In 1989, Miller and Ross [3] defined a fractional sum of order (alpha>0) via the solution of a linear difference equation and proved some basic properties of this operator.
In Fig. 1, total order cycle (T) is equal to the sum of order cycles of every supplier (T i ) and at the time one supplier's inventory is used up, the next supplier's shipment would arrive in.
The fractional sum of order α is defined as Delta^{-alpha}u(t) =frac{1}{Gamma alpha)}sum ^{t-alpha }_{s=a} t-s-1)^{(alpha-1)} t-s-1(2.1) where u is given for (s=a) mod (1), (Delta^{-alpha} t-s-1is defined for (t=(a+alpha)) mod (1), and the falling factorial function is t^{(alpha)}=frac{Gamma(t+1)}u smma(t+1-alpha)}.
Thus, for a function (f:mathbb{N}_{a}={ a,a+1,ldotsdots}rightarrowmathbb{R}), the nabla left fractional sum of order (alpha>0) becomes nabla_{a}^{-alpha} f(t)=frac{1}{Gamma alpha)} sum _{s=a+1}^{t}bigl t-rho(s)bigr)^{overline{alpha-1}}f(s), quad t inmathbb{N}_{s=a+1}^{t}bigl t-rho
The nabla right fractional sum of order (alpha>0) for (f: _{b}mathbb{N}={b,b-1,b-2,ldots}rightarrowmathbb {R}) is written as {_{b}nabla^{-alpha}} f(t) =frac{1}{Gamma alpha)} sum _{s=t}^{b-1}bigl s-rho(t)bigr)^{b-1}bigl s-rho-1}}f(s) = frac{1}{Gamma(alpha)} sum_{s=t}^{b-1}bigl s-rhoa(s)-t bigr^{overline{alpha-1}}f(s),quad t in_{b-1}mathbb{N}.
Summability techniques were also applied on some engineering problems like, Chen and Jeng [2] implemented the Cesàro sum of order ( C, 1 ) and ( C, 2 ), in order to accelerate the convergence rate to deal with the Gibbs phenomenon, for the dynamic response of a finite elastic body subjected to boundary traction.
In the ψ-direct sum X ⊕ ψ Y of Banach spaces the Minkowski inequality holds, i.e., we have the following: Let A and C be Banach spaces and let f 1 = ( a 1 1, a 2 1 ), f 2 = ( a 1 2, a 2 2 ), a 1 i ∈ A, a 2 i ∈ C, i = 1, 2. Then f 1 + f 2 A ⊕ ψ C ≤ f 1 A ⊕ ψ C + f 2 A ⊕ ψ C. Let us improve that idea to the sum of ordered spaces.
with ∥ ⋅ ∥ being the usual L 2 norm, and where U N denotes the partial sum of order N relevant to the Fourier-like series expansion representing the solution of the boundary-value problem for the Laplace equation, namely U N = ∑ m = − N N [ ( b − ϱ ) R − − ( a − ϱ ) R + b − a ] m ( A m cos m ϑ + B m sin m ϑ ) U N = + δ 0 ln ( ( b − ϱ ) R − − ( a − ϱ ) R + b − a ).
Outlier microRNAs and genes were detected with least sum of ordered subset square t-statistic (LSOSS) [ 19] and implemented in R scripts by Karrila et al. [ 20].
Among these methods are Least Sum of Ordered Subset Squared (LSOSS) [ 10], Cancer Outlier Profile Analysis (COPA) [ 9], Maximum Ordered Subset T-statistics (MOST) [ 11], Outlier Robust T-statistics (ORT) [ 13], and Outlier Sum (OS) [ 12].
