Sentence examples for there exists uniquely from inspiring English sources

Then for there exists uniquely a sequence of positive numbers (4.11).

From lim x → ∞ G ( x, a ) = ∞, there exists uniquely a number x 3 > x 2 satisfying G ( x 3, a ) = 0.

Since lim a → + 0 g ′ ( a ) = ∞ and g ′ ( 1 2 ) = 2 ln 2 − 2 < 0, there exists uniquely a number c ∈ ( 0, 1 2 ) such that g ′ ( c ) = 0. Since g ′ ( a ) > 0 for 0 < a < c and g ′ ( a ) < 0 for c < a < 1 2, g is strictly increasing on the interval ( 0, c ) and strictly decreasing on the interval ( c, 1 2 ).

From Proposition 2.8, for fixed a, if G x ( 1, a ) ≥ 0, then G x ( x, a ) > 0 for x > 1 and if G x ( 1, a ) < 0, then there exists uniquely a number x ˜ > 1 such that G x ( x ˜, a ) = 0. From Propositions 2.5 and 2.6, we notice that G ( 1, a ) < 0 when G x ( 1, a ) < 0. (1) and (2) play an important role in the proof of Theorem 1.1.

Since g ( 0 ) = 3 and g ( 1 2 ) = − 23 4, there exists uniquely a number c 1 ∈ ( 0, 1 2 ) such that g ( c 1 ) = 0. Since g ( t ) > 0 for 0 < t < c 1 and g ( t ) < 0 for c 1 < t < 1 2, we have f ‴ ( t ) > 0 for 0 < t < c 1 and f ‴ ( t ) < 0 for c 1 < t < 1 2. It follows that f ″ is strictly increasing on the interval ( 0, c 1 ) and strictly decreasing on the interval ( c 1, 1 2 ).

If is a Dunkl polyharmonic function in of degree, then there exist uniquely Dunkl harmonic functions such that (1.14).

Then the following results hold: (1) In the case of d a = 1, for any given ( x, f ) ∈ H a , there exist uniquely ( y ˜ 0, f ˜ 0 ) ∈ H ˆ a, 0 and c ˜ 1, c ˜ 2 ∈ C such that x ( t ) = y ˜ 0 ( t ) + c ˜ 1 z ˜ 1 ( t ) + c ˜ 2 z ˜ 2 ( t ), a − 1 ≤ t ≤ c 0. (3.27)   (2) In the case of d a = 2, let ϕ ˜ 1 and ϕ ˜ 2 be two linearly independent solutions of equation (1.1) restricted on I 1.

there exist uniquely harmonic functions such that (1.2).

Then there exist uniquely Dunkl harmonic homogeneous polynomials of degree such that (4.4).

If for some, (3.2). for all and, then there exists a uniquely determined mapping such that and (3.3).

Let be a mapping satisfying (3.2). for all, then there exists a uniquely determined mapping such that and (3.3).