forcing extensions of partial lattices abstract we prove the following result let k be a lattice let d be a distributive lattice with zero and let con c k d be a 0 homomorphism where con c k denotes the 0 semilattice of all finitely generated congruences of k then there are a lattice l a lattice homomorphism f k l and an isomorphism con c l d such that con c f furthermore l and f satisfy many additional properties for example i l is relatively complemented ii l has definable principal congruences iii if the range of is cofinal in d then the convex sublattice of l generated by f k equals l we mention the following corollaries that extend many results obtained in the last decades in that area 1 every lattice k such that con c k is a lattice admits a congruence preserving extension into a relatively complemented lattice 2 every 0 direct limit of a countable sequence of distributive lattices with zero is isomorphic to the semilattice of compact congruences of a relatively complemented lattice with zero

improved derivative bounds of the rational quadratic bezier curves a new derivative bound of the rational quadratic bezier curves is obtained it is an improvement over the bound which was regarded as a sharp bound in a recent paper also the new bound is superior to the existing bounds given by some other authors

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

k π 1 and word problems for infinite type artin tits groups and applications to virtual braid groups let be a coxeter graph let w s be its associated coxeter system and let a be its associated artin tits system we regard w as a reflection group acting on a real vector space v let i be the tits cone and let e be the complement in i iv of the reflecting hyperplanes recall that salvetti charney and davis have constructed a simplicial complex having the same homotopy type as e we observe that if t subset s then t naturally embeds into we prove that this embedding admits a retraction pi t omega gamma to omega gamma t and we deduce several topological and combinatorial results on parabolic subgroups of a from a family mathcal s of subsets of s having certain properties we construct a cube complex we show that has the same homotopy type as the universal cover of e and we prove that is cat 0 if and only if mathcal s is a flag complex we say that x subset s is free of infinity if x has no edge labeled by we show that if e gamma x is aspherical and a x has a solution to the word problem for all x subset s free of infinity then e is aspherical and a has a solution to the word problem we apply these results to the virtual braid group vb n in particular we give a solution to the word problem in vb n and we prove that the virtual cohomological dimension of vb n is n 1

the integer cohomology algebra of toric arrangements abstract we compute the cohomology ring of the complement of a toric arrangement with integer coefficients and investigate its dependency from the arrangement s combinatorial data to this end we study a morphism of spectral sequences associated to certain combinatorially defined subcomplexes of the toric salvetti category in the complexified case and use a technical argument in order to extend the results to full generality as a byproduct we obtain a combinatorial version of brieskorn s lemma in terms of salvetti complexes of complexified arrangements a uniqueness result for realizations of arithmetic matroids with at least one basis of multiplicity 1

boundary manifolds of line arrangements abstract while the boundary 3 manifold of a line arrangement in the complex plane depends only on the incidence correspondence of the line arrangement the homotopy type of the complement depends on the relative positions of incidences in this paper we describe the homotopy type of line arrangement complements in terms of an associated plumbed graph for pseudo real line arrangements this method provides an explicit description of the homotopy type and fundamental group of the complement in terms of its ordered incidence graph the method also extends to a larger class of unknotted line arrangements

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

the equivariant orlik solomon algebra abstract given a real hyperplane arrangement a the complement m a of the complexification of a admits an action of the group z 2 by complex conjugation we define the equivariant orlik solomon algebra of a to be the z 2 equivariant cohomology ring of m a with coefficients in the field f 2 we give a combinatorial presentation of this ring and interpret it as a deformation of the ordinary orlik solomon algebra into the varchenko gelfand ring of locally constant f 2 valued functions on the complement m r a of a in r n we also show that the z 2 equivariant homotopy type of m a is determined by the oriented matroid of a as an application we give two examples of pairs of arrangements a and a such that m a and m a have the same nonequivariant homotopy type but are distinguished by the equivariant orlik solomon algebra

