some discrete maximum principles arising for nonlinear elliptic finite element problems the discrete maximum principle dmp is an important measure of the qualitative reliability of the applied numerical scheme for elliptic problems this paper starts with formulating simple sufficient conditions for the matrix case and for nonlinear forms in banach spaces then a dmp is derived for finite element solutions for certain nonlinear partial differential equations we address nonlinear elliptic problems with mixed boundary conditions and interface conditions allowing possibly degenerate nonlinearities and thus extending our previous results

the discrete maximum principle for linear simplicial finite element approximations of a reaction diffusion problem abstract this paper provides a sufficient condition for the discrete maximum principle for a fully discrete linear simplicial finite element discretization of a reaction diffusion problem to hold it explicitly bounds the dihedral angles and heights of simplices in the finite element partition in terms of the magnitude of the reaction coefficient and the spatial dimension as a result it can be computed how small the acute simplices should be for the discrete maximum principle to be valid numerical experiments suggest that the bound which considerably improves a similar bound in p g ciarlet p a raviart maximum principle and uniform convergence for the finite element method comput methods appl mech eng 2 1973 17 31 is in fact sharp

smoothed projections in finite element exterior calculus the development of smoothed projections constructed by combining the canonical interpolation operators defined from the degrees of freedom with a smoothing operator have proved to be an eective tool in finite element exterior calculus the advan tage of these operators is that they are l 2 bounded projections and still they commute with the exterior derivative in the present paper we generalize the construction of these smoothed projections such that also non quasi uniform meshes and essential boundary conditions are covered the new tool introduced here is a space dependent smoothing operator which commutes with the exterior derivative

finite element solution of the fundamental equations of semiconductor devices in part i of the paper see zlamal 13 finite element solutions of the nonstationary semiconductor equations were constructed two fully discrete schemes were proposed one was nonlinear the other partly linear in this part of the paper we justify the nonlinear scheme we consider the case of basic boundary conditions and of constant mobilities and prove that the scheme is unconditionally stable further we show that the approximate solution extended to the whole time interval as a piecewise linear function converges in a strong norm to the weak solution of the semiconductor equations these results represent an extended and corrected version of results announced without proof in zlamal 14

dissection of the path simplex in r n into n path subsimplices abstract we review properties of acute and non obtuse simplices and of ortho simplices and path simplices dissection of path simplices is considered which leads to a new result generalization of coxeter s trisection of a path tetrahedron into three path subtetrahedra to arbitrary spatial dimension n moreover following earlier results by korotov and k i ek we show that applying this procedure recursively in the proper way leads to a self similar path simplicial refinement towards a chosen vertex of the original path simplex

averaging of directional derivatives in vertices of nonobtuse regular triangulations for a shape regular triangulation mathcal t h without obtuse angles of a bounded polygonal domain omega subset re 2 let mathcal l h be the space of continuous functions linear on the triangles from mathcal t h and h the interpolation operator from c overline omega to mathcal l h this paper is devoted to the following classical problem find a second order approximation of the derivative partial u partial z a in a direction z of a function u in c 3 overline omega in a vertex a in the form of a linear combination of the constant directional derivatives partial pi h u partial z on the triangles surrounding a an effective procedure for such an approximation is presented its error is proved to be of the size o h 2 an operator mbox w h mathcal l h longrightarrow mathcal l h times mathcal l h relating a second order approximation wh h u of nabla u to every u in c 3 overline omega is constructed and shown to be a so called recovery operator the accuracy of the presented approximation is compared with the accuracies of the local approximations by other known techniques numerically

a monotone finite element scheme for convection diffusion equations a simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection diffusion problems in any spatial dimension by properly averaging the pde coefficients on element edges the resulting finite element stiffness matrix is an m matrix under some mild assumption for the underlying generally unstructured finite element grids as a consequence the proposed edge averaged finite element scheme is particularly interesting for the discretization of convection dominated problems this scheme admits a simple variational formulation it is easy to analyze and it is also suitable for problems with a relatively smooth flux variable some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems

