Devil's attractors and chaos of a driven impact oscillator

The motion of a driven elastic impact oscillator: $\text{x}\text{?}\text{+ 0.4}\text{x}\text{?}\text{+ x =}\text{cos}\text{(ωt)}$, x > 0 and $\text{x}\text{?}\text{(t}{}^{\mathrm{+}}\text{) = -}\text{x}\text{?}\text{(t}{}^{\mathrm{?}}\text{)}$ at x = 0, is studied for ω ≈ 2?4. The oscillator exhibits Feigenbaum's bifurcations (computed δ ≈ 4.70), the Feigenbaum and intermittent transitions to chaos, crises in chaos and a strong hysteresis region for ω ≈ 3.18–3.20 where the impact/period ratios of a group of attractors show the Devil's staircase behaviour with locking values between $\text{3}\text{5}$ and $\text{3}\text{4}$.     

Systems of ordinary differential equations with nonlinear superposition principles

This work is concerned with the derivation of superposition rules which express the general solution of ordinary differential equations.

$\text{x}\text{?}\text{= η(x,t).}\text{(x, η ?}\text{R}{}^{\mathrm{n}}\text{, t ?}\text{R}\text{)}$

. in terms of a finite number of particular solutions. The point of departure is Lie's criterion according to which such a rule exists if and only if the vector fields η(x,t). ? generate a finite dimensional Lie algebra. We provide three different constructive methods for deriving superposition rules and apply them to systems of coupled Riccati equations of the projective and conformal types based, respectively, on the Lie algebra sl(n + 1, R) and o(p + 1, n ? p + 1).     

Perturbation theory for the effective interaction in nuclei

Starting from a decomposition of the Hamiltonian H(x) of the nuclear many-body problem in the form H(x) = H₀ + xV, where H₀ is a shell-model Hamiltonian, V the residual interaction, and x a strength parameter, we introduce a general effective interaction W(x) describing the interaction of nucleons within a shell, and the associated effective operators $\text{A}\text{?}\text{(x)}$. We display some properties of these operators. From a particular choice of W(x) we obtain the expressions introduced earlier by several authors. The convergence of the expansions for W(x) and $\text{A}\text{?}\text{(x)}$ in powers of x is investigated. It is shown that W(x) and $\text{A}\text{?}\text{(x)}$ are holomorphic in a domain of the complex x-plane including the point x = 0. With the help of a generalization of the von Neumann-Wigner noncrossing rule, we exhibit the nature of the common singularity of W(x) and $\text{A}\text{?}\text{(x)}$ which is closest to the origin and thus defines the radius r₀ of convergence of the expansions of W and $\text{A}\text{?}$. It is shown that r₀ is unaffected by the cancellation of unlinked diagrams. A criterion of consistency is established, which shows that most of the practical calculations of W lead to results which are inconsistent with the definition of W.     

Electronic transition-moment variation in the B 2ϵ+−X 2Пr system of the AsO molecule

The spectrum of AsO was excited in a quartz discharge tube and the relative band intensities were measured by using photographic photometry. It was found that the electronic transition moment varies as R_e(r) = constant x (1–0.575r) in the range $\text{1.46}\text{A}\text{?}\text{< r < 1.66}\text{A}\text{?}$. Smoothed relative band strengths are derived for the 11 bands.     

Gauge invariance and renormalization

The renormalization of Abelian and non-Abelian local gauge theories is discussed. It is recalled that whereas Abelian gauge theories are invariant to local c-number gauge transformations δA_μ(x) = ?_μ,…, with □Λ = 0, and to the operator gauge transformation δA_μ(x) = ?_μφ(x), …, δφ(x) = α^?1?·A(x), with □φ = 0, non-Abelian gauge theories are invariant only to the operator gauge transformations $\text{δA}{}_{\mathrm{\mu }}\text{(x) ～}\textcolor{red}{}{}_{\mathrm{\mu }}\text{C(x)}$, …, introduced by Becchi, Rouet and Stora, where

