If (2.5) takes the form of the equality, then there exists a pair of non-zero constants and such that (2.6).
If this statement is false, then there exist a pair of constants T > 0 and ε ∈ ( 0, 1 ) such that P { τ ∞ ⩽ T } > ε.
If this statement is false, then there exists a pair of constants (T>0) and (varepsilonin 0,1)) such that (P{tau_{infty}leq T}>varepsilon), hence there exists an integer (k_{1}>k_{0}) such that P{tau_{k}leq T}geqvarepsilon quad mbox{for all } k>k_{1}.
(see, [5]) If φ : X ⊸ Y and ψ : Y ⊸ T are admissible, then the composition ψ ○ φ : X ⊸ T is admissible and for every (p1, q1) ⊂ φ and (p2, q2) ⊂ ψ there exists a pair (p, q) ⊂ ψ ○ φ such that q 2 * p 2 * - 1 ∘ q 1 * p 1 * - 1 = q * p * - 1. Let φ : X ⊸ X be an admissible map.
Constraints (1d -(1e) force variables z ij s to be one if in the optimal solution to the problem there exist a pair of adjacent vertices i j ∈ V having a different value at the s-th coordinate.
By introducing a definition for the coupled lower and upper solutions of BVP (1.1) and (1.2), we obtain the existence of solutions of the problem based on the assumption that there exists a pair of coupled lower and upper solutions.
For example, it is shown that for ducted shear flows, there exist a pair of even and odd eigenfunctions, in the absence of critical levels.
For each case, there exist a pair of differential equations coupled in terms of the flexural displacement and the angle of rotation due to bending.
We also provide an analytical proof of the D-optimality when there exist a pair of blocks of odd size and remaining blocks are of even size.
Let J∗T v) denote the set of all integers s such that there exists a pair of disjoint S 2,4,v s intersecting in s triangles.
The flower intersection problem for Steiner systems is the determination of all pairs (v,suchuch that there exists a pair of Steiner systems (X,B1) and (X,B2) of order v having a common flower F satisfying |(B1∖F)∩(B2∖F)|="s.
