some performances of local bivariate quadratic c 1 quasi interpolating splines on nonuniform type 2 triangulations we investigate spline quasi interpolants defined by c1 bivariate quadratic b splines on nonuniform type 2 triangulations and by discrete linear functionals based on a fixed number of triangular mesh points either in the support or close to the support of such b splines we show they can approximate a real function and its partial derivatives up to an optimal order and we derive local and global upper bounds we also present some numerical and graphical results

on smooth multivariate spline functions in this paper the dimensions of bivariate spline spaces with simple cross cut grid partitions are determined and expressions of their basis functions are given consequently the closures of these spaces over all partitions of the same type can be determined a somewhat more detailed study on bivariate splines with rectangular grid partitions is included the results in this paper can be applied to problems on interpolation and approximation by bivariate spline functions

elliptic surfaces and linear systems with fat points we investigate the expected dimensionality of linear systems with general fat points on certain surfaces using an approach by specialization to elliptic surfaces for the projectivization of the atiyah bundle over an elliptic curve with a certain polarization we observe that the special case of only one fat point implies the general case of arbitrarily many fat points as well as results concerning other surfaces we conjecture that this special case holds in characteristic 0 but prove that it fails in any positive characteristic

on the linear independence of multivariate b splines i triangulations of simploids this paper is concerned with the construction of certain collections of linearly independent multivariate b splines whose linear span is appropriate for approximation by constructing linear projectors onto these spline spaces we are able to estimate local and global approximation errors as well as the condition number of the b spline basis our approach is based upon some combinatorial properties of certain triangulations of a class of n polytopes which we call simploids

integral error formulae for the scale of mean value interpolations which includes kergin and hakopian interpolation in this paper we provide an integral error formula for a certain scale of mean value interpolations which includes the multivariate polynomial interpolation schemes of kergin and hakopian this formula involves only derivatives of order one higher than the degree of the interpolating polynomial space and from it we can obtain sharp l infty estimates these l infty estimates are precisely those that numerical analysts want to guarantee that a scheme based on such an interpolation has the maximum possible order

the generic dimension of the space of c 1 splines of degree d 8 on tetrahedral decompositions this paper considers the linear space of globally differentiable piecewise polynomial functions defined on a three dimensional polyhedral domain which has been partitioned into tetrahedra combining bernstein bezier methods and combinatorial and geometric techniques from rigidity theory this paper gives an explicit expression for the generic dimension of this space for sufficiently large polynomial degrees d geq 8 this is the first general dimension statement of its kind

a bivariate c 3 interpolation scheme on the twice ct type split of a triangle we prove that there exists a c3 piecewise polynomial of degree 7 interpolating arbitrarily given values and derivatives of orders up to 3 at the vertices and on the edges of the triangle the method available is a mixture of b forms of polynomials and the serendipity of bivariate splines

rational shape functions for c1 interpolation on quadrilaterals the aim of this paper is to establish c1 rational spline functions on any regular quadrilateral partition of a simply or multiply connected and bounded domain and to get new c1 quadrilateral shape functions for finite element analysis c1 generalized wedge functions on any quadrilateral of are defined and constructed a c1 interpolating operator with cubic precision on any quadrilateral is given

systems of p laplacian differential inclusions with large diffusion abstract in this paper we consider coupled systems of p laplacian differential inclusions and we prove under suitable conditions that a homogenization process occurs when diffusion parameters become arbitrarily large in fact we obtain that the attractors are continuous at infinity on l 2 l 2 topology with respect to the diffusion coefficients and the limit set is the attractor of an ordinary differential problem

asymptotically autonomous multivalued cauchy problems with spatially variable exponents abstract we study the asymptotic behavior of a non autonomous multivalued cauchy problem of the form u t t div d t u t p x 2 u t u t p x 2 u t f t u t 0 on a bounded smooth domain in r n n 1 with a homogeneous neumann boundary condition where the exponent p c satisfies p min p x 2 we prove the existence of a pullback attractor and study the asymptotic upper semicontinuity of the elements of the pullback attractor a a t t r as t for the non autonomous evolution inclusion in a hilbert space h under the assumptions amongst others that f is a measurable multifunction and d l t is bounded above and below and is monotonically nonincreasing in time the global existence of solutions is obtained through results of papageorgiou and papalini

