Some twisted self-dual solutions for the Yang-Mills equations on a hypertorus

TheSU(N) Yang-Mills equations are considered in a four-dimensional Euclidean box with periodic boundary conditions (hypertorus). Gauge-invariant twists can be introduced in these boundary conditions, to be labeled with integersn (= –n ), defined moduloN. The Pontryagin number in this space is often fractional. Whenever this number is zero there are solutions to the equationsG =0 HereG is the covariant curl. When this number is not zero we find a set of solutions to the equations , provided that the periodsa of the box satisfy certain relations.Work supported in part by the US Department of Energy under Contract No. DE-AC-03-76ER 00068 and by the Fairchild FoundationOn leave from the Institute for Theoretical Physics, University of Utrecht, P.O. Box 80.006, NL-3508 TA Utrecht, The Netherlands     

Band-gap structure of the spectrum of periodic Maxwell operators

We investigate the band-gap structure of some second-order differential operators associated with the propagation of waves in periodic two-component media. Particularly, the operator associated with the Maxwell equations with position-dependent dielectric constant (x),xR ³, is considered. The medium is assumed to consist of two components: the background, where (x) = _b , and the embedded component composed of periodically positioned disjoint cubes, where (x) = _a . We show that the spectrum of the relevant operator has gaps provided some reasonable conditions are imposed on the parameters of the medium. Particularly, we show that one can open up at least one gap in the spectrum at any preassigned point provided that the size of cubesL, the distancel=L betwen them, and the contrast = _b / _a are chosen in such a way thatL ^–2, and quantities ^-1-3/2 and ² are small enough. If these conditions are satisfied, the spectrum is located in a vicinity of widthw(^3/2)^-1 of the set {² L ^-2 k ²:kZ³}. This means, in particular, that any finite number of gaps between the elements of this discrete set can be opened simultaneously, and the corresponding bands of the spectrum can be made arbitrarily narrow. The method developed shows that if the embedded component consists of periodically positioned balls or other domains which cannot pack the space without overlapping, one should expect pseudogaps rather than real gaps in the spectrum.     

Semi-infinite Potts model and percolation at surfaces

The effects of surfaces on percolation are investigated near the bulk percolation threshold ind=6– dimensions. Using field-theoretic methods, this is done within the framework of a semi-infinite continuousq-state Potts model withq1. Renormalization-group equations are obtained which imply that the usual scaling laws for surface and bulk exponents are valid to all orders in , and the surface exponents at the ordinary and special transition are computed to order . Our result for ₁ ^ord is in conformity with the one by Carton.     

A derivation of the Broadwell equation

We consider a stochastic system of particles in a two dimensional lattice and prove that, under a suitable limit (i.e.N, 0,N²const, whereN is the number of particles and is the mesh of the lattice) the one-particle distribution function converges to a solution of the two-dimensional Broadwell equation for all times for which the solution (of this equation) exists. Propagation of chaos is also proven.Research partially supported by CNR-PS-MMAIT     

Cooperative plasticity due to the motion of disorientation and phase separation boundaries

Conclusions As has already been noted above, the theory of planar defects organically includes the mechanics of twinning, grain boundaries, Somigliani dislocations, translational dislocations, disclination, and dispiration. The fundamental propositions of the theory and methods of giving the tensor T are listed in Table 4. The mathematical formalism remains the same throughout, and it is applicable to both discrete objects (it is then necessary to conserve the -function apparatus), and to a continuous (then appropriate smoothing is needed, which usually reduces to replacement of the multiplication procedure by the normal n or by the direction , to operations of finding the gradient, divergence, and curl of regular expressions, and discarding the -functional), In particular, the problem of thermoelasticity is formulated successfully by such a method in the terminology of the present theory.In a broad sence of the word, the development of the theory should be perceived as an extension of the concept of imperfection to defects of sufficiently arbitrary origin. A completely developed formalism was worked out earlier for just linear defects; in the symbols used here, for the case b=b₀ + X (r – r₀) for constant b₀,, and r₀, and without taking account of processes on the boundary S if the linear defect contained such a feature. Let us emphasize that to describe three-dimensional defects occurring because of homogeneous distortion = (V)., it is sufficient to use the apparatus of just the theory of planar defects since the fundamental phenomena are associated with precisely the presence of boundaries and in a formal plane, with the spatial derivatives of , they are always expressed in terms of the functional (S), while in the case of finite surface gradients in terms of (L). The time derivatives of the distortion T, i.e., is written down in the developed representations in terms of the form with all the resulting consequences.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 83–102, June, 1981.     

