c2 quadratic trigonometric polynomial curves with local bias quadratic trigonometric polynomial curves with local bias are presented in this paper the quadratic trigonometric polynomial curves have c 2 continuity with a non uniform knot vector and any value of the bias while the quadratic b spline curves have c 1 continuity the changes of a local bias parameter will only affect two curve segments with the bias parameters the quadratic trigonometric polynomial curves can move locally toward or against a control vertex a quadratic trigonometric bezier curve is also introduced as special case of the given trigonometric polynomial curves

explicit representations of changeable degree spline basis functions changeable degree spline cd spline for short basis functions defined by an iterative integral method are extensions of b spline basis functions a cd spline basis function is a piecewise function made up of polynomials of different degrees in this paper we will give the explicit representations of cd spline basis functions from which the spanned linear space can be seen clearly our method is also feasible for the explicit representations of the other basis functions given in analogous integral ways

bernstein operators for extended chebyshev systems abstract let u n c n a b be an extended chebyshev space of dimension n 1 suppose that f 0 u n is strictly positive and f 1 u n has the property that f 1 f 0 is strictly increasing we search for conditions ensuring the existence of points t 0 t n a b and positive coefficients 0 n such that for all f c a b the operator b n c a b u n defined by b n f k 0 n f t k k p n k satisfies b n f 0 f 0 and b n f 1 f 1 here it is assumed that p n k k 0 n is a bernstein basis defined by the property that each p n k has a zero of order k at a and a zero of order n k at b

on monotone and convex spline interpolation this paper is concerned with the problem of existence of monotone and or convex splines having degree n and order of continuity k which interpolate to a set of data at the knots the interpolating splines are obtained by using bernstein polynomials of suitable continuous piecewise linear functions they satisfy the inequality k n k the theorems presented here are useful in developing algorithms for the construction of shape preserving splines interpolating arbitrary sets of data points earlier results of mcallister passow and roulier can be deduced from those given in this paper

frobenius and cartier algebras of stanley reisner rings we study the generation of the frobenius algebra of the injective hull of a complete stanley reisner ring over a field with positive characteristic in particular by extending the ideas used by m katzman to give a counterexample to a question raised by g lyubeznik and k e smith about the finite generation of frobenius algebras we prove that the frobenius algebra of the injective hull of a complete stanley reisner ring can be only principally generated or infinitely generated also by using our explicit description of the generators of such algebra and applying the recent work by m blickle about cartier algebras and generalized test ideals we are able to show that the set of f jumping numbers of generalized test ideals associated to complete stanley reisner rings form a discrete subset inside the non negative real numbers

total positivity and its applications preface i spline functions total positivity of the spline kernel and its applications b bojanov total positivity properties of lb splines t lyche l l schumaker total positivity and splines k morken totally positive systems and interpolation by subspaces of generalized splines m sommer h strauss ii matrix theory preservation of total nonnegativity under the hadamard product and related topics j garloff d g wagner vertex implications for totally nonnegative under the hadamard product and related topics j garloff on factorizations of totally positive matrices j garloff on factorizations of totally positive matrices m gasca j m pena iii geometric modeling total positivity and optimal bases j m carnicer j m pena total positivity and the shape of curves t n t goodman tchebycheff curves m l mazure h pottmann iv probability and mathematical biology analytical and probabilistic methods for totally positive matrices g s goodman some inequalities of total positivity in pure and applied mathematics s karlin z ziegler new directions and problems of multivariate total positivity motivated by genetic recombination probabilities s karlin v approximation theory tchebycheff spaces and total positivity j m carnicer j m pena cebysev and weak cebysev systems r a zalik vi complex analysis zero sets of entire fourier transforms of multiply positive density functions o m katkova vii statistics total nonnegativity of moment matrices and its application to e optimal designs in polynomial regression b heiligers analysis of regression dependence in 2 xn bivariate distributions and some applications in contingency tables s kasala m b rao viii real analysis total positivity and nonlinear analysis r q jia solutions of certain functional equations s karlin alternance and snakes in the nonlinear finite moment problem k tatalian ix combinatorics the applications of total positivity to combinatorics and conversely f brenti x integral equations spectral properties of totally positive kernels and matrices a pinkus index

isogeometric collocation methods with generalized b splines we introduce isogeometric collocation methods based on generalized b splines and we analyze their performance through numerical examples for univariate and multivariate scalar and vector valued problems in particular advection diffusion and linear elasticity model problems are addressed the resulting method combines the favorable properties of isogeometric collocation and the geometrical and analytical problem oriented advantages of generalized b splines

convergence analysis of c2 hermite interpolatory subdivision schemes by explicit joint spectral radius formulas abstract subdivision schemes are popular iterative processes to build graphs of functions curves and surfaces we analyze the 2 point hermite c 2 subdivision scheme introduced by merrien in 26 for the analysis of its convergence and its smoothness properties we are concerned with the computation of the joint spectral radius of a family of 2 matrices associated with the scheme in this paper by an explicit computation of the joint spectral radius of such pairs of matrices we determine necessary and sufficient conditions for the scheme to be c 2 convergent whenever it reproduces cubic polynomials in addition we present two one parameter families of convergent subdivision schemes belonging to the class in 26 possessing interesting properties from the shape control point of view

analyzing lattice networks through substructures abstract analyzing the topology of network structures is an important topic studied from many different aspects of science and mathematics the wiener polarity index number of unordered pairs of vertices at distance 3 from each other is one of the representative descriptors of graph structures it was computed for several lattice networks by chen et al 11 in an effort to understand the properties of these networks the wiener polarity index is a variation of the classic distance based graph invariant the wiener index sum of distances between all pairs of vertices which is known to be closely related to the number of substructures in this paper we examine the numbers of various subgraphs of order 4 for these lattice networks in addition to confirming their symmetric nature comparing the numbers of various substructures leads to insights on other less trivial characteristics of these network structures of common interest

