the determinants of gcd matrices abstract let s x1 x2 xn be an ordered set of distinct positive integers and s the gcd matrix defined on s the value of the determinant of s is obtained it is shown that det s x1 x2 xn iff s is factor closed

scattered data interpolation using c 2 supersplines of degree six we show how c2 supersplines of degree six can be used to interpolate hermite data at the vertices of a quadrangulation we also present error bounds that show that our method has full approximation order seven and compare its efficiency with other c2interpolation methods in the literature

higher dimensional gcd matrices abstract we evaluate the higher dimensional determinants of the greatest common divisor matrix defined on a gcd closed set and on an arbitrary set of distinct positive integers

divisibility of determinants of power gcd matrices and power lcm matrices on finitely many quasi coprime divisor chains abstract let a n 1 be integers and s x1 xn be a set of n distinct positive integers the matrix having the ath power xi xj a of the greatest common divisor of xi and xj as its i j entry is called ath power greatest common divisor gcd matrix defined on s denoted by sa similarly we can define the ath power lcm matrix sa we say that the set s consists of finitely many quasi coprime divisor chains if we can partition s as s s1 sk where k 1 is an integer and all si 1 i k are divisor chains such that max si max sj gcd s for 1 i j k in this paper we first obtain formulae of determinants of power gcd matrices sa and power lcm matrices sa on the set s consisting of finitely many quasi coprime divisor chains with gcd s s using these results we then show that det sa det sb det sa det sb and det sa det sb if a b and s consists of finitely many quasi coprime divisor chains with gcd s s but such factorizations fail to be true if such divisor chains are not quasi coprime

a c r trivariate macro element based on the worsey farin split of a tetrahedron we construct trivariate c r macro elements for any r ge 1 over the worsey farin refinement of any tetrahedral partition this extends the construction of c 1 cubic worsey farin elements and c 2 elements of degree nine to the c r situation with r 2 we also show that the degree of polynomials used for our macro elements is the lowest

nonsingularity of matrices associated with classes of arithmetical functions let s x1 xn be a set of n distinct positive integers let f be an arithmetical function let f xi xj denote the n n matrix having f evaluated at the greatest common divisor xi xj of xi and xj as its i j entry and f xi xj denote the n n matrix having f evaluated at the least common multiple xi xj of xi and xj as its i j entry the set s is said to be gcd closed if xi xj s for all 1 i j n for an integer x let x denote the number of distinct prime factors of x in this paper by using the concept of greatest type divisor introduced by s hong in adv math china 25 1996 566 568 j algebra 218 1999 216 228 we obtain a new reduced formula for detf xi xj if s is gcd closed then we show that if s x1 xn is a gcd closed set satisfying maxx s x 2 and if f is a strictly increasing respectively decreasing completely multiplicative function or if f is a strictly decreasing respectively increasing completely multiplicative function satisfying 0 f p 1p respectively f p p for any prime p then the matrix f xi xj respectively f xi xj defined on s is nonsingular as a corollary we show the following interesting result the lcm matrix xi xj defined on a gcd closed set is nonsingular if maxx s x 2

mesh generation using unstructured computational meshes and elliptic partial differential equation smoothing a novel appro ach for generating unstructured meshes using elliptic smoothing is presented like structured mesh generation methods the approach begins with the construction of a computational mesh the computational mesh is used to solve elliptic partial differential equations that control grid point distributions and improve mesh quality two types of elliptic partial differential equations are employed modified linear elastic theory and winslow equation s with or without forcing functions results are included to il lustrate the use of these methods for unstructured mesh generation mesh boundary shape modification mesh untangling and mesh movement

divisibility properties of power gcd matrices and power lcm matrices abstract let a b and n be positive integers and the set s x 1 x n of n distinct positive integers be a divisor chain i e there exists a permutation on 1 n such that x 1 x n in this paper we show that if a b then the a th power gcd matrix s a having the a th power x i x j a of the greatest common divisor of x i and x j as its i j entry divides the b th power gcd matrix s b in the ring m n z of n n matrices over integers we show also that if a b and n 2 then the a th power gcd matrix s a does not divide the b th power gcd matrix s b in the ring m n z similar results are also established for the power lcm matrices

