Consistency proof via pointwise induction

We show that the consistency of the first order arithmetic follows from the pointwise induction up to the Howard ordinal. Our proof differs from U. Schmerl [Sc]: We do not need Girard's Hierarchy Comparison Theorem. A modification on the ordinal assignment to proofs by Gentzen and Takeuti [T] is made so that one step reduction on proofs exactly corresponds to the stepping down in ordinals. Also a generalization to theories of finitely iterated inductive definitions is proved. Received May 30, 1996     

Synthesis,Characterization, and Properties of [4]Cyclo‐2,7‐pyrenylene: Effects of Cyclic Structure on the Electronic Properties of Pyrene Oligomers

A cyclic tetramer of pyrene, [4]cyclo‐2,7‐pyrenylene ([4]CPY), was synthesized from pyrene in six steps and 18 % overall yield by the platinum‐mediated assembly of pyrene units and subsequent reductive elimination of platinum. The structures of the two key intermediates were unambiguously determined by X‐ray crystallographic analysis. DFT calculations showed that the topology of the frontier orbitals in [4]CPY was essentially the same as those in [8]cycloparaphenylene ([8]CPP), and that all the pyrene units were fully conjugated. The electrochemical analyses proved the electronic properties of [4]CPY to be similar to those of [8]CPP. The results are in sharp contrast to those obtained for the corresponding linear oligomers of pyrene in which each pyrene unit was electronically isolated. The results clearly show a novel effect of the cyclic structure on cyclic π‐conjugated molecules.     

Molecular shape of palytoxin in aqueous solution

Palytoxin, one of the most toxic non-peptide substances, formed an associated dimer of 5 nm length in aqueous solution.     

Probability Distribution on Full Rooted Trees

The recursive and hierarchical structure of full rooted trees is applicable to statistical models in various fields, such as data compression, image processing, and machine learning. In most of these cases, the full rooted tree is not a random variable; as such, model selection to avoid overfitting is problematic. One method to solve this problem is to assume a prior distribution on the full rooted trees. This enables the optimal model selection based on Bayes decision theory. For example, by assigning a low prior probability to a complex model, the maximum a posteriori estimator prevents the selection of the complex one. Furthermore, we can average all the models weighted by their posteriors. In this paper, we propose a probability distribution on a set of full rooted trees. Its parametric representation is suitable for calculating the properties of our distribution using recursive functions, such as the mode, expectation, and posterior distribution. Although such distributions have been proposed in previous studies, they are only applicable to specific applications. Therefore, we extract their mathematically essential components and derive new generalized methods to calculate the expectation, posterior distribution, etc.     

Stochastic Model of Block Segmentation Based on Improper Quadtree and Optimal Code under the Bayes Criterion

Most previous studies on lossless image compression have focused on improving preprocessing functions to reduce the redundancy of pixel values in real images. However, we assumed stochastic generative models directly on pixel values and focused on achieving the theoretical limit of the assumed models. In this study, we proposed a stochastic model based on improper quadtrees. We theoretically derive the optimal code for the proposed model under the Bayes criterion. In general, Bayes-optimal codes require an exponential order of calculation with respect to the data lengths. However, we propose an algorithm that takes a polynomial order of calculation without losing optimality by assuming a novel prior distribution.     

Alternative Synthesis of Dibenzo-and Dipyrido-[1,3]Diazepines from Thioamides and o,o′-Diaminobiaryls

Thioamides were treated with iodomethane, and then o,o′-diaminobiphenyl or bipyridyl to afford 6-substituted-5H-dibenzo- or dipyrido [1,3]diazepines in good yields.     

Selective and random syntheses of [n]cycloparaphenylenes (n=8-13) and size dependence of their electronic properties

[n]Cycloparaphenylenes (n = 8-13, CPPs) were synthesized, and their physical properties were systematically investigated. [8] and [12]CPPs were selectively prepared from the reaction of 4,4'-bis(trimethylstannyl)biphenyl and 4,4'-bis(trimethylstannyl)terphenyl, respectively, with Pt(cod)Cl(2) (cod = 1,5-cyclooctadiene) through square-shaped tetranuclear platinum intermediates. A mixture of [8]-[13]CPPs was prepared in good combined yields by mixing biphenyl and terphenyl precursors with platinum sources. Products were easily separated and purified by using gel permeation chromatography. In (1)H NMR spectra, the proton of the CPPs shifts to a lower field as n increased due to an anisotropic effect from the nearby PP moieties. Although the UV-vis spectra were rather insensitive to the size of the CPPs, the fluorescence spectra changed significantly in relation to their size. A larger Stokes shift was observed for the smaller CPPs. Redox properties of the CPPs were measured for the first time by using cyclic voltammetry, and the smaller CPPs had lower oxidation potentials. The results are consistent with the HOMO energies of CPPs, of which the smaller CPPs had higher energies.