The Holy Alignment: Geodesic and Astronomical Fundamentals for Calculating the Adjustment of Medieval Naves

It is commonly believed that the longitudinal axes of churches extend exactly in an east-west direction. However, thorough investigations have shown that this is not always correct; rather, both southern and northern deviations of up to about 25° can occur. The angular deviation between the church axis and true east is called the Holy Alignment. This present study presents the possibility that the nave is oriented towards the direction of the sun-rising point on the name day of the patron saint of the church. If several saints share the patronage, the Holy Alignment equals the algebraic sum of the angular distances for each saint. The orientation of a nave can be analyzed by means of common mathematical relations used in geodesy, astronomy and gnomonics. In order to perform such an analysis, it is necessary to know the history of the patronage of the church; the Gauss-Krüger coordinates of the ground plan; and characteristic astronomical quantities at the time the church was built. A calculated example with the saints Andrew, James and Philip for the year 980 illustrates the analysis. An appendix deals with the influence of atmospheric refraction on the apparent altitude of the sun near the horizon.     

Mathematical models of formation of rational organizational hierarchies

A survey of Russian and foreign papers devoted to one of the burning trends of the theory of control of social and economic systems, i.e., mathematical models of formational of rational organization hierarchies is made. The following lines of investigations are emphasized: multilevel symmetrical hierarchies, hierarchies of knowledge, multilevel hierarchies of data processing, hierarchies and command theory, hierarchies of decision-making, hierarchies and contracts theory, and general model of optimal hierarchies search.     

Estimating parameters of the synchronous twofold-stochastic flow of events

Consideration was given to estimation of the parameters of the synchronous twofold-stochastic flow of events which makes up a mathematical model of the information flow of demands circulating in the queuing systems and networks. Two variants were studied. For the first variant, the problem of optimal estimation of the parameters of a synchronous flow of events with a finite arbitrary number of states was solved. The second variant differs in that the event flow operates in the environment where part of events is lost during the so-called dead time. Consideration was given to a synchronous flow with two states. The problem of estimating the length of the dead time in the conditions of continued dead time was solved. The results of numerical estimation based on the simulation model of the synchronous flow were presented for both variants.     

Decomposition of problems of optimal control and estimation for discrete systems with fast and slow variables

Consideration was given to the linear-quadratic problem of optimal control for the discrete linear system with fast and slow variables under incomplete information about system state. Decomposition of the discrete matrix Riccati equations was carried out. The proposed decomposition algorithm relies on a geometrical approach using the properties of the invariant manifolds of slow and fast motions of the nonlinear multirate discrete systems as basis. The splitting transformation was constructed in the form of asymptotic decomposition in the degrees of a small parameter.     

Markov model of a digital stochastic generator

A stochastic process generator based on a chain of modulo-two adders with asynchronous feedback loops is investigated. The generator dynamics is described by the Erlang equations. A mathematical model of the generated stochastic process is proposed. The implementation of the generator using modern electronic devices is considered.     

Output control algorithm with the compensation of biased harmonic disturbances

The paper develops the methods of harmonic disturbance compensation using the measurements of plant output variable. An approach for biased harmonic disturbance compensation is proposed. This approach is superior to known ones in some characteristics.     

Accounting of heterogeneity in the estimation of radiation risks

The problem is considered of taking account of heterogeneity in the individual radiosensitivity in evaluating the radiation risk of the origin of a disease in a group of persons subjected to the radiation action. Mathematical models are described for the estimation of the radiation risk of the origin of the disease with and without regard for heterogeneity. The use of the described method is demonstrated with an example of the estimation of the risk of the origin of radiation-induced diseases from the class of “solid cancers” among the persons taking part in the liquidation of consequences of the accident at the Chernobyl’ Atomic Power Station (CAPS) in the years 1986–1991. It is shown that the heterogeneity disregard leads to understated estimates of the radiation risk of the origin of radiation-induced solid cancers. The effect is investigated of the length of the latent period in the origin of radiation-induced “solid cancers” in taking account of heterogeneity.     

Petersen’s lemma on matrix uncertainty and its generalizations

Various generalizations and refinements are proposed for a well-known result on robust matrix sign-definiteness, which is extensively exploited in quadratic stability, design of robustly quadratically stabilizing controllers, robust LQR-problem, etc. The main emphasis is put on formulating the results in terms of linear matrix inequalities.     

Multidimensional analogs of the kendall equation for priority queueing systems: computational aspects

An analogy between celebrated Kendall equation for busy periods in the system M|GI|1 and analytical results for busy periods in the priority systemsM _r |GI _r |1 is drawn. These results can be viewed as generalizations of the functional Kendall equation. The methodology and algorithms of numerical solution of recurrent functional equations which appear in the analysis of such queueing systems are developed. The efficiency of the algorithms is achieved by acceleration of the numerical procedure of solving the classical Kendall equation. An algorithm of calculation of the system workload coefficient calculation is given.     

Adaptive algorithms to estimate parameters of the optimal policies of inventory control under limited stock-out

Consideration was given to the inventory control by the criterion for minimum of the cumulative mean costs over a sufficiently long planning horizon in conditions of uncertain demand characteristics. Adaptive algorithms to estimate the parameters of the optimal two-level control policies oriented to the supply systems with high level of customer servicing were constructed for the stationary mode of operation.