On the functional central limit theorem and the law of the iterated logarithm for Markov processes

Summary Let X _tt0 be an ergodic stationary Markov process on a state space S. If  is its infinitesimal generator on L ²(S, dm), where m is the invariant probability measure, then it is shown that for all f in the range of } } 0)$$ " align="middle" border="0"> converges in distribution to the Wiener measure with zero drift and variance parameter ² =–2f, g=–2Âg, g where g is some element in the domain of  such that Âg=f (Theorem 2.1). Positivity of ² is proved for nonconstant f under fairly general conditions, and the range of  is shown to be dense in 1. A functional law of the iterated logarithm is proved when the (2+)th moment of f in the range of  is finite for some >0 (Theorem 2.7(a)). Under the additional condition of convergence in norm of the transition probability p(t, x, d y) to m(dy) as t , for each x, the above results hold when the process starts away from equilibrium (Theorems 2.6, 2.7 (b)). Applications to diffusions are discussed in some detail.This research was partially supported by NSF Grants MCS 79-03004, CME 8004499     

Stereoselective reduction of arteannuin B and its chemical transformations

Absolute stereochemistry of dihydroarteannuin B 5 obtained by the reduction of arteannuin B 3 with Ni₂B, NaBH₄ or CdCl₂-Mg-MeOH-H₂O has been established by 2D NMR and single crystal X-ray diffraction studies. Some experiments aimed at the synthesis of dihydrodeoxyarteannuin B [C-4, 5 double bond isomer of 11] are also discussed.     

Energies of Charge Transfer for the Supramolecular Complexes of [60]- and [70]Fullerenes with a Series of meso-Tetraphenylporphyrins in the Solution State

Supramolecular complexes of [60]- and [70]fullerenes with various meso-tetraphenylporphyrins in toluene solutions have been studied by electronic absorption spectroscopy. Charge transfer (CT) absorption bands are observed in the visible region. Vertical ionization potentials (I _D ^V) of the meso-tetraphenylporphyrins are reported from a study of EDA interaction of these porphyrins with a number of electron acceptors like o-chloranil, p-chloranil, 2,3-dichloro-5,6-dicyanobenzoquinone (DDQ) and vitamin K. The dependence of the CT transition energy on the donor ionization potential has been utilized to estimate the vertical electron affinities (E _A ^V) of [60]- and [70]fullerenes in solution. The value of E _A ^V for [60]fullerene is found to be 0.10 eV lower in magnitude than that of [70]fullerene. We have extracted degrees of CT, and oscillator and transition dipole strengths of the fullerenes/meso-tetraphenylporphyrins complexes. The experimental results show that the CT complexes studied here have a neutral character in their ground states. Electronic coupling elements have been determined for fullerene/meso-tetraphenylporphyrin complexes. Values of the solvent reorganization energy indicate that the electron transfer process takes place at a faster rate in the case of [70]fullerene/meso-tetraphenylporphyrin complexes.     

On Some Properties of (Θ+) Excited States

The mass splitting and the decay width of pentaquark (Θ⁺) at the higher angular momentum states have been investigated in the framework of flux tube model considering the diquark-diquark-antiquark configuration for Θ⁺ along with statistical model. A quasi particle model for diquark has been considered. The Regge trajectory of Θ⁺ has also been investigated with interesting some observations.     

On the two dimensional effective electron mass in quantum wells,inversion layers and NIPI superlattices of Kane type semiconductors in the presence of strong light waves: Simplified theory and relative comparison

An attempt is made to study the two dimensional (2D) effective electron mass (EEM) in quantum wells (Qws), inversion layers (ILs) and NIPI superlattices of Kane type semiconductors in the presence of strong external photoexcitation on the basis of a newly formulated electron dispersion laws within the framework of k.p. formalism. It has been found, taking InAs and InSb as examples, that the EEM in Qws, ILs and superlattices increases with increasing concentration, light intensity and wavelength of the incident light waves, respectively and the numerical magnitudes in each case is band structure dependent. The EEM in ILs is quantum number dependent exhibiting quantum jumps for specified values of the surface electric field and in NIPI superlattices; the same is the function of Fermi energy and the subband index characterizing such 2D structures. The appearance of the humps of the respective curves is due to the redistribution of the electrons among the quantized energy levels when the quantum numbers corresponding to the highest occupied level changes from one fixed value to the others. Although the EEM varies in various manners with all the variables as evident from all the curves, the rates of variations totally depend on the specific dispersion relation of the particular 2D structure. Under certain limiting conditions, all the results as derived in this paper get transformed into well known formulas of the EEM and the electron statistics in the absence of external photo-excitation and thus confirming the compatibility test. The results of this paper find three applications in the field of microstructures.     

Delineating incoherent non-Markovian dynamics using quantum coherence

We introduce a method of characterization of non-Markovianity using coherence of a system interacting with the environment. We show that under the allowed incoherent operations, monotonicity of a valid coherence measure is affected due to non-Markovian features of the system–environment evolution. We also define a measure to quantify non-Markovianity of the underlying dynamics based on the non-monotonic behavior of the coherence measure. We investigate our proposed non-Markovianity marker in the behavior of dephasing and dissipative dynamics for one and two qubit cases. We also show that our proposed measure captures the back-flow of information from the environment to the system and compatible with well known distinguishability criteria of non-Markovianity.     

Existence of Surface Waves and Band Gaps in Periodic Heterogeneous Half-spaces

We find a sufficient condition for the existence of surface (Rayleigh) waves based on the Rayleigh-Ritz variational method. When specialized to a homogeneous half-space, the sufficient condition recovers the known criterion for the existence of subsonic surface waves. A simple existence criterion in terms of material properties is obtained for periodic half-spaces of general anisotropic materials. Further, we numerically compute the dispersion relation of the surface waves for a half-space of periodic laminates of two materials and demonstrate the existence of surface wave band gaps.     

Application of Polyphosphoric Acid‐Mediated Acyl Migration for Regiospecific Synthesis of Diverse 2‐Acylpyrroles from Chalcones

A metal‐free approach for the synthesis of 2‐acylpyrroles is reported in this paper. Synthesis of the target molecule started from chalcones and was carried out in two steps. Initial step involved the conversion of chalcones to corresponding 4‐substituted‐3‐acylpyrroles by reaction with TosMIC. In the subsequent step, target molecules were obtained in modest to good yields by polyphosphoric acid‐mediated acyl rearrangement of 3‐acylpyrroles to their 2‐acyl congeners. The crucial final step was amenable to diverse substitutions on pyrrole ring. Preliminary experiment for the determination of mechanism indicated the involvement of acylium ion.