local orders in jordan algebras abstract making use of results on general algebras of quotients of jordan algebras we study a notion of local order based on the version for linear jordan algebras of the ideas of fountain and gould 14 as adapted to the jordan context by fernandez lopez and garcia rus in 7 in particular we characterize the set of lesieur croisot elements of a nondegenerate jordan algebra as those elements of the jordan algebra lying in the socle of its maximal algebra of quotients and apply this relationship to extend to quadratic jordan algebras the results of fernandez lopez and garcia rus on local orders in nondegenerate jordan algebras satisfying the descending chain condition on principal inner ideals and not containing ideals which are nonartinian quadratic factors

stabilized galerkin approximation of convection diffusion reaction equations discrete maximum principle and convergence we analyze a nonlinear shock capturing scheme for h1 conforming piecewise affine finite element approximations of linear elliptic problems the meshes are assumed to satisfy two standard conditions a local quasi uniformity property and the xu zikatanov condition ensuring that the stiffness matrix associated with the poisson equation is an m matrix a discrete maximum principle is rigorously established in any space dimension for convection diffusion reaction problems we prove that the shock capturing finite element solution converges to that without shock capturing if the cell peclet numbers are sufficiently small moreover in the diffusion dominated regime the difference between the two finite element solutions super converges with respect to the actual approximation error numerical experiments on test problems with stiff layers confirm the sharpness of the a priori error estimates

the tate conjecture for k3 surfaces over finite fields artin s conjecture states that supersingular k3 surfaces over finite fields have picard number 22 in this paper we prove artin s conjecture over fields of characteristic p 5 this implies tate s conjecture for k3 surfaces over finite fields of characteristic p 5 our results also yield the tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields with some restrictions on the characteristic as a consequence we prove the tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p 5

simulation of stochastic reaction diffusion processes on unstructured meshes we model stochastic chemical systems with diffusion by a reaction diffusion master equation on a macroscopic level the governing equation is a reaction diffusion equation for the averages of the chemical species on a mesoscopic level the master equation for a well stirred chemical system is combined with a discretized brownian motion in space to obtain the reaction diffusion master equation the space is covered in our method by an unstructured mesh and the diffusion coefficients on the mesoscale are obtained from a finite element discretization of the laplace operator on the macroscale the resulting method is a flexible hybrid algorithm in that the diffusion can be handled either on the meso or on the macroscale level the accuracy and the efficiency of the method are illustrated in three numerical examples inspired by molecular biology

a study on nonnegativity preservation in finite element approximation of nagumo type nonlinear differential equations preservation of nonnengativity and boundedness in the finite element solution of nagumo type equations with general anisotropic diffusion is studied linear finite elements and the backward euler scheme are used for the spatial and temporal discretization respectively an explicit an implicit and two hybrid explicitimplicit treatments for the nonlinear reaction term are considered conditions for the mesh and the time step size are developed for the numerical solution to preserve nonnegativity and boundedness the effects of lumping of the mass matrix and the reaction term are also discussed the analysis shows that the nonlinear reaction term has significant effects on the conditions for both the mesh and the time step size numerical examples are given to demonstrate the theoretical findings

convergence of finite volume schemes for semilinear convection diffusion equations the topic of this work is the discretization of semilinear elliptic problems in two space dimensions by the cell centered finite volume method dirichlet boundary conditions are considered here a discrete poincare inequality is used and estimates on the approximate solutions are proven the convergence of the scheme without any assumption on the regularity of the exact solution is proven using some compactness results which are shown to hold for the approximate solutions

force stepping integrators in lagrangian mechanics we present a class of integration schemes for lagrangian mechanics referred to as energy stepping integrators that are momentum and energy conserving symplectic and convergent in order to achieve these properties we replace the original potential energy by a piecewise constant or terraced approximation at steps of uniform height by taking steps of diminishing height an approximating sequence of energies is generated the trajectories of the resulting approximating lagrangians can be characterized explicitly and consist of intervals of piecewise rectilinear motion we show that the energy stepping trajectories are symplectic exactly conserve all the momentum maps of the original system and subject to a transversality condition converge to trajectories of the original system when the energy step is decreased to zero these properties the excellent long term behavior of energy stepping and its automatic time step selection property are born out by selected examples of application including the dynamics of a frozen argon cluster the spinning of an elastic cube and the collision of two elastic spheres

