Nontrivial extensions of a representation of the Poincaré group with mass and helicity zero by its tensorial product

Let U(a, ) be a representation of the Poincaré group with mass and helicity zero, realized in the space of C -functions with compact support on ³, without the origin. Let U ⁽²⁾(a, ) denote the tensorial product of U(a, ) by itself. We explicitly determine the cocycles of extension of U(a, ) by U ⁽²⁾(a, ) and we prove that the nontrivial cohomology is indexed by (u(),),u()D ¹ _]0,3\,.     

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.     

Analysis of the order parameter for uniform nearest particle system

The uniform nearest particle system (UNPS) is studied, which is a continuoustime Markov process with state space . The rigorous upper bound ^(mf) = ( – 1)/ for the order parameter ₂, is given by the correlation identity and the FKG inequality. Then an improvement of this bound ^(mf) is shown in a similar fashion; C( – 1)/|log( – 1) for >1. Recently, Mountford proved that the critical value _c=1. Combining his result and our improved bound implies that if the critical exponent exists, it is strictly greater than the mean-field value 1 in the weak sense.     

Kac-Moody symmetry of generalized non-linear Schrödinger equations

The classical non-linear Schrödinger equation associated with a symmetric Lie algebra =km is known to possess a class of conserved quantities which from a realization of the algebrak []. The construction is now extended to provide a realization of the Kac-Moody algebrak[, ^–1] (with central extension). One can then define auxiliary quantities to obtain the full algebra [, ^–1]. This leads to the formal linearization of the system.     

Baryon decays and gamma-spectroscopy of hypernuclei

The applicability of (K ^–,^–) - and (K ^–,^– N)-reactions on^13,14C and^14,15N nuclei to the study of -transitions in primary and daughter -hypernuclei is discussed. The intensity of -deexcitation of ¹³C state |S ¹²C(15·11 MeV; 1⁺1): 1/2⁺ has been shown to be comparable with the intensity of baryon decay. Isospin selection rules are used to distinguish excitation energy ranges of primary hypernuclei, where the identification of the secondary -lines is probable.Presented at the symposium Mesons and Light Nuclei, Bechyn, Czechoslovakia, May 27–June 1, 1985.     

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.     

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.     

On the equivalence of boundary conditions

We show that ifb andb are two boundary conditions (b.c.) for general spin systems on ^d such that the difference in the energies of a spin configuration in ^d is uniformly bounded, |H _,b ()–H _,b()|C < , then any infinite-volume Gibbs states and obtained with these b.c. have the same measure-zero sets. This implies that the decompositions of and into extremal Gibbs states are equivalent (mutually absolutely continuous). In particular, if is extremal,=. Application of this observation yields in an easy way (among other things) (a) the uniqueness of the Gibbs states for one-dimensional systems with forces that are not too long-range; (b) the fact that various b.c. that are natural candidates for producing non-translation-invariant Gibbs states cannot lead to such an extremal Gibbs state in two dimensions.Supported in part by NSF Grant PHY 78–15920 and by the Swiss National Foundation For Scientific Research.     

ФИЗИЧЕСКИЕ ОСНОВЫ ИЗ МЕРЕНИЯ НАПРАВЛЕНИЯ ВОЗДУШНОГО ПОТОКА МЕ ТОДОМ НАКАЛЕННОЙ ПРО ВОЛОКИ ПРИ ВЫСОКНХ ПОДЗВУКО ВЫХ СКОРОСТЯХ

d - l - t _d R _max - J , ( ) - - , . . - ( ) - - R R _max —R. ,      

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.     

Linking numbers of closed manifolds at random in ℝ n,inductances and contacts

We give here new results of topology and integral geometry concerning the Gauss linking number I of closed manifolds inn-dimensional space. The rigid manifolds have arbitrary shapes and dimensions, and are statistically at random positions in ⁿ . Generalizing Pohl's work, for two closed manifoldsC ₁ ^r ,C ₂ ^s , of respective dimensionsr ands, with 0rn–1, andr+s+1=n, we consider the kinematic linking integralI=<I ²(x,O)d ⁿ x>, of the square linking number I ofC ₁ ^r andC ₂ ^s , over the group of Euclidean motions of one manifold (translationsx, rotationsO). Introducing a new tensorial method, and using group theory, we show quite generally thatI=num. fact. , where is a length variable and whereA , (=1, 2) are characteristic functions associated with the manifoldC only. We study functionsA and of a manifoldC ^r , of dimensionr, in all cases 0rn–1.A always exists.A(0) givesC's area, whereas equals the interior volume of a hypersurfaceC. is found to exist and not to vanish only if 2 dimC+1=n andn=3+4q=3, 7, 11 ...A and are explicitly calculated for segments andr-spheresS ^r . As an application the topological excluded volume of a gas of nonlinked spheresS ^r moving in ^2r+1 is calculated. We generalize toN manifoldsC , =1, ...,N, linked successively to each other and forming a ring. The cyclic product of their linking numbers is integrated over the group of motions of the manifolds. It is shown to factorize completely in Fourier space, with special algebraic rules, over the set of 2N characteristic functionsA , , associated with theC 's. The same algebra of characteristic functions is shown to describe a larger class of topology and electromagnetism properties: a new theorem is given for a family of Euclidean group integrals involving the random linking numbers, mutual inductances and contact distributions ofN manifolds.     

