faces of the cone of euclidean distance matrices characterizations structure and induced geometry in this paper we obtain new characterizations of the faces of the cone of euclidean distance matrices in one case the characterization is based on a special subspace associated with each distance matrix in other case on linear restrictions of coordinate matrices we also relate the faces to the supporting hyperplanes of the cone and we show how the supporting hyperplanes induce certain geometry on the configurations that belong to that face

simplical volume of closed locally symmetric spaces of non compact type we show that closed locally symmetric spaces of non compact type have positive simplicial volume this gives a positive answer to a question that was first raised by gromov in inst hautes etudes sci publ math 56 5 1982

c1 weierstrass for compact sets in hilbert space abstract the c 1 weierstrass approximation theorem is proved for any compact subset x of a hilbert space h the same theorem is also proved for whitney 1 jets on x when x satisfies the following further condition there exist finite dimensional linear subspaces h 1 h 2 h such that n 1 h n is dense in span x and n x x h n for each n 1 here n h h n is the orthogonal projection it is also shown that when x is compact convex with span x h and satisfies the above condition then c 1 x is complete if and only if the c 1 whitney extension theorem holds for x finally for compact subsets of h an extension of the c 1 weierstrass approximation theorem is proved for c 1 maps h h with compact derivatives

an alternative alpha finite element method aαfem for free and forced structural vibration using triangular meshes an alternative alpha finite element method email protected using triangular elements is proposed that significantly improves the accuracy of the standard triangular finite elements and provides a superconvergent solution in the energy norm for the static analysis of two dimensional solid mechanics problems in the email protected the piecewise constant strain field of linear triangular fem models is enhanced by additional strain terms with an adjustable parameter a which results in an effectively softer stiffness formulation compared to a linear triangular element the element is further extended to the free and forced vibration analyses of solids several numerical examples show that the email protected achieves high reliability compared to other existing elements in the literature

self similar groups and the zig zag and replacement products of graphs abstract every finitely generated self similar group naturally produces an infinite sequence of finite d regular graphs n we construct self similar groups whose graphs n can be represented as an iterated zig zag product and graph powering n 1 n k z k 1 we also construct self similar groups whose graphs n can be represented as an iterated replacement product and graph powering n 1 n k r k 1 this gives simple explicit examples of self similar groups whose graphs n form an expanding family and examples of automaton groups whose graphs n have linear diameters diam n o n and bounded girth

quadratic serendipity finite elements over convex polyhedra summary the conventional approach to construct quadratic elements for an n sided polygon will yield n n 1 2 shape functions which increases the computational effort it is well known that the serendipity elements based on isoparametric formulation suffers from mesh distortion floater and lai proposed a systematic way to construct higher order approximations over arbitrary polygons using the generalized barycentric coordinates and triangular coordinates this approach ensures 2n shape functions with nodes only on the boundary of the polygon in this paper we extend the polygonal splines approach to three dimensions and construct serendipity shape functions over hexahedra and convex polyhedra this is done by expressing the shape functions using the barycentric coordinates and the local tetrahedral coordinates the quadratic shape functions possess kronecker delta property and satisfy constant linear and quadratic precision the accuracy and the convergence properties of the quadratic serendipity shape elements are demonstrated with a series of standard patch tests the numerical results show that the quadratic serendipity elements pass the patch test yield optimal convergence rate and can tolerate extreme mesh distortion this article is protected by copyright all rights reserved

moore penrose inverse of a euclidean distance matrix abstract we obtain expressions for the moore penrose inverse of a euclidean distance matrix edm that are determined only by the positive semidefinite matrix associated with the edm the results complement formulas for the moore penrose inverse of an edm given in balaji and bapat 2007 2 a formula for the inverse of a principal submatrix of an edm is also derived whose expression uses the schur complement of the laplacian of the edm as an application we obtain an expression for the terminal wiener index of a tree

a smoothed coupled ns nes fem for dynamic analysis of 2d fluid solid interaction problems abstract a node based smoothed finite element method ns fem t3 using triangular elements and a n sided polygonal edge based smoothed fem n es fem using polygonal elements is combined to give the smoothed coupled ns n es fem for dynamic analysis of two dimensional 2d fluid solid interaction problems based on the pressure displacement formulation in the present method the ns fem t3 is used for the fluid domain and the gradient of pressure is smoothed while the n es fem is used for the solid domain and the gradient of displacement is smoothed this gradient smoothing technique can provide proper softening effect which will effectively relieve the overly stiff behavior of the fem model and thus improve significantly the solution of coupled system some numerical examples have been presented to illustrate the effectiveness of the coupled ns n es fem compared with some existing methods for 2d fluid solid interaction problems

on cartesian product of euclidean distance matrices abstract if a r m m and b r n n we define the product a b as a b a j n j m b where denotes the kronecker product and j n is the n n matrix of all ones we refer to this product as the cartesian product of a and b since if d 1 and d 2 are the distance matrices of graphs g 1 and g 2 respectively then d 1 d 2 is the distance matrix of the cartesian product g 1 g 2 we study cartesian products of euclidean distance matrices edms we prove that if a and b are edms then so is the product a b we show that if a is an edm and u is symmetric then a u is an edm if and only if u c j n for some c it is shown that for edms a and b a b is spherical if and only if both a and b are spherical if a and b are edms then we derive expressions for the rank and the moore penrose inverse of a b in the final section we consider the product a b for arbitrary matrices for a r m m b r n n we show that all nonzero minors of a b of order m n 1 are equal an explicit formula for a nonzero minor of order m n 1 is proved the result is shown to generalize the familiar fact that the determinant of the distance matrix of a tree on n vertices does not depend on the tree and is a function only of n

