Connected subgraphs with small degree sums in 3‐connected planar graphs

It is well‐known that every planar graph has a vertex of degree at most five. Kotzig proved that every 3‐connected planar graph has an edge xy such that deg(x) + deg (y) ≤ 13. In this article, considering a similar problem for the case of three or more vertices that induce a connected subgraph, we show that, for a given positive integer t, every 3‐connected planar graph G with |V(G)| ≥ t has a connected subgraph H of order t such that Σ_x∈V(H) deg_G(x) ≤ 8t − 1. As a tool for proving this result, we consider decompositions of 3‐connected planar graphs into connected subgraphs of order at least t and at most 2t − 1. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 191–203, 1999     

Determination of trace levels of iron in a seawater sample using isotope dilution/inductively coupled plasma mass spectrometry.

An analytical method for trace levels of iron in a seawater sample using isotope dilution ICP-MS was developed. Preconcentration of iron and the removal of major elements in seawater such as alkali and alkaline-earth elements can be carried out quickly using a chelating resin disk by adjusting the sample pH to 3. The collision cell option of the ICP-MS instrument method was used to improve the performance of the instrument for iron measurements since ArO and ArN interferences could be reduced using this analytical method. About 4 ml min(-1) helium, as the collision gas, were introduced into the cell. 40Ar14N and 40Ar16O which interfere with 54Fe and 56Fe in water had their amounts decreased by 5 orders of magnitude. Then, the isotope dilution method was used for iron determination below ng g(-1) level of trace iron in four environmental reference materials (river water standard sample JAC-0031 (Japan Soc. for Analytical Chemistry), estuarine standard sample SLEW-2 (NRC Canada) and seawater standard samples CASS-3 and NASS-5 (NRC Canada)) were measured. Good agreement between analytical results and certified values of reference materials was obtained, which confirmed the effectiveness of this method.     

Derivatives of Dedekind sums and their reciprocity law

In this paper derivatives of Dedekind sums are defined, and their reciprocity laws are proved. They are obtained from values at non-positive integers of the first derivatives of Barnes’ double zeta functions. As special cases, they give finite product expressions of the Stirling modular form and the double gamma function at positive rational numbers.     

Every Graph Is an Integral Distance Graph in the Plane

We prove that every finite simple graph can be drawn in the plane so that any two vertices have an integral distance if and only if they are adjacent. The proof is constructive.     

An improved synthesis of pyrido[2,3-d]pyrimidines

An improved and efficient synthesis of 1,3-dialkylpyrido[2,3-d]pyrimidine-2,4-(1H,3H)-diones from 6-methylaminouracils and methyl propiolate or diethyl ethoxymethylenemalonate is described.     

Influences of serum magnesium on measurements of serum parathyroid hormone

The amino sequence recognizable parathyroid hormones (PTH) in lower and non-lower magnesemic groups in randomized samples were measured by using various kinds of the well established PTH kits. The levels of carboxy terminal, mid-region and intact PTH in lower magnesemic group were more decreased than these in non-lower magnesemic group. It is likely that the shortage of magnesium in serum makes a suppression of PTH secretion.     

Selective C-3 lithiation of 2,3-dibromo- and 2,3-diiodo-1-methylindoles

Reaction of 2,3-dibromo-1-methylindole with tert-butyllithium at −78 °C followed by treatment with tropylium ion gave 2-bromo-3-cycloheptatrienyl-1-methylindole in a moderate yield, indicating that 2-bromo-3-lithio-1-methylindole would be involved as an intermediate instead of 3-bromo-2-lithio-1-methylindole. The structure of 2-bromo-3-cycloheptatrienyl-1-methylindole was unequivocally determined by a single X-ray analysis. 2,3-Diiodo-1-methylindole also gave a 3-lithio derivative exclusively.