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Vestnik St. Petersburg University: Mathematics
Journal Prestige (SJR): 0.219 Number of Followers: 0 Hybrid journal (It can contain Open Access articles) ISSN (Print) 1934-7855 - ISSN (Online) 1063-4541 Published by Springer-Verlag [2653 journals] |
- Synthesis of Stabilization Control on Outputs for a Class of Continuous
and Pulse-Modulated Undefined Systems- Abstract: Consider system $$\left\{ {\begin{array}{*{20}{c}} {{{\dot x}_1} = {\varphi _1}(.) + {\rho _1}{x_{l + 1}},} \\ {{{\dot x}_m} = {\varphi _m}(.) + {\rho _m}{x_n},} \\ {{{\dot x}_{m + 1}} = {\varphi _{m + 1}}(.) + {\mu _1},} \\ {{{\dot x}_n} = {\varphi _n}(.) + {\mu _1},} \end{array}} \right.$$ where x1, …, and xn is the state of the system, u1, …, and ul are controls, n/l is not an integer, and l ≥ 2. It is supposed that only outputs x1, …, and xl are measurable, (l > n) and ϕi(·) are non-anticipating arbitrary functionals, and 0 < ρ–≤ ρi (t, x1, …, and xl) ≤ ρ+. Using the backstepping method, we construct the square Lyapunov function and stabilize the control for the global exponential stability of the closed loop system. The stabilization by means of synchronous modulators with a sufficiently high impulsion frequency is considered as well.
PubDate: 2018-10-01
DOI: 10.3103/S1063454118040143
- Abstract: Consider system $$\left\{ {\begin{array}{*{20}{c}} {{{\dot x}_1} = {\varphi _1}(.) + {\rho _1}{x_{l + 1}},} \\ {{{\dot x}_m} = {\varphi _m}(.) + {\rho _m}{x_n},} \\ {{{\dot x}_{m + 1}} = {\varphi _{m + 1}}(.) + {\mu _1},} \\ {{{\dot x}_n} = {\varphi _n}(.) + {\mu _1},} \end{array}} \right.$$ where x1, …, and xn is the state of the system, u1, …, and ul are controls, n/l is not an integer, and l ≥ 2. It is supposed that only outputs x1, …, and xl are measurable, (l > n) and ϕi(·) are non-anticipating arbitrary functionals, and 0 < ρ–≤ ρi (t, x1, …, and xl) ≤ ρ+. Using the backstepping method, we construct the square Lyapunov function and stabilize the control for the global exponential stability of the closed loop system. The stabilization by means of synchronous modulators with a sufficiently high impulsion frequency is considered as well.
- On One Property of Bounded Complexes of Discrete $$\mathbb{F}_p[\pi]$$
-modules- Abstract: The aim of this paper is to prove the following assertion: let π be a profinite group and K* be a bounded complex of discret \(\mathbb{F}_p[\pi]\) -modules. Suppose that Hi(K*) are finite Abelian groups. Then, there exists a quasi-isomorphism L* → K*, where L* is a bounded complex of discrete \(\mathbb{F}_p[\pi]\) -modules such that all Li are finite Abelian groups. This is an analog for discrete \(\mathbb{F}_p[\pi]\) -modules of the wellknown lemma on bounded complexes of A-modules (e.g., concentrated in nonnegative degrees), where A is a Noetherian ring, which states that any such complex is quasi-isomorphic to a complex of finitely generated A-modules, that are free with a possible exception of the module lying in degree 0. This lemma plays a key role in the proof of the base-change theorem for cohomology of coherent sheaves on Noetherian schemes, which, in turn, can be used to prove the Grothendieck theorem on the behavior of dimensions of cohomology groups of a family of vector bundles over a flat family of varieties.
PubDate: 2018-10-01
DOI: 10.3103/S1063454118040131
- Abstract: The aim of this paper is to prove the following assertion: let π be a profinite group and K* be a bounded complex of discret \(\mathbb{F}_p[\pi]\) -modules. Suppose that Hi(K*) are finite Abelian groups. Then, there exists a quasi-isomorphism L* → K*, where L* is a bounded complex of discrete \(\mathbb{F}_p[\pi]\) -modules such that all Li are finite Abelian groups. This is an analog for discrete \(\mathbb{F}_p[\pi]\) -modules of the wellknown lemma on bounded complexes of A-modules (e.g., concentrated in nonnegative degrees), where A is a Noetherian ring, which states that any such complex is quasi-isomorphic to a complex of finitely generated A-modules, that are free with a possible exception of the module lying in degree 0. This lemma plays a key role in the proof of the base-change theorem for cohomology of coherent sheaves on Noetherian schemes, which, in turn, can be used to prove the Grothendieck theorem on the behavior of dimensions of cohomology groups of a family of vector bundles over a flat family of varieties.
