on the construction of equiangular frames from graphs abstract we give details of the 1 1 correspondence between equiangular frames of n vectors for r d and graphs with n vertices this has been studied recently for tight equiangular frames because of applications to signal processing and quantum information theory the nontight examples given here which correspond to graphs with more than 2 eigenvalues have the potential for similar applications e g the frame corresponding to the 5 cycle graph is the unique grassmannian frame of 5 vectors in openr 3 further the associated canonical tight frames have a small number of angles in many cases

z4 kerdock codes orthogonal spreads and extremal euclidean line sets when m is odd spreads in an orthogonal vector space of type omega 2m 2 2 are related to binary kerdock codes and extremal line sets in rr 2 m 1 with prescribed angles spreads in a 2m dimensional binary symplectic vector space are related to kerdock codes over zz 4 and extremal line sets in cc 2 m with prescribed angles these connections involve binary real and complex geometry associated with extraspecial 2 groups a geometric map from symplectic to orthogonal spreads is shown to induce the gray map from a corresponding zz 4 kerdock code to its binary image these geometric considerations lead to the construction for any odd composite m of large numbers of zz 4 kerdock codes they also produce new zz 4 linear kerdock and preparata codes 1991 mathematics subject classification primary 94b60 secondary 51m15 20c99

an efficient flexible order model for 3d nonlinear water waves the flexible order finite difference based fully nonlinear potential flow model described in h b bingham h zhang on the accuracy of finite difference solutions for nonlinear water waves j eng math 58 2007 211 228 is extended to three dimensions 3d in order to obtain an optimal scaling of the solution effort multigrid is employed to precondition a gmres iterative solution of the discretized laplace problem a robust multigrid method based on gauss seidel smoothing is found to require special treatment of the boundary conditions along solid boundaries and in particular on the sea bottom a new discretization scheme using one layer of grid points outside the fluid domain is presented and shown to provide convergent solutions over the full physical and discrete parameter space of interest linear analysis of the fundamental properties of the scheme with respect to accuracy robustness and energy conservation are presented together with demonstrations of grid independent iteration count and optimal scaling of the solution effort calculations are made for 3d nonlinear wave problems for steep nonlinear waves and a shoaling problem which show good agreement with experimental measurements and other calculations from the literature

the eisenbud koh stillman conjecture on linear syzygies de nition 0 1 a variety x and line bundle l has property np for p 0 an integer if 1 p 0 it is normally generated i e skh0 x l h0 x l k is surjective for all k 0 2 p 1 it is normally generated and the kernel of k 0sh x l k 0h x l k is generated by quadrics 3 p 2 it has properties n0 and n1 and the k th syzygies among the quadrics are generated by linear syzygies for all 1 k p y 1

equiangular tight frames and signature sets in groups abstract quiangular frames are an important class of finite dimensional frames because of their superior performance and numerous applications the objective of this paper is to present a new tool to construct equiangular tight frames from groups we prove that many equiangular tight frames arise from subsets of groups which we call signature sets subsequently we define quasi signature sets and examine real equiangular tight frames associated with these subsets of groups this approach yields further results and establishes new correspondences we extend these results to complex equiangular tight frames where the inner product between any pair of vectors is a common multiple of a cube root of unity and exhibit equiangular tight frames that arise from groups in this manner

the symmetry group of a finite frame abstract we define the symmetry group of a finite frame as a group of permutations on its index set this group is closely related to the symmetry group of vale and waldron 2005 12 for tight frames they are isomorphic when the frame is tight and has distinct vectors the symmetry group is the same for all similar frames in particular for a frame its dual and canonical tight frames it can easily be calculated from the gramian matrix of the canonical tight frame further a frame and its complementary frame have the same symmetry group we exploit this last property to construct and classify some classes of highly symmetric tight frames

an energy stable fourth order finite difference scheme for the cahn hilliard equation abstract in this paper we propose and analyze an energy stable numerical scheme for the cahn hilliard equation with second order accuracy in time and the fourth order finite difference approximation in space in particular the truncation error for the long stencil fourth order finite difference approximation over a uniform numerical grid with a periodic boundary condition is analyzed via the help of discrete fourier analysis instead of the standard taylor expansion this in turn results in a reduced regularity requirement for the test function in the temporal approximation we apply a second order bdf stencil combined with a second order extrapolation formula applied to the concave diffusion term as well as a second order artificial douglas dupont regularization term for the sake of energy stability as a result the unique solvability energy stability are established for the proposed numerical scheme and an optimal rate convergence analysis is derived in the l 0 t l 2 l 2 0 t h h 2 norm a few numerical experiments are presented which confirm the robustness and accuracy of the proposed scheme

construction of robust frames in erasure recovery abstract erasures occur often in signal transmission in 10 the authors introduced robust frames with which one can recover erased frame coefficients even if the locations are unknown in this paper we give some explicit construction of robust frames

