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## An analytic study of the Ornstein–Uhlenbeck process with time-varying coefficients in the modeling of anomalous diffusions

We consider the problem of modeling anomalous diffusions with the Ornstein–Uhlenbeck process with time-varying coefficients. An anomalous diffusion is defined as a process whose mean-squared displacement non-linearly grows in time which is nonlinearly growing in time. We classify diffusions into types (subdiffusion, normal diffusion, or superdiffusion) depending on the parameters of the underlying process. We solve the problem of finding the coefficients of dynamics equations for the Ornstein–Uhlenbeck process to reproduce a given mean-squared displacement function.

In this study we investigate the processes of diffusion of impurity particles in the media with stochastic characteristics. The exact equation with fractal erivation for middle admixture concentration in fibers with telegraph-type of curve angel is obtained. Exact solution is found. Admixtures anomalous diffusion effect is hown.

The book presents the most important aspects of safe digital image workflows, starting from the basic practical implications and gradually uncovering the underlying concepts and algorithms. With an easy-to-follow, down-to-earth presentation style, the text helps you to optimize your diagnostic imaging projects and connect the dots of medical informatics.

Measurements of velocity in navigation receivers are performed in two stages. At the primary (after-satellite) processing stage each of received signals is synchronized using a separate PLL, after that an estimation block (EB) estimates nonenergy (phase and frequency) and energy (SNR) parameters of the received signal. Doppler primary estimations can be subject to after-satellite filtration to obtain secondary frequency estimates. A number of Doppler estimates are conversed into primary estimates of velocity vector projections (for example, onto axes of the local Cartesian coordinate system) using the least square method (LSM). Primary estimates of velocity can be filtered at the secondary (after-coordinate) processing. Secondary velocity vector coordinates are outputted to users.

The present paper considers different methods of measuring velocity, they being different from each other by different tracking filters of primary and secondary processing and different EB. Primary filters operate at the same control frequency Fc as PLL (for instance, at Fc= 200 Hz), and LSM and secondary filters – at lower frequency FE < Fc (for example, at FE=100 Hz or FE=10 Hz). To shift from Fc to FE, some samples are rejected (intermediate samples are thrown). EB generates either primary estimates of instantaneous frequency or instantaneous phase of the input signal, or primary estimates of average input phase over control period Tc=Fc^-1. These primary estimates are fed to the filters of primary processing. At the outputs of these filters either secondary estimates of instantaneous frequency or estimates of averaged frequency and it’s derivative over period Tc are outputted which further are recalculated in estimates of instantaneous frequency. Based on thinned instantaneous phase estimates sometimes there are generated increments of these phase estimates over period TE = FE^-1. Primary estimates of either coordinates of the instantaneous velocity vector or averaged over period TE are fed to the input of secondary processing filters. In the first case, secondary estimates of instantaneous coordinates of the velocity vector are obtained at filter outputs at once. In the second case, at the filter outputs there are estimates of averaged velocities and accelerations over period TE which are further calculated in estimates of instantaneous velocity vector coordinates.

It has been shown that frequency estimation typically used in analog systems brings about a biased frequency (and hence, velocity) estimate when a receiver with digital PLLs has constant non-zero acceleration. Various algorithms of non-biased estimation have been also considered.

This paper is a continuation of the author’s previous study on methods of velocity measurements of navigation receivers and devoted to their comparative analysis.

Each method of measuring velocity has a few parameters. Let us fix all parameters except for one (main) and vary this parameter. The value of each varied parameter corresponds to some noise error of velocity measurements which can be characterized by standard deviation, or SD (cm/s). A dynamic model of GNSS receiver motion determines dynamic errors. Maximal dynamic error (MDE) (cm/s) is of interest in this case. This error depends on “maneuver phase”, i.e., a shift of the maneuver start time from the starting point of PLL control period and also the starting point of the secondary processing period. The maximal value of MDE is of interest in these shifts.

