Synthesis of perfluorobicyclic ethers

The electrochemical fluorination of cycloalkyl-substituted acetic and propionic acids were conducted to obtain several kinds of perfluorobicyclic ethers [reaction conditions: Anodic current density: 3.5 A/dm², Volt: 5～8V, Temp: 5～6 °C] . For example, from cyclopentyl-substituted acetic acids, perfluoro(4-alkyl-2-oxabicyclo[3.3.0]octane)s (I) were formed as the cyclization products in yields of 10～19% together with the corresponding perflouroalkanoyl flourides (II) (Y=3～13%).

While, perfluorospiro-ethers were obtained from the fluorination of 3-cycloalkyl-substituted propionic acids. The characterization of these perfluorobicyclic ethers which consists of the reaction with anhydrous aluminum chloride will be reported also.     

Asymmetric synthesis of permethric acid. stereochemistry of chiral copper carbenoid reaction

The most effective optical isomer (1$\text{R}$-$\text{cis}$) of permethric acid ($\text{2}$, R = H), a potent intermediate in the production of synthetic pyrethroid, was enantioselectively prepared.     

Organoaluminium reagents induced isomerization of 5-hydroxy-3,3,6-trimethyl-5-heptenoic acid δ-lactone to 2,2,5,5-tetramethyl-1,3-cyclohexanedione

Treatment of the title lactone and the related δ or ε-lactones with Et₂AlSR (R = Et or Ph) gave 1,3-cyclohexanediones or 1,3-cycloheptanediones in excellent yields.     

Edge proximity and matching extension in projective planar graphs

A graph $G$ with at least $2m+2$ vertices is said to be distance $d$ $m$-extendable if, for any matching $M$ of $G$ with $m$ edges in which the edges lie at distance at least $d$ pairwise, there exists a perfect matching of $G$ containing $M$. In this paper we prove that every 5-connected triangulation on the projective plane of even order is distance 3 7-extendable and distance 4 $m$-extendable for any $m$.     

Hausdorff measures and packing measures of limit sets of CIFSs of generalized complex continued fractions

ABSTRACT

We consider a certain family of CIFSs of the generalized complex continued fractions with a complex parameter space. We show that for each CIFS of the family, the Hausdorff measure of the limit set of the CIFS with respect to the Hausdorff dimension is zero and the packing measure of the limit set of the CIFS with respect to the Hausdorff dimension is positive (main result). This is a new phenomenon of infinite CIFSs which cannot hold in finite CIFSs. We prove the main result by showing some estimates for the unique conformal measure of each CIFS of the family and by using some geometric observations.     

Oxidation of alcohols with oxoperoxobis(N-phenylbenzohydroxamato)molybdenum(VI)

The title complex oxidizes primary and secondary alcohols to the corresponding carbonyl compounds. Stereoselective epoxidation of allylic alcohols is also described.     

Fluorination of 2-alkyl-substituted oxanes. The synthesis and purification of perfluoro(2-alkyl-substituted oxane)s

Five kinds of 2-alkyl-substituted oxanes like 2-ethyloxane, 2-n-propyloxane, 2-iso-propyloxane, 2-n-butyloxane and 2-n-amyloxane were fluorinated electrochemically to give the corresponding perfluoro(2-alkyloxane)s. The perfluoro(2-alkyloxane)s were obtained in good yields from these starting materials together with isomeric perfluoro(2-alkyloxolane)s, perfluoro(2-alkyl-5-methyloxolane)s and perfluoro(dialkyl ether)s. The purification of the perfluoro(2-alkyloxane)s which contained small amounts of isomeric perfluoro(2-alkyloxolane)s was successfully achieved by recovering the former unreacted after treating these mixture with anhydrous aluminum chloride at 150 /sR 160 °C during /sR 48 hrs in order to convert the latter into the easy-separable perfluoro(2,5,5- trichloro-2-alkyloxolane)s. Small quantities of new perfluoro(5,5-dichloroalkanoyl chloride)s were also among the chlorination products. The spectroscopic data as well as the physical properties of these new fluorination products, and perfluoro(2,5,5-trichloro-2-alkyloxolane)s and perfluoro(5,5-dichloroalkanoyl chloride)s are presented.     

Interaction cohomology of forward or backward self-similar systems

We investigate the dynamics of forward or backward self-similar systems (iterated function systems) and the topological structure of their invariant sets. We define a new cohomology theory (interaction cohomology) for forward or backward self-similar systems. We show that under certain conditions, the space of connected components of the invariant set is isomorphic to the inverse limit of the spaces of connected components of the realizations of the nerves of finite coverings U of the invariant set, where each U consists of (backward) images of the invariant set under elements of finite word length. We give a criterion for the invariant set to be connected. Moreover, we give a sufficient condition for the first cohomology group to have infinite rank. As an application, we obtain many results on the dynamics of semigroups of polynomials. Moreover, we define postunbranched systems and we investigate the interaction cohomology groups of such systems. Many examples are given.     

Transversality family of expanding rational semigroups

We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a d$d$-parameter family of such semigroups satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the Julia set is the minimum of 2 and the zero of the pressure function. Moreover, the Hausdorff dimension of the exceptional set of parameters is estimated. We also show that if the zero of the pressure function is greater than 2$2$, then typically the 2-dimensional Lebesgue measure of the Julia set is positive. Some sufficient conditions for a family to satisfy the transversality conditions are given. We give non-trivial examples of families of semigroups of non-linear polynomials with the transversality condition for which the Hausdorff dimension of the Julia set is typically equal to the zero of the pressure function and is less than 2$2$. We also show that a family of small perturbations of the Sierpinski gasket system satisfies that for a typical parameter value, the Hausdorff dimension of the Julia set (limit set) is equal to the zero of the pressure function, which is equal to the similarity dimension. Combining the arguments on the transversality condition, thermodynamical formalisms and potential theory, we show that for each a∈C$a\in \mathbb{C}$ with |a|≠0,1$\left|a\right|\ne 0,1$, the family of small perturbations of the semigroup generated by {z²,az²}$\{z^2,a{z}^2\}$ satisfies that for a typical parameter value, the 2-dimensional Lebesgue measure of the Julia set is positive.     

Complex genetic evolution of artificial self‐replicators in cellular automata

It is widely believed that evolutionary dynamics of artificial self‐replicators realized in cellular automata (CA) are limited in diversity and adaptation. Contrary to this view, we show that complex genetic evolution may occur within simple CA. The evolving self‐replicating loops (“evoloops”) we investigate exhibit significant diversity in macro‐scale morphologies and mutational biases, undergoing nontrivial genetic adaptation by maximizing colony density and enhancing sustainability against other species. Nonmutable subsequences enable genetic operations that alter fitness differentials and promote long‐term evolutionary exploration. These results demonstrate a unique example of genetic evolution hierarchically emerging from local interactions between elements much smaller than individual replicators. © 2004 Wiley Periodicals, Inc. Complexity 10: 33–39, 2004