A poset hierarchy

This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy ^κℌ* which characterises the closure of the class of all κ-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This includes a broader class of “scattered” posets that we call κ-scattered. These posets cannot embed any order such that for every two subsets of size < κ, one being strictly less than the other, there is an element in between. If a linear order has this property and has size κ it is unique and called ℚ(κ). Partial orders such that for every a < b the set {x: a < x < b} has size ≥ κ are called weakly κ-dense, and posets that do not have a weakly κ-dense subset are called strongly κ-scattered. We prove that ^κℌ* includes all strongly κ-scattered FAC posets and is included in the class of all FAC κ-scattered posets. For κ = ℵ₀ the notions of scattered and strongly scattered coincide and our hierarchy is exactly aug(ℌ) from the Abraham-Bonnet theorem. The authors warmly thank Uri Abraham for his many useful suggestions and comments. Mirna Džamonja thanks EPSRC for their support on an EPSRC Advanced Fellowship.     

Two-tuple balance of non-binary sequences with ideal two-level autocorrelation

Let p be a prime, q=p^m and F_q be the finite field with q elements. In this paper, we will consider q-ary sequences of period qⁿ-1 for q>2 and study their various balance properties: symbol-balance, difference-balance, and two-tuple-balance properties. The array structure of the sequences is introduced, and various implications between these balance properties and the array structure are proved. Specifically, we prove that if a q-ary sequence of period qⁿ-1 is difference-balanced and has the “cyclic” array structure then it is two-tuple-balanced. We conjecture that a difference-balanced q-ary sequence of period qⁿ-1 must have the cyclic array structure. The conjecture is confirmed with respect to all of the known q-ary sequences which are difference-balanced, in particular, which have the ideal two-level autocorrelation function when q=p.     

Polling systems with state-dependent setup times

We analyze a polling system with multiple stations (queues) attended by a cycling server, in which a setup occurs only when the queue that is polled by the server has one or more customers present. Although such systems are appropriate for modeling numerous manufacturing and telecommunication systems, their analysis is not well developed in the literature. We provide an exact analysis for the 2 station model and present two approximation schemes to determine the mean station waiting times for models with 3 or more stations. We show that some approximate models which have been proposed in the literature for providing upper bounds on the mean station waiting times do not always yield upper bounds. Extensive numerical tests indicate that a simple average of the two approximation schemes yields a close estimate of the true mean station waiting time. This average-of-approximations procedure appears to be robust for a large range of parameter values.Research supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP0045904.Research supported in part by the National Science Foundation under grant DMI-9500471.     

Orbital connections for perturbation-dependent basis sets

Summary The use of perturbation-dependent basis sets is analysed with emphasis on the connection between the basis sets at different values of the perturbation strength. A particular connection, the natural connection, that minimizes the change of the basis set orbitals is devised and the second quantization realization of this connection is introduced. It is shown that the natural connection is important for the efficient evaluation of molecular properties and for the physical interpretation of the terms entering the calculated properties. For example, in molecular Hessian calculations the natural connection reduces the size of the relaxation term, leading to faster convergence of the response equations. The physical separation of the terms also means that first-order non-adiabatic coupling matrix elements can be obtained in a very simple way from a molecular Hessian calculation.     

Linearized impulsive rendezvous problem

The solution of the problem of impulsive minimization of a weighted sum of characteristic velocities of a spacecraft subject to linear equations of motion is presented without the use of calculus of variations or optimal control theory. The geometric structure of the set of boundary points associated with an optimal primer vector is found to be a simplex composed of convex conical sets. Eachk-dimensional open face of the simplex consists of boundary points having nondegeneratek-impulse solutions. This geometric structure leads to a simple proof that at mostn-impulses are required to solve a problem inn-dimensional space. This work is applied to the problem of planar rendezvous of a spacecraft with a satellite in Keplerian orbit using the Tschauner-Hempel equations of motion, with special emphasis on four-impulse solutions. Primer vectors representing four-impulse solutions are sought out and found for elliptical orbits, but none were found for orbits of higher eccentricity. For highly eccentric elliptical orbits, degenerate fiveimpulse solutions were found. In this situation, computer simulations reveal vastly different optimal trajectories having identical boundary conditions and cost.     

MATHER SETS FOR SUBLINEAR DUFFING EQUATIONS

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Minimization approach to membership evaluation in fuzzy sets and error analysis

Evaluation of the degree of membership in fuzzy sets is a fundamental topic in fuzzy set theory. Saaty (Ref. 1) proposes a method for solving this problem that has been widely accepted. In this paper, we examine the problem from an error minimization point of view that attempts to reflect the real intentions of the decision maker. When this approach is used, the findings reveal that fuzzy sets of different cardinalities have dramatically different requirements in the consistency level of the input data as far as the error minimization criterion is concerned.     

On the existence of a finite invariant measure

A necessary and sufficient condition is given for a Borel automorphism on a standard Borel space to admit an invariant probability measure.     

基于粗糙集与层次分析法的组合预测方法

本文在改进的属性约简启发式算法 ,即应用加权平均的属性重要度对属性进行约简的基础上 ,将粗糙集理论和层次分析相结合 ,给出了一种全新的组合预测方法 .     

The Index Set of Injectively Enumerable Classes of Recursively Enumerable Sets in ∑5-Complete

I introduce an effective enumeration of all effective enumerations of classes of r. e. sets and define with this the index set IE of injectively enumerable classes. It is easy to see that this set is ∑₅ in the Arithmetical Hierarchy and I describe a proof for the ∑₅-hardness of IE. Mathematics Subject Classification: 03D25, 03D45.