quadratic pencils and least squares piecewise polynomial approximation for a partition o o m denotes the space of piecewise polynomials of order k and of smoothness m 1 this space can be represented as the graph of the appropriate linear operator between two finite dimensional hilbert spaces it gives an approach to the c de boor problem 1972 on uniform boundedness with respect to in the l norm of the orthogonal projections onto stm and we give the detailed analysis of a quadratic pencil matrix valued polynomial of the second degree which appears in the case of a geometric mesh if 2m k the explicit calculations and estimates of zeros of the characteristic polynomial show that in the case sbx x the geometric mesh with the parameter x 0 m 0 and any partition ax n of the unit interval 0 1 01 0 1 0 1 n n i 1 we define the piecewise polynomial subspace 0 2 s t km n c m1 0 11 where p fel 01vfia k i pk e f ej l il f ia is a polnom1ial caxi of order k 0 s a s n each interior breakpoint a 1 a n generates m continuity conditions 0 3 ia c af ia ya i 0 in to make a functionf e pkt be an element of scm let q qs be the orthogonal projection onto s in l2 0 1 i e with respect to the inner product f g f x g x dx we are interested in q as a map in c 0 1 or l 0 1 and we would like to get the estimates of its norm iiqiioo sup iiqfli lflfloo f where ilf 11k0 esssupo x if x i f e l o 1 received september 29 1980 1980 mathematics subject classification primary 41a1 5 secondary 47a68 this paper was partly supported by nsf grant 7906079 1983 american mathematical society 0025 571 8 82 0000 0750 05 25

weyl groups of fine gradings on matrix algebras octonions and the albert algebra given a grading a g gag on a nonassociative algebra a by an abelian group g we have two subgroups of aut a the automorphisms that stabilize each component ag as a subspace and the automorphisms that permute the components by the weyl group of we mean the quotient of the latter subgroup by the former in the case of a cartan decomposition of a semisimple complex lie algebra this is the automorphism group of the root system i e the so called extended weyl group a grading is called fine if it cannot be refined we compute the weyl groups of all fine gradings on matrix algebras octonions and the albert algebra over an algebraically closed field of characteristic different from 2 in the case of the albert algebra

the dixmier map for nilpotent super lie algebras in this article we prove that there exists a dixmier map for nilpotent super lie algebras in other words if we denote by mathrm prim mathcal u mathfrak g the set of graded primitive ideals of the enveloping algebra mathcal u mathfrak g of a nilpotent lie superalgebra mathfrak g and mathcal a d 0 the adjoint group of mathfrak g 0 we prove that the usual dixmier map for nilpotent lie algebras can be naturally extended to the context of nilpotent super lie algebras i e there exists a bijective map i mathfrak g 0 mathcal a d 0 rightarrow mathrm prim mathcal u mathfrak g defined by sending the equivalence class of a functional to a primitive ideal i of mathcal u mathfrak g and which coincides with the dixmier map in the case of nilpotent lie algebras moreover the construction of the previous map is explicit and more or less parallel to the one for lie algebras a major difference with a previous approach cf 18 one key fact in the construction is the existence of polarizations for super lie algebras generalizing the concept defined for lie algebras as a corollary of the previous description we obtain the isomorphism mathcal u mathfrak g i lambda simeq rm cliff q k otimes a p k where p q rm dim mathfrak g 0 mathfrak g 0 lambda 2 rm dim mathfrak g 1 mathfrak g 1 lambda we get a direct construction of the maximal ideals of the underlying algebra of mathcal u mathfrak g and also some properties of the stabilizers of the primitive ideals of mathcal u mathfrak g

voronoi delaunay analysis of voids in systems of nonspherical particles the voronoi network is known to be a useful tool for the structural description of voids in the packings of spheres produced by computer simulations in this article we extend the voronoi delaunay analysis to packings of nonspherical convex objects main properties of the voronoi network which are known for systems of spheres are valid for systems of any convex objects a general numerical algorithm for calculation of the voronoi network in three dimensions is proposed it is based on the calculation of the trajectory of the imaginary empty sphere of variable size moving inside a system the delaunay empty sphere method analysis of voids is presented for an ensemble of random straight lines and for a molecular dynamics model of liquid crystal the spatial distribution of voids and a simple percolation analysis are obtained the distributions of the bottleneck radii and the radii of spheres inscribed in the voids are calculated

