on the negative k theory of schemes in finite characteristic abstract we show that if x is a d dimensional scheme of finite type over an infinite perfect field k of characteristic p 0 then k i x 0 and x is k i regular for i d 2 whenever the resolution of singularities holds over k this proves the k dimension conjecture of weibel c weibel k theory and analytic isomorphisms invent math 61 1980 177 197 2 9 except for d 1 i d 2 in all characteristics assuming the resolution of singularities

some least squares galerkin procedures for first order time dependent convection diffusion system abstract four least squares galerkin finite element schemes are formulated to solve first order time dependent convection diffusion systems and the convergence of these schemes is analyzed

nil k theory maps to cyclic homology algebraic k theory breaks into two pieces nil k theory and karoubi villamayor k theory karoubi has constructed chern classes from the latter groups into cyclic homology we construct maps from nil k theory to cyclic homology which are compatible with karoubi s maps but with a degree shift several recent results show that in characteristic zero our map is often an isomorphism ever since a connes discovered cyclic homology in 1982 it has been widely held that cyclic homology is closely related to algebraic k theory supporting this belief karoubi defined chern classes limep kn a limhcn 2p a in k1 and goodwillie defined a related map ey kn a hcn a in g other maps between k theory and variants of cyclic homology have been constructed by loday unpublished burghelea bur 1 ogle ogle 1 and others of course all of these maps are related to the dennis trace map to hochschild homology d kn a hhn a which r k dennis discovered in 1975 d in this paper we prove three results the first is that karoubi s map factors through goodwillie s theorem a for every associative algebra there is a commutative diagram kn a

implicit time discretization schemes for mixed least squares finite element formulations abstract this work is an extension of the ideas in averweg et al 2019 with the focus on a detailed investigation of implicit time discretization schemes to model instationary fluid flow based on the incompressible navier stokes equations and linear elastodynamic structural behavior the variational approaches for fluid and solid mechanics are based on a mixed least squares finite element method the l 2 norm minimization of the residuals of the constructed first order systems of the governing differential equations is based on two field stress velocity sv functionals for the time discretization of the sv fluid formulation four different types of implicit integration schemes are investigated namely the houbolt method the crank nicolson method and two explicit singly diagonally implicit runge kutta methods esdirk the sv formulation for the solid is discretized applying the houbolt method the presented time integration schemes are validated investigating an unsteady fluid flow and an elastodynamic structural benchmark since both fluid and solid sv formulations are discretized using conforming finite element spaces in h div and h 1 respectively the inherent fulfillment of coupling conditions when modeling fluid structure interaction problems is given a priori therefore the applicability is also examined by two simplified fsi problems for small deformations in order to represent the main characteristics of the presented approach

handling borders in systolic architectures for the 1 d discrete wavelet transform for perfect reconstruction we propose a modified systolic architecture that implements the 1 d discrete wavelet transform dwt on the basis of the recursive pyramid algorithm rpa while correctly managing the border problem and obtaining perfect reconstruction all the architectures so far described in the literature do not explicitly address the handling of borders and usually assume a zero padding extension the rpa makes heavy use of this assumption which produces very efficient systolic architectures it is however well known that the zero padding extension does not allow perfect recovery of the original signal if we compute exactly n coefficients more coefficients are required either in the forward or in the inverse transform and their production within the rpa scheme lowers the efficiency of the systolic architecture to a minimum we introduce a modified rpa working on an extended transform with a compressed schedule that restores good values of efficiency ec rpa we also propose a second version of rpa based on a periodic extension of the signal and of the transform pe rpa that achieves perfect reconstruction by computing exactly n dwt coefficients a reduced version of the second algorithm r pe rpa is tailored to the efficient computations of a fixed number of levels in the transform the vlsi complexity of the management of borders is analyzed through synthesis from a vhdl description of the three algorithms synthesis results show that the controller required for pe rpa can be as large as a ten tap array r pe rpa instead requires half this area in both cases the controller complexity scales sublinearly with the length of the input

generalized self similarity abstract we prove the existence of l pfunctions satisfying a kind of self similarity condition this is achieved by solving a functional equation by means of the construction of a contractive operator on an appropriate functional space the solution a fixed point of the operator can be obtained by an iterative process making this model very suitable to use in applications such as fractal image and signal compression on the other hand this generalized self similarity equation includes matrix refinement equations of the typef x ckf ax k which are central in the construction of wavelets and multiwavelets the results of this paper will therefore yield conditions for the existence of l p refinable functions in a very general setting