spanning trees on graphs and lattices in d dimensions the problem of enumerating spanning trees on graphs and lattices is considered we obtain bounds on the number of spanning trees nst and establish inequalities relating the numbers of spanning trees of different graphs or lattices a general formulation is presented for the enumeration of spanning trees on lattices in d 2 dimensions and is applied to the hypercubic body centred cubic face centred cubic and specific planar lattices including the kagome diced 4 8 8 bathroom tile union jack and 3 12 12 lattices this leads to closed form expressions for nst for these lattices of finite sizes we prove a theorem concerning the classes of graphs and lattices with the property that nst exp nz as the number of vertices n where z is a finite non zero constant this includes the bulk limit of lattices in any spatial dimension and also sections of lattices whose lengths in some dimensions go to infinity while others are finite we evaluate z exactly for the lattices we consider and discuss the dependence of z on d and the lattice coordination number we also establish a relation connecting z to the free energy of the critical ising model for planar lattices

quasi extended chebyshev spaces and weight functions we recently showed that the class of quasi extended chebyshev spaces is the largest class of sufficiently differentiable functions permitting design in previous articles we mentioned a simple procedure to build such spaces by means of both generalised derivatives associated with non vanishing weight functions and two dimensional chebyshev spaces in the present one we prove that conversely on a closed bounded interval any quasi extended chebyshev space can be obtained via the latter procedure we then draw a few interesting consequences from the latter result

curve and surface construction using hermite subdivision schemes in this paper we present a very efficient hermite subdivision scheme based on rational functions and outline its potential applications with special emphasis on the construction of cubic like b splines well suited for the design of constrained curves and surfaces

a note on the optimal stability of bases of univariate functions this note is concerned with the characterizations and uniqueness of bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation with respect to bases whose elements have no sign changes

a characterization of rational singularities in terms of injectivity of frobenius maps the notions of f rational and f regular rings are defined via tight closure which is a closure operation for ideals in a commutative ring of positive characteristic the geometric significance of these notions has persisted and k e smith proved that f rational rings have rational singularities we now ask about the converse implication the answer to this question is yes and no for a fixed positive characteristic there is a rational singularity which is not f rational so the answer is no in this paper however we aim to show that the answer is yes in the following sense if a ring of characteristic zero has rational singularity then its modulo p reduction is f rational for almost all characteristic p this result leads us to the correspondence of f regular rings and log terminal singularities

a family of non uniform subdivision schemes with variable parameters for curve design in this paper we present non uniform subdivision schemes with variable parameter sequences a locally different tension parameter is set at each edge of the initial control polygon to control locally the shape of the resulting curve such that the scheme becomes non uniform due to the variable parameters the scheme can reproduce locally different analytic curves such as conics lissajous trigonometric and catenary curves hence blending curves including such analytic components can be successfully generated we discuss the convergence and smoothness of the proposed non uniform schemes and present some numerical results to demonstrate their advantages in geometric modeling furthermore as an application we propose a chamfering algorithm which can be used in designing automobile and mechanical products

the g2 and c2 rational quadratic trigonometric bezier curve with two shape parameters with applications the rational quadratic trigonometric bezier curve with two shape parameters is presented in this paper which is new in literature the purposed curve inherits all the geometric properties of the traditional rational quadratic bezier curve the presence of shape parameters provides a control on the shape of the curve more than that of traditional bezier curve moreover the weight offers an additional control on the curve simple constraints for shape parameters are derived using the end points curvature so that their values always fall within the defined range the composition of two segments of curve using g 2 and c 2 continuity is given the new curves can accurately represent some conics and best approximates the traditional rational quadratic bezier curve

which spaces for design we determine the largest class of spaces of sufficient regularity which are suitable for design in the sense that they do possess blossoms it is the class of all spaces containing constants of which the spaces derived under differentiation are quasi extended chebyshev spaces i e they permit hermite interpolation taylor interpolation excepted it is also the class of all spaces which possess bernstein bases or of all spaces for which any associated spline space does possess a b spline basis note that blossoms guarantee that such bases are normalised totally positive bases they even are the optimal ones

the structure of f pure rings for a reduced f finite ring r of characteristic p 0 and q p e one can write open image in new window where m q has no free direct summands over r we investigate the structure of f finite f pure rings r by studying how the numbers a q grow with respect to q this growth is quantified by the splitting dimension and the splitting ratios of r which we study in detail we also prove the existence of a special prime ideal open image in new window r of r called the splitting prime that has the property that r open image in new window r is strongly f regular we show that this ideal captures significant information with regard to the f purity of r

mixed hyperbolic trigonometric spaces for design we investigate the class of five dimensional null spaces of linear differential operators with constant coefficients and odd characteristic polynomials one of the advantages of this class is that it permits to mix trigonometric and hyperbolic functions within the same space and we will more specially focus on this interesting blending whenever necessary we determine the critical lengths for design this yields the largest possible intervals on which existence of bernstein bases is guaranteed such bases being then automatically the optimal normalised totally positive bases this also enables us to show the interest of this class of spaces for geometric design

signless laplacians of finite graphs we survey properties of spectra of signless laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix for regular graphs the whole existing theory of spectra of the adjacency matrix and of the laplacian matrix transfers directly to the signless laplacian and so we consider arbitrary graphs with special emphasis on the non regular case the results which we survey old and new are of two types a results obtained by applying to the signless laplacian the same reasoning as for corresponding results concerning the adjacency matrix b results obtained indirectly via line graphs among other things we present eigenvalue bounds for several graph invariants an interpretation of the coefficients of the characteristic polynomial a theorem on powers of the signless laplacian and some remarks on star complements

