The Statistics of the Trajectory of a Certain Billiard in a Flat Two-Torus

We consider a billiard in the punctured torus obtained by removing a small disk of radius >0 from the flat torus ², with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity only. Let (), and respectively R(), denote the first exit time (length of the trajectory), and respectively the number of collisions with the side cushions when ² is being identified with [0,1)². We prove that the probability measures on [0, ) associated with the random variables and R are weakly convergent as 0⁺ and explicitly compute the densities of the limits. Research partially supported by ANSTI grant C6189/2000.     

On the Surface Pressure for the Edwards-Anderson Model

For the Edwards-Anderson model we introduce an integral representation for the surface pressure (per unit surface) in terms of a quenched moment of the bond-overlap on the surface. We consider free , periodic and antiperiodic ^* boundary conditions (by symmetry ⁽⁾=^(*)), and prove the bounds We show moreover that at high temperatures ⁽⁾ is close to ^2/4 and ⁽⁾ is close to ^2/4 uniformly in the volume .     

R andRτ ratios at the five-loop level of perturbative QCD

We study perturbative QCD at the five-loop level. In particular we considerR = _tot(e ⁺ e ^– hadrons)/(e ⁺ e ^{– + –}) andR = ( v+hadrons)/( ev). We use our method to estimate the five-loop coefficients. As a result, we obtain _s (M _z ) = 0.1186(11) and _s (34 GeV) = 0.1396(16), which are accurate at the 1% level. We also findR = 3.8350(18), which is consistent withR and is accurate to 0.05%.     

Decay of Re186

The decay of Re 186 was investigated with a double focusing spectrometer and scintillation spectrometer. Four transitions were observed with energies of 122·7±0·1 keV, 137·2 keV, 632·2±1·5 keV and 768·2±1·5 keV and the relative-ray intensities 18, 246, 0·9 and 1 respectively. TheK-conversion coefficients determined for low-energy transitions are in good agreement with the theoretical values ( _K ^122·7 =0·53±0·05, _K ^137·2 =0·44±0·02). The relative intensities of theK, L, M andN conversion lines were also determined for these transitions:KL _I +IIL _III M=1·20±0·201·32±0·20 1±0·150·57±0·08 for 122·7 keV transition,KL _I +IIL _III MN=1·57±0·081· 72±0080·70±0·030·20±0·01 for 137·2 keV transition. In the beta spectrum two groups were observed with the end points of 1076±3 keV and 939±3 keV and relative intensities 74±4% and 21±4% respectively. For the branching ofK-capture it was found that 1·5% populates the 122·7 keV level and 3·5% the ground state. The decay scheme from the present results is proposed.The authors would like to thank M. Buriánek and V. Kopiva for help in the measurements.     

Four-wave mixing in CO via the C 1Σ+ (υ′=0) state

Coherent vacuum ultraviolet (VUV) radiation was generated by four-wave difference frequency mixing (_VUV=2₁–₂) of pulsed dye laser radiation in carbon monoxide (CO). The frequency ₁ was tuned to the C ¹⁺(=0)X ¹⁺(=0) two-photon transition, while the dye laser frequency ₂ was scaned around 17650 cm^–1 which corresponds to the A ¹(=7)«C ¹⁺(=0) transition energy. The VUV intensity was found to be strongly wavelength dependent. The analysis of the spectrum revealed (i) that the VUV intensity was enhanced by the rotational levels of the A ¹(=7) state and (ii) that the off-resonance excitation in the C ¹⁺(=0)X ¹⁺(=0) two-photon transition greatly contributed to the present four-wave mixing process. The effects of pumping laser detuning, saturation and foreign gases are briefly discussed.     

Massive spin-two particle in a gravitational field

The spin-two particle is described by a symmetric tensorh subject to the subsidiary conditionsh = h =0. Their covariant generalization and the wave equation have been obtained directly from the Eulerian variational equations by algebraic methods only. In addition to the tensor fieldh a symmetric third-rank tensor = as well as a vector fieldA have been added, neither of which enter in the final result. The Lagrangian function is taken as a linear sum of all combinations which can be constructed from these functions, as well as terms involving the curvature tensor and its two possible contractions. Variation with respect toh , andA independently gives the Euler equations. Combining the various trace equations and choice of arbitrary constants yields the subsidiary conditions, while the Euler equations themselves give the connection between the auxiliary functions and the tensorh as well as the generalization of the wave equationD D h + 2R h -R h -R h +g R h +Rh =m ² h Finally, variation with respect tog yields the energy-momentum tensor.     

