Symmetry Breaking and Other Phenomena in the Optimization of Eigenvalues for Composite Membranes

We consider the following eigenvalue optimization problem: Given a bounded domain Ω⊂ℝ and numbers α > 0, A∈[ 0, |Ω|], find a subset D⊂Ω of area A for which the first Dirichlet eigenvalue of the operator −Δ+αχ _D is as small as possible. We prove existence of solutions and investigate their qualitative properties. For example, we show that for some symmetric domains (thin annuli and dumbbells with narrow handle) optimal solutions must possess fewer symmetries than Ω on the other hand, for convex Ω reflection symmetries are preserved. Also, we present numerical results and formulate some conjectures suggested by them. Received: 22 November 1999/ Accepted: 31 March 2000     

On the Point-Particle (Newtonian) Limit¶of the Non-Linear Hartree Equation

We consider the nonlinear Hartree equation describing the dynamics of weakly interacting non-relativistic Bosons. We show that a nonlinear M?ller wave operator describing the scattering of a soliton and a wave can be defined. We also consider the dynamics of a soliton in a slowly varying background potential W(ɛx). We prove that the soliton decomposes into a soliton plus a scattering wave (radiation) up to times of order ɛ⁻¹. To leading order, the center of the soliton follows the trajectory of a classical particle in the potential W(ɛx). Received: 30 June 2000 / Accepted: 25 June 2001     

Construction of the Incipient Infinite Cluster for Spread-out Oriented Percolation Above 4 + 1 Dimensions

 We construct the incipient infinite cluster measure (IIC) for sufficiently spread-out oriented percolation on ℤ ^d × ℤ₊, for d +1 > 4+1. We consider two different constructions. For the first construction, we define ℙ ⁿ (E) by taking the probability of the intersection of an event E with the event that the origin is connected to (x,n)  ℤ ^d × ℤ₊, summing this probability over x  ℤ ^d , and normalising the sum to get a probability measure. We let n → ∞ and prove existence of a limiting measure ℙ_∞, the IIC. For the second construction, we condition the connected cluster of the origin in critical oriented percolation to survive to time n, and let n → ∞. Under the assumption that the critical survival probability is asymptotic to a multiple of n ⁻¹, we prove existence of a limiting measure ℚ_∞, with ℚ_∞ = ℙ_∞. In addition, we study the asymptotic behaviour of the size of the level set of the cluster of the origin, and the dimension of the cluster of the origin, under ℙ_∞. Our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super-Brownian motion, for d+1 > 4+1. Received: 13 December 2001 / Accepted: 11 July 2002 Published online: 29 October 2002 RID="*" ID="*" Present address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: rhofstad@win.tue.nl     

On the Stability of Double Homoclinic Loops

We consider 2-degrees of freedom Hamiltonian systems with an involutive symmetry and a pair of orbits bi-asymptotic (homoclinic) to a saddle-center equilibrium (related to pairs of pure real, ±ν, and pure imaginary eigenvalues, ±ω i). We show that the stability of this double homoclinic loop is determined by the reflection coefficient of a one-dimensional scattering problem and ω/ν. We also show that the mechanism for losing stability is the creation of an infinite heteroclinic chain connecting a sequence of periodic orbits that accumulates at the double loop. Received: 10 November 1995 / Accepted: 5 June 1996     

Invasion Percolation and the Incipient Infinite Cluster in 2D

 We establish two links between two-dimensional invasion percolation and Kesten's incipient infinite cluster (IIC). We first prove that the k ^th moment of the number of invaded sites within the box [−n, n]×[−n, n] is of order (n ²π _n ) ^k , for k≥1, where π _n is the probability that the origin in critical percolation is connected to the boundary of a box of radius n. This improves a result of Y. Zhang. We show that the size of the invaded region, when scaled by n ²π _n , is tight. Secondly, we prove that the invasion cluster looks asymptotically like the IIC, when viewed from an invaded site v, in the limit |v|→∞. We also establish this when an invaded site v is chosen at random from a box of radius n, and n→∞. Received: 3 December 2000 / Accepted: 3 December 2002 Published online: 18 February 2003 RID="⋆" ID="⋆" Present address: CWI, PNA 3, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands. E-mail:jarai@cwi.nl Communicated by M. Aizenman     