Correlation tensors of the blackbody radiation in rectangular cavities

Electromagnetic equilibrium fluctuations in finite cavities filled with a dissipative medium (dielectric function ()=+i) and bounded by walls of infinite conductivity are considered. Expanding the fields in terms of a complete and orthonormal set of functions and solving the Maxwell equations the response of the EM field to external forces (polarization and magnetization) is obtained. With the aid of the fluctuation dissipation theorem and the linear response functions the 2nd order correlation tensors of the EM field are derived.For rectangular cavities explicit considerations are made. In the case of transparent media (=0) the spectral energy density of the EM radiation is calculated.     

A generalized Kubo-Toyabe formula for muon spin relaxation in crystals with uniaxial symmetry

The Kubo-Toyabe semiclassical formula, describing the time development of the polarization of a particle in zero external field at a lattice site with cubic local environment, is generalized for uniaxial site symmetry. The relaxation function and, in particular, its first moments and long time asymptotics obtained in a closed form depend on the angle between polarization and the crystalc-axis and are shown to vary sensitively with the asymmetry of the field distribution at the particular muon site. Besides the exact uniaxial variant of the Kubo-Toyabe relaxation function, an approximate simple interpolation formula is also derived, which is correct for both short times and in its long time asymptotics. The two parameters (, ₁) in the uniaxial formulae can be determined by using the observed values of the second momentM ₂ for two different crystal orientations.     

Honeycomb domain structure

A theory is given of a new structure produced when demagnetizing uniaxial single crystals (e.g. magnetoplumbite) in a field normal to thec axis. The experimental results fully support the theory on the assumption that domain structures produced under normal conditions are metastable. Methods are given by which stable structures can be prepared from such metastable ones. On the basis of these results opinions are expressed as to the nucleation of plate and honeycomb structures, produced either by reducing the field from saturation or by cooling below the Curie point.

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, (, ) , c. , , , , , . , . , .

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In conclusion the authors would like to thank C. Novák from the Institute of Technical Physics, Czechoslovak Academy of Sciences, for checking the calculation of the harmonic analysis and Z. Málek and J. Eousek for carefully reading the paper and for valuable remarks.     

Double quantum transitions in paramagnetic resonance

The theory of double quantum transitions of the M=±2 type, with regard to inhomogeneously broadened spin systems is studied in this paper with the approximation ²T₂T₃ 1. We suppose that the inhomogeneous broadening is formed by an inhomogeneous crystal field. The obtained results describe the magnitude of absorption as a function of the h.f. power and also describe the shape of the absorption curve. It is demonstrated that in inhomogeneously broadened spin systems the absorption curve of double quantum transitions has the form of the difference of two different Lorentz's curves and that at the saturation ( ²T₂T₁ 1) the absorption increases with the cube of the h.f. field intensity. The shape of the curves is expressed by means of phenomenological relaxation constants of the system.

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M=±2 ²T₂T₃ 1. , - . . , ( ²T₂T₁ 1) . .

     

Новый метод измерени я магнитокалорическ ого эффекта у ферромагни тных полупроводнико в

, .     

Electron velocity and momentum density

A null 4-vector ^°, based on Dirac's relativistic electron equation, is shown explicitly for a plane wave and various Coulomb states. This 4-vector constitutes a mechanical model for the electron in those states, and expresses the important spinor quantities represented conventionally byn, f, g, m, j, ,1, ands. The model for a plane wave agrees precisely with the relation between velocity and phase gradient customarily used in quantum theory, but the models for Coulomb states contradict that relation.     

On the probability of double beta decay

Double beta decay is discussed in relation to parity non-conservation. Two possible ways of neutrino-less double beta decay (allowed and forbidden) are investigated and the half-life of decay is calculated. For allowed transitions we obtain for Ca⁴⁸ an estimatedT1/2=2×10¹⁹ years. The negative results of the experiments by Lukjanov et al., who give the valueT1/2=0.7×10¹⁹ years for the lower limit of the half-life of double beta decay of Ca⁴⁸, cannot therefore be regarded as a definitive solution of the question, whether the neutrino is a Dirac or Majorana particle. Further study of double beta decay, aimed at finding higher values of the lower limit of half-life, are of considerable importance for theory.

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- . - ( ) . ⁴⁸ T1/2210¹⁹ . - . [1] ( - ⁴⁸ T1/2 0,710¹⁹ ), , . - .