a modified chebyshev pseudospectral dd algorithm for the gbh equation in this paper a chebyshev spectral collocation domain decomposition dd semi discretization by using a grid mapping derived by kosloff and tal ezer in space is applied to the numerical solution of the generalized burger s huxley gbh equation to reduce roundoff error in computing derivatives we use the above mentioned grid mapping in this work we compose the chebyshev spectral collocation domain decomposition and kosloff and tal ezer grid mapping elaborately firstly the theory of application of the chebyshev spectral collocation method with grid mapping and dd on the gbh equation is presented this method yields a system of ordinary differential algebraic equations daes secondly we use a fourth order runge kutta formula for the numerical integration of the system of daes application of this modified method to the gbh equation show that this method m dd is faster and more accurate than the standard chebyshev spectral collocation dd s dd method

a time dependent neutron transport method of characteristics formulation with time derivative propagation a new time dependent method of characteristics moc formulation for nuclear reactor kinetics was developed utilizing angular flux time derivative propagation this method avoids the requirement of storing the angular flux at previous points in time to represent a discretized time derivative instead an equation for the angular flux time derivative along 1d spatial characteristics is derived and solved concurrently with the 1d transport characteristic equation this approach allows the angular flux time derivative to be recast principally in terms of the neutron source time derivatives which are approximated to high order accuracy using the backward differentiation formula bdf this approach called source derivative propagation sdp drastically reduces the memory requirements of time dependent moc relative to methods that require storing the angular flux an sdp method was developed for 2d and 3d applications and implemented in the computer code decart in 2d decart was used to model two reactor transient benchmarks a modified twigl problem and a c5g7 transient the sdp method accurately and efficiently replicated the solution of the conventional time dependent moc method using two orders of magnitude less memory

directed strongly regular graphs obtained from coherent algebras abstract the notion of a directed strongly regular graph was introduced by a duval in 1988 as one of the possible generalizations of classical strongly regular graphs to the directed case we investigate this generalization with the aid of coherent algebras in the sense of d g higman we show that the coherent algebra of a mixed directed strongly regular graph is a non commutative algebra of rank at least 6 with this in mind we examine the group algebras of dihedral groups the flag algebras of a steiner 2 designs in search of directed strongly regular graphs as a result a few new infinite series of directed strongly regular graphs are constructed in particular this provides a positive answer to a question of duval on the existence of a graph with certain parameter set having 20 vertices one more open case with 14 vertices listed in duval s paper is ruled out while new interpretations in terms of coherent algebras are given for many of duval s results

artin level modules we introduce level modules and show that these form a natural class of modules over a polynomial ring we prove that there exist compressed level modules i e level modules with the expected maximal hilbert function given socle type and the number of generators we also show how to use the theory of level modules to compute minimal free resolutions of the coordinate ring of points from the back which reveals new examples where random sets of points fail to satisfy the minimal resolution conjecture

higher order and higher floating point precision numerical approximations of finite strain elasticity moduli two real domain numerical approximation methods for accurate computation of finite strain elasticity moduli are developed and their accuracy and computational efficiency investigated with reference to hyperelastic constitutive models with known analytical solutions the methods are higher order and higher floating point precision numerical approximation the latter being novel in this context a general formula for higher order approximation finite difference schemes is derived and a new procedure is proposed to implement increased floating point precision the accuracy of the approximated elasticity moduli is investigated numerically using higher order approximations in standard double precision and increased quadruple precision it is found that as the order of the approximation increases the elasticity moduli tend towards the analytical solution using higher floating point precision the approximated elasticity moduli for all orders of approximation are found to be more accurate than the standard double precision evaluation of the analytical moduli application of the techniques to a finite element problem shows that the numerically approximated methods obtain convergence equivalent to the analytical method but require greater computational effort it is concluded that numerical approximation of elasticity moduli is a powerful and effective means of implementing advanced constitutive models in the finite element method without prior derivation of difficult analytical solutions

complex hadamard matrices and equiangular tight frames abstract in this paper we give a new construction of parametric families of complex hadamard matrices of square orders and connect them to equiangular tight frames the results presented here generalize some of the recent ideas of bodmann et al 3 and extend the list of known equiangular tight frames in particular a 36 21 frame coming from a nontrivial cube root signature matrix is obtained for the first time

euler maclaurin and gregory interpolants let a sufficiently smooth function f f on 1 1 1 1 be sampled at n 1 n 1 equispaced points and let k ge 0 k 0 be given an euler maclaurin interpolant to the data is defined consisting of a sum of a degree k k algebraic polynomial and a degree n n trigonometric polynomial which deviates from f f by o n k o n k and whose integral is equal to the order k k euler maclaurin approximation of the integral of f f this interpolant makes use of the same derivatives f j pm 1 f j 1 as the euler maclaurin formula a variant gregory interpolant is also defined based on finite difference approximations to the derivatives whose integral for k k odd is equal to the order k k gregory approximation to the integral