Production of φ mesons in near-threshold πN and NN reactions

We analyze the production of φ mesons in πN and NN reactions in the near-threshold region, using throughout the conventional “non-strange” dynamics based on such processes which are allowed by the non-ideal ω-φ mixing. We show that the occurrence of the direct φNN interaction may show up in different unpolarized and polarization observables in πN→Nφ reactions. We find a strong non-trivial difference between observables in the reactions pp→ppφ and pn→pnφ caused by the different role of the spin singlet and triplet states in the entrance channel. A series of predictions for the experimental study of this effect is presented. Received: 27 January 2000     

Conformal Subnets and Intermediate Subfactors

 Given an irreducible local conformal net 𝒜 of von Neumann algebras on S ¹ and a finite-index conformal subnet ℬ⊂𝒜, we show that 𝒜 is completely rational iff ℬ is completely rational. In particular this extends a result of F. Xu for the orbifold construction. By applying previous results of Xu, many coset models turn out to be completely rational and the structure results in [27] hold. Our proofs are based on an analysis of the net inclusion ℬ⊂𝒜; among other things we show that, for a fixed interval I, every von Neumann algebra  intermediate between ℬ(I) and 𝒜(I) comes from an intermediate conformal net ℒ between ℬ and 𝒜 with ℒ(I)=. We make use of a theorem of Watatani (type II case) and Teruya and Watatani (type III case) on the finiteness of the set ℑ(𝒩,ℳ) of intermediate subfactors in an irreducible inclusion of factors 𝒩⊂ℳ with finite Jones index [ℳ:𝒩]. We provide a unified proof of this result that gives in particular an explicit bound for the cardinality of ℑ(𝒩,ℳ) which depends only on [ℳ:𝒩]. Received: 21 December 2001 / Accepted: 28 February 2002 Published online: 14 March 2003 RID="⋆" ID="⋆" Supported in part by MIUR and INDAM-GNAMPA. Communicated by K. Fredenhagen     

The Stability of Magnetic VorticesRID="*"ID="*"Research on this paper was supported by NSERC under grant N7901

We study the linearized stability of n-vortex (n∈ℤ) solutions of the magnetic Ginzburg–Landau (or Abelian Higgs) equations. We prove that the fundamental vortices (n = ± 1) are stable for all values of the coupling constant, λ, and we prove that the higher-degree vortices (|n|≥ 2) are stable for λ < 1, and unstable for λ > 1. This resolves a long-standing conjecture (see, eg, [JT]). Received: 16 November 1998 / Accepted: 3 January 2000 RID="*" ID="*"Research on this paper was supported by NSERC under grant N7901 RID="**" ID="**"Present address: Courant Institute, 251 Mercer St., New York, NY 10012, USA.&para;E-mail: gustaf&commat;cims.nyu.edu     

Exponential Mixing of the 2D Stochastic Navier-Stokes Dynamics

 We consider the Navier-Stokes equation on a two dimensional torus with a random force which is white noise in time, and excites only a finite number of modes. The number of excited modes depends on the viscosity ν, and grows like ν ^-3 when ν goes to zero. We prove that this Markov process has a unique invariant measure and is exponentially mixing in time. Received: 14 March 2002 / Accepted: 7 May 2002 Published online: 22 August 2002     

On Convergence to Equilibrium Distribution,I.¶The Klein–Gordon Equation with Mixing

