the five dimensional complete left symmetric algebra structures compatible with an abelian lie algebra structure abstract the classification of five dimensional complete left symmetric algebras is known to be a hard problem in this paper we establish an explicit classification of the five dimensional complete left symmetric algebras over r having an abelian associated lie algebra a detailed list of all possible isomorphism types is presented

optimal domain splitting for interpolation by chebyshev polynomials polynomial interpolants defined using chebyshev extreme points as nodes converge uniformly at a geometric rate when sampling a function that is analytic on an interval however the convergence rate can be arbitrarily close to unity if the function has a singularity close to the interval when extended to the complex plane in such cases splitting the interval and doing piecewise interpolation may be more efficient in the total number of nodes than the global interpolant because the convergence rate is determined by bernstein ellipses obtained through a joukowski conformal map relative efficiency of splitting at any point in the interval can be calculated and then optimized over the interval the optimal splitting may be applied recursively the chebfun software project uses a simple rule of thumb without prior singularity information to create a binary search that can be shown to do an excellent job of finding the optimal splitting in most cases however the process can use a large number of intermedi

translations in simply transitive affine actions of heisenberg type lie groups let g be a 2 step nilpotent lie group with a 1 dimensional commutator subgroup we prove that for any simply transitive and affine action of g there exists a non trivial subgroup of g acting as pure translations this result no longer holds in case the commutator subgroup is higher dimensional

on nonsmooth solutions of linear hyperbolic systems initial value problems for linear hyperbolic systems with smooth 2 periodic coefficients are solved numerically by a modified fourier galerkin method when the initial values are nonsmooth the described approach is seen to give substantially improved accuracy compared to more traditional methods the discontinuities are accurately resolved already on coarse grids and the fine structure of structured solutions is resolved on relatively coarse grids as well the accuracy is seen to be of high order and even for very long term integrations the global error can be kept very small if the grid is sufficiently refined

space time spectral method for the cattaneo equation with time fractional derivative abstract this paper introduces a high order accurate numerical method for solving the cattaneo equation with time fractional derivative it is based on the galerkin legendre spectral method in space and the chebyshev collocation method in time arbitrarily high order accurate can be made in both space and time optimal priori error bound of the semi discrete method and the stability and convergence of the full discrete method are strictly given extensive experimental results confirm the theoretical claims of this method in both space and time

on a high order numerical method for functions with singularities by splitting a given singular function into a relatively smooth part and a specially structured singular part it is shown how the traditional fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals singular functions with various types of singularities of importance in applications are considered relations between the discrete and the continuous fourier series for the singular functions are established of particular interest are piecewise smooth functions for which various important applications are indicated and for which numerous numerical results are presented

approximating the jump discontinuities of a function by its fourier jacobi coefficients in the present paper we generalize eckhoff s method i e the method for approximating the locations of discontinuities and the associated jumps of a piecewise smooth function by means of its fourier chebyshev coefficients a new method enables us to approximate the locations of discontinuities and the associated jumps of a discontinuous function which belongs to a restricted class of the piecewise smooth functions by means of its fourier jacobi coefficients for arbitrary indices approximations to the locations of discontinuities and the associated jumps are found as solutions of algebraic equations it is shown as well that the locations of discontinuities and the associated jumps are recovered exactly for piecewise constant functions with a finite number of discontinuities in addition we study the accuracy of the approximations and present some numerical examples

on fermionic novikov algebras novikov algebras were introduced in connection with the poisson brackets of hydrodynamic type and hamiltonian operators in formal variational calculus they are a class of left symmetric algebras with commutative right multiplication operators which can be viewed as bosonic fermionic novikov algebras are a class of left symmetric algebras with anti commutative right multiplication operators they correspond to a certain hamiltonian superoperator in a supervariable in this paper we commence a study on fermionic novikov algebras from the algebraic point of view we will show that any fermionic novikov algebra in dimension 3 must be bosonic moreover we give the classification of real fermionic novikov algebras on four dimensional nilpotent lie algebras and some examples in higher dimensions as a corollary we obtain kinds of four dimensional real fermionic novikov algebras which are not bosonic all of these examples will serve as a guide for further development including the application in physics

polynomial structures for nilpotent groups if a polycyclic by finite rank k group admits a faithful affine representation making it acting on rk properly discontinuously and with compact quotient we say that admits an affine structure in 1977 john milnor questioned the existence of affine structures for such groups very recently examples have been obtained showing that even for torsion free finitely generated nilpotent groups n affine structures do not always exist it looks natural to consider affine structures as examples of polynomial structures of degree one we introduce the concept of a canonical type polynomial structure for polycyclic by finite groups using the algebraic framework of the seifert fiber space construction and a nice cohomology vanishing theorem we prove the existence and uniqueness up to conjugation of canonical type polynomial structures for virtually finitely generated nilpotent groups applying this uniqueness to a result going back to mal cev it follows that for torsion free finitely generated nilpotent groups each canonical polynomial structure is expressed in polynomials of limited degree the minimal degree needed for obtaining a polynomial structure will determine the affine defect number we prove that the known counterexamples to milnor s question have the smallest possible affine defect i e affine defect number equal to one

polycyclic by finite groups admit a bounded degree polynomial structure for a polycyclic by finite group gamma of hirsch length h an affine resp polynomial structure is a representation of gamma into rm aff bbb r h resp rm p bbb r h the group of polynomial diffeomorphisms letting gamma act properly discontinuously on bbb r h recently it was shown by counter examples that there exist groups gamma even nilpotent ones which do not admit an affine structure thus giving a negative answer to a long standing question of john milnor we prove that every polycyclic by finite group gamma admits a polynomial structure which moreover appears to be of a special simple type called canonical and as a consequence of this consists entirely of polynomials of a bounded degree the construction of this polynomial structure is a special case of an iterated seifert fiber space construction which can be achieved here because of a very strong and surprising cohomology vanishing theorem

