The Role of Aging and Wind in Inducing Death and/or Growth Reduction in Korean Fir (Abies Koreana Wilson) on Mt. Halla, Korea

The purpose of this study was to investigate the role of strong winds and aging in the death and/or decline in the growth of Korean fir on Mt. Halla in Korea. Bangeoreum (BA-S), Jindalrebat (JD-E), and Youngsil (YS-W) on the southern, eastern, and western slopes of Mt. Halla (ca. 1600 and 1700 m a.s.l.) were selected for the study. The site chronologies were established using more than 10 living Korean firs at each site. Additionally, to date the years and seasons of death of standing/fallen dead Korean firs, 15/15, 14/15, and 10/10 trees were selected at BA-S, JD-E, and YS-W, respectively. After adjusting the age with the period of growth up to the sampling point, the oldest Korean fir found among the living trees was 114 years old at JD-E and the oldest fir among the dead trees was 131 years old at JD-E. Besides this, most of the trees at BA-S and JD-E were found to have died between 2008 and 2015, and at irregular intervals between 1976 and 2013 at YS-W. Also, the maximum number of trees, that is, 62.7% died between spring and summer, followed by 20.9% between summer and autumn, and 16.4% between autumn of the current year and spring of the following year. Abrupt growth reductions occurred at BA-S and JD-E, and have become more significant in recent years, whereas at YS-W, the abrupt growth reduction and recovery occur in a cyclic order. The intensity and frequency of the typhoons increased from 2012, and this trend was in-line with the increased number of abrupt growth reductions at BA-S and JD-E. Therefore, the typhoons of 2012 are considered as the most likely influencing factor in death and/or growth reduction in Korean firs. In contrast, the decline in the growth of the Korean firs located on the windward slope (YS-W) showed a relationship with winds stronger than 25–33 m/s.

Coloring Properties of Mixed Cycloids

In this paper, we investigate partitions of highly symmetrical discrete structures called cycloids. In general, a mixed hypergraph has two types of hyperedges. The vertices are colored in such a way that each C-edge has two vertices of the same color, and each D-edge has two vertices of distinct colors. In our case, a mixed cycloid is a mixed hypergraph whose vertices can be arranged in a cyclic order, and every consecutive p vertices form a C-edge, and every consecutive q vertices form a D-edge in the ordering. We completely determine the maximum number of colors that can be used for any p≥3 and any q≥2. We also develop an algorithm that generates a coloring with any number of colors between the minimum and maximum. Finally, we discuss the colorings of mixed cycloids when the maximum number of colors coincides with its upper bound, which is the largest cardinality of a set of vertices containing no C-edge.

On a Fire Fighter’s Problem

Suppose that a circular fire spreads in the plane at unit speed. A single fire fighter can build a barrier at speed [Formula: see text]. How large must [Formula: see text] be to ensure that the fire can be contained, and how should the fire fighter proceed? We contribute two results. First, we analyze the natural curve [Formula: see text] that develops when the fighter keeps building, at speed [Formula: see text], a barrier along the boundary of the expanding fire. We prove that the behavior of this spiralling curve is governed by a complex function [Formula: see text], where [Formula: see text] and [Formula: see text] are real functions of [Formula: see text]. For [Formula: see text] all zeroes are complex conjugate pairs. If [Formula: see text] denotes the complex argument of the conjugate pair nearest to the origin then, by residue calculus, the fire fighter needs [Formula: see text] rounds before the fire is contained. As [Formula: see text] decreases towards [Formula: see text] these two zeroes merge into a real one, so that argument [Formula: see text] goes to 0. Thus, curve [Formula: see text] does not contain the fire if the fighter moves at speed [Formula: see text]. (That speed [Formula: see text] is sufficient for containing the fire has been proposed before by Bressan et al. [6], who constructed a sequence of logarithmic spiral segments that stay strictly away from the fire.) Second, we show that for any curve that visits the four coordinate half-axes in cyclic order, and in increasing distances from the origin the fire can not be contained if the speed [Formula: see text] is less than 1.618…, the golden ratio.

Extensions of Partial Cyclic Orders, Euler Numbers and Multidimensional Boustrophedons

We enumerate total cyclic orders on $\left\{x_1,\ldots,x_n\right\}$ where we prescribe the relative cyclic order of consecutive triples $(x_i,x_{i+1},x_{i+2})$, with indices taken modulo $n$. In some cases, the problem reduces to the enumeration of descent classes of permutations, which is done via the boustrophedon construction. In other cases, we solve the question by introducing multidimensional versions of the boustrophedon. In particular we find new interpretations for the Euler up/down numbers and the Entringer numbers.

Representations of cyclically ordered semigroups

SupposeGis a group. We equip the groupGwith a cyclic order in such a way so thatP(G), the positive cone under the cyclic order ofGis a semigroup.We construct a representation of the semigroupP(G) as isometries acting on a certainHilbert space.

Genus Ranges of 4-Regular Rigid Vertex Graphs

A rigid vertex of a graph is one that has a prescribed cyclic order of its incident edges. We study orientable genus ranges of 4-regular rigid vertex graphs. The (orientable) genus range is a set of genera values over all orientable surfaces into which a graph is embedded cellularly, and the embeddings of rigid vertex graphs are required to preserve the prescribed cyclic order of incident edges at every vertex. The genus ranges of 4-regular rigid vertex graphs are sets of consecutive integers, and we address two questions: which intervals of integers appear as genus ranges of such graphs, and what types of graphs realize a given genus range. For graphs with $2n$ vertices ($n>1$), we prove that all intervals $[a, b]$ for all $a<b \leq n$, and singletons $[h, h]$ for some $h\leq n$, are realized as genus ranges. For graphs with $2n-1$ vertices ($n\geq 1$), we prove that all intervals $[a, b]$ for all $a<b \leq n$ except $[0,n]$, and $[h,h]$ for some $h\leq n$, are realized as genus ranges. We also provide constructions of graphs that realize these ranges.