Consider the Klein–Gordon equation (KGE) in ℝ ⁿ , n≥ 2, with constant or variable coefficients. We study the distribution μ _t of the random solution at time t∈ℝ. We assume that the initial probability measure μ₀ has zero mean, a translation-invariant covariance, and a finite mean energy density. We also assume that μ₀ satisfies a Rosenblatt- or Ibragimov–Linnik-type mixing condition. The main result is the convergence of μ _t to a Gaussian probability measure as t→∞ which gives a Central Limit Theorem for the KGE. The proof for the case of constant coefficients is based on an analysis of long time asymptotics of the solution in the Fourier representation and Bernstein's “room-corridor” argument. The case of variable coefficients is treated by using an “averaged” version ofthe scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay. Received: 4 January 2001 / Accepted: 2 July 2001     

q-quantum plane,ψ(q)-umbral calculus, and all that

It is known that one can formulateq-extended finite operator calculus with help of “quantumq-plane”q-commuting variablesA, B : AB − qBA ≡ [A, B]q=0. We shall recall this simple fact in its natural entourage which is the so-calledψ(q)-extension of Rota’s finite operator calculus. We aim to convince the audience that this is a natural and elementary method for formulation and treatment ofq-extended and possiblyR-extended orψ(q)-extended models for quantum-likeψ(q)-deformed oscillators. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001.     

One Dimensional Behavior of Singular N Dimensional Solutions of Semilinear Heat Equations

We consider u(x,t) a solution of u _t =Δu+|u| ^{p − 1} u that blows up at time T, where u:ℝ ^N ×[0, T)→ℝ, p>1, (N−2)p<N+2 and either u(0)≥ 0 or (3N−4)p<3N+8. We are concerned with the behavior of the solution near a non isolated blow-up point, as T−t→ 0. Under a non-degeneracy condition and assuming that the blow-up set is locally continuous and N−1 dimensional, we escape logarithmic scales of the variable T−t and give a sharper expansion of the solution with the much smaller error term (T−t)^{1, 1/2−η} for any η>0. In particular, if in addition p>3, then the solution is very close to a superposition of one dimensional solutions as functions of the distance to the blow-up set. Finally, we prove that the mere hypothesis that the blow-up set is continuous implies that it is C ^{1, 1/2−η} for any η>0. Received: 20 June 2001 / Accepted: 6 October 2001     

Quasiclassical Expansion for Tr { (−1) F e − β H }

We start with some methodic remarks referring to purely bosonic quantum systems and then explain how corrections to the leading-order quasiclassical result for the fermion-graded partition function Tr { (−1) ^F e ^{− β H} } can be calculated at small β. We perform such a calculation for certain supersymmetric quantum mechanical systems where such corrections are expected to appear. We consider in particular supersymmetric Yang–Mills theory reduced to (0 + 1) dimensions and were surprised to find that the correction ∝ β² vanishes in this case. We discuss also a nonstandard N=2 supersymmetric σ-model defined on S ³ and show that the quasiclassical expansion breaks down for this system. Received: 12 December 2001 / Accepted: 29 January 2002?Published online: 11 September 2002     

Study of FCNC mediated Z boson effectin the semileptonic rare baryonic decays Λ b → Λl + l -

We study the effect of the FCNC mediated Z boson in the rare semileptonic baryonic decays Λ_b → Λl⁺l^-. We consider the model where the standard model fermion sector is extended by an extra vector-like down quark, as a consequence of which it allows for CP-violating Z mediated flavor changing neutral current at the tree level. We find that due to this non-universal Zbs coupling, the branching ratios of the rare semileptonic Λ_b decays are enhanced reasonably from their corresponding standard model values and the zero point of the forward-backward asymmetry for Λ_b → Λ_μ⁺_μ^- is shifted to the left. Received: 2 June 2005, Published online: 26 October 2005 PACS: 13.30.Ce, 12.60.-i, 11.30.Hv     

