Dynamical properties of families of discrete delay advection–reaction operators

A family of discrete delay advection–reaction operators is introduced along with an infinite matrix formulation in order to investigate the asymptotic behaviour of the orbits of their iterates. The infinite matrices obtained are triangular matrices with only one non-zero subdiagonal. We show that the elements of powers of these matrices can be written as distinctive products of two factors, one of them involving derivatives of the Lagrange polynomials of basic functions with the diagonal elements as nodes. The other factor consists of products of the subdiagonal elements. Consequently the convergence of the iterates of the operators depends on their eigenvalues and the products of their subdiagonal elements.     

Two-point Padé approximants for the expansions of Stieltjes functions in real domain

The fundamental inequalities for the sequences of subdiagonal and diagonal one-point Padé approximants to Stieltjes function has been extended to the case of certain two-point Padé approximants. The results can be applied to the theory of inhomogeneous media for calculating the bounds for the effective transport coefficients of two-components heterogeneous materials.     

Quasi-diagonal exponent symmetry model for square contingency tables with ordered categories

For square contingency tables, we propose a quasi-symmetry model with an exponential form along subdiagonal and give the theorem that Tomizawa’s (1992) diagonal exponent symmetry model holds if and only if the proposed model and marginal means equality model hold with the orthogonality of test statistics.     

A note on Radau and Lobatto formulae for O.D.E.: s

There exist Runge-Kutta methods based on Radau and Lobatto quadrature formulae. One class gives the set of all first and second above diagonal Padé approximations and another class gives the set of all first and second subdiagonal Padé approximations to the exponential function. A new short proof of the strongA-stability of the latter class of methods and a connection between these two classes are presented.     

On tridiagonal matrices unitarily equivalent to normal matrices

In this article the unitary equivalence transformation of normal matrices to tridiagonal form is studied.It is well-known that any matrix is unitarily equivalent to a tridiagonal matrix. In case of a normal matrix the resulting tridiagonal inherits a strong relation between its super- and subdiagonal elements. The corresponding elements of the super- and subdiagonal will have the same absolute value.In this article some basic facts about a unitary equivalence transformation of an arbitrary matrix to tridiagonal form are firstly studied. Both an iterative reduction based on Krylov sequences as a direct tridiagonalization procedure via Householder transformations are reconsidered. This equivalence transformation is then applied to the normal case and equality of the absolute value between the super- and subdiagonals is proved. Self-adjointness of the resulting tridiagonal matrix with regard to a specific scalar product is proved. Properties when applying the reduction on symmetric, skew-symmetric, Hermitian, skew-Hermitian and unitary matrices and their relations with, e.g., complex symmetric and pseudo-symmetric matrices are presented.It is shown that the reduction can then be used to compute the singular value decomposition of normal matrices making use of the Takagi factorization. Finally some extra properties of the reduction as well as an efficient method for computing a unitary complex symmetric decomposition of a normal matrix are given.     

Full algebras of matrices

A classification of full algebras of matrices is given. All such algebras are permutation-isomorphic to block lower-triangular matrices with corresponding subdiagonal blocks being either zero-blocks or full. Two full algebras are isomorphic if and only if they are permutation-isomorphic. A one-to-one correspondence is provided between the full algebras and transitive directed graphs. It is also proven that such algebras, if endowed with a lattice order, can be almost f- or d-algebras only if they are diagonal.     

Absolute Continuity for Unbounded Jacobi Matrices with Constant Row Sums

We investigate the spectral properties of a class of Jacobi matrices in which the subdiagonal entries are quadratics and the row sums are constants.     

On parallel algorithms for semidiscretized parabolic partial differential equations based on subdiagonal Padé approximations

It is well known that methods for solving semidiscretized parabolic partial differential equations based on the second-order diagonal [1/1] Padé approximation (the Crank–Nicolson or trapezoidal method) can produce poor numerical results when a time discretization is imposed with steps that are “too large” relative to the spatial discretization. A monotonicity property is established for all diagonal Padé approximants from which it is shown that corresponding higher-order methods suffer a similar time step restriction as the [1/1] Padé. Next, various high-order methods based on subdiagonal Padé approximations are presented which, through a partial fraction expansion, are no more complicated to implement than the first-order implicit Euler method based on the [0/1] Padé approximation; moreover, the resulting algorithms are free of a time step restriction intrinsic to those based on diagonal Padé approximations. Numerical results confirm this when various test problems from the literature are implemented on a Multiple Instruction Multiple Data (MIMD) machine such as an Alliant FX/8. © 1993 John Wiley & Sons, Inc.     

A noncommutative Szegö theorem for subdiagonal subalgebras of von Neumann algebras

For almost forty years now the most frustrating open problem regarding the theory of finite maximal subdiagonal algebras has been the question regarding the universal validity of a non-commutative Szegö theorem and Jensen inequality (Arveson, 1967). These two properties are known to be equivalent. Despite extensive efforts by many authors, their validity has to date only been established in some very special cases. In the present note we solve the general problem in the affirmative by proving the universal validity of Szegö's theorem for finite maximal subdiagonal algebras.