optimal decomposition of the domain in spectral methods for wave like phenomena a strategy for determining the optimal number of grid points and subdomains in a spectral method with domain decomposition on a serial computer is presented the rapidly growing computational cost for large numbers of grid points in each subdomain is balanced against the exponential convergence for spectral approximation of smooth functions and the optimum is found as the number of grid points and subdomains that gives the minimal computational cost for a given accuracy the typical length scale of the problem is found to influence the number of subdomains but not the number of grid points within each subdomain

exterior algebra methods for the minimal resolution conjecture if r 6 r 6d 9 we show that the minimal resolution conjecture mrc fails for a general set of points in p r for almost 1 2 p r values of this strengthens the result of d eisenbud and s popescu ep1 who found a unique such for each r in the given range our proof begins like a variation of that of eisenbud and popescu but uses exterior algebra methods as explained by eisenbud g floystad and f o schreyer efs to avoid the degeneration arguments that were the most difficult part of the eisenbud popescu proof analogous techniques show that the mrc fails for linearly normal curves of degree d and genus g when d 3g 2 g 4 re proving results of schreyer m green and r lazarsfeld

minimal free resolution for points on surfaces a long standing problem in algebraic geometry and commutative algebra is to determine the minimal graded free resolution of a 0 dimensional scheme z in pn or in an arbitrary projective variety x in 18 m musta 1998 predicted the graded betti numbers of the minimal free resolution of a general set of distinct points z in x in this paper we state a refined version of musta s conjecture mrc and we predict the existence of a non empty open subset u hilbs x such that any z u has a minimal graded free resolution without ghost terms wmrc in this paper we are going to prove 1 for any s d 33 1 there exists a d 22 dimensional family of irreducible generically smooth surfaces of degree d in p3 satisfying the wmrc for s and 2 any smooth cubic surface satisfies the mrc for any s 19

harmonic equiangular tight frames comprised of regular simplices abstract an equiangular tight frame etf is a sequence of unit norm vectors in a euclidean space whose coherence achieves equality in the welch bound and thus yields an optimal packing in a projective space a regular simplex is a simple type of etf in which the number of vectors is one more than the dimension of the underlying space more sophisticated examples include harmonic etfs which equate to difference sets in finite abelian groups recently it was shown that some harmonic etfs are comprised of regular simplices in this paper we continue the investigation into these special harmonic etfs we begin by characterizing when the subspaces that are spanned by the etf s regular simplices form an equi isoclinic tight fusion frame eitff which is a type of optimal packing in a grassmannian space we shall see that every difference set that produces an eitff in this way also yields a complex circulant conference matrix next we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set it turns out that these relative difference sets lend themselves to a second related and yet distinct construction of complex circulant conference matrices finally we provide explicit infinite families of etfs to which this theory applies

mimetic seismic wave modeling including topography on deformed staggered grids abstractfinite difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography as a consequence alternatives to finite difference methods have been proposed to circumvent these issues we present a new numerical scheme for 3d elastic wave propagation in the presence of strong topography this finite difference scheme is based upon a staggered grid of the lebedev type or fully staggered grid fsg it uses a grid deformation strategy to make a regular cartesian grid conform to a topographic surface in addition the scheme uses a mimetic approach to accurately solve the free surface condition and hence allows for a less restrictive grid spacing criterion in the computations the scheme can use high order operators for the spatial derivatives and obtain low dispersion results with as few as six points pe

quick and accurate q parameterization in viscoelastic wave modeling we introduce a procedure for including the attenuation factor q in a consistent manner in seismic modeling and show 3d examples the q fitting over a chosen frequency band involves two algorithms the first creates starting values of relaxation times and the second does nonlinear inversion using the results of the first as initial values the resulting q function gives a good approximation to a constant q over the chosen frequency band the algorithm is combined with a finite difference f d code that includes topographies in 3d seismic media the velocity stress formulation for viscoelastic wave modeling is used with an arbitrary number of relaxation mechanisms to model a desired q behavior these equations are discretized by high order f ds in the interior of the medium and we gradually reduce the f d order to two at the stress free surface where we implement our free surface boundary conditions the seismic f d algorithm is applied to a marine seismic experiment with and without viscoelasticity to emphasize the importance of including physical attenuation and dispersion in seismic modeling their inclusion even for marine surveys is clearly important for lossy ocean bottoms our procedure for more accurate modeling of physical dispersion and attenuation may increase future motivation to include viscoelasticity in seismic inversion

