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Abstract: The classical scheme of independent Bernoulli trials, starting with Jacob Bernoulli’s work, has been one of the most popular topics in probability theory for more than three centuries. It is ideally suited for setting and solving various practical problems. A variety of results have been obtained related to modifications of this scheme. However, new situations and problems emerge that require further advancement in the study of various random variables associated to some extent with independent Bernoulli trials. Solutions to some of these problems are presented here. PubDate: 2022-06-01

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Abstract: Boundary conditions for fluid-dynamic parameters of a strongly nonequilibrium single-component rarefied gas flow in the slip regime are obtained using the methods of kinetic theory. The gas flow is described in the frame of the state-to-state approach assuming vibrational energy exchange as the slow relaxation process. The set of governing equations, including conservation equations, coupled with additional relaxation equations for vibrational state populations is presented. The gas-solid surface interaction is considered on the basis of the specular-diffusive model, and possible vibrational deactivation/excitation processes on the wall are taken into account. The obtained boundary conditions depend on the accommodation and deactivation coefficients along with the transport coefficients, such as the multicomponent vibrational energy diffusion and thermal diffusion coefficients, the thermal conductivity, the bulk and shear viscosity coefficients and the relaxation pressure. The dependence of boundary conditions on the normal mean stress has been obtained for the first time. In the particular case of the gas without internal degrees of freedom, the slip velocity and the temperature jump can be reduced to the well-known in the literature expressions. Implementation of the state-specific boundary conditions should not cause additional computational costs in numerical simulations of viscous flows in the state-to-state approach since the slip/jump equations depend on the transport coefficients that have to be evaluated regardless of the boundary conditions used in the code. PubDate: 2022-06-01

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Abstract: A generalization of Newton’s formula for shear stress in a fluid is carried out by giving it a power-law form and the corresponding rheological relation is written in tensor form. Depending on the exponent in this rheological ratio, one can come to a description of either a laminar or turbulent flow regime. In the latter case, there is a system of differential equations with the no-slip boundary condition. The proposed set of equations for turbulent fluid motion can be useful, at least, for obtaining preliminary, estimated characteristics of turbulent flow before starting numerical modeling using modern differential turbulence models. For some values of the exponent, this system can be used to describe the behavior of power-law fluids as well as fluids with small additives of polymers in the manifestation of the Toms effect. PubDate: 2022-06-01

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Abstract: In this paper, various types of behavior of reaction forces and Lagrange multipliers for the motion of mechanical systems with a configuration space singularity are studied. The motion of a one-dimensional double pendulum (or a singular pendulum) with a transversal singular point or a first-order tangency singular point is considered. Depending on the properties of the curve along which the free vertex of the double pendulum moves, the configuration space of the mechanical system is two smooth curves on the torus without common points, two transversely intersecting smooth curves, or two curves with first-order tangency. The reaction forces on a two-dimensional torus are found to study the pendulum motion. The expressions for the reaction forces in angular coordinates are obtained analytically. In the case of a transverse intersection, it is proven that the reaction forces must be zero at the singular point. In the case of a first-order tangency singularity, the reaction forces are nonzero at the singular point. The Lagrange multiplier, which depends on the motion along the ellipse, becomes unlimited near the singular point. Two mechanisms with a different type of singular points in the configuration space are described: a nonsmooth singular pendulum and a broken singular pendulum. There are no smooth regular curves passing through a singular point in the configuration spaces of these mechanical systems. For a nonsmooth singular pendulum, the Lagrange multiplier, which depends on the motion along the ellipse, becomes undefined while passing the singular point. For a broken singular pendulum, the Lagrange multiplier makes a jump from a finite value to an infinite one. PubDate: 2022-06-01

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Abstract: The paper discusses the issues of optimal damping of oscillations of a spatial double pendulum with noncollinear joint axes. Both simple passive damping associated with the influence of viscous friction and combined passive and active damping are considered as options, and active influences are formed according to the principle of collinear control. The analytical solution of the system motion equations is given for both cases in the framework of the linear model, and it clearly demonstrates the motion damping for eigenmodes of the original conservative model. The optimization criteria characterizing the efficiency of the damping processes of system motions are considered. It is noted that, in order to obtain the strongest damping modes, either the degree of stability should be maximized or the integral energy-temporal criterion should be minimized. In addition, the main advantages and disadvantages of these optimization criteria are discussed. This article is the basis for further research, which will be presented as a separate article “Optimization of Oscillation Damping Modes of a Spatial Double Pendulum: II. Solution of the Problem and Analysis of the Results.” PubDate: 2022-06-01

