Similar dissection of sets

In 1994, Martin Gardner stated a set of questions concerning the dissection of a square or an equilateral triangle in three similar parts. Meanwhile, Gardner’s questions have been generalized and some of them are already solved. In the present paper, we solve more of his questions and treat them in a much more general context. Let \({D\subset \mathbb{R}^d}\) be a given set and let f ₁, . . . , f _k be injective continuous mappings. Does there exist a set X such that \({D = X \cup f_1(X) \cup \cdots \cup f_k(X)}\) is satisfied with a non-overlapping union? We will prove that such a set X exists for certain choices of D and {f ₁, . . . , f _k }. The solutions X will often turn out to be attractors of iterated function systems with condensation in the sense of Barnsley. Coming back to Gardner’s setting, we use our theory to prove that an equilateral triangle can be dissected in three similar copies whose areas have ratio 1 : 1 : a for \({a \ge (3+\sqrt{5})/2}\).     

Synthesis, structures and inclusion properties of tetranaphthalides: new macrocyclic clathrate hosts

Novel tetranaphthalide host compounds 3 and 4 bearing isomeric naphthalene moieties have been synthesized and their inclusion properties were investigated. These host compounds enclathrated several kinds of ketones, cyclic ethers, amides, sulfoxides and aromatic compounds. The structures of two representative inclusion compounds containing different host molecules and a common guest (dimethyl sulfoxide) were investigated by X-ray diffraction to determine the nature of guest inclusion and to rationalize their distinctly different thermal decomposition profiles.     

Occurrence of C-glucoside of resveratrol oligomers in Hopea parviflora

Investigation of the highly polar chemical constituents in the stem of Hopea parviflora (Dipterocarpaceae) resulted in the isolation of four new resveratrol derivatives, hopeasides A and B (1, 2) (resveratrol pentamers), C (3) (resveratrol trimer), and D (4) (resveratrol dimer) together with nine known resveratrol oligomers (5-13). The new structures have a common partial structure of the 1-hydroxy-1-(3,5-dihydroxy-2-C-glucopyranosylphenyl)-2-(4-hydroxyphenyl)ethane-2-yl group after oxidative condensation of (E)-resveratrol-10-C-β-glucopyranoside (14). The structures were determined by spectroscopic analysis including 2D-NMR and computer-aided molecular modeling. The biogenetic relationship of the isolates and NMR characteristics caused by steric hindrance are also discussed in this paper.     

Electrical orientation behaviour of a nematic liquid crystal with negative dielectric anisotropy above the microelectrodes

Factors for the electrical orientation on interdigitated microelectrodes deposited on a substrate, a microdielectrometric sensor (or a sensor), were discussed experimentally using a model nematic liquid crystal with negative dielectric anisotropy. The electrical orientation behaviour of the nematic liquid crystal was investigated using microdielectrometry and polarised optical microscopy. The experimental results were discussed in relation to electric-field line, surface topology, anchoring of the liquid crystalline molecules to the microelectrodes, and transversal rotation of a rode-like molecule of the liquid crystal with negative dielectric anisotropy.     

Analysis of Photoexcitation Energy Dependence in the Photoluminescence of Firefly Luciferin

The whole pathways for photoluminescence, which include absorption, relaxation and emission, of firefly luciferin in aqueous solutions of different pH values with different photoexcitation energies were theoretically investigated by considering protonation/deprotonation. It is experimentally known that the color of fluorescence changes from green to red with a decrease in the photoexcitation energy. We confirmed with the theoretical analysis that the peak energy shift in the fluorescence spectra with varying photoenergies is due to a change in photoluminescence pathway. When the photoexcitation energy is decreased, the red emission from a monoanion form of firefly luciferin with carboxylate and phenolate groups and N‐protonated thiazoline ring occurs irrespective of the pH values. However, because the species abundant in the solution and those excited by the photon depend on the solution pH, the pathway leading to the monoanion form changes with the solution pH.     

Preparation of highly photosensitizing liposomes with fullerene-doped lipid bilayer using dispersion-controllable molecular exchange reactions

Fullerene-containing liposomes with high photosensitization ability were prepared. Disaggregated fullerenes were efficiently injected into the bilayer of liposomes by a phototriggered molecular exchange reaction. These liposomes showed far higher photoreactivity than liposomes thermally produced by heating and microwave irradiation. This result indicates that control of self-aggregation of fullerene leads to a high quantum yield for the photoreaction because of the suppression of self-quenching of photoexcited fullerenes.     

Theoretical study of absorption and fluorescence spectra of firefly luciferin in aqueous solutions

The absorption and fluorescence spectra of firefly luciferin, which is an analog of oxyluciferin, are investigated by performing the density functional theory (DFT) calculations, especially focusing on the experimentally unassigned peaks. Time-dependent DFT calculations are performed for the excited states of firefly luciferin and its conjugate acids and bases. We find that (1) the peaks in the experimental absorption spectra correspond to the excited states of not only (6'O(-), 4COO(-)) and (6'OH, 4COO(-)), but also (6'OH, 4COOH) and (6'OH, 3H(+), 4COOH); (2) the peaks in the experimental fluorescence spectra correspond to the excited states of not only (6'O(-), 4COO(-)), but also (6'OH, 4COO(-)), (6'O(-), 4COOH), (6'OH, 4COOH) and (6'OH, 3H(+), 4COOH); (3) the unassigned peak near 400 nm in the experimental absorption spectra at pH 1 is assigned to the absorption from the equilibrium ground state to the first excited state of (6'OH, 3H(+), 4COOH); and (4) the unassigned peak at 610 nm in the experimental fluorescence spectra corresponds to the transition from the equilibrium first excited state to the ground state of (6'OH, 4COO(-)).     

Absolute structure of shoreaketone: a rotational isomeric resveratrol tetramer in Dipterocarpaceaeous plants

A rotational isomeric shoreaketone (1), identified as a skeletal member of resveratrol tetramers, was isolated from three species of Dipterocarpaceaeous plants: Shorea uliginosa, Shorea hemsleyana, and Vateria indica. The structure was elucidated by spectroscopic analysis including NMR experiments and their absolute configurations determined based on circular dichroism data. Shoreaketone has 10 asymmetric carbons and a framework of fused heptacyclic ring system including a spiro ring and an α,β-unsaturated carbonyl group that has not been reported in any other natural product. NMR experiments using shoreaketone indicate the presence of two conformers due to restricted rotation of a C–C bond in solution. The complex stereochemistry is due to its skeleton, 10 asymmetric carbons, and a chiral axis. The conformations of rotational isomeric stilbenoid were studied by variable-temperature NMR, ROESY, a skeletal conversion. The coexistence of two conformers for shoreaketone (1) was confirmed to be 1a and 1b, in which the diaryl-dihydrobenzofuran moiety (unit 1B) is extended below or above the plane of the cyclopentane ring (unit 1A), respectively.