Interval Linear Algebra by W. B. Vasantha Kandasamy, Florentin Smarandache

By W. B. Vasantha Kandasamy, Florentin Smarandache

Period mathematics, or period arithmetic, was once built within the Nineteen Fifties and Sixties as an method of rounding error in mathematical computations. even if, there has been no methodical improvement of period algebraic buildings to this date. This publication presents a scientific research of period algebraic constructions, viz. period linear algebra, utilizing durations of the shape [0, a].

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The map T: R o S is defined by T([0, n]) = [0, n], n  Z+ ‰ {0} is a semigroup interval linear transformation. 26: Let R = {[n, 5n] | n  Z+ ‰ {0}} and S = {[n, 5n] | n  R+ ‰ {0}} be two semigroup interval linear algebras defined over the semigroup F = Z+ ‰ {0}. Define T: Ro S by T {[n, 5n]} = [n, 5n], for all [n, 5n]  R. It is easily verified T is a semigroup interval linear transformation of R to S and infact T is an embedding. We will give an example of a semigroup interval linear operator.

Let S = {[0, n] | n  Z36} be a semigroup interval vector space of S over the semigroup Z36 = F. Choose P1 = {[0, 2n] |n  Z36} = {[0, 0], [0, 2], [0, 4], [0, 6], I ; P1 is a semigroup interval vector subspace …, [0, 34]} Ž Z36 of S over the semigroup F = Z36. P2 = {[0, 4n] | n  Z36} = {[0, 0], [0, 4], [0, 8], [0, 12], [0, 16], [0, 20], [0, 24], [0, 28]} Ž S is a semigroup interval vector subspace of S over the semigroup F 32 = Z36. P3 = {[0, 3n] / n  Z36} Ž S is a semigroup interval vector subspace of S over F = Z36.

Now we will proceed onto define the notion of semigroup linearly independent linearly dependent interval subset of a semigroup interval vector space. 3: Let S be a semigroup interval vector space over the semigroup F. A set of interval elements B = {s1, s2, …, sn} of S is a said to be a semigroup linearly independent interval subset if si z csj; for all c  F and si, sj  B; i z j; 1 d i, j d n. If for some si = csj, c  F; izj; si, sj  B then we say the semigroup interval subset is linearly dependent or not linearly independent.

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