tate s conjecture for k3 surfaces of finite height this paper extends the proof 16 of the tate conjecture for ordinary k3 surfaces over a finite field to the more general case of all k3 s of finite height as in 16 our method is to find a lifting of the k3 to characteristic zero with sufficiently many hodge cycles in the ordinary case the so called canonical lifting of deligne and illusie 7 did the job and a study of the galois action on p adic etale cohomology revealed the hodge cycles here we use more general quasi canonical liftings and the action of the crystalline weil group on

trisecting an orthoscheme a tetrahedron having two right angles at each of two vertices was investigated by lobachevsky who called it a pyramid schlafli who called it an orthoscheme wythoff who called it double rectangular and schoute who called its theory polygonometry there is a simple procedure for dissecting such a tetrahedron into three smaller orthoschemes the two cutting planes meet three of the four faces which are right angled triangles along lines which can easily be described when the tetrahedron is unfolded so as to put all the faces in one plane the arrangement of lines suggests an interesting theorem of absolute geometry when a particular spherical orthoscheme of known volume is dissected into three pieces and the volumes of these smaller orthoschemes are expressed as definite integrals the result is a peculiar identity which has not been verified directly there is a one parameter family of orthoschemes for which the three smaller orthoschemes are all congruent the euclidean member of this family turns out to be related to a very simply frieze pattern of integers

estimation of the effect of numerical integration in finite element eigenvalue approximation finite element approximations of the eigenpairs of differential operators are computed as eigenpairs of matrices whose elements involve integrals which must be evaluated by numerical integration the effect of this numerical integration on the eigenvalue and eigenfunction error is estimated specifically for 2nd order selfadjoint eigenvalue problems we show that finite element approximations with quadrature satisfy the well known estimates for approximations without quadrature provided the quadrature rules have appropriate degrees of precision

failure of the discrete maximum principle for an elliptic finite element problem there has been a long standing question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem this is related to knowing whether the discrete green s function is positive for triangular meshes allowing sufficiently good approximation of h 1 functions we study this question for the poisson problem in two dimensions discretized via the galerkin method with continuous piecewise linears we give examples which show that in general the answer is negative and furthermore we extend the number of cases where it is known to be positive our techniques utilize some new results about discrete green s functions that are of independent interest

a simplicial gauge theory we provide an action for gauge theories discretized on simplicial meshes inspired by lattice gauge theory and finite element methods the action is discretely gauge invariant and we give a proof of consistency a discrete noether s theorem that can be applied to our setting is also proved

on nonobtuse simplicial partitions this paper surveys some results on acute and nonobtuse simplices and associated spatial partitions these partitions are relevant in numerical mathematics including piecewise polynomial approximation theory and the finite element method special attention is paid to a basic type of nonobtuse simplices called path simplices the generalization of right triangles to higher dimensions in addition to applications in numerical mathematics we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics

tight bounds on angle sums of nonobtuse simplices it is widely known that the sum of the angles of a triangle equals two right angles far less known are the answers to similar questions for tetrahedra and higher dimensional simplices in this paper we review some of these less known results and look at them from a different point of view then we continue to derive tight bounds on the dihedral angle sums for the subclass of nonobtuse simplices all the dihedral angles of such simplices are less than or equal to right they have several important applications brandts et al 2009 the main conclusion is that when the spatial dimension n is even the range of dihedral angle sums of nonobtuse simplices is n times smaller than the corresponding range for arbitrary simplices when n is odd it is n 1 times smaller

on the shafarevich and tate conjectures for hyperkahler varieties 1 problems and results 2 polarized hyperk ihler varieties and cubic fourfoids 3 the period mapping 4 the kuga satake construction 5 the kuga satake construction in the relative setting 6 proof of theorem 1 5 1 7 proof of theorem 1 6 1 8 descent 9 proof of theorem 1 3 1 appendix 1 spinorial shimura varieties appendix 2 a short proof of the hodge conjecture for cubic fourfolds 205 210 212 216 220 224 229 233 240 244 245

discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions one of the most important problems in numerical simulations is the preservation of qualitative properties of solutions of the mathematical models by computed approximations for problems of elliptic type one of the basic properties is the continuous maximum principle in our work we present several variants of the maximum principles and their discrete counterparts for scalar second order nonlinear elliptic problems with mixed boundary conditions the problems considered are numerically solved by the continuous piecewise linear finite element approximations built on simplicial meshes sufficient conditions providing the validity of the corresponding discrete maximum principles are presented geometrically they mean that the employed meshes have to be of acute or nonobtuse type depending of the type of the problem finally some examples of real life problems where the preservation of maximum principles plays an important role are presented