f purity and rational singularity we investigate singularities which are f pure respectively f pure type a ring r of characteristic p is f pure if for every r module m 0 m r m r is exact where 1r denotes the r algebra structure induced on r via the frobenius map if r e r and s e 1r then r s rps in 1r f pure type is defined in characteristic 0 by reducing to characteristicp it is proven that when r s i is the quotient of a regular local ring s r is f pure at the prime ideal q if and only if i pi i q p here j p denotes the ideal ap i a e j several theorems result from this criterion if f is a quasihomogeneous hypersurface having weights r1 r and an isolated singularity at the origin 1 17 1ri i implies k x1 x f has f pure type at m x1 x 2 1n i ri i implies k x1 xn f does not have f pure type at m 3 1 1ri i remains unsolved but does connect with a problem that number theorists have studied for many years this theorem parallels known results about rational singularities it is also proven that classifying f pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces and that the f pure locus in the maximal spectrum of k x1 xn i where k is a perfect field of characteristic p is zariski open an important conjecture is that r fr is f pure type should imply r is f pure type whenever r is a cohen macauley normal local ring it is proven that ext1 1r r 0 is a sufficient though not necessary condition a local ring r m of characteristic p is f injective if the frobenius map induces an injection on the local cohomology modules h r h 1r an example is constructed which is f injective but not f pure from this a counterexample to the conjecture that r fr is f pure implies r is f pure is constructed however it is not a domain much less normal moreover it does not lead to a counterexample to the characteristic 0 version of the conjecture 0 introduction let r be a ring of characteristic p and let r denote the ring r viewed as an r module via the frobenius map f r rp r is f pure if for every r module m 0 r m 4 r m is exact a notion of f pure type is then defined in characteristic 0 by reduction to characteristicp f pure rings are connected with invariant theory and appear in the proof that the ring of invariants of a linearly reductive affine linear group acting on a regular ring is cohen macaulay 3 it has also been demonstrated that f purity measures good singularities in the sense that it implies a great deal of simplification in the computation of local cohomology 1 received by the editors march 30 1982 and in revised form august 3 1982 1980 mathematics subject classification primary 13h99 1983 american mathematical society 0002 9947 82 0000 0829 05 25

unified and extended form of three types of splines the three types refer to polynomial trigonometric and hyperbolic splines in this paper we unify and extend them by a new kind of spline ue spline for short defined over the space cos wt sin wt 1 t t l where l is an arbitrary nonnegative integer w is a frequency sequence w i a i a i r existing splines such as usual polynomial b splines cb splines hb splines nuat splines ah splines fb splines and the third form fb splines etc are all special cases of ue splines ue splines inherit most properties of usual polynomial b splines and enjoy some other advantageous properties for modelling they can exactly represent classical conics the catenary the helix and even the eight curve a kind of snake like curves etc

optimal bases for a class of mixed spaces and their associated spline spaces we consider spaces of the form span where the functions u i i 1 4 are algebraic polynomials or trigonometric or hyperbolic functions we find intervals 0 a where we can guarantee that the spaces possess normalized totally positive bases and so shape preserving representations we construct their normalized b bases with optimal shape preserving properties we also present a unified approach to deal with the associated spline spaces a recursive procedure to obtain bases with optimal shape preserving and stability properties is presented

on the quartic curve of han the quartic curve of han x han piecewise quartic polynomial curves with shape parameter journal of computational and applied mathematics 195 2006 34 45 can be considered as the generalization of the cubic b spline curve incorporating shape parameters into the polynomial basis functions we show that this curve can be considered as the linear blending of the original cubic b spline curve and a fixed quartic curve moreover we present the bezier form of the curve which is useful in terms of incorporating the curve into existing cad systems geometric effects of the alteration of shape parameters is also discussed including design oriented computational methods for constrained shape control of the curve

asymptotic energy of lattices abstract the energy of a simple graph g arising in chemical physics denoted by e g is defined as the sum of the absolute values of eigenvalues of g as the dimer problem and spanning trees problem in statistical physics in this paper we propose the energy per vertex problem for lattice systems in general for a type of lattice in statistical physics to compute the entropy constant with toroidal cylindrical mobius band klein bottle and free boundary conditions are different tasks with different hardness and may have different solutions we show that the energy per vertex of plane lattices is independent of the toroidal cylindrical mobius band klein bottle and free boundary conditions in particular the asymptotic formulae of energies of the triangular 3 3 4 2 and hexagonal lattices with toroidal cylindrical mobius band klein bottle and free boundary conditions are obtained explicitly

generalized spline spaces over t meshes univariate generalized splines are smooth piecewise functions with sections in certain extended tchebycheff spaces they are a natural extension of univariate algebraic polynomial splines and enjoy the same structural properties as their polynomial counterparts in this paper we consider generalized spline spaces over planar t meshes and we deepen their parallelism with polynomial spline spaces over the same partitions first we extend the homological approach from polynomial to generalized splines this provides some new insights into the dimension problem of a generalized spline space defined on a prescribed t mesh for a given degree and smoothness second we extend the construction of lr splines to the generalized spline context

