The Miura transformations of Kaup's equation and of Mikhailov's equation

Two types of Miura transformations have been found. One relates Kaup's equation to Sawada-Kotera's equation and another relates the generalized Mikhailov equation to a model equation for a shallow water wave.     

Dirac's extended electron model

Dirac's extended electron model is elaborated here both on the classical and quantum level. The classical equations of motion are deduced from Dirac's action principle. It is shown that the model is free of the troublesome runaway solutions in the classical theory. The quantum theory of the radial oscillations is worked out in detail and the spectrum is discussed. The stability of the model is studied and it is found that Dirac's extended electron is unstable against quadrupole deformations.     

Freundlich's adsorption isotherm

We present a brief derivation of Freundlich's adsorption isotherm using straightforeward methods of statistical mechanics. The assumption that the adsorbed atoms move in Morse potential wells of different depths leads to the observed compensation effect in the energy dependence of the occupation probability of surface sites, essential for the understanding of Halsey's form of the Freundlich isotherm.     

't Hooft's consistency condition as a consequence of analyticity and unitarity

We derive 't Hooft's consistency condition on the bound-state spectrum of a confining field theory from the principles of analytic S-matrix theory.     

MacDonald's theorem and Milatz's theorem for multivariate stochastic processes

MacDonald's theorem, which expresses the spectral density of a randomly fluctuating variable α(t) in terms of the finite time average of that variable, α_θ(t), is generalized for multivariate processes. For purely random processes, having a white spectrum, this also yields the corresponding generalization of Milatz's theorem.     

Contribution of meron-like configurations to 't Hooft's disorder parameter

t' Hooft's disorder parameter is written as a Euclidean functional integral over fields with singular boundary conditions. Meron-like configurations are constructed and their contributions to the disorder parameter is evaluated approximately. The calculated disorder parameter exhibits a confinement phase behaviour.     

A note on Keldysh's perturbation formalism

The expansion theorem of quantum field theoy relating Heisenberg operators to asymptotic free-field operators is rewritten by means of the time-path technique, originally due to Schwinger, which to date has only found application in statistical mechanics. The theorem is combined with Bogoliubov's initial condition of vanishing correlations in the infinite past to rederive Keldysh's perturbation scheme for non-equilibrium statistical Green's functions.     

Geometric dependence of casimir energy and scalar green's functions

It is well known that the Casimir Energy depends on the geometry of the conducting cavity. In this note, scalar Green's functions are used to determine the Casimir energy's dependence on terms bilinear in the extrinsic curvatures of the cavity's surface, and thus to resolve the controversy over the Casimir energy's finiteness.     

Kirchhoff's and Planck's radiation laws for small particles in scattering atmospheres or flames

Size and shape effects that modify Planck's radiation law for small enclosures and the emission cross section for small particles do not affect the validity of Kirchhoff's radiation laws when applied to macroscopic collections of small particles such as in scattering atmospheres or flames.     

Relation between generalized idempotency of Dirac's density operator and semigroup property of Bloch's density operator

It is shown that a generalized form of idempotency of Dirac's density operator is a necessary and sufficient condition for semigroup property of Bloch's density operator.     

Slavnov-Taylor identities in Weinberg's renormalization scheme

Weinberg's renormalization scheme, although more cumbersome from the computational point of view, has a more immediate physical interpretation than 't Hooft's minimal renormalization scheme. It is expected to lead to smaller higher-order coefficients in a perturbative approach to QCD. However, it a priori violates the Slavnov-Taylor identities. A complete study of this problem is performed, both theoretically and for the practitioner's sake. The ambiguities in the choice of the tensorial basis of some of the QCD vertices, as well as the dependence in the gauge parameter are used for substantiating, eventually, the Slavnov-Taylor identities in this renormalization scheme.     

Bell's inequality and non-contextual dispersion-free states

With a view to exploring the possibility of wider implication of Bell's theorem, we argue that Bell's inequality is derivable as a general consequence of non-contextual hidden-variable theories. We formulate a new type of gedanken example indicating incompatibility of Bell's inequality with quantum mechanical predictions concerning simultaneous measurement of commuting observables associated with a system having no spatially separated components, where the locality condition is not at issue. Significance of this example is pointed out.     

