monte carlo study of the point for collapsing trees we investigate the collapse transition of lattice trees with nearest neighbor attraction in two and three dimensions two methods are used 1 a stochastic optimization process of the robbins monro type which is designed solely to locate the maximum value of the specific heat and 2 umbrella sampling which is designed to sample data over a wide temperature range as well as to combat the quasiergodicity of metropolis algorithms in the collapsed phase we find good evidence that the transition is second order with a divergent specific heat and that the divergence of the specific heat coincides with the metric collapse

3d finite element forward modeling of electromagnetic data using vector and scalar potentials and unstructured grids abstractwe present a finite element solution to the 3d electromagnetic forward modeling problem in the frequency domain the method is based on decomposing the electric field into vector and scalar potentials in the helmholtz equation and in the equation of conservation of charge edge element and nodal element basis functions were used respectively for the vector and scalar potentials this decomposition was performed with the intention of satisfying the continuity of the tangential component of the electric field and the normal component of the current density across the interelement boundaries therefore finding an efficient solution to the problem the computational domain was subdivided into unstructured tetrahedral elements the system of equations was discretized using the galerkin variant of the weighted residuals method with the approximated vector and scalar potentials as the unknowns of a sparse linear system a generalized minimum residual solver with an incomplete lu preconditioner was use

algebraic invariants for right angled artin groups a finite simplicial graph determines a right angled artin group g with generators corresponding to the vertices of and with a relation w w for each pair of adjacent vertices we compute the lower central series quotients the chen quotients and the first resonance variety of g directly from the graph

on the error of linear interpolation and the orientation aspect ratio and internal angles of a triangle in this paper we attempt to reveal the precise relation between the error of linear interpolation on a general triangle and the geometric characters of the triangle taking the model problem of interpolating quadratic functions we derive two exact formulas for the h 1 seminorm and l 2 norm of the interpolation error in terms of the area aspect ratio orientation and internal angles of the triangle these formulas indicate that 1 for highly anisotropic triangular meshes the h 1 seminorm of the interpolation error is almost a monotonically decreasing function of the angle between the orientations of the triangle and the function 2 maximum angle condition is not essential if the mesh is aligned with the function and the aspect ratio is of magnitude sqrt lambda 1 lambda 2 or less where lambda 1 and lambda 2 are the eigenvalues of the hessian matrix of the function with these formulas we identify the optimal triangles which produce the smallest h 1 seminorm of the interpolation error to be the acute isosceles aligned with the solution and of an aspect ratio about 0 8 frac lambda 1 lambda 2 the l 2 norm of the interpolation error depends on the orientation and the aspect ratio of the triangle but not directly on its maximum or minimum angles the optimal triangles for the l 2 norm are those aligned with the solution and of an aspect ratio sqrt lambda 1 lambda 2 these formulas can be used to formulate more accurate mesh quality measures and to derive tighter error bounds for interpolations

the varieties and the cohomology ring of a module let g be a finite group and let k be a field of characteristic p 0 recently several studies have focused on the homological invariants of finitely generated kg modules in l alperin proposed the study of the complexity of a module the complexity is related to the degree of the polynomial rate of growth of the terms in a projective resolution of the module see definition 2 4 in 2 alperin and evens proved that the complexity of a kg module m is equal to the maximum of the complexities of the restrictions of m to elementary abelian p subgroups of g one of the roots of the alperin evens theorem is quillen s dimension theorem see 21 or 22 which with some difficulty can be interpreted as saying the same thing for the special case of the trivial kg module alperin and evens 3 and avrunin 4 have further studied the annihilator in ext k k of the cohomology of m and have produced theorems of a similar nature these are related to the work in 9 on the structure of the cohomology ring ext m m the purpose of this paper is to investigate the relationship between these homological invariants and the structure of the module we concentrate on the case in which g is an elementary abelian p group however using such results as those mentioned above we can apply the present work to more general groups even as the earlier results indicate that much of the structure and cohomology of a module is revealed in the restrictions to elementary abelian p subgroups we show here that in the case of an elementary abelian group much of this information can be found by looking at the restrictions to cyclic p subgroups of the group of units of kg some of the results in this paper were announced in s let g xl xn be an elementary abelian group of order p suppose that k is algebraically closed to any finitely generated kg module m we