two graphs and skew two graphs in finite geometries abstract we describe the use of two graphs and skew oriented two graphs as isomorphism invariants for translation planes and m systems including ovoids and spreads of polar spaces in odd characteristic

optimal frames for erasures we study frames from the viewpoint of coding theory we introduce a numerical measure of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting

a note on stability of pseudospectral methods for wave propagation in this paper we deal with the effects on stability of subtle differences in formulations of pseudospectral methods for solution of the acoustic wave equation we suppose that spatial derivatives are approximated by chebyshev pseudospectral discretizations through reformulation of the equations as first order hyperbolic systems any appropriate ordinary differential equation solver can be used to integrate in time however the resulting stability and hence efficiency properties of the numerical algorithms are drastically impacted by the manner in which the absorbing boundary conditions are incorporated specifically mathematically equivalent well posed approaches are not equivalent numerically an analysis of the spectrum of the resultant system operator predicts these properties

exact wavefield reconstruction on finite difference grids with minimal memory requirements abstractwavefield injection in finite difference fd grids can be described by the method of multiple point sources the method teaches how synthetically generated wavefields and wavefield constituents can be reconstructed from surface recordings using a combination of monopole and dipole sources on an injection surface surrounding the model we show how to properly record surface wavefields and inject point sources in staggered fd grids in a way that is consistent with the order of spatial accuracy of the fd scheme the description is general and can be used for schemes of any order only one or two surface wavefields are required to reconstruct the original wavefields or wavefield constituents to numerical precision independent of the order of spatial accuracy of the fd stencil we have applied the method for the separation of up and downgoing wavefields and for source wavefield reconstruction for reverse time migration our implementation enables accurate source wavefield reconstruction with optima

high order finite difference schemes for the solution of second order bvps we introduce new methods in the class of boundary value methods bvms to solve boundary value problems bvps for a second order ode these formulae correspond to the high order generalizations of classical finite difference schemes for the first and second derivatives in this research we carry out the analysis of the conditioning and of the time reversal symmetry of the discrete solution for a linear convection diffusion ode problem we present numerical examples emphasizing the good convergence behavior of the new schemes finally we show how these methods can be applied in several space dimensions on a uniform mesh

equiangular tight frames that contain regular simplices abstract an equiangular tight frame etf is a type of optimal packing of lines in euclidean space a regular simplex is a special type of etf in which the number of vectors is one more than the dimension of the space they span in this paper we consider etfs that contain a regular simplex that is have the property that a subset of its vectors forms a regular simplex as we explain such etfs are characterized as those that achieve equality in a certain well known bound from the theory of compressed sensing we then consider the so called binder of such an etf namely the set of all regular simplices that it contains we provide a new algorithm for computing this binder in terms of products of entries of the etf s gram matrix in certain circumstances we show this binder can be used to produce a particularly elegant naimark complement of the corresponding etf other times an etf is a disjoint union of regular simplices and we show this leads to a certain type of optimal packing of subspaces known as an equichordal tight fusion frame we conclude by considering the extent to which these ideas can be applied to numerous known constructions of etfs including harmonic etfs

an asymptotic numerical method for a class of singular sturm liouville problems a class of sturm liouville problems has been discovered for which the method of approximating the coefficients by piecewise constant functions has considerable difficulty we characterize this class give examples and derive a modification of the method based on local asymptotic approximations which surmounts the difficulties

periodic resolutions over exterior algebras in this paper we study modules with periodic free resolutions that is periodic modules over an exterior algebra we show that any module with bounded betti numbers that is a module whose syzygy modules have a bounded number of generators must have periodic free resolution of period 2 and that for graded modules the period is 1 we also show that any module with a linear tate resolution is periodic we give a criterion of exactness for periodic complexes and a parameterization of the set of periodic modules 2002 elsevier science usa all rights reserved

linearly connected sequences and spectrally optimal dual frames for erasures abstract in the case that a frame is prescribed for applications and erasures occur in the process of data transmissions we examine optimal dual frames for the recovery from single erasures in contrast to earlier papers we consider the spectral radius of the error operator instead of its operator norm as a measure of optimality this notion of optimality is natural when the neumann series is used to recover the original data in an iterative manner we obtain a complete characterization of spectrally one erasure optimal dual frames in terms of the redundancy distribution of the prescribed frame our characterization relies on the connection between erasure optimal frames and the linear connectivity property of the frame we prove that the linear connectivity property is equivalent to the intersection dependent property and is also closely related to the well known concept of a k independent set additionally we also establish several necessary and sufficient conditions for the existence of an alternate dual frame to make the iterative reconstruction work

derivative formulae and errors for non uniformly spaced points lagrange interpolation estimates a function at a reference position from known values of the function at distinct non uniformly spaced points x 1 x n here the corresponding n point finite difference formulae are derived to estimate derivatives up to order n 1 at a recurrence relation is derived that permits the errors to be determined as a taylor series to any accuracy the error coefficient multiplying the n j th derivative is a polynomial of order j 1 in the elementary symmetric functions for the displacements x 1 x n appendices state finite difference formulae to estimate the derivatives and the first four error terms for n 1 5

