Characterization and representation problems for intersection betweennesses

For a set system M=(M_v)_v∈V indexed by the elements of a finite set V, the intersection betweenness B(M) induced by M consists of all triples (u,v,w)∈V³ with M_u∩M_w⊆M_v. Similarly, the strict intersection betweenness B_s(M) induced by M consists of all triples (u,v,w)∈B(M) such that u, v, and w are pairwise distinct. The notion of a strict intersection betweenness was introduced by Burigana [L. Burigana, Tree representations of betweenness relations defined by intersection and inclusion, Math. Soc. Sci. 185 (2009) 5-36]. We provide axiomatic characterizations of intersection betweennesses and strict intersection betweennesses. Our results yield a simple and efficient algorithm that constructs a representing set system for a given (strict) intersection betweenness. We study graphs whose strict shortest path betweenness is a strict intersection betweenness. Finally, we explain how the algorithmic problem related to Burigana’s notion of a partial tree representation can be solved efficiently using well-known algorithms.     

The Erdős–Pósa Property for Long Circuits

For an integer ? at least 3, we prove that if G is a graph containing no two vertex‐disjoint circuits of length at least ?, then there is a set X of at most vertices that intersects all circuits of length at least ?. Our result improves the bound due to Birmelé, Bondy, and Reed (The Erd?s–Pósa property for long circuits, Combinatorica 27 (2007), 135–145) who conjecture that ? vertices always suffice.     

The separation potential of pervaporation : Part 2. Process design and economics

Pervaporation is a relatively complex process compared to other membrane processes like reverse osmosis for two reasons:— the process is sensitive to pressure losses at the permeate side— the evaporation enthalpy has to be transferred to the membrane surface (permeate side).Selectivity and flux can decrease markedly in case of hindered permeate flow. This is demonstrated by numerical design calculations of hollow-fiber modules. The calculations indicate that optimal fiber dimensions of hollow-fiber pervaporation modules should be much larger than those employed in RO and gas permeation modules.In principle, several alternatives exist for the supply of the evaporation enthalpy. The most economical solution seems to be to draw this energy from the liquid and to maintain the operating temperature level by a combination of modules and heat-exchangers in series. An alternative is the sweeping of the permeate by a partially condensing, and with respect to the permeate immiscible, carrier vapor. Since the latent heat of the carrier vapor can be utilized only partially, this concept will not be economically competitive.Pervaporation has a wide range of possible applicatio for this reasons it is impossible to discuss the economics of pervaporation in general. In this paper the separation of benzene/cyclohexane, an azeotropic system with similar vapor pressures of the components, has been chosen as an example. But even such a limited discussion reveals tendencies which seem to be generally valid:1.Pervaporation processes consisting of several stages (cascade) cannot compete with conventional separation processes like extractive distillation.2. Hybrid processes like a combination of extractive distillation, pervaporation are very promising, especially in cases where high product purities required     

Structure and synthesis of a derivative of fasciculiferin,a novel peltogynoid from Acacia fasciculifera

The structure and synthesis of the 5-O-ethyl-2,3,10-tril-O-methyl derivative of fasciculiferin (5-hydroxypeltogynin) a natural peltogynoid which exhibits an intermediate oxidation state of the C-5 methylene function in the D-ring are described.     

Maximally distance-unbalanced trees

Journal of Mathematical Chemistry - For a graph G, and two distinct vertices u and v of G, let $$ n_{{G(u,v)}} $$ be the number of vertices of G that are closer in G to u than to v. Miklavi?...     

Approximately locating an invisible agent in a graph with relative distance queries

In a pursuit evasion game on a finite, simple, undirected, and connected graph $G$, a first player visits vertices $m_1,m_2,\dots$ of $G$, where $m_{i+1}$ is in the closed neighborhood of $m_i$ for every $i$, and a second player probes arbitrary vertices $c_1,c_2,\dots$ of $G$, and learns whether or not the distance between $c_{i+1}$ and $m_{i+1}$ is at most the distance between $c_i$ and $m_i$. Up to what distance $d$ can the second player determine the position of the first? For trees of bounded maximum degree and grids, we show that $d$ is bounded by a constant. We conjecture that $d=O(logn)$ for every graph $G$ of order $n$, and show that $d=0$ if $m_{i+1}$ may differ from $m_i$ only if $i$ is a multiple of some sufficiently large integer.     

Large values of the clustering coefficient

A prominent parameter in the context of network analysis, originally proposed by Watts and Strogatz (1998), is the clustering coefficient of a graph $G$. It is defined as the arithmetic mean of the clustering coefficients of its vertices, where the clustering coefficient of a vertex $u$ of $G$ is the relative density $m\left(G\left[N_G\left(u\right)\right]\right)∕\left(\genfrac{}{}{0ex}{}{d_G\left(u\right)}{2}\right)$ of its neighborhood if $d_G\left(u\right)$ is at least $2$, and $0$ otherwise. It is unknown which graphs maximize the clustering coefficient among all connected graphs of given order and size.We determine the maximum clustering coefficients among all connected regular graphs of a given order, as well as among all connected subcubic graphs of a given order. In both cases, we characterize all extremal graphs. Furthermore, we determine the maximum increase of the clustering coefficient caused by adding a single edge.