Excess 1/f noise in systems with an exponentially wide spectrum of resistances and dual universality of the percolation-like noise exponent

The excess 1/f noise in a random lattice with bond resistances r∼exp(−λx), where x is a random variable and λ≪1, is studied theoretically. It is shown that if the correlation function {δr ²}∼r r ^θ+2, then the relative spectral density of the noise in the system is expressed as C _e∼λ^m exp(−λ(1−p _c)), where p _c is the percolation threshold and m=νd (ν is the critical exponent of the correlation length and d is the dimensionality of the problem). It is hypothesized that the exponent m possesses a dual universality: It is independent of 1) the geometry of the lattice and 2) the θ-mechanism responsible for the generation of the local noise. Numerical modeling in a three-dimensional lattice gives m=52.3 for θ=1 and θ=0, in agreement with the hypothesis. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 8, 614–618 (25 April 1996)     

Modular conformal scaling group evidenced in lepton-quark mass

Lepton-quark mass may reflect a correspondence in spacetime structure described by a modular conformal scaling group. Stemming in part from a spacetime line element correspondenceds → (expλ _n)ds in which the eight quantitiesλ ₀,λ ₁, ...,λ ₇ constitute a closed set under a modular addition, the associated formula for lepton-quark mass (yielding values at the 1 GeV scale for the leptons and lighter quarks and at the physical pole for the top) is conjectured to bem=m _fQ²(exp −λ _n), wherem _f=10.245 TeV is the progenitor fermion mass,Q is the charge number of the lepton or quark, and the modular group parameterλ _n is indexed by a fermion principal quantum numbern that depends on three mutually independent projection operators.     

Continuity Conditions for the Radial Distribution Function of Square-Well Fluids

The continuity properties of the radial distribution function g(r) and its close relative the cavity function y(r) are studied in the context of the Percus–Yevick (PY) integral equation for 3D square-well fluids. The cases corresponding to a well width (–1) equal to a fraction of the diameter of the hard core /m, with m=1, 2, 3, have been considered. In these cases, it is proved that the function y(r) and its first derivative are everywhere continuous, but eventually the derivative of some order becomes discontinuous at the points (n+1)/m, n=0, 1,.... The order of continuity [the highest order derivative of y(r) being continuous at a given point] _n is found to be _nn in the first case (m=1) and _n2n in the other two cases (m=2, 3), for n1. Moreover, derivatives of y(r) up to third order are continuous at r= and r= for =3/2 and =4/3, but only the first derivative is continuous for =2. This can be understood as a nonlinear resonance effect.     

Higher Dimensional Extension of Charged and Neutral Fluid Spheres with Pressure

General exact higher-dimensional (n+2), n>2 solutions in general theory of relativity of Einstein-Maxwell field equations for spherically symmetric distribution of charged pressure perfect fluid are expressed in terms of pressure extending 4-dimensional solutions presented by Bijalwan (Astrophys. Space Sci. 2011, doi:). Subsequently, metrics (e ^λ and e ^υ ), matter density and electric intensity are expressible in terms of pressure. Consequently, Pressure is found to be an invertible arbitrary function of ω (=c ₁+c ₂ r ²), where c ₁ and c ₂ (≠0) are arbitrary constants, and r is the radius of star, i.e. p=p(ω). We present a general solution for charged pressure fluid in terms for ω. We list and discuss some old and new solutions which fall in this category. Also, these solutions satisfy barotropic equation of state relating the radial pressure to the energy density. But we noticed that none of these solutions in terms of pressure for charged fluids has a well behaved neutral counter part for a spatial component of metric e ^λ i.e. choosing same spatial component for charged and neutral fluid. To illustrate the approach, we discovered a new solution for extended charged analogues of Schwarzschild interior solution in higher dimensions which is found to be well behaved only for n=2. The maximum mass found to be 1.512 M _Θ with linear dimension 14.964 km. Physical quantities pressure, energy density, red-shift, velocity of sound and p/c ² ρ are well behaved and monotonically decreasing towards the surface while adiabatic index and charge density are monotonically increasing. For brevity we don’t discuss the numerical results in detailed.     