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

free abelian covers short loops stable length and systolic inequalities we explore the geometry of the abel jacobi map f from a closed orientable riemannian manifold x to its jacobi torus h 1 x mathbb r h 1 x mathbb z mathbb r applying m gromov s filling inequality to the typical fiber of f we prove an interpolating inequality for two flavors of shortest length invariants of loops the procedure works provided the lift of the fiber is non trivial in the homology of the maximal free abelian cover overline x classified by f we show that the finite dimensionality of the rational homology of overline x is a sufficient condition for the homological non triviality of the fiber when applied to nilmanifolds our fiberwise inequality typically gives stronger information than the filling inequality for x itself in dimension 3 we present a sufficient non vanishing condition in terms of massey products this condition holds for certain manifolds that do not fiber over their jacobi torus such as 0 framed surgeries on suitable links our systolic inequality applies to surface bundles over the circle provided the algebraic monodromy has 1 dimensional coinvariants even though the massey product invariant vanishes for some of these bundles

two theorems on euclidean distance matrices and gale transform abstract we present a characterization of those euclidean distance matrices edms d which can be expressed as d e c for some nonnegative scalar and some correlation matrix c where e is the matrix of all ones this shows that the cones cone e e n cone e e n d n where e n is the elliptope set of correlation matrices and d n is the closed convex cone of edms the characterization is given using the gale transform of the points generating d we also show that given points p 1 p 2 p n r r for any scalars 1 2 n such that j 1 n j p j 0 j 1 n j 0 we have j 1 n j p i p j 2 for all i 1 n for some scalar independent of i

volumes of balls in large riemannian manifolds we prove two lower bounds for the volumes of balls in a riemannian manifold if m n g is a complete riemannian manifold with lling radius at least r then it contains a ball of radius r and volume at least n r n if m n hyp is a closed hyperbolic manifold and if g is another metric on m with volume no greater than n vol m hyp then the universal cover of m g contains a unit ball with volume greater than the volume of a unit ball in hyperbolic n space

a node based smoothed finite element method ns fem for upper bound solution to visco elastoplastic analyses of solids using triangular and tetrahedral meshes abstract a node based smoothed finite element method ns fem was recently proposed for the solid mechanics problems in the ns fem the system stiffness matrix is computed using the smoothed strains over the smoothing domains associated with nodes of element mesh in this paper the ns fem is further extended to more complicated visco elastoplastic analyses of 2d and 3d solids using triangular and tetrahedral meshes respectively the material behavior includes perfect visco elastoplasticity and visco elastoplasticity with isotropic hardening and linear kinematic hardening a dual formulation for the ns fem with displacements and stresses as the main variables is performed the von mises yield function and the prandtl reuss flow rule are used in the numerical procedure however the stress variables are eliminated and the problem becomes only displacement dependent the numerical results show that the ns fem has higher computational cost than the fem however the ns fem is much more accurate than the fem and hence the ns fem is more efficient than the fem it is also observed from the numerical results that the ns fem possesses the upper bound property which is very meaningful for the visco elastoplastic analyses which almost have not got the analytical solutions this suggests that we can use two models ns fem and fem to bound the solution and can even estimate the global relative error of numerical solutions

connections between the real positive semidefinite and distance matrix completion problems abstract though the real symmetric positive semidefinite psd matrices and the euclidean distance matrices are closely related exact relationships between the corresponding completion problems are not apparent we establish strong partial relationships of two types this permits the transfer of some insights from one problem to the other and allows computational tools for the psd problem to be used for the distance problem

isometries rigidity and universal covers the goal of this paper is to describe all closed aspherical riemannian manifolds m whose universal covers m have have a nontrivial amount of symmetry by this we mean that isom m is not discrete by the well known theorem of myers steenrod ms this condition is equivalent to isom m 1 m also note that if any cover of m has a nondiscrete isometry group then so does its universal cover m our description of such m is given in theorem 1 2 below the proof of this theorem uses methods from lie theory harmonic maps large scale geometry and the homological theory of transformation groups the condition that m have nondiscrete isometry group appears in a wide variety of problems in geometry since theorem 1 2 provides a taxonomy of such m it can be used to reduce many general problems to verifications of specific examples actually it is not always theorem 1 2 which is applied directly but the main subresults from its proof after explaining in 1 1 the statement of theorem 1 2 we give in 1 2 a number of such applications these range from new characterizations of locally symmetric manifolds to the classification of contractible manifolds covering both compact and finite volume manifolds to a new proof of the nadel frankel theorem in complex geometry

