Some examples of random walks on free products of discrete groups

Summary We consider the random walk (X_n) associated with a probability p on a free product of discrete groups. Knowledge of the resolvent (or Green's function) of p yields theorems about the asymptotic behaviour of the n-step transition probabilities p^*n(x)=P(X_n= x¦ X₀=e) as n. Woess [15], Cartwright and Soardi [3] and others have shown that under quite general conditions there is behaviour of the type p^*n(x)C_x^– n n^– 3/2. Here we show on the other hand that if G is a free product of m copies ofZ ^r and if (X_n) is the « average » of the classical nearest neighbour random walk on each of the factorsZ ^r, then while it satisfies an « n^–3/2 — law » for r small relative to m, it switches to an n^– r/2 -law for large r. Using the same techniques, we give examples of irreducible probabilities (of infinite support) on the free groupZ ^*m which satisfyn ^– for .     

Quantum relativistic action at a distance

A well-known relativistic action at a distance interaction of two unequal masses is altered so as to yield purely Newtonian radial forces with fixed particle rest masses in the system center-of-momentum inertial frame. Although particle masses experience no kinematic mass increase in this frame, speeds are naturally restricted to less than the speed of light. We derive a relation between the center-of-momentum frame total Newtonian energy and the composite rest mass. In a new proper time quantum formalism, we obtain an L²(R⁴ R⁴, C) Hilbert space by varying individual particle rest masses. We propose the use of density operators, recognizing that the auxiliary proper time parameter is not an observable. The quantum formalism is applied to our altered version of the relativistic harmonic oscillator. Our generalized coherent states yield four-dimensional wave packets which follow the correct classical world lines. Appendices contain reviews of classical Hamiltonian reparametrization (incorporating our notion of manifest covariance), and a comparison of this work with the literature.     

Ground state of a spin-phonon system. I. Variational estimates

A study is made of the ground-state energy of a spin-one-half particle in a fieldB and interacting with a phonon bath. The infrared-sensitive case of acoustic phonons with point coupling in three dimensions is characterized by two parameters, a coupling constant andB. Units are used where the high-momentum phonon cutoff is unity. There is a curve (B) separating a symmetry-breaking region with a long-range phonon field from a normal region. Two simple, well-known, approximations are compared. The source theory yields discontinuities in the first derivatives of the energy with respect toB and whenB>e ^–1 and an infinite-order transition whenB<e ^–1, but is trivial in the large- region. The classical theory yields discontinuities in the second derivatives but is trivial in the small- region. An improved variationally fixed ground-state wave function is analyzed. It gives a new (B) curve with an infinite-order transition with continuous energy derivatives whenB<e/(e ²–1/4) and with discontinuous derivatives whenB is larger than this value. It is nontrivial in the entire (B) plane. The crossover to classical behavior occurs near =1/2 forB1. But the wave function does not describe quantum fluctuations in the large- phase. A second way of combining source and classical effects is described. It yields a second-order transition (near =1/2 forB1) everywhere. These theories are special cases of a symmetry-breaking transformation together with a one-mode treatment of quantum fluctuations. The transition is viewed in terms of a single mode with a variable length, coupled dynamically to the spin.     

Thermodynamic properties of transition metals in aqueous solution. 2. The enthalpies of mixing of aqueous solutions of CoCl2, CuCl2, and MnCl2 with NaCl at varying ionic strength at 25°C

Enthalpies of mixing (_m H) aqueous solutions of CoCl₂, CuCl₂, and MnCl₂ with NaCl solutions were measured at constant ionic strengths of 0.5, 1.0, and 3.0 molal at 25°C. The excess enthalpy equations of Pitzer were then fit to the resulting _m H data. The resulting parameters are the temperature derivatives of the activity coefficient mixing parameters in the Pitzer system. The heat of mixing data for CoCl₂ and CuCl₂ were in agreement with earlier isomolal results by other workers.     

