If ∗ be a t-norm and { x n } n ≥ 1 is a sequence of numbers in [ 0, 1 ], one defines recurrently ∗ i = 1 n x i by ∗ i = 1 1 x i = x 1 and ∗ i = 1 n x i = ∗ ( ∗ i = 1 n − 1 x i, x n ), ∀ n ≥ 2. ∗ i = 1 ∞ x i is defined as lim n → ∞ ∗ i = 1 n x i and ∗ i = n ∞ x i as ∗ i = 1 ∞ x n + i.
(ii) If ∗ be a t-norm and { x n } n ≥ 1 is a sequence of numbers in [ 0, 1 ], one defines recurrently ∗ i = 1 n x i by ∗ i = 1 1 x i = x 1 and ∗ i = 1 n x i = ∗ ( ∗ i = 1 n − 1 x i, x n ), ∀ n ≥ 2. ∗ i = 1 ∞ x i is defined as lim n → ∞ ∗ i = 1 n x i and ∗ i = n ∞ x i as ∗ i = 1 ∞ x n + i. .
If T is a t-norm and ( x 1, x 2, …, x n ) ∈ [ 0, 1 ] n ( n ∈ N ∗ ), one can define recurrently ⊤ i = 1 n x i = T ( ⊤ i = 1 n − 1 x i, x n ) for all n ≥ 2. One can also extend T to a countable infinitary operation by defining ⊤ i = 1 ∞ x i for any sequence ( x i ) i ∈ N ∗ as lim n → ∞ ⊤ i = 1 n x i.
Since my release eight years ago I have thought recurrently that while prison is a human hell, it offers exceptional opportunities for inner peace by creating, if forcibly, a haven from the distractions of ordinary life.
It is a recurrent – and recurrently urgent – theme in English football.
The standard approach to identify the driver genes is to identify recurrently mutated genes, i.e. those genes whose observed frequency of mutations is significantly higher than the expected passenger mutation probability[ 1- 3].
I would like to replace this recurrently hamstrung, self-defeating and mostly isolated self with the more promising image: the person in communion with other persons.
Although vegetation changes in African drylands, i.e. desertification, have been recurrently discussed [1] [3], changes in hyper-arid areas have long been neglected [4].
If is a and, then is defined recurrently by 1, if and for all.
If T is a t-norm and ( x 1, x 2,..., x n ) ∈ [ 0, 1 ] n ( n ≥ 1 ), ⊤ i = 1 ∞ x i is defined recurrently by ⊤ i = 1 1 x i = x 1 and ⊤ i = 1 n x i = T ⊤ i = 1 n - 1 x i, x n for all n ≥ 2. T can be extended to a countable infinitary operation by defining ⊤ i = 1 ∞ x i for any sequence ( x i ) i ∈ N * as lim n → ∞ ⊤ i = 1 n x i. Definition 2.3.
Recall (see [38, 39]) that if T is a t-norm and {x n } is a given sequence of numbers in [0, 1], then T i = 1 n x i is defined recurrently by T i = 1 1 x i = x 1 and T i = 1 n x i = T ( T i = 1 n - 1 x i, x n ) for n ≥ 2. T i = n ∞ x i is defined as T i = 1 ∞ x n + i - 1.
