Mutually Singular implicit functions
Year: 2024
Type: article
Abstract: A continuously differentiable (or smooth or in ๐ถ โฒ (๐1 )) function ๐1from an open set ๐ค which is a subset of a linear space X1 of dimension 2n into a linear space X2 of dimension n has an inverse onto a subspace upto equal dimension. If (s, t) is in ๐ค where ๐1 (๐ ,๐ก) = ๐, ๐1 โฒ (๐ ,๐ก) = ๐invertible at (s, t), then ๐1is invertible in the neighbourhood of (s, t) ๐๐ค. Upto bijection, the invertibility of a multi variate function ๐1is possible and for this ๐1 from X1 is mapped onto X2 so as to get invertible for the 1st n components of the 2n โ tuple. In the present discussion, we focus on two multi variate functions ๐1 and ๐2 or linear mappings from X1 into X2 such that both are smooth in two open sets ๐ค1 and ๐ค2 subsets of ๐1 taking the 1st n โ components of 2n โ tuples in X1 onto ๐2 while ๐2 is from the open set ๐ค2 of ๐1 taking the 2 nd set of n โ components of 2n โ tuples in X1 such that ๐1 โฒ is invertible at (๐ 1,๐ก1 )๐๐ค1 and ๐2โฒ is invertible at (๐ 2,๐ก2 )๐๐ค2 . The sum of derivative of the implicit functions is a continuously differentiable function gives ๐1โฒ(๐ก) + ๐2โฒ(๐ ) = - {๐1๐ โ1๐1ฮฎ + ๐2ฮฎ โ1๐2๐ } ...... 0.1 with ๐1 and ๐2 are the implicit functions, ๐1 = ๐1๐ + ๐1ฮฎ and ๐2 = ๐2๐ + ๐2ฮฎ in which ๐1๐ and ๐2ฮฎ are the invertible parts while ๐1ฮฎ and ๐2๐ are singular parts. Further, the null space of ๐1 is the range space of ๐2 and vice versa leading to ๐1 and ๐2 are mutually singular, denoted by ๐1โด๐2 ......0.2 Key words: linear space, multi variate function or a vector valued function, nonsingular mapping, mutually singular mapping, invertible mapping, null space ๐(๐), range space ๐พ(๐), dimension of a linear space, span, rank and nullity of a linear transformation, contraction, upto invertibility, subset, open mapping.
Authors: T.Srinivasarao, Dr.B.Venkateswarlu, D.Vasubabu
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6893028848
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