Let the matrix of operator (1) belong to the class O n +, n ≥ 0. Assume that at least one of the conditions (44) and (45) is valid.
Suppose that the matrix of operator (1) belongs to the class O n +, n ≥ 0. Let the operator (1) be compact from l p,v into l q,u.
In general case, if matrices ( a i, j k ) of operators ( A k + f ) j = ∑ j = 1 i a i, j k f j belong to the classes O m k +, k = 1, …, n, then the matrix of operator A n + ≡ A 1 + ∘ A 2 + ∘ ⋯ ∘ A n + belongs to the class O m +, where m = ∑ k = 1 n m k + n - 1.
Finally, we obtain generalizations of many of these results to matrices of operators, which we apply to the study of representations of certain subalgebras of the n × n matrices.
A matrix representation of operators with respect to a nice wavelet base plays an important role in the formulation.
The natural frequencies are the eigenvalues of a matrix of differential operators.
The general FE formulations [ 18] are used to obtain the stiffness matrix, K, such as (4) K = ∑ e ∫ V e B T · D · B d V, where e is the number of elements, V e is a typical volume element, B is the strain-displacement matrix, and D is the matrix of differential operators that convert displacement to strain.
Suppose that the matrix of the operator (2) belongs to the class O m -, m ≥ 0 and (3) holds.
By the definition of O 0 -, the matrix of the operator (2) has the form a i, j ( 0 ) = β i ∀ i ≥ j ≥ 1.
In the G matrix of the operator longest paths in the dataflow graph, the value g i, j must be less than or equals to Tstage in order for the i and j operators to be included in one stage.