toric complexes and artin kernels abstract a simplicial complex l on n vertices determines a subcomplex t l of the n torus with fundamental group the right angled artin group g l given an epimorphism g l z let t l be the corresponding cover with fundamental group the artin kernel n we compute the cohomology jumping loci of the toric complex t l as well as the homology groups of t l with coefficients in a field k viewed as modules over the group algebra k z we give combinatorial conditions for h r t l k to have trivial z action allowing us to compute the truncated cohomology ring h r t l k we also determine several lie algebras associated to artin kernels under certain triviality assumptions on the monodromy z action and establish the 1 formality of these not necessarily finitely presentable groups

monomial golod quotients of exterior algebras it is proved that for any set m of squarefree monomials in the variables x1 xn the algebra a k x1 xn m is golod if and only if the algebra b e x1 xn m is golod where e is the exterior algebra this is proved by showing the equivalence of the extremality of the poincare series of a and b

exact evaluation of the collapse phase boundary for two dimensional directed polymers the phase boundary of the collapsed phase for a directed polymer on a square lattice in the presence of an attractive wall and monomer monomer interactions is calculated exactly using transfer matrix techniques the position of the multicritical point is also identified

a variational formula for the free energy of the partially directed polymer collapse long linear polymers in dilute solutions are known to undergo a collapse transition from a random coil expand itself to a compact ball fold itself up when the temperature is lowered or the solvent quality deteriorates a natural model for this phenomenon is a 1 1 dimensional self interacting and partially directed self avoiding walk in this paper we develop a new method to study the partition function of this model from which we derive a variational formula for the free energy this variational formula allows us to prove the existence of the collapse transition and to identify the critical temperature in a simple way we also provide a probabilistic proof of the fact that the collapse transition is of second order with critical exponent 3 2

a collapse transition in a directed walk model the authors consider a directed walk model of linear polymers in dilute solution with an energy associated with the number of near neighbour contacts in the walk for this model they can derive an exact expression for the generating function in two variables conjugate to the number of steps and the number of contacts they discuss the analytic structure of this generating function and identify the transition corresponding to collapse

h1 parametrizations of complex planar physical domains in isogeometric analysis abstract isogeometric analysis is a method for solving geometric partial differential equations pdes generating parametrizations of a pde s physical domain is a basic and important issue within isogeometric analysis framework in this paper we present a global h 1 parametrization method for a complex planar physical domain

exact scaling behavior of partially convex vesicles we solve analytically for the perimeter area generating functions for two models of vesicles while from the solution of the first model staircase polygons one can easily extract the asymptotic scaling behavior the exact solution of the second column convex polygons is difficult to analyze this leads us to apply a recently developed method for deriving the scaling behavior indirectly utilizing a set of nonlinear differential equations one result of this work is a nontrivial confirmation of the scaling universality hypothesis

lower central series and free resolutions of hyperplane arrangements if m is the complement of a hyperplane arrangement and a h m k is the cohomology ring of m over a field of characteristic 0 then the ranks k of the lower central series quotients of 1 m can be computed from the betti numbers bii dimtor a k k i of the linear strand in a minimal free resolution of k over a we use the cartan eilenberg change of rings spectral sequence to relate these numbers to the graded betti numbers b dimtor e a k j of a minimal resolution of a over the exterior algebra e from this analysis we recover a formula of falk for 3 and obtain a new formula for 4 the exact sequence of low degree terms in the spectral sequence allows us to answer a question of falk on graphic arrangements and also shows that for these arrangements the algebra a is koszul if and only if the arrangement is supersolvable we also give combinatorial lower bounds on the betti numbers b 1 of the linear strand of the free resolution of a over e if the lower bound is attained for i 2 then it is attained for all i 2 for such arrangements we compute the entire linear strand of the resolution and we prove that all components of the first resonance variety of a are local for graphic arrangements which do not attain the lower bound unless they have no braid sub arrangements we show that b 1 is determined by the number of triangles and k4 subgraphs in the graph

