Equilibrium ring formation in polymer solutions

This paper delivers a flexible formalism for handling equilibrium ring formation. Based on graphical models of polymerization, it includes as special cases the Flory-StockmayerRA _f model, the FloryA _f RB _g model, and Gordon's branching process formalism. When simple ring formation occurs in equireactive systems, it also includes the Jacobson-StockmayerRA ₂ and HoeveRA _f models. The formalism is built from first principles in statistical mechanics and all assumptions are clearly stated. All parameters are given in terms of thermodynamic variables. With ring weights generalizing the Jacobson-Stockmayer Gaussian random walk, the formalism yields results for branchingRA _f ,A _f RB _g , andRA _f -RB _g polymer models. Equireactivity then gives explicit solutions. The equireactiveRA _f -RB _g model compares favorably with data from gel-point vs. dilution experiments. With the exception of the Spanning Tree Approximation, graphical models of polymerization suffer from combinations of the following defects: equireactivity assumptions, restrictions to one type of monomer or bond, absence of rings, or absence of fused rings. This paper provides a promising exact approach to handling all of these problems simultaneously.     

Fractional noise

Fractional noiseN(t),t 0, is a stochastic process for every , and is defined as the fractional derivative or fractional integral of white noise. For = 1 we recover Brownian motion and for = 1/2 we findf ^–1-noise. For 1/2 1, a superposition of fractional noise is related to the fractional diffusion equation.     

Investigation of nematic order by coherent neutron scattering

The mathematical relationship between the orientational order parameters and the coherent neutron scattering cross section for a nematic liquid crystal is given. For deuterated para-azoxyanisole the single-molecule part of the cross section is evaluated within the meanfield approximation and combined with experimental results to give information about molecular orientational order in terms of P ₂, P ₄ and P ₆. Both P ₂ and P ₄ are found necessary for describing the molecular order. Discrepancies between experimental and theoretical results are interpreted as possibly reflecting the inadequacy of the meanfield theory of Maier and Saupe.     

Observed difference between the probabilities of recoilless resonance absorptionf′ and reemissionf″ of gamma-quanta in single crystals of Na2[Fe(CN)5NO]·2 H2O

The recoilless absorption probability factor,f, and recoilless reemission,f, both measured on Na₂[Fe(CN)₅NO]·2 H₂O single crystals using the black filter technique, were found to be different. Unexpectedly, the results found weref>f. In the calculation off, selfabsorption in the scatterer, non-ideality of the black filter and the influence of non-resonant scattering processes have all been taken into account. By varying the scattering geometry for the incoming and outgoing -beam relative to the crystallographic axes only a change in the reemitted valuesf _a, f_b, f_c could be detected because of the long lifetime of the excited nucleus (10^–7 s) relative to the lattice vibration frequencies (10¹² Hz).     

Symmetries and correlation inequalities for classicaln-vector models

We describe a new class of single spin measures on then-dimensional sphereS _r ⁿ of radiusr (n 4) for which Lebowitz-type [J. Lebowitz,J. Stat. Phys. 16:463 (1977)] inequalities hold. This is achieved by an appropriate parametrization ofS _r ⁿ . The above class includes the uniform measures onxs Rⁿ ¦x¦ r for any 0 p r. The second topic of this paper is an abstract formulation of the first Griffiths inequality [R. B. Griffiths,J. Math. Phys. 8:478 (1967)] and the underlying symmetry property.     

Metastable states in the van der Waals-Maxwell theory

A slight modification of the recent Penrose and Lebowitz treatment of thermodynamic metastable states is presented. For the case of periodic boundary conditions, this modification allows the condition of metastability to be extended to all the metastable states in the van der Waals-Maxwell theory of the liquid-vapor phase transition, that is, for all states satisfyingf ₀()+1/2 ²>f(, 0+) andf₀()+x>0 wheref(, 0+) is the (stable) Helmholtz free energy density of the generalized van der Waals-Maxwell theory andf ₀() is the Helmholtz free energy density of a reference system with no long-range interaction, is a mean field-type term arising from a long-range Kac interaction, is the overall mean particle density, andx is any positive number. For the case of rigid-wall boundary conditions, a more restrictive condition is placed onx.     

Density of states for intermediate valence and Kondo systems

We calculate the density of states for the nondegenerate Anderson model for various values ofu=U/ andn _f using the perturbation theory withu as the expansion parameter. Summing all the -independent self-energy diagrams, we use the Friedel sum rule and Ward identities to express the physical quantities in terms of the remaining -dependent part of the self-energy, which we evaluate to the 2nd order. The results for the spin and charge susceptibilities obtained in such a way compare rather well with the Bethe-ansatz results. The density of states exhibits different features in different parts of the parameter space. In Kondo region (u>1,n _f 1, i.e., – _f ~U/2), we obtain a many-body resonance (half-width T _K ) around the Fermi level and two broad peaks () at about _f +n _f U and _f +U. In the VF region (u>1, and | _f |) we obtain only two peaks (), one at about _f and one between _f +n _f U and _f +U. The consequences regarding the shape of the photoemission and inverse photoemission spectra of Ce intermetallics are discussed.     

Far-Infrared Laser Stark Spectroscopy of CH3OD

The high resolution Stark spectra of the singly deuterated methanol isotope, CH₃OD, have been studied using the HCN laser with Stark fields up to approximately 60 000 V/cm. Numerous families of absorption lines have been observed in both parallel and perpendicular polarizations, resulting in the following assignments: with the 311 m line – J _K = 18₁ 18₀ E ₂, _t = 1; and with the 337 m line – J _k = 6₄ 5₃ E ₂, _t = 0 and J _K = 14₆ 13₅ A, _t = 1. Zero-field frequencies for the assigned transitions are in agreement with Fourier transform measurements and those calculated from the available molecular constants.     

Unification of electromagnetism and gravitation: The equations of electrodynamics derived from the Riemann tensor in general relativity

The equations of free-space electrodynamics are derived directly from the Riemann curvature tensor and the Bianchi identity of general relativity by contracting on two indices to give a novel antisymmetric Ricci tensor. Within a factore/h, this is the field-strength tensor G of free-space electrodynamics. The Bianchi identity for G describes free-space electrodynamics in a manner analogous to, but more general than, Maxwell's equations for electrodynamics, the critical difference being the existence in general and special relativity of the Evans-Vigier fieldB ⁽³⁾.     

Duality in parameter space and approximation of measures for mixing repellers

For one-dimensional expanding mapsT with an invariant measure we consider, in a parameter space, the envelope _n of the real lines associated to any couple of points of the orbit, connected byn iterations ofT. If the map hass inverses and is piecewise linear, then the sets _n are just the union ofs ⁿ points and converge to the invariant Cantor set ofT. A correspondence between all the sets and their measures is established and allows one to associate the atomic measure on ₁ to the completly continuous measure on the Cantor set. If the map is nonlinear, hyperbolic, and hass inverses, the sets _n are homeomorphic to the Cantor set; they converge to the Cantor set ofT and their measures converge to the measure of the Cantor set whenn. The correspondence between the sets _n allows one to define converging approximation schemes for the map an its measure: one replaces each of thes ⁿ disjoint sets with a point in a convenient neighborhood and a probability equal to its measure and transforms it back to the original set ₁. All the approximations with linear Cantor systems previously proposed are recovered, the converging proprties being straightforward in the present scheme. Moreover, extensions to higher dimensionality and to nondisconnected repellers arte possible and are briefly examined.