Absolutely continuous invariant measures for one-parameter families of one-dimensional maps

Given a one-parameter familyf (x) of maps of the interval [0, 1], we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to Lebesgue measure. We show that this set has positive measure, for two classes of maps: i)f (x)=f(x) where 0<4 andf(x) is a functionC ³-near the quadratic mapx(1–x), and ii)f (x)=f(x) (mod 1) wheref isC ³,f(0)=f(1)=0 andf has a unique nondegenerate critical point in [0, 1].     

Mean-Field Critical Behavior for the Contact Process

The contact process is a model of spread of an infectious disease. Combining with the result of ref. 1, we prove that the critical exponents take on the mean-field values for sufficiently high dimensional nearest-neighbor models and for sufficiently spread-out models with d>4:()– _c as _c and ()( _c–)^–1 as _c, where () and () are the spread probability and the susceptibility of the infection respectively, and _c is the critical infection rate. Our results imply that the upper critical dimension for the contact process is at most 4.     

The asymmetric contact process on a finite set

The contact process onZ has one phase transition; let _c be the critical value at which the transition occurs. Let _N be the extinction time of the contact process on {0,...,N}. Durrett and Liu (1988), Durrett and Schonmann (1988), and Durrett, Schonmann, and Tanaka (1989) have respectively proved that the subcritical, supercritical, and critical phases can be characterized using a large finite system (instead ofZ) in the following way. There are constants ₁() and ₂() such that if < _c , lim _{N N} /logN = 1/₁(); if > _c , lim _N log _N /N = ₂(); if = _c , lim _{N N} /N= and lim _{N N} /N ⁴=0 in probability. In this paper we consider the asymmetric contact process onZ when it has two distinct critical values _c1< _c2. The arguments of Durrett and Liu and of Durrett and Schonmann hold for < _c1 and > _c2. We show that for [ _c1< _c2), lim _{N N} /N=-1/, (where _i is an edge speed) and for = _c2, lim _N log _N /logN=2 in probability.     

Energy level crossings in quantum mechanics

From the eigenvalue equationH \ _n () =E _n ()\ _n () withH H ₀ +V one can derive an autonomous system of first order differential equations for the eigenvaluesE _n () and the matrix elementsV _mn () where is the independent variable. To solve the dynamical system we need the initial valuesE _n ( = 0) and \ _n ( = 0). Thus one finds the motion of the energy levelsE _n (). We discuss the question of energy level crossing. Furthermore we describe the connection with the stationary state perturbation theory. The dependence of the survival probability as well as some thermodynamic quantities on is derived. This means we calculate the differential equations which these quantities obey. Finally we derive the equations of motion for the extended caseH =H ₀ +V ₁ + ² V ₂ and give an application to a supersymmetric Hamiltonian.     

Painlevé test and energy level motion

From the eigenvalue H|_n()=E_n() |_n(), where HH₀+V, one can derive an autonomous system of first-order differential equations for the eigenvaluesE _n() and the matrix elements V_mn(), where is the independent variable. We perform a Painlevé test for this system and discuss the connection with integrability. It turns out that the equations of motion do not pass the Painlevé test, but a weaker form. The first integrals are polynomials and can be related to the Kowalewski exponents.     

Three body asymptotic completeness forP(Φ)2 models

We consider weakly coupled even P()₂ models that do not have a two-body bound state, and prove asymptotic completeness on the subspace of states with mass between 3m+a() and 4m–b(), wherea andb are positive functions tending to zero with . The analytic structure of the six point function, integrated over the three incoming momenta, shows only two Landau singular manifolds (plus normal thresholds) associated to three particle processes.Laboratoire associé au Centre National de la Recherche ScientifiqueGroupe de Recherche du C.N.R.S. No. 48     

Soliton mass and surface tension in the (λ|Ø|4)2 quantum field model

The spectrum of the mass operator on the soliton sectors of the anisotropic (|ø|⁴)₂—and the (ø⁴)₂—quantum field models in the two phase region is analyzed. It is proven that, for small enough >0, the mass gapm _s() on the soliton sector is positive, andm _s()=0(^–1). This involves estimatingm _s() from below by a quantity () analogous to the surface tension in the statistical mechanics of two dimensional, classical spin systems and then estimating () by methods of Euclidean field theory. In principle, our methods apply to any two dimensional quantum field model with a spontaneously broken, internal symmetry group.A Sloan Foundation Fellow; Research supported in part by the U.S. National Science Foundation under Grant No. MPS 75-11864.Supported in part by the National Science Foundation under Grant No. PHY 76-17191     

