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Kosambi, D. D.
- On Differential Equations with the Group Property
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1 Muslim University, Aligarh, IN
1 Muslim University, Aligarh, IN
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The Journal of the Indian Mathematical Society, Vol 19 (1932), Pagination: 215-219Abstract
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.- Statistics in Function Space
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1 Poona, IN
1 Poona, IN
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The Journal of the Indian Mathematical Society, Vol 7 (1943), Pagination: 76-88Abstract
The main purpose of this note is to develop statistical methods for discrimination between samples consisting of whole curves.- On the Zeros and Closure of Orthogonal Functions
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The Journal of the Indian Mathematical Society, Vol 6 (1942), Pagination: 16-24Abstract
This note deals primarily with infinite sets of continuous functions Φn(x) uniformly bounded and orthonormal over (0,I), i.e. the sequence being Φ1, Φ2, ........., Φn,
∫Φ,Φ1dx = δn, |Φn|≤M for all n, 0≤x≤I.
- Path Equations Admitting the Lorentz Group-II
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1 Poona, IN
1 Poona, IN
Source
The Journal of the Indian Mathematical Society, Vol 5 (1941), Pagination: 62-72Abstract
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.- The Concept of Isotropy in Generalized Path-Spaces
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1 Poona, IN
1 Poona, IN
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The Journal of the Indian Mathematical Society, Vol 4 (1940), Pagination: 80-88Abstract
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.
- The Tensor Analysis of Partial Differential Equations
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1 Fergusson College, Poona, IN
1 Fergusson College, Poona, IN
Source
The Journal of the Indian Mathematical Society, Vol 3 (1939), Pagination: 249-253Abstract
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.- Differential Geometry of the Laplace Equation
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1 Poona, IN
1 Poona, IN
Source
The Journal of the Indian Mathematical Society, Vol 2 (1937), Pagination: 141-143Abstract
Given any n - dimensional Euclidean space in general co-ordinates, with a metric ds2=gijdxidxj, Laplace's equation for the given system.- Collineations in Path-Space
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1 Poona, IN
1 Poona, IN