Affine PBW bases and MV polytopes in rank 2

Mirkovi?–Vilonen (MV) polytopes have proven to be a useful tool in understanding and unifying many constructions of crystals for finite-type Kac-Moody algebras. These polytopes arise naturally in many places, including the affine Grassmannian, pre-projective algebras, PBW bases, and KLR algebras. There has recently been progress in extending this theory to the affine Kac-Moody algebras. A definition of MV polytopes in symmetric affine cases has been proposed using pre-projective algebras. In the rank-2 affine cases, a combinatorial definition has also been proposed. Additionally, the theory of PBW bases has been extended to affine cases, and, at least in rank-2, we show that this can also be used to define MV polytopes. The main result of this paper is that these three notions of MV polytope all agree in the relevant rank-2 cases. Our main tool is a new characterization of rank-2 affine MV polytopes.     

The crystal commutor and Drinfeld’s unitarized <Emphasis Type="Italic">R</Emphasis>-matrix

Drinfeld defined a unitarized R-matrix for any quantum group . This gives a commutor for the category of representations, making it into a coboundary category. Henriques and Kamnitzer defined another commutor which also gives representations the structure of a coboundary category. We show that a particular case of Henriques and Kamnitzer’s construction agrees with Drinfeld’s commutor. We then describe the action of Drinfeld’s commutor on a tensor product of two crystal bases, and explain the relation to the crystal commutor. P. Tingley was supported by the RTG grant DMS-0354321.     

On the Origins of Some Spectroscopic Properties of “Purple Iron” (the Tetraoxoferrate(VI) Ion) and Its Pourbaix Safe-Space

In this work, the authors attempt to interpret the visible, infrared and Raman spectra of ferrate(VI) by means of theoretical physical-inorganic chemistry and historical highlights in this field of interest. In addition, the sacrificial decomposition of ferrate(VI) during water treatment will also be discussed together with a brief mention of how Rayleigh scattering caused by the decomposition of Fe^VIO₄²⁻ may render absorbance readings erroneous. This work is not a compendium of all the instrumental methods of analysis which have been deployed to identify ferrate(VI) or to study its plethora of reactions, but mention will be made of the relevant techniques (e.g., Mössbauer Spectroscopy amongst others) which support and advance this overall discourse at appropriate junctures, without undue elaboration on the foundational physics of these techniques.     

Direct, loss-tolerant characterization of nonclassical photon statistics

We experimentally investigate a method of directly characterizing the photon-number distribution of nonclassical light beams that is tolerant to losses and makes use of only standard binary detectors. This is achieved in a single measurement by calibrating the detector using some small amount of prior information about the source. We demonstrate the technique on a freely propagating heralded two-photon-number state created by conditional detection of a two-mode squeezed state generated by parametric down-conversion.     

Autoconfocal microscopy with a cw laser and thermionic detection

We introduce an application of thermionic emission in a PMT photocathode. Because of the nonlinear dependence of thermionic emission on absorbed laser power, a conventional PMT is found to produce a virtual pinhole effect that rejects unfocused light at least as strongly as a physical pinhole. This virtual pinhole effect is exploited in a scanning transmission confocal microscope equipped with a cw laser source. Because the area of the PMT photocathode is large, signal descanning is not required and thermionic detection acts as a self-aligned pinhole. Our technique of thermionic-detection autoconfocal microscopy is further implemented with graded-field contrast to obtain enhanced phase-gradient sensitivity in unlabeled samples, such as rat hippocampal brain slices.     

Derivatives of molecular Vibrational force constants with a singular Jacobian

Counter examples are presented that negate the generality of Gans' claim that a singular jacobian demands the existence of one off-diagonal force constant F_st possessing the extremal property ?F_rr/?F_st = ∞ for all of the n diagonal force constants F_rr (r = 1, > 62; n) in a symmetry species containing n vibrations.     

Rigid core vinamidinium salts and their N,N'-rotamers

The power of proton magnetic resonance spectroscopy to unravel stereochemical details is amply demonstrated. O-Methylation of 3-methylamino-5,5-dimethyl-2-cylohexen-1-one (1a) produces stable diastereomers, (Z)- and (E)-N-(3-methoxy-5,5-dimethyl-2-cyclohexen-1-ylidine)-N-methylaminium iodide (2a). As predicted by computation and confirmed by spectroscopy, the (Z)-vinylogous imidate salt predominates. Reaction of 2a with primary and secondary amines furnished a number of vinamidinium salts, including N-(3-methylamino-5,5-dimethyl-2-cyclohexen-1-ylidene)-N-methylaminium iodide (3a). Two rotamers of 3a were identified and characterized. A substantial number of additional compounds 2 and 3 are included in the study.     

A new route to highly functionalized heterocyclic rings

A novel cascade reaction has been developed for the rapid construction of heterocyclic rings. The cyclization is thermally induced and does not involve the use of metal ions. This highly efficient construction of furans has been developed during studies directed toward the synthesis of the antibiotic lactonamycin 1.