group gradings on associative algebras abstract let r g g rg be a g graded ring we describe all types of gradings on r if g is torsion free and r is artinian semisimple if r is a matrix algebra over an algebraically closed field f then we give a description of all g gradings on r provided that g is an abelian group in the case of an abelian group g we also classify all finite dimensional graded simple algebras and finite dimensional graded division algebras over an algebraically closed field of characteristic zero

two stage approximation methods with extended b splines we develop and analyze a framework for two stage methods with eb splines applicable to continuous and discrete approximation problems in particular we propose a weighted discrete least squares fit that yields optimal convergence rates for sufficiently dense data on lipschitz domains in rd

generalizations of the quaternion algebra and lie algebras natural generalizations of the quaternion algebra called quaternionlike algebras ql algebras are considered the notion of a lie algebra induced by a ql algebra is introduced and a classification of such lie algebras is presented it is shown that local lie groups defined by lie algebras induced by ql algebras exhaust with accuracy to local isomorphisms all local lie groups endowed with some simple composition laws of their local parameters

gradings on semisimple algebras abstract the classification of gradings by abelian groups on finite direct sums of simple finite dimensional nonassociative algebras over an algebraically closed field is reduced by means of the use of loop algebras to the corresponding problem for simple algebras this requires a good definition of free products of group gradings

a multigrid method with unstructured adaptive grids for steady euler equations the multigrid method based on multi stage jacobi relaxation earlier developed by the authors for structured grid calculations with euler equations is extended to unstructured grid applications the meshes are generated with delaunay triangulation algorithms and are adapted to the flow solution

on the cartan jacobson theorem abstract the well known cartan jacobson theorem claims that the lie algebra of derivations of a cayley algebra is central simple if the characteristic is not 2 or 3 in this paper we have studied these two cases with the following results if the characteristic is 2 the theorem is also true but if the characteristic is 3 the derivation algebra is not simple we have also proved that in this last case there is a unique nonzero proper seven dimensional ideal which is a central simple lie algebra of type a2 and the quotient of the derivation algebra modulo this ideal turns out to be isomorphic as a lie algebra to the ideal itself the original motivation of this work was a series of computer aided calculations which proved the simplicity of derivation algebras of cayley algebras in the case of characteristic not 3 these computations also proved the existence of a unique nonzero proper ideal which turns out to be seven dimensional in the algebra of derivations of split cayley algebras in characteristic 3

error estimates for scattered data interpolation on spheres we study sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n sphere the interpolation knots are scattered our approach partly follows the general theory of golomb and weinberger and related estimates these error estimates are then based on series expansions of smooth functions in terms of spherical harmonics the markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics

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

fine gradings on simple classical lie algebras abstract the fine abelian group gradings on the simple classical lie algebras including d 4 over algebraically closed fields of characteristic 0 are determined up to equivalence this is achieved by assigning certain invariant to such gradings that involve central graded division algebras and suitable sesquilinear forms on free modules over them

weighted extended b spline approximation of dirichlet problems we describe a new finite element method which uses weighted extended b splines on a regular grid as basis functions for solving dirichlet problems on bounded domains in arbitrary dimensions this web method does not require any grid generation and can be implemented very efficiently it yields smooth high order accurate approximations with relatively low dimensional subspaces

notes on a cross product of vectors the relation between quaternionlike algebras and cross products of vectors is demonstrated a classification of all cross products of vectors is given

inverses of infinite sign regular matrices let a be an infinite sign regular sr matrix which can be viewed as a bounded linear operator from lx to itself it is proved here that if the range of a contains the sequence 1 1 1 1 then a is onto if a exists then da d is also sr where d is the diagonal matrix with diagonal entries alternately 1 and 1 in case a is totally positive tp then da d is also tp under additional assumptions on a 0 introduction if the problem of spline interpolation is expressed in terms of b splines then the question of existence of a bounded spline interpolant to bounded data is seen to be equivalent to the question of whether a certain bounded band matrix has all bounded sequences in its range in 5 c a micchelli conjectured that there exists a unique bounded spline interpolant of a given order and a given knot sequence to any data sequence t y z in the plane with t strictly increasing and y bounded provided only that it is possible to interpolate the particular data sequence t l ez by sucn a sp ine there is apparently nothing special about the particular spline problem other than that it leads to a banded totally positive matrix therefore one of us quoted this conjecture in 3 p 319 prob lem 4 as a bi infinite banded totally positive matrix a is boundedly invert ible if and only if the linear system ax 1 has a bounded solution 2 micchelli gave a simple argument for the case when a is a toeplitz matrix cavaretta dahmen micchelli and smith 2 recently proved the conjecture in case a is a block toeplitz matrix this is all the more remarkable since it is easy to see in hindsight that the conjecture is faulty even in the original context of spline interpolation for example interpolation by bounded broken lines with breakpoint sequence z at the sequence t z 0 is possible to any bounded ordinate sequence y but not uniquely so since the value of the interpolant at 0 is freely choosable in matrix terms this corresponds to the matrix obtained from the bi infinite identity matrix by dropping one row but with the condition changed to has a unique bounded solution the conjecture was proved in 1