Reactions of [NiIII(cyclam)] with aqueous sulfur dioxide: Exhibition of a deuterium isotope effect and catalysis by chloride and bromide

One unit of S(IV) (SO₂ or SHO₃^?) is oxidized per 2 units of [Ni^III(cyclam)] species to obtain sulfate. Kinetic analyses have been done by varying the acidities (0.013 ? [H⁺] ? 1.0 M) and halide concentrations (0.000 ? [X^?] ? 0.012 M; X=Cl and Br) at constant ionic strength (μ = 1.0 M). The rate law that incorporates the [X^?] and [H⁺] dependence is ?d[Ni^III]_T/dt=2k[Ni^III]_T[S(IV)]_T where 2k={k_a[H⁺] + k_bK + kK′_X[H⁺] [X^?] + kK′_XK[X^?]} {[H⁺] + K}^?1 {1 + K′_X[X^?]}^?1, here k_a=87 ± 7 M^?1 s^?1, k_b=(2.5 ± 0.5)×10³ M^?1 s^?1 and pK = 1.8 ± 0.2. Rate constants k_a and k_b are attributed to the reactions of [Ni^III(cyclam) (H₂O)₂]³⁺ with SO₂ and SHO₃^?, respectively. Monohalo species apparent equilibrium constants K′_Cl=(1600 ± 400) M^?1 and K′_Br=(190 ± 20) M^?1 and rate constants k=80 ± 8 M^?1 s^?1 and k = 140 ± 15 M^?1 s^?1 are ascribed to the protonated pathway, involving the [Ni^III(cyclam) (H₂O)X]²⁺ and SO_2(aq) reaction pairs. The other two rate constants of k=(5 ± 1)×10³ M^?1 s^?1 and k=(3.1 ± 0.5)×10⁴ M^?1 s^?1, refer to the deprotonated pathway and are assigned to the [Ni^III(cyclam) (H₂O)X]²⁺ /SHO₃^? redox couple. A deuterium H₂O / D₂O isotope effect of 2.1–2.8 can be attributed partially to an equilibrium isotope effect at low acidity though a small kinetic isotope (2.5 ± 0.5) effect is evident for the dihydrogen sulfito pathway, k_a. The kinetic isotope effect and the absence of sulfite radical scavenging effects are explained by a mechanism entailing migration of a hydride from sulfur to the Ni^III center to produce a Ni^III—H species, which rapidly comproportionates, and S(VI). © 1993 John Wiley & Sons, Inc.     

Classification and identification of mass spectra of toxic compounds with an inductive rule-building expert system and information theory

An expert system for classifying and identifying low-resolution mass spectra of toxic and related compounds was developed with an expert shell program. The shell system used was an inexpensive, rule-building software package with an implementation of the ID3 algorithm. Seventy-eight target compounds were used to establish classes previously found by SIMCA class modeling. The six classes included nonhalobenzenes; chlorobenzenes; bromoalkanes and bromoalkenes; mono- and di-chloroalkanes and the analogous alkenes; tri-, tetra- and penta-chloroalkanes and the analogous alkenes; and unknowns. Identification modules for the target compounds were forward-chained to the classification modules. An expert system based on binary-encoded mass spectra, with 17 masses selected on the basis of information content, gave 97 and 86% classification accuracy for training and test spectra, respectively. Identification accuracy was 77 and 80%, respectively. An expert system was also developed which was based on ternary encoding of the mass spectra of 108 training compounds using 25 masses. Ternary encoding has many of the advantages of binary encoding, without the disadvantages. This latter system was tested with the spectra of thirty compounds found in field samples or potential air pollutants. The classification accuracy for training and test spectra was 99 and 97%, respectively. The identification accuracy was 96 and 93%, respectively. With proper precautions, the rule-building expert system can be very effective in spectral classification and identification problems.     

Mechanism of the glutathione transferase-catalyzed conversion of antitumor 2-crotonyloxymethyl-2-cycloalkenones to GSH adducts

Human glutathione (GSH) transferase (hGSTP1-1) processes with similar kinetic efficiencies the antitumor agents 2-crotonyloxymethyl-2-cyclohexenone (COMC-6), 2-crotonyloxymethyl-2-cycloheptenone (COMC-7), and 2-crotonyloxymethyl-2-cyclopentenone (COMC-5) to 2-glutathionylmethyl-2-cyclohexenone, 2-glutathionylmethyl-3-glutathionyl-2-cycloheptenone, and 2-glutathionylmethyl-2-cyclopentenone, respectively. This process likely involves initial enzyme-catalyzed Michael addition of GSH to the COMC derivative to give a glutathionylated enol(ate), which undergoes nonstereospecific ketonization, either while bound to the active site or free in solution, to a glutathionylated exocyclic enone. Free in solution, GSH reacts at the exomethylene carbon of the exocyclic enone, displacing the first GSH to give the final product. This mechanism is supported by the observation of multiphasic kinetics in the presence of high concentrations of hGSTP1-1 and the ability to trap kinetically competent exocyclic enones in aqueous acid using COMC-6 and COMC-7 as substrates. That the exocyclic enone is formed by nonstereospecific ketonization of an enol(ate) species is indicated by the observation that COMC-6 (chirally labeled with deuterium at the exomethylene carbon) gives stereorandomly labeled exocyclic enone. The isozymes hGSTP1-1, hGSTA1-1, hGSTA4-4, and hGSTM2-2 catalyze the conversion of COMC-6 to final product with similar efficiencies (K(m) = 0.08-0.34 mM, k(cat) = 1.5-6.1 s(-)(1)); no activity was detected with the rat rGSTT2-2 isozyme. Molecular docking studies indicate that in hGSTP1-1, the hydroxyl group of Tyr108 might serve as a general acid catalyst during substrate turnover. The possible significance of these observations with respect to the metabolism of COMC derivatives in multidrug resistant tumors is discussed.     

