semigroup algebras and noetherian maximal orders abstract in this paper we describe when a monoid algebra k s is a noetherian pi domain which is a maximal order our work relies on the study of the height one primes of k s and of the minimal primes of the monoid s and leads to a characterization purely in terms of s it turns out that the primes p intersecting s plays a crucial role and therefore we reduce the problem to certain local monoids s p that is monoids with only one minimal prime however we illustrate by examples that such monoids and their algebras are much more complicated than the discrete valuation rings arising in the commutative case our work is based on the results of brown concerning the group ring case and of anderson and chouinard on commutative monoid rings we also rely on the description of noetherian monoid algebras obtained earlier by the authors

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

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

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

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

quadratic algebras of skew type and the underlying monoids abstract we consider algebras over a field k defined by a presentation k x1 xn r where r consists of n 2 square free relations of the form xixj xkxl with every monomial xixj i j appearing in one of the relations certain sufficient conditions for the algebra to be noetherian and pi are determined for this we prove more generally that right noetherian algebras of finite gelfand kirillov dimension defined by homogeneous semigroup relations satisfy a polynomial identity the structure of the underlying monoid defined by the same presentation is described this is used to derive information on the prime radical and minimal prime ideals some examples are described in detail earlier gateva ivanova and van den bergh and jespers and okninski considered special classes of such algebras in the contexts of noetherian algebras grobner bases finitely generated solvable groups semigroup algebras and set theoretic solutions of the yang baxter equation

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

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

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

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

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

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

set theoretic solutions of the yang baxter equation rc calculus and garside germs abstract building on a result by w rump we show how to exploit the right cyclic law x y x z y x y z in order to investigate the structure groups and monoids attached with involutive nondegenerate set theoretic solutions of the yang baxter equation we develop a sort of right cyclic calculus and use it to obtain short proofs for the existence both of the garside structure and of the i structure of such groups we describe finite quotients that play for the considered groups the role that coxeter groups play for artin tits groups

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

semigroups ofi type abstract assume thatsis a semigroup generated by x1 xn and let u be the multiplicative free commutative semigroup generated by u1 un we say thatsis ofi typeif there is a bijectivev u ssuch that for alla u v u1a v una x1v a xnv a this condition appeared naturally in the work on sklyanin algebras by john tate and the second author in this paper we show that the condition for a semigroup to be ofi type is related to various other mathematical notions found in the literature in particular we show that semigroups ofi type appear in the study of the set theoretic solutions of the yang baxter equation in the theory of bieberbach groups and in the study of certain skew binomial polynomial rings which were introduced by the first author

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

height one prime ideals in semigroup algebras satisfying a polynomial identity abstract for a submonoid s of a torsion free abelian by finite group we describe the height one prime ideals of the semigroup algebra k s as an application we investigate when such algebras are unique factorization rings

monoids of ig type and maximal orders let g be a finite group that acts on an abelian monoid a i f a g is a map so that a a b a b f or alla b a then the submonoid s a a a a of the associated semidirect product ag is said to be a monoid of ig type if a is a finitely generated free abelian monoid of rank n and g is a subgroup of the symmetric group symn of degree n then these monoids first appeared in the work of gateva ivanova and van den bergh they are called monoids of i type and later in the work of jespers and okni nski it turns out that their associated semigroup algebras share many properties with polynomial algebras in finitely many commuting variables in this paper we first note that finitely generated monoids s of ig type are epimorphic images of monoids of i type and their algebras k s are noetherian and satisfy a polynomial identity in case the group of fractions ss 1 of s is torsion free abelian then it is characterized when k s also is a maximal order it turns out that they often are and hence these algebras again share arithmetical properties with natural classes of commutative algebras the characterization is in terms of prime ideals of s in particular g orbits of minimal prime ideals in a play a crucial role hence we first describe the prime ideals of s i t also is described when the group ss 1 is torsion free

monocular 3d tracking of deformable surfaces using sequential second order cone programming reconstructing structures of deformable objects from monocular image sequences is important for applications like visual servoing and augmented reality in this paper we propose a method to recover 3d shapes of deformable surfaces using sequential second order cone programming socp the key of our approach is to represent the surface as a triangulated mesh and introduce two sets of constraints one for model to image keypoint correspondences which are socp constraints another for retaining the original lengths of the mesh edges which are non convex constraints in the process of tracking the surface structure is iteratively updated by solving sequential socp feasibility problems in which the non convex constraints are replaced by a set of convex constraints over a local convex region the shape constraints used in our approach is more generic than previous methods that enables us to reliably recover surface shapes with smooth sharp and other complex deformations the capability and efficiency of our approach are evaluated quantitatively with synthetic image sequences and qualitatively with real image sequences

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