Thesis (Ph. D.)--University of Chicago, 1918. / Vita. "Private edition, distributed by the University of Chicago Libraries, Chicago, Illinois." "Reprinted from American journal of mathematics, Vol. XLIV, No. 3, July, 1922." Includes bibliographical references. Also available on the Internet.

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Some of the most accurate and economical of the known numerical methods for solving the initial-value problem [Formula omitted] are of the predictor-corrector type. For systems of equations, the predictor-corrector procedures are defined in the same manner as they are for single equations. For a given problem and domain of t , a plot of the maximum error in the numerical approximation to x(t) obtained by a predictor-corrector procedure, versus the step-size, can be divided into three general regions - round-off, truncation, and instability. The most practical procedures are stable and have a small truncation error. The stability of a method depends on the magnitudes of the eigenvalues of a certain matrix that is associated with the matrix [Formula omitted] When the functions f[subscript i] are complicated, predictor-corrector procedures involving two evaluations per step seem to be the most efficient for general-purpose applications. / Science, Faculty of / Mathematics, Department of / Graduate

This paper is concerned with the relationships between certain covering systems useful for set differentiation and with their application to density theorems and approximate continuity. The covering systems considered are the Vitali systems (which we call V-systems), the systems introduced by Sion (which we call S-systems), and a modification of the tile systems (which we call T-systems). It is easily checked from the definitions that V-systems are S-systems, and under slight restrictions, T-systems. We show also that under certain conditions S-systems are T-systems, and that in general the converses do not hold. Density theorems and the relationships between approximate continuity and measurability of functions are discussed for these systems. In particular, we prove that for T-systems measurable functions are approximately continuous and hence a density theorem holds. / Science, Faculty of / Mathematics, Department of / Graduate

Families of three- and four-point corrector formulae are derived, which differ from standard formulae in that they express yո in terms of more than one previously computed ordinate. It is shown that the standard formulae are special cases of the more general formulae derived here. By theoretical argument and by numerical experiments it is shown that the standard formulae are often inferior to others which are developed in this thesis. The three-point family, with its associated truncation error, is given in (7) and (9) of Chapter 2 on page 12. The four-point family is given in (41) on page 25. With the help of Rutishauser's method each family is examined for stability. In the four-point case a procedure is described, whereby the magnitude of the coefficient in the error term can be minimized subject to the restriction that the formula shall remain stable. Also a theorem is proved, which states that no stable four-point formula can have a truncation error of degree higher than fifth in the step-size h. / Science, Faculty of / Mathematics, Department of / Graduate

After a brief discussion of numerical methods for the solution of the ordinary differential equation x'= f(t, x) the problem of finding optimum methods is considered. The thesis then deals with this problem for the family of Runge-Kutta methods. Criteria for optimum methods are discussed and then the derivation of third-order methods is examined in detail. The next part of the thesis deals with possible approaches to finding optimum methods. The first approach consists of finding some sort of estimate for f and its derivatives contained in the truncation error T[subscript n]. The resulting expression is then dependent on some free parameter or parameters (depending on the order of the method) which are chosen in order to minimize this expression. The second approach assumes the independence of terms in the truncation error and minimizes, in some sense, the coefficients of these terms. Several procedures based on these approaches, are used to predict optimum second-order and third-order methods and comparisons are made with experimental results. While no definite conclusions could be drawn it was seen that one particular procedure gave a good prediction. This result encourages further studies in this area. I hereby certify that this abstract is satisfactory. / Science, Faculty of / Mathematics, Department of / Graduate

The theory of Poincaré and Bendixson is applied to establish the existence of periodic solutions of the differential equation ẍ + f(x, ẋ)ẋ + g(x) x = 0 One part of the work is concerned with those equations which can be considered as arising from small perturbations of other equations of the same type, already possessing periodic solutions. Two existence theorems are demonstrated and the stability and uniqueness of the periodic solutions is also discussed. The other part contains several, theorems stating sufficient conditions for existence of periodic solutions which cannot be treated by perturbation methods. / Science, Faculty of / Mathematics, Department of / Graduate

A method is given by which a-differential equation with initial conditions can be converted into an integral equation. This procedure is used to derive the Multiplication Theorems for Bessel functions, and to obtain an expansion of the confluent hyper geometric function in terms, of Bessel functions. The method is adapted to find approximate eigenvalues: and eigenfunctions of bounded quantum mechanical problems, and to obtain an approximate solution of a non-linear differential equation. / Science, Faculty of / Mathematics, Department of / Graduate

There are now engulfing theorems for topological, piecewise linear, and differentiable manifolds. Differentiable engulfing so far was reduced to piecewise linear engulfing using the J. H. C. Whitehead triangulation of a differentiable manifold and J. R. Munkres' theory of obstructions to the smoothing of piecewise-differentiable homeomorphisms. In the first part of the thesis we observe that the method of proof of M. H. A. Newman's topological engulfing theorem applies, up to a local lemma, simultaneously to all three categories of manifolds. We prove this local lemma in the differentiable case and thus obtain a differentiable engulfing theorem which has a direct proof. Then we solve the problem of the existence of a stretching diffeomorphism between complementary subcomplexes of a simplicial complex in Euclidean space which is crucial for all applications of engulfing. Next we prove a theorem concerning the uniqueness of open differentiable cylinders which is the differentiable analogue of the uniqueness theorem for open cones. A consequence of this theorem is that if M₁ and M₂ are compact differentiable manifolds with diffeomorphic interiors then M₁x R and M₂xR are diffeomorphic, where (R denotes the real line. Another consequence is that if a differentiable manifold is the monotone union of open differentiable cells it is diffeomorphic to Euclidean space. We present several applications of differentiable engulfing which actually hold in all three categories of manifolds. Our methods are such that they apply also to noncompact manifolds. Theorem: [formulae omitted] This theorem has several corollaries. For instance, if M is a k-connected differentiable manifold of dimension n without boundary, k ≤ n - 3 if k>0, and if [formula omitted] then M may be covered by m open differentiable n-cells. Using this result, we give a new and direct proof of the uniqueness of the differentiable structure of Euclidean n-space for n ≥ 5. Finally, we prove a general h-cobordism theorem. Theorem: Let M be a connected differentiable manifold of dimension n, n ≥ 5, with two connected boundary components N₁, and N₂ such that the inclusion of N₁ into M is a homotopy equivalence, i = 1,2. Then there is a diffeomorphism of N₁x(0,oo) onto M - N₂. / Science, Faculty of / Mathematics, Department of / Graduate