基于Riesz-表示算子非协调稳定化有限元法

Following the framework of the finite element methods based on Riesz-representing operators developed by Duan Huoyuan in 1997,through discrete Riesz representing-operators on some virtual(non-) conforming finite-dimensional subspaces,a stabilization formulation is presented for the Stokes problem by employing nonconforming elements.This formulation is uniformly coercive and not subject to the Babu ka-Brezzi condition,and the resulted linear algebraic system is positive definite with the spectral condition number O(h-2). Quasi-optimal error bounds are obtained,which is consistent with the interpolation properties of the finite elements used.     

Minimal number of generators of the lattice of subspaces of a finite-dimensional linear space

It is proved that a minimal generating system of the lattice of all subspaces of a finite-dimensional vector space over a finite field of q elements contains at most max(q+3) elements. This bound does not depend on the dimension of the space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 148–149, 1982.     

Layer-projective lattices. I

The class of layer-projective lattices is singled out. For example, it contains the lattices of subgroups of finite Abelianp-groups, finite modular lattices of centralizers that are indecomposable into a finite sum, and lattices of subspaces of a finite-dimensional linear space over a finite field that are invariant with respect to a linear operator with zero eigenvalues. In the class of layer-projective lattices, the notion of type (of a lattice) is naturally introduced and the isomorphism problem for lattices of the same type is posed. This problem is positively solved for some special types of layer-projective lattices. The main method is the layer-wise lifting of the coordinates. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 170–182, February, 1998.     

Minimum number of generators of the lattice of submodules of a semisimple module

It is proved that the minimum number of generators of the lattice of all subspaces of a finite-dimensional vector space over a field is finite if and only if the field is finitely generated over the prime field. An upper bound is given for this number, which does not depend on the dimension of the space and is linearly dependent on the number of elements generating the field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 110–113, 1983.     

Orthorecursive Fourier-Stieltjes expansions

Orthorecursive Fourier-Stieltjes expansions are defined, and two examples of expansions are considered. The first example deals with orthogonal systems of functions (which include the Haar system as a particular case), and properties of Fourier-Stieltjes expansions in these systems are proved. It is pointed out that in the case of the Haar system, the integrated Fourier-Stieltjes expansion of a continuous function coincides, up to a constant, with the Faber-Schauder series expansion. The second example deals with nonorthogonal systems of functions that are structurally related to the earlier considered orthogonal systems. Properties of orthorecursive Fourier-Stieltjes expansions in these systems are established.     

Orthorecursive expansions in Hilbert spaces

Orthorecursive expansions over systems of closed subspaces of a Hilbert space are considered. Sufficient conditions for convergence of these expansions to the expanded elements are proved. The results obtained are illustrated on systems of contractions and translations of fixed functions.     

The finite-dimensional decomposition property in non-Archimedean Banach spaces

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property（OFDDP） if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces.This property has an influence in the non-Archimedean Grothendieck＇s approximation theory,where an open problem is the following： Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E.Does D have the OFDDP？ In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP.Next we prove that,however,for certain classes of Banach spaces of countable type,the OFDDP is preserved by taking finite-codimensional subspaces.     

Strong proximinality and renormings

We characterize finite-dimensional normed linear spaces as strongly proximinal subspaces in all their superspaces. A connection between upper Hausdorff semi-continuity of metric projection and finite dimensionality of subspace is given.

     

Uniformly Conditioned Bases of Spectral Subspaces

A condition number of an ordered basis of a finite-dimensional normed space is defined in an intrinsic manner. This concept is extended to a sequence of bases of finite-dimensional normed spaces, and is used to determine uniform conditioning of such a sequence. We address the problem of finding a sequence of uniformly conditioned bases of spectral subspaces of operators of the form T _n  = S _n  + U _n , where S _n is a finite-rank operator on a Banach space and U _n is an operator which satisfies an invariance condition with respect to S _n . This problem is reduced to constructing a sequence of uniformly conditioned bases of spectral subspaces of operators on ? ^n×1. The applicability of these considerations in practical as well as theoretical aspects of spectral approximation is pointed out.     

On the Rate of Convergence of Projection-Difference Methods for Smoothly Solvable Parabolic Equations

A linear parabolic problem in a separable Hilbert space is solved approximately by the projection-difference method. The problem is discretized in space by the Galerkin method orientated towards finite-dimensional subspaces of finite-element type and in time by using the implicit Euler method and the modified Crank-Nicolson scheme. We establish uniform (with respect to the time grid) and mean-square (in space) error estimates for the approximate solutions. These estimates characterize the rate of convergence of errors to zero with respect to both the time and space variables.     

