Symmetric spaces with maximal projection constants

In this paper we introduce a special class of finite-dimensional symmetric subspaces of L₁, so-called regular symmetric subspaces. Using this notion, we show that for any k?2, there exist k-dimensional symmetric subspaces of L₁ which have maximal projection constant among all k-dimensional symmetric spaces. Moreover, L₁ is a maximal overspace for these spaces (see Theorems 4.4 and 4.5.) Also a new asymptotic lower bound for projection constants of symmetric spaces is obtained (see Theorem 5.3). This result answers the question posed in [12, p. 36] (see also [15, p. 38]) by H. Koenig and co-authors. The above results are presented both in real and complex cases.     

Spaces with a uniformly continuous metric projection

This paper contains a characterization of spaces in which the metric projection is uniformly continuous on the class of convex existence sets.     

Spaces of generalized operators with bounded projection trace

We construct a theory of Banach spaces of “generalized” operators with bounded projection trace over a given Hilbert space. This theory can be efficient in the investigation of evolution problems for quantum systems with infinitely many particles.     

Fields with large Kronecker constants

For any algebraic number field K there is a positive number ? such that if α is a nonzero integer of K other than a root of unity, then at least one conjugate of α has absolute value ≥ 1 + ?. It has been conjectured that ? can be taken as $\text{2}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{? 1}$, where n is the degree of K over the field of rationals. In this paper various conditions are discussed under which the validity of this conjecture can be established.     

Spaces with large weak extent

In this paper, we show the following statements:

(1)

For any cardinal κ, there exists a pseudocompact centered-Lindelöf Tychonoff space X such that we(X)?κ.

(2)

Assuming ℵ₀²=ℵ₁², there exists a centered-Lindelöf normal space X such that we(X)?ω₁.

     

Spaces with large relative extent

In this paper, we prove the following statements: (1) For every regular uncountable cardinal κ, there exist a Tychonoff space X and Y a subspace of X such that Y is both relatively absolute star-Lindelöf and relative property (a) in X and e(Y, X) ? κ, but Y is not strongly relative star-Lindelöf in X and X is not star-Lindelöf. (2) There exist a Tychonoff space X and a subspace Y of X such that Y is strongly relative star-Lindelöf in X (hence, relative star-Lindelöf), but Y is not absolutely relative star-Lindelöf in X.     

An ergodic theorem with large normalising constants

We generalise W. Feller’s limit theorem for (independent) random variables with infinite moments.     

Asymmetry and projection constants of Banach spaces

We discuss various asymmetry constants of finite-dimensional Banach spaces in a more generalized frame than that of [2], and solve a problem raised in [7] by finding an increasing sequence of Banach spaces whose diagonal asymmetry constants tend to infinity. We investigate the question of whether the projection constant of everyn-dimensional Banach space is strictly less than , and show that this is so whenn=2. The research for this paper was partially supported by NSF-GP-34193.     

On computing smooth solutions of problems with large lipschitz constants

We investigate how one can construct numerical methods for computing smooth solutions of ODE's which potentially possess fast growing or decaying solutions. We do not want to use a global method (which computes a solution on the entire relevant interval first), but rather a procedure that obtains numerical values in a marching algorithm. It is shown how this can be achieved by both implicit and explicit integrators, for which some detailed analysis is given. Some numerical examples are also included.     

On the projection and macphail constants ofl n p spaces

We prove that the projection and Macphail constants ofl _n ^p (1≦p≦2) are asymptotically equivalent ton ^1/2 andn ^−1/2 respectively. We also obtain some relations linking certain parameters of general finite dimensional real Banach spaces. This note is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. J. Lindenstrauss, to whom the author wishes to express his thanks and appreciation.     

Composite orthogonal projection methods for large matrix eigenproblems

For classical orthogonal projection methods for large matrix eigenproblems, it may be much more difficult for a Ritz vector to converge than for its corresponding Ritz value when the matrix in question is non-Hermitian. To this end, a class of new refined orthogonal projection methods has been proposed. It is proved that in some sense each refined method is a composite of two classical orthogonal projections, in which each refined approximate eigenvector is obtained by realizing a new one of some Hermitian semipositive definite matrix onto the same subspace. Apriori error bounds on the refined approximate eigenvector are established in terms of the sine of acute angle of the normalized eigenvector and the subspace involved. It is shown that the sufficient conditions for convergence of the refined vector and that of the Ritz value are the same, so that the refined methods may be much more efficient than the classical ones.     

Composite orthogonal projection methods for large matrix eigenproblems

For classical orthogonal projection methods for large matrix eigenproblems, it may be much more difficult for a Ritz vector to converge than for its corresponding Ritz value when the matrix in question is non-Hermitian. To this end, a class of new refined orthogonal projection methods has been proposed. It is proved that in some sense each refined method is a composite of two classical orthogonal projections, in which each refined approximate eigenvector is obtained by realizing a new one of some Hermitian semipositive definite matrix onto the same subspace. Apriori error bounds on the refined approximate eigenvector are established in terms of the sine of acute angle of the normalized eigenvector and the subspace involved. It is shown that the sufficient conditions for convergence of the refined vector and that of the Ritz value are the same, so that the refined methods may be much more efficient than the classical ones. Project supported by the China State Major Key Projects for Basic Researches, the National Natural Science Foundation of China (Grant No. 19571014), the Doctoral Program (97014113), the Foundation of Excellent Young Scholors of Ministry of Education, the Foundation of Returned Scholars of China and the Liaoning Province Natural Science Foundation.     

Harmonic projection methods for large non-symmetric eigenvalue problems

The problem of finding interior eigenvalues of a large nonsymmetric matrix is examined. A procedure for extracting approximate eigenpairs from a subspace is discussed. It is related to the Rayleigh–Ritz procedure, but is designed for finding interior eigenvalues. Harmonic Ritz values and other approximate eigenvalues are generated. This procedure can be applied to the Arnoldi method, to preconditioning methods, and to other methods for nonsymmetric eigenvalue problems that use the Rayleigh–Ritz procedure. The subject of estimating the boundary of the entire spectrum is briefly discussed, and the importance of preconditioning for interior eigenvalue problems is mentioned. © 1998 John Wiley & Sons, Ltd.     

Numerical identification of selected rate constants in large chemical reaction systems

Significant progress in the development of user-oriented software for parameter identification in large chemical reaction systems is reported. The progress includes new theoretical ideas and an increased domain of applicability of the codes. The solution of several realistic large scale problems is demonstrated.