MATHEMATICAL INDUCTION
... of the axioms was so designed as to incorporate induction as a method of proof. In other words, the intuitive way to deal with induction below is actually a legitimate technique. In what follows, the theory is presented in short sections, each with its own problems. These are rather easy especially ...
... of the axioms was so designed as to incorporate induction as a method of proof. In other words, the intuitive way to deal with induction below is actually a legitimate technique. In what follows, the theory is presented in short sections, each with its own problems. These are rather easy especially ...
A Fixed Point Theorem for G-Monotone Multivalued - PMF-a
... Corollary 2.4, reduces to following corollary by taking V = X, q = 1 and E1 = X × X. Corollary 2.5. Let (X, d) be a complete metric space and let f : X → CB(X) be a multivalued mapping. Also suppose that there exist α ∈ [0, 1) and a lower semi-continuous function ϕ : X → [0, ∞) such that H( f x, f y ...
... Corollary 2.4, reduces to following corollary by taking V = X, q = 1 and E1 = X × X. Corollary 2.5. Let (X, d) be a complete metric space and let f : X → CB(X) be a multivalued mapping. Also suppose that there exist α ∈ [0, 1) and a lower semi-continuous function ϕ : X → [0, ∞) such that H( f x, f y ...
Direct-sum decompositions over local rings
... for some positive integer t. But then tγ ∈ G ∩ Nn = Λ(M ). Now let N = c1 L1 ⊕ · · · ⊕ cn Ln (where the Li are as in the second paragraph of this section). Since tγ ∈ Λ(M ), tN has constant rank by (2) of (1.7). Then N has constant rank, and γ ∈ Λ(M ) by (2) of (1.7). ¤ §2. Realization of expanded s ...
... for some positive integer t. But then tγ ∈ G ∩ Nn = Λ(M ). Now let N = c1 L1 ⊕ · · · ⊕ cn Ln (where the Li are as in the second paragraph of this section). Since tγ ∈ Λ(M ), tN has constant rank by (2) of (1.7). Then N has constant rank, and γ ∈ Λ(M ) by (2) of (1.7). ¤ §2. Realization of expanded s ...