the probability that a slightly perturbed numerical analysis problem is difficult we prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis our probability estimates depend only on geometric invariants of the corresponding sets of ill posed inputs several applications to linear and polynomial equation solving show that the estimates obtained in this way are easy to derive and quite accurate the main theorem is based on a volume estimate of varepsilon tubular neighborhoods around a real algebraic subvariety of a sphere intersected with a spherical disk of radius sigma besides varepsilon and sigma this bound depends only on the dimension of the sphere and on the degree of the defining equations

on the distortion of knots on embedded surfaces our main result is a nontrivial lower bound for the distortion of some specic knots in particular we show that the distortion of the torus knot tp q satises tp q 1 160 min p q this answers a 1983 question of gromov

on the essence and the evaluation of the shape functions for the smoothed finite element method sfem this paper is written in response to the recently published paper int j numer meth engng 2008 76 1285 1295 at ijnme entitled on the smoothed finite element method sfem by zhang hh liu sj li lx in this paper we 1 repeat briefly the important essence of the original sfem presented in comp mech 2007 39 859 877 int j numer meth engng 2007 71 902 930 int j numer meth engng 2008 74 175 208 finite elem anal des 2007 43 847 860 j sound vib 2007 301 803 820 and 2 examine further issues in the evaluation of the shape functions used in the sfem it will be shown that the sfem presented in paper int j numer meth engng 2008 76 1285 1295 is not at all our original sfem presented in comp mech 2007 39 859 877 int j numer meth engng 2007 71 902 930 int j numer meth engng 2008 74 175 208 finite elem anal des 2007 43 847 860 j sound vib 2007 301 803 820 therefore all these theorems corollaries and remarks presented in paper int j numer meth engng 2008 76 1285 1295 have nothing to do with our original sfem the properties of the original sfem stand as they were presented in our original papers comp mech 2007 39 859 877 int j numer meth engng 2007 71 902 930 int j numer meth engng 2008 74 175 208 finite elem anal des 2007 43 847 860 j sound vib 2007 301 803 820 finally we brief on our advancements made far beyond our original sfem and our visions on future numerical methods copyright 2009 john wiley sons ltd

on euclidean distance matrices abstract if a is a real symmetric matrix and p is an orthogonal projection onto a hyperplane then we derive a formula for the moore penrose inverse of pap as an application we obtain a formula for the moore penrose inverse of an euclidean distance matrix edm which generalizes formulae for the inverse of a edm in the literature to an invertible spherical edm we associate a laplacian matrix which we define as a positive semidefinite n n matrix of rank n 1 and with zero row sums and prove some properties known results for distance matrices of trees are derived as special cases in particular we obtain a formula due to graham and lovasz for the inverse of the distance matrix of a tree it is shown that if d is a nonsingular edm and l is the associated laplacian then d 1 l is nonsingular and has a nonnegative inverse finally infinitely divisible matrices are constructed using edms

local boundary rigidity of a compact riemannian manifold with curvature bounded above this paper considers the boundary rigidity problem for a compact convex riemannian manifold m g with boundary o9m whose curvature satisfies a general upper bound condition this includes all nonpositively curved manifolds and all sufficiently small convex domains on any given riemannian manifold it is shown that in the space of metrics g on m there is a c3 neighborhood of g such that g is the unique metric with the given boundary distance function i e the function that assigns to any pair of boundary points their distance as measured in m more precisely given any metric g in this neighborhood with the same boundary distance function there is diffeomorphism o which is the identity on o9m such that g g there is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance function 1 statement of the result the general boundary rigidity problem reads to which extent is a riemannian metric on a compact manifold with boundary determined from the distances between boundary points more precisely it can be formulated as follows let m g be a compact riemannian manifold with boundary om let g be another riemannian metric on m we say that g and g have the same boundary distance function if d1 x y d9 x y for arbitrary boundary points x y e om where d1 resp d9 represents distance in m with respect to g resp g it is easy to give examples of pairs of metrics with the same boundary distance function indeed if m m is an arbitrary diffeomorphism of m onto itself which is the identity on the boundary then the metrics g and g g have the same boundary distance function here g og is the pull back of g under i e for arbitrary vectors 1 e txm we have r1 71 x i where p txm t x m is the differential of at x and resp is the inner product with respect to the metric g resp g we say that a compact riemannian manifold is boundary rigid if this is the only type of nonuniqueness more precisely m g is boundary rigid if for every received by the editors august 10 1998 2000 mathematics subject classification primary 53c65 53c22 53c20 secondary 58c25 58c40 the first and third authors were supported by crdf grant rm2 143 the first author was also supported by nsf grants mds 9505175 and mds 9626232 the third author was also supported by intas rfbr grant 95 0763 2000 american matlhematical society 3937 this content downloaded from 207 46 13 167 on thu 21 jul 2016 04 20 00 utc all use subject to http about jstor org terms 3938 c b croke n s dairbekov and v a sharafnjtdinov riemannian metric g on m1with the same boundary distance function as g there is a diffeomorphism o m m which is the identity on the boundary and for which g 0 of in ward and outward vectors here v is the unit outward norn al to om1 given x c qmvi we denote by k x the maximum of the sectional curvatures of all two planes oc t1 m such that c a for m g a cdrm we define the following invariant

