Distributive lattices determined by weighted double skeletons

Related to his S-glued sum construction, the skeleton S(L) of a finite lattice L was introduced by C. Herrmann in 1973. Our theorem asserts that if D is a finite distributive lattice and its second skeleton, S(S(D)), is the trivial lattice, then D is characterized by its weighted double skeleton, introduced by the second author in 2006. The assumption on the second skeleton is essential.     

Designing and scaling up an innovation in abstract algebra

In this paper, we describe the process of designing and scaling up the TAAFU group theory curriculum. This work unfolded in three overlapping stages of research and design. The initial designs emerged along with local instructional theories as the result of small-scale design experiments conducted with pairs of students. A second stage of the research and design process focused on generalizing from the initial laboratory design context to an authentic classroom setting. The third (ongoing) stage involves generalizing to instructors (mathematicians) who were not involved in the design process. We describe each of these stages, and our efforts to investigate the efficacy of the resulting curriculum, in order to provide an illustrative example of the process of scaling up an innovation.     

How to Exhibit Toroidal Maps in Space

Steinitz's theorem states that a graph is the 1-skeleton of a convex polyhedron if and only if it is 3-connected and planar. The polyhedron is called a geometric realization of the embedded graph. Its faces are bounded by convex polygons whose points are coplanar. A map on the torus does not necessarily have such a geometric realization. In this paper we relax the condition that faces are the convex hull of coplanar points. We require instead that the convex hull of the points on a face can be projected onto a plane so that the boundary of the convex hull of the projected points is the image of the boundary of the face. We also require that the interiors of the convex hulls of different faces do not intersect. Call this an exhibition of the map. A map is polyhedral if the intersection of any two closed faces is simply connected. Our main result is that every polyhedral toroidal map can be exhibited. As a corollary, every toroidal triangulation has a geometric realization.     

Fluorescence Enhancements on Silver Colloid Coated Surfaces

We observed a strong, more than 16-fold, enhancement of Texas Red-labeled BSA fluorescence emission when deposited on silver colloid coated surfaces (SCCS). The same labeled protein deposited on silver island films (SIFs) showed an approximate 8-fold fluorescence enhancement. The lifetimes of Texas Red-BSA fluorescence are significantly shorter on silvered surfaces than on uncoated quartz substrate indicating a strong change in radiative decay rate of the dyes. We also observed a 36-fold increased brightness of overlabeled fluorescein-HSA deposited on silver colloid coated surface. Stronger enhancement observed for overlabeled Fl-HSA protein indicates that presence of silver particles partially decreased self-quenching. Our results indicate that surfaces coated with silver colloids are valuable substrates for metal-enhanced fluorescence.     

Diagram,valence-to-core,and hypersatellite Kβ X-ray transitions in metallic chromium

We report on measurements of the Kβ diagram, valence-to-core (VtC), and hypersatellite X-ray spectra induced in metallic Cr by photon single and double K-shell ionization. The experiment was carried out at the Stanford Synchrotron Radiation Lightsource using the seven-crystal Johann-type hard X-ray spectrometer of the beamline 6-2. For the Kβ diagram and VtC transitions, the present study confirms the line shape features observed in previous works, whereas the K^hβ hypersatellite transition was found to exhibit a complex spectral line shape and a characteristic low-energy shoulder. The energy shift of the hypersatellite relative to the parent diagram line was deduced from the measurements and compared with the result of extensive multiconfiguration Dirac–Fock (MCDF) calculations. A very good agreement between experiment and theory was found. The MCDF calculations were also used to compute the theoretical line shape of the hypersatellite. A satisfactory agreement was obtained between the overall shapes of the experimental and theoretical spectra, but deviations were observed on the low- and high-energy flanks of the hypersatellite line. The discrepancies were explained by chemical effects, which were not considered in the MCDF calculations performed for isolated atoms.     

On Raman optical activity sign-switching between the ground and excited states leading to an unusual resonance ROA induced chirality

Raman optical activity (ROA) spectra recorded for a chiral naphthalene diimide derivative (nBu-NDI–BINAM) dissolved in a series of solvents exhibit strong solute to solvent induced chirality with: (1) dominating bands of solvents, (2) nBu-NDI–BINAM resonance ROA (RROA) bands which are barely visible, (3) monosignate RROA Solvent spectra with an unexpected sign concordant with that of the ECD band of the resonant electronic state, (4) bisignate RROA bands for a few solvents, and (5) superposition of non-resonant and resonant ROA bands of the chiral solvents. The unusual ROA enhancement was explained in terms of resonance energy transfer with resonant Raman emission. The surprising RROA sign-switching was found to be due to specific conformational equilibria where one solute conformer dominates in the ground and the other in the first excited singlet state, and, the signs of the related ECD bands of these two conformers are opposite.

Unusual solute to solvent induced chirality in ROA comes from specific conformer equilibria in the ground and the excited states.     

Supravital fluorometric apoptosis detection in a single mouse embryo using lab-on-a-chip

Detection of apoptosis is one of the main criteria of preimplantation embryo growth potential assessment. Recent developments in lab-on-a-chip techniques has led to apoptosis detection and monitoring on a single cell or embryo level. However, single embryo apoptosis detection without a change in embryo developmental competence and post-examination "recovery" still remains a challenge. In this paper we present a lab-on-a-chip, co-working with miniaturized optical instrumentation, which allows supravital examination of single embryos for the presence of apoptotic blastomers with full after lab-on-a-chip study "recovery" and maintenance of their further developmental capacity.