a fast adaptive diffusion wavelet method for burger s equation abstract a fast adaptive diffusion wavelet method is developed for solving the burger s equation the diffusion wavelet is developed in 2006 coifman and maggioni 2006 and its most important feature is that it can be constructed on any kind of manifold classes of operators which can be used for construction of the diffusion wavelet include second order finite difference differentiation matrices the efficiency of the method is that the same operator is used for the construction of the diffusion wavelet as well as for the discretization of the differential operator involved in the burger s equation the diffusion wavelet is used for the construction of an adaptive grid as well as for the fast computation of the dyadic powers of the finite difference matrices involved in the numerical solution of burger s equation in this paper we have considered one dimensional and two dimensional burger s equation with dirichlet and periodic boundary conditions for each test problem the cpu time taken by fast adaptive diffusion wavelet method is compared with the cpu time taken by finite difference method and observed that the proposed method takes lesser cpu time we have also verified the convergence of the given method

an adaptive wavelet collocation method for the solution of partial differential equations on the sphere a dynamic adaptive numerical method for solving partial differential equations on the sphere is developed the method is based on second generation spherical wavelets on almost uniform nested spherical triangular grids and is an extension of the adaptive wavelet collocation method to curved manifolds wavelet decomposition is used for grid adaption and interpolation an o n hierarchical finite difference scheme based on the wavelet multilevel decomposition is used to approximate laplace beltrami jacobian and flux divergence operators the accuracy and efficiency of the method is demonstrated using linear and nonlinear examples relevant to geophysical flows although the present paper considers only the sphere the strength of this new method is that it can be extended easily to other curved manifolds by considering appropriate coarse approximations to the desired manifold here we used the icosahedral approximation to the sphere at the coarsest level

spectra of modules abstract this manuscript solves the problem that the so called stable category mod of an artin algebra is in general not triangulated the method is to mimick topology and hence first form the spanier whitehead category stab mod and then construct a category spectra of spectra of modules which completes the compact part of stab mod under small coproducts spectra is then a triangulated substitute for mod the main results are that spectra is a compactly generated triangulated category which contains the compact part of stab mod as a full subcategory and even admits a precise description of its compact objects which only form a small set of isomorphism classes as an application it is proved that over an artin algebra the gorenstein projective modules form a pre covering class this was previously only known for rings satisfying strong homological conditions

on the category of modules of gorenstein dimension zero ii abstract let r be a commutative noetherian henselian non gorenstein local ring the author has conjectured in r takahashi on the category of modules of gorenstein dimension zero preprint 2003 that there exist infinitely many isomorphism classes of indecomposable r modules of gorenstein dimension zero if there exists a non free module of gorenstein dimension zero and has proved that the conjecture holds when r has depth zero in this paper we prove that the conjecture also holds when r has depth one

a comparison of differential quadrature methods for the solution of partial differential equations abstract discrete ordinate methods for the solution of partial differential equations are examined and compared a novel approach for the discrete representation of differential operators based on split range polynomial expansions is introduced the utility of the method is demonstrated for the case of differentiation of functions involving steep gradients the solution of burgers equation is presented to illustrate the effectiveness of the technique for the solution of nonlinear partial differential equations exhibiting nearly discontinuous solutions comparisons are made with other differential quadrature methods

space time spectral element methods for one dimensional nonlinear advection diffusion problems abstract the following space time galerkin spectral element methods are developed and applied to solve the burgers equation with small viscosity a coupled methods consisting of an explicit method for hyperbolic dominated equations and an implicit method for parabolic dominated equations b two splitting methods which solve the hyperbolic substep explicitly and the parabolic one implicitly one uses spectral elements in the explicit part and the other uses the adams bashforth multistep method a subcycling technique in which several convective steps are taken for each implicit viscous step was also investigated for the two splitting methods a stability analysis of the four methods is performed and subsequent results are debated a convergence study and a comparison of computer execution time for the four methods is made and the results are discussed comparative study leads to the conclusion that the space time spectral element splitting method with subcycling is superior to the other methods presented in terms of robustness and computer execution time the number of subcycles should be kept low 2 3 in order to avoid significant loss of accuracy the coupled explicit method is also applied to the solution of the one dimensional coupled continuity momentum and energy equations for non isothermal flow of an ideal gas with temperature dependent properties in a cylindrical duct of variable radius