a finite element method with mesh separation based approximation technique and its application in modeling crack propagation with adaptive mesh refinement summary this paper presents a fem with mesh separation based approximation technique that separates a standard element into three geometrically independent elements a dual mapping scheme is introduced to couple them seamlessly and to derive the element approximation the novel technique makes it very easy for mesh generation of problems with complex or solution dependent varying geometry it offers a flexible way to construct displacement approximations and provides a unified framework for the fem to enjoy some of the key advantages of the hansbo and hansbo method the meshfree methods the semi analytical fems and the smoothed fem for problems with evolving discontinuities the method enables the devising of an efficient crack tip adaptive mesh refinement strategy to improve the accuracy of crack tip fields both the discontinuities due to intra element cracking and the incompatibility due to hanging nodes resulted from the element refinement can be treated at the elemental level the effectiveness and robustness of the present method are benchmarked with several numerical examples the numerical results also demonstrate that a high precision integral scheme is critical to pass the crack patch test and it is essential to apply local adaptive mesh refinement for low fracture energy problems copyright 2014 john wiley sons ltd

a smoothed finite element method for plate analysis a quadrilateral element with smoothed curvatures for mindlin reissner plates is proposed the curvature at each point is obtained by a non local approximation via a smoothing function the bending stiffness matrix is calculated by a boundary integral along the boundaries of the smoothing elements smoothing cells numerical results show that the proposed element is robust computational inexpensive and simultaneously very accurate and free of locking even for very thin plates the most promising feature of our elements is their insensitivity to mesh distortion

theoretical aspects of the smoothed finite element method sfem this paper examines the theoretical bases for the smoothed finite element method sfem which was formulated by incorporating cell wise strain smoothing operation into standard compatible finite element method fem the weak form of sfem can be derived from the hu washizu three field variational principle for elastic problems it is proved that 1d linear element and 2d linear triangle element in sfem are identical to their counterparts in fem while 2d bilinear quadrilateral elements in sfem are different from that of fem when the number of smoothing cells scs of the elements equals 1 the sfem solution is proved to be variationally consistent and has the same properties with those of fem using reduced integration when sc approaches infinity the sfem solution will approach the solution of the standard displacement compatible fem model when sc is a finite number larger than 1 the sfem solutions are not variationally consistent but energy consistent and will change monotonously from the solution of sfem sc 1 to that of sfem sc it is suggested that there exists an optimal number of sc such that the sfem solution is closest to the exact solution the properties of sfem are confirmed by numerical examples copyright 2006 john wiley sons ltd

effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for gfem xfem stress approximations around brittle cracks abstract the convergence rates of the conventional generalized extended finite element method gfem xfem in crack modeling are similar to the convergence rates of the finite element method fem laborde et al 2005 and bechet et al 2005 unless the crack tip enrichment functions are applied in a subdomain with fixed dimension independently of the mesh parameter h and also demanding some special care for blending along the transition zone chahine et al 2006 and taracon et al 2009 thus to improve convergence rates more degrees of freedom dof are generated due to the larger quantity of enriched nodes this work seeks to identify and understand the advantages of better capturing the information provided by singular enrichments over mesh based smooth partitions of unity pou such pou with higher regularity can be built through the so called c k gfem framework following duarte et al 2006 based on a moving least square of degree zero and considering mesh based smooth weighting functions associated with arbitrary polygonal clouds the purpose herein is to investigate some possible advantages of mesh based smooth pou for modeling discontinuities and singularities in two dimensional problems of linear elastic fracture mechanics in such a fashion that the discretization error associated to stress discontinuities inherent in standard c 0 continuous gfem xfem approximations is eliminated the procedure shares features similar to the standard fem regarding domain partition and numerical integration but as neither the pou nor the enrichment functions are defined in natural domains the integrations are performed using only global coordinates the approximation capabilities of c k gfem discretizations with different patterns of singular enrichment distribution are investigated by analyzing the convergence rates of the h and p versions considering global measures in terms of strain energy and l 2 norm of displacements the effects on stability are also verified by analyzing the evolution of the condition number the effect of smoothness on conditioning is investigated and the eigenvalues distributions are used to identify the several aspects involved the smoothness of the pou the different types of enrichment functions and the pattern of enrichment the performance of the smooth approximations is compared to the c 0 counterparts built using conventional c 0 fem based pou it is shown that smoothness in the presence of extrinsically applied singular enrichments is important as the information provided by such enrichment is better captured in addition there are no stress jumps around singularities reducing error propagation beyond the neighborhood of the singularity

logarithmic growth of systole of arithmetic riemann surfaces along congruence subgroups we apply a study of orders in quaternion algebras to the differential geometry of riemann surfaces the least length of a closed geodesic on a hyperbolic surface is called its systole and denoted sys 1 p buser and p sarnak constructed riemann surfaces x whose systole behaves logarithmically in the genus g x the fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field we generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces the key tool is a new trace estimate valid for an arbitrary ideal in a quaternion algebra we obtain a particularly sharp bound for a principal congruence tower of hurwitz surfaces pch namely the 4 3bound sys 1 xpch 4 log g xpch similar results are obtained for the systole of hyperbolic 3 manifolds relative to their simplicial volume

