On Hausdorff Dimension of Unimodal Attractors

There exists a universal constant σ<1 such that every attractor of every C⁴ unimodal map with a non-degenerate critical point is an analytic manifold or its Hausdorff dimension is equal to or less than σ.     

Billiards in Tubular Neighborhoods of Manifolds¶of Codimension 1

I prove that in (sufficiently small) tubular ρ$-neighborhoods of a given C ³ manifold of codimension 1, any two points can be connected by a billiard trajectory, and that in addition there exists such a trajectory having at most collision points, for a suitable H>0, provided the manifold is of class C ³. Received: 19 June 1998 / Accepted: 7 April 1999     

Covariant tensor formalism for partial-wave analyses of ψ decays into γB¯, γγV and ψ(2s)↦γχ<Subscript>c0,1,2</Subscript> with χ<Subscript>c0,1,2</Subscript>↦K¯πand 2π<Superscript>+</Superscript>2π<Superscript>-</Superscript>

With accumulation of high statistics data at BES and CLEO-c, many new interesting channels can get enough statistics for partial-wave analysis (PWA). Among them, ψ↦γpˉ,γΛˉ,γΣˉ,γΞˉ channels provide a good place for studying baryon-antibaryon interactions; the double radiative decays ψ↦γγV with V ≡ ρ,ω,φ have a potential to provide information on the flavor content of any meson resonances (R) with positive charge parity (C = +) and mass above 1 GeV through ψ↦γR↦γγV; ψ(2s)↦γχ_c0,1,2 with χ_c0,1,2↦Kˉπ⁺π^- and 2π⁺2π^- decays are good processes to study χ_cJ charmonium decays. Using the covariant tensor formalism, here we provide theoretical PWA formulae for these channels.     

Generalized q-Hermite Polynomials

We consider two operators A and A ⁺ in a Hilbert space of functions on the exponential lattice , where 0<q<1. The operators are formal adjoints of each other and depend on a real parameter . We show how these operators lead to an essentially unique symmetric ground state ψ₀ and that A and A ⁺ are ladder operators for the sequence . The sequence (ψ _n /ψ₀) is shown to be a family of orthogonal polynomials, which we identify as symmetrized q-Laguerre polynomials. We obtain in this way a new proof of the orthogonality for these polynomials. When γ=0 the polynomials are the discrete q-Hermite polynomials of type II, studied in several papers on q-quantum mechanics. Received: 6 December 1999 / Accepted: 21 May 2001     

Calculation of the Eigenstates and Eigenvalues for the Power and Inverse-Power Hypercentral Potentials

The bound-state solutions to the hyperradial Schr?dinger equation is constructed for any general case comprising any hypercentral power and inverse-power potentials. The hypercentral potential depends only on the hyperradius which itself is a function of Jacobi relative coordinates that are functions of particle positions (r ₁,r ₂, … , and r _N ). This paper is mainly devoted to the demonstration of the fact that any ψ of the form ψ = power series × exp(polynomial) = [f(x) exp (g(x))] is potentially a solution of the Schr?dinger equation, where the polynomial g(x) is an ansatz depending on the interaction potential.     

Existence of Noncompact Static Spherically Symmetric Solutions of Einstein SU(2)-Yang–Mills Equations

We consider static spherically symmetric solutions of the Einstein equations with cosmological constant Λ coupled to the SU(2)-Yang–Mills equations that are smooth at the origin r=0. We prove that all such solutions have a radius r _c at which the solution in Schwarzschild coordinates becomes singular. However, for any positive integer N, there exists a small positive Λ _N such that whenever 0<Λ<Λ _N , there exist at least N distinct solutions for which the singularity is only a coordinate singularity and the solution can be extended to r≥r _c . Received: 5 June 2000 / Accepted: 13 March 2001     

Theψ-particles in an SU4 scheme with anomalous currents

The recently discovered narrow peaks (theψ-particles) in e⁺e⁻ system at 3.105 and 3.695 GeV are interpreted as hadrons in a broken SU₄ symmetry scheme. A new additional additive quantum number, parachargeZ, is combined with the usual SU₃ quantum numbers in the group SU₄. Theψ (3.1) is assigned to a near ideally mixed15⊕1 multiplet of vector mesons (containing theρ) as theI=Y=0, charge conjugationC=−combination ofZ=±1.members. Theψ (3.7) is assigned correspondingly to another mixed15⊕1 multiplet containing theρ′ (1600). The hadronic electromagnetic interactions are modified by the addition of (non-minimal) anomalous pieces that can changeZ. The decays of theψ-particles are discussed. New enlarged SU₄ multiplets of other hadrons are proposed. Tests of our scheme are put forward. The most crucial test will be the observation of two rather broad resonances in e⁺ e⁻ collisions with masses around 4.2 GeV and 5.1 GeV. Another prediction is the presence of energetic photons in the decays of theψ-particles. Important results concerning the recently observed phenomena in the process e⁺e⁻→hadrons follow in this scheme.     