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Abstract: By a Poisson stochastic index process (PSI-process), is meant a continuous-time random process obtained by the discrete-time randomization of a random sequence. The case when this randomization is generated by a doubly stochastic Poisson process, i.e., a Poisson process with a random intensity, is considered. Under the condition of existence of the second moment, stationary PSI-processes have a covariance coinciding with the Laplace transform of a random intensity. In this paper, distributions for extremes of one PSI-process that are expressed in terms of the Laplace transform of a random intensity are obtained. The second problem that is solved here is the convergence of the maximum Gaussian limit for normalized sums of independent identically distributed stationary PSI-processes. Necessary and sufficient conditions imposed on the random intensity for a suitably centered and normalized maximum of this Gaussian limit to converge in distribution to the double exponential law are found. For this purpose, we essentially use a Tauberian theorem in Feller’s form and the results of the monograph by Leadbetter et al. (1983), “Extremes and Relative Properties of Random Sequences and Processes.” PubDate: 2022-06-01

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Abstract: This article is the first part of a work whose main result is the following statement: if, for functions \({{\gamma }_{1}} \in {{L}^{{{{p}_{1}}}}}({{\mathbb{R}}^{n}})\) , …, \({{\gamma }_{m}} \in {{L}^{{{{p}_{m}}}}}({{\mathbb{R}}^{n}})\) , where m \( \geqslant \) 2 and the numbers p1, …, pm ∈ (1, +∞] are such that \(\frac{1}{{{{p}_{1}}}}\) + … + \(\frac{1}{{{{p}_{m}}}}\) < 1, a nonresonant condition is met (the concept introduced by the author for functions from \({{L}^{p}}({{\mathbb{R}}^{n}})\) ), p ∈ (1, +∞]) then, \({{\sup }_{{a,b \in {{\mathbb{R}}^{n}}}}}\left {\int_{[a,b]} {\prod\nolimits_{k = 1}^m {[{{\gamma }_{k}}(\tau ) + \Delta {{\gamma }_{k}}(\tau )]d\tau } } } \right \) \(\leqslant \) \(C\prod\nolimits_{k = 1}^m {{{{\left\ {{{\gamma }_{k}} + \Delta {{\gamma }_{k}}} \right\ }}_{{L_{{{{h}_{k}}}}^{{{{p}_{k}}}}({{\mathbb{R}}^{n}})}}}} \) , where [a, b] is an n-dimensional parallelepiped, the constant C > 0 does not depend on functions Δγk ∈ \(L_{{{{h}_{k}}}}^{{{{p}_{k}}}}({{\mathbb{R}}^{n}})\) , while \(L_{{{{h}_{k}}}}^{{{{p}_{k}}}}({{\mathbb{R}}^{n}})\) ⊂ \(L_{{}}^{{{{p}_{k}}}}({{\mathbb{R}}^{n}})\) , 1 \(\leqslant \) k \(\leqslant \) m are specially constructed normalized spaces. In this article, for any spaces \(L_{{}}^{{{{p}_{0}}}}({{\mathbb{R}}^{n}})\) , \(L_{{}}^{p}({{\mathbb{R}}^{n}})\) , p0, p ∈ (1, +∞] and any function γ ∈ \(L_{{}}^{{{{p}_{0}}}}({{\mathbb{R}}^{n}})\) , the concept of the set of resonant points of the function γ with respect to \(L_{{}}^{p}({{\mathbb{R}}^{n}})\) is introduced. This set is a subset of { \({{\mathbb{R}}^{1}}\) ∪ {∞}}n and for any trigonometric polynomial of n variables with respect to any \(L_{{}}^{p}({{\mathbb{R}}^{n}})\) represents the spectrum of the considered polynomial. Theorems are provided on the representation of each function γ ∈ \(L_{{}}^{{{{p}_{0}}}}({{\mathbb{R}}^{n}})\) such that its resonant set is nonempty by a sum of two functions such that the first of them belongs to PubDate: 2022-06-01

