Sentence examples for provided the expression from inspiring English sources
for a ∈ w provided the expression on the right-hand side exists and is finite, which is the case whenever X is a BK-space and a ∈ X β [9].
The fractional integral of order q > 0 of y is given by I q y ( t ) = 1 Γ ( q ) ∫ 0 t ( t − s ) q − 1 y ( s ) d s. provided the expression on the right-hand side is defined.
If (Xsupset{phi}) is a BK-space and (a=(a_{k})in{w}) then we write |{a}|_{X}^=sup_{xin{S_{X}}} Bigglvert sum_{k=0}^{infty }{a_{k}x_{k}} Biggrvert, (3) provided the expression on the right-hand side exists and is finite.
If (Xsupset{phi}) is a BK-space and (a=(a_{nk})in {w}), then we write |{a}|_{X}^=sup_{xin{S_{X}}} Bigglvert sum_{k=0}^{infty}a_{k}x_{k} Biggrvert, (2) provided the expression on the right is defined and finite which is the case whenever (ain{X^{beta}}) [17].
The Riemann-Liouville fractional integral (I^{alpha}u) of order (alpha >0) of (u:mathbb{R}_tomathbb{R}) is defined by I^{alpha}u(t)=frac{1}{Gamma alpha)} int_{0}^{t} t-s)^{alpha-1} t-s,ds, provided the expression on right hand side is defined.
If (Xsubset w) is a normed space and (ain w) then we write Vert aVert ^{ast}=Vert aVert _{X}^{ast }=sup Biggl{ Bigglvert sum _{k=0}^{infty}a_{k}x_{k}Biggrvert : Vert xVert leq1 Biggr}, (1.2) provided the expression on the right-hand side exists and is finite which is the case whenever (Xsupsetphi) is a BK space and (ain X^{beta}) [2], p.35.
In the following, we provide the expression of the block-size b such that d = N p L h.
The following two theorems provide the expression for evaluating two performance measures of interest: system throughput and channel access delay.
We also establish the condition for the disease free equilibrium to be asymptotically stable and provide the expression of the basic reproduction number.
In Section II, we describe the multicell downlink transmission environment, and we provide the expression of the interference power for which we want to find a statistical model.
We next present original analytical results, which exactly specify under which conditions Rabiner's estimator is unbiased, and we also provide the expression of its variance.