on lie algebras with modular lattices of subalgebras the aim of the article is to classify lie algebras over an arbitrary field with modular lattices of subalgebras the study of various classes of universal algebras all subalgebras of which have a lattice satisfying the modular identity m algebras has a long history the first works which dealt the group case appeared at the end of the 1920s and the beginning of the 1930s a rottender 0 ore r baer there are a great number of papers devoted to the study of modular semi groups associative and nonassociative rings and algebras quasigroups modules over commutative rings and other algebraic structures however a full classification of the universal algebras with a modular lattice of subalgebras for any class of the classical universal algebras does not exist in the middle of the 1960s an attempt was made to describe lie algebras with various restrictions imposed on the lattice of subalgebras and in particular to describe the lie algebras with a modular lattice of subalgebras kolman cl 23 goto 3 barnes 4 a few years later the papers of towers s amayo 6 amayo and schwarz 7 gein s appeared in which they studied lie algebras with modular lattices of subalgebras similar problems were considered in 9 10 111 the above papers classified the lie algebras with modular lattices of subalgebras when certain restrictions are imposed on the ring of operators on 80 0021 8693186 53 00

on maximal subalgebras of central simple malcev algebras let f be a field of characteristic different from two and a a central sim ple non lie malcev algebra over f our purpose is to determine the maximal subalgebras of a with respect to inclusion maximal subalgebras of simple lie algebras over an algebraically closed field have been studied by dynkin 3 41 and maximal subalgebras of central simple associative alternative and special jordan algebras have been studied by racine 121 racine s results on octonion algebras will be relevant in our investigation 1

macro elements and stable local bases for splines on powell sabin triangulations macro elements of arbitrary smoothness are constructed on powell sabin triangle splits these elements are useful for solving boundary value problems and for interpolation of hermite data it is shown that they are optimal with respect to spline degree and we believe they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over powell sabin refinements these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

on locally complex algebras and low dimensional cayley dickson algebras the paper begins with short proofs of classical theorems by frobenius and resp zorn on associative and resp alternative real division algebras these theorems characterize the first three resp four cayley dickson algebras then we introduce and study the class of real unital nonassociative algebras in which the subalgebra generated by any nonscalar element is isomorphic to c we call them locally complex algebras in particular we describe all such algebras that have dimension at most 4 our main motivation however for introducing locally complex algebras is that this concept makes it possible for us to extend frobenius and zorn s theorems in a way that it also involves the fifth cayley dickson algebra the sedenions

scattered data interpolation by bivariate splines with higher approximation order given a set of scattered data we usually use a minimal energy method to find a lagrange interpolation in a bivariate spline space over a triangulation of the scattered data locations it is known that the approximation order of the minimal energy spline interpolation is only 2 in terms of the size of triangulation to improve this order of approximation we propose several new schemes in this paper mainly we follow the ideas of clamped cubic interpolatory splines and not a knot interpolatory splines in the univariate setting and extend them to the bivariate setting in addition instead of the energy functional of the second order we propose to use higher order versions we shall present some theoretical analysis as well as many numerical results to demonstrate that our new interpolation schemes indeed have a higher order of approximation than the classic minimal energy interpolatory spline

maximal subalgebras of associative superalgebras abstract the maximal subalgebras of the finite dimensional central simple associative superalgebras possibly endowed with a superinvolution are determined this relies on the corresponding description by m racine in the ungraded case which is completed here too

gradings on tensor products of composition algebras and on the smirnov algebra abstract we give classifications of group gradings up to equivalence and up to isomorphism on the tensor product of a cayley algebra c and a hurwitz algebra over a field of characteristic different from 2 we also prove that the automorphism group schemes of c n and c n are isomorphic on the other hand we prove that the automorphism group schemes of a smirnov algebra t c a 35 dimensional simple exceptional structurable algebra constructed from a cayley algebra c and c are isomorphic this is used to obtain classifications up to equivalence and up to isomorphism of the group gradings on smirnov algebras

