The Journal of the Indian Mathematical Society

D. D. Kosambi ¹

Affiliations
1 Muslim University, Aligarh, IN

Abstract

The purpose of the present work is the discussion of the possible types of homogeneous linear differential equations of which the solutions generate continuous groups of transformations in a space of n dimensions. The geometrical interpretations of the results form perhaps the sole interesting element in a tedious formal presentation. A more elegant form of reasoning is greatly to be wished for, but seems to be impossible at present. The entire paper considers problems inverse to the usual discussions of continuous groups of deformation of a space into itself.

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D. D. Kosambi ¹

Affiliations
1 Poona, IN

Abstract

The main purpose of this note is to develop statistical methods for discrimination between samples consisting of whole curves.

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D. D. Kosambi

Abstract

This note deals primarily with infinite sets of continuous functions Φ_n(x) uniformly bounded and orthonormal over (0,I), i.e. the sequence being Φ₁, Φ₂, ........., Φ_n,

∫Φ,Φ₁dx = δ_n, |Φ_n|≤M for all n, 0≤x≤I.

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D. D. Kosambi ¹

Affiliations
1 Poona, IN

Abstract

In the first paper with the same title, it was shown that the fundamental ideas of the theory of relativity could be applied directly to the trajectories of particles, without assuming the existence of a Riemann metric. The results of that paper are valid also for more than four dimensions, with the corresponding extended Lorentz group; in particular, to the Kaluza-Klein theory in five and the Proca-Goudsmit in six dimensions.

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D. D. Kosambi ¹

Affiliations
1 Poona, IN

Abstract

This note attempts a generalization of the concept of isotropy for spaces defined by "paths", i.e. the solution curves of

xⁱ+αⁱ(x,x',t)=0; i=1,.......,n. xⁿ=dxⁱ/dt etc.                             (I)

Besides the tensor summation convention for repeated indices.

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D. D. Kosambi ¹

Affiliations
1 Fergusson College, Poona, IN

Abstract

After the comprehensive works of Bortolotti on partial differential equations of the second order from the differential geometer's point of view, and the equally comprehensive memoir of Kawaguchi and Hombu on systems of higher order, the present note serves only to show that slightly different results can be obtained by keeping to the point of view that I have used in my former papers. The method, in particular, is to handle such systems as obey the following postulates and for the special transformation groups under which the postulates hold: (1) the system of equations transforms according to the tensor law; (2) the equations of variation of the given system are also tensorial when the variation itself is a vector; (3) there exists at least one operator which is vectorial in character and corresponds to total differentiation with respect to one of the independent variables.

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D. D. Kosambi ¹

Affiliations
1 Poona, IN

Abstract

Given any n - dimensional Euclidean space in general co-ordinates, with a metric ds²=g_ijdxⁱdx^j, Laplace's equation for the given system.

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D. D. Kosambi ¹

Affiliations
1 Poona, IN

Abstract

Curves representing solutions of (1) can be regarded as the generalized autoparallel lines or paths of a space, and the intrinsic differential geometry thereof is developed from two main assumptions: (a) the tensor invariance of all fundamental equations, including (1), and (b) the existence of a vectorial operator, the vanishing of which defines a parallelism making solutions of (1) autoparallel lines.

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