Toposes for semigroups: an invitation

This is an expository paper in which we explain the basic ideas of topos theory in connection with semigroup theory. We focus mainly on the classifying topos of an inverse semigroup or pseudogroup, and to some extent on creating a dictionary between the language of semigroups and topos theory. We begin with the algebraic theory having to do with inverse semigroups, and then turn to an analysis of pseudogroups using sheaf theory. Our work includes some new material on wide semigroup homomorphisms and their geometric morphisms.

     

Pentameric circular iron(II) double helicates and a molecular pentafoil knot

We report on the synthesis of 11 pentameric cyclic helicates formed by imine condensation of alkyl monoamines with a common bis(formylpyridine)bipyridyl-derived building block and iron(II) and chloride ions. The cyclic double-stranded helicates were characterized by NMR spectroscopy, mass spectrometry, and in the case of a 2,4-dimethoxybenzylamine-derived pentameric cyclic helicate, X-ray crystallography. The factors influencing the assembly process (reactant stoichiometry, concentration, solvent, nature and amount of anion) were studied in detail: the role of chloride in the assembly process appears not to be limited to that of a simple template, and larger circular helicates observed with related tris(bipyridine) ligands with different iron salts are not produced with the imine ligands. Using certain chiral amines, pentameric cyclic helices of single handedness could be isolated and the stereochemistry of the helix determined by circular dichroism. By employing a particular diamine, a closed-loop molecular pentafoil knot was prepared. The pentafoil knot was characterized by NMR spectroscopy, mass spectrometry, and X-ray crystallography, confirming the topology and providing insights into the reasons for its formation.     

Excited Random Walks with Non-nearest Neighbor Steps

Let W be an integer-valued random variable satisfying \(E[W] =: \delta \ge 0\) and \(P(W<0)>0\), and consider a self-interacting random walk that behaves like a simple symmetric random walk with the exception that on the first visit to any integer \(x\in \mathbb Z\), the size of the next step is an independent random variable with the same distribution as W. We show that this self-interacting random walk is recurrent if \(\delta \le 1\) and transient if \(\delta >1\). This is a special case of our main result which concerns the recurrence and transience of excited random walks (or cookie random walks) with non-nearest neighbor jumps.     

Free volume studies in miscible polymer blend systems

This paper summarises the currently available literature concerned with measurement of free volume in miscible, amorphous polymer blends using positron annihilation lifetime spectroscopy (PALS) which probes excluded volume at the angstrom level. Previously reported data is compared with new data from a range of different blend systems. Miscible blends tend to show a negative deviation of free volume size (and to a lesser degree free volume fraction) on mixing due to the intimacy of packing of the blend component macromolecules. A largely immiscible system is also reported and shows a different behaviour (positive deviation of free volume size) and this is ascribed to additional free volume at the interface.     

Promoting Equitable Biology Lab Instruction by Engaging All Students in a Broad Range of Science Practices: An Exploratory Study

This study examines what students enrolled in the honors and general sections of a high school biology course offered at the same school learn when they have an opportunity to participate in a broad or narrow range of science practices during their laboratory experiences. The results of our analysis suggest that the students enrolled in the general sections of the course made similar or larger gains than the students enrolled in the honors section of the course in their abilities to plan and carry out an investigation, argue from evidence, and write a science‐specific persuasive essay when these students had an opportunity to participate in a broad range of science practices. These findings suggest that laboratory experiences that give students an opportunity to participate in a broad range of science practices, although considered challenging by many teachers, have the potential to help all students become more proficient in science. The article concludes with a discussion of the implications of this study for classroom instruction and educational policy.     

Optical stimulation of neural tissue in vivo

For more than a century, the traditional method of stimulating neural activity has been based on electrical methods, and it remains the gold standard to date. We report a technological breakthrough in neural activation in which low-level, pulsed infrared laser light is used to elicit compound nerve and muscle potentials in mammalian peripheral nerve in vivo. Optically induced neural action potentials are spatially precise, artifact free, and damage free and are generated by use of energies well below tissue ablation threshold. Thus optical stimulation presents a simple yet novel approach to contact-free in vivo neural activation that has major implications for clinical neurosurgery, basic neurophysiology, and neuroscience.     

Substituent effects in homoleptic iron(II) and ruthenium(II) complexes of 4′-hydrazone derivatives of 2,2′:6′,2″-terpyridine

The synthesis and characterization of eight homoleptic iron(II) and ruthenium(II) complexes containing 4′-hydrazone-substituted 2,2′:6′,2″-terpyridine ligands are described. ¹H NMR spectroscopic data illustrate that the coordinated ligands undergo facile rotation about the C_pyridine–N_hydrazone bond when the N atom is methylated, and hindered bond rotation when the hydrazone NH unit is available for hydrogen bonding to solvent molecules. Detailed structural studies illustrate how the flexibility of the backbone of the complexes leads to significant variation in packing. Throughout the series of solid structures, the packing is dictated by a combination of face-to-face aromatic π-stacking, edge-to-face aromatic interactions and classical and non-classical hydrogen bonding.