Multiresolution weighted norm equivalences and applications

Summary. We establish multiresolution norm equivalences in weighted spaces L ² _w ((0,1)) with possibly singular weight functions w(x)≥0 in (0,1). Our analysis exploits the locality of the biorthogonal wavelet basis and its dual basis functions. The discrete norms are sums of wavelet coefficients which are weighted with respect to the collocated weight function w(x) within each scale. Since norm equivalences for Sobolev norms are by now well-known, our result can also be applied to weighted Sobolev norms. We apply our theory to the problem of preconditioning p-Version FEM and wavelet discretizations of degenerate elliptic and parabolic problems from finance. Revised version received March 19, 2003 Mathematics Subject Classification (2000): 65F35, 65F50, 65N22, 65N35, 65N30, 65T60, 60H10, 60H35 An erratum to this article is available at .     

On the stability of <Emphasis Type="Italic">BDMS</Emphasis> and <Emphasis Type="Italic">PEERS</Emphasis> elements

Summary. Using Fortin operators we give a new proof of stability for Stenbergs family of BDMS elements in linear elasticity. Our approach yields the inf-sup condition with respect to the standard norms, which is indispensable for a posteriori error analysis. Furthermore our technique allows the construction of another family of finite elements strongly related to the classical PEERS element. The given estimates are robust for nearly incompressible materials.Mathematics Subject Classification (2000): 65N30, 65N15, 65F15     

Analytical description of strong extremum in a convex set

Let E, F be normed linear partially ordered spaces and E ⁺,F ⁺ the cones of positive elements in E and F. The mapping G: E F is given and X=G ¹ (F ⁺ ). Necessary and sufficient conditions are derived for x ₀ X to be a strong minimum (the least element). The conditions are applied to establish the existence of optimal correction in an improper linear programming problem.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 107–112, 1988.     

Automorphisms of Hyperbolic Groups and Graphs of Groups

Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups G_e are rigid (i.e. Out(G_e) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.     

POLARITIES FOR NEARRINGS

Abstract

We show that the left and right polar sets of a nearring N form complete bounded lattices P_R(N) and P_L(N). We consider properties of P_L(N) and its elements, which are ideals, when N is 3-semiprime. If in addition N is zero-symmetric we strengthen these results and use them to describe a class in which 3-semiprime nearrings are strongly 3-semiprime.     

A posteriori error estimates and domain decomposition with nonmatching grids

Let F be a nonlinear mapping defined from a Hilbert space X into its dual X, and let x be in X the solution of F(x)=0. Assume that, a priori, the zone where the gradient of the function x has a large variation is known. The aim of this article is to prove a posteriori error estimates for the problem F(x)=0 when it is approximated with a Petrov–Galerkin finite element method combined with a domain decomposition method with nonmatching grids. A residual estimator for a model semi-linear problem is proposed. We prove that this estimator is asymptotically equivalent to a simplified one adapted to parallel computing. Some numerical results are presented, showing the practical efficiency of the estimator. AMS subject classification 65J10, 65N55, 65M60T. Sassi: Present address: Université de Caen, Laboratoire de Mathématiques Nicolas Oresme, UFR Sciences Campus II, Bd Maréchal Juin, 14032 Caen Cedex, France.     

The independence number condition for the existence of a spanning f‐tree

Let G be a graph and f be a mapping from V(G) to the positive integers. A subgraph T of G is called an f‐tree if T forms a tree and d_T(x)≤f(x) for any x∈V(T). We propose a conjecture on the existence of a spanning f‐tree, and give a partial solution to it. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 173–184, 2010     

Estimates for the L 2-Projection onto Continuous Finite Element Spaces in a Weighted L p -Norm

We consider the orthogonal L₂-projection P onto continuous finite element spaces. We prove estimates for P in a weighted L_p-norm and use these to prove corresponding approximation properties. AMS subject classification (2000) 65N60, 35K85     

