Ordinary differential operators in Hilbert spaces and Fredholm pairs

Let be a path of bounded operators on a real Hilbert space, hyperbolic at . We study the Fredholm theory of the operator . We relate the Fredholm property of to the stable and unstable linear spaces of the associated system . Several examples are included to point out the differences with respect to the finite dimensional case, in particular concerning the role of the spectral flow. We define a general class of paths A for which many properties typical of the finite dimensional framework still hold. Our motivation is to develop the linear theory which is necessary for the set-up of Morse homology on Hilbert manifolds. Received: 9 March 2001; in final form: 1 March 2002 / Published online: 2 December 2002     

Minimal entropy and collapsing with curvature bounded from below

We show that if a closed manifold M admits an ℱ-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S ¹-action. As a corollary we obtain that the simplicial volume of a manifold admitting an ℱ-structure is zero.?We also show that if M admits an ℱ-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative.?We show that ℱ-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold.?We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S ⁴, ℂP ², ²,S ²×S ²and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁴,ℂP ²,S ²×S ²,ℂP ²#  ² or ℂP ²# ℂP ². Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁵,S ³×S ², then on trivial S ³-bundle over S ² or the Wu-manifold SU(3)/SO(3). Oblatum 13-III-2002 & 12-VIII-2002?Published online: 8 November 2002 G.P. Paternain was partially supported by CIMAT, Guanajuato, México.?J. Petean is supported by grant 37558-E of CONACYT.     

Consistent values and the core in continuum market games with two types of players

The consistent value is an extension of the Shapley value to the class of games with non-transferable utility.? In this paper, the consistent value will be characterized for market games with a continuum of players of two types. We will show that for such games the consistent value need not belong to the core, and provide conditions under which there is equivalence between the two concepts. Received: October 1998 RID="*" ID="*"  This thesis was completed under the supervision of Professor Sergiu Hart, The Center for Rationality and Interactive Decision Theory, Department of Mathematics, Department of Economics, The Hebrew University of Jerusalem. I would like to thank Professor Hart for introducing me to this area of research, for his help and guidance, and, especially, for all his patience.? I would also like to thank Michael Borns for improving the style, and an anonymous referee for helpful comments.     

Cost benefit analysis of series systems with warm standby components

This paper deals with the cost benefit analysis of series systems with warm standby components. The time-to-repair and the time-to-failure for each of the primary and warm standby components is assumed to have the negative exponential distribution. We develop the explicit expressions for the mean time-to-failure, MTTF, and the steady-state availability, A _T () for three configurations and perform a comparative analysis. Under the cost/benefit (C/B) criterion, comparisons are made based on assumed numerical values given to the distribution parameters, and to the cost of the components. The configurations are ranked based on: MTTF, A _T (), and C/B where B is either MTTF or A _T ().     

Mann-Whitney test for associated sequences

Let {X ₁, ...,X _m } and {Y ₁, ...,Y _n } be two samples independent of each other, but the random variables within each sample are stationary associated with one dimensional marginal distribution functionsF andG, respectively. We study the properties of the classical Wilcoxon-Mann-Whitney statistic for testing for stochastic dominance in the above set up.     

Variations on the Theorem of Birkhoff – von Neumann and Extensions

 The theorem of Birkhoff – von Neumann concerns bistochastic matrices (i.e., matrices with nonnegative real entries such that all row sums and all column sums are equal to one). We consider here real matrices with entries unrestricted in sign and we extend the notion of permutation matrices (integral bistochastic matrices); some generalizations of the theorem are derived by using elementary properties of graph theory. Received: October 10, 2000 Final version received: April 11, 2002     

On Positive and Weakly Positive Implicative BCI-Algebras

Prove that the notion of positive implicative BCI-algebras coincides with that of weakly positive implicative BCI-algebras, thus the whole results in the latter are still true in the former, in particular, one of these results answers definitely the first half of J. Meng and X.L. Xin’s open problem: Does the class of positive implicative BCI-algebras form a variety? The second half of the same problem is: What properties will the ideals of such an algebra have? Here, some further properties are obtained.     

On distributive modules and rings

In this paper we study ω-distributive modules, where ω is a cardinal number. We extend a characterization of distributive modules to ω-distributive modules. In particular, the case in which ω = n is a finite cardinal is considered. We apply the results to the case n = 2, obtaining new characterizations for distributive modules and rings. Special attention is given to saturated submodules and ideals.     

Machine assignment in a nonlinear multi-product flowline

A group of machines for processing a set of jobs in a manufacturing system is often located in a serial line. An efficient strategy for locating these machines such that the total travel distance or the cost of transporting the jobs is minimized is desired. In this research, the assumption of a linear line with equally spaced machine location is relaxed. This research addressed problems of locating unique machines. It is found that the machine distances possess unique properties in this type of a problem. Utilizing these properties, heuristic strategies are proposed to obtain efficient solution where optimal methods are expected to be computationally prohibitive. A lower bound for the optimum solution is also proposed. Results are encouraging.     

Tuning inventory policy parameters in a small chemical company

A project to improve inventory management in a small UK chemical company is described. A research group comprising university academics and company managers for logistics and information technology examined current practices and analysed a database of historical records of business operations of the company. Based on the analysis, a scheme to categorise stock and set ordering policies to optimise inventory costs was developed. Some comments are made on process issues and the learning that took place.