applications of mapping cones over clements lindstrom rings abstract we prove that gotzmann s persistence theorem holds over every clements lindstrom ring we also construct the infinite minimal free resolution of a square free borel ideal over such a ring

conditioning of finite element equations with arbitrary anisotropic meshes this is the published version also available here http dx doi org 10 1090 s0025 5718 2014 02822 6 first published in math comput in 2014 published by the american mathematical society

constrained centroidal voronoi tessellations for surfaces centroidal voronoi tessellations are useful for subdividing a region in euclidean space into voronoi regions whose generators are also the centers of mass with respect to a prescribed density function of the regions their extensions to general spaces and sets are also available for example tessellations of surfaces in a euclidean space may be considered in this paper a precise definition of such constrained centroidal voronoi tessellations ccvts is given and a number of their properties are derived including their characterization as minimizers of an energy deterministic and probabilistic algorithms for the construction of ccvts are presented and some analytical results for one of the algorithms are given computational examples are provided which serve to illustrate the high quality of ccvt point sets finally ccvt point sets are applied to polynomial interpolation and numerical integration on the sphere

adaptive finite element methods with convergence rates adaptive finite element methods for numerically solving elliptic equations are used often in practice only recently 12 17 have these methods been shown to converge however this convergence analysis says nothing about the rates of convergence of these methods and therefore does in principle not guarantee yet any numerical advantages of adaptive strategies versus non adaptive strategies the present paper modifies the adaptive method of morin nochetto and siebert 17 for solving the laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm which is equivalent to the h1 norm namely it is shown that whenever s 0 and the solution u is such that for each n 1 it can be approximated to accuracy o n s in the energy norm by a continuous piecewise linear function on a triangulation with n cells using complete knowledge of u then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process moreover the number of arithmetic computations in the proposed method is also of order o n for each n 1 the construction and analysis of this adaptive method relies on the theory of nonlinear approximation

on balanced moving mesh methods moving mesh methods are a widely used approach in the numerical solution of pdes where the original pdes are transformed from a physical domain to a computational domain the objective is to utilize a uniform mesh in the computational domain to get a non uniform physical mesh that better captures the behavior of the solution the movement of the physical mesh points can be governed by a moving mesh pde associated with a corresponding monitor function and both the original pdes and the moving mesh pdes must be solved simultaneously the motivation for this paper is to study a balanced moving mesh method where the aim is to strike a balance between the properties of the solution of the original pde and that of the moving mesh pde we focus on particular choices of the monitor function that give both a well behaved mesh transformation and a well behaved solution in the computational domain both theoretical analysis and numerical experiments are presented as illustrations

multigrid methods for the computation of singular solutions and stress intensity factors i corner singularities we consider the poisson equation u f with homogeneous dirichlet boundary condition on a two dimensional polygonal domain with cracks multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed the convergence rate for the stress intensity factors is mathcal o h 3 2 in whenfel2 and mathcal o h 2 in whenfeh1 the convergence rate in the energy norm is mathcal o h 1 in in the first case and mathcal o h in the second case the costs of these multigrid methods are proportional to the number of elements in the triangulation the general case wherefehm is also discussed

a posteriori error analysis of finite element solutions for one dimensional problems the paper develops a theory of a posteriors error estimates under the l p energy norm for 2 leqq p leqq infty the theory is based on a general concept of error indicators and error estimators several specific examples of these quantities are introduced and analyzed in detail the results provide a variety of easily computable error estimates under all these important norms for finite element solutions of one dimensional problems

