The components x 1, x 2 : 0, ∞ → R are continuous for t ∈ 0, ∞ with the possible exception of the points t ∈ 0, ∞, 2.
( x 1 ′ ( t ), x 2 ′ ( t ) ) exists for every t ∈ 0, ∞ with the possible exception of the points t ∈ 0, ∞ where one-sided derivates exist, 4.
(x t)) is continuous on (mathbb{R}^); the derivative (x'(t)) exists for (tinmathbb{R}^) with the possible exception of the points (theta_{k}), (kinmathbb{N}), where one-sided derivatives exist; (5) is satisfied by (x t)) on each interval ((theta_{k},theta_{k+1})), (kinmathbb{N}), and it holds for the right derivative of (x t)) at the points (theta_{k}), (kinmathbb{N}).
Here u ( x, y ) is an unknown nonnegative continuous function with the exception of the points ( x i, y i ) where there is a finite jump: u ( x i − 0, y i − 0 ) ≠ u ( x i + 0, y i + 0 ), i = 1, 2, … .
(u(t)) is continuous on ([0, infty)), the derivative (u'(t)) exists at each point t in ([0, infty)), with the possible exception of the points (t=2n-1) for (n inmathbf{N}), where one-sided derivatives exist, (1) is satisfied on each interval ([2n-1,2n+1 [2n-1,2n+1nmathbfor}). [12].
A solution of (1) on ([0, infty)) is a function (u(t)) which satisfies the following conditions: (i) (u(t)) is continuous on ([0, infty)), (ii) the derivative (u'(t)) exists at each point t in ([0, infty)), with the possible exception of the points (t=2n-1) for (n inmathbf{N}), where one-sided derivatives exist, (iii) (1) is satisfied on each interval ([2n-1,2n+1 [2n-1,2n+1nmathbfor}). .
The MI-based analyses yielded comparable findings, with the exception of the point estimate corresponding to whether the plant was one of the original founding plants.
Using those parameter values, the differences between the experimental and calculated points are never larger that the dispersion in the experimental data, perhaps with the exception of the point corresponding to XPOPC = 0.20 in Fig. 6.
And there is reason to think that this approach won't work in any case: some recent work suggests that there is some atomless measure for any point in the Cantor space, with the exception of those points falling in a poorly understood countable set known as NCR1 (Reimann and Slaman 2008).
Crossrail and HS2 are Britain's most expensive public infrastructure projects (with the possible exception of the Hinkley Point nuclear power station, whose eventual cost to the public purse is hard to quantify).
No significant differences were found in vertical and sagittal relationships, with the exception of the A point, which proved to be more forward-projected in the MGBM group (Δ PTV-A, 2.5 mm MGBM vs. 0.6 mm Pendulum; p = 0.04).
