Some applications of the discrete fourier transform to problems of crystal lattice deformation I

The theory of the discrete Fourier transform [1], [2] is applied in solving a system of difference equations describing the positions of atoms in a deformed crystal lattice. The crystal lattice is approximated by the Born-Kármán model modified to include the internal energy of the undeformed crystal.

---

, , , [1] [2]. - , , .

     

On the geometrization of electrodynamics

This paper develops the conjecture that the electromagnetic interaction is the manifestation of the torsion of spacetime. This conjecture is made feasible by the natural separation of the connection ^v into gravitational and electromagnetic parts ^v and ^v , respectively, related to the metric and to the torsion. When ^v is neglected in front of ^v , the affine geodesics are shown to become the equations of motion of charged particles with Lorentz force, for an appropriate choice of . Since ^v contains the factor q/m, neutral particles do not see the torsional part of the connection and behave as if were zero, i.e., as in Einstein's theory of gravity (the same effect is obviously obtained for charged particles when ^{v v} ).In addition to the factor q/m, the velocity of the test particle appears in . This indicates that the appropriate context for this problem is to be found in velocity-dependent connections. The velocities are now coordinates and become the actual velocities of the test particles only in the system of equations that one solves for obtaining the affine geodesics in connections of this type.When written with differential forms, the combination of Maxwell's equations and of the pertinent form of the torsion suggests geometric field equations for electrodynamics. As for the gravitational part of the connection, it can be made to obey equations similar in form to the Einstein field equations. A unified geometric theory of electrodynamics and gravitation spontaneously emerges. The present state of the theory does not yet permit us to ascertain whether the right-hand side of the fully geometric, gravitational field equations corresponds to the energy-momentum tensor.     

On the Nature of the “Low-Temperature Anomaly” in Metal–Semiconductor Schottky-Barrier Contacts

It is shown that the low-temperature anomaly of the I-V characteristics of homogeneous metal–semiconductor Schottky-barrier contacts (an increase in the ideality factor n and a decrease in the barrier height _bm, measured using the saturation current, with decreasing temperature, as well as the fact that their product _bn n_bm coincides very closely with the actual barrier height _b(V)) is a consequence of two factors: (1) the nonlinear dependence of the actual barrier height on the bias, resulting in an increase in the ideality factor with increasing bias voltage (current) and (2) measurement of the parameters n and _bm for the same current for all temperatures. A new expression for the flat-band barrier height _bf is derived.     

Symmetry vector fields and similarity solutions of a nonlinear field equation describing the relaxation to a maxwell distribution

In the study of the formulation of Maxwellian tails the nonlinear partial differential equation ² u/x +u/x+u ²=0 arises. We determine the Lie point symmetry vector fields and calculate the similarity ansätze. Then we discuss the resulting ordinary differential equations. Finally, the existence of Lie Bäcklund vector fields is studied and a Painlevé analysis is performed.     

Calculation of the Tautomeric Properties of 3‐Formyl‐Tetrinic Acid by the ab initio and DFT Methods

We have carried out a nonempirical quantumchemical calculation with full optimization of the geometry of all theoretically possible tautomeric forms of 3formyltetrinic acid (FTRA) according to the Møller–Plesset secondorder perturbation theory with the use of a 6–31G(d) double zeta basis set. The correlation corrections to the total energy of molecules for optimized geometric configurations were calculated in the 631G(d,p) basis. All the possible tautomeric forms of FTRA and the harmonic vibrational frequencies were also calculated within the density functional theory (DFT) with the use of the Perdew–Burke–Ernzerhof (PBE) functional in a threeexponent basis. It is shown that FTRA in vapors (in the perfect gas approximation) exists in the form of a mixture of three enol forms, among which two exoforms predominate. The frequencies and forms of normal vibrations for each cisenol tautomer in the region of vibrations of ketogroups and double bonds differ widely, which permits identification of the tautomers present in the mixture. The possible mechanisms of enolenol transformations of 3acyltetrinic acids are discussed.     