fine gradings on simple exceptional jordan pairs and triple systems abstract we give a classification up to equivalence of the fine group gradings by abelian groups on the jordan pairs and triple systems of types bi cayley and albert under the assumption that the base field is algebraically closed of characteristic different from 2 the associated weyl groups are computed we also determine for each fine grading on the bi cayley and albert pairs the induced grading on the exceptional simple lie algebra given by the tits kantor koecher construction

models of the octonions and g2 different models of the cayley algebras and of their lie algebras of derivations are given based on some distinguished subalgebras of the later ones

hierarchic finite element bases on unstructured tetrahedral meshes the problem of constructing hierarchic bases for finite element discretization of the spaces h1 h curl h div and l2 on tetrahedral elements is addressed a simple and efficient approach to ensuring conformity of the approximations across element interfaces is described hierarchic bases of arbitrary polynomial order are presented it is shown how these may be used to construct finite element approximations of arbitrary non uniform local order approximation on unstructured meshes of curvilinear tetrahedral elements

classification of group gradings on simple lie algebras of types a b c and d abstract for a given abelian group g we classify the isomorphism classes of g gradings on the simple lie algebras of types a n n 1 b n n 2 c n n 3 and d n n 4 in terms of numerical and group theoretical invariants the ground field is assumed to be algebraically closed of characteristic different from 2

stability of optimal order approximation by bivariate splines over arbitrary triangulations let a be a triangulation of some polygonal domain in r2 and skr a the space of all bivariate cr piecewise polynomials of total degree 3r 2 that can be used to provide the highest order of approximation with the property that the approximation constant of this order is independent of the geometry of a with the exception of the smallest angle in the partition this result is obtained by means of a careful choice of locally supported basis functions which however require a very technical proof to justify their stability in optimal order approximation a new formulation of smoothness conditions for piecewise polynomials in terms of their b net representations is derived for this purpose

gradings on symmetric composition algebras abstract the group gradings on the symmetric composition algebras over arbitrary fields are classified applications of this result to gradings on the exceptional simple lie algebras are considered too

division composition algebras through their derivation algebras abstract composition algebras are algebras with a nondegenerate quadratic multiplicative form in this paper we construct five families of division composition algebras and prove that any division composition algebra with nonabelian derivation algebra belongs to one of these families as a by product we obtain new examples of real division algebras

a series of algebras generalizing the octonions and hurwitz radon identity we study non associative twisted group algebras over mathbb z 2 n with cubic twisting functions we construct a series of algebras that extend the classical algebra of octonions in the same way as the clifford algebras extend the algebra of quaternions we study their properties give several equivalent definitions and prove their uniqueness within some natural assumptions we then prove a simplicity criterion

the metric of a space and quadratic algebras a collection of algebras with product that preserves the metric of the underlying vector space is investigated it is shown that this collection includes the generalized quaternions the composition algebras and the algebra of color all the algebras in this collection that are either commutative or alternative are determined the ramifications of nondegenerate metrics are detailed if the algebra is alternative and not of dimension 1 2 4 or 8 over the base field the metric must be degenerate

the inverse of a totally positive biinfinite band matrix it is shown that a bounded bi infinite banded totally positive matrix a is boundedly invertible iff there is one and only one bounded sequence mapped by a to the sequence the argument shows that such a matrix has a main diagonal i e the inverse of a is the bounded pointwise limit of inverses of finite sections of a principal with respect to a particular diagonal hence 4 j 7 or its inverse is again totally positive 0 introduction this paper is a further step in a continuing effort to understand certain linear spline approximation schemes convergence of such processes is intimately tied to their stability i e to their boundedness as maps on c say use of the b spline basis shows this question to be equivalent to bounding the inverse of certain totally positive band matrices the calculation of bounds on the inverse of a given matrix is in general a difficult task it is hoped that the present investigation into the consequences of bandedness and total positivity for the structure of the inverse may ultimately prove helpful in obtaining such bounds the results in this paper were obtained in the study of a conjecture due to c a micchelli 7 in connection with his work on the specific approximation scheme of interpolation at a strictly increasing point sequence t by elements of sm t i e by splines of some order m with some knot sequence t micchelli became convinced that every bounded function has one and only one bounded spline interpolant iff the particular function which satisfies t all has a bounded spline interpolant in sm if ni ni mt denotes the corresponding b spline basis for sm then micchelli s conjecture can be phrased thus the matrix

