Volterra Theory Research Articles

Limits to growth from Volterra theory of population.

An integro-differential equation, proposed by Volterra (1931), is used to reproduce the results of MIT model of growth of population. The effects of pollution are taken into account assuming an exponential weighting function and a delay time. Predominant non-linear effects are produced by the dependence of life-time of pollutants on the pollution load.

Conditions for the design of shock-absorbing and vibration-proofing systems

The regions are given for the parameters of shock-absorbing and vibration-proofing systems leading to final overloads that do not exceed the design values. Shock-absorbing and vibration-proofing systems are considered as viscoelastic systems with hereditary properties. The stresses and displacements of the system as a function of time are based on the linear theory of Boltzmann and Volterra, and on the nonlinear main square theory of Ilyushin. The nucleus which characterizes the viscoelastic properties of the system is assumed to be singular; the regions of parameter variation of this function, obtained by minimization of the vibration overloads, represent the solution of the problem.

Averaging method applicable to the dynamics of not fully elastic bodies

Using well-known methods, various problems of dynamics are reduced to a standard system of integral differential equations, accounting for heredity effects in the form of multiple integrals according to the theory of Volterra. An averaging scheme is applied to this standard system, converting it to a system of differential equations which are much simpler that the initial equations. A theorem is proved which establishes the similarity of the solutions of these systems.

Stationary mode of a nonlinear elastically hereditary oscillator

The steady-state forced oscillations of a single-mass system subject to an external pure harmonic force are considered. The role of restoring force is played by a nonlinear function, which takes account of hereditary effects as a sum of multiple integrals in accordance with the Volterra theory. The problem is solved by the method of equivalent linearization, the discussion being confined to a triple integral of the hereditary type. The influence of the hereditary nonlinearity on the system dynamic characteristics, namely, its amplitude, phase, dynamic rigidity, hysteresis loop area, and Q, is investigated. In particular, the reciprocal of the Q, which can be taken as a measure of the internal friction, is shown to be independent of the amplitude of oscillation and to be the same as that obtained by linear theory. The other dynamic characteristics prove sensitive to the nonlinear properties. The exponential rational fractions proposed by Yu. N. Rabotnov are used as concrete hereditary functions.

Erratum: Kinetic Equations for Orientational and Shear Relaxation and Depolarized Light Scattering in Liquids

The application of Mori's theory of fluctuations to the calculation of depolarized light scattering spectra of liquids is described. Two sets of kinetic equations are derived. The first is capable of describing the shear wave doublet observed by Fabelinskii and co‐workers and Stegeman and Stoicheff and reduces to the theory of Leontovich when the antisymmetric part of the stress tensor is ignored. The second set describes the doublet as well as the broad background observed in several laboratories and reduces to the theory of Volterra in the limit where antisymmetric stress is ignored. The angular dependence of the spectrum predicted by the first set of equations is compared with the experimental results of Stegeman and Stoicheff for aniline and quinoline. Good agreement is obtained for quinoline at all angles and for aniline at all but the highest scattering angle at which doublet structure was observed. Our results indicate that the relaxation of the antisymmetric part of the stress tensor may be import...

Kinetic Equations for Orientational and Shear Relaxation and Depolarized Light Scattering in Liquids

The application of Mori's theory of fluctuations to the calculation of depolarized light scattering spectra of liquids is described. Two sets of kinetic equations are derived. The first is capable of describing the shear wave doublet observed by Fabelinskii and co-workers and Stegeman and Stoicheff and reduces to the theory of Leontovich when the antisymmetric part of the stress tensor is ignored. The second set describes the doublet as well as the broad background observed in several laboratories and reduces to the theory of Volterra in the limit where antisymmetric stress is ignored. The angular dependence of the spectrum predicted by the first set of equations is compared with the experimental results of Stegeman and Stoicheff for aniline and quinoline. Good agreement is obtained for quinoline at all angles and for aniline at all but the highest scattering angle at which doublet structure was observed. Our results indicate that the relaxation of the antisymmetric part of the stress tensor may be important for discussing viscoelastic relaxation in aniline.

On the existence and uniqueness of solution of the Cauchy problem for a class of linear integro-differential equations

Some problems in the theory of viscoelasticity may be described by means of integro-differential equations. In the paper a class of initial-value problems is considered which includes these physical examples, covering also their analogues - equations of the second order in time coordinate. The theory is restricted to the equations only, possessing in the same term both the highest spatial and the highest derivatives. The weak solution is defined on the base of variational principles, introduced in a previous article, and its existence, uniqueness and continuous dependence on the given data is proved, using the theory of integral Volterra's equations in Banach spaces.

