Mathematical Modeling of Population Dynamics Based on Recurrent Equations: Results and Prospects. Part I

Approaches to modeling population dynamics using discrete-time models are described in this two-part review. The development of the scientific ideas of discrete time models, from the Malthus model to modern population models that take into account many factors affecting the structure and dynamics, is presented. The most important and interesting results of recurrent equation application to biological system analysis obtained by the authors are given. In the first part of this review, the population dynamic effects that result from density-dependent regulation of population, the age and sex structures, and the influence of external factors are considered.

Mathematical Modelling of Sayano-Shushenskaya Dam Displacement Process after 2009 Accident

The research presented in the article is of cutting-edge importance because it proves the necessity to develop prognostic mathematical models with the view to studying the behavior of high-head dams for identifying the regularities of their deformations development process and thus providing quantitative definition for the set criteria values of the diagnostic indices to ensure safe operation of such structures. The paper focuses on the peculiarities of building prognostic mathematical models of the dynamic type on the basis of recurrent equations of the 1stand 2ndorders of different types depending on the order of the mathematical model, number of the principal acting factors and discreteness of the input data, with decorrelation of the input actions and sequence of transport delay introduction. It is shown that the properties of the recurrent equation solution in the form of two first conditional moment generating functions of the displacement process of the observed points in the structure delineate a prognostic model which allows predicting the displacements of the observed points. The paper describes the sequence of estimation stages during the creation of prognostic mathematical models in respect to the character of the predictive problem for various time periods of the structure operation. Different combinations of input actions and discreteness of input data, as well as their decorrelation, have been used. We also applied transport delay in order to correctly consider the inertial delay of the dam under different loads. To account for the residual part of the inertial delay, which is affected by random and unaccounted for factors, we used the autoregression model of the process development regularity. To determine the order of the autoregression model, we calculated asymptotically unbiased ratings of the correlation function for the residual error as a difference between the actual and predicted displacements. Methodological specifics of constructing prognostic models have been established in the context of the factors above. Prognostic mathematical models of different types have been developed for the selected period of the dam operation and the results of prediction have been discussed.

Structurally-Parametrical Optimization Technological Process by Dijkstra's Method in System Mathcad

The article describes the procedure of structurally-parametrical optimization of technological process by Dijkstra's method with using of Dijkstra’s functional (recurrent) equation and Mathcad system. The technological process includesnof operations and each operation can be executed by various types of equipment. Expenses (cost, time, ...) on execution ofioperation bykequipment after execution byjequipment (i-1) operation are known -c(i, j, k). The algorithm of the decision of a problem by Dijkstra’s method includes two phases.The firstphase is calculations of the minimum expenses for execution of all partial technological processes, from first operation of process to the last.The secondphase is definition of the required optimum set of equipment which is carrying out all technological process with the minimum expenses from last operation of process to the first. The proposed procedure of structurally-parametrical optimization of technological process using Dijkstra’s method and Mathcad software significantly decreases time and labour costs on execution of such calculations and efficiently to execute investigations related with change of equipment parameters.

Automation of optimization of discrete technological processes

The article describes the procedure automation of optimization discrete technological processes with using of Bellman’s functional (recurrent) equation and system Mathcad. As rule the technological processes includes n of operations and each operation can be executed by various types of equipment. Expenses (cost, time, …) on execution of i operation by k equipment after execution by j equipment (i-1) operation are known - c (i, j, k). Expenses for execution by k equipment i operation can depend on the equipment - j, which executed previous (i-1) operation. It is necessary to execute automation of optimization technological process with the minimum expenses. The algorithm of the decision of a problem by Bellman’s method includes two phases. The first phase is calculations of the minimum expenses for execution of all partial technological processes, from last operation of process to the first. The second phase is definition of the required optimum set of equipment which is carrying out all technological process with the minimum expenses. The proposed procedure of automation of optimization technological process using Bellman’s method and system Mathcad significantly decreases time and labour costs on execution of such calculations and efficiently to execute investigations related with change of equipment parameters.

Urn models and differential algebraic equations

The aim of this paper is to study the distribution of colours, { X n }, in a generalized Pólya urn model with L colours, an urn function and a random environment. In this setting, the number of actions to be taken can be greater than L, and the total number of balls added in each step can be random. The process { X n } is expressed as a stochastic recurrent equation that fits a Robbins—Monro scheme. Since this process evolves in the (L—1)-simplex, the stability of the solutions of the ordinary differential equation associated with the Robbins—Monro scheme can be studied by means of differential algebraic equations. This approach provides a method of obtaining strong laws for the process { X n }.

Urn models and differential algebraic equations

The aim of this paper is to study the distribution of colours, {Xn}, in a generalized Pólya urn model with L colours, an urn function and a random environment. In this setting, the number of actions to be taken can be greater than L, and the total number of balls added in each step can be random. The process {Xn} is expressed as a stochastic recurrent equation that fits a Robbins—Monro scheme. Since this process evolves in the (L—1)-simplex, the stability of the solutions of the ordinary differential equation associated with the Robbins—Monro scheme can be studied by means of differential algebraic equations. This approach provides a method of obtaining strong laws for the process {Xn}.

TRANSFORMATION OF NESTED LOOPS INTO UNIFORM RECURRENCES AND THEIR MAPPING TO REGULAR PROCESSOR ARRAYS

A methodology for transforming a class of iterative algorithms, expressed in nested loops, into Uniform Recurrent Equation (URE) forms and their mapping into regular processor array architectures is presented. The propagation space of the variables of the original nested loop is specified by a system of linear equations formed by their index functions. The data flow within the index space, is then localized by the derivation of a set of parametric dependence vectors, which eventually can be used to transform the initial algorithm into a set of UREs. The mapping of the UREs is accomplished by decomposition of the index space into independent subsets of variable instances using the derived dependence vectors. The dependence graphs of the subsets are normalized and subsequently, are mapped on the processor array architecture. The exploitation of the independent subsets leads to significant improvement of the efficiency of the processor array compared to architectures derived by using linear transformations of the entire index space. Under certain conditions, only local interconnections in the processor array are required. The proposed methodology is illustrated by the design of alternative processor arrays implementing the convolution algorithm.