Sentence examples for on a larger interval from inspiring English sources

two real distinct roots and b 12 ∗ = b 21 ∗ = 0 ; a pair of complex conjugate roots; a double real root and b 12 ∗ = b 21 ∗ = 0. Proof We will carefully go through all the theorems considered (Theorems 2-6) adding the case of a pair of complex conjugate roots and our conclusion will hold at least on Z m + 2 ∞ (some of the statements hold on a larger interval).

The range of Fourier methods can be significantly increased by extending a nonperiodic function f(x) to a periodic function f̃ on a larger interval.

We now show that if (gin V_{N,a}) and (age N/3), then condition (A3) automatically holds on a larger interval.

In 2000, Gesztesy and Simon [7] gave several important generalizations of the Hochstadt-Lieberman theorem to the case where the (L^{1}[0,1]) potential q is known on a larger interval ([0,a]) with (ain[1/2,1)) and the set of common eigenvalues is sufficiently large.

Lemma C.0.1 Under hypotheses of Theorem C.0.1, if V 1 ∈ C 1 ( J 1, O ) and V 2 ∈ C 1 ( J 2, O ) are two solutions and if there exists t 0 ∈ J 1 ∩ J 2 such that V 1 ( t 0 ) = V 2 ( t 0 ) then: V 1 ( t ) = V 2 ( t ) for all t ∈ J 1 ∩ J 2. This lemma shows the existence of a larger interval J 0 on which the initial value problem (34) has a unique solution.

The number of stationary points on the potential energy surface increases, and their energies get spread over a larger interval.

qTBN1 was located in a large interval of 56.6 Mb on chromosome 1, and explained 2.1% of the phenotypic variation.

We now assume a stronger condition, in comparison with (3.1), to approximate the given function by a solution of the Kummer's equation on a larger (punctured) interval.

Indeed, it is well known that univariate polynomials of degree at most (n) bounded by 1 on ([-1,1]) are also bounded by an absolute constant on the larger interval ([-1-1/n^2,1+1/n^2]), this easily follows, for instance, from the classical Remez inequality.

It thus appeared that there were no marked effects of multicollinearity on those findings with a large confidence interval.

Candidate genes in these intervals are often colocated with one or other of these peaks, as can be seen for QTL4.1, a large QTL interval on Setaria chromosome IV.