Stochastic operators,information, and entropy

For a stochastic operatorU on andL ₁-space, i.eU is linear, positive, and norm preserving on the positive cone ofL ₁, it is shown thatU decreases relative information between two nonnegativeL ₁-functions. Furthermore it is shown that the following properties ofU are closely related:U is energy decreasing (energy preserving),U isH-decreasing, whereH is Boltzmann'sH-functional, and the Maxwell distributions are fixed points ofU.     

Double Quantization on Some Orbits in the Coadjoint Representations of Simple Lie Groups

Let ? be the function algebra on a semisimple orbit, M, in the coadjoint representation of a simple Lie group, g, with the Lie algebra ?. We study one and two parameter quantizations ? _h and ? _t,h of ? such that the multiplication on the quantized algebra is invariant under action of the Drinfeld–Jimbo quantum group, U _{h (?)}. In particular, the algebra ? _t,h specializes at h= 0 to a U(?)-invariant ($G$-invariant) quantization, %Ascr; _{t ,0}. We prove that the Poisson bracket corresponding to ? _h must be the sum of the so-called r-matrix and an invariant bracket. We classify such brackets for all semisimple orbits, M, and show that they form a dim H ²(M) parameter family, then we construct their quantizations. A two parameter (or double) quantization, $? _t,h , corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-Kostant-Souriau bracket on M. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization of these pairs in some of the cases. Received: 15 August 1998 / Accepted: 13 January 1999     

Compound windows of the Hénon-map

For the two-parameter second-order Hénon map, the shapes and locations of the periodic windows-continua of parameter values for which solutions x₀,x₁,… can be stably periodic, embedded in larger regions where chaotic solutions or solutions of other periods prevail-are found by a random searching procedure and displayed graphically. Many windows have a typical shape, consisting of a central “body” from which four narrow “antennae” extend. Such windows, to be called compound windows, are often arranged in bands, to be called window streets, that are made up largely of small detected but poorly resolved compound windows.For each fundamental subwindow-the portion of a window where a fundamental period prevails-a stability measure U is introduced; where the solution is stable, |U|<1. Curves of constant U are found by numerical integration. Along one line in parameter space the Hénon-map reduces to the one-parameter first-order logistic map, and two antennae from each compound window intersect this line. The curves where U=1 and U=−1 that bound either antenna are close together within these intersections, but, as either curve with U=−1 leaves the line, it diverges from the curve where U=1, crosses the other curve where U=−1, and nears the other curve where U=1, forming another antenna. The region bounded by the numerically determined curves coincides with the subwindow as found by random searching. A fourth-degree equation for an idealized curve of constant U is established.Points in parameter space producing periodic solutions where x₀=x_m=0, for given values of m, are found to lie on Cantor sets of curves that closely fit the window streets. Points producing solutions where x₀=x_m=0 and satisfying a third condition, approximating the condition that x_n be bounded as n→−∞, lie on curves, to be called street curves of order m, that approximate individual members of the Cantor set and individual window streets. Compound windows of period m+m^′ tend to occur near the intersections of street curves of orders m and m^′.Some exceptions to what appear to be fairly general results are noted. The exceptions render it difficult to establish general theorems.     

Coadjoint Orbits of Certain Motion Groups and Their Coherent States

Let ?_n, n ≥ 1, be the (2n+1)-dimensional Heisenberg group and let K be a closed connected subgroup of the unitary group U(n) acting on ?_n by automorphisms. Using the moment map, we provide in this paper a dequantization procedure for all generic admissible coadjoint orbits of the semidirect product G = K ? ?_n. In the opposite direction, we show that Gilmore-Perelomov's coherent states define “pure state quantizations” of such orbits.     

Some new four-parameter potentials and their use in the study of vibrational thermodynamical quantities of diatomic molecules

