Saturation of Berge hypergraphs

Given a graph $F$, a hypergraph is a Berge- $F$ if it can be obtained by expanding each edge in $F$ to a hyperedge containing it. A hypergraph $H$ is Berge-$F$-saturated if $H$ does not contain a subhypergraph that is a Berge-$F$, but for any edge $e\in E\left(\overline{H}\right)$, $H+e$ does. The $k$-uniform saturation number of Berge-$F$ is the minimum number of edges in a $k$-uniform Berge-$F$-saturated hypergraph on $n$ vertices. For $k=2$ this definition coincides with the classical definition of saturation for graphs. In this paper we study the saturation numbers for Berge triangles, paths, cycles, stars and matchings in $k$-uniform hypergraphs.     

Addressing the role of working memory in mathematical word-problem solving when designing intervention for struggling learners

ZDM – Mathematics Education - The focus of this article is the well documented association between low working memory capacity and difficulty with mathematical word-problem solving. We begin...     

Enantioselective Total Synthesis of Brevetoxin A: Unified Strategy for the B,E, G,and J Subunits

Brevetoxin A is a decacyclic ladder toxin that possesses 5‐, 6‐, 7‐, 8‐, and 9‐membered oxacycles, as well as 22 tetrahedral stereocenters. Herein, we describe a unified approach to the B, E, G, and J rings based upon a ring‐closing metathesis strategy from the corresponding dienes. The enolate technologies developed in our laboratory allowed access to the precursor acyclic dienes for the B, E, and G medium‐ring ethers. The strategies developed for the syntheses of these four monocycles ultimately provided multigram quantities of each of the rings, supporting our efforts toward the completion of a convergent synthesis of brevetoxin A.     

Eliminating false positive C4 sugar tests on New Zealand Manuka honey

Carbon isotope analyses (δ¹³C) of some New Zealand Manuka honeys show that they often fail the internationally recognised Association of Official Analytical Chemists sugar test (AOAC method 998.12) which detects added C₄ sugar, although these honeys are from unadulterated sources. Failure of these high value products is detrimental to the New Zealand honey industry, not only in lost export revenue, but also in brand and market reputation damage. The standard AOAC test compares the carbon isotope value of the whole honey and corresponding protein isolated from the same honey. Differences between whole honey and protein δ¹³C values should not be greater than +1.0‰, as it indicates the possibility of adulteration with syrups or sugars from C₄ plants such as high fructose corn syrup or cane sugar. We have determined that during the standard AOAC method, pollen and other insoluble components are isolated with the flocculated protein. These non‐protein components have isotope values which are considerably different from those of the pure protein, and can shift the apparent δ¹³C value of protein further away from the δ¹³C value of the whole honey, giving a false positive result for added C₄ sugar. To eliminate a false positive C₄ sugar test for Manuka honey, prior removal of pollen and other insoluble material from the honey is necessary to ensure that only the pure protein is isolated. This will enable a true comparison between whole honey and protein δ¹³C isotopes. Furthermore, we strongly suggest this modification to the AOAC method be universally adopted for all honey C₄ sugar tests. Copyright © 2010 John Wiley & Sons, Ltd.