Cyclic Subset Research Articles

On the peripheral spectrum of bounded positive semigroups on atomic Banach lattices

Generalizing a recent result of E.B. Davies [4], we show that generators of bounded positive C0-semigroups on atomic Banach lattices with order continuous norm have trivial peripheral point spectrum. Moreover, we give examples that the peripheral spectrum can be any closed cyclic subset of $$i\mathbb{R}$$ .

The Spectral Analysis of Frobenius-Perron Operators

It is known that the Frobenius-Perron operator Ps:L1(0,1)→L1(0,1) associated with a transformation S from [0,1] to itself with inf|S′|>1 is quasi-compact as an operator on the Banach space BV[0,1] of functions of bounded variation in L1(0,1), and thus Ps: BV[0,1]→BV[0,1] possesses only the finite peripheral spectrum and in particular 1 is an isolated eigenvalue of Ps. In this paper, we show that under mild conditions on S, the spectrum of Ps:L1(X)→L1(X) is either the closed unit disk {λϵC:|λ|≤1} or a cyclic subset of {λϵC:|λ|=1}.

A generalized form of hajós' theorem

Hajos proved that if a finite abelian group is a direct product of cyclic subsets, then at least one of the factors is a subgroup of the group. A cyclic subset consists of the “first some consecutive elements” of a cyclic subgroup. We give a generalization for Hajos' theorem. The proof of this generalization is simpler and the steps are better motivated than the proof of the original result.

An extension for Hajós' theorem

Hajós theorem asserts that if a finite abelian group is expressed as a direct product of cyclic subsets of prime cardinality, then at least one of the factors must be a subgroup. (A cyclic subset is a ‘front end’ of a cyclic subgroup.) A.D. Sands proved that if a finite cyclic group is the direct product of subsets each of which has cardinality a power of a prime, then at least one of the factors is a direct product of some subset and a nontrivial subgroup. We prove that the same conclusion holds if a general finite abelian group is factored as a direct product of cyclic subsets of prime cardinalities and general subset of cardinalities that are powers of primes provided that the components of the group corresponding to these latter primes are cyclic.

Computer-aided design: Part 3 decision variable selection to avoid hidden singularities in resulting recycle calculations

The chemical engineer is often faced with the task of choosing an appropriate set of variables to be used as input information in the analysis of a process design. The problem can be considered as solving a set of N equations with M variables ( M ≥ N). The engineer must therefore choose MN variables to be decision variables, and these must then be externally fixed to allow calculation of the remaining N state variables. In design each equation will contain only a few of the system variables. Such sparse systems of equations lend themselves to precedence ordering and tearing techniques as effective solution procedures. The concept of an output variable assignment for each equation is basic to these techniques. This paper presents an algorithm for automatically choosing an appropriate set of decision variables. It chooses a set which avoids leaving singular or nearly singular N × N equation sets for solution. Simultaneously the algorithm also chooses an output set for each N × N system so that the solution of any cyclic subset of equations by the iterative method of successive substitutions has good convergence properties. This technique extends earlier work by the authors on locating output sets for N × N cyclic equation sets. This extension is an effective method for evaluating sparse embedded systems of determinants thus permitting one to avoid singularities. The paper concludes with two examples including a binary flash distillation system.