invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

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

the bezout number for linear piecewise algebraic curves a piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function this paper discusses the bezout number the maximum number of intersections between two linear piecewise algebraic curves whose intersections are finite on regular triangulations we give an upper bound of the bezout number for linear piecewise algebraic curves bn 1 0 1 0 d on the triangulation with an odd interior vertex for the triangulations which satisfy a vertex coloring condition we compute the exact value of the bezout number bn 1 0 1 0 d

on the dimension of bivariate spline spaces of smoothnessr and degreed 3r 1 we consider the well known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessingr continuous derivatives globally to date dimension formulae for such spaces have been established only whend 3r 2 except for the special case wherer 1 andd 4 in this paper we establish dimension formulae for allr 1 andd 3r 1 for almost all triangulations

dimensions of spline spaces over t meshes a t mesh is basically a rectangular grid that allows t junctions in this paper we propose a method based on bezier nets to calculate the dimension of a spline function space over a t mesh when the order of the smoothness is less than half of the degree of the spline functions a dimension formula is derived which involves only the topological quantities of the t mesh the construction of basis functions is briefly discussed furthermore the dimension formulae for t meshes after mesh operations such as edge insertion and mesh merging are also obtained

inequalities for the generalized elliptic integrals and modular functions abstract in this paper the authors study monotonicity and convexity of the generalized elliptic integrals and certain combinations of these special functions such as m a r and a r making use of these results the authors obtain some sharp inequalities for the so called ramanujan s generalized modular functions

adaptive thinning algorithm of nonsingular triangulation this paper studies adaptive thinning strategies for the non singular triangulation of scattered data by c 1 rational spline function given a set of points in r 2 luo liu and chen have presented a triangulation algorithm which ensures the non singularity of s 2 1 and s 3 1 spaces in this paper we improve the algorithm to reduce the number of knots of the triangulation within a given tolerance while the non singularity of s 2 1 and s 3 1 spaces is ensured our strategies presented here depend on both the locations of the data points in the plane and the data values at these points we give the definition of discrete norm for c 1 rational spline function by using its coefficients then a weight is assigned to each knot which is a measure of the importance of knot in the representation of spline when the weight of the knot is less than the given tolerance its influence is regarded negligible then it can be removed it s a discrete method in the end of this paper several numerical examples are presented to show the feasibility and validity of our algorithm

the quadrilateral mindlin plate elements using the spline interpolation bases abstract by the mindlin reissner plate theory the displacement and rotations x y are interpolated by independent functions for which only c 0 continuity condition is required the difficulty for constructing interpolation bases on the quadrilateral element without isoparametric transformation can be overcome by using the spline method in this paper two sets of spline interpolation bases are adopted to construct two quadrilateral spline mindlin plate elements qsmp1 and qsmp2 with 12 degrees of freedom the spline elements can be applied for both thick and thin plates and can converge for the very thin case numerical examples are discussed to show that the mindlin plate element combined with the spline interpolation bases can possess good accuracy

triangulation based isogeometric analysis of the cahn hilliard phase field model abstract this paper presents triangulation based isogeometric analysis of the cahn hilliard phase field model the cahn hilliard phase field model is governed by a time dependent fourth order partial differential equation the corresponding primal variational form involves second order operators making it difficult to be directly analyzed with traditional c 0 finite element analysis in this paper we construct c 1 bernstein bezier simplicial elements through macro element techniques including various triangle split based macro elements in both 2d and 3d space we extend triangulation based isogeometric analysis to solving the primal variational form of the cahn hilliard equation we validate our method by convergence analysis showing the nodal and degree of freedom advantages over c 0 finite element analysis we then demonstrate detailed system evolution from randomly perturbed initial conditions in periodic two dimensional squares and three dimensional cubes we incorporate an adaptive time stepping scheme in these numerical experiments our numerical study demonstrates that triangulation based isogeometric analysis offers optimal convergence and time step stability is applicable to complex geometry and allows local refinement