the minimal resolution conjecture abstract by extending the ideal generation conjecture we formulate the minimal resolution conjecture for a sufficiently generic set of points in p n and we explicitly write out the conjectured values of the betti numbers we then relate the minimal resolution conjecture to the ideal generation conjecture and the cohen macaulay type conjecture and we prove the minimal resolution conjecture for d nn 1 n points for n 2 points and for d 1 nn points we also prove the cohen macaulay type conjecture for s points with n 1 s 2 nn finally we recover information about the resolution of s 1 or s 1 points from that of s points

gale duality and free resolutions of ideals of points what is the shape of the free resolution of the ideal of a general set of points in p this question is central to the programme of connecting the geometry of point sets in projective space with the structure of the free resolutions of their ideals there is a lower bound for the resolution computable from the known hilbert function and it seemed natural to conjecture that this lower bound would be achieved this is the minimal resolution conjecture lorenzini 1987 1993 although the conjecture has been shown to hold in many cases three examples discovered computationally by frank olaf schreyer in 1993 strongly suggested that it would fail in general in this paper we shall describe a novel structure inside the free resolution of a set of points which accounts for the failure and provides a counterexample in p for every r 6 r 6 9 we begin by reviewing the conjecture and its status consider a set of c points in the projective r space over a eld k say c pk let s k x0 xr let ic be the homogeneous ideal of c and let sc denote the homogeneous coordinate ring of c let

a pseudospectral fictitious point method for high order initial boundary value problems when pseudospectral approximations are used for space derivatives one often encounters spurious eigenvalues these can lead to severe time stepping difficulties for pdes this is especially the case for equations with high order derivatives in space requiring multiple conditions at one or both boundaries we note here that a very simple to implement fictitious point approach circumvents most of these difficulties the new approach is tested on the kuramoto sivashinsky equation and on a dispersive linear pde featuring a time space corner singularity

classroom note calculation of weights in finite difference formulas the classical techniques for determining weights in finite difference formulas were either computationally slow or very limited in their scope e g specialized recursions for centered and staggered approximations for adams bashforth adams moulton and bdf formulas for odes etc two recent algorithms overcome these problems for equispaced grids such weights can be found very conveniently with a two line algorithm when using a symbolic language such as mathematica reducing to one line in the case of explicit approximations for arbitrarily spaced grids we describe a computationally very inexpensive numerical algorithm

matrix methods for radial schrodinger eigenproblems defined on a semi infinite domain in this paper we discuss numerical approximation of the eigenvalues of the one dimensional radial schrodinger equation posed on a semi infinite interval the original problem is first transformed to one defined on a finite domain by applying suitable change of the independent variable the eigenvalue problem for the resulting differential operator is then approximated by a generalized algebraic eigenvalue problem arising after discretization of the analytical problem by the matrix method based on high order finite difference schemes numerical experiments illustrate the performance of the approach

equiangular tight frames from paley tournaments abstract we prove the existence of equiangular tight frames having n 2 d 1 elements drawn from either c d or c d 1 whenever n is either 2 k 1 for k n or a power of a prime such that n 3 mod 4 we also find a simple explicit expression for the prime power case by establishing a connection to a 2 d element equiangular tight frame based on quadratic residues

minimal resolutions chow forms and ulrich bundles on k3 surfaces the minimal resolution conjecture mrc for points on a projective varietyx p predicts that the minimal graded free resolution of a general set x of points is as simple as the geometry of x allows originally the most studied case has been that when x p see epsw the general form of the mrc for subvarieties x p was formulated in mus and fmp the betti diagram of a large enough set x consisting of general points is obtained from the betti diagram of x by adding two rows indexed by u 1 and u where u is an integer depending on all differences bi 1 u 1 bi u are known and depend on the hilbert polynomial px and i u and see fmp the minimal resolution conjecture for general points on x predicts that

equiangular tight frames from complex seidel matrices containing cube roots of unity we derive easily verifiable conditions which characterize when complex seidel matrices containing cube roots of unity have exactly two eigenvalues the existence of such matrices is equivalent to the existence of equiangular tight frames for which the inner product between any two frame vectors is always a common multiple of the cube roots of unity we also exhibit a relationship between these equiangular tight frames complex seidel matrices and highly regular directed graphs we construct examples of such frames with arbitrarily many vectors