a sharp upper bound on the incidence energy of graphs in terms of connectivity abstract let g v g e g be a simple undirected graph with vertex set v g v 1 v 2 v n and edge set e g e 1 e 2 e m the incidence matrix i g of g is the n m matrix whose i j entry is 1 if v i is incident to e j and 0 otherwise the incidence energy ie g of g is the sum of the singular values of i g let k s be the complete graph on s vertices in this paper we derive an upper bound for the incidence energy of the graphs g on n vertices having a vertex connectivity less than or equal to k this upper bound is attained if and only if g k k k 1 k n k 1 obtained from the graphs k k and k 1 k n k 1 and the edges connecting each vertex of k k with every vertex of k 1 k n k 1

constructing totally positive piecewise chebyshevian b spline bases abstract we consider piecewise chebyshevian splines in the sense of splines with pieces taken from any different five dimensional extended chebyshev spaces and with connection matrices at the knots in this large context we establish necessary and sufficient conditions for the existence of totally positive refinable b spline bases these conditions are applied in many important special cases e g symmetric cardinal geometrically continuous quartic b spline parametrically continuous mixed l splines the great variety of illustrations provided proves the richness of this class of splines for design this richness can be exploited in various other fields as well

test ideals in local rings it is shown that certain aspects of the theory of tight closure are well behaved under localization let j be the parameter test ideal for r a complete local cohen macaulay ring of positive prime characteristic for any multiplicative system u c r it is shown that ju 1r is the parameter test ideal for u 1 r this is proved by proving more general localization results for the here introduced classes of f ideals of r and f submodules of the canonical module of r which are annihilators of r modules with an action of frobenius it also follows that the parameter test ideal cannot be contained in any parameter ideal of r tight closure has produced surprising new results for instance that the absolute integral closure of a complete local domain r of prime characteristic is a cohen macaulay algebra for r as well as tremendously simple proofs for otherwise difficult theorems such as the fact that the ring of invariants of a linearly reductive group acting on a regular ring is cohen macaulay the definition of tight closure is recalled in section 1 but we refer the reader to the papers of hochster and huneke listed in the bibliography for more about its applications while primarily a prime characteristic notion tight closure offers insight into arbitrary commutative rings containing q by fairly standard reduction to characteristic p techniques despite its successes the tight closure operation which in its principal setting is a closure operation performed on ideals in a commutative noetherian ring of prime characteristic remains poorly understood for example tight closure is not known to behave well under localization given an ideal i of a ring r letting i denote the tight closure of i it is not known whether or not

how to build all chebyshevian spline spaces good for geometric design in the present work we determine all chebyshevian spline spaces good for geometric design by chebyshevian spline space we mean a space of splines with sections in different extended chebyshev spaces and with connection matrices at the knots we say that such a spline space is good for design when it possesses blossoms to justify the terminology let us recall that in this general framework existence of blossoms defined on a restricted set of tuples makes it possible to develop all the classical geometric design algorithms for splines furthermore existence of blossoms is equivalent to existence of a b spline bases both in the spline space itself and in all other spline spaces derived from it by insertion of knots we show that chebyshevian spline spaces good for design can be described by linear piecewise differential operators associated with systems of piecewise weight functions with respect to which the connection matrices are identity matrices many interesting consequences can be drawn from the latter characterisation as an example all chebsyhevian spline spaces good for design can be built by means of integral recurrence relations

effortless quasi interpolation in hierarchical spaces we present a general and simple procedure to construct quasi interpolants in hierarchical spaces such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement the proposed hierarchical quasi interpolants are described in terms of the so called truncated hierarchical basis assuming a quasi interpolant is selected for each space associated with a particular level in the hierarchy the hierarchical quasi interpolants are obtained without any additional manipulation the main properties like polynomial reproduction of the quasi interpolants selected at each level are locally preserved in the hierarchical construction we show how to construct hierarchical local projectors and the local approximation order of the underling hierarchical space is also investigated the presentation is detailed for the truncated hierarchical b spline basis and we discuss its extension to a more general framework

gb splines of arbitrary order explicit formulae and recurrence relations for the calculation of generalized b splines gb splines of arbitrary order are given we derive main properties of gb splines and their series i e partition of unity shape preserving properties invariance with respect to ane transformations etc it is shown that such splines have the variation diminishing property and are chebyshevian splines c 1999 elsevier science b v all rights reserved

interpolation with slackness and continuity control and convexity preservation using singular blending this paper presents a new interpolation method that enables the construction of c2 cubic polynomial spline curves without solving a global system of equations while providing slackness continuity control and convexity preserving ability the basic idea is to blend a cubic b spline curve with a singularly parametrized sequence of connected line segments a global slackness parameter controls the tautness specifically the distance between the interpolating curve and the linear interpolant the order of continuity at each knot is controlled via multiple knot insertions so that cusps and straight line segments can be conveniently prescribed in addition a method for selecting local slackness values to produce g1 convexity preserving curve is presented with the low degree polynomials and direct computation of control vertices this local method is computationally simple and is useful for interactive shape design and computer graphics applications

the hyers ulam stability constant for chebyshevian bernstein operators abstract on a closed bounded interval a given extended chebyshev space which possesses a bernstein basis generates infinitely many operators of the bernstein type we show that all these operators share the same hyers ulam stability constant this constant is the maximum in absolute value of the bezier coefficients of the generalised chebyshev polynomial associated with the given space we establish an optimality property of these bernstein operators with respect to the hyers ulam stability constant numerical computations of these constants are investigated in two cases rational and muntz bernstein operators with special emphasis on their behaviour with respect to the concerned interval and to dimension elevation

on a class of weak tchebycheff systems in this paper we study the approximation power the existence of a normalized b basis and the structure of a degree raising process for spaces of the form requiring suitable assumptions on the functions u and v the results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of cagd