- Inverse Shadowing in Actions of a Baumslag–Solitar Group
- Abstract: In parallel with the shadowing theory (which is now very well-developed), the theory of inverse shadowing has been advanced. The main difference between the two theories is that the shadowing property means that we can find an exact trajectory near an approximate one while inverse shadowing means that, given a family of approximate trajectories, we can find a member of this family that is close to any chosen exact trajectory. We generalize the property of inverse shadowing for group actions and prove the absence of this property for some linear actions of the Baumslag–Solitar group, which is often considered as a source of counterexamples.
PubDate: 2018-10-01
DOI: 10.3103/S1063454118040064
- Abstract: In parallel with the shadowing theory (which is now very well-developed), the theory of inverse shadowing has been advanced. The main difference between the two theories is that the shadowing property means that we can find an exact trajectory near an approximate one while inverse shadowing means that, given a family of approximate trajectories, we can find a member of this family that is close to any chosen exact trajectory. We generalize the property of inverse shadowing for group actions and prove the absence of this property for some linear actions of the Baumslag–Solitar group, which is often considered as a source of counterexamples.
- Honda Formal Module in an Unramified p -Extension of a Local Field as a
Galois Module- Abstract: For a fixed rational prime number p, consider a chain of finite extensions of fields K0/ℚp, K/K0, L/K, and M/L, where K/K0 is an unramified extension and M/L is Galois extension with Galois group G. Suppose that a one-dimensional Honda formal group F over the ring \(\mathcal{O}_K\) relative to the extension K/K0 and a uniformizing element π ∈ K0 is given. This paper studies the structure of \(F(\mathfrak{m}_M)\) as an \(\mathcal{O}_{K_0}\) [G]-module for an unramified p-extension M/L provided that \(W_F\cap{F({\frak{m}}_L)}=W_F\cap{F({\frak{m}}_M)}=W_F^s\) for some s ≥ 1, where W F s is the πs-torsion and WF = ∪n=1∞WFn is the complete π-torsion of a fixed algebraic closure Kalg of the field K.
PubDate: 2018-10-01
DOI: 10.3103/S1063454118040027
- Abstract: For a fixed rational prime number p, consider a chain of finite extensions of fields K0/ℚp, K/K0, L/K, and M/L, where K/K0 is an unramified extension and M/L is Galois extension with Galois group G. Suppose that a one-dimensional Honda formal group F over the ring \(\mathcal{O}_K\) relative to the extension K/K0 and a uniformizing element π ∈ K0 is given. This paper studies the structure of \(F(\mathfrak{m}_M)\) as an \(\mathcal{O}_{K_0}\) [G]-module for an unramified p-extension M/L provided that \(W_F\cap{F({\frak{m}}_L)}=W_F\cap{F({\frak{m}}_M)}=W_F^s\) for some s ≥ 1, where W F s is the πs-torsion and WF = ∪n=1∞WFn is the complete π-torsion of a fixed algebraic closure Kalg of the field K.
- Existence of Liouvillian Solutions in the Problem of Rolling Motion of a
Dynamically Symmetric Ball on a Perfectly Rough Sphere- Abstract: The problem of rolling without sliding of a rotationally symmetric rigid body on a motionless sphere is considered. The rolling body is assumed to be subjected to forces whose resultant force is applied to the center of mass G of the body, directed to center O of the sphere, and depends only on the distance between G and O. In this case, the process of solving this problem is reduced to integrating the second-order linear differential equation with respect to the projection of the angular velocity of the body onto its axis of dynamic symmetry. Using the Kovacic algorithm, we search for Liouvillian solutions of the corresponding second-order linear differential equation. We prove that all solutions of this equation are Liouvillian in the case when the rolling rigid body is a nonhomogeneous dynamically symmetric ball. The paper is organized as follows. In the first paragraph, we briefly discuss the statement of the general problem of motion of a rotationally symmetric rigid body on a perfectly rough sphere. We prove that this problem is reduced to solving the second-order linear differential equation. In the second paragraph, we find Liouvillian solutions to this equation for the case when the rolling rigid body is a dynamically symmetric ball.
