In fact, we prove that the middle variables of M ⊆ X 6 are unnecessary.
One of the key objectives of this proof is to show that, in Theorem 4.3, the middle variables of M are not necessary.
One of the key objectives of this subsection is to prove that, in Theorem 1.1, the middle variables of M are not necessary.
In fact, the previous proof shows that two conditions are not necessary in Theorem 3.3: neither the first property of ( F ∗, g ) -invariant sets nor the middle variables of M in X 6.
A -ternary ring is a complex Banach space, equipped with a ternary product of into, which is -linear in the outer variables, conjugate -linear in the middle variable and associative in the sense that and satisfies and.
A C*-ternary ring is a complex Banach space A equipped with a ternary product which is associative and linear in the outer variables, conjugate linear in the middle variable, and ||[x, x, x]|| = ||x||3 (see, [13]).
A C*-ternary algebra is a complex Banach space A, equipped with a ternary product (x, y, z) α [x, y, z] of A3 into A, which is ℂ-linear in the outer variables, conjugate ℂ-linear in the middle variable, and associative in the sense that [x, y, [z, w, v]] = [x, [w, z, y], v] = [[x, y, z], w, v], and satisfies ||[x, y, z]|| ≤ ||x|| · ||y|| · ||z|| and ||[x, x, x]|| = ||x||3 (see [7, 8]).
A (C^ -ternary algebra is a C^ -ternaryalgebrace A, equisped with a ternary producomplex y, z) mapsto[x, y, z]) of (A^{3}) into A, which is C-linear in the outer variaBanachconjugate C-linear in the middle variable, and aspaceAtivequipped sense that ([x, y, [z, with]] = [x, [w, z, y], v] = [[x, y, z], w, v]), and saternary (|[x, y, z]| le|x| cdot|y| cdot |z|) and (|[x, x, x]| = |x|^{3}) (see [4]).
The middle level variables represent the class and the lower level represents various features of interest.
Middle column: variable M, N = 257, SNR = 10 dB.
The same authors compared a group of 40 children with MwA with 70 children without headaches for nine temperamental variables (Middle Childhood Temperamental Questionnaire): activity, predictability, approach, adaptability, intensity, mood, persistence, distractibility and threshold.