Communication activity in social networks: growth and correlations

We investigate the timing of messages sent in two online communities with respect to growth fluctuations and long-term correlations. We find that the timing of sending and receiving messages comprises pronounced long-term persistence. Considering the activity of the community members as growing entities, i.e. the cumulative number of messages sent (or received) by the individuals, we identify non-trivial scaling in the growth fluctuations which we relate to the long-term correlations. We find a connection between the scaling exponents of the growth and the long-term correlations which is supported by numerical simulations based on peaks over threshold. In addition, we find that the activity on directed links between pairs of members exhibits long-term correlations, indicating that communication activity with the most liked partners may be responsible for the long-term persistence in the timing of messages. Finally, we show that the number of messages, M, and the number of communication partners, K, of the individual members are correlated following a power-law, K ~ M ^λ , with exponent λ ≈ 3 / 4.     

The Existence and Stability of Noncommutative Scalar Solitons

 We establish existence and stability results for solitons in noncommutative scalar field theories in even space dimension 2d. In particular, for any finite rank spectral projection P of the number operator 𝒩 of the d-dimensional harmonic oscillator and sufficiently large noncommutativity parameter θ we prove the existence of a rotationally invariant soliton which depends smoothly on θ and converges to a multiple of P as θ→∞. In the two-dimensional case we prove that these solitons are stable at large θ, if P=P _N , where P _N projects onto the space spanned by the N+1 lowest eigenstates of 𝒩, and otherwise they are unstable. We also discuss the generalisation of the stability results to higher dimensions. In particular, we prove stability of the soliton corresponding to P=P ₀ for all θ in its domain of existence. Finally, for arbitrary d and small values of θ, we prove without assuming rotational invariance that there do not exist any solitons depending smoothly on θ. Received: 13 July 2001 / Accepted: 9 July 2002 Published online: 10 January 2003     

Entropy Production in Nonequilibrium Statistical Mechanics

We consider systems of nonequilibrium statistical mechanics, driven by nonconservative forces and in contact with an ideal thermostat. These are smooth dynamical systems for which one can define natural stationary states μ (SRB in the simplest case) and entropy production e(μ) (minus the sum of the Lyapunov exponents in the simplest case). We give exact and explicit definitions of the entropy production e(μ) for the various situations of physical interest. We prove that e(μ)≥0 and indicate cases where e(μ)>0. The novelty of the approach is that we do not try to compute entropy production directly, but make it depend on the identification of a natural stationary state for the system. Received: 15 July 1996 / Accepted: 30 October 1996     

Possible Loss and Recovery of Gibbsianness¶During the Stochastic Evolution of Gibbs Measures

We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a reversible Gibbs measure μ≠ν. Both ν and μ are assumed to have a translation-invariant finite-range interaction. We study the Gibbsian character of the measure νS(t) at time t and show the following: (1) For all ν and μ, νS(t) is Gibbs for small t. (2) If both ν and μ have a high or infinite temperature, then νS(t) is Gibbs for all t > 0. (3) If ν has a low non-zero temperature and a zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t and non-Gibbs for large t. (4) If ν has a low non-zero temperature and a non-zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t. The regime where μ has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios. Received: 26 April 2001 / Accepted: 10 October 2001     

Infinite slabs and other weird plane symmetric space–times with constant positive density

We present the exact solution of Einstein’s equation corresponding to a static and plane symmetric distribution of matter with constant positive density located below z = 0. This solution depends essentially on two constants: the density ρ and a parameter κ. We show that these space–times finish down below at an inner singularity at finite depth. We show that for κ ≥ 0.3513 . . . the dominant energy condition is satisfied all over the space–time. We match this solution to the vacuum one and compute the external gravitational field in terms of slab’s parameters. Depending on the value of κ, these slabs can be attractive, repulsive or neutral. In the first case, the space–time also finishes up above at an empty repelling singular boundary. In the other cases, they turn out to be semi-infinite and asymptotically flat when z → ∞. We also find solutions consisting of joining an attractive slab and a repulsive one, and two neutral ones. We also discuss how to assemble a “gravitational capacitor” by inserting a slice of vacuum between two such slabs.