Infinite Spectrum Research Articles

The nature of σ, κ, a0(980) and f0(980)

Abstract Masses and widths of the four light scalar mesons σ , κ , a 0 ( 980 ) and f 0 ( 980 ) may be reproduced in a model where mesons scatter via a q q ¯ loop. A transition potential is used to couple mesons to q q ¯ at a radius of ∼ 0.57 fm . Inside this radius, there is an infinite bare spectrum of confined q q ¯ states, for which a harmonic oscillator is chosen here. The coupled-channel system approximately reproduces the features of both light and heavy meson spectroscopy. The generation of σ , κ , a 0 ( 980 ) and f 0 ( 980 ) is a balance between attraction due to the q q ¯ loop and suppression of the amplitudes at the Adler zeros. Phase shifts increase more rapidly as the coupling constant to the mesons increases. This leads to resonant widths which decrease with increasing coupling constant—a characteristically non-perturbative effect.

Higher Harmonics In Vacuum From Nonlinear QED Effects without Low-Mass Intermediate Particles

We show that in the presence of a slowly rotating strong transverse magnetic field there is an infinite spectrum of harmonic wave functions An due to the first order QED correction (in alpha2) given by the Euler-Heisenberg Lagrangian. The frequency shifts are integer multiples +/- omega(0)n of the magnetic field angular frequency rotation omega(0)=2pinu(m), and the several modes n are coupled to the nearest harmonics n+/-1. This is a new effect due to QED vacuum fluctuations, not exploited before, that can explain, both qualitatively and quantitatively, the recent experimental results of the PVLAS collaboration without the need of a low-mass intermediate particle, hence may dismiss the recent claim of the discovery of the axion.

Low-Dimensional Manifolds in Reaction−Diffusion Equations. 2. Numerical Analysis and Method Development

Calculations are undertaken to study the approach to equilibrium for systems of reaction-diffusion equations on bounded domains. It is demonstrated that a number of systems approach equilibrium along attractive low-dimensional manifolds over significant ranges of parameter space. Numerical methods for generating the manifolds are adapted from methods that were developed for systems of ordinary differential equations. The truncation of the infinite spectrum of the partial differential equations makes it necessary to devise a new version of one of these methods, the well-known algorithm of Maas and Pope.

Power-Law Rheology of Isolated Nuclei with Deformation Mapping of Nuclear Substructures

Force-induced changes in genome expression as well as remodeling of nuclear architecture in development and disease motivate a deeper understanding of nuclear mechanics. Chromatin and green fluorescent protein-lamin B dynamics were visualized in a micropipette aspiration of isolated nuclei, and both were shown to contribute to viscoelastic properties of the somatic cell nucleus. Reversible swelling by almost 200% in volume, with changes in salt, demonstrates the resilience and large dilational capacity of the nuclear envelope, nucleoli, and chromatin. Swelling also proves an effective way to separate the mechanical contributions of nuclear elements. In unswollen nuclei, chromatin is a primary force-bearing element, whereas swollen nuclei are an order of magnitude softer, with the lamina sustaining much of the load. In both cases, nuclear deformability increases with time, scaling as a power law—thus lacking any characteristic timescale—when nuclei are either aspirated or indented by atomic force microscopy. The nucleus is stiff and resists distortion at short times, but it softens and deforms more readily at longer times. Such results indicate an essentially infinite spectrum of timescales for structural reorganization, with implications for regulating genome expression kinetics.

Finite‐bandwidth Kramers–Kronig relations for the determination of acoustic group velocity and frequency derivative of attenuation

Kramers–Kronig (K–K) analyses of experimental data are complicated by the extrapolation problem, which is concerned with how the unexamined spectral bands impact K–K calculations. In this talk the causal linkages in resonant‐type data are demonstrated without extrapolation through acoustic K–K relations for the group velocity and the derivative of the attenuation coefficient (components of the derivative of the complex wave number). Due to their shape‐invariant predictions with respect to subtraction constants, these new relations provide stricter tests of causal consistency than previously established K–K relations for the phase velocity and attenuation coefficient. For both the group velocity and attenuation derivative, three forms of the relations are derived. These relations are equivalent for bandwidths covering the infinite spectrum, but differ when restricted to finite spectral windows. Using experimental data from microsphere suspensions, the accuracy of finite‐bandwidth K–K predictions for group velocity, and the attenuation derivative are demonstrated. Of the multiple methods, the most accurate forms were found to be those whose integrals were expressed only in terms of the phase velocity, and attenuation coefficient themselves requiring no differentiated quantities.