remarks on generalized elliptic integrals we study the monotonicity for certain combinations of generalized elliptic integrals thus generalizing analogous well known results for classical complete elliptic integrals and prove a conjecture put forward by heikkala vamanamurthy and vuorinen

a c 1 rational spline in range restricted interpolation of scattered data a range restricted c1 interpolation local scheme to scattered data is developed c rational spline function classes over triangles and quadrilaterals have been investigated by the first author in this paper we concentrate on the c1 rational spline over triangles and represent it as piecewise a convex combination of three cubic bezier triangular patches further with such an equivalent form of c1 rational spline the construction of a range restricted c1 interpolant subject to polynomial constraint surface of degree up to three is considered in the end of this paper some numerical examples are illustrated

accurate approximations for the complete elliptic integral of the second kind abstract in this paper we prove that the double inequality s 11 4 7 4 1 r e r s 11 4 7 4 1 r holds for all r 0 1 if and only if 2 1 570796 and 11 7 1 571428 where r 1 r 2 1 2 e r 0 2 1 r 2 sin 2 t d t is the complete elliptic integral of the second kind and s p q a b q a p b p p a q b q 1 p q is the stolarsky mean of a and b

turan type inequalities for generalized complete elliptic integrals in this paper our aim is to establish some turan type inequalities for gaussian hypergeometric functions and for generalized complete elliptic integrals these results complete the earlier result of p turan proved for legendre polynomials moreover we show that there is a close connection between a turan type inequality and a sharp lower bound for the generalized complete elliptic integral of the first kind at the end of this paper we prove a recent conjecture of t sugawa and m vuorinen related to estimates of the hyperbolic distance of the twice punctured plane

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

on multivariate e splines abstract in a recent interesting paper a ron 14 initiated the study of cardinal exponential splines in several variables beyond settling some open problems which were suggested by his results this paper addresses a variety of additional questions of an algebraic nature thus some of the results we obtained elsewhere on cube spline surfaces appear now as a special case of the present general setting our approach is critically based on some new bounds and formulas for the dimension of certain intersections of null spaces of commuting linear operators where the particular way of forming these intersections is determined by a matroid structure specifically this allows us to characterize the local linear independence of translates of the exponential cube spline as well as to study local algebraic properties of functions spanned by them only special cases were covered by ron s investigation furthermore we consider in detail the construction of dual bases linear projectors linear dependence relations among these exponential splines and subdivision algorithms for their approximate evaluation the analysis of these latter algorithms is based on the concept of discrete exponential splines we investigate to what extent these discrete objects possess analogous properties to their continuous counterparts finally the combinatorial interpretation of the discrete splines their relationship to their continuous counterparts and the machinery we develop leads to interesting combinatorial applications for enumerating the number of solutions to linear diophantine systems and corresponding reciprocity relations an announcement of these results appeared in 10

asymptotical bounds for complete elliptic integrals of the second kind abstract in this paper we establish several asymptotical bounds for the complete elliptic integrals of the second kind e r and improve the well known conjecture e r 1 1 r 2 3 4 2 2 3 2 for all r 0 1 proposed by m vuorinen

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

highest weight uq sl n modules and invariant integrable n state models with periodic boundary conditions weights are computed for the bethe vectors of an rsos type model with periodic boundary conditions obeying uq sl n q exp i pi r invariance they are shown to be highest weight vectors the q dimensions of the corresponding irreducible representations are obtained

a note on the dimension of the bivariate spline space over the morgan scott triangulation in d diener siam j numer anal 27 1990 pp 543 551 a conjecture on the dimension of the bivariate spline space s 2r r triangle over the morgan scott triangulation was posed in this paper it is proved that the conjecture should be modified for all even r 2

difference equations and highest weight modules of based on the trigonometric r matrix of a family of q deformed discrete knizhnik zamolodchikov equations is formulated solutions to them are constructed by using the generalization of the algebraic bethe ansatz developed to solve difference equations these solutions are shown to be of highest weight with respect to the underlying quantum group structure using the variant of the nested bethe ansatz method these results are extended to the higher ranked symmetry of

concavity of the complete elliptic integrals of the second kind with respect to holder means abstract in this paper we establish a necessary and sufficient condition for the concavity of the complete elliptic integrals of the second kind with respect to holder means

