Let α(n) be the least number k for which there exists a simple graph with k vertices having precisely n ≥ 3 spanning trees. Similarly, define β(n) as the least number k for which there exists a simple graph with k edges having precisely n ≥ 3 spanning trees. As an n-cycle has exactly n spanning trees, it follows that α(n), β(n) ≤ n. In this paper, we show that α(n) ≤ 1⁄3 (n + 4) and β(n) ≤ 1⁄3 (n + 7) if and only if n /∈ {3, 4, 5, 6, 7, 9, 10, 13, 18, 22}, which is a subset of Euler’s idoneal numbers. Moreover, if n 6≡ 2 (mod 3) and n ≠ 25 we show that α(n) ≤ 1⁄4 (n + 9) and β(n) ≤ 1⁄4 (n + 13). This improves some previously estabilished bounds.
Presented paper is focused on description of algorithm for fast analysis of leveling process in detail and on verification of obtained results. The main idea is based on using Finite Element Method (FEM) and solving its fundamental equation. The algorithm is programmed using software MATLAB. The program has modular structure, so it is not complicated to modify any setting or utility. Leveling process of long semi products respecting the Eulerian approach is considered. Curved material passes through laterally offset rollers so repeated elasto-plastic bending occurs. Plastic deformation causes redistribution of residual stress. The semi product is straightened when rollers are appropriately positioned. Given problem is complicated due to high inherent nonlinearity and sensitivity, caused by cyclic plasticity. Useful setting of the leveling machine is found by iterative process, based on input measured geometrical data and material characteristics. and Obsahuje seznam literatury
Euparyphium albuferensis and Echinostoma friedi cercarial infectivity to four species of sympatric snails was examined under single- or multiple-choice laboratory conditions to show the level of parasite-snail host compatibility. Radix peregra, Lymnaea fuscus, Physella acuta and Gyraulus chinensis act as second intermediate hosts of both parasite species although different cercarial transmission success (CTS) was observed. In single-host experiments, R. peregra and P. acuta showed a high degree of compatibility with E. albuferensis, while only P. acuta in the case of E. friedi. In two-choice snail communities, a snail with high CTS increased the values of another with low compatibility, in both parasite species. In multiple-choice snail communities, high CTS of some hosts decreased, while low CTS of other hosts increased. The degree of parasite-host compatibility of each snail species could be determined by the presence of other snails in the community.