The application of the moment method to a class of Fredholm integral equations leads to the study of both an infinite system of ordinary differential equations and a functional differential …
The application of the moment method to a class of Fredholm integral equations leads to the study of both an infinite system of ordinary differential equations and a functional differential equation. Sufficient conditions are given that the solution to a truncated system of differential equations approximates the solutions to the original problem. A numerical algorithm for constructing approximate solutions is discussed.
The nominal trajectory used in a modified Encke special perturbation method can be written r = FM+&N, where F and G are the classical two-body functions, and the vectors M …
The nominal trajectory used in a modified Encke special perturbation method can be written r = FM+&N, where F and G are the classical two-body functions, and the vectors M and N rotate in space so that they are always in the nominal orbital plane and maintain constant angles with the precessing line of apsides of the nominal effipse. This representation of the nominal trajectory reduces initilaization and rectification errors and is computationally more efficient than that used earlier.
The purpose of this paper is to give infinite dimensional generalizations of the classical Borsuk and Borsuk-Ulam theorems of algebraic topology. In addition, a topological property of compact linear transformations …
The purpose of this paper is to give infinite dimensional generalizations of the classical Borsuk and Borsuk-Ulam theorems of algebraic topology. In addition, a topological property of compact linear transformations is established. Let E be an infinite dimensional real linear topological space with a bounded convex neighborhood of the origin. The fact that E can be made into a normed linear space will be used, but the results depend on the given topology rather than on the chosen norm. All functions considered are continuous and are either defined on V, a convex body symmetric with respect to the origin, or on its boundary, S. The ranges of these functions are in E. A function is compact if the closure of its range is compact, and nonzero if the origin is not in its range. If I is the identity map and F is a compact map defined on S, then the function f = I-F is a compact displacement of S. If F(S) is finite dimensional, then f= I-F is a finite displacement of S. And finally, a function g is antipodal on S if for all x in S, g(-x) =-g(x). The results of this paper follow from the fact that, after a consistent norm has been chosen, a compact antipodal map can be approximated by finite antipodal maps.
Leontovich has proved that the triangular equilibrium positions in the planar restricted problem of three bodies are stable for almost all admissible mass ratios. It is shown here that the …
Leontovich has proved that the triangular equilibrium positions in the planar restricted problem of three bodies are stable for almost all admissible mass ratios. It is shown here that the set of exceptional mass ratios for which stability remains to be proved or invalidated contains only one point besides the critical mass ratios of order two and three.
This harmony that human intelligence believes it discovers in nature -does it exist apart from that intelligence?No, without doubt, a reality completely independent of the spirit which conceives it, sees …
This harmony that human intelligence believes it discovers in nature -does it exist apart from that intelligence?No, without doubt, a reality completely independent of the spirit which conceives it, sees it or feels it, is an impossibility.A world so exterior as that, even if it existed, would be forever inaccessible to us.But what we call objective reality is, in the last analysis, that which is common to several thinking beings, and could be common to all; this common part, we will see, can be nothing but the harmony expressed by mathematical laws.H. Poincaré, La valeur de la science, p. 9 ... ignorance of the roots of the subject has its price-no one denies that modern formulations are clear, elegant and precise; it's just that it's impossible to comprehend how any one ever thought of them.
e I small, where the vector R and its partial derivatives with respect to xi up to the third order are continuous and periodic in t of period T and …
e I small, where the vector R and its partial derivatives with respect to xi up to the third order are continuous and periodic in t of period T and continuous in e for small I I I when x is near the closed orbit. For the moment let us take the case where n = 2 and discuss (1.2) geometrically. The periodic solution of (1.0) is a limit-cycle in the (xl, x2) plane. In (xI, x2, t) space the limit cycle becomes a cylinder parallel to the t axis. The cylinder is generated by all the solutions of (1.0) which start, at t = 0, on the limit cycle and which are regarded as curves in (x1, x2, t) space. All solutions of (1.0) near the cylinder tend to it as t -X o so that the cylinder is stable. Now because of the stability of the cylinder it appears plausible that the solutions of (1.2) emanating at t = 0 from a closed curve near the limit cycle will tend as t -X o to generate a surface S in (x1, x2, t) space which lies near the cylinder. The solutions of (1.2) starting at any point of S remain in S. If there is only one such surface then the intersections of S with t = to and t = to + T must be congruent (since the translation of S by T units parallel to the t axis gives another surface with the same properties as S so far as (1.2) is concerned). Indeed we 727