on ad semisimple lie algebras the question whether any ad semisimple lie algebra is commutative is related to the existence of simple semiabelian lie algebras these algebras determine the brauer group of a field and are in addition to that significant for the study of lie algebras all whose proper subalgebras are nilpotent in this paper we consider a class of lie algebras related to several topics of lie algebra theory and field theory we begin with the study of finite dimensional lie algebras g having the property that the adjoint representation sends every element of g to a semisimple derivation we call such a lie algebra ad semisimple our study of these may be regarded as a complement to the classical investigation of ad nilpotent lie algebras due to engel however the results of the first section show that in contrast with engels theory the theory of ad semisimple lie algebras is highly sensitive to the choice of the base field this feature becomes more evident if we confine our studies to the subclass of simple semiabelian lie algebras sections 3 5 are devoted to a brief discussion of some aspects of the structure of this interesting class a more thorough investigation can be found in 15 1 theorem 5 2 and section 6 are closely related to a paper by towers 11 i after discussing the relationship between simple semiabelian lie algebras and restricted lie algebras having a nonsingular p map i e a p map whose only zero is 0 we illustrate in the final section some aspects of the linkage between the brauer group of the underlying field and the isomorphism classes of simple semiabelian lie algebras the brauer group of the field admitting no simple semiabelian lie algebras is trivial it is conceivable that the brauer group determines the isomorphism classes of simple semiabelian lie algebras the cases of ir and 6 for example indicate the existence of a more general yet discerning relationship

about the delaunay voronoi tesselation abstract the topological description of the neighborhood of the constituing atoms in a dense random material liquids glasses is of a major importance for their characterization we discuss an algorithm for the delaunay construction which is global and independent of the dimension of the space and derive from it the full dual voronoi tesselation by an original realization of duality

detecting structure in two dimensions combining voronoi tessellation and percolation conventional source detection algorithms in high energy astrophysics and other fields mostly use spherical or quadratic sliding windows of varying size on two dimensionally binned representations of spatial event distributions in order to detect statistically significant event enhancements sources within a given field while this is a reasonably reliable technique for nearly pointlike sources with good statistics poor and extended sources are likely to be incorrectly assessed or even missed at all as the calculations are governed by nonphysical parameters like the bin size and the window geometry rather than by the actual data the approach presented here does not introduce any artificial bias but makes full use of the unbinned two dimensional event distribution a voronoi tessellation on a finite plane surface yields individual densities or fluxes for every single event the distribution of which allows the determination of the contribution from a random poissonian background field noise the application of a nonparametric percolation to the tessellation cells exceeding this noise level leads directly to a source list which is free of any assumptions about the source geometry high density fluctuations from the random background field will still be included in this tentative source list but can be easily eliminated in most cases by setting a lower threshold to the required number of events per source since no finite size detection windows or the like have been used this analysis yields automatically straightforward fluxes for every source finally accepted the main disadvantage of this approach is the considerable cpu time required for the construction of the voronoi tessellation it is thus applicable only to either small fields or low event density regions

simple graded commutative algebras we study the notion of graded commutative algebra for an arbitrary abelian group the main examples are the clifford algebras already treated in h albuquerque and s majid 2002 2 we prove that the clifford algebras are the only simple finite dimensional associative graded commutative algebras over r or c our approach also leads to non associative graded commutative algebras extending the clifford algebras

a generalization of the construction of ilamed and salingaros ilamed and salingaros construct the real and complex algebras with three anticommuting elements which can arise in physics it is shown here that the algebra of color can be similarly constructed with six anticommuting elements as a consequence of this construction these algebras are all simple quadratic algebras

primitive superalgebras with superinvolution abstract our main purpose is to provide for primitive associative superalgebras a structure theory analogous to that for algebras 5 6 10 and to classify primitive superrings with superinvolution having a minimal one sided superideal we were led to this problem by our work on finite dimensional central simple jordan superalgebras over fields of characteristic not 2 9 see also 7 of course just as symmetric elements give rise to jordan superalgebras skewsymmetric elements give rise to lie superalgebras 8 4 the results and methods are closely related to those of structure theory of associative rings and central simple associative algebras with involution 5 chap i 6 chaps ii iii 1 chap x 10 chap 2 some of the results have been announced in 13

