Numerical solution of the two-dimensional Poincaré equation

This paper deals with numerical approximation of the two-dimensional Poincaré equation that arises as a model for internal wave motion in enclosed containers. Inspired by the hyperbolicity of the equation we propose a discretisation particularly suited for this problem, which results in matrices whose size varies linearly with the number of grid points along the coordinate axes. Exact solutions are obtained, defined on a perturbed boundary. Furthermore, the problem is seen to be ill-posed and there is need for a regularisation scheme, which we base on a minimal-energy approach.     

Thermal,Raman and dielectric study of 0.5K0.5Bi0.5TiO3–0.5PbTiO3 ceramics

The 0.5K_0.5Bi_0.5TiO₃–0.5PbTiO₃ ceramics were prepared by following a standard solid-state method. The Raman, thermal and dielectric properties of these ceramics were investigated. The X-ray measurements showed that samples have single perovskite-type structure with tetragonal symmetry. Dielectric study revealed that the dielectric behaviour of the investigated ceramics is rather of normal ferroelectrics with large thermal hysteresis. The transition temperature observed by means of differential scanning calorimetry measurements is in good agreement with that obtained from dielectric study.     

Trace norm estimates for products of integral operators and diffusion semigroups

We give trace norm estimates for products of integral operators and for diffusion semigroups. These are applied to differences of heat semigroups. A natural example of an integral operator with finite trace which is not trace class is given.     

Magnetic Study of Doped Ba Ferrites Prepared by Citrate Method

The M-type barium ferrite is doped with either Co²⁺-Ti⁴⁺ or Co²⁺-Zr⁴⁺ pairs to reduce its intrinsically high uniaxial magnetic anisotropy in order to make fine particles for magnetic recording. The magnetic parameters were investigated by magnetic measurements and the Mössbauer spectroscopy. Compounds (BaF) obtained from BaCo _x (Ti,Zr) _x Fe_12-2x O₁₉ with 0x1.0 have been prepared by the modified citrate method with the initial ratio of Ba:Fe equal to 1:10.8. The substitution of Co - Ti or Co - Zr for Fe³⁺ ions affects mainly the positions 2a and 4f ₂. Ba ferrite with the substitution x 0.8 is promising for perpendicular recording media applications.     

A Note on the Dimension Theory of ℤ n -Graded Rings

In this note we want to generalize some of the results in [1 Brewer , J. , Montgomery , P. , Rutter E. , Heinzer , W. ( 1973 ). Krull dimension of polynomial rings in “Conference on Commutative Algebra, Lawrence 1972.” . Springer Lecture Notes in Mathematics 311 : 26 – 45 .[Crossref] , [Google Scholar]] from polynomial rings in several indeterminates to arbitrary ? ⁿ -graded commutative rings. We will prove an analogue of Jaffard's Special Chain Theorem and a similar result for the height of a prime ideal 𝔭 over its graded core 𝔭*.     

Computation of connection coefficients and measure modifications for orthogonal polynomials

We observe that polynomial measure modifications for families of univariate orthogonal polynomials imply sparse connection coefficient relations. We therefore propose connecting L ² expansion coefficients between a polynomial family and a modified family by a sparse transformation. Accuracy and conditioning of the connection and its inverse are explored. The connection and recurrence coefficients can simultaneously be obtained as the Cholesky decomposition of a matrix polynomial involving the Jacobi matrix; this property extends to continuous, non-polynomial measure modifications on finite intervals. We conclude with an example of a useful application to families of Jacobi polynomials with parameters (γ,δ) where the fast Fourier transform may be applied in order to obtain expansion coefficients whenever 2γ and 2δ are odd integers.     

Field of linear frames as a fundamental self-interacting system

We present some philosophical and physical arguments supporting the hypothesis that the most fundamental self-interacting field in an amorphous space-time is the field of linear frames, i.e. the quadruple of vector bosons. We construct a wide class of Lagrangian dynamical models invariant under the total group of diffeomorphisms and under the natural action of the proper linear group GL⁺ (4, R) on the tetrad field. There exist some links between these models and the Hamiltonian dynamical systems on GL⁺ (3, R) (the mechanics of affinely-rigid bodies [23] [27]). We present the general form of field equations, conservation laws and Bianchi identities. There exist some formal similarities between our Lagrangians and those used in non-linear electrodynamics, in particular in the Born–Infeld theory [21]. We also give a few rough remarks concerning models invariant under natural subgroups of GL⁺ (4, R), i.e. under SL(4, R) and SO(1, 3; R) (special linear group and Lorentz group). The latter class includes the conventional Einstein relativity and the more general metrical-parallelism models. It turns out that there are GL⁺ (4, R)-invariant Lagrangians which are structurally alike the conventional Einstein Lagrangian.We have not derived as yet either mathematical or physical consequences of the presented model. Nevertheless, it seems to follow from our discussion that, a priori, the GL⁺ (4, R)-invariant tetrad models could be competitive with the Einstein theory. The next thing to be done would be a careful mathematical analysis of these models and attempts to compare their consequences with those of the Einstein relativity and of other field theories.     

Bi(Pb)–Sr–Ca–Cu–O layers prepared by a new method of laser ablation in air

Bi(Pb)–Sr–Ca–Cu–O layers were prepared by laser ablation in air, that is without using a vacuum chamber. At this method a much higher density of pulse energy of laser beam is required (10–30 J cm^–2) in comparison with the standard method (1–3 J cm^–2). The mentioned condition has specific effects on the nature of the transport of material from the target. As follows from the electron microprobe analysis of the layers, the starting composition of the transported material is identical with that of the superconducting phase sintered in the target; one exception represents Pb, the content of which in the transported material is lower. The image of the process is complemented by UV/VIS spectroscopy scans of the plasma plume.