Random Matrix Theory and L-Functions at s= 1/2

Recent results of Katz and Sarnak [8, 9] suggest that the low-lying zeros of families of L-functions display the statistics of the eigenvalues of one of the compact groups of matrices U(N), O(N) or USp(2N). We here explore the link between the value distributions of the L-functions within these families at the central point s= 1/2 and those of the characteristic polynomials Z(U,θ) of matrices U with respect to averages over SO(2N) and USp(2N) at the corresponding point θ= 0, using techniques previously developed for U(N) in [10]. For any matrix size N we find exact expressions for the moments of Z(U,0) for each ensemble, and hence calculate the asymptotic (large N) value distributions for Z(U,0) and log Z(U,0). The asymptotic results for the integer moments agree precisely with the few corresponding values known for L-functions. The value distributions suggest consequences for the non-vanishing of L-functions at the central point. Received: 1 February 2000 / Accepted: 24 March 2000     

Mixing Time of Critical Ising Model on Trees is Polynomial in the Height

In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuous-time chain exhibits the following behavior. For some critical inverse-temperature β _c , the inverse-gap is O(1) for β < β _c , polynomial in the surface area for β = β _c and exponential in it for β > β _c . This has been proved for \mathbbZ²{\mathbb{Z}^2} except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph. Recently, the dynamics for the Ising model on a regular tree, also known as the Bethe lattice, has been intensively studied. The facts that the inverse-gap is bounded for β < β _c and exponential for β > β _c were established, where β _c is the critical spin-glass parameter, and the tree-height h plays the role of the surface area. In this work, we complete the picture for the inverse-gap of the Ising model on the b-ary tree, by showing that it is indeed polynomial in h at criticality. The degree of our polynomial bound does not depend on b, and furthermore, this result holds under any boundary condition. We also obtain analogous bounds for the mixing-time of the chain. In addition, we study the near critical behavior, and show that for β > β _c , the inverse-gap and mixing-time are both exp[Θ((β − β _c )h)].     

Temperature of the superconducting transition in the TiV alloy at pressures up to 60.8 GPa

The dependence of the temperature of the transition to the superconducting state T _c on the pressure up to 60.8 GPa is measured for the TiV alloy. The dependence T _c (P) is increasing except for an anomaly in the form of a local minimum near P = 10 GPa. At the maximum pressure of 60.8 GPa, the superconducting transition temperature T _c reaches 18.2 K. The obtained curve T _c (P) is compared with the known dependences for pure vanadium, for which T _c (P) increases to 17.2 K at P = 120 GPa, and for pure niobium and the ZrNb alloy, for which the dependences T _c (P) also have anomalies in the form of local maxima at pressures of 5–10 GPa.     

Annealedn-vectorp-spin model

A disorderedn-vector model withp spin interactions is introduced and studied in mean field theory for the annealed case. We present complete solutions for the casesn=2 andn=3, and have obtained explicit order parameter equations for all the stable solutions for arbitraryn. For alln andp we find one stable high-temperature phase and one stable low-temperature phase. The phase transition is of first order. Forn=2, it is continuous in the order parameters for p4 and has a jump discontinuity in the order parameters ifp>4. Forn=3, it has a jump discontinuity in the order parameters for allp.     

Single Particle Distributions from Veneziano Amplitudes II

The x- and p_⟂²-dependence of the distribution function for the inclusive reaction A + B → C + anything has been investigated for some typical cases using the dual six point function in the tree approximation as a model for T₃₃. It is found that the form the x-distribution depends critically on the trajectory intercepts in the AC̄ and BC̄ channels, respectively. The p_⟂²-dependence is nearly exponential for large x. For small x-values a steeper behaviour in the small p_⟂² region is obtained which is especially pronounced for small masses of the observed particles. Diagrams with twists contributing to the distribution function for inclusive reactions with neutral pions lead to oscillations and dips near the nonsense wrong signature points of the AC̄ and BC̄ channel trajectories.     

The Carter constant for inclined orbits about a massive Kerr black hole: near-circular,near-polar orbits

In an extreme mass-ratio binary black hole system, a non-equatorial orbit will list (i.e. increase its angle of inclination, i) as it evolves in Kerr spacetime. The abutment, a set of evolving, near-polar, retrograde orbits, for which the instantaneous Carter constant (Q) is at its maximum value (Q _X ) for given values of latus rectum (l̃) and eccentricity (e), has been introduced as a laboratory in which the consistency of dQ/dt with corresponding evolution equations for d l̃/dt and de/dt might be tested independently of a specific radiation back-reaction model. To demonstrate the use of the abutment as such a laboratory, a derivation of dQ/dt, based only on published formulae for d l̃/dt and de/dt, was performed for elliptical orbits on the abutment. The resulting expression for dQ/dt matched the published result to the second order in e. We believe the abutment is a potentially useful tool for improving the accuracy of evolution equations to higher orders of e and l̃⁻.     

