Low frequency limits of third-order harmonic coefficients for atomic hydrogen in any level of discrete energy spectrum

In this paper the analytical expression is derived for the third-order optical harmonic coefficients x _abcd ^v (,,) of the atomic hydrogen at very low frequencies 0 in many level of the discrete energy spectrum.The author expresses his thank to Prof. Nguyen Van Hieu for suggesting the problem and fruitful discussions.     

A posteriori error estimation and basis adaptivity for reduced-basis approximation of nonaffine-parametrized linear elliptic partial differential equations

In this paper, we extend the earlier work [M. Barrault, Y. Maday, N. C. Nguyen, A.T. Patera, An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations, C.R. Acad. Sci. Paris, Série I 339 (2004) 667–672; M.A. Grepl, Y. Maday, N.C. Nguyen, A.T. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, M2AN Math. Model. Numer. Anal. 41 (3) (2007) 575–605.] to provide a posteriori error estimation and basis adaptivity for reduced-basis approximation of linear elliptic partial differential equations with nonaffine parameter dependence. The essential components are (i) rapidly convergent reduced-basis approximations – (Galerkin) projection onto a space spanned by N global hierarchical basis functions which are constructed from solutions of the governing partial differential equation at judiciously selected points in parameter space; (ii) stable and inexpensive interpolation procedures – methods which allow us to replace nonaffine parameter functions with a coefficient-function expansion as a sum of M products of parameter-dependent coefficients and parameter-independent functions; (iii) a posteriori error estimation – relaxations of the error-residual equation that provide inexpensive yet sharp error bounds for the error in the outputs of interest; (iv) optimal basis construction – processes which make use of the error bounds as an inexpensive surrogate for the expensive true error to explore the parameter space in the quest for an optimal sampling set; and (v) offline/online computational procedures – methods which decouple the generation and projection stages of the approximation process. The operation count for the online stage – in which, given a new parameter value, we calculate the output of interest and associated error bounds – depends only on N, M, and the affine parametric complexity of the problem; the method is thus ideally suited for repeated and reliable evaluation of input–output relationships in the many-query or real-time contexts.     

Phi-symmetric effect algebras

The notion of a Sasaki projectionon an orthomodular lattice is generalized to a mapping : E × E E, where E is an effect algebra. If E is lattice ordered and is symmetric, then E is called a -symmetric effect algebra.This paper launches a study of such effect algebras. In particular, it is shown that every interval effect algebra with a lattice-ordered ambient group is -symmetric, and its group is the one constructed by Ravindran in his proof that every effect algebra that has the Riesz decomposition property is an interval algebra. It is shown that the doubling construction introduced in the paper is connected to the conditional event algebrasof Goodman, Nguyen, and Walker.     

Properties of long superconducting variable thickness bridges

We have studied the properties of long (lengthL20m) superconducting threads (Pb of Sn) formed by pulse electrical breakdown in monocrystalline silicon. These variable thickness bridges have well defined dc and ac Josephson effects in wide temperature range. Some interesting results on the quasiparticle subharmonic gap structures are also reported.The authors would like to express their thanks to J.Móza and A.Neasová for technical assistance.     

Kramers time in weakly nonpotential systems

We consider a bistable Fokker-Planck system with a known stationary distribution and a small nonpotential part in the drift force. We perform a perturbation calculation of its Kramers time, _K, and compare it with the corresponding time, _K ⁽⁰⁾ , for the potential system which has the same stationary distribution. We show that _K/ _K ⁽⁰⁾ depends only on the properties of the drift force close to the saddle-point.The authors would like to dedicate this work to their colleagues Y. Orlov, R. Nazarian, and V. Brailovski.     

Analytic and numerical evidence from quantum field theory for the hyperscaling relation dv = 2Δ−γ in the d=3 ising model

The hyperscaling relationdv = 2 - (d=3) for the Ising model has been shown to follow from a constructive approach proposed by one of the authors (R.S.) of a relativistic theory of self-interacting Bosons in d space-time dimensions. We present evidence that the two assumptions made in this approach are valid: On a finite Euclidean (hyper-) cubical lattice in d dimensions the renormalization map from the bare to the renormalized parameters should have nonvanishing Jacobian everywhere. We show this analytically and numerically on the boundary set of the parameters. The numerical analysis involves Monte Carlo calculations in the region where the bare coupling constantg ₀ is infinite, giving the Ising model. The linear sizen of the lattice (with periodic boundary conditions) was taken to be 5, 6, and 10. There we also checked the second assumption saying that the correlation length for the Ising model is a monotonic function of the temperature. We also comment on the possible numbers of zeros of the Callan-Symanzik function of this theory.     

