Phases of the number-theoretic spin chain

We present numerical and analytical evidence for a first-order phase transition of the ferromagnetic spin chain with partition functionZ()=(–1)/() at the inverse temperature _cr=2.     

Large deviations for the 2D ising model: A lower bound without cluster expansions

We show that a lower large-deviation bound for the block-spin magnetization in the 2D Ising model can be pushed all the way forward toward its correct Wulff value for all >_c.     

Discontinuity of the magnetization in one-dimensional 1/¦x−y¦2 Ising and Potts models

Results from percolation theory are used to study phase transitions in one-dimensional Ising andq-state Potts models with couplings of the asymptotic formJ _x,y const/¦x–y¦². For translation-invariant systems with well-defined lim _x x ² J _x =J ⁺ (possibly 0 or ) we establish: (1) There is no long-range order at inverse temperatures withJ ⁺1. (2) IfJ ⁺>q, then by sufficiently increasingJ ₁ the spontaneous magnetizationM is made positive. (3) In models with 0<J ⁺< the magnetization is discontinuous at the transition point (as originally predicted by Thouless), and obeysM( _c )1/( _c J ⁺)^1/2. (4) For Ising (q=2) models withJ ⁺<, it is noted that the correlation function decays as xy()c()/|x–y|² whenever< _c . Points 1–3 are deduced from previous percolation results by utilizing the Fortuin-Kasteleyn representation, which also yields other results of independent interest relating Potts models with different values ofq.     

Vacuum Einstein spaces with axial symmetry

Four classes of solutions are found to the equations R=–2_{; ;} and g _;=0 in three-dimensional space with metric gdxdx and signature (+ ––), equivalent to the Einstein equations R_ij=0 in a vacuum for the metric . The metric ds² assumes axial symmetry and symmetry with respect to the reflection .Translated from Izvestiya Uchebnykh Zavedenii, Fizika, No. 9, pp. 43–45, September, 1976.     

Calculation of characteristics of benzene derivatives

The C₆H₅X compounds are considered as regards the energy and wave function _x as functions of within limits of –3 and +3, and also as functions of _cx within limits of 0. 5 and 1. 5. Convenient numerical tables are compiled.We are indebted to N.A. Prilezhaev and V.I. Danilov for extensive collaboration in this work.     

The energy-momentum gravitational pseudotensor and Kerr-Schild metrics

It is shown that the Landau-Lifshitz pseudotensor vanishes for any Kerr-Schild metricg =n + 2Hl l satisfying the conditionS l =0. Some solutions verifying this condition are indicated.     

Random surfaces with two-sided constraints: An application of the theory of dominant ground states

We consider some models of classical statistical mechanics which admit an investigation by means of the theory of dominant ground states. Our models are related to the Gibbs ensemble for the multidimensional SOS model with symmetric constraints _x m/2. The main result is that for ₀, where ₀ does not depend onm, the structure of thermodynamic phases in the model is determined by dominant ground states: for an evenm a Gibbs state is unique and for an oddm the number of space-periodic pure Gibbs states is two.     

Magnetization at corners in two-dimensional Ising models

We investigate the corner spin magnetization of two-dimensional ferromagnetic Ising models in various wedge geometries. Results are obtained for triangular and square lattices by numerical studies on finite wedges using the star-triangle transformation, as well as analytic calculations using corner transfer matrices and a fermionic representation of the row-to-row transfer matrix. The corner magnetizations vanish at the bulk critical temperature T_c with an exponent _c differing from the bulk exponent. For isotropic systems with free edges we find that _c is given simply by _c =/2, where is the angle at the corner. For apex magnetizations of conical lattices we obtain the strikingly similar result _a=/4. These formulas apply equally well to anisotropic lattices if the angle is interpreted as an effective angle, ^eff, determined by the relative strengths of the interactions.     

