Small business is an important part of modern Russian economy. We give a wide panorama developed by us of possible approaches to the construction of economic-mathematical models that may be useful to describe the dynamics of small businesses, as well as management. As for the description of certain problems of small business can use a variety of types of economic-mathematical and econometric models, we found it useful to consider a fairly wide range of such models, which resulted in quite a short description of the specific models. In this description of the models brought to such a level that an experienced professional in the field of economic-mathematical modeling could, if necessary, to develop their own specific model to the stage of design formulas and numerical results. Particular attention is paid to the use of statistical methods of non-numeric data, the most pressing at the moment. Are considered the problems of economic-mathematical modeling in solving problems of small business marketing. We have accumulated some experience in application of the methodology of economic-mathematical modeling in solving practical problems in small business marketing, in particular in the field of consumer goods and industrial purposes, educational services, as well as in the analysis and modeling of inflation, taxation and others. In marketing models of decision making theory we apply rankings and ratings. Is considered the problem of comparing averages. We present some models of the life cycle of small businesses - flow model projects, model of capture niches, and model of niche selection. We discuss the development of research on economic-mathematical modeling of small businesses
Some estimators of the probability density function in spaces of arbitrary nature are used for various tasks in statistics of non-numerical data. Systematic exposition of the theory of such estimators had a start in our work [2]. This article is a direct continuation of the article [2]. We will regularly use references to conditions and theorems of the article [2], in which we introduced several types of nonparametric estimators of the probability density. We studied more linear estimators. In this article we consider particular cases - kernel density estimates in spaces of arbitrary nature. When estimating the density of the one-dimensional random variable, kernel estimators become the Parzen-Rosenblatt estimators. Asymptotic behavior of kernel density estimators in the general case of an arbitrary nature spaces are devoted to Theorem 1 - 8. Under different conditions we prove the consistency and asymptotic normality of kernel density estimators. We have studied uniform convergence. We have introduced the concept of "preferred rate differences" and studied nuclear density estimators based on it. We have also introduced and studied natural affinity measures which are used in the analysis of the asymptotic behavior of kernel density estimators. We have found the asymptotic behavior of dispersions of kernel density estimators and considered the examples including kernel density estimators in finite-dimensional spaces and in the space of square-integrable functions
In this article we propose a method that uses the apparatus of the theory of fuzzy sets, together with the five-factor model of Altman in assessing the creditworthiness of an enterprise. Altman's model works in two ways: It applies the root mean square (RMS) integral approximation for the exact calculation of quantitative assessment of creditworthiness (probability of bankruptcy), and using the device of fuzzy sets for ordered sets by the degree of confidence in the resulting probability. In this paper we conducted simulation procedure for the credit assessment and showed the capabilities of the model. The model input parameters , forms system inputs (input variables), allowing you to get the value of the parameter z of Altman. With the help of Altman's model, approximating function L6, the decision function I(p) and the algorithm for calculating preference we obtain the number of the set i to which belongs a number of ordered sets as fuzzy logic . On the selected simulation parameters, stable statistics can be obtained. Altman's model with the use of computational function allows real values of the input parameters of the enterprise replaced by random values of the simulation model. This technique allows, as shown by the results of computational experiments, the creditor to obtain additional information on the creditworthiness of the investigated enterprise and make a more informed conclusion about its financial condition, which speeds up the decision on the possibility of issuing the required credit. The development of method of estimating fuzzy logic can be applied to other models of assessing the creditworthiness of a company: Davydov's model, Zaitseva's, Saifullina's, Kadykova's and others with appropriate modification
Metric describing the accelerated and rotating reference system in general relativity in the case of an arbitrary dependence of acceleration and angular velocity on time has been proposed. It is established that the curvature tensor in such metrics is zero, which corresponds to movement in the flat spaces. It is shown that the motion of test bodies in the metric accelerated and rotating reference system in general relativity is similarly to the classical motion in non-inertial reference frame. Maxwell's equations and Yang-Mills theory are converted to the moving axes in metric describes the acceleration and rotating reference frame in the general relativity in the case of an arbitrary dependence of acceleration and angular velocity of the system from time. The article discusses the known effects associated with acceleration and (or) the rotation of the reference frame - the Sagnac effect, the effect of the Stewart-Tolman and other similar effects. The numerical model of wave propagation in non-inertial reference frames in the case when potential depending of one, two and three spatial dimensions has been developed. It has been shown in numerical experiment that the acceleration of the reference system leads to retardation effects, as well as to a violation of the symmetry of the wave front, indicating that there is local change of wave speed
In the article we consider integrative codes of the elements of discrete systems for the first time. It is shown that these codes in the general case divided into group and system parts. The group part of the code characterizes a set of elements with identical value of the sign as a whole. System part of the code appears when different sets are combined into the system. We have established that in using the weighted average of these parts of integrative code we can express information measures of combinatorial, probabilistic and synergistic approaches to determine the quantity of information. It is concluded that there is an integrative coding relationship between these approaches, and the existing types of information have genetic relationship. It is shown that the information considered in the synergetic approach is genetically of primary in relation to the information, which operates on the combinatorial and probabilistic approaches. Also, we have answered the question why the different conceptions of information lead to identical formulas to measure it