Band-mapping of InP (100) along the ΓX-line

We have prepared InP (100) surfaces from wafer-material by cycles of argon ion sputtering and gentle annealing. From these samples normal-emission photoelectron spectra were recorded using photons in the energy range 10eV<h<75eV. From the results we derive the initial-state band disperison along the X-line of the 3-dimensional Brillouin zone. Our results are fully consistent with initial-state bands mapped by other authors along the KX-line. From the good agreement we conclude that the quality of the (100)-surfaces as used in our work is at least adequate for bulk band investigations.     

Gauge theory in deformed $$ \mathcal{N} $$ = (1, 1) superspace

We review the non-anticommutative Q-deformations of = (1, 1) supersymmetric theories in four-dimensional Euclidean harmonic superspace. These deformations preserve chirality and harmonic Grassmann analyticity. The associated field theories arise as a low-energy limit of string theory in specific backgrounds and generalize the Moyal-deformed supersymmetric field theories. A characteristic feature of the Q-deformed theories is the half-breaking of supersymmetry in the chiral sector of the Euclidean superspace. Our main focus is on the chiral singlet Q-deformation, which is distinguished by preserving the SO(4) ∼ Spin(4) “Lorentz” symmetry and the SU(2) R-symmetry. We present the superfield and component structures of the deformed = (1, 0) supersymmetric gauge theory as well as of hypermultiplets coupled to a gauge superfield: invariant actions, deformed transformation rules, and so on. We discuss quantum aspects of these models and prove their renormalizability in the Abelian case. For the charged hypermultiplet in an Abelian gauge superfield background we construct the deformed holomorphic effective action. The text was submitted by the authors in English.     

Spectral properties and scaling relations in off diagonally disordered chains

We obtain the localization lengthL as a function of the energyE and the disorder widthW for an off-diagonally disordered chain. This is done performing numerical simulations involving the continued fraction representations of the transfer matrix. The scaling relationL=W ^s is obtained with values of the exponents in agreement with calculations of other authors. We also obtain the relationL|E| ^v forE0, and use it in the Herbert-Spencer-Thouless formula forL to describe the singularity of the density of states nearE=0. We show that the slightest diagonal disorder obliterates this singularity. A practical method is presented to calculate the Green function by exploiting its continued fraction expansion.     

Nuclearg-factors of the 1229 and 2911 keV isomers in143Nd

Nuclearg-factors of the 1229 keV 13⁺/2 and 2911 keV 21⁺/2 level and the half-life of the 1229 keV level have been measured to beg(1229)=0.058(5),g(2911)=0.69(12) andT _1/2 (1229)=6.79(2) ns. The experimental g-factors show that the 1229 keV level has a dominant configuration of (f _7/23^–), and the 2911 keV level has an almost pure (f _7/27^–), configuration.The authors would like to thank the crew of CYRIC for the operation of the cyclotron.     

On vector-tensor minimally coupled field theories

We consider vector-tensor minimally coupled Lagrangians, i.e., scalar densities of the form = g ^1/2 R +L(g _ij ; _i ; _i,j ). We prove that the gauge invariance of any of the sets of Euler-Lagrange expressions implies the gauge invariance of the Lagrangian itself forn even, and an almost gauge invariance forn odd. We also find those for whichE ⁱ () = 0 orE ^ij (L) = 0, generalizing well-known results by Lovelock and a result by the authors.     

Reflection equations and exactly solvable lattice spin models

Reflection equations are used to obtain families of commuting double-row transfer matrices for interaction-round-a-face (IRF) models with fixed and free boundary conditions. We illustrate our methods for the Andrews-Baxter-Forrester (ABF) models which areL-state models associated with the quantum groupU _q (su(2)) at a root of unity. We construct elliptic solutions to the reflection equations for the ABF models by a procedure which uses fusion to build the solutions starting from a trivial solution.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.On leave from Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia.The authors thank Vladimir Rittenberg for his kind hospitality at Bonn. This work was supported by the Australian Research Council.     

Towards a unified dynamical theory of the Brownian particle in an ideal gas

We consider the trajectoryQ ^M(t) of a Brownian particle of massM in an ideal gas of identical particles of mass 1 and of density 1 in equilibrium at inverse temperature 1 (the dynamics is uniform motion plus elastic collisions with the Brownian particle). Our theory, in dimension one, describes a variety of limiting processes — containing the Wiener process and the Ornstein-Uhlenbeck process — forA ^–1/2 Q ^M(A)(At) depending on the asymptotic behaviour ofM(A). Part of the theory is hypothetical while another part relies upon known results. We also prove that, ifA ^1/2+M(A)A, thenA ^–1/2 Q ^M(A) (At) converges to a Wiener process whose variance is known from papers of Sinai-Soloveichik and of the present authors.     