on support varieties for lie superalgebras and finite supergroup schemes abstract we study the spectrum of the cohomology rings of cocommutative hopf superalgebras restricted and non restricted lie superalgebras and finite supergroup schemes we also investigate support varieties in these settings and demonstrate that they have the desirable properties of such a theory we completely characterize support varieties for finite supergroup schemes over algebraically closed fields of characteristic zero while for non restricted lie superalgebras in positive characteristic we obtain descriptions that are suggestive of varieties previously defined by duflo and serganova in characteristic zero but which have no known cohomological interpretation our computations for restricted lie superalgebras and infinitesimal supergroup schemes provide natural generalizations of foundational results of friedlander and parshall and of bendel friedlander and suslin in the classical setting

bounds on spectral condition numbers of matrices arising in the p version of the finite element method we estimate condition numbers of p version matrices for tensor product elements with two choices of reference element degrees of freedom in one case lagrange elements the condition numbers grow exponentially in p whereas in the other hierarchical basis functions based on tchebycheff polynomials the condition numbers grow rapidly but only algebraically in p we conjecture that regardless of the choice of basis the condition numbers grow like p 4d or faster where d is the dimension of the spatial domain

on the analyticity properties of scaling functions in models of polymer collapse we consider the mathematical properties of the generating and partition functions in the two variable scaling region about the tricritical point in some models of polymer collapse we concentrate on models that have a similar behaviour to that of interacting partially directed self avoiding walks ipdsaw in two dimensions however we do not restrict the discussion to that model after describing the properties for a general class of models and stating exactly what we mean by scaling we prove the following theorem if the generating function of finite size partition functions has a tricritical cross over scaling form around the theta point and the associated tricritical scaling function g has a finite radius of convergence then the partition function has a finite size scaling form and importantly the finite size scaling function f is an entire function in the ipdsaw case we have an explicit representation of the finite size scaling function we point out that given our description of tricritical scaling this theorem should apply mutatis mutandis to a wider class of theta point models we discuss the result in relation to the edwards model of polymer collapse for which it has recently been argued that the finite size scaling functions are not entire

scaling behaviour of two dimensional polygon models exactly solvable two dimensional polygon models counted by perimeter and area are described by q algebraic functional equations we provide techniques to extract the scaling behaviour of these models up to arbitrary order and apply them to some examples these are then used to analyze the unsolved model of self avoiding polygons where we numerically confirm predictions about its scaling function and its first two corrections to scaling

multigrid methods for the computation of singular solutions and stress intensity factors iii interface singularities it is shown in this paper that two dimensional interface problems with large jumps in the coefficients can be solved effectively by the piecewise linear finite element method on quasi uniform grids this is achieved by combining the full multigrid methodology the singular function representation of the solution and the extraction formulas for stress intensity factors the robustness of the algorithm is demonstrated by numerical experiments

mesh dependent stability for finite element approximations of parabolic equations with mass lumping in this paper based on some mesh dependent estimates on the extreme eigenvalues of a general finite element system defined on a simplicial mesh novel and sharp bounds on the permissible time step size are derived for the mass lumping finite element approximations of parabolic equations the bounds are dependent not only on the mesh size but also on the mesh shape these results provide guidance to the stability of numerical solutions of parabolic problems in relation to the unstructured geometric meshing numerical experiments on both uniform meshes and adaptive meshes are presented to validate the theoretical analysis

almost regular sequences and betti numbers let be a simplicial complex and s the shifted simplicial complex of as defined by kalai it is shown that the stanley reisner ideals i and i s have the same extremal betti numbers this theorem is the squarefree analogue of the corresponding result by bayer charalambous and popescu which asserts that a graded ideal and its generic ideal have the same extremal betti numbers our theorem implies well known results by kalai according to which and s have the same simplicial homology and is cohen macaulay if and only if its shifted complex is cohen macaulay

