Sentence examples for let for a from inspiring English sources

Let be the group of unitary elements in and let For a given mapping and a given we define and by.

Suppose that ((X,mathbb{A},d)) is a (C^ -algebra-valued metriC^ -algebra-valuedr a mapping (T:Xto X) there exists (ainmathbb{A}) with (|a|<1) suC^ -algebra-valuedeceq a^d(x,y) a,quad textit{for all }x,yin X. (3.1) Then T has a unique fixed point in X.

Suppose now that X is a subset in a normed space E, and let, for a given (rho>0), (X_{rho}) be the set (Xcap D_{rho}), where (D_{rho}) is the closed disk of radius ρ centered at the origin in E. Assuming that (X_{rho}) is non-empty, denote by (Psi_{rho}) the compression of Ψ restricted to (X_{rho}) given by (Psi_{rho} u):=Psi u cap D_{rho }), (uin X_{rho}).

The definition of d is as follows: P 0) is a set consisting of the single pair 〈root  tube   d0, 1〉; that is, d0 0) = 1; let for a nonroot tube d = (s1, s2) hold τ = t(s1); then, by induction from root to leaves assume that d equals d′, where d′ is the parent tube for d; determine the change of d in a small time interval δt of the argument such that d contains τ + δt.

Definition 1.3 Let, for a constant ρ < 1, a function μ ( r ) be a continuous and positive function on the interval ( 0, ρ ), and assume that lim r → 0 + μ ( r ) = μ ( 0 ) = 0. We say that a function f ∈ L l o c 1 ( R n ) is μ-smooth of order μ ( r ) at x ∈ R n if D μ ( x ) = sup 0 < r < 1 1 r n μ ( r ) ∫ | t | < r | f ( x − t ) − f ( x ) | d t < ∞. (1.5).

Let, for an integer,.

Let for an interaction effect, such that.

Let for all Let be a solution of (P1).

Let for where is a given constant.

(a) Let for all.

Let for and let M be a constant such that.