procedures for computing one and two dimensional integrals of functions with rapid irregular oscillations a collocation procedure for efficient integration of rapidly oscillatory functions is presented the integration problem is transformed into a certain o d e problem and this is solved by a collocation technique the method is also extended to two dimensional integration and some numerical results are appended showing the efficiency of the method in handling difficult cases of rapid irregular oscillations 1 the procedure for one dimensional integrals we consider integrals of the form a 1 1 1 f x e iq x dx where f is smooth and nonoscillatory and iq x i b a 1 two practical methods for evaluating rapidly oscillatory integrals are described in 1 the use of approximation as in filon s method 3 4 and the speedup method of longman 5 formally both methods are applicable to any integral of the form 1 1 but their best performance is for the case of a constant frequency q w in this note we present an efficient method which is applicable for cases of varying frequency q using only a small number of values of f and q in a b and the values q a and q b the proposed method follows the spirit of filon s method it is based upon the fact that if f were of the form 1 2 f x iq x p x p x l l1p x a x b then the integral could be evaluated directly as 1 3 j b iq x p x pt x eiq x dx f d p x eiq x dx b e qbp a eiqa equation 1 2 can be considered as a differential equation for p x and any solution of this equation can be used in 1 3 for evaluating i the general solution of this equation is 1 4 p x e iq x f t eiq t dt cj received march 24 1981 revised july 29 1981 1980 mathematics subject classification primary 65d30 secondary 65d32 1982 american mathematical society 0025 5718 8 1 ooo 0 1 23 03 25 531 this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 36 01 utc all use subject to http about jstor org terms

uniqueness and estimation of three dimensional motion parameters of rigid objects with curved surfaces two main results are established in this paper first we show that seven point correspondences are sufficient to uniquely determine from two perspective views the three dimensional motion parameters within a scale factor for the translations of a rigid object with curved surfaces the seven points should not be traversed by two planes with one plane containing the origin nor by a cone containing the origin second a set of essential parameters are introduced which uniquely determine the motion parameters up to a scale factor for the translations and can be estimated by solving a set of linear equations which are derived from the correspondences of eight image points the actual motion parameters can subsequently be determined by computing the singular value decomposition svd of a 3 3 matrix containing the essential parameters

aspherical manifolds with virtually 3 step nilpotent fundamental group a closed aspherical manifold of dimension 3 4 with virtually 3 step nilpotent fundamental group supports a complete affinely flat structure topological and seifert bundle classifications are given for a special case called seifert relatives of flat riemannian manifolds we use the seifert fiber space construction as a main tool 0 introduction it is well known that given a torsion free virtually poly z group ir there exists a closed smooth k r 1 manifold eg a double coset space of a lie group a question we are interested in is whether such a manifold is homeomorphic to a quotient of r by an action of a discrete group of affine motions the best known result seems to be j scheuneman s 18 stating that the answer is yes if ir is 3 step nilpotent in this paper we generalize this to virtually 3 step nilpotent groups in dimension 3 4 the main idea is to refine the seifert fiber space construction of p conner and f raymond 4 this gives rise to an embedding of such ir into the group of affine motions so that rm ir is compact then we apply t farrell and w c hsiang s theorem 6 stating that any two closed aspherical manifolds with isomorphic fundamental groups which are virtually nilpotent are homeomorphic in dimension 3 4 this paper is organized as follows in section 1 we refine the seifert fiber space construction by restricting the coefficient module to some submodules of m w rk the main result is that if the first coboundary homomorphism 6 of certain cohomology exact sequence is surjective then one can convert the algebraic object group into a geometric one group action manuscript received november 23 1981 manuscript revised may 14 1981 this is a part of the author s ph d thesis at the university of michigan 1981 the second part geometric realization will appear as a separate article elsewhere the author wishes to express his sincere appreciation to professor frank raymond for his guidance in this work

fast numerical solution of the biharmonic dirichlet problem on rectangles a new method for the numerical solution of the first biharmonic dirichlet problem in a rectangular domain is presented for an n times n mesh the complexity of this algorithm is on the order of

detection of edges in spectral data ii nonlinear enhancement we discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities where f x f x f x neq 0 our approach is based on two main aspects localization using appropriate concentration kernels and separation of scales by nonlinear enhancement to detect such edges one employs concentration kernels k epsilon cdot depending on the small scale epsilon it is shown that odd kernels properly scaled and admissible in the sense of having small w 1 infty moments of order cal o epsilon satisfy k epsilon f x f x cal o epsilon thus recovering both the location and amplitudes of all edges as an example we consider general concentration kernels of the form k sigma n t sum sigma k n sin kt to detect edges from the first 1 epsilon n spectral modes of piecewise smooth f s here we improve in generality and simplicity over our previous study in a gelb and e tadmor appl comput harmon anal 7 1999 pp 101 135 both periodic and nonperiodic spectral projections are considered we identify in particular a new family of exponential factors sigma exp cdot with superior localization properties the other aspect of our edge detection involves a nonlinear enhancement procedure which is based on separation of scales between the edges where k epsilon f x sim f x neq 0 and the smooth regions where k epsilon f cal o epsilon sim 0 numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors

spectral tau approximation of the two dimensional stokes problem we analyse the spectral tau method for the approximation of the stokes system on a square with dirichlet boundary conditions we provide an error estimate in the norm of the sobolev spaceh s for the approximation of a divergence free vector field with polynomial divergence free vector fields we apply this result to prove some convergence estimates for the solution of the discrete stokes problem

the real cohomology of virtually nilpotent groups abstract in this paper we present a method to compute the real cohomology of any finitely generated virtually nilpotent group the main ingredient in our setup consists of a polynomial crystallographic action of this group as any finitely generated virtually nilpotent group admits such an action which can be constructed quite easily the approach we present applies to all these groups our main result is an algorithmic way of computing these cohomology spaces as a first application we prove a kind of poincare duality also in the nontorsion free case and we derive explicit formulas in the virtually abelian case

a gpu parallelized spectral method for elliptic equations in rectangular domains we design and implement a polynomial based spectral method on graphic processing units gpus the key to success lies in the seamless integration of the matrix diagonalization technique and the new generation cuda tools the method is applicable to elliptic equations in rectangular domains with general boundary condition we show remarkable speedups of up to 15 times in the 2 d case and more than 35 times in the 3 d case

etale representations for reductive algebraic groups with one dimensional center abstract a complex vector space v is a prehomogeneous g module if g acts rationally on v with a zariski open orbit the module is called etale if dim v dim g we study etale modules for reductive algebraic groups g with one dimensional center for such g even though every etale module is a regular prehomogeneous module its irreducible submodules have to be non regular for these non regular prehomogeneous modules we determine some strong constraints on the ranks of their simple factors this allows us to show that there do not exist etale modules for g gl 1 s s with s simple

on the convergence rate of a modified fourier series the rate of convergence for an orthogonal series that is a minor modification of the fourier series is proved this series converges pointwise at a faster rate than the fourier series for nonperiodic functions we present the error as an asymptotic expansion where the lowest term in this expansion is of asymptotic order two subtracting out the terms from this expansion allows us to increase the order of convergence though the terms of this expansion depend on derivatives alternatively we can employ extrapolation methods which achieve higher convergence rates using only the coefficients of the series we also present a method for the efficient computation of the coefficients in the series

second kind integral equations for the first kind dirichlet problem of the biharmonic equation in three dimensions a fredholm second kind integral equation skie formulation is constructed for the dirichlet problem of the biharmonic equation in three dimensions a fast numerical algorithm is developed based on the constructed skie its performance is illustrated via several numerical examples

quasi filiform lie algebras of maximum length revisited abstract we revisit the problem of classification of quasi filiform lie algebras of maximum length the lists found in the literature are either incomplete or non explicit en route we touch upon the classification of graded filiform lie algebras generated by two elements of degree one and three respectively and as a byproduct obtain a classification of n graded infinite dimensional lie algebras with one dimensional homogeneous components of coclass two

three dimensional affine crystallographic groups those groups r which act properly discontinuously and aillnely on ii with compact fundamental domain are classified first it is shown that such a group f contains a solvable subgroup of finite index thus establishing a conjecture of auslander in dimension three then unimodular simply transitive altine actions on ir are classified this leads to a classification of attine crystallographic groups acting on ir3 a characterization of which abstract groups admit such an action is given moreover it is proved that every isomorphism between virtually solvable atline crystallographic groups respectively simply transitive afline groups is induced by conjugation by a polynomial automorphism of the affrne space a characterization is given of which closed 3 manifolds can be represented as quotients of ir by groups of afftne transformations a closed 3 manifold m admits a complete atline structure if and only if m has a finite covering homeomorphic or homotopy equivaient to a 2 torus bundle over the circle

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

lie admissible algebras and operads abstract a lie admissible algebra gives a lie algebra by anticommutativity in this work we describe remarkable types of lie admissible algebras such as vinberg algebras pre lie algebras or lie algebras we compute the corresponding binary quadratic operads and study their duality considering lie algebras as lie admissible algebras we can define for each lie algebra a cohomology with values in an lie admissible module this leads to the study some deformations of lie algebras in the classes of lie admissible algebras

novikov algebras and novikov structures on lie algebras we study ideals of novikov algebras and novikov structures on finite dimensional lie algebras we present the first example of a three step nilpotent lie algebra which does not admit a novikov structure on the other hand we show that any free three step nilpotent lie algebra admits a novikov structure we study the existence question also for lie algebras of triangular matrices finally we show that there are families of lie algebras of arbitrary high solvability class which admit novikov structures

multipole expansions and pseudospectral cardinal functions a new generalization of the fast fourier transform the polynomial or trigonometric interpolant of an arbitrary function f x may be presented as a critical function series whose coefficients are the values of f x at the interpolation points we show that the cardinal series is identical to the sum of the forces due to a set of n point charges with appropriate force laws it follows that the cardinal series can be summed via the fast multipole method fmm in o nlog sub 2 n operations which is much cheaper than the o n sup 2 cost of direct summation the fmm is slower than the fast fourier transform fft so the latter should always be used where applicable however the multipole expansion succeeds where the fft fails in particular the fmm can be used to evaluate fourier and chebyshev series on an irregular grid as is needed when adaptively regridding in a time integration also the multipole expansion can be applied to basis sets for which the fft is inapplicable even on the canonical grid including legendre polynomials hermite and laguerre functions spherical harmonics and zinc functions 12 refs