orthogonal and biorthogonal fir hexagonal filter banks with sixfold symmetry recently hexagonal image processing has attracted attention the hexagonal lattice has several advantages in comparison with the rectangular lattice the conventionally used lattice for image sampling and processing for example a hexagonal lattice needs less sampling points it has better consistent connectivity it has higher symmetry and its structure is plausible to human vision systems the multiresolution analysis method has been used for hexagonal image processing since the hexagonal lattice has high degree of symmetry it is desirable that the hexagonal filter banks designed for multiresolution hexagonal image processing also have high order of symmetry which is pertinent to the symmetry structure of the hexagonal lattice the orthogonal or prefect reconstruction pr hexagonal filter banks that are available in the literature have only threefold symmetry in this paper we investigate the construction of orthogonal and pr finite impulse response fir hexagonal filter banks with sixfold symmetry we obtain block structures of 7 size refinement seven channel two dimensional orthogonal and pr fir hexagonal filter banks with sixfold rotational symmetry radic7 refinement orthogonal and biorthogonal wavelets based on these block structures are constructed in this paper we also consider fir hexagonal filter banks with axial line symmetry and we present a block structure of fir hexagonal filter banks with pseudo sixfold axial symmetry

on the implementation of nonseparable two dimensional haar wavelet transforms we introduce a transformation to localize the equations defining the successive levels of the mallat 1989 pyramid for two dimensional 2 d haar wavelets we propose a methodology for implementing these wavelet transforms in parallel architectures like systolic arrays more specifically we show that there is a perfect match between the wavelet algorithms and the multiphase multirate array mpra architectures

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

stabilized finite element methods for the velocity pressure stress formulation of incompressible flows abstract formulated in terms of velocity pressure and the extra stress tensor the incompressible navier stokes equations are discretized by stabilized finite element methods the stabilized methods proposed are analyzed for a linear model and extended to the navier stokes equations the numerical tests performed confirm the good stability characteristics of the methods these methods are applicable to various combinations of interpolation functions including the simplest equal order linear and bilinear elements

a mixed finite element method for the stokes problem an acceleration pressure formulation the existing finite element methods 6 9 11 for the stokes equation lead to a saddle point problem for removing certain restrictions on the elements and improving the accuracy this paper develops a new method the stokes equation is treated as a first order linear system using a least squares method to minimize the defect in the differential operator we find that the result converges asymptotically to optimal order in sobolev spaces in the 2 d case if the domain is subdivided into regular triangles and the piecewise linear functions in the space h 1 are chosen to be trial functions for both velocity and pressure then we have error estimates p p h 0 u u h 0 ch 2

quasiperiodic spectra and orthogonality for iterated function system measures we extend classical basis constructions from fourier analysis to attractors for affine iterated function systems ifss this is of interest since these attractors have fractal features e g measures with fractal scaling dimension moreover the spectrum is then typically quasi periodic but non periodic i e the spectrum is a small perturbation of a lattice due to earlier research on ifss there are known results on certain classes of spectral duality pairs also called spectral pairs or spectral measures it is known that some duality pairs are associated with complex hadamard matrices however not all ifss x admit spectral duality when x is given we identify geometric conditions on x for the existence of a fourier spectrum serving as the second part in a spectral pair we show how these spectral pairs compose and we characterize the decompositions in terms of atoms the decompositions refer to tensor product factorizations for associated complex hadamard matrices

cyclic homology cdh cohomology and negative k theory we prove a blow up formula for cyclic homology which we use to show that infinitesimal k theory satisfies cdh descent combining that result with some computations of the cdh cohomology of the sheaf of regular functions we verify a conjecture of weibel predicting the vanishing of algebraic k theory of a scheme in degrees less than minus the dimension of the scheme for schemes essentially of finite type over a field of characteristic zero

the recursive pyramid algorithm for the discrete wavelet transform the recursive pyramid algorithm rpa is a reformulation of the classical pyramid algorithm pa for computing the discrete wavelet transform dwt the rpa computes the n point dwt in real time running dwt using just l log n spl minus 1 words of storage as compared with o n words required by the pa l is the length of the wavelet filter the rpa is combined with the short length fir filter algorithms to reduce the number of multiplications and additions

analysis of least squares finite element methods for the navier stokes equations in this paper we study finite element methods of least squares type for the stationary incompressible navier stokes equations in two and three dimensions we consider methods based on velocity vorticity pressure form of the navier stokes equations augmented with several nonstandard boundary conditions least squares minimization principles for these boundary value problems are developed with the aid of the agmon douglis nirenberg adn elliptic theory among the main results of this paper are optimal error estimates for conforming finite element approximations and analysis of some nonstandard boundary conditions results of several computational experiments with least squares methods which illustrate among other things the optimal convergence rates are also reported

space time least squares spectral elements for convection dominated unsteady flows legendre polynomials are employed in a space time least squares spectral element formulation applied to linear and nonlinear hyperbolic scalar equations no stabilization techniques are required to render a stable high order accurate scheme in parts of the domain where the underlying exact solution is smooth the scheme exhibits exponential convergence with polynomial enrichment whereas in parts of the domain where the underlying exact solution contains discontinuities the solution displays a gibbs like behavior numerical results will be given in which the capabilities of the space time formulation to capture discontinuities will be demonstrated