Three four-parameter potentials,U _I,U _II andU _III have been proposed and their accuracy has been demonstrated by finding the mean square deviation from the true RKR potential curve for 15 electronic states of 12 diatomic molecules. Their percentage average mean square deviations from RKR curve have been found to be 1.45, 1.86 and 2.89, respectively. These compare favourably with the value 2.67 for the recently suggested four-parameter potential of Wei Hua which itself yields better results than the commonly employed three-parameter potentials. The superiority of the new potentials (especially ofU _I andU _II has been further established by using these potentials to calculate the molecular constants α_e and ω_ex_e following Dunham’s method. The corresponding percentage average mean deviations, for α_e, turn out to be 3.75, 5.13 and 15.43 and for ω_ex_e 8.73, 17.23 and 27.49, respectively, against the respective values of 7.97 and 18.88 with Wei Hua’s four-parameter potential. Also included are the values of dissociation energy determined with these potentials and these too corroborate the better performance ofU _I andU _II. The relative worth of various potential functions has been further tested by carrying out numerical study of vibrational partition function (evaluated by sum over states method), entropy and thermal capacity for the ground state of 7 molecules and comparing these with the corresponding findings based on the RKR data.     

A variational problem for a system of magnetic monopoles joined by Abrikosov vortices

An action functional, related to the Higgs model to field theory, depending on a complex scalar field and aU(1) connection is defined. The complex scalar field is a section of a line bundle associated to a principalU(1)-bundle with base space ³\{x ₁,...,x _n }. The pointsx ₁,...,x _n are the positions ofn magnetic monopoles of magnetic chargesm ₁,...,m _n, with . The existence of minimizers of the action functional is proven using direct methods of the calculus of variation. Regularity and decay properties of the minimizers are obtained. By constructing explicit comparison field configurations, we establish accurate upper and lower bounds for the action of the minimizers in a variety of special situations, e.g.n=2 andm ₁=–m ₂.     

119Sn Mössbauer spectroscopy of U x Th1−x CoSn and UCoAl1−x Sn x

Magnetization and¹¹⁹Sn Mössbauer spectroscopy studies were performed on two pseudoternary systems, namely U _x Th_1?x CoSn and UCoAl_1?x Sn _x . Magnetic hyperfine fields on¹¹⁹Sn nuclei were found to be due to local surrounding of U atoms which carry magnetic moments. In pure UCoSn the valuesB _hf=8.66 T and μ_∪ satisfy the relationB _hf=6.75 μ_B found for a number of UTX compounds studied. Local surrounding effects are indicated by multicomponent spectra obtained for the two pseudoternary series.     

The influence of actinides on the superconducting properties of molybdenum chalcogenides

The results of the investigations of molybdenum chalcogenides: U_xPb_1?xMo₆S₈, U_xPb_1?xMo_6.4S₈, Th_xPb_1?xMo₆S₈, U_x1?xMo₆S₈ with actinides as a component, are presented. As a rule the decrease of superconducting parameters with x was observed. Possible influence of magnetic interaction in this system is noted.     

Global properties of radial wave functions in Schwarzschild's space-time

The radial factorR(x, x _s ) of a scalar field in Schwarzschild's space-time satisfies a second order ordinary differential equation with two regular singular points atx=0 andx=x _s and one irregular singular point atx=∞. The analytical properties of four solutions ?₁, ?₂, ?₃, and ?₄ (defined by their power series expansions aboutx=0 andx=x _s ) with respect tox _s are studied. An analytical continuation is given for each solution outside its circle of convergence. Relations to the flat-space solutions are established. Finally the coefficients relating linearly any three of these solutions are determined and studied as functions of the parameterx _s .     

Transport properties in non-oxide perovskite ferroelectric relaxors

Electric transport in a Cu-doped Cd salt [(CH₂)₃(NH₃)₂Cd_{1? x} Cu _x Cl₄, x?=?0.0, 0.07, 0.395 and 0.69] was investigated in the frequency range 60?Hz–100?kHz and the temperature range 290–450?K. The conductivity increases with increasing copper doping. Samples with x?=?0.0 and 0.07 undergo phase transitions at 374?K and 369?K, respectively. Ferroelectric relaxor-like behaviour appears for x?=?0.395 and 0.69. The conduction mechanism of the samples with x?=?0.0 and 0.07 depends on the temperature region. Below the transition temperatures chlorine vacancy and proton hopping prevails, whereas above the transition temperatures mainly proton conduction dominates. Transport in the new non-oxide ferroelectric relaxors, where x?=?0.395 and 0.69, can be explained by the jump relaxation model where proton and ionic hopping contribute to the conductivity throughout the whole temperature range.     