construction of quintic powell sabin spline quasi interpolants based on blossoming abstract by blossoming marsden s identity we investigate local quasi interpolation schemes for c 2 continuous quintic powell sabin splines represented with a normalized b spline basis as applications various families of discrete and differential quasi interpolants reproducing quintic polynomials are presented

on meet and join matrices associated with incidence functions we study recently meet matrices on meet semilattices as an abstract generalization of greatest common divisor gcd matrices analogously in this paper we consider join matrices on lattices as an abstract generalization of least common multiple lcm matrices a formula for the determinant of join matrices on join closed sets bounds for the determinant of join matrices on all sets and a formula for the inverse of join matrices on join closed sets are given the concept of a semi multiplicative function gives us formulae for meet matrices on join closed sets and join matrices on meet closed sets finally we show what new the study of meet and join matrices contributes to the usual gcd and lcm matrices

on the factorization of lcm matrices on gcd closed sets abstract let s x1 xn be a set of n distinct positive integers the matrix having the greatest common divisor gcd xi xj of xi and xj as its i j entry is called the greatest common divisor matrix denoted by s n the matrix having the least common multiple lcm xi xj of xi and xj as its i j entry is called the least common multiple matrix denoted by s n the set is said to be gcd closed if xi xj s for all 1 i j n in this paper we show that if n 3 then for any gcd closed set s x1 xn the gcd matrix on s divides the lcm matrix on s in the ring m n z of n n matrices over the integers for n 4 there exists a gcd closed set s x1 xn such that the gcd matrix on s does not divide the lcm matrix on s in the ring m n z this solves a conjecture raised by the author in 1998

c 1 spline wavelets on triangulations in this paper we investigate spline wavelets on general triangulations in particular we are interested in c 1 wavelets generated from piecewise quadratic polynomials by using the powell sabin elements we set up a nested family of spaces of c 1 quadratic splines which are suitable for multiresolution analysis of besov spaces consequently we construct c 1 wavelet bases on general triangulations and give explicit expressions for the wavelets on the three direction mesh a general theory is developed so as to verify the global stability of these wavelets in besov spaces the wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations

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

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

smooth macro elements on powell sabin 12 splits macro elements of smoothness c r are constructed on powell sabin 12 splits of a triangle for all r 0 these new elements complement those recently constructed on powell sabin 6 splits and can be used to construct convenient superspline spaces with stable local bases and full approximation power that can be applied to the solution of boundary value problems and for interpolation of hermite data

composite finite elements of class ck let be a triangulation of d r2 with vertices a ai 1 i n and pkn d v ck d t v t pn where pn polynomials of total degree n for u cr d r k and v finite dimensional subspace of ck d we consider the hermite problem hk a v find v v d v ai d u ai 1 i n k this problem is solvable in v pkn d iff n 4k 1 enisek when v pkn d 3 where 3 is a subtriangulation of obtained by subdividing each t into 3 triangles the problem has a solution for n 4k 1 e g the c1 cubic hct triangle and a p27 triangle with 31 parameters when v pkn d 6 where 6 is a subtriangulation of obtained by subdividing each t into 6 triangles the problem has a solution for n 3k 1 e g the c1 quadratic ps triangle and a p25 triangle with 31 parameters in both cases the construction needs the partial derivatives d u ai for 2k 1 the domain d is assumed to be polygonal compact and connected