gradings on the real forms of the albert algebra of g2 and of f4 we find all the fine group gradings on the real forms of the albert algebra and of the exceptional lie algebras g2 and f4

algebras with three anticommuting elements i spinors and quaternions a general construction of alternative algebras with three anticommuting elements and a unit is given as an exhaustive result over the real and complex fields we obtain the clifford algebras h quaternions n1 dihedral clifford algebra which is related to real 2 spinors and s1 algebra of pauli matrices which is related to complex 2 spinors what is important is that the algebras n1 and s1 possess inverses everywhere except on a region akin to the light cone of the minkowski space while the quaternion algebra h has inverses everywhere except at the zero element we discuss the reasons why the three algebras n1 h and s1 are so difficult to distinguish in the representation space of 2 2 complex matrices

spherical splines for data interpolation and fitting we study minimal energy interpolation and discrete and penalized least squares approximation problems on the unit sphere using nonhomogeneous spherical splines several numerical experiments are conducted to compare approximating properties of homogeneous and nonhomogeneous splines our numerical experiments show that nonhomogeneous splines have certain advantages over homogeneous splines

maximal subalgebras of lie superalgebras of cartan type over fields of characteristic zero this paper aims to characterize the maximal graded subalgebras of lie superalgebras of cartan type w s s and h over fields of characteristic zero all maximal graded subalgebras are completely determined by virtue of a constructive method and their isomorphism classes dimension formulas are given except for maximal s subalgebras in the exceptional situation the classification of maximal s subalgebras is reduced to the classification of the maximal irreducible subalgebras for the classical lie algebras

hierarchic hp edge element families for maxwell s equations on hybrid quadrilateral triangular meshes we construct and study a set of hierarchic basis functions for the galerkin discretisation of the space h curl 169 suitable for hybrid meshes containing both quadrilateral and triangular elements with arbitrary non uniform order polynomial approximation we investigate the conditioning and dispersive behaviour of the elements in addition numerical examples are shown which demonstrate the accuracy of the space for computing solutions of the time harmonic maxwell s equations that have singularities

spin models on random lattices under certain conditions phase transitions in systems with quenched disorder are expected to exhibit a dierent behaviour than in the corresponding pure system here we discuss a series of monte carlo studies of a special type of such disordered systems namely spin models dened on quenched random lattices exhibiting geometrical disorder in the connectivity of the lattice sites in two dimensions we present results for the q state potts model on random tri valent 3 planar graphs which appear quite naturally in the dynamically triangulated random surface dtrs approach to quantum gravity as well as on poissonian random lattices of voronoi delaunay type both cases q64 and 4 are discussed which in the pure model without disorder give rise to second and rst order phase transitions respectively in three dimensions results for the ising model on poissonian random lattices are briey described we conclude with a comparison of the two types of connectivity disorder with the more standard case of bond disorder and a discussion of the distinguishing dierences c 2000 elsevier science b v all rights reserved

on the construction of the voronoi mesh on a sphere abstract a new construction of the voronoi mesh on the sphere is presented the main feature is that the algorithm adds points one at a time until the final voronoi mesh is built up by adding one point to an existing voronoi mesh of k points only local changes are needed to construct a voronoi mesh of k 1 points this construction is particularly well suited to time dependent problems since using information from the voronoi mesh at the previous time step allows us to reduce the construction to o n operations when the two configurations are close while the algorithm does not break down when they are far apart numerical experiments are presented to substantiate the o n operation count for a typical case

voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids voronoi cells and the notion of natural neighbours are used to develop a finite difference method for the diffusion operator on arbitrary unstructured grids natural neighbours are based on the voronoi diagram which partitions space into closest point regions the sibson and the laplace non sibsonian interpolants which are based on natural neighbours have shown promise within a galerkin framework for the solution of partial differential equations in this paper we focus on the laplace interpolant with a two fold objective first to unify the previous developments related to the laplace interpolant and to indicate its ties to some well known numerical methods and secondly to propose a voronoi cell finite difference scheme for the diffusion operator on arbitrary unstructured grids a conservation law in integral form is discretized on voronoi cells to derive a finite difference scheme for the diffusion operator on irregular grids the proposed scheme can also be viewed as a point collocation technique a detailed study on consistency is conducted and the satisfaction of the discrete maximum principle stability is established owing to symmetry of the laplace weight a symmetric positive definite stiffness matrix is realized which permits the use of efficient linear solvers on a regular rectangular or hexagonal grid the difference scheme reduces to the classical finite difference method numerical examples for the poisson equation with dirichlet boundary conditions are presented to demonstrate the accuracy and convergence of the finite difference scheme copyright 2003 john wiley sons ltd