convergence acceleration of modified fourier series in one or more dimensions modified fourier series have recently been introduced as an adjustment of classical fourier series for the approximation of nonperiodic functions defined on d variate cubes such approximations offer a number of advantages including uniform convergence however like fourier series the rate of convergence is typically slow in this paper we extend eckhoff s method to the convergence acceleration of multivariate modified fourier series by suitable augmentation of the approximation basis we demonstrate how to increase the convergence rate to an arbitrary algebraic order moreover we illustrate how numerical stability of the method can be improved by utilising appropriate auxiliary functions in the univariate setting it is known that eckhoff s method exhibits an auto correction phenomenon we extend this result to the multivariate case finally we demonstrate how a significant reduction in the number of approximation coefficients can be achieved by using a hyperbolic cross index set

polynomiality properties of group extensions with a torsion free abelian kernel abstract in studying torsion free nilpotent groups of class 2 it is a key fact that each central group extension z n e z mis representable by abilinearcocycle so for investigating nilpotent groups of higher class it is natural to ask for a generalization of this fact namely when z mis replaced by sometorsion free nilpotentgroupgand z nby sometorsion free abeliangroupbwithnilpotent g action in this paper we study a suitable notion of bi polynomialcocycles in the strong sense of polynomiality introduced by passi and prove the desiredrepresentability theorem 4 3 this was known before only forcentralextensions withdivisiblekernel and with kernel z ifgis abelian nilpotent of class 2 or if the quotients of the lower central series ofgare torsion free our representability result implies aconvergence theoremfor an approximation ofh2 g b bypolynomial cohomology groups 4 4 since the latter ones are well accessible to computation one obtains a formula forh2 g b in terms of a presentation ofgwhich can be evaluated by integral matrix calculus more precisely a given presentation ofgamounts to a three term cochain complex consisting offinitely generated free z modulesif the groupsgandbare finitely generated the cohomology of this complex is identified withh2 g b in such a way that representing 2 cocycles are explicitly given in terms of integer valued rational polynomial functions 6 4 as an application we establish an explicitbijectionbetween torsion free nilpotentgroupsand z torsion free nilpotentlie ringsboth finitely generated and nilpotent ofclass 3 7 2 in case a given group extension is representable by a polynomial cocycle we also determine theminimal degreeof polynomiality for which this holds indeed dropping the assumption thatgis torsion free nilpotent we give anintrinsic characterizationof all group extensions with torsion free abelian kernel which are representable by a polynomial cocycle of degree n provided that the cokernel is finitely generated and acts nilpotently on the kernel 4 2 thus a polynomiality theory for group extensions as asked for by passi is now achieved in case the kernel istorsion free a motivation for this apart from calculatingh2 g b explicitly comes from a question posed by j milnor in 1977 namely whether all finitely generated torsion free virtually polycyclic groups arise as fundamental groups of compact complete affinely flat manifolds even the case of torsion free nilpotent groups is interesting but far from being understood notably since counterexamples of this type have recently been discovered a connection of milnor s question for nilpotent groups with polynomial constructions was first indicated in the work of p igodt and k b lee recently a very close connection of this type was established and anobstruction theorydeveloped for the problem in terms of polynomial cohomology on the other hand polynomiality properties of extensions with atorsionkernel are closely related todimension subgroups as was observed by passi in the case ofcentralextensions an extension of this idea to thenoncentralcase is provided by theorem 3 4 below this might be of interest in connection with the recent discovery of gupta and kuz min that the quotient groupdn g n g is an abelian normal but in general noncentral subgroup ofg n g moreover we obtain afunctorial equivalencebetween extensions oftorsion free nilpotent groupsand of z torsion free nilpotent modules 5 1 this improves a general result of reiner and roggenkamp in the special situation of torsion free nilpotent groups as applications it yields an inductive cohomological description ofautomorphism groupsof torsion free nilpotent groups 9 and a generalization of the classicaldold kan equivalence between simplicial abelian groups and chain complexes of abelian groups to simplicial groups of class 2 applications in localization theory are also to be expected namely to questions concerning the genus of torsion free nilpotent groups all the above mentioned results are based on the more technical work of the first section there also generalizations of theorems of witt and quillen are obtained concerning certain lie algebras associated with groups these results might be of independent interest

on discontinuous solutions of hyperbolic equations abstract a modified fourier method which is applicable to numerical simulations of discontinuous solutions of hyperbolic equations is presented truncated fourier series representations are de truncated at every timestep by utilising step functions in the reconstruction of the discontinuous solutions the de truncated expansions are used to eliminate the numerical dispersion and diffusion which normally ruin the discontinuous solutions in a few timesteps when traditional fourier methods are applied

acceleration of algebraically converging fourier series when the coefficients have series in powers of 1 n if the coefficients in a fourier cosine series f x f email protected n 0 a ncos nx decrease as a small negative power of n then one may need millions of terms to sum the series to high accuracy we show that if the a n are known analytically and have a power series in 1 n then it is straightforward to approximate f x as a series of what we shall the lanczos krylov lk functions we describe the similar methodology for sine series general fourier series are merely the sum of a cosine series with a sine series and thus are implicitly handled too for cosine coefficients that involve only even powers of n and sine coefficients that are functions of odd powers of n the lk functions may be expressed in terms of bernoulli polynomials the lk functions for cosine coefficients involving odd powers of n and for sine coefficients in even powers of n are not known explicitly these are also known as clausen functions we provide rapidly convergent series to compute these clausen functions to high accuracy our method includes the endpoint subtraction ideas of lanczos and krylov but is more general the sum n 1 1 n 1 1 email protected cos nx where l 0 is a constant arises in phase transitions in absorbed monolayers on metal surfaces it is easily summed by our method which correctly incorporates the logarithmic singularities at x email protected

some results about the pseudospectral approximation of one dimensional fourth order problems we analyze the pseudospectral approximation of fourth order problems we give convergence results in the one dimensional case numerical experiments are shown in two dimensions for the approximation of the rhombic plate bending problem eigenvalues and preconditioning are also investigated

