Sentence examples for by the linear operator from inspiring English sources

Denote by the linear operator from onto defined by (2.4).

(HA):  The (C_{0}) semigroup ({T t)}_{tgeq0}) generated by the linear operator A is equicontinuous.

(i) If and, where is the first eigenvalue of with Dirichlet condition, or if, and then, under the hypotheses (H1)–(H3), the semigroup generated by the linear operator of the system is exponentially stable.

The considered classes are defined by using the convolution φ ∗ f or equivalently by the linear operator J φ : M ( p, k ) → M ( p, k ), J φ ( f ) = φ ∗ f.

Specifically, we prove that under some conditions over the coefficients, the semigroup generated by the linear operator of the system is exponentially stable, and under other conditions we prove that for all the system is approximately controllable on.

Due to the representation of the solution (6), it follows that a solution can only be as spatially regular as the stochastic convolution ∫ 0 t e − α ( t − s ) d W s. In the present case, the semigroup generated by the linear operator is not smoothing in contrast to, e.g., the semigroup generated by the Laplacian in the heat equation.

The purpose of this paper is to introduce two novel subclasses and of meromorphic valent functions by using the linear operator.

We use two methods 1) the computation of the eigenvalues of the linear operator defined by the linearized equations and 2) the formulation of the problem as a fixed point problem.

Let,,,, the identity operator and, The operator is the trace operator such that is well defined and belongs to for each and the operator is given by, where and are usual Sobolev spaces on We define the linear operator by where is the unique solution to the Dirichlet boundary value problem.

Then, by applying Proposition 4.5 to the linear operator (T^{(r)}_{k}) and replacing (f(x)) by (ln f(x)), we obtain the following important example.

Let be the linear operator defined by.