_μ is the covariant derivative matrix and C is the vector of ghost fields. The renormalization of these gauge transformation is discussed in a formal way, assuming that a gauge-invariant regularization is present. The naive renormalized local non-Abelian c-number gauge transformation $\text{δA}{}_{\mathrm{\mu }}\text{(x) = (Z}{}_1\text{/Z}{}_3\text{)gA}{}_{\mathrm{\mu }}\text{(x) ×}\text{Λ}\text{(x)+?}{}_{\mathrm{\mu }}\text{Λ}\text{(x)}$, …, is never a symmetry transformation and is never finite in perturbation theory. Only for $\text{Λ}\text{(x) = (Z}{}_3\text{/Z}{}_1\text{)L}$ with L finite constants or for $\text{Λ}\text{(x) = Ω}\text{z}\text{?}{}_3\text{C(x)}$ with Ω a finite constant does it become a finite symmetry transformation, where $\text{z}\text{?}{}_3$ is the ghost field renormalization constant. The renormalized non-Abelian Ward-Takahashi (Slavnov-Taylor) identities are consequences of the invariance of the renormalized gauge theory to this formation. It is also shown how the symmetry generators are renormalized, how photons appear as Goldstone bosons, how the (non-multiplicatively renormalizable) composite operator A_μ × C is renormalized, and how an Abelian c-number gauge symmetry may be reinstated in the exact solution of many asymptotically fr ee non-Abelian gauge theories.     

Excited vibrational state microwave spectra,structure, and force-field of trifluorosilane

The J = 2?1 microwave spectrum of six isotopic species of HSiF₃ has been observed and assigned in excited states of five of the six fundamental vibrations. The assignment is based on relative intensities, double resonance experiments, and trial anharmonic force constant calculations. Analysis of the spectra leads to experimental values for five of the $\text{α}{}_{\mathrm{r}}{}^{\mathrm{B}}$ constants, all three l-doubling constants q_t, one Fermi resonance constant φ₂₃₃, and one zeta constant $\text{ζ}{}_{\mathrm{6,\; 6}}{}^{\mathrm{(z)}}$.The harmonic force field has been refined to all the available data on vibration wavenumbers, centrifugal distortion constants, and zeta constants. The cubic anharmonic force field has been refined to the data on $\text{α}{}_{\mathrm{r}}{}^{\mathrm{B}}$ and q_t constants, using two models: a valence force model with two cubic force constants for SiH and SiF stretching, and a more sophisticated model. With the help of these calculations, the following equilibrium structure has been determined: $\text{r}{}_{\mathrm{e}}\text{(}\text{SiH}\text{) = 1.4468(±5)}\text{A}\text{?}$, $\text{r}{}_{\mathrm{e}}\text{(}\text{SiF}\text{) = 1.5624(±1)}\text{A}\text{?}$, ∠HSiF = 110.64(±3)°,     

A new proof of the invariance principle in scattering theory

For free and interacting Hamiltonians, H₀ and H = H₀ + V(r) acting in L²(R³, dx) with V(r) a radial potential satisfying certain technical conditions, and for ? a real function on R with ?′ > 0 except on a discrete set, we prove that the Moller wave operators

exist and are independent of ?. The scattering operator

$\text{S = (Ω}{}^{\mathrm{+}}\text{)1Ω}{}^{\mathrm{?}}$

is shown to be unitary. Our proof utilizes time independent methods (eigenfunction expansions) and is effective in cases not previously analyzed, e.g. $\text{V(r) =}\text{sin}\text{r}\text{r}$ and many others.     

