What determinates the chemical changes of the internal conversion coefficients for inner atomic shells?

Calculations of internal conversion coefficients (ICC) of the E1–E4 and M1–M4 transitions for nuclei in ions show that the relative changes _i / _i of the ICC _i for deep inner subshells can differ significantly from the relative changes _i/_i of the electron densities _i at the nucleus. For the K conversion _i/ _i are many times greater than _i/_i. Especially large deviations of _i/ _i are characteristic of transitions of high multipolarity; however, for the M1 transitions they can also be significant. Illustrations of various dependencies of _i/ _iare presented for the conversion in the regionZ-50.     

A multidimensional continued fraction and some of its statistical properties

The problem of simultaneously approximating a vector of irrational numbers with rationals is analyzed in a geometrical setting using notions of dynamical systems theory. We discuss here a (vectorial) multidimensional continued-fraction algorithm (MCFA) of additive type, the generalized mediant algorithm (GMA), and give a geometrical interpretation to it. We calculate the invariant measure of the GMA shift as well as its Kolmogorov-Sinai (KS) entropy for arbitrary number of irrationals. The KS entropy is related to the growth rate of denominators of the Euclidean algorithm. This is the first analytical calculation of the growth rate of denominators for any MCFA.Glossary L ⁺ set of positive integers - [.] Gauss integer symbol (Section 2) - h entropy - I of irrationals to be simultaneously approximated - d dimension of the vector of convergents (equal to I+1) - ^P unit hypercube inp dimensions - support of the invariant measure (see Section 5) - E_ij elementary matrix, with klth component _kl + _{ik jl} - E-string product of elementary matrices given by the algorithm - verticesV _i corners of the elementary simplex adjoined to the origin (Section 3) - mediantsM _ik a direct sum of any two of the vertices (Section 3) - focus sum of all the vertices (Section 3) - Euclidean reverse of the E-string procedure (see Section 2) algorithm - OCF ordinary continued-fraction algorithm - GMA generalized mediant algorithm: the subject of this paper - JP Jacobi-Perron: the most well-studied MCFA - MCFA Multidimensional continued-fraction algorithm - KS entropy Kolmogorov-Sinai entropy - T _OCF ordinary continued-fraction shift map - FS Farey shift map - (a,..., z) irrational vector withI components; each element is an irrational - d(x) invariant measure - (x) invariant density =d(x)/dx] - ₁, ₂,..., _d thed Oseledec eigenvalues of the E-string (see Section 4) ordered ₁>1>₂₃... - ₁,...,_d–1 Oseledec eigenvalues of the shift map (Section 4) ordered greatest to smallest; all the _i>1, and _i=₁/_d i+1 - ln ₁,..., In _{d– 1} Oseledecexponents of the shift map (Section 4) - Perm a permutation matrix (Section 4)     

Analytic study of power spectra of the tent maps near band-splitting transitions

Successive band-splitting transitions occur in the one-dimensional map x_i+1=g(x_i),i=0, 1, 2,... withg(x)=x, (0 x 1/2) –x +, (1/2 <x 1) as the parameter is changed from 2 to 1. The transition point fromN (=2ⁿ) bands to 2Nbands is given by=(2)^1/N (n=0, 1,2,...). The time-correlation function _i=x_ix₀/(x₀)²,x_i x_i–x_i is studied in terms of the eigenvalues and eigenfunctions of the Frobenius-Perron operator of the map. It is shown that, near the transition point=2, _i–[(10–42)/17] _i,0-[(102-8)/51]_i,1 + [(7 + 42)/17](–1)ⁱe^–yi, where2(–2) is the damping constant and vanishes at=2, representing the critical slowing-down. This critical phenomenon is in strong contrast to the topologically invariant quantities, such as the Lyapunov exponent, which do not exhibit any anomaly at=2. The asymptotic expression for _i has been obtained by deriving an analytic form of _i for a sequence of which accumulates to 2 from the above. Near the transition point=(2)^1/N, the damping constant of _i fori N is given by _N=2(^N-2)/N. Numerical calculation is also carried out for arbitrary a and is shown to be consistent with the analytic results.     

