Bounds to frequencies of shafts in torsional vibration with restraints and attached masses

In this paper upper and lower bounds are given to eigenvalues of torsional vibration of the following uniform structures: free shaft elastically restrained at a point; fixed shaft elastically restrained at a point; free shaft with masses elastically attached at points; fixed shaft with a mass elastically attached over an interval. Upper bounds were obtained by using the well-known Rayleigh-Ritz procedure. The more difficult problem of obtaining lower bounds was resolved by using recently developed methods of Bazley and Fox. The various problems discussed in this paper illustrate the use of these lower bound methods and their effectiveness in producing excellent bounds.     

Percus-Yevick virial relation and scaled particle theory

The conditional probability of finding a cavityG(r) devoid of molecular centers of hard sphere fluid is modified by making use of the discontinuity atr=a/2 (a is the diameter of a molecule). The new equation of state thus obtained is bounded by the Percus-Yevick compressibility and virial pressure equations of state, which may be the upper and lower bounds to the results of machine calculation.     

Nonrelativistic potential model calculations for three-quark systems in the integro-differential equation approach

Employing a recently derived integro-differential equation which is equivalent to the Faddeev equation, in an adiabatic approximation, we obtain accurate lower bounds to the binding energy of three-quark systems close to the upper bounds provided by the hypercentral approximation to the hyperspherical expansion method. Calculations have been performed for the Martin and the spin-dependent Ono-Schöberl potential.     

On the use of variational methods for solving Boltzmann equations involving non-Hermitian operators

Variational principles yielding upper and lower bounds on transport coefficients can readily be applied to the Boltzmann equation, provided it has the form of a linear, inhomogeneous integrodifferential equation with a Hermitian operator acting on the deviation from equilibrium of the distribution function. In transport problems involving a magnetic field or an alternating electric field, this operator is non-Hermitian. By suitably transforming the transport equation, we show how Variational principles may still give upper and lower bounds. The bounds are used for considering the frequency-dependent conductivity associated with a general scattering operator, and the longitudinal magnetoresistivity in the relaxation time approximation for the scattering operator. Explicit results are presented for (1) the frequency-dependent conductivity of a charged Fermi liquid and (2) the longitudinal magnetoresistivity for a weakly anisotropic Fermi surface.     

Derivation and low density solution of a nonlinear Kirkwood-Salsburg equation

Nonlinear Kirkwood-Salsburg equations which are parametrized by the density ? are derived from the linear ones by elimination of the activity z. Upper bounds on ? are derived below which the solution of these equations is unique. Narrow upper and lower bounds on z(?) are obtained as well as upper bounds on the pair distribution function. Nonlinear Mayer-Montroll equations are also briefly discussed.     

Upper bounds on the transition temperature of classical n.n. ferromagnetic systems for general n and d

A differential equation is obtained for the spin-spin correlation function of a system of n-component classical spins interacting via a nearest neighbour ferromagnetic interaction. The differential equation makes it possible to obtain upper bounds on the correlation length and the transition temperature. It also makes it possible to relate the correlation length with the energy per spin and to obtain a necessary condition for the existence of the phase transition in terms of the energy per spin.     

Bounds on the effective thermal conductivity of two-phase systems

The weighted geometric mean of resistors considered for determining the effective thermal conductivityK _E of two-phase systems has been optimised. Solutions of the equations lead to a useful set of bounds. When compared with other bounds the present bounds give the better results in estimating the upper and lower values of the effective thermal conductivity of a two-phase system.     

In-plane vibrations of an arch of variable cross section elastically restrained against rotation at one end and carrying a concentrated mass at the other

This study deals with the determination of upper and lower bounds for the fundamental frequency of the structural system described in the title. The upper bound is determined by approximating the fundamental mode shape by means of a polynomial coordinate function in the angular coordinate, which includes an exponential optimization parameter. The fundamental frequency equation is generated by means of the Rayleigh-Ritz method, and the resulting upper bound is minimized with respect to the previously mentioned exponential parameter. The lower bound for the frequency coefficient is obtained by means of an extension of Dunkerley's method.     

Limits on massive Majorana neutrinos from neutrinoless double β-decay

It is pointed out that present lower bounds on the lifetime for neutrinoless double β-decay put stringent upper limits on the mixing of a heavy Majorana neutrino (< a few MeV) with the dominantly coupled light neutrino of β-decay.     

On the free energy in systems with separable interactions. III

In this paper we extend the calculation of the free energy in systems with separable interactions of the ferromagnetic type as given in paper I of this series to a more general class of systems characterized by a hamiltonian, in which the one-particle operator Σ_kT(k) is replaced by a more general operator T which may contain interactions as well. This extension enables us to simplify a great deal the derivation given in paper II of the lower bound to the free energy for systems with both ferromagnetic and antiferromagnetic interactions. We also discuss in some detail under which conditions the upper and the lower bounds can be equal. (The conditions include in particular the hamiltonian treated in paper II.)     

Population inversion in optically thick,recombining hydrogen plasmas

The plasma condition is investigated theoretically for population inversion between the first two excited states of hydrogen atoms in a recombining plasma. The rate equation, including atom-atom collision terms, is solved consistently with the optical escape factors. The upper bound of the ground level population density (n₁)_max necessary for inversion in the optically thick plasma at specified electron density and temperature is nearly inversely proportional to the mean radius of the plasma r_O. With a decrease in the atom temperature, the upper bounds increase in the optically thin plasma but decrease in the optically thick plasma.     

