Geometric algebra techniques for general relativity

Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the straightforwardness of coordinate methods. We focus our attention on orthonormal frames and the associated connection bivector, using them to find the Schwarzschild and Kerr solutions, along with a detailed exposition of the Petrov types for the Weyl tensor.     

Mathematical modeling of the Magnus effect in the geometrical optics approximation

The Magnus effect in multimode fibers with triangular and parabolic refractive index profiles (RIPs) is simulated in the geometrical optics approximation. The calculations confirm the linear relation between the angle of rotation Δ? and the fiber length z. The results of calculations for a spiral path with constant radius are compared with the analytical solution obtained. For a fiber with a parabolic RIP, the value of Δ? obtained in this work is one-half the result obtained in the wave approach.     

Computer processing of electronic absorption spectra of rare-earth elements in inorganic laser media

Algorithms and program modules are developed for calculation of the Judd-Ofelt parameters on modern computers. These programs can be used to perform numerical analysis of the energy absorption by rareearth ions in laser media.     

Some inhomogeneous magnetized viscous-fluid cosmological models with varyingΛ

Some cylindrically symmetric inhomogeneous viscous-fluid cosmological models with electromagnetic field are obtained. To get a solution a supplementary condition between metric potentials is used. The viscosity coefficient of bulk viscous fluid is assumed to be a power function of mass density. Without assuming anyad hoc law, we obtain a cosmological constant as a decreasing function of time. The behaviour of the electromagnetic field tensor together with some physical aspects of the model are also discussed.     

Hysteresis and Eddy-current Relaxation in SiFe Materials

The rate-dependent effects in metallic ferromagnetic materials with magnetization processes attributed to domain-walls motion were studied. The experimental method for eddy-current relaxation time determination was proposed. Some discrepancies between the data and theoretical predictions were observed and discussed. The origin of these discrepancies is accounted for by the decrease of viscous-type relaxation process as the rate of change of averaged magnetization is approaching zero.     

Branes,Quantum Nambu Brackets and the Hydrogen Atom

The Nambu-bracket quantization of the hydrogen atom is worked out as an illustration of the general method. The dynamics of topological open branes is controlled classically by Nambu brackets. Such branes then may be quantized through the consistent quantization of the underlying Nambu brackets: properly defined, the quantum Nambu-brackets comprise an associative structure, although the naive derivation property is mooted through operator entwinement. For superintegrable systems, such as the hydrogen atom, the results coincide with those furnished by Hamiltonian quantization - but the method is not limited to Hamiltonian systems.     

An interference method for studying a substance in the vicinity of its critical point

An interference method for measuring certain parameters of a substance in the vicinity of the critical state is proposed. The essence of the method is to decipher the interference pattern arising upon grazing incidence of a laser beam on a curved liquid-vapor interface. The deciphering allows one to determine the difference between the refractive indices of the two phases and the scaling factor of the wetting meniscus shape. This, in turn, makes it possible to calculate the critical exponents for the temperature dependences of the surface tension and the difference in the densities of the two phases. The ratio of these critical exponents for xenon, measured in the paper, is 3.81±0.03.     

On two functionals connected to the Laplacian in a class of doubly connected domains in space-forms

LetB ₁ be a ball of radiusr ₁ inS ⁿ (ℍⁿ), and letB ₀ be a smaller ball of radiusr ₀ such thatB ₀ ⊂B ₁ . ForS ⁿ we considerr ₁ π. Let u be a solution of the problem- δm = 1 in Ω :=B ₁ /B ₀ vanishing on the boundary. It is shown that the associated functionalJ (Ω) is minimal if and only if the balls are concentric. It is also shown that the first Dirichlet eigenvalue of the Laplacian on Ω is maximal if and only if the balls are concentric.     

Sequent calculi for propositional star-free likelihood logic

We consider classical, multisuccedent intuitionistic, and intuitionistic sequent calculi for propositional likelihood logic. We prove the admissibility of structural rules and cut rule, invertibility of rules, correctness of the calculi, and completeness of the classical calculus with respect to given semantics.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 3–21, January–March, 2005.