A Stochastic Model Associated with Enskog Equation and Its Kinetic Limit

 We examine a system of particles in which the particles travel deterministically in between stochastic collisions. The collisions are elastic and occur with probability ɛ ^d when two particles are at a distance σ. When the number of particles N goes to infinity and Nɛ ^d goes to a nonzero constant, we show that the particle density converges to a solution of the Enskog Equation. Received: 29 January 2002 / Accepted: 30 July 2002 Published online: 14 November 2002 RID="*" ID="*" Research supported in part by NSF Grant DMS-0072666     

Torus Chiral <Emphasis Type="Italic">n</Emphasis>-Point Functions for Free Boson and Lattice Vertex Operator Algebras

 We obtain explicit expressions for all genus one chiral n-point functions for free bosonic and lattice vertex operator algebras. We also consider the elliptic properties of these functions. Received: 6 May 2002 / Accepted: 4 October 2002 Published online: 24 January 2003 RID="*" ID="*" Partial support provided by NSF DMS-9709820 and the Committee on Research, University of California, Santa Cruz RID="⋆⋆" ID="⋆⋆" Supported by an Enterprise Ireland Basic Research Grant and the Millenium Fund, National University of Ireland, Galway Communicated by L. Takhtajan     

Experimental investigation of non-linear selective reflection at the interface of Cs atomic vapor and quartz

We present an experimental study of non-linear selective reflection (SR) at a quartz–Cs-vapor interface in a V-type three-level scheme. The non-linear selective reflection at the Cs D₂ resonance line (⁶ S _1/2F=4→⁶ P _3/2) is monitored with and without pumping. The sub-Doppler reflection spectrum is observed and the effect of pumping on the signal of the selective reflection is investigated. The experimental result is in agreement with the theoretical calculation. Received: 16 April 2002 / Revised version: 12 June 2002 / Published online: 25 September 2002 RID="*" ID="*"Corresponding author. Fax: +86-351/701-1500, E-mail: zhaojm@sxu.edu.cn     

Micromagnetism of two-dimensional permalloy particles with different aspect ratios

Micromagnetic properties of the Fe₁₉Ni₈₁ (5 nm)/NiO (50 nm)/Fe₁₉Ni₈₁ (30 nm) structured system have been investigated in a photoemission electron microscope in the magnetic X-ray circular dichroism operating mode. The microstructured Fe₁₉Ni₈₁ (5 nm) film contained two-dimensional islands with the aspect ratio varying from 1:1 to 10:1, and the linear size of their long axis comprised 24, 12 and 6 μm. It is shown that the magnetic domains have the direction of magnetization preferentially parallel and antiparallel to the magnetic field direction in which this system was prepared. Their number is determined by the particles’ sizes, their shape as well as by the direction of the external magnetizing field and can be characterized by a non-monotonic size dependence. The magnetization of domains with different lateral sizes was found to be 0.4 T with an accuracy better than 20%. Received: 29 May 2002 / Accepted: 17 July 2002 / Published online: 22 November 2002 RID="*" ID="*"Corresponding author. Fax: +49-6131/392-3807, E-mail: nepijko@mail.uni-mainz.de     

Quantum Spheres and Projective Spaces as Graph Algebras

 The C ^* -algebras of continuous functions on quantum spheres, quantum real projective spaces, and quantum complex projective spaces are realized as Cuntz-Krieger algebras corresponding to suitable directed graphs. Structural results about these quantum spaces, especially about their ideals and K-theory, are then derived from the general theory of graph algebras. It is shown that the quantum even and odd dimensional spheres are produced by repeated application of a quantum double suspension to two points and the circle, respectively. Received: 31 January 2001 / Accepted: 29 July 2002 Published online: 7 November 2002 RID="*" ID="*" Supported by grant No. R04–2001–000–00117–0 from the Korea Science & Engineering Foundation. RID="**" ID="**" Partially supported by the Research Management Committee of the University of Newcastle.     