the possible values of critical points between strongly congruence proper varieties of algebras abstract we denote by con c a the 0 semilattice of all finitely generated congruences of an universal algebra a and we define con c v as the class of all isomorphic copies of all con c a for a v for any variety v of algebras let v and w be locally finite varieties of algebras such that for each finite algebra a v there are up to isomorphism only finitely many b w such that con c a con c b and every such b is finite if con c v con c w then there exists a 0 semilattice of cardinality 2 in con c v con c w our result extends to quasivarieties of first order structures with finitely many relation symbols and relative congruence lattices in particular if w is a finitely generated variety of algebras then this occurs in case w omits the tame congruence theory types 1 and 5 which in turn occurs in case w satisfies a nontrivial congruence identity the bound 2 is sharp

generalized bezier curves and surfaces based on lupas q analogue of bernstein operator in this paper a new generalization of bezier curves with one shape parameter is constructed it is based on the lupas q analogue of bernstein operator which is the first generalized bernstein operator based on the q calculus the new curves have some properties similar to classical bezier curves moreover we establish degree evaluation and de casteljau algorithms for the generalization furthermore we construct the corresponding tensor product surfaces over the rectangular domain and study the properties of the surfaces as well as the degree evaluation and de casteljau algorithms compared with q bezier curves and surfaces based on phillips q bernstein polynomials our generalizations show more flexibility in choosing the value of q and superiority in shape control of curves and surfaces the shape parameters provide more convenience for the curve and surface modeling

the fundamental group of the complement of a union of complex hyperplanes let m be the complement of the union of n linear subspaces vii of and let n be the quotient space of m by the t topf action of c c t 0 on m in this paper we determine nl m and hi n if q t p are linear forms determining v 1 v then n is just the complement of the projective variety w f c p i p 0 iw 1 furthermore the milnor fiber f f i c g is a cyclic n fold cover of n and we also determine nt f such spaces and groups are of interest in several areas of mathematics including algebraic geometry lie algebras combinatorics and topology in 1929 zariski showed 9 that if the v were planes in 3 in general position then i n is abelian we note that h m or h i n is easily computed by duality so that this determines n this was then used to show that if c c i p 2 is a curve which degenerates to such a configuration then 1 ip2 c is abelian as well much more recently arnold 1 brieskorn 2 and deligne 5 considered the situation in which g is a finite group acting by reflections on ir e g the weyl group acting on a lie algebra by complexification one gets the above situation with the v just the reflecting hyperplanes in this case it was shown that m is a k n i where zl m is a generalized colored braid group in the case that the lie group is at i so that the weyl group is the symmetric group s l one has an exact sequence

extending fundamental formulas from classical b splines to quantum b splines we derive a collection of fundamental formulas for quantum b splines analogous to known fundamental formulas for classical b splines starting from known recursive formulas for evaluation and quantum differentiation along with quantum analogues of the marsden identity we derive quantum analogues of the de boor fix formula for the dual functionals explicit formulas for the quantum b splines in terms of divided differences of truncated power functions formulas for computing divided differences of arbitrary functions by quantum integrating certain quantum derivatives of these functions with respect to the quantum b splines closed formulas for the quantum integral of the quantum b splines over their support and finally a 1 q convolution formula for uniform q b splines

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

universal cover of salvetti s complex and topology of simplicial arrangements of hyperplanes let v be a real vector space an arrangement of hyperplanes in v is a finite set a of hyperplanes through the origin a chamber of a is a connected component of v h a h the arrangement a is called simplicial if h a h 0 and every chamber of a is a simplicial cone for an arrangement a of hyperplanes in v we set formule where v c c v is the complexification of v and for h a h c is the complex hyperplane of v c spanned by h let a be an arrangement of hyperplanes of v salvetti constructed a simplicial complex sal a and proved that sal a has the same homotopy type as m a in this paper we give a new short proof of this fact

improved bounds on the magnitude of the derivative of rational bezier curves abstract in this paper we derive some new derivative bounds of rational bezier curves according to some existing identities and inequalities the comparison of the new bounds with some existing ones is also presented