PubDate: 2018-10-01
DOI: 10.3103/S1063454118040106
- Abstract: The problem of rolling without sliding of a rotationally symmetric rigid body on a motionless sphere is considered. The rolling body is assumed to be subjected to forces whose resultant force is applied to the center of mass G of the body, directed to center O of the sphere, and depends only on the distance between G and O. In this case, the process of solving this problem is reduced to integrating the second-order linear differential equation with respect to the projection of the angular velocity of the body onto its axis of dynamic symmetry. Using the Kovacic algorithm, we search for Liouvillian solutions of the corresponding second-order linear differential equation. We prove that all solutions of this equation are Liouvillian in the case when the rolling rigid body is a nonhomogeneous dynamically symmetric ball. The paper is organized as follows. In the first paragraph, we briefly discuss the statement of the general problem of motion of a rotationally symmetric rigid body on a perfectly rough sphere. We prove that this problem is reduced to solving the second-order linear differential equation. In the second paragraph, we find Liouvillian solutions to this equation for the case when the rolling rigid body is a dynamically symmetric ball.
- The Problem of Selfish Parking
- Abstract: One of the models of discrete analog of the Rényi problem known as the “parking problem” has been considered. Let n and i be integers, n ≥ 0, and 0 ≤ i ≤ n–1. Open interval (i, i + 1), where i is a random variable taking values 0, 1, 2, …, and n–1 for all n ≥ 2 with equal probability, is placed on interval [0, n]. If n < 2, we say that the interval cannot be placed. After placing the first interval, two free intervals [0, i] and [i + 1, n] are formed, which are filled with intervals of unit length according to the same rule, independently of each other, etc. When the filling of [0, n] with unit intervals is completed, the distance between any two neighboring intervals does not exceed 1. Let Xn be the number of placed intervals. This paper analyzes the asymptotic behavior of moments of random variable Xn. Unlike the classical case, exact expressions for the first moments can be found.
PubDate: 2018-10-01
DOI: 10.3103/S1063454118040039
- Abstract: One of the models of discrete analog of the Rényi problem known as the “parking problem” has been considered. Let n and i be integers, n ≥ 0, and 0 ≤ i ≤ n–1. Open interval (i, i + 1), where i is a random variable taking values 0, 1, 2, …, and n–1 for all n ≥ 2 with equal probability, is placed on interval [0, n]. If n < 2, we say that the interval cannot be placed. After placing the first interval, two free intervals [0, i] and [i + 1, n] are formed, which are filled with intervals of unit length according to the same rule, independently of each other, etc. When the filling of [0, n] with unit intervals is completed, the distance between any two neighboring intervals does not exceed 1. Let Xn be the number of placed intervals. This paper analyzes the asymptotic behavior of moments of random variable Xn. Unlike the classical case, exact expressions for the first moments can be found.
- Relation of the Böttcher Equation with the Parametrized Poisson
Integral- Abstract: The Böttcher functional equation and one of its real generalizations are considered. It is shown that, in some situations, after finding a solution of the generalized equation, other solutions can also be obtained. For example, a three-parameter family of real functional equations for a function of two arguments is described, for which solutions are found. The generalization described has wide application. Many quantities after an appropriately introduced parameterization satisfy the generalized Böttcher equation as functions of parameters. As an illustration, two-parametric families generated by the determinant of a linear combination of second-order matrices are presented. It is shown that the parameterized Poisson integral as a function of its parameters satisfies the generalized Böttcher equation. This made it possible to calculate the Poisson integral and the Euler integral in a new way. In addition, the calculation of the Poisson integral by the method of integral sums is described.
PubDate: 2018-10-01
DOI: 10.3103/S106345411804009X
- Abstract: The Böttcher functional equation and one of its real generalizations are considered. It is shown that, in some situations, after finding a solution of the generalized equation, other solutions can also be obtained. For example, a three-parameter family of real functional equations for a function of two arguments is described, for which solutions are found. The generalization described has wide application. Many quantities after an appropriately introduced parameterization satisfy the generalized Böttcher equation as functions of parameters. As an illustration, two-parametric families generated by the determinant of a linear combination of second-order matrices are presented. It is shown that the parameterized Poisson integral as a function of its parameters satisfies the generalized Böttcher equation. This made it possible to calculate the Poisson integral and the Euler integral in a new way. In addition, the calculation of the Poisson integral by the method of integral sums is described.
- Examples of the Best Piecewise Linear Approximation with Free Nodes
- Abstract: The monograph published by E.Ya. Remez in 1957 addressed numerical methods with Chebyshev approximations. Particularly, the problem of the best uniform approximation of a function that is convex on an interval with continuous piecewise linear functions with free nodes was considered. In 1975, A.M. Vershik, V.N. Malozemov, and A.B. Pevnyi developed a general approach for constructing the best piecewise polynomial approximations with free nodes. The notion of partition with equal deviations was introduced, and it was found that such partition exists and generates the best piecewise polynomial approximation. In addition, a numerical method for constructing a partition with equal deviations was proposed. This paper gives three examples to describe the general approach to solving the problem of the best piecewise linear approximation with free nodes. In the case of an arbitrary continuous function, its best piecewise linear approximation in general is not continuous. It is continuous when approximating strictly convex and strictly concave functions.