Structural Transformations in a Bi2O3 Crystal and in Bi2O3‐Based Solid Solutions in the Temperature Interval 25–750°C

The Raman scattering spectra of isostructural Bi₂O₃ and Bi_1.8Tm_0.2O₃ in the course of heating have been investigated. It is shown that the sequences of structural changes with increase in temperature differ: and ^* , respectively. In the hightemperature region, the structure takes the form of a disordered cube irrespective of the previous history of specimens.     

Weakly Bound States in Light Hypernuclei

From the viewpoint of critical stability, we discuss the three- and four-body structure of ⁶He, ⁷He, ⁴He, and ⁴H. With the + + N three-body model, ⁶He is found to have a three-layer structure of the matter distribution: core, a skin and neutron halo. Also the level structure of ⁷He with the three-body model of ⁵He + n + n is predicted. This stimulates a new study of neutron-rich and proton-rich hypernuclei. By performing a four-body calculation with both NNN and NNN channels and with both NN and NNN channels, we show that the N-N and -N couplings are very important in critical stability of few-body hypernuclear systems.     

Macroscopic short-time electrodynamics for conductors

In the macroscopic electrodynamics (MED) of good conductors (metals) based on Ohm's law j=E, the momentum relaxation time =m/ne² of the electrons limits the application to electromagnetic (EM) processes with characteristic timest. An interesting physical difficulty occurs in MED since the EM field damping time _R=/ of metals is very small compared with the minimum macroscopic time scale, _R. Consequently, the damping and propagation of EM waves and pulses in good conductors cannot be correctly described within the frame of conventional MED. New hyperbolic EM wave equations with relaxation and memory are proposed, which no longer exhibit the _R deficiency. The latter is caused by Ohm's law, which breaks down for short-time processes, due to neglect of electron inertia. The advantages of the proposed and the disadvantages of the conventional EM wave equations for good conductors are discussed in applications.     

Spectral superbroadening of self-focused picosecond laser pulses in D2O

Intense picosecond light pulses of a mode-locked Nd: glass laser at _L =1054 nm (fundamental wavelength) and _SH =527 nm (second harmonic wavelength) are passed through a sample of D₂O under self-focusing conditions. Spectrally structured superbroadened, spatially bell-shaped emission in the forward direction is obtained. Primary generation processes are pump-pulse-degenerate stimulated parametric four-photon interaction (₁ + _{1 3} + ₄) and stimulated Raman scattering (_{1 R} + ), which occur concurrently (₁= _L or _SH angular pump frequency, #x03C9; _R first Stokes frequency, #x03C9; signal frequency, #x03C9;₃ signal frequency, #x03C9;₄ idler frequency). The parametric four-photon interaction occurs under collinear non-phase-matched conditions and under longitudinally phase-matched, transversally non-phase-matched (erenkov-like) conditions. Subsequent interaction processes are pump-pulse-nondegenerate four-photon interaction of the type ₁ + _{R 3} + ₄, coherent antiStokes Raman scattering (CARS, ₁ + ₁ – _{4 3}), inverse Raman scattering ( _A + ₁ + ), and cascading light up-conversion of the type ₁ + _(i) – _{R (i+1)}.     

The form factors of the decays π− → e−¯veγ and K− → e−¯veγ

In the special type of the quark model we obtain the ratio=h _A/h_V of the axial (h_A) and vector (h_V) form factors for the decays ^– e^– ¯v_e and K^– e^–¯v_e different from unity. The low-energy theorem, relating the electric polarizability of the charged pion with the ratio, is analyzed. It is shown that < 1 corresponds to , calculated by accounting the contribution of the scalar meson(700) into the amplitude of the Compton effect on the pion. In the absence of the(700) contribution we have=1.Dedicated to Academician Václav Votruba on the occasion of his eightieth birthday.     

Equivalence of vacuum Yang-Mills gravitation and vacuum Einstein gravitation

In the Yang-Mills formulation of gravitational dynamics based uponSL(2,C) spin transformations acting on Dirac spinors, the vacuum field equations are R +C R = 0 and and . HereR is the Ricci curvature andC is the Weyl conformal curvature; is a coupling constant. We show the equivalence between solutions of these equations and the vacuum Einstein equationsR = 0. The proof uses the Newman-Penrose formalism.Supported by a NATO fellowship.Supported by a SRC fellowship.     

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.     

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.     