Causal determination of acoustic group velocity and frequency derivative of attenuation with finite-bandwidth Kramers-Kronig relations

Kramers-Kronig (KK) analyses of experimental data are complicated by the extrapolation problem, that is, how the unexamined spectral bands impact KK calculations. This work demonstrates the causal linkages in resonant-type data provided by acoustic KK relations for the group velocity (c(g)) and the derivative of the attenuation coefficient (alpha') (components of the derivative of the acoustic complex wave number) without extrapolation or unmeasured parameters. These relations provide stricter tests of causal consistency relative to previously established KK relations for the phase velocity (c(p)) and attenuation coefficient (alpha) (components of the undifferentiated acoustic wave number) due to their shape invariance with respect to subtraction constants. For both the group velocity and attenuation derivative, three forms of the relations are derived. These relations are equivalent for bandwidths covering the entire infinite spectrum, but differ when restricted to bandlimited spectra. Using experimental data from suspensions of elastic spheres in saline, the accuracy of finite-bandwidth KK predictions for c(g) and alpha' is demonstrated. Of the multiple methods, the most accurate were found to be those whose integrals were expressed only in terms of the phase velocity and attenuation coefficient themselves, requiring no differentiated quantities.

Mathematical functions for the analysis of luminescence decays with underlying distributions: 2. Becquerel (compressed hyperbola) and related decay functions

The Becquerel (compressed hyperbola) decay law is analyzed in detail and shown to be an interesting approach for the analysis of complex luminescence decays. A decay function unifying the modified Kohlrausch and Becquerel decay laws is also introduced. It is proposed that the analysis of luminescence decays with a sum of Becquerel functions is a powerful alternative to the usual sum of exponentials. It is also shown that some complex decay laws can be written as a sum of an infinite number of exponentials and have for this reason an infinite but discrete spectrum of rate constants.

Inside charged black holes. I. Baryons

An extensive investigation is made of the interior structure of self-similar accreting charged black holes. In this, the first of two papers, the black hole is assumed to accrete a charged, electrically conducting, relativistic baryonic fluid. The mass and charge of the black hole are generated self-consistently by the accreted material. The accreted baryonic fluid undergoes one of two possible fates: either it plunges directly to the spacelike singularity at zero radius, or else it drops through the Cauchy horizon. The baryons fall directly to the singularity if the conductivity either exceeds a certain continuum threshold kappa_oo, or else equals one of an infinite spectrum kappa_n of discrete values. Between the discrete values kappa_n, the solution is characterized by the number of times that the baryonic fluid cycles between ingoing and outgoing. If the conductivity is at the continuum threshold kappa_oo, then the solution cycles repeatedly between ingoing and outgoing, displaying a discrete self-similarity reminiscent of that observed in critical collapse. Below the continuum threshold kappa_oo, and except at the discrete values kappa_n, the baryonic fluid drops through the Cauchy horizon, and in this case undergoes a shock, downstream of which the solution terminates at an irregular sonic point where the proper acceleration diverges, and there is no consistent self-similar continuation to zero radius. As far as the solution can be followed inside the Cauchy horizon, the radial direction is timelike. If the radial direction remains timelike to zero radius (which cannot be confirmed because the self-similar solutions terminate), then there is presumably a spacelike singularity at zero radius inside the Cauchy horizon, which is distinctly different from the vacuum solution for a charged black hole.

