On the existence of 3-way k-homogeneous Latin trades

A $\mu$-way Latin trade of volume $s$ is a collection of $\mu$ partial Latin squares $T_1,T_2,\dots ,T_{\mu }$, containing exactly the same $s$ filled cells, such that, if cell $(i,j)$ is filled, it contains a different entry in each of the $\mu$ partial Latin squares, and such that row $i$ in each of the $\mu$ partial Latin squares contains, set-wise, the same symbols, and column $j$ likewise. It is called a $\mu$-way$k$-homogeneous Latin trade if, in each row and each column, $T_r$, for $1\le r\le \mu$, contains exactly $k$ elements, and each element appears in $T_r$ exactly $k$ times. It is also denoted as a $(\mu ,k,m)$ Latin trade, where $m$ is the size of the partial Latin squares.We introduce some general constructions for $\mu$-way $k$-homogeneous Latin trades, and specifically show that, for all $k\le m$, $6\le k\le 13$, and $k=15$, and for all $k\le m$, $k=4,5$ (except for four specific values), a $3$-way $k$-homogeneous Latin trade of volume $km$ exists. We also show that there is no $(3,4,6)$ Latin trade and there is no $(3,4,7)$ Latin trade. Finally, we present general results on the existence of $3$-way $k$-homogeneous Latin trades for some modulo classes of $m$.     

On Lundh’s percolation diffusion

A collection of spherical obstacles in the unit ball in Euclidean space is said to be avoidable for Brownian motion if there is a positive probability that Brownian motion diffusing from some point in the ball will avoid all the obstacles and reach the boundary of the ball. The centres of the spherical obstacles are generated according to a Poisson point process while the radius of an obstacle is a deterministic function. If avoidable configurations are generated with positive probability, Lundh calls this percolation diffusion. An integral condition for percolation diffusion is derived in terms of the intensity of the point process and the function that determines the radii of the obstacles.     

On Defining Sets of Full Designs with Block Size Three

A defining set of a t-(v, k, λ) design is a subcollection of its blocks which is contained in no other t-design with the given parameters, on the same point set. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set {|M| | M is a minimal defining set of D}. We show that if a t-(v, k, λ) design D is contained in a design F, then for every minimal defining set d _D of D there exists a minimal defining set d _F of F such that \({d_D = d_F\cap D}\). The unique simple design with parameters \({{\left(v,k, {v-2\choose k-2}\right)}}\) is said to be the full design on v elements; it comprises all possible k-tuples on a v set. Every simple t-(v, k, λ) design is contained in a full design, so studying minimal defining sets of full designs gives valuable information about the minimal defining sets of all t-(v, k, λ) designs. This paper studies the minimal defining sets of full designs when t = 2 and k = 3. Several families of non-isomorphic minimal defining sets of these designs are found. For given v, a lower bound on the size of the smallest and an upper bound on the size of the largest minimal defining set are given. The existence of a continuous section of the spectrum comprising approximately v values is shown, where just two values were known previously.     

Mechanotropic Elastomers

Liquid crystal elastomers (LCEs) are anisotropic polymeric materials. When subjected to an applied stress, liquid crystalline (LC) mesogens within the elastomeric polymer network (re)orient to the loading direction. The (re)orientation during deformation results in nonlinear stress‐strain dependence (referred to as soft elasticity). Here, we uniquely explore mechanotropic phase transitions in elastomers with appreciable mesogenic content and compare these responses to LCEs in the polydomain orientation. The isotropic (amorphous) elastomers undergo significant directional orientation upon loading, evident in strong birefringence and x‐ray diffraction. Functionally, the mechanotropic displacement of the elastomers to load is also nonlinear. However, unlike the analogous polydomain LCE compositions examined here, the isotropic elastomers rapidly recover after deformation. The mechanotropic orientation of the mesogens in these materials increase the toughness of these thiol‐ene photopolymers by nearly 1300 % relative to a chemically similar elastomer prepared from wholly isotropic precursors.     

Assessment of oscillator strengths with multiconfigurational short-range density functional theory for electronic excitations in organic molecules

ABSTRACT

We have in a series of recent papers investigated electronic excited states with a hybrid between a complete active space self-consistent field (CASSCF) wave function and density functional theory (DFT). This method has been dubbed the CAS short-range DFT method (CAS–srDFT). The previous papers have primarily focused on the excitation energies, and not on the oscillator strengths, although they comprise an important part of the absorption spectrum. In this study, we have carried out a quantitative analysis of oscillator strengths obtained with CAS–srDFT. As target molecules, we have considered the large collection of organic molecules whose excited states were investigated with a range of electronic structure methods by Thiel et al. As a by-product of our calculations of oscillator strengths, we also obtain electronic excitation energies, which enable us to compare the performance of CAS–srPBE for excitation energies, using a larger set of chromophores compared to previous studies.     

Determination of the composition of Ultra‐thin Ni‐Si films on Si: constrained modeling of electron probe microanalysis and x‐ray reflectivity data

The homogeneous bulk assumption used in traditional electron probe microanalysis (EPMA) can be applied for thin‐layered systems with individual layers as thick as 50 nm provided the penetration depth of the lowest accelerating voltage exceeds the total film thickness. Analysis of an NIST Ni‐Cr thin film standard on Si using the homogeneous model yielded certified compositions and application of the same model to ultra‐thin Ni‐Si layers on GaAs yielded their expected compositions. In cases where the same element is present in multiple layers or in the substrate as well as the film, the homogeneous assumption in EPMA alone is not sufficient to determine composition. By combining x‐ray reflectivity (XRR) thickness and critical angle data and using an iterative approach, quantitative compositional data in EPMA can be achieved. This technique was utilized to determine the composition of Ni‐Si ultra‐thin films grown on silicon. The Ni‐Si composition determined using this multi‐instrumental technique matched that of Ni‐Si films simultaneously deposited on GaAs. Copyright © 2008 John Wiley & Sons, Ltd.