zelmanov s prime theorem for quadratic jordan algebras e i zelmanov has recently shown that any prime linear jordan algebra without nil ideals but no finiteness conditions imposed is either a homomorphic image of a special algebra or is a form of the 27 dimensional exceptional algebra in particular all division algebras and all simple unital algebras are either derived from associative algebras or are 27 dimensional over their centers this effectively ends the search for infinite dimensional exceptional algebras in the present paper zelmanov s result is extended to quadratic jordan algebras jordan algebras were invented in the 1930s in the search for an exceptional algebraic setting for quantum mechnics an algebraic system which behaved like the usual algebra of quantum mechanical observables hermitian operators on hilbert space but was exceptional in the sense that its structure was not determined behind the scenes by some unobservable associative algebra in their pioneering paper of 1933 91 jordan von neumann and wigner classified the finite dimensional formally real linear jordan algebras and a a albert showed 11 that the only simple algebra in the list which was exceptional was a certain 27 dimensional algebra of 3 x 3 hermitian matrices with entries from an gdimensional cayley algebra all subsequent investigations of exceptional jordan algebras led back to this same albert algebra the next breakthrough in the structure of jordan algebras was n jacobson s 1965 introduction 6 j of inner ideals and the resultant quadratilication of the theory in terms of the quadratic product xyx instead of the linear product xy vx the final breakthrough was the astounding 1979 result of a young russian mathematician e i zelmanov that the only simple exceptional linear jordan are albert algebras thus the search for an infinite dimensional setting for exceptional quantum mechanics is doomed to failure the only simple exceptional structure allotted to mortals is the 27 dimensional albert algebra this came as a complete surprise to western researchers who hoped the free jordan algebra might lead to an 291 002 i 8693 82 060297 30 02 00 0

acute type refinements of tetrahedral partitions of polyhedral domains we present a new technique to perform refinements on acute type tetrahedral partitions of a polyhedral domain provided that the center of the circumscribed sphere around each tetrahedron belongs to the tetrahedron the resulting family of partitions is of acute type thus all the tetrahedra satisfy the maximum angle condition both these properties are highly desirable in finite element analysis

weakened acute type condition for tetrahedral triangulations and the discrete maximum principle we prove that a discrete maximum principle holds for continuous piecewise linear finite element approximations for the poisson equation with the dirichlet boundary condition also under a condition of the existence of some obtuse internal angles between faces of terahedra of triangulations of a given space domain this result represents a weakened form of the acute type condition for the three dimensional case

some remarks concerning the grothendieck period conjecture we discuss various results and questions around the grothendieck period conjecture which is a counterpart concerning the de rham betti realization of algebraic vari eties over number fields of the classical conjectures of hodge and tate these results give new evidence towards the conjectures of grothendieck and kontsevich zagier concerning transcen dence properties of the torsors of periods of varieties over number fields let q be the algebraic closure of q in c let x be a smooth projective variety over q and let x an c denote the compact complex analytic manifold that it defines the grothendieck period conjecture in codimension k on x denoted gpc k x asserts that any class in the

the weil petersson geometry of the moduli space of su n 3 calabi yau manifolds i the weil petersson metric is defined on the moduli space of calabi yau manifolds the curvature of this weil petersson metrics is computed and its potential is explicitely defined it is proved that the moduli space of calabi yau manifolds is unobstructed see tian