Reduced qKZ Equation and Correlation Functions of the XXZ Model

Correlation functions of the XXZ model in the massive and massless regimes are known to satisfy a system of linear equations. The main relations among them are the difference equations obtained from the qKZ equation by specializing the variables (λ₁, . . . ,λ_2n) as (λ₁, . . . ,λ_n,λ_n+1, . . . ,λ₁+1). We call it the reduced qKZ equation. In this article we construct a special family of solutions to this system. They can be written as linear combinations of products of two transcendental functions ,ω with coefficients being rational functions. We show that correlation functions of the XXZ model in the massive regime are given by these formulas with an appropriate choice of ,ω. We also present a conjectural formula in the massless regime. On leave of absence from the Institute for High Energy Physics, Protvino, 142281, Russia Membre du CNRS     

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     

Infinite Operator-Sum Representation of Density Operator for a Dissipative Cavity with Kerr Medium Derived by Virtue of Entangled State Representation

By using the thermo entangled state representation we solve the master equation for a dissipative cavity with Kerr medium to obtain density operators’ infinite operator-sum representation ρ(t)=∑ _m,n,l=0^∞ M _m,n,l ρ ₀ℳ _m,n,l ^† . It is noticeable that M _m,n,l is not Hermite conjugate to ℳ _m,n,l ^† , nevertheless the normalization ∑ _m,n,l=0^∞ℳ _n,m,l ^† M _m,n,l =1 still holds, i.e., they are trace-preserving in a general sense. This example may stimulate further studying if general superoperator theory needs modification.     

Localization of Surface Spectra

We study spectral properties of the discrete Laplacian H on the half-space with random boundary condition ; the V(n) are independent random variables on a probability space and λ is the coupling constant. It is known that if the V(n) have densities, then on the interval [-2(d+1), 2(d+1)] (=σ(H ₀), the spectrum of the Dirichlet Laplacian) the spectrum of H is P-a.s. absolutely continuous for all λ [JL1]. Here we show that if the random potential P satisfies the assumption of Aizenman–Molchanov [AM], then there are constants λ _d and Λ _d such that for |λ|<lambda; _d and |λ|> Λ _d the spectrum of H outside σ(H ₀) is P-a.s. pure point with exponentially decaying eigenfunctions. Received: 3 December 1998 / Accepted: 27 May 1999     

Analytical time-like geodesics in Schwarzschild space-time

Time-like orbits in Schwarzschild space-time are presented and classified in a very transparent and straightforward way into four types. The analytical solutions to orbit, time, and proper time equations are given for all orbit types in the form r = r(λ), t = t(χ), and τ= τ (χ), where λ is the true anomaly and χ is a parameter along the orbit. A very simple relation between λ and χ is also shown. These solutions are very useful for modelling temporal evolution of transient phenomena near black holes since they are expressed with Jacobi elliptic functions and elliptic integrals, which can be calculated very efficiently and accurately.     

Exact self-consistent plane-symmetric solutions of the spinor-field equation with a nonlinear term dependent on the invariant P2

Exact self-consistent plane-symmetric solutions of the spinor-field equation with zero mass parameter and a nonlinear term that is an arbitrary function of the invariant , are obtained in gravitation theory. An equation with power-law nonlinearity in which the nonlinear term in the spinor-field Lagrangian has the form L_N=λP²ⁿ, where λ is the nonlinearity parameter and n=const, is investigated in detail. It is shown that λ=−Λ²<0, n>1, the original system of Einstein and nonlinear spinor-field equations has regular solutions with a localized spinor-field energy density. Here the soliton-like configuration of the fields possesses a negative energy. Exact solutions are also obtained for the above spinor-field equation in flat spacetime, and it is demonstrated that there are no soliton-like solutions in that case. Thus it is established that the proper gravitational field plays a decisive, controlling role in the formation of soliton-type solutions of the above nonlinear spinor-field equation. Russian International Friendship University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 48–53, July, 1997.     