a geometric approach for hermite subdivision we present a non stationary non uniform scheme for two point hermite subdivision the novelty of this approach relies on a geometric interpretation of the subdivision steps related to generalized bernstein bases which permits to overcome the usually unavoidable analytical difficulties the main advantages consist in extra smoothness conditions which in turn produce highly regular limit curves and in an elegant structure of the subdivision described by three de casteljau type matrices as a by product the scheme is inherently shape preserving

graded annihilators and tight closure test ideals abstract let r be a commutative noetherian local ring of prime characteristic p with maximal ideal m the main purposes of this paper are to show that if the injective envelope e of r m has a structure as an x torsion free left module over the frobenius skew polynomial ring over r in the indeterminate x then r has a tight closure test element for modules and is f pure and to relate the test ideal of r to the smallest e special ideal of r of positive height a byproduct is an analogue of a result of janet cowden vassilev she showed in the case where r is an f pure homomorphic image of an f finite regular local ring that there exists a strictly ascending chain 0 0 1 t r of radical ideals of r such that for each i 0 t 1 the reduced local ring r i is f pure and its test ideal has positive height and is exactly i 1 i this paper presents an analogous result in the case where r is complete but not necessarily f finite and e has a structure as an x torsion free left module over the frobenius skew polynomial ring whereas cowden vassilev s results were based on r fedder s criterion for f purity the arguments in this paper are based on the author s work on graded annihilators of left modules over the frobenius skew polynomial ring

fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines a method is presented for fitting a function defined on a general smooth spherelike surfaces given measurements on the function at a set of scattered points lying ons the approximating surface is constructed by mapping the surface onto a rectangle and using a tensor product of polynomial splines with periodic trigonometric splines the use of trigonometric splines allows a convenient solution of the problem of assuring that the resulting surface is continuous and has continuous tangent planes at all points ons two alternative algorithms for computing the coefficients of the tensor fit are presented one based on global least squares and the other on the use of local quasi interpolators the approximation order of the method is established and the numerical performance of the two algorithms is compared

understanding recurrence relations for chebyshevian b splines via blossoms the purpose of this article is to show how naturally recurrence relations for most general chebyshevian b splines emerge from blossoms in particular this work gives a new insight into previous results by lyche a recurrence relation for chebyshevian b splines constr approx 1 1985 155 178 the importance of which it underlines

shape preserving properties of generalized bernstein operators on extended chebyshev spaces we study the existence and shape preserving properties of a generalized bernstein operator b n fixing a strictly positive function f 0 and a second function f 1 such that f 1 f 0 is strictly increasing within the framework of extended chebyshev spaces u n the first main result gives an inductive criterion for existence suppose there exists a bernstein operator b n c a b u n with strictly increasing nodes fixing f 0 f 1 in u n if u n subset u n 1 and u n 1 has a non negative bernstein basis then there exists a bernstein operator b n 1 c a b u n 1 with strictly increasing nodes fixing f 0 and f 1 in particular if f 0 f 1 f n is a basis of u n such that the linear span of f 0 f k is an extended chebyshev space over a b for each k 0 n then there exists a bernstein operator b n with increasing nodes fixing f 0 and f 1 the second main result says that under the above assumptions the following inequalities hold b n f geq b n 1 f geq f furthermore b nf is f0 f1 convex for all f0 f1 convex functions f in c a b

centers of f purity in this paper we study a positive characteristic analogue of the centers of log canonicity of a pair r we call these analogues centers of f purity we prove positive characteristic analogues of subadjunction like results prove new stronger subadjunction like results and in some cases lift these new results to characteristic zero using a generalization of centers of f purity which we call uniformly f compatible ideals we give a characterization of the test ideal which unifies several previous characterizations finally in the case that 0 we show that uniformly f compatible ideals coincide with the annihilators of the mathcal f e r k submodules of e r k as defined by lyubeznik and smith

the triangular kagome lattices revisited abstract the dimer problem ising spins and bond percolation on the triangular kagome lattice have been studied extensively by physicists in this paper based on the fact the triangular kagome lattice with toroidal boundary condition can be regarded as the line graph of 3 12 12 lattice with toroidal boundary condition we derive the formulae of the number of spanning trees the energy and the kirchhoff index of the triangular kagome lattice with toroidal boundary condition

spaces of generalized splines over t meshes we consider a class of non polynomial spaces namely a noteworthy case of extended chebyshev spaces and we generalize the concept of polynomial spline space over t mesh to this non polynomial setting in other words we focus on a class of spaces spanned in each cell of the t mesh both by polynomial and by suitably chosen non polynomial functions which we will refer to as generalized splines over t meshes for such spaces we provide under certain conditions on the regularity of the space a study of the dimension and of the basis based on the notion of minimal determining set as well as some results about the dimension of refined and merged t meshes finally we study the approximation power of the just constructed spline spaces

asymptotic incidence energy of lattices abstract the energy of a graph g arising in chemical physics denoted by e g is defined as the sum of the absolute values of the eigenvalues of g as an analogue to e g the incidence energy ie g defined as the sum of the singular values of the incidence matrix of g is a much studied quantity with well known applications in chemical physics in this paper based on the results by yan and zhang 2009 we propose the incidence energy per vertex problem for lattice systems and present the closed form formulae expressing the incidence energy of the hexagonal lattice triangular lattice and 3 3 4 2 lattice respectively moreover we show that the incidence energy per vertex of lattices is independent of the toroidal cylindrical and free boundary conditions in particular the explicit asymptotic values of the incidence energy in these lattices are obtained by utilizing the applications of analysis approach with the help of calculational software

