A note on the acyclicity of the Koszul complex of a module

We prove the vanishing of the Koszul homology group H _μ(Kos(M)_μ), where μ is the minimal number of generators of M. We give a counterexample that the Koszul complex of a module is not always acyclic and show its relationship with the homology of commutative rings.     

Corrigendum to the paper ``Adjoining an Order Unit to a Matrix Ordered Space'

An error has been detected (and also corrected) in Theorem 2.8 of the paper entitled “Adjoining an Order Unit to a Matrix Ordered Space” (Positivity, (2005)9: 207–223; DOI ). Accordingly, some of the results of the paper have been modified. Also, a notion of C*-matricially, Riesz normed spaces has been introduced.     

Designing a majorization scheme for the recourse function in two-stage stochastic linear programming

We discuss issues pertaining to the domination from above of the second-stage recourse function of a stochastic linear program and we present a scheme to majorize this function using a simpler sublinear function. This majorization is constructed using special geometrical attributes of the recourse function. The result is a proper, simplicial function with a simple characterization which is well-suited for calculations of its expectation as required in the computation of stochastic programs. Experiments indicate that the majorizing function is well-behaved and stable.     

On conservative approximation by linear polynomial operators an extension of the Bernstein's operator

In this work we study linear polynomial operators preserving some consecutive i-convexities and leaving invariant the polynomials up to a certain degree. First, we study the existence of an incom patibility between the conservation of certain i-convexities and the invariance of a space of polynomials. Interpolation properties are obtained and a theorem by Berens and DeVore about the Bernstein's operator is extended. Finally, from these results a generalized Bernstein's operator is obtained. This work was supported by Junta de Andalucia. Grupo de investigación: Matemática Aplicada. Código: 1107     

1,3-diallyl-5-[ω-(diphenylphosphino)alkyl] isocyanurates in reactions of complex formation with palladium(ii) dichloride

The reactions of phosphine derivatives of diallyl isocyanurates with palladium(ii) dichloride lead to the formation of complexes, whose structure, composition, and stability depend on the length of the methylene chain between the isocyanurate and diphenylphosphine fragments in the ligand. 1,3-Diallyl-5-[5′-(diphenylphosphino)pentyl and 10′-(diphenyl-phosphino)decyl] isocyanurates with PdCl₂ form monomeric L₂PdCl₂ trans-complexes in which P atoms of the ligands participate in coordination with the metal. 1,3-Diallyl-5-[2′-(diphenylphosphino)ethyl] isocyanurate with PdCl₂ forms a dimeric (LPdCl₂)₂ complex, which decomposes in a solution to the monomer including solvent molecule into the coordination sphere of the metal. The reactions of 1,3-diallyl-5-[4′-(diphenylphosphino)butyl] isocyanurate and 1,3-diallyl-5-[6′-(diphenylphosphino)hexyl] isocyanurate with PdCl₂ give monomeric chelate LPdCl₂ complexes in which one of the allyl groups of the isocyanurate cycle participates in coordination with the central ion along with the phosphorus atom. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 1859–1865, September, 1998.     

Density estimation and adaptive control of markov processes: Average and discounted criteria

We consider a class of discrete-time Markov control processes with Borel state and action spaces, and ^d i.i.d. disturbances with unknown distribution . Under mild semi-continuity and compactness conditions, and assuming that is absolutely continuous with respect to Lebesgue measure, we establish the existence of adaptive control policies which are (1) optimal for the average-reward criterion, and (2) asymptotically optimal in the discounted case. Our results are obtained by taking advantage of some well-known facts in the theory of density estimation. This approach allows us to avoid restrictive conditions on the state space and/or on the system's transition law imposed in recent works, and on the other hand, it clearly shows the way to other applications of nonparametric (density) estimation to adaptive control.Research partially supported by The Third World Academy of Sciences under Research Grant No. MP 898-152.     

A strange pigeon-hole principle

Using Ramsey theory, we establish the following pigeon-hole type principle: From a large number of random variables (functions, vectors, etc.) one can always select two, X and Y, such that P(X < Y) 1/2. We apply the principle for a poset problem.