Hitting Time and Dimension in Axiom A Systems, Generic Interval Exchanges and an Application to Birkoff Sums

In this note we prove that for equilibrium states of axiom A systems having positive dimension the time τ _B (x) needed for a typical point x to enter for the first time in a typical ball B with radius r behaves for small r as τ _B (x)∼ r ^−d where d is the local dimension of the invariant measure at the center of the ball. A similar relation is proved for a full measure set of interval exchanges. Some applications to Birkoff averages of unbounded (and not L ¹) functions are shown.     

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     

Boundary Layer Stability¶in Real Vanishing Viscosity Limit

In the previous paper [20], an Evans function machinery for the study of boundary layer stability was developed. There, the analysis was restricted to strongly parabolic perturbations, that is to an approximation of the form u _t +(F(u)) _x =ν(B(u)u _x ) _x $ (ν≪1) with an “elliptic” matrix B. However, real models, like the Navier–Stokes approximation of the Euler equations for a gas flow, involve incompletely parabolic perturbations: B is not invertible in general. We first adapt the Evans function to this realistic framework, assuming that the boundary is not characteristic, neither for the hyperbolic first order system u _t +(F(u)) _x = 0, nor for the perturbed system. We then apply it to the various kinds of boundary layers for a gas flow. We exhibit some examples of unstable boundary layers for a perfect gas, when the viscosity dominates heat conductivity. Received: 27 November 2000/ Accepted: 16 March 2001     

Fluctuation Bounds for Chaos Plus Noise in Dynamical Systems

We are interested in time series of the form y _n =x _n +ξ _n where {x _n } is generated by a chaotic dynamical system and where ξ _n models observational noise. Using concentration inequalities, we derive fluctuation bounds for the auto-covariance function, the empirical measure, the kernel density estimator and the correlation dimension evaluated along y ₀,…,y _n , for all n. The chaotic systems we consider include for instance the Hénon attractor for Benedicks-Carleson parameters.     

On the Algebraic Structure of Globally or Locally SU(2) Invariant Lattice Field Theories

The local contribution to the action of the O(3) σ model in D = 2 or pure SU(2) gauge models in D ≧ 3 dimensions are expanded and integrated on the group. There results a field of variables j, a 3nj coefficient W({j}) with n → ∞ and dynamical factors f(j, β). We prove that for the gauge models a local decomposition of W({j}) into a product of 3nj coefficients with n = 2D(D – 2) exists. We study generating functions for W({j}) or the 3nj coefficients and develop an algorithm for their computation. Some of these generating functions are explicitly calculated.     

Self-diffusion in liquid metals

A survey is made of all published data on self-diffusion in liquid metals and an examination is carried out on the temperature-dependence of D for studies covering a wide range of temperatures. Log D versus 1/T and D versus T represent the best data equally well, while D versus T ^1/2 and D versus T ² do not. The proper functional form is taken to be D=CT experimentally, and a simple free diffusion theory is developed from the standpoint of vibrational atom motion. An equation having no adjustable parameters results (D=k ² ? _D T/hk), in which κ is the vibrational force constant. Values calculated for D at the melting point are in good agreement with experimental values for potassium, sodium, copper and silver, but are below the measured values for lead, indium, zinc and tin.     

Analytical representation of the density derivative of the Percus–Yevick hard-sphere radial distribution function

ABSTRACT

Explicit analytical expressions are presented for the density derivative, ?g_HS(R; ρ)/?ρ, of the Percus–Yevick approximation to the hard-sphere radial distribution function for R ≤ 6σ, where σ is the hard-sphere diameter and ρ = (N/V)σ³ is the reduced density, where N is the number of particles and V is the volume. A FORTRAN program is provided for the implementation of these for R ≤ 6σ, which includes code for the calculation of g_HS(R; ρ) itself over this range. We also present and incorporate within the program code convenient analytical expressions for the numerical extrapolation of both quantities past R = 6σ. Our expressions are numerically tested against exact results.     

