Hassler Whitney

Congruent Graphs and the Connectivity of Graphs

We give here conditions that two graphs be congruent and some theorems on the connectivity of graphs, and we conclude with some applications to dual graphs. These last theorems might also be proved by topological methods. The definitions and results of a paper by the author on “Non-separable and planar graphs,” † will be made use of constantly. We shall refer to this paper as N. For convenience, we shall say two arcs touch if they have a common vertex.

On the Abstract Properties of Linear Dependence

Let C1 , C2 ,· · · ,Cm be the columns of a matrix M. Any subset of these columns is either linearly independent or linearly dependent; the subsets thus fall into two classes. These classes are not arbitrary; for instance, the two following theorems must hold

Non-Separable and Planar Graphs*

Introduction. In this paper the structure of graphs is studied by purely combinatorial methods. The concepts of rank and nullity are fundamental. The first part is devoted to a general study of non-separable graphs. Conditions that a graph be non-separable are given ; the decomposition of a separable graph into its non-separable parts is studied; by means of theorems on circuits of graphs, a method for the construction of non-separable graphs is found, which is useful in proving theorems on such graphs by mathematical induction. In the second part, a dual of a graph is defined by combinatorial means, and the paper ends with the theorem that a necessary and sufficient condition that a graph be planar is that it have a dual. The results of this paper are fundamental in papers by the author on Congruent graphs and the connectivity of graphs^ and on The coloring of graphs. X

Tangents to an Analytic Variety

The purpose of this paper is to study the structure of the sets of tangent vectors and tangent planes to a complex analytic variety, particularly in the neighborhood of singular points of the variety. We prove the existence of a stratification of the variety which has nice properties relative to tangent planes. Corresponding properties of real analytic varieties (which are the real parts of of complex ones) may be found by considering the corresponding complex analytic variety.

Elementary Structure of Real Algebraic Varieties

A real (or complex) algebraic variety V is a point set in real n-space R n (or complex n-space C n ) which is the set of common zeros of a set of polynomials. The general properties of a real V as a point set have not been the subject of much study recently (but see for instance [2], [3] and [4]); attention has turned more to the complex case, the complex projective case, and especially the abstract algebraic theory. Facts about the real case are sometimes needed in the applications; proofs are commonly very difficult to locate.

On the Extension of Differentiable Functions

The author has shown previously how to extend the definition of a function of class C m defined in a closed set A so it will be of class C m throughout space (see [l]).1 Here we shall prove a uniformity property: If the function and its derivatives are sufficiently small in A, then they may be made small throughout space.

A characterization of the closed 2-cell

* Presented to the Society, October 31, 1931; received by the editors April 13, 1932. t National Research Fellow. I That is, a point set homeomorphic with the surface of a sphere. See L. Zippin, American Journal of Mathematics, vol. 52 (1931), pp. 331-350; these Transactions, vol. 31 (1929), pp. 744-770; C. Kuratowski, Fundamenta Mathematicae, vol. 13 (1929), pp. 307-318; also references in these papers. ? See Lemma A. II p(p, q) = distance from p to q, or in general, distance between two point sets; 6(S) = diameter of S; VE(S) = those points p for which p(p, S) < e; W,(S) = those points p for which p(p, S) ? e.

Coming Alive in School Math and Beyond.

In spite of ever-increasing efforts, the failure of schools to help children learn mathematics in a relevant and useful manner continues. We know that the enforced rote learning is a direct cause of this, so we try to promote better teaching methods. But having given up on the children, believing most of them incapable, we continue teaching everything with drill and testing, thus ensuring the continued rote learning and confirming our beliefs. Thus the childrens power of thought, vitality and responsibility remain wiped out.A strong shift in attitudes of students to “I can explore, I can control my study and learning” is not difficult to obtain, and we have seen many examples of this, with excellent results. In any community where there is sufficient concern, cooperation and communication, this can be achieved. The need is to let better ways come in in little bits, not by trying to stop standard methods. We describe some ways in which such a process may be carried out.

Origins and basic concepts

There are many notions of dependence in algebra. Besides linear dependence of vectors, there are algebraic and p-dependence of elements in a field extension. (For a definition of p-dependence, see the commentary on Mac Lane [I. 4] in §1.4.) One of the historical forces behind the discovery of the concept of a matroid in the thirties was the recognition that these notions of dependence share many common properties, the most striking being the fact that the maximal independent sets all have the same cardinality. It was natural, in a decade when the axiomatic method was still a fresh idea, to attempt to find the fundamental properties of dependence common to these notions, postulate them as axioms, and derive their common properties from the axioms in a purely abstract manner. This was done by many. (See §1.5 for a complete survey; it was an early testimony to the naturalness and inevitability of the concept of a matroid that all these axiomatizations, discovered independently by very different mathematicians, are all equivalent.) However, except for Whitney’s work, there was no attempt to go beyond the elementary facts and equivalences. This was perhaps due to the fact that a key example, independence of a set of edges in a graph, and hence a key concept, that of a dual graph, were not available to those approaching matroids from an algebraic point of view. Thus, while future historians of mathematics may debate when the definition of a matroid first appeared, there is no doubt that the theory of matroids began in Whitney’s 1935 paper, “On the abstract properties of linear dependence”. This paper is the first paper reprinted in our anthology.