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

temporal spatial and thermal features of 3 d rayleigh benard convection by a least squares finite element method numerical solutions of 3 d time dependent rayleigh benard convection are presented in this work the temporal spatial and thermal features of convective patterns are studied for four different geometric aspect ratios 2 1 2 4 1 4 5 1 5 and 3 5 1 2 1 at supercritical rayleigh numbers ra 8 103 2 4 104 and prandtl numbers pr 0 71 2 5 several physical phenomena such as multicellular flow pattern oscillatory transient solution t shaped rolls at the ends of a rectangular box and roll alignment are observed in our simulations the numerical technique is based on an implicit fully coupled and time accurate method which consists of the crank nicolson scheme for time integration newton s method for the convective terms with extensive linearization steps and a least squares finite element method a matrix free algorithm of the jacobi conjugate gradient method is implemented to solve the symmetric positive definite linear system of equations

least squares mixed generalized multiscale finite element method abstract in this paper we present an approximation of elliptic problems with multiscale and high contrast diffusion coefficients a mixed formulation is considered such that both pressure and velocity are approximated simultaneously this formulation arises naturally in many applications such as flows in porous media due to the multiscale nature of the solutions using model reduction is required to efficiently obtain approximate solutions there are many multiscale approaches for elliptic problems in mixed formulation these approaches include numerical homogenization and mixed multiscale finite element method which aim to obtain a coarse accurate representation of the velocity without using an accurate representation for pressure it has been a challenging task to construct a method giving accurate representation for both pressure and velocity the goal in this paper is to construct multiscale basis functions for both pressure and velocity we will apply the framework of generalized multiscale finite element method gmsfem and design systematic strategies for the construction of basis the construction involves snapshot spaces and dimension reduction via local spectral problems the mixed formulation is minimized in the sense of least squares the compatibility condition for multiscale finite element spaces of the pressure and velocity is not required in the least squares mixed form this gives more flexibility for the construction of multiscale basis functions for velocity and pressure convergence analysis is carried out for the least squares mixed gmsfem several numerical examples for various permeability fields are presented to show the performance of the presented method the numerical results show that the least squares mixed gmsfem can give accurate approximation for both pressure and velocity using only a few basis functions per coarse element

non conforming least squares spectral element method for stokes equations on non smooth domains abstract in this paper we present a discontinuous least squares spectral element method for stokes equations with primitive variable formulation on both smooth and non smooth domains we propose an exponentially accurate numerical scheme based on the stability estimates and implement it on different polygonal domains preconditioned conjugate gradient method is used to obtain the numerical solution

issues related to least squares finite element methods for the stokes equations least squares finite element methods have become increasingly popular for the approximate solution of first order systems of partial differential equations here after a brief review of some existing theories a number of issues connected with the use of such methods for the velocity vorticity pressure formulation of the stokes equations in two dimensions in realistic settings are studied through a series of computational experiments finite element spaces that are not covered by existing theories are considered included in these are piecewise linear approximations for the velocity mixed boundary conditions which are also not covered by existing theories are also considered as is enhancing mass conservation next problems in nonconvex polygonal regions and the resulting nonsmooth solutions are considered with a view toward seeing how accuracy can be improved a conclusion that can be drawn from this series of computational experiments is that the use of appropriate mesh dependent weights in the least squares functional almost always improves the accuracy of the approximations concluding remarks concerning three dimensional problems the nonlinear navier stokes equations and the conditioning of the discrete systems are provided

least squares methods for elliptic systems a weighted least squares method is given for the numerical solution of elliptic partial differential equations of agmon douglis nirenberg type and an error analysis is provided some examples are given 1 introduction the use of least squares methods for the approximate solution of equations dates back at least to gauss the modern theory of least squares methods in the numerical solution of elliptic boundary value problems starts in 1970 with the papers of bramble and schatz 5 6 this work uses a finite dimensional space s of approximating functions similar to the spaces used in finite element methods the approximate solution is defined to be the minimizer of a least squares functional that is a weighted sum of the least squares residual in the differential equation and the least squares residual in the boundary condition the paper 5 has an historical importance for the following reason it appeared during the time when numerical analysts were shifting attention from finite difference methods to finite element methods and it provided for the first time a family of approximation methods for the solution of the dirichlet problem whose order of accuracy could be made arbitrarily large the paper 6 provided an extension to an elliptic equation of order 2m and 3 gave important simplifications in the analysis the principal advantages of the method are that one need not satisfy exactly the dirichlet boundary condi tions and that the mathematical analysis dictates in a natural way the relative weights that are given to the boundary and interior terms in the least squares functional also the method provides in a quasioptimal sense as good a solution as can be expected from the space s on the other hand the method requires that s consist of functions which are smooth enough to lie in the domain of the elliptic operator also the method seems to produce matrices with large condition number for various reasons it is of interest to extend the theory of least squares methods to include elliptic systems first if a second order elliptic equation is written as a first order system it would seem and this is borne out by our analysis that the smoothness requirements for the spaces of approximating functions would be reduced thus eliminating one of the disadvantages of the method a second motivation for extending the least squares method to elliptic systems is that elliptic systems occur frequently in applications an example of an elliptic system is the system of equations for stokes flow for this system the least squares method does not require the space of approximating vector fields to be incompressible instead

