Sentence examples for there exist two points from inspiring English sources

There exist two points such that (3.31).

Since f and g are surjective maps and hence there exist two points y and (y^{prime}) in X such that (x^=fy) and (x^=gy ^{prime}).

For any λ j ∈ ( 0, λ j ∗ ), by the intermediate value theorem, there exist two points b 1 ∈ ( 0, r 0 ), b 2 ∈ ( r 0, ∞ ) such that p j ( b 1 ) = p j ( b 2 ) = λ j.

For any λ ∈ ( 0, λ ∗ ), by means of the intermediate value theorem, there exist two points l 1, l 2 ( 0 < l 1 < r 0 < l 2 < ∞ ) such that q ( l 1 ) = q ( l 2 ) = λ.

Then there exist two points (overline {x},widehat {x}in X) such that (overline {x}preceq widehat {x}), (g overline {x},widehat {x})=overline {x}), and (g(widehat {x},overline {x})=widehat {x}).

The system has an order-one periodic solution if there exist two points ({C}in{M_{I}^) and ({D}in{M_{I}^) such that the successor function ({f {C} f({D})}<0).

P3: There exist four points such that no three of them are collinear.

Then there exist three points (a,b,cin X) such that ([a,b]_Lcap ([a,c]_Lcup [b,c]_L)={a,b}.) Since ([a,b]_L) contains points other than (a) and (b,) this shows the failure of disjunctivity.

P1: If P and Q are distinct points, then there is a unique line passing through P and Q (denoted by P ∨ Q or PQ); P2: If l and m are any lines, then there exist at least one point on both l and m; P3: There exists four points such that no three of them are collinear.

If 1 < R ≤ ( 1 + 1 c ) b, then there exist two fixed points: the extinction fixed point ( 0, 0 ) and the exclusion fixed point ( R 1 b − 1, 0 ).

If R > ( 1 + 1 c ) b, then there exist three fixed points: extinction fixed point ( 0, 0 ), exclusion fixed point ( R 1 b − 1, 0 ), and a coexistence fixed point.