PubDate: 2018-10-01
DOI: 10.3103/S1063454118040118
- Abstract: The monograph published by E.Ya. Remez in 1957 addressed numerical methods with Chebyshev approximations. Particularly, the problem of the best uniform approximation of a function that is convex on an interval with continuous piecewise linear functions with free nodes was considered. In 1975, A.M. Vershik, V.N. Malozemov, and A.B. Pevnyi developed a general approach for constructing the best piecewise polynomial approximations with free nodes. The notion of partition with equal deviations was introduced, and it was found that such partition exists and generates the best piecewise polynomial approximation. In addition, a numerical method for constructing a partition with equal deviations was proposed. This paper gives three examples to describe the general approach to solving the problem of the best piecewise linear approximation with free nodes. In the case of an arbitrary continuous function, its best piecewise linear approximation in general is not continuous. It is continuous when approximating strictly convex and strictly concave functions.
- Energy Dissipation during Vibrations of Heterogeneous Composite
Structures: 2. Method of Solution- Abstract: This paper describes the method of numerical solution of decaying vibration equations for heterogeneous composite structures. The system of algebraic equations is generated by applying the Ritz method with Legendre polynomials as coordinate functions. First, real solutions are found. To find complex natural frequencies of the system, the obtained real natural frequencies are taken as initial values, and then, by means of the third-order iteration method, complex natural frequencies are calculated. The paper discusses the convergence of numerical solution of the differential equations describing the motion of layered heterogeneous structures, obtained for an unsupported rectangular two-layered plate. The bearing layer of the plate is made of unidirectional CRP, its elastic and dissipation properties within the investigated band of frequencies and temperatures are independent of vibration frequency. The bearing layer has one of its outer surfaces covered with a layer of “stiff” isotropic viscoelastic polymer characterized by a temperature-frequency relationship for the real part of complex Young’s modulus and loss factor. Validation of the mathematical model and numerical solution performed through comparison of calculation results for natural frequencies and loss factor versus test data (for two composition variants of a two-layered unsupported beam) has shown good correlation.
PubDate: 2018-10-01
DOI: 10.3103/S106345411804012X
- Abstract: This paper describes the method of numerical solution of decaying vibration equations for heterogeneous composite structures. The system of algebraic equations is generated by applying the Ritz method with Legendre polynomials as coordinate functions. First, real solutions are found. To find complex natural frequencies of the system, the obtained real natural frequencies are taken as initial values, and then, by means of the third-order iteration method, complex natural frequencies are calculated. The paper discusses the convergence of numerical solution of the differential equations describing the motion of layered heterogeneous structures, obtained for an unsupported rectangular two-layered plate. The bearing layer of the plate is made of unidirectional CRP, its elastic and dissipation properties within the investigated band of frequencies and temperatures are independent of vibration frequency. The bearing layer has one of its outer surfaces covered with a layer of “stiff” isotropic viscoelastic polymer characterized by a temperature-frequency relationship for the real part of complex Young’s modulus and loss factor. Validation of the mathematical model and numerical solution performed through comparison of calculation results for natural frequencies and loss factor versus test data (for two composition variants of a two-layered unsupported beam) has shown good correlation.
- A Numerical Method for Finding the Optimal Solution of a Differential
Inclusion- Abstract: In the paper, we study a differential inclusion with a given continuous convex multivalued mapping. For a prescribed finite time interval, it is required to construct a solution to the differential inclusion, which satisfies the prescribed initial and final conditions and minimizes the integral functional. By means of support functions, the original problem is reduced to minimizing some functional in the space of partially continuous functions. When the support function of the multivalued mapping is continuously differentiable with respect to the phase variables, this functional is Gateaux differentiable. In the study, the Gateaux gradient is determined and the necessary conditions for the minimum of the functional are obtained. Based on these conditions, the method of steepest descent is applied to the original problem. The numerical examples illustrate the implementation of the constructed algorithm.