multivariate spline functions b spline basis and polynomial interpolations the main result of this paper is a formula relating a multivariate b spline and a certain directional derivative to a lower order b spline the structure of discontinuous b splines is given a formula relates the discontinuous k variate b spline to a k 1 variate b spline with the help of these formulas we have succeeded in determining the exact class of local and global smoothness of a multivariate b spline depending on the configuration of the knots

approximation by cubic c 1 splines on arbitrary triangulations in this paper we present an efficient representation for bivariate piecewise cubicc 1 splines on arbitrary triangulations a numerical method is discussed for computing the dimension of the spaces 3 1 of these splines we consider subspaces ofs 3 1 satisfying certain boundary conditions some applications are given where piecewise cubicc 1 functions are used to solve interpolation problems and least squares approximation problems

on the linear independence of multivariate b splines ii complete configurations the first part of this paper is concerned with global characterizations of both the multivariate b spline and the multivariate truncated power function as smooth piecewise polynomials in the second part of the paper we establish combinatorial criteria for the linear independence of multivariate b splines corresponding to certain configurations of knot sets

abstract reaction diffusion systems with nonlocal initial conditions abstract we prove some sufficient conditions for the existence and global uniform asymptotic stability of c 0 solutions for a class of nonlinear delay reaction diffusion systems subjected to nonlocal initial conditions some applications to a specific reaction diffusion system are included

a general method for constructing quasi interpolants from b splines a general method for constructing quasi interpolation operators based on b splines is developed given a b spline f in r s s 1 normalized by i z email protected i 1 the classical structure q f email protected i z email protected i f i for a quasi interpolation operator q is considered a minimization problem is derived from an estimate of the quasi interpolation error associated with q when lf is a linear combination of values of f at points in some neighbourhood of the support of f or a linear combination of values of f and some of its derivatives at some points in this set or a linear combination of weighted mean values of the function to be approximated that linear functional is defined to produce a quasi interpolant exact on the space of polynomials of maximal total degree included in the space spanned by the integer translates of f the solution of that minimization problem is characterized in terms of specific splines which do not depend on l but only on f

optimal order approximation by mixed three directional spline elements this paper is concerned with a study of approximation order and construction of locally supported elements for the space s14 of c1 quartic pp piecewise polynomial functions on a triangulation of a connected polygonal domain in r2 it is well known that when is a three directional mesh 1 the order of approximation of s14 1 is only 4 not 5 though a local clough tocher refinement procedure of an arbitrary triangulation yields the optimal fifth order of approximation from the space s14 see 1 it needs more data points in addition to the vertex set of the triangulation in this paper we will introduce a particular mixed three directional mesh 3 and construct so called mixed three directional elements we prove that the space s14 3 achieves its optimal order of approximation by constructing an interpolation scheme using mixed three directional elements

homogeneous simplex splines homogeneous simplex splines also known as cone splines or multivariate truncated power functions are discussed from a perspective of homogeneous divided differences and polar forms this makes it possible to derive the basic properties of these splines in a simple and economic way in addition a construction of spaces of homogeneous simplex splines is considered which in the nonhomogeneous setting is due to dahmen micchelli and seidel a proof for this construction is presented based on knot insertion restricting the homogeneous splines to a sphere gives rise to spaces of spherical simplex splines

on a wildland fire model with radiation summary the 2 d simplied wildlandre model presented here is based on conservation laws and takes into account radiation as the dominant thermal transfer mechanism as well as convection which represents the eect of the wind and the slope the non dimensional equations are obtained using the frank kamenestkii change of variables the existence of weak solutions of the non linear reaction diusion problem obtained is established as a particular case of more general existence results using a compactness method the uniqueness of the weak solution is shown for a given initial data andxed parameters of the equations the approximate solution is obtained using a mixednite element method this preserves the continuity of theux through the inter element boundaries and allows to represent high gradients in the solution semi discrete error estimation is presented the convective term is solved by a splitting technique using godunov s method the numerical examples show the eciency of the algorithm in capturing there front even for critical values of the parameters copyright 2002 john wiley sons ltd