gradings on octonions abstract all the different ways of grading any cayley dickson or octonion algebra as well as the associated central simple malcev algebras will be obtained

on the voronoi estimator for the intensity of an inhomogeneous planar poisson process summary the voronoi estimator may be defined for any location as the inverse of the area of the corresponding voronoi cell we investigate the statistical properties of this estimator for the intensity of an inhomogeneous poisson process and demonstrate it is approximately unbiased with a gamma sampling distribution we also introduce the centroidal voronoi estimator a simple extension based on spatial regularization of the point pattern simulations show the voronoi estimator has remarkably low bias while the centroidal voronoi estimator has slightly more bias but much less variability the performance is compared to kernel estimators using two simulated data sets and a dataset consisting of earthquakes within the continental united states

quadratic alternative algebras quadratic alternative algebras are completely classified up to isomorphism by means of the associated set of vectors they include the quaternion and octonion algebras but also many other nonassociative algebras used in physics

generalized quaternions and spacetime symmetries the construction of a class of associative composition algebras qn on r4 generalizing the well known quaternions q provides an explicit representation of the universal enveloping algebra of the real three dimensional lie algebras having tracefree adjoint representations class a bianchi type lie algebras the identity components of the four dimensional lie groups gl qn 1 qn general linear group in one generalized quaternion dimension which are generated by the lie algebra of this class of quaternion algebras are diffeomorphic to the manifolds of spacetime homogeneous and spatially homogeneous spacetimes having simply transitive homogeneity isometry groups with tracefree lie algebra adjoint representations in almost all cases the complete group of isometries of such a spacetime is isomorphic to a subgroup of the group of left and right translations and automorphisms of the appropriate generalized quaternion algebra similar results hold for the single class b lie algebra of bianchi type v characteri

macro elements and stable local bases for splines on clough tocher triangulations macro elements of arbitrary smoothness are constructed on clough tocher triangle splits these elements can be used for solving boundary value problems or for interpolation of hermite data and are shown to be optimal with respect to spline degree we conjecture they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over clough tocher refinements of arbitrary triangulations these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

algebras with anticommuting basal elements space time symmetries and quantum theory a collection a of algebras with anticommuting basal elements is investigated it is shown that the collection a includes the quaternions the octonions the algebra of color as well as other algebras familiar to the physicist each algebra a is a quadratic jordan admissible algebra and possesses a norm that is a generalization of the minkowski metric using a cayley dickson like process each such algebra a in a can be embedded into a larger algebra a that is also in the collection a these algebras should provide candidates for models to describe observables color and other phenomena encountered in particle physics

on the variety determined by symmetric quadratic algebras abstract we consider some polynomial identities of degree 5 which are satisfied by all symmetric quadratic algebras we call rings satisfying these identities generalized quadratic rings or gq rings we show that when the ring is not flexible these identities are enough to make the ring quadratic over its center therefore simple nonflexible gq rings are symmetric quadratic algebras over their center which is a field for prime gq rings the center has no nonzero zero divisors prime gq rings which are not flexible are subrings of the quadratic algebra formed by extending the center to its field of quotients flexible gq rings are noncommutative jordan rings which satisfy a quadratic condition over their commutative center we show that any semi prime gq ring is a subdirect sum of a noncommutative jordan ring which satisfies a quadratic condition over its commutative center and a nonflexible ring which is symmetric quadratic over its center

error estimates in w 2 semi norms for discrete interpolating d 2 splines we derive error estimates in w 2 semi norms for multivariate discrete d 2 splines that interpolate an unknown function at the vertices of given triangulations these results are widely based on the construction of approximation operators and linear projectors onto piecewise polynomial spaces having weakly stable local bases

