For given real-valued scalars a and b, (text {Pr}left (a {le } b right)), (text {Pr}left (a b right)), (text {Pr}left (a b | mathcal {C} right)), ⌈a⌉, and (log _{2} (a)), denote the probability of a being smaller than b, the probability of a being equal to b, the probability of a being equal to b given condition (mathcal {C}), the ceiling function of a, and the base 2 logarithm of a, respectively.
where m is the ceiling function of ν.
Note that the transmit powers are selected from this discrete power level set which corresponds to ceiling function of (10).
The μth fractional difference is defined as Delta^{mu}f(t)=Delta^{k-nu}f(t)=Delta^{k} Delta^{-nu}f(t), (7) where (lceilmurceil) is the ceiling function of μ.
The μth fractional difference is defined as Delta^{mu}f(t)=Delta^{m-nu}f(t)= Delta^{m}Delta^{-nu}f(t), (3) where (lceilmurceil) is the ceiling function of μ. ([7]).
The Caputo fractional derivative of order ν is defined as D_{c}^{nu}f(x)=frac{1}{Gamma m-nu)} int_{0}^{x}{(x-zeta )}^{m-nu-1} frac{d^{m}}{dt^{m}}f(zeta),dzeta, quad m-1< nuleq m, x>0, (2.4) where m is the ceilinGamma m-nu of ν.
The next equation defines the Riemann-Liouville fractional derivative (D^{nu}) of order ν D^{nu}f(x)=frac{1}{Gamma m-nu)} frac{d^{m}}{dx^{m}} biGamma m-nu{0}^{x}{(x-t)}^{m-nu-1} frac{d^{m}ggr), quad m-1< nuleq m, x>0, (2.3) where m is the ceiling function of ν.
The expression ⌈ x ⌉ corresponds to the ceiling function of x.
where ceil denotes the ceiling function.
Here, is a real valued parameter that controls the release time of the ceiling function.