Depolarized light scattering from liquid bromine

The depolarized light scattering spectrum of liquid bromine is reported for frequency shifts of 0.4 to 150 cm -1. The line is composed of two lorentzian components with differing width, both centered at zero frequency shift. The wings of the line decay exponentially. A simple modification of the Volterra theory accounts for the data.

The Asymptotic behaviour Of μ(z, β, α)

The function μ(z, β, α) defined bywhereplays an important role in Volterra's theory of convolution-logarithms, and also in the Paley-Wiener inversion formula for the Laplace transformation. These and other properties of pi are briefly described in (3).Our aim in this paper is to find the asymptotic behaviour of μ as |z| < ∞, a result which, as far as we are aware, has not been obtained. It is of interest to note that the procedure to be used also gives, with some minor modification, the asymptotic behaviour of μ as z → 0, a result that is well known (3, p. 219). When this behaviour becomes known, it becomes possible to write a significant generalization of Watson's Lemma.In 1906, Barnes (1) published a significant paper containing many asymptotic results of major importance. It is of passing interest to note that this paper contains a general theorem from which the result of Watson's Lemma, published in 1918, can easily be obtained. It would seem that this latter result is misnamed in mathematical literature, and might well be called Barnes’ Lemma.

A note on crime control.

The Volterra theory of two competing populations is extended to the contemporary social problem of crime control. Domains of stability for the time dependence of the numbers in the criminal and enforcement groups are exposed by a numerical example. Both augmentation and reduction of enforcement can produce a stable system. Average values of the ratio of members in each group show great sensitivity to the control policies adopted by the remaining sector of the total population.

On the Volterra-Lotka principle

The supplantation of one of two closely similar competing species by the other (Volterra-Lotka principle) is studied as an example, according to Volterra's general theory of population dynamics, of the decay of a biological association of an odd number 2n+1 of species into one with an even number 2n. The three-species association (n=1) is worked out in detail, with a demonstration of how a Volterra predator-prey cycle gets gradually deformed—when a slightly superior predator is introduced—into another new-predator-prey cycle, spelling a steady eclipse of the original predator. Whenn is made large, the close competitors being embedded in an association of many other species, a statistical treatment of the supplantation process can be given through the author's statistical-mechanical theory of Volterra's dynamics. The result is a probability law, changing systematically as time goes on, for the chance that the successful competitor's population has any given amplitude; explicitly time-dependent measures of mean amplitude and mean frequency of oscillation of all populations can then be worked out. Throughout, the simplifying assumption is made that the competitors differ only as regards intrinsic rate of self-growth. Two things are accomplished by viewing the competition as a decay of 2n+1 into 2n: the competitors are not abstracted from the rest of the biological world, and their population variations are always oscillatory (with long-term rises and falls of amplitude); this is in contrast to the original Volterra-Lotka analysis in which purely static, and therefore ecologically unrealistic, population levels of but two species eventuate.

三次元弾性論におけるCastiglianoの定理について

Abstract Southwell showed the relation between the stress functions-Maxwell's and Morera's-and the so-called“equations of compatibility”. His treatment of the surface integral, however, does not appear to the author to be very clear. Moreover, the consideration of the “dislocation”in an elastic body occupying a multiply-connected region, is lacking. These deficiencies are corrected in this paper. Maxwell's and Morera's stress functions are united to form a “stress functions tensor”(3). Then the “incompatibility tensor”is derived as the variational coefficient of the “stress energy” with respect to the “stress functions tensor.” Incidently, the surface-inregral appears, and is transformed into the sum of the porducts of line-ire egrals and the remaining surface-integral. The variation of the “stress-energy” can be written, finally in the form 28) from which the conditions of compatibilit y-both in the small and in the large-are derived, the latter being coincident with Volterra's theory.(9)

The Double Refraction produced by the Distortions of Elastic Bodies according to Volterra's Theory

THE interesting experiments of Prof. E. G. Coker on the application of optical methods to technical problems of stress distribution have been described in an article in NATURE of December 5, 1912. This article suggests that this would be an opportune moment to publish, outside of Italy, the results of researches which Sig. Trabacchi and I undertook some years ago for a similar purpose (Rend. Lincei, vol. xviii., 1909). There is this essential difference from Prof. Coker's experiments, that our object at that time was the experimental verification of precise calculations deduced from Volterra's theory of elastic “distortions” (Ann. de l'Ecole Normale de Paris, 1907).