Sentence examples for it exists a vector from inspiring English sources

First, if τ=1 is an eigenvalue of M, then it exists a vector a≠0 such that M a−a=0.

Then b i − x ≻ 0 and x − a i ≻ 0 for i = 1, 2. It follows from (S12) that there exists a vector c ≻ 0 such that c ≺ b i − x and c ≺ x − a i for i = 1, 2. Hence, a i ≺ x − c and x + c ≺ b i for i = 1, 2. This implies that ( x − c, x + c ) ⊂ ( a 1, b 1 ) ∩ ( a 2, b 2 ).

Then d ( x, x i ) ≺ c i for i = 1, 2. From (S3), we get c i − d ( x, x i ) ≻ 0 for i = 1, 2. It follows from (S12) that there exists a vector c ∈ Y with c ≻ 0 such that c ≺ c i − d ( x, x i ) for i = 1, 2. By (S3), we obtain d ( x, x i ) ≺ c i − c for i = 1, 2. Now using the triangle inequality and (S10), it easy to show that U ( x, c ) ⊂ U ( x 1, c 1 ) ∩ U ( x 2, c 2 ).

Thus, there exists a vector such that for,.

Sufficiency: Assume that there exists a vector λ ∈ ℝ + n such that (8) holds.

Note that there exists a vector (vsucc0) in (Re^{n}) with ((A_{0}-I vprec0).

Say there exists a vector Π≠0 in the null space of both the operators.

From the definition of, for, there exists a vector function such that (317).

Then there exists a vector x ∈ Y such that x ≺ x.

Then there exists a vector x ∈ H such that { ( T ∗ ) n x } n = 0 ∞ ¯ = H.

So, by Remark 2.1, there exists a vector solution of the vector matrix game B i, i = 1, …, p. □.