stability for the lens rigidity problem let g be a riemannian metric for mathbb r d d geqq 3 which differs from the euclidean metric only in a smooth and strictly convex bounded domain m the lens rigidity problem is concerned with recovering the metric g inside m from the corresponding lens relation on the boundary partial m in this paper the stability of the lens rigidity problem is investigated for metrics which are a priori close to a given non trapping metric satisfying the strong fold regular condition a metric g is called strong fold regular if for each point x in m there exists a set of geodesics passing through x whose conormal bundle covers t x m moreover these geodesics contain either no conjugate points or only fold conjugate points with a non degeneracy condition examples of strong fold regular metrics are constructed our main result gives the first stability result for the lens rigidity problem in the case of anisotropic metrics which allow conjugate points the approach is based on the study of the linearized inverse problem of recovering a metric from its induced geodesic flow which is a weighted geodesic x ray transform problem for symmetric 2 tensor fields a key ingredient is to show that the kernel of the x ray transform on symmetric solenoidal 2 tensor fields is of finite dimension it remains open whether the kernel space is trivial or not

a new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces let m be a compact riemannian manifold with a fixed conformal structure then we introduce the concept of conformal volume of m in the following manner for each branched conformal immersion q9 of m into the unit sphere s n we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of s we let vc n qg be the max imum volume of these branched immersions the conformal volume of m is defined to be the infimum of v n q0 where qo ranges over all branched conformal immersions of m into the unit sphere s in this paper we study the case when m is a compact surface and we call the conformal volume of m to be the conformal area of m we demonst ra te that this conformal invariant is non trivial in fact we prove that if there exists a minimal immersion of m into s where coordinate functions are first eigenfunctions then the conformal area of m is given by the area of m with respect to the induced metric this enables us to compute the conformal area for several surfaces for example the conformal area of r p z is 6n and the conformal area of the square torus is 27c 2 we believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces we demonstrate this claim by applying the concept of conformal area to two different branches of surface theory the first application is to study the total curvature of a compact surface in r this problem has a long history fenchel and fary 9 proved that for a closed curve o in r slk 27c where k is its curvature then milnor 12

on the spread of positively curved alexandrov spaces it was proved by f wilhelm that gromov s filling radius of closed positively curved manifolds with a uniform lower bound on sectional curvature attains the maximum with the round sphere recently the author proved that this is also the case for closed finite dimensional alexandrov spaces with a positive lower curvature bound these were proved as a corollary of a comparison theorem for the invariant called spread of those spaces in this paper we extend the latter result to infinite dimensional alexandrov spaces

volume of minimal hypersurfaces in manifolds with nonnegative ricci curvature we establish a min max estimate on the volume width of a closed riemannian manifold with nonnegative ricci curvature more precisely we show that every closed riemannian manifold with nonnegative ricci curvature admits a pl morse function whose level set volume is bounded in terms of the volume of the manifold as a consequence of this sweep out estimate there exists an embedded closed possibly singular minimal hypersurface whose volume is bounded in terms of the volume of the manifold

linear smoothed polygonal and polyhedral finite elements the strain smoothing technique over higher order elements and arbitrary polytopes yields less accurate solutions than other techniques such as the conventional polygonal finite element method in this work we propose a linear strain smoothing scheme that improves the accuracy of linear and quadratic approximations over convex polytopes the main idea is to subdivide the polytope into simplicial subcells and use a linear smoothing function in each subcell to compute the strain this new strain is then used in the computation of the stiffness matrix the convergence properties and accuracy of the proposed scheme are discussed by solving a few benchmark problems numerical results show that the proposed linear strain smoothing scheme makes the approximation based on polytopes able to deliver the same optimal convergence rate as traditional quadrilateral and hexahedral approximations the accuracy is also improved and all the methods tested pass the patch test to machine precision

circum euclidean distance matrices and faces abstract we study the structure of circum euclidean distance matrices those euclidean distance matrices generated by points lying on a hypersphere we show for example that such euclidean distance matrices are characterized as having constant row sums and they constitute the interior of the cone of all euclidean distance matrices also we provide a formula for computing the radius of a representing configuration in the smallest embedding dimension r and show that rk d r 1 finally we obtain a geometric characterization of the faces of this cone given a configuration of points and its euclidean distance matrix d any matrix in the minimal face containing d comes from a configuration that is a linear perturbation of the points that generate d

bounding the distance of quantum surface codes homological quantum codes also called topological codes are low density parity check error correcting codes that come from surfaces and higher dimension manifolds homological codes from surfaces i e surface codes have also been suggested as a possible way to construct stable quantum memory and fault tolerant computation it has been conjectured that all homological codes have a square root bound on there distance and therefore cannot produce good codes this claim has been disputed in dimension four using the geometric property of systolic freedom we will show in this paper that the conjecture holds in dimension two due to the negation of systolic freedom i e systolic rigidity

