Network Geometry and Complexity

Higher order networks are able to characterize data as different as functional brain networks, protein interaction networks and social networks beyond the framework of pairwise interactions. Most notably higher order networks include simplicial complexes formed not only by nodes and links but also by triangles, tetrahedra, etc. More in general, higher-order networks can be cell-complexes formed by gluing convex polytopes along their faces. Interestingly, higher order networks have a natural geometric interpretation and therefore constitute a natural way to explore the discrete network geometry of complex networks. Here we investigate the rich interplay between emergent network geometry of higher order networks and their complexity in the framework of a non-equilibrium model called Network Geometry with Flavor. This model, originally proposed for capturing the evolution of simplicial complexes, is here extended to cell-complexes formed by subsequently gluing different copies of an arbitrary regular polytope. We reveal the interplay between complexity and geometry of the higher order networks generated by the model by studying the emergent community structure and the degree distribution as a function of the regular polytope forming its building blocks. Additionally, we discuss the underlying hyperbolic nature of the emergent geometry and we relate the spectral dimension of the higher-order network to the dimension and nature of its building blocks.     

Julius Petersen''s theory of regular graphs

In 1891 the Danish mathematician Julius Petersen (1839–1910) published a paper on the factorization of regular graphs. This was the first paper in the history of mathematics to contain fundamental results explicitly in graph theory. In this report Petersen's results are analysed and their development in subsequent decades are followed.     

Interval-regular graphs

An interval-regular graph is a connected graph in which, for any two vertices u and v, the number of neighbours of u on all shortest (u, v)-paths equals d(u, v). It is proved that in an interval-regular graph the shortest (u, v)-paths induce a hypercube of dimension d(u, v), for any two vertices u and v. The products of complete graphs are characterized as interval-regular graphs satisfying some extra conditions. The extended odd graphs are introduced as critical example with respect to the results proved.     

Solution of the excluded volume problem for biaxial particles

We report the solution of the excluded volume problem for a pair of biaxial hard molecules; namely, sphero-platelets. As an application of this result we study the isotropic to nematic liquid crystal transition for a fluid composed of these particles in the Onsager limit (length δ breadth or width). We show that the range of stability of the isotropic phase decreases with increasing particle biaxiality.     

Penalization in non-classical convex programming via variational convergence

A penalty method for convex functions which cannot necessarily be extended outside their effective domains by an everywhere finite convex function is proposed and combined with the proximal method. Proofs of convergence rely on variational convergence theory.     

Liquid chromatographic determination of diastereomeric glutathione conjugates and further derivatives of alpha-bromoisovalerylurea in rat bile and urine by electrochemically generated bromine

To study the glutathione conjugation of alpha-bromoisovalerylurea in the rat in vivo, a reversed-phase liquid chromatographic assay of the thioether metabolites in bile and urine was developed. Since alpha-bromoisovalerylurea has a chiral centre, two diastereomeric glutathione conjugates (in bile) and two diastereomeric mercapturates (in urine) can be expected. The separation characteristics of these metabolites and the corresponding cysteine conjugates were investigated. Whereas all thioether metabolites could be separated in one run, optimal separation of the diastereomers required different mobile phases for the glutathione conjugates (in bile) and the mercapturates (in urine). The glutathione conjugates were analysed with the ion-pairing agent sodium decanesulphonate in the mobile phase, but the mercapturates were analysed without an ion-pair-forming agent. For detection, on-line generation of a constant bromine level (100%) was used; bromine-reactive compounds result in a decrease of the amperometric response from the 100% baseline. This technique could be used in continuous automated operation and required little clean-up of the sample. Thus, the diastereomeric glutathione conjugates and mercapturates were quantified in rat bile and urine samples, respectively, by direct injection of the (centrifuged and diluted) samples on the column. The limit of determination of the respective metabolites was 9 and 2.6 ng in bile and urine, respectively. Incubation mixtures of alpha-bromoisovalerylurea with a rat liver cytosolic fraction or with isolated rat hepatocytes were chromatographed after deproteinization with a double volume of methanol. The limit of determination of the diastereomeric glutathione conjugates in the deproteinized incubation samples was 2.0 ng.     