Group theoretical aspects ofU B(6) ⊗U F(20) symmetry limits of IBFM related to theU B(5) andO B(6) limits of IBM

TheU ^B(6)⊗U ^F(20) Bose-Fermi dynamical symmetry of interacting boson-fermion model arises when the odd nucleon occupies single particle orbits withj=1/2, 3/2, 5/2, and 7/2. The subgroup structures ofU ^B(6)⊗U ^F(20) related to theU ^B(5) andO ^B(6) limits of sdIBM (U ^B(6)) are analysed. Broadly speaking,U ^B(6)⊗U ^F(20) admitsU ^BF(5)⊗U _s ^F (4), Spin^BF(5)⊗U _k ^F (5) andU ^BF(5)⊗U _s ^F (2) limits withU ^B(5) core and Spin^BF(6),O ^BF(5)⊗U _s ^F (4), Spin^BF(6)⊗U _k ^F (5) andO ^BF(6)⊗U _s ^F (2) limits withO ^B(6) core respectively. For each of these seven symmetry limits, group chains, quantum numbers labelling the basis states, generators and Casimir operators for the various subgroups and energy formulas are given. Recoupling coefficients (reduced Wigner coefficients) for constructing wavefunctions of low-lying states are tabulated and these will allow (together with sdIBMU ^B(5) andO ^B(6) limit results) one to calculateB(E2)’s,B(M1)’s, one and two nucleon transfer strengths etc. in the seven symmetry limits. Experimental examples for theU ^B(6)⊗U ^F(20) symmetry limits are briefly discussed.     

Non-stoichiometry in uranium sesquinitride

A statistical model is proposed for analyzing thermodynamic properties of non-stoichiometric uranium sesquinitride. It consists of the combination of the site exclusion mechanism and cluster formation reactions. The model was applied to the three ionic possibilities of U_3/2⁴⁺U_1/2⁶⁺N₃^3?, U⁴⁺U⁵⁺N₃^3? and U₂³⁺N₃^2?. By fitting the theoretical equations to the experimentally observed data, it was found that the set U_3/2⁴⁺U_1/2⁶⁺U₃^3?, β=2, 1:2 cluster gave the most satisfactory results. From β=2, the upper limit of non-stoichiometry is expected to be N/U=1·75. The result that the 1:2 clusters predominantly exist in U₂N_3+x suggests the existence of UN_1·75 phase. The energy required for nitrogen atom for occupying the lattice site of the crystal, E, was calculated to be ?120·5 kcal/mole, and the interaction energy to form clusters, E_c, was ?11·07 kcal/mole.     

Existence of standing waves for dirac fields with singular nonlinearities

We prove the existence of stationary states for nonlinear Dirac equations of the form (E) $$i\sum\limits_{\mu = 0}^3 {\gamma ^\mu \partial _\mu \psi - M\psi + F\left( {\bar \psi \psi } \right)\psi = 0,} $$ whereM>0 andF is a singular self-interaction. In particular, in the model case whereF(s)=?s ^?α, for some 0<α<1, and for every ω>M, there exists a solution of (E) of the form ψ(t, x)=e ^iωt?(x), wherex ₀=t andx=(x ₁,x ₂,x ₃), such that ? has compact support. IF 0<α<1/3, then ? is of classC ¹. If 1/3<α<1, then ? is continuously differentiable, except on some sphere {|x|=R}, where |??| is infinite.     

Recovering Asymptotics of Short Range Potentials

Any compact smooth manifold with boundary admits a Riemann metric of the form near the boundary, where x is the boundary defining function and h' restricts to a Riemannian metric, h, on the boundary. Melrose has associated a scattering matrix to such a metric which was shown by he and Zworski to be a Fourier integral operator. It is shown here that the principal symbol of the difference of the scattering matrices for two potentials at fixed energy determines a weighted integral of the lead term of V ₁ - V ₂ over all geodesics on the boundary. This is used to prove that the entire Taylor series of the potential at the boundary is determined by the scattering matrix at a non-zero fixed energy for certain manifolds including Euclidean space. Received: 3 January 1997 / Accepted: 15 August 1997     

Special points on the phase diagrams of langbeinites. The case of (x,T)-phase diagram of mixed K2xRb2(1−x)Cd2(SO4)3 crystals

We have studied special points that appear on the (x, T)-phase diagram of mixed K_2xRb_2(1?x)Cd₂(SO₄)₃ crystals. This phase diagram has been constructed issuing from the changes in domain structure occurring in the course of phase transitions. We have found that two triple points on the (x, T)-phase diagram of the above solid solutions are very close to each other in the phase space and can change their places or transform into a quadruple point. These special points are associated with disappearance of ferroelectric phases with the symmetries P1 and P2₁ taking place when Rb₂⁺ are substituted with K₂⁺ ions.     

