Entropy for Symbolic Dynamics with Overlapping Alphabets

We consider shift spaces in which elements of the alphabet may overlap nontransitively. We define a notion of entropy for such spaces and show that it is equal to a limit of entropies of standard (non-overlapping) shifts when the underlying shift is of finite type. When a shift space with overlaps arises as a model for a discrete dynamical system with a finite set of overlapping neighborhoods, the entropy gives a lower bound for the topological entropy of the dynamical system.     

John’s Ellipsoid and the Integral Ratio of a Log-Concave Function

We extend the notion of John’s ellipsoid to the setting of integrable log-concave functions. This will allow us to define the integral ratio of a log-concave function, which will extend the notion of volume ratio, and we will find the log-concave function maximizing the integral ratio. A reverse functional affine isoperimetric inequality will be given, written in terms of this integral ratio. This can be viewed as a stability version of the functional affine isoperimetric inequality.     

Cytotoxic and NF-κB inhibitory sesquiterpene lactones from Piptocoma rufescens

Six new (1-6) and eight known germacranolide-type sesquiterpene lactones, along with several known phenylpropanol coumarates and methylated flavonoids, were isolated from the leaves of Piptocoma rufescens, collected in the Dominican Republic. The new compounds were identified by analysis of their spectroscopic data, with the molecular structure of 3 being established by single-crystal X-ray diffraction. The absolute configurations of the sesquiterpene lactones isolated were determined from their CD and NOESY NMR spectra, together with the analysis of Mosher ester reactions. Bioassay screening results showed the majority of the sesquiterpene lactones isolated (1-13) to be highly cytotoxic toward the HT-29 human colon cancer cell line, with the most potent compound being 15-deoxygoyazensolide (10, IC(50), 0.26 μM). In addition, several of the sesquiterpene lactones exhibited NF-κB (p65) inhibitory activity.     

Similarities and differences in the structures of 5‐bromo‐6‐hydroxy‐7,8‐dimethylchroman‐2‐one and 6‐hydroxy‐7,8‐dimethyl‐5‐nitrochroman‐2‐one

The title compounds, C₁₁H₁₁BrO₃, (I), and C₁₁H₁₁NO₅, (II), respectively, are derivatives of 6‐hydroxy‐5,7,8‐trimethylchroman‐2‐one substituted at the 5‐position by a Br atom in (I) and by a nitro group in (II). The pyranone rings in both molecules adopt half‐chair conformations, and intramolecular O—H...Br [in (I)] and O—H...O_nitro [in (II)] hydrogen bonds affect the dispositions of the hydroxy groups. Classical intermolecular O—H...O hydrogen bonds are found in both molecules but play quite dissimilar roles in the crystal structures. In (I), O—H...O hydrogen bonds form zigzag C(9) chains of molecules along the a axis. Because of the tetragonal symmetry, similar chains also form along b. In (II), however, similar contacts involving an O atom of the nitro group form inversion dimers and generate R₂²(12) rings. These also result in a close intermolecular O...O contact of 2.686 (4) Å. For (I), four additional C—H...O hydrogen bonds combine with π–π stacking interactions between the benzene rings to build an extensive three‐dimensional network with molecules stacked along the c axis. The packing in (II) is much simpler and centres on the inversion dimers formed through O—H...O contacts. These dimers are stacked through additional C—H...O hydrogen bonds, and further weak C—H...O interactions generate a three‐dimensional network of dimer stacks.     

The Use of tert-Butyl Vinyl Ether in Stepwise Electrophilic Addition Reactions

tert-Butyl vinyl ether (1) reacts with p-TolSCl to give 2-tert-butoxy-2-chloroethyl p-tolyl sulfide (2). In the presence of SnCl₄, 2 reacts with silyl enol ethers, allyltrimethylsilane, and vinyl ethers to form a C-C bond. In the case of vinyl ethers, the reaction proceeds through the formation of the 5-membered sulfonium salt intermediate which in turn can react with H₂O, TMSCN, allyltrimethylsilane, and Grignard reagents.