In this paper we present eight-component values "octons", generating associative noncommutative algebra. It is shown that the electromagnetic field in a vacuum can be described by a generalized octonic equation, which leads both to the wave equations for potentials and fields and to the system of Maxwell's equations. The octonic algebra allows one to perform compact combined calculations simultaneously with scalars, vectors, pseudoscalars and pseudovectors. Examples of such calculations are demonstrated by deriving the relations for energy, momentum and Lorentz invariants of the electromagnetic field. The generalized octonic equation for electromagnetic field in a matter is formulated.

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We report on results of computer micromodelling of anti-vortex states in asymmetrical cross-like ferromagnetic nanostructures and their practical realization. The arrays of cobalt crosses with 1 μm branches, 100 nm widths of the branches and 40 nm thicknesses were fabricated using e-beam lithography and ion etching. Each branch of the cross was tapered at one end and bulbous at the other. The stable formation of anti-vortex magnetic states in these nanostructures during magnetization reversal was demonstrated experimentally using magnetic force microscopy.

We report on the results of a magnetic force microscopy investigation of remagnetization processes in arrays of CoPt nanodisks with diameters of 35 and 200 nm and a thickness of 9.8 nm fabricated by e-beam lithography and ion etching. The controllable magnetization reversal of individual CoPt nanodisks by the magnetic force microscope ͑MFM͒ tip-induced magnetic field was demonstrated. We observed experimentally two essentially different processes of tip-induced remagnetization. Magnetization reversal of 200 nm disks was observed when the probe moved across the particle while in case of 35 nm nanodisks one-touch remagnetization was realized. Micromagnetic modeling based on the Landau-Lifshitz-Gilbert ͑LLG͒ equation demonstrated that the tip-induced magnetization reversal occurs through the essentially inhomogeneous states. Computer simulations confirmed that in case of 200 nm disks the mechanism of embryo nucleation with reversed magnetization and further dynamic propagation following the probe moving across the particle was realized. On the other hand one-touch remagnetization of 35 nm disks occurs through the inhomogeneous vortexlike state. Micromagnetic LLG simulations showed that magnetization reversal in an inhomogeneous MFM probe field has a lower energy barrier in comparison with the mechanism of coherent rotation, which takes place in a homogeneous external magnetic field.

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This Letter presents a rigorous mathematical proof of the applicability of phase approximation in the Huygens-Kirchhoff method for the problems of laser-beam propagation in a turbulent atmosphere. The calculated results are in good agreement with experimental data.