A model for Hopfions on the space–time S3×R

We construct static and time dependent exact soliton solutions for a theory of scalar fields taking values on a wide class of two dimensional target spaces, and defined on the four dimensional space–time S3×R. The construction is based on an ansatz built out of special coordinates on S3. The requirement for finite energy introduce boundary conditions that determine an infinite discrete spectrum of frequencies for the oscillating solutions. For the case where the target space is the sphere S2, we obtain static soliton solutions with nontrivial Hopf topological charges. In addition, such Hopfions can oscillate in time, preserving their topological Hopf charge, with any of the frequencies belonging to that infinite discrete spectrum.

Convergence of the Generalized Volume Averaging Method on a Convection-Diffusion Problem: A Spectral Perspective

This paper proposes a thorough investigation of the convergence of the volume averaging method described by Whitaker [The Method of Volume Averaging, Kluwer Academic, Norwell, MA, 1999] as applied to convection-diffusion problems inside a cylinder. A spectral description of volume averaging brings to the fore new perspectives about the mathematical analysis of those approximations. This spectral point of view is complementary with the Lyapunov--Schmidt reduction technique and provides a precise framework for investigating convergence. It is shown for convection-diffusion inside a cylinder that the spectral convergence of the volume averageddescription depends on the chosen averaging operator, as well as on the boundary conditions. A remarkable result states that only part of the eigenmodes among the infinite discrete spectrum of the full solution can be captured by averaging methods. This leads to a general convergence theorem (which was already examined with the use of the center manifold theorem [G. N. ...

Solvable scalar and spin models with near-neighbors interactions

We construct new solvable rational and trigonometric spin models with near-neighbors interactions by an extension of the Dunkl operator formalism. In the trigonometric case we obtain a finite number of energy levels in the center of mass frame, while the rational models are shown to possess an equally spaced infinite algebraic spectrum. For the trigonometric and one of the rational models, the corresponding eigenfunctions are explicitly computed. We also study the scalar reductions of the models, some of which had already appeared in the literature, and compute their algebraic eigenfunctions in closed form. In the rational cases, for which only partial results were available, we give concise expressions of the eigenfunctions in terms of generalized Laguerre and Jacobi polynomials.

Applications of non-orthogonal Laguerre function basis in helium atom

We present an L 2 discretization of the helium atom using a non-orthogonal Laguerre function basis. The frozen-core approximation is used to calculate the helium atom Hamiltonian. The resulting three-term recurrence relation is a special case of the recurrence relation of the Pollaczek polynomials which is a set of orthogonal polynomials having a non-empty continuous spectrum in addition to an infinite discrete spectrum. The completeness of the helium atom wave functions obtained is studied in terms of weights of the Gauss quadrature.

The Electrodynamic 2-Body Problem and the Origin of Quantum Mechanics

We numerically solve the functional differential equations (FDEs) of 2-particle electrodynamics, using the full electrodynamic force obtained from the retarded Lienard–Wiechert potentials and the Lorentz force law. In contrast, the usual formulation uses only the Coulomb force (scalar potential), reducing the electrodynamic 2-body problem to a system of ordinary differential equations (ODEs). The ODE formulation is mathematically suspect since FDEs and ODEs are known to be incompatible; however, the Coulomb approximation to the full electrodynamic force has been believed to be adequate for physics. We can now test this long-standing belief by comparing the FDE solution with the ODE solution, in the historically interesting case of the classical hydrogen atom. The solutions differ. A key qualitative difference is that the full force involves a ‘delay’ torque. Our existing code is inadequate to calculate the detailed interaction of the delay torque with radiative damping. However, a symbolic calculation provides conditions under which the delay torque approximately balances (3rd order) radiative damping. Thus, further investigations are required, and it was prematurely concluded that radiative damping makes the classical hydrogen atom unstable. Solutions of FDEs naturally exhibit an infinite spectrum of discrete frequencies. The conclusion is that (a) the Coulomb force is not a valid approximation to the full electrodynamic force, so that (b) the n-body interaction needs to be reformulated in various current contexts such as molecular dynamics.

