A fast algorithm for scalar Nevanlinna-Pick interpolation

Summary In this paper, we derive a fast algorithm for the scalar Nevanlinna-Pick interpolation. Givenn distinct pointsz _i in the unit disk |z|<1 andn complex numbersw _i satisfying the Pick condition for 1in, the new Nevanlinna-Pick interpolation algorithm requires onlyO(n) arithmetic operations to evaluate the interpolatory rational function at a particular value ofz, in contrast to the classical algorithm which requiresO(n ²) arithmetic operations to compute the so-called Fenyves array (which is inherent in the classical algorithm). The new algorithm bypasses the generation of the Fenyves array to speed up the computation, and also yields a parallel scheme requiring onlyO(logn) arithmetic operations on a concurrent-read, exclusive-write parallel random access machine withn processors. We must remark that the rational functionf(z) computed by the new algorithm is one degree higher than the function computed by the classical algorithm.Supported in part by the US Army Research Office Grant No. DAAL03-91-G-0106     

The cohomology of the moduli space of controllable linear systems

In this paper we explicitly describe, by generators and relations, the cohomology ring of the manifold _n,m (F) of controllable linear systems having m inputs and state-space dimension n. It is shown that the cohomology ring of _n,m (F) is isomorphic to the invariant cohomology ring of a product of projective spaces. Estimates for the cup length of the cohomology ring are obtained.     

The canonical Cauchy problem for linear systems of partial difference equations with constant coefficients over the completer-dimensional integral lattice ℕ2r

We canonically define and algorithmically solve the problem of the title. Such algorithms are of great significance for the method of finite differences for the solution of partial differential equations and for many technical applications such as image processing. In contrast to the wide (system theoretic) literature for ordinary difference equations and in spite of the great theoretical and practical significance of this problem, until now, there was no systematic theory of these systems and in particular of the corresponding Cauchy problem, let alone an algorithm. In this paper, we give both. The method consists in a transformation of this problem into a naturally associated problem which is defined over the 2r-dimensional natural number lattice ^2r (the upper quadrant in ^2r ) and for which the canonical initial value or Cauchy problem was defined and constructively solved by the second author.     

Controllability of linear systems,differential geometry curves in grassmannians and generalized grassmannians,and riccati equations

We study the linear system =Ax+Bu from a differential geometric point of view. It is well-known that controllability of the system is related to the one-parameter family of operators e^t B. We use this to give a proof of the classical controllability conditions in terms of the differential geometry of certain curves in ⁿ. We then view (t)=Im(e^t B) as a curve in appropriate Grassmannian and see that, in local coordinates, is an integral curve of the flow induced by a matrix Riccati equation. We obtain qualitative geometric conditions on that are equivalent to the controllability of the system. To get quantitiative results, we lift to a curve l' in a splitting space, a generalized Grassmannian, which has the advantage of being a reductive homogeneous space of the general linear group, GL(ⁿ). Explicit and simple expressions concerning the geometry of are computed in terms of the Lie algebra of GL(ⁿ), and these are related to the controllability of the system.James Wolper was a visiting professor in the Department of Mathematics at Texas Tech University while much of this research was conducted. He would like to express appreciation for the hospitality he received during his visit.     

On trivial and non-trivial n-homogeneousC * algebras

LetA be aC ^* — algebra for which all irrèducible representations are of dimensional n. Then ([F], [TT], [V]) algebraA is isomorphic to algebra of all continuous sections of an appropriate algebraic bundle _A . The basisX of this bundle coincides with the compact of all maximal two-sided ideals ofA. We obtain some conditions which provide that _A is trivial and this yields thatA is isomorphic to the algebra of alln×n matrix functions continuous onX. In the case whenX=S ⁿ is a sphere we describe the set of algebraic bundles overX and algebraic structures on this set. Some applications to algebras generated by idempotents are suggested.     

Quasicomplete factorizations of rational matrix functions

It is shown that within the class ofn×n rational matrix functions which are analytic at infinity with valueW()=I _n, any rational matrix functionW is the productW=W ₁...W _p of rational matrix functionsW ₁,...,W _p of McMillan degree one. Furthermore, such a factorization can be established with a number of factors not exceeding 2(W)–1, where (W) denotes the McMillan degree ofW.     

Intrinsic stochasticity of dynamical systems

For the concept of intrinsic stochasticity as introduced by Prigogineet al., a general mathematical approach is outlined. It usesW ^*-algebras. A with a trace of dynamical observables, identifying the state space with =L ²(A,). The main result is that the incorporation of Lyapunov processes in leads necessarily to the larger algebra (). This induces a strictly ascending chain of algebras of observables of increasing complexity.     

Existence and Stability of Periodic Solutions of a Duffing Equation by Using a New Maximum Principle

The purpose of this paper is to obtain new criteria for existence and asymptotic stability of periodic solutions of a Duffing equation x + cx + g(t, x) = 0, taking advantage of a new maximum principle with L^p-conditions combined with known relations between upper and lower solutions, topological degree and stability.Partially supported by D.G.I. BFM2002-01308, Ministerio Ciencia y Tecnología, Spain.     