Green's functions for gravitational multi-instantons

The Green's functions for scalar fields propagating on the self-dual gravitational multi-instantons and multi-Taub-NUT metrics are given explicitly in closed form. The special cases for flat space, Taub-NUT and the Eguchi-Hanson instanton are listed. A construction is described for obtaining the Green's functions for fields of arbitrary spin.     

The bound particle spatial density distribution in Appelbaum-Kondo's trial ground state of Kondo's Hamiltonian

In view of incomplete treatments the charge (spin) density of the bound particle in Appelbaum-Kondo's trial ground state of Kondo's Hamiltonian is recalculated. The asymptotic behaviour at large distances from the impurity is at variance with earlier findings.     

The vacuum violates Bell's inequalities

It is found that the vacuum state in any Bose or Fermi free quantum field theory violates Bell's inequalities maximally, i.e. in principle, with suitable detectors maximal violations of Bell's inequalities may be obtained without setting up a source. We explain, however, why it would be difficult to measure such violations.     

Weyl's association,Wigner's function and affine geometry

According to Weyl one may associate a function with a dynamical operator; these functions depend on the parameters p and q and can be displayed in a p, q manifold, the W space. In the classical limit the W space becomes the phase space parametrised by the canonical variables. The function associated in this manner with the density operator is Wigner's function. It turns out that if Wigner's function is a delta function it cannot represent the density operator of a physically realisable state unless the argument of the delta-function is linear in the parameters a and q. In all other cases Wigner's function associated with a physically realisable state has a finite width, proportional to $\text{h}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$. Consequently straightness (linear combination of p and q) has a fundamental significance in the W space. Since this property is preserved under linear inhomogeneous transformations the W space will have a geometry generated by these transformations, the affine geometry of Euler, Moebius and Blaschke. In the present note we show how this comes about, how it simplifies the semiclassical approximations of Wigner's function, and makes one understand how in the classical limit this geometry is lost, allowing to be replaced by the geometry of canonical transformations.     

New constants of motion for systems of free gravitating particles in Newton's and Einstein's theories of gravitation

New quantities have been found which are constants of motion in Newton's gravitational theory. Analogous but different quantities exist in Einstein's theory. The difference between the Newtonian and the relativistic quantities may be used to distinguish experimentally between the theories.     

Shear and rotatory inertia effect on Beck's column

In this paper the influence of transverse shear deformation and rotatory inertia upon the flutter load of Beck's column with various support characteristics for a variety of slenderness ratios and cross-sectional shapes is presented. The analysis is based on Cowper's formulae for establishing Timoshenko's shear coefficient K'. From this investigation it is found that the inclusion of these parameters may have an appreciable destabilizing effect in the case of a fully fixed cantilever, and particularly in the case of a partially fixed cantilever with an attached mass at the support. This occurs especially in columns with low critical slenderness ratios and thin cross-sections. Moreover, it is noticed that the flutter frequency— for flutter loads obtained by coalescing either of the first and second or second and third flexural eigenfrequencies-never exceeds the precise value 11·01l… of Beck's column.     

Superposition de deux ondes d'indices d'heterogeneite differents

Heterogeneous plane and uniform electromagnetic waves are particular solutions of Maxwell's equations. The sum of two of these solutions of the same frequency is generally not a wave of the same kind. A new type of interference appears for two heterogeneous plane and uniform components of different heterogeneity. The experimental proof of this phenomenon would show the dependence of the phase velocity on the heterogeneity. In the general case, the equal-amplitude surfaces of the various fields components are different; the same holds for the equal-phase surfaces. It is not always possible to define an eikonal and a wave surface for such a solution of Maxwell's equations. These difficulties disappear for waves of equal or opposite heterogeneity index.     

Nonlinear superposition rule for Abel's equation

A nonlinear superposition rule for Abel's cubic differential equation involving four particular solutions can be replaced by a superposition rule involving only two particular solutions. Chiellini's restrictions on the particular solutions are found to be satisfied.