The strength of Einstein's equations

The strength of Einstein's empty-space field equations is computed anew and shown to be equal to the amount of initial data required for a local solution of the equations. This same amount of initial data is shown to be precisely that required for a set of 16 unknown first-order differential equations containing 10 field variables and having six identities of second order. The 10 field variables must be functions of second order in the metric coefficients. The 16 field equationsC , = 0 whereC is Weyl's conformal tensor, are shown to have the same properties as those of the unknown equations, suggesting thatC = 0 is a satisfactory local first-order formulation of Einstein's second-order empty-space field equations.     

The use of time-dependent projection operators in the derivation of master equations

In this paper we generalize our previous work on the use of time-dependent projection operators for the derivation of master equations for general systems. Previously we had generalized the usual time-independent projection operator approach to include time-dependent projection operators, in which the relevant part of the full density operator is considered to be the uncorrelated part of the full density operator. The irrelevant part of the density operator was then the part describing the correlations between the coupled systems. In the present work we present new time-dependent projections operators which have the property that some correlations between the interacting subsystems are placed in the relevant part of the distribution function and the remaining correlations are placed in the irrelevant part of the distribution function.     

More inequalities for critical exponents

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv +2 2–. Others are 1 1 +v, 1 1 , 1,d 1 + 1/ (for d),dv, ₃ + (for d), ₄ , and _{2m 2m+2} (form 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.NSF Predoctoral Fellow (1976–1979). Research supported in part by NSF Grant PHY 78-23952.     

The derivation of the quantum kinetic equation and the two-time resolvent method

Van Hove's partial density matrix, _E (t), in his generalized master equation is interpreted as a Wigner representation of two-time dyad for energy E and time t. This interpretation enables us to integrate the energyE in Van Hove's master equation. The resultant equation is of non-Markov type on two time parameters. Starting with this master equation, the derivation of quantum kinetic equations, including the second-order approximation in the density expansion, is discussed. The scaling of the quantum kinetic equation is examined in detail for a system in which particles interact through the delta shell potential. It is shown that the quantum kinetic equation, including three-particle scattering, may exist for the physical situations of low-energy scattering,high-energy scattering, and for resonance scattering for time scales of the system sufficiently separated. In deriving the quantum kinetic equation, a factorization theorem form-particle distribution functions is proved to arbitrary order in perturbation expansion.     

Solutions without phase-slip for the Ginsburg-Landau equation

We consider the Ginsburg-Landau equation for a complex scalar field in one dimension and consider initial data which have two different stationary solutions as their limits in space asx±. If these solutions are not very different, then we show that the initial data will evolve to a stationary solution by a phase melting process which avoids phase slips, i.e., which does not go through zero amplitude.     

On Poisson brackets and symplectic structures for the classical and quantum zitterbewegung

The symplectic structures (brackets, Hamilton's equations, and Lagrange's equations) for the Dirac electron and its classical model have exactly the same form. We give explicitly the Poisson brackets in the dynamical variables (x ,p ,v ,S ^v). The only difference is in the normalization of the Dirac velocities =4 which has significant consequences.Dedicated to David Hestenes, whose work profoundly connects geometry (spacetime), algebra (Clifford), and physics (electron).     

Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two dimensional fluids

We construct a family of probability spaces,P ), <0 associated with the Euler equation for a two dimensional inviscid incompressible fluid which carries a pointwise flow _t (time evolution) leavingP globally invariant. _t is obtained as the limit of Galerkin approximations associated with Euler equations.P is also in invariant measure for a stochastic process associated with a Navier-Stokes equation with viscosity, , stochastically perturbed by a white noise force. Dedication. After completion of this work the terrible news of the sudden death of Raphael Høgh-Krohn reached us. In deep sorrow we mourn his departure. The present work has its roots in previous inspiring work by him and we dedicate it to him as a small sign of our gratitude.     

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.     

