Estimates for Functionals of Solutions to Higher-Order Heat-Type Equations with Random Initial Conditions

In the present paper we continue the investigation of solutions to higher-order heat-type equations with random initial conditions, which play the important role in many applied areas. We consider the random initial conditions given by harmonizable \(\varphi \)-sub-Gaussian processes. The main results are the bounds for the distributions of the suprema over bounded and unbounded domains for solutions of such equations. The results obtained in the paper hold, in particular, for the case of Gaussian initial condition.     

Random walks with imperfect trapping in the decoupled-ring approximation

We investigate random walks on a lattice with imperfect traps. In one dimension, we perturbatively compute the survival probability by reducing the problem to a particle diffusing on a closed ring containing just one single trap. Numerical simulations reveal this solution, which is exact in the limit of perfect traps, to be remarkably robust with respect to a significant lowering of the trapping probability. We demonstrate that for randomly distributed traps, the long-time asymptotics of our result recovers the known stretched exponential decay. We also study an anisotropic three-dimensional version of our model. We discuss possible applications of some of our findings to the decay of excitons in semiconducting organic polymer materials, and emphasize the crucial influence of the spatial trap distribution on the kinetics. Received 23 July 2001 / Received in final form 14 May 2002 Published online 13 August 2002     

Existence and Uniqueness of Stationary Solutions for 3D Navier–Stokes System with Small Random Forcing via Stochastic Cascades

We consider the 3D Navier–Stokes system in the Fourier space with regular forcing given by a stationary in time stochastic process satisfying a smallness condition. We explicitly construct a stationary solution of the system and prove a uniqueness theorem for this solution in the class of functions with Fourier transform majorized by a certain function h. Moreover we prove the following “one force—one solution” principle: the unique stationary solution at time t is presented as a functional of the realization of the forcing in the past up to t. Our explicit construction of the solution is based upon the stochastic cascade representation.     

On the Initial Conditions and Solutions of the Semiclassical Einstein Equations in a Cosmological Scenario

In this paper we shall discuss the backreaction of a massive quantum scalar field on the curvature, the latter treated as a classical field. Furthermore, we shall deal with this problem in the realm of cosmological spacetimes by analyzing the Einstein equations in a semiclassical fashion. More precisely, we shall show that, at least on small intervals of time, solutions for this interacting system exist. This result will be achieved providing an iteration scheme and showing that the series, obtained starting from the massless solution, converges in the appropriate Banach space. The quantum states with good ultraviolet behavior (Hadamard property), used in order to obtain the backreaction, will be completely determined by their form on the initial surface if chosen to be lightlike. Furthermore, on small intervals of time, they do not influence the behavior of the exact solution. On large intervals of time the situation is more complicated but, if the spacetime is expanding, we shall show that the end-point of the evolution does not depend strongly on the quantum state, because, in this limit, the expectation values of the matter fields responsible for the backreaction do not depend on the particular homogeneous Hadamard state at all. Finally, we shall comment on the interpretation of the semiclassical Einstein equations for this kind of problems. Although the fluctuations of the expectation values of pointlike fields diverge, if the spacetime and the quantum state have a large spatial symmetry and if we consider the smeared fields on regions of large spatial volume, they tend to vanish. Assuming this point of view the semiclassical Einstein equations become more reliable.     

On the Unification of Geometric and Random Structures through Torsion Fields: Brownian Motions,Viscous and Magneto-fluid-dynamics

We present the unification of Riemann–Cartan–Weyl (RCW) space-time geometries and random generalized Brownian motions. These are metric compatible connections (albeit the metric can be trivially euclidean) which have a propagating trace-torsion 1-form, whose metric conjugate describes the average motion interaction term. Thus, the universality of torsion fields is proved through the universality of Brownian motions. We extend this approach to give a random symplectic theory on phase-space. We present as a case study of this approach, the invariant Navier–Stokes equations for viscous fluids, and the kinematic dynamo equation of magnetohydrodynamics. We give analytical random representations for these equations. We discuss briefly the relation between them and the Reynolds approach to turbulence. We discuss the role of the Cartan classical development method and the random extension of it as the method to generate these generalized Brownian motions, as well as the key to construct finite-dimensional almost everywhere smooth approximations of the random representations of these equations, the random symplectic theory, and the random Poincaré–Cartan invariants associated to it. We discuss the role of autoparallels of the RCW connections as providing polygonal smooth almost everywhere realizations of the random representations.     

Ergodicity for the Dissipative Boussinesq Equations with Random Forcing

We study the stationary measure for the two-dimensional Boussinesq equation with random forcing. We prove the ergodicity for the two-dimensional stochastically forced Boussinesq equation. We also study the Galerkin truncations of the three-dimensional Boussinesq equations under degenerate stochastic forcing. We follow closely the previous results on the stochastically forced Navier–Stokes equations.     