The pair distributions and the osmotic coefficient in anomalous electrolytes up to concentrations c≈ 1 mol/1

The osmotic coefficient of anomalous electrolytes up to concentrations c ≈ 1 mol/l is explained by the pair distributions $\text{n(r) =}\text{exp}\text{[-β(}\text{V}{}_{\mathrm{c}}\text{(r) + V(}{}^{\mathrm{hs}}\text{)(r) + V}{}_1\text{(r))]}$. Here $\text{V}$_c(r) is a screened Coulomb potential, V(^hs)(r) a hard sphere potential and V₁(r) = ?A/r⁶ a short range attractive potential. For the contact distances R₊₊, R_?? and R_+? of the hard sphere potentials between ions with the same sign of their charges (++,??) and ions of opposite charges (+?) the relations R₊₊ = R_?? = R and R_+? = q₁R with 0 < q₁ < 1 are assumed. In contrast to a previous paper the parameter q₁ takes a fixed value q₁ ≈ 0,8. The constant A is determined by the fraction q₂ defined by A/R⁶ = q₂(Z²e²/DR) where the positive integer Z is the charge number of the ions and D the dielectric constant of the solvent. The numerical calculation of the osmotic coefficient of 1-1-valent hydrous electrolytes in the range of temperature 273 K ? T ? 293 K shows that the anomalous electrolytes are described by fractions q₂ in the range 0,25 ? q₂ ? 0,5 if the contact distances R are in the range $\text{3}\text{A}\text{?}\text{? R ? 7}\text{A}\text{?}$.     

Molecular structure of phosgene as studied by gas electron diffraction and microwave spectroscopy: The rz structure and isotope effect

The r_z structure of phosgene has been determined by a joint analysis of the electron diffraction intensity and the rotational constants as follows: $\text{r}{}_{\mathrm{z}}\text{(}\text{CO}\text{) = 1.1785 ± 0.0026}\text{A}\text{?}$, $\text{r}{}_{\mathrm{z}}\text{(}\text{CCl}\text{) = 1.7424 ± 0.0013}\text{A}\text{?}$, ∠_z;ClCCl = 111.83 ± 0.11°, where uncertainties represent estimated limits of experimental error. The effective constants representing bond-stretching anharmonicity have been obtained from an analysis of the isotopic differences in the r_z structure: $\text{a}{}_3\text{(}\text{CO}\text{) = 2.9 ± 0.9}\text{A}\text{?}{}^{\mathrm{?1}}$, $\text{a}{}_3\text{(}\text{CCl}\text{) = 1.6 ± 0.4}\text{A}\text{?}{}^{\mathrm{?1}}$. The equilibrium bond distances have been estimated from the r_z structure for the normal species and from the anharmonic constants to be $\text{r}{}_{\mathrm{e}}\text{(}\text{CO}\text{) = 1.1756 ± 0.0032}\text{A}\text{?}$, $\text{r}{}_{\mathrm{e}}\text{(}\text{CCl}\text{) = 1.7381 ± 0.0019}\text{A}\text{?}$.     

A convergent expansion about mean field theory: I. The expansion

We give a convergent expansion for nearly Gaussian quantum field theory in the multiphase region. The expansion combines (1) an expansion in phase boundaries, (2) a cluster expansion, and (3) a perturbation expansion to isolate dominant behavior. We study in detail the ground state of the $\text{P}$(φ)₂ = (λφ⁴ ? φ² ? μφ)₂ model, with ∥ μ ∥ ? λ² ? 1. The ground state is close to the classical free field, obtained by replacing $\text{P}$(φ) by the quadratic mean field polynomial $\text{P}$_c(φ), tangent to $\text{P}$ at a global minimum. Selecting one minimum gives a pure phase (ergodic ground state) satisfying the Wightman-Osterwalder-Schrader axioms with a positive mass. We also establish analyticity in λ for μ = 0 in the sector ∥ Im λ ∥ < ? Re λ ? 1, for ? ? 1.     

Evolution of the density for a chaotic map

For the discrete-time map x_t+11 = 4x_t(1?x_t) an exact, explicit expression is given for the time-dependent density r_t (x) evolving from a uniform initial density on (0,1). As t → ∞, r_t(x) approaches the known invariant density $\text{r(x) = 1/[π}\text{x(1?x)}\text{]}$.     