Sum Rules and the Szegő Condition for Orthogonal Polynomials on the Real Line

We study the Case sum rules, especially $C_0$, for general Jacobi matrices. We establish situations where the sum rule is valid. Applications include an extension of Shohat's theorem to cases with an infinite point spectrum and a proof that if $\lim n (a_n -1)=\alpha$ and $\lim nb_n =\beta$ exist and $2\alpha <\abs{\beta}$, then the Szeg\H{o} condition fails.

Decay out of superdeformed bands

A consistent theory of the decay out of superdeformed rotational levels is presented. It is based on exact equations for Green's functions, taking into account simultaneously both the residual and the electromagnetic interactions. In the weak-coupling limit, we generalized the two-level model of Stafford and Barrett to the case where the superdeformed level is coupled with an infinite equidistant spectrum of normal compound states. For nonoverlapping levels, the general equations of Vigezzi et al. are rederived. The possibility of the nonexponential decay law is discussed.

On the chromatic tower

We fix a prime p and work in the p - local stable homotopy category. Then, Hopkins' chromatic splitting conjecture essentially predicts the information of a p - completed finite spectrum, obtained using the first ( n + 1) Morava K - theories K (0), K (1), . . . , K ( n ), may be obtained using a single higher Morava K - theory K ( m ). ( m ≥ n +1). However, in spite of its importance, this conjecture is very difficult and subtle. Actually, Devinatz noted the conjecture is false, as soon as we omit the finiteness assumption to include such a nice infinite spectrum as the p - completed BP spectrum. In this paper, we prove a result which reconciles Hopkins' chromatic splitting conjecture and Devinatz' observation about the p - completed BP spectrum. For the p - completion of "nice" spectra, including finite spectra and the BP - spectrum, our result essentially claims that the information obtained using the first ( n +1) Morava K - theories K (0), K (1), . . . , K ( n ), may be obtained using any m - k consecutive higher Morava K - theories K ( k + 1), K ( k + 2), . . . , K ( m - 1), K ( m ) with m - k ≥ n + s 0 + 1. Here, n 0 is the Hopkins-Ravenel (Hovey-Sadofsky) uniform horizontal vanishing line for the E ( n )-based standard Adams-Novikov spectral sequence.

A note on limit laws for minimal Cantor systems with infinite periodic spectrum

Recently in [6] Y. Lacroix proved that any distribution function can be obtained as a limit law of return time for any ergodic aperiodic system. In this note we provide an alternative construction, based on Bratteli-Vershik representations of systems, which works for any minimal Cantor system having an infinite periodic spectrum. The construction is especially simple for odometers.

Degree Spectra of Relations on Boolean Algebras

We show that every computable relation on a computable Boolean algebra \(\mathfrak{B}\) is either definable by a quantifier-free formula with constants from \(\mathfrak{B}\) (in which case it is obviously intrinsically computable) or has infinite degree spectrum.

Resonances of the Rayleigh waves in an elastic semi-infinite strip

Kirchhoff’s theory of plates is used to study forced harmonic vibrations of a semi-infinite strip when the latter is in the generalized stressed state or experiences flexural deformation. The forced vibrations are excited by a load applied to the strip end. Cross-boundary conditions are imposed on the strip’s sides, which allows one to obtain a closed solution. The presence of an infinite real frequency spectrum corresponding to the edge resonances is revealed. The relation of these resonances to the Rayleigh planar and flexural waves is established.

The Analytical (Spectral) Representation of the Solution of Delay Algebraic-Differential Equations

A new approach to finding analytical solutions of linear delay algebraic-differential equations is suggested. The analytical form of the solution is determined in terms of the infinite set of eigenvalues of a parametric matrix whose entries are the delay-time operators exp(–pτ), where p is the Laplace operator. In order to compute constants in the solution of the homogeneous equations, one must analytically find higher derivatives at the input of the delay operator. The problem of stopping the computation of the infinite spectrum upon determining a certain number of its components is discussed. Bibliography: 5 titles.