a nonlinear weighted least squares finite element method for stokes equations the paper concerns a nonlinear weighted least squares finite element method for the solutions of the incompressible stokes equations based on the application of the least squares minimization principle to an equivalent first order velocity pressure stress system model problem considered is the flow in a planar channel the least squares functional involves the l 2 norms of the residuals of each equation multiplied by a nonlinear weighting function and mesh dependent weights using linear approximations for all variables by properly adjusting the importance of the mass conservation equation and a carefully chosen nonlinear weighting function the least squares solutions exhibit optimal l 2 norm error convergence in all unknowns numerical solutions of the flow pass through a 4 to 1 contraction channel will also be considered

a finite element method for first order elliptic systems in three dimensions abstract we consider finite element approximation schemes for boundary value problems for linear first order elliptic systems in three dimensions we also discuss div curl type systems in three independent variables both a least squares method and a new subdomain collocation least squares method are considered the analysis yield optimal asymptotic h 1 and l 2 error estimates

fast algorithms for discrete and continuous wavelet transforms several algorithms are reviewed for computing various types of wavelet transforms the mallat algorithm 1989 the a trous algorithm and their generalizations by shensa the goal of this work is to develop guidelines for implementing discrete and continuous wavelet transforms efficiently and to compare the various algorithms obtained and give an idea of possible gains by providing operation counts most wavelet transform algorithms compute sampled coefficients of the continuous wavelet transform using the filter bank structure of the discrete wavelet transform although this general method is already efficient it is shown that noticeable computational savings can be obtained by applying known fast convolution techniques such as the fft fast fourier transform in a suitable manner the modified algorithms are termed fast because of their ability to reduce the computational complexity per computed coefficient from l to log l within a small constant factor for large filter lengths l for short filters smaller gains are obtained fast running fir finite impulse response filtering techniques allow one to achieve typically 30 savings in computations

efficient stress velocity least squares finite element formulations for the incompressible navier stokes equations abstract in this contribution three different mixed least squares finite element methods lsfems are investigated with respect to accuracy and efficiency with simultaneous consideration of regular and adaptive meshing strategies the deliberations are made for the incompressible navier stokes equations the generally known first order div grad system in terms of the total stress velocity and pressure svp formulation is the basis for two further div grad least squares formulations in terms of total stress and velocity sv whereby both formulations are derived from different fountainheads the extended sv formulation is an enrichment of the known stress velocity formulation proposed by cai et al 2004 the second sv formulation is based on the substitution of the pressure in the svp formulation thus it is based on a discontinuous pressure interpolation advantage of the sv formulations is a smaller system matrix size due to the reduction of the relevant degrees of freedom nevertheless the approximation quality has to last the demand of the established least squares formulations the drawback of a poor mass conservation is well investigated for the lsfem and known to overcome by using high order interpolations for all solution variables therefore the main attention of this contribution is focused on accuracy while aiming for high efficiency which is intrinsic for the fundamental idea in here in the light of efficient lsfem solutions the advantage of the inherent a posteriori error estimator of the method is used for deliberations on different marking strategies in an h type adaptive mesh refinement

a technique for the construction of compactly supported biorthogonal wavelets of l2 rn n 2 abstract in this paper a technique for the concrete construction of compactly supported biorthogonal wavelet bases of l2 rn is given this technique does not depend on the dimension n and it gives rise to non separable multidimensional wavelet bases of special interest is the study of the stability of the constructed wavelet frames it is assumed that the dilation matrices associated with the wavelet bases are given by d 2in

dual mixed finite element approximation of stokes and nonlinear stokes problems using trace free velocity gradients in this work a finite element method for a dual mixed approximation of stokes and nonlinear stokes problems is studied the dual mixed structure which yields a twofold saddle point problem arises in a formulation of this problem through the introduction of unknown variables with relevant physical meaning the method approximates the velocity its gradient and the total stress tensor but avoids the explicit computation of the pressure which can be recovered through a simple postprocessing technique this method improves an existing approach for these problems and uses raviart thomas elements and discontinuous piecewise polynomials for approximating the unknowns existence uniqueness and error results for the method are given and numerical experiments that exhibit the reduced computational cost of this approach are presented