Present and future experiments with stored exotic nuclei at the FRS-ESR facility

High accuracy mass and lifetime measurements are performed with the combination of the in-flight separator FRS and the cooler-storage ring ESR at GSI. Exotic nuclei are produced via projectile fragmentation and fission at (400–900) MeV/u. This energy range is the basis for unique experimental conditions with bare and few-electron fragments and allows for the first time investigations for decay channels which are rare or strictly forbidden in neutral atoms. New experimental developments and data including the discovery of new isotopes are presented. The comparison of the experimental results with theoretical predictions indicates the potential for improvements of nuclear models. A new generation of experiments will be possible with the advent of the proposed international Facility for Antiproton and Ion Research (FAIR).     

Penning trap mass measurements of rare isotopes produced by projectile fragmentation with LEBIT at NSCL

The Low-Energy Beam and Ion Trap facility LEBIT at the NSCL at MSU has demonstrated that rare isotopes produced by fast-beam fragmentation can be slowed down and prepared for precision experiments with low-energy beams. High-pressure gas-stopping was combined with advanced ion manipulation techniques to carry out these studies with a high-precision 9.4-Tesla Penning trap mass spectrometer. The spectrometer has been used for a series of high precision mass measurements of short-lived neutron- and proton-rich isotopes during the past year. This paper presents an overview of the LEBIT facility and summarizes the first mass measurement results. The mass measurements of ⁸¹Se, where ground and isomeric states have been resolved, and of ⁸⁰As will be discussed in detail.     

Gibbs Distributions for Random Partitions Generated by a Fragmentation Process

In this paper we study random partitions of {1,…,n} where every cluster of size j can be in any of w _j possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly among such partitions with k clusters. We provide conditions on the weight sequence w allowing construction of a partition valued random process where at step k the state has the Gibbs (n,k,w) distribution, so the partition is subject to irreversible fragmentation as time evolves. For a particular one-parameter family of weight sequences w _j , the time-reversed process is the discrete Marcus–Lushnikov coalescent process with affine collision rate K _i,j = a+b(i+j) for some real numbers a and b. Under further restrictions on a and b, the fragmentation process can be realized by conditioning a Galton–Watson tree with suitable offspring distribution to have n nodes, and cutting the edges of this tree by random sampling of edges without replacement, to partition the tree into a collection of subtrees. Suitable offspring distributions include the binomial, negative binomial and Poisson distributions. Research supported in part by N.S.F. Grant DMS-0405779.     

Experimental details of the Stopped Beam RISING campaign

The RISING (Rare ISotope INvestigations at GSI) project constitutes a major pan-european initiative to study nuclear structure in exotic nuclei. A brief outline of the technical details specific to studies of isomeric decays following relativistic projectile fragmentation reactions is presented.     

Random quantum magnets with broad disorder distribution

We study the critical behavior of Ising quantum magnets with broadly distributed random couplings (J), such that P(ln J) ∼ | ln J|^{-1 - α}, α > 1, for large | ln J| (Lévy flight statistics). For sufficiently broad distributions, α < , the critical behavior is controlled by a line of fixed points, where the critical exponents vary with the Lévy index, α. In one dimension, with = 2, we obtained several exact results through a mapping to surviving Riemann walks. In two dimensions the varying critical exponents have been calculated by a numerical implementation of the Ma-Dasgupta-Hu renormalization group method leading to ≈ 4.5. Thus in the region 2 < α < , where the central limit theorem holds for | ln J| the broadness of the distribution is relevant for the 2d quantum Ising model. Received 6 December 2000 and Received in final form 22 January 2001     

A Noncommutative Generalization of the Free-Field Yang–Mills Equations

The purpose of this paper is to propose a noncommutative generalization of a gauge connection and the free-field Yang–Mills equations. The paper draws upon the techniques proposed by Heller et al. for the noncommutative generalization of the Einstein field equations.     

On the quest of production of superheavy nuclei in reactions of 48Ca with the heaviest cctinide targets

