It has been shown that for any nonsingular matrix M, there exists a finite set of 'unmixing' matrices S such that at least one member Si ϵ S will exhibit the property that MSi will be stable, i.e. MSi will be a Hurwitz Matrix.
This means there exists a finite set of points { x i } i = 1 n in X such that a = x 1, b = x n and d ( x i, x i + 1 ) < ε for all i = 1, …, n − 1.
Moreover, as M has finite dual Goldie dimension and M is square-free (because it is semiboolean and idempotents in ({{mathrm{End}}}_R(M /J({{mathrm{End}}}_R(M))) lift to idempotents of ({{mathrm{End}}}(M)) [11]), we deduce that there exists a finite set of non-isomorphic indecomposable projective modules ({P_i}_{i=1}^n) such that (P=oplus _{i=1}^n P_i).
(2) There exists a finite set of non-isomorphic indecomposable injective modules ({E_i}_{i=1}^n), with (nge 2), such that (E(M =oplus _{i=1}^n E_i) and ({{mathrm{End}}}(E_i)/J({{mathrm{End}}}(E_i cong mathbb {F}_2) for every (i=1,ldots,n). .
There exists a finite set of non-isomorphic indecomposable injective modules ({E_i}_{i=1}^n), with (nge 2), such that (E(M =oplus _{i=1}^n E_i) and ({{mathrm{End}}}(E_i)/J({{mathrm{End}}}(E_i cong mathbb {F}_2) for every (i=1,ldots,n).
We say that ({x,y}) is ε-chainable in S if there exists a finite set of points ({x_{0} = x, x_{1}, ldots, x_{p-1}, x_{p} = y} subset S) ((p geq1)) such that for all (lambdain 0, 1]), F(x_{i-1}, x_{i}) (varepsilon) > 1-lambdaquad mbox{for all }i= 1,2, ldots, p. Let ((S,F,T)) be a Menger PM-space, (varepsilon> 0), (kappain (0, 1)) and (f: S to S) be a multivalued mapping.
We say that a linear operator a on (C(X^omega, Bbbk )) is automatic if there exists a finite set (mathfrak {A}) of operators such that (ain mathfrak {A}), and for every (a'in mathfrak {A}) all entries of the matrix (Xi _{mathsf {B}}(a')) belong to (mathfrak {A}).
Since is compact, there exists a finite set such that for each, there exists with.
Since is compact, there exists a finite set such that (3.3).
If it is false, then there exist a finite set and with and such that (2.17).
Let X be a finite set of labels.
