Sentence examples for and is not empty from inspiring English sources

Assume that and is not empty.

If (lim C_{n}) exists and is not empty, then ({ P_{c_{n}}x}) converges weakly to (P_{lim C_{n}}x) for every (x in E).

Also note that (partial F x)) is always a convex set, and is not empty whenever F is real-valued (or x is in the interior of ({x,F x(x)<+infty })), and we denote by (partial ^circ F x)) its element of minimal norm.

If the voids were just an illusion, and were not empty but just dark, cosmologists reasoned, there would be no gravitational fields tugging at the galaxies, which would explain why their so-called peculiar velocities were so low.

So,,,, which means that and are not empty.

We used the guidelines described to assess the qualitative degree of violation based on dumpster type and securing methods, where a dumpster was considered in violation if it had high or medium violation and was not empty.

The set of common fixed points of A and B is not empty due to Theorem 2.1; suppose now that x and y are two common fixed points of A and B, i.e., (Ax=Bx=x), (Ay=By=y).

if (bandwidth_available(Region II) and queue is not empty).

Due to Theorem 2.2, the set of tripled coincidence points of F and g is not empty.

Proof The set of quadruple coincidence points of F and g is not empty due to Theorem 2.1.

Theorem 2.1 (or Theorem 2.2 respectively) implies that The set of quadruple coincidence points of F and g is not empty.