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Abstract: In the present work, we consider discretization of continuous distributions and study relations between continuous record values, weak record values and discrete record values obtained from continuous record values by discretization. We first compare the numbers of continuous record values registered in real intervals and the numbers of discrete record values and weak record values located on the sets of non-negative integers. Further in the paper, we introduce the so-called strong continuous record values that hold out after discretization and discuss their distributional properties. We also analyze what happens with record values after discretization of the standard exponential distribution. Finally, we present a simulation experiment, which supports the theoretical results of the paper. Ref. [7] items. PubDate: 2022-06-01

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Abstract: The paper investigates the nonlinear modal interaction of longitudinal and flexural vibrations of a beam resonator under periodic thermal loading. The mode of parametric oscillations is investigated under conditions of internal multiple resonance between some flexural and longitudinal forms of free oscillations of the resonator. The possibility of the longitudinal-bending mode generation in the system has been found, the frequency of the slow envelope of which essentially depends on the parameter of the internal frequency detuning, which is directly related to the magnitude of external disturbances subject to high-precision measurement. PubDate: 2022-06-01

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Abstract: The notion of irregularity of formal modules in one-dimensional local fields is considered. A connection is obtained between the irregularity of all unramified extensions M/L and the ramification index eL/K for a sufficiently wide class of formal groups. The notion of s-irregularity for a natural s is introduced (generalization of the notion of irregularity to the case of roots [πs]), and similar criteria for irregularity are proven for it for the case of generalized and relative formal Lubin–Tate modules. PubDate: 2022-06-01

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Abstract: The paper considers a model problem of one-dimensional small transverse vibrations of a hinged web moving at a constant speed. The oscillatory process is described by a linear differential equation of the fourth order with constant coefficients. The considered model takes into account the Coriolis force, which produces a term with a mixed derivative in the differential equation. This effect makes it impossible to use the classical method of separating variables. However, families of exact solutions for the oscillation equation in the form of a progressing wave have been constructed. For the initial boundary value problem, it was established that the solution can be constructed in the form of a Fourier series according to the system of eigenfunctions of the auxiliary problem on beam vibrations. For the considered oscillatory process, the law of conservation of energy is established and the solution to the initial boundary value problem is shown to be unique. PubDate: 2022-06-01

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Abstract: The motion problem for a heavy rigid body on a perfectly rough horizontal plane, which is a classical problem of the nonholonomic system dynamics, is considered. The effect from the loss of stability of a body’s permanent rotation at a certain critical value of its angular velocity is discussed. It is proven that this effect is accompanied by the occurrence of periodic motions of the body with a frequency close to the critical value; that is, the Hopf bifurcation takes place. It is proven by means of the direct calculation of the first Lyapunov coefficient that the corresponding periodic motions are unstable. PubDate: 2022-06-01

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Abstract: The general theoretical approach to the asymptotic extraction of the signal series from the additively perturbed signal with the help of singular spectrum analysis (SSA) was already outlined in Nekrutkin (2010, Stat. Its Interface 3, 297–319). In this paper, the example of such an analysis applied to the linear signal and the additive sinusoidal noise is considered. It is proven that, in this case, the so-called reconstruction errors ri(N) of SSA uniformly tend to zero as the series length N tends to infinity. More precisely, we demonstrate that maxi ri(N) = O(N−1) as N → ∞ if the “window length” L equals (N + 1)/2. It is important to mention that a completely different result is valid for the increasing exponential signal and the same noise. As is proven in Ivanova and Nekrutkin (2019, Stat. Its Interface 12(1), 49–59), no finite number of last terms of the error series tends to any finite or infinite values in this case. PubDate: 2022-06-01

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Abstract: The need to work with unbounded operators is a long-standing problem that arises when constructing the mathematical apparatus of quantum mechanics. Since the space of nuclear operators is preconjugate for the algebra of all bounded operators, we can consider nuclear operators as the states of a quantum system and the bounded operators as observables. In this case, taking the trace for the product of a nuclear operator (a quantum state) and a bounded operator (a quantum observable) yields the average value of an observable in the fixed state of the quantum system. The existence of such an average for unbounded operators is not guaranteed. If we want to define a space of observables that includes such naturally occurring unbounded operators as the position and momentum, for which average values are always determined, we should consider a space of states smaller than all nuclear operators. Recently, this approach has been accurately implemented mathematically in the Hilbert space \(\mathcal{H}\) = \({{L}^{2}}({{\mathbb{R}}^{N}})\) . The so-called space of Schwarz operators equipped with a system of seminorms and being a Frechet space was chosen as the space of states. Schwarz operators are integral operators whose kernels are functions belonging to the ordinary Schwarz space. The space dual to the space of Schwarz operators should be considered as the space of quantum observables, and it does include such standard observables as polynomials of the products of the position and momentum operators. In this work, we transfer this approach to the symmetric Fock space \(\mathcal{H}\) = \(F(\mathfrak{H})\) over an infinite dimensional separable Hilbert space \(\mathfrak{H}\) . We introduce the space of Schwartz operators in \(F(\mathfrak{H})\) and find out which standard operators of quantum white noise belong to the space dual to the space of Schwartz operators. PubDate: 2022-06-01