A resistometric investigation of the decomposition kinetics of Al-21·8 at.% Zn solid solution at 276 °C

Under the condition of nearly equilibrium concentration of vacancies, time dependence of the amount of isothermal transformation given byy=R/R _f was investigated whereR _f is the total structural change of resistivity on completion of the whole process andR is the measured resistivity change. The investigation was done on the 21·8 at.% (40·3wt.%) Zn alloy under the condition of relatively low supersaturation of a few degrees centigrade below the metastable _R solvus line. The total transformation involves four kinetic stages: the first two stages correspond probably to diffusion-controlled growth of the _R particles from the supersaturated solid solution and to the ripening of these particles till their conversion to the cubic phase takes place. The last two kinetic stages account analogously for the particles growth and ripening. Both _R and phases were identified by the transmission electron microscopy. When separating the individual stages, the approximation byy=1–exp [–(mt) ⁿ] of the amount of transformationy was used. The approximation allowed to get the starting values of both the time and the change of the structural part of the electrical resistance for individual stages and also to derive the parametersm _i, n_i which had to be redetermined for the individual separated stages. These data made it possible to synthetize the experimental curves ofR andy vs. time for the total transformation.It is a pleasure to thank Doc. Dr. V.Syneek CSc. for stimulating the author's interest in this problem and for providing helpful discussions. I also would like to express my thanks to Ing. P.Bartuka CSc. for the transmission electron microscopic study carried out to identify the particular phases. The author is indebted to Ing.V. íma for the preparation of the investigated alloy and to Mrs. A.Mendlová and Mr. P.Vyhlídka for technical assistance.     

Critical (Φ4)3,ε

The Euclidean (⁴)_3, model in R³ corresponds to a perturbation by a ⁴ interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter in the range 01. For =1 one recovers the covariance of a massless scalar field in R³. For =0, ⁴ is a marginal interaction. For 0<1 the covariance continues to be Osterwalder-Schrader and pointwise positive. We consider the infinite volume critical theory with a fixed ultraviolet cutoff at the unit length scale and we prove that for >0, sufficiently small, there exists a non-gaussian fixed point (with one unstable direction) of the Renormalization Group iterations. We construct the stable critical manifold near this fixed point and prove that under Renormalization Group iterations the critical theories converge to the fixed point. Partially supported by NSERC of CanadaLaboratoire Associé au CNRS. UMR 5825Partially supported by CNR, G.N.F.M. and MURST     

Precipitation sequences and transformation kinetics in the Al-30 wt.% Zn alloy aged below 160°C

The transformation sequences in the Al-30 wt.% Zn alloy aged below 160 °C have been investigated by X-ray diffraction and transmission electron microscopy. The sequence G.-P. zones _R has been observed between 85 and 161 °C, whereas the direct G.-P. zones to-transformation occurs below 85 °C. The identical isothermal growth kinetics of both the G.-P. zones and the _R -precipitates was found. The log-log plots yielding a set of parallel straight lines from 50 to 150 °C indicate no change in the activation energy of growth of both these precipitates. The change in the precipitation sequence led to the change in the transformation kinetics. The T-T-T curves have been measured for the formation of metastable _R -phase and for the completion of ellipsoidal G.-P. zones or of _R to-transformation. The retardation in reaching equilibrium found below 85 °C was associated with the observed change in the transformation sequence. This was explained by the different accommodation of the misfit at the interfaces between G.-P. zones or _R -precipitates and the-matrix.The authors are much indebted to Doc. Dr. V.Syneek, CSc. for stimulating discussions. Our thanks are also due to Ing. V.íma and Mr. P.Vyhlídka for the preparation and careful chemical analyses of the investigated samples. The assistance of Mr. Z.iký and Mr. V.Petr in X-ray diffraction measurements and the heat-treatment of samples is appreciated.     

Field of View Increase for Optical Heterodyne Receivers

The field of view of optical heterodyne systems are usually limited by a simple criterion- namely, that A_RR ², where A_R is an effective aperture, _R the corresponding field of view and the optical wavelength. This paper will demonstrate that with a simple change of systems geometry and the possible use of array detectors, this severe restriction can be alleviated considerably and, in some cases, eliminated. Both coherent sources and thermal radiation sources are considered. This might suggest that heterodyne systems that were eliminated in the past because of that criterion alone, could turn out to be, not only possible, but even advantageous compared to other techniques.     

Differential calculi on quantum vector spaces with Hecke-type relations

From a vector spaceV equipped with a Yang-Baxter operatorR one may form the r-symmetric algebraS _R V=TV/v w–R(v w), which is a quantum vector space in the sense of Manin, and the associated quantum matrix algebraM _R V=T(End(V))/f g–R(f g)R ^-1. In the case whenR satisfies a Hecke-type identityR ²=(1–q)R+q, we construct a differential calculus _R V forS _R V which agrees with that constructed by Pusz, Woronowicz, Wess, and Zumino whenR is essentially theR-matrix of GL _q (n). Elements of _R V may be regarded as differential forms on the quantum vector spaceS _R V. We show that _R V isM _R V-covariant in the sense that there is a coaction _*: _R V M _R V _R V with _*d=(1 d)_* extending the natural coaction :S _R V M _R V S _R V.     

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.