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In conclusion the author thanks Prof. I. S. apiro for suggesting this work and help in elaborating it.     

A Rigorous Derivation of a Linear Kinetic Equation of Fokker–Planck Type in the Limit of Grazing Collisions

We rigorously derive a linear kinetic equation of Fokker–Planck type for a 2-D Lorentz gas in which the obstacles are randomly distributed. Each obstacle of the Lorentz gas generates a potential V( ), where V is a smooth radially symmetric function with compact support, and >0. The density of obstacles diverges as ^– , where >0. We prove that when 0< <1/8 and =2+1, the probability density of a test particle converges as 0 to a solution of our kinetic equation.     

Hermiticity of Quantum Observables Versus Commutation Relations

In order to obtain sum rules and spectral representations the Hermiticity property , A = A, of observables is used. It is shown that for certain and the property turns out to be inconsistent with the commutation relations that contain A. The known Schwinger paradox is explained by this inconsistency.     

ДАЛЬНЕЙШИЕ ИССЛЕДОВ АНИЯ ФИЛЬТРА ДВОЙНОГ О ЛУЧЕПРЕЛОМЛЕНИЯ

, 4, (1954), 53. . .     

Ising-Type and Other Transitions in One-Dimensional Coupled Map Lattices with Sign Symmetry

We consider a one-dimensional lattice of expanding antisymmetric maps [–1, 1][–1, 1] with nearest neighbor diffusive coupling. For such systems it is known that if the coupling parameter is small there is unique stationary (in time) state, which is chaotic in space-time. A disputed question is whether such systems can exhibit Ising-type phase transitions as grows beyond some critical value _c. We present results from computer experiments which give definite indication that such a transition takes place: the mean square magnetization appears to diverge as approaches some critical value, with a critical exponent around 0.9. We also study other properties of the coupled map system.     

Exponent inequalities for the bulk conductivity of a hierarchical model

The bulk conductivity ^*(p) of the bond lattice in ^d is considered, where the bonds have conductivity 1 with probabilityp or 0 with probability 1-p Various representations of the derivatives of ^*(p) are developed. These representations are used to analyze the behavior of ^*(p) for =0 near the percolation thresholdp _c , when the conducting backbone is assumed to have a hierarchical node-link-blob (NLB) structure. This model has loops on arbitrarily many length scales and contains both singly and multiply connected bonds. Exact asymptotics of for the NLB model are proven under some technical assumptions. The proof employs a novel technique whereby for the NLB model with =0 andp nearp _c is computed using perturbation theory for ^*(p) (for two- and three-component resistor lattices) aroundp=1 with a sequence of s converging to 1 as one goes deeper in the hierarchy. These asymptotics establish convexity of ^*(p) (for the NLB model) nearp _c , and that its critical exponentt obeys the inequalities 1t2 ford=2,3, while 2t3 ford4. The upper boundt=2 ind=3, which is realizable in the NLB class, virtually coincides with two very recent numerical estimates obtained from simulation and series expansion for the original model.Supported in part by NSF Grant DMS-8801673 and AFOSR Grant AFOSR-90-0203     

Absolute Continuity of Vitali–Hahn–Saks Measure Convergence Theorems

In this paper, we prove the following improved Vitali–Hahn–Saks measure convergence theorem: Let (L, 0, 1) be a Boolean algebra with the sequential completeness property, (G, ) be an Abelian topological group, be a nonnegative finitely additive measure defined on L, {_n: n N} be a sequence of finitely additive s-bounded G-valued measures defined on L, too. If for each a L, {_n(a)}_{n N} is a -convergent sequence, for each nN, when { (a)} convergent to 0, {_n(a)} is -convergent, then when { (a)} convergent to 0, {_n(a)} are -convergent uniformly with respect to nN     

s-Polarized nonlinear surface polaritons

An exact solution of Maxwell's equations is found, corresponding to ans-polarized nonlinear surface polarition, at the planar interface between two dielectric media, one of which is optically unaxial and is characterized by a diagonal dielectric tensor whose elements depend on the amplitude of the electric field according to ₁₁=₂₂=₀()+a() (|E ₁|²+|E ₂|²), ₃₃=(). Such modes have no counterpart in the corresponding linear system.     

Exact expressions for row correlation functions in the isotropicd=2 ising model

We present exact explicit expressions for the row spin-spin correlation functions _{00 n}0 in the isotropicd= 2 Ising model, in terms of elliptic integrals, forn 5. We also give a general structural formula for _{00 n}0.