Let K be a nonempty proper subset of a real Banach space E. A map (A Krightarrow K) is called a strict contraction if there exists (k in[0,1)) such that (|Ax-Ay| leq k|x-y|) for all (x,y in K), and A is called nonexpansive if, for arbitrary (x, y in K), (|A x - Ay| leq|x - y|).
Let (mathcal{A}=(a_{i_{1}cdots i_{m}})inmathbb{R}^{[m, n]}) be an irreducible M-tensor, S be a nonempty proper subset of N, S̅ be the complement of S in N. Then begin{aligned} tau(mathcal{A} geqmin Bigl{ min_{iin S}max _{jinoverline{S}}L_{ij}(mathcal{A}), min_{iinoverline{S}} max_{jin S}L_{ij}(mathcal{A}) Bigr}.
Next, a comparison theorem is given for Theorems 1, 2 and 4. Let (mathcal{A}=(a_{i_{1}cdots i_{m}})in{mathbb{C}}^{[m,n]}), S be a nonempty proper subset of N. Then begin{aligned} Delta^{cap}(mathcal{A})subseteqDelta^{S}(mathcal{A}) subseteqDelta (mathcal{A}).
Let S be a nonempty proper subset of V. We use (G[S]) to denote the subgraph of G induced by S. The edge cut of G, denoted by (partial(S)), is a subset of (E(G)) of the form ([S,bar{S}]), where (bar{S}=Vbackslash S).
Let (mathcal{A}) be a tensor with order m and dimension (ngeq2), and S be a nonempty proper subset of N. Then sigma(mathcal{A})subseteqmathcal{G}^{S}(mathcal{A})subseteq mathcal{K}(mathcal{A}), where (mathcal{K}(mathcal{A})) is a Z-eigenvalue inclusion set in Lemma 1.
Let (mathcal{A}) be a ((p,q)) th order (ntimes n) dimensional nonnegative rectangular tensor, S be a nonempty proper subset of N, S̄ be the complement of S in N. Then Psi^{S}(mathcal{A} leq U^{S}(mathcal{A} leqPhi( mathcal{A} leqmax_{i,jin N}bigl{ R_{i}( mathcal{A}),C_{j}(mathcal{A} bigr}.
Scott proved that if A is a nonempty proper subset of λ-terms that is closed under equality then A is not recursive.
If is a nonempty proper subset of then contains the product of at least one matrix and one matrix Thus, using the rotation property, we can write (3.10).
(mathcal{A}) is called reducible if there is a nonempty proper subset (Ksubset N) such that a_{i_{1}i_{2}cdots i_{m}}=0,quad forall i_{1}in K, forall i_{2},ldots, i_{m}notin K.
By the assumption that there exists a finite m-closed set F such that x ∈ F, one can easily check that S ∪ F c is a nonempty proper cofinite m-open set.
Suppose ( X, M ) Open image in new window and ( Y, N ) Open image in new window are two m-spaces, S is a nonempty proper cofinite m-open subset of X, and T is a nonempty proper cofinite m-open subset of Y, then there exist at least two (cofinite) maximal m-open sets A × Y and X × B in product m-space such that S × T ⊆ A × Y and S × T ⊆ X × B.
