Regular simplices passing through holes

What is the smallest circular or square wall hole that a regular tetrahedron can pass? This problem was solved by Itoh et al. (Rend Circ Mat Palermo 2(77):349–354, 2006). Then, we settled the case of equilateral triangular hole in Bárány et al. (Tetrahedra passing through a triangular hole, 2009). Motivated by these results, we consider the corresponding problems in higher dimensions. Among other results, we determine the minimum (n ? 1)-dimensional ball hole that a unit regular n-simplex can pass. The diameter of the minimum hole goes to \({3\sqrt{2}/4}\) as n tends to infinity.     

Crack-free processing of glasses by 1 μm-YAG laser and translucent adhesive tape and its numerical analyses by CIP

We succeeded in drilling a crack-free high-aspect-ratio hole through glass with a YAG laser of 1 m wavelength, 5 ns pulse width and 30–630 mJ energies. The hole is 25.5 mm long with 500 m radius, and no crack is observed even in atmospheric condition. This success is due to the translucent adhesive tape pasted on glass surface in which the temperature is controlled and thermal stress is reduced. Computer simulation revealed the occurring phenomena. PACS 61.43.Fs; 61.80.; 68.08.De     

Application of protein-coupled liposomes to effective affinity screening from phage library

For effective screening by biopanning, we propose a new affinity screening method utilizing protein-coupled liposomes (proteoliposomes) as adsorbents. With multilamellar vesicles (MLVs) composed of dipalmitoylphosphatidylcholine (DPPC): dicetylphosphate (DCP) = 10: 1 (molar ratio), adsorption of nonspecific phage VCSM13 to the liposomes without any blocking was comparable to that on polystyrene tube wall coated with blocking protein. Phages displaying octapeptides specific to an anti-peptide antibody against a peptide antigen (FVNQHLCK) were screened from an octapeptide-displayed phage library by biopanning utilizing liposomes coupled with the antibody (AB-MLVs) or a conventional immunotube coated with the antibody (AB-tube). After four rounds of biopanning, all selected phages displayed homological peptides to the antigen peptide by use of AB-MLVs, while only 15% of the selected phages displayed homological peptides in the conventional biopanning. The octapeptide selected by AB-MLVs against the anti-peptide antibody showed comparable binding affinity, which were determined by the competitive ELISA and an immunoaffinity chromatography, to that of the peptide antigen. Thus, protein-coupled liposomes are useful as adsorbents for screening from combinatorial phage libraries.     

Stereoselective oxidation at C-4 of flavans by the endophytic fungus Diaporthe sp. isolated from a tea plant

The microbial transformation of five flavans (1-5) by endophytic fungi isolated from the tea plant Camellia sinensis was investigated. It was found that the endophytic filamentous fungus Diaporthe sp. oxidized stereoselectively at C-4 position of (+)-catechin (1) and (-)-epicatechin (2) to give the correspondent 3,4-cis-dihydroxyflavan derivatives (6, 10), respectively. (-)-Epicatechin 3-O-gallate (3) and (-)-epigallocatechin 3-O-gallate (4) were also oxidized by the fungus into 3,4-dihydroxyflavan derivatives (10, 12) via (-)-epicatechin (2) and (-)-epigallocatechin (11), respectively. Meanwhile, (-)-gallocatechin 3-O-gallate (5), (-)-catechin (ent-1) and (+)-epicatechin (ent-2), which possess a 2S-phenyl substitution, resisted the biotransformation.     

Simplices whose dihedral angles are all rational multiples of <Emphasis Type="Italic">π</Emphasis>, and related topics

We prove the following: (1) For every \({n \geq 2}\), there are infinitely many, mutually non-similar n-dimensional simplices in \({\mathbb{R}^n}\) whose dihedral angles are all rational multiples of π. (2) For every \({n \geq 3}\), there are uncountably many, mutually non-similar n-simplices whose dihedral angles and π are linearly independent over the rational field. Moreover, in the set of all n-simplices in \({\mathbb{R}^n}\) the subset of such n-simplices is everywhere dense with respect to the Hausdorff distance.     

Element number of the Platonic solids

Let Σ be a set of polyhedra. A set Ω of polyhedra is said to be an element set for Σ if each polyhedron in Σ is the union of a finite number of polyhedra in Ω. We call each polyhedron of the element set Ω an element for Σ. In this paper, we determine one element set for the set Π of the Platonic solids, and prove that this element set is, in fact, best possible; it achieves the minimum in terms of cardinality among all the element sets for Π. We also introduce the notion of indecomposability of a polyhedron and present a conjecture in Sect. 3.