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## Четырехфакторный вычислительный эксперимент для задачи случайного блуждания на двумерной решетке

Nowadays the random search became a widespread and effective tool for solving different complex optimization and adaptation problems. In this work, the problem of an average duration of a random search for one object by another is regarded, depending on various factors on a square field. The problem solution was carried out by holding total experiment with 4 factors and orthogonal plan with 54 lines. Within each line, the initial conditions and the cellular automaton transition rules were simulated and the duration of the search for one object by another was measured. As a result, the regression model of average duration of a random search for an object depending on the four factors considered, specifying the initial positions of two objects, the conditions of their movement and detection is constructed. The most significant factors among the factors considered in the work that determine the average search time are determined. An interpretation is carried out in the problem of random search for an object from the constructed model.The important result of the work is that the qualitative and quantitative influence of initial positions of objects, the size of the lattice and the transition rules on the average duration of search is revealed by means of model obtained. It is shown that the initial neighborhood of objects on the lattice does not guarantee a quick search, if each of them moves. In addition, it is quantitatively estimated how many times the average time of searching for an object can increase or decrease with increasing the speed of the searching object by 1 unit, and also with increasing the field size by 1 unit, with different initial positions of the two objects. The exponential nature of the growth in the number of steps for searching for an object with an increase in the lattice size for other fixed factors is revealed. The conditions for the greatest increase in the average search duration are found: the maximum distance of objects in combination with the immobility of one of them when the field size is changed by 1 unit. (that is, for example, with 4x4 at 5x5) can increase the average search duration in e^1,69≈5,42. The task presented in the work may be relevant from the point of view of application both in the landmark for ensuring the security of the state, and, for example, in the theory of mass service.

According to the currently prevalent theory, hippocampal formation constructs and maintains cognitive spatial maps. Most of the experimental evidence for this theory comes from the studies on navigation in laboratory rats and mice, typically male animals. While these animals exhibit a rich repertoire of behaviors associated with navigation, including locomotion, head movements, whisking, sniffing, raring and scent marking, the contribution of these behavioral patterns to hippocampal activity has not been sufficiently studied. Instead, many publications have considered animal position in space as the single variable that affects the firing of hippocampal place cells and entorhinal grid cells. Here we argue that future work should focus on a more detailed examination of different behaviors exhibited during navigation in order to interpret the cause of spatial tuning in hippocampal neurons. As a step in this direction, we have analyzed data from two datasets, shared online, containing recordings from rats navigating in square and round arenas. Our analyses revealed structured, grid-like navigation patterns, evident from the spatial maps of animal position, velocity and acceleration. Moreover, grid cells available in the datasets exhibited the same spatial periodicity as the navigation parameters. These findings cast doubt on the cognitive-map interpretation of grid cells, since they suggest that neuronal spatial patterns could be caused by behaviors associated with navigation instead of representing a hierarchically high spatial map. Additionally, we speculate that scent marks left by navigating animals could contribute to neuronal responses while rats and mice sniff their environment.

This note states several results on the exponential functionals of the Brownian motion and their approximations by Markov chains. Starting from M.Yor, such functionals were studied in mathematical finance. At the same time, they play a significant role in different settings: the analysis of diffusions on the class of solvable Lie groups, in particular on the group of (2 X 2) upper triangular matrices, with positive diagonal elements. The discrete random walks cannot properly describe the local structure of diffusion. However, instead of the usual local limit theorem (which is not applicable) its weaker form, namely quasi-local form is given.

This study is an attempt to obtain reliable data on the natural history of breast cancer growth. The opportunities for using classical mathematical models (exponential and logistic tumor growth models, Gompertz and von Bertalanffy tumor growth models) were analysed in order to describe growth of the primary tumor and the secondary distant metastases of human breast cancer. Our results suggest a new «Consolidated mathematical growth Model of the Primary tumor and the Secondary distant metastases» (CoMPaS). The CoMPaS is based on exponential tumor growth model and consists of a system of determinate nonlinear and linear equations. The CoMPaS describes correctly the primary tumor growth (parameter T) and the secondary distant metastases growth (parameter M). Also, CoMPaS associates with data of 10–15-year survival in patients with the different tumor stage. Analysis of the metastases «nonvisible period» growth indicate the case of discrepancy between 15-year survival depending on tumor stage. In conclusion, the CoMPaS and supporting computer program were build to improve the accuracy of the forecast on survival of breast cancer and facilitate the optimisation of diagnosing secondary distant metastases. This led to completely original results that show how the growth rate of the metastases can change in relation to the growth rate of the primary tumour, taking into consideration its size and diameter of the tumour.

