Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions

Combinatorial proofs are given for certain entries in Ramanujan's lost notebook. Bijections of Sylvester, Franklin, and Wright, and applications of Algorithm Z of Zeilberger are employed. A new bijection, involving the new concept of the parity sequence of a partition, is used to prove one of Ramanujan's fascinating identities for a partial theta function.     

What Is the Mechanism Driving the Reduction of Cardiovascular Events from Glucagon-like Peptide-1 Receptor Agonists?—A Mini Review

Glucagon-like peptide-1 receptor agonists (GLP-1 RAs) are considered the standard of care for type 2 diabetes in many countries worldwide. These molecules have profound anti-hyperglycaemic actions with a favourable safety profile. They are now being considered for their robust cardiovascular (CV) protective qualities in diabetic patients. Most recent CV outcome trials have reported that GLP-1 RAs reduce major adverse cardiovascular events (MACE). Furthermore, the GLP-1 RAs seem to target the atherosclerotic CV disease processes preferentially. GLP-1 RAs also improve a wide range of routinely measured surrogate markers associated with CV risk. However, mediation analysis suggests these modest improvements may contribute indirectly to the overall anti-atherogenic profile of the molecules but fall short in accounting for the significant reduction in MACE. This review explores the body of literature to understand the possible mechanisms that contribute to the CV protective profile of GLP-1 RAs.     

Iron in a Cage: Fixation of a Fe(II)tpy2 Complex by Fourfold Interlinking

The coordination sphere of the Fe(II) terpyridine complex 1 is rigidified by fourfold interlinking of both terpyridine ligands. Profiting from an octa‐aldehyde precursor complex, the ideal dimensions of the interlinking structures are determined by reversible Schiff‐base formation, before irreversible Wittig olefination provided the rigidified complex. Reversed‐phase HPLC enables the isolation of the all‐trans isomer of the Fe(II) terpyridine complex 1 , which is fully characterized. While temperature independent low‐spin states were recorded with superconducting quantum interference device (SQUID) measurements for both, the open precursor 8 and the interlinked complex 1 , evidence of the increased rigidity of the ligand sphere in 1 was provided by proton T₂ relaxation NMR experiments. The ligand sphere fixation in the macrocyclized complex 1 even reaches a level resisting substantial deformation upon deposition on an Au(111) surface, as demonstrated by its pristine form in a low temperature ultra‐high vacuum scanning tunneling microscope experiment.     

Sequence‐Based Prediction of Promiscuous Acyltransferase Activity in Hydrolases

Certain hydrolases preferentially catalyze acyl transfer over hydrolysis in an aqueous environment. However, the molecular and structural reasons for this phenomenon are still unclear. Herein, we provide evidence that acyltransferase activity in esterases highly correlates with the hydrophobicity of the substrate‐binding pocket. A hydrophobicity scoring system developed in this work allows accurate prediction of promiscuous acyltransferase activity solely from the amino acid sequence of the cap domain. This concept was experimentally verified by systematic investigation of several homologous esterases, leading to the discovery of five novel promiscuous acyltransferases. We also developed a simple yet versatile colorimetric assay for rapid characterization of novel acyltransferases. This study demonstrates that promiscuous acyltransferase activity is not as rare as previously thought and provides access to a vast number of novel acyltransferases with diverse substrate specificity and potential applications.     

Coverage-Controlled Superstructures of C3-Symmetric Molecules: Honeycomb versus Hexagonal Tiling

The competition between honeycomb and hexagonal tiling of molecular units can lead to large honeycomb superstructures on surfaces. Such superstructures exhibit pores that may be used as 2D templates for functional guest molecules. Honeycomb superstructures of molecules that comprise a C₃ symmetric platform on Au(111) and Ag(111) surfaces are presented. The superstructures cover nearly mesoscopic areas with unit cells containing up to 3000 molecules, more than an order of magnitude larger than previously reported. The unit cell size may be controlled by the coverage. A fairly general model was developed to describe the energetics of honeycomb superstructures built from C₃ symmetric units. Based on three parameters that characterize two competing bonding arrangements, the model is consistent with the present experimental data and also reproduces various published results. The model identifies the relevant driving force, mostly related to geometric aspects, of the pattern formation.     

The Cologne Carbon Cluster experiment: ro-vibrational spectroscopy on C8 and other small carbon clusters

We report on our ongoing efforts in obtaining the IR-spectra of the linear carbon cluster molecules Cn with n=8-13. So far C8, C9, C10, and C13 have been recorded at Cologne. With the exception of C8 all assignments have been secured. For C8 a tentative assignment could be derived with the bandcenter of the sigmau antisymmetric stretching mode located at nu0=2067.9779 cm(-1) and a preliminary rotational constant in the vibrational ground state of B"=0.02068 cm(-1). The measured signal to noise ratio of the ro-vibrational band is fairly weak and thus the lower J ro-vibrational transitions can not be assigned with certainty. As a consequence the band center remains uncertain by 4 J or 0.17 cm(-1). For a more reliable assignment the sensitivity of the system has to be increased by at least one order of magnitude. The envisaged sensitivity increase of our experiment will be discussed along with the intention to perform terahertz observations of the low energetic bending ro-vibrational spectra. These sub-mm wave measurements will be carried out simultaneously with the IR measurements.     

Internal structure of C60 fullerence molecules as revealed by low-temperature STM

We have performed Scanning Tunneling Microscopy (STM) in ultra-high vacuum at low temperatures (5 and 50K) of unordered and ordered C₆₀ layers adsorbed on a Au(110) surface. STM topographs of the frozen C₆₀ molecules reveal four symmetric patterns within single molecules, which may be associated with different orientations of the fullerenes on a highly corrugated gold substrate.     

What is the temperature in heavy ion collisions?

We consider the Tsallis distribution as a source of the apparent slope of one-particle spectra in heavy-ion collisions and investigate the equation of state of this special kind of quark matter in the framework of non-extensive thermodynamics. We relate the energy per particle to the power-law tail of spectra at a given temperature.     

A mortar mimetic finite difference method on non-matching grids

We consider mimetic finite difference approximations to second order elliptic problems on non-matching multiblock grids. Mortar finite elements are employed on the non-matching interfaces to impose weak flux continuity. Optimal convergence and, in certain cases, superconvergence is established for both the scalar variable and its flux. The theory is confirmed by computational results. Supported by the US Department of Energy, under contractW-7405-ENG-36. LA-UR-04-4740. Partially supported by NSF grants EIA 0121523 and DMS 0411413, by NPACI grant UCSD 10181410, and by DOE grant DE-FGO2-04ER25617. Partially supported by NSF grants DMS 0107389, DMS 0112239 and DMS 0411694 and by DOE grant DE-FG02-04ER25618.