If the limits x ∞ and y ∞ of sequences (19) and (20), respectively, exist uniformly on [ 0, 1 2 p ] and [ 1 2 p, p ], then: 1.
We introduce an azimuthal dependent fracture envelope and illustrate that two competing fracture mechanisms, governed by the pressure and the Lode angle of the stress tensor, respectively, exist ahead of the crack tip.
Initial laboratory investigation indicated that microbial groups of lactic acid bacteria, yeast and total bacteria count of 1.6 × 107, 1.0 × 104 and 1.2 × 107 cfu/mL, respectively, exists in the solutions.
The left and right Riemann-Liouville fractional derivatives of order γ for function f denoted by (_{a}D_{t}^{gamma}f(t)) and (_{t}D_{b}^{gamma}f(t)), respectively, exist almost everywhere on ([a,b]).
If γ ∈ ( n − 1, n ) and f ∈ A C n ( [ a, b ], R N ), then the left and right Caputo fractional derivatives of order γ for function f, denoted by D t γ a c f ( t ) and D b γ t c f ( t ) respectively, exist almost everywhere on [ a, b ].
If (gammain n-1,n)) and (fin mathit{AC}^{n}([a,b],{mathbb{R}}^{mathrm {N}})), then the left and rigammain n-1,nandional derivatives ofinrder γ for function f denoted by (_{a}^{c} D_{t}^{gamathit{AC}^{n (_{t}^{c} D_{b}^{gamma}f(t)), respectively, exist almost everywhere on ([a,b]).
If (gammain n-1,n)) and (fin mathit{AC}^{n}([a,b],mathbb{R})), then the left and rigammain n-1,nandional derivatives ofinrder γ for function f denoted by (_{a}^{c} D_{t}^{gamathit{AC}^{n (_{t}^{c} D_{b}^{gamma}f(t)), respectively, exist almost everywhere on ([a,b]).
Let γ ≥ 0 and n ∈ N. (i) If γ ∈ ( n − 1, n ) and f ∈ A C n ( [ a, b ], R N ), then the left and right Caputo fractional derivatives of order γ for function f, denoted by D t γ a c f ( t ) and D b γ t c f ( t ) respectively, exist almost everywhere on [ a, b ].
A positive correlation between SCC and fat, and protein contents (r = 0.07 and r = 0.15, respectively) exists in composite samples from Holstein cows [ 5].
If var (Y s, t)) < ∞ for all (s, t) ∈ ℛ × ℛ, then the covariance, cov (Y s, t), Y(s + u, t + v)), where u and v are spatial and temporal lags, respectively, exists.