on the derivation of highest order compact finite difference schemes for the one and two dimensional poisson equation with dirichlet boundary conditions the primary aim of this paper is to answer the question what are the highest order five or nine point compact finite difference schemes to answer this question we present several simple derivations of finite difference schemes for the one and two dimensional poisson equation on uniform quasi uniform and nonuniform face to face hyperrectangular grids and directly prove the existence or nonexistence of their highest order local accuracies our derivations are unique in that we do not make any initial assumptions on stencil symmetries or weights for the one dimensional problem the derivation using the three point stencil on both uniform and nonuniform grids yields a scheme with arbitrarily high order local accuracy however for the two dimensional problem the derivation using the corresponding five point stencil on uniform and quasi uniform grids yields a scheme with at most second order local accuracy and on nonuniform grids yields at most first order local accuracy when expanding the five poin

mesh independence of the generalized davidson algorithm abstract we give conditions under which the generalized davidson algorithm for eigenvalue computations is mesh independent in this case mesh independence means that the iteration statistics residual norms convergence rates for example of a sequence of discretizations of a problem in a banach space converge the statistics for the infinite dimensional problem we illustrate the result with several numerical examples

reconstruction of signals from frame coefficients with erasures at unknown locations we propose new approaches to the problems of recovering signals from the rearranged frame coefficients or frame coefficients with erasures at either known or unknown locations these problems naturally arise from applications where the encoded information needs to be transmitted for example in signal image processing information and coding theory and communications we show that with the appropriate choices of the frames that are used for encoding the signal with erasures occurring at known locations can be easily recovered without inverting the sub frame operators each time our new easy to implement and cost efficient algorithm provides perfect reconstruction of the original signal to address the problem of recovering erased coefficients from unknown locations we propose to use a class of frames that are almost robust with respect to m erasures we prove that every frame with uniform excess can be rescaled to an almost robust frame and the locations of erased data can be perfectly recovered for almost all the signals similar results are obtained for recovering the original order of a disordered rearranged set of frame coefficients numerical examples are presented to test the main results whenever the received data are noise free we can recover the original signal exactly from frame coefficients with erasures at unknown locations or from a disordered set of frame coefficients

the projective geometry of the gale transform abstract the gale transform an involution on sets of points in projective space appears in a multitude of guises and in subjects as diverse as optimization coding theory theta functions and recently in our proof that certain general sets of points fail to satisfy the minimal free resolution conjecture see eisenbud and popescu 1999 invent math 136 419 449 in this paper we reexamine the gale transform in the light of modern algebraic geometry we give a more general definition in the context of finite locally gorenstein subschemes we put in modern form a number of the more remarkable examples discovered in the past and we add new constructions and connections to other areas of algebraic geometry we generalize goppa s theorem in coding theory and we give new applications to castelnuovo theory we also give references to classical and modern sources

numerical solution of calcium mediated dendritic branch model the standard algorithms for spatial discretizations of calcium mediated dendritic branch models via finite difference methods are quite accurate but they are also extremely slow to improve computational efficiency we apply spatial discretization using a spectral collocation method simulations using the spectral collocation method are compared to the finite difference approach using a model for calcium mediated restructuring with spine pruning we find that the spectral collocation method is about fifteen times more efficient to achieve similar accuracy than the finite difference approach even though spectral collocation requires more steps

adaptive solution of bvps in singularly perturbed second order odes by the extended numerov method combined with an iterative local grid h refinement the extended numerov scheme of chawla is one of high order compact finite difference discretisations applicable to bvps in second order odes and to pdes the use of such discretisations in connection with adaptive grids remains largely unexplored in order to make up this deficiency we study the combination of the extended numerov discretisation with the iterative local adaptive grid h refinement previously tested by the present author at the assumption of the conventional finite difference discretisation a detailed formalism enabling a posteriori error analysis of the extended numerov scheme is developed together with several alternative choices for error estimators grid refinement indicators and regridding strategies the adaptive algorithms obtained are examined in calculations on a collection of 15 examples of singularly perturbed odes including equations relevant for electrochemistry the most satisfactory algorithm involves deferred approach to error estimation based on virtual grid coarsening for truncation error evaluation local discretisation errors as refinement indicators and regridding strategy that uses the mean indicator values as the indicator thresholds for refinement the extended numerov scheme proves more efficient than the conventional discretisation when used together with the iterative h refinement

complex equiangular tight frames and erasures in this paper we demonstrate that there are distinct dierences between real and complex equiangular tight frames etfs with regards to erasures for example we prove that there exist arbitrarily large non trivial complex equiangular tight frames which are robust against three erasures and that such frames come from a unique class of complex etfs in addition we extend certain results in 1 to complex vector spaces as well as show that other results regarding real etfs are not valid for complex etfs

