However with z > max { z 0, 0 }, Theorems 1.1, and are still applicable provided the function p → - 1 - | p | 2 / Z ( x, u, p ) is convex as there exist arbitrarily small g -affine functions.
Furthermore by taking z sufficiently small, we see that there exist arbitrarily large g -affine functions and hence all the hypotheses of Theorems 1.1 and 1.2 are satisfied by the generating function g if Z is concave in p. Note that when Φ is constant, then Z = ( 1 - | p | 2 ) / 2 ( u - Φ ), u > Φ, | p | < 1 and the strict condition A3 holds, [16].
Since locally uniformly on, there exists, for arbitrarily large positive, an such that, for, on.
From Theorem 2.1, when (R_{2}<0) one has lim_{trightarrowinfty}I_{2}(t)=0 quad mbox{a.s.} (24) Therefore, there exists an arbitrarily small constant (varepsilon>0) such that when t is large enough, we have (I_{2}(t)<varepsilon).
Case (i): Note that limsup_{trightarrow+infty}frac{ln I_{1}(t)}{t}leq0, there exists an arbitrarily small constant (varepsilon>0) such that when t is large enough, we have frac{ln I_{1}(t)}{t}< varepsilon.
Since no threshold for interpreting the QHES exists, we arbitrarily set a score of 90 and above as excellent quality, and a score of 75 to 90 as good quality.
Given that large number of attributes, it is likely that some attributes are correlated, therefore clusters might exist in arbitrarily oriented affinity subspaces.
Intuitively, a map possesses sensitive dependence on initial conditions if there exist points arbitrarily close to which eventually separate from by at least under iteration of.
The increasingly complex rules for who you admit to Arstotzka don't just exist to arbitrarily increase the game's difficulty.
In addition, a usual problem in fiber angle optimization methods is that because fibers can be arbitrarily oriented, structural problems may exist at the intersection of discontinuous fiber paths.