The nonlinear complementarity problem with applications,part 1

The main result in this paper is an existence and uniqueness theorem for the following nonlinear complementarity problem: Given a mapping from then-dimensional Euclidean spaceE ⁿ into itself, find a nonnegative vector inE ⁿ whose image, under the given mapping, is also nonnegative, the two vectors being orthogonal to each other. It is shown that the above problem has a unique solution if the given mapping is continuous and strongly monotone on the nonnegative orthantE ₊ ⁿ ofE ⁿ . It is also shown that a sufficient condition for a differentiable mapping to be strongly monotone on an open set is that all the eigenvalues of the symmetric part of its Jacobian be bounded below by a positive constant on the given set.This research constituted a part of the author's Ph.D. dissertation at the University of California at Berkeley, Berkeley, California. The author would like to express his appreciation to Professor G. B. Dantzig, who brought this problem to his attention and guided his research with his several suggestions and helpful criticism. Also, he thanks the referee for several important comments and recommendations.     

Uniform convergence and a posteriori error estimators for the enhanced strain finite element method

Summary. Enhanced strain elements, frequently employed in practice, are known to improve the approximation of standard (non-enhanced) displacement-based elements in finite element computations. The first contribution in this work towards a complete theoretical explanation for this observation is a proof of robust convergence of enhanced element schemes: it is shown that such schemes are locking-free in the incompressible limit, in the sense that the error bound in the a priori estimate is independent of the relevant Lamé constant. The second contribution is a residual-based a posteriori error estimate; the L ² norm of the stress error is estimated by a reliable and efficient estimator that can be computed from the residuals. Mathematics Subject Classification (2000):65N30     

Dirac operators on the quantum group SU(2) and the quantum sphere

The problem of constructing the Dirac operators on the quantum groupSU(2) and the quantum sphere S_qμ ² are discussed. In both cases, the constructions presented have the sameSU _q(2)-invariant form and are directly connected with the corresponding Laplace operators. Bibliography:16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 245, 1997, pp. 49–65. Translated by P. N. Bibikov.     

A Neumann-Neumann algorithm for a mortar discretization of elliptic problems with discontinuous coefficients

Summary. Neumann-Neumann algorithm have been well developed for standard finite element discretization of elliptic problems with discontinuous coefficients. In this paper, an algorithm of this kind is designed and analyzed for a mortar finite element discretization of problems in three dimensions. It is established that its rate of convergence is independent of the discretization parameters and jumps of coefficients between subregions. The algorithm is well suited for parallel computations.Mathematics Subject Classification (1991): 65N55, 65N10, 65N30, 65N22.The work was supported in part by the U.S. Department of Energy under contract DE-FG02-92ER25127 and in part by Polish Science Foundation under grant 2P03A00524.AcknowledgmentThe author would like to thank Olof Widlund for many fruitful discussions and valuable remarks and suggestions on how to improve the presentation of our results.     

Finite element approximation of spectral acoustic problems on curved domains

Summary This paper deals with the finite element approximation of the displacement formulation of the spectral acoustic problem on a curved non convex two-dimensional domain . Convergence and error estimates are proved for Raviart-Thomas elements on a discrete polygonal domain _h in the framework of the abstract spectral approximation theory. Similar results have been previously proved only for polygonal domains. Numerical tests confirming the theoretical results are reported.Mathematics Subject Classification (2000):65N25, 65N30, 70J30Supported by FONDECYT 2000114 (Chile)Supported by FONDAP in Applied Mathematics (Chile)     

The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space

LetE andF be Hilbert spaces with unit spheresS ₁(E) andS ₁(F). Suppose thatV ₀ S₁(E) →S ₁(F) is a Lipschitz mapping with Lipschitz constantk=1 such that −V ₀[S ₁(E)] ⊂V ₀[S ₁(E)]. Then V₀ can be extended to a real linear isometric mappingV fromE intoF. In particular, every isometric mapping fromS ₁(E) ontoS ₁(F) can be extended to a real linear isometric mapping fromE ontoF.     

