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.     

Perturbations of flows on Banach spaces and operator algebras

For automorphism groups of operator algebras we show how properties of the difference _t – ' _t are reflected in relations between the generators , . Indeed for a von Neumann algebraM with separable predual we show that if _t – '_t 0.28 for smallt, then = 0(+)°^-1 where is an inner automorphism ofM and is a bounded derivation ofM. If the difference _t – ' _t =O(t) ast ; 0, then = + and if _t – ' _t 0.28 for allt then =. We prove analogous results for unitary groups on a Hilbert space andC ₀,C ₀ ^* groups on a Banach space.This paper subsumes an earlier work of the same title which appeared as a report from Z.I.F. der Universität BielefeldWith partial support of the U.S. National Science Foundation     

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.     

On the pointwise behavior of semi-classical measures

In this paper we concern ourselves with the small asymptotics of the inner products of the eigenfunctions of a Schrödinger-type operator with a coherent state. More precisely, let _j and E _j denote the eigenfunctions and eigenvalues of a Schrödinger-type operator H with discrete spectrum. Let _(x,) be a coherent state centered at the point (x, ) in phase space. We estimate as 0 the averages of the squares of the inner products ( _a ^(x,) , _j ) over an energy interval of size around a fixed energy, E. This follows from asymptotic expansions of the form for certain test function and Schwartz amplitudes a of the coherent state. We compute the leading coefficient in the expansion, which depends on whether the classical trajectory through (x, ) is periodic or not. In the periodic case the iterates of the trajectory contribute to the leading coefficient. We also discuss the case of the Laplacian on a compact Riemannian manifold.Research supported in part by NSF grant DMS-9303778     

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.     

Inverse Problem for Harmonic Oscillator Perturbed by Potential,Characterization

Consider the perturbed harmonic oscillator Ty=-y+x²y+q(x)y in L²(), where the real potential q belongs to the Hilbert space H={q, xq L²()}. The spectrum of T is an increasing sequence of simple eigenvalues _n(q)=1+2n+_n, n 0, such that _n 0 as n. Let _n(x,q) be the corresponding eigenfunctions. Define the norming constants _n(q)=lim_xlog |_n (x,q)/_n (-x,q)|. We show that for some real Hilbert space and some subspace Furthermore, the mapping :q(q)=({_n(q)}₀, {_n(q)}₀) is a real analytic isomorphism between H and is the set of all strictly increasing sequences s={s_n}₀ such that The proof is based on nonlinear functional analysis combined with sharp asymptotics of spectral data in the high energy limit for complex potentials. We use ideas from the analysis of the inverse problem for the operator -ypy, p L²(0,1), with Dirichlet boundary conditions on the unit interval. There is no literature about the spaces We obtain their basic properties, using their representation as spaces of analytic functions in the disk.     

Enhancement of activated decay of metastable states by resonant pumping

We consider the effect of a high-frequency pumping cost on the escape rate of a classical underdamped Brownian particle out of a deep potential well. The energy dependence of the oscillation frequency(E) is assumed to be weak on the scale of thermal energy,_E(0)T(0)T/V₀ (0)[_E(0) is the derivative of(E) atE= 0,V ₀ is the barrier height,V ₀ T]. The quadratic-in- contribution to the decay rate is calculated in two different regimes: (1) for the case of resonance of the pumping frequency with the nth harmonic of the internal motion at an energye, when = n(e); (2) for a rollout region of the basic resonance near the bottom of the potential well, when ¦-(0)¦ and is the damping coefficient. In the latter case the absorption spectrum and the enhancement of the decay rate are calculated as functions of two reduced parameters, the anharmonicity of the potential,v _E (0)T/, and the resonance mismatch, [ —(0)]/. It is shown that the effect of the pumping increases with diminishing ¦v¦ and at small v is proportional tov ^–1. In this regime, the dependence on is stepwise: the pumping contribution is large for v > 0 and small for v < 0. In the frame of our theory, the decay rate is invariant against the simultaneous alternation of the signs of andv. The spectrum of the energy absorption has the standard Lorentzian shape in the absence of anharmonicity,v=0, and with increasing of ¦v¦ shifts and widens retaining its bell-shape form.     