transport schemes in spherical geometries using spline based rbf fd with polynomials abstract this work presents a numerical algorithm for using radial basis function generated finite differences rbf fd to solve partial differential equations pdes on s 2 using polyharmonic splines with added polynomials defined in a 2d plane phs poly we introduce a novel method for calculating rbf fd phs poly differentiation weights on s 2 using first a householder reflection and then a projection onto the tangent plane the new phs poly rbf fd method is implemented on two standard test cases 1 solid body rotation on s 2 and 2 3d tracer transport within earth s atmosphere compared to existing methods including those in the rbf literature at similar resolutions this approach requires fewer degrees of freedom and is algorithmically much simpler a matlab code to implement the method is included in the appendix

an improved immersed boundary finite difference method for seismic wave propagation modeling with arbitrary surface topography abstractto accurately simulate seismic wave propagation for the purpose of developing modern land data processing tools especially full waveform inversion fwi we have developed an efficient high order finite difference forward modeling algorithm with the capability of handling arbitrarily shaped free surface topography unlike most existing forward modeling algorithms using curvilinear grids to fit irregular surface topography this finite difference algorithm based on an improved immersed boundary method uses a regular cartesian grid system without suffering from staircasing error which is inevitable in a conventional finite difference method in this improved immersed boundary finite difference ibfd algorithm arbitrarily curved surface topography is accounted for by imposing the free surface boundary conditions at exact boundary locations instead of using body conforming grids or refined grids near the boundaries thus greatly reducing the complexity of its preprocessing procedures and the com

steiner equiangular tight frames abstract we provide a new method for constructing equiangular tight frames etfs the construction is valid in both the real and complex settings and shows that many of the few previously known examples of etfs are but the first representatives of infinite families of such frames it provides great freedom in terms of the frame s size and redundancy this method also explicitly constructs the frame vectors in their native domain as opposed to implicitly defining them via their gram matrix moreover in this domain the frame vectors are very sparse the construction is extremely simple a tensor like combination of a steiner system and a regular simplex this simplicity permits us to resolve an open question regarding etfs and the restricted isometry property rip we show that the rip behavior of some etfs is unfortunately no better than their coherence indicates

generation of pseudospectral differentiation matrices i we present a simple method for computing n times n pseudospectral differentiation matrices of order p in cal o pn 2 operations for the case of quasi polynomial approximation the algorithm is based on fornberg s finite difference algorithm and is numerically stable a fortran implementation is included a necessary and sufficient condition for d p d p 1 is also given

geometry of the welch bounds abstract a geometric perspective involving grammian and frame operators is used to derive the entire family of welch bounds this perspective unifies a number of observations that have been made regarding tightness of the bounds and their connections to symmetric k tensors tight frames homogeneous polynomials and t designs in particular a connection has been drawn between sampling of homogeneous polynomials and frames of symmetric k tensors it is also shown that tightness of the bounds requires tight frames the lack of tight frames of symmetric k tensors in many cases however leads to consideration of sets that come as close as possible to attaining the bounds the geometric derivation is then extended in the setting of generalized or continuous frames the welch bounds for finite sets and countably infinite sets become special cases of this general setting

3d weak dispersion reverse time migration using a stereo modeling operator reliable 3d imaging is a required tool for developing models of complex geologic structures reverse time migration rtm as the most powerful depth imaging method has become the preferred imaging tool because of its ability to handle complex velocity models including steeply dipping interfaces and large velocity contrasts finite difference methods are among the most popular numerical approaches used for rtm however these methods often encounter a serious issue of numerical dispersion which is typically suppressed by reducing the grid interval of the propagation model resulting in large computation and memory requirements in addition even with small grid spacing numerical anisotropy may degrade images or worse provide images that appear to be focused but position events incorrectly recently stereo operators have been developed to approximate the partial differential operator in space these operators have been used to develop several weak dispersion and efficient stereo modeling methods that have been found to be superior to conventional algorithms in suppressing numerical dispersion and numerical anisotropy we generalized one stereo modeling method fourth order nearly analytic central difference nacd from 2d to 3d and applied it to 3d rtm the rtm results for the 3d seg eage phase a classic data set 1 and the seg advanced modeling project model demonstrated that even when using a large grid size the nacd method can handle very complex velocity models and produced better images than can be obtained using the fourth order and eighth order lax wendroff correction lwc schemes we also applied 3d nacd and fourth order lwc to a field data set and illustrated significant improvements in terms of structure imaging horizon layer continuity and positioning we also investigated numerical dispersion and found that not only does the nacd method have superior dispersion characteristics but also that the angular variation of dispersion is significantly less than for lwc

complex equiangular cyclic frames and erasures we derive various interesting properties of complex equiangular cyclic frames for many pairs n k using gauss sums and number theory we further use these results to study the random and burst errors of some special classes of complex equiangular cyclic n k frames

