According to RPA [25], the operator products in the last two lines of (10) can be replaced by their average values in ground state |0〉 of the system: a p ′ γ ′ + a p γ → ⟨ 0 a p ′ γ ′ + a p γ 0 ⟩ = δ p p ′ δ γ γ ′ n p γ, (12).
The same problem is also studied for the potential operator with product kernels I α, β defined on a product of two homogeneous groups, where only the right-hand side weight is of product type.
The operator tensor product structure provides a convenient representation of the operators in Eq. (1).
Using this result, we also prove the boundedness of the bilinear fractional integral operator on product Wiener amalgam spaces.
Then, Breaz and Breaz [2] introduced a new integral operator using product n-multiply analytic functions and gave another univalence criteria for such analytic integral operators.
SMSs were (and in some cases, remain) a key profit center for mobile operators, a product which costs almost nothing for the operator yet costs consumers several dollars for tiny "buckets" of messages.
There are basic unary operators (multiplication by a constant scalar, absolute value, logarithm, exponential, exponentiation and filter), binary operators (sum, product, ratio and difference), statistics on sets (mean, median, standard deviation, variance, minimum, maximum) and statistics on pairs of sets (Welch's t test, Mann–Whitney U).
After the application of a search operator, repeated products are removed from the generated individuals.