hermite sobolev and closely connected orthogonal polynomials abstract in this paper a bilinear functional a g m o n m c m m g m which corresponds to the hermite sobolev and closely connected orthogonal polynomials is studied the domain of m m 1 n in which a is positive definite is given and the hermite sobolev and closely connected formal orthogonal polynomials are given in the nonnormal case

semi dualizing complexes and their auslander categories let r be a commutative noetherian ring we study r modules and complexes of such with excellent duality properties while their common properties are strong enough to admit a rich theory we count among them such potentially diverse objects as dualizing complexes for r on one side and on the other the ring itself in several ways these two examples constitute the extremes and their well understood properties serve as guidelines for our study however also the employment in recent studies of ring homomorphisms of complexes lying between these extremes is incentive

a fully implicit finite difference scheme for two dimensional burgers equations the two dimensional burgers equations are discretized in fully implicit finite difference form this scheme leads to a system of nonlinear difference equations to be solved at each time step newton s method is used to solve this nonlinear system the linear system is solved by a direct method at each iteration of newton s method the accuracy of the proposed numerical scheme is examined by comparison with other analytical and numerical results the present method performs well

second generation wavelet collocation method for the solution of partial differential equations an adaptive numerical method for solving partial differential equations is developed the method is based on the whole new class of second generation wavelets wavelet decomposition is used for grid adaptation and interpolation while a new o n hierarchical finite difference scheme which takes advantage of wavelet multilevel decomposition is used for derivative calculations the treatment of nonlinear terms and general boundary conditions is a straightforward task due to the collocation nature of the algorithm in this paper we demonstrate the algorithm for one particular choice of second generation wavelets namely lifted interpolating wavelets on an interval with uniform regular sampling the main advantage of using second generation wavelets is that wavelets can be custom designed for complex domains and irregular sampling thus the strength of the new method is that it can be easily extended to the whole class of second generation wavelets leaving the freedom and flexibility to choose the wavelet basis depending on the application

orthogonal polynomials on sobolev spaces old and new directions abstract during the last years orthogonal polynomials on sobolev spaces have attracted considerable attention algebraic properties distribution of their zeros and fourier expansions as well as their relevance in the analysis of spectral methods for partial differential equations provide a very large field to explore and to compare with the standard case in this paper we present an introductory survey about the subject the origin of the problems and their development show the interest and the promising future of this field

general sobolev orthogonal polynomials abstract in this paper we study orthogonal polynomials with respect to the inner product f g s n u fg m 1 n m u f m g m where m 0 for m 1 n and u is a semiclassical positive definite linear functional for these non standard orthogonal polynomials algebraic and differential properties are obtained as well as their representation in terms of the standard orthogonal polynomials associated with u

variational problems and partial differential equations on implicit surfaces a novel framework for solving variational problems and partial differential equations for scalar and vector valued data defined on surfaces is introduced in this paper the key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed cartesian coordinate system using this new embedding function the equations are then both intrinsic to the surface and defined in the embedding space this approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces as is commonly done in the literature we describe the framework and present examples in computer graphics and image processing applications including texture synthesis flow field visualization and image and vector field intrinsic regularization for data defined on 3d surfaces

a wavelet optimized very high order adaptive grid and order numerical method wavelets detect information at different scales and at different locations throughout a computational domain furthermore wavelets can detect the local polynomial content of computational data numerical methods are most efficient when the basis functions of the method are similar to the data present by designing a numerical scheme in a completely adaptive manner around the data present in a computational domain one can obtain optimal computational efficiency this paper extends the numerical wavelet optimized finite difference wofd method to arbitrarily high order so that one obtains in effect an adaptive grid and adaptive order numerical method which can achieve errors equivalent to errors obtained with a spectrally accurate numerical method