geodesic loops and periodic geodesics on a riemannian manifold diffeomorphic to s 3 let m n be a closed 2 connected riemannian manifold such that 3 m n 0 in this paper we prove that either there exists a periodic geodesic on m n of length 6d where d is the diameter of m n or at each point p m n there exists a geodesic loop of length 2d

fundamental groups of manifolds with positive isotropic curvature a central theme in riemannian geometry is understanding the relationships between the curvature and the topology of a riemannian manifold positive isotropic curvature pic is a natural and much studied curvature condition which includes manifolds with pointwise quarter pinched sectional curvatures and manifolds with positive curvature operator by the results of micallef and moore there is only one topological type of compact simply connected manifold with pic namely any such manifold must be homeomorphic to the sphere on the other hand there is a large class of nonsimply connected manifolds with pic an important open problem has been to understand the fundamental groups of manifolds with pic in this paper we prove a new result in this direction we show that the fundamental group of a compact manifold m n with pic n 5 does not contain a subgroup isomorphic to the techniques used involve minimal surfaces

on the approximation in the smoothed finite element method sfem this letter aims at resolving the issues raised in the recent short communication int j numer meth engng 2008 76 8 1285 1295 doi 10 1002 nme 2460 and answered by int j numer meth engng 2009 doi 10 1002 nme 2587 by proposing a systematic approximation scheme based on non mapped shape functions which both allows to fully exploit the unique advantages of the smoothed finite element method sfem comput mech 2007 39 6 859 877 doi 10 1007 s00466 006 0075 4 commun numer meth engng 2009 25 1 19 34 doi 10 1002 cnm 1098 int j numer meth engng 2007 71 8 902 930 comput meth appl mech engng 2008 198 2 165 177 doi 10 1016 j cma 2008 05 029 comput meth appl mech engng 2007 submitted int j numer meth engng 2008 74 2 175 208 doi 10 1002 nme 2146 comput meth appl mech engng 2008 197 13 16 1184 1203 doi 10 1016 j cma 2007 10 008 and resolve the existence linearity and positivity deficiencies pointed out in int j numer meth engng 2008 76 8 1285 1295 we show that wachspress interpolants a rational basis for function approximation academic press inc new york 1975 computed in the physical coordinate system are very well suited to the sfem especially when elements are heavily distorted obtuse interior angles the proposed approximation leads to results that are almost identical to those of the sfem initially proposed in comput mech 2007 39 6 859 877 doi 10 1007 s00466 006 0075 4 these results suggest that the proposed approximation scheme forms a strong and rigorous basis for the construction of sfems copyright 2009 john wiley sons ltd

volume growth of some uniformly contractible riemannian manifolds it is shown that if a uniformly contractible riemannian n manifold m g is k quasi isometric to an n dimensional normed space v n cdot k 1 then liminf r rightarrow infty frac vol g ball r r n omega n geq frac 1 k 2n where n is the volume of the unit euclidean ball in particular if m is uniformly contractible and d gh m d g v n cdot infty then m has at least euclidean volume growth this corollary covers an earlier result by burago and ivanov our results are motivated by a volume growth theorem contained in gromov s book gromov in progress in mathematics vol 152 birkhauser boston 1999 p 256 we give a detailed proof of this theorem using the same argument we also derive a generalization of the theorem which is pointed out by gromov

non recursive functions knots with thick ropes and self clenching thick hyperspheres we introduce an approach to certain geometric variational problems based on the use of the algorithmic unrecognizability of the n dimensional sphere for n 5 sometimes this approach allows one to prove the existence of infinitely many solutions of a considered variational problem this recursion theoretic approach is applied in this paper to a class of functionals on the space of c1 1 smooth hypersurfaces diffeomorphic to sn in rn 1 where n is any fixed number 5 the simplest of these functionals kv is defined by the formula kv n vol n 1 n r n where r n denotes the radius of injectivity of the normal exponential map for n rn l we prove the existence of an infinite set of distinct locally minimal values of kv on the space of c1 1 smooth topological hyperspheres in rn 1 for any n 5 the functional kv naturally arises when one attempts to generalize knot theory in order to deal with embeddings and isotopies of thick circles and more generally thick spheres into euclidean spaces we introduce the notion of knot with thick rope types the theory of knot with thick rope types turns out to be quite different from the classical knot theory because of the following result there exists an infinite set of non trivial knot with thick rope types in codimension one for every dimension greater than or equal to five

linear smoothed extended finite element method the extended finite element method was introduced in 1999 to treat problems involving discontinuities with no or minimal remeshing through appropriate enrichment functions this enables elements to be split by a discontinuity strong or weak and hence requires the integration of discontinuous functions or functions with discontinuous derivatives over elementary volumes a variety of approaches have been proposed to facilitate these special types of numerical integration which have been shown to have a large impact on the accuracy and the convergence of the numerical solution the smoothed extended finite element method xfem for example makes numerical integration elegant and simple by transforming volume integrals into surface integrals however it was reported in the literature that the strain smoothing is inaccurate when non polynomial functions are in the basis in this paper we investigate the benefits of a recently developed linear smoothing procedure which provides better approximation to higher order polynomial fields in the basis some benchmark problems in the context of linear elastic fracture mechanics are solved and the results are compared with existing approaches we observe that the stress intensity factors computed through the proposed linear smoothed xfem is more accurate than that obtained through smoothed xfem