Periodic inverse problem for a new hierarchy of coupled evolution equations

We study the inverse problem with periodic boundary condition for a new class of integrable nonlinear evolution equations. The multiphase periodic solutions for the nonlinear fields (p, q, r) are expressed in terms of the Riemann theta function, which is obtained via the linearization of the flows of the set of auxiliary variables _j on a Riemann surface. An explicit case is evaluated to obtain the form of the algebraic curve on which the variables _j move.     

Quasigroups Connected with Clifford Groups

In Clifford groups, a nonassociative product is defined which leads to the definition of nonassociative groups. These nonassociative groups have matrix representations on the condition that the row by column product of two matrices is replaced by the column by column product. A nonassociative group of transformations connected with the Lorentz group is determined, together with its irreducible, double-valued matrix representation, whose matrices undergo the column by column product.     

Diode-pumped Tm: YAG solid-state lasers with indirect and direct manifold pumping

We study the continuous-wave (cw) characteristics of both two-manifold and three-manifold Tm: YAG laser pumped at _p 1.8 µm or _p = 0.785 µm and lasing at ₁ = 2.02 µm. The three-manifold rate equations are adiabatically reduced to their two-manifold form. For each pumping scheme, the steady-state rate equations are combined with the cw differential equations for the forward- and reverse-lasing fields and the pump-depletion differential equation. These three coupled cw differential equations are solved analytically. This gives the linear flux-conservation law between the input pump and the laser output, the minimum crystal length, and optimal output couplings. We show that the major difference between these two pumping schemes is due to the different pump effective absorption cross sections and not the two-for-one cross relaxation. Our example shows that the minimum intensity threshold and optimal crystal length are smaller for pumping at _tp = 0.785 µm than pumping at _p 1.8 µm.     

A classical theory of hard squares

A simple phenomenological theory of the hard-square lattice gas is obtained by analyzing a low-order corner transfer matrix variational approximation. The free energy is of Landau type and expressions are obtained for the order parameter and densities. In this approximation, the model exhibits a critical point atz _c =4(3 + 23)/9 with critical exponents given by the classical values: =0_disc,=1/2, =1, =3.     

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.     

Characteristic Properties of the Scattering Data for the mKdV Equation on the Half-Line

In this paper we describe characteristic properties of the scattering data of the compatible eigenvalue problem for the pair of differential equations related to the modified Korteweg-de Vries (mKdV) equation whose solution is defined in some half-strip or in the quarter plane (0<x<)×[0,T), T. We suppose that this solution has a C initial function vanishing as x, and C boundary values, vanishing as t when T=. We study the corresponding scattering problem for the compatible Zakharov-Shabat system of differential equations associated with the mKdV equation and obtain a representation of the solution of the mKdV equation through Marchenko integral equations of the inverse scattering method. The kernel of these equations is valid only for x0 and it takes into account all specific properties of the pair of compatible differential equations in the chosen half-strip or in the quarter plane. The main result of the paper is the collection A–B–C of characteristic properties of the scattering functions given below.     

Characterizations of the Kerr metric

For stationary, asymptotically flat solutions of Einstein's equations, covariant functionals of the metric variables are defined which characterize the Kerr metric uniquely. For instance, we obtain a generalization of the Bach tensor to stationary metrics, which vanishes if and only if the solution is Kerr. We also give a new interpretation of the Schwarzschild-to-Kerr-transformation. Our results might be applicable to simplify the proof of the uniqueness theorem for stationary black holes.     

Determination of the Lie groups of equations of motion. I

A practical method is described for determining the symmetry groups of the differential equations of physics via a discussion of all possible operators in space of coordinates {x} and derivatives {/x}. The Lie-algebra operators on the solutions are sought. The problem is solved by successively complicating the form of the operators.     

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 .     

