Sentence examples for for distinct points from inspiring English sources

For distinct points u0, u1, u2 ∈ I we suppose.

Then for distinct points u i ∈ I, i = 0, 1, 2, the divided differences in first and second order are defined as follows: u i, u i + 1, f = f ( u i + 1 ) - f ( u i ) u i + 1 - u i i = 0, 1, u 0, u 1, u 2, f = u 1, u 2, f - u 0, u 1, f u 2 - u 0. (13).

A function is said to be -convex on,, if and only if for all choices of distinct points in, (52).

For every pair of distinct points a, b ∈ X, there exists a point c ∈ X such that d ( a, b, c ) ≠ 0.   2.

For every pair of distinct points x, y ∈ X, there exists a point z ∈ X such that d ( x, y, z ) ≠ 0.   2.

For every pair of distinct points x, y ∈ X, there exists a point z ∈ X such that d ( x, y, z ) ≠ 0. If at least two of three points x, y, z are the same, then d ( x, y, z ) = 0.

For every pair of distinct points a, b ∈ X, there exists a point c ∈ X such that d ( a, b, c ) ≠ 0. d ( a, b, c ) = 0 only if at least two of three points are the same.

(W) For each pair of (distinct) points u, v, there is a number r u, v > 0 such that for every z ∈ X, r u, v < d ( u, z ) + d ( z, v ).

For each pair of (distinct) points a, b ∈ X, there is a number r a, b > 0 such that for every c ∈ X, r a, b ≤ d ( a, c ) + d ( c, b ).

For each pair of (distinct) points u, v, there is a number (r_{u,v} > 0) such that for every (z in X), (r_{u,v} < d u, z) + d z, v)).

We say that f is expansive if there is e > 0 such that for any pair of distinct points x, y ∈ M, d ( f n ( x ), f n ( y ) ) > e for some n ∈ Z.