multivariate b splines on triangulated rectangles although many results in univariate spline theory have been extended to the higher dimensional settings by taking tensor products very little is known on the general theory of multivariate spline functions since a univariate spline function is a smooth piecewise polynomial separated by a set of points which are called knots a bivariate spline function is a smooth piecewise polynomial in two variables separated by a grid of curves and so on in the two dimensional setting for example if a domain g in ifi is divided into a finite or countable number of cells by a grid partition a then the space s a of multivariate or more precisely bivariate spline functions is the collection of all functions in p u such that the restriction of every s e s a to each cell of the grid partition is a polynomial p u 4 of total degree k namely

a cubic structure on an arbitrary quadrilateral abstract this paper presents a cubic structure on an arbitrary quadrilateral by means of spline interpolations the element has continuity of class c 1 some properties are also discussed

on mixed finite element methods for first order elliptic systems a physically based duality theory for first order elliptic systems is shown to be of central importance in connection with the galerkin finite element solution of these systems using this theory in conjunction with a certain hypothesis concerning approximation spaces optimal error estimates for galerkin type approximations are demonstrated an example of a grid which satisfies the hypothesis is given and numerical examples which illustrate the theory are provided

homogenization results for a coupled system of reaction diffusion equations abstract the macroscopic behavior of the solution of a coupled system of partial differential equations arising in the modeling of reaction diffusion processes in periodic porous media is analyzed our mathematical model can be used for studying several metabolic processes taking place in living cells in which biochemical species can diffuse in the cytosol and react both in the cytosol and also on the organellar membranes the coupling of the concentrations of the biochemical species is realized via various properly scaled nonlinear reaction terms these nonlinearities which model at the microscopic scale various volume or surface reaction processes give rise in the macroscopic model to different effects such as the appearance of additional source or sink terms or of a non standard diffusion matrix

minimizing the quasi interpolation error for bivariate discrete quasi interpolants we define a class of discrete quasi interpolants based on bivariate box splines by imposing the exactness on a space of polynomials of total degree depending on the box spline and minimizing a constant appearing in the leading term of an appropriate quasi interpolation error estimate we give some c 1 quadratic and c 2 quartic examples and compare them with other well known quasi interpolants

degeneration of linear systems through fat points on k3 surfaces in this paper we introduce a technique to degenerate k3 surfaces and linear systems through fat points in general position on k3 surfaces using this degeneration we show that on generic k3 surfaces it is enough to prove that linear systems with one fat point are non special in order to obtain the non speciality of homogeneous linear systems through n 4 u 9 w fat points in general position moreover we use this degeneration to obtain a result for homogeneous linear systems through n 4 u 9 w fat points in general position on a general quartic surface in p 3

regularity of linear systems of plane curves the subject of this paper is the study of linear systems of plane curves which are given by assigning a set of fixed multiple points this study not only has an interest per se but is connected to other subjects for example results on rational surfaces and curves on them can be found in ro a and gil a linear system of plane curves of projective dimension r is a subvector space i s

existence in the large for nonlinear delay evolution inclusions with nonlocal initial conditions in this paper we provide a sufficient condition for the existence of c0 solutions for a class of nonlinear functional differential evolution equation of the form u t au t f t t r f t f t u t ut t r u t g u t t 0 where x is a real banach space a is the infinitesimal generator of a nonlinear compact semigroup f r x c 0 d a x is a nonempty convex and weakly compact valued multi function and g cb d a c 0 d a

on the optimal approximation rates for criss cross finite element spaces abstract let denote the triangulation of the plane obtained by multi integer translates of the four lines x 0 y 0 x y and x y by l k h we mean the space of all piecewise polynomials of degree k with respect to the scaled triangulation h having continuous partial derivatives of order on r 2 we show that the approximation properties of l k h are completely governed by those of the space spanned by the translates of all so called box splines contained in l k h combining this fact with fourier analysis techniques allows us to determine the optimal controlled approximation rates for the above subspace of box splines where is the largest degree of smoothness for which these spaces are dense as h tends to zero furthermore we study the question of local linear dependence of the translates of the box splines for the above criss cross triangulations