Geometry and combinatorics of Julia sets of real quadratic maps

For real a correspondence is made between the Julia setB forz(z–)², in the hyperbolic case, and the set of-chains±(±(±..., with the aid of Cremer's theorem. It is shown how a number of features ofB can be understood in terms of-chains. The structure ofB is determined by certain equivalence classes of-chains, fixed by orders of visitation of certain real cycles; and the bifurcation history of a given cycle can be conveniently computed via the combinatorics of-chains. The functional equations obeyed by attractive cycles are investigated, and their relation to-chains is given. The first cascade of period-doubling bifurcations is described from the point of view of the associated Julia sets and-chains. Certain Julia sets associated with the Feigenbaum function and some theorems of Lanford are discussed.Supported by NSF grant No. MCS-8104862.Supported by NSF grant No. MCS-8203325.     

On the maximization of the central gravitational red-shift of a schwarzschild sphere with a given surface gravitational red-shift

Following Bondi static, spherically symmetric equilibrium configurations with a core and an envelope have been considered. It has been shown that for any configurations with nonnegative pressure and density and with a surface red-shiftz _s 4.77 arbitrarily large central red-shiftsz _c are possible in the limiting case of arbitrarily large radius. The effects of imposition of further constraints in the form of a real speed of sound not exceeding the speed of light are also examined. It is seen that for a given limiting sound-to-light-speed ratio . (i) There exists a limiting surface red-shiftz _s() 1.71. (ii) A configuration withz _s >z _s() is not possible, (iii) A configuration withz _s=z _s() has a unique and finitez _c=z _c(). (iv) Forz _s<z _s() arbitrarily large central red-shifts can be obtained for configurations with arbitrarily large radii.     

Zero measure spectrum for the almost Mathieu operator

We study the almost Mathieu operator: (H _{, ,} u)(n)=u(n+1)+u(n-1)+ cos (2n+)u(n), onl ² (Z), and show that for all ,, and (Lebesgue) a.e. , the Lebesgue measure of its spectrum is precisely |4–2|. In particular, for ||=2 the spectrum is a zero measure cantor set. Moreover, for a large set of irrational 's (and ||=2) we show that the Hausdorff dimension of the spectrum is smaller than or equal to 1/2.Work partially supported by the GIF     

A Double Infinity of Oscillator Massless Boson Resonances

In the simplest coupling of a harmonic oscillator with a massless boson field, we show that a family of coupling functions leads to resonances or bound-states of the form E _{n1 n0}()=n ₁ z ₁()+n ₀ z ₀(), where z ₁(), z ₀() are in and n ₁, n ₀ are any nonnegative integers. This holds for arbitrary values of the coupling constant.     

The S-matrix for abstract scattering systems

Let S() be the S-matrix at energy for an abstract scattering system. We derive a bound, in terms of the interaction, on integrals of the form h () S()- _HS ² d, where denotes the Hilbert-Schmidt norm.Supported by the Swiss National Science Foundation.     

Scaling behavior of permeability and conductivity anisotropy near the percolation threshold

We use the finite-size scaling method to estimate the critical exponent that characterizes the scaling behavior of conductivity and permeability anisotropy near the percolation thresholdp _c . Here is defined by the scaling lawk _l /k _t –1(p–p _c ), wherek _t andk _t are the conductivity or permeability of the system in the direction of the macroscopic potential gradient and perpendicular to this direction, respectively. The results are (d=2)0.819±0.011 and (d=3)0.518±0.001. We interpret these results in terms of the structure of percolation clusters and their chemical distance. We also compare our results with the predictions of a scaling theory for due to Straley, and propose that (d=2)=t- _B , wheret is the critical exponent of the conductivity or permeability of the system, and _B is the critical exponent of the backbone of percolation clusters.     