Convexity of internal energy of cubic crystal deformed to orthorhombic structure

A generalized set of strain variablesq _r ^N , has been defined to develop the expression for a generalized set of second order and third-order elastic moduliC _rs ^N andC _rst ^N for a cubic crystal deformed to orthorhombic structure. The HessainC _rs ^N δq_rδq_s andC _rst ^N δq_rδq_sδq_t (r=1, 2……6; summation convention) are calculated in the new variables and compared withG-strength andS-strength, for both positive and negative loading environment. The convexity of the internal energy relative to various choice of strain measure is examined considering up to third degree terms in the internal energy expression. The computational results forbcc iron is presented according to the new moduli. The stable ranges thus obtained for iron under hydrostatic compressive and tensile stresses is found to generate the classical stable range, green-stable range and stretch-stable range as the specific cases. However,bcc iron does not seem to follow any conventional stable ranges under hydrostatic compression, where the present generalized stable range is found satisfactory.     

Kähler-Einstein metrics on complex surfaces withC 1>0

Various estimates of the lower bound of the holomorphic invariant (M), defined in [T], are given here by using branched coverings, potential estimates and Lelong numbers of positive,d-closed (1, 1) currents of certain type, etc. These estimates are then applied to produce Kähler-Einstein metrics on complex surfaces withC ₁>0, in particular, we prove that there are Kähler-Einstein structures withC ₁>0 on any manifold of differential type .Dedicated to Walter Thirring on his 60^th birthdayResearch supported in part by Alfred P. Sloan Fellowship for doctoral dissertationResearch supported in part by NSF grant # DMS 84-08447 and ONR contract # N-00014-85-K-0367     

Stability of naked singularity arising in gravitational collapse of Type I matter fields

Considering gravitational collapse of Type I matter fields, we prove that, given an arbitrary C²-mass functionM(r, v) and a C¹-functionh(r, v) (through the corresponding C¹-metric functionν(t, r)), there exist infinitely many choices of energy distribution functionb(r) such that the ’true’ initial data(M, h(r,v)) leads the collapse to the formation of naked singularity. We further prove that the occurrence of such a naked singularity is stable with respect to small changes in the initial data. We remark that though the initial data leading to both black hole (BH) and naked singularity (NS) form a ’big’ subset of the true initial data set, their occurrence is not generic. The terms ’stability’ and ’genericity’ are appropriately defined following the theory of dynamical systems. The particular case of radial pressurep _r (r) has been illustrated in details to get a clear picture of how naked singularity is formed and how, it is stable with respect to initial data.     

Numerical study of the oscillatory convergence to the attractor at the edge of chaos

This paper compares three different types of “onset of chaos” in the logistic and generalized logistic map: the Feigenbaum attractor at the end of the period doubling bifurcations; the tangent bifurcation at the border of the period three window; the transition to chaos in the generalized logistic with inflection 1/2 (x_n+1 = 1-μx_n^1/2), in which the main bifurcation cascade, as well as the bifurcations generated by the periodic windows in the chaotic region, collapse in a single point. The occupation number and the Tsallis entropy are studied. The different regimes of convergence to the attractor, starting from two kinds of far-from-equilibrium initial conditions, are distinguished by the presence or absence of log-log oscillations, by different power-law scalings and by a gap in the saturation levels. We show that the escort distribution implicit in the Tsallis entropy may tune the log-log oscillations or the crossover times.     

Rotational analysis of the ultraviolet bands of PS

The 2-0, 1-0 and 0-0 bands of the ultraviolet system of PS have been analysed for their rotational structure. It is shown that they involve the transitionC ² Σ -X ² Π _r (a). TheC ² Σ state shows a significant spin doubling.     