on the performance of strain smoothing for quadratic and enriched finite element approximations xfem gfem pufem by using the strain smoothing technique proposed by chen et al comput mech 2000 25 137 156 for meshless methods in the context of the finite element method fem liu et al comput mech 2007 39 6 859 877 developed the smoothed fem sfem although the sfem is not yet well understood mathematically numerical experiments point to potentially useful features of this particularly simple modification of the fem to date the sfem has only been investigated for bilinear and wachspress approximations and is limited to linear reproducing conditions the goal of this paper is to extend the strain smoothing to higher order elements and to investigate numerically in which condition strain smoothing is beneficial to accuracy and convergence of enriched finite element approximations we focus on three widely used enrichment schemes namely a weak discontinuities b strong discontinuities c near tip linear elastic fracture mechanics functions the main conclusion is that strain smoothing in enriched approximation is only beneficial when the enrichment functions are polynomial cases a and b but that non polynomial enrichment of type c lead to inferior methods compared to the standard enriched fem e g xfem copyright c 2011 john wiley sons ltd

a polytree based adaptive polygonal finite element method for topology optimization summary we propose a highly effective approach using a novel adaptive methodology to perform topology optimization with polygonal meshes called polytree meshes polytree is a hierarchical data structure based on the principle of recursive spatial decomposition of each polygonal element with n nodes into n 1 arbitrary new polygonal elements enabling more efficient utilization of unstructured meshes and arbitrary design domains in topology optimization in order to treat hanging nodes after each optimization loop we define the wachspress coordinate on a reference element and then utilize an affine map to obtain shape functions and their gradients on arbitrary polygons with n vertices and m hanging nodes called side nodes as the polygonal element with n m vertices the polytree meshes do not only improve the boundary description quality of the optimal result but also reduce the computational cost of optimization process in comparison with the use of uniformly fine meshes several numerical examples are investigated to show the high effectiveness of the proposed method this article is protected by copyright all rights reserved

systoles des surfaces plates singulieres de genre deux the extremal metrics for the isosystolic problem on surfaces of genus two are studied in this paper we show that contrary to e calabi s conjecture in genus three no flat metric with conical singularities is extremal for this problem in genus two

on the smoothed finite element method recently liu et al proposed the smoothed finite element method by using the non mapped shape functions and then introducing the strain smoothing operator when evaluating the element stiffness in the framework of the finite element method however the theories and examples by liu et al are not sufficient for general quadrilateral elements this paper shows that the non mapped shape functions used in the smoothed finite element have disadvantages in existence linearity non negativity and patch test copyright 2008 john wiley sons ltd

properties of euclidean and non euclidean distance matrices abstract a distance matrix d of order n is symmetric with elements 1 2 d ij 2 where dii 0 d is euclidean when the 1 2 n n 1 quantities dij can be generated as the distances between a set of n points x n p in a euclidean space of dimension p the dimensionality of d is defined as the least value of p rank x of any generating x in general p 1 and p 2 are also acceptable but may include imaginary coordinates even when d is euclidean basic properties of euclidean distance matrices are established in particular when rank d it is shown that depending on whether etd e is not or is zero the generating points lie in either p 1 dimensions in which case they lie on a hypersphere or in p 2 dimensions in which case they do not the notation e is used for a vector all of whose values are one when d is non euclidean its dimensionality p r s will comprise r real and s imaginary columns of x and r s are invariant for all generating x of minimal rank higher ranking representations can arise only from p 1 r 1 s or p 1 r s 1 or p 2 r 1 s 1 so that not only are r s invariant but they are both minimal for all admissible representations x

a polytree based adaptive polygonal finite element method for multi material topology optimization abstract this study presents a polytree based adaptive methodology for multi material topology optimization mmtop polytree data structure is introduced as a general recursive multi level mesh that is automatically refined in processing based on error analysis in order to resolve hanging nodes in element edges the wachspress coordinate is employed on a reference element before using a mapping scheme to obtain shape functions and their derivatives for any polygons a new definition of filter radius is also proposed to improve the efficiency of filters and optimized results the combination of polytree meshes and adaptive filters not only clarifies the interfaces between material phases including void phase but also decreases the computing time of the overall process in comparison to using the regular fine meshes several benchmark and practical problems are considered to show distinct features of the proposed method

ends of finite volume nonpositively curved manifolds we study complete finite volume n manifolds m of bounded nonpositive sectional curvature a classical theorem of m gromov says that if such m has negative curvature then it is homeomorphic to the interior of a compact manifold with boundary and we denote this boundary partial m if n ge 3 we prove that the universal cover of the boundary widetilde partial m and also the pi 1m cover of the boundary partial widetilde m have vanishing n 2 dimensional homology for n 4 the first of these recovers a result of nguy tilde hat mathrm e n phan saying that each component of the boundary partial m is aspherical for any n ge 3 the second of these implies the vanishing of the first group cohomology group with group ring coefficients h 1 b pi 1m mathbb z pi 1m 0 a consequence is that pi 1m is freely indecomposable these results extend to manifolds m of bounded nonpositive curvature if we assume that m is homeomorphic to the interior of a compact manifold with boundary our approach is a form of homological collapse for ends of finite volume manifolds of bounded nonpositive curvature this paper is very much influenced by earlier yet still unpublished work of nguy tilde hat mathrm e n phan

