Denumerable System Research Articles

Stochastic Geometry for Population-Dynamic Modeling: A Dieckmann Model with Immovable Individuals

A study is performed of the main approaches to investigating the stochastic process of population dynamics. Continuous time and space and immovable individuals are used to derive a denumerable system of integrodifferential equations corresponding to the dynamics of the spatial momentum of this process. A way to find an approximate solution using the momentum approach is described.

Modelling and analysis of the natural oscillations of a prismatic elastic beam based on a projection approach

A regular approach to the construction of mathematical models describing the natural motions of beam-type elastic bodies within the limits of the linear theory of elasticity is developed using the method of integrodifferential relations. By employing the integral form of the equations of state, relating the stresses and strains and also the velocities and momenta, the system of partial differential equations is reduced to a denumerable system of ordinary differential-algebraic equations. A polynomial representation of the unknown functions of the displacements, stresses and momentum density along two spatial coordinates is used for this purpose. The effect of the geometric and mechanical parameters of the system on the frequencies and modes of the natural oscillations of a rectilinear elastic beam is investigated.

Optimal singular and chattering modes in the problem of controlling the vibrations of a string with clamped ends

The problem of minimizing the root mean square deviation of a uniform string with clamped ends from an equilibrium position is investigated. It is assumed that the initial conditions are specified and the ends of the string are clamped. The Fourier method is used, which enables the control problem with a partial differential equation to be reduced to a control problem with a denumerable system of ordinary differential equations. For the optimal control problem in the l 2 space obtained, it is proved that the optimal synthesis contains singular trajectories and chattering trajectories. For the initial problem of the optimal control of the vibrations of a string it is also proved that there is a unique solution for which the optimal control has a denumerable number of switchings in a finite time interval.

ON THE GROSS–PITAEVSKII EQUATION WITH STRONGLY ANISOTROPIC CONFINEMENT: FORMAL ASYMPTOTICS AND NUMERICAL EXPERIMENTS

The three-dimensional (3D) Gross–Pitaevskii equation with strongly anisotropic confining potential is analyzed. The formal limit as the ratio of the frequencies ε tends to zero provides a denumerable system of two-dimensional Gross–Pitaevskii equations, strongly coupled through the cubic nonlinearities. To numerically solve the asymptotic approximation only a finite number of limiting equations is considered. Finally, the approximation error for a fixed number of equations is compared for different ε tending to zero. On the other hand, the approximation error for an increasing number of terms in the approximation is observed.

The stability of the cylindrical precession of a viscoelastic body undergoing oscillations along an axis of symmetry

The relative motion of a viscoelastic body undergoing longitudinal oscillations along an axis of symmetry in a Newtonian central gravitational field is considered. The linear theory of viscoelasticity is used. The stability of a particular solution of the equations of motion is studied. This solution corresponds to the uniform rotation of the body about an axis of symmetry perpendicular to the plane of the circular orbit. Using the reduction principle of the theory of denumerable systems of differential equations and Kamenkov's criterion, the stability in the whole domain of variation of the parameters in the non-linear problem is investigated. It is shown that the “quasi-static approach” to studying stability is correct.

HYBRID ANALYSIS OF TRANSIENT NON‐LINEAR CONVECTION‐DIFFUSION PROBLEMS

A hybrid numerical‐analytical approach, based on recent developments in the generalized integral transform technique, is presented for the solution of a class of non‐linear transient convection‐diffusion problems. The original partial differential equation is integral transformed into a denumerable system of coupled non‐linear ordinary differential equations, which is numerically solved for the transformed potentials. The hybrid analysis convergence is illustrated by considering the one‐dimensional non‐linear Burgers equation and numerical results are presented for increasing truncation orders of the infinite ODE system.

Integral transform solutions of diffusion problems with nonlinear equation coefficients

A hybrid numerical-analytical procedure is described for the accurate and reliable solution of nonlinear diffusion problems due to potential dependent equation coefficients. A sufficiently general formulation of a transient multidimensional problem is first considered, and formal solutions provided in terms of the related transformed potentials, obtained from the numerical solution of a denumerable system of coupled nonlinear ordinary differential equations. An application related to heat conduction with temperature dependent thermal conductivity is then more closely studied, and numerical results presented to illustrate convergence behavior for increasing truncation order of the associated infinite O.D.E. system.

Laminar flow inside hexagonal ducts

Analytical solutions are obtained for fully developed laminar flow inside hexagonal ducts and, as a special case, rhombic ducts, through application of the recently advanced generalized integral transform technique for the solution of diffusion-type problems within irregular domains. First, the formal complete solution of the original problem is obtained, in terms of a denumerable system of coupled ordinary differential equations. Then, an explicit and quite straightforward lowest order solution results from the appropriate simplification of the coupled system. Quantities of particular interest in the field of heat exchanger thermohydraulics are then systematically presented, and utilized to establish the accuracy and suggest application ranges for the approximate solution.

HYBRID NUMERICAL/ANALYTICAL APPROACH TO NONLINEAR DIFFUSION PROBLEMS

A class of nonlinear diffusion-type problems is handled through a hybrid method. This method incorporates the ideas in the generalized integral transform technique to reduce the original partial differential equation into a denumerable system of coupled ordinary differential equations. These equations can then be solved through standard numerical techniques, once the system is truncated to a finite order. Sufficient conditions for the convergence of the truncated finite system are then examined. An application is considered that deals with a transient radiative fin problem, which is quite suitable for illustrating the solution methodology and convergence behavior.