on the vanishing of negative k groups we show that for a d dimensional scheme x essentially of finite type over an infinite perfect field k of characteristic p 0 the negative k groups k q x vanish for q d provided that strong resolution of singularities holds over the field k

decoupled algorithm for solving phan thien tanner viscoelastic fluid by finite element method in this paper we present a decoupled algorithm for the viscoelastic fluid flow obeying the phan thien tanner ptt differential model the method consists to solve alternatively a stokes like problem by weighted least squares wls finite element method and the constitutive equation by streamline upwind petrov galerkin supg method a priori error estimate for the wls supg finite element method is derived the existence and uniqueness of the approximate solution are obtained using fixed point techniques we show that the mapping of the decoupled algorithm is locally contracting the planar channel flow problem is used to illustrate our theoretical results we encounter no limit of the weissenberg number for reasonable values of e a dimensionless material parameter in our calculations

error analysis of some galerkin least squares methods for the elasticity equations the recent technique of stabilizing mixed finite element methods by augmenting the galerkin formulation with least squares terms calculated separately on each element is considered the error analysis is performed in a unified manner yielding improved results for some methods introduced earlier in addition a new formulation is introduced and analyzed method is used within two different contexts the first is in connection with the plain strain or three dimensional problems near the incompressible limit for which the poisson ratio equals one half it is well known that for these problems a standard lower order displacement finite element method will break down by now it is also well known how a working method can be obtained a new unknown the pressure is introduced and discretized with a finite element space different from that used for the displacements the finite element method obtained falls into the class of saddlepoint problems for which there is a general theory developed by brezzi and babuska 5 7 the application of this theory shows that the method converges optimally if the discrete spaces for the displacement and pressure satisfy a stability condition for a particular choice of displacement and pressure approximations it can be a difficult task to verify this babuska brezzi or inf sup condition during the last decade this problem has been intensively studied and the performances of many combinations are now known cf 18 and 23 26 for some recent extensions in addition there exists a general technique utilizing bubble functions by which a stable method can

multiresolution analysis haar bases and self similar tilings of r sup n orthonormal bases for l sup 2 r sup n are constructed that have properties that are similar to those enjoyed by the classical haar basis for l sup 2 r for example each basis consists of appropriate dilates and translates of a finite collection of piecewise constant functions the construction is based on the notion of multiresolution analysis and reveals an interesting connection between the theory of compactly supported wavelet bases and the theory of self similar tilings

two fast architectures for the direct 2 d discrete wavelet transform we propose two architectures for the direct two dimensional 2 d discrete wavelet transform dwt the first one is based on a modified recursive pyramid algorithm mrpa and performs a nonstandard decomposition i e mallat s 1989 tree of an n spl times n image in approximately 2n sup 2 3 clock cycles ccs this result consistently speeds up other known architectures that commonly need approximately n sup 2 ccs furthermore the proposed architecture is simpler than others in terms of hardware complexity subsequently we show how symmetric anti symmetric properties of linear phase wavelet filter bases can be exploited in order to further reduce the vlsi area this is used to design a second architecture that provides one processing unit for each level of decomposition pipelined approach and performs a decomposition in approximately n sup 2 2 ccs in many practical cases even this architecture is simpler than general mrpa based devices having only one processing unit

complete to overcomplete discrete wavelet transforms theory and applications a new transform is proposed that derives the overcomplete discrete wavelet transform odwt subbands from the critically sampled dwt subbands complete representation this complete to overcomplete dwt codwt has certain advantages in comparison to the conventional approach that performs the inverse dwt to reconstruct the input signal followed by the a spl grave trous or the lowband shift algorithm specifically the computation of the input signal is not required as a result the minimum number of downsampling operations is performed and the use of upsampling is avoided the proposed codwt computes the odwt subbands by using a set of prediction filter matrices and filtering and downsampling operators applied to the dwt this formulation demonstrates a clear separation between the single rate and multirate components of the transform this can be especially significant when the codwt is used in resource constrained environments such as resolution scalable image and video codecs to illustrate the applicability of the proposed transform in these emerging applications a new scheme for the transform calculation is proposed and existing coding techniques that benefit from its usage are surveyed the analysis of the proposed codwt in terms of arithmetic complexity and delay reveals significant gains as compared with the conventional approach

quasi least squares finite element method for steady flow and heat transfer with system rotation two quasi least squares finite element schemes based on l2 inner product are proposed to solve a steady navier stokes equations coupled to the energy equation by the boussinesq approximation and augmented by a coriolis forcing term to account for system rotation the resulting nonlinear systems are linearized around a characteristic state resulting in linearized least squares models that yield algebraic systems with symmetric positive definite coefficient matrices existence of solutions are investigated and a priori error estimates are obtained the performance of the formulation is illustrated by using a direct iteration procedure to treat the nonlinearities and shown theoretical convergent rate for general initial guess