Percolation of the minority spins in high-dimensional Ising models

We present some new results on the region in the-h plane where the + spins percolate for the nearest neighbor Ising model. In particular, it is shown that in high enough dimensionsd there is percolation of the minority spins at inverse temperatures< ₊ with some ₊>_c, for which ₊/gb_c1/2log(cd),c a constant.On leave from Rutgers University.     

The room-temperature decomposition of metastable phases precipitated in the Al-Zn alloys at 200 °C

The room-temperature decomposition of metastable phases in the Al-Zn alloys (from 25 to 50 wt. % Zn) was studied by the transmission electron microscopy and X-ray diffraction. Metastable phases, i.e. G.-P. zones, _R-and -phases, were grown at 200 °C and their decomposition into equilibrium -phase at 20 °C was investigated. Ageing times comprised 1 to 999 days.Both the decomposition mechanism and the rate of decomposition of coherent phases were found to be dependent on the particle sizes and their density reached at 200 °C. The local vacancy supersaturation around the -nucleus in a dense system of G.-P. zones leads to an enhanced growth rate of such nucleus and thus to the formation of one large -precipitate at the expense of several neighbouring G.-P. zones. The elastic stress field around this -particle promotes the further nucleation and growth of -precipitates and leads to their gradual spread throughout the matrix. The decomposition of intermediately sized _Rprecipitates results in the development of -precipitates of comparable sizes nucleated on the array of misfit dislocations at the periphery of _R-precipitates. The cooperative effect between neighbouring particles does not influence the decomposition of large _R-precipitated which split then into several smaller -particles. The rate of G.-P. zones or _R to -decomposition increases with the increasing sizes of transition precipitates and with the zinc content of the alloy. The kinetics of to -decomposition was found to be independent both on the annealing time at 200 °C and on the investigated alloy composition. This can be attributed to the constant density of misfit dislocations as nucleation sites for -precipitates along the -matrix interface and to the large mutual separation of -precipitates in all these alloys.In conclusion we would like to express our thanks to Doc. Dr. V.Syneek, CSc. for his valuable discussions and to Ing. V.íma for the preparation of Al-Zn alloys. Our thanks are also due to Mr. Z.iký for his help in the X-ray diffraction measurement and to P.Vyhlídka for the careful chemical analyses of the investigated alloys.     

Density Oscillations at the Interface Between Vapor and Liquid

We study the interface between liquid and vapor in the context of the van der Waals theory, considering the non-local free energy functional recently derived by Lebowitz, Mazel, and Presutti from a system of particles in the continuum with Kac potentials. We prove that the density profile between vapor and liquid is monotone when the inverse temperature is between the critical value _c and a second critical value *> _c , becoming oscillatory after * and overshooting the equilibrium density of the liquid phase infinitely often.     

Unified fields of analytic space-time-energy

An analytic gravitational fieldZ (Z ^y ) is shown to include electromagnetic phenomena. In an almost flat and almost static complex geometryds ² =zdzdz of four complex variables z=t, x, y, x the field equationsR Rz = –(U U – Z ) imply the conventional equations of motion and the conventional electromagnetic field equations to first order if =(Z ^v) and =(z ) are expressed in terms of the conventional mass density function , the conventional charge density function , and a pressurep as follows: v=const=p/c ²–10^–29 gm/cm³.     

Statistically periodic processes and toeplitz(α by β) matrices

If each element in a Toeplitz matrix is replaced by an by matrix and the original constraints preserved, the result is a doubly infinite matrix with periodic structure called a toeplitz_{( by)} matrix. Such matrices are a basic tool for describing, generating, estimating, filtering, synchronizing, and analyzing information-theoretic functions for statistically periodic processes.     

Zero value of the Schwarzschild mass for an asymptotically Euclidian system of gravitational waves

A class of the asymptotically Euclidian space-times is shown to exist for which the Schwarzschild mass is equal to zero. The coordinate atlases of these space-times satisfy two additional conditions: _k (-gg ^0k )=0 and _ik ⁰ _0g ^ik - _ik ^k _0g ⁰ⁱ =0. In aT-orthogonal metricgs ²=g ₀₀ dt ² -g dx dx these conditions take a simple form: ₀(detg )=0 and (₀ g )(₀ g )=0.     

