Singular perturbed Markov chains and exact behaviors of simulated annealing processes

We study asymptotic properties of discrete and continuous time generalized simulated annealing processesX(·) by considering a class of singular perturbed Markov chains which are closely related to the large deviation of perturbed diffusion processes. Convergence ofX(t) in probability to a setS ₀ of desired states, e.g., the set of global minima, and in distribution to a probability concentrated onS ₀ are studied. The corresponding two critical constants denoted byd and withd are given explicitly. When the cooling schedule is of the formc/logt, X(t) converges weakly forc>0. Whether the weak limit depends onX(0) or concentrates onS ₀ is determined by the relation betweenc, d, and . Whenc>, the expression for the rate of convergence for each state is also derived.     

Annular Functions and Gap Series

If f(z) = c_kz^nk, where n_k+1/n_k q > 1, and f(z) is analyticin |z| < 1, the f(z) is an annular function if and only ifsup |c_k| = . This answers a question posed by L. R. Sons andD.M. Campbell simplifies the proofs of many known examples ofannular functions. Present address: Dept. of Mathematical Sciences, McMaster University,Hamilton, Ontario, Canada L8S4K1     

Development of β-hydroxyamide/titanium complexes for catalytic enantioselective silylcyanation of aldehydes

A highly enantioselective addition of trimethylsilylcyanide to aldehydes catalyzed by chiral titanium complexes is described. The chiral titanium complexes were prepared in situ from Ti(OⁱPr)₄ and β-hydroxyamide ligands, that could easily be synthesized from ketopinic acid and C₂ symmetrical chiral diamines in a small number of steps.     

Molecular structure of meso-tetraphenylporphyrinatothallium(III) cyanide: Tl(tpp) (CN)

meso-Tetraphenylporphyrinatothallium(III) cyanide, Tl(tpp)(CN), was previously assumed to be monomeric and has been confirmed by X-ray analysis to exist as two independent molecules in one asymmetric unit. This unit displays two square-pyramidal coordination geometries for the thallium atoms with the cyano ligand coordinated to both Tl atoms. It crystallizes in the triclinic space group P , with a 10.003(3), b 16.231(7), c 21.277(8) Å, 89.98(3), β 90.57(3), γ 90.31(3)°' and z = 4. The structure was solved by direct methods. A total of 7995 unique reflections having I > 3σ(I) was measured with an automated diffractometer and used to refine the crystal structure to a conventional R factor of 6.05 %. The thallium-cyanide distances are 2.140(14) Å (for thallium(I)) and 2.277(14) Å (for thallium(2)) respectively, with thallium(1) situated 0.908 Å above the porphyrin ring and thallium(2) located 1.027 Å below the ring. IR and NMR spectroscopy p rovide complementary methods for investigation of the CN ligand. The characteristic band observed at 2160 cm⁻¹ in the FTIR spectrum is assigned to the CN stretching in the Tl(tpp)(CN) complex. The ¹³C resonance of axial cyano ligand is observed with a pulse delay of 3.5 s at 24°C at 139.2 ppm (with ¹J(²⁰⁵Tl-¹³C) 5394 and ¹J(²⁰³Tl-¹³C) 5344 Hz). This observation disagrees with the conclusion, drawn from previous work, in that an exchange process involving the apical ligand explains the invisibility due to line broadening at 35°C of the ¹³C signal.     

Excitation-energy migration in self-assembled cyclic zinc(II)-porphyrin arrays: a close mimicry of a natural light-harvesting system

The excitation-energy-hopping (EEH) times within two-dimensional cyclic zinc(II)-porphyrin arrays 5 and 6, which were prepared by intermolecular coordination and ring-closing metathesis reaction of olefins, were deduced by modeling the EEH process based on the anisotropy depolarization as well as the exciton-exciton annihilation dynamics. Assuming the number of energy-hopping sites N = 5 and 6, the two different experimental observables, that is, anisotropy depolarization and exciton-excition annihilation times, consistently give the EEH times of 8.0 +/- 0.5 and 5.3 +/- 0.6 ps through the 1,3-phenylene linkages of 5 and 6, respectively. Accordingly, the self-assembled cyclic porphyrin arrays have proven to be well-defined two-dimensional models for natural light-harvesting complexes.     

Porphyrin boxes constructed by homochiral self-sorting assembly: optical separation, exciton coupling, and efficient excitation energy migration

meso-Pyridine-appended zinc(II) porphyrins Mn and their meso-meso-linked dimers Dn assemble spontaneously, in noncoordinating solvents such as CHCl3, into tetrameric porphyrin squares Sn and porphyrin boxes Bn, respectively. Interestingly, formation of Bn from Dn proceeds via homochiral self-sorting assembly, which has been verified by optical separations of B1 and B2. Optically pure enantiomers of B1 and B2 display strong Cotton effects in the CD spectra, which reflect the length of the pyridyl arm, thus providing evidence for the exciton coupling between the noncovalent neighboring porphyrin rings. Excitation energy migration processes within Bn have been investigated by steady-state and time-resolved spectroscopic methods in conjunction with polarization anisotropy measurements. Both the pump-power dependence on the femtosecond transient absorption and the transient absorption anisotropy decay profiles are directly associated with the excitation energy migration process within the Bn boxes, where the exciton-exciton annihilation time and the polarization anisotropy rise time are well described in terms of the F?rster-type incoherent energy hopping model by assuming a number of hopping sites of N = 4 and an exciton coherence length of L = 2. Consequently, the excitation energy hopping rates between the zinc(II) diporphyrin units have been estimated for B1 (48 ps)(-1), B2 (98 +/- 3 ps)(-1), and B3 (361 +/- 6 ps)(-1). Overall, the self-assembled porphyrin boxes Bn serve as a well-defined three-dimensional model for the light-harvesting complex.     

Amperometric Determination of Cr(III) and Cr(VI) by Flow Injection Analysis

Speciation of Cr(III) and Cr(VI) can be attained by flow injection analysis with amperometric detection. Cr(VI) is reduced in an acidic medium to Cr(III) with a glassy carbon electrode at —0.1 V vs. Ag/AgCl and the current is recorded. Cr(III) is oxidised on-line to Cr(VI) with alkaline hydrogen peroxide solution. From the difference of the total chromium and Cr(VI), the amount of Cr(III) was obtained. A linear calibration curve for Cr(VI) was obtained for the concentration ranges 0.01-5.0ppm of Cr(VI) and we have calculated the limit of determination to be about 0.5ppb. We have studied the degree of reproducibility obtained using the solid electrodes under various conditions. The influence of flow rate, coil length, interfenences and the extent of reaction were studied.     

Convenient Synthesis of Phenyl Pyridyl-Ethers Utilizing Fluoride Ion

Aryl pyridyl ethers were prepared from phenols and active halopyridines under essentially neutral condition via fluoride ion assisted reaction.