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

local pi theory of jordan systems ii abstract in this paper we study jordan systems having nonzero local algebras that satisfy a polynomial identity we prove that in nondegenerate jordan systems the set of elements at which the local algebra is pi is an ideal and that if it is nonzero it coincides with the socle when the system is primitive

simplicial gauge theory on spacetime we define a discrete gauge invariant yang mills higgs action on spacetime cylindrical meshes with simplicial spatial base the formulation is a generalization of classical lattice gauge theory and we prove consistency of the action in the finite element sense in addition we perform numerical tests of convergence towards exact continuum results for several choices of gauge fields in pure gauge theory

supersingular k3 surfaces are unirational we show that supersingular k3 surfaces in characteristic p 5a re related by purely inseparable isogenies this implies that they are unirational which proves conjectures of artin rudakov shafarevich and shioda as a byproduct we exhibit the moduli space of rigidified k3 crystals as an iterated p 1 bundle over f p2 to complete the picture we also establish shioda inose type isogeny theorems for k3 surfaces with picard rank 19 in positive characteristic mathematics subject classification 14j28 14g17 14m20 14d22

stable finite elements for problems with wild coefficients we consider solving an elliptic boundary value problem in the case that the coefficients vary by many orders of magnitude over the domain a linear finite element method is used it is shown that the standard method for solving the resulting linear equations in finite precision arithmetic can give an arbitrarily inaccurate answer because of ill conditioning in the stiffness matrix a new method for solving the linear equations is proposed this method is based on a mixed formulation and gives a numerically accurate answer independent of the variation in the coefficients the numerical error in the soution of the linear system for the new method is shown to depend on the aspect ratio of the triangulation

simulation of stochastic diffusion via first exit times in molecular biology it is of interest to simulate diffusion stochastically in the mesoscopic model we partition a biological cell into unstructured subvolumes in each subvolume the number of molecules is recorded at each time step and molecules can jump between neighboring subvolumes to model diffusion the jump rates can be computed by discretizing the diffusion equation on that unstructured mesh if the mesh is of poor quality due to a complicated cell geometry standard discretization methods can generate negative jump coefficients which no longer allows the interpretation as the probability to jump between the subvolumes we propose a method based on the mean first exit time of a molecule from a subvolume which guarantees positive jump coefficients two approaches to exit times a global and a local one are presented and tested in simulations on meshes of different quality in two and three dimensions

local lesieur croisot theory of jordan algebras in this paper we study jordan algebras having nonzero local algebras that satisfy the property of being lesieur croisot i e being orders in nondegenerate jordan algebras of finite capacity we will prove that the set of the elements of a nondegenerate jordan algebra at which the local algebra is lesieur croisot is an ideal

the tate conjecture for k3 surfaces in odd characteristic we show that the classical kuga satake construction gives rise away from characteristic 2 to an open immersion from the moduli of primitively polarized k3 surfaces of any fixed degree to a certain regular integral model for a shimura variety of orthogonal type this allows us to attach to every polarized k3 surface in odd characteristic an abelian variety such that divisors on the surface can be identified with certain endomorphisms of the attached abelian variety in turn this reduces the tate conjecture for k3 surfaces over finitely generated fields of odd characteristic to a version of the tate conjecture for certain endomorphisms on the attached kuga satake abelian variety which we prove as a by product of our methods we also show that the moduli stack of primitively polarized k3 surfaces of degree 2d is quasi projective and when d is not divisible by p 2 is geometrically irreducible in characteristic p we indicate how the same method applies to prove the tate conjecture for co dimension 2 cycles on cubic fourfolds

the structure of strongly prime quadratic jordan algebras abstract we extend the structure theory for strongly prime jordan algebras of arbitrary dimension to quadratic algebras of characteristic 2 we show that such j are either 1 hermitian forms h 0 a j h q a lying between an ample subspace of hermitian elements in a prime associative algebra a and those in its martindale ring of quotients q a 2 clifford forms with a scalar extension j j which is an algebra j j q 1 of a nondegenerate quadratic from q with basepoint 1 or 3 albert forms with a scalar extension j j which is a 27 dimensional split albert algebra j h 3 k of 3 3 hermitian matrices over an 8 dimensional split octonion algebra k as a consequence the simple jordan algebras of arbitrary dimension are either hermitian h 0 a for simple a or clifford forms in j q 1 or are albert algebras in particular our methods give an idempotent free classification of all simple finite dimensional algebras which are i special the key to this characterization is the existence of zel manov polynomials in the free special jordan algebra which are both hermitian polynomials whose values always look like hermitian elements and at the same time clifford polynomials nontrivial on 3 3 hermitian matrices hence on special algebras with 3 interconnected idempotents if a zel manov polynomial does not vanish on j then j is of hermitian type and if it does vanish on j then j is of clifford type one example of such a polynomial is q48 p16 x1 y1 z1 w1 p16 x2 y2 z2 w2 p16 x3 y3 z3 w3 for p16 d2x y z 2 dx y w dx y w dx y z x y z thus the structure of jordan algebras depends very directly on polynomial identities and commutators

