'Alright, let 's give Mr Safran a ring-a-ling,' Jim says.
Ring ling, which had led virtually all the way, was third.
Moreover, the ODE for ({mathscr {G}}), that is, (2.20) can be integrated to give the solution ({mathscr {G}}(r) = (2pi m + z) mathscr{N}(r)) for (a le r le b) where the profile (mathscr{N}) is described by (cf. (2.18)) mathscr{N}(r) = frac{(r/a)^{2-n} - 1}{(b/a)^{2-n} - 1}, quad ngeq3.
Note that upon setting (s=sup r^{2} dot{{mathscr {G}}}^{2}(r)) for (a le r le b), a sufficient condition implying (3.13) is the strict inequality gamma ( n-1 + s ) < frac{ n-2)^{2}}{4} + 2. (3.17) In case ofrac{ n-2ichlet energy (i.e., (F(r, t) equiv t)) it is easily seen that (overline{d}{ underline{d}) and so (gamma=1).
Δ-a.e. (t inmathbb{T}_{0}) and for every (x in mathbb{R}^{n}) such that (|x-v sigma(t))|x-v sigmaa(t))), there exists (y in F(t,x)) such that bigllangle x-vbigl(sigma(t)bigr),y-v^{Delta}(t)bigrrangle le r bigl(sigma(t)bigr)r^{Delta}(t); (|v(b)-v(a)| le r(a)- r(b)).
We say that ((v,r)) is a solution-tube of (1.1) if (i) Δ-a.e. (t inmathbb{T}_{0}) and for every (x in mathbb{R}^{n}) such that (|x-v sigma(t))|x-v sigmaa(t))), there exists (y in F(t,x)) such that bigllangle x-vbigl(sigma(t)bigr),y-v^{Delta}(t)bigrrangle le r bigl(sigma(t)bigr)r^{Delta}(t); (ii) (|v(b)-v(a)| le r(a)- r(b)).
Boidot R, Vegran F, Meulle A, Le Breton A, Dessy C, Sonveaux P et al. Regulation of monocarboxylate transporter MCT1 expression by p53 mediates inward and outward lactate fluxes in tumors.
Pick (varepsilon >0) and find B so large that begin{aligned} (1-varepsilon )f(r)Phi (vartheta )le a(rvartheta )le (1+varepsilon )f(r)Phi (vartheta ) end{aligned}for all (vartheta in mathbb {S}^{2d-1}) and (r>B).
Suppose that for each (varepsilon in (0,1)) there are positive constants (c_{epsilon },C_{epsilon },B_{epsilon }>0) such that begin{aligned} c_{varepsilon }fleft( (1-varepsilon ) r Phi (vartheta )right) le a(rvartheta )le C_{varepsilon }fleft( (1+varepsilon ) r Phi (vartheta )right), end{aligned} (5.3 for all (rge B_{varepsilon }) and (vartheta in mathbb {S}^{2d-1}).
end{aligned}A hyperbolic space (X, d, W) is uniformly convex [16] if for any (r > 0) and (varepsilon in (0, 2],) there exists (delta in (0, 1]) such that for all (a, x, y in X,) begin{aligned} dbigg (frac{1}{2} x oplus frac{1}{2} y, a bigg )le & ( 1- delta ) r. end{aligned}provided (d x, a) le r,~ d y, a) le r) and (d x, y) ge varepsilon r).