Typicality of Pure States Randomly Sampled According to the Gaussian Adjusted Projected Measure

Consider a mixed quantum mechanical state, describing a statistical ensemble in terms of an arbitrary density operator ρ of low purity, tr ρ ² ≪1, and yielding the ensemble averaged expectation value tr (ρ A) for any observable A. Assuming that the given statistical ensemble ρ is generated by randomly sampling pure states |ψ〉 according to the corresponding so-called Gaussian adjusted projected measure (Goldstein et al. in J. Stat. Phys. 125:1197, 2006), the expectation value 〈ψ|A|ψ〉 is shown to be extremely close to the ensemble average tr (ρ A) for the overwhelming majority of pure states |ψ〉 and any experimentally realistic observable A. In particular, such a ‘typicality’ property holds  whenever the Hilbert space ℋ of the system contains a high dimensional subspace ℋ₊⊂ℋ with the property that all |ψ〉∈ℋ₊ are realized with equal probability and all other |ψ〉∈ℋ are excluded.     

Zeros of the Jones Polynomial for Multiple Crossing-Twisted Links

Let D be a general connected reduced alternating link diagram, C be the set of crossings of D and C′ be the nonempty subset of C. In this paper we first define a multiple crossing-twisted link family {D ⁿ (C′)|n=1,2,…} based on D and C′, which produces (2,2n+1)-torus knot family, the link family A _n defined in Chang and Shrock (Physica A 301:196–218, 2001) and the pretzel link family P(n,n,n) as special cases. Then by applying Beraha-Kahane-Weiss’s Theorem we prove that limits of zeros of Jones polynomials of {D ⁿ (C′)|n=1,2,…} are the unit circle |z|=1 (It is independent of the selections of D and C′) and several isolated limits, which can be determined by computing flow polynomials of subgraphs of G corresponding to D. Furthermore, we use the method of Brown and Hickman (Discrete Math. 242:17–30, 2002) to prove that, for any ε>0, all zeros of Jones polynomial of the link D ⁿ (C) lie inside the circle |z|=1+ε, provided that n is large enough. Our results extend results of F.Y. Wu, J. Wang, S.-C. Chang, R. Shrock and the present authors and refine partial result of A. Champanerkar and L. Kofman.     

Critical Slowing Down in One-Dimensional Maps and Beyond

This is a brief review on critical slowing down near the Feigenbaum period-doubling bifurcation points and its consequences. The slowing down of numerical convergence leads to an “operational” fractal dimension D=2/3 at a finite order bifurcation point. There is a cross-over to D ₀=0.538... when the order goes to infinity, i.e., to the Feigenbaum accumulation point. The problem of whether there exists a “super-scaling” for the dimension spectrum D _q ^W that does not depend on the primitive word W underlying the period-n-tupling sequence seems to remain open     

Quark-antiquark states and their radiative transitions in terms of the spectral integral equation: Charmonia

Earlier by the authors (Yad. Fiz. 70, 68 (2007)), the bƃ states were treated in the framework of the spectral integral equation, together with simultaneous calculations of radiative decays of the considered bottomonia. In the present paper, such a study is carried out for the charmonium states. We reconstruct the interaction in the c-c sector on the basis of the data for the charmonium levels with J _PC = 0⁻⁺, 1⁻⁻, 0⁺⁺, 1⁺⁺, 2⁺⁺, 1⁺⁻ and radiative transitions ψ(2S) → γχ _c0(1P), γχ _c1(1P), γχ _c2(1P), γχ _c(1S) and χ _c0(1P), χ _c1(1P), χ _c2(1P) → γJ/ψ. The c-c levels and their wave functions are calculated for the radial excitations with n ≤ 6. Also, we determine the c-c component of the photon wave function using the e ⁺ e ⁻-annihilation data: e ⁺ e ⁻ → J/ψ(3097), ψ(3686), ψ(3770), ψ(4040), ψ(4160), ψ(4415) and perform the calculations of the partial widths of the two-photon decays for the n = 1 states η _c0(1S), χ _c0(1P), χ _c2(1P) → γγ and n = 2 states η _c0(2S) → γγ, χ _c0(2P) → γγ. We discuss the status of the recently observed c-c states X(3872) and Y(3941): according to our results, the X(3872) can be either χ _c1(2P) or η _c2(1D), while Y(3941) is χ _c2(2P). The text was submitted by the authors in English.     