A topological obstruction to the geodesibility of a foliation of odd dimension

Let M be a compact Riemannian manifold of dimension n, and let be a smooth foliation on M. A topological obstruction is obtained, similar to results of R. Bott and J. Pasternack, to the existence of a metric on M for which is totally geodesic. In this case, necessarily that portion of the Pontryagin algebra of the subbundle must vanish in degree n if is odd-dimensional. Using the same methods simple proofs of the theorems of Bott and Pasternack are given.     

Some remarks on the Hardy-Littlewood maximal function on variable <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

We show that any pointwise multiplier for BMO(ℝⁿ) generates a function p from the class (ℝⁿ) of those functions for which the Hardy-Littlewood maximal operator is bounded on the variable L^p space. In particular, this gives a positive answer to Diening's conjecture saying that there are discontinuous functions which nevertheless belong to (ℝⁿ).     

On Bergman-Toeplitz operators with commutative symbol algebras

Let be the unit disk in, be the Bergman space, consisting of all analytic functions from , and be the Bergman projection of onto . We constructC ^*-algebras , for functions of which the commutator of Toeplitz operators [T _a ,T _b ]=T _a T _b –T _b T _a is compact, and, at the same time, the semi-commutator [T _a ,T _b )=T _a T _b –T _ab is not compact.It is proved, that for each finite set =n ₀,n ₁, ...,n _m , where 1=n _{0 1} <... _m , andn _k {}, there are algebras of the above type, such that the symbol algebras Sym of Toeplitz operator algebras arecommutative, while the symbol algebras Sym of the algebras , generated by multiplication operators and , haveirreducible representations exactly of dimensions n ₀,n ₁,..., n _m .This work was partially supported by CONACYT Project 3114P-E9607, México.     

The Nehari problem for the Pritchard-Salamon class of infinite-dimensional linear systems: a direct approach

A complete solution is obtained to the Nehari problem for symbols which have a realization as an exponentially stable Pritchard-Salamon system (A, B, C). This allows for the possibility thatB andC be unbounded and have infinite rank. The approach is to solve an equivalentJ-spectral factorization problem for this particular realization.     

The Nehari problem for infinite-dimensional linear systems of parabolic type

A complete solution is obtained to the suboptimal Nehari extension problem for transfer functions of parabolic systems with Dirichlet boundary control and smooth observations. The solutions are given in terms of the realization (–A, B, C), whereA is a uniformly strongly elliptic operator of order two with smooth coefficients defined on a bounded open domain ofR _d ,B=AB _D andB _D is the Dirichlet map associated with Dirichlet boundary conditions andC is a bounded observation map fromL ₂() to the output spaceY. The approach is to solve an equivalentJ-spectral factorization problem for this particular realization.     

The graph induced by affine <Emphasis Type="Italic">M</Emphasis>-flats over finite fields II

Let AG(n, F _q) be the n-dimensional affine space over F _q, where F _q is a finite field with q elements. Denote by Γ ^(m) the graph induced by m-flats of AG(n, F _q). For any two adjacent vertices E and F of is studied. In particular, sizes of maximal cliques in Γ ^(m) are determined and it is shown that Γ ^(m) is not edge-regular when m<n−1. Supported by the National Natural Science Foundation of China (19571024) and Hunan Provincial Department of Education (02C512).     

The infinite-time admissibility of observation operators and operator Lyapunov equations

Let (A, –, C) be an abstract dynamical system withA being the generator of aC ₀-semigroup on a Hilbert spaceH, C:D(A)Y a linear operator,Y another Hilbert space. In this paper, some sufficient and necessary conditions are obtained for the observation operatorC to be infinite-time admissible. For a control system (A, B, –), due to duality argument, some sufficient and necessary conditions are also given for the control operatorB to be extended admissible. It is wellknown that observation operatorC is admissible if and only if the operator Lyapunov equation associated with the system has a nonnegative solution. In this paper, all nonnegative solutions to this equation are represented parametrically.This project is supported by the NNSF of China, and the Youth Science and Technique Foundation of Shanxi Province.     

Covering the hypercube with a bounded number of disjoint snakes

We present a construction of an induced cycle in then-dimensional hypercubeI[n] (n2), and a subgroup _n ofI[n] considered as the group ₂ ⁿ , such that | _n |16 and the induced cycle uses exactly one element of every coset of _n . This proves that for anyn2 the vertices ofI[n] can be covered using at most 16 vertex-disjoint induced cycles.     

Stability inL 1 of some integral operators

A class of Markov operators appearing in biomathematics is investigated. It is proved that these operators are asymptotic stable inL ¹, i.e. lim _n P ⁿ f=0 forfL ¹ and f(x) dx=0.     

The numerical stability of evaluation schemes for polynomials based on the lagrange interpolation form

For evaluation schemes based on the Lagrangian form of a polynomial with degreen, a rigorous error analysis is performed, taking into account that data, computation and even the nodes of interpolation might be perturbed by round-off. The error norm of the scheme is betweenn ² andn ²+(3n+7) _n , where _n denotes the Lebesgue constant belonging to the nodes. Hence, the error norm is of least possible orderO(n ²) if, for instance, the nodes are chosen to be the Chebyshev points or the Fekete points.     

Attached primes of the top local cohomology modules with respect to an ideal

Let be a Noetherian local ring and let be an ideal of R. Let M be an R-module of dimension n. In this paper we study the attached primes of the top local cohomology module Received: 12 May 2004