The asymmetric contact process on a finite set

The contact process onZ has one phase transition; let _c be the critical value at which the transition occurs. Let _N be the extinction time of the contact process on {0,...,N}. Durrett and Liu (1988), Durrett and Schonmann (1988), and Durrett, Schonmann, and Tanaka (1989) have respectively proved that the subcritical, supercritical, and critical phases can be characterized using a large finite system (instead ofZ) in the following way. There are constants ₁() and ₂() such that if < _c , lim _{N N} /logN = 1/₁(); if > _c , lim _N log _N /N = ₂(); if = _c , lim _{N N} /N= and lim _{N N} /N ⁴=0 in probability. In this paper we consider the asymmetric contact process onZ when it has two distinct critical values _c1< _c2. The arguments of Durrett and Liu and of Durrett and Schonmann hold for < _c1 and > _c2. We show that for [ _c1< _c2), lim _{N N} /N=-1/, (where _i is an edge speed) and for = _c2, lim _N log _N /logN=2 in probability.     

On the spectrum of Schrödinger operators with a random potential

We investigate the spectrum of Schrödinger operatorsH of the type:H =–+q _i ()f(x–x _i + _i ())(q _i () and _i () independent identically distributed random variables,i ^d ). We establish a strong connection between the spectrum ofH and the spectra of deterministic periodic Schrödinger operators. From this we derive a condition for the existence of forbidden zones in the spectrum ofH . For random one- and three-dimensional Kronig-Penney potentials the spectrum is given explicitly.     

A central limit theorem for the Fourth Wick power of the free lattice field

LetG _a be the free lattice field measure of massm ₀ onaZ ^d , and: _x ⁴ : be the corresponding fourth Wick power of the lattice field _x . LetgC ₀ (R ^d ),g0, be a given function anda=a(a)a satisfy: lim _a0+a=0 andaZ ^d aZ ^d . We prove that ifd3, ord=2 and lim _a0+ a|loga|²=, then satisfies the central limit theorem: there isV(a, a) with lim _a0+ V(a, a)= such that the distribution of underG _a is convergent to the standard normal distribution, asa0⁺.     

Magnetic moments of proton drip-line nuclei13O and9C

The magnetic moments of the proton drip-line nuclei¹³O(I = 3/2^–,T _1/2 = 8.6 ms) and 9C(I = 3/2^–,T _1/2 = 126 ms) have been determined for the first time through the combined techniques of polarized radioactive nuclear beams and-NMR detection. The observed magnetic moments are ¦(¹³O)¦ = 1.3891 ±0.0003 _N and ¦(⁹C)¦ = 1.3914 ±0.0005 _N. Spin expectation values are deduced to be 0.76 and 1.44 for¹³O and⁹C, respectively. While the of¹³O is consistent with the systematics from isospinT= 1/2 mirror pairs, the of⁹C is unusually large, even far larger than the single particle value, = 1.     

