# American Journal of Mathematics (Google eBook)

Johns Hopkins University Press, 1919 - Mathematics

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Page 302 - ... through any two given points. I have not however determined whether every such space is descriptive. Definitions. — A point P is said to be a limit point of a point-set M if, and only if, every region that contains P contains at least one point of M distinct from P. The boundary of a point-set M is the set of all points [X\ such that every region that contains X contains at least one point of M and at least one point that does not belong to M. If M is a set of points, M' denotes the point-set...
Page 325 - XVIII in progress. \$5 per volume. (Foreign postage, fifty cents.) Johns Hopkins University Circular, including the President's Report, Annual Register, and Medical Department Catalogue.
Page 325 - XLI in progress. \$6 per volume. (Foreign postage, fifty cents.) American Journal of Philology. Edited by BL GILDERSLEEVE and CWE MILLET;, managing Editor.
Page 267 - A most useful feature is that the determinant of a product of matrices is equal to the product of...
Page 266 - В 1 a. HB PHILLIPS. Functions of matrices. It is the purpose of the present paper to study the functions represented by polynomials or convergent series in a matrix or a finite number of matrices. As the work is concerned mainly with the roots of the matrices, the fundamental facts about the roots are first briefly developed (p.
Page 302 - ... that satisfies 22 satisfies also 23, but not conversely. In every space satisfying 22 there exists infinitely many open curves through any two given points. I have not however determined whether every such space is descriptive. Definitions. — A point P is said to be a limit point of a point-set M if, and only if, every region that contains P contains at least one point of M distinct from P. The boundary of a point-set M is the set of all points [X] such that every region that contains X contains...
Page 238 - Hence any modular invariant can be expressed in one and but one way as a linear homogeneous function of the characteristic invariants. Moreover, the number of linearly independent modular invariants equals the number of classes. For example, using (19), we see that a complete set of linearly independent modular invariants of the quadratic form qm modulo p (p > 2) is given by (23) /0, Ar, A* (r= 1, ...,m-1), Dk (k = 1, . . ., p-1).
Page 133 - Mathieu1 who showed how the problem could be reduced to the solution of certain ordinary linear differential equations. But he found these equations to be so unmanageable that he contented himself with approximating to their solutions for the special case of an ellipsoid of revolution.
Page 301 - Hilbert's hypotheses are far more general. Without doubt this is still not entirely satisfactory, since though the form of the group is supposed any whatever, its matter, that is to say the plane which undergoes the transformations, is still subjected to being a number-manifold in Lie's sense. Nevertheless, this is a step in advance, and besides Hilbert analyzes better than anyone before him the idea of number-manifold and gives outlines which may become the germ of an assumptional theory of analysis...
Page 183 - ... to coincide with x. This remark applies also to similar situations below. ' The functions u(g, x) and l(g, x) are often, though not quite unobjectionably, called the 'maximum' and the 'minimum