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     

sh-Lie Algebras Induced by Gauge Transformations

Traditionally symmetries of field theories are encoded via Lie group actions, or more generally, as Lie algebra actions. A significant generalization is required when “gauge parameters” act in a field dependent way. Such symmetries appear in several field theories, most notably in a “Poisson induced” class due to Schaller and Strobl [SS94] and to Ikeda [Ike94], and employed by Cattaneo and Felder [CF99] to implement Kontsevich's deformation quantization [Kon97]. Consideration of “particles of spin > 2” led Berends, Burgers and van Dam [Bur85,BBvD84,BBvD85] to study “field dependent parameters” in a setting permitting an analysis in terms of smooth functions. Having recognized the resulting structure as that of an sh-Lie algebra (L _∞-algebra), we have now formulated such structures entirely algebraically and applied it to a more general class of theories with field dependent symmetries. Received: 14 December 2000 / Accepted: 8 February 2002?Published online: 2 October 2002     

Global Regularity of Wave Maps from R 3+1 to Surfaces

 We consider Wave Maps with smooth compactly supported initial data of small -norm from R ³⁺¹ to certain 2-dimensional Riemannian manifolds and show that they stay smooth globally in time. Our methods are based on the introduction of a global Coulomb Gauge as in [17], followed by a dynamic separation as in [8]. We then rely on an adaptation of T. Tao's methods used in his recent breakthrough result [24]. Received: 16 March 2002 / Accepted: 31 January 2003 Published online: 5 May 2003 Communicated by P. Constantin     

Dielectric losses in statistical mixtures

The specific features of the dielectric spectra of statistical mixtures in the form of heterogeneous systems with spherical particles chaotically arranged in the space have been investigated. The distribution function of relaxation times f(τ) has been restored. It has been established that the relaxation times are continuously distributed within a wide interval [τ₁, τ₂]. Different methods for broadening the relaxation time distribution interval and approximating the relaxation time distribution function f(τ) have been analyzed. It has been demonstrated that f(τ) is a nonmonotonic function with two maxima at the boundaries and a minimum in the vicinity of the midpoint of the interval [τ₁, τ₂]. These features of the relaxation time distribution function are responsible for the large difference between the average relaxation frequencies of the permittivity and the dielectric loss (electrical conductivity).     

Stability and Asymptotic Stability for Subcritical gKdV Equations

 We prove in this paper the stability and asymptotic stability in H ¹ of a decoupled sum of N solitons for the subcritical generalized KdV equations The proof of the stability result is based on energy arguments and monotonicity of the local L ² norm. Note that the result is new even for p=2 (the KdV equation). The asymptotic stability result then follows directly from a rigidity theorem in [16]. Received: 8 October 2001 / Accepted: 2 July 2002 Published online: 14 October 2002     

Twisted Index Theory on Good Orbifolds, II:¶Fractional Quantum Numbers

This paper uses techniques in noncommutative geometry as developed by Alain Connes [Co2], in order to study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group, continuing our earlier work [MM]. We also compute the range of the higher cyclic traces on K-theory for cocompact Fuchsian groups, which is then applied to determine the range of values of the Connes–Kubo Hall conductance in the discrete model of the quantum Hall effect on the hyperbolic plane, generalizing earlier results in [Bel+E+S], [CHMM]. The new phenomenon that we observe in our case is that the Connes–Kubo Hall conductance has plateaux at integral multiples of a fractional valued topological invariant, namely the orbifold Euler characteristic. Moreover the set of possible fractions has been determined, and is compared with recently available experimental data. It is plausible that this might shed some light on the mathematical mechanism responsible for fractional quantum numbers. Received: 4 November 1999 / Accepted: 22 September 2000     