The quality of a system is seen as an emergent property of systems, due to their composition and structure, and it reflects their functionality, reliability and cost. Therefore, when we speak about quality management, the purpose of management is the formation of pre-defined system properties of the object of management. The stronger the object of the control expresses its system properties, the stronger the nonlinearity manifests of the object: both the dependence of the management factors from each other, and the dependence of the results of the action of some factors from the actions of others. Therefore, the problem of quality management is that in the management process the management object itself changes qualitatively, i.e. it changes its level of consistency, the degree of determinism and the transfer function itself. This problem can be viewed as several tasks: First is the system identification of the condition of the object of management, 2nd – making decisions about controlling influence that changes the composition of the control object in a way its quality maximally increases at minimum costs. To solve the 2nd problem we have proposed an application of the component selection of the object by functions based on the resources allocated for the implementation of different functions; costs associated with the choice of the components and the degree of compliance of various components to their functional purpose. In fact, we have proposed a formulation and a solution of the new generalization of a variant of the assignment problem: "multi backpack", which differs from the known with the fact that the selection has been based not only on the resources and costs, but also with taking into account the degree of compliance of the components to their functional purpose. A mathematical model, which provides a solution to the 1st problem, and reflecting the degree of compliance of the components to their functionality, as well as the entire decision-making process for selections, i.e. 2nd task, has been implemented in the ASC-analysis and in the system called "Eidos" X++". The article also provides a simplified numerical example of the proposed approach with the selection of staff members
According to the new paradigm of applied mathematical statistics one should prefer non-parametric methods and models. However, in applied statistics we currently use a variety of parametric models. The term "parametric" means that the probabilistic-statistical model is fully described by a finite-dimensional vector of fixed dimension, and this dimension does not depend on the size of the sample. In parametric statistics the estimation problem is to estimate the unknown value (for statistician) of parameter by means of the best (in some sense) method. In the statistical problems of standardization and quality control we use a three-parameter family of gamma distributions. In this article, it is considered as an example of the parametric distribution family. We compare the methods for estimating the parameters. The method of moments is universal. However, the estimates obtained with the help of method of moments have optimal properties only in rare cases. Maximum likelihood estimation (MLE) belongs to the class of the best asymptotically normal estimates. In most cases, analytical solutions do not exist; therefore, to find MLE it is necessary to apply numerical methods. However, the use of numerical methods creates numerous problems. Convergence of iterative algorithms requires justification. In a number of examples of the analysis of real data, the likelihood function has many local maxima, and because of that natural iterative procedures do not converge. We suggest the use of one-step estimates (OS-estimates). They have equally good asymptotic properties as the maximum likelihood estimators, under the same conditions of regularity that MLE. One-step estimates are written in the form of explicit formulas. In this article it is proved that the one-step estimates are the best asymptotically normal estimates (under natural conditions). We have found OS-estimates for the gamma distribution and given the results of calculations using data on operating time to limit state for incisors
In this article we have proposed an intersectoral mathematical model of self-organization of the labor market. This model is the system of balance equations of the dynamics of the labor force. The model contains parameters that show where workers were employed in previous times. Therefore it is possible to monitor the dynamics of intersectoral labor force over a long period of time. It has been shown that the model allows to solve the problem of forecasting the number of employed and unemployed in the labor market under the assumption that the parameters of the probabilistic model are constant for a certain period. The use of the model is illustrated on the example in which the probabilities of hiring and firing of employees were calculated, as well as the probabilities that the employees in the analyzed period (2011 – 2012 years) leave the labor market. The forecast of the number of employed and unemployed at the end of the next period (2013) is based on the calculated probabilities. It has been demonstrated that the deviation of the predicted values from the statistical data is not significant, which witnesses about the efficiency of the forecast
The aim of this work is to carry out numerical analysis of time behavior of electroconvection in membrane systems, such as a desalting channel of electrodialysis apparatus. The current-voltage curve and the solution flaw were analyzed theoretically using mathematical models of ions transfer taking into account electroconvection in smooth desalting channel consisting of ideally selective anion- and cation-exchange membranes. The Hurst numbers for different parts of the current-voltage curve were calculated in order to determine whether the parts were persistent. The Fourier-analysis of the oscillating term of the current-voltage curve was carried out for the first time so as to determine predominant frequencies in the signal. Frequencies of passing of complexes of vortexes through the cross-section of the desalting channel were calculated. Frequencies of oscillations of concentration profiles were determined. It was found that the frequencies of oscillations of the concentration profiles coincide with the frequencies of passing of complexes of vortexes through the cross-section of the desalting channel. The oscillations of the current-voltage curve were physically interpreted. Namely, it was shown that the main frequency of oscillations of the current-voltage curve corresponds to the frequency of oscillations of the concentration profiles. The oscillations of the concentration profiles, produced by the passing of complexes of vortexes, cause oscillations of conductivity and, consequently, cause oscillations of resistance and of the current density. It was shown that the main frequency of the signal corresponds to the frequency of passing of complexes of vortexes through the cross-section of the desalting channel
The article discusses the excitation of electromagnetic radiation, nuclear reactions and decays of particles by the acceleration of charges, atomic nuclei and the macroscopic volumes of matter. The motion of charged particles in a magnetic trap used for plasma confinement was computed. We propose a model of the electromagnetic radiation of a charge moving in a non-inertial reference frame in general relativity. We have also constructed a theory of perturbation with using a wave equation with small parameters, taking into account a characteristic radius of the trajectory of the electrons as they move in a magnetic field. It was found that in the first approximation, the radiation back-reaction force depends on the acceleration of the charge. For the simulating of processes in hadrons and nuclei we used Yang-Mills theory and the metric, describes the acceleration and rotating reference frame in general relativity. We consider the scalar glueball model for an arbitrary dependence of acceleration and angular velocity of the system on time. The numerical model of wave propagation in non-inertial reference frame for the geometry of system of one, two or three spatial dimensions was tested. In the numerical experiments shown that the acceleration of the system leads to instability, leading to an unlimited increase in the amplitude of waves, which is interpreted as a decay of system. It was found that there are critical values of acceleration above which the instability develops