multivariate spline approximation of the signed distance function the signed distance function can effectively support many geometry processing tasks such as decimates smoothing and shape reconstruction since it provides efficient access to distance estimates in this paper we present an adaptive method to approximate the signed distance function of a smooth curve by using polynomial splines over type 2 triangulation the trimmed offsets are also studied

on multi level bases for elliptic boundary value problems abstract we study the multi level method for preconditioning a linear system arising from a galerkin discretization method of an elliptic boundary value problem of order 2r the solution is approximated in the spline space s10 n when r 1 and s3r 1r 1 n or s 3r 3 r 1 n when r 2 where sd denotes a spline space of smoothness and degree d n is the nth either uniform or nonuniform refinement of a given triangulation 0 and n is the triangulation obtained from the nth refinement either uniform or nonuniform of a given quadrangulation we show that we can construct a multi level basis in these spline spaces which preconditions the linear system so that its condition number is o n 1 2

a class of nonlinear evolution equations subjected to nonlocal initial conditions abstract we prove the existence of c 0 solutions for a class of nonlinear evolution equations subjected to nonlocal initial conditions of the form u t a u t f t f t f t u t u 0 g u where a d a x x is an m accretive operator acting on the infinite dimensional banach space x f 0 2 d a x is an almost strongly weakly u s c multi function which satisfies an appropriate sign condition while g c 0 2 d a d a is a continuous function

developments in bivariate spline interpolation the aim of this survey is to describe developments in the field of interpolation by bivariate splines we summarize results on the dimension and the approximation order of bivariate spline spaces and describe interpolation methods for these spaces moreover numerical examples are given

lagrange interpolation by bivariate splines on cross cut partitions the piecewise algebraic curve is a generalization of the classical algebraic curve in this paper we give the nother type theorem of piecewise algebraic curves on cross cut partitions by using interpolation along a piecewise algebraic curve we also present a method to construct lagrange interpolation sets for bivariate spline spaces on cross cut partitions

cardinal interpolation by multivariate splines the purpose of this paper is to investigate cardinal interpolation using locally supported piecewise polynomials in particular the notion of a commutator is introduced and its connection with the marsden identity is observed the order of a commutator is shown to be equivalent to the strang and fix conditions that arise in the study of the local approxima tion orders using quasi interpolants we also prove that scaled cardinal interpolants give these local approximation orders introduction cardinal interpolation by bivariate box splines was first studied by de boor hollig and riemenschneider 3 the purpose of our paper is to investigate the cardinal interpolation problem from a different point of view in particular the notion of a commutator is introduced it will be shown that this notion generalizes the marsden identity for univariate splines to the multivariate setting the order of a commutator will be shown to be equivalent to the strang and fix conditions used in the study of the order of controlled approximation by dahmen and micchelli 8 or local approximation by de boor and jia 4 an application to obtain approximation orders through the constructive method of scaled cardinal interpolation will also be studied in this paper 1 preliminaries this section consists of preliminary material for multivariate cardinal interpolation our approach is motivated by the work in 5 and 13 where cardinal interpolation in l2 was connected with certain convolution operators lr let z denote the set of integers and z the nonnegative ones for any given complex sequence by

progressive iterative approximation for regularized least square bivariate b spline surface fitting abstract recently the use of progressive iterative approximation pia to fit data points has received a deal of attention benefitting from its simplicity flexibility and generality in this paper we present a novel progressive iterative approximation for regularized least square bivariate b spline surface fitting rlspia rlspia extends the pia property of univariate ntp normalized totally positive bases to linear dependent non tensor product bivariate b spline bases which leads to a lower order fitting result than common tensor product b spline surface during each iteration the weights for generating fairing updating surface are obtained by solving an energy minimization problem with box constraints iteratively furthermore an accelerating term is introduced to speed up the convergence rate of rlspia which is comparable favourably with the theoretical optimal one several examples are provided to illustrate the efficiency and effectiveness of the proposed method

