It is shown that for every pair of natural numbers $m\geq n\geq 1$, there exists a compact Fréchet space $X_{m,n}$ such that \item {(a)} $\mathop{\rm dim}X_{m,n}=n$, $\mathop{\rm ind}X_{m,n}=\mathop{\rm Ind}X_{m,n}=m$, and \item {(b)} every component of $X_{m,n}$ is homeomorphic to the $n$-dimensional cube $I^n$. \endgraf \noindent This yields new counter-examples to the theorem on dimension-lowering maps in the cases of inductive dimensions.
In this study, we presented the most commonly employed net photosynthetic light-response curves (PN/I curves) fitted by the Solver function of Microsoft Excel. Excel is attractive not only due to its wide availability as a part of the Microsoft Office suite but also due to the increased level of familiarity of undergraduate students with this tool as opposed to other statistical packages. In this study, we explored the use of Excel as a didactic tool which was built upon a previously published paper presenting an Excel Solver tool for calculation of a net photosynthetic/chloroplastic CO2-response curve. Using the Excel spreadsheets accompanying this paper, researchers and students can quickly and easily choose the best fitted PN/I curve, selecting it by the minimal value of the sum of the squares of the errors. We also criticized the misuse of the asymptotic estimate of the maximum gross photosynthetic rate, the light saturation point estimated at a specific percentile of maximum net photosynthetic rate, and the quantum yield at zero photosynthetic photon flux density and we proposed the replacement of these variables by others more directly linked to plant ecophysiology. and F. de A. Lobo ... [et al.].
If $G$ is a connected graph with distance function $d$, then by a step in $G$ is meant an ordered triple $(u, x, v)$ of vertices of $G$ such that $d(u, x) = 1$ and $d(u, v) = d(x, v) + 1$. A characterization of the set of all steps in a connected graph was published by the present author in 1997. In Section 1 of this paper, a new and shorter proof of that characterization is presented. A stronger result for a certain type of connected graphs is proved in Section 2.