This paper is concerned with Random walk approximations of the Brownian motion on the Affine group Aff(R). We are in particular interested in the case where the innovations are discrete. In this framework, the return probabilities of the walk have fractional exponential decay in large time, as opposed to the polynomial one of the continuous object. We prove that in tegrating those return probabilities on a suitable neighborhood of the origin, the expected polynomial decay is restored. This is what we call a Quasi-local theorem.

We present a comparative study of several algorithms for an in-plane random walk with a variable step. The goal is to check the efficiency of the algorithm in case where the random walk terminates at some boundary. We recently found that a finite step of the random walk produces a bias in the hitting probability and this bias vanishes in the limit of an infinitesimal step. Therefore, it is important to know how a change in the step size of the random walk influences the performance of simulations. We propose an algorithm with the most effective procedure for the step-length-change protocol.

Proceedings of the III International Conference in memory of V.I. Zubov "Stability and Control Processes (SCP 2015)".

Symmetric random walks in $R^d$ and $Z^d$ are considered. It is assumed that the jump distribution density has moderate tails, i.e., several density moments are finite, including the second one. The global (for all $x$ and $t$) asymptotic behavior at infinity of the transition probability (fundamental solution of the corresponding parabolic convolution operator) is found. Front propagation of ecological waves in the corresponding population dynamics models is described.

This book deals with mathematical problems arising in the context of meteorological modelling. It gathers and presents some of the most interesting and important issues from the interaction of mathematics and meteorology. It is unique in that it features contributions on topics like data assimilation, ensemble prediction, numerical methods, and transport modelling, from both mathematical and meteorological perspectives.

The derivation and solution of all kinds of numerical prediction models require the application of results from various mathematical fields. The present volume is divided into three parts, moving from mathematical and numerical problems through air quality modelling, to advanced applications in data assimilation and probabilistic forecasting.

The book arose from the workshop “Mathematical Problems in Meteorological Modelling” held in Budapest in May 2014 and organized by the ECMI Special Interest Group on Numerical Weather Prediction. Its main objective is to highlight the beauty of the development fields discussed, to demonstrate their mathematical complexity and, more importantly, to encourage mathematicians to contribute to the further success of such practical applications as weather forecasting and climate change projections. Written by leading experts in the field, the book provides an attractive and diverse introduction to areas in which mathematicians and modellers from the meteorological community can cooperate and help each other solve the problems that operational weather centres face, now and in the near future.

Readers engaged in meteorological research will become more familiar with the corresponding mathematical background, while mathematicians working in numerical analysis, partial differential equations, or stochastic analysis will be introduced to further application fields of their research area, and will find stimulation and motivation for their future research work.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

The problem of minimizing the root mean square deviation of a uniform string with clamped ends from an equilibrium position is investigated. It is assumed that the initial conditions are specified and the ends of the string are clamped. The Fourier method is used, which enables the control problem with a partial differential equation to be reduced to a control problem with a denumerable system of ordinary differential equations. For the optimal control problem in the l2 space obtained, it is proved that the optimal synthesis contains singular trajectories and chattering trajectories. For the initial problem of the optimal control of the vibrations of a string it is also proved that there is a unique solution for which the optimal control has a denumerable number of switchings in a finite time interval.

For a class of optimal control problems and Hamiltonian systems generated by these problems in the space *l *2, we prove the existence of extremals with a countable number of switchings on a finite time interval. The optimal synthesis that we construct in the space *l *2 forms a fiber bundle with piecewise smooth two-dimensional fibers consisting of extremals with a countable number of switchings over an infinite-dimensional basis of singular extremals.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

In this paper, we construct a new distribution corresponding to a real noble gas as well as the equation of state for it.