Existence of positive solutions of delay dynamic equations

In this article we study the existence of positive solutions for a dynamic equations on time scales. The main tool employed here is the Schauder’s fixed point theorem. The asymptotic properties of solutions are also treated. The results obtained here extend the work of Dorociakova and Olach (Tatra Mt Math Publ 43:63–70, 2009). Three examples are also given to illustrate this work.     

Stability and convergence of difference scheme for nonlinear evolutionary type equations

A finite difference scheme is derived for the initial-boundary problem for the nonlinear equation system $$\frac{\partial u}{\partial t}=A\frac{\partial^{2}u}{\partial x^{2}}+f(u),$$ where A is a complex diagonal matrix, f is a complex vector function. The stability and convergence in discrete L ^∞-norm of proposed Crank-Nicolson type finite difference schemes is proved. No restrictions on the ratio of time and space grid steps are assumed. Some numerical experiments have been conducted in order to validate the theoretical results.     

Methods for the numerical solution of the Benjamin‐Bona‐Mahony‐Burgers equation

L^∞‐error estimates for finite element for Galerkin solutions for the Benjamin‐Bona‐Mahony‐Burgers (BBMB) equation are considered. A priori bound and the semidiscrete Galerkin scheme are studied using appropriate projections. For fully discrete Galerkin schemes, we consider the backward Euler method and analyze the corresponding error estimates. For a second order accuracy in time, we propose a three‐level backward method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Temperature dependence of the spin-lattice relaxation time for quadrupole nuclei under conditions of NMR line saturation

The contributions of different mechanisms of nuclear spin-lattice relaxation are experimentally separated for ⁶⁹Ga and ⁷¹Ga nuclei in GaAs crystals (nominally pure and doped with copper and chromium), ²³Na nuclei in a nominally pure NaCl crystal, and ²⁷Al nuclei in nominally pure and lightly chromium-doped Al₂O₃ crystals in the temperature range 80–300 K. The contribution of impurities to spin-lattice relaxation is separated under the condition of additional stationary saturation of the nuclear magnetic resonance (NMR) line in magnetic and electric resonance fields. It is demonstrated that, upon suppression of the impurity mechanism of spin-lattice relaxation, the temperature dependence of the spin-lattice relaxation time T₁ for GaAs and NaCl crystals is described within the model of two-phonon Raman processes in the Debye approximation, whereas the temperature dependence of T₁ for corundum crystals deviates from the theoretical curve for relaxation due to the spin-phonon interaction.     

Towards Magnetic Carbo‐meric Molecular Materials

Numerous studies have underlined the putative diradical character of π‐conjugated molecules that can be described by closed‐shell Lewis structures, for instance, p‐dimethylene p–n phenylenes, or long polyacenes. In the latter compounds, the only way to save the aromaticity of the six‐membered rings is to give up the Lewis electron pairing in the singlet biradical ground state. The present work considers the possibility of doing the same by using the basic C₂ units of carbo‐meric architectures. A series of acyclic and cyclic carbo‐meric architectures is studied by using UB3LYP DFT broken‐symmetry calculations, including spin decontaminations and subsequent geometry optimization of the singlet diradical. The C₂ units are shown to stabilize the singlet biradical by spin delocalization, two of them playing approximately the same role as one radical‐insulating 1,4 phenylene moiety. The results are generalized to the investigation of open‐shell polyradical singlet states of rigid hydrocarbon structures, the symmetry and rigidity of which can assist cooperativity and self spin polarization effect. Several synthesis targets with challenging magnetic/spin properties are suggested in the carbo‐mer series.     

Separation of the impurity and lattice components of the spin-lattice relaxation of Al27 nuclei in corundum crystals under magnetic saturation conditions

The impurity and lattice contributions to the spin-lattice relaxation of Al²⁷ nuclei in nominally pure and lightly chromium-doped corundum (Al₂O₃) crystals are separated experimentally under the conditions of additional stationary resonance magnetic saturation of a pulsed NMR signal. The relaxation time T _i ^lat due to the lattice mechanism of the spin-phonon interaction is determined. Fiz. Tverd. Tela (St. Petersburg) 39, 1041–1043 (June 1997)     

Magnetogenesis in Water Induced by a Chemical Analyte

This Minireview aims to shed light on the emergent field of inducing a change in the magnetic properties of a solution‐phase sample by exposing it to a chemical analyte. A considerable body of knowledge exists on materials that alter their magnetic characteristics after a change in the surrounding physical conditions and a number of cases even exist of solution‐phase samples that do so under these same circumstances. However, examples of dissolved molecules or particles that react in this fashion under constant conditions and in response to an analyte are limited. Although some cases in organic solvents are discussed, the emphasis of this Minireview is on water. Our aim is to provide the reader with guidelines for designing new magnetogenic probes for the detection of the desired chemical analyte.