A new approach to Euler splines

Starting from the exponential Euler polynomials discussed by Euler in “Institutions Calculi Differentialis,” Vol. II, 1755, the author introduced in “Linear operators and approximation,” Vol. 20, 1972, the so-called exponential Euler splines. Here we describe a new approach to these splines. Let t be a constant such that t=|t|e^iα, −π<α<π,t≠0,t≠1.. Let S₁(x:t) be the cardinal linear spline such that S₁(v:t) = t^v for all v ε Z. Starting from S₁(x:t) it is shown that we obtain all higher degree exponential Euler splines recursively by the averaging operation . Here S_n(x:t) is a cardinal spline of degree n if n is odd, while is a cardinal spline if n is even. It is shown that they have the properties S_n(v:t) = t^v for v ε Z.     

Bemerkungen zu den eisensteinschen reihen und ihren anwendungen in der arithmetik

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg -     

Oxygen - 17 superhyperfine structure of transition metal ions in MgO

EPR measurements of the oxygen-17 superhyperfine structure of Fe³⁺ Cr³⁺, Ni³⁺ and Rh²⁺ impurities in isotopically labeled single crystals of MgO have been made. In connection with the interpretation of the spectra of the Jahn-Teller ions Ni³⁺ and Rh²⁺, expressions are derived for the superhyperfine Hamiltonian for the ²E state in octahedral symmetry.     

Wave propagation in alternating solid and fluid layers

The exact solution for harmonic plane wave propagation through periodically stratified media composed of alternating elastic solid and ideal fluid layers is derived for all frequencies and wavenumbers. The modal structure and the structure of the pass and stop bands is shown in the plane of slowness parallel to the stratification, s₁, and frequency, ω. In the low frequency limit, the slowness component perpendicular to the stratification, s₃, is found as a function of s₁, defining slowness surfaces in the s₁ and s₃ plane. At any angle of phase propagation through the medium (except perpendicular) there are two modes, a fast wave for which the motion in the solid and fluid layers is in phase and a slow wave for which the motion is 180° out of phase. An equivalent acoustic medium is found which has an anisotropic density tensor and a nonlocal constitutive relation between pressure and dilatation whose terms are fourth derivatives in time and in the space coordinate parallel to the stratification. This equivalent homogeneous anisotropic medium exhibits the indentical wave behavior as the layered solid fluid medium in the long wavelength limit as it has the same slowness surfaces.     

Two dimensional wave problems in rotating elastic media

The rotation of an elastic medium makes it act anisotropically and dispersively. The eigenvectors for plane wave propagation are in general complex and thus the waves are elliptically polarized. In general the waves are neither pure shear nor pure compressional waves, and their speeds depend on the ratio of rotational frequency of the medium and the angular frequency of the wave.The class of problems discussed here involves waves propagating perpendicularly to the axis of rotation and in particular we discuss plane strain modes. The reflection and refraction of plane waves is considered.The plane waves are used to construct a general solution in cylindrical coordinates. The solution is given in terms of Bessel functions. The cylindrical solution is applied to scattering by circular cylinders. The problem of free oscillations is mentioned briefly.