The depth of a reflexive polytope

Given arbitrary integers d and r with $$d \ge 4$$ and $$1 \le r \le d + 1$$ , a reflexive polytope $${\mathscr {P}}\subset {\mathbb R}^d$$ of dimension d with $$\mathrm{depth}\,K[{\mathscr {P}}] = r$$ for which its dual polytope $${\mathscr {P}}^\vee $$ is normal will be constructed, where $$K[{\mathscr {P}}]$$ is the toric ring of $${\mathscr {P}}$$ .     

Wet deposition of titania-apatite composite in cotton fibrils

Titania consisting of rutile and anatase was precipitated on and inside of cotton fibrils when cotton was soaked in an aqueous solution of 30 mM TiOSO₄ and 30 mM H₂O₂ kept at 80°C for 24 h. As the resultant titania-cotton fibrils were soaked in a simulated body fluid (Kokubo solution), apatite was deposited on the fibrils to yield cotton-titania-apatite composites within 1 d. Thus, the present study provided a simple technique to prepare titania-apatite composites and coatings on organic materials at low temperature. Presented at the Sol-gel 2005 in Los Angeles     

A bimodal structure of solution-grown isotactic polypropylene with orthogonally crossed lamellae

Isotactic polypropylene (iPP) was crystallized from solution on a uniaxially-oriented iPP film. Small-angle x-ray scattering patterns obtained from the sample showed two perpendicularly crossed lameliae 9.3 nm thick that overgrew flat-on and edge-on on the substrate. In the through wide-angel x-ray diffraction pattern (taken with incident x-rays normal to the iPP film surface), strong hkO reflections were arranged in an hkO net pattern indicating that the a-axis of the monoclinic α unit cell was oriented parallel to the chain direction of the substrate. From this, it was concluded that the flat-on lamellae grew with the a-axis parallel to the chain axis of the substrate and with the b-axis parallel to its surface. In the edge wide-angle x-ray diffraction pattern (X-rays incident on the edge of the film), arced, strong 110 and 220 reflections from overgrown crystals were observed on the equator of the fiber pattern of the substrate. This indicated that the edge-on lamellae epitaxially grew with the c-axis aligned parallel to the chain axis of the substrate. The homoepitaxy explains the correlated growth mode between the orthogonally crossed lamellae: they grew epitaxially, the a-axis of one lamella coinciding with the c-axis of the other and the {010} planes in contact. © 1993 John Wiley & Sons, Inc.     

The effect of fused heterocycles on the liquid crystalline properties of di(alkoxy-substituted-phenyl)pyridazines and di(alkoxy-substituted-phenyl)pyridines

Di(alkoxy-substituted-phenyl)pyridines and di(alkoxy-substituted-phenyl)pyridazines annelated with heterocycles such as imidazole, triazole, pyrazine, and 1,2,5-oxa(thia)diazole were prepared and the effect of the fused heterocycles on the liquid crystalline properties were investigated.     

Dijkstra’s algorithm and L-concave function maximization

Dijkstra’s algorithm is a well-known algorithm for the single-source shortest path problem in a directed graph with nonnegative edge length. We discuss Dijkstra’s algorithm from the viewpoint of discrete convex analysis, where the concept of discrete convexity called L-convexity plays a central role. We observe first that the dual of the linear programming (LP) formulation of the shortest path problem can be seen as a special case of L-concave function maximization. We then point out that the steepest ascent algorithm for L-concave function maximization, when applied to the LP dual of the shortest path problem and implemented with some auxiliary variables, coincides exactly with Dijkstra’s algorithm.     

Construction of peptide-vehicles,bioconjugates having modules for cancer cell surface capture and cell-penetrating peptide with anticancer agents

Recently molecular targeting therapy has been applied to cancer chemotherapy, although in some cases side-effects are not negligible. Based on our bio-detection concept, that is, protein–protein interaction can be mimicked by using peptides, a novel cell-targeting concept designated peptide-vehicle has been proposed, which has conjugates consisting of the cancer cell recognition and cell penetrating peptides with anticancer drugs. The cancer cell surface protein can be captured by a cyclotide, containing protease resistant d-cystine. A library of cell penetrating peptides has been prepared and conjugated to the cyclotide. Anticancer molecules were recovered after clinical use, which were pooled, purified, and derivatized for loading into the vehicle. The present Letter describes construction of peptide-vehicles, bioconjugates focusing on more efficiency and cancer cell selective delivery for anticancer drugs.     

One-pot synthesis of helical aromatics: stereoselectivity, stability against racemization, and assignment of absolute configuration assisted by experimental and theoretical circular dichroism

Helical aromatics (1) were synthesized via one step in good quantity by solvent-free condensation of N,N'-p-phenylenediamine (2) and various carboxylic acids in the presence of a Lewis acid. Microwave irradiation greatly facilitated the condensation reaction to furnish 1 with a 100% diastereo- and a 50% enantioselectivity, when a chiral carboxylic acid was utilized. 1f, derived from 2-methylglutaric acid, was quite stable, no racemization taking place even at 200 degrees C. The assignment of the absolute configurations to the helical aromatics has been achieved by experimental and theoretical CD spectra calculated by time-dependent density functional theory.