On the Number of 5‐Cycles in a Tournament

We find a formula for the number of directed 5‐cycles in a tournament in terms of its edge scores and use the formula to find upper and lower bounds on the number of 5‐cycles in any n‐tournament. In particular, we show that the maximum number of 5‐cycles is asymptotically equal to , the expected number 5‐cycles in a random tournament (), with equality (up to order of magnitude) for almost all tournaments.     

Integral operators commuting with dilations and rotations in generalized Morrey-type spaces

We find conditions for the boundedness of integral operators $K$ commuting with dilations and rotations in a local generalized Morrey space. We also show that under the same conditions, these operators preserve the subspace of such Morrey space, known as vanishing Morrey space. We also give necessary conditions for the boundedness when the kernel is non-negative. In the case of classical Morrey spaces, the obtained sufficient and necessary conditions coincide with each other. In the one-dimensional case, we also obtain similar results for global Morrey spaces. In the case of radial kernels, we also obtain stronger estimates of $Kf$ via spherical means of $f$. We demonstrate the efficiency of the obtained conditions for a variety of examples such as weighted Hardy operators, weighted Hilbert operator, their multidimensional versions, and others.     

A penalty method for numerically handling dispersive equations with incompatible initial and boundary data

This article is the numerical counterpart of a theoretical work in progress Qin and Teman, Applicable Anal (2011), 1–19, related to the approximation of evolution hyperbolic equations with incompatible data. The Korteweg‐de Vries and Schrödinger equations with incompatible initial and boundary data are considered here. For hyperbolic equations, the lack of regularity (compatibility) is known to produce large numerical errors which propagate throughout the spatial domain, destroying convergence. In this article, we numerically test the effectiveness of the penalty‐based method proposed in Qin and Teman, Applicable Anal (2011), 1–19, which replaces the hyperbolic equations with incompatible data by a system with compatible data. We observe that convergence is increased. As explained in the text, in the case of the Schrödinger equation, the impact of incompatible (nonregular) data is most severe, and the authors are not aware of any other method that can handle such severe incompatible data. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Homogenization methods to approximate the effective response of random fibre-reinforced Composites

In this article a fibre-reinforced composite material is modelled via an approach employing a representative volume element with periodic boundary conditions. The effective elastic moduli of the material are thus derived. In particular, the method of asymptotic homogenization is used where a finite number of fibres are randomly distributed within the representative periodic cell. The study focuses on the efficacy of such an approach in representing a macroscopically random (hence transversely isotropic) material. Of particular importance is the sensitivity of the method to cell shape, and how this choice affects the resulting (configurationally averaged) elastic moduli. The averaging method is shown to yield results that lie within the Hashin–Shtrikman variational bounds for fibre-reinforced media and compares well with the multiple scattering and (classical) self-consistent approximations with a deviation from the latter in the larger volume fraction cases. Results also compare favourably with well-known experimental data from the literature.     

An Investigation of Photo‐ and Pressure‐Induced Effects in a Pair of Isostructural Two‐Dimensional Spin‐Crossover Framework Materials

Two new isostructural iron(II) spin‐crossover (SCO) framework (SCOF) materials of the type [Fe(dpms)₂(NCX)₂] (dpms=4,4′‐dipyridylmethyl sulfide; X=S ( SCOF‐6(S) ), X=Se ( SCOF‐6(Se) )) have been synthesized. The 2D framework materials consist of undulating and interpenetrated rhomboid (4,4) nets. SCOF‐6(S) displays an incomplete SCO transition with only approximately 30 % conversion of high‐spin (HS) to low‐spin iron(II) sites over the temperature range 300–4 K (T_1/2=75 K). In contrast, the NCSe^? analogue, SCOF‐6(Se) , displays a complete SCO transition (T_1/2=135 K). Photomagnetic characterizations reveal quantitative light‐ induced excited spin‐state trapping (LIESST) of metastable HS iron(II) sites at 10 K. The temperature at which the photoinduced stored information is erased is 58 and 50 K for SCOF‐6(S) and SCOF‐6(Se) , respectively. Variable‐pressure magnetic measurements were performed on SCOF‐6(S) , revealing that with increasing pressure both the T_1/2 value and the extent of spin conversion are increased; with pressures exceeding 5.2 kbar a complete thermal transition is achieved. This study confirms that kinetic trapping effects are responsible for hindering a complete thermally induced spin transition in SCOF‐6(S) at ambient pressure due to an interplay between close T_1/2 and T(LIESST) values.     

Biologically Relevant Molecular Transducer with Increased Computing Power and Iterative Abilities

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