Easy preparation of cobalt corrole and hexaphyrin and isolation of new oligopyrroles in the solvent-free condensation of pyrrole with pentafluorobenzaldehyde

[reaction: see text] Following the discovery that meso-substituted corroles are formed in solvent-free condensation of pyrrole with aldehydes, we demonstrate that a small variation in the methodology is suitable for facile synthesis of cobalt(III) corrole and hexaphyrin. These compounds, as well as three noncyclic products, were fully characterized by spectroscopy and X-ray crystallography.     

Macrocyclization on the fullerene core: Direct regio- and diastereoselective multi-functionalization of [60]fullerene,and synthesis of fullerene-dendrimer derivatives

The macrocyclization between buckminsterfullerene, C₆₀, and bis-malonate derivatives in double Bingel reaction provides a versatile and simple method for the preparation of covalent bis-adducts of C₆₀ with high regio- and diastereoselectivity. A combination of spectral analysis, stereochemical considerations, and X-ray crystallography (Fig. 2) revealed that out of the possible in-in, in-out, and out-out stereoisomers, the reaction of bis-malonates linked by o-, m-, or p-xylylene tethers afforded only the out-out ones (Scheme 1). In contrast, the use of larger tethers derived from 1,10-phenanthroline also provided a first example, (±)- 19 (Scheme 2), of an in-out product. Starting from optically pure bis-malonate derivatives, the new bis-functionalization method permitted the diastereoselective preparation of optically active fullerene derivatives (Schemes 4 and 5) and, ultimately, the enantioselective preparation (enantiomeric excess ee > 97%) of optically active cis-3 bis-adducts whose chirality results exclusively from the addition pattern (Fig. 6). The macrocyclic fixation of a bis-malonate with an optically active, 9,9′-spirobi[9H-fluorene]-derived tether to C₆₀ under generation of 24 and ent- 24 with an achiral addition pattern (Scheme 4) was found to induce dramatic changes in the chiroptical properties of the tether chromophore such as strong enhancement and reversal of sign of the Cotton effects in the circular dichroism (CD) spectra (Figs. 4 and 5). By the same method, the functionafized bis-adducts 50 and 51 (Schemes 10 and 11) were prepared as initiator cores for the synthesis of the fullerene dendrimers 62 , 63 , and 66 (Schemes 12 and 13) by convergent growth. Finally, the new methodology was extended, to the regio- and diastereoselective construction of higher cyclopropanated adducts. Starting from mono-adduct 71 , a clipping reaction provided exclusively the all-cis-2 tris-adduct (±)- 72 (Scheme 14), whereas the similar reaction of bis-adduct 76 afforded the all-cis-2 tetrakis-adduct 77 (Scheme 15). Electrochemical investigations by steady-state voltammetry (Table 2) in CH₂Cl₂ (+0.1M Bu₄NPF₆) showed that all macroeyciic bis(methano)fullerenes underwent multiple reduction steps, and that regioisomerism was not much influencing the redox potentials, All cis-2 bis-adducts gave an instable dianion which decomposed during the electrochemical reduction. In CH₂Cl₂, the redox potential of the fullerene core in dendrimers 62, 63 , and 66 is not affected by differences in size and density of the surrounding poly(ether-amide) dendrons. The all-cis-2 tris- and tetrakis(meihano)fullercnes (±) -72 and 77 , respectively, are reduced at more negative potential than previously reported all-e tris- and tetrakis-adducts with methane bridges that are also located along an equatorial belt. This indicates a larger perturbation of the original fullerene π-chromophore and a larger raise in LUMO energy in the former derivatives.