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>ck7/3 suffices. 相似文献
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 DS and (a – F)S, respectively, of the total spin number S are constructed. The single-ion second-order axial constant Dc is well described by a linear dependence on S. The constant (a – F)S has a quadratic dependence. 相似文献
Treatment of 1-bromo-2,2-dichloro(trimethylsilyl)cyclopropane (2) with tetra--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 -butoxide in tetrahydrofuran. 相似文献
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. 相似文献
1,3,6,8-tetrabenzoylpyrene (1,3,6,8-Bz4PY) and 1,3,6-tribenzoylpyrene (1,3,6-Bz3PY) were synthesized and their crystal structures were determined. The Friedel–Crafts deacylations in PPA of 1,3,6,8-Bz4PY (at 120–200 °C) and of 1,3,6-Bz3PY (at 80–160 °C) have been studied. The mono-deacylation of 1,3,6-Bz3PY was regioselective and led to three dibenzoylpyrenes in the following order of relative amounts: 1,8-Bz2PY > 1,6-Bz2PY > 1,3-Bz2PY. 1,3,6,8-Bz4PY was resistant to deacylation at 120–160 °C. The deacylations of 1,3,6,8-Bz4PY 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-Bz4PY 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-Bz3PY and 1,3,6,8-Bz4PY was performed. The estimated energy barriers of formation of dibenzoylpyrenes by deacylation of 1,3,6-Bz3PY increase in the following order: 1,8-Bz2PY < 1,3-Bz2PY < 1,6-Bz2PY. A mechanism of the Friedel–Crafts deacylation of 1,3,6-Bz3PY in PPA via the respective O-protonated ketone and σ-complexes is presented. 相似文献
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 Dn (?, ?, 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. 相似文献
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. 相似文献
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. 相似文献