diffusion generated motion of curves on surfaces we present a new method for computing the curvature driven motion of a curve constrained to move on a given surface it is based on the diffusion generated motion algorithm and retains both the novel simplicity of that method as well as the natural extension to curves with junctions to general geometric motion laws and to higher dimensions the result is an extremely simple algorithm for curvature dependent motion on surfaces wherein the only evolution operation is linear diffusion in three dimensional euclidean space

the implicit closest point method for the numerical solution of partial differential equations on surfaces many applications in the natural and applied sciences require the solutions of partial differential equations pdes on surfaces or more general manifolds the closest point method is a simple and accurate embedding method for numerically approximating pdes on rather general smooth surfaces however the original formulation is designed to use explicit time stepping this may lead to a strict time step restriction for some important pdes such as those involving the laplace beltrami operator or higher order derivative operators to achieve improved stability and efficiency we introduce a new implicit closest point method for surface pdes the method allows for large stable time steps while retaining the principal benefits of the original method in particular it maintains the order of accuracy of the discretization of the underlying embedding pde it works on sharply defined bands without degrading the accuracy of the method and it applies to general smooth surfaces it also is very simple and may be applied to a rather general class of surface pdes convergence studies for the in surface heat equation and a fourth order biharmonic problem are given to illustrate the accuracy of the method we demonstrate the flexibility and generality of the method by treating flows involving diffusion reaction diffusion and fourth order spatial derivatives on a variety of interesting surfaces including surfaces of mixed codimension

on the category of modules of gorenstein dimension zero let r be a commutative noetherian henselian non gorenstein local ring of depth zero denote by modr the category of all finitely generated r modules and by open image in new window the full subcategory of modr consisting of all r modules of gorenstein dimension zero we prove in this paper that if open image in new window contains a non free module then it is not precovering in modr in particular there exist infinitely many isomorphism classes of indecomposable r modules of gorenstein dimension zero

ring homomorphisms and finite gorenstein dimension the local structure of homomorphisms of commutative noetherian rings is investigated from the point of view of dualizing complexes a concept of finite gorenstein dimension which substantially weakens the notion of finite flat dimension is introduced for homomorphisms it is shown to impose structural constraints due to a remarkable equivalence of subcategories of the derived category of all modules an essential part of this study is the development of relative notions of dualizing complexes and bass numbers it is proved that the bass numbers of local homomorphisms are rigid extending a known result for local rings quasi gorenstein homomorphisms are introduced as local homomorphisms that base change a dualizing complex for the source ring into one for the target they are shown to have the stability properties of the gorenstein homomorphism that they generalize 1991 mathematics subject classification primary 13h10 13d23 14e40 secondary 13c15

coherent pairs of measures and markov bernstein inequalities abstract all the coherent pairs of measures associated to linear functionals c 0 and c 1 introduced by iserles et al in 1991 have been given by meijer in 1997 there exist seven kinds of coherent pairs all these cases are explored in order to give three term recurrence relations satisfied by polynomials the smallest zero 1 n of each of them of degree n has a link with the markov bernstein constant m n appearing in the following markov bernstein inequalities c 1 p 2 m n 2 c 0 p 2 p p n where m n 1 1 n the seven kinds of three term recurrence relations are given in the case where c 0 e x d x 0 and c 1 e x d x explicit upper and lower bounds are given for 1 n and the asymptotic behavior of the corresponding markov bernstein constant is stated except in a part of one case lim n 1 n 0 is proved in all the cases

laguerre sobolev orthogonal polynomials abstract in this paper polynomials that are orthogonal with respect to the inner product f g s 0 f x g x x e x dx 0 f x g x x e x dx where 1 and 0 are studied for these nonstandard orthogonal polynomials algebraic and differential properties as well as the relation with the classical laguerre polynomials are obtained finally some properties concerning the localization and separation of the zeros of these polynomials are deduced

on the positivity of some bilinear functionals in sobolev spaces abstract the positivity of a bilinear functional a a f g n m 0 m c m f m g m is studied as a function of coefficients m the concerned cases are those of laguerre gegenbauer and jacobi for c m c 0 m 1 n the domain m m 1 n where a is positive definite is given as a consequence when n 1 the corresponding markov bernstein inequalities are given