On the localization of binding for Schrödinger operators and its extension to elliptic operators

In this paper we study the asymptotic behavior of the ground state energyE(R) of the Schrödinger operatorP _R=?Δ+V ₁(x)+V ₂(x-R),x,R ∈?ⁿ, where the potentialsV _i are small perturbations of the Laplacian in ?ⁿ,n≥3. The methods presented here apply also in the investigation of the ground state energyE(g) of the operatorPg=P+V ₁(x)+V ₂(gx), x ∈X,g ∈G, whereP _g is an elliptic operator which is defined on a noncompact manifoldX, G is a discrete group acting onX by diffeomorphismsG×X∈(g, x)→gx∈X, andP is aG-invariant elliptic operator which is subcritical inX.     

Ozonolysis by microwave discharge of oxygen of naphthalene adsorbed on florisil

Microwave discharge of O₂ produces O(³P) atoms which form O₃ on florisil surface. Reaction with adsorbed naphthalene results in $\text{o}$-formylcinnamaldehydes as the main product.     

Rates of desorption from solid surfaces: Coverage dependence

The recently developed Classical Stochastic Diffusion Theory is applied to obtain the coverage dependence of desorption rates for Xe on W(111). Using the attractive Xe-Xe potential from gas phase experiments, we find a strong coverage dependence in the desorption rates and calculate Temperature Programmed Desorption Spectra (for a potential with reduced attractiveness) that are in excellent qualitative agreement with experimental results. We also investigated the effect of purely repulsive adsorbate-adsorbate interactions where we find, for some coverage ranges, that two different adsorption configurations can be stable (at different temperatures) leading to a marked change in the corresponding desorption rates and to distinct non-Arrhenius behavior.     

Numerical analysis of large elastic-plastic deformation of beams due to dynamic loading

Numerical calculations were performed for two examples of the response of elastic-plastic beams subjected to dynamic loads. These were a simply supported, axially restrained beam under suddenly applied uniform pressure, and an axially restrained, clamped beam with a central mass that is impacted by a projectile. Large elastic-plastic deflections were considered, and the method of finite differences was used. Two different constitutive equations were assumed: the elástic-perfectly plastic relation, and a special elastic-viscoplastic, strain hardening model. Analysis of the results included examining the interaction between the bending moment and the axial force, the variation of the axial force, bending moment and deflection with time, and the propagation velocities of the various phenomena during motion. Experiments were carried out in which a rifle projectile hit a central mass which had been fastened to a clamped beam. Comparison between the theoretical and experimental dynamic deflections shows good agreement for relatively short response times.     

A ground state alternative for singular Schrödinger operators

Let a be a quadratic form associated with a Schrödinger operator L=-∇·(A∇)+V on a domain Ω⊂R^d. If a is nonnegative on , then either there is W>0 such that for all , or there is a sequence and a function ?>0 satisfying L?=0 such that a[?_k]→0, ?_k→? locally uniformly in Ω?{x₀}. This dichotomy is equivalent to the dichotomy between L being subcritical resp. critical in Ω. In the latter case, one has an inequality of Poincaré type: there exists W>0 such that for every satisfying there exists a constant C>0 such that for all .     

A Finitary Version of Gromov’s Polynomial Growth Theorem

We show that for some absolute (explicit) constant C, the following holds for every finitely generated group G, and all d > 0: If there is some R ₀ > exp(exp(Cd ^C )) for which the number of elements in a ball of radius R ₀ in a Cayley graph of G is bounded by R₀^d{R_0^d} , then G has a finite-index subgroup which is nilpotent (of step < C ^d ). An effective bound on the finite index is provided if “nilpotent” is replaced by “polycyclic”, thus yielding a non-trivial result for finite groups as well.     

On the best constant for Hardy's inequality in

Let be a domain in and . We consider the (generalized) Hardy inequality , where . The inequality is valid for a large family of domains, including all bounded domains with Lipschitz boundary. We here explore the connection between the value of the Hardy constant and the existence of a minimizer for this Rayleigh quotient. It is shown that for all smooth -dimensional domains, , where is the one-dimensional Hardy constant. Moreover it is shown that for all those domains not possessing a minimizer for the above Rayleigh quotient. Finally, for , it is proved that if and only if the Rayleigh quotient possesses a minimizer. Examples show that strict inequality may occur even for bounded smooth domains, but for convex domains.

     

Axiplyns A-E, new sesquiterpene isothiocyanates from the marine sponge Axinyssa aplysinoides

Five new isothiocyanate sesquiterpenes, designated axiplyns A-E (1-5) have been isolated, together with two known isothiocyanate sesquiterpenes (6, 7), from the sponge Axinyssa aplysinoides collected at Misali Island, Tanzania. Axiplyns 4 and 5 embody a new indane sesquiterpene skeleton, and compounds 1, 2, and 5 contain unprecedented ring systems, namely a 6,8-dioxabicyclo[3.2.1]octane and a 2-oxabicyclo[2.2.1]heptane. Axiplyns A, B, and C are potent brine shrimp toxins with LD₅₀ values between 1.5 and 1.8 μg/mL.