Zur Ausmittelung des Quecksilbers

Ohne Zusammenfassung     

L^p -uniqueness for Dirichlet operators with singular potentials

We study the problem of strong uniqueness in L^p for the Dirichlet operator perturbed by a singular complex-valued potential. First we construct the generator -H_p of a C₀-semigroup in L^p, with H_p extending the restriction of the perturbed Dirichlet operator to the set of smooth functions. The corresponding sesquilinear form in L² is not assumed to be sectorial. Then we reveal sufficient conditions on the logarithmic derivative # of the measure rdx \rho dx and the potential q which ensure that -H_p is the only extension of D+b·?-q \upharpoonright_C0^￥ \Delta +\beta \cdot \nabla -q \upharpoonright_{C_0^{\infty}} which generates a C₀-semigroup on L^p. The method of a priori estimates of solutions to corresponding differential equations is employed.     

A Critical Phenomenon for Sublinear Elliptic Equations in Cone-Like Domains

The authors of this paper study positive supersolutions to theelliptic equation -u = c|x|^–su^p in Cone-like domains ofR^N (N 2), where p, s R and c > 0. They prove that in thesublinear case p < 1 there exists a critical exponent p_*> 1 such that the equation has a positive supersolution ifand only if – < p < p_*. The value of p_* is determinedexplicitly by s and the geometry of the cone. 2000 MathematicsSubject Classification 35J60 (primary), 35B05, 35R45 (secondary).     

Restrictions on collapsing with a lower sectional curvature bound

We obtain new topological information about the local structure of collapsing under a lower sectional curvature bound. As an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every Alexandrov space can be obtained as a limit of a sequence of Riemannian manifolds with sectional curvature bounded from below.     

Kondo effect of molecular complexes at surfaces: ligand control of the local spin coupling

The spin state of single magnetic atoms and molecules at surfaces is of fundamental interest and may play an important role in future atomic-scale technologies. We demonstrate the ability to tune the coupling between the spin of individual cobalt adatoms with their surroundings by controlled attachment of molecular ligands. The strength of the coupling is determined via the Kondo resonance by low-temperature scanning tunneling spectroscopy. Spatial Kondo resonance mapping is introduced as a novel imaging tool to localize spin centers in magnetic molecules with atomic precision.     

Amorphous-polycrystal transition induced by laser pulse in self-ion implanted silicon

Reflection high energy electron diffraction has been used to investigate the amorphous to polycrystalline structure transition in silicon induced by laser pulse. The power density of the ruby laser pulse, in the free generation mode, has been maintained below the threshold to induce surface damage. Depth analysis has been carried out in 〈100〉 silicon crystal using the channeling effect technique.     

Convenient and efficient reduction of 1,1'-binaphthyls to H8-1,1'-binaphthyl derivatives with Pd and Ru catalysts on solid support

Hydrogenation of chiral 2,2'-functionalized 1,1'-binaphthyls with Pd and Ru solid-supported metal catalysts was found to be a clean and convenient pathway to 5,5',6,6',7,7',8,8'-octahydro-1,1'-dinaphthyl derivatives. In most cases no racemization was observed in the course of the reaction.     

The derivation equation for C k -functions: stability and localization

For a given integer k ∈ ? we determine the possible forms of operators T: C ^k (?) → C(?) satisfying a generalized Leibniz rule operator equation T(f · g)(x) = Tf(x) · g(x)+f(x) · Tg(x)+S(f, g)(x), f,g ∈ C ^k (?), x ∈ ? for two different types of perturbations S(f, g). In the first case, S is given by a function B in localized form $$S(f,g)(x) = B(x,({f^{(j)}}(x))_{j = 0}^{k - 1},({g^{(j)}}(x))_{j = 0}^{k - 1})$$ involving only derivatives of lower order. We show that, if in addition T annihilates the polynomials of degree ≤ k ? 1, T is a multiple of the k-th derivative. For k = 2 and functions on ? ⁿ , we give a characterization of the Laplacian by a similar equation, orthogonal invariance and annihilation of affine functions. In the second setting, we assume S to have the form S(f, g)(x) = Af(x) · Ag(x) where A: C ^k (?) → C(?) is a general operator. Thus here, S has a product form, but the factor Af(x) is not assumed to depend only on the jet of f at x. We describe the possible forms of T and A satisfying the generalized Leibniz rule; T and A turn out to be closely related. Here, T and A need not to be localized, i.e., Tf(x) and Af(x) may depend on values f(y) for y ≠ x.