(H,*) and (K,*') are called isomorphic [H.sub.v]-groups, and written as H [congruent to] K, if there exists a bijective
function f: R [right arrow] S that is also a homomorphism.
Let E : [F.sup.n.sub.2] [right arrow] [F.sup.n.sub.2] be a encryption algorithm of a block cipher, whose non-linear components are the bijective
Let [phi] : X [right arrow] Y be a bijective
function, (K, [K.sup.*]) be an (L, M)-dffb on X, and (B, [B.sup.*]) be an (L, M)-dffb on Y.
relation between the refractive index of pure sucrose solution and its concentration can be used to titrate sucrose in aqueous solutions .
Then [phi] is injective and so bijective
. In fact, if [rho], [sigma] [member of] S(H) with [phi]([rho]) = [phi]([sigma]), then, for t [member of] (0,1),
With this observation, we may as well say that x [member of] [[u].sub.W] if and only if there are multisimilar constant [c.sub.W](x, u) and a bijective
function f on W defined as
By [18, Example 3.1.27]; the restriction continuous map R : [C.sub.p]([[omega].sub.1]) + 1) [right arrow] [C.sub.p]([[omega].sub.1]) is bijective
. However R is not open because the set
His topics include the q-binomial theorem, Heine's transformation, the Jacobi triple product identity, the Rogers-Fine identity, Bailey chains, WP-Bailey pairs and chains, further results on Bailey/WP-Bailey pairs and chains, bijective
proofs of basic hypergeometric identities, q-continued fractions, Lambert series, and mock theta functions.
(2) Let [u.sub.n] : [P.sub.n] [right arrow] [Z.sub.N(n)] be a bijective
function such that [u.sub.n]([R.sub.1]) [less than or equal to] [u.sub.n]([R.sub.2]) if [o.sub.n]([R.sub.1]) [[less than or equal to].sub.n] [o.sub.n]), where [R.sub.1], [R.sub.2] [member of] [P.sub.n]
But any bijective
linear operator in the Minkowski space-time, preserving the Lorentz-Minkowski pseudo-metric, belongs to the general Lorentz group , and it can not be coordinate transform for superluminal reference frame.