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

the bivariate quadratic c1 spline spaces with stable dimensions on the triangulations abstract the dimension is a basic problem in the theory of the bivariate spline space in general the dimension of the bivariate spline space defined on the triangulation is unstable or with singularity in this paper we consider the dimensions of the bivariate quadratic c 1 spline spaces defined on the triangulations our main result is when the degree of each interior vertex of the non degenerate triangulation is at least 6 the dimension of the corresponding bivariate quadratic c 1 spline space is stable and equal to the number of the boundary vertices plus 3 we also give an example to show that the non degenerate condition is necessary

adaptive thinning for bivariate scattered data this paper studies adaptive thinning strategies for approximating a large set of scattered data by piecewise linear functions over triangulated subsets our strategies depend on both the locations of the data points in the plane and the values of the sampled function at these points adaptive thinning all our thinning strategies remove data points one by one so as to minimize an estimate of the error that results by the removal of a point from the current set of points this estimate is termed anticipated error the thinning process generates subsets of most significant points such that the piecewise linear interpolants over the delaunay triangulations of these subsets approximate progressively the function values sampled at the original scattered points and such that the approximation errors are small relative to the number of points in the subsets we design various methods for computing the anticipated error at reasonable cost and compare and test the performance of the methods it is proved that for data sampled from a convex function with the strategy of convex triangulation the actual error is minimized by minimizing the best performing measure of anticipated error it is also shown that for data sampled from certain quadratic polynomials adaptive thinning is equivalent to thinning which depends only on the locations of the data points nonadaptive thinning based on our numerical tests and comparisons two practical adaptive thinning algorithms are proposed for thinning large data sets one which is more accurate and another which is faster

an explicit basis for c 1 quartic by various bivariate splines we establish the dimension of the space of c 1 bivariate piecewise quartic polynomials defined on a triangulation of a connected polygonal domain our approach is to construct a minimal determining set and an associated explicit basis for the space for general triangulations the minimal determining set must be defined globally

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

upper and lower bounds on the dimension of multivariate spline spaces we give upper and lower bounds on the dimensions of multivariate spline spaces defined on triangu lations in mathbb r k the bounds are optimal in a certain sense

multivariate weak spline function space in this paper b net method for studying multivariate weak spline is discussed and the dimension of w21 i1 1 is presented several b splines on type 1 triangulation are also obtained

dimensions of biquadratic spline spaces over t meshes this paper discusses the dimensions of spline spaces over t meshes of low degree two new concepts are proposed an extension of t meshes and spline spaces with homogeneous boundary conditions in the dimensional analysis the key strategy is linear space embedding with the operator of the mixed partial derivative a lower bound on the dimension of the biquadratic spline spaces over general t meshes is provided furthermore by making full use of the level structure of hierarchical t meshes a dimension formula of biquadratic spline space over hierarchical t meshes is proved additionally a topological explanation of the dimension formula is provided

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

generalized hersch pfluger distortion function and complete elliptic integrals abstract in this paper we study the relationship between the generalized hersch pfluger distortion function k a r and complete elliptic integrals of the second kind e r and establish several inequalities for k a r and e r

recent researches on multivariate spline and piecewise algebraic variety the multivariate splines as piecewise polynomials have become useful tools for dealing with computational geometry computer graphics computer aided geometrical design and image processing it is well known that the classical algebraic variety in algebraic geometry is to study geometrical properties of the common intersection of surfaces represented by multivariate polynomials recently the surfaces are mainly represented by multivariate piecewise polynomials i e multivariate splines so the piecewise algebraic variety defined as the common intersection of surfaces represented by multivariate splines is a new topic in algebraic geometry moreover the piecewise algebraic variety will be also important in computational geometry computer graphics computer aided geometrical design and image processing the purpose of this paper is to introduce some recent researches on multivariate spline piecewise algebraic variety curve and their applications