spline interpolations over type 2 triangulations this paper presents a structure of bivariate spline interpolation bases on nonuniform type 2 triangulations the approximation properties of spline interpolations are also considered

linear systems of plane curves with a composite number of base points of equal multiplicity in this article we study linear systems of plane curves of degree d passing through general base points with the same multiplicity at each of them these systems are known as homogeneous linear systems we especially investigate for which of these systems the base points with their multiplicities impose independent conditions and which homogeneous systems are empty such systems are called non special we extend the range of homogeneous linear systems that are known to be non special a theorem of evain states that the systems of curves of degree d with 4 h base points with equal multiplicity are non special the analogous result for 9 h points was conjectured both of these will follow as corollaries from the main theorem proved in this paper also the case of 4 h 9 k points will follow from our result the proof uses a degeneration technique developed by c ciliberto and r miranda

multivariate spline spaces in principle all of the properties and applications of multivariate spline functions can be derived from the basic theories in chapter 1 the expressions of the multivariate spline functions in 2 of chapter 1 are inconvenient to be applied directly because the expressions also rely on the solutions of the global conformality conditions so as to determine all the smooth cofactors

approximation by smooth multivariate splines abstract one of the important properties of univariate splines is that in most senses smooth splines approximate just as well as do piecewise polynomials on the same mesh this report shows this to be untrue in the multivariate setting in particular it details the cost in approximating power one may have to pay for the luxury of a smooth piecewise polynomial approximant in an extreme case piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or rho continuous will fail to approximate certain smooth functions at all as the grid goes to zero unless rho is kept below r 3 2 during the analysis of approximation on a certain regular triangular grid a novel kind of bivariate b spline is introduced this b spline in contrast to the established multivariate b spline derived from a simplex can be made to have all its breaklines in a given regular grid this makes it a prime candidate for use in the construction of smooth multivariate piecewise polynomial approximation and its properties will be explored further

homology of smooth splines generic triangulations and a conjecture of strang for a a triangulated d dimensional region in rd let sr a denote the vector space of all cr functions f on a that restricted to any simplex in a are given by polynomials of degree at most m we consider the problem of computing the dimension of such spaces we develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds a in the plane getting lower bounds on the dimension of sr a for all r for r 1 we prove a conjecture of strang concerning the generic dimension of the space of cl splines over a triangulated manifold in r2 finally we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region

vector bundles on complex projective spaces and systems of partial differential equations i on etablit et on etudie une relation entre l espace des solutions d un systeme d equations aux derivees partielles a coefficient constant et la cohomologie d un fibre vectoriel associe faisceau reflexif sur un espace projectif complexe

the correspondence between multivariate spline ideals and piecewise algebraic varieties as a piecewise polynomial with a certain smoothness the spline plays an important role in computational geometry the algebraic variety is the most important subject in classical algebraic geometry as the zero set of multivariate splines the piecewise algebraic variety is a generalization of the algebraic variety in this paper the correspondence between piecewise algebraic varieties and spline ideals is discussed furthermore hilbert s nullstellensatz for the piecewise algebraic variety is also studied

translates of multivarlate splines abstract we study spaces generated by translations of a fixed function over lattice points we provide some new algebraic and approximation properties for these spaces which show their applicability for finite element analysis

a note on bivariate box splines on a k direction mesh abstract we determine the dimension of the polynomial subspace of the linear space spanned by the translates over lattice points of a bivariate box spline on a k direction mesh

structure and dimension of multivariate spline space of lower degree on arbitrary triangulation in this paper we discuss the structure of multivariate spline spaces on arbitrary triangulation by using the methods and results of smoothing cofactor and generator basis of modules on the base of analyzing the algebraic and geometric results about singularity of s21 ms we build the structure of triangulation and give some useful geometric conditions such that s 1 space is singular and we obtain an algebraic condition which is necessary and sufficient for the singularity of s 1 spaces as well as their dimension formulae moreover the structure matrix of spline spaces over any given partition is defined which has been used to discuss the structure of s31 and s21 spaces over arbitrary triangulation and to prove the nonsingularity of s31 spaces this partially settles a conjecture on the singularity of spline spaces in wang et al multivariate spline and its applications kluwer press dordrecht 2002 academic press beijing 1994 in chinese meanwhile the dimension formulae of s31 s21 spaces and the dimension formulae of s 1 1 spaces are also given in this paper

