Dimensional Kolmogorov Research Articles

Optimal Approximation of Periodic Analytic Functions with Integrable Boundary Values

LetS=[z∈C: |Im(z)|<β] be a strip in the complex plane.Hq, 1⩽q<∞, denotes the space of functions, which are analytic and 2π-periodic inS, real-valued on the real axis, and possessq-integrable boundary values. Letμbe a positive measure on [0, 2π] andLp(μ) be the corresponding Lebesgue space of periodic real-valued functions on the real axis. The even dimensional Kolmogorov, Gel'fand, and linear widths of the unit ball ofHqinLp(μ) are determined exactly, when 1⩽p⩽q<∞ or when=q<p<∞ andβis sufficiently large. It is shown that all threen-widths coincide and a characterization of the widths in terms of Blaschke products is established.

Bifurcations in the three dimensional Kolmogorov model

The behavior of the perturbed Kolmogorov model dN i dt=N iF i(N 1, N 2, N 3, ϵ) , i= 1,2,3, where N i is the size of the i-th population and ϵ is a small parameter, is investigated in the critical case when the unperturbed model ( ϵ=0) has a multiple equilibrium E 0. Conditions are found under which E 0 bifurcates into simple equilibria of the perturbed model.

Perturbations in the three dimensional Kolmogorov model

The perturbed Kolmogorov model dN i dt = N iF i(N 1, N 2, N 3, ϵ), i = 1,2,3 , where N i is the size of the ith species and ϵ is a small positive parameter, is investigated qualitatively in the three dimensional phase space. The equilibrium points are studied and stability conditions are presented. The behavior of the perturbed model is compared with that of the unperturbed system ( ϵ = 0). The nature of a hyperbolic equilibrium of the unperturbed system is preserved under small perturbations, while a simple nonhyperbolic equilibrium usually generates a hyperbolic equilibrium of the perturbed model.