Picturing the lattice of invariant subspaces of a nilpotent complex matrix

Let Q be a nilpotent transformation acting on a finite-dimensional complex vector space. A method is given by which a diagrammatic representation of Lat Q, the lattice of invariant subspaces of Q, can be obtained. Basically the method consists of adding to the Hasse diagram of Hyperlat Q the finite lattice of hyperinvariant subspaces of Q in a specified way. Some applications are given.     

On the convergence of orthorecursive expansions in nonorthogonal wavelets

The present paper is concerned with orthorecursive expansions which are generalizations of orthogonal series to families of nonorthogonal wavelets, binary contractions and integer shifts of a given function φ. It is established that, under certain not too rigid constraints on the function φ, the expansion for any function f ∈ L ²(?) converges to f in L ²(?). Such an expansion method is stable with respect to errors in the calculation of the coefficients. The results admit a generalization to the n-dimensional case.     

A condition for almost everywhere convergence of orthorecursive expansions

An almost everywhere convergence condition with the Weyl multiplier W111111111(n) = v n is obtained for orthorecursive expansions that converge to the expanded function in L².     

On the Spectral-Lyapunov Approach to Parametric Optimization of Distributed-Parameter Systems

The paper is devoted to the discussion of the Riesz-bases applicationin the analysis of distributed-parameter systems controlledby a finite-dimensional or conventional controller. Both boundarycontrol and boundary observation are allowed. The Riesz basesare constructed from a system of eigenfunctions of the closed-loopsystem operator. On the basis of the results obtained, theyare expected to be an effective tool in proving that the closed-loopsystem is well-posed; i.e. they give rise to a C₀-semigroup.These bases also enable us to construct Lyapunov functionalsin the form of series expansions. The analysis is illustratedby a completely worked-out example where the proportional controllersetting is optimized with respect to the ISE criterion. Thecontroller is applied to a parabolic plant modelling a resistive—capacitative(noninductive) direct-current transmission line.     

On a class of finite dimensional contractive perturbations of restricted shifts of finite multiplicity

We study a class of finite-dimensional contractive perturbations of shift operators of finite multiplicity restricted to left invariant subspaces of vectorialH ² spaces. We determine their spectra in terms of the characteristic function of the unperturbed operator and the perturbation. Partially supported by the Batsheva de Rothschild Fund for the Advancement of Science and Technology.     

Perturbation analysis of singular subspaces and deflating subspaces

Summary. Perturbation expansions for singular subspaces of a matrix and for deflating subspaces of a regular matrix pair are derived by using a technique previously described by the author. The perturbation expansions are then used to derive Fr\'echet derivatives, condition numbers, and th-order perturbation bounds for the subspaces. Vaccaro's result on second-order perturbation expansions for a special class of singular subspaces can be obtained from a general result of this paper. Besides, new perturbation bounds for singular subspaces and deflating subspaces are derived by applying a general theorem on solution of a system of nonlinear equations. The results of this paper reveal an important fact: Each singular subspace and each deflating subspace have individual perturbation bounds and individual condition numbers. Received July 26, 1994     

Incidence modules for symplectic spaces in characteristic two

We study the permutation action of a finite symplectic group of characteristic 2 on the set of subspaces of its standard module which are either totally isotropic or else complementary to totally isotropic subspaces with respect to the alternating form. A general formula is obtained for the 2-rank of the incidence matrix for the inclusion of one-dimensional subspaces in the distinguished subspaces of a fixed dimension.     

Gabor analysis over finite Abelian groups

Gabor frames for signals over finite Abelian groups, generated by an arbitrary lattice within the finite time–frequency plane, are the central topic of this paper. Our generic approach covers both multi-dimensional signals as well as non-separable lattices, and in fact the multi-window case as well. Our generic approach includes most of the fundamental facts about Gabor expansions of finite signals for the case of product lattices, as they have been given by Qiu, Wexler–Raz or Tolimieri–Orr, Bastiaans and Van-Leest and others. In our presentation the spreading representation of linear operators between finite-dimensional Hilbert space as well as a symplectic version of Poisson's summation formula over the finite time–frequency plane are essential ingredients. They bring us to the so-called Fundamental Identity of Gabor Analysis. In addition, we highlight projective representations of the time–frequency plane and its subgroups and explain the natural connection to twisted group algebras. In the finite-dimensional setting discussed in this paper these twisted group algebras are just matrix algebras and their structure provides the algebraic framework for the study of the deeper properties of finite-dimensional Gabor frames, independent of the structure theory theorem for finite Abelian groups.