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

numerical solution of the quasilinear poisson equation in a nonuniform triangle mesh a finite difference method using a nonuniform triangle mesh is described for the numerical solution of the nonlinear two dimensional poisson equation s 0 where is a function of or its derivatives sis a function of position and or its normal derivative is specified on the boundary the finite difference equations are solved by successive overrelaxation the triangle mesh which is constructed numerically by solving laplace s equation is easily adapted to nonrectangular boundaries and interfaces examples of numerical results are given for the magnetostatic problem with iron and other possible applications are mentioned

non divisibility of lcm matrices by gcd matrices on gcd closed sets abstract in this paper we consider the divisibility problem of lcm matrices by gcd matrices in the ring m n z proposed by shaofang hong in 2002 and in particular a conjecture concerning the divisibility problem raised by jianrong zhao in 2014 we present some certain gcd closed sets on which the lcm matrix is not divisible by the gcd matrix in the ring m n z this could be the first theoretical evidence that zhao s conjecture might be true furthermore we give the necessary and sufficient conditions on the gcd closed set s with s 8 such that the gcd matrix divides the lcm matrix in the ring m n z and hence we partially solve hong s problem finally we conclude with a new conjecture that can be thought as a generalization of zhao s conjecture

on the divisibility of meet and join matrices abstract let p p be a lattice let s x 1 x 2 x n be a meet closed subset of p and let f p z be a function we characterize the matrix divisibility of the join matrix s f f x i x j by the meet matrix s f f x i x j in the ring z n n in terms of the usual divisibility in z and we present two algorithms for constructing certain classes of meet closed sets s such that s f divides s f as an example we present the lattice theoretic structure of all meet closed sets with at most five elements possessing the matrix divisibility property finally we show that our methods solve some open problems in the divisor lattice concerning the divisibility of gcd and lcm matrices

on the stability of normalized powell sabin b splines in this paper we show that the normalized powell sabin b splines form a stable basis for the max norm the approximation constants depend only on the smallest angle in the underlying triangulation since the b splines refer to the size of the powell sabin triangles we find that small powell sabin triangles yield better approximation constants than big powell sabin triangles next in addition to the max norm we treat the lp norm here the approximation constants depend also on a fraction proper to the triangulation thus the b splines are not stable for the lp norm finally as a special case we consider the b spline bases obtained from powell sabin triangles with minimal area and pay extra attention to the approximation constants for the max norm

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

smooth macro elements based on clough tocher triangle splits summary macro elements of smoothness c r on clough tocher triangle splits are constructed for all r ge 0 these new elements are improvements on elements constructed in 11 in that disproving a conjecture made there certain unneeded degrees of freedom have been removed numerical experiments on hermite interpolation with the new elements are included

divisibility among power gcd matrices and among power lcm matrices on finitely many coprime divisor chains let a b and h be positive integers and s x1 xh be a set of h distinct positive integers the set s is called a divisor chain if there is a permutation on 1 h such that x 1 x h we say that the set s consists of finitely many coprime divisor chains if there is a positive integer k such that we can partition s as s s1 sk where all the si are divisor chains and each element of si is coprime to each element of sj for 1 i j k the matrix having the ath power xi xj a of the greatest common divisor of xi and xj as its i j entry is called ath power greatest common divisor gcd matrix defined on s denoted by sa similarly we can define the ath power lcm matrix sa in this paper we show that if a b and s consists of finitely many coprime divisor chains with 1 s then in the ring mh z of h h matrices over integers we have sa sb sa sb and sa sb but such results fail to be true if a b these results confirm partially hong s conjectures raised in 2008

rational finite elements of class ck over a quadrangulation abstract in this paper we give a constructive method for rational c k finite element method in r 2 for quadrilateral for k 2 the usual polynomials splines require some supersmoothness at the vertices of the quadrilateral even when dealing with macro elements this is no more a requirement for our rational approach the process of the construction is illustrated with the help of two examples namely the c 1 case and the c 2 case

determinants and divisibility of power gcd and power lcm matrices on finitely many coprime divisor chains let a b and h be positive integers and s x 1 x h be a set of h distinct positive integers the hxh matrix s a x i x j a having the ath power x i x j a of the greatest common divisor of x i and x j as its i j entry is called the ath power gcd matrix on s the ath power lcm matrix on s can be defined similarly in this paper we first obtain the formulae for determinants of power gcd and power lcm matrices on the set s consisting of finitely many coprime divisor chains i e there is a positive integer k such that we can partition s as s s 1 s k where s i and s j are divisor chains and each element of s i is coprime to each element of s j for any 1 j