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

bivariate spline spaces and minimal determining sets abstract the paper describes bivariate spline spaces and minimal determining sets shows applications of these concepts presents two new triangular finite elements describes public domain software for the analysis of spline spaces and concludes with a concise history of these subjects the software described in this paper is available on http www math utah edu alfeld mds that page also contains links to colored versions of the figures in this paper

inequalities for means abstract a monotone form of l hospital s rule is obtained and applied to derive inequalities between the arithmetic geometric mean of gauss the logarithmic mean and stolarsky s identric mean some related inequalities are given for complete elliptic integrals

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

nonstandard coproducts and the izergin korepin open spin chain corresponding to the izergin korepin a 2 2 r matrix there are three diagonal solutions k matrices of the boundary yang baxter equation using these r and k matrices one can construct transfer matrices for open integrable quantum spin chains the transfer matrix corresponding to the identity matrix k is known to have uq o 3 symmetry we argue here that the transfer matrices corresponding to the other two k matrices also have uq o 3 symmetry but with a nonstandard coproduct we briefly explore some of the consequences of this symmetry

spline element methods allowing multiple level hanging nodes abstract in this paper a systematic algorithm of spline element method allowing multiple level hanging nodes on triangular mesh is proposed the classical smoothness condition of the piecewise polynomials on triangular mesh is generalized to the case of meshes with hanging nodes and is treated as the linear constraints of the system equations we then derive a concise formulation for such linear constraints according to the special hierarchical configurations of the hanging nodes moreover an efficient linear equation solver is proposed when applied to solve the elliptic partial differential equations numerical examples are illustrated to show the accuracy of the proposed methods and the comparisons are made between different levels of hanging nodes

on approximating the arithmetic geometric mean and complete elliptic integral of the first kind abstract in the article we prove that the double inequalities 1 6 p 7 r p 5 p 6 r tanh 1 r 2 r k r 1 6 q 7 r q 5 q 6 r tanh 1 r 2 r q a 1 r 5 q 6 g 1 r a 1 r 6 q 7 g 1 r l 1 r a g m 1 r p a 1 r 5 p 6 g 1 r a 1 r 6 p 7 g 1 r l 1 r hold for all r 0 1 if and only if p 2 1 570796 and q 89 69 1 289855 where k r 0 2 1 r 2 sin 2 t 1 2 d t is the complete elliptic integral of the first kind tanh 1 r log 1 r 1 r 2 is the inverse hyperbolic tangent function r 1 r 2 and a 1 r 1 r 2 g 1 r r l 1 r r 1 log r and a g m 1 r are the arithmetic geometric logarithmic and gaussian arithmetic geometric means of 1 and r respectively

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

on the dimensions of bivariate spline spaces and the stability of the dimensions in this paper the dimensions of bivariate spline spaces are studied using the smoothing cofactor conformality method based on the analysis on the conformality condition at one interior vertex the stability or singularity to the contrary of the dimensions of general spline spaces is discussed in detail by the aid of directed partition some new results on dimensions are obtained with the corresponding constraints depending on the degree the smoothness order of the spline spaces and the structure of the partition as well

bethe ansatz solution of a closed spin 1 xxz heisenberg chain with quantum algebra symmetry a quantum algebra invariant integrable closed spin 1 chain is introduced and analyzed in detail the bethe ansatz equations as well as the energy eigenvalues of the model are obtained the highest weight property of the bethe vectors with respect to uq sl 2 is proved

multivariate spline space over cross cut partition in this paper we mainly study the generator bases of the prime module determined by the global conformality condition of multivariate spline over star cross cut partition the results can help us to get the dimension and construct the spline bases for multivariate spline space over cross cut partition quasi cross cut partition and super cross cut partition

monotonicity theorems and inequalities for the complete elliptic integrals we prove monotonicity properties of certain combinations of complete elliptic integrals of the first and second kind k and e these results lead to sharp symmetrical bounds for k and e which improve recently discovered inequalities