К ВОПРОСУ O КРИТИЧЕСКО Й ЧАСТОТЕ ПРИ ПРЕРЫВИСТОМ ОСВЕ ЩЕНИИ

c , c n , n AgBr. .     

Kinetic equation in the kinetic region of the dilute and nonuniform electron plasma

Mori's scaling method is used to derive the kinetic equation for a dilute, nonuniform electron plasma in the kinetic region where the space-time cutoff (b, t _c) satisfies _Dbl _f , _D t _{c f} , with _D the Debye length, _D ^–1= _p the plasma frequency, andl _f and _f the mean free path and time, respectively. The kinetic equation takes account of the nonuniformity of the order ofl _f and _D for the single-and the two-particle distribution function, respectively. Thus the Vlasov term associated with the two-particle distribution function is retained. This kinetic equation is deduced from the kinetic equation in the coherent region obtained by Morita, Mori, and Tokuyama, where the space-time cutoff of the coherent region satisfies _Dbr ₀, _Dt _{c 0}, withr ₀ the Landau length and ₀ the corresponding time scale.     

The temperature vibrations of urea molecules

The structural and dynamic parameters of urea at 112°K and 295°K were determined by the least squares method. The characteristic temperature of the torsional optical vibrations of a molecule about a C-O bond was determined and is in good agreement with the value determined by Raman scattering. The fractional X-coordinate of the nitrogen atom corrected for torsional vibrations was determined and it was found that the magnitude of the projection of the C-N bond in the given temperature range changes only within the limits of observational errors. A new method, called temperature difference synthesis, is described and it is shown that it is suitable for rapid qualitative determination of the thermal anisotropy of the vibrations of atoms in a crystal lattice.

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112°K 295°K. C-O , , . X- , C-N . , , , .

     

On the equivalence of the relativistic theories of gravitation

Einstein's equations are rewritten in terms of a certain torsionless linear connection which differs, in general, from the Levi-Civita metric connection . The torsionless connection appears in a natural way as the canonical momentum of the gravitational field g . Einstein's equations have a simple interpretation in terms of the connection . The equivalence of the so-calledpurely metric, purely affine, andmetricaffine theories of gravitation is proved.This work has been written under the financial support of Gruppo Nazionale per la Fisica Matematica of the Italian National Research Council.     

Comments on the mean spherical approximation for hard-core and soft potentials and an application to the one-component plasma

The thermodynamic properties of the mean spherical (MSA), Percus-Yevick (PY), and hypernetted-chain (HNC) approximations are derived by a simple and unified approach by considering the RPA free-energy functionalF and employing an Ewald-type identity. It is demonstrated that with decreasing relative contribution of the hard-core insertion to the thermodynamic functions, the MSA changes its nature from PY-like to HNC-like, withF changing its role from excess pressure to excess free energy, respectively. It is found that the condition of continuity of the MSA pair functions is equivalent to a stationarity condition forF and leads to thermodynamic consistency between the virial and energy equations of state for the (thus defined) soft-MSA (SMSA), withF playing the role of the excess free energy. It is shown that the PY-compressibility and virial equations of state forD-dimensional hard spheres may be simply obtained one from the other without knowing any details of the solution of the model. Using this relation we find an indication that the PY approximation for hard spheres becomes less accurate with increasing dimensionality. A general variational formulation is presented for the application of the MSA for soft potentials, and results for the one-component plasma are discussed and extended.On sabbatical leave from the Nuclear Research Center-Negev, P.O. Box 9001, Beer Sheva, Israel.     

Hyperbolic initial-boundary-value problem for magnetic flux compression by plane liners

Based on the (relativistic) Maxwell equations with displacement current E/t, the initial-boundary-value problem for the compression of an initially homogeneous magnetic fieldB={0,B(x,t),0} between a fixed liner atx=0 and a detonation-driven liner atx=s(t) is solved analytically. By homogenizing the boundary conditions at the moving boundary, the transient electromagnetic fields are shown to be a superposition of quasistatic elliptic (E/t=0) and hyperbolic (E/t0) wave solutions. The wave equation is solved by a Fourier expansion in time-dependent eigenfunctionsf _n =f _n [nx/s(t)] for the variable region 0xs(t), where the Fourier amplitudes _n (t) are determined by coupled differential equations of second order. It is concluded that the conventional elliptic flux compression theories (E/t=0) hold approximately for nonrelativistic liner speeds , whereas the hyperbolic theory (E/t0) is valid for arbitrary liner speeds .     