Self-diffusion of V-48 in the FeV σ-phase and in an Fe-47 wt.% V solid solution

Self-diffusion coefficients of vanadium in the FeV -phase and in the corresponding -solid solution (Fe-47 wt.% V) measured in the temperature ranges 1002–1115 °C (-phase) and 1230–1320 °C are reported. The found results differ fundamentally and significantly from the relations in ordered and disordered solid solutions [9]. The diffusivity in -phase at the transition temperature (T _/=1200 °C) is cca 14 times lower than the diffusivity in the b.c.c. solid solution, the chemical composition of which is the same. The lowering is caused by the different values of frequency factors,D _O=0.11 cm²/s andD _O=45 cm²/s. The effect of the corresponding activation enthalpiesH =252 kJ/mole andH =293 kJ/mole is small and quite opposite. The occurence of higher activation enthalpyH in the -solid solution at temperaturesT>T _/ may be attributed to a certain amount of the f.c.c. phase coexisting in the b.c.c. matrix at concentrationsc _v>27 wt.% at sufficiently high temperatures [7]. A comparison of vanadium self-diffusion characteristics measured in the -phase to the extrapolated values obtained on the basis of the previous measurements [1] in the Fe-V primary solid solutions ₁ shows that the diffusivity ratioD ₁/D (1473 K)=33 and that the activation enthalpyH is by about 3% higher than the valuesH ₁ (eq. (5)) measured in the uniphase b.c.c. solid solutions.     

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)}.     

Critical Region for Droplet Formation in the Two-Dimensional Ising Model

We study the formation/dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence. Specifically, we consider the 2D Ising model in volumes of size L ² , inverse temperature > _c and overall magnetization conditioned to take the value m L ² –2m v _L , where _c ^–1 is the critical temperature, m =m () is the spontaneous magnetization and v _L is a sequence of positive numbers. We find that the critical scaling for droplet formation/dissolution is when v _L ^3/2 L ^–2 tends to a definite limit. Specifically, we identify a dimensionless parameter , proportional to this limit, a non-trivial critical value _c and a function such that the following holds: For < _c , there are no droplets beyond log L scale, while for > _c , there is a single, Wulff-shaped droplet containing a fraction _c =2/3 of the magnetization deficit and there are no other droplets beyond the scale of log L. Moreover, and are related via a universal equation that apparently is independent of the details of the system.     

Energy and momentum in general relativity

In general relativity, conservation of energy and momentum is expressed by an equation of the form /x= 0, where –gT represents the total energy, momentum, and stress. This equation arises from the divergence formula dV _v = (/x ^v )d ⁴ d. Here we show that this formula fails to account properly for the system of basis vectors e(x). We obtain the (invariant) divergence formula e dV _v = e (/x ^v + )d ⁴ d. Conservation of energy and momentum is therefore expressed by the covariant equation (/x ^v ) + = 0. We go on to calculate the variation of the action under uniform displacements in space-time. This calculation yields the covariant equation of conservation, as well as the fully symmetric energy tensor . Finally, we discuss the transfer of energy and momentum, within the context of Einstein's theory of gravitation.     

EC and α decays of 235Am

EC and decays of ²³⁵Am have been studied using a gas-jet coupled on-line isotope separator. Excited states in ²³⁵Pu have been established for the first time by means of -ray spectroscopy following the EC decay of ²³⁵Am. The deduced log ft value suggests that the ground state of ²³⁵Am should have the 5/2^-[523] configuration. The - coincidence result has revealed that the 5/2^-[523] state in ²³¹Np populated by the favored transition of ²³⁵Am is located at < 15 keV, which allows us to precisely determine the Q value of ²³⁵Am.     

ЧАстИЧНО пОгРУжЕННы И БлОк В кВАДРАтНОИ РЕ шЕткЕ

( ) , . , . [1] , , .

---

Partially inserted block in square lattice

The paper gives a theory of the partially inserted block (control rod) in a reactor of finite height, consisting of a square lattice of fuel blocks and a moderator. The partially inserted block replaces the block of the original lattice which was located in the axis of cylindrical symmetry of the reactor. The results obtained agree with the relations for a homogeneous medium in paper [1] and are thereby justified for homogeneized reactors.

     

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.

---

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

---

---

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.     

A Double Infinity of Oscillator Massless Boson Resonances

In the simplest coupling of a harmonic oscillator with a massless boson field, we show that a family of coupling functions leads to resonances or bound-states of the form E _{n1 n0}()=n ₁ z ₁()+n ₀ z ₀(), where z ₁(), z ₀() are in and n ₁, n ₀ are any nonnegative integers. This holds for arbitrary values of the coupling constant.     

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.