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

periodic resolutions over exterior algebras in this paper we study modules with periodic free resolutions that is periodic modules over an exterior algebra we show that any module with bounded betti numbers that is a module whose syzygy modules have a bounded number of generators must have periodic free resolution of period 2 and that for graded modules the period is 1 we also show that any module with a linear tate resolution is periodic we give a criterion of exactness for periodic complexes and a parameterization of the set of periodic modules 2002 elsevier science usa all rights reserved

linearity defects of face rings abstract let s k x 1 x n be a polynomial ring over a field k and e y 1 y n an exterior algebra the linearity defect ld e n of a finitely generated graded e module n measures how far n departs from componentwise linear it is known that ld e n for all n but the value can be arbitrary large while the similar invariant ld s m for an s module m is always at most n we will show that if i resp j is the squarefree monomial ideal of s resp e corresponding to a simplicial complex 2 1 n then ld e e j ld s s i moreover except some extremal cases ld e e j is a topological invariant of the geometric realization of the alexander dual of we also show that when n 4 ld e e j n 2 this is the largest possible value if and only if is an n gon

bounding the influence of domain parameterization and knot spacing on numerical stability in isogeometric analysis abstract isogeometric analysis iga was introduced by hughes et al 2005 1 as a new method to bridge the gap between the geometry description and numerical analysis similar to the finite element approach the iga concept to solve a partial differential equation leads to a linear system of equations the condition number of the coefficient matrix is a crucial factor for the stability of the system it depends strongly on the domain parameterization which provides the isogeometric discretization in this paper we derive a bound for the condition number of the stiffness matrix of the poisson equation in particular we investigate the influence of the domain parameterization and the knot spacing on the stability of the numerical system the factors appearing in our bound reflect the stability properties of a given domain parameterization

hyperplane arrangement cohomology and monomials in the exterior algebra we show that if x is the complement of a complex hyperplane arrangement then the homology of x has linear free resolution as a module over the exterior algebra on the first cohomology of x we study invariants of x that can be deduced from this resolution a key ingredient is a result of aramova avramov and herzog 2000 on resolutions of monomial ideals in the exterior algebra we give a new conceptual proof of this result

tensor triangular geometry for classical lie superalgebras abstract tensor triangular geometry as introduced by balmer 3 is a powerful idea which can be used to extract the ambient geometry from a given tensor triangulated category in this paper we provide a general setting for a compactly generated tensor triangulated category which enables one to classify thick tensor ideals and the balmer spectrum for the general linear lie superalgebra g g 0 g 1 we construct a zariski space from a detecting subalgebra of g and demonstrate that this topological space governs the tensor triangular geometry for the category of finite dimensional g modules which are semisimple over g 0

on the maximum angle condition for linear tetrahedral elements synge s maximum angle condition for triangular elements is generalized to tetrahedral elements for the generalized condition it is proved that tetrahedra may degenerate in a certain way and the error of the standard linear interpolation remains o h in the w p 1 omega norm for sufficiently smooth functions and p in 1 infty

critical exponents from nonlinear functional equations for partially directed cluster models we present a method for the derivation of the generating function and computation of critical exponents for several cluster models staircase bar graph and directed column convex polygons as well as partially directed self avoiding walks starting with nonlinear functional equations for the generating function by linearizing these equations we first give a derivation of the generating functions the nonlinear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz alternatively taking the continuum limit leads to nonlinear differential equations from which one can extract the scaling function we find that all the above models are in the same universality class with exponents y 1 2 7 1 3 and b 2 3 all models have as their scaling function the logarithmic derivative of the airy function