PubDate: 2018-10-01
DOI: 10.3103/S1063454118040076
- Abstract: In the paper, we study a differential inclusion with a given continuous convex multivalued mapping. For a prescribed finite time interval, it is required to construct a solution to the differential inclusion, which satisfies the prescribed initial and final conditions and minimizes the integral functional. By means of support functions, the original problem is reduced to minimizing some functional in the space of partially continuous functions. When the support function of the multivalued mapping is continuously differentiable with respect to the phase variables, this functional is Gateaux differentiable. In the study, the Gateaux gradient is determined and the necessary conditions for the minimum of the functional are obtained. Based on these conditions, the method of steepest descent is applied to the original problem. The numerical examples illustrate the implementation of the constructed algorithm.
- An Estimate for the Number of Periodical Trajectories of the Given Period
for Mapping of an Interval, Lucas Numbers, and Necklaces- Abstract: In 1964, A.N. Sharkovskii published an article in which he introduced a special ordering on the set of positive integers. This ordering has the property that if p ◃ q and a mapping of a closed bounded interval into itself has a point of period p, then it has a point of period q. The least number with respect to this ordering is 3. Thus, if a mapping has a point of period 3, then it has points of any periods. In 1975, the latter result was rediscovered by Li and Yorke, who published the paper “Period Three Implies Chaos.” In the present paper, an exact lower bound for the number of trajectories of a given period for a mapping of a closed bounded interval into itself having a point of period 3 is given. The key point of the reasoning consisted in solution of a combinatorial problem the answer to which is expressed in terms of Lucas numbers. As a consequence, an explicit formula for the number of necklaces of a special type is obtained. We also consider a piecewise linear unimodular mapping of [0, 1] into itself for which it is possible to find points of an arbitrary given period.
PubDate: 2018-10-01
DOI: 10.3103/S1063454118040088
- Abstract: In 1964, A.N. Sharkovskii published an article in which he introduced a special ordering on the set of positive integers. This ordering has the property that if p ◃ q and a mapping of a closed bounded interval into itself has a point of period p, then it has a point of period q. The least number with respect to this ordering is 3. Thus, if a mapping has a point of period 3, then it has points of any periods. In 1975, the latter result was rediscovered by Li and Yorke, who published the paper “Period Three Implies Chaos.” In the present paper, an exact lower bound for the number of trajectories of a given period for a mapping of a closed bounded interval into itself having a point of period 3 is given. The key point of the reasoning consisted in solution of a combinatorial problem the answer to which is expressed in terms of Lucas numbers. As a consequence, an explicit formula for the number of necklaces of a special type is obtained. We also consider a piecewise linear unimodular mapping of [0, 1] into itself for which it is possible to find points of an arbitrary given period.
- Two-Dimensional Homogeneous Cubic Systems: Classifications and Normal
Forms – V- Abstract: The present article is the fifth in a cycle of papers dedicated to two-dimensional homogeneous cubic systems. It considers a case when the homogeneous polynomial vector in the right-hand part of the system has a linear common factor. A set of such systems is divided into classes of linear equivalence, wherein the simplest system being a third-order normal form is distinguished based on properly introduced principles. Such a form is defined by the matrix of its right-hand part coefficients, which is called the canonical form (CF). Each CF has its own arrangement of non-zero elements, their specific normalization, and canonical set of permissible values for the unnormalized elements, which relates the CF to the selected class of equivalence. In addition to classification, each CF is provided with: (a) conditions on the coefficients of the initial system, (b) non-singular linear substitutions that reduce the right-hand part of the system under these conditions to the selected CF, (c) obtained values of CF’s unnormalized elements.
PubDate: 2018-10-01
DOI: 10.3103/S1063454118040040
- Abstract: The present article is the fifth in a cycle of papers dedicated to two-dimensional homogeneous cubic systems. It considers a case when the homogeneous polynomial vector in the right-hand part of the system has a linear common factor. A set of such systems is divided into classes of linear equivalence, wherein the simplest system being a third-order normal form is distinguished based on properly introduced principles. Such a form is defined by the matrix of its right-hand part coefficients, which is called the canonical form (CF). Each CF has its own arrangement of non-zero elements, their specific normalization, and canonical set of permissible values for the unnormalized elements, which relates the CF to the selected class of equivalence. In addition to classification, each CF is provided with: (a) conditions on the coefficients of the initial system, (b) non-singular linear substitutions that reduce the right-hand part of the system under these conditions to the selected CF, (c) obtained values of CF’s unnormalized elements.
- On the History of the St. Petersburg School of Probability and Statistics.
