Multivariate Hörmander-Type Multiplier Theorem for the Hankel transform

Let $\mathcal{H}(f)(x)=\int_{(0,\infty)^{d}} f(\lambda) E_{x}(\lambda) d\nu(\lambda )$ , be the multivariate Hankel transform, where $E_{x}(\lambda)=\prod_{k=1}^{d} (x_{k} \lambda_{k})^{-\alpha _{k}+1/2}J_{\alpha_{k}-1/2}(x_{k} \lambda_{k})$ , with dν(λ)=λ ^2α dλ, α=(α ₁,…,α _d ). We give sufficient conditions on a bounded function m(λ) which guarantee that the operator $\mathcal{H}(m\mathcal{H} f)$ is bounded on L ^p (dν) and of weak-type (1,1), or bounded on the Hardy space H ¹((0,∞) ^d ,dν) in the sense of Coifman-Weiss.     

Microstructure of a composite with a quasicrystalline phase fraction obtained by directional solidification of Al61Cu27Fe12 alloy

The microstructure of a composite containing a quasicrystal phase, i.e. so-called crystal–quasicrystal (CQ) composite, was studied. The CQ composite was obtained by the Bridgman method via solidification of Al₆₁Cu₂₇Fe₁₂ alloy (numbers indicate at%). The process was conducted at a pull out rate of v = 0.07 mm/min. The average temperature gradient in the heating zone was 43 K/cm. The composite matrix consisted of cubic β phase Al(Fe, Cu), with reinforcement of λ-phase rod-shaped fibres surrounded by a quasicrystal icosahedral ψ phase, which also existed in the fibre core. The fibres were rhomboidal in cross-section. The composite was studied using X-ray and electron diffraction, light-optical and scanning electron microscopy (SEM), X-ray topography and Laue diffraction.     

Reliability in reconstruction of the axisymmetric refractive index profiles from transverse interferograms. Accurate versus approximate methods: A comparative study

A mathematical formalism related to the approximate as well as the accurate interferometric data reduction schemes is briefly outlined and subsequently applied in the reconstruction of the assumed (Luneburg) axisymmetric refractive index distribution. The presented comparative analysis confirms the generally held opinion that the approximate methods (“straight-line” and Kokubun-Iga approaches) may provide quite reasonable results in the examination of refractively inhomogeneous disturbances characterized by weak gradient and moderate index variation. On the other hand, the accurate scheme for transverse interferometric data interpretation and reduction, based on the close-form correlation between the fringe shift and the deflection function, [1] remains the only method for reliable reconstruction of strongly refracting index profiles, independently of magnitude of refraction and ambiguity of the fringes. This fact is of considerable importance for transverse interferometry of fibers (preform rods) and/or GRIN-rod lenses characterized by high numerical apertures.     

Paraxial designing of planar waveguide variable-index focusing elements: Part 1—lenses of circular symmetry

A paraxial ray equation is used to evaluate refractive index profiles for a family of radially symmetric planar waveguide lenses. Two cases are considered: the (theoretically) full-aperture solution, and the solution with smooth index variation (non-full-aperature case). In deriving the latter case a new method has been applied that consists of the proper choice of a continuous deflection function. Both types of index profiles are described by exceptionally simple formulas.     

Scattering of small solutions to generalized benjamin-bona-mahony equation in several space dimensions

We show that the supremum norm of solutions with small initial data of the generalized Benjamin-Bona-Mahony equation u_t-△u_t=(b,▽u)+u^p(a,▽u)in x?Rⁿ,n≥2, with integer p≥3 , decays to zero like t^-2/3 if n=2 and like t^-1+6, for any δ0, if n≥3, when t tends to infinity. The proofs of these results are based on an analysis of the linear equation u_t-△=(b,▽u)) and the associated oscillatory integral which may have nonisolated stationary points of the phase function.     

New developments in the primal–dual column generation technique

The optimal solutions of the restricted master problems typically leads to an unstable behavior of the standard column generation technique and, consequently, originates an unnecessarily large number of iterations of the method. To overcome this drawback, variations of the standard approach use interior points of the dual feasible set instead of optimal solutions. In this paper, we focus on a variation known as the primal–dual column generation technique which uses a primal–dual interior point method to obtain well-centered non-optimal solutions of the restricted master problems. We show that the method converges to an optimal solution of the master problem even though non-optimal solutions are used in the course of the procedure. Also, computational experiments are presented using linear-relaxed reformulations of three classical integer programming problems: the cutting stock problem, the vehicle routing problem with time windows, and the capacitated lot sizing problem with setup times. The numerical results indicate that the appropriate use of a primal–dual interior point method within the column generation technique contributes to a reduction of the number of iterations as well as the running times, on average. Furthermore, the results show that the larger the instance, the better the relative performance of the primal–dual column generation technique.     

Study of Dependence for Some Stochastic Processes

Abstract

This article is concerned with studying the following problem: Consider a multivariate stochastic process whose law is characterized in terms of some infinitesimal characteristics, such as the infinitesimal generator in case of finite Markov chains. Under what conditions imposed on these infinitesimal characteristics of this multivariate process, the univariate components of the process agree in law with given univariate stochastic processes. Thus, in a sense, we study a stochastic processe' counterpart of the stochastic dependence problem, which in case of real valued random variables is solved in terms of Sklar's theorem.     

High peak power Nd:YAG laser pumped by 600-W diode laser stack

The Q-switched laser with triangle slab made of Nd:YAG crystal side pumped by 600-W quasi-cw diode laser stack has been designed. The multimode (M²≈2.6) output energy of about 42 mJ was demonstrated in free running mode for110-mJ pump energy. In Q-switch experiments, the KDDP Pockels cell was placed between the slab and rear mirror in plane-plane cavity with output coupler of 84% transmission. The energy of 8 mJ in 2.1-ns pulse duration was obtained for near TEM₀₀ output beam. For passive Q-switching by means of Cr:YAG crystal of 12.6% unsaturated transmission, the energy of 5.1 mJ in 2.5-ns pulse duration was obtained for output beam close to TEM₀₀ mode.     

Winding up of the wave-function phase by an insulator-to-superfluid transition in a ring of coupled Bose-Einstein condensates

We study phase transition from the Mott insulator to superfluid in a periodic optical lattice. Kibble-Zurek mechanism predicts buildup of winding number through random walk of BEC phases, with the step size scaling as a third root of transition rate. We confirm this and demonstrate that this scaling accounts for the net winding number after the transition.     

The temperature dependence of the magnetoelastic characteristics of cores for force sensors utilizing Fe<Subscript>70</Subscript>Ni<Subscript>8</Subscript>Si<Subscript>10</Subscript>B<Subscript>12</Subscript> amorphous alloy

This paper presents the results of investigation on the influence of temperature on magnetoelastic characteristics of the two ring-shaped cores, made of Fe₇₀Ni₈Si₁₀B₁₂ amorphous alloy. The cores were annealed for 1 h at 350 and 400°C, respectively. The compressive force F was applied perpendicular to the direction of the magnetizing field H in the sample. Special cylindrical backing enables application of the uniform compressive stress σ to the wound ring sample. A resistive furnace heated the experimental set-up. Results presented in the paper indicate a significant influence of the temperature on the magnetoelastic characteristics of Fe₇₀Ni₈Si₁₀B₁₂ amorphous alloy. Information about the magnetoelastic characteristics of this material may be useful in the magnetoelastic sensor development. Also this will create new possibilities in the development of physical model of magnetoelastic effect.