Another Look at the Invariance Principle for Wave Operators

 We give a new proof of the invariance principle for wave operators based on certain well-known properties of Calderón commutators. This method extends the invariance principle to a larger class of admissible functions. In particular, admissible functions need not be monotonic on any interval. Received: 18 March 2002 / Accepted: 5 July 2002 Published online: 22 November 2002 RID="*" ID="*" This work was supported in part by National Science Foundation grant DMS-0100249     

Decay Rates and Probability Estimates¶for Massive Dirac Particles¶in the Kerr–Newman Black Hole Geometry

The Cauchy problem is considered for the massive Dirac equation in the non-extreme Kerr–Newman geometry, for smooth initial data with compact support outside the event horizon and bounded angular momentum. We prove that the Dirac wave function decays in at least at the rate t ^−5/6. For generic initial data, this rate of decay is sharp. We derive a formula for the probability p that the Dirac particle escapes to infinity. For various conditions on the initial data, we show that p = 0, 1 or 0 < p < 1. The proofs are based on a refined analysis of the Dirac propagator constructed in [4]. Received: 20 August 2001 / Accepted: 22 January 2002 RID="*" ID="*"Present address: NWF I – Mathematik, Universit?t Regensburg, 93040 Regensburg, Germany.?E-mail: felix.finster@mathematik.uni-regensburg.de RID="**" ID="**"Research supported by NSERC grant # RGPIN 105490-1998. RID="***" ID="***"Research supported in part by the NSF, Grant No. DMS-0103998. RID="****" ID="****"Research supported in part by the NSF, Grant No. 33-585-7510-2-30.     

Heteroclinic Connections Between Periodic Orbits in Planar Restricted Circular Three-Body Problem – A Computer Assisted Proof

 The restricted circular three-body problem is considered for the following parameter values C=3.03, μ=0.0009537 – the values for the Oterma comet in the Sun-Jupiter system. We present a computer assisted proof of an existence of a homo- and heteroclinic cycle between two Lyapunov orbits and an existence of symbolic dynamics on four symbols built on this cycle. Received: 15 April 2002 / Accepted: 16 May 2002 Published online: 20 January 2003 RID="*" ID="*" Research supported in part by Polish KBN grant 2 P03A 011 18, NSF grant DGE-98-04459 and PRODYN. Communicated by G. Gallavotti     

Gluing and Wormholes for the Einstein Constraint Equations

 We establish a general gluing theorem for constant mean curvature solutions of the vacuum Einstein constraint equations. This allows one to take connected sums of solutions or to glue a handle (wormhole) onto any given solution. Away from this handle region, the initial data sets we produce can be made as close as desired to the original initial data sets. These constructions can be made either when the initial manifold is compact or asymptotically Euclidean or asymptotically hyperbolic, with suitable corresponding conditions on the extrinsic curvature. In the compact setting a mild nondegeneracy condition is required. In the final section of the paper, we list a number ways this construction may be used to produce new types of vacuum spacetimes. Received: 4 October 2001 / Accepted: 26 July 2002 Published online: 29 October 2002 RID="*" ID="*" Supported by the NSF under Grant PHY-0099373 RID="**" ID="**" Supported by the NSF under Grant DMS-9971975 and at MSRI by NSF grant DMS-9701755 RID="***" ID="***" Supported by the NSF under Grant DMS-9704515     

On the Structure of Stationary Solutions of the Navier-Stokes Equations

 This paper is a supplementary section to [1]. We show that without any additional hypothesis the main result in [1] (Theorem 1) can be considerably strengthened. Note. This paper cannot be read independently of [1]. The numbering of equations, theorems and propositions as well as cross-references used here have to be understood as if this paper were an additional section to [1]. Received: 7 May 2002 / Accepted: 15 October 2002 Published online: 10 February 2003 RID="*" ID="*" Supported in part by the Fonds National Suisse. Communicated by A. Kupiainen     

Potts Model on Infinite Graphs and the Limit of Chromatic Polynomials

 Given an infinite graph 𝔾 quasi-transitive and amenable with maximum degree Δ, we show that reduced ground state degeneracy per site W _r (𝔾, q) of the q-state antiferromagnetic Potts model at zero temperature on 𝔾 is analytic in the variable 1/q, whenever |2Δe ³ /q|<1. This result proves, in an even stronger formulation, a conjecture originally sketched in [12] and explicitly formulated in [16 and 19], based on which a sufficient condition for W _r (𝔾, q) to be analytic at 1/q=0 is that 𝔾 is a regular lattice. Received: 16 January 2002 / Accepted: 17 October 2002 Published online: 18 February 2003 RID="*" ID="*" Partially supported by CNPq (Brazil) RID="**" ID="**" Partially supported by CNR, G.N.F.M. (Italy) Communicated by H. Spohn     