bivariate spline interpolation at grid points we describe algorithms for constructing point sets at which interpolation by spaces of bivariate splines of arbitrary degree and smoothness is possible the splines are defined on rectangular partitions adding one or two diagonals to each rectangle the interpolation sets are selected in such a way that the grid points of the partition are contained in these sets and no large linear systems have to be solved our method is to generate a net of line segments and to choose point sets in these segments which satisfy the schoenberg whitney condition for certain univariate spline spaces such that a principle of degree reduction can be applied in order to include the grid points in the interpolation sets we give a sufficient schoenberg whitney type condition for interpolation by bivariate splines supported in certain cones this approach is completely different from the known interpolation methods for bivariate splines of degree at most three our method is illustrated by some numerical examples

on p laplacian differential inclusions global existence compactness properties and asymptotic behavior abstract in this work we consider coupled systems of p laplacian differential inclusions we obtain the global existence of solutions and good compactness properties in such way that the set of all possible solutions compose a generalized semiflow which has a global attractor for each pair of positive diffusion coefficients we also prove that the attractors are upper semicontinuous on positive finite diffusion parameters

on structure of bivariate spline space of lower degree over arbitrary triangulation the structure of bivariate spline space over arbitrary triangulation is complicated because the dimension of a multivariate spline space depends not only on the topological property of the triangulation but also on its geometric property a new vertex coding method to a triangulation is introduced in this paper to further study structure of the spline spaces the upper bound of the dimension of spline spaces over triangulation given by l l schumaker is slightly improved via the new vertex coding method the structure of multivariate spline spaces s 2 1 and s 3 1 over arbitrary triangulation are studied via the method of smoothness cofactor and the structure matrix of multivariate spline ring by luo and wang a kind of sufficient conditions on judging non singularity of the s 2 1 and s 3 1 spaces over arbitrary triangulation is given which only depends on the topological property of the triangulation from the sufficient conditions a triangulation strategy is presented at the end of the paper the strategy ensures that the constructed triangulation is non singular or generic for s 2 1 and s 3 1

degenerations of planar linear systems fixing n general points pi in the plane what is the dimension of the space of plane curves of degree d having multiplicity mi at pi for each i in this article we propose an approach to attack this problem and demonstrate it by successfully computing this dimension for all n and for mi constant at most 3 this application while previously known see hi1 demonstrates the utility of our approach which is based on an analysis of the corresponding linear system on a degeneration of the plane itself leading to a simple recursion for these dimensions we also obtain results in the quasi homogeneous case when all the multiplicities are equal except one this is the natural family to consider in the recursion

multivariate spline and algebraic geometry abstract the purpose of this survey is to emphasize the special relationship between multivariate spline and algebraic geometry we will not only point out the algebraic geometric method of multivariate spline but also the algebraic geometric background and essence of multivariate spline especially we have made an introduction to the so called piecewise algebraic curve piecewise algebraic variety and some of their properties

topological degree methods for perturbations of operators generating compact c 0 semigroups abstract the main aim of this paper is to construct a topological degree for maps a f m d a e where a densely defined closed operator a d a e of a banach space e is such that a is the generator of a compact c 0 semigroup and f m e is a locally lipschitz map defined on a neighborhood retract m e if m is a closed convex cone then a degree formula allowing an effective computation of the degree is proved this formula provides an infinite dimensional counterpart of the well known krasnosel skii theorem by the use of the introduced topological degree and an abstract result concerning branching of fixed points the bifurcation of periodic points of the parameterized boundary value problem u au f t u 0 u t m u 0 u t is studied examples of applications to partial differential equations are discussed

a viability result for a class of fully nonlinear reaction diffusion systems abstract we consider a class of abstract evolution systems governed by continuous perturbations of m dissipative operators and we prove a necessary and sufficient condition for a given locally closed set to be c 0 viable we conclude with a comparison result for a fully nonlinear reaction diffusion system