trigonometric generalized t splines abstract the paper s main aim consists of extending the t spline approach to trigonometric generalized b splines a particularly relevant case of non polynomial splines such goal can be achieved by a careful revision of some results concerning the basic properties of the trigonometric generalized b splines and by a formalization of the concept of t splines in the trigonometric setting moreover fundamental for the use of this new tool is the study of the noteworthy case with constant frequencies and of the linear independence of the corresponding blending functions which can be proved to be strongly linked to the linear independence of the polynomial blending functions associated to the same t mesh

an extension of the bezier model abstract in this paper we first construct a new kind of basis functions by a recursive approach based on these basis functions we define the bezier like curve and rectangular bezier like surface then we extend the new basis functions to the triangular domain and define the bernstein bezier like surface over the triangular domain the new curve and surfaces have most properties of the corresponding classical bezier curve and surfaces moreover the shape parameter can adjust the shape of the new curve and surfaces without changing the control points along with the increase of the shape parameter the new curve and surfaces approach the control polygon or control net in addition the evaluation algorithm for the new curve and triangular surface are provided

on dimension elevation in quasi extended chebyshev spaces via blossoms we analyse the dimension elevation process from mathcal e n p q to mathcal e n 1 p q where mathcal e n p q is spanned over 0 1 by 1 x xn 2 xp 1 x q p q being any convenient real numbers such spaces are not extended chebyshev spaces but quasi extended chebyshev spaces they were recently introduced in cagd for shape preservation purposes costantini in math comp 46 203 214 1986 costantini in advanced course on fairshape pp 87 114 in 1996 costantini in curves and surfaces with applications in cagd pp 85 94 1997 our results give a new insight into the special case p q for which dimension elevation had already been considered first when p q was supposed to be an integer goodman and mazure in j approx theory 109 48 81 2001 then without the latter requirement costantini et al in numer math 101 333 354 2005 the question of dimension elevation in more general quasi extended chebyshev spaces is also addressed

generalized b splines as a tool in isogeometric analysis the concept of isogeometric analysis has been proposed in 13 where nurbs are considered as basis of the analysis thanks to their ability to construct an exact geometric model in several practical applications and to their popularity in commercial cad systems in this paper we propose an alternative to the rational model presenting an isogeometric analysis approach based on generalized b splines geometric models exactly represented by generalized b splines include those generated by nurbs moreover generalized b splines possess all fundamental properties of algebraic b splines and nurbs including classical refinement processes as h p k refinements finally since generalized b splines are not confined to rational functions they behave completely similar to algebraic b splines with respect to differentiation and integration this seems to be of interest in the treatment of some relevant problems

overlapping schwarz methods for isogeometric analysis based on generalized b splines abstract generalized b splines gb splines are a special class of tchebycheff b splines that are smooth piecewise function with sections in more general spaces gb splines allow for an exact representation of conic sections as well as transcendental curves and thus they become very attractive for geometrical modeling and numerical simulation in this paper we present overlapping schwarz preconditioners for elliptic problems discretized with isogeometric analysis based on gb splines an h analysis of the proposed preconditioners shows an optimal convergence rate bound that is scalable in the number of subdomains and that is linear in the ratio between subdomain and overlap sizes numerical results in two and three dimensional tests confirm this analysis and also illustrate the good convergence properties of the preconditioner with respect to the discretization parameters the domain deformation and the jumping coefficients

shape analysis of cubic trigonometric bezier curves with a shape parameter abstract for the cubic trigonometric polynomial curves with a shape parameter tb curves for short the effects of the shape parameter on the tb curve are made clear the shape features of the tb curve are analyzed the necessary and sufficient conditions are derived for these curves having single or double inflection points a loop or a cusp or be locally or globally convex the results are summarized in a shape diagram of tb curves which is useful when using tb curves for curve and surface modeling furthermore the influences of shape parameter on the shape diagram and the ability for adjusting the shape of the curve are shown by graph examples respectively

a novel extension to the polynomial basis functions describing bezier curves and surfaces of degree n with multiple shape parameters the construction of bezier curves using shape control parameters is one of the most popular areas of research in computer aided geometric design cagd a class of new polynomial basis functions with n 1 local shape control parameters is presented here to allow the construction of bezier curves with n local shape control parameters which is an extension to the classical bernstein basis functions of degree n the properties of the proposed basis functions and the corresponding piecewise polynomial curve with n 1 local shape control parameters are analyzed this analysis shows that the new class of polynomial functions meets the conditions required for both c 0 c 1 and c 2 continuity as well as g 0 g 1 and g 2 continuity some curve design applications are then discussed and an extended application for surface design is also presented the modeling examples illustrate that the new extension provides not only a better approximation and mathematical description of bezier curves but allows the shape parameters to be altered making it a valuable method for the design of curves and surfaces

curve construction based on four αβ bernstein like basis functions a b s t r a c t four new bernstein like basis functions with two exponential shape parameters are constructed in this paper which include the cubic said ball basis functions and the cubic bernstein basis functions within the general framework of quasi extended chebyshev space we prove that the proposed bernstein like basis is an optimal normalized totally positive basis in order to compute the corresponding bezier like curves stably and efficiently a new corner cutting algorithm is developed necessary and sufficient conditions are derived for the planar bezier like curve having single or double inflection points a loop or a cusp or be locally or globally convex in terms of the relative position of its control polygons side vectors based on the new proposed bernstein like basis a class of b spline like basis functions with two local exponential shape parameters is constructed their totally positive property is also proved the associated b splinelike curves have c 2 continuity at single knots and include the cubic non uniform b spline

