Description of growth and oscillation of solutions of complex LDE's

It is known that, equally well in the unit disc as in the whole complex plane, the growth of the analytic coefficients $A_0,\dotsc,A_{k-2}$ of \begin{equation*} f^{(k)} + A_{k-2} f^{(k-2)} + \dotsb + A_1 f'+ A_0 f = 0, \quad k\geq 2, \end{equation*} determines, under certain growth restrictions, not only the growth but also the oscillation of its non-trivial solutions, and vice versa. A uniform treatment of this principle is given in the disc $D(0,R)$, $0<R\leq \infty$, by using several measures for growth that are more flexible than those in the existing literature, and therefore permit more detailed analysis. In particular, results obtained are not restricted to cases where solutions are of finite (iterated) order of growth in the classical sense. The new findings are based on an accurate integrated estimate for logarithmic derivatives of meromorphic functions, which preserves generality in terms of three free parameters.Comment: 24 pages. This is a revision of a previously announced preprint. There are many changes throughout the manuscrip

Oscillation of solutions of LDE's in domains conformally equivalent to unit disc

Oscillation of solutions of $f^{(k)} + a_{k-2} f^{(k-2)} + \dotsb + a_1 f' +a_0 f = 0$ is studied in domains conformally equivalent to the unit disc. The results are applied, for example, to Stolz angles, horodiscs, sectors and strips. The method relies on a new conformal transformation of higher order linear differential equations. Information on the existence of zero-free solution bases is also obtained.Comment: 14 page

Dynamics measured in a non-Archimedean field

We study dynamical systems using measures taking values in a non-Archimedean field. The underlying space for such measure is a zero-dimensional topological space. In this paper we elaborate on the natural translation of several notions, e.g., probability measures, isomorphic transformations, entropy, from classical dynamical systems to a non-Archimedean setting.Comment: 12 page

Functional planning and occupational safety of milk production in cold loose housing barns

The study charted the most serious occupational hazards of cold loose housing barns. Furthermore, the study gave a good general picture of cold loose housing barns (CLHB) in dairy production. The study was a thesis for the Department of Agricultural Engineering and Household Technology in the University of Helsinki

Converse growth estimates for ODEs with slowly growing solutions

Let $f_1,f_2$ be linearly independent solutions of $f''+Af=0$, where the coefficient $A$ is an analytic function in the open unit disc $\mathbb{D}$ of $\mathbb{C}$. It is shown that many properties of this differential equation can be described in terms of the subharmonic auxiliary function $u=-\log\, (f_1/f_2)^{\#}$. For example, the case when $\sup_{z\in\mathbb{D}} |A(z)|(1-|z|^2)^2 < \infty$ and $f_1/f_2$ is normal, is characterized by the condition $\sup_{z\in\mathbb{D}} |\nabla u(z)|(1-|z|^2) < \infty$. Different types of Blaschke-oscillatory equations are also described in terms of harmonic majorants of $u$. Even if $f_1,f_2$ are bounded linearly independent solutions of $f''+Af=0$, it is possible that $\sup_{z\in\mathbb{D}} |A(z)|(1-|z|^2)^2 = \infty$ or $f_1/f_2$ is non-normal. These results relate to sharpness discussion of recent results in the literature, and are succeeded by a detailed analysis of differential equations with bounded solutions. Analogues results for the Nevanlinna class are also considered, by taking advantage of Nevanlinna interpolating sequences. It is shown that, instead of considering solutions with prescribed zeros, it is possible to construct a bounded solution of $f''+Af=0$ in such a way that it solves an interpolation problem natural to bounded analytic functions, while $|A(z)|^2(1-|z|^2)^3\, dm(z)$ remains to be a Carleson measure.Comment: 29 page