The surface layer on a ferromagnetic. The energy of some periodic domain structures

Explicit solutions are found for a number of special cases of the potential problem of periodic charge distribution. The effect of uneven distribution of the charges of a Bloch wall on the magnetostatic energy is studied, the effective width of the Bloch wall is determined, an analytical expression is found for this distribution of poles and finally the effective height of the surface layer is also determined. An expression is derived for the orientation of the vectors of magnetization in the surface layer with respect to the easy directions of magnetization. The stability of the basic domain structure up to an inclination of 6°of the plane under observation to the direction of easy magnetization and the dimensions of the domains in periodic closure structures follow from the calculations.

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.

. , , , , . . , 6°,, , .

     

On the Effect of Increasing the Radiative-Transition Rate Near a Nanosize Point

A short review of theoretical and experimental studies concerning the photoexcited florescence and Raman scattering of light for a substance in a space containing small material bodies is presented. Calculations of the radiativetransition probability for atoms (molecules) in the vicinity of bodies with a size much smaller than the light wavelength are performed. The probabilities of the singlephoton and doublephoton transitions are shown to increase by factors of 9 and 81 in the vicinity of a nanosize sphere with dielectric constant ||\ 1. The probability of a radiative transition in the vicinity of the vertex of a conic needle bearing up against a plane (both with || 1) increases by factors of (/R _in)² and (/R _in)⁴ for singlephoton and doublephoton transitions, respectively (R _in is the curvature radius for the needle vertex). This theoretical result is suggested as an explanation of the effect of increasing the radiation process intensity in the experiments carried out in the studies cited below.     

Probabilistic approach to homoclinic chaos

Three-dimensional systems possessing a homoclinic orbit associated to a saddle focus with eigenvalues ±i, – and giving rise to homoclinic chaos when the Shil'nikov condition < is satisfied are studied. The 2D Poincaré map and its 1D contractions capturing the essential features of the flow are given. At homoclinicity, these 1D maps are found to be piecewise linear. This property allows one to reduce the Frobenius—Perron equation to a master equation whose solution is analytically known. The probabilistic properties such as the time autocorrelation function of the state variablex are explicitly derived.     

The distribution of ions and their valencies in manganese ferrites I. MnFe2O4+γ ferrites

A model for the distribution of cations in the spinel lattice of manganese ferrites MnFe₂O₄₊ was elaborated on the basis of the experimental studies of the basic magnetic quantities, electrical conductivity and magnetic relaxation of the given ferrites, taking into consideration their crystallographic properties. The conclusions following from this model are in good agreement with the experimental results obtained by us and by other authors both for stoichiometric manganese ferrite and for a ferrite where 0.

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I. MnFe₂O₄₊

, MnFe₂O₄₊ . , , , , , 0.

     

An Equation for the Dissipation Rate Correlation and Its Implications for the Intermittency Exponent μ in Turbulence

We derive an equation satisfied by the dissipation rate correlation function, for the homogeneous, isotropic state of fully-developed turbulence from the the Navier–Stokes equation. In the equal time limit we show that the equation leads directly to two intermittency exponents ₁=2– ₆ and ₂=z₄– ₄, where the 's are exponents of velocity structure functions and z₄ is a dynamical exponent characterizing the fourth order structure function. We discuss the contributions of the pressure terms to the equation and the consequences of hyperscaling.     

Properties of theL p -solutions of linear impulsive equations in a banach space

Conditions for the existence ofL _p-solutions (1p) of linear impulsive equations in a Banach space are found.