Vertex Operator Solutions to the Discrete KP-Hierarchy

Vertex operators, which are disguised Darboux maps, transform solutions of the KP equation into new ones. In this paper, we show that the bi-infinite sequence obtained by Darboux transforming an arbitrary KP solution recursively forward and backwards, yields a solution to the discrete KP-hierarchy. The latter is a KP hierarchy where the continuous space $x$-variable gets replaced by a discrete $n$-variable. The fact that these sequences satisfy the discrete KP hierarchy is tantamount to certain bilinear relations connecting the consecutive KP solutions in the sequence. At the Grassmannian level, these relations are equivalent to a very simple fact, which is the nesting of the associated infinite-dimensional planes (flag). The discrete KP hierarchy can thus be viewed as a container for an entire ensemble of vertex or Darboux generated KP solutions. It turns out that many new and old systems lead to such discrete (semi-infinite) solutions, like sequences of soliton solutions, with more and more solitons, sequences of Calogero–Moser systems, having more and more particles, just to mention a few examples; this is developed in [3]. In this paper, as another example, we show that the q-KP hierarchy maps, via a kind of Fourier transform, into the discrete KP hierarchy, enabling us to write down a very large class of solutions to the q-KP hierarchy. This was also reported in a brief note [4]. Received: 27 August 1998 / Accepted: 24 November 1998     

Log-Infinitely Divisible Multifractal Processes

 We define a large class of multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal Multifractal Random Walk processes (MRW) [33, 3] and the log-Poisson ``product of cylindrical pulses' [7]. Their construction involves some ``continuous stochastic multiplication' [36] from coarse to fine scales. They are obtained as limit processes when the finest scale goes to zero. We prove the existence of these limits and we study their main statistical properties including non-degeneracy, convergence of the moments and multifractal scaling. Received: 8 July 2002 / Accepted: 17 December 2002 Published online: 14 April 2003 Communicated by A. Connes     

Scaling Limits of Random Skew Plane Partitions with Arbitrarily Sloped Back Walls

The paper studies scaling limits of random skew plane partitions confined to a box when the inner shapes converge uniformly to a piecewise linear function V of arbitrary slopes in [−1, 1]. It is shown that the correlation kernels in the bulk are given by the incomplete Beta kernel, as expected. As a consequence it is established that the local correlation functions in the scaling limit do not depend on the particular sequence of discrete inner shapes that converge to V. A detailed analysis of the correlation kernels at the top of the limit shape, and of the frozen boundary is given. It is shown that depending on the slope of the linear section of the back wall, the system exhibits behavior observed in either Okounkov and Reshetikhin (Commun Math Phys 269(3):571–609, 2007) or Boutillier et al. ( [math-ph], 2009).     

Schlesinger Transformations and Quantum R-Matrices

 Schlesinger transformations are discrete monodromy preserving symmetry transformations of a meromorphic connection which shift by integers the eigenvalues of its residues. We study Schlesinger transformations for twisted -valued connections on the torus. A universal construction is presented which gives the elementary two-point transformations in terms of Belavin's elliptic quantum R-matrix. In particular, the role of the quantum deformation parameter is taken by the difference of the two poles whose residue eigenvalues are shifted. Elementary one-point transformations (acting on the residue eigenvalues at a single pole) are constructed in terms of the classical elliptic r-matrix. The action of these transformations on the τ-function of the system may completely be integrated and we obtain explicit expressions in terms of the parameters of the connection. In the limit of a rational R-matrix, our construction and the τ-quotients reduce to the classical results of Jimbo and Miwa in the complex plane. Received: 19 December 2001 / Accepted: 20 May 2002 Published online: 14 October 2002     

Brownian Motion with Restoring Drift: Micro-canonical Ensemble and the Thermodynamic Limit

 We take up the old problem of micro-canonical conditioning in the context of diffusion. Starting with a potential , the Schr?dinger operator with ground state is carried by a conjugation into the diffusion generator with invariant density . The latter motion is made micro-canonical by first conditioning the path to be periodic, , and then further conditioning on the empirical mean-square or ``particle number' . The thermodynamics are then studied by taking while D remains fixed. The problem in this form owes its inception to McKean-Vaninsky \cite{MV2} who obtained the following result. For with , they showed the same type of diffusion appears in the thermodynamic limit, but with drift arising from the shifted potential being such that the limiting mean-square equals D. Their method of proof predicts the same outcome for , so long as D is smaller than the canonical mean-square , while if , the matter was unresolved. The purpose of this note is to show a type of phase transition takes place in this case: the conditioning is overcome in the limit and one sees the original (stationary) diffusion on the line. The proof employs an entropy inequality due to Csiszár [1]. Received: 9 January 2001 / Accepted: 17 July 2002 Published online: 14 October 2002     

Passive Advection and the Degenerate Elliptic Operators <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