detecting discontinuity points from spectral data with the quotient difference qd algorithm this paper introduces a new technique for the localization of discontinuity points from spectral data through this work we will be able to detect discontinuity points of a 2 p periodic piecewise smooth function from its fourier coefficients this could be useful in detecting edges and reducing the effects of the gibbs phenomenon which appears near discontinuities and affects signal restitution our approach consists in moving from a discontinuity point detection problem to a pole detection problem then adapting the quotient difference qd algorithm in order to detect those discontinuity points

accurate reconstructions of functions of finite regularity from truncated fourier series expansions kowledge of a truncated fourier series expansion for a 2 periodic function of finite regularity which is assumed to be piecewise smooth in each period is used to accurately reconstruct the corresponding function an algebraic equation of degree m is constructed for the m singularity locations in each period for the function in question the m coefficients in this algebraic equation are obtained by solving an algebraic system of m equations determined by the coefficients in the known truncated expansion if discontinuities in the derivatives of the function are considered in addition to discontinuities in the function itself that algebraic system will be nonlinear with respect to the m unknown coefficients the degree of the algebraic system will depend on the desired order of accuracy for the reconstruction i e a higher degree will normally lead to a more accurate determination of the singularity locations by solving an additional linear algebraic system for the jumps of the function and its derivatives up to the arbitrarily specified order at the calculated singularity locations we are able to reconstruct the 2 periodic function of finite regularity as the sum of a piecewise polynomial function and a function which is continuously differentiab1e up to the specified order

error estimates of spectral legendre galerkin methods for the fourth order equation in one dimension we employ spectral legendre galerkin and mixed legendre galerkin approximations to solve the first bi harmonic equation in one dimension respectively by orthogonal properties of legendre polynomials we obtain an explicit a posteriori error indicator for spectral legendre galerkin methods furthermore in virtue of an auxiliary variable we present spectral mixed legendre galerkin methods and study the a priori estimate and a posteriori error indicator especially these indicators only depend on the expansions of the right hand item numerical examples are presented to verify our theoretical analysis

a stability barrier for reconstructions from fourier samples we prove that any stable method for resolving the gibbs phenomenon that is recovering high order accuracy from the first m fourier coefficients of an analytic and nonperiodic function can converge at best root exponentially fast in m any method with faster convergence must also be unstable and in particular exponential convergence implies exponential ill conditioning this result is analogous to a recent theorem of platte trefethen and kuijlaars concerning recovery from pointwise function values on an equispaced m grid the main step in our proof is an estimate for the maximal behavior of a polynomial of degree n with bounded m term fourier series this estimate is related to a conjecture of hrycak and grochenig in the final part of the paper we discuss the implications of our main theorem to polynomial based interpolation and least squares approaches for overcoming the gibbs phenomenon

on the resolution power of fourier extensions for oscillatory functions functions that are smooth but non periodic on a certain interval possess fourier series that lack uniform convergence and suffer from the gibbs phenomenon however they can be represented accurately by a fourier series that is periodic on a larger interval this is commonly called a fourier extension when constructed in a particular manner fourier extensions share many of the same features of a standard fourier series in particular one can compute fourier extensions which converge spectrally fast whenever the function is smooth and geometrically fast if the function is analytic much the same as the fourier series of a smooth analytic and periodic function with this in mind the purpose of this paper is to describe analyse and explain the observation that fourier extensions much like classical fourier series also have excellent resolution properties for representing oscillatory functions the resolution power or required number of degrees of freedom per wavelength depends on a user controlled parameter and as we show it varies between 2 and p the former value is optimal and is achieved by classical fourier series for periodic functions for example the latter value is the resolution power of algebraic polynomial approximations thus fourier extensions with an appropriate choice of parameter are eminently suitable for problems with moderate to high degrees of oscillation

uniform approximation of piecewise r smooth and globally continuous functions we study the uniform chebyshev approximation of continuous and piecewise r smooth r ge2 functions f 0 t to mathbb r with a finite number of singular points the approximation algorithms use only n function values at adaptively or nonadaptively chosen points we construct a nonadaptive algorithm mathcal a r n rm non that for the functions with at most one singular point enjoys the best possible convergence rate n r this is in sharp contrast to results concerning discontinuous functions for r ge3 this optimal rate of convergence holds only in the asymptotic sense i e it occurs only for sufficiently large n that depends on f in a way that is practically impossible to verify however it is enough to modify mathcal a r n rm non by using r 1 lfloor r 1 2 rfloor extra 1pt function evaluations to obtain an adaptive algorithm mathcal a r n rm ada with error satisfying f mathcal a r n rm ada f c le c rt r f r l infty n r for all n ge n 0 and n 0 independent of f this result cannot be achieved for functions with more than just one singular point however the convergence rate n r can be recovered asymptotically by a nonadaptive algorithm overline mathcal a r n rm non that is a slightly modified mathcal a r n rm non specifically limsup n to infty f overline mathcal a r n rm non f c cdot n r le c rt r f r l infty for all r smooth functions f with finitely many singular points

polynomial structures on polycyclic groups we know by recent work of benoist and of burde grunewald that there exist polycyclic by finite groups g of rank h the examples given were in fact nilpotent admitting no properly discontinuous affine action on rh on the other hand for such g it is always possible to construct a properly discontinuous smooth action of g on rh our main result is that any polycyclic by finite group g of rank h contains a subgroup of finite index acting properly discontinuously and by polynomial diffeomorphisms of bounded degree on rh moreover these polynomial representations always appear to contain pure translations and are extendable to a smooth action of the whole group g

