Asymptotic properties of constrained Markov Decision Processes

We present in this paper several asymptotic properties of constrained Markov Decision Processes (MDPs) with a countable state space. We treat both the discounted and the expected average cost, with unbounded cost. We are interested in (1) the convergence of finite horizon MDPs to the infinite horizon MDP, (2) convergence of MDPs with a truncated state space to the problem with infinite state space, (3) convergence of MDPs as the discount factor goes to a limit. In all these cases we establish the convergence of optimal values and policies. Moreover, based on the optimal policy for the limiting problem, we construct policies which are almost optimal for the other (approximating) problems. Based on the convergence of MDPs with a truncated state space to the problem with infinite state space, we show that an optimal stationary policy exists such that the number of randomisations it uses is less or equal to the number of constraints plus one. We finally apply the results to a dynamic scheduling problem.This work was partially supported by the Chateaubriand fellowship from the French embassy in Israel and by the European Grant BRA-QMIPS of CEC DG XIII     

On the connection between the one-dimensionalS=1/2 Heisenberg chain and Haldane-Shastry model

Extra integrals of motion and the Lax representation are found for interacting spin systems with the HamiltonianH = (J/2) _{j, k=1,j k} ^N P(j – k) _{j k} , where one of the periods of the WeierstrassP function is equal toN. The Heisenberg and Haldane-Shastry chains appear as limiting cases of these systems at some values of the second period. The simplest eigenvectors and eigenvalues ofH corresponding to the scattering of two spin waves are presented explicitly for these finite-dimensional systems and for their infinite-dimensional version.     

Random walks generated by affine mappings

This article is concerned with Markov chains on ^m constructed by randomly choosing an affine map at each stage, and then making the transition from the current point to its image under this map. The distribution of the random affine map can depend on the current point (i.e., state of the chain). Sufficient conditions are given under which this chain is ergodic.     

Syntheses and Crystal Structures of Two Coordination Polymers with Aromatic Dioxide as Bridge Ligands

Two coordination polymers, namely, two-dimensional complex 1 {[Cu(μ- L)1.5(ClO4)2(H2O)].(H2O)0.5}n (L = pyrazine-1,4-dioxide) and one-dimensional complex 2 [Co(μ-L)Br2(H2O)2]n, have been synthesized with pyrazine-1,4-dioxide as bridging ligands, and their crystal structures were determined by X-ray crystallography. Crystal data for complex 1: monoclinic system, space group C2/c, with a = 23.310(3), b = 12.2338(17), c = 10.6075(15) , β = 110.487(2)°, V = 2833.6(7) 3, Z = 8, C6H9Cl2CuN3O12.5, Mr = 457.60, Dc = 2.145 g/cm3, F(000) = 1832 and μ = 1.998 mm-1; and those for 2: monoclinic system, space group C2/c, with a = 11.012(3), b = 7.483(2), c = 11.451(3) , β = 101.654(4)°, V = 924.2(4) 3, Z = 4, C4H8Br2CoN2O4, Mr = 366.87, Dc = 2.637 g/cm3, F(000) = 700 and μ = 10.487 mm-1. 1 shows a two-dimensional sheet structure on the ac plane through the coordination of μ-L bridging ligands with Cu(II) ions, while 2 displays a zigzag one-dimensional chain along the c axis via the coordination of μ-L bridging ligand with Co(II) ions. Hydrogen bonds in 1 and 2 make the sheets (or chains) connect each other to form a three-dimensional structure.     

Synthesis and Crystal Structure of a Novel Layered Lead(Ⅱ) Carboxylate-sulfonate Compound Templated by 4,4''-Bipyridine

A new lead(Ⅱ) carboxylate-sulfonate has been hydrothermally synthesized and characterized by X-ray single-crystal and elemental analyses. It crystallizes in triclinic, space group P,(1) with a=8.1573(6), b=9.4663(7), c=12.7488(9) (A), α=89.2320(10), β=80.7380(10), γ=77.9760(10)°, Z=2, V=950.10(12) (A)3, Mr=599.57, Dc=2.096 g/cm3, μ=9.032 mm-1, F(000)=572, the final R=0.0412 and wR=0.1035. It has a 1D chain structure, and the Pb(Ⅱ) is six-coordinated. Two PbO6 polyhedra are interconnected via edge-sharing into a two-core unit. Such units are further interconnected by 3-sulfonato-benzoic acids into 1D chains which are linked by hydrogen bonds into a layer structure. The 4,4'-bipyridines are located at the interlayer space and link the layers into a supramolecular structure by π-π stacking interactions.     