     

On close eigenvalues of tridiagonal matrices

Summary. A symmetric tridiagonal matrix with a multiple eigenvalue must have a zero subdiagonal element and must be a direct sum of two complementary blocks, both of which have the eigenvalue. Yet it is well known that a small spectral gap does not necessarily imply that some is small, as is demonstrated by the Wilkinson matrix. In this note, it is shown that a pair of close eigenvalues can only arise from two complementary blocks on the diagonal, in spite of the fact that the coupling the two blocks may not be small. In particular, some explanatory bounds are derived and a connection to the Lanczos algorithm is observed. The nonsymmetric problem is also included. Received April 8, 1992 / Revised version received September 21, 1994     

Lipschitz estimates for multilinear commutator of Littlewood-Paley operator

In this paper, we will study the continuity of multilinear commutator generated by Littlewood-Paley operator and Lipschitz functions on Triebel-Lizorkin space, Hardy space and Herz-Hardy space.      

Continuity Properties of Integral Kernels Associated with Schrödinger Operators on Manifolds

For Schr?dinger operators (including those with magnetic fields) with singular scalar potentials on manifolds of bounded geometry, we study continuity properties of some related integral kernels: the heat kernel, the Green function, and also kernels of some other functions of the operator. In particular, we show the joint continuity of the heat kernel and the continuity of the Green function outside the diagonal. The proof makes intensive use of the Lippmann–Schwinger equation. Submitted: September 20, 2005. Revised: July 20, 2006. Accepted: October 31, 2006.     

Smoothing with positivity‐preserving Padé schemes for parabolic problems with nonsmooth data

We introduce a new class of higher order numerical schemes for parabolic partial differential equations that are more robust than the well‐known Rannacher schemes. The new family of algorithms utilizes diagonal Padé schemes combined with positivity‐preserving Padé schemes instead of first subdiagonal Padé schemes. We utilize a partial fraction decomposition to address problems with accuracy and computational efficiency in solving the higher order methods and to implement the algorithms in parallel. Optimal order convergence for nonsmooth data is proved for the case of a self‐adjoint operator in Hilbert space as well as in Banach space for the general case. Numerical experiments support the theorems, including examples in pricing options with nonsmooth payoff in financial mathematics. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

An extension of the Vitali-Hahn-Saks Theorem

The Vitali-Hahn-Saks theorem on the absolute continuity of the setwise limit of a sequence of bounded measures is extended to allow unbounded measures and convergence of integrals of continuous functions vanishing at infinity.

     

On the Commutator Width of Perfect Groups

This paper proves bounds for the commutator width of a wreathproduct of two groups. As a corollary, it is shown that thecommutator width of finite perfect linear groups of dimension15 is unbounded. It follows that the covering number of thesegroups is unbounded. On the other hand, the commutator widthof iterated wreath products of nonabelian finite simple groupsis bounded by an absolute constant. 2000 Mathematics SubjectClassification 20E22, 20E45.     

一般介质下的超过程的研究

本文考虑一般介质作用下的分枝粒子系统，进而建立了一类比较广泛分枝机制下的超布朗运动．同时本文研究了这类超过程的轨道性质．亦即证明了其轨道连续性和局部绝对连续性     

Special Additive and Multiplicative Commutator Representations for Diagonal Operators

Let ?? be a dense linear subspace of a separable Hilbert space and let ??⁺(??) be the maximal Op*-Algebra on ??. The paper deals with a class of diagonal operators of ??⁺(??), for which we can prove some results concerning special commutator representations. For this end ideas of [1] are generalized.     

Sine transform based preconditioners for elliptic problems

We consider applying the preconditioned conjugate gradient (PCG) method to solving linear systems Ax = b where the matrix A comes from the discretization of second-order elliptic operators with Dirichlet boundary conditions. Let (L + Σ)Σ⁻¹(L^t + Σ) denote the block Cholesky factorization of A with lower block triangular matrix L and diagonal block matrix Σ. We propose a preconditioner M = (Lˆ + Σˆ)Σˆ⁻¹(Lˆ^t + Σˆ) with block diagonal matrix Σˆ and lower block triangular matrix Lˆ. The diagonal blocks of Σˆ and the subdiagonal blocks of Lˆ are respectively the optimal sine transform approximations to the diagonal blocks of Σ and the subdiagonal blocks of L. We show that for two-dimensional domains, the construction cost of M and the cost for each iteration of the PCG algorithm are of order O(n² log n). Furthermore, for rectangular regions, we show that the condition number of the preconditioned system M⁻¹A is of order O(1). In contrast, the system preconditioned by the MILU and MINV methods are of order O(n). We will also show that M can be obtained from A by taking the optimal sine transform approximations of each sub-block of A. Thus, the construction of M is similar to that of Level-1 circulant preconditioners. Our numerical results on two-dimensional square and L-shaped domains show that our method converges faster than the MILU and MINV methods. Extension to higher-dimensional domains will also be discussed. © 1997 John Wiley & Sons, Ltd.     

On flows associated to Sobolev vector fields in Wiener spaces: An approach à la DiPerna-Lions

In this paper we extend the DiPerna-Lions theory of flows associated to Sobolev vector fields to the case of Cameron-Martin-valued vector fields in Wiener spaces E having a Sobolev regularity. The proof is based on the analysis of the continuity equation in E, and on uniform (Gaussian) commutator estimates in finite-dimensional spaces.     

Regular Carathéodory-type selectors under no convexity assumptions

We prove the existence of Carathéodory-type selectors (that is, measurable in the first variable and having certain regularity properties like Lipschitz continuity, absolute continuity or bounded variation in the second variable) for multifunctions mapping the product of a measurable space and an interval into compact subsets of a metric space or metric semigroup.