On the solution of periodic multidimensional diffusion problems

Analytical solutions are obtained for periodic multidimensional diffusion problems through extension of the ideas in the generalized integral transform technique, completely avoiding the consideration of complex Sturm-Liouville systems. Exact solutions are obtained for a class of problems as a special case and, for a more general formulation, a coupled system of linear ordinary differential equations is generated, that can be solved by conventional algorithms and provide a complete solution to the periodic problem. Also, approximate analytical solutions for practical purposes are readily obtained from appropriate simplification of the coupled, denumerable system.

A Method for the Delay Analysis of Random Multiple-Access Algorithms Whose Delay Process is Regenerative

Random multiple-access algorithms are used to control the accessing of a common communication channel by a large population of bursty channel users. For such algorithms, the induced transmission delay is a key performance measure. A systematic method for finding the delay characteristics of random multiple-access algorithms, whose delay process is regenerative, is presented. The method uses a powerful result from the theory of regenerative processes, in effect, to reduce the problem of determining the delay moments to the problem of solving denumerable dimensional systems of linear equations. Techniques for finding tight bounds on the solutions of such systems are presented. The "0.487" algorithm is used to exemplify the method.

Certain particular cases of stability in first approximation of difference systems: PMM vol. 40, n≗1, 1976, pp. 174–176

The results of this paper can be regarded as a transposition of the results of Chetaev obtained for the finite systems of differential equations [1] to the denumerable systems of the finite difference equations. We use the concepts of [2].

An application of Markov potential theory to Markovian decision processes

Abstract The aim of this paper is to identify the denumerable stochastic systems which have an optimal strategy, namely, to give general sufficient conditions of the existence of an optimal strategy for denumerable Markovian decision processes. The aim is accomplished by using Markov potential theory and showing the range of the validity of a method to find an optimal strategy called Howard's technique. Since Markov potential theory helps to characterize the properties of evaluations of strategies on such processes, it plays an important role to find the sufficient conditions mentioned above For Markov potential theory, new concepts are introduced such as absorbable chains, quasi-potentials, etc., and using them, sufficient conditions for the existence of a solution of the equation x = f+Px are generalized to the one's for cyclic chains and multi-chains The results obtained in this paper are as follows : if a system represents a strong Markov chain for any strategy, then there exists an optimal strategy w...

Construction of approximate solutions of denumerable systems and their stability

Construction of approximate solutions of denumerable systems and their stability

Locally convex spaces with denumerable systems of weakly compact subsets

Let E be a locally convex vector space, and let (At) be a sequence of subsets of E, which cover E. Dieudonné (3) has proved the two following results.

On the stability “in the large” of the solutions of a denumerable system of differential equations under continuously acting disturbances

On the stability “in the large” of the solutions of a denumerable system of differential equations under continuously acting disturbances

Denumerability conditions in locally convex vector spaces

1. Let E be a locally convex Hausdorff vector space,1 Z its topology, e a set of bounded subsets of E; we say that a subset W CQ3 is a fundamental system of sets of Q3 if any set of e3 is contained in a set of ~. When e3 is the set of all bounded subsets of E, the condition that there exist a denumerable fundamental system of bounded sets has been extensively studied [1; 3; 4]: coupled with mild restrictions on the topology 5, it narrows considerably the possibilities for E; as an example, let us only recall that if 3 is metrizable, then the existence of a denumerable fundamental system of bounded sets implies that in fact E is normable [1, p. 15, exercise 20]. I propose to show in this note that even more stringent restrictions are imposed by the existence of a denumerable system 0 when e8 is either the set of all compact, or of all convex compact subsets of E.

The formal structure of a denumerable system

Introduction. A strictly formal mathernatical system comprising definite and variable number and function signs, logical constants and universal and existential operators, and employing the sentential calculus, may be termed classical if the application of the operators to the variables is without restriction; as is well known classical analysis is formalisable in such a system [1](1). If the application of the operators is restricted to variables whose range of values is a finite class, the system is said to be finitist. A limited part only of classical analysis can be formulated in a system finitist in this sense [2]. A denumerable system is one in which the universal and existential operators are restricted to variables whose range of values is a class of cardinal No. The concept of a denumerable system was introduced by Weyl [4]. The purpose of this paper is to indicate how wide a part of classical function theory may be formulated in a denumerable system. Formal characterisation. The primary elements of the system are numerals, numeral variables and numeral functions, the universal, existential, and minimal operators, and the logical constants. The calculus of the primary elements will be taken to be the system Z, of Hilbert-Bernays [1], so that at the first level, a denumerable system differs in no respect from the classical. (In particular the tertium non datur is valid in the form . (Ax) (f(x) 0) . V . (Ex) (f (x)-O0)., where x is a numeral variable.) We use different kinds of variables to distinguish numeral variables from function variables (a denumerable infinity of both kinds being included amongst the primary elements) and we rely on Russell's ramified theory of types [5] for avoiding paradoxes. The secondary elements are integral and rational number and function signs, which may be eliminated from the formulas of the system by means of the following definitions: An integer is a pair of numerals [p, q]; we define [p, q]_ [p', q'], [p, q] p'+q, p+q' q or p <q; it follows that [p, q] is positive or negative according as [p, q] [0, O]. An integer function f([p, q]) is a pair of numeral functions [P(p, q), Q(p, q) ] such that