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.     

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.     

Experimental survey of the sticking problem in muon catalyzed dt fusion

The sticking process dt + n, which constitutes the most severe limit to the number of fusions which a muon can catalyze, is reviewed. Many attempts were made to determine by calculations and measurements the probability for initial sticking _s ⁰ (immediately after dt fusion) and for final sticking _s (after the came to rest). Previous results based on neutron disappearance rates and on the observation of -X-rays were controversial and also in some disagreement with theory. New data are reported from PSI on direct observation of final sticking, using a setup with the St. Petersburg ionization chamber. These data mark a significant improvement in reliability and may clarify questions concerning previous discrepancies. The new results is _s(0.56±0.04)%, lower than the theory prediction _s=(0.65±0.03)%, at medium density.     

Decay of Nd140→Pr140→Ce140

An intermediate image magnetic spectrometer was used to study the continuous ⁺-spectrum of Pr¹⁴⁰ and the Auger electrons of the decay chain Nd¹⁴⁰Pr¹⁴⁰Ce¹⁴⁰. The maximum energy of the continuous ⁺-spectrum is 2318±10 keV. The ratio of the intensities of the different Auger groupsK–LLK–LXK–MX=1 (0.58±0.05) (0.09±0.02). The ratio of the number of Auger electrons of the K-series to the electrons of the continuous spectrume _K/ ⁺ is 0.292.

---

Nd¹⁴⁰Pr¹⁴⁰Ce¹⁴⁰

Pr¹⁴⁰ Nd¹⁴⁰ Pr¹⁴⁰¹⁴⁰. ⁺- — 2318±10 keV. -:K–LLK–LXK–MX=1(0,58±0,05) (0,09±0,02). K e _K/ ⁺=0,292.

---

---

We should like to thank Z. Macháek and Z. Dubcová for help in the measurements and J. Vlachý, graduate of the Faculty of Technical and Nuclear Physics, for help in the measurements and elaboration of the experimental results.     

Inertial effects on the escape rate of a particle driven by colored noise: An instanton approach

A recent calculation, in the weak-noise limit, of the rate of escape of a particle over a one-dimensional potential barrier is extended by including an inertial term in the Langevin equation. Specifically, we consider a system described by the Langevin equation , where is a Gaussian colored noise with mean zero and correlator (t)(t')=(D/)exp(–|t–t'|/). A pathintegral formulation is augmented by a steepest descent calculation valid in the weak-noise (D0) limit. This yields an escape rateexp(–S/D), where the actionS is the minimum, over paths characterizing escape over the barrier, of a generalized Onsager-Machlup functional, the extremal path being an instanton of the theory. The extremal actionS is calculated analytically for smallm and for general potentials, and numerical results forS are displayed for various ranges ofm and for the typical case of the quartic potentialV(x)=–x ₂/2+x ₄/4.     

Feynman integral and complex classical trajectories

Let _t be an analytic solution of the Schrödinger equation with the initial condition . Let _t be the solution of the Schrödinger equation with the initial condition =, where is an analytic function. When 0, then _t (x) _t (x)¹ ( _t (x)), where _t (x) trajectory starting from x. We relate this result to Feynman's sum over trajectories and complex stochastic differential equations.     

Anisotropic fiber with cylindrical polar axes

The newly proposed anisotropic fiber structures with cylindrical polar principal axes appear to be an interesting novel class of special lightguides. In this paper, some interesting results relating to such fibers are derived which, to the knowledge of this author, have not yet been reported in the literature. It is found that, ifn _c0 ( )^1/2>n _zc0( _r )^1/2, TE₀₁ will be the fundament mode with a range of single-mode operation given by 2.61n _zc0 (2 _r )^1/2 /a<2.61n _c0(2 )^1/2. On the other hand, ifn _zc0 ( _r )^1/2>n _c0( )^1/2, then TM₀₁ becomes the fundamental mode whose single-mode operation range is 2.61n _c0 (2 )^1/2 /a<2.61n _zc0(2 _r )^1/2.     

Un phénomène de concentration évanescente pour des flots non-stationnaires incompressibles en dimension deux

We consider a sequencev of non-stationary solutions of the incompressible 2D-Euler equation, locally bounded inL ². We prove that if the defect measure is supported in a one-dimensional set (³) of some special type (which we call finite type), the weak limitv ofv is a solution of the Euler equations: our theorem is of the type concentration-cancellation.     

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