a priori and a posteriori error analyses of a velocity pseudostress formulation for a class of quasi newtonian stokes flows abstract in this paper we introduce and analyze new mixed finite element schemes for a class of nonlinear stokes models arising in quasi newtonian fluids the methods are based on a non standard mixed approach in which the velocity the pressure and the pseudostress are the original unknowns however we use the incompressibility condition to eliminate the pressure and set the velocity gradient as an auxiliary unknown which yields a twofold saddle point operator equation as the resulting dual mixed variational formulation in addition a suitable augmented version of the latter showing a single saddle point structure is also considered we apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well posed in particular we show that raviart thomas elements of order k 0 for the pseudostress and piecewise polynomials of degree k for the velocity and its gradient ensure the well posedness of the associated galerkin schemes moreover we prove that any finite element subspace of the square integrable tensors can be employed to approximate the velocity gradient in the case of the augmented formulation then we derive a reliable and efficient residual based a posteriori error estimator for each scheme finally we provide several numerical results illustrating the good performance of the resulting mixed finite element methods confirming the theoretical properties of the estimator and showing the behaviour of the associated adaptive algorithms

least squares finite element approximations to solutions of interface problems a least squares finite element method for second order elliptic boundary value problems having interfaces due to discontinuous media properties is proposed and analyzed both dirichlet and neumann boundary data are treated the boundary value problems are recast into a first order formulation to which a suitable least squares principle is applied among the advantages of the method are that nonconforming with respect to the interface approximating subspaces may be used moreover the grids used on each side of an interface need not coincide along the interface error estimates are derived that improve on other treatments of interface problems and a numerical example is provided to illustrate the method and the analyses

least squares finite element method with high continuity nurbs basis for incompressible navier stokes equations modern least squares finite element method lsfem has advantage over mixed finite element method for non self adjoint problem like navier stokes equations but has problem to be norm equivalent and mass conservative when using c 0 type basis in this paper lsfem with non uniform b splines nurbs is proposed for navier stokes equations high order continuity nurbs is used to construct the finite dimensional spaces for both velocity and pressure variational form is derived from the governing equations with primitive variables and the dofs due to additional variables are not necessary there is a novel k refinement which has spectral convergence of least squares functional the method also has same advantages as isogeometric analysis like automatic mesh generation and exact geometry representation several benchmark problems are solved using the proposed method the results agree well with the benchmark solutions available in literature the results also show good performance in mass conservation

bass nk groups and cdh fibrant hochschild homology the k theory of a polynomial ring r t contains the k theory of r as a summand for r commutative and containing q we describe k r t k r in terms of hochschild homology and the cohomology of kahler differentials for the cdh topology we use this to address bass question whether kn r kn r t implies kn r kn r t1 t 2 the answer to this question is affirmative when r is essentially of finite type over the complex numbers but negative in general

certain discrete dynamical systems number systems and related integral self affine sets we consider certain discrete dynamical systems conventional number systems and redundant number systems for z 2 which are associated with certain 2x2 integral expansive matrices and collinear digit sets for them using integral self affine sets in r n we also give a geometric characterization to attractors of certain discrete dynamical systems and associated redundant number systems for z n

first order system least squares for the stokes equations with application to linear elasticity following our earlier work on general second order scalar equations here we develop a least squares functional for the two and three dimensional stokes equations generalized slightly by allowing a pressure term in the continuity equation by introducing a velocity flux variable and associated curl and trace equations we are able to establish ellipticity in an h1 product norm appropriately weighted by the reynolds number this immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems both estimates are naturally uniform in the reynolds number moreover our pressure perturbed form of the generalized stokes equations allows us to develop an analogous result for the dirichlet problem for linear elasticity where we obtain the more substantive result that the estimates are uniform in the poisson ratio

analysis of least squares mixed finite element methods for nonlinear nonstationary convection diffusion problems some least squares mixed finite element methods for convection diffusion problems steady or nonstationary are formulated and convergence of these schemes is analyzed the main results are that a new optimal a priori l 2 error estimate of a least squares mixed finite element method for a steady convection diffusion problem is developed and that four fully discrete least squares mixed finite element schemes for an initial boundary value problem of a nonlinear nonstationary convection diffusion equation are formulated also some systematic theories on convergence of these schemes are established

a weighted least squares finite element method for phan thien tanner viscoelastic fluid abstract in this paper we present the weighted least squares finite element approximations to the steady phan thien tanner ptt viscoelastic fluid we approximate the nonlinear viscoelastic model by linearizing the constitutive equation and apply a least squares method to solve the linearized system the least squares functional involves the residuals of each equation in l 2 norm multiplied by mesh dependent weights the method is tested on a planar channel flow problem numerical results exhibit the convergence rates of the proposed scheme

algebraic k theory and descent for blow ups we prove that algebraic k theory satisfies pro descent for abstract blow up squares of noetherian schemes as an application we derive weibel s conjecture on the vanishing of negative k groups