arbitrary level hanging nodes for adaptive hp fem approximations in 3d in this paper we discuss constrained approximation with arbitrary level hanging nodes in adaptive higher order finite element methods hp fem for three dimensional problems this technique enables using highly irregular meshes and it greatly simplifies the design of adaptive algorithms as it prevents refinements from propagating recursively through the finite element mesh the technique makes it possible to design efficient adaptive algorithms for purely hexahedral meshes we present a detailed mathematical description of the method and illustrate it with numerical examples

an algebraic discrete maximum principle in hilbert space with applications to nonlinear cooperative elliptic systems discrete maximum principles are derived for finite element discretizations of nonlinear elliptic systyems with cooperative and weakly diagonally dominant coupling the results are achieved via an algebraic discrete maximum principle organized in a hilbert space framework and in the case of simplicial elements are obtained under weakened acute type conditions for the finite element fe meshes

discrete maximum principle and adequate discretizations of linear parabolic problems in this paper we analyze the connections between the different qualitative properties of numerical solutions of linear parabolic problems with dirichlet type boundary condition first we formulate the qualitative properties for the differential equations and shed light on their relations then we show how the well known discretization schemes can be written in the form of a one step iterative process we give necessary and sufficient conditions of the main qualitative properties of these iterations we apply the results to the finite difference and galerkin finite element solutions of linear parabolic problems in our main result we show that the nonnegativity preservation property is equivalent to the maximum minimum principle and they imply the maximum norm contractivity in one two and three dimensions we list sufficient a priori conditions that ensure the required qualitative properties finally we demonstrate the above results on numerical examples

geometric aspects of the symmetric inverse m matrix problem abstract we investigate the symmetric inverse m matrix problem from a geometric perspective the central question in this geometric context is which conditions on the k dimensional facets of an n simplex s guarantee that s has no obtuse dihedral angles the simplest of such conditions is that if all triangular facets of s are equilateral then s is regular and thus nonobtuse first we study the properties of an n simplex s whose k facets are all nonobtuse and generalize some classical results by fiedler 14 we prove that if all n 1 facets of an n simplex s are nonobtuse each makes at most one obtuse dihedral angle with another facet this helps to identify a special type of tetrahedron which we will call sub orthocentric with the property that if all tetrahedral facets of s are sub orthocentric then s is nonobtuse rephrased in the language of linear algebra this constitutes a purely geometric proof of the fact that each symmetric ultrametric matrix 28 is the inverse of a weakly diagonally dominant m matrix the geometric proof provides valuable insights that supplement the discrete measure theoretic 24 and the linear algebraic 25 proof the review papers 20 21 support our believe that the linear algebraic perspective on the inverse m matrix problem dominates the literature the geometric perspective however connects sign properties of entries of inverses of a symmetric positive definite matrix to the dihedral angle properties of an underlying simplex and enables an explicit visualization of how these angles and signs can be manipulated this will serve to formulate purely geometric conditions on the k facets of an n simplex s that may render s nonobtuse also for k 3 for this we generalize the class of sub orthocentric tetrahedra that gives rise to the class of ultrametric matrices to sub orthocentric simplices that define symmetric positive definite matrices a with special types of k k principal submatrices for k 3 each sub orthocentric simplices is nonobtuse and we conjecture that any simplex with sub orthocentric facets only is sub orthocentric itself along the way several additional new concepts will be introduced such as vertex gramians simplicial matrix classes the dual hull and the sub orthocentric set of a nonobtuse simplex and nonblocking matrices these concepts may also be of use in a different linear algebraic setting in particular in the context if completely positive matrices for which we will prove some auxiliary results as well

