By W. B. Vasantha Kandasamy, Florentin Smarandache

Period mathematics, or period arithmetic, was once constructed within the Nineteen Fifties and Sixties as an method of rounding blunders in mathematical computations. notwithstanding, there has been no methodical improvement of period algebraic constructions to this date.

This ebook presents a scientific research of period algebraic buildings, viz. period linear algebra, utilizing periods of the shape [0, a].

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Let ª[0,1] [0, 2] [0,3] º x= « » and y = ¬[0, 4] [0,1] [0, 2]¼ be elements of V. 44 ª[0, 4] [0,8] [0,12]º «[0,1] [0, 4] [0,8] » ¬ ¼ We see {x, y} forms a linearly dependent subset of V. For we see x = 4y where 4 Z15 = G. 45: Let V be a group interval vector space over a group G. Let H be a proper subgroup of G. If W V is such that W is a group interval vector space over the subgroup H of G then we define W to be a subgroup interval vector subspace of V over the subgroup H of G. If W happens to be both a group interval vector subspace as well a subgroup interval vector subspace then we define W to be duo subgroup interval vector subspace.

37: Let ° ° ° V= ® ° ° °¯ ½ ª[0, a1 ] º ° «[0, a ]» 2 » « a i Z90 °° ª[0, a1 ] [0, a 2 ]º « » , [0, a ],[0, a ] , [0, a ] ¾ 1 2 3 «[0, a ] [0, a ]» « » 1 d i d 5° 3 4 ¼ ¬ «[0, a 4 ]» ° «¬[0, a 5 ]»¼ °¿ be a group interval vector space over the group Z90 = G , under addition modulo 90. 38: Let a Z14 °½ ° ª[0, a1 ] º V= ®« , [0, a1 ],[0, a 2 ][0, a 3 ] i ¾ » 1 d i d 3¿° ¯° ¬[0, a 2 ]¼ be the group interval vector space over the group Z14 = G under addition modulo 14. Now we proceed on to define substructures of group interval vector spaces.

Define IK : V o I [0, 1] by 59 1 °[0, ] if a i z 0 IK [0, ai] = ® a i ; °[0,1] if a 0. i ¯ IKV is a set fuzzy interval vector space. 4: Let V be a set interval linear algebra over the set S. A set fuzzy interval linear algebra (or fuzzy set interval linear algebra) (V, KI) or VKI is a map KI: V o I [0, 1] such that KI(a + b) t min(KI(a), KI(b)) for every a, b V. We will illustrate this situation by some examples. 4: Let f ½ V = ®¦ [0,a i ]x i a i Z {0}¾ ¯i 0 ¿ be a set interval linear algebra over the set S = {0, 1, 2, 5, 7, 13, 16}.