on multivariate polynomials in bernstein bezier form and tensor algebra the bernstein bezier representation of polynomials is a very useful tool in computer aided geometric design in this paper we make use of multilinear tensors to describe and manipulate multivariate polynomials in their bernstein bezier form as an application we consider hermite interpolation with polynomials and splines

quadrilateral finite elements of fvs type and class c rho let cal p be some partition of a planar polygonal domain omega into quadrilaterals given a smooth function u we construct piecewise polynomial functions upsilon in c rho omega of degree n 3 rho for rho odd and n 3 rho 1 for rho even on a subtriangulation tau 4 of cal p the latter is obtained by drawing diagonals in each q in cal p and upsilon vert q is a composite quadrilateral finite element generalizing the classical c 1 cubic fraeijs de veubeke and sander or fvs quadrilateral the function upsilon interpolates the derivatives of u up to order rho rho 2 at the vertices of cal p polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements

on c 2 quintic spline functions over triangulations of powell sabin s type abstract given a triangulation of a polygonal domain we find a refinemet of of by choosing t in a neighborhood of the center of the inscribed circle of each triangle of t connecting t to the vertices of the triangle t and connecting t to t if t shares an interior edge with t or to the midpoint e of any boundary edge e of t the resulting triangulation is a triangulation of powell sabin s type we investigate a c 2 quintic spline space s 5 2 whose elements are c 3 only at t s we give a dimension formula for this spline space show how to construct a locally supported basis display an interpolation scheme and prove that this spline space has the full approximation order

greatest common divisor matrices abstract let s x 1 x 2 x n be a set of distinct positive integers then n n matrix s s ij where s ij x i x j the greatest common divisor of x i and x j is call the greatest common divisor gcd matrix on s we initiate the study of gcd matrices in the direction of their structure determinant and arithmetic in z n several open problems are posed

on the bourque ligh conjecture of least common multiple matrices abstract let s x1 xn be a set of n distinct positive integers the matrix s n having the least common multiple xi xj of xi and xj as its i j entry is called the least common multiple lcm matrix on s a set s is gcd closed if xi xj s for 1 i j n bourque and ligh conjectured that the lcm matrix s n defined on a gcd closed set s is nonsingular in this paper we prove that the conjecture is true for n 7 and is not true for n 8 so the conjecture is solved completely

an explicit local basis for c 1 cubic spline spaces over a triangulated quadrangulation let s31 be the bivariate c1 cubic spline space over a triangulated quadrangulation in this paper an explicit representation of a locally supported basis of s31 is given using the interpolation conditions at vertices

from nurbs to nurps geometries abstract quadratic powell sabin splines and their rational extension the so called nurps surfaces are an interesting alternative for classical tensor product nurbs in the context of isogeometric analysis because they allow the use of local refinements while retaining a b spline like representation and exact description of conic sections in this paper we present a simple and effective strategy to convert a given planar geometry defined by a quadratic nurbs representation into a nurps representation suitable for the analysis

approximation order from certain spaces of smooth bivariate splines on a three direction mesh determination de l ordre exact d approximation de certains espaces de fonctions splines regulieres a deux variables

on gcd and lcm matrices let s x1 x2 xn be a set of distinct positive integers the matrix s having the greatest common divisor xi xj of xi and xj as its i j entry is called the greatest common divisor gcd matrix on s the matrix s having the least common multiple of xi and xj as its i j entry is called the least common multiple lcm matrix on s the set s is factor closed if it contains every divisor of each of its elements if s is factor closed we calculate the inverses of the gcd and lcm matrices on s and show that s s 1 is an integral matrix we also extend a result of h j s smith by calculating the determinant of s when xi xj s for 1 i j n

a bivariate c1 cubic super spline space on powell sabin triangulation a bivariate c 1 cubic super spline is constructed on powell sabin type 1 split with the additional smoothness at vertices in the original triangulation being c 2 which permits the hermite interpolation up to the second order partial derivatives exactly on all the vertices in the original triangulation the locally supported dual basis and computational details using derivatives around each vertex are given

