Naphthalene-based calixarenes: unusual regiochemistry of a Friedel-Crafts alkylation

[reaction: see text] In the pursuit of naphthalene-based calixarenes, a Friedel-Crafts alkylation with unusual regiochemistry was observed. Treatment of carbinol 14 with catalytic triflic acid was expected to produce calixarenes of the class represented by 16. Instead, the major product was cyclic trimer 15, in which alkylation of each naphthalene ring occurred at the electronically deactivated position. The structure of compound 15 was assigned by 2-D NMR studies.     

Weakly abelian lattice-ordered groups

Every nilpotent lattice-ordered group is weakly Abelian; i.e., satisfies the identity . In 1984, V. M. Kopytov asked if every weakly Abelian lattice-ordered group belongs to the variety generated by all nilpotent lattice-ordered groups [The Black Swamp Problem Book, Question 40]. In the past 15 years, all attempts have centred on finding counterexamples. We show that two constructions of weakly Abelian lattice-ordered groups fail to be counterexamples. They include all preiously considered potential counterexamples and also many weakly Abelian ordered free groups on finitely many generators. If every weakly Abelian ordered free group on finitely many generators belongs to the variety generated by all nilpotent lattice-ordered groups, then every weakly Abelian lattice-ordered group belongs to this variety. This paper therefore redresses the balance and suggests that Kopytov's problem is even more intriguing.

     

Paroxysmal starting and stopping of circulating waves in excitable media

Levels of intracellular Ca2+ were monitored using fluorescence from Ca2+-sensitive dyes in chick embryonic heart cells cultured in an annular geometry. There was spontaneous starting and stopping of reentrant waves of activity. The results are modeled using modified FitzHugh-Nagumo equations representing pacemakers embedded in a conducting medium. These results provide a potential mechanism for spontaneous abnormal cardiac rhythms in which there are rapid heart beats (tachycardias) that repetitively start and stop.     

Subtle white matter volume differences in children treated for medulloblastoma with conventional or reduced dose craniospinal irradiation

Medulloblastoma is the most common malignant brain tumor in children, and approximately seventy percent of average-risk patients will achieve long-term survival. Craniospinal irradiation (CSI), combined with chemotherapy and surgery, is currently the mainstay of treatment but places children who survive at risk for serious neurocognitive sequelae. These sequelae are intensified with a younger age at treatment, greater elapsed time following treatment, and an increased radiation dose. Many newer treatment approaches have attempted to address this problem by reducing the dose of the CSI component of radiation therapy while maintaining the current survival rates. This study evaluates longitudinal MR imaging during therapy to assess the impact of the two CSI doses (conventional [36 Gy] and reduced [23.4 Gy]) on normal appearing white matter volumes (NAWMV) evaluated in a single index slice. Twenty-six children and young adults at least three years of age enrolled on an institutional protocol for newly diagnosed, previously untreated primary medulloblastoma had at least four MR examinations over a minimum nine month period following CSI. These serial volumes were evaluated as a function of time since CSI in three analyses: 1) all subjects, 2) subjects stratified by age at CSI, and 3) subjects stratified by CSI dose. The first analysis demonstrated that medulloblastoma patients treated with CSI have a significant loss of NAWMV in contradistiction to normally expected maturation. Stratifying the patients by age at CSI found no significant differences in the rate of NAWMV loss. The final analysis stratified the patients by CSI dose and revealed that the rate of NAWMV loss was 23% slower in children receiving reduced-dose. Serial quantitative MR measures of NAWMV may provide a neuroanatomical substrate for assessing functional impact of CSI on normal brain function following treatment for medulloblastoma.     

Combinatorial explosion in model gene networks

The explosive growth in knowledge of the genome of humans and other organisms leaves open the question of how the functioning of genes in interacting networks is coordinated for orderly activity. One approach to this problem is to study mathematical properties of abstract network models that capture the logical structures of gene networks. The principal issue is to understand how particular patterns of activity can result from particular network structures, and what types of behavior are possible. We study idealized models in which the logical structure of the network is explicitly represented by Boolean functions that can be represented by directed graphs on n-cubes, but which are continuous in time and described by differential equations, rather than being updated synchronously via a discrete clock. The equations are piecewise linear, which allows significant analysis and facilitates rapid integration along trajectories. We first give a combinatorial solution to the question of how many distinct logical structures exist for n-dimensional networks, showing that the number increases very rapidly with n. We then outline analytic methods that can be used to establish the existence, stability and periods of periodic orbits corresponding to particular cycles on the n-cube. We use these methods to confirm the existence of limit cycles discovered in a sample of a million randomly generated structures of networks of 4 genes. Even with only 4 genes, at least several hundred different patterns of stable periodic behavior are possible, many of them surprisingly complex. We discuss ways of further classifying these periodic behaviors, showing that small mutations (reversal of one or a few edges on the n-cube) need not destroy the stability of a limit cycle. Although these networks are very simple as models of gene networks, their mathematical transparency reveals relationships between structure and behavior, they suggest that the possibilities for orderly dynamics in such networks are extremely rich and they offer novel ways to think about how mutations can alter dynamics. (c) 2000 American Institute of Physics.     

A circle map in a human heart

A circle is divided into two regions, a black one and a white one. Successive iterates of an invertible nonlinear circle map generate a symbolic string indicating whether each iterate is in the black or white region. A number of remarkable properties of the symbolic sequences are described. These properties were previously described for a linear circle map corresponding to a rigid rotation in the “gaps and steps” problem. These results have direct application to a cardiac arrhythmia, parasystole, that results from the competition between two pacemakers in the heart, one in the sinus mode and the other in the ventricles. The theoretical results are directly applicable to a clinical case of a young man who had frequent extra heartbeats.     

Controllability and asymptotic stabilization of the Camassa-Holm equation

We investigate the problems of exact controllability and asymptotic stabilization of the Camassa-Holm equation on the circle, by means of a distributed control. The results are global, and in particular the control prevents the solution from blowing up.