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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenThu, 30 Nov 2023 19:43:17 +0000Differential geometries of function space
https://resolver.caltech.edu/CaltechAUTHORS:MICpnas30a
Authors: Michal, Aristotle D.
Year: 1930
In a previous paper (1) the author initiated the study of a species of functional differential geometries. These geometries are the function space analogues of the n-dimensional theories of affinely connected manifolds. An attempt to develop a projective theory in function space was instrumental in showing that the functional geometries which were developed in the cited paper (1) were a truncated form of a much more general situation.https://authors.library.caltech.edu/records/fz1zb-44p57The differential geometry of a continuous infinitude of contravariant functional vectors
https://resolver.caltech.edu/CaltechAUTHORS:MICpnas30b
Authors: Michal, Aristotle D.
Year: 1930
1. Introduction. - A general theory of function space affinely connected manifolds has been developed by the author in several publications (2). In this paper I propose to give a number of new results pertaining to the differential geometry and invariant theory of a continuous infinitude of contravariant functional vectors. An application is made of these results to the differential geometry of functional group vectors of infinite groups of functional transformations. It is my intention to publish the complete results and proofs elsewhere.https://authors.library.caltech.edu/records/9zg5c-5cg44Projective functional tensors and other allied functionals
https://resolver.caltech.edu/CaltechAUTHORS:MICpnas30d
Authors: Michal, Aristotle D.
Year: 1930
Introduction. - Some time ago I discovered a sequence of functionals (1) that is the correspondent in function space (2) of Weyl's projective curvature tensor in n dimensions (3). Since then, I have succeeded in finding a sequence of non-tensor functionals (4) with the property that each functional of the sequence remains unaltered under an arbitrary projective functional transformation (1) of the functional affine connection. It is the object of this note to outline briefly the results obtained in the above studies. A detailed presentation with proofs as well as an account of numerous results and theorems that flow out of the ideas and methods of the present note is reserved for a series of papers to be published elsewhere.https://authors.library.caltech.edu/records/skwgf-vy335Quadratic functional forms in a composite range
https://resolver.caltech.edu/CaltechAUTHORS:MICpnas30c
Authors: Michal, A. D.; Kennison, L. S.
Year: 1930
Transformations of Third Kind in a Composite Range. – Let y(1), y(2), …y(n) be n independent variables and y(α) a real continuous function of a real variable α, defined for a ≤ a ≤ b.https://authors.library.caltech.edu/records/41xmr-4qw47"Riemannian" differential geometry in abstract spaces
https://resolver.caltech.edu/CaltechAUTHORS:MICpnas35
Authors: Michal, A. D.
Year: 1935
1. Introduction. - In this note brief indications are given of a set of postulates for "Riemannian" differential geometry in abstract spaces. The detailed treatment of this geometry has been included in a comprehensive memoir that I intend to publish elsewhere. An introduction to more general abstract differential geometries with one or several linear connections has been given by me in another comprehensive memoir.(2)https://authors.library.caltech.edu/records/r6fct-che90Differentials of Functions with Arguments and Values in Topological Abelian Groups
https://resolver.caltech.edu/CaltechAUTHORS:20150811-112829955
Authors: Michal, A. D.
Year: 1940
PMCID: PMC1078188
By a topological abelian group T (t.a.g. T) we shall mean an abstract abelian group-written additively-such that (a) the function x + y and the inverse function -x are continuous functions (neighborhood continuity) of both variables x and y and of the variable x, respectively, with respect to a postulated Hausdorff topology; (b) given any y ε T and any Hausdorff neighborhood U of 0 ε T, there exists a "positive integer" n such that y ε nU.
In this note we shall give brief indications of a differential calculus for functions f(x) with x e t.a.g. T_1 and values in a t.a.g. T_2. Proofs and further
developments will appear elsewhere.https://authors.library.caltech.edu/records/40g6x-hgm53