Relationship between (2 + 1) and (3 + 1)-Friedmann–Robertson–Walker cosmologies; linear,non-linear,and polytropic state equations

It is shown that Friedmann–Robertson–Walker (FRW) cosmological models coupled to a single scalar field and to a perfect fluid fitting a wide class of matter perfect fluid state equations, determined in (3+1) dimensional gravity can be related to their (2+1) cosmological counterparts, and vice-versa, by using simple algebraic rules relating gravitational constants, state parameters, perfect fluid and scalar field characteristics. It should be pointed out that the demonstration of these relations for the scalar fields and potentials does not require the fulfilment of any state equation for the scalar field energy density and pressure. As far as to the perfect fluid is concerned, one has to demand the fulfilment of state equations of the form p+ = f(). If the considered cosmologies contain the inflation field alone, then any (3+1) scalar field cosmology possesses a (2+1) counterpart, and vice-versa. Various families of solutions are derived, and we exhibited their correspondence; for instance, solutions for pure matter perfect fluids and single scalar field fulfilling linear state equations, solutions for scalar fields coupled to matter perfect fluids, a general class of solutions for scalar fields subjected to a state equation of the form p + = are reported, in particular Barrow–Saich, and Barrow–Burd–Lancaster–Madsen solutions are exhibited explicitly, and finally perfect fluid solutions for polytropic state equations are given.     

Effect of the Λ n interaction on the cross section of the reaction γd → K+Λ n

Taking into account the n interaction, a nonrelativistic graphical technique is used to calculate the differential cross section of the reaction d K⁺ n as a function of photon energy, K⁺-momentum, and angle of emission of the K⁺. The kinematic region in which variation of the n-scattering parameters gives the most noticeable effect is separated out.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 104–106, August, 1978.The author thanks G. M. Radutskii and V. A. Filimonov for many useful discussions.     

Gravitation and universal Fermi coupling in general relativity

The generally covariant Lagrangian densityG = + 2K _matter of the Hamiltonian principle in general relativity, formulated by Einstein and Hilbert, can be interpreted as a functional of the potentialsg _ikand of the gravitational and matter fields. In this general relativistic interpretation, the Riemann-Christoffel form _kl ⁱ = _kl ⁱ for the coefficients _kl ⁱ of the affine connections is postulated a priori. Alternatively, we can interpret the LagrangianG as a functional of , g_ik, and the coefficients _kl ⁱ . Then the _kl ⁱ are determined by the Palatini equations. From these equations and from the symmetry _kl ⁱ = _lk ⁱ for all matter fields with /=0 the Christoffel symbols again result. However, for Dirac's bispinor fields, / becomes dependent on the Dirac current, essentially with a coupling factor Khc. In this case, the Palatini equations define a new transport rule for the spinor fields, according to which a second universal interaction results for the Dirac spinors, besides Einstein's gravitation. The generally covariant Dirac wave equations become the general relativistic nonlinear Heisenberg wave equations, and the second universal interaction is given by a Fermi-like interaction term of the V-A type. The geometrically induced Fermi constant is, however, very small and of the order 10^–81erg cm³     

Renormalization group equations with several length scales and crossover scaling functions for axial anisotropy

The critical behaviour of axially anisotropicn-vector models is characterized by two distinct length scales, the correlation lengths and for the easy and hard axes. In order to handle the full range of anisotropics from to partial differential renormalization group equations are derived, depending on and . The anisotropicX-Y model is studied in detail near four dimensions. The crossover scaling functions for the susceptibilities are calculated to first order in=4–d. Two distinct crossover regions are found for weak and dominant anisotropy, respectively.     

Correlation functionals of infinite volume quantum spin systems

The existence and analyticity of the correlation functionals of a quantum lattice in the infinite volume limit is proved. The result is valid at sufficiently high temperatures and for a large class of interactions. Our method estimates the kernelK for a set of Kirkwood-Salzburg equations. While a naive estimate would indicate that K =, we take into account cancellations between different contributions toK in order to show that for sufficiently high temperatures K <1, and this estimate is independent of the volume of the system.Supported in part by the Air Force Office of Scientific Research.     

Periodic and solitary wave solutions of the three-wave problem. A different approach

We consider the inverse problem for a three-wave interaction system in a manner different from Zakharovet al. and of Kaup. Our method is an adaptation of the technique due to Date to a 3 × 3 Lax pair. The analysis leads to a system of ordinary nonlinear equations for the _ivariables linearizable through a suitable definition of differential on a Riemann surface. Next, in the degenerate case, when the _iare equal in pairs, we prove that such a set of equations is exactly integrable and leads to solitary solutions.