III. Distributions of Functionals of Processes, Stochastic Geometry, and
Extrema- Abstract: This is the third paper in a series of reviews devoted to the scientific achievements of the Leningrad–St. Petersburg School of Probability and Statistics in 1947–2017. The paper deals with the studies on functionals of random processes, some problems of stochastic geometry, and problems associated with ordered systems of random variables. The first sections of the paper are devoted to the problems of calculating the distributions of various functionals of Brownian motion and consider the so-called invariance principles for Brownian local times and random walks. The second part is dedicated to limit theorems for weakly dependent random variables and local limit theorems for stochastic functionals. It provides information about the stratification method and the local invariance principle. The asymptotic behavior of the convex hulls of random samples of increasing size and limit theorems for random zonotopes are also considered. An important relation between Poisson point processes and stable distributions is explained. The final part presents extensive information on research related to ordered systems of random variables. The maxima of sequential sums, order statistics, and record values are analyzed in detail.
PubDate: 2018-10-01
DOI: 10.3103/S1063454118040052
- Abstract: This is the third paper in a series of reviews devoted to the scientific achievements of the Leningrad–St. Petersburg School of Probability and Statistics in 1947–2017. The paper deals with the studies on functionals of random processes, some problems of stochastic geometry, and problems associated with ordered systems of random variables. The first sections of the paper are devoted to the problems of calculating the distributions of various functionals of Brownian motion and consider the so-called invariance principles for Brownian local times and random walks. The second part is dedicated to limit theorems for weakly dependent random variables and local limit theorems for stochastic functionals. It provides information about the stratification method and the local invariance principle. The asymptotic behavior of the convex hulls of random samples of increasing size and limit theorems for random zonotopes are also considered. An important relation between Poisson point processes and stable distributions is explained. The final part presents extensive information on research related to ordered systems of random variables. The maxima of sequential sums, order statistics, and record values are analyzed in detail.
- Conditions for the Existence of Two Limit Cycles in a System with
Hysteresis Nonlinearity- Abstract: This work deals with a two-dimensional automatic control system containing a single nonlinear hysteretic element in the general form. The conditions sufficient for the existence of at least two limit cycles in the system are presented. To prove the existence of cycles, three closed contours embedded into each other are constructed on the phase manifold by “sewing” together pieces of the level lines of various Lyapunov functions. System trajectories cross the inner contour “from outside inwards” and the middle contour “from inside outwards.” The outer contour is crossed by system trajectories “from outside inwards.” The existence of these contours proves the presence of at least two limit cycles in the system. This paper is a continuation of our earlier published work “Conditions for the Global Stability of a Single System with Hysteresis Nonlinearity,” in which the conditions of global stability in this system are formulated.
PubDate: 2018-07-01
DOI: 10.3103/S1063454118030135
- Abstract: This work deals with a two-dimensional automatic control system containing a single nonlinear hysteretic element in the general form. The conditions sufficient for the existence of at least two limit cycles in the system are presented. To prove the existence of cycles, three closed contours embedded into each other are constructed on the phase manifold by “sewing” together pieces of the level lines of various Lyapunov functions. System trajectories cross the inner contour “from outside inwards” and the middle contour “from inside outwards.” The outer contour is crossed by system trajectories “from outside inwards.” The existence of these contours proves the presence of at least two limit cycles in the system. This paper is a continuation of our earlier published work “Conditions for the Global Stability of a Single System with Hysteresis Nonlinearity,” in which the conditions of global stability in this system are formulated.
- On the History of St. Petersburg School of Probability and Mathematical
Statistics: II. Random Processes and Dependent Variables- Abstract: This is the second paper in a series of reviews devoted to the scientific achievements of the Leningrad and St. Petersburg school of probability and mathematical statistics from 1947 to 2017. This paper is devoted to the works on limit theorems for dependent variables (in particular, Markov chains, sequences with mixing properties, and sequences admitting a martingale approximation) and to various aspects of the theory of random processes. We pay particular attention to Gaussian processes, including isoperimetric inequalities, estimates of the probabilities of small deviations in various norms, and the functional law of the iterated logarithm. We present a brief review and bibliography of the works on approximation of random fields with a parameter of growing dimension and probabilistic models of systems of sticky inelastic particles (including laws of large numbers and estimates for the probabilities of large deviations).
PubDate: 2018-07-01
DOI: 10.3103/S1063454118030123
- Abstract: This is the second paper in a series of reviews devoted to the scientific achievements of the Leningrad and St. Petersburg school of probability and mathematical statistics from 1947 to 2017. This paper is devoted to the works on limit theorems for dependent variables (in particular, Markov chains, sequences with mixing properties, and sequences admitting a martingale approximation) and to various aspects of the theory of random processes. We pay particular attention to Gaussian processes, including isoperimetric inequalities, estimates of the probabilities of small deviations in various norms, and the functional law of the iterated logarithm. We present a brief review and bibliography of the works on approximation of random fields with a parameter of growing dimension and probabilistic models of systems of sticky inelastic particles (including laws of large numbers and estimates for the probabilities of large deviations).