characteristic polynomials of subspace arrangements and finite fields let a be any subspace arrangement in rndefined over the integers and let fqdenote the finite field withqelements letqbe a large prime we prove that the characteristic polynomial a q of a counts the number of points in fnqthat do not lie in any of the subspaces ofa viewed as subsets of fnq this observation which generalizes a theorem of blass and sagan about subarrangements of the bnarrangement reduces the computation of a q to a counting problem and provides an explanation for the wealth of combinatorial results discovered in the theory of hyperplane arrangements in recent years the basic idea has its origins in the work of crapo and rota 1970 we find new classes of hyperplane arrangements whose characteristic polynomials have simple form and very often factor completely over the nonnegative integers

convexity of rational curves and total positivity we analyze convexity preserving properties of curves from a geometric point of view we also characterize totally positive systems of functions in terms of geometric convexity preserving properties of the rational curves rational bezier and nonuniform rational b spline curves are included in this setting

distributive congruence lattices of congruence permutable algebras abstract we prove that every distributive algebraic lattice with at most 1 compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module the 1 bound is optimal as we find a distributive algebraic lattice d with 2 compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences hence neither of any group nor of any module thus solving negatively a problem of e t schmidt from 1969 furthermore d may be taken as the congruence lattice of the free bounded lattice on 2 generators in any non distributive lattice variety some of our results are obtained via a functorial approach of the semilattice valued distances used by b jonsson in his proof of whitman s embedding theorem in particular the semilattice of compact elements of d is not the range of any distance satisfying the v condition of type 3 2 on the other hand every distributive 0 semilattice is the range of a distance satisfying the v condition of type 2 this can be done via a functorial construction

the deligne complex for the four strand braid group this paper concerns the homotopy type of hyperplane arrangements associated to infinite coxeter groups acting as reflection groups on c n a long standing conjecture states that the complement of such an arrangement should be aspherical some partial results on this conjecture were previously obtained by the author and m davis in this paper we extend those results to another class of coxeter groups the key technical result is that the spherical deligne complex for the 4 strand braid group is cat 1

a solution to dilworth s congruence lattice problem we construct a distributive algebraic lattice d that is not isomorphic to the congruence lattice of any lattice this solves a long standing open problem traditionally attributed to r p dilworth from the forties the lattice d has compact top element and aleph omega 1 compact elements our results extend to all algebras possessing a polynomially definable structure of a join semilattice with a largest element

total positivity and neville elimination abstract neville elimination is described in terms of schur complements of matrices and used to improve some well known characterizations of totally positive and strictly totally positive matrices

semilattices of finitely generated ideals of exchange rings with finite stable rank we find a distributive v 0 1 semilattice s of size aleph 1 that is not isomorphic to the maximal semilattice quotient of any riesz monoid endowed with an order unit of finite stable rank we thus obtain solutions to various open problems in ring theory and in lattice theory in particular there is no exchange ring thus no von neumann regular ring and no c algebra of real rank zero with finite stable rank whose semilattice of finitely generated idempotent generated two sided ideals is isomorphic to s there is no locally finite modular lattice whose semilattice of finitely generated congruences is isomorphic to s these results are established by constructing an infinitary statement denoted here by urpsr that holds in the maximal semilattice quotient of every riesz monoid endowed with an order unit of finite stable rank but not in the semilattice s

accurate computations with lupa matrices lupa q analogues of the bernstein functions play an important role in approximation theory and computer aided geometric design their collocation matrices are called lupa matrices in this paper we provide algorithms for computing the bidiagonal decomposition of these matrices and their inverses to high relative accuracy it is also shown that these algorithms can be used to perform to high relative accuracy several algebraic calculations with these matrices such as the calculation of their inverses their eigenvalues or their singular values numerical experiments are included

iterated monoidal categories abstract we develop a notion of an n fold monoidal category and show that it corresponds in a precise way to the notion of an n fold loop space specifically the group completion of the nerve of such a category is an n fold loop space and free n fold monoidal categories give rise to a finite simplicial operad of the same homotopy type as the classical little cubes operad used to parametrize the higher h space structure of an n fold loop space we also show directly that this operad has the same homotopy type as the n th smith filtration of the barratt eccles operad and the n th filtration of berger s complete graph operad moreover this operad contains an equivalent preoperad which gives rise to milgram s small model for 2 2 x when n 2 and is very closely related to milgram s model of n n x for n 2