the lifting problem for affine structures in nilpotent lie groups affine manifolds occur in several situations in pure and applied mathematics e g leaves of lagrangian foliations completely integrable hamiltonian systems linear representations of virtually polycyclic groups geometric quantization and so on this work is devoted to left invariant affinely flat structures in lie groups we are mainly concerned with the following situation let g and go be nilpotent lie groups of dimension n 1 and n respectively and let h g p go be a continuous homomorphism from g onto go given a left invariant affinely flat structure go vo the lifting problem is to discover whether g has a left invariant affinely flat structure g v such that h becomes an affine morphism in the present work we answer positively when go vo is normal therefore the existence problem for a left invariant complete affinely flat structure in nilpotent lie groups is solved by applying the following subsequent results let tf go be the set of left invariant affinely flat structures in the nilpotent lie group go 10 tf go 0 implies the existence of normal structure go vo e tf go 20 h g go being as above every normal structure go v0 has a normal lifted in atf g

on the evaluation of highly oscillatory integrals by analytic continuation we consider the integration of one dimensional highly oscillatory functions based on analytic continuation rapidly converging quadrature rules are derived for a general class of oscillatory integrals with an analytic integrand the accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency and for the case of an increasing number of points and fixed frequency these results are then used to obtain quadrature rules for more general oscillatory integrals i e for functions that exhibit some smoothness but that are not analytic the approach described in this paper is related to the steepest descent method but it does not employ asymptotic expansions it can be used for small or moderate frequencies as well as for very high frequencies the approach is compared with the oscillatory integration techniques recently developed by iserles and norsett

efficient energy stable numerical schemes for a phase field moving contact line model in this paper we present two efficient energy stable schemes to solve a phase field model incorporating moving contact line the model is a coupled system that consists of incompressible navier stokes equations with a generalized navier boundary condition and cahn hilliard equation in conserved form in both schemes the projection method is used to deal with the navier stokes equations and stabilization approach is used for the non convex ginzburg landau bulk potential by some subtle explicit implicit treatments we obtain a linear coupled energy stable scheme for systems with dynamic contact line conditions and a linear decoupled energy stable scheme for systems with static contact line conditions an efficient spectral galerkin spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes numerical results show that the proposed schemes are very efficient and accurate we proposed two linear energy stable numerical schemes for a phase field moving contact line model discrete energy laws that are consistent to the continuous ones are proved the efficiency and fast convergence of the proposed schemes with spectral spatial approximations are numerically verified

how well does the hermite pade approximation smooth the gibbs phenomenon in order to reduce the gibbs phenomenon exhibited by the partial fourier sums of a periodic function f defined on discontinuous at 0 driscoll and fornberg considered so called singular fourier pade approximants constructed from the hermite pade approximants of the system of functions 1 g1 z g2 z where g1 z log 1 z and g2 z is analytic such that re g2 e it f t convincing numerical experiments have been obtained by these authors but no error estimates have been proven so far in the present paper we study the special case of nikishin systems and their hermite pade approximants both theoretically and numerically we obtain rates of convergence by using orthogonality properties of the functions involved along with results from logarithmic potential theory in particular we ad dress the question of how to choose the degrees of the approximants by considering diagonal and row sequences as well as linear hermite pade approximants our theoretical findings and numerical experiments confirm that these hermite pade approximants are more efficient than the more elementary pade approximants par ticularly around the discontinuity of the goal function f

left invariant affine structures on reductive lie groups we describe left invariant affine structures that is left invariant flat torsion free affine connections on reductive linear lie groupsg they correspond bijectively to lsa structures on the lie algebra g ofg here lsa stands for left symmetric algebra if g has trivial or one dimensional center z then the affine representation 1 of g induced by any lsa structure g on g isradiant i e the radiance obstructionc h1 g g vanishes if dim z 1 we prove that g s z where s is split simple admits lsa structures if and only if s is of typeal that is g gln here we have the associative lsa structure given by ordinary matrix multiplication corresponding to the bi invariant affine structure on gl n which was believed to be essentially the only possible lsa structure on gln we exhibit interesting lsa structures different from the associative one they arise as certain deformations of the matrix algebra then we classify all lsa structures on glnusing a result of baues forn 2 we compute all structures explicitly over the complex numbers

on the uniqueness of some affinely flat infra nilmanifolds i p irn irk u p irn irk since pi irn irk pi irn ir k each element of i 1 ir irk is completely determined by k polynomials of total degree at most i in n variables let g be a group and suppose we have a morphism 4 g g l k ir and a homomorphism p g aff ir we can provide a g module structure for p ir f k in the following way for g e g and pepi ir irk we define

reduction of the gibbs phenomenon for smooth functions with jumps by the e algorithm recently brezinski has proposed to use wynn s e algorithm in order to reduce the gibbs phenomenon for partial fourier sums of smooth functions with jumps by displaying very convincing numerical experiments in the present paper we derive analytic estimates for the error corresponding to a particular class of hypergeometric functions and obtain the rate of column convergence for such functions possibly perturbed by another sufficiently differentiable function we also analyze the connection to pade fourier and pade chebyshev approximants including those recently studied by kaber and maday

efficient quadrature of highly oscillatory integrals using derivatives in this paper we explore quadrature methods for highly oscillatory integrals generalizing the method of stationary phase we expand such integrals into asymptotic series in inverse powers of the frequency the outcome is two families of methods one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of filon filon 1928 proc r soc edinb 49 38 47 1 both kinds of methods approximate the integral as a linear combination of function values and derivatives with coefficients that may depend on frequency we determine asymptotic properties of these methods proving perhaps counterintuitively that their performance drastically improves as frequency grows the paper is accompanied by numerical results that demonstrate the potential of this set of ideas 1 ref 4

