Narrow Results

Diffusion governs the movement of molecules in gases and liquids and consequently affects the supply of living organisms with nutrients and oxygen and the distribution within bodies. In the absence of active transport processes, it is generally the only supply mechanism. Models of dynamic diffusion where the diffusing substance can react can be complex, as shown here with models for a non-steady-state diffusion process of oxygen into respiring tissue such as wood. In the simplest case, approximating the dependence of oxygen consumption on oxygen concentration by a linear function, we are able to establish an explicit analytic solution of the modeling partial differential equation. This is applied to analyze the oxygen distribution in tree stems, but is also relevant for other media such as aquatic sediments. More complex models analyze different assumptions about the dependence of oxygen consumption and limit the extension of the respiring layer. For these models it was not possible to derive an explicit analytic solution and Euler's method was used.

Diffusion is an essential component of gas exchange at the cellular and tissue level, and a mathematical analysis of diffusion is therefore important to model biological processes in many systems. When several factors affect diffusion, finding an explicit non-steady-state equation can be difficult or impossible. In an earlier work (J Biol Systems15:63–72), we described such a function for a system where oxygen diffuses from the air into a body that consumes oxygen, assuming that the exchange surface is flat. Here, an explicit solution is limited to the case where tissue oxygen consumption decreases linearly with oxygen concentration and reaches 0 only when all oxygen has been consumed. The objective of this article is the analysis of gas diffusion into a respiring tissue that is cylindrical, which applies to tree stems and is a more realistic approximation for many other organs. This approach differs from diffusion along a flat surface, resulting in formally completely different explicit solutions, and is more flexible allowing for different relationships between oxygen concentration and tissue oxygen consumption.

Let ℤ_m be the group of residue classes modulo m. Let s(m, n) denote the total number of subgroups of the group ℤ_m × ℤ_n, where m and n are arbitrary positive integers. We derive asymptotic formulas for the sum ∑_m,n≤x s(m, n) and for the corresponding sum restricted to gcd(m, n) > 1 which concerns the groups ℤ_m × ℤ_n having rank two.

In the sense of the theory of lattice points in large bodies, a fairly general body of rotation ${{ℬ}}_{\mathrm{\text{rot}}}$ in ${ℝ}^3$ is considered whose convexity is being hampered by a “dent” (see Fig. 1). It turns out that, for $t$ a large real parameter, the true order of magnitude of the lattice discrepancy of $t{{ℬ}}_{\mathrm{\text{rot}}}$ essentially can be evaluated, contrary to the convex case.