2-Cell Embeddings with Prescribed Face Lengths and Genus

Let n be a positive integer, let d ₁, . . . , d _n be a sequence of positive integers, and let ${{q = \frac{1}{2}\sum^{n}_{i=1} d_{i}\cdot}}$ . It is shown that there exists a connected graph G on n vertices, whose degree sequence is d ₁, . . . , d _n and such that G admits a 2-cell embedding in every closed surface whose Euler characteristic is at least n ? q?+?1, if and only if q is an integer and q ?? n ? 1. Moreover, the graph G can be required to be loopless if and only if d _i ?? q for i = 1, . . . , n. This, in particular, answers a question of Skopenkov.     

Simplices and Spectra of Graphs

In this note we show that the (n−2)-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent quantities of the volumes of its edge-lengths. The proof involves computation of the eigenvalues of Kneser graphs. We also show examples of families of simplices (of dimension 4 or greater) which show that the set of (n−2)-dimensional volumes of (n−2)-dimensional faces of a simplex do not determine its volume.     

An Eberhard-Like Theorem for Pentagons and Heptagons

Eberhard proved that for every sequence (p _k ), 3≤k≤r, k≠6, of nonnegative integers satisfying Euler’s formula ∑ _k≥3(6−k)p _k =12, there are infinitely many values p ₆ such that there exists a simple convex polyhedron having precisely p _k faces of size k for every k≥3, where p _k =0 if k>r. In this paper we prove a similar statement when nonnegative integers p _k are given for 3≤k≤r, except for k=5 and k=7 (but including p ₆). We prove that there are infinitely many values p ₅,p ₇ such that there exists a simple convex polyhedron having precisely p _k faces of size k for every k≥3. We derive an extension to arbitrary closed surfaces, yielding maps of arbitrarily high face-width. Our proof suggests a general method for obtaining results of this kind.     

Maps with the Unique Extension Property and $$\hbox {C}^{*}$$-Extreme Points

We investigate boundary representations in the context where Hilbert spaces are replaced by \(\hbox {C}^{*}\)-modules over abelian von Neumann algebras and apply this to study \(\hbox {C}^{*}\)-extreme points. We present an (unexpected) example of a weak* compact \(\mathcal {B}\)-convex subset of \({\mathbb {B}}(\mathcal {H})\) without \(\mathcal {B}\)-extreme points, where \(\mathcal {B}\) is an abelian von Neumann algebra on a Hilbert space \(\mathcal {H}\). On the other hand, if \(\mathcal {A}\) is a von Neumann algebra with a separable predual and whose finite part is injective, we show that each weak* compact \(\mathcal {A}\)-convex subset of \(\ell ^{\infty }(\mathcal {A})\) is generated by its \(\mathcal {A}\)-extreme points.     

Photoacoustic study of CO2-laser coincidences with absorption of some organic solvent vapours

Detection of specific molecules in the atmosphere is motivated by the need for monitoring the emission of industrial pollutants. CO₂-laser photoacoustic (PA) spectroscopy was used here to obtain coincidence spectra of the vapours of three organic solvents: trichloroethylene (TCE), benzene, and dioxan. Spectra of coincidences were obtained as a function of total pressure, adding dry air to the vapour of the solvents up to 1000 mbar.A narrow cavity photoacoustic cell with window acoustic buffers was built for improved sensitivity, and is reported here.The vapour molecules studied do have some significant absorption in the CO₂-laser region, or close to it, but that fact does not necessarily lead to usable coincidences with the narrow laser lines. The results reveal several prominent coincidences that can be used for practical purposes, especially in the case of trichloroethylene.     

Cyanoamino compounds in synthesis syntheses of some heterocycles

Transformations of some heterocyclic cyanoamino compounds leading to various heterocyclic systems are described.s-Triazolo (1,5-α)azines are obtained either in a direct synthetic approach or via the substituted aminotetrazoles, substituted 3-amino-5-oxo-1,2,4-oxadiazolines, and fromN-ethoxycarbonylN′-heteroaryl thioureas orN-heteroarylN′-hydroxyguanidine. The cyanoamino group reacts also witho-difunctional benzenes to give the corresponding substituted derivatives of benzimidazole, benzoxazole or benzothiazole.     

Correlation of ring nitrogen substituents with carbon-13 nuclear magnetic resonance data in azoloazines

Carbon–13 nuclear magnetic resonance data have been acquired on 22 azoloazines. Chemical shifts have been correlated by a step–wise linear multiple regression with nitrogen substituents in both the 5- and 6-membered rings using pyrrolo[1,2-α]pyridine as the reference for chemical shift correlation. The data demonstrate that a highly correlated set of chemical shift parameters exist. Nitrogen substitution in the five–membered ring produces larger cross–ring effects than are oberved in the five–membered ring when substitution occurs in the six–membered ring. Within the six–membered ring a constant para- substituent parameter is noted. The meta- and para- parameters are more complex and fall into two groups for each parameter. Within the five–membered ring, a highly regular chemical shift pattern is observed which reflects an attenuated perturbation from nitrogen substitution in the six–membered ring.     

Folding–Unfolding Equilibrium of a Methylidene‐Substituted β‐Peptide

β‐Peptides possess the ability to fold into secondary structure elements, and this property, together with resistance to biodegradation, makes these compounds interesting for pharmaceutical applications. Recently, a novel class of β‐peptides containing methylidene moieties was described. The GROMOS 53A6 force field was used to simulate the folding equilibrium of a β³‐hexapeptide with methylidene (CH₂?) groups at all six CA‐atoms. Due to the rotational barriers induced by these methylidene groups, the helical secondary‐structure elements, normally found in β³‐peptides, are disfavored in this molecule. Simulations, started from fully extended and 3₁₄‐helical conformations, showed that the molecule adopts a complete 2₈‐helix for ca. 5% of the time and partial 2₈‐helical conformations for ca. 20% of the time. Yet, as suggested by experiments, the folding equilibrium is dominated by unfolded conformations.     

The Trp cage: folding kinetics and unfolded state topology via molecular dynamics simulations

Using over 75 mus of molecular dynamics simulation, we have generated several thousand folding simulations of the 20-residue Trp cage at experimental temperature and solvent viscosity. A total of 116 independent folding simulations reach RMSDcalpha values below 3 A RMSDcalpha, some as close as 1.4 A RMSDcalpha. We estimate a folding time of 5.5+/-3.5 mus, a rate that is in reasonable agreement with experimental kinetics. Finally, we characterize both the folded and unfolded ensemble under native conditions and note that the average topology of the unfolded ensemble is very similar to the topology of the native state.     

Coordination compounds from the planar tridentate Schiff-base ligand 2-methoxy-6-((quinolin-8-ylimino)methyl)phenol (mqmpH) with several transition metal ions: Use of [Fe(mqmp)(CH3OH)Cl2] in the catalytic oxidation of alkanes and alkenes

Four coordination compounds were synthesized in high yields from different transition metal ions (Fe^III, Co^II, and Cu^II) and an in situ generated Schiff-base ligand, i.e. 2-methoxy-6-((quinolin-8-ylimino)methyl)phenol (mqmpH). The compounds were characterized by single-crystal X-ray diffraction, ESI-MS, IR spectroscopy, and ligand-field spectroscopy. The iron(III) complex is an efficient catalyst for the oxidation of alkanes and alkenes, under relatively mild conditions and with dihydrogen peroxide as terminal oxidant.