Study of the reactions 118Sn(d,t) and98Mo(d,t): (I). Spectroscopy of 117Sn and 97Mo

The vector analyzing power and differential cross section have been measured at a deuteron energy of 12.0 MeV for ${}^{118}\text{Sn}\text{(}\text{d}\text{,}\text{t}\text{)}$ transitions to six states of ¹¹⁷Sn (E_x = 0.0, 0.16, 0.31, 0.71, 1.02 and 1.18 MeV), for ${}^{98}\text{Mo}\text{(}\text{d}\text{,}\text{t}\text{)}$ transitions to eight states of ⁹⁷Mo (E_x = 0.0, 0.68, 0.72, 0.89, 1.12, 1.28, 2.39 and 2.52 MeV), and for ${}^{118}\text{Sn}\text{(}\text{d}\text{,}\text{d}\text{)}\text{and}{}^{98}\text{Mo}\text{(}\text{d}\text{,}\text{d}\text{)}$. Deuteron optical model potentials were obtained from analysis of the elastic scattering measurements, and were used in a DWBA analysis of the $\text{(}\text{d}\text{,}\text{t}\text{)}$ results. Comparison of the measurements and DWBA predictions for σ(θ) and for iT₁₁(θ) allows unambiguous determination of tl_n and j_n for all ¹¹⁸Sn(d, t) and most ⁹⁸Mo(d, t) transitions. Differences in the triton energy relative to the Coulomb barrier cause marked qualitative differences in the measured cross sections and analyzing powers between ${}^{118}\text{Sn}\text{(}\text{d}\text{,}\text{t}\text{)}\text{and}{}^{98}\text{Mo}\text{(}\text{d}\text{,}\text{t}\text{)}$ transitions of the same l_n and j_n.     

Solving the linearized Φ4 equation in external driving fields

The Φ⁴ equation, linearized about a kink at rest, is solved for external driving fields F(t) = Σ_i=n^∞f_itⁱ. The x-dependent part of the solution $\text{u}\text{?}\text{(x, t)}$ is proportional to the translation mode. This implies that also for large times the kink's dynamics is adiabatic, if the external field is sufficiently weak and $\text{|}\text{u}\text{?}\text{|? z1}$|ν| ? 1.     

Chaotic behaviour and limit cycle behaviour of anharmonic systems with periodic external perturbations

The chaotic behaviour and limit cycle behviaiour of the dynamical system $\text{x}\text{?}\text{+ α}\text{x}\text{?}\text{+}\text{d}\text{V(x)}\text{d}\text{x}\text{= ?}\text{cos}\text{(Ωt)}$ is investigated for various potentials V and parameters values α, $\text{?}$ and $\text{Ω}$.     

Magnetovolume effect in transition metal-metalloid amorphous alloys

The thermal expansion, spontaneous volume magnetostriction ω_s, forced volume magnetostriction $\text{(}\text{?ω}\text{?H}\text{)}$ and Young's modulus of amorphous Fe-B, Fe-P, Co-B and (Fe-M)₇₇Si₁₀B₁₃ (M = Cr, Mn, Co, Ni) alloys have been measured to make clear the magnetovolume effect in transition metal-metalloid amorphous alloys. The thermal expansion coefficient α, ω_s and $\text{(}\text{?ω}\text{?H}\text{)}$ are dependent on the number of d-electrons per transition metal atom n_eff calculated based on the charge transfer model. The n_eff vs. α, ω_s and $\text{(}\text{?ω}\text{?H}\text{)}$ curves are quite similar to the corresponding curves in fcc alloys. The maxima in those curves are, however, found at n_eff ≈ 8.2 for the amorphous alloys in contrast with n_eff ≈ 8.7 for the fcc Fe-Ni alloys. On the other hand, Young's modulus measured under the saturation of magnetization is governed by the molar volume, irrespective of n_eff. The magnetovolume effect in transition metal-metalloid amorphous alloys is discussed in connection with the instability of ferromagnetism of amorphous Fe.     