an efficient and rapid numerical quadrature to generate element matrices for quadrilateral and hexahedral elements in functionally graded materials fgms abstract this research paper introduces a robust efficient and stabilized midpoint quadrature to generate element stiffness matrices for quadrilateral and hexahedral elements in finite element method fem this new quadrature is established based on passive richardson extrapolation which has the inherent ability to stabilize and improve the accuracy of the extrapolated results the new quadrature includes a simple artificial stabilizing function with gauss one point quadrature 1gp as an anti hourglass effect for the first approximation the 1gp sampling point to be considered at origin 0 0 is done to evaluate element matrices thermal conductivity matrices capacitance matrices etc in the 4 square element 2 2 x 2 2 in the plane as a stabilizing function sampling points are considered to be either at the center of the four element edges or center of the four quadrants in the reference square element with appropriate constant weighting functions for hexahedral elements also the element stiffness coefficients are calculated only in the center of the element and center of all the octants of the reference cubic element 2 2 x 2 2 x 2 2 the proposed method deals mostly with integer variables and sampling at 0 0 0 whereas gauss quadrature uses lengthy float variables corresponding to irrational numbers in terms of decimals in the stiffness matrix calculations thus consuming time a number of standard example problems are demonstrated to assess the efficiency of the proposed method the results of all the numerical examples are found to be in good agreement with the exact results and conventional gauss quadrature results

quantitative null cobordism author s chambers gregory r dotterrer dominic manin fedor weinberger shmuel abstract for a given null cobordant riemannian n manifold how does the minimal geometric complexity of a null cobordism depend on the geometric complexity of the manifold in gro99 gromov conjectured that this dependence should be linear we show that it is at most a polynomial whose degree depends on n this construction relies on another of independent interest take x and y to be sufficiently nice compact metric spaces such as riemannian manifolds or simplicial complexes suppose y is simply connected and rationally homotopy equivalent to a product of eilenberg maclane spaces for example any simply connected lie group then two homotopic l lipschitz maps f g x rightarrow y are homotopic via a cl lipschitz homotopy we present a counterexample to show that this is not true for larger classes of spaces y

smooth finite element methods convergence accuracy and properties a stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented the integration of the stiffness matrix is performed on the boundaries of the finite elements a rigorous variational framework based on the hu washizu assumed strain variational form is developed we prove that solutions yielded by the proposed method are in a space bounded by the standard finite element solution infinite number of subcells and a quasi equilibrium finite element solution a single subcell we show elsewhere the equivalence of the one subcell element with a quasi equilibrium finite element leading to a global a posteriori error estimate we apply the method to compressible and incompressible linear elasticity problems the method can always achieve higher accuracy and convergence rates than the standard finite element method especially in the presence of incompressibility singularities or distorted meshes for a slightly smaller computational cost it is shown numerically that the one cell smoothed four noded quadrilateral finite element has a convergence rate of 2 0 in the energy norm for problems with smooth solutions which is remarkable for problems with rough solutions this element always converges faster than the standard finite element and is free of volumetric locking without any modification of integration scheme copyright 2007 john wiley sons ltd

boundary rigidity with partial data we study the boundary rigidity problem with partial data consisting of determining locally the riemannian metric of a riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary we show that one can recover uniquely and in a stable way a conformal factor near a strictly convex point where we have the information in particular this implies that we can determine locally the isotropic sound speed of a medium by measuring the travel times of waves joining points close to a convex point on the boundary the local results lead to a global lens rigidity uniqueness and stability result assuming that the manifold is foliated by strictly convex hypersurfaces

homological and homotopical dehn functions are different the homological and homotopical dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space the homological dehn function measures fillings of cycles by chains whereas the homotopical dehn function measures fillings of curves by disks because the two definitions involve different sorts of boundaries and fillings there is no a priori relationship between the two functions however before this work there were no known examples of finitely presented groups for which the two functions differ this paper gives such examples constructed by amalgamating a free by cyclic group with several bestvina brady groups

a polytree based adaptive polygonal finite element method for topology optimization of fluid submerged breakwater interaction abstract the polytree based adaptive polygonal finite element method was recently proved to be very efficient for a wide class of mechanics problems in this study we further exploit its advantages for topology optimization of fluid submerged breakwater interaction due to the inherent limitations of triangular and quadrilateral elements for topology optimization an alternative formulation of using arbitrary polygons based on the wachspress coordinate is employed for structural domain in addition we use a polytree based adaptive technique to enhance the accuracy for topology problem in structural domain with a minimum number of degrees of freedom dof we apply this method to fluid structure interaction fsi problems and then finding an optimal shape of submerged breakwater sbw which is as a special kind of coastal structure located in front of the beach and underwater to protect the landside from the erosion several numerical results demonstrate the effectiveness of the present method it is observed that the obtained results are in good agreement with reference solutions