Construction of trial functions for nuclear groundstate problems

Variational calculations yield upper bounds on the groundstate energy which often become approximately stationary when the flexibility of the trial function is increased. It is shown that this apparent convergence is no safe criterion for a good approximation of the groundstate energyE ₀. To avoid this uncertainty it is proposed to expand the trial function with respect to the function basis {H ^v · ?₀} where ?₀ is a suitable zeroth order approximation of the wave function. The application of this method to a threeparticle system with the central potential of Bakeret al. yields an upper bound¯E ₀=?12.3 MeV which is lower than the upper bounds published so far except that obtained by Appel. The computational effort is very small.     

Pricing bounds for discrete arithmetic Asian options under Lévy models

Analytical bounds for Asian options are almost exclusively available in the Black-Scholes framework. In this paper we derive bounds for the price of a discretely monitored arithmetic Asian option when the underlying asset follows an arbitrary Lévy process. Explicit formulas are given for Kou’s model, Merton’s model, the normal inverse Gaussian model, the CGMY model and the variance gamma model. The results are compared with the comonotonic upper bound, existing numerical results, Monte carlo simulations and in the case of the variance gamma model with an existing lower bound. The method outlined here provides lower and upper bounds that are quick to evaluate, and more accurate than existing bounds.     

Lattice Yang-Mills theory at nonzero temperature and the confinement problem

We discuss finite temperature lattice Yang-Mills theory with special attention to the confinement problem. The relationship between the confinement criteria of Wilson, Polyakov, and 't Hooft is clarified by establishing a string of inequalities between the corresponding string tensions. The close connection between finite temperature Yang-Mills models and spin models is exploited to obtain new and rather sharp upper bounds for the critical coupling constant above which there is confinement. This same analogy also allows us to establish infrared bounds for the gauge models that yield a lower bound for this critical coupling and thereby show the existence of a weak coupling regime without confinement at nonzero temperature in three or more space dimensions. Finally we discuss extension of our results to other forms of the lattice action, the Hamiltonian lattice models of Kogut and Susskind and 't Hooft'sN → ∞ limit.     

Bounds on phase velocities of guided modes in open waveguides

Guiding of waves along cylinders with a surface impedance or along low velocity sound channels lead to eigenvalue problems in unbounded domains. Variational and comparison methods are used to find bounds on the discrete eigenvalues, which are related to the phase velocities of the guided modes. The variational methods yield upper bounds only, but they can be applied systematically to a large class of problems. The comparison methods yield upper and lower bounds, but are restricted to a smaller class of problems.     

Ambartsumian's principle of invariance and the reflection of radio waves from plane inhomogeneous slabs

Using Ambartsumian's principle of invariance we investigate for radio waves the reflection coefficient of a plane inhomogeneous slab. We find that the reflection coefficient, as a function of slab thickness, satisfies the Riccati equation. From this equation we deduce a geometric theorem on the upper and lower bounds of the reflection coefficient. We illustrate the theorem by applying it to several special cases.     

On the critical properties of the Edwards and the self-avoiding walk model of polymer chains

We study the two- and three-dimensional, superrenormalizable Edwards model and the self-avoiding walk model of polymers. Using a Schwinger-Dyson equation and upper and lower bounds on correlations in terms of “skeleton diagrams” [6] we establish the existence of a non-trivial continuum limit in the two- and three-dimensional, superrenormalizable Edwards model. We also prove that perturbation theory is asymptotic for the continuum correlations of these models.A fairly detailed analysis of the approach to the critical point in the self-avoiding walk model is presented. In particular, we show that η<1. In dimension d?4, we discuss rigorous consequences of the conjecture that η is non-negative: among other implications, we derive that the continuum limit is trivial and that γ=1, in d?5 dimensions, and that corrections to mean-field scaling laws are at most logarithmic in four dimensions.     

Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy

Pinsker’s and Fannes’ type bounds on the Tsallis relative entropy are derived. The monotonicity property of the quantum f -divergence is used fot its estimation from below. For order $\alpha \in (0,1)$ , a family of lower bounds of Pinsker type is obtained. For $\alpha >1$ and the commutative case, upper continuity bounds on the relative entropy in terms of the minimal probability in its second argument are derived. Both the lower and upper bounds presented are reformulated for the case of Rényi’s entropies. The Fano inequality is extended to Tsallis’ entropies for all $\alpha >0$ . The deduced bounds on the Tsallis conditional entropy are used to obtain inequalities of Fannes’ type.     

Unitary bounds on elastic absorptive amplitudes involving total cross section,forward slope and curvature

We obtain rather tight unitarity upper and lower bounds on the absorptive part A(s, t) of elastic amplitudes with given total cross section, slope and curvature at t = 0 and compare them with experiment.     

Hydrodynamic limit for particle systems with nonconstant speed parameter

We establish the hydrodynamic limit for a class of particle systems on ℤ ^d with nonconstant speed parameter, assuming that the speed parameter is continuously differentiable in the spatial variable. If the particle system is on the one-dimensional latticeℤ and totally asymmetric, we derive the hydrodynamic equation for continuous speed parameters. We obtain nontrivial upper and lower bounds when either the speed parameter is discontinuous or there is a blockage at a fixed site.