So, for each value of the varied parameter there is a pair of numbers: SD and MDE. Let us arrange these numbers in plane of the coordinate system: x-axis is MDE, and y-axis is SD. Connect nearest points and obtain a curve which is called an exchange diagram. Since SD and MDE vary within a wide range, the diagrams should be built in logarithmic scale, that is in dB relative to 1 cm/s. Let us call them logarithmic exchange diagrams (LED). Different LED were plotted for tough and soft dynamic scenarios for different methods of velocity measurements including the conventional one frequently discussed in the literature.

As a result of the analysis, a method of generating frequency estimates of the input signal and their further filtering using an after-satellite second order tracking filter, and a method based on quasi-optimal estimates of the input signal phase and further after-satellite filtration using the third order tracking filter have been recommended for tougher dynamic conditions. Under more favorable conditions in addition to the two above, a method of generating coordinate increments over one period with further after-coordinate filtration using the second order tracking filter, and a method of generating local coordinates with further aftercoordinate filtration using the third order tracking filter have been also recommended. In conclusion, a law of velocity estimate SD variation for one of the best recommended methods was investigated in the process of varying method parameters.

Nowadays, production control problems has been widely studied and a lot of valuable approaches have been implemented. Some work addresses the problem of tracking the uncertain demand in case of uncertain production speeds. The uncertainties are described by deterministic inequalities and the performance is analyzed in from of the worst-case scenario. First, simple mathematical models are introduced and the control problem is formulated. In continuous-time, the cumulative output of a manufacturing machine is the integral of the production speed over time. At the same time, the production speed is bounded from below and above, and hence the manufacturing process can be modeled as an integrator with saturated input. Since the cumulative demand (which is the reference signal to track) is a growing function of time, it is natural to consider control policies that involve integration of the mismatch between the current output and current demand. In the simplest consideration it results in models similar to a double integrator closed by saturated linear feedback with an extra input that models disturbances of a different nature. This model is analyzed and particular attention is devoted to the integrator windup phenomenon: lack of global stability of the system solutions that correspond to the same input signal.

The short-time dynamics of bacterial chromosomal loci is a mixture of subdiffusive and active motion, in the form of rapid relocations with near-ballistic dynamics. While previous work has shown that such rapid motions are ubiquitous, we still have little grasp on their physical nature, and no positive model is available that describes them. Here, we propose a minimal theoretical model for loci movements as a fractional Brownian motion subject to a constant but intermittent driving force, and compare simulations and analytical calculations to data from high-resolution dynamic tracking in *E. coli*. This analysis yields the characteristic time scales for intermittency. Finally, we discuss the possible shortcomings of this model, and show that an increase in the effective local noise felt by the chromosome associates to the active relocations.

The task of designing the control actions for a heavy water reactor under uncertainty changes its parameters considered in the key differential game. The possibility of representing nonlinear dynamics of the object in the form of a system with parameters depending on the state (State Dependent Coefficients) and quadratic functional qualities allow you to go from having to solve a scalar partial differential equation (the Hamilton-Jacobi-Bellman) to the Riccati equation with parameters depending on the state. Feasible solution obtained by applying the min-max method. The results of mathematical modeling system in the shutdown of a nuclear reactor.

The idea of this paper appeared after the workshop on ‘Human Rights on the Internet: Legal Frames and Technological Implications’, organized by the Higher School of Economics on 7th Meeting of the Internet Governance Forum in Baku (Azerbaijan) on November 2012. This paper shows importance of the trilateral Internet Governance model in context of the example of governmental insufficiency to control the Internet.

Internet technologists contribute to the practical realization of human rights. First of all, they can improve effectiveness of existing institutions. Unfortunately in the same time Internet technologies give rise to new mechanisms of human rights violations. So we need to create new means, new technologies for human rights protection. We need new technological means, identification and classification of violations, based on predictive analytics. But to improve the situation, we should improve the existing means, and build new models of communication. Perhaps such models could be based on the concept of Web 2.0 and Web 3.0.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.