Hydromagnetic laminar flow of a conducting fluid in an annulus in presence of a radial magnetic field

The object of the present paper is to study the MHD effects on the laminar flow of a viscous, incompressible and conducting fluid in an annulus with arbitrary time-varying pressure gradient and arbitrary initial velocity in presence of a radial magnetic field. Using finite Hankel transform, solutions for both the unsteady and steady flows under different prescribed pressure gradients have been found out.Notation H a constant characterising the intensity of the magnetic field - p hydrostatic pressure - _e magnetic permeability - coefficient of viscosity - kinematic coefficient of voscosity - conductivity of the medium - density - a radius of the inner cylinder - b radius of the outer cylinder - parameter - s positive root - J (sr) Bessel's function of first kind of ordergl - Y (sr) Bessel's function of second kind of order     

The vacuum energy forP(ø)2: Infinite volume limit and coupling constant dependence

LetE(g) be the vacuum energy for theP(ø)₂ Hamiltonian with space cutoffg(x)0 and coupling constant 0. For suitable families of cutoffsg1, the vacuum energy per unit volume converges; i.e., –E(g)/g(x)dx (). We obtain bounds on the dependence of () for large and small . These lead to estimates forE(g) as a functional ofg that permit a weakening of the standard regularity conditions forg. Typical of such estimates is the linear lower bound, –E(g)const g(x)² dx, valid for allg0 provided thatP is normalized so thatP(0)=0. Finally we show that the perturbation series for () is asymptotic to second order.Research partially supported by AFOSR under Contract No. F 44620-71-C-0108.Postal address after September 30, 1972: via A. Falcone 70, 80127, Napoli, Italy.     

Anderson localization for the almost Mathieu equation: A nonperturbative proof

We prove that for any diophantine rotation angle and a.e. phase the almost Mathieu operator (H()) _n = _n–1 + _n+1 +cos(2(+n)) _n has pure point spectrum with exponentially decaying eigenfunctions for 15. We also prove the existence of some pure point spectrum for any 5.4.     

Modified London model for the determination of λ and ζ in a high κ superconductor

We present a modified London model suggested by Brandt [1–3] which introduces a finite vortex core size appropriate for isotropic superconductors in which the average internal field is less than approximately (1/4)H _c2. TheSR lineshape resulting from this model possesses a distinctive shape due to the magnetic penetration depth and the vortex core diameter (approximately equal to twice the coherence length ). However, for a given lineshape, there is a large range of values of and which produce nearly the same lineshape. Lineshape smearing caused by disorder in the vortex lattice increases uncertainty in values for and . If well-determined values of either (T) or (T) are not available from another technique, both of them can be determined bySR measurements alone if runs in more than one applied field at the same temperature are fit with and as shared parameters. We also present our method of estimating the degree of disorder in the vortex lattice.     

Dynamical bifurcation with noise

It was shown by A. Neishtadt that dynamical bifurcation, in which the control parameter is varied with a small but finite speed , is characterized by adelay in bifurcation, here denoted _j and depending on . Here we study dynamical bifurcation, in the framework and with the language of Landau theory of phase transitions, in the presence of a Gaussian noise of strength . By numerical experiments at fixed = ₀, we study the dependence of _j on a for order parameters of dimension 3; an exact scaling relation satisfied by the equations permits us to obtain for this the behavior for general . We find that in the smallnoise regime _j() a(^–b ), while in the strong-noise regime _j() – ce(^–d); we also measure the parameters in these formulas.     

Differential equations in the spectral parameter,Darboux transformations and a hierarchy of master symmetries for KdV

We study a certain family of Schrödinger operators whose eigenfunctions (, ) satisfy a differential equation in the spectral parameter of the formB(, )=(x). We show that the flows of a hierarchy of master symmetries for KdV are tangent to the manifolds that compose the strata of this class ofbispectral potentials. This extends and complements a result of Duistermaat and Grünbaum concerning a similar property for the Adler and Moser potentials and the flows of the KdV hierarchy.     

On the invariant sets of a family of quadratic maps

The Julia setB for the mappingz (z–)² is considered, where is a complex parameter. For 2 a new upper bound for the Hausdorff dimension is given, and the monic polynomials orthogonal with respect to the equilibrium measure onB are introduced. A method for calculating all of the polynomials is provided, and certain identities which obtain among coefficients of the three-term recurrence relations are given. A unifying theme is the relationship betweenB and -chains ± (± (± ...), which is explored for –1/42 and for with ||1/4, with the aid of the Böttcher equation. ThenB is shown to be a Hölder continuous curve for ||<1/4.Supported by NSF Grant MCS-8104862Supported by NSF Grant MCS-8002731