A sharpened form of the inverse function theorem

In this note we establish an advanced version of the inverse function theorem and study some local geometrical properties like starlikeness and hyperbolic convexity of the inverse function under natural restrictions on the numerical range of the underlying mapping.

Solvable and Unsolvable Problems (1954)

In Chapter 1 Turing proves the existence of mathematical problems that cannot be solved by the universal Turing machine. There he also advances the thesis, now called the Church–Turing thesis, that any systematic method for solving mathematical problems can be carried out by the universal Turing machine. Combining these two propositions yields the result that there are mathematical problems which cannot be solved by any systematic method—cannot, in other words, be solved by any algorithm. In ‘Solvable and Unsolvable Problems’ Turing sets out to explain this result to a lay audience. The article first appeared in Science News, a popular science journal of the time. Starting from concrete examples of problems that do admit of algorithmic solution, Turing works his way towards an example of a problem that is not solvable by any systematic method. Loosely put, this is the problem of sorting puzzles into those that will ‘come out’ and those that will not. Turing gives an elegant argument showing that a sharpened form of this problem is not solvable by means of a systematic method (pp. 591–2). The sharpened form of the problem involves what Turing calls ‘the substitution type of puzzle’. An typical example of a substitution puzzle is this. Starting with the word BOB, is it possible to produce BOOOB by replacing selected occurrences of the pair OB by BOOB and selected occurences of the triple BOB by O? The answer is yes: . . . BOB → BBOOB → BBOBOOB → BOOOB . . .Turing suggests that any puzzle can be re-expressed as a substitution puzzle. Some row of letters can always be used to represent the ‘starting position’ envisaged in a particular puzzle, e.g. in the case of a chess problem, the pieces on the board and their positions. Desired outcomes, for example board positions that count as wins, can be described by further rows of letters, and the rules of the puzzle, whatever they are, are to be represented in terms of permissible substitutions of groups of letters for other groups of letters.

A General and Sharpened form of Opial's Inequality

Z. Opial [11] proved in 1960 the following theorem:Theorem 1. If u is a continuously differentiable function on [0, b], and if u(0)= u(b)=0 and u(x) > 0 for x ∊ (0, b), then1where the constant b/4 is the best possible.

Infinitesimal Holonomy Groups of Bundle Connections

In Introduction In differential geometry of linear connections, A. Nijenhuis has introduced the concepts of local holonomy group and infinitesimal holonomy group and obtained many interesting results [6].The purpose of the present note is to generalize his results to the case of connections in arbitrary principal fiber bundles with Lie structure groups. The concept of local holonomy group can be immediately generalized and has been already utilized by S. Kobayashi [4]. Our main results are Theorems 4 and 5 on infinitesimal holonomy groups. The proofs depend on a little sharpened form of a theorem of Ambrose-Singer [1]. In the case of linear connections, our infinitesimal holonomy group coincides with that of Nijenhuis, as we shall show in Section 6.

A theorem on cardinal numbers

A classical theorem of Cantor states that the class of all sub-classes of a given class has a cardinal greater than that of the given class. This theorem is here established in a sharpened form, which was suggested to me by a question set by Professor J. M. Whittaker, F.R.S, in the 1950 examination for the Honours B.Sc. Degree at Liverpool.