Extremality and the global Markov property II: The global markov property for non-FKG maximal Gibbs measures

We give a condition on a Gibbs measure for an attractive Markov specification, which assures extremality and the global Markov property. As an example of application we consider the class of attractive Markov specifications defined on a compact configuration space over a two-dimensional lattice by the interaction Hamiltonians (assumed to have a finite set of periodic ground configurations) satisfying Peierl's condition. We prove that each extremal Gibbs measure for such a specification, at sufficiently low temperature, has the global Markov property.On leave of absence from the Institute of Theoretical Physics, University of Wrocaw, Poland.     

Synthesis, structure and magnetic properties of the S=1/2, one-dimensional antiferromagnet, Y5Re2O12

Single crystals of Y₅Re₂O₁₂ have been grown, and the crystal structure has been determined by X-ray diffraction. This compound crystallizes in space group C2/m with cell dimensions of a=12.4081(10) Å b=5.6604(5)Å, c=7.4951(6) Å, β=107.837(3)°, Z=2. The final refinement led to R1=0.0238, WR2=0.0459 for 1053 observed reflections with F>4σ(F₀). Edge-sharing ReO₆ octahedra form infinite linear [ReO₂O_4/2]_n chains along the b direction with alternating short and long Re-Re distances. Three crystallographically independent yttrium atoms surround O2 to form OY₄ tetrahedra, which share edges and corners in the ab plane to form a two-dimensional Y₅O₄ network which separates the [ReO₂O_4/2]_n magnetic chains. This compound is therefore isostructural with the series Ln₅Re₂O₁₂Ln=Gd-Lu, which have been known since 1969. The average Re oxidation state is +4.5 in the chains and a reasonable, if qualitative MO scheme results in one unpaired electron per Re dimer. Consistent with this, magnetic susceptibility data can be fitted to the one-dimensional antiferromagnetic Heisenberg model with S=1/2 and parameters J^intra/k=−89(1)K, g=2.15(4) and χ(TIP)=5(1)×10⁻⁴ emu/mol. There is no sign of long-range magnetic order down to 2 K. These results are contrasted with those for the isostructural Y₅Mo₂O₁₂.     

Enzymatic degradation of highly phenolic lignin-based polymers (lignophenols)

Enzymatic degradation of two lignin-based polymers (lignophenols), lignocatechol and lignocresol, prepared by selectively grafting catechol and p-cresol to C_α positions of lignin, respectively, were carried out in aqueous organic solvents. Both lignophenols showed high reactivity in the peroxidase-catalyzed oxidation. Structural analyses by NMR spectroscopies revealed that the degraded lignophenols contained aliphatic chain content, which might be mainly formed in the reduction of the intermediate initially generated by the aromatic ring cleavage. Lower amount of aromatic units in the lignophenols after degraded by peroxidase also indicted the cleavage of aromatic rings. Due to the substitution of phenols at C_α positions of lignin, the degraded lignophenols did not have carbonyl structure, which was abundant in the biodegradation products of native lignin. The two lignophenols were also degraded by Rhus vernicifera laccase. But the degree of degradation was lower than that of the degradation by peroxidase, which might be due to the low activity of laccase on the lignin moieties in lignophenols.     

XPS study of interactions between linear carbon chains and colloidal Au nanoparticles

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Residence times in kink sites and Markov chain model of alloy and intermetallic compound deposition

The concept of the residence time τ_ksp of an atom in a kink site has recently been suggested to understand the processes in electrochemical deposition of alloys and intermetallic compounds. Different kink sites with different residence times must be defined for alloys and intermetallic compounds. Based on this model, the finite Markov chain theory is applied to describe the selectivity of the growth process. An analytical relationship between the alloy composition and the metal ion concentrations in the electrolyte is derived. General model parameters are ratios g _i = K _ii/K _ij of equilibrium constants of the reaction of electrolyte ions with different kink sites on the surface (i, j representing different alloy components). These ratios are called selectivity constants. For simple conditions, the equation g _i ≈ τ_ii/τ_ij connects the g _i values with the residence times. The theory is tested in the deposition of alloys Co-Ni (anomalous co-deposition) and Ni-Mo (induced co-deposition). Additionally, Bi₂Te₃, an example of stoichiometric semiconductor deposition, is treated. Published in Russian in Elektrokhimiya, 2006, Vol. 42, No. 10, pp. 1216–1223. The text was submitted by the authors in English.