Discrete dynamic programming with outcomes in random variable structures

Multiobjective approach is the common way of generalization single-criterion dynamic programming models. Another way is to consider partially ordered criteria structures. That approach is rather rare. The aim of the paper is to present such a model. Generalization of Bellman’s principle of optimality is employed to create a forward procedure to find the set of all maximal elements. As this set is usual large, the second problem under consideration is to find its subsets. To reduce the number of solutions presented to decision maker we propose to apply a family of narrowing relations. That approach is similar to scalarization in multiobjective programming. Ordered structures of random variables based on mean–variance, stochastic dominance and inverse stochastic dominance are considered. Numerical illustration is given at the end of the paper.     

Dihydroxyacetone (DHA) monomer complexes with CaBr2 and CdCl2

Two hydrated complexes of monomeric dihydroxyacetone (DHA; the simplest ketose), viz. the calcium bromide complex bis(μ‐dihydroxyacetone)bis[tetraaquacalcium(II)] tetrabromide (isomorphous with the chloride compound reported previously), [Ca₂(C₃H₆O₃)₂(H₂O)₈]Br₄, (2e), and the cadmium chloride complex poly[[bis(μ‐dihydroxyacetone)bis[bis(dihydroxyacetone)cadmium(II)]] [diaquatetradeca‐μ‐chlorido‐dichloridohexacadmium(II)] tetrahydrate], {[Cd₂(C₃H₆O₃)₆][Cd₆Cl₁₆(H₂O)₂]·4H₂O}_n, (2f), are described. The Ca²⁺ or Cd²⁺ ions are bridged by the carbonyl O atoms from two DHA molecules to form centrosymmetric dimers, with Ca...Ca distances of 4.334 (2) and 4.300 (2) Å in (2e), and a Cd...Cd distance of 4.195 (1) Å in (2f). Almost identical in shape, the eight‐coordinate polyhedra of the Ca²⁺ and Cd²⁺ ions are composed of 2n O atoms from n DHA molecules [n = 2 in (2e) and n = 4 in (2f)] and are completed by four water molecules in (2e). DHA molecules chelate the cations via both the hydroxyl and carbonyl groups and exist in an extended conformation, with both hydroxyl groups being synperiplanar to the carbonyl O atom. The crystal structures are stabilized by similar extensive O—H...X (X = Cl or Br) and O—H...O hydrogen‐bond networks involving all hydroxyl groups and the water molecules.     

Mean-field approximation for two- and three-person Prisoner’s Dilemmas

We consider the mean-field approximation (MFA) to the system of interacting agents playing two- and three-person Prisoner’s Dilemma games (2-PD and 3-PD). The agents have three available strategies: All-C, All-D, and a third one, which we choose in various ways. Long-time distribution of the strategies is compared for both games. The latter game admits coexistence of larger number of strategies in the long run. In the case of external pressure for cooperation, more pressure is in general necessary for 3-PD than for 2-PD to guarantee cooperation for almost all initial compositions of the strategies. Results for different types of tit-for-tat strategies for 3-PD are discussed. Unlike the 2-PD case, the more forgiving tit-for-tat strategy can survive in the long run in 3-PD.     

Estimates and structure of α-harmonic functions

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set D. This yields a unique representation of such functions as integrals against measures on D ^c ∪ {∞} satisfying an integrability condition. The corresponding Martin boundary of D is a subset of the Euclidean boundary determined by an integral test. K. Bogdan was supported by KBN grant 1 P03A 026 29 and RTN contract HPRN-CT-2001-00273-HARP. T. Kulczycki was supported by KBN grant 1 P03A 020 28 and RTN contract HPRN-CT-2001-00273-HARP. M. Kwaśnicki was supported by KBN grant 1 P03A 020 28 and RTN contractHPRN-CT-2001-00273-HARP.     

Convergence theorems for the Birkhoff integral

We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence (f _n ) of functions from a measure space to a Banach space. In one result the equi-integrability of f _n ’s is involved and we assume f _n → f almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of (f _n ) to f is assumed.     

Nonlinear boundary value problems for second order differential equations with causal operators

In this paper we deal with second order differential equations with causal operators. To obtain sufficient conditions for existence of solutions we use a monotone iterative method. We investigate both differential equations and differential inequalities. An example illustrates the results obtained.     

A Functional Integral Representation for Many Boson Systems II: Correlation Functions

We derive functional integral representations for the partition function and correlation functions of many Boson systems for which the configuration space consists of finitely many points. Submitted: August 20, 2007. Accepted: July 1, 2008.