Magnetic studies on synthetic phosphoferrite Fe 3 2+ (H2O)3(PO4)2

Synthetic crystals of phosphoferrite were studied by Mössbauer spectroscopy for temperatures ranging from 4.2 to 17.5 K. The analysis of the behaviour of the hyperfine fields for the two unequivalent sites near the Néel temperature (17.4 K) allowed the determination of the critical exponent for each site as ₁=0.171 and ₂=0.314.     

Bosons in thermal contact: AC*-algebraic model

A gas of two Boson systems coexisting inR ³, and interacting only mutually, is analyzed. The interaction is quadratic, so that the dynamical problem may be solved completely and exactly.The initial state is taken to be the mutually uncorrelated Gibbs states: ^{(1) (2)} = . We find the time evolved state, and its projections onto the separate species and the subvolumes.The principle consequences of this model are discussed. In particular we examine the possible occurrence of harmonic oscillations between the species.On Study Leave at the Department of Physics and Astronomy, The University of Rochester.This research was partially supported by the National Science Foundation under Contract No. 5-28501.     

Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems

We consider a ferromagnetic Ising spin system isomorphic to a lattice gas with attractive interactions. Using the Fortuin, Kasteleyn and Ginibre (FKG) inequalities we derive bounds on the decay of correlations between two widely separated sets of particles in terms of the decay of the pair correlation. This leads to bounds on the derivatives of various orders of the free energy with respect to the magnetic fieldh, and reciprocal temperature . In particular, if the pair correlation has an upper bound (uniform in the size of the system) which decays exponentially with distance in some neighborhood of (,h) then the thermodynamic free energy density (,h) andall the correlation functions are infinitely differentiable at (,h). We then show that when only pair interactions are present it is sufficient to obtain such a bound only ath=0 (and only in the infinite volume limit) for systems with suitable boundary conditions. This is the case in the two dimensional square lattice with nearest neighbor interactions for 0<₀, where ₀ ^–1 is the Onsager temperature at which (,h=0) has a singularity. For >₀, (,h)/h is discontinuous ath=0, i.e. ₀=_c, where _c ^–1 is the temperature below which there is spontaneous magnetization.Research supported by AFOSR Contract # F 44620-71-C-0013.     

Cosmological Relativity: Determining the Universe by the Cosmological Redshift as Infinite and Curved

Using cosmological relativity theory, we derive the formula for the cosmological redshift written explicitly in terms of 1 – , where = /_c is the ratio of the average mass density to the critical closure density. Based on the present day data of observed redshifts, we conclude that < 1, which means the universe is infinite and curved, and expands forever.     

Decay of the Bethe–Salpeter Kernel and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We consider lattice classical ferromagnetic spin systems at high temperature (1) with nearest neighbor interactions and even single-spin distributions (ssd). Associated with each system is an imaginary time lattice quantum field theory. It is known that there is a particle of mass m–ln in the energy-momentum spectrum. If s ⁴–3s ²²<0, where s ^k is the kth moment of the ssd, and is sufficiently small, we show that in the two-particle subspace there is no mass spectrum up to 2m. For >0 we show that the only mass spectrum in (m, 2m) is a bound state of mass m _b=2m+ln(1–)+O(), where =(+2s ²²)^–1. A bound on the decay of the kernel of a Bethe–Salpeter equation is obtained and used to prove these results.     

Gentle perturbations

By introducing a specific type of perturbation,A, in the Hamiltonian, we define a class of gently perturbed states, ,A, of a canonical ensemble, . The perturbations are chosen so as to preserve a relationship of the form ,A constant ×. Applications in ergodic theory and phase transitions are described.