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Abstract: We consider integral equations of the first kind, which are associated with the class of ill-posed problems. This class also includes the problem of inversing the integral Laplace transform, which is used to solve a wide class of mathematical problems. Integral equations are reduced to ill-conditioned systems of linear algebraic equations (in which unknowns represent the coefficients of expansion in a series in shifted Legendre polynomials of some function that is simply expressed in terms of the sought original; this function is found as a solution of a certain finite moment problem in a Hilbert space). To obtain a reliable solution of the system, regularization methods are used. The general strategy is to use the Tikhonov stabilizer or its modifications. A specific type of stabilizer in the regularization method is indicated; this type is focused on an a priori low degree of smoothness of the desired original. The results of numerical experiments are presented; they confirm the efficiency of the proposed inversion algorithm. PubDate: 2022-03-01

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Abstract: The system \(\dot {x}\) = M(·)x + enu, u = sTx, where M(·) ∈ \({{R}^{{n \times n}}}\) , s ∈ Rn, and the pair (M(·), en) is completely controllable, is considered. The elements of the matrix M(·) are nonanticipating functionals of arbitrary nature. The object matrix is assumed to have the form M(·) = A(·) + D(·), where A(·) is a generalized Frobenius matrix and D(·) is a disturbance matrix. The quadratic Lyapunov function V(x) with a constant matrix of special form and the number α > 0, which is an estimate for \(\dot {V}\) under the condition D(·) = 0, are brought into consideration. For an arbitrary α > 0, the vector s and the estimate of the norm of the matrix D(·) are determined in such a way that the system under consideration becomes globally exponentially stable. PubDate: 2022-03-01

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Abstract: The free descent of a finned body in a resisting medium is studied. Fins are installed in such a way that there exists the regime of translational descent with constant speed. Earlier, the descent of a heavy body in the autorotation mode was studied. It arises when the pitch angles of all fins are equal to each other. For descent with autorotation, the general properties and tendencies of motion are described depending on the initial conditions and parameters. In the current paper, the descent of a body is discussed for the case when fins have equal absolute values of the pitch angles, but the signs of these angles alternate. The asymptotic stability of translational descent with constant speed is studied for such fin orientation. Domains of stability are constructed in the plane of the following parameters: pitch angle of the fins and displacement of the center of mass. Fins represented either by circular or rectangular blades are discussed. The domains of stability are compared with that for the autorotation mode. Trajectories of the center of mass are constructed. Varied types of such trajectories are obtained when the parameters of the model correspond to the case of unstable translation descent with constant speed. PubDate: 2022-03-01

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Abstract: As is known, regression-analysis tools are widely used in machine-learning problems to establish the relationship between the observed variables and to store information in a compact manner. Most often, a regression function is described by a linear combination of some given functions fj(X), j = 1, …, m, X ∈ D ⊂ Rs. If the observed data contain a random error, then the regression function reconstructed from the observations contains a random error and a systematic error depending on the selected functions fj. This article indicates the possibility of an optimal, in the sense of a given functional metric, choice of fj, if it is known that the true dependence obeys some functional equation. In some cases (a regular grid, s ≤ 2), close results can be obtained using a technique for random-process analysis. The numerical examples given in this work illustrate significantly broader opportunities for the assumed approach to regression problems. PubDate: 2022-03-01

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Abstract: The paper considers the quadratic programming problem with a strictly convex separable objective function, a single linear constraint, and two-sided constraints on variables. This problem is commonly called the convex separable quadratic knapsack problem (CQKnP). We are interested in an algorithm for solving the CQKnP with linear complexity. The works devoted to this topic contain inaccuracies in the description of algorithms and inefficient implementations. In this work, the existing difficulties are overcome. PubDate: 2022-03-01