Sentence examples for depending additionally on from inspiring English sources

Then the estimate (2.13) holds for any vector field (tau ) with skew symmetric Jacobian, with the constant C depending additionally on (|tau |_{1 Omega }).

5.2 in order to prove (5.38) with a constant depending additionally on the volume of the set E, but possibly blowing up as (vartheta downarrow 0).

We remark that for the Monge Ampère operator, in the form (F r) =log (det r)), we can take (tau ) to be any (C^2) vector field in (2.13) with the constant C now depending additionally on (|tau |_{2 Omega }).

Then Theorems 1.1 and 1.2 will continue to hold provided the one jet (J_1 =J_1[u](Omega ) = (cdot, u, Du)(Omega )) is strictly contained in ({mathcal {U}}), with the constants C in the estimates (1.14) and (1.18) depending additionally on dist((J_1, partial {mathcal {U}})).

If "o" is extended to "O" in (1.22) and (3.33), then there exists a positive constant (theta ) depending on the same quantities as C, such that the estimates (3.55) and (3.56) continue to hold provided (mathop {mathrm{osc}}limits _{ Omega cap B_R} u < theta ) and ({mathcal {F}}) satisfies (3.54) in case (ii) with C depending additionally on (mu _0), (mu _1) and (mu _2).

If the maximum of (v-Phi ) takes its maximum over (overline{(B_Rbackslash B_rho )cap Omega }) at a point (x_0in (B_Rbackslash B_rho )cap Omega ), by choosing a larger constant a depending additionally on (kappa ), (|phi |_{2 Omega }), we can obtain the same inequality as in (3.90) and get a contradiction with (3.87).

The amount of incoming solar radiation, and thus heat, depends additionally on the incident angle of the rays and therefore decreases inversely with latitude to reach a minimum at the geographic poles.

The variation of Collembola abundance depended additionally on microbial biomass.

A GSIP problem is characterized by an infinite number of inequality constraints, and the corresponding index set depends additionally on the decision variables.

The coupling loss factor depends additionally on an equipartition parameter which is significant when the coupling transmission coefficient is of the order of 1.

The structure coefficients depend on the fixed value of the power n ∈ N 0 on the left-hand side of (16) as well as on the fixed value of the power j ∈ N 0 of the polynomial index of P j + 1 - moreover they depend additionally on the counting index k.