Sentence examples for be an arbitrary domain from inspiring English sources

Let be an arbitrary domain on the unit sphere.

Let be an arbitrary domain with nonempty boundary.

Let, be an arbitrary domain with a piecewise smooth boundary on the unit sphere.

Let D be an arbitrary domain in ({mathbf{R}}^{n}) and (mathscr{A}_{a}) denote the class of nonnegative radial potentials (a(P)), i.e. (0leq a(P =a(r)), (P= r,Theta)in D), such that (ain L_{mathrm{loc}}^{b}(D)) with some (b> {n}/{2}) if (ngeq4) and with (b=2) if (n=2) or (n=3).

Let (C_{n}(Omega)) be an arbitrary domain in (mathbf{R}^{n}) and A a denote the class of nonnegative radial potentials (a(P)), i.e. (0leq a(P =a(r)), (P= r,Theta)in C_{n}(Omega)), such that (ain L_{mathrm{loc}}^{b}(C_{n}(Omega))) with some (b> {n}/{2}) if (ngeq4) and with (b=2) if (n=2) or (n=3).

Let D be an arbitrary domain in R n and A a denote the class of non-negative radial potentials a ( P ), i.e. 0 ≤ a ( P ) = a ( r ), P = ( r, Θ ) ∈ D, such that a ∈ L loc b ( D ) with some b > n / 2 if n ≥ 4 and with b = 2 if n = 2 or n = 3 (see [[1], p.354] and [2]).

Let now (Omega ) be an arbitrary convex domain.

Let (Omegasubsetmathbb{R}^{N}, Ngeq2) be a bounded domain and (chi>0) be an arbitrary positive constant.

Let Ω be a horizontally periodic domain with finite height, ω be an arbitrary real number, and (m_{3}neq0).

Let be an arbitrary function.

(xiv) Let be an arbitrary.