 We prove estimates for the stationary state n-point functions at zero molecular diffusivity in the Kraichnan model [13]. This is done by proving upper bounds for the heat kernels and Green's functions of the degenerate elliptic operators M _n that occur in the Hopf equations for the n-point functions. Received: 25 August 2001 / Accepted: 30 September 2002 Published online: 20 January 2003 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     

Risk analysis in investment appraisal based on the Monte Carlo simulation technique by A. Hacura, M. Jadamus-Hacura and A. Kocot

Comment on Eur. Phys. J. B 20, 551 (2001) Since Hertz major work on investment appraisal using the Monte Carlo Simulation technique, the so called “Risk Analysis” has become a standard tool for supporting investment decisions [1,2]. A main problem in investment appraisal is to consider and specify the risk of investment projects in an appropriate way, for enabling consistent project evaluation. In calculating a risky project's net present value (NPV) the major difficulty is to quantify the project's risk for quantifying an appropriate risk adjusted discount rate (RADR). Theoretically not founded risk adjusted discount rates face a lot of critique. Furthermore it is discussed that the incorporation of a constant risk factor into the discount rate makes a certain assumption about the resolution of uncertainty over time [3] and finally that a single net present value could not in general reflect risk properly. Especially in consequence of the last point the proponents of simulation argue that a whole distribution of net present values shows a project's risk better than a single number. In the special issue “Econophysics” of this journal Hacura et al. tried to describe the methodology and use of Monte Carlo Simulation in investment appraisal [4]. The purpose of this comment is to point out three fundamental flaws in that article. Received 29 April 2002 Published online 19 December 2002 RID="a" ID="a"e-mail: robert.obermaier@wiwi.uni-regensburg.de     

High refractive index and transparent heavy metal oxide glassy thin films

Heavy metal oxide thin films of the ternary system Nb₂O₅–GeO₂–PbO have been prepared by pulsed laser deposition in an O₂ environment from either glassy or crystalline bulk samples. The range of ([Pb]+[Nb]) content in which the films are optically homogeneous and transparent is much broader (0.5–1.0) than that of the bulk samples considered in the present work (0.55–0.62). The imaginary part of the refractive index is very low in all cases (k<10^-3), whereas the real part increases linearly with the ([Pb]+[Nb]) content up to values as high as 2.35. The optical energy gap has been found to be strongly dependent on [Pb], whereas it is almost independent of [Nb]. This dependence is discussed in terms of the role of Pb and Nb as network modifiers or formers. Received: 5 August 2002 / Accepted: 8 August 2002 / Published online: 17 December 2002 RID="*" ID="*"Corresponding author. Fax: +34-91/564-5557, E-mail: j.gonzalo@io.cfmac.csic.es     

Symmetric Simple Exclusion Process:¶Regularity of the Self-Diffusion Coefficient

We prove that the self-diffusion coefficient of a tagged particle in the symmetric exclusion process in Z ^d , which is in equilibrium at density α, is of class C ^∞ as a function of α in the closed interval [0,1]. The proof provides also a recursive method to compute the Taylor expansion at the boundaries. Received: 6 December 2000 / Accepted: 6 April 2001     

Identification of the new isotope 241Bk

A new neutron-deficient berkelium isotope ²⁴¹Bk produced in the ²³⁹Pu(⁶Li, 4n) reaction has been identified using a gas-jet coupled on-line isotope separator. Cm K and L X-rays associated with the EC decay of ²⁴¹Bk were observed in the mass-241 fraction, and three γ transitions were attributed to the EC decay of ²⁴¹Bk through X-γ coincidences. The half-life of ²⁴¹Bk was determined to be 4.6±0.4 min which is 1/2-1/4 of that of theoretical predictions. The half-life value and the observed γ transitions can be consistently explained as a consequence of the allowed EC transition of π7/2⁺[633] → ν7/2⁺[624]. Received: 14 October 2002 / Accepted: 6 November 2002 / Published online: 17 January 2003 RID="a" ID="a"e-mail: asai@tandem.tokai.jaeri.go.jp Communicated by J. ?yst?     

Power Law Subordinacy and Singular Spectra.¶II. Line Operators

We study Hausdorff-dimensional spectral properties of certain “whole-line” quasiperiodic discrete Schr?dinger operators by using the extension of the Gilbert–Pearson subordinacy theory that we previously developed in [19]. Received: 21 October 1999 / Accepted: 21 December 1999