Asymptotic Statistics of Zeroes for the Lamé Ensemble

The Lamé polynomials naturally arise when separating variables in Laplace's equation in elliptic coordinates. The products of these polynomials form a class of spherical harmonics, which are joint eigenfunctions of a quantum completely integrable (QCI) system of commuting, second-order differential operators P ₀=Δ, P ₁,…,P ^{N −1} acting on C ^∞(? ^N ). These operators naturally depend on parameters and thus constitute an ensemble. In this paper, we compute the limiting level-spacings distributions for the zeroes of the Lamé polynomials in various thermodynamic, asymptotic regimes. We give results both in the mean and pointwise, for an asymptotically full set of values of the parameters. Received: 17 January 2001 / Accepted: 14 May 2001     

Static charged spheres with anisotropic pressure in general relativity

We report a generalization of our earlier formalism [Pramana, 54, 663 (1998)] to obtain exact solutions of Einstein-Maxwell’s equations for static spheres filled with a charged fluid having anisotropic pressure and of null conductivity. Defining new variables: w=(4π/3)(ρ+ε)r ², u=4πξr ², v _r=4πp _r r ², v _⊥=4πp _⊥ r ²[ρ, ξ(=−(1/2)F ₁₄ F ¹⁴), p _r, p _⊥ being respectively the energy densities of matter and electrostatic fields, radial and transverse fluid pressures whereas ε denotes the eigenvalue of the conformal Weyl tensor and interpreted as the energy density of the free gravitational field], we have recast Einstein’s field equations into a form easy to integrate. Since the system is underdetermined we make the following assumptions to solve the field equations (i) u=v _r=(a ²/2κ)r ⁿ⁺², v _⊥=k ₁ v _r, w=k ₂ v _r; a ², n(>0), k ₁, k ₂ being constants with κ=((k ₁+2)/3+k ₂) and (ii) w+u=(b ²/2)r ⁿ⁺², u=v _r, v _⊥−v _r=k, with b and k as constants. In both cases the field equations are integrated completely. The first solution is regular in the metric as well as physical variables for all values of n>0. Even though the second solution contains terms like k/r ² since Q(0)=0 it is argued that the pressure anisotropy, caused by the electric flux near the centre, can be made to vanish reducing it to the generalized Cooperstock-de la Cruz solution given in [14]. The interior solutions are shown to match with the exterior Reissner-Nordstrom solution over a fixed boundary. Dedicated to Prof. F A E Pirani.     

The ground state problem in the infinite-U Hubbard model

The problem of the ground state of the electronic system in the Hubbard model for U=∞ is discussed. The author investigates the normal (singlet or nonmagnetic) N state of the electronic system over the entire range of electron densities n⩽1. It is shown that the energy of the N state ɛ ₀ ⁽¹⁾ (n) in a one-particle approximation, such as (e.g.) the extended Hartree-Fock approximation, is lower than the energy of the saturated ferromagnetic FM state ɛ _FM(n) for all n. The dynamic magnetic susceptibility is calculated in the random phase approximation, and it is shown that the N state is stable over the entire range of electron densities: The static susceptibility (ω=0) does not have a band singularity in the zero-wave vector limit q→0. A formally exact representation is obtained for the mass operator of the one-particle Green’s function, and an approximation of this operator is proposed: M _k(E)⋍λF(E), where λ=n(1−n)/(1−n/2)z is the kinematic interaction parameter, z is the number of nearest neighbors, and F(E) is the total single-site Green’s function. For an elliptical density of states the integral equation for F(E) is solved exactly, ad it is shown that the spectral intensity rigorously satisfies the sum rule. The calculated energy of the strongly correlated N state ɛ ₀(n)<ɛ _FM(n) for all n, and in light of this relationship the author discusses the hypothesis that the ground state of the system is the normal (singlet) state in the thermodynamic limit. The electron distribution function at T=0 differs significantly from the Fermi step; it is “smeared” along the entire energy spectrum, and discontinuities do not occur in the region of the chemical potential m. Fiz. Tverd. Tela (St. Petersburg) 39, 193–203 (February 1997)     

Crystal structure of N-m-tolyl phthalimide

N-m-tolyl phthalimide, C₁₅NO₂H₁₁ crystallizes in the space group Cc with unit cell dimensions,a=8·54(1),b=19·89(2),c = 7·59(1)A, β=114·53(1)° andZ=4.V=1173(2)A³,D _m =1·35(1),D _c = 1·344 mg.m⁻³,M _r =237 λCoK_a=1·7903A. The structure was solved byMULTAN and refined to an R-factor of 0·116 for 632 counter reflections. The molecules are held together by van der Waal’s forces. The angle between the tolyl plane and the plane through the phthalimide group is 53·4(4)°. Contribution No. 607.     

