By W. B. Vasantha Kandasamy: Florentin Smarandache, K. Ilanthenral

ISBN-10: 1931233977

ISBN-13: 9781931233972

Within the smooth age of improvement, it has turn into crucial for any algebraic constitution to get pleasure from higher attractiveness and examine value purely whilst it has large purposes to different fields. This new algebraic notion, Linear Bialgebra, is person who will locate functions to a number of fields like bigraphs, algebraic coding/communication conception (bicodes, most sensible biapproximations), Markov bichains, Markov bioprocess and Leonief financial bimodels: those also are introduced out during this ebook. the following, the linear bialgebraic constitution is given sub-bistructures and super-structures known as the smarandache neutrosophic linear bialgebra so as to simply yield itself to the above purposes.

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3: Let G = G1 ∪ G2 be a bigroup with G1 = {set of all n × n matrices under '+' over the field of reals} and G2 = { set of all n × n matrices A with |A| ≠ {0} with entries from Q}, (G1, +) and (G2, ×) are groups and G = G1 ∪ G2 is a non-commutative bigroup of infinite order. In this section we introduce the concept of bivector spaces and S-bivector spaces. The study of bivector spaces started only in 1999 [106]. Here we recall these definitions and extend it to the Smarandache bivector spaces. 2: Let V = V1 ∪ V2 where V1 and V2 are two proper subsets of V and V1 and V2 are vector spaces over the same field F that is V is a bigroup, then we say V is a bivector space over the field F.

V = V1 ∪ V2 is a weak linear bialgebra over F1 ∪ F2. Now consider the subbigroup, W = W1 ∪ W2 where W1 = {Set of all polynomials in x with coefficients from the ( 2 ) } and W = {set of all 3 × 3 matrices with entries from Q ( 3, 7 ) }. F = Q ( 2 ) ∪ Q ( 3, 7 ) is a field Q 2 bifield and the subbigroup W = W1 ∪ W2 is a strong linear bialgebra over the bifield F = Q 2 ∪ Q 3, 7 . Thus ( ) ( ) V has a subset W such that W is a strong linear bialgebra so V is a strong special weak linear bialgebra. Now we give condition for a weak linear bialgebra to be strong special weak linear bialgebra.

T = T1 ∪ T2 ‘∪’ is just only a notational convenience. iii. T(ν) = T1 (ν) if ν ∈ V1 and T (ν) = T2 (ν) if ν ∈ V2 . Yet another marked difference between bivector spaces and vector spaces are the associated matrix of an operator of bivector spaces which has m1 + n1 rows and m + n columns 42 where dimension of V is m + n and dimension of W is m1 + n1 and T is a linear transformation from V to W. If A is the associated matrix of T then. ⎡ Bm1×m A= ⎢ ⎣⎢ O m1×n O n1×m ⎤ ⎥ C n1×n ⎦⎥ where A is a (m1 + n1 ) × (m + n) matrix with m1 + n1 rows and m + n columns.

### Introduction to Linear Bialgebra by W. B. Vasantha Kandasamy: Florentin Smarandache, K. Ilanthenral

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