ZFC的模型的上限序数的一个性质

S1 引言 Forcing方法假设存在ZFC的一个可数可传的模型M。记满足αM的最小序数α为,显然M中一切序数所成的集合即。由于M是ZFC的模型,故应具有某些性质。本文证明了它满足关系,故为ε数或1级关键数,进而证明了是H级关键数(H为任意自然数)。文中的记号等引用。     

The Equation of Motion of a Vortex Layer of Small Thickness

The equation governing the evolution of a vortex layer whose thickness is small compared to its radius of curvature and in which the vorticity is everywhere the same is obtained, viscous and compressibility effects being neglected. The method of matched asymptotic expansions is applied and results in an extension, which is unexpectedly simple, of Birkhoff's integro-differential equation for vortex sheets. The equation is applied to long waves on a straight vortex layer of uniform thickness.     

Impact of the cost of the time resource on efficiency of economic processes

The objective of the paper is to simulate behavior of economic agents with a special emphasis on the use of the time resource in economic processes. The case studied is from the fishing industry. With simulations, the study analyzes the behavior of economic agents as they lead their processes through time. The paper estimates the value of time in a fishery operation, and shows that the time cost is essential in clarifying some phenomena in the economic behavior of agents. Among the conclusions is that the difference in economic efficiency between strong and weak ownership is due to difference in time costs. The secondary goal of the paper is to develop a simulation tool for analyzing a range of problems in the fishing industry, ranging from decision situations onboard fishing vessels to serving as a tool for fishery managers and legislators.     

Asymptotic solution of the problem of the optimal control of non-linear oscillations in the neighbourhood of a resonance

A weakly controllable system with two rotating phases is considered in a regime of resonance oscillations. The characteristic rate of change of the slow variables in the system is of the order of ε, and the control is contained in the terms of the equations of order ε^3/2. This order of magnitude of the control makes it possible to control a resonance regime over time intervals of the order of 1/ε. The purpose of the control is to minimize a functional representing the deviation from a steady resonance regime. It is shown that there is a hierarchy of fast and slow motions in the equations of the maximum principle. An algorithm is described for the asymptotic integration of these equations using successive averaging. The problem of vibrational maintenance of the steady rotation of an unbalanced rotor is considered as an example.     

Analysis of models for calculating the motion of solids of revolution of minimum resistance in soil media

The solution of problems of searching for the optimal shape of a body when it penetrates into dense media is considered using local interaction models (LIMs) and Grigoryan's model of a soil medium in an axisymmetric formulation. A new LIM is obtained that is improved by taking account of the non-linear compressibility and shear strength in the analytical solution of a problem on the expansion of a spherical cavity. The applicability of an LIM that is quadratic with respect to the velocity in determining the forces resisting penetration of sharp bodies into soft soil is justified theoretically and experimentally and the violation of the conditions for the model to be applicable in the case of blunt bodies is established. It is shown that a solution taking account of non-linear flow effects in a two-dimensional formulation enables both the shape as well as power and kinematic characteristics of the optimal blunt bodies as they pass through soil media to be improved considerably. The ratio of the finite depths of penetration of solids of revolution into soft ground taking account of internal friction is estimated by the ratio of the coefficients in the Rankine–Resal formulae.     

Modelling and analysis of the spread of AIDS epidemic with immigration of HIV infectives

We propose and analyze, a nonlinear mathematical model of the spread of HIV/AIDS in a population of varying size with immigration of infectives. It is assumed that susceptibles become infected via sexual contacts with infectives (also assumed to be infectious) and all infectives ultimately develop AIDS. The model is studied using stability theory of differential equations and computer simulation. Model dynamics is also discussed under two particular cases when there is no direct inflow of infectives. On analyzing these situations, it is found that the disease is always persistent if the direct immigration of infectives is allowed in the community. Further, in the absence of inflow of infectives, the endemicity of the disease is found to be higher if pre-AIDS individuals also interact sexually in comparison to the case when they do not take part in sexual interactions. Thus, if the direct immigration of infectives is restricted, the spread of infection can be slowed down. A numerical study of the model is also carried out to investigate the influence of certain key parameters on the spread of the disease.     

The method of determining all real nonmultiple roots of systems of nonlinear equations

A method for determining all nonmultiple roots of the system of nonlinear equations in an n-dimensional parallelepiped is proposed. The main idea of the method is that the original set, in which the roots are sought, is divided into subsets where either the system of equations does not have solutions or its Jacobian matrix is nonsingular. A partition algorithm is presented and its convergence is proved. The application of the method is demonstrated using several examples.     

