The mathematical theory of classification contains a large number of approaches, models, methods, algorithms. This theory is very diverse. We distinguish three basic results in it - the best method of diagnosis (discriminant analysis), an adequate indicator of the quality of discriminant analysis algorithm, the statement about stopping after a finite number of steps iterative algorithms of cluster analysis. Namely, on the basis of Neyman - Pearson Lemma we have shown that the optimal method of diagnosis exists and can be expressed through probability densities corresponding to the classes. If the densities are unknown, one should use non-parametric estimators of training samples. Often, we use the quality indicator of diagnostic algorithm as "the probability (or share) the correct classification (diagnosis)" - the more the figure is the better algorithm is. It is shown that widespread use of this indicator is unreasonable, and we have offered the other - "predictive power", obtained by the conversion in the model of linear discriminant analysis. A stop after a finite number of steps of iterative algorithms of cluster analysis method is demonstrated by the example of k-means. In our opinion, these results are fundamental to the theory of classification and every specialist should be familiar with them for developing and applying the theory of classification
When considering the ecological safety of industrial productions, territory, etc., we usually allocate the constant (permanent) risk and the accident (emergency) risk. Permanent risk is given by the used technology, and cannot be changed substantially. Emergency risks are associated with uncertainty, in contrast to the constant risks. Let in adopted mathematical model the uncertainty is probabilistic in nature, and the loss describes as one-dimensional random variable. The distribution function of the loss, as a rule, is not normal. We have discussed in detail the seven characteristics of accidental loss: expectation; median and, more generally, quantile; dispersion; standard deviation; coefficient of variation; a linear combination of the expectation and standard deviation; the expectation of the loss function. Risk management may be to minimize these characteristics and their combinations (in different variants of multicriteria optimization). For example, in the two-criteria formulation it is required to minimize the expectation of loss and the standard deviation. Two-criteria formulation one way or another is reduced to a one-criteria formulation. In addition to probabilistic methods of risk modeling, sometimes we consider methods for describing risk using by means of objects of non-numeric nature, in particular qualitative characteristics, concepts of the theory of fuzzy sets, interval mathematical and econometric models and other mathematical tools. The main problems of the theory and practice of ecological insurance have been discussed
We propose a mathematical model of ion transport binary salt in electroosmotic flow in a capillary. The capillary is open on one side and immersed in a vessel of large volume, in which the concentration of the solution is maintained constant, and the other side closed ion exchange membrane. The walls are considered wettable, i.e. the solution adheres to the walls. This means that the mathematical modeling used to rate the condition of sticking. We study the boundary value problem for a coupled system of equations Nernst, Planck, Poisson and Navier-Stokes equations. Used boundary conditions of general form. The mathematical model is based on the general laws of transport and contains no adjustable parameters. Using this model, the basic laws of ion transport salt solution liquid flow, the emergence and development electroconvection, distribution of concentration of salt ions in the capillary with a small change in time, ie, in the initial (transitional) regime. We have identified the presence of ion-exchange membrane surface electroconvective vortices and their influence on the mechanisms of ion transport of salt and fluid movement in different areas of the capillary. A feature of the capillary transport is to the right of the vortex region stagnant areas with a higher concentration of ions
In the article we have derived mathematical models of non-stationary transport binary electrolyte in EMS (electromembrane systems: electrodialysis apparatus, electromembrane cell, etc.) for the galvanostatic mode. To be specific, as EMS viewed channel of desalting of EDA (electrodialysis apparatus) and EMS with RMD (rotating membrane disk). We present a formula expressing the intensity of the electric field through the current density and concentration. Also, we have received the differential equation for the current density. The fundamental point here is derived new equation for the unknown vector function of current density of the initial system of equations of Nernst-Planck. In addition, the article shows the output equation for the current density in three dimensions; we have proposed various methods for solving the equation of the current density and the boundary conditions for the current density. The proposed mathematical models of transport binary electrolyte are easy to be generalized to an arbitrary electrolyte. However, the corresponding equations are cumbersome. It should be also noted that the boundary conditions can be varied and depend on the purpose of a particular study in this regard, in this work are just the equation having the general form
The article presents a new approach to 2D modeling of transport of salt ions in EMC (electro systems: electrodialysis devices, electro-cells, etc.) under the condition of electrical neutrality with limiting and overlimiting current density. For definiteness as seen half of EMS channel EDA desalting (electrodialysis apparatus), the right border, which serves as a CEM (cation exchange membrane). The new approach in the use of partial differential equations of the first order, instead of equations of convective diffusion. A common method of transport modeling binary electrolyte in the EMS under the condition of electrical neutrality, is to use the equation of convective diffusion (partial differential equations of the second order). The article presents a new approach to modeling 2D transfer binary electrolyte in EMS under the same conditions, using partial differential equation of the first order for the decision, which does not require a boundary condition for concentration on the membrane surface. This allows you to simulate the transport of salt ions, as in prelimit and exorbitant current density and to determine the boundaries of the field of electrical neutrality
