"Proponents" implies "for"; make it "proponents of".
Note that (H3.1) and (H3.3) imply, for every ((t,x,x' in[0,T]times mathbb{R}^{n} timesmathbb{R}^{n}), that the following function u rightarrowsigma bigl t,x,x',u bigr) is a bijection on (mathbb{R}^{ntimes d}).
imply for every k (in ) ({mathbb N}), t (in ) [0, T), (h ge 0) (with (t+h) (le ) T) and any "less regular" coefficients ({mathbf g}), u, w (as in Proposition 29) Open image in new window with a constant (C = {mathrm{const}} eta, rho, C_rho ) > 0).
The zero solution of (2.1) is said to be strictly stable (SS), if for any and, there exists a such that implies for and for every, there exists an such that (2.4).
The Shapley Folkman lemma implies, for example, that every point in [0, 2] is the sum of an integer from {0, 1} and a real number from [0, 1].
which can alternatively be written as Integrating from to and using we obtain for every which together with implies that for every Thus, is nondecreasing on which is a contradiction as has a positive maximum value at Hence is an isolated maximum point.
Assumption (ii) implies that for every, there exists, such that for every, (5.6).
The existence theorem for ordinary differential equation implies that for every v ∈ T M, there exist an open interval J ( v ) containing 0 and exactly one geodesic γ v : J ( v ) → M with d γ v ( 0 ) / d t = v.
The correspondence F is upper increasing if for every x, y ∈ X, x ≤ y implies that for every a ∈ F ( x ), there is some b ∈ F ( y ) such that a ≤ b.