Computation of parabolic cylinder functions by means of a tricomi expansion

Fast convergence expansion of the parabolic cylinder functions U(a, x), V(a, x), W(a, ± x) is obtained in terms of the Tricomi functions E_v(z). The numerical results are quite accurate for a large interval of values of “a” and for |x| 7. Tables are given for U and V in order to compare our results with other recent works on the same functions.     

A classification of the unitary irreducible representations ofSU(N, 1)

All inequivalent continuous unitary irreducible representations ofS U(N, 1) (N2) have been determined and classified. The matrix elements of the infinitesimal generators realized on a certain Hilbert space have been derived. Representations of the groups ,S U(N, 1)/Z _N+1, andU(N, 1) are classified in a similar manner.     

Mössbauer studies of valence fluctuations

In the last ten years the Mössbauer technique has made a considerable contribution to the research of the phenomena of mixed valencies, valence instabilities, valence fluctuations and intermediate valencies of transition elements. The sensitivity of the hyperfine interaction parameters and in particular the isomer shift, to the valency of the Mössbauer ion, enabled the research of dynamics of valence fluctuations and the temperature, pressure and composition dependence of the ionic intermediate valence state. Studies of¹⁴⁹Sm,¹⁵²Sm and¹⁵³Eu in Sm_1?xR_xS contributed to the understanding of the outstanding insulator-metal phase transition that occurs in these systems. Studies of⁵⁷Fe and¹⁵¹Eu in mixed valent systems yield the charge fluctuation rates of the “hopping” mechanism, contributing to the conductivity in these systems. Studies of¹⁵¹Eu in EuRh₂, EuCu₂Si₂, EuPd₂Si₂, EuFe₄Al₈, and EuPd₆B₄ as a function of temperature and pressure reveal many aspects of the thermodynamics of intermediate valencies. Studies of systems like Eu_1?xLa_xRh₂, EuA_2?xB_x, EuA_5?xB_x reveal strong local environment dependence of the intermediate valency. Mössbauer spectra of¹⁶⁹Tm in TmSe,¹⁷⁰Yb in YbAl₃ and²³⁷Np in NpOs₂ also display phenomena associated with fluctuating valencies.     

Two interacting particles in a disordered chain III: Dynamical aspects of the interplay disorder-interaction

The interplay between the quantum interferences responsible for one particle localization over a length L₁, and the partial dephasing induced by a local interaction of strength U with another particle leading to partial delocalization over a length L ₂ > L ₁ , is illustrated by a study of the motion of two particles put close to each other at the initial time. Localization is reached in two steps. First, before the time t₁ necessary to propagate over L₁, the interaction slows down the ballistic motion. On the contrary, after t₁ the interaction favors a very slow delocalization, characterized by a spreading of the center of mass, until L₂ is reached. This slow motion is related to the absence of quantum chaos in this one dimensional model, the interaction being only able to induce weaker chaos with critical spectral statistics. Under appropriate initial conditions, the motion remains invariant under the duality transformation mapping the behavior at small U onto the behavior at large U. Received 24 August 1998     

Superconducting and magnetic properties of Nb/Pd 1-xFex/Nb triple layers

The superconducting and magnetic properties of Nb/Pd_1-xFe_x/Nb triple layers with constant Nb layer thickness d_Nb=200 ? and different interlayer thicknesses 3 ?≤ d_PdFe ≤ ? are investigated. The thickness dependence of the magnetization and of the superconducting transition temperature shows that for small iron concentration x the Pd_1-xFe_x layer is likely to be in the paramagnetic state for very thin films whereas ferromagnetic order is established for x ≥ 0.13. The parallel critical field B_c2||(T){B_{c2||}}(T) exhibits a transition from two-dimensional (2D) behavior where the Nb films are coupled across the interlayer, towards a 2D behavior of decoupled Nb films with increasing d_PdFeand/or x. This transition allows a determination of the penetration depth x_F{\xi _F} of Cooper pairs into the Pd_1-xFe_x layer as a function of x. For samples with a ferromagnetic interlayer x_F{\xi _F} is found to be independent of x.