Sentence examples for defined and there exists from inspiring English sources

Let q ⩾ 2. If f is a measurable function on Rn such that f(x) ¦x¦n(1 − 2q) ϵ Lq(Rn), then its Fourier transform ɞ can be defined and there exists a constant Aq such that the inequality ∥fɞ∥q ∥ f ¦ · ¦n(1 − 2q ∥q holds. This result is called the Hardy-Littlewood theorem.

In this section, we show that the process can be defined and there exists a bounded uniform absorbing set for the family of processes.

A subset G of E is said to have the countable concatenation property if every countable concatenation (sum_{n in N}tilde{I}_{A_{n}}x_{n}) with (x_{n}in G) for each (nin N) still belongs to G, namely (sum_{n in N}tilde {I}_{A_{n}}x_{n}) is well defined and there exists (xin G) such that (x=sum_{n in N}tilde{I}_{A_{n}}x_{n}). ([17]).

f ∈ L s is a given function, ϑ is the exterior normal derivative of Γ, g : Ω × R → R is a given function satisfying Carathéodory's conditions such that the mapping u ∈ L s → g ( x, u ( x ) ) ∈ L s is defined and there exists a function T ( x ) ∈ L s such that g ( x, t ) t ≥ 0, for | t | ≥ T ( x ) and x ∈ Ω. β x is the subdifferential of a proper, convex, and semi-lower-continuous function.

for all By virtue of (4.6), is well defined, and there exists a constant independent of such that (423).

where is a probability density function defined on (see [9, 10]) and induces an invertible operator defined on and there exists a positive constant and such that and.

In sum, gene conversion is not defined by mechanism, and there exist multiple mechanisms leading to conversion.

The research design takes advantage of a situation where many agents are evaluating the same thing, private incentives are clearly defined and measurable, and there exists an alternative source of computer rankings that is bias free.

Let (inftyleq a< bleqinfty), f and g be differentiable on ((a,b)) with (f(b^)=g(b^)=0) and (g^{prime}(x)<0) on ((a,b)), (H_{f, g}) be defined by (2.1), and there exists (lambdain a, b)) such that (f^{prime}(x)/g^{prime}(x)) is strictly increasing on ((a, lambda)) and strictly decreasing on ((lambda, b)).

The function (delta(i)) is a real-valued non-negative function defined on (iin Z^), and there exists a constant scalar (xi>0) such that sum_{i=1}^{+infty}delta(i =xi< + infty,qquad sum_{i=1}^{+infty}delta(i i< + infty.

(H1): a i ( t ) does not vanish identically on any subinterval of ( 0, 1 ), 0 < ∫ 0 1 G i ( s ) a i ( s ) d s < + ∞, i = 1, 2, where G i ( s ) is defined by Lemma 2.6 and there exists t 0 ∈ ( 0, 1 ) such that a i ( t 0 ) > 0. By (H1), we can choose a subinterval ⊂ ( 0, 1 ) such that t 0 ∈.