analysis and design of jump coefficients in discrete stochastic diffusion models in computational systems biology the mesoscopic model of reaction diffusion kinetics is described by a continuous time discrete space markov process to simulate diffusion stochastically the jump coefficients are obtained by a discretization of the diffusion equation using unstructured meshes to represent complicated geometries may lead to negative coefficients when using piecewise linear finite elements several methods have been proposed to modify the coefficients to enforce the nonnegativity needed in the stochastic setting in this paper we present a method to quantify the error introduced by that change we interpret the modified discretization matrix as the exact finite element discretization of a perturbed equation the forward error the error between the analytical solutions to the original and the perturbed equations is bounded by the backward error the error between the diffusion of the two equations we present a backward analysis algorithm to compute the diffusion coefficient from a given discretization matrix the analysis suggests a new way of deriving nonnegative jump coefficients that minimizes the backward error the theory is tested in numerical experiments indicating that the new method is superior and also minimizes the forward error

on the generalization of the synge křižek maximum angle condition for d simplices abstract in this note we present a generalization of the maximum angle condition proposed by j l synge in 1957 and m k i ek in 1992 for triangular and tetrahedral elements respectively for the case of higher dimensional simplicial finite elements its relations to the other angle type conditions commonly used in interpolation theory and in finite element analysis are discussed

local nonobtuse tetrahedral refinements around an edge in this note we show how to generate and conformly refine nonobtuse tetrahedral meshes locally around and towards an edge so that all dihedral angles of all resulting tetrahedra remain nonobtuse the proposed technique can be used e g for a numerical treatment of solution singularities and also for various mesh adaptivity procedures near the reentrant corners of cylindric type 3d domains

goldie theory for jordan algebras it is shown that zelmanov s version of goldie s conditions still characterizes quadratic jordan algebras having an artinian algebra of quotients which is nondegenerate at the same time jordan versions of the main notions of the associative theory such as those of the uniform ideal uniform element singular ideal and uniform dimension are studied moreover it is proved that the nondegenerate unital jordan algebras of finite capacity are precisely the algebras of quotients of nondegenerate jordan algebras having the property that an inner ideal is essential if and only if it contains an injective element 2002 elsevier science usa

discrete maximum principle for fe solutions of the diffusion reaction problem on prismatic meshes in this paper we analyse the discrete maximum principle dmp for a stationary diffusion reaction problem solved by means of prismatic finite elements we derive geometric conditions on the shape parameters of the prismatic partitions which guarantee validity of the dmp the presented numerical tests show the sharpness of the obtained conditions

on continuous and discrete maximum principles for elliptic problems with the third boundary condition in this work we present and discuss some continuous and discrete maximum principles for linear elliptic problems of the second order with the third boundary condition almost never addressed to in the available literature in this context solved by finite element and finite difference methods numerical tests are given

the maximum angle condition is not necessary for convergence of the finite element method we show that the famous maximum angle condition in the finite element analysis is not necessary to achieve the optimal convergence rate when simplicial finite elements are used to solve elliptic problems this condition is only sufficient in fact finite element approximations may converge even though some dihedral angles of simplicial elements tend to

tautological classes on moduli spaces of hyper kahler manifolds we study algebraic cycles on moduli spaces f h of h polarized hyperkahler manifolds following previous work of marian oprea and pandharipande on the tautological conjecture on moduli spaces of k3 surfaces we first define the tautological ring on f h we then study the images of these tautological classes in the cohomology groups of f h and prove that most of them are linear combinations of noether lefschetz cycle classes in particular we prove the cohomological version of the tautological conjecture on moduli space of k3 n type hyperkahler man ifolds with n 2 secondly we prove the cohomological generalized franchetta conjecture on universal family of these hyperkahler mani folds

the optimal refinement strategy for 3 d simplicial meshes 3 d mesh refinement strategies are the object of interest in the present paper the dependence of finite element solutions on various refinement procedures is discussed a new refinement strategy which is optimal with respect to the measure of degeneracy for all canonical domains is obtained comparisons with known refinement techniques are made some isoparametric examples are presented

a gauge invariant discretization on simplicial grids of the schrodinger eigenvalue problem in an electromagnetic field we propose a method to compute approximate eigenpairs of the schrodinger operator on a bounded domain in the presence of an electromagnetic field formulated for simplicial meshes the method combines techniques from lattice gauge theory and finite elements retaining the discrete gauge invariance of the former but allowing for noncongruent space elements as in the latter the error of the method is studied in the framework of strang s variational crimes comparing with a standard galerkin approach when the meshes are quasi uniform and satisfy the discrete maximum principle we prove that for a smooth electromagnetic field the crime is of the order of the mesh width h for a coulomb potential it is of order h log h and for a general finite energy electromagnetic field it is of order h 1 2

