Absolutely Continuous Invariant Measures¶for Piecewise Real-Analytic Expanding Maps¶on the Plane

We prove the existence of absolutely continuous invariant measures for piecewise real-analytic expanding maps on bounded regions in the plane. Received: 5 June 1998 / Accepted: 11 May 1999     

Laplace–Beltrami Operator on a Riemann Surface¶and Equidistribution of Measures

We consider the Laplace–Beltrami operator on a compact Riemann surface of a constant negative curvature. For any eigenvalue of the Laplace–Beltrami operator there is an associated sequence of measures on the Riemann surface. These measures naturally appear in Quantum Chaos type questions in the theory of electro-magnetic flow on a Riemann surface. The main result of the paper is the claim that this sequence of measures has the Liouville measure as the (weak^*) limit. We prove a quantitative version of this equidistribution claim. Received: 12 March 2001 / Accepted: 23 April 2001     

No or Infinitely Many A.C.I.P.¶for Piecewise Expanding Cr Maps¶in Higher Dimensions

We modify Tsujii's example [9] to show that in contrast with the one-dimensional case, piecewise uniformly expanding and C ^r maps of the plane may: (1) either have no absolutely continuous invariant probability measures (a.c.i.p. for short) and be such that {\bf every point} is statistically attracted to a fixed repelling point;? (2) or have infinitely many ergodic a.c.i.p. Received: 6 September 2000 / Accepted: 15 May 2001     

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

In this paper we further develop the theory of α-induction for nets of subfactors, in particular in view of the system of sectors obtained by mixing the two kinds of induction arising from the two choices of braiding. We construct a relative braiding between the irreducible subsectors of the two “chiral” induced systems, providing a proper braiding on their intersection. We also express the principal and dual principal graphs of the local subfactors in terms of the induced sector systems. This extended theory is again applied to conformal or orbifold embeddings of SU(n WZW models. A simple formula for the corresponding modular invariant matrix is established in terms of the two inductions, and we show that it holds if and only if the sets of irreducible subsectors of the two chiral induced systems intersect minimally on the set of marked vertices, i.e. on the “physical spectrum” of the embedding theory, or if and only if the canonical endomorphism sector of the conformal or orbifold inclusion subfactor is in the full induced system. We can prove either condition for all simple current extensions of SU _{( n )} and many conformal inclusions, covering in particular all type I modular invariants of SU(2) and SU(3), and we conjecture that it holds also for any other conformal inclusion of SU _{( n )} as well. As a by-product of our calculations, the dual principal graph for the conformal inclusion SU(3)₅⊂SU(6)₁ is computed for the first time. Received: 24 December 1998 / Accepted: 22 February 1999     

Discussion notes on “Measures of wettability of solid surfaces”, by A. Marmur

Comments are provided for [7].     

The Action of Outer Automorphisms on Bundles¶of Chiral Blocks

On the bundles of WZW chiral blocks over the moduli space of a punctured rational curve we construct isomorphisms that implement the action of outer automorphisms of the underlying affine Lie algebra. These bundle-isomorphisms respect the Knizhnik–Zamolodchikov connection and have finite order. When all primary fields are fixed points, the isomorphisms are endomorphisms; in this case, the bundle of chiral blocks is typically a reducible vector bundle. A conjecture for the trace of such endomorphisms is presented; the proposed relation generalizes the Verlinde formula. Our results have applications to conformal field theories based on non-simply connected groups and to the classification of boundary conditions in such theories. Received: 11 May 1998 / Accepted: 17 April 1999     

A Spin-Statistics Theorem for Quantum Fields¶on Curved Spacetime Manifolds¶in a Generally Covariant Framework

A model-independent, locally generally covariant formulation of quantum field theory over four-dimensional, globally hyperbolic spacetimes will be given which generalizes similar, previous approaches. Here, a generally covariant quantum field theory is an assignment of quantum fields to globally hyperbolic spacetimes with spin-structure where each quantum field propagates on the spacetime to which it is assigned. Imposing very natural conditions such as local general covariance, existence of a causal dynamical law, fixed spinor- or tensor type for all quantum fields of the theory, and that the quantum field on Minkowski spacetime satisfies the usual conditions, it will be shown that a spin-statistics theorem holds: If for some of the spacetimes the corresponding quantum field obeys the “wrong” connection between spin and statistics, then all quantum fields of the theory, on each spacetime, are trivial. Received: 1 March 2001 / Accepted: 28 May 2001     