An universal relation between fractal and Euclidean (topological) dimensions of random systems

It is shown that a dimension-invariant form for fractal dimension D of random systems (where d is Euclidean dimension of the embedding space) is in good agreement with results of numerical simulations performed by different authors for critical (p=p _c ) and subcritical (p<p _c ) percolation, for lattice animals, and for different aggregation processes. Received: 9 July 1998 / Revised and Accepted: 12 July 1998     

Lazy structure-activity relationships (lazar) for the prediction of rodent carcinogenicity and Salmonella mutagenicity

lazar is a new tool for the prediction of toxic properties of chemical structures. It derives predictions for query structures from a database with experimentally determined toxicity data. lazar generates predictions by searching the database for compounds that are similar with respect to a given toxic activity and calculating the prediction from their activities. Apart form the prediction, lazar provides the rationales (structural features and similar compounds) for the prediction and a reliable condence index that indicates, if a query structure falls within the applicability domain of the training database. Leave-one-out (LOO) crossvalidation experiments were carried out for 10 carcinogenicity endpoints ({female|male} {hamster|mouse|rat} carcinogenicity and aggregate endpoints {hamster|mouse|rat} carcinogenicity and rodent carcinogenicity) and Salmonella mutagenicity from the Carcinogenic Potency Database (CPDB). An external validation of Salmonella mutagenicity predictions was performed with a dataset of 3895 structures. Leave-one-out and external validation experiments indicate that Salmonella mutagenicity can be predicted with 85% accuracy for compounds within the applicability domain of the CPDB. The LOO accuracy of lazar predictions of rodent carcinogenicity is 86%, the accuracies for other carcinogenicity endpoints vary between 78 and 95% for structures within the applicability domain.     

Scaling assumption for lattice animals in percolation theory

A scaling assumption for the numberg _ns of different cluster configurations with perimeters and sizen leads to the desired cluster numbers near the percolation threshold. The perimeter distribution function has a mean square width proportional ton for largen. The relation between the average perimeter and the cluster sizen for percolation has three different forms atp _c, belowp _c, and abovep _c and is closely related to the shape of the cluster size distribution.     

The Mass of the Ninth Pseudoscalar Meson in the Frame of Broken Chiral SU(3) ⊗ SU(3)

Using GELL -MANN 's ansatz for the SU(3)?SU(3) symmetry breaking part H_SB = -u⁰ -cu⁸ in the strong HAMILTONIAN density, where the operators u^j (j = 0, 1,…8) are the scalar part of a basis for the {(3,3) ⊕ (3,3)} representation of chiral SU(3)?SU(3) and where the constant c is a measure for SU(3) breaking within the SU(3)?SU(3) breaking, a sum rule for the spin zero spectral functions of the pseudoscalar axial vector current octet is derived. Saturating the sum rule with the lowest lying states, the mass of the ninth pseudoscalar meson can be estimated as m_η1 = 950 MeV.     

The W N Minimal Model Classification

We first rigourously establish, for any N ≥ 2, that the toroidal modular invariant partition functions for the (not necessarily unitary) W _N (p, q) minimal models biject onto a well-defined subset of those of the SU(N) × SU(N) Wess-Zumino-Witten theories at level (p − N, q − N). This permits considerable simplifications to the proof of the Cappelli-Itzykson-Zuber classification of Virasoro minimal models. More important, we obtain from this the complete classification of all modular invariants for the W ₃(p, q) minimal models. All should be realised by rational conformal field theories. Previously, only those for the unitary models, i.e. W ₃(p, p + 1), were classified. For all N our correspondence yields for free an extensive list of W _N (p, q) modular invariants. The W ₃ modular invariants, like the Virasoro minimal models, all factorise into SU(3) modular invariants, but this fails in general for larger N. We also classify the SU(3) × SU(3) modular invariants, and find there a new infinite series of exceptionals.     

New results for virial coefficients of hard spheres inD dimensions

We present new results for the virial coefficientsB _k for κ<- 10 for hard spheres in dimensionsD = 2,..., 8.     

Effects of time delay on the statistical fluctuations for a bistable system driven by cross-correlated noises

We study the effects of time delay on the normalized correlation function C(s) and the associated relaxation time T _c for a bistable system with correlations between multiplicative and additive white noises under the condition of small time delay. Using the projection operator method, the expressions of T _c and C(s) are obtained. Based on numerical computations, it is found that the delay time τ slows down the rate of fluctuation decay of dynamical variable for the presence of positive feedback intensity (∈ > 0), while speeds up the rate of fluctuation decay of dynamical variable for the presence of negative feedback intensity (∈ < 0). The effects of the delay time τ on the T _c and C(s) are entirely opposite for ∈ 〉 0 and ∈ < 0.