For a symmetric operator or relation A with infinite deficiency indices in a Hilbert space we develop an abstract framework for the description of symmetric and self-adjoint extensions A_Θ of A as restrictions of an operator or relation T which is a core of the adjoint A^*. This concept is applied to second order elliptic partial differential operators on smooth bounded domains, and a class of elliptic problems with eigenvalue dependent boundary conditions is investigated

Abstract. Self-adjoint Schrödinger operators with δ and δ ′ -potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman-Schwinger principle and a variant of Krein's formula are shown. Furthermore, Schatten-von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ ′ -potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity.

The notion of quasi boundary triples and their Weyl functions is reviewed and applied to self-adjointness and spectral problems for a class of elliptic, formally symmetric, second order partial differential expressions with variable coefficients on bounded domains.

We discuss the general design of the ANTARES code which is intended for simulations in stellar hydrodynamics with radiative transfer and realistic microphysics in 1D, 2D and 3D. We then compare the quality of various numerical methods. We have applied ANTARES in order to obtain high resolution simulations of solar granulation which we describe and analyze. In order to obtain high resolution, we apply grid refinement to a region predominantly occupied by an exploding granule. Strong, rapidly rotating vortex tubes of small diameter (∼ 100 km) generated by the downdrafts and ascending into the photosphere near the granule boundaries evolve, often entering the photosphere from below in an arclike fashion. They essentially contribute to the turbulent velocity field near the granule boundaries.

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Abstract. The notion of quasi boundary triples and their Weyl functions is an abstract concept to treat spectral and boundary value problems for elliptic partial differential equations. In the present paper the abstract notion is further developed, and general theorems on resolvent differences belonging to operator ideals are proved. The results are applied to second order elliptic differential operators on bounded and exterior domains, and to partial differential operators with δ and δ ′ -potentials supported on hypersurfaces.

The attachment pads of fly legs are covered with setae, each ending in small terminal plates coated with secretory fluid. A cluster of these terminal plates contacting a substrate surface generates strong attractive forces that hold the insect on smooth surfaces. Previous research assumed that cohesive forces and molecular adhesion were involved in the fly attachment mechanism. The main elements that contribute to the overall attachment force, however, remained unknown. Multiple local force-volume measurements were performed on individual terminal plates by using atomic force microscopy. It was shown that the geometry of a single terminal plate had a higher border and considerably lower centre. Local adhesion was approximately twice as strong in the centre of the plate as on its border. Adhesion of fly footprints on a glass surface, recorded within 20 min after preparation, was similar to adhesion in the centre of a single attachment pad. Adhesion strongly decreased with decreasing volume of footprint fluid, indicating that the layer of pad secretion covering the terminal plates is crucial for the generation of a strong attractive force. Our data provide the first direct evidence that, in addition to Van der Waals and Coulomb forces, attractive capillary forces, mediated by pad secretion, are a critical factor in the fly's attachment mechanism.