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

stabilized mixed finite element methods based on riesz representing operators for solving saddle point problems abstract based on riesz representing operators a new stabilized finite element method is presented for saddle point problems it is proved that this method is not subject to the discrete babuska brezzi condition and that the corresponding finite element approximation problem yields a symmetrically positively definite linear system error bounds are obtained which are agree with the interpolation properties as an application the stationary stokes problem is analyzed

analysis of the l2 least squares finite element method for the velocity vorticity pressure stokes equations with velocity boundary conditions a theoretical analysis of the l 2 least squares finite element method lsfem for solving the stokes equations in the velocity vorticity pressure vvp first order system formulation with the dirichlet velocity boundary condition is given the least squares energy functional is defined to be the sum of the squared l 2 norms of the residuals in the partial differential equations weighted appropriately by the viscosity constant n it is shown that with many advantages the method is stable and convergent without requiring extra smoothness of the exact solution and the piecewise linear finite elements can be used to approximate all the unknowns furthermore with respect to the order of approximation for smooth exact solutions the method exhibits an optimal rate of convergence in the h 1 norm for velocity and a suboptimal rate of convergence in the l 2 norm for vorticity and pressure some numerical experiments in two and three dimensions are given which confirm the a priori error estimates since the boundary of the bounded domain under consideration is polygonal in r 2 or polyhedral in r 3 instead of c 1 smooth the authors adopt the more direct technique of bramble pasciak and cai manteuffel mccormick which departs from the agmon douglis nirenberg theory in showing the coercivity bound of the least squares functional

special paraunitary matrices cayley transform and multidimensional orthogonal filter banks we characterize and design multidimensional md orthogonal filter banks using special paraunitary matrices and the cayley transform orthogonal filter banks are represented by paraunitary matrices in the polyphase domain we define special paraunitary matrices as paraunitary matrices with unit determinant we show that every paraunitary matrix can be characterized by a special paraunitary matrix and a phase factor therefore the design of paraunitary matrices and thus of orthogonal filter banks becomes the design of special paraunitary matrices which requires a smaller set of nonlinear equations moreover we provide a complete characterization of special paraunitary matrices in the cayley domain which converts nonlinear constraints into linear constraints our method greatly simplifies the design of md orthogonal filter banks and leads to complete characterizations of such filter banks

the k theory of toric varieties recent advances in computational techniques for k theory allow us to describe the k theory of toric varieties in terms of the k theory of fields and simple cohomological data

infinitesimal cohomology and the chern character to negative cyclic homology there is a chern character from k theory to negative cyclic homology we show that it preserves the decomposition coming from adams operations at least in characteristic zero

k regularity cdh fibrant hochschild homology and a conjecture of vorst it is a well known fact that algebraic if theory is homotopy invariant as a functor on regular schemes if x is a regular scheme then the natural map kn x kn x x 1 is an isomorphism for all n ez this is false in general for nonregular schemes and rings to express this failure bass introduced the terminology that for any contravari ant functor v defined on schemes a scheme x is called v regular if the pullback maps v x v x x ar are isomorphisms for all r 0 if x spec i we also say that r is p regular thus regular schemes are ifn regular for every n in contrast it was observed as long ago as in 2 that a nonreduced affine scheme can never be k regular in particular if a is an artinian ring that is a o dimensional noetherian ring then a is regular that is reduced if and only if a is ifi regular in 17 vorst conjectured that for an affine scheme x of finite type over a field f and of dimension d regularity and k i regularity are equivalent vorst proved this conjecture for d 1 by proving that regularity implies normality in this paper we prove vorst s conjecture in all dimensions provided the char acteristic of the ground field f is zero in fact we prove a stronger statement we say that x is regular in codimension n in x note that for all n z if a ring r is ifn regular then it is kn i regular this is proved in 17 for n 1 and in 6 4 4 for n kh x where k x is the algebraic if theory spectrum of x and kh x is the homotopy if theory of x defined in 19 we write fk r for fk spec r theorem 0 1 let r be a commutative ring which is essentially of finite type over a field f of characteristic 0 then the following hold a if tk r is n connected then r is regular in codimension khi x is an isomorphism for i n and a surjection for i n 1 so that fk x