On the Sutherland Spin Model of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">N</Emphasis></Subscript> Type and Its Associated Spin Chain

 The B _N hyperbolic Sutherland spin model is expressed in terms of a suitable set of commuting Dunkl operators. This fact is exploited to derive a complete family of commuting integrals of motion of the model, thus establishing its integrability. The Dunkl operators are shown to possess a common flag of invariant finite-dimensional linear spaces of smooth scalar functions. This implies that the Hamiltonian of the model preserves a corresponding flag of smooth spin functions. The discrete spectrum of the restriction of the Hamiltonian to this spin flag is explicitly computed by triangularization. The integrability of the hyperbolic Sutherland spin chain of B _N type associated with the dynamical model is proved using Polychronakos's ``freezing trick'. Received: 14 February 2002 / Accepted: 19 June 2002 Published online: 10 December 2002 RID="*" ID="*" Corresponding author. E-mail: artemio@fis.ucm.es RID="**" ID="**" On leave of absence from Institute of Mathematics, 3 Tereschenkivska St., 01601 Kyiv-4 Ukraine Communicated by L. Takhtajan     

Sub-Doppler fluorescence on the atomic <Emphasis Type="Italic">D</Emphasis><Subscript>2</Subscript> line of a submicron rubidium-vapor layer

An extremely thin cell (ETC) with the thickness of a Rb atomic vapor layer in the range of 100–300 nm was fabricated. It is demonstrated that a simple laser-diode technique with a single resonant light beam is sufficient to observe separately all of the atomic hyperfine transitions of the D ₂ line of Rb (780 nm) and also allows us to measure the relative transition probabilities of the hyperfine transitions. The onset of collisional self-broadening of the hyperfine transitions as the number density of atoms increases was studied. The detrimental role of the atoms with slow longitudinal velocity in the sub-Doppler response of the Rb ETC is demonstrated by studies in which the cell is tilted from normal incidence of the laser beam. It is also shown that using an ETC allows us to resolve in a moderate external magnetic field the Zeeman splitting of the hyperfine transitions of the ⁸⁷Rb D ₁ transition F _g=1F _e=1,2. Received: 19 February 2003 / Revised version: 4 April 2003 / Published online: 2 June 2003 RID="*" ID="*"Corresponding author. Fax: +374/32-31172, E-mail: david@ipr.sci.am     

Asymptotics of Determinants of Bessel Operators

 For aL ^∞ (ℝ₊)∩L ¹ (ℝ₊) the truncated Bessel operator B _τ (a) is the integral operator acting on L ² [0,τ] with the kernel

Interfaces under ion irradiation: growth and taming of nanostructures

We have investigated the synthesis of nanostructures, as well as the control of their size and location by means of ion beams. The phase separation and interface kinetics under ion irradiation give new possibilities for controlling the growth of nanostructures. Additionally, the chemical decomposition of the host matrix by collisional mixing can contribute to the self-organization of nanostructures, especially at interfaces. It is shown how collisional mixing during ion implantation affects nanocrystal (NC) synthesis and how ion irradiation through NCs modifies their size and size distribution. An analytical expression for solute concentration around an ion-irradiated NC was found, which may be written like the well-known Gibbs–Thomson relation. However, parameters have modified meanings, which has a significant impact on the evolution of NC ensembles. “Inverse Ostwald ripening” of NCs, resulting in an unimodal NC size distribution, is predicted, which has been confirmed experimentally for Au NCs in SiO₂ and by kinetic lattice Monte Carlo simulations. At interfaces, the same ion-irradiation-induced mechanism may result in self-organization of NCs into a thin δ-layer. Collisional decomposition of SiO₂ may enhance the NC δ-layer formation in SiO₂ at the Si/SiO₂ interface. The distance of the self-organized NC δ-layer from the SiO₂/Si interface renders the structure interesting for non-volatile memory applications. Received: 11 November 2002 / Accepted: 12 November 2002 / Published online: 4 April 2003 RID="*" ID="*"Corresponding author. Fax: +49-351-260-3285, E-mail: K.-H.Heinig@fz-rossendorf.de     