efficient spectral galerkin method ii direct solvers of second and fourth order equations using chebyshev polynomials efficient direct solvers based on the chebyshev galerkin methods for second and fourth order equations are presented they are based on appropriate base functions for the galerkin formulation that lead to discrete systems with special structured matrices that can be efficiently inverted numerical results indicate that the direct solvers presented in this paper are significantly more accurate and efficient than those based on the chebyshev tau method

some studies on nonpolynomial interpolation and error analysis abstract in this paper we propose nonpolynomial and hermite nonpolynomial interpolation with multiple parameters and present method to determine optimal value of parameters which generate minimum error in approximation the generalized error analysis results for nonpolynomial and hermite nonpolynomial interpolations are derived we establish theoretical relationship among nonpolynomial polynomial interpolation and the fourier series and propose solution to runge s phenomenon through nonpolynomial interpolation the hermite nonpolynomial cubic spline nonpolynomial cubic spline interpolation methods and their error analysis are presented numerical simulations are carried out for the analysis of error in cubic spline interpolations proposed method is applied to the analysis of various time series to show comparison in errors between polynomial and nonpolynomial spline interpolations and to empirical mode decomposition emd to illustrate practical usefulness of the results

left symmetric algebras for mathfrak gl n we study the classification problem for left symmetric algebras with commutation lie algebra gr n in characteristic 0 the problem is equivalent to the classification of etale affine representations of g n algebraic invariant theory is used to characterize those modules for the algebraic group sl n which belong to affine etale representations of gr n from the classification of these modules we obtain the solution of the classification problem for gl n as another application of our approach we exhibit left symmetric algebra structures on certain reductive lie algebras with a one dimensional center and a non simple semisimple ideal

counterexample to a conjecture about braces abstract we find an example of a finite solvable group in fact a finite p group in which is not possible to define a left brace structure or equivalently which is not an iyb group this answers a question posed by cedo jespers and del rio related to the yang baxter equation our argument is an improvement of an argument of rump using results about hopf galois extensions and lsa structures we explain explicitly the relation between these two areas of mathematics and brace theory hoping that it will be useful in the future

a stable high order method for two dimensional bounded obstacle scattering a stable and high order method for solving the helmholtz equation on a two dimensional domain exterior to a bounded obstacle is developed in this paper the method is based on a boundary perturbation technique transformed field expansions coupled with a well conditioned high order spectral galerkin solver the method is further enhanced with numerical analytic continuation implemented via pade approximation ample numerical results are presented to show the accuracy stability and versatility of the proposed method

approximation of the discontinuities of a function by its classical orthogonal polynomial fourier coefficients in the present paper we generalize the method suggested in an earlier paper by the author and overcome its main deficiency first we modify the well known prony method which subsequently will be utilized for recovering exactly the locations of jump discontinuities and the associated jumps of a piecewise constant function by means of its fourier coefficients with respect to any system of the classical orthogonal polynomials next we will show that the method is applicable to a wider class of functions namely to the class of piecewise smooth functions for functions which piecewise belong to c 2 1 1 the locations of discontinuities are approximated to within o 1 n by means of their fourier jacobi coefficients unlike the previous one the generalized method is robust since its success is independent of whether or not a location of the discontinuity coincides with a root of a classical orthogonal polynomial in addition the error estimate is uniform for any c d c 1 1 to the end we discuss the accuracy stability and complexity of the method and present numerical examples

adi spectral collocation methods for parabolic problems we discuss the crank nicolson and laplace modified alternating direction implicit legendre and chebyshev spectral collocation methods for a linear variable coefficient parabolic initial boundary value problem on a rectangular domain with the solution subject to non zero dirichlet boundary conditions the discretization of the problems by the above methods yields matrices which possess banded structures this along with the use of fast fourier transforms makes the cost of one step of each of the chebyshev spectral collocation methods proportional except for a logarithmic term to the number of the unknowns we present the convergence analysis for the legendre spectral collocation methods in the special case of the heat equation using numerical tests we demonstrate the second order accuracy in time of the chebyshev spectral collocation methods for general linear variable coefficient parabolic problems

trouble with gegenbauer reconstruction for defeating gibbs phenomenon runge phenomenon in the diagonal limit of gegenbauer polynomial approximations to defeat gibbs phenomenon in fourier and chebyshev series gottlieb et al d gottlieb c w shu a solomonoff h vandeven on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function j comput appl math 43 1992 81 98 developed a gegenbauer reconstruction the partial sums of the fourier or other spectral series are reexpanded as a series of gegenbauer polynomials c n m x recovering spectral accuracy even in the presence of shock waves or other discontinuities to achieve a rate of convergence which is exponential in n however gegenbauer reconstruction requires increasing the order m of the polynomials linearly with the truncation n of the series m bn for some constant b 0 when the order m is fixed it is well known that the gegenbauer series converges as n everywhere on x 1 1 if f x the function being expanded is analytic on the interval but what happens in the diagonal limit in which m n tend to infinity simultaneously we show that singularities of f x off the real axis can destroy convergence of this diagonal approximation process in the sense that the error diverges for subintervals of x 1 1 gegenbauer reconstruction must therefore be constrained to use a sufficiently small ratio of order m to truncation n this off axis singularity constraint is likely to impair the effectiveness of the reconstruction in some applications