Any ?-state solutions of the Eckart potential via asymptotic iteration method

The asymptotic iteration method is employed to calculate the any ℓ-state solutions of the Schr?dinger equation with the Eckart potential by proper approximation of the centrifugal term. Energy eigenvalues and corresponding eigenfunctions are obtain explicitly. The energy eigenvalues are calculated numerically for some values of ℓ and n. Our results are in excellent agreement with the findings of other methods for short potential ranges.     

Five Dimensional Axially Symmetric String Cosmological Models with Bulk Viscous Fluid

Bulk viscous fluid distribution with massive strings in LRS Bianchi type-1 space time is studied. The exact solutions of the field equations are obtained by using the equation of state ρ=−λ and ρ=λ. We observed that the bulk viscous fluid does not survive for ρ=−λ whereas it survives for ρ=λ. Some physical and geometrical properties of the models are discussed.     

Lattice Dislocations in a 1-Dimensional Model

The spectral properties of the Schr?dinger operator T(t)=−d ²/dx ²+q(x,t) in L ²(ℝ) are studied, where the potential q is defined by q=p(x+t), x>0, and q=p(x), x<0; p is a 1-periodic potential and t∈ℝ is the dislocation parameter. For each t the absolutely continuous spectrum σ _ac (T(t))=σ _ac (T(0)) consists of intervals, which are separated by the gaps γ _n (T(t))=γ _n (T(0))=(α _n ⁻,α _n ⁺), n≥1. We prove: in each gap γ _n ≠?, n≥ 1 there exist two unique “states” (an eigenvalue and a resonance) λ _n ^±(t) of the dislocation operator, such that λ _n ^±(0)=α _n ^± and the point λ _n ^±(t) runs clockwise around the gap γ _n changing the energy sheet whenever it hits α _n ^±, making n/2 complete revolutions in unit time. On the first sheet λ _n ^±(t) is an eigenvalue and on the second sheet λ _n ^±(t) is a resonance. In general, these motions are not monotonic. There exists a unique state λ₀(t) in the basic gap γ₀(T(t))=γ₀(T(0))=(−∞ ,α₀ ⁺). The asymptotics of λ _n ^±(t) as n→∞ is determined. Received: 5 April 1999 / Accepted: 3 March 2000     

Quasi-Stable Configurations of Torus Vortex Knots and Links

The dynamics of torus vortex configurations V_{n, p, q} in a superfluid liquid at zero temperature (n is the number of quantum vortices, p is the number of turns of each filament around the symmetry axis of the torus, and q is the number of turns of the filament around its central circle; radii R₀ and r₀ of the torus at the initial instant are much larger than vortex core width ξ) has been simulated numerically based on the regularized Biot–Savart law. The lifetime of vortex systems till the instant of their substantial deformation has been calculated with a small step in parameter B₀ = r₀/R₀ for various values of parameter Λ = ln(R₀/ξ). It turns out that for certain values of n, p, and q, there exist quasi-stability regions in the plane of parameters (B₀, Λ), in which the vortices remain almost invariable during dozens and even hundreds of characteristic lengths.

     

The Geometry of the Semiclassical Wave Front Set for Schrödinger Eigenfunctions on the Torus

This paper deals with the phase space analysis for a family of Schrödinger eigenfunctions ψ _? on the flat torus ?? ⁿ = (?/2π?) ⁿ by the semiclassical Wave Front Set. We study those ψ _? such that WF_?(ψ _?) is contained in the graph of the gradient of some viscosity solutions of the Hamilton-Jacobi equation. It turns out that the semiclassical Wave Front Set of such Schrödinger eigenfunctions is stable under viscous perturbations of Mean Field Game kind. These results provide a further viewpoint, and in a wider setting, of the link between the smooth invariant tori of Liouville integrable Hamiltonian systems and the semiclassical localization of Schrödinger eigenfunctions on the torus.