hilbert schemes and maximal betti numbers over veronese rings macaulay s theorem macaulay in proc lond math soc 26 531 555 1927 characterizes the hilbert functions of graded ideals in a polynomial ring over a field we characterize the hilbert functions of graded ideals in a veronese ring r the coordinate ring of a veronese embedding of pr 1 we also prove that the hilbert scheme which parametrizes all graded ideals in r with a fixed hilbert function is connected this is an analogue of hartshorne s theorem hartshorne in math ihes 29 5 48 1966 that hilbert schemes over a polynomial ring are connected furthermore we prove that each lex ideal in r has the greatest betti numbers among all graded ideals with the same hilbert function

mesh smoothing using a posteriori error estimates we develop a simple mesh smoothing algorithm for adaptively improving finite element triangulations the algorithm makes use of a posteriori error estimates which are now widely used in finite element calculations in this paper we derive the method present some numerical illustrations and give a brief analysis of the issue of uniqueness

adaptive finite element methods for elliptic pdes based on conforming centroidal voronoi delaunay triangulations a new triangular mesh adaptivity algorithm for elliptic pdes that combines a posteriori error estimation with centroidal voronoi delaunay tessellations of domains in two dimensions is proposed and tested the ability of the first ingredient to detect local regions of large error and the ability of the second ingredient to generate superior unstructured grids result in a mesh adaptivity algorithm that has several very desirable features including the following errors are very well equidistributed over the triangles at all levels of refinement the triangles remain very well shaped even if the grid size at any particular refinement level when viewed globally varies by several orders of magnitude and the convergence rates achieved are the best obtainable using piecewise linear finite elements this methodology can be easily extended to higher order finite element approximations or mixed finite element formulations although only the linear approximation is considered in this paper

an optimal triangulation for second order elliptic problems abstract let be a polygonal domain in r n h an associated triangulation and uh the finite element solution of a well posed second order elliptic problem on h let m mi p qi 1 be the set of nodes which defines the vertices of the triangulation h for each i m i x i l 1 l n in rn the object of this paper is to provide a computational tool to approximate the best set of positions m of the nodes and hence the best triangulation gt h which minimizes the solution error in the natural norm associated with the problem the main result of this paper are theorems which provide explicit expressions for the partial derivatives of the associated energy functional with respect to the coordinates xil 1 l n of each of the variable nodes mi i 1 p

mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions in this paper we study the linear bond based nonlocal peridynamic models with a particular focus on problems associated with nonstandard nonlocal displacement loading conditions both stationary and time dependent problems are considered for a one dimensional scalar equation defined on a finite bar and for a two dimensional system defined on a square the related peridynamic operators and associated functional spaces are analyzed applications to the numerical analysis of the finite dimensional approximations to peridynamic models are also illustrated

the tricritical behavior of self interacting partially directed walks we present the thermodynamics of two variations of the interacting partially directed self avoiding walk problem by discussing versions where the length of the walks assume real as well as a integral values while the discrete model has been considered previously to varying degrees of success the continuous model we now define has not the examination of the continuous model leads to theexact derivation of several exponents for the discrete model some of these exponents can be calculated using a continued fraction representation for both models the crossover exponent is found to be 2 3 moreover we confirm the tricritical nature of the collapse transition in the generalized ensemble and calculate the full scaling form of the generating function additionally the similarities noticed previously to other models but left unexplored are explained with the aid of necklacing arguments

exact solution of the discrete 1 1 dimensional sos model with field and surface interactions we present the solution of a linear solid on solid sos model configurations are partially directed walks on a two dimensional square lattice and we include a linear surface tension a magnetic field and surface interaction terms in the hamiltonian there is a wetting transition at zero field and as expected the behavior is similar to a continuous model solved previously the solution is in terms ofq series most closely related to theq hypergeometric functions1 1

