Solutions of the Allen-Cahn equation which are invariant under screw-motion

Let ${\phi(x)}$ be a rational function of degree >?1 defined over a number field K and let ${\Phi_{n}(x,t) = \phi^{(n)}(x)-t \in K(x,t)}$ where ${\phi^{(n)}(x)}$ is the nth iterate of ${\phi(x)}$ . We give a formula for the discriminant of the numerator of Φ _n (x, t) and show that, if ${\phi(x)}$ is postcritically finite, for each specialization t ₀ of t to K, there exists a finite set ${S_{t_0}}$ of primes of K such that for all n, the primes dividing the discriminant are contained in ${S_{t_0}}$ .     

Modification of HPTLC-plates by in situ chemical bonding with non-polar and polar organosilanes

A method has been developed for the preparation of modified silica plates for high performance thin-layer chromatography (HPTLC). Some typical organosilanes were thus allowed to react in situ with the silica of Merck HPTLC-plates. This method was found to be highly reproducible, simple and cheap. Non-polar plates were prepared and compared with commercial plates from Merck, Whatman and Macherey-Nagel. Modification with cyanodecyltrichlorosilane resulted in plates that showed good coverage, efficiency and low residual silica activity. Silica modified with a multifunctional silane has different properties in different organic solvents. It will appear to be non-polar in a polar solvent and vice versa. New advantageous separation systems are thus made feasible by the presence of cyano groups on the plate. The utility of modified thin-layer plates is demonstrated by the separation of some homologues of p-hydroxybenzoic acid esters and of some polycyclic aromatic hydrocarbons.     

Radial Solutions of¶Superlinear Equations on ℝN¶Part II: The Forced Case

We study the effect of a forcing term in the context of the search of multiple nodal solutions u∈h ¹(? ^N ) to a class of elliptic equations of type¶?Δu(x)=f(|x|,u(x))+h(|x|), x∈? ^N ,¶where f(|x|≡0 and f is superlinear but subcritical at infinity.     

Bootstrap for estimating the MSE of the Spatial EBLUP

This work assumes that the small area quantities of interest follow a Fay–Herriot model with spatially correlated random area effects. Under this model, parametric and nonparametric bootstrap procedures are proposed for estimating the mean squared error of the empirical best linear unbiased predictor (EBLUP). A simulation study based on the Italian Agriculture Census 2000 compares bootstrap and analytical estimates of the MSE and studies their robustness to non-normality. Results indicate lower bias for the non-parametric bootstrap under specific departures from normality.      

A Maximum Principle for an Explicit Finite Difference Scheme Approximating the Hodgkin-Huxley Model

We analyze an explicit finite difference scheme for the general form of the Hodgkin-Huxley model, which is a nonlinear partial differential equation coupled to a set of ODEs. The system of equations describes propagation of an electrical signal in excitable cells. We prove that the numerical solution is bounded in the L^∞-norm and L² converges to a unique solution. The L^∞-bound, which is the key point of our analysis, is proved by showing that the discrete solutions are invariant in a physically relevant bounded region. For the convergence proof we use the compactness method. AMS subject classification (2000) 65F20     

The Brezis-Nirenberg problem near criticality in dimension 3

We consider the problem of finding positive solutions of Δu+λu+u^q=0 in a bounded, smooth domain Ω in , under zero Dirichlet boundary conditions. Here q is a number close to the critical exponent 5 and 0<λ<λ₁. We analyze the role of Green's function of Δ+λ in the presence of solutions exhibiting single and multiple bubbling behavior at one point of the domain when either q or λ are regarded as parameters. As a special case of our results, we find that if , where λ∗ is the Brezis-Nirenberg number, i.e., the smallest value of λ for which least energy solutions for q=5 exist, then this problem is solvable if q>5 and q−5 is sufficiently small.     

Multispike solutions for the Brezis–Nirenberg problem in dimension three

We consider the problem $\mathrm{\Delta }u+\lambda u+u^5=0$, $u>0$, in a smooth bounded domain Ω in ${\mathbb{R}}^3$, under zero Dirichlet boundary conditions. We obtain solutions to this problem exhibiting multiple bubbling behavior at k different points of the domain as λ tends to a special positive value ${\lambda }_0$, which we characterize in terms of the Green function of $?\mathrm{\Delta }?\lambda$.     

Existence of the Equilibrium in Choice

In this paper, we prove the existence of the equilibrium in choice for games in choice form. Thus, we add to the research recently appeared in the scientific literature. In fact, our results link the most recent research to the older approaches of the games in normal-form and the qualitative games.