some remarks on weighted lupas q bezier curves abstract recently han et al 2016 introduced weighted lupas q bezier curves as a novel generalization of the classical rational bezier curves in this note we make some remarks on their article in particular we revisit the rational quadratic case for an illustrative exposition

hyperplane complements of large type let f2 be an open convex subset of a finite dimensional real vectorspace v and m a collection of linear hyperplanes we shall suppose that m is locally finite on f2 this means that each point of f2 has a neighborhood meeting only finitely many hyperplanes in m as in 2 chap 5 f2 is partitioned by m in facets recalling that a plinth is a codimension 2 facet we shall say that m is of large type on f2 if for each plinth p its multiplicity m m p c m 3 the main result of the paper is the following

homological stability for hurwitz spaces and the cohen lenstra conjecture over function fields we prove a homological stabilization theorem for hurwitz spaces moduli spaces of branched covers of the complex projective line this has the following arithmetic consequence let 2 be prime and a a nite abelian group then there exists q q a such that for q greater than q a positive fraction of quadratic extensions of fq t have the part of their class group isomorphic to a

a nice labelling for tree like event structures of degree 3 we address the problem of finding nice labellings for event structures of degree 3 we develop a minimum theory by which we prove that the index of an event structure of degree 3 is bounded by a linear function of the height the main theorem of the paper states that event structures of degree 3 whose causality order is a tree have a nice labelling with 3 colors we exemplify how to use this theorem to construct upper bounds for the index of other event structures of degree 3

parallel subspace correction methods for nearly singular systems abstract in this paper we consider the parallel subspace correction psc methods for the nearly singular systems we apply the psc methods as the preconditioners when we solve the nearly singular systems by the conjugate gradient methods our focus is to estimate the condition number of the preconditioned systems we deduce the parameter independent estimates on the psc preconditioners for the nearly singular systems under appropriate assumptions on subspace decomposition the main assumption is that the kernel of the singular part of the system can be decomposed into a sum of local kernel subspaces our estimates can be applied into actual problems and two examples are analyzed in this paper one is the elliptic problem with large jumps in the coefficients the other is the planar nearly incompressible elasticity problem with the scott vogelius finite element discretization we prove that the related parallel multilevel methods for both examples are convergent uniformly with respect to the coefficients and the mesh size

on the bounds of the derivative of rational bezier curves according to some identities and inequalities about the intermediate weights and control points of the de casteljau algorithm for rational bezier curve we derive a new bound of the derivative of rational bezier curves the comparison of the new bounds with some existing ones is also presented based on some previous bounds the bound presented in this paper and many numerical examples we propose a conjecture of the bound for the derivative of rational bezier curves

the multivariate arithmetic tutte polynomial we introduce an arithmetic version of the multivariate tutte polynomial recently studied by sokal and a quasi polynomial that interpolates between the two we provide a generalized fortuin kasteleyn representation for representable arithmetic matroids with applications to arithmetic colorings and flows we give a new proof of the positivity of the coefficients of the arithmetic tutte polynomial in the more general framework of pseudo arithmetic matroids in the case of a representable arithmetic matroid we provide a geometric interpretation of the coefficients of the arithmetic tutte polynomial

on the lupas q transform the lupasq transform emerges in the study of the limit q lupas operator the latter comes out naturally as a limit for a sequence of the lupas q analogues of the bernstein operator lately it has been studied by several authors from different perspectives in mathematical analysis and approximation theory this operator is closely related to the q deformed poisson probability distribution which is used widely in the q boson operator calculus given email protected 0 1 email protected c 0 1 the q lupas transform of f is defined by l qf z 1 z q email protected email protected k 0 f 1 q k q k k 1 2 q q kz k in this paper we study some analytic properties of l qf z in particular we examine the conditions under which l qf can either be an entire function or a rational one