In the article on the basis of numbers of the specific form, where the parameter elements, which form a semigroup under multiplication we have presented a method for determination and distribution of composite numbers and the prime numbers, and accurate calculation of the values of pi in the interval from 1 to N. We present a new algorithm for the distribution of primes. We have reached the law of distribution parameters of composite numbers and prime numbers (Distribution of the parameters of composite numbers and prime numbers (DCPN)). We have given a formula for of finding prime numbers by serial number in the set DCPN. Due to the law of distribution of parameters of composite numbers and prime numbers it becomes apparent disintegration set of prime numbers. We have also introduced a proposal that each element of the plurality of composite numbers can be represented by one of the specific types of works. The proof of Proposition 2 allows us to give one of the most effective ways of recognizing primes. The description of the algorithm for numbers of twins and proof of their infinity. All algorithms presented in the article is a listing of programs in Software Module ACCESS
The article contains results of information research of acute myeloid leukemia (AML) as complicated multiple systems. The purpose of the research is creation an information presentation of AML and algorithms for determining the temporal characteristics of the disease. For describing the development of the disease we used the system of equations describing the growth of cells in populations of acute leukemia and considering decrease of protective forces of organism. A distinctive feature of this presentation is a more detailed description of the disease. For describing the processes of the division we used logistic equation. From the moment of an initiation of treatment the new parameters have been added into the system of equations, they are in charge of action of the applied preparations and responsive mutations the leukemic cells. On the basis of the submission of the information, we presented algorithms for calculating the temporal characteristics of the disease, namely, the development time of an irreversible condition in which the body is not able to destroy the leukemic clone of yourself, and the duration of remission. Also, as a result of the research we have made an evaluation of opportunities of the obtained algorithms. The article showed the wide range of possible solutions of the algorithm of determination the duration of remission
Following the absence of a definite treatment for the human immunodeficiency virus (HIV) or the acquired immune deficiency syndromes (AIDS) since their appearance, many scientific studies with the help of mathematical models have been formulated to the extent possible to prevent and eradicate the disease. In this article we have formulated a mathematical model that explores the dynamics of the impact of the use of condom and therapeutic treatment simultaneously, as a means (tools) against the spread of HIV/AIDS in the heterosexual population. The proposed model uses a nonlinear differential equation system consisting of seven (7) differential equations in seven (7) groups of the population. The model takes into account natural birth rate of the studied population, and the proportion of infected males, which simultaneously uses condom and antiretroviral therapy. The model explores the behavioral change of proportion of infected individuals in the population following the application of control measures (condom use and antiretroviral therapy). It is proved that the effectiveness of preventive measures greatly depends on a number of parameters described. In addition, the results of numerical experiments showed that in the absence of both preventive measures, the entire population is contaminated with the infection. The interaction of the model parameters show that the population with high levels of condom use in the presence of significant adherence to antiretroviral therapy as prophylaxis significantly reduces the level of HIV/AIDS. Thus, prevention of infection is significantly improved with the increasing number of the infected population using condoms and antiretroviral therapy simultaneously
This article discusses the mathematical and numerical modeling of the immune system of the course of HIV infection without treatment. Presently a significant number of scientific papers are devoted to the study of this problem. However, HIV infection is highly volatile and there is no effective drug, in that HIV has the ability to mutate and reproduce itself in the presence of chemical substances that are meant to inhibit or destroy it. The mathematical models used in this paper are conceptual and exploratory in nature. The proposed mathematical model allow us to obtain a complete description of the dynamics of HIV infection, and also an understanding of the progression to AIDS.
Thus, the results of the numerical solution of differential equations in this work show that: the disease develops, and at low concentration of the virus, a certain level of stability does not depend on the initial concentration of infestation. In the absence of treatment, for interesting competition between virus and the loss of virus caused by immune response should be strictly greater than the rate of multiplication of the virus in the blood; the reproduction rate of the uninfected cells should be stricly greater than the mortality rate of the uninfected cells
It is known that transmission coefficient of quartz glass containing the same amount of 28Si and 30Si in the silicon optical fiber is lesser than in commercial LEDs for telecommunications. Therefore it is topical to develop the method of optical glass formation with specified isotope composition in the core and in the shell. The article provides an analysis of physical and chemical processes occurring at the formation of quartz optical fiber blanks by vapor deposition from the gas phase. It is shown that the part of the silicon tetrachloride oxidation stages passes through the radical processes. Therefore for quartz glass formation with specified isotope composition it is possible to use the paramagnetic phenomena caused by the external magnetic field in a high-temperature flow at the quartz glass chemical deposition from the vapor phase. In this case alloy additive using is not necessary. Alloy additives can form density inhomogeneities in the glass. Simultaneous silicon glass formation and silicon isotope separation process bring to significant reduction of the fiber cost in comparison with isotope-enriched materials using. The permanent magnets can be used for magnetic field formation at existing process units