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Verification of the physical-statistical approach within reliability theory for the simplest cases was carried out, which showed its validity. An analytical solution of the one-dimensional basic equation of the physicalstatistical approach is presented under the assumption of a stationary degradation rate. From a mathematical point of view this equation is the well-known continuity equation, where the role of density is played by the density distribution function of goods in its characteristics phase space, and the role of fluid velocity is played by intensity (rate) degradation processes. The latter connects the general formalism with the specifics of degradation mechanisms. The cases of coordinate constant, linear and quadratic degradation rates are analyzed using the characteristics method. In the first two cases, the results correspond to physical intuition. At a constant rate of degradation, the shape of the initial distribution is preserved, and the distribution itself moves equably from the zero. At a linear rate of degradation, the distribution either narrows down to a narrow peak (in the singular limit), or expands, with the maximum shifting to the periphery at an exponentially increasing rate. The distribution form is also saved up to the parameters. For the initial normal distribution, the coordinates of the largest value of the distribution maximum for its return motion are obtained analytically.

In the quadratic case, the formal solution demonstrates counterintuitive behavior. It consists in the fact that the solution is uniquely defined only on a part of an infinite half-plane, vanishes along with all derivatives on the boundary, and is ambiguous when crossing the boundary. If you continue it to another area in accordance with the analytical solution, it has a two-humped appearance, retains the amount of substance and, which is devoid of physical meaning, periodically over time. If you continue it with zero, then the conservativeness property is violated. The anomaly of the quadratic case is explained, though not strictly, by the analogy of the motion of a material point with an acceleration proportional to the square of velocity. Here we are dealing with a mathematical curiosity. Numerical calculations are given for all cases. Additionally, the entropy of the probability distribution and the reliability function are calculated, and their correlation is traced.

Image denoising is one of the fundamental problems in digital image processing. This problem usually refers to the reconstruction of an image from an observed image degraded by noise. There are many factors that cause this degradation such as transceiver equipment, or environmental influences, etc. In order to obtain higher quality images, many methods have been proposed for image denoising problem. Most image denoising method are based on total variation (TV) regularization to develop efficient algorithms for solving the related optimization problem. TV-based models have become a standard technique in image restoration with the ability to preserve image sharpness.

In this paper, we focus on Poisson noise usually appearing in photon-counting devices. We propose an effective regularization model based on combination of first-order and fractional-order total variation for image reconstruction corrupted by Poisson noise. The proposed model allows us to eliminate noise while edge preserving. An efficient alternating minimization algorithm is employed to solve the optimization problem. Finally, provided numerical results show that our proposed model can preserve more details and get higher image visual quality than recent state-of-the-art methods.

The biological structure is considered as an open nonequilibrium system which properties can be described on the basis of kinetic equations. New problems with nonequilibrium boundary conditions are introduced. The nonequilibrium distribution tends gradually to an equilibrium state. The region of spatial inhomogeneity has a scale depending on the rate of mass transfer in the open system and the characteristic time of metabolism. In the proposed approximation, the internal energy of the motion of molecules is much less than the energy of translational motion. Or in other terms we can state that the kinetic energy of the average blood velocity is substantially higher than the energy of chaotic motion of the same particles. We state that the relaxation problem models a living system. The flow of entropy to the system decreases in downstream, this corresponds to Shrödinger’s general ideas that the living system “feeds on” negentropy. We introduce a quantity that determines the complexity of the biosystem, more precisely, this is the difference between the nonequilibrium kinetic entropy and the equilibrium entropy at each spatial point integrated over the entire spatial region. Solutions to the problems of spatial relaxation allow us to estimate the size of biosystems as regions of nonequilibrium. The results are compared with empirical data, in particular, for mammals we conclude that the larger the size of animals, the smaller the specific energy of metabolism. This feature is reproduced in our model since the span of the nonequilibrium region is larger in the system where the reaction rate is shorter, or in terms of the kinetic approach, the longer the relaxation time of the interaction between the molecules. The approach is also used for estimation of a part of a living system, namely a green leaf. The problems of aging as degradation of an open nonequilibrium system are considered. The analogy is related to the structure, namely, for a closed system, the equilibrium of the structure is attained for the same molecules while in the open system, a transition occurs to the equilibrium of different particles, which change due to metabolism. Two essentially different time scales are distinguished, the ratio of which is approximately constant for various animal species. Under the assumption of the existence of these two time scales the kinetic equation splits in two equations, describing the metabolic (stationary) and “degradative” (nonstationary) parts of the process.

The model of the process of using machinery and equipment is considered, which takes into account the probabilistic nature of the process of their operation and sale. It takes into account the possibility of random hidden failures, after which the condition of the machine deteriorates abruptly, as well as the randomly arising need for premature (before the end of its service life) sale of the machine, which requires, generally speaking, random time. The model is focused on assessing the market value and service life of machines in accordance with International Valuation Standards. Strictly speaking, the market value of a used machine depends on its technical condition, but in practice, appraisers only take into account its age, since generally accepted measures of the technical condition of machines do not yet exist. As a result, the market value of a used machine is assumed to be equal to the average market value of similar machines of the corresponding age. For these purposes, appraisers use coefficients that reflect the influence of the age of machines on their market value. Such coefficients are not always justified and do not take into account either the degradation of the machine or the probabilistic nature of the process of its use. The proposed model is based on the anticipation of benefits principle. In it, we characterize the state of the machine by the intensity of the benefits it brings. The machine is subjected to a complex Poisson failure process, and after failure its condition abruptly worsens and may even reach its limit. Situations also arise that preclude further use of the machine by its owner. In such situations, the owner puts the machine up for sale before the end of its service life (prematurely), and the sale requires a random timing. The model allows us to take into account the influence of such situations and construct an analytical relationship linking the market value of a machine with its condition, and calculate the average coefficients of change in the market value of machines with age. At the same time, it is also possible to take into account the influence of inflation and the scrap cost of the machine. We have found that the rate of prematurely sales has a significant impact on the cost of new and used machines. The model also allows us to take into account the influence of inflation and the scrap value of the machine. We have found that the rate of premature sales has a significant impact on the service life and market value of new and used machines. At the same time, the dependence of the market value of machines on age is largely determined by the coefficient of variation of the service life of the machines. The results obtained allow us to obtain more reasonable estimates of the market value of machines, including for the purposes of the system of national accounts.