Showing papers on "Green's theorem published in 1976"

Direct and inverse estimates for a singular Cauchy integral along a closed curve

Abstract: A new metric characteristicθ(δ) of rectifiable Jordan curves is introduced. We will find an estimate of the type of the Zygmund estimate for an arbitrary rectifiable closed Jordan curve in its terms. It is shown that the Plemel'-Privalov theorem on the invariance of Holder's spaces is true for the class of curves satisfying the conditionθ(gd)∼δ, which is much wider than the class of piecewise smooth curves (the presence of cusps is admissible). The Bari-Stechkin theorem on the necessary conditions of action of a singular operator in the spaces Hω is generalized. It is shown that this theorem is valid for every curve which has a continuous tangent at least at one point and θ(δ)∼δ.

World dynamics: An optimization study of the pollution subsystem

Abstract: In a simplified version of the pollution subsystem of Forrester's world dynamics model, it is proposed to choose the capital investment policy for maximizing the integral of the quality of life. The optimization is carried out by using Green's theorem and the maximum principle. The optimal control is characterized by a combination of bang-bang and singular control, with the singular are forming a turnpike, corresponding to a global equilibrium policy.

An Existence Theorem for Waves in a Channel

Abstract: It is shown that infinitesimal waves can propagate without change of shape on the surface of a lengthwise-uniform channel of sufficiently smooth, but otherwise arbitrary, cross section. The proof is based on the Krein–Rutman theorem, which also gives an upper bound for the wave speed.