On the approximability and the selection of particle shape functions

Summary. Particle methods, also known as meshless or meshfree methods, have become popular in approximating solutions of partial differential equations, especially in the engineering community. These methods do not employ a mesh, or use it minimally, in the construction of shape functions. There is a wide variety of classes of shape functions that can be used in particle methods. In this paper, we primarily address the issue of selecting a class of shape functions, among this wide variety, that would yield efficient approximation of the unknown solution. We have also made several comments and observations on the order of convergence of the interpolation error, when these shape functions are used; specifically, we have shown that the interpolation error estimate, for certain classes of shape functions, may not indicate the actual order of convergence of the approximation error. Mathematics Subject Classification (2000):65N15, 65N30, 41A30This research was supported by NSF Grant # DMS-98-02367 and ONR Grant # N00014-99-1-0724     

Microlocal Analysis of the Geometric Separation Problem

Image data are often composed of two or more geometrically distinct constituents; in galaxy catalogs, for instance, one sees a mixture of pointlike structures (galaxy superclusters) and curvelike structures (filaments). It would be ideal to process a single image and extract two geometrically “pure” images, each one containing features from only one of the two geometric constituents. This seems to be a seriously underdetermined problem but recent empirical work achieved highly persuasive separations. We present a theoretical analysis showing that accurate geometric separation of point and curve singularities can be achieved by minimizing the ?₁ norm of the representing coefficients in two geometrically complementary frames: wavelets and curvelets. Driving our analysis is a specific property of the ideal (but unachievable) representation where each content type is expanded in the frame best adapted to it. This ideal representation has the property that important coefficients are clustered geometrically in phase space, and that at fine scales, there is very little coherence between a cluster of elements in one frame expansion and individual elements in the complementary frame. We formally introduce notions of cluster coherence and clustered sparsity and use this machinery to show that the underdetermined systems of linear equations can be stably solved by ?₁ minimization; microlocal phase space helps organize the calculations that cluster coherence requires. © 2012 Wiley Periodicals, Inc.     

Representation of the space spanned by a sequence in a Banach space

Summary Let (x _n ) be a general sequence of a Banach space, there is given a particular representation of the elements of the space spanned by (x _n ), by means of the elements of (x _n ). Some properties of this representation are shown, also if (x _n is minimal or normingM-basis. Moreover some open questions are solved.     

A GENERALIZATION OF COMPLETE GROUPS

Let G be a finite group and let G be the semi-direct product of a normal subgroup N and a subgroup K. In [1], conditions were found which are equivalent to the existence of a normal complement to N in G. We consider the structure of groups N for which the above condition always holds. Thus we use Bechtell's results to gain information on groups N such that if G is a semi-direct product of N and a subgroup K, then N is a direct factor of G, for all G. It is an old result that a group N is complete if and only if whenever N is a normal subgroup of G, then N is a direct factor of G, [4]. Hence it is not surprising that complete groups are part of our result. Moreover a group N is complete if and only if N is isomorphic to Aut(N) under the mapping σ(n) = σ _n , where σ _n is the inner automorphism induced by n. This remark leads us to consider groups N which contain a subgroup H such that H is isomorphic to Aut(N) under σ: H → Aut(N). All groups considered here are finite. The results found here do not parallel the results found in the author's dissertation for Lie algebras. There it is shown that only complete Lie algebras have the desired property. Thus, these results provide an example of when the theory of Lie algebras diverges from that of groups.     

Convergence of finite elements enriched with mesh-less methods

Summary A combined hierarchical approximation based on finite elements and mesh-less methods is proposed and studied. Finite Elements are enriched adding hierarchical shape functions based on a particle distribution. Convergence results are presented and proved. Mathematics Subject Classification (2000):65N15, 65N30, 65N99Grant sponsor: Ministerio de Ciencia y Tecnologí a. Grant number: REN2001-0925-C03-01