules with support in the ideal of maximal minors of a generic matrix as well as of those with support in the ideal of 2 n 2 n pfaffians of a 2 n 1 2 n 1 generic skew symmetric matrix as an application we characterize the cohen macaulay modules of covariants for the action of the special linear group sl g on g m the main tool we develop is a method for computing certain ext modules based on the geometric technique for computing syzygies and on matlis duality