Novel perturbation expansion for the Langevin equation

We discuss the randomly driven systemdx/dt= -W(x) +f(t), wheref(t) is a Gaussian random function or stirring force withf(t)f(t)=2(t–t), andW(x) is of the formgx ¹⁺². The parameter is a measure of the nonlinearity of the equation. We show how to obtain the correlation functionsx(t)f(t)···x(t( n)) _f as a power series in. We obtain three terms in the expansion and show how to use Padé approximants to analytically continue the answer in the variable. By using scaling relations, we show how to get a uniform approximation to the equal-time correlation functions valid for allg and.     

A theory of 1/f noise

Letu() be an absolutely integrable function and define the random process where thet _i are Poisson arrivals and thes _i, are identically distributed nonnegative random variables. Under routine independence assumptions, one may then calculate a formula for the spectrum ofn(t), S _n(), in terms of the probability density ofs, p_s(). If any probability density p_s() having the property p_s() I for small is substituted into this formula, the calculated S_n() is such that S_n() 1 for small . However, this is not a spectrum of a well-defined random process; here, it is termed alimit spectrum. If a probability density having the property p_s() for small , where > 0, is substituted into the formula instead, a spectrum is calculated which is indeed the spectrum of a well-defined random process. Also, if the latter p_s is suitably close to the former p_s, then the spectrum in the second case approximates, to an arbitrary, degree of accuracy, the limit spectrum. It is shown how one may thereby have 1/f noise with low-frequency turnover, and also strict 1/f ^1– noise (the latter spectrum being integrable for > 0). Suitable examples are given. Actually, u() may be itself a random process, and the theory is developed on this basis.     

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     

Some jacobi matrices with decaying potential and dense point spectrum

We discuss doubly infinite matrices of the formM _ij= _i,j+1+ _i,j–1+V _{i ij} as operators on ². We present for each >0, examples of potentialsV _n with |V _n|=O(n ^–1/2+) and whereM has only point spectrum. Our discussion should be viewed as a remark on the recent work of Delyon, Kunz, and Souillard.Research partially supported by USNSF under grant MCS 81-20833     

The slow passage through a steady bifurcation: Delay and memory effects

We consider the following problem as a model for the slow passage through a steady bifurcation: dy/dt = (t) y – y³ +, where is a slowly increasing function oft given by= _i + t ( ⁱ,<0). Both and are small parameters. This problem is motivated by laser experiments as well as theoretical studies of laser problems. In addition, this equation is a typical amplitude equation for imperfect steady bifurcations with cubic nonlinearities. When=0, we have found that=0 is not the point where the bifurcation transition is observed. This transition appears at a value = ^j > 0. We call ^j the delay of the bifurcation transition. We study this delay as a function of ⁱ, the initial position of, and, the imperfection parameter. To this end, we propose an asymptotic study of this equation as 0, small but fixed. Our main objective is to describe this delay in terms of the relative magnitude of and. Since time-dependent imperfections are always present in experiments, we analyze in the second part of the paper the effect of a small-amplitude but time-periodic imperfection given by (t) = cos(t).     

Derivations commuting with abelian gauge actions on lattice systems

Let be an action of a compact abelian groupG on aC*-algebraA, and assume that the fixed-point subalgebraA is an AF-algebra. We show that if is a closed *-derivation onA commuting with , and the restriction of toA generates a one-parameter group of *-automorphisms, then itself is a generator. In particular, the result applies if is an infinite product action ofG on a UHF algebra. Furthermore, if in this situation ₁ and ₂ are two derivations both satisfying the hypotheses on , and ₁ and ₂ have the same restriction toA , then there exists a one-parameter subgroup of the action with generator ₀ such thatD(₁)D(₂)D(₀) is a joint core for the three derivations, and ₂=₁+₀ on this core.     

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     

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.     

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.     

The hydrodynamic limit of a one-dimensional nearest neighbor gradient system

We study the hydrodynamic behavior of a one-dimensional nearest neighbor gradient system with respect to a positive convex potential . In the hydrodynamic limit the density distribution is shown to evolve according to the nonlinear diffusion equation ,(q)/t= (²/dq²){F([1/₁(q)]), with F= –.     