l boundedness of l 2 projections on splines for a multiple geometric mesh this paper concerns the l2 projectors from loo to the normed linear space of polynomial splines it is shown that for the multiple geometric meshes the loo norms of the corresponding l2 projectors are bounded independently of the mesh ratio 0 introduction let us begin with some notation let k be a positive integer and x x i z a real nondecreasing knot sequence with xi xi k all i set x 00 lim xi i x o lim xji i x oo x00 i 0 i o00o fori e z and x e i define mi k x k xi xi k x i ni k x xi k xi mi k x k where po pr f denotes the rth divided difference of the function f at the points po 0 pr then ni k is called the i th b spline of order k on the knot sequence x for a e rz the rule f x a i niak x defines a function on i if we take the sum to be pointwise every such function is called a polynomial spline of order k with the knot sequence x and their collection is denoted by skx further let denote the normed linear space of bounded polynomial splines of order k with the knot sequence x and norm fll ii sup i f x ix xei we shall be concerned with p the orthogonal projector onto with respect to the ordinary inner product f g ff x g x dx but restricted to l i we want to bound its norm 11 p ii sup iipsf ii 1 iii o fel j received march 26 1984 revised december 16 1985 1980 mathematics subject classification primary 41a15 15a60 this is a part of the author s thesis which was submitted to the graduate school university of wisconsin madison in partial fullfillment of the requirement for ph d degree present address department of mathematics zhejiang university hangzhou zhejiang the people s republic of china 1987 american mathematical society 0025 5718 87 1 00 25 per page 675 this content downloaded from 207 46 13 178 on wed 15 jun 2016 07 00 20 utc all use subject to http about jstor org terms

pauli gradings on lie superalgebras and graded codimension growth abstract we introduce grading on certain finite dimensional simple lie superalgebras of type p t by elementary abelian 2 group this grading gives rise to pauli matrices and is a far generalization of z 2 z 2 grading on lie algebra of 2 2 traceless matrices we use this grading for studying numerical invariants of polynomial identities of lie superalgebras in particular we compute graded pi exponent corresponding to pauli grading

norms of projectors onto spaces with riesz bases abstract many important linear operators p x s of a linear space x onto a subspace s of x are defined by the minimum problem f p f min f u u s f x where the norm on x is induced by an inner product we study the l norm of such operators p when s has a riesz basis b then we apply the results in detail to the best l2 and l2 approximations for univariate polynomial splines spaces and to box spline series

minimal numerical differentiation formulas we investigate numerical differentiation formulas on irregular centers in two or more variables that are exact for polynomials of a given order and minimize an absolute seminorm of the weight vector error bounds are given in terms of a growth function that carries the information about the geometry of the centers specific forms of weighted ell 1 and weighted least squares minimization are proposed that produce numerical differentiation formulas with particularly good performance in numerical experiments the results are of interest in particular for meshless generalized finite difference methods as they provide a consistency error analysis for such methods

on the approximation power of splines on triangulated quadrangulations we study the approximation properties of the bivariate spline spaces srt cal s r 3r hbox diamondsuit kern 0 78em raise0 10786ex hbox of smoothness r and degree 3r defined on triangulations lai hbox diamondsuit kern 0 78em raise0 10786ex hbox which are obtained from arbitrary nondegenerate convex quadrangulations by adding the diagonals of each quadrilateral

quasi interpolation projectors for box splines we consider box spline quasi interpolants based on local linear functionals of point evaluator and integral type the approximations are easy to compute and reproduce the whole spline space in question

a domain decomposition method for computing bivariate spline fits of scattered data a domain decomposition method for solving large bivariate scattered data fitting problems with bivariate minimal energy discrete least squares and penalized least squares splines is described the method is based on splitting the domain into smaller domains solving the associated smaller fitting problems and combining the coefficients to get a global fit explicit error bounds are established for how well our locally constructed spline fits approximate the global fits some numerical examples are given to illustrate the effectiveness of the method

a new algorithm for three dimensional voronoi tessellation abstract a new efficient algorithm for computing the voronoi tessellation in three dimensions is presented the voronoi tessellation is obtained by constructing delaunay tetrahedra the complete set of these tetrahedra corresponds to that of a single voronoi polyhedron

on the lie structure of a prime associative superalgebra abstract in this paper some results on the lie structure of prime superalgebras are discussed we prove that with the exception of some special cases for a prime superalgebra a over a ring of scalars with 1 2 if l is a lie ideal of a and w is a subalgebra of a such that w l w then either l z or w z likewise if v is a submodule of a and v l v then either v z or l z or there exists an ideal of a m such that 0 m a v this work extends to prime superalgebras some results of i n herstein c lanski and s montgomery on prime algebras

gradings on the simple real lie algebras of types g2 and d4 abstract we classify group gradings on the simple lie algebras of types g 2 and d 4 over the field of real numbers or any real closed field fine gradings up to equivalence and g gradings with a fixed group g up to isomorphism