- The Spectrum of a Separable Dynkin Algebra and the Topology Defined on It
- Abstract: The author continues his previous works on preparation to develop generalized axiomatics of the probability theory. The approach is based on the study of set systems of a more general form than the traditional set algebras and their Boolean versions. They are referred to as Dynkin algebras. The author introduces the spectrum of a separable Dynkin algebra and an appropriate Grothendieck topology on this spectrum. Separable Dynkin algebras constitute a natural class of abstract Dynkin algebras, previously distinguished by the author. For these algebras, one can define partial Boolean operations with appropriate properties. The previous work found a structural result: each separable Dynkin algebra is the union of its maximal Boolean subalgebras. In the present note, leaning upon this result, the spectrum of a separable Dynkin algebra is defined and an appropriate Grothendieck topology on this spectrum is introduced. The corresponding constructions somewhat resemble the constructions of a simple spectrum of a commutative ring and the Zariski topology on it. This analogy is not complete: the Zariski topology makes the spectrum of a commutative ring an ordinary topological space, while the Grothendieck topology, which, generally speaking, is not a topology in the usual sense, turns the spectrum of a Dynkin algebra into a more abstract object (site or situs, according to Grothendieck). This suffices for the purposes of the work.
PubDate: 2018-07-01
DOI: 10.3103/S106345411803010X
- Abstract: The author continues his previous works on preparation to develop generalized axiomatics of the probability theory. The approach is based on the study of set systems of a more general form than the traditional set algebras and their Boolean versions. They are referred to as Dynkin algebras. The author introduces the spectrum of a separable Dynkin algebra and an appropriate Grothendieck topology on this spectrum. Separable Dynkin algebras constitute a natural class of abstract Dynkin algebras, previously distinguished by the author. For these algebras, one can define partial Boolean operations with appropriate properties. The previous work found a structural result: each separable Dynkin algebra is the union of its maximal Boolean subalgebras. In the present note, leaning upon this result, the spectrum of a separable Dynkin algebra is defined and an appropriate Grothendieck topology on this spectrum is introduced. The corresponding constructions somewhat resemble the constructions of a simple spectrum of a commutative ring and the Zariski topology on it. This analogy is not complete: the Zariski topology makes the spectrum of a commutative ring an ordinary topological space, while the Grothendieck topology, which, generally speaking, is not a topology in the usual sense, turns the spectrum of a Dynkin algebra into a more abstract object (site or situs, according to Grothendieck). This suffices for the purposes of the work.
- The Strong Continuity of Convex Functions
- Abstract: A convex function defined on an open convex set of a finite-dimensional space is known to be continuous at every point of this set. In fact, a convex function has a strengthened continuity property. The notion of strong continuity is introduced in this study to show that a convex function has this property. The proof is based on only the definition of convexity and Jensen’s inequality. The definition of strong continuity involves a constant (the constant of strong continuity). An unimprovable value of this constant is given in the case of convex functions. The constant of strong continuity depends, in particular, on the form of a norm introduced in the space of arguments of a convex function. The polyhedral norm is of particular interest. It is straightforward to calculate the constant of strong continuity when it is used. This requires a finite number of values of the convex function.
PubDate: 2018-07-01
DOI: 10.3103/S1063454118030056
- Abstract: A convex function defined on an open convex set of a finite-dimensional space is known to be continuous at every point of this set. In fact, a convex function has a strengthened continuity property. The notion of strong continuity is introduced in this study to show that a convex function has this property. The proof is based on only the definition of convexity and Jensen’s inequality. The definition of strong continuity involves a constant (the constant of strong continuity). An unimprovable value of this constant is given in the case of convex functions. The constant of strong continuity depends, in particular, on the form of a norm introduced in the space of arguments of a convex function. The polyhedral norm is of particular interest. It is straightforward to calculate the constant of strong continuity when it is used. This requires a finite number of values of the convex function.
- Linear Kalman–Bucy Filter with Autoregressive Signal and Noise
- Abstract: In the Kalman—Bucy filter problem, the observed process consists of the sum of a signal and a noise. The filtration begins at the same moment as the observation process and it is necessary to estimate the signal. As a rule, this problem is studied for the scalar and vector Markovian processes. In this paper, the scalar linear problem is considered for the system in which the signal and noise are not Markovian processes. The signal and noise are independent stationary autoregressive processes with orders of magnitude higher than 1. The recurrent equations for the filter process, its error, and its conditional cross correlations are derived. These recurrent equations use previously found estimates and some last observed data. The optimal definition of the initial data is proposed. The algebraic equations for the limit values of the filter error (the variance) and cross correlations are found. The roots of these equations make possible the conclusions concerning the criterion of the filter process convergence. Some examples in which the filter process converges and does not converge are given. The Monte Carlo method is used to control the theoretical formulas for the filter and its error.
PubDate: 2018-07-01
DOI: 10.3103/S1063454118030093
- Abstract: In the Kalman—Bucy filter problem, the observed process consists of the sum of a signal and a noise. The filtration begins at the same moment as the observation process and it is necessary to estimate the signal. As a rule, this problem is studied for the scalar and vector Markovian processes. In this paper, the scalar linear problem is considered for the system in which the signal and noise are not Markovian processes. The signal and noise are independent stationary autoregressive processes with orders of magnitude higher than 1. The recurrent equations for the filter process, its error, and its conditional cross correlations are derived. These recurrent equations use previously found estimates and some last observed data. The optimal definition of the initial data is proposed. The algebraic equations for the limit values of the filter error (the variance) and cross correlations are found. The roots of these equations make possible the conclusions concerning the criterion of the filter process convergence. Some examples in which the filter process converges and does not converge are given. The Monte Carlo method is used to control the theoretical formulas for the filter and its error.
- Generating Large Sequences of Normal Maxima via Record Values
- Abstract: In our recent paper [4], algorithms for generating normal record values were developed. The developed algorithms were faster and more efficient than currently existing algorithms for generating normal record values. Algorithm 2.2, presented in this paper, is the most efficient algorithm among the algorithms studied in [4]. It allows generating “long” sequences of record values (up to two billion record values). In the present paper, two algorithms for generating normal maxima are proposed, one of which is based on algorithm 2.2. It also allows the generation of maxima taken from “large” samples. An algorithm for generating record times in a general continuous case is also proposed in the present paper.
PubDate: 2018-07-01
DOI: 10.3103/S106345411803007X
- Abstract: In our recent paper [4], algorithms for generating normal record values were developed. The developed algorithms were faster and more efficient than currently existing algorithms for generating normal record values. Algorithm 2.2, presented in this paper, is the most efficient algorithm among the algorithms studied in [4]. It allows generating “long” sequences of record values (up to two billion record values). In the present paper, two algorithms for generating normal maxima are proposed, one of which is based on algorithm 2.2. It also allows the generation of maxima taken from “large” samples. An algorithm for generating record times in a general continuous case is also proposed in the present paper.
- Two-Sided Estimates of Fourier Sums Lebesgue Functions with Respect to
Polynomials Orthogonal on Nonuniform Grids- Abstract: Let Ω = {t0, t1, …, tN} and ΩN = {x0, x1, …, xN–1}, where xj = (tj + tj + 1)/2, j = 0, 1, …, N–1 be arbitrary systems of distinct points of the segment [–1, 1]. For each function f(x) continuous on the segment [–1, 1], we construct discrete Fourier sums Sn, N( f, x) with respect to the system of polynomials {p̂k,N(x)} k=0 N–1 , forming an orthonormal system on nonuniform point systems ΩN consisting of finite number N of points from the segment [–1, 1] with weight Δtj = tj + 1–tj. We find the growth order for the Lebesgue function Ln,N (x) of the considered partial discrete Fourier sums Sn,N ( f, x) as n = O(δ N −2/7 ), δN = max0≤ j≤N−1 Δtj More exactly, we have a two-sided pointwise estimate for the Lebesgue function Ln, N(x), depending on n and the position of the point x from [–1, 1].
PubDate: 2018-07-01
DOI: 10.3103/S1063454118030068
- Abstract: Let Ω = {t0, t1, …, tN} and ΩN = {x0, x1, …, xN–1}, where xj = (tj + tj + 1)/2, j = 0, 1, …, N–1 be arbitrary systems of distinct points of the segment [–1, 1]. For each function f(x) continuous on the segment [–1, 1], we construct discrete Fourier sums Sn, N( f, x) with respect to the system of polynomials {p̂k,N(x)} k=0 N–1 , forming an orthonormal system on nonuniform point systems ΩN consisting of finite number N of points from the segment [–1, 1] with weight Δtj = tj + 1–tj. We find the growth order for the Lebesgue function Ln,N (x) of the considered partial discrete Fourier sums Sn,N ( f, x) as n = O(δ N −2/7 ), δN = max0≤ j≤N−1 Δtj More exactly, we have a two-sided pointwise estimate for the Lebesgue function Ln, N(x), depending on n and the position of the point x from [–1, 1].