The spectrum of a Schrödinger operator inL p ℝ v isp-independent

LetH _p =–1/2+V denote a Schrödinger operator, acting inL _p ^v , 1p. We show that (H _p )=(H ₂) for allp[1, ], for rather general potentialsV.     

Probability distribution of internal distances of a single polymer in good solvent

IfP _ij(x) is the probability distribution function of the scaled distancex between two elementsi andj of a long polymer in a good solvent, it is shown by Monte Carlo calculations that is in good agreement with out data for allx (B is a normalization constant). As a model we consider the freely jointed chain consisting ofN=160 rigid links. We estimate the exponents to ₀=0.27±0.01, ₀=2.44±0.02 (fori=1,j=N); ₁=0.55±0.06, ₁=2.60±0.15 (fori=1,j=N/2); ₂=0.9±0.1, ₂=2.48±0.06 (fori=N/4,j=3N/4). ₀ and ₀ are in agreement with ₀=1/(1-v) and ₀=(-1)/v proposed by Fisher and des Cloiseaux respectively, but we find concerning ₁ and ₂ that our estimates differ from recent -expansion calculations, by an amount of 20–30%. We analyse the crossover between the various exponents.     

Temperature Dependence of the Gibbs State in the Random Energy Model

We consider the problem of temperature dependence of the Gibbs states in two spin-glass models: Derrida's Random Energy Model and its analogue, where the random variables in the Hamiltonian are replaced by independent standard Brownian motions. For both of them we compute in the thermodynamic limit the overlap distribution ^N _{i=1 i i} /N[–1,1] of two spin configurations , under the product of two Gibbs measures, which are taken at temperatures T,T respectively. If TT are fixed, then at low temperature phase the results are different for these models: for the first one this distribution is D _{0 0}+D _{1 1}, with random weights D ₀, D ₁, while for the second one it is ₀. We compute consequently the overlap distribution for the second model whenever T–T0 at different speeds as N.     

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 exponents for long-range interactions

Long-range components of the interaction in statistical mechanical systems may affect the critical behavior, raising the system's effective dimension. Presented here are explicit implications to this effect of a collection of rigorous results on the critical exponents in ferromagnetic models with one-component Ising (and more genrally Griffiths=Simon class) spin variables. In particular, it is established that even in dimensions d<4 if a ferromagnetic Ising spin model has a reflection-positive pair interaction with a sufficiently slow decay, e.g. as J _x=1/|x| ^d+ with 0<d/2, then the exponents , , and ₄ exist and take their mean-field values. This proves rigorously an early renormalization-group prediction of Fisher, Ma and Nickel. In the converse direction: when the decay is by a similar power law with >-2, then the long-range part of the interaction has no effect on the existent critical exponent bounds, which coincide then with those obtained for short-range models.Also in the Physics Department. Research supported in part by the National Science Foundation Grant PHY 86-05164.     

Two singular integrals containing a partial wave of the two-body coulomb T-matrix

The operatorsT _C,l E+i0)[–G ₀(E+i0)]^1–i andT _C,l(E+i)G ₀[–iG ₀(E+i)]ⁱ acting on spaces of Hölder continuous, differentiable and analytic functions are investigated. The results of their action are expressed in terms of explicit singular factors and terms and Hölder (differentiable, analytic) functions. The most singular part of these operators is shown to be determined by a simple functional.     

Singular continuous spectrum on a cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian

It is rigorously proven that the spectrum of the tight-binding Fibonacci Hamiltonian,H _mn= _{m, n+1}+ _{m, n–1}+ _{m, n} [(n+1)]–[n]) where =(5–1)/2 and [·] means integer part, is a Cantor set of zero Lebesgue measure for all real nonzero, and the spectral measures are purely singular continuous. This follows from a recent result by Kotani, coupled with the vanishing of the Lyapunov exponent in the spectrum.On leave from the Central Research Institute for Physics, Budapest, Hungary.