Grand unified models including extraZ bosons

The grand unified theories (GUT) of the simple Lie groups including extraZ bosons are discussed. There are onlySU _5+m,SO _6+4n, andE ₆ under our hypothesis. First we give a general discussion forSU _5+m, then forSU ₆ andSU ₇ for illustration. We use15 +6 ^* +6 ^* fermion representations inSU ₆ but not with the fermion content, Yukawa coupling, and the hierarchy of other authors. We suggest that there is a series of clans of particles. These clans consist of the extraZ bosons and the corresponding fermions of the scale.     

Rightfulness of Summation Cut-Offs in the Albedo Problem with Gaussian Fluctuations of the Density of Scatterers

The one-dimensional version of the radiative transfer problem (i.e. the so-called rod model) is analysed with a Gaussian random extinction function (x). Then the optical length X = ₀ ^Ldx(x) is a Gaussian random variable. The transmission and reflection coefficients, T(X) and R(X), are taken as infinite series. When these series (and also when the series representing T ²(X), T ²(X), R(X)T(X), etc.) are averaged, term by term, according to the Gaussian statistics, the series become divergent after averaging. As it was shown in a former paper by the authors (in Acta Physica Slovaca (2003)), a rectification can be managed when a `modified' Gaussian probability density function is used, equal to zero for X > 0 and proportional to the standard Gaussian probability density for X > 0. In the present paper, the authors put forward an alternative, showing that if the m.s.r. of X is sufficiently small in comparison with , the standard Gaussian averaging is well functional provided that the summation in the series representing the variable T ^m-j (X)R ^j (X) (m = 1,2,..., j = 1,...,m) is truncated at a well-chosen finite term. The authors exemplify their analysis by some numerical calculations.     

On nonlinear stability of general undercompressive viscous shock waves

We study the nonlinear stability of general undercompressive viscous shock waves. Previously, the authors showed stability in a special case when the shock phase shift can be determined a priori from the total mass of the perturbation, using new pointwise methods. By examining time invariants associated with the linearized equations, we can now overcome a new difficulty in the general case, namely, nonlinear movement of the shock. We introduce a coordinate transformation suitable to treat this new aspect, and demonstrate our method by analyzing a model system of generic type. We obtain sharp pointwise bounds andL ^p behavior of the solution for allp, 1p.     

Kinetic formulation of the isentropic gas dynamics andp-systems

We consider the 2×2 hyperbolic system of isentropic gas dynamics, in both Eulerian or Lagrangian variables (also called thep-system). We show that they can be reformulated as a kinetic equation, using an additional kinetic variable. Such a formulation was first obtained by the authors in the case of multidimensional scalar conservation laws. A new phenomenon occurs here, namely that the advection velocity is now a combination of the macroscopic and kinetic velocities. Various applications are given: we recover the invariant regions, deduce newL estimates using moments lemma and proveL –w* stability for 3.     

On representations of the elliptic quantum groupE τ,η(sl 2)

We describe representation theory of the elliptic quantum groupE _,(sl ₂). It turns out that the representation theory is parallel to the representation theory of the YangianY(sl ₂) and the quantum loop group .We introduce basic notions of representation theory of the elliptic quantum groupE _,(sl ₂) and construct three families of modules: evaluation modules, cyclic modules, one-dimensional modules. We show that under certain conditions any irreducible highest weight module of finite type is isomorphic to a tensor product of evaluation modules and a one-dimensional module. We describe fusion of finite dimensional evaluation modules. In particular, we show that under certain conditions the tensor product of two evaluation modules becomes reducible and contains an evaluation module, in this case the imbedding of the evaluation module into the tensor product is given in terms of elliptic binomial coefficients. We describe the determinant element of the elliptic quantum group. Representation theory becomes special ifN=m+l, whereN,m,l are integers. We indicate some new features in this case.The authors were supported in part by NSF grants DMS-9400841 and DMS-9501290.     

Proton-proton and deuteron-deuteron correlations in interactions of relativistic helium nuclei with protons

The reactions⁴Hep pp+X,³Hep pp+X and⁴Hep ddp have been investigated and the correlation function has been measured for protons and deuterons with small relative momenta. Strong positive correlation has been observed for protons related mainly to the final state interactions in¹S₀ state. The root mean square radius of the proton source calculated from the correlation function has been found to be equal to (1.7±0.3) fm and (2.1±0.3) fm for⁴He and³He respectively. It agrees with the known radii of these nuclei.We would like to thank Dr R. Lednicky for discussion, helpful suggestions and for making available the computer program that calculates the theoreticalpp correlation function. The authors also thank Dr D.H. Boal for providing the results of the two-deuteron correlation function calculations.