In this paper we discuss some issues related to Poincar´e’s inequality for a
special class of weighted Sobolev spaces. A common feature of these spaces is that they can be naturally associated with differential operators with variable diffusion coefficients that are not uniformly elliptic. We give a classification of these spaces in the 1-D case bases on a measure of degeneracy of the corresponding weight coefficient and study their key properties.
Initial boundary value problems in space-time rectangle for the following linear inhomogeneous degenerate wave equation of the second order smooth coefficient function a(x) vanishes in single points of segment.
The well-posedness of the initial boundary value problems is achieved using some approaches to regularization of the equation and the theory of characteristics. The problems for wave equation are then reduced to problems for hyperbolic balance laws 2 and 3 of partial differential equations of the first order. Weak solutions to the problems are obtained using proper numerical methods.
Results obtained for some approaches to regularization are presented.
Robust chaos is determined by the absence of periodic windows in bifurcation diagrams and coexisting attractors with parameter values taken from some regions of the parameter space of a dynamical system. Reliable chaos is an important characteristic of a dynamic system when it comes to its practical application. This property ensures that the chaotic behavior of the system will not deteriorate or be adversely affected by various factors. There are many methods for creating chaotic systems that are generated by adjusting the corresponding system parameters. However, most of the proposed systems are functions of well-known discrete mappings. In view of this, in this paper we consider a continuous system that illustrates some robust chaos properties.
In this paper we propose a new technique for the solution of the image segmentation problem which is based on the concept of a piecewise smooth approximation of some target functional. We discuss in details the consistency of the new statement of segmentation problem and its solvability. We focus our main intension on the rigor mathematical substantiation of the proposed approach, deriving the corresponding optimality conditions, and show that the new optimization problem is rather flexible and powerful model to the study of variational image segmentation problems. We illustrate the accuracy and efficiency of the proposed algorithm by numerical experiences.
Some implications concerning an unobservable medium whose micro-motion (rotational motion in Faraday vortex tubes) produces magnetic phenomena after Maxwell, used in a previous article on the origin of Maxwell’s equation for the magnetic induction (JODEA, 26 (1), 29 – 44) are clarified and explained.
The Fourier problem or, in other words, the problem without initial conditions for evolution equations and inclusions arise in modeling different nonstationary processes in nature, that started a long time ago and initial conditions do not affect on them in the actual time moment. The Fourier problem for evolution variational inequalities (inclusions) with functionals is considered in this paper. The conditions for existence and uniqueness of weak solutions of the problem are set. Also the estimates of weak solutions are obtained.
A new autonomous 4D nonlinear model with two nonlinearities describing the dynamics of change of voltage and current in the contact railway electric network is offered. This model is a connection of two 2D oscillatory circuits for current and voltage in the contact electric network. In the found system for the defined values of parameters an existence of limit cycles is proved. By introduction of new variables this system can be reduced to 5D system only with one quadratic nonlinearity. The constructed model may be used for the control by voltage stability in a direct current power supply system.
We consider the problem of existence and location of a solution of a nonlinear operator equation with a Fr´echet differentiable operator in a Banach space and present the convergence results for a projection-iteration method based on a Newton-like method under the Cauchy’s conditions, which generalize the results for the projection-iteration realization of the Newton-Kantorovich method. The proposed method unlike the traditional interpretation is based on the idea of whatever approximation of the original equation by a sequence of approximate operator equations defined on subspaces of the basic space with the subsequent application of the Newton-like method to their approximate solution. We prove the convergence theorem, obtain the error estimate and discuss the advantages of the proposed approach and some of its modifications.
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