An overlooked property of plot methods

An interesting and often overlooked property of the MLR least-squares method is discussed. Here it is shown how the observed vs. calculated and calculated vs. observed plots, which are usually presented as equivalent, in fact they are not. Due to the inherent properties of the MLR procedure, it is shown that the slope of the calculated vs. observed plot is r ², as the slope of the experimental vs. calculated plot is just 1. This behaviour also has direct consequences to the corresponding residual plots.     

Supramolecular systems constructed by crown ether-based cryptands

Cryptands, seen as a kind of crown ether derivatives, have attracted much attention because of their three dimensional spatial structures and additional binding sites. Moreover, by changing functional groups on the third arm, various stimuli-responsive host–guest systems were obtained, which have played key roles in the construction of functional supramolecular systems. The recent advances in supramolecular systems constructed by crown ether-based cryptands, and further applications, especially in supramolecular polymers are summarized in this digest paper.     

Effective permittivity of a fiber-reinforced composite with transversely isotropic constituents

Simple closed-form expressions for effective permittivity of fiber-reinforced composites with transversely isotropic constituents are found using asymptotic homogenization. Circular cylindrical fibers are distributed in a square array. The analysis considers four orientations of constituents transverse symmetry axis relative to fibers direction. Local problems defined on a periodic square unit cell are solved by means of complex potential theory using Weierstrassian and Natanzon's functions, assuming that the contrast of permittivities is small. Derived closed-form formulas are compared with finite element calculations and, in the isotropic case, with standard mixture rules and classical bounds, obtaining excellent results even when contrast is large.     

On the Sobolev trace Theorem for variable exponent spaces in the critical range

In this paper, we study the Sobolev trace Theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. Then, we give local conditions on the exponents and on the domain (in the spirit of Adimurthy and Yadava) in order to satisfy such conditions and therefore to ensure the existence of extremals.     

High rank elliptic curves with prescribed torsion group over quadratic fields

There are 26 possibilities for the torsion groups of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with a given torsion group which set the current rank records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for \(\mathbb {Z}/15\mathbb {Z}\) , there exists an elliptic curve over some quadratic field with this torsion group and with rank \(\ge 2\) .     

Competing through altering the environment: A cross-diffusion population model coupled to transport–Darcy flow equations

We consider an evolution model describing the spatial population distribution of two salt tolerant plant species, such as mangroves, which are affected by inter- and intra-specific competition (Lotka–Volterra), population pressure (cross-diffusion) and environmental heterogeneity (environmental potential). The environmental potential and the Lotka–Volterra terms are assumed to depend on the salt concentration in the root region, which may change as a result of mangroves’ ability to uptake fresh water and leave the salt of the solution behind, in the saturated porous medium. Consequently, partial differential equations modelling the population dynamics on the surface are coupled with Darcy–transport equations modelling the salt and pressure-velocity distribution in the subsurface. We prove the existence of weak solutions of the coupled problem and provide a numerical discretization based on a stabilized mixed finite element method, which we use to numerically demonstrate the behaviour of the system.