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

minimizing kirchhoff index among graphs with a given vertex bipartiteness the resistance distance between any two vertices of a graph g is defined as the effective resistance between them if each edge of g is replaced by a unit resistor the kirchhoff index kf g is the sum of the resistance distances between all the pairs of vertices in g the vertex bipartiteness vb of a graph g is the minimum number of vertices whose deletion from g results in a bipartite graph in this paper we characterize the graph having the minimum kf g values among graphs with a fixed number n of vertices and fixed vertex bipartiteness 1 v b n 3

limit curve of h bezier curves and rational bezier curves in standard form with the same weight the basis of h bezier curves of degree n is 1 t t n 2 sinh t and cosh t for t 0 1 we find the limit curve of h bezier curves of degree n as a parameter goes to which is the bezier curve of degree n 2 and prove it using mathematical induction and special properties of h basis functions we also compare it to the limit curve of rational bezier curves of degree n in standard form with the same weight w as it goes to which is the rational bezier curve of degree n 2

a class of algebraic trigonometric blended splines this paper presents a new kind of algebraic trigonometric blended spline curve called xyb curves generated over the space 1 t sint cost sin 2t sin 3t cos 3t the new curves not only inherit most properties of usual cubic b spline curves in polynomial space but also enjoy some other advantageous properties for modeling for given control points the shape of the new curves can be adjusted by using the parameters x and y when the control points and the parameters are chosen appropriately the new curves can represent some conics and transcendental curves in addition we present methods of constructing an interpolation xyb spline curve and an xyb spline curve which is tangent to the given control polygon the generation of tensor product surfaces by these new spline curves is straightforward many properties of the curves can be easily extended to the surfaces the new surfaces can exactly represent the rotation surfaces as well as the surfaces with elliptical or circular sections

piecewise quartic polynomial curves with a local shape parameter piecewise quartic polynomial curves with a local shape parameter are presented in this paper the given blending function is an extension of the cubic uniform b splines the changes of a local shape parameter will only change two curve segments with the increase of the value of a shape parameter the curves approach a corresponding control point the given curves possess satisfying shape preserving properties the given curve can also be used to interpolate locally the control points with gc2 continuity thus the given curves unify the representation of the curves for interpolating and approximating the control polygon as an application the piecewise polynomial curves can intersect an ellipse at different knot values by choosing the value of the shape parameter the given curve can approximate an ellipse from the both sides and can then yield a tight envelope for an ellipse some computing examples for curve design are given

normalized b basis of the space of trigonometric polynomials and curve design a normalized b basis of the space of trigonometric polynomials of degree n is presented some interesting properties of the basis functions are given based on the basis symmetric trigonometric polynomial curves like bezier curves are constructed the trigonometric polynomial curves present the shape of their control polygons well thus the theoretics and methods are proposed for curve representation of the trigonometric polynomial space by adding additional control points the given graph examples show that the trigonometric polynomial curves are nearer their control polygons than the bezier curves for the same parametric variable and the same degree

test ideals in quotients of f finite regular local rings let s be an f finite regular local ring and i an ideal contained in s let r s i fedder proved that r is f pure if and only if i p i m p we have noted a new proof for his criterion along with showing that i q i q where is the pullback of the test ideal for r combining the the f purity criterion and the above result we see that if r s i is f pure then r is also f pure in fact we can form a filtration of r i 0 1 i that stabilizes such that each r i is f pure and its test ideal is i 1 to find examples of these filtrations we have made explicit calculations of test ideals in the following setting let r t i where t is either a polynomial or a power series ring and i p1 pn is generated by monomials and the r pi are regular set j p1 pi pn then j par this paper concerns the study of the test ideal in f finite quotients of regular local rings test elements play a key role in tight closure theory once known they make computing tight closures of ideals and modules easier in fact in an excellent gorenstein local ring with an isolated singularity r hom i i e where i is generated by a system of parameters that are test elements and e is the injective hull of r see hu1 and s1 we also know for parameter ideals i that i i thus knowing is basically equivalent to knowing the tight closure of a system of parameters which is contained in the test ideal a recent paper of huneke and smith hs links tight closure to kodaira vanishing for graded rings r with characteristic either 0 or p where p 0 recall that the a invariant for a graded ring r denoted a is equal to min i r i 6 0 where r is the canonical module for r if r is gorenstein then r r a huneke and smith prove that the test ideal is exactly the ideal generated by elements of degree greater than the a invariant of r if and only if a strong kodaira vanishing holds on r a recent paper of hara ha confirms that this strong kodaira vanishing holds in finitely generated algebras over a field of characteristic zero in this paper we study test ideals of f finite rings which are reduced quotients of a regular local ring reduced quotients of f finite regular local rings have been studied by both fedder fe and glassbrenner gl fedder s work concerns f purity aspects of these rings and glassbrenner s results use fedder s techniques to examine strong f regularity the object that plays a key role in their work is received by the editors november 4 1996 1991 mathematics subject classification primary 13a35

monotonicity preserving representations of non polynomial surfaces we prove that in contrast to the case for rational surfaces some tensor product representations through spaces containing algebraic trigonometric and hyperbolic polynomials are monotonicity preserving the surface representations provided in this paper are the only known monotonicity preserving surfaces in addition to the tensor product bezier and tensor product b spline surfaces

the signless laplacian coefficients and incidence energy of bicyclic graphs abstract let q g x det x i q g i 1 n 1 i i x n i be the characteristic polynomial of the signless laplacian matrix of a graph g of order n this paper investigates how the signless laplacian coefficients i e coefficients of q g x change after some graph transformations these results are used to prove that the set b n of all bicyclic graphs of order n has exactly two minimal elements with respect to the partial ordering of their coefficients furthermore we present a sharp lower bound for the incidence energy of bicyclic graphs of order n and characterize all extremal graphs