the auslander conjecture for nil affine crystallographic groups let n be a simply connected connected real nilpotent lie group of finite dimension n we study subgroups in aff n n aut n acting properly discontinuously and cocompactly on n this situation is a natural generalization of the so called affine crystallographic groups we prove that for all dimensions 1 n 5 the generalized auslander conjecture holds i e that such subgroups are virtually polycyclic

polynomial alternatives for the group of affine motions it is known that every polycyclic by finite group even if it admits no affine structure allows a polynomial structure of bounded degree a major obstacle to a further development of the theory of these polynomial structures is that the group of the polynomial diffeomorphisms of mathbb r n in contrast to the group of affine motions is no longer a finite dimensional lie group in this paper we construct a family of finite dimensional lie groups even linear algebraic groups of polynomial diffeomorphisms which we call weighted groups of polynomial diffeomorphisms it turns out that every polycyclic by finite group admits a polynomial structure via these weighted groups in the nilpotent and other case s we can sharpen by specifying a nice set of weights the existence results obtained in earlier work we introduce unipotent polynomial structures of nilpotent groups and show how the existence of such polynomial structures is closely related to the existence of simply transitive actions of the corresponding mal cev completion this and other properties provide a strong analogy with the situation of affine structures and simply transitive affine actions considered e g in the work of fried goldman and hirsch

multivariate modified fourier series and application to boundary value problems in this paper we analyse the approximation theoretic properties of modified fourier series in cartesian product domains with coefficients from both full and hyperbolic cross index sets we show that the number of expansion coefficients may be reduced significantly whilst retaining comparable error estimates in doing so we extend the univariate results of iserles norsett and s olver we then demonstrate that these series can be used in the spectral galerkin approximation of second order neumann boundary value problems which offers some advantages over standard chebyshev or legendre polynomial discretizations

spectral elements on infinite domain a spectral element technique to approximate partial differential equations pdes on an infinite domain is examined the method is based on boyd s mapping of a semiinfinite interval to a finite interval and it is extended to a variational setting which allows for an implementation using a spectral element method by extending the method to a variational form a straightforward implementation allows for high order approximations over an infinite computational domain methods to find a high order approximation have been proposed by funaro maday mavriplis and many others the proposed methods all have drawbacks including restrictions on the behavior of the approximation at infinity because the proposed method is based on a mapping to a finite domain a much wider variety of boundary conditions can be enforced on the approximation

space time spectral method for the two dimensional generalized sine gordon equation abstract in this paper we propose a space time spectral method for solving the generalized two dimensional sine gordon equation with nonhomogeneous dirichlet boundary conditions and initial conditions the proposed method is based on the legendre galerkin spectral method in space and the spectral collocation method or block spectral collocation method in time we derive a priori error estimates in both l 2 and h 1 norms for the semidiscrete formulation our formulation has spectral accuracy in both space and time our numerical results confirm the exponential convergence of the proposed method in both space and time

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

recovery of the three dimensional shape of an object from a single view given a single picture which is a projection of a three dimensional scene onto the two dimensional picture plane we usually have definite ideas about the 3 d shapes of objects to do this we need to use assumptions about the world and the image formation process since there exist a large number of shapes which can produce the same picture the purpose of this paper is to identify some of these assumptions mostly geometrical ones by demonstrating how the theory and techniques which exploit such assumptions can provide a systematic shape recovery method the method consists of two parts the first is the application of the origami theory which models the world as a collection of plane surfaces and recovers the possible shapes qualitatively the second is the technique of mapping image regularities into shape constraints for recovering the probable shapes quantitatively actual shape recovery from a single view is demonstrated for the scenes of an object such as a box and a chair given a single image the method recovers the 3 d shapes of an object in it and generates images of the same object as we would see it from other directions

chebyshev rational spectral and pseudospectral methods on a semi infinite interval a weighted orthogonal system on the half line based on the chebyshev rational functions is introduced basic results on chebyshev rational approximations of several orthogonal projections and interpolations are established to illustrate the potential of the chebyshev rational spectral method a model problem is considered both theoretically and numerically error estimates for the chebyshev rational spectral and pseudospectral methods are established preliminary numerical results agree well with the theoretical estimates and demonstrate the effectiveness of this approach

interpretation of straight line correspondences using angular relations abstract formal analysis of the structure from motion problem has produced computational schemes that are often unstable and sensitive to image measurement errors this is in part due to the fact that the formal analysis is mainly concerned as it should be with the most general cases treating special cases with marginal interest however special cases abound in man made environments their occurrence if ascertained simplifies drastically the task of 3d interpretation after examining the problem of interpreting general configurations of lines in space this paper proposes a number of simple and efficient computational units that are tuned to the occurrence of special line configurations the process of propagation which allows interpretation to spread over the image is also advocated such computational units and processes can be organized in a rule based framework where the occurrence of special configurations can either be tested directly or hypothesized and used until an inconsistency is demonstrated

optimal filter and mollifier for piecewise smooth spectral data we discuss the reconstruction of piecewise smooth data from its pseudo spectral information spectral projections enjoy superior resolution provided the function is globally smooth while the presence of jump discontinuities is responsible for spurious o 1 gibbs oscillations in the neighborhood of edges and an overall deterioration of the convergence rate to the unacceptable first order classical filters and mollifiers are constructed to have compact support in the fourier frequency and physical time spaces respectively and are dilated by the projection order or the width of the smooth region to maintain this compact support in the appropriate region here we construct a noncompactly supported filter and mollifier with optimal joint time frequency localization for a given number of vanishing moments resulting in a new fundamental dilation relationship that adaptively links the time and frequency domains not giving preference to either space allows for a more balanced error decomposition which when minimized yields an optimal filter and mollifier that retain the robustness of classical filters yet obtain true exponential accuracy