sheaf cohomology and free resolutions over exterior algebras in this paper we derive an explicit version of the bernstein gel fand gel fand bgg correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free reso lutions over its koszul dual exterior algebra among the facts about the bgg correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the linear part of a resolution over the exterior algebra we explore the structure of free resolutions over an exterior algebra for example we show that such resolutions are eventually dominated by their linear parts in the sense that erasing all terms of degree 1 in the complex yields a new complex which is eventually exact as applications we give a construction of the beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms the explicitness of our version allows us to to prove two conjectures about the morphisms in the monad and we get an efficient method for machine computation of the cohomology ofsheaves we also construct all the monads for a sheaf that can be built from sums of line bundles and show that they are often characterized by numerical data let v be a finite dimensional vector space over a field k and let w v be the dual space in this paper we will study complexes and resolutions over the exterior algebra e v and their relation to modules over s sym w and sheaves on projective space p w in this paper we study the bernstein gel fand gel fand bgg correspondence 1978 usually stated as an equivalence between the derived category of bounded complexes of coherent sheaves on p w and the stable category of finitely generated graded modules over e its essential content is a functor r from complexes of graded s modules to complexes of graded e modules and its adjoint l for example if m imi is a graded s module regarded as a complex with just one term then the first and third authors are grateful to the nsf for partial support during the preparation of this paper the third author wishes to thank msri for its hospitality

a spectral collocation method based on integrated chebyshev polynomials for two dimensional biharmonic boundary value problems this paper reports a new spectral collocation method for numerically solving two dimensional biharmonic boundary value problems the construction of the chebyshev approximations is based on integration rather than conventional differentiation this use of integration allows i the imposition of the governing equation at the whole set of grid points including the boundary points and ii the straightforward implementation of multiple boundary conditions the performance of the proposed method is investigated by considering several biharmonic problems of first and second kinds more accurate results and higher convergence rates are achieved than with conventional differential methods

a modified crank nicolson scheme with incremental unknowns for convection dominated diffusion equations a modified crank nicolson scheme based on one sided difference approximations is proposed for solving time dependent convection dominated diffusion equations in two dimensional space the modified scheme is consistent and unconditionally stable a priori l 2 error estimate for the fully discrete modified scheme is derived with the use of the incremental unknowns preconditioner at each time step a comparison among several classical numerical schemes has been made and numerical results confirm stability and efficiency of the modified crank nicolson scheme

automatic solution of sturm liouville problems using the pruess method abstract we develop algorithms based on coefficient approximation for the automatic solution of regular and singular sturm liouville problems

equiangular tight frames and fourth root seidel matrices in this paper we construct complex equiangular tight frames etfs in particular we study the grammian associated with an etf whose off diagonal entries consist entirely of fourth roots of unity these etfs are classified and we also provide some computational techniques which give rise to previously undiscovered etfs

conditional sic povms in this paper we examine a generalization of the symmetric informationally complete povms sic povms are the optimal measurements for full quantum tomography but if some parameters of the density matrix are known then the optimal sic povm should be orthogonal to a subspace this gives the concept of the conditional sic povm the existence is not known in general but we give a result in the special case when the diagonal is known of the density matrix

some observations on a static spatial remeshing method based on equidistribution principles this paper proposes a method of lines solution procedure with time and space adaptation for one dimensional systems of partial differential equations whose solutions display steep moving fronts the spatial remeshing algorithm which is a variation of the method published by sanz serna and christie and an extension suggested by revilla is a static remeshing method based on equidistribution principles the selection of the several algorithm components i e grid placement criterion spatial discretization scheme time integrator adaptation frequency and parameter tuning are investigated and illustrated with several test examples i e the cubic schrodinger equation a model of a single step reaction with diffusion and a model of flame propagation

generation of finite difference formulas on arbitrarily spaced grids on etablit des recurrences simples pour le calcul des poids dans les formules aux differences finies compactes pour les derivees de tous ordres avec une precision d ordre arbitraire sur des grilles a une dimension d espacement arbitraire

frames of uniform subframe bounds with applications to erasures abstract we introduce a class of frames called usb frames frames of uniform subframe bounds which mean that any subsequence of the frame is a frame for the subspace it spans and the frame bounds depend only on the cardinality of the subsequence we give several necessary and sufficient conditions for a frame to be a usb frame as an application we show that a tight usb frame is optimal for erasures in the sense that the reconstruction error is independent of the position of erasures we give several explicit construction of usb frames

approximation of eigenvalues of sturm liouville problems defined on a semi infinite domain abstract in this paper we describe how to approximate numerically the eigenvalues of a sturm liouville problem defined on a semi infinite interval the key idea is to transform the problem in such a way as to compress the semi infinite interval in a finite interval by applying a suitable change of the independent variable then we approximate each derivative in the sturm liouville equation thus obtained with finite difference schemes consequently we convert the sturm liouville problem into an algebraic eigenvalue problem the numerical results of the experiments show that the proposed approach is promising