Density of States for Random Band Matrices

 By applying the supersymmetric approach we rigorously prove smoothness of the averaged density of states for a three dimensional random band matrix ensemble, in the limit of infinite volume and fixed band width. We also prove that the resulting expression for the density of states coincides with the Wigner semicircle with a precision 1/W ² , for W large but fixed. Received: 6 February 2002 / Accepted: 17 July 2002 Published online: 7 November 2002 RID="*" ID="*" Supported by NSF grant DMS 9729992     

Thermodynamic and kinetic stability with respect to hole trapping. of silanic bonds at the Si–SiO<Subscript>2</Subscript> interface

The thermodynamic and kinetic stabilities of the Si–H bonds at the Si–SiO₂ interface are studied on the basis of high-level quantum-mechanical calculations in the framework of density-functional theory. In the absence of an applied electric field, the silanic bond is shown to be stable with respect to both hole capture from the top of the silicon valence band and electron loss to the bottom of the silicon conduction band, but unstable with respect to hole capture from the top of the SiO₂ conduction band. The positively charged hydrogen does not shift spontaneously to protonate a neighbouring siloxanic bridge unless it contains one adsorbed water molecule at least. The protonated siloxanic site thus formed may restore the original silanic site (via simultaneous electron capture from the conduction band and hydron shift to silicon) but also evolve spontaneously to a hydrogen atom via simple electron capture. Received: 2 November 2001 / Accepted: 6 January 2002 / Published online: 20 March 2002 / Published online: 20 March 2002 RID="*" ID="*"Also at: STMicroelectronics, 20041 Agrate MI, Italy (E-mail: gianfranco.cerofolini@st.com)     

Non-Topological Multi-Vortex Solutions to the Self-Dual Chern-Simons-Higgs Equation

 In this article, we construct self-dual N-vortex solutions with a large magnetic flux Φ of (2+1)-dimensional relativistic Chern-Simons model, provided that the coupling constant κ is small and the cites of vorticity satisfies

Low-threshold field electron emission of Si micro-tip arrays produced by laser ablation

Low-threshold field electron emission (FEE) is reported for periodic arrays of micro-tips produced by laser ablation of Si wafers. The best samples show emission at threshold fields as low as 4–5 V/μm for n-type Si substrates and of 1–2 V/μm for p-doped Si substrates, as measured with a flat-screen technique. Auger electron spectroscopy and X-ray electron spectroscopy reveal island-like deviation of the SiO₂ stoichiometry on the tip surfaces, with lateral dimensions of less than 100 nm. Microscopic studies using a special field-emission STM show that the emission originates from well-conducting regions of sub-micron size. The experimental data suggest FEE from the tip arrays by a geometric field enhancement of both the individual micro-tip and the narrow conducting channels in the tip body. Received: 3 May 2002 / Accepted: 1 July 2002 / Published online: 28 October 2002 RID="*" ID="*"Corresponding author. Fax: +7-095/135-82-34, E-mail: shafeev@kapella.gpi.ru     

Equilibrium Fluctuations for Interacting Ornstein-Uhlenbeck Particles

 We study the hydrodynamic density fluctuations of an infinite system of interacting particles on ℝ ^d . The particles interact between them through a two body superstable potential, and with a surrounding fluid in equilibrium through a random viscous force of Ornstein-Uhlenbeck type. The stationary initial distribution is the Gibbs measure associated with the potential and with a given temperature and fugacity. We prove that the time-dependent density fluctuation field converges in law, under diffusive scaling of space and time, to the solution of a linear stochastic partial differential equation driven by white noise. Received: 10 July 2001 / Accepted: 9 September 2002 Published online: 8 January 2003 RID="*" ID="*" We thank J. Fritz for fruitful discussions, in particular about the existence of the infinite dynamics. A special thanks to L. Bertini for help in the proof of the spectral gap estimate (cf. Appendix B). Communicated by H. Spohn