Simple high-performance liquid chromatographic assay for melphalan in perfusate, rat liver and tumour tissue

A simple, sensitive and selective reversed-phase liquid chromatographic assay has been developed and validated for the anti-cancer agent melphalan in perfusate, liver and tumour tissue originating from isolated rat liver perfusion studies. Melphalan was extracted from the matrix using ice-cold methanol. The drug and the internal standard, propylparaben, were detected using ultraviolet absorbance at 262 nm. The assay has been validated in the 0.05-25 microg/mL range for perfusate; the lower limit of quantification (LLQ) is 0.05 microg/mL in perfusate and 0.25 ng/mg in liver and tumour tissues. Accuracies ranged from 89 to 110% and the inter-assay precisions were all below 15% (20% at the LLQ). Melphalan in a biological matrix has to be processed between 0 and 4 degrees C and is stable under all relevant processing and storage conditions tested. The assay has been exhaustively used in isolated liver perfusion studies with the drug demonstrating its applicability.     

Photocontrolled release and uptake of a porphyrin guest by dithienylethene-tethered beta-cyclodextrin host dimers

Two photoswitchable dithienylethene-tethered beta-cyclodextrin dimers were synthesized to function as host molecules with an externally controllable binding affinity. The cyclodextrin cavities of these dimers are linked through their secondary sides by a photochromic dithienylethene unit that is connected to the secondary rim either directly (4) or through propyl spacers (9). Irradiation with light switches these dimers between a relatively flexible (open) and a rigid (closed) form. The binding properties of the dimers depend on the configuration of the dithienylethene spacer, as is shown by microcalorimetry performed with tetrakis-sulfonatophenyl porphyrin (TSPP) as a guest molecule. The differences in binding properties are most pronounced for the more rigid dimer 4, which binds TSPP 35 times more strongly in the open form (4 a) than in the closed form (4 b). The values found for the enthalpy of binding (deltaH degrees ) indicate that this difference in binding is due to the loss of cooperativity between the two beta-cyclodextrin cavities in the closed form. Molecular modeling shows that 4 b is not able to bind TSPP effectively in both cyclodextrin cavities. The open and closed forms of the more flexible dimer 9 show no substantial difference in their binding of TSPP. Thermodynamic values indicative of strong binding of TSPP by two beta-cyclodextrin cavities were measured for both forms of the dimer, and molecular modeling confirms that both are flexible enough to tightly bind TSPP. The binding differences between the forms of dimer 4 allow the photocontrolled release and uptake of TSPP, which renders control of the ratio of complexed to free TSPP in solution possible.     

O-H bond dissociation enthalpies in oximes: order restored

The O-H bond dissociation enthalpies (BDEs) of 13 oximes, RR'C=NOH, having R and/or R' = H, alkyl, and aryl are reported. Experimental anchor points used to validate the results of theoretical calculations include (1) the O-H BDEs of (t-Bu)2C=NOH, t-Bu(i-Pr)C=NOH, and t-Bu(1-Ad)C=NOH determined earlier from the heat released in the reaction of (t-Bu)2C=NO* with (PhNH)2 in benzene and EPR spectroscopy (Mahoney, L. R.; Mendenhall, G. D.; Ingold, K. U. J. Am. Chem. Soc. 1973, 95, 8610), all of which were decreased by 1.7 kcal/mol to reflect a revision to the heat of formation of (E)-azobenzene (which has significant ramifications for other BDEs) and to correct for the heat of hydrogen bonding of (t-Bu)2C=NOH (alphaH2 = 0.43 measured in this work) to benzene, and (2) the measured rates of thermal decomposition of six RR'C=NOCH2Ph at 423 or 443 K, which were used to derive O-H BDEs for the corresponding RR'C=NOH. Claims (Bordwell, F. G.; Ji, G. Z. J. Org. Chem. 1992, 57, 3019; Bordwell, F. G.; Zhang, S. J. Am. Chem. Soc. 1995, 117, 4858; and Bordwell, F. G.; Liu, W.-Z. J. Am. Chem. Soc. 1996, 118, 10819) that the O-H BDEs in mono- and diaryloximes are significantly lower than those for alkyloximes due to delocalization of the unpaired electron into the aromatic ring have always been inconsistent with the known structures of iminoxyl radicals as are the purported perpendicular structures, i.e., phi(Calpha-C=N-O*) = 90 degrees, for sterically hindered dialkyl iminoxyl radicals. The present results confirm the 1973 conclusion that simple steric effects, not electron delocalization or dramatic geometric changes, are responsible for the rather small differences in oxime O-H BDEs.     

Alternant and non-alternant systems; valence bond analysis

The properties of the complete CI-matrix in a VB-formalism based on orthogonal (ized) orbitals allow for a direct demonstration of the difference between alternant and non-alternant systems, including the radical ions, carbenions and carbanions derived from them.