The sequence of radioactive decays of an unknown isotope produced in a rare fusion reaction to known lighter isotopes is used to identify mass and atomic number of the mother isotope, which has been separated before from the bulk of other reaction products by an in-flight recoil separator. By this technique the elements 107 to 112 were produced by single atom decay-chain analysis. Such a correlation technique reaches its limit by the occurrence of accidental sequences and it collapses beyond a maximum possible correlation time, at which a true event cannot be distinguished anymore from a random event. ⁴⁸Ca-induced fusion reactions with actinides are discussed. In 1983 at GSI, Darmstadt and LBL, Berkeley, ⁴⁸Ca/²⁴⁸Cm-experiments (II) were performed, which are compared to recent ⁴⁸Ca-experiments at FLNR-Dubna (I) irradiating ²⁴⁴Pu, ²⁴²Pu, and ²³⁸U. In these experiments production of isotopes of superheavy elements 112 and 114 is claimed. Our analysis of accidental sequences in ⁴⁸Ca-induced reactions is presented, which is at variance with the published analysis from FLNR-Dubna. We find that the maximum correlation time using continuous beams at today existing separation systems is not in the one-hour regime, but in the few-minute regime. The five spontaneous fission events observed in the FLNR experiments are preceded by signals in the (1–16)-minute range. These times are shown to be longer than the maximum possible correlation times. The preceding signals are decoupled from the spontaneous fission signal and carry no information on the spontaneous fission events observed. Moreover, random probabilities of 0.2 to 0.6 for the signals preceding the fission events indicate that the correlations are of random origin. The evidence to have discovered element 114 in the reported experiments is classified “very weak”. Received: 13 October 1999     

New low-energy measurement of the 18F(p,α)15O reaction

The ¹⁸F, produced during nova explosions, is the main responsible for the 511 keV γ-ray emitted during the outburst that could be detected with satellite missions or future γ-ray telescopes. In spite of many experimental efforts, the amount of ¹⁸F synthesized still suffers from large uncertainties concerning mainly the ¹⁸F(p,α)¹⁵O reaction. We report here on a new direct measurement of the ¹⁸F(p,α)¹⁵O cross section recently carried out at the RIB facility at Louvain-la-Neuve. Cross section data down to a center-of-mass energy of 400 keV have been obtained using an isobarically pure ¹⁸F beam (averaged intensity of the order of 10⁶ pps) and a thin polyethylene target. The reaction products were detected using the multi-strip detector array LEDA. The cross section data are analysed in the framework of the R-matrix model. Our main aim is to determine the interference sign between three 3/2⁺ resonances above threshold that can significantly affect the extrapolation of the ¹⁸F(p,α)¹⁵O cross section to the energies relevant for novae, and thus, the modeling of nova explosions.     

Model-dependent nature of blowtorch effect on the escape and equilibration rates

We analyze the relaxation behavior of a bistable system when the background temperature profile is inhomogeneous due to the presence of a localized hot region (blowtorch) on one side of the potential barrier. Since the diffusion equation for inhomogeneous medium is model-dependent, we consider two physical models to study the kinetics of such system. Using a conventional stochastic method, we obtain the escape and equilibration rates of the system for the two physical models. For both models, we find that the hot region enhances the escape rate from the well where it is placed while it retards the escape rate from the other well. However, the value of the escape rate from the well where the hot region is placed differs for the two models while that of the escape rate from the other well is identical for both. This work, for the first time, gives a detailed report of the similarities and differences of the escape rates and, hence, exposes the common and distinct features of the two known physical models in determining the way the bistable system relaxes. Received 25 September 2001     

On the slow dynamics of density fluctuations near the colloidal glass transition

Slow dynamics of density fluctuations near the colloidal glass transition is discussed from a new viewpoint by numerically solving a nonlinear stochastic diffusion equation for the density fluctuations recently proposed by one of the present authors (MT). The effects of spatial heterogeneities on the dynamics of density fluctuations are then investigated in an equilibrium system. The spatial heterogeneities are generated by the nonlinear density fluctuations, while in a nonequilibrium system they are described by a nonlinear deterministic equation for the average number density. The dynamics of equilibrium density fluctuations is thus shown to be quite different from that of nonequilibrium ones, leading to a logarithmic decay followed by less distinct α- and β-relaxation processes. Received 9 March 2002 and Received in final form 19 September 2002     

On the Behavior of Solutions of a Class of Nonlinear Partial Differential Equations

The behavior of the steady-state (or the traveling wave) solutions for a class of nonlinear partial differential equations is studied. The nonlinearity in these equations is expressed by the presence of the convective term. It is shown that the steady-state (or the traveling wave) solution may explode at a finite value of the spatial (or the characteristic) variable. This holds whatever the order of the spatial derivative term in these equations. Furthermore, new special solutions of a set of equations in this class are also found.     

On the theory of the relativistic cross-sections for stimulated bremsstrahlung on an arbitrary electrostatic potential in the strong electromagnetic field

On the base of relativistic generalized eikonal approximation wave function the multiphoton cross-sections of a Dirac particle bremsstrahlung on an arbitrary electrostatic potential and strong laser radiation field are presented. In the limit of the Born approximation the ultimate analytical formulas for arbitrary polarization of electromagnetic wave have been obtained and numerically analyzed. Received 5 April 2001 / Received in final form 18 March 2002 Published online 28 June 2002