Hidden structure in the randomness of the prime number sequence?

We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes, that would eventually appear in applications. Finally, our theory allows us to link with three different but important topics: the Hardy–Littlewood conjecture, the statistical mechanics of spin systems, and the celebrated Sierpinski fractal.     

Large deviation principle for enhanced Gaussian processes

We study large deviation principles for Gaussian processes lifted to the free nilpotent group of step N. We apply this to a large class of Gaussian processes lifted to geometric rough paths. A large deviation principle for enhanced (fractional) Brownian motion, in Hölder- or modulus topology, appears as special case.     

Maximal Lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise

Asymptotic expansions for the exponential growth rate, known as the Lyapunov exponent, and rotation numbers for two coupled oscillators driven by real noise are constructed. Such systems arise naturally in the investigation of the stability of steady-state motions of nonlinear dynamical systems and in parametrically excited linear mechanical systems. Almost-sure stability or instability of dynamical systems depends on the sign of the maximal Lyapunov exponent. Stability conditions are obtained under various assumptions on the infinitesimal generator associated with real noise provided that the natural frequencies are noncommensurable. The results presented here for the case of the infinitesimal generator having a simple zero eigenvalue agree with recent results obtained by stochastic averaging, where approximate ItÔ equations in amplitudes and phases are obtained in the sense of weak convergence.Dedicated to Thomas K. Caughey on the occasion of his 65th birthday.     

Finding numerical solution of linear partial differential equations of first order with real coefficients

Applying Bittner's operational calculus we present a method to give approximate solutions of linear partial differential equations of first order

Irreversible deformation of macromolecular networks

Based upon a thermodynamical approach, the generalized Onsager type of relaxation of van der Waals networks is presented. By linearly and identically coupling the set hidden variables to the network, the memory function of the system can be related to the equilibrium strain-energy function. The relaxation behavior of real networks on stretching can quantitatively be described by means of a distribution of relaxation times known from small strain experiments. Some new and interesting conclusions are discussed as to how the macroscopically non-linear visco-elastic response might be interpreted.     

The effectivity of solvents as electron pair donors

The effectivity of solvents as electron pair donors, their donicity as expressed by their donor numbers DN, is reexamined. The linear dependence of the enthalpy and the Gibbs free energy for the reaction of donor solvents with antimony pentachloride is affirmed. Extension of the DN scale by other measured quantities, via their linear correlations for sets of solvents where both kinds of data are known is applied to Drago's E-C scale, Kamlet's scale, and Koppel's B scale. This extension, added to previous extensions employing Selbin's D_{I, II} and Popov's ²³Na, produces a DN scale for 170 solvents. The relation of DN measured for isolated solvent molecules in 1,2-dichloroethane to DN values measured for bulk solvents is examined by means of the scaled particle theory and solute-solvent interaction terms. A scale of normalized donor numbers DN ^N is presented. These values are practically the same as the scale, and for those solvents for which DN values are presented here and values are lacking, they can serve in their place for those preferring to use the scale for donor solvents.     

Fine Structure of Meniscus of a Wetting Liquid

Equilibrium of a capillary meniscus near a wetting film on a solid in a gravitational field is considered. Unlike previous studies, the present study proves that the fine meniscus structure in a gravitational field is a universal feature—it takes place in a wide variety of problems. In the general case, the capillary meniscus is at a certain distance from the wetting film and does not intersect it. The relation for the minimum distance from the arbitrary meniscus to the solid generalizes the Derjaguin formula for a flat slit. An equation that optimally approximates the meniscus with due account of the contribution of the meniscus/film transition region is derived. A refined solution to the problem of a meniscus on a vertical plate is derived within the perturbation theory. Both gravity and nonuniformity of the vertical static film above a capillary–gravitational meniscus do not affect the minimum distance (the influence is less than 0.0001). A general method for solving sophisticated problems of capillary equilibrium in gravitational field is proposed.     

Bestimmung der Oberfl?chenzusammensetzung von Calcit in Mg2+-haltigen Ca-45-markierten L?sungen

Zusammenfassung Über den Ca²⁺-Mg²⁺-Ionenaustausch wurde der Mg²⁺-Ionengehalt in Calcitoberflächen in Abhängigkeit von der Mg²⁺-Konzentration in Ca-45-markierten Lösungen bestimmt. Es wurde gefunden, daß der Anstieg des Molverhältnisses Mg/Ca in der Calcit-Oberfläche bei steigendem Mg²⁺-Gehalt in der Lösung jeweils bei Erreichen von Mg/Ca 1 und 3 deutlich verringert wird. Die ermittelte Austauschisotherme kann benutzt werden, um über eine Bestimmung des Mg/Ca-Verhältnisses in der Lösung auf dasjenige in der Oberfläche zu schließen.Vortrag anläßlich der Tagung Spurenanalyse, 2. bis 5. April 1973 in Erlangen.