analysis of piecewise linear approximations to the generalized stokes problem in the velocity stress pressure formulation in this paper we study continuous piecewise linear polynomial approximations to the generalized stokes equations in the velocity stress pressure first order system formulation by using a cell vertex finite volume least squares scheme this method is composed of a direct cell vertex finite volume discretization step and an algebraic least squares step where the least squares procedure is applied after the discretization process is accomplished this combined approach has the advantages of both finite volume and least squares approaches an error estimate in the h1 product norm for continuous piecewise linear approximating functions is derived it is shown that with respect to the order of approximation for h2 regular exact solutions the method exhibits an optimal rate of convergence in the h1 norm for all unknowns velocity stress and pressure

the advanced oca for 2 d discrete periodized wavelet transformation we develop a novel computation method for the 2 d discrete periodized wavelet transformation dpwt the new algorithm is based on the operator correlation algorithm oca compared with the classical 2 d pyramid algorithm the new advanced operator correlation algorithm aoca has two major advantages it requires half the number of multiplications and can yield the same output snr with half the precision of the dpwt filter coefficients furthermore the modular structure of the adca makes it particularly suitable for a vlsi implementation

least squares methods for the velocity pressure stress formulation of the stokes equations we formulate and study finite element methods for the solution of the incompressible stokes equations based on the application of least squares minimization principle to an equivalent first order velocity pressure stress system our least squares functional involves the l norms of the residuals of each equation multiplied by a mesh dependent weight each weight is determined according to the agmon douglis nirenberg index of the corresponding equation as a result the approximating spaces are not subject to the lbb condition and conforming discretizations are possible with merely continuous finite element spaces moreover the resulting discrete problems involve only symmetric positive definite systems of linear equations i e assembly of the discretization matrix is not required even at the element level we prove that the least squares approximadons converge to the solutions of the stokes problem at the best possible rate and then present some numerical examples illustrating our theoretical results among other things these numerical examples indicate that the method is not optimal without the weights in the least squares functional

a nonlinear weighted least squares finite element method for the oldroyd b viscoelastic flow this paper studies a nonlinear weighted least squares finite element method for the steady oldroyd b viscoelastic flow our least squares functional involves the l 2 norm of the residuals of each equation multiplied by a proper weight the weights include mass conservation constant mesh dependent weight and a nonlinear weighting function we prove that the least squares approximation converges to the solutions of the linearized versions of the viscoelastic fluid model at the best possible rate and then present the planar channel flow problem illustrating our theoretical results results of the least squares approach indicate that with carefully chosen nonlinear weighting functions and linear basis functions the numerical solution exhibits a second order convergence rate for velocity and viscous stress and superlinear convergence rate in polymeric stress and pressure the method is applied to the 4 to 1 planar contraction problem the effects of weissenberg numbers are also presented in the work

analysis of a velocity pressure pseudostress formulation for the stationary stokes equations abstract we consider a non standard mixed approach for the stokes problem in which the velocity the pressure and the pseudostress are the main unknowns alternatively the pressure can be eliminated from the original equations thus yielding an equivalent formulation with only two unknowns in this paper we develop a priori and a posteriori error analyses of both approaches we first apply the babuska brezzi theory to prove the well posedness of the continuous and discrete formulations in particular we show that raviart thomas elements of order k 0 for the pseudostresses and piecewise polynomials of degree k for the velocities and the pressures ensure unique solvability and stability of the associated galerkin schemes then we derive reliable and efficient residual based a posteriori error estimators for both schemes without and with pressure unknown finally we provide several numerical results illustrating the good performance of the resulting mixed finite element methods confirming the theoretical properties of the estimators and showing the behaviour of the associated adaptive algorithms

large scale computation of incompressible viscous flow by least squares finite element method abstract the least square finite element method lsfem based on the velocity pressure vorticity formulation is applied to large scale three dimensional steady incompressible navier stokes problems this method can accommodate equal order interpolations and results in a symmetric positive definite algebraic system which can be solved effectively by simple iterative methods the first order velocity bernoulli pressure vorticity formulation for incompressible viscous flows is also tested the first order velocity pressure stress formulation is not elliptic in the ordinary sense so we do not recommend its use for newtonian flows for three dimensional flows a compatibility equation i e zero divergence of vorticity vector is included to make the first order system elliptic as a by product of proving the ellipticity of first order systems a rigorous mathematical technique has been developed to justify the number of permissible boundary conditions for the navier stokes equations the simple substitution or newton s method is employed to linearize the partial differential equations the lsfem is used to obtain discretized equations and the system of algebraic equations is solved using the jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices matrix free to achieve high efficiency to show the validity of this method for large scale computation we give numerical results for the 2d driven cavity problem at re 10000 with 408 400 bilinear elements the flow in a 3d cavity is calculated at re 100 400 and 1000 with 50 52 50 trilinear elements the taylor gortler like vortices are observed for re 1000

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed