Hamiltonicity, independence number, and pancyclicity

A graph on n vertices is called pancyclic if it contains a cycle of length ? for all 3≤?≤n. In 1972, Erd?s proved that if G is a Hamiltonian graph on n>4k⁴ vertices with independence number k, then G is pancyclic. He then suggested that n=Ω(k²) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such that n>ck^7/3 suffices.     

Pairwise Interactions of Fe3+ Ions in InBO3 Crystals

Measurements of the electron paramagnetic resonance (EPR) of Fe³⁺ ion pairs in InBO₃ crystals are conducted. Some parts of the procedure of spectrum identification without measuring the temperature dependence of the resonant line intensities have been developed. Experimental curves of the second- and fourth-order axial Hamiltonian constants D _S and (a – F)_S, respectively, of the total spin number S are constructed. The single-ion second-order axial constant D _c is well described by a linear dependence on S. The constant (a – F)_S has a quadratic dependence.     

1-bromo-2-chlorocyclopropene--A new cycloproparene synthon. Synthesis of 1H-Cyclopropa[b]phenanthrene

Treatment of 1-bromo-2,2-dichloro(trimethylsilyl)cyclopropane (2) with tetra-$\text{n}$-butylammonium fluoride in tetrahydrofuran yields 1-bromo-2-chlorocyclopropene (1). 1H-Cyclopropa[b]phenanthrene (3) can be prepared by aromatization of the Diels-Alder adduct of 1 and 1,2-dimethylene-3,5,6,7,8,9-hexahydronaphthalene (6) using DDQ followed by potassium $\text{t}$-butoxide in tetrahydrofuran.     

The effect of aryl nitration on gas sorption and permeation in polysulfone

CO_2, CH_4, O_2, and N₂ permeability and solubility of unmodified and aryl-nitrated polysulfone were determined at 35°C and pressures up to 20 atm. The degree of nitration was varied from 0 to 2 nitro groups per repeat unit. The permeability and diffusion coefficients for all gases decreased with increasing degree of nitro substitution. The decrease in gas diffusivity is attributed to a combination of decreased fractional free volume and decreased torsional mobility with increasing degree of substitution. The solubilities of N_2, O_2, and CH₄ do not show a systematic dependence on degree of substitution. However, CO₂ solubility apparently goes through a minimum as the degree of substitution is increased. CO₂ solubility may be influenced by a competition between increases in polymer polarity (favoring higher solubility) and lower free volume (favoring lower solubility) that accompanies increases in the polar nitro substituent concentration. CO₂/CH₄ solubility selectivity increases monotonically as the degree of substitution increases. CO₂/CH₄ permselectivity and diffusivity selectivity increased with increasing degree of substitution. © 1995 John Wiley & Sons, Inc.     

On a problem of A. Pleijel

In 1955, Arne Pleijel proposed the following problem which remains unsolved to this day: Given a closed plane convex curve C and a point x() at a fixed distance above the plane, as the point x() varies, characterize the point for which the conical surface with vertex x() and base C attains its minimum, and determine the limits as 0 and of this minimum point. The purpose of this paper is to solve the cases where approach its extremities and in the course of the solution, we obtain an interesting characterization of the limit points, which we shall call the Pleijel points of C. A consequence is that the inner Pleijel point provides an upper bound for the isoperimetric defect of C. We also generalize the problem to higher dimensional spaces, and obtain the corresponding characterizations of the limiting points for convex surfaces.     

Regioselective Friedel–Crafts deacylations of polycyclic aromatic ketones in the pyrene series

1,3,6,8-tetrabenzoylpyrene (1,3,6,8-Bz ₄ PY) and 1,3,6-tribenzoylpyrene (1,3,6-Bz ₃ PY) were synthesized and their crystal structures were determined. The Friedel–Crafts deacylations in PPA of 1,3,6,8-Bz ₄ PY (at 120–200 °C) and of 1,3,6-Bz ₃ PY (at 80–160 °C) have been studied. The mono-deacylation of 1,3,6-Bz ₃ PY was regioselective and led to three dibenzoylpyrenes in the following order of relative amounts: 1,8-Bz ₂ PY > 1,6-Bz ₂ PY > 1,3-Bz ₂ PY. 1,3,6,8-Bz ₄ PY was resistant to deacylation at 120–160 °C. The deacylations of 1,3,6,8-Bz ₄ PY at 200 °C gave the polycyclic aromatic ketone (PAK) 8H-dibenzo[def,qr]chrysen-8-one (DBCO) via an intramolecular Scholl reaction. Two plausible pathways of the Friedel–Crafts deacylation of 1,3,6,8-Bz ₄ PY to give DBCO are proposed. A density functional theory (DFT) B3LYP/6-311(d,p) computational study of the conformational spaces of 1,3,6-Bz ₃ PY and 1,3,6,8-Bz ₄ PY was performed. The estimated energy barriers of formation of dibenzoylpyrenes by deacylation of 1,3,6-Bz ₃ PY increase in the following order: 1,8-Bz ₂ PY < 1,3-Bz ₂ PY < 1,6-Bz ₂ PY. A mechanism of the Friedel–Crafts deacylation of 1,3,6-Bz ₃ PY in PPA via the respective O-protonated ketone and σ-complexes is presented.     

Disjoint systems

A disjoint system of type (?, ?, k, n) is a collection ?? = {??₁,…, ??_m} of pairwise disjoint families of k-subsets of an n-element set satisfying the following condition. For every ordered pair ??_i and ??_j of distinct members of ?? and for every A ? ??_i there exists a B ? ??_j that does not intersect A. Let D_n (?, ?, k) denote the maximum possible cardinality of a disjoint system of type (?, ?, k, n). It is shown that for every fixed k ? 2,. This settles a problem of Ahlswede, Cai, and Zhang. Several related problems are considered as well.     

A rotationally invariant procedure for density functional calculations

A simple method which is rigorously invariant under molecular rotations is presented for evaluation of the density functional exchange—correlation energy by numerical quadrature. The corresponding expressions for the first and second derivatives of the energy with respect to nuclear displacement are presented. In particular, such a scheme is necessary to remove the difficulties previously encountered in calculating Kohn—Sham harmonic vibrational frequencies for low-lying modes.     

The relative contribution of adsorption to the overall sorption and transport of small molecules in amber

In 1987, Berner and Landis reported that upon vacuum grinding of 80 million year old amber, a gas mixture was released which suggested an oxygen-rich prehistoric environment. Fundamental to their argument was the assumption that amber, an organic glass formed during the fossilization of tree sap, is a perfect sealant. Their assumption was challenged by three technical comments which collectively concluded that gases diffuse readily through amber. In order to defend their key assumption that gases are perfectly trapped in amber, Berner and Landis dismissed the data obtained through gravimetric sorption experiments as only a measure of surface adsorption rather than bulk absorption in and concomitant diffusion through the amber matrix. The validity of interpreting these gravimetric experiments as a measure of bulk diffusion is demonstrated by exploring the physical basis for interpreting gravimetric sorption data. Most importantly, new experimental gravimetric sorption data are presented which demonstrate an explicit separation of adsorption from diffusion-controlled absorption and also reveal that adsorption accounts for a very small fraction of the total sorption in amber.