Internal waves in a compressible two-layer model atmosphere: Hamiltonian description

We consider slow, compared to the speed of sound, motions of an ideal compressible fluid (gas) in a gravitational field in the presence of two isentropic layers with a small specific-entropy difference between them. Assuming the flow to be potential in each of the layers (v _{1, 2} = ▿ϕ_{1, 2}) and neglecting the acoustic degrees of freedom (div($ \bar \rho $ \bar \rho (z)▿ϕ_{1, 2}) ≈ 0, where $ \bar \rho $ \bar \rho (z) is the average equilibrium density), we derive the equations of motion for the boundary in terms of the shape of the surface z = η(x, y, t) itself and the difference between the boundary values of the two velocity field potentials: ψ(x, y, t) = ψ₁ − ψ₂. We prove the Hamilto nian structure of the derived equations specified by a Lagrangian of the form ℒ = ∫$ \bar \rho $ \bar \rho (η)η _t ψdxdy − ℋ{η, ψ}. The system under consideration is the simplest theoretical model for studying internal waves in a sharply stratified atmosphere in which the decrease in equilibrium gas density due to gas compressibility with increasing height is essentially taken into account. For plane flows, we make a generalization to the case where each of the layers has its own constant potential vorticity. We investigate a system with a model dependence $ \bar \rho $ \bar \rho (z) ∝ e ^−2αz with which the Hamiltonian ℋ{η, ψ} can be represented explicitly. We consider a long-wavelength dynamic regime with dispersion corrections and derive an approximate nonlinear equation of the form u _t + auu _x − b[−$ \hat \partial _x^2 $ \hat \partial _x^2 + α²]^1/2 u _x = 0 (Smith’s equation) for the slow evolution of a traveling wave.     

A Variational Theory for Light Rays in Stably Causal Lorentzian Manifolds: Regularity and Multiplicity Results

This paper is dedicated to the study of light rays joining an event p with a timelike curve γ in a light–convex subset &\Lambda; of a stably causal Lorentzian manifold . We set up a functional framework, defined intrinsically, consisting of a family of manifolds and a positive functional Q defined on them. The critical points of Q on approach, as , the lightlike, future pointing geodesics joining p and γ. We prove some regularity results, including the C ¹–regularity of , the C ²–regularity of Q on and the C ²–regularity of its critical points. Using them, we develop a Ljusternik–Schnirelman theory for light rays, obtaining some multiplicity results, depending on the topology of the space of all lightlike curves joining p and γ. Received: 9 April 1996 / Accepted: 27 December 1996     

Construction of Fredholm representations and a modification of the Higson-Roe corona

The Fredholm representation theory is well adapted to the construction of homotopy invariants of non-simply-connected manifolds by means of the generalized Hirzebruch formula [σ(M)] = 〈L(M)ch _A f*ξ, [M]〉 ∈ K _A ⁰(pt) ⊗ Q, where A = C*[π] is the C*-algebra of the group π, π = π ₁(M). The bundle ξ ∈ K _A ⁰(Bπ) is the canonical A-bundle generated by the natural representation π → A. Recently, the first author constructed a natural family of Fredholm representations that lead to a symmetric vector bundle on the completion of the fundamental group with a modification of the Higson-Roe corona, provided that the completion is a closed manifold.     

On Geometric Problems Related to Brown-York and Liu-Yau Quasilocal Mass

We discuss some geometric problems related to the definitions of quasilocal mass proposed by Brown and York (Contemporary mathematics, vol 132, American Mathematical Society, Providence, pp 129–142, 1992; Phys Rev D (3) 47(4):1407–1419, 1993) and Liu and Yau (Phys Rev Lett 90(23):231102, 2003; J Am Math Soc 19(1):181–204, 2006). Our discussion consists of three parts. In the first part, we propose a new variational problem on compact manifolds with boundary, which is motivated by the study of Brown-York mass. We prove that critical points of this variation problem are exactly static metrics. In the second part, we derive a derivative formula for the Brown-York mass of a smooth family of closed two dimensional surfaces evolving in an ambient three dimensional manifold. As a by-product, we are able to write the ADM mass (Arnowitt et al. in Phys. Rev. (2), 122:997–1006, 1961) of an asymptotically flat 3-manifold as the sum of the Brown-York mass of a coordinate sphere S _r and an integral of the scalar curvature plus a geometrically constructed function Φ(x) in the asymptotic region outside S _r . In the third part, we prove that for any closed, spacelike, 2-surface Σ in the Minkowski space \mathbb R^3,1{\mathbb {R}^{3,1}} for which the Liu-Yau mass is defined, if Σ bounds a compact spacelike hypersurface in \mathbb R^3,1{\mathbb {R}^{3,1}} , then the Liu-Yau mass of Σ is strictly positive unless Σ lies on a hyperplane. We also show that the examples given by ó Murchadha et al. (Phys Rev Lett 92:259001, 2004) are special cases of this result.