Moment Problems in Hereditary Function Spaces

We introduce a concept of hereditary set of multi-indices, and consider vector spaces of functions generated by families associated to such sets of multi-indices, called hereditary function spaces. Existence and uniquenes of representing measures for some abstract truncated moment problems are investigated in this framework, by adapting the concept of idempotent and that of dimensional stability, and using some techniques involving C*-algebras and commuting self-adjoint multiplication operators.

AN HYPOTHESIS CONCERNING TURBULENT DIFFUSION

It is shown that the Goldstein equation for turbulent diffusion implies diffusion currents dependent upon the history of the concentration gradient. An analysis of the stochastic model upon which the Goldstein equation is based reveals that the hereditary function, by which the history is weighted, is the ensemble autocorrelation function of velocity. Heuristic arguments and an appeal to the theory of irreversible thermodynamic processes lead to the postulation of an integro-differential equation for turbulent diffusion involving the velocity autocorrelations of the diffusate particles.

Vibrations of Elastic Systems Having Hereditary Characteristics

Results of experiments carried out on plastics and rubberlike materials at high rate of straining are given. It is shown that the dynamic stress-strain (σ, ϵ) relationship for those materials can be expressed by the formula σ=f(ϵ)+∫0tϕ(t-τ)dϵ(τ)dτdτ The first term represents the static stress-strain relationship, while the second depends on the rate of straining dedt. As a first approximation it is supposed that the materials follow Hooke’s law when statically stressed. Equation [1] then becomes σ=Eϵ+∫0tϕ(t-τ)dϵ(τ)dτdτ Materials which follow the second equation are called materials with “hereditary characteristics.” Vibrations of single-degree-of-freedom systems having hereditary characteristics are considered. Methods of finding the hereditary function ϕ(t) from forced vibrations are given. Free and forced vibrations of simply supported beams having hereditary characteristics are studied.