A Hodge decomposition interpretation for the coefficients of the chromatic polynomial

Let be a simple graph with nodes. The coloring complex of , as defined by Steingrimsson, has -faces consisting of all ordered set partitions, in which at least one contains an edge of . Jonsson proved that the homology of the coloring complex is concentrated in the top degree. In addition, Jonsson showed that the dimension of the top homology is one less than the number of acyclic orientations of .

In this paper, we show that the Eulerian idempotents give a decomposition of the top homology of into components . We go on to prove that the dimensions of the Hodge pieces of the homology are equal to the absolute values of the coefficients of the chromatic polynomial of . Specifically, if we write , then .

     

The Space of Bounded Spherical Functions on the Free 2-Step Nilpotent Lie Group

Let N be a connected and simply connected 2-step nilpotent Lie group and let K be a compact subgroup of Aut(N). We say that (K, N) is a Gelfand pair when the set of integrable K-invariant functions on N forms an abelian algebra under convolution. In this paper we construct a one-to-one correspondence between the set Δ(K, N) of bounded spherical functions for such a Gelfand pair and a set of K-orbits in the dual of the Lie algebra for N. The construction involves an application of the Orbit Method to spherical representations of K ⋉ N. We conjecture that the correspondence is a homeomorphism. Our main result shows that this is the case for the Gelfand pair given by the action of the orthogonal group on the free 2-step nilpotent Lie group. In addition, we show how to embed the space Δ(K, N) for this example in a Euclidean space by taking eigenvalues for an explicit set of invariant differential operators. These results provide geometric models for the space of bounded spherical functions on the free 2-step group.     

1,3-Diol synthesis via controlled, radical-mediated C-H functionalization

The invention of a method for the synthesis of 1,3-diols from the corresponding alcohols is described, via controlled, radical-mediated C-H functionalization. The sequence described herein entails near quantitative conversion to the corresponding trifluoroethyl carbamate, followed by a variant of the Hofmann-L?ffler-Freytag reaction, cyclization, and hydrolysis to provide the 1,3-diols. In addition to the 10 examples presented herein, the syntheses of four natural products are facilitated by this directed oxyfunctionalization. This methodology is demonstrated to be orthogonal to other known C-H oxidations. Finally, this sequence is efficient, practical, inexpensive, and scalable.     

Nickel‐Catalyzed Cross‐Coupling of Redox‐Active Esters with Boronic Acids

A transformation analogous in simplicity and functional group tolerance to the venerable Suzuki cross‐coupling between alkyl‐carboxylic acids and boronic acids is described. This Ni‐catalyzed reaction relies upon the activation of alkyl carboxylic acids as their redox‐active ester derivatives, specifically N‐hydroxy‐tetrachlorophthalimide (TCNHPI), and proceeds in a practical and scalable fashion. The inexpensive nature of the reaction components (NiCl₂?6 H₂O—$9.5 mol^?1, Et₃N) coupled to the virtually unlimited commercial catalog of available starting materials bodes well for its rapid adoption.     

Genetic microheterogeneity of human transthyretin detected by IEF

Mutations of the human transthyretin (TTR) gene have attracted medical interest as a cause of amyloidosis. Recently, we have described in detail an electrophoretic procedure with PAGE followed by IEF in urea gradients for the study of the microheterogeneity of TTR monomers (Altland, K., Winter, P., Sauerborn, M. K., Electrophoresis 1999, 20, 1349-1364). In this paper, we present a study on 49 different mutations of TTR including 33 that result in electrically neutral amino acid substitutions. The aims of the investigation were to test the sensitivity of the procedure to detect TTR variants in patients with TTR amyloidosis and their relatives and to identify some common characteristics that could explain the amyloidogenicity of these variants. We found that all tested amyloidogenic mutations could be detected by our method with the exception of those for which the corresponding variant was absent in plasma samples. Most of the electrically neutral amyloidogenic TTR variants had in common a reduced conformational stability of monomers by the activity of protons and urea. For three variants, e.g. TTR-F64L, TTR-I107V and TTR-V122I, the monomers had a conformational stability close to that of normal monomers but we found experimental and structural arguments for a weakening of the monomer-monomer contact. All types of amyloidogenic mutations affected the stability of TTR tetramers.     

Protein-nanoparticle labelling probed by surface enhanced resonance Raman spectroscopy

A benzotriazole dye has been attached to a heme protein via a Michael addition and the unique potential of surface enhanced resonance Raman scattering (SERRS) to provide informative in situ recognition of more than one label on one protein demonstrated.     

Investigation into the factors affecting accuracy of mass measurements on a time-of-flight mass spectrometer using Design of Experiment

The results of an investigation of the parameters which have the most significant effect on the accuracy of mass measurements on a quadrupole orthogonal acceleration time-of-flight mass spectrometer (q-oaToF) are reported. The influence of eight factors is investigated: ion abundances of reference and analyte compounds, mass difference between analyte and reference compounds, quality of calibration, number of reference acquisitions averaged and TDC (time-to-digital converter) settings (resolution, Np multiplier (number of pushes correction factor), minimum number of points, i.e. minimum acquisition width which defines a peak). To extract the maximum information from as few experiments as possible, a Design of Experiment approach was used. The data will be used as a basis for developing guidance on accurate mass measurement on q-oaToF instruments.     

Time correlations and the second entropy

The authors study the transport of mass and heat in simulations of a Lennard-Jones fluid and demonstrate the calculation of transport coefficients, and of both the first and second entropies. These entropies are calculated from time correlation functions, as are the transport coefficients. They discuss the role of the second entropy in providing a physical explanation for the link between dynamic fluctuations and response. They illustrate the physical significance of the various contributions to the second entropy and how they simplify in the case of relaxation by steady-state flow. Certain approximations proposed for the calculation of the first entropy, common in the literature, are shown to break down under certain circumstances, and they give an improved method of calculation. They pay particular attention to the coupling between variables of opposite time parity in the transport matrix, and show that in general this cannot be neglected.     

Asymmetric synthesis of mono- and dinuclear bis(dipyrrinato) complexes

The diastereoselective syntheses of Zn(II) bis(dipyrrinato) helicates is reported, involving ligands templated by the incorporation of homochiral binol within the linker joining the two dipyrrinato units. The most diastereoselective formation of dinuclear bis(dipyrrinato) helicates to date is reported. The formation of either mononuclear or dinuclear helicates can be tuned by varying the length of the linker between the dipyrrinato units and by varying the complexation procedure. The neutral dipyrrinato helicates were readily analyzed by HPLC to ascertain diastereoselectivity, and circular dichroism studies revealed the helical nature of the complexes. The molar ellipticities of the helicates produced by diastereoselective complexation are very large in the visible region and typically correspond to binol moieties in the UV region. Extensive X-ray crystallographic investigations further confirmed the helicity of the mononuclear Zn(II) helicates and identified significant interlayer displacement and bending within crystals.