on the approximation power of generalized t splines the paper presents some properties of generalized t splines gt splines which are crucial to their actual application in particular we construct a dual basis for a noteworthy class of gt splines which allows to show that under suitable conditions they form a partition of unity moreover we study the approximation properties of the gt spline space by constructing a class of quasi interpolants which belong to it and are defined by giving a dual basis

graph energy change due to edge deletion the energy of a graph is the sum of the singular values of its adjacency matrix we are interested in how the energy of a graph changes when edges are deleted examples show that all cases are possible increased decreased unchanged our goal is to find possible graph theoretical descriptions and to provide an infinite family of graphs for each case the main tool is a singular value inequality for complementary submatrices and its equality case

a general class of bernstein like bases this paper presents a unified approach to deal with spaces containing simultaneously algebraic and trigonometric or hyperbolic polynomials bases with optimal shape preserving and stability properties are constructed evaluation and subdivision algorithms for them are provided bases for the corresponding and mixed spline spaces are also constructed some nice properties of these bases and the generated curves are shown

a unified approach to the asymptotic topological indices of various lattices we present a unified approach to the asymptotic topological indices of various lattices we propose the topological indices per vertex problem for lattice systems the explicit asymptotic values of laplacian energies for various lattices are obtained we deduce the laplacian energies of many types of lattices are independent of various boundary conditions in this paper we present a unified approach to the asymptotic topological indices of various lattices moreover we propose the various topological indices per vertex problem for lattice systems and show that the various topological indices per vertex of lattices are independent of the toroidal cylindrical and free boundary conditions our result is a generalization of some earlier results

asymptotic laplacian energy like invariant of lattices let 1 2 n denote the laplacian eigenvalues of a graph g with n vertices the laplacian energy like invariant denoted by lel g i 1 n 1 i is a novel topological index in this paper we show that the laplacian energy like per vertex of various lattices is independent of the toroidal cylindrical and free boundary conditions simultaneously the explicit asymptotic values of the laplacian energy like in these lattices are obtained moreover our approach implies that in general the laplacian energy like per vertex of other lattices is independent of the boundary conditions

spectral analysis of matrices in galerkin methods based on generalized b splines with high smoothness we present a first step towards the spectral analysis of matrices arising from iga galerkin methods based on hyperbolic and trigonometric gb splines second order differential problems with constant coefficients are considered and discretized by means of sequences of both nested and non nested spline spaces we prove that there always exists an asymptotic eigenvalue distribution which can be compactly described by a symbol just like in the polynomial case there is a complete similarity between the symbol expressions in the hyperbolic trigonometric and polynomial cases this results in similar spectral features of the corresponding matrices we also analyze the iga discretization based on trigonometric gb splines for the eigenvalue problem related to the univariate laplace operator we prove that for non nested spaces the phase parameter can be exploited to improve the spectral approximation with respect to the polynomial case as part of the analysis we derive several fourier properties of cardinal gb splines

changeable degree spline basis functions a b spline basis function is a piecewise function of polynomials of equal degree on its support interval this paper extends b spline basis functions to changeable degree spline cd spline for short basis functions each of which may consist of polynomials of different degrees on its support interval the cd spline basis functions possess many b spline like properties and include the b spline basis functions as subcases their corresponding parametric curves called cd spline curves are like b spline curves and also have many good properties if we use the cd spline basis functions to design a curve made up of polynomial segments of different degrees the number of control points may be decreased

degree elevation of changeable degree spline changeable degree spline cd spline a direct extension of the b spline allows different segments to possess different degrees in this paper we present a method for degree elevation of a cd spline the method is flexible as it applies to any segment of any cd spline elevated by any degree a cd spline can be transformed into a b spline and vice versa in various orders by using this method moreover this degree elevation method is presented as a corner cutting process

a novel generalization of bezier curve and surface a new formulation for the representation and designing of curves and surfaces is presented it is a novel generalization of bezier curves and surfaces firstly a class of polynomial basis functions with n adjustable shape parameters is present it is a natural extension to classical bernstein basis functions the corresponding bezier curves and surfaces the so called quasi bezier i e q bezier for short curves and surfaces are also constructed and their properties studied it has been shown that the main advantage compared to the ordinary bezier curves and surfaces is that after inputting a set of control points and values of newly introduced n shape parameters the desired curve or surface can be flexibly chosen from a set of curves or surfaces which differ either locally or globally by suitably modifying the values of the shape parameters when the control polygon is maintained the q bezier curve and surface inherit the most properties of bezier curve and surface and can be more approximated to the control polygon it is visible that the properties of end points on q bezier curve and surface can be locally controlled by these shape parameters some examples are given by figures

the construction of λµ b spline curves and its application to rotational surfaces in order to provide more flexible approaches for designers the b spline curves are constructed as a generalization of the traditional cubic uniform b spline curves possessing multiple local shape control parameters b spline curves not only inherit the properties of cubic uniform b spline curves but also exhibit better performance when adjusting its local shapes through two local shape control parameters particularly to adjust and control the shapes of rotational surfaces more elegantly the b spline rotational surfaces with two local shape parameters are presented and utilized a rotational surface is produced by combining b spline with a transfinite vector valued rational interpolation function further the properties of rotational surfaces as well as its applications in rotational surface designs are explored finally the modeling examples are supplied to illustrate the proposed method in admitting the easy control of the shape of a surface which suggest the much wider applications to the pattern design system of apparel cad cam

isogeometric analysis in advection diffusion problems tension splines approximation we present a novel approach within the new paradigm of isogeometric analysis introduced by hughes et al 2005 6 to deal with advection dominated advection diffusion problems the key ingredient is the use of galerkin approximating spaces of functions with high smoothness as in iga based on classical b splines but particularly well suited to describe sharp layers involving very strong gradients