Correlation functions of the transverse Ising chain at the critical field for large temporal and spatial separations

We study the behavior of 〈σ₀^x(t)σ_n^x(0)〉 and 〈σ₀^y(t)σ_n^y(0)〉 for the transverse Ising chain at the critical magnetic field at T = 0. Explicit results are obtained for the three distinct regions where t → ∞ and n → ∞with $\text{0 ?}\text{n}\text{t}\text{<1}$, $\text{1 <}\text{n}\text{t}$, or $\text{t = n + n}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{(}\text{z}\text{2}\text{)}$ where z is fixed of order one. In this latter region the general Painlevé V solution is shown to reduce to a Painlevé II function. We use our results to discuss the general problem of long-time behavior of Toda equations with slowly decaying initial values.     

Elastic and inelastic φ photoproduction

The differential cross section of the reaction (γp → pφ) has been measured in the t range 0 ? t ? 0.4 GeV² and for photon energies from 3.0 to 6.7 GeV. In particular for the small t region the measurement accuracy was better than 10%. We obtained for the slope parameter B in an exponential parametrization of the differential cross section dσ/dt = Ae^?Bt values of $\text{B ? 6 ± 0.5}\text{GeV}{}^{\mathrm{?2}}$ which are significantly larger than the slopes obtained by most other experiments at higher t values. This indicates a t dependence of B particularly in the small t region.An energy dependence of the optical point (dσ/dt)_t=0, observed in our measurements, has been explained as a kinematic effect due to the VDM relation. A fit of our measurements is in excellent agreement with all other published values of (dσ/dt)_t=0(γp → φp), this implies that σ_tot(φp) must be essentially energy independent in this energy range.Spin density matrix elements of the φ have been evaluated and an analysis of the helicity amplitudes has been carried out. This analysis confirmed s-channel helicity conservation. Moments of spherical harmonics of the $\text{K}\text{K}$ angular decay distribution have been computed for 10 MeV $\text{K}\text{K}$ mass-bins from threshold to 1.3 GeV. The mass dependence of the normalized moments is generally smooth. Contributing amplitudes have essentially only even moments. The moment 〈Y₂⁰〉/〈Y₀⁰〉 changes sign above the φ mass.Differential cross sections for the inelastic φ production γp → φX have been evaluated for the first time both with respect to t?t_min and M_K. The integrated inelastic cross sections are comparable in size with the elastic ones. The slopes of the differential cross sections dσ/dt appear to become flatter with increasing M_X.     

Supersymmetry and the division algebras

We show how spinors in space-times of dimension D = t + s, where t is the time dimension, are associated for s - t = 1, 2, 4, 8 (and if t = 0, 1, 2) with the number systems (division algebras), $\text{|R, C, H, O}$. For t = 1 and s - t = 1, 2, 4 this association is “realized” by the sequence of Lorentz groups S1(2,$\text{|R}$), S1(2;$\text{|C}$), S1(2;$\text{|H}$) for D = 3, 4, 6 respectively. We discuss how octonions may be related to D = 10. For D = 6 we give details of S1(2; $\text{|H}$) spinors and construct supersymmetric models with them. These results explain various “empirical” observations in the literature relating quaternions and supersymmetry.     

New form of dynamics and current algebra

We study a new form of dynamical system, in which the commutation relations for the dynamical variables of a quantized field are defined on a “lightlike surface $\text{τ≡}\text{(t+z)}\text{2}\text{=0}$ rather than at one instant of time t = 0. We clarify the physical implications of the use of the new variables x¹ = x, x² = y, $\text{x}{}^{\mathrm{+}}\text{=}\text{(t+z)}\text{2}\text{, x}{}^{\mathrm{?}}\text{=}\text{(t?z)}\text{2}$ and explore its significance as a new form of relativistic dynamics, which holds in any Lorentz frame but not in the so-called “infinite momentum frame.” Using the quark model, we build up a new algebra of currents, in which the current commutators are defined at equal τ. The sum rules and other results of the usual current algebra can be obtained without taking the unjustifiable limit of infinite momentum. In particular, we obtain the Gell-Mann-Okubo mass formulas in quadratic form for both mesons and baryons without the trouble due to momentum dependence. We derive the reduction formula and find the physical high energy limit (not the Bjorken limit) of an amplitude is determined by the equal τ commutator.