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

c 1 hierarchical riesz bases of lagrange type on powell sabin triangulations in this paper we construct c1 continuous piecewise quadratic hierarchical bases on powell sabin triangulations of arbitrary polygonal domains in r2 our bases are of lagrange type instead of the usual hermite type and under some weak regularity assumptions on the underlying triangulations we prove that they form strongly stable riesz bases for the sobolev spaces hs with s 1 5 2 especially the case s 2 is of interest because we can use the corresponding hierarchical basis for preconditioning fourth order elliptic equations leading to uniformly well conditioned stiffness matrices compared to the hierarchical riesz bases by davydov and stevenson hierarchical riesz bases for hs 1 s 5 2 constructive approximation to appear our construction is simpler

nonsingularity of matrices associated with classes of arithmetical functions on lcm closed sets abstract let s x1 xn be a set of n distinct positive integers and f be an arithmetical function let f xi xj denote the n n matrix having f evaluated at the greatest common divisor xi xj of xi and xj as its i j entry and f xi xj denote the n n matrix having f evaluated at the least common multiple xi xj of xi and xj as its i j entry the set s is said to be lcm closed if xi xj s for all 1 i j n for an integer x 1 let x denote the number of distinct prime factors of x define 1 0 in this paper we show that if s x1 xn is an lcm closed set satisfying max x s lcm s x 2 and if f is a strictly increasing resp decreasing completely multiplicative function or if f is a strictly decreasing resp increasing completely multiplicative function satisfying 0 f p 1 p resp f p p for any prime p then the matrix f xi xj resp f xi xj defined on s is nonsingular by using the concept of least type multiple introduced in s hong j algebra 281 2004 1 14 we also obtain reduced formulas for det f xi xj and det f xi xj when f is completely multiplicative and s is lcm closed we also establish several results about the nonsingularity of lcm matrices and reciprocal gcd matrices

vibro acoustic radiation of rigid bodies oscillating at large amplitude and low mach number modelling and experiments the acoustic radiation of vibrating bodies can usually be considered as a linear phenomenon as in most cases the vibration amplitudes of oscillating bodies are small compared to their dimension however in case of large amplitude and small mach number the large displacement amplitude of a vibrating body gives rise to geometric nonlinear acoustic effects in this paper it is shown that the nonlinearity causes an asymmetry in the time signal of the radiated pressure as observed at a fixed position resulting in harmonic distortion of the radiated sound field these nonlinear effects can be analysed by the finite element method using a linear fluid behaviour in combination with a moving mesh approach this approach is implemented and used to predict the nonlinear sound radiation of lithographic stages oscillating at large amplitude and low mach number acoustic experiments prove the validity of the analysis technique for the geometrical nonlinear phenomena in addition the structural response of a nearby body excited by the nonlinear acoustic radiation of the stage is calculated and verified experimentally

cr rational splines over a triangulation abstract we propose a construction of c r bivariate rational splines over a triangulation via a finite element approach for r 2 the usual bivariate polynomial splines require some supersmoothness at the vertices of the triangles even when dealing with macro elements this is no more a requirement for our rational approach in this paper we give a constructive method for rational c r finite elements in r 2 defined over triangles beside the general case the process of construction is illustrated with two examples namely the c 1 case and the c 2 case

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

isogeometric analysis with powell sabin splines for advection diffusion reaction problems abstract this paper presents the use of powell sabin splines in the context of isogeometric analysis for the numerical solution of advection diffusion reaction equations powell sabin splines are piecewise quadratic c 1 functions defined on a given triangulation with a particular macro structure we discuss the galerkin discretization based on a normalized powell sabin b spline basis we focus on the accurate detection of internal and boundary layers and on local refinements we apply the approach to several test problems and we illustrate its effectiveness by a comparison with classical finite element and recent isogeometric analysis procedures