on mesh geometry and stiffness matrix conditioning for general finite element spaces the performance of finite element computation depends strongly on the quality of the geometric mesh and the efficiency of the numerical solution of the linear systems resulting from the discretization of partial differential equation pde models it is common knowledge that mesh geometry affects not only the approximation error of the finite element solution but also the spectral properties of the corresponding stiffness matrix in this paper for typical second order elliptic problems some refined relationships between the spectral condition number of the stiffness matrix and the mesh geometry are established for general finite element spaces defined on simplicial meshes the derivation of such relations for general high order elements is based on a new trace formula for the element stiffness matrix it is shown that a few universal geometric quantities have the same dominant effect on the stiffness matrix conditioning for different finite element spaces these results provide guidance to the studies of both linear algebraic solvers and the unstructured geometric meshing

grid generation and optimization based on centroidal voronoi tessellations centroidal voronoi tessellations cvts are voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding voronoi regions such tessellations are of use in very diverse applications including data compression clustering analysis cell biology territorial behavior of animals and optimal allocation of resources in this paper we explore the use of cvts in grid generation in connection with finite element approximations of partial differential equations we being by describing these tessellations and methods for their determination we then discuss their application to mesh generation and finish with some examples of their use for the solution of partial differential equations

resolutions of a stable ideals abstract let a a 1 a 2 a n be a sequence of integers or we introduce a stable ideals in a polynomial ring and study their homological properties our results generalize results on square free monomial ideals by aramova avramov herzog hibi and srinivasan

modules with small loewy length a module of complexity c for e z p r in characteristic p has loewy length at least p 1 r c 1 we study the case of equality if p is odd the only rank varieties possible are finite unions of linear subspaces of dimension c and every such rank variety occurs if p 2 the variety has to be equidimensional if such a variety is a finite union of set theoretic complete intersections then it occurs for such a module but otherwise the situation is unclear exterior algebras in any characteristic are also treated and follow the same behaviour as the case p 2 above

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

local problems on stars a posteriori error estimators convergence and performance a new computable a posteriori error estimator is introduced which relies on the solution of small discrete problems on stars it exhibits built in flux equilibration and is equivalent to the energy error up to data oscillation without any saturation assumption a simple adaptive strategy is designed which simultaneously reduces error and data oscillation and is shown to converge without mesh pre adaptation nor explicit knowledge of constants numerical experiments reveal a competitive performance show extremely good effectivity indices and yield quasi optimal meshes

optimal finite element mesh for elliptic equation of divergence form we consider an optimal triangular mesh minimizing the condition number of the finite element stiffness matrix for an elliptic equation email protected email protected email protected x i a i email protected email protected x j f u w g using a sharp bound for the condition number of the stiffness matrix it is shown that the element of the optimal uniform triangular mesh is equilateral with respect to the metric which is the inverse of the coefficient matrix in the equation it is verified by numerical examples that such a mesh is really effective in reducing the condition number of the stiffness matrix in addition we suggest an algorithm generating a mesh in which every element is almost equilateral with respect to a metric

tetrahedral mesh generation and optimization based on centroidal voronoi tessellations the centroidal voronoi tessellation based delaunay triangulation cvdt provides an optimal distribution of generating points with respect to a given density function and accordingly generates a high quality mesh in this paper we discuss algorithms for the construction of the constrained cvdt from an initial delaunay tetrahedral mesh of a three dimensional domain by establishing an appropriate relationship between the density function and the specified sizing field and applying the lloyd s iteration the constrained cvdt mesh is obtained as a natural global optimization of the initial mesh simple local operations such as edges faces flippings are also used to further improve the cvdt mesh several complex meshing examples and their element quality statistics are presented to demonstrate the effectiveness and efficiency of the proposed mesh generation and optimization method copyright 2003 john wiley sons ltd

