Simple Explicit Formulas for the Frame-Stewart Numbers

Several different approaches to the multi-peg Tower of Hanoi problem are equivalent. One of them is Stewart's recursive formula ¶¶ S (n, p) = min {2S (n₁, p) + S (n-n₁, p-1) | n₁, n-n₁ ? \mathbbZ⁺}. S (n, p) = min \{2S (n_1, p) + S (n-n_1, p-1)\mid n_1, n-n_1 \in \mathbb{Z}^+\}. ¶¶In the present paper we significantly simplify the explicit calculation of the Frame-Stewart's numbers S(n, p) and give a short proof of the domain theorem that describes the set of all pairs (n, n₁), such that the above minima are achieved at n₁.     

On graphs with largest possible game domination number

Let $\gamma \left(G\right)$ and ${\gamma }_g\left(G\right)$ be the domination number and the game domination number of a graph $G$, respectively. In this paper ${\gamma }_g$-maximal graphs are introduced as the graphs $G$ for which ${\gamma }_g\left(G\right)=2\gamma \left(G\right)?1$ holds. Large families of ${\gamma }_g$-maximal graphs are constructed among the graphs in which their sets of support vertices are minimum dominating sets. ${\gamma }_g$-maximal graphs are also characterized among the starlike trees, that is, trees which have exactly one vertex of degree at least $3$.     

Microtomography of Imbibition Phenomena and Trapping Mechanism

Water imbibition during the waterflooding process of oil production only sweeps part of the oil present. After water disrupts the oil continuity, most oil blobs are trapped in porous rock by capillary forces. Developing an efficient waterflooding scheme is a difficult task; therefore, an understanding of the oil trapping mechanism in porous rock is necessary from a microscopic viewpoint. The development of microfocused X-ray CT scanner technology enables the three-dimensional visualization of multiphase phenomena in a pore-scale. We scanned packed glass beads filled with a nonwetting phase (NWP) and injected wetting phase (WP) in upward and downward injections to determine the microscopic mechanism of immiscible displacement in porous media and the effects of buoyancy forces. We observed the imbibition phenomena for small capillary numbers to understand the spontaneous imbibition mechanism in oil recovery. This study is one of the first attempts to use a microfocused X-ray CT scanner for observing the imbibition and trapping mechanisms. The trapping mechanism in spontaneous imbibition is determined by the pore configuration causing imbibition speed differences in each channel; these differences can disrupt the oil continuity. Gravity plays an important role in spontaneous imbibition. In upward injection, the WP flows evenly and oil is trapped in single or small clusters of pores. In downward injection, the fingering phenomena determine the amount of trapped oil, which is usually in a network scale. Water breakthrough causes dramatic decrease in the oil extraction rate, resulting in lower oil production efficiency.     

Counting hypercubes in hypercubes

Subcubes of a hypercube are counted in three different ways, yielding a new graph theory interpretation of a known combinatorial identity. From this and the binomial inversion some additional combinatorial identities related to hypercubes are obtained.     

Three methods for calculation of the hyper-Wiener index of molecular graphs

The hyper-Wiener index WW of a graph G is defined as WW(G) = (summation operator d (u, v)(2) + summation operator d (u, v))/2, where d (u, v) denotes the distance between the vertices u and v in the graph G and the summations run over all (unordered) pairs of vertices of G. We consider three different methods for calculating the hyper-Wiener index of molecular graphs: the cut method, the method of Hosoya polynomials, and the interpolation method. Along the way we obtain new closed-form expressions for the WW of linear phenylenes, cyclic phenylenes, poly(azulenes), and several families of periodic hexagonal chains. We also verify some previously known (but not mathematically proved) formulas.     

Strontium isotope ratios (87Sr/86Sr) of tooth enamel: a comparison of solution and laser ablation multicollector inductively coupled plasma mass spectrometry methods

Strontium isotope ratios (87Sr/86Sr) in tooth enamel provide a means to investigate migration and landscape use in humans and other animals. Established methods for measuring (87)Sr/(86)Sr in teeth use bulk sampling (5-20 mg) and labor-intensive elemental purification procedures before analysis by either thermal ionization mass spectrometry (TIMS) or multicollector inductively coupled plasma mass spectrometry (MC-ICP-MS). Another method for measuring 87Sr/86Sr is laser ablation MC-ICP-MS, but concerns have been expressed about its accuracy for measuring tooth enamel. In this study we test the precision and accuracy of the technique by analyzing 30 modern rodent teeth from the Sterkfontein Valley, South Africa by laser ablation MC-ICP-MS and solution MC-ICP-MS. The results show a mean difference in 87Sr/86Sr measured by laser ablation and by solution of 0.0003 +/- 0.0002. This degree of precision is well within the margin necessary for investigating the potential geographic origins of humans or animals in many areas of the world. Because laser ablation is faster, less expensive, and less destructive than bulk sampling solution methods, it opens the possibility for conducting 87Sr/86Sr analyses of intra-tooth samples and small and/or rare specimens such as micromammal and fossil teeth.     

Oxy-Polybrominated Diphenyl Ethers from the Indonesian Marine Sponge,Lamellodysidea herbacea: X-ray,SAR, and Computational Studies

Polybrominated diphenyl ether (PBDE) compounds, derived from marine organisms, originate from symbiosis between marine sponges and cyanobacteria or bacteria. PBDEs have broad biological spectra; therefore, we analyzed structure and activity relationships of PBDEs to determine their potential as anticancer or antibacterial lead structures, through reactions and computational studies. Six known PBDEs (1–6) were isolated from the sponge, Lamellodysdiea herbacea; ¹³C NMR data for compound 6 are reported for the first time and their assignments are confirmed by their theoretical ¹³C NMR chemical shifts (RMSE < 4.0 ppm). Methylation and acetylation of 1 (2, 3, 4, 5-tetrabromo-6-(3′, 5′-dibromo-2′-hydroxyphenoxy) phenol) at the phenol functional group gave seven molecules (7–13), of which 10, 12, and 13 were new. New crystal structures for 8 and 9 are also reported. Debromination carried out on 1 produced nine compounds (1, 2, 14, 16–18, 20, 23, and 26) of which 18 was new. Debromination product 16 showed a significant IC₅₀ 8.65 ± 1.11; 8.11 ± 1.43 µM against human embryonic kidney (HEK293T) cells. Compounds 1 and 16 exhibited antibacterial activity against Gram-positive Staphylococcus aureus and Gram-negative Klebsiella pneumoniae with MID 0.078 µg/disk. The number of four bromine atoms and two phenol functional groups are important for antibacterial activity (S. aureus and K. pneumoniae) and cytotoxicity (HEK293T). The result was supported by analysis of frontier molecular orbitals (FMOs). We also propose possible products of acetylation and debromination using analysis of FMOs and electrostatic charges and we confirm the experimental result.