Linear Associative Algebras

Algebras Having Linear Multiplicative Complexities

“…The polynomial $(t) = t z + t + 1 is irreducible over Zz since neither 0 nor 1 is a root; thus K = Z2[t]/(ck(t)) = z2(a) is the (unique up to isomorphism) Galois field of four elements, with multiplication table 1 Now arK --> 3 since K is a 2-dimensional division algebra over Zz; however, Theorem 7 does not apply since Zz does not contain at least 2(2) -1 = 3 elements. In spite of this, arK ~ 3 (and so arK = 3), as is shown by the "one-liner" (x~ + XzCQ(yl + yza) = x2Y2 + x~y~(1 + a) + (x~ + x2)(y~ + y2)a, which we derive from the following decomposition of the multiplication table (viewed as a matrix), keeping in mind that in any algebraA over Z,z,a +a =0:…”

“…The familiar equation xy = (x~ya -x2Y2) + t(x2yl + x~y2) Is obtained by first using bilinearity to yield the purely formal expansion xy = xly~ll + xlyzli + x2yltI + x2yzii and then simplifying via the multiplication table 1 t t -1 defined only for the basis elements. Equating coefficients of linearly independent elements yields z~ = xly~ -x2y2 and z2 = x2yl + xly2, which in matrix form can be written as…”

The Application of a Linear Algebra to the Analysis of Mutation Rates

“…We therefore introduce in Section 3.1 a truncated clone size distribution and in Section 3.2 we formalize the operation of truncated discrete convolution in terms of which we can develop the probability distributions for the number of colonies arising from multiple clones. We note in Section 3.3 that, with the addition and scalar multiplication by reals the resulting structure is a commutative linear associative algebra (see, for instance, Abian, 1971). Because the number of clones in a culture is a Poisson-distributed random variable, the number of colonies detected will be compound Poisson.…”

Spectral representation of 2D NMR spectra by hypercomplex numbers

“…[3] With these definitions of sum and product, it can been shown that H forms a linear associative and commutative algebra of order 4 on the field of the real numbers (8).…”

“…Hypercomplex algebras are linear algebras (8) which are presented as a generalization of complex-number algebra. Hypercomplex algebras have already been successfully 120 MARC A. DELSUC used in gravitation theory (9, IO), in particle classification theory (II), and in NMR for the representation of composite pulses (12).…”