Comments on “A new analysis of the paradox of nonsymmetry for the volume relaxation of glasses…,” by J. P. Ibar

The paper of Ibar [1] is mainly devoted to the simulation of the asymmetrical aspects of some volume recovery isotherms reported by us previously [2], using an ad hoc mathematical model. The present comments deal (1) with the possible relationship of this model to that recently proposed by Kovacs, Aklonis, Hutchinson, and Ramos (KAHR) [3]; and (2) with the asymmetry and the autocatalytic nature of the expansion isotherms.     

New Erdős-Kac Type Theorems for Signed Measures on Square-Free Integers

We consider a family of signed measures supported on the set of square-free numbers. We prove some local limit theorems for the prime divisor counting function ω(n) and establish new Erd?s-Kac type results.     

Comment on “The growth of Fe overlayers on Ag(100)” by G.C. Smith,H.A. Padmore and C. Norris

Smith, Padmore and Norris have claimed that the first three monolayers of Fe grown at room temperature on Ag(100) form a strained fcc γ-Fe layer for which the mismatch to the substrate is 11%. We point out a simple error in their calculations of mismatch, and suggest instead that such growth produces a bcc α-Fe(100) film, for which the mismatch to the substrate is only 0.8%.     

Comment on ‘Structure and diffusion of clusters on Ni surfaces by C.-L. Liu and J.B. Adams’

We calculate the dissociation energy of Ni clusters containing up to 13 atoms adsorbed on Ni(111), in order to elucidate its dependence on the cluster size. As opposed to what is claimed by Liu and Adams [Surf. Sci. 268 (1992) 73] the dissociation energy of Ni/Ni(111) does not display a marked peak for N =5, its value for this cluster size being indeed very close to what is found for N = 4 and N = 6.     

Trace Construction of a Basis for the Solution Space¶of sl N qKZ Equation

The trace of intertwining operators over the level one irreducible highest weight modules of the quantum affine algebra of type A_Nу⁽¹⁾ is studied. It is proved that the trace function gives a basis of the solution space of the qKZ equation at a generic level. The highest-highest matrix elements of the composition of intertwining operators are explicitly determined as rational functions up to an overall scalar function. The integral formula for the trace is presented.     

On the Hausdorff Dimension of Invariant Measures of Weakly Contracting on Average Measurable IFS

We consider measures which are invariant under a measurable iterated function system with positive, place-dependent probabilities in a separable metric space. We provide an upper bound of the Hausdorff dimension of such a measure if it is ergodic. We also prove that it is ergodic iff the related skew product is.     

Comment on¶“On the relationship between the entropy and enthalpy of interfacial segregation” by L. Rubinovich and M. Polak

The paper of L. Rubinovich and M. Polak [Eur. Phys. J. B 22, 267 (2001)] contains some controversial items. The notation “excess” in connection with various thermodynamic quantities describing solute segregation on the basis of the Langmuir-McLean isotherm is used in a meaning contradictory to general thermodynamics. The approach used to interpret the linear relationship between segregation entropy and enthalpy results in rather strange conclusions. These items are discussed, and it is shown how a formulation with standard terms is in accordance with general thermodynamics. Received 14 January 2002 / Received in final form 27 September 2002 Published online 14 February 2003 RID="a" ID="a"e-mail: lejcekp@fzu.cz     

Comments on “the J.C. Phillips model for vitreous SiO2: A critical appraisal”

Under the above title Galeener and Wright (GW) have criticized [1] a lengthy review article by me [2]. Their six-page discussion includes some cogent observations but also a number of misinterpretations of my model which I wish to correct here. I also wish to draw the reader's attention to recent refinements of my model not discussed by GW.