stable and unstable cross grid pkql mixed finite elements for the stokes problem in this paper we develop and analyze a family of mixed finite element methods for the numerical solution of the stokes problem in two space dimensions in these schemes the pressure is interpolated on a mesh of rectangular elements while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals continuous interpolations of degrees k for the velocity and l for the pressure are considered so the new finite elements are called cross grid p kq l a stability analysis of these approximations is provided based on the macroelement technique of stenberg the lowest order p 1q 1 and p 2q 1 cases are analyzed in detail in the first case a global spurious pressure mode is shown to exist so this element is unstable in the second case however stability is rigorously proved numerical results obtained in these two cases are also presented which confirm the existence of the spurious pressure mode for the p 1q 1 element and the stability of the p 2q 1 element

on the solution of certain systems of partial difference equations and linear dependence of translates of box splines on resout des systemes d equations aux differences partielles qui apparaissent dans l etude de la dependance lineaire des translatees de splines en boite

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

semi cardinal interpolation and difference equations from cubic b splines to a three direction box spline construction this paper considers the problem of interpolation on a semi plane grid from a space of box splines on the three direction mesh building on a new treatment of univariate semi cardinal interpolation for natural cubic splines the solution is obtained as a lagrange series with suitable localization and polynomial reproduction properties it is proved that the extension of the natural boundary conditions to box spline semi cardinal interpolation attains half of the approximation order of the cardinal case

existence for nonlinear evolution inclusions with nonlocal retarded initial conditions abstract in this paper we prove a sufficient condition for the global existence of bounded c 0 solutions for a class of nonlinear functional differential evolution equation of the form u t a u t f t t 0 f t f t u t u t 1 u t n t 0 u t g u t t 0 where x is a real banach space a is the infinitesimal generator of a nonlinear compact semigroup f r d a n 1 x is a nonempty convex weakly compact valued and almost strongly weakly u s c multi function and g c b d a c 0 d a is nonexpansive

existence for reaction diffusion systems a compactness method approach the authors study the existence of weak solutions for the following system ut u f u v vt v g u v in 0 t u v 0 on 0 t u 0 x u0 x v 0 x v0 x in the region rn with smooth boundary the functions r r are assumed to be continuous and nondecreasing with 0 0 0 u0 v0 l f g r2 2r with f an upper semicontinuous mapping u s c the following local existence results are shown 1 for the diffusive case i e when both and are strictly increasing with u s c g 2 for the semi diffusive case only one function is strictly increasing with g being either with separated variables i e having the form of the product or of the sum of two functions g u and h v or globally lipschitz with respect to its second variable i e g u v g u v l v v for each u b r v v r and some l l b additional conditions of linear form on the growth of f and g are indicated to guarantee global existence results

compactness results with applications for nonautonomous coupled inclusions abstract in this work we provide compactness results which are generalizations of baras theorem for the case when the main operator is time dependent and we give applications for nonautonomous coupled inclusions proving that these multivalued problems define exact generalized processes

the dimension and basis of spaces of multivariate splines abstract the purpose of this paper is to study the recent development of certain aspects of multivariate spline spaces more specifically we will survey the existence of bivariate spline the dimension and basis of the bivariate spline spaces s k with various partitions furthermore we will also introduce some results on the higher dimensional splines

bivariate box splines and smooth pp functions on a three direction mesh abstract let s denote the space of bivariate piecewise polynomial functions of degree k and smoothness on the regular mesh generated by the three directions 1 0 0 1 1 1 we construct a basis for s in terms of box splines and truncated powers this allows us to determine the polynomials which are locally contained in s and to give upper and lower bounds for the degree of approximation for 2k 2 3 k 2 3 the case of minimal degree k for given smoothness we identify the elements of minimal support in s and give a basis for s loc f s supp f with a convex subset of r 2

well posed p laplacian problems with large diffusion abstract in this work we study the asymptotic behavior of p laplacian parabolic problems of the form u t d p u u p 2 u b u in a bounded smooth domain in r n and under neumann boundary conditions when the diffusion coefficient d becomes large we prove under suitable assumptions that the family of attractors behaves lower and upper semicontinuously as the diffusion increases to infinity