No 4 (2016)

The problem of existence of Global Half-Geodesic Surface Parameterization is considered. The problem is well known and it is yet unsolved in general case. It is known that for the twice-differentiable surfaces it has local solution. At the same time example of paraboloid of revolution proves that it is not possible in the general case to use local nets in order to construct the global halfgeodesic ones. In order to solve the problem the authors follow the way leading to the construction of isothermal parameterization for the surfaces with positive first quadratic form. To this end they deduce partial differential equation for the mappings giving necessary parameterization. In the contrast with the case of isothermal parameterization when the equation is Beltrami equation corresponding to the homogeneous elliptic system this equation is essentially non-linear one. Besides the new system admits degeneration at the points where the Jacobian of the solution is equal to zero or infinity. The speed of degeneration strongly affecting properties of the solutions is also unknown. In order to surpass these difficulties the authors change the challenge. Instead of the geodesics covering the whole surface they propose to find the geodesics covering the surface up to the set of Hausdorff null measure. Using the theory of -quasiconformal mappings they construct nonregular generalized solutions of non-linear Beltrami equation that nevertheless detect the necessary family of the geodesics. The constructed theory permits to study non-classical equilibrium forms of liquid drops.

Internet of Things (IoT) is thought to become the third wave of the Internet and to bring important changes into both technological and business aspects of telecommunications. However, for this to happen, an infrastructure should be developed in order to provide network access and management functions to millions, if not billions of IoT-enabled devices. LTE networks could play the key role in the IoT communications landscape, provided that their capabilities are enhanced to efficiently support IoT devices and provide massive machine-to-machine connections without hampering human-to-human communications. Our paper addresses resource allocation in an LTE cell with both human-to-human and machine-to-machine connections. The cell is modeled as a multiservice queuing system with streaming and elastic jobs flows. Resources for machine-to-machine connections are allocated in batches of fixed size; requests for them arrive according to a Markovian arrival process. We obtain the stationary probability distribution of the system and formulas for request blocking probabilities.

The problem of receiving points with high curvature (singular points) of contours for identification of the shape of objects on images is solved. Analysis of existing methods of numerical differentiation in the given aspect is held. The new method of differentiation of the flat discretely defined curves, which are dots (pixels) of circuits, based on variations of Arch Height method is considered. Features of such method of differentiation are shown using various formulas of calculation of a derivative. Dependency aspects of the accuracy of the derivative on the chord length are analyzed. It is shown, that with an increase in its length differentiation accuracy degrades, and the result tends to the module of curvature of a curve at the given point. Comparison of the developed method with other known methods is made. The analysis of area of applicability and variability of parameters of differentiation is made. The accuracy aspects of calculation of derivatives for various parameters of differentiation are investigated. Examples of differentiation of various curves, both set analytically, and the functions-contours received from real images are considered. It is shown, that the offered method allows to get rid of the ambiguity in position of points of a contour with high curvature and consequently to raise quality of recognition of the shape of objects. Possible scopes of the given method in various areas of science and technics are stated.

In solving field problems, in particular problems of electrodynamics, we commonly use the Lagrangian and Hamiltonian formalisms. Hamiltonian formalism of field theory has the advantage over the Lagrangian, which inherently contains a gauge condition. While the gauge condition is introduced ad hoc from some external reasons in the Lagrangian formalism. However, the use of the Hamiltonian formalism in the field theory is difficult due to the non-regularity of the field Lagrangian. We must use such variant of the Lagrangian and the Hamiltonian formalism, which would allow us to work with the field models, in particular, to solve the problem of electrodynamics. We suggest using the modern differential geometry and the algebraic topology, in particular the theory of fiber bundles, as a mathematical apparatus. This apparatus leads to greater clarity in the understanding of mathematical structures, associated with physical and technical models. Using the fiber bundles theory allows us to deepen and expand both the Lagrangian and the Hamiltonian formalism. We can detect a wide range of these formalisms. We can select the most appropriate formalism. Actually just using the fiber bundles formalism we can adequately solve the problems of the field theory, in particular the problems of electrodynamics.