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 \(\ge 2\) .     

Equivalence of the Pecka-Ponec Correlation Probability and the Statistical F Significance for MLR Models

In an article of this journal Pecka and Ponec [J. Math. Chem. 27 (2000) 13] have proposed, by means of a probability calculation, a method to evaluate the statistical importance of correlations obtained from multilinear regression equations involving an arbitrary number of experimental points and parameters. Here, it is demonstrated how this probability exactly coincides with a more general concept: the confidence probability of an F distribution having the appropriate degrees of freedom.     

Electrical double layer and rheological properties of yttria-stabilized zirconia suspensions in solutions of high molecular weight polyacrylic acid polymers

This work deals with the effect of the adsorption of two high molecular weight polyacrylic acid polymers (Carbopol) on the interfacial properties, and the rheology of aqueous zirconia suspensions. Since the Carbopol-covered particles can be thought of as soft colloids, Ohshimas theory was used to gain information on the surface potential and the charge density of the polymer layer (Ohshima H (1995) Electrophoretic mobility of soft particles. Colloids Surf A Physicochem Eng Aspects 103:249–255). The effect of the pH of the solution on the double layer characteristics is related to the different conformations of the adsorbed molecules provoked by the dissociation of the acrylic groups present in polymer molecules. The electrokinetic properties of the suspensions are studied for different pH and Carbopol concentrations in solution in order to investigate the possible stabilization of the suspensions by electrostatic repulsion between the particles. The rheological behavior of the suspensions was investigated in steady-state and dynamic conditions, and the corresponding yield stress and storage modulus were obtained in absence and presence of polymer in solution. The competition between bridging flocculation provoked by polymer adsorption and electrosteric stabilization determines the rheological properties of the suspensions. In the pH range investigated, bridging flocculation predominates at the neutral pH because of the graft of the uncoiled polymer to more than one particle, while at the extreme pH values (pH 3, pH 9) steric or electrosteric stabilization seems to be the predominant mechanism that explains the rheological results. These facts were confirmed by estimating the zirconia particle (or aggregate) diameter in the liquid medium by means of light scattering measurements.     

Shape‐Persistent and Adaptive Multivalency: Rigid Transgeden (TGD) and Flexible PAMAM Dendrimers for Heparin Binding

This study investigates transgeden (TGD) dendrimers (polyamidoamine (PAMAM)‐type dendrimers modified with rigid polyphenylenevinylene (PPV) cores) and compares their heparin‐binding ability with commercially available PAMAM dendrimers. Although the peripheral ligands are near‐identical between the two dendrimer families, their heparin binding is very different. At low generation (G1), TGD outperforms PAMAM, but at higher generation (G2 and G3), the PAMAMs are better. Heparin binding also depends strongly on the dendrimer/heparin ratio. We explain these effects using multiscale modelling. TGD dendrimers exhibit “shape‐persistent multivalency”; the rigidity means that small clusters of surface amines are locally well optimised for target binding, but it prevents the overall nanoscale structure from rearranging to maximise its contacts with a single heparin chain. Conversely, PAMAM dendrimers exhibit “adaptive multivalency”; the flexibility means individual surface ligands are not so well optimised locally to bind heparin chains, but the nanostructure can adapt more easily and maximise its binding contacts. As such, this study exemplifies important new paradigms in multivalent biomolecular recognition.     

Asymmetric Catalytic Michael Reaction by Means of Natural Proteins

The addition of thiophenol to α, β, unsaturated ketones in the presence of different proteins gives optically active adducts with enantiomeric excess up to 29%.     

Pseudotetrahedral Rhodium(I) Complexes

A combination of four‐electron donors, such as alkynes, with strongly donating and strong‐field scorpionate ligands is appropriate to create pseudotetrahedral rhodium(I) environments, as found in [Rh(PhBP₃)(HC?CPh)], which promotes H?C bond activation and C?C coupling reactions under very smooth conditions.     

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.