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

q bernstein polynomials and bezier curves we define q bernstein polynomials which generalize the classical bernstein polynomials and show that the difference of two consecutive q bernstein polynomials of a function f can be expressed in terms of second order divided differences of f it is also shown that the approximation to a convex function by its q bernstein polynomials is one sided a parametric curve is represented using a generalized bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve we study the nature of degree elevation and degree reduction for this basis and show that degree elevation is variation diminishing as for the classical bernstein basis

weighted lupas q bezier curves this paper is concerned with a new generalization of rational bernstein bezier curves involving q integers as shape parameters a one parameter family of rational bernstein bezier curves weighted lupas q bezier curves is constructed based on a set of lupas q analogue of bernstein functions which is proved to be a normalized totally positive basis the generalized rational bezier curve is investigated from a geometric point of view the investigation provides the geometric meaning of the weights and the representation for conic sections we also obtain degree evaluation and de casteljau algorithms by means of homogeneous coordinates numerical examples show that weighted lupas q bezier curves have more modeling flexibility than classical rational bernstein bezier curves and lupas q bezier curves and meanwhile they provide better approximations to the control polygon than rational phillips q bezier curves

projective wonderful models for toric arrangements abstract in this paper we illustrate an algorithmic procedure which allows to build projective wonderful models for the complement of a toric arrangement in a n dimensional algebraic torus t the main step of the construction inspired by 9 is a combinatorial algorithm that produces a toric variety by subdividing in a suitable way a given smooth fan

on the derivative formula of a rational bezier curve at a corner in this paper we develop the exact arbitrary order derivative formula of a rational bezier curve at its corner some examples are presented to demonstrate the computation of the exact derivatives

bezier curves based on lupas p q analogue of bernstein functions in cagd abstract this paper deals with the extension of rational lupas bernstein functions lupas bezier curves and surfaces involving p q integers as shape parameters for all p 0 and q 0 two different techniques such as de casteljau s algorithm and korovkin s type approximation based on p q integers are used a two parameter family for lupas p q bernstein functions is constructed and their degree elevation and reduction properties have been studied for lupas p q bezier curves some of their basic properties as well as degree elevation and de casteljau algorithm have been discussed the new curves have some properties similar to lupas q bezier curves similarly the corresponding tensor product for lupas bezier surfaces over the rectangular domain u v 0 1 0 1 depending on four parameters is constructed the de casteljau algorithm and degree evaluation properties of the surfaces for these generalizations and some fundamental properties are discussed we get lupas q bezier surfaces for u v 0 1 0 1 when we set the parameter p 1 p 2 1 with the help of rational lupas bernstein functions p q lupas bernstein operators are constructed based on korovkin s type approximation it has been shown that the sequence of p q analogue of lupas bernstein operators l p n q n n f x converges uniformly to f x c 0 1 if and only if 0 q n p n 1 such that lim n q n 1 lim n p n 1 and lim n p n n 1 lim n q n n 1 on the other hand for any p 0 fixed and p 1 the sequence l p q n f x converges uniformly to f x c 0 1 if and only if f x a x b for some a b r furthermore in comparison to q bezier curves and surfaces based on lupas q bernstein rational functions our generalization gives us more flexibility in controlling the shapes of curves and surfaces

symmetrical subgroups of artin groups we show that the subgroup fixed by a group of symmetries of an artin system a s is itself an artin group under the hypothesis that the deligne complex associated to a admits a suitable cat 0 metric such a metric is known to exist for all artin groups of type fc which include all the finite type artin groups as well as many infinite types we also recover the previously known analogous result for an arbitrary coxeter system w s

derivative bound estimations on rational conic bezier curves abstract in this paper we derive a bound on derivatives of rational conic bezier curves both theoretical analysis and numerical examples show that our bound is sharper than some existed ones discussions on optimal bounds for the derivatives of rational conic bezier curves are also provided

bernstein schurer kantorovich operators based on q integers in this paper we introduce a new kantorovich type generalization of the q bernstein schurer operators defined in muraru 2011 first we give the basic convergence theorem and then obtain the local direct results for these operators estimating the rate of convergence by using the modulus of smoothness and the lipschitz type maximal function respectively we also obtain a voronovskaja type theorem and investigate the statistical approximation properties of these operators with the help of a korovkin type statistical approximation theorem given in duman 2008