interpolation with c2 quartic macro elements based on 10 splits abstract in this paper interpolation methods for the construction of quartic splines on triangulations refined with 10 splits are proposed after examining the c 2 macro structure on a single triangle in terms of the bernstein bezier representation three methods that can be applied on general triangulations are developed the methods make use of hermite interpolation data prescribed at the vertices and optionally on the edges of the triangulation the first two approaches lead to splines that are c 2 continuous on triangles and c 1 continuous across the interior edges of the triangulation the third method gives rise to splines with overall c 2 continuity which is an exceptionally high order of smoothness for splines of degree four but comes at the cost of solving a global system of linear equations the derived results are accompanied with a few numerical examples that show an interesting behavior of splines in dependence of interpolated data

isogeometric analysis with bezier tetrahedra abstract this paper presents an approach for isogeometric analysis of 3d objects using rational bezier tetrahedral elements in this approach both the geometry and the physical field are represented by trivariate splines in bernstein bezier form over the tetrahedrangulation of a 3d geometry given a nurbs represented geometry either untrimmed or trimmed we first convert it to a watertight geometry represented by rational triangular bezier splines rtbs for trimmed geometries a compatible subdivision scheme is developed to guarantee the watertightness the rtbs geometry preserves exactly the original nurbs surfaces except for an interface layer between trimmed surfaces where controlled approximation occurs from the watertight rtbs geometry a bezier tetrahedral partition is generated automatically by imposing continuity constraints on bezier ordinates of the elements we obtain a set of global c r smooth basis functions and use it as the basis for analysis numerical examples demonstrate that our method achieves optimal convergence in c r spaces and can handle complicated geometries

sharp approximations for the complete elliptic integrals of the second kind by one parameter means abstract let e r be the complete elliptic integrals of the second kind and j p a b be the one parameter mean of two distinct positive numbers a and b defined by j p a b p p 1 a p 1 b p 1 a p b p for p 1 0 in this study using the recurrence method with the sign rule of a special power series we prove that the double inequality 7 22 j 2 2 1 r 2 e r j 2 2 1 r holds for r 1 r 2 0 1 where the upper bound is sharp while the lower bound is related to the archimedes approximation 22 7 for using the same method we also easily show that both of the functions r 2 2 e r j 7 4 1 r r 12 and r 2 1 r 10 1 1 2 e r 1 j 7 4 1 r are absolutely monotonic on 0 1 which greatly improves the known results moreover it should be emphasized that our method may be used to explore special functions related to hypergeometric functions

continuity and convergence in rational triangular bezier spline based isogeometric analysis abstract this paper presents a method for isogeometric analysis using rational triangular bezier splines rtbs where optimal convergence rates are achieved in this method both the geometry and the physical field are represented by bivariate splines in bernstein bezier form over the triangulation of a domain from a given physical domain bounded by nurbs curves a parametric domain and its triangulation are constructed by imposing continuity constraints on bezier ordinates we obtain a set of global c r smooth basis functions convergence analysis shows that isogeometric analysis with such c r rtbs basis can deliver the optimal rate of convergence provided that the c r geometric map remains unchanged during the refinement process this condition can be satisfied by constructing a pre refinement geometric map that is sufficiently smooth numerical experiments verify that optimal rates of convergence are achieved for poisson and linear elasticity problems

inequalities for zero balanced hypergeometric functions the authors study certain monotoneity and convexity properties of the gaussian hypergeometric function and those of the euler gamma function

instability in the dimension of spaces of bivariate piecewise polynomials of degree 2 r and smoothness order r the spaces of bivariate splines on a particular triangulation are considered the dimension of the space of splines with degree 2 and with smoothness order 1 on this triangulation has long been known to depend on the geometry of the triangulation this result is extended to the space of splines of degree 2r and smoothness order r the dimension of the spline space is shown to have a similar dependence on geometry for all values of r geqq 1

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