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

sheaf cohomology and free resolutions over exterior algebras in this paper we derive an explicit version of the bernstein gel fand gel fand bgg correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free reso lutions over its koszul dual exterior algebra among the facts about the bgg correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the linear part of a resolution over the exterior algebra we explore the structure of free resolutions over an exterior algebra for example we show that such resolutions are eventually dominated by their linear parts in the sense that erasing all terms of degree 1 in the complex yields a new complex which is eventually exact as applications we give a construction of the beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms the explicitness of our version allows us to to prove two conjectures about the morphisms in the monad and we get an efficient method for machine computation of the cohomology ofsheaves we also construct all the monads for a sheaf that can be built from sums of line bundles and show that they are often characterized by numerical data let v be a finite dimensional vector space over a field k and let w v be the dual space in this paper we will study complexes and resolutions over the exterior algebra e v and their relation to modules over s sym w and sheaves on projective space p w in this paper we study the bernstein gel fand gel fand bgg correspondence 1978 usually stated as an equivalence between the derived category of bounded complexes of coherent sheaves on p w and the stable category of finitely generated graded modules over e its essential content is a functor r from complexes of graded s modules to complexes of graded e modules and its adjoint l for example if m imi is a graded s module regarded as a complex with just one term then the first and third authors are grateful to the nsf for partial support during the preparation of this paper the third author wishes to thank msri for its hospitality

resonance linear syzygies chen groups and the bernstein gelfand gelfand correspondence if a is a complex hyperplane arrangement with complement x we show that the chen ranks of g 1 x are equal to the graded betti numbers of the linear strand in a minimal free resolution of the cohomology ring a h x k viewed as a module over the exterior algebra e on a k g dim k tor e k 1 a k k for k 2 where k is a field of characteristic 0 the chen ranks conjecture asserts that for k sufficiently large k g k 1 r 1 r k 1 k where h r is the number of r dimensional components of the projective resonance variety r 1 a our earlier work on the resolution of a over e and the above equality yield a proof of the conjecture for graphic arrangements using results on the geometry of r 1 a and a localization argument we establish the inequality k g k 1 r 1 hr r k 1 k for k 0 for arbitrary a finally we show that there is a polynomial p t of degree equal to the dimension of r 1 a such that k g p k for all k 0

two dimensional self avoiding walk with hydrogen like bonding phase diagram and critical behaviour the phase diagram for a two dimensional self avoiding walk model on the square lattice incorporating attractive short ranged interactions between parallel sections of walk is derived using numerical transfer matrix techniques the model displays a collapse transition in contrast to the standard point model the transition is first order the phase diagram in the full fugacity temperature plane displays an additional transition line when compared to the point model as well as a critical transition at finite temperature in the hamiltonian walk limit

the statistical mechanics of random copolymers random copolymers are polymers with two or more types of monomer where the monomer sequence is determined by some random process once determined the sequence is fixed so random copolymers are an example of a system with quenched randomness we review the statistical mechanics of random copolymers focusing on self avoiding walk models where there are two types of monomers a and b which are randomly distributed along the polymer chain theoretical approximate and numerical results are reviewed for models of the random copolymer adsorption localization and collapse phase transitions we concentrate on what is known about the existence of phase transitions the morita approximation and results about self averaging we also discuss in less detail the replica trick and numerical methods including monte carlo methods exact enumeration and transfer matrix methods important open problems are identified throughout and highlighted in the conclusions

orthogonal polynomials and cubature formulae on balls simplices and spheres we report on recent developments on orthogonal polynomials and cubature formulae on the unit ball bd the standard simplex td and the unit sphere sd the main result shows that orthogonal structures and cubature formulae for these three regions are closely related this provides a way to study the structure of orthogonal polynomials for example it allows us to use the theory of h harmonics to study orthogonal polynomials on bd and on td it also provides a way to construct new cubature formulae on these regions

exact solution of the staircase and row convex polygon perimeter and area generating function an explicit expression is obtained for the perimeter and area generating function g y z sigma n or 2 sigma m or 1 cn mynzm where cn m is the number of row convex polygons with area m and perimeter n a similar expression is obtained for the area perimeter generating function for staircase polygons both expressions contain q series

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments