Ашық рұқсат
Рұқсат берілді
Тек жазылушылар үшін
Том 49, № 4 (2016)
- Жылы: 2016
- Мақалалар: 14
- URL: https://journal-vniispk.ru/1063-4541/issue/view/11599
Mathematics
Generalizations of the parking problem
Аннотация
The well-known parking problem of the Hungarian mathematician Rényi is about the asymptotic behavior of the mathematical expectation of the number of open unit intervals randomly filling a long interval. The length of the interval being filled increases without bound.
The paper studies generalizations of the parking problem in two directions. The first is the case where the length of the placed intervals is a random variable. Unlike in the original setting of the problem, the behavior of the expectations of both the number of placed intervals and the measure of the occupied part of the long interval are studied. The second direction is the case where the distribution of the position of a placed unit interval is nonuniform, unlike in the classical parking problem.
299-304
On some statistical properties of the “Book Stack” transformation
Аннотация
This paper is devoted to studying the statistical properties of the “Book Stack” transformation proposed by B.Ya. Ryabko (Probl. Inf. Transm., 1980, vol. 16, no. 4) as a data compression method. The same transformation has been used by Ryabko and A.I. Pestunov (Probl. Inf. Transm., 2004, vol. 40, no. 1) to construct the similarly named statistical test. This test is designed for the verification of the null hypothesis that an available input i.i.d. sample corresponds to a discrete uniform distribution with a known support. They propose to verify this hypothesis for a new sample obtained via the Book Stack transformation instead of the input sample. This gives rise to the natural problem of comparing the results given by the same statistical test in the application to input and output samples. If the null hypothesis is true, these procedures prove to be equivalent; however, this is actually not the case anymore when there are some violations of this hypothesis. The results of comparing the criteria surely depend on the class of the alternatives considered. This paper deals with the natural alternative consisting of the fact that the initial replicated sample corresponds to a discrete, albeit, nonuniform, distribution with a fixed support. It has been demonstrated that some standard criteria for the verification of the null hypothesis prove to be more powerful for an input sample in comparison with a transformed sample. In particular, this takes place for the likelihood ratio criterion and (with some formal constraints) the χ2-criterion.
305-312
Regular formal modules in one-dimensional local fields
Аннотация
This paper considers the problem of the description of unramified extensions of a local field which, together with the main field, do not contain nontrivial roots of isogeny of the corresponding formal group defined over a ring of integers of this field. This problem originated from investigation of extensions without higher ramification for multiplicative formal groups in the paper by Z.I. Borevich (1962).
313-319
On the norm property of the Hilbert symbol for polynomial formal modules in a multidimensional local field
Аннотация
In a two-dimensional local field K containing the pth root of unity, a polynomial formal group Fc(X, Y) = X + Y + cXY acting on the maximal ideal M of the ring of integers бK and a constructive Hilbert pairing {·, ·}c: K2(K) × Fc(M) → <ξ>c, where <ξ>c is the module of roots of [p]c (pth degree isogeny of Fc) with respect to formal summation are considered. For the extension of two-dimensional local fields L/K, a norm map of Milnor groups Norm: K2(L) → K2(K) is considered. Its images are called norms in K2(L). The main finding of this study is that the norm property of pairing {·, ·}c: {x,β}c: = 0 ⇔ x is a norm in K2(K([p]c-1(β))), where [p]c-1(β) are the roots of the equation [p]c = β, is checked constructively.
320-324
On Monte Carlo methods in distributed memory systems
Аннотация
The way for solving a system of linear algebraic equations (SLAEs) with computers with distributed memory is presented. It is assumed that there are M computing nodes, each of which has a limited fast memory, and communication between nodes takes considerable time.If the matrix elements and the right side vectors cannot be placed in their entirety in the one node memory, the problem of using equipment efficiently between the exchange, i.e., whether each node is able to use the available information to reduce the total residual, appears. The answer to this question is negative under general assumptions on the system’s matrix and the example presented in the Appendix verifies this fact. We examine the case when the system is of sufficiently high order and it is reasonable to use the Monte Carlo method. In this case the matrix is divided between computing nodes on blocks of rows that do not overlap with the same partition into blocks of indices of rows and columns. We also consider a modification of the method of simple iteration based on this partition consisting of two nested iterative processes so that messaging between nodes takes place only in the outer iterations. This iterative process naturally results in a similar process, where the Monte Carlo method is used, and where it is not necessary to save a matrix’s full copy at each computing node. The unbiased estimations of linear algebraic equations’ solutions for the examined case are studied in the present paper. Under certain additional conditions imposed on the matrix, we prove the sufficient convergence conditions.
325-333
Orthogonal triangulation of polygons
Аннотация
In this study, the question whether any convex polygon can be divided using an orthogonal grid into right-angled triangles is answered in the negative. Moreover, it is demonstrated that there exists a convex pentagon which cannot be even approximated by divisible pentagons.
334-339
Using tropical optimization to solve minimax location problems with a rectilinear metric on the line
Аннотация
Methods of tropical (idempotent) mathematics are applied to the solution of minimax location problems under constraints on the feasible location region. A tropical optimization problem is first considered, formulated in terms of a general semifield with idempotent addition. To solve the optimization problem, a parameter is introduced to represent the minimum value of the objective function, and then the problem is reduced to a parametrized system of inequalities. The parameter is evaluated using existence conditions for solutions of the system, whereas the solutions of the system for the obtained value of the parameter are taken as the solutions of the initial optimization problem. Then, a minimax location problem is formulated to locate a single facility on a line segment in the plane with a rectilinear metric. When no constraints are imposed, this problem, which is also known as the Rawls problem or the messenger boy problem, has known geometric and algebraic solutions. For the location problems, where the location region is restricted to a line segment, a new solution is obtained, based on the representation of the problems in the form of the tropical optimization problem considered above. Explicit solutions of the problems for various positions of the line are given both in terms of tropical mathematics and in the standard form.
340-349
On a cubic variational problem
Аннотация
An extremal curve of the simplest variational problem is a continuously differentiable function. Hilbert’s differentiability theorem provides a sufficient condition for the existence of the second derivative of an extremal curve. It is desirable to have a simple example in which the condition of Hilbert’s theorem is violated and an extremal curve is not twice differentiable.
In this paper, a cubic variational problem with the following properties is analyzed. The functional of the problem is bounded neither above nor below. There exists an extremal curve for this problem which is obtained by sewing together two different extremal curves and not twice differentiable at the sewing point. Despite this unfavorable situation, an attempt to apply the method of steepest descent (in the form proposed by V.F. Dem’yanov) to this problem is made. It turns out that the method converges to a stationary curve provided that a suitable step size rule is chosen.
350-358
Planar sections of three-dimensional cylinders
Аннотация
Is it true that every interior point of a three-dimensional convex body lies on its planar section with an inscribed regular hexagon and the center of a centrally symmetric convex body lies on a planar section with an inscribed regular octagon? In this paper, we prove these propositions for cylinders of a special type.
359-360
Representations of ordered random variables in the terms of the sums or products of independent variables
Аннотация
This is a survey of the available representations capable of expressing dependent ordered variables like order statistics and records as sums of independent summands or as products of independent random multipliers. Some new similar representations are obtained that in a similar manner express the common distributions of representatives of these two types of ordered random variables.
361-370
The strong law of large numbers for a stationary sequence
Аннотация
General results on the applicability of the strong law of large numbers to a sequence of dependent random variables, as formulated in terms of estimates for the moments of sums of such variables, are applied to give new conditions of the applicability of this law to (in a wide sense) a stationary sequence of random variables.
371-372
Approximations by entire functions on countable unions of segments of the real axis
Аннотация
The problem to approximate functions continuous on subsets of the real line by entire functions has a long history that started from the Jackson–Bernstein theorem on the approximation of 2- periodic functions by trigonometric polynomials naturally treated as exponential-type entire functions. In this paper, we deal with the problem referring to the concept of this theorem describing classes of functional spaces via the rate of their possible approximation by entire functions. A key example is the Bernstein theorem describing the class of bounded functions from Holder classes over the whole axis by exponential-type entire functions. The key point is that the approximation rate at a neighborhood of the segment edge exceeds the one that originally appeared in the theory of approximation functions from Holder classes on segments (this allows us to coordinate the direct and inverse theorems for that case, i.e., to recover the holder smoothness from the approximation rate in the said scale). In the present paper, we present a direct theorem on the possibility of a prescribed-rate approximation of functions from Holder classes on countable unions of segments by entire functions. Earlier, such approximations were not considered. Also, we provide general definitions and important lemmas used for further constructing approximating functions. In the second part of the work, we provide a proof of the direct theorem. In our further papers, to obtain a constructive description of the smoothness class by means of the approximation rate, we will prove the corresponding inverse theorem. Usually, to deduce such assertions, one needs a fact similar to the Bernstein theorem on the estimate of the norm of an entire function via the norm of the function itself. In our case, we need an assertion similar to the Akhiezer–Levin theorem estimating an entire function on the axis via its values on a subset of the axis.
373-378
On inequalities for conditional probabilities of unions of events and the conditional Borel–Cantelli lemma
Аннотация
New sharp upper and lower bounds for conditional (given a σ-algebra A) probabilities of unions of events and for a generalization of the conditional Borel–Cantelli lemma are obtained. Averaging the left- and right-hand sides of the corresponding inequalities yields bounds better than those obtained by directly estimating the probabilities of events. An example is given. New generalizations of the conditional Borel–Cantelli lemma are also obtained. Averaging yields new versions of this lemma under conditions different from the classical ones.
379-388
Mechanics
Inverse Laplace transformation for evaluation of state-specific cross sections for dissociation reaction and vibrational energy transitions
Аннотация
In the present work, we study the cross sections of VV (vibration-vibration) and VT (vibration-translation) energy exchanges in nitrogen and oxygen, as well as the vibrational state-specific cross sections of the dissociation reaction in N2 molecules. For VV- and VT-transitions the original approximations of the rate coefficients have been modified to make it possible to apply the inverse Laplace transformation in the analytical form. A satisfactory approximation of the state-resolved dissociation rate coefficient allowing for the application of the inverse Laplace transformation is also proposed. For all the considered reactions, analytical expressions for the cross sections are obtained. The results are analyzed in the wide range of energies and vibrational levels. It is shown that the cross sections of VV transitions increase almost linearly with the energy of the colliding particles. VT-exchanges and the dissociation reaction manifest threshold behavior and their cross sections are nonmonotonic. The dissociation threshold shifts significantly towards the low-energy region for high vibrational states. Using the hard sphere model for the dissociation cross section results in significant inaccuracy. The results of our work can be applied in nonequilibrium fluid dynamics, while simulating rarefied gas flows using the direct Monte-Carlo methods.
389-397