polyhedra and parameter spaces for matroids over valuation rings abstract in this paper we address two of the major foundational questions in the theory of matroids over rings first we provide a cryptomorphic axiomatisation by introducing an analogue of the base polytope for matroids second we describe a parameter space for matroids over a valuation ring which turns out to be a tropical version of the bott samelson varieties for the full flag variety thus a matroid over a valuation ring is a sequence of flags of tropical linear spaces a k a valuated matroids

new minimal bounds for the derivatives of rational bezier paths and rational rectangular bezier surfaces new minimal bounds are derived for the magnitudes of the derivatives of the rational bezier paths and the rational rectangular bezier surface patches of arbitrary degree which improve previous work of this type in many cases moreover our new bounds are explicitly given by simple and closed form expressions an important advantage of the closed form expressions is that they allow us to prove that our bounds are sharp under certain well defined conditions some numerical examples highlighting the potential of the new bounds in providing improved estimates are given in an appendix

higher homotopy groups of complements of complex hyperplane arrangements abstract we generalize results of hattori on the topology of complements of hyperplane arrangements from the class of generic arrangements to the much broader class of hypersolvable arrangements we show that the higher homotopy groups of the complement vanish in a certain combinatorially determined range and we give an explicit z 1 module presentation of p the first non vanishing higher homotopy group we also give a combinatorial formula for the 1 coinvariants of p for affine line arrangements whose cones are hypersolvable we provide a minimal resolution of 2 and study some of the properties of this module for graphic arrangements associated to graphs with no 3 cycles the algorithm for computing 2 is purely combinatorial the fitting varieties associated to 2 may distinguish the homotopy 2 types of arrangement complements with the same 1 and the same betti numbers in low degrees

the tilting theory of contraction algebras abstract to every minimal model of a complete local isolated cdv singularity donovan wemyss associate a finite dimensional symmetric algebra known as the contraction algebra we construct the first known standard derived equivalences between these algebras and then use the structure of an associated hyperplane arrangement to control the compositions obtaining a faithful group action on the bounded derived category further we determine precisely those standard equivalences which are induced by two term tilting complexes and show that any standard equivalence between contraction algebras up to algebra isomorphism can be viewed as the composition of our constructed functors thus for a contraction algebra we obtain a complete picture of its derived equivalence class and in particular of its derived autoequivalence group

arithmetic matroids the tutte polynomial and toric arrangements abstract we introduce the notion of an arithmetic matroid whose main example is a list of elements of a finitely generated abelian group in particular we study the representability of its dual providing an extension of the gale duality to this setting guided by the geometry of generalized toric arrangements we provide a combinatorial interpretation of the associated arithmetic tutte polynomial which can be seen as a generalization of crapo s formula for the classical tutte polynomial

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

combinatorial stratification of complex arrangements we present a method for discretizing complex hyperplane arrange ments by encoding their topology into a finite partially ordered set of sign vectors this is used in the following ways 1 a general method is given for constructing regular cell complexes having the homotopy type of the complement of the arrangement 2 for the case of complexified arrangements this specializes to the con struction of salvetti s we study the combinatorial structure of complexified arrangements and the salvetti complex in some detail 3 this general method simultaneously produces cell decompositions of the singularity link this link is shown to have the homotopy type of a wedge of d spheres for arrangements in cd d 4 4 the homology of the link and the cohomology of the complement are computed in terms of explicit bases which are matched by alexander duality this gives a new more elementary and more generally valid proof for the brieskorn orlik solomon theorem and some related results 5 our setup leads to a more general notion of 2 pseudoarrangements which can be thought of as topologically deformed complex arrangements re taining only the essential topological and combinatorial structure we show that all of the above remains true in this generality except for the sign patterns of the orlik solomon relations department of mathematics royal institute of technology s10044 stockholm sweden e mail address andersb nada kth se institut fur mathematik universitat augsburg d 8900 augsburg germany current address institut mittag leffler auravagen 17 s 18262 djursholm sweden address after 1 april 1992 konrad zuse zentrum fur informationstechnik berlin heilbronner str 10 w 1000 berlin 31 germany e mail addresses ziegler ml kva se ziegler zib berlin de this content downloaded from 157 55 39 211 on fri 24 jun 2016 06 44 45 utc all use subject to http about jstor org terms