well centered triangulation meshes composed of well centered simplices have nice orthogonal dual meshes the dual voronoi diagram this is useful for certain numerical algorithms that prefer such primal dual mesh pairs we prove that well centered meshes also have optimality properties and relationships to delaunay and minmax angle triangulations we present an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed the cost function is a direct result of a new characterization of well centeredness in arbitrary dimensions that we present ours is the first optimization based heuristic for well centeredness and the first one that applies in both two and three dimensions we show the results of applying our algorithm to small and large two dimensional meshes some with a complex boundary and obtain a well centered tetrahedralization of the cube we also show numerical evidence that our algorithm preserves gradation and that it improves the maximum and minimum angles of acute triangulations created by the best known previous method

sign preserving of principal eigenfunctions in p1 finite element approximation of eigenvalue problems of second order elliptic operators this paper is concerned with the p1 finite element approximation of the eigenvalue problem of second order elliptic operators subject to the dirichlet boundary condition the focus is on the preservation of the basic properties of the principal eigenvalue and eigenfunctions of the continuous problem it is shown that when the stiffness matrix is an irreducible m matrix the discrete eigenvalue problem maintains almost all of the basic properties such as the smallest eigenvalue being real and simple and the corresponding eigenfunctions being either positive or negative inside the physical domain mesh conditions leading to such a stiffness matrix are also studied a sufficient condition is that the mesh is simplicial interiorly connected and acute when measured in the metric specified by the inverse of the diffusion matrix the acute requirement can be replaced by the delaunay condition in two dimensions numerical results show that when the stiffness matrix is not an m matrix a finite element approximation can be structurally different from the continuous eigenvalue problem the eigenfunctions corresponding to the smallest eigenvalue can change sign over the physical domain and the smallest eigenvalue in modulus can even be complex for the case with nonsymmetric operators

a nonconforming finite element method for the biot s consolidation model in poroelasticity a stable finite element scheme that avoids pressure oscillations for a three field biot s model in poroelasticity is considered the involved variables are the displacements fluid flux darcy velocity and the pore pressure and they are discretized by using the lowest possible approximation order crouzeix raviart finite elements for the displacements lowest order raviart thomas nedelec elements for the darcy velocity and piecewise constant approximation for the pressure mass lumping technique is introduced for the raviart thomas nedelec elements in order to eliminate the darcy velocity and therefore reduce the computational cost we show convergence of the discrete scheme which is implicit in time and use these types of elements in space with and without mass lumping finally numerical experiments illustrate the convergence of the method and show its effectiveness to avoid spurious pressure oscillations when mass lumping for the raviart thomas nedelec elements is used

triangulations of cayley and tutte polytopes cayley polytopes were defined recently as convex hulls of cayley compositions introduced by cayley in 1857 in this paper we resolve braun s conjecture which expresses the volume of cayley polytopes in terms of the number of connected graphs we extend this result to a two variable deformations which we call tutte polytopes the volume of the latter is given via an evaluation of the tutte polynomial of the complete graph our approach is based on an explicit triangulation of the cayley and tutte polytope we prove that simplices in the triangulations correspond to labeled trees and forests the heart of the proof is a direct bijection based on the neighbors first search graph traversal algorithm r esum

on the characteristic polynomial of the frobenius on etale cohomology let x be a smooth projective variety of even dimension d over a finite field then we establish a restriction on the value at 1 of the characteristic polynomial of the frobenius on the middle dimensional etale cohomology of x with coefficients in ql d 2

global and local refinement techniques yielding nonobtuse tetrahedral partitions preservation of basic qualitative properties for example the maximum principle ofthe solution of partial differential equations by its finite element approximations is an important goal in mathematical modelling and simulation nonobtuse tetrahedral partitions and linear finite elements guarantee the validity of the discrete analogues of the maximum principle for a wide class of parabolic and elliptic problems in order to get more accurate approximation we often need to refine the used partitions globally or locally in this paper we first propose two variants of global refinement techniques which produce nonobtuse face to face tetrahedral partitions second we present a new local refinement technique which generates nonobtuse face to face tetrahedral partitions in a neighbourhood of a given vertex