Non-singular locally optimal ellipsoidal approximation of the estimate of the states of linear systems

A problem which arises when estimating the attainability domains of linear dynamical systems by ellipsoids is investigated in a short time interval in the case when the initial position of the system in phase space is known precisely for some at least coordinates. A method is proposed which allows one to avoid problems associated with the degeneracy of the right-hand sides of the differential equations of the locally optimal ellipsoidal approximation. The mathematical meaning of these equations is made more precise in the case of the minimization of the phase volume. An example is given.     

Zeros of solutions of functional equations in the space of discontinuous functions of two variables

In this paper distribution of zeros of solutions of functional equations in the space of functions of two variables is studied. A zero of a solution in the space of noncontinuous functions is defined. It is demonstrated that oscillatory properties of functional equations are determined by the spectral radius of a corresponding operator acting in the space of essentially bounded functions. Zones of solution positivity are estimated. Various exact oscillation and non-oscillation tests are proposed. A necessary and sufficient condition of oscillation is obtained.     

Formulation of the problem of the flutter of a shell of revolution and a shallow shell in a high-velocity supersonic gas flow

Aero-elastic vibration is investigated in the case of a shallow shell of revolution or a cylindrical panel, which respectively occupy a part of a thin cylindrical body or a thin profile, in a high-velocity supersonic gas flow at zero angle of attack. Particular attention is paid to finding the pressure interaction and this problem is solved within the framework of the law of plane sections in boundary-layer theory. An expression is obtained which refines and supplements the well-known formula of “piston” theory. A linearized formulation of the problem of the panel flutter of a shallow shell is presented. Using the example of a plate located on one of the sides of a wedge, it is shown that the formula of “piston” theory is complemented with a term which has the meaning of a compressive force in the plane of the plate. It is shown that, when account is taken of this term, there is a reduction in the critical flow velocity.     

Stabilization of the motion of a dynamic system under conditions of uncertainty

The problem of the stabilization of the unperturbed motion of a dynamic system when there is incomplete information about the system parameters is considered. The solution is sought by Lyapunov's second method in the class of dynamic controllers and generalizes the result obtained in /1/ to controlled dynamic systems. Similar control problems were considered, in particular, in /2/.

The solution is used to stabilize the permanent rotation of a rigid body by a controlling moment with zero x-component /3/.     

Variational formulations of certain problems of the theory of the flow of rigid-plastic media

The mathematical content of variational problems of the mechanics of rigid-plastic media reduces to minimizing convex functionals in non-reflexive spaces of solenoidal vector fields. Different formulations are presented in /1/. Problems are examined there in which discontinuous velocity fields occur. The initial functional is undetermined in such fields. In this connection, the problem is posed of constructing that extension of the original set of kinematically allowable velocity fields as would contain all their possible discontinuities allowed by the mechanics of rigidly plastic media, and of continuing the original functional into the extension obtained. This problem is solved in this paper for the case when the velocity field is given on the whole surface of the rigid-plastic body.     

D'Alembert's proof of the fundamental theorem of algebra

D'Alembert's proof of the fundamental theorem of algebra (FTA), the first published, is still widely misunderstood. Typical of d'Alembert, his work is bold and imaginative but in need of significant repair. The proof is examined in detail, in both the 1746 and 1754 versions, along with commentary over 250 years and recent efforts to revive d'Alembert's reputation. A particular challenge is to work with algebraic equations while avoiding dependence on the FTA itself. A repaired version is offered.     

Study of a special model of resource allocation over an infinite interval of time

A special model of resource allocation over an infinite interval of time is studied. Using the Pontryagin maximum principle, an extreme solution is constructed whose optimality is proven with the help of a theorem on sufficient conditions, in the form of constructions of Pontryagin??s maximum principle. A concrete example in which the classical maximum principle is inapplicable is considered.     

Differential properties of solutions of variational problems for functionals of linear growth

The differential properties of the vector extremals of convex functionals of linear growth, depending only on the modulus of the gradient of the desired function, are investigated. It is proved that if the integrand is strictly convex and its derivative is concave for large values of the argument, then, under some additional conditions, the generalized solution is regular in an open set of full measure. Another result consists in the fact that if the integrand is linear for all large values of the argument, then, under some additional conditions, there exists an open set in which the gradient of the solution is continuous and its modulus is strictly smaller than the value of the parameter, starting from which the integrand becomes a linear function.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 51–79, 1990.     

Stability of solutions of monotone systems of impulsive differential equations

We obtain sufficient conditions for the stability of solutions of nonlinear systems of impulsive differential equations in a cone. The main tool in our study is the comparison principle suggested in the paper. This approach permits one to study critical cases; it is illustrated by examples of nonlinear systems for which the trivial orbit of the continuous and discrete components is simultaneously unstable.