Frequency-dependent properties of the hydrogen atom: A generalized eigenvalue equation approach

The time-independent first-order perturbation equations which arise on considering the response of a system to a harmonic time-dependent perturbation are solved by using an expansion in terms of a complete discrete set of solutions of a generalized eigenvalue equation. Explicit formulae are obtained for several frequency-dependent properties of the hydrogen atom.     

Untersuchung von Mehl

Ohne Zusammenfassung     

Über Dicarbonylderivate von Phenolen

Ohne ZusammenfassungAls I. Mitteilung zählen wir die Arbeit: Dialdehyde des Phloroglucins und seiner Homologen in Ber. dtsch. chem. Ges.75, 29 (1942).     

Ueber die Zusammensetzung des Cacao?les

Ohne Zusammenfassung     

Ergodic theory and the duality principle on homogeneous spaces

We prove mean and pointwise ergodic theorems for the action of a lattice subgroup in a connected algebraic Lie group on infinite volume homogeneous algebraic varieties. Under suitable necessary conditions, our results are quantitative, namely we establish rates of convergence in the mean and pointwise ergodic theorems, which can be estimated explicitly. Our results give a precise and in most cases optimal quantitative form to the duality principle governing dynamics on homogeneous spaces. We illustrate their scope in a variety of equidistribution problems.     

Dynamics of digging in wet soil

Numerous animals live in, and locomote through, subsea soils. To move in a medium dominated by frictional interactions, many of these animals have adopted unique burrowing strategies. This paper presents a burrowing model inspired by the Atlantic razor clam (Ensis directus), which uses deformations of its body to cyclically loosen and re-pack the surrounding soil in order to locally manipulate burrowing drag. The model reveals how an anisotropic body - composed of a cylinder and sphere varying sinusoidally in size and relative displacement - achieves unidirectional motion through a medium with variable frictional properties. This net displacement is attained even though the body kinematics are reciprocal and inertia of both the model organism and the surrounding medium are negligible. Our results indicate that body aspect ratio has a strong effect on burrowing velocity and efficiency, with a well-defined maximum for given kinematics and soil material properties.     

Prime and almost prime integral points on principal homogeneous spaces

We develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space. In particular, we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable where the orbit consists of the integers. When the orbit is the set of integral matrices of a fixed determinant, we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups, and sharp and uniform counting of points on such orbits when ordered by various norms.