Specifically, o_{ij}=frac{vert n_{i} cap n_{j}vert }{vert n_{i} cup n_{j}vert }, (2) where (n_{i}) and (n_{j}) are respectively the set of neighbors of nodes i and j and (vert n_{i}vert ) indicates the number of them.
}Ax=b, end{aligned} (1) where (Vert x Vert _{0}) indicates the number of nonzero elements of x.
Then, for all (xinvartheta(F)), we have x^bigl H-F^HFbigr)x^bigl H-F^HFbigrgl(1-vert lambda vert ^{2}bigr)x=x^Hx-x^F^HFx=bigl 1-vert this lemma is true.
The fact that (vert mathit{Ared}_{k}(d_{k}) - mathit{Pred}_{k}(d_{k})vert = O Vert d_{k}Vert ^{2})) indicates that lim_{krightarrowinfty} r_{k} = 1, which shows that, for sufficiently large k and (kin K), Delta_{k+1} geqDelta_{k}.
end{aligned} Besides, we obtain that begin{aligned} vert nabla u_{k+2} vert ^{p-1}approx biggl( 1- frac{1}{p-1} biggr) ^{p-1}vert nabla u_{k+1} vert ^{p-1}, end{aligned} which indicates that the gradient moduli of iterative functions decrease with a fixed rate.
If C and C ′ are two disjoint cycles (i.e., Vert ( C ) ∩ Vert ( C ′ ) = ∅ ), the symbol C ⇒ C ′ indicates that there is a path which starts in Vert ( C ) and ends in Vert ( C ′ ).
All of this indicates that there exists ({widehat{C}_{lambda}} > 0) with { vert u vert ^{p}} ge{widehat{C}_{lambda}} Vert u Vert quad text{for } u in{widehat{E}_{lambda}}, (3.1) where ({widehat{C}_{lambda}} ) is a constant dependent on λ.
end{aligned}Since (hslash in zeta ( mathfrak {R}times chi, chi, m ),) we have begin{aligned} lim _{r rightarrow infty } frac{1}{m([r,-r])} int _{[r,-r]} Vert hslash _{K} (t) Vert mathrm{d} m (t) =0,quad K = 1,ldots, n. end{aligned}It indicates that (lim _{r rightarrow infty } frac{1}{m([r,-r])} int _{[r,-r]} Vert H (t) Vert mathrm{d} m (t) =0).
If X is a real Hilbert space with inner product ((cdot,cdot)_{X}), we will denote the induced norm by (vert cdot vert _{X}) while (X^) will indicate its dual with (langle cdot,cdot rangle ) for the duality between (X^) and X and the norm by (Vert cdot Vert _).
end{aligned} These indicate that frac{partial^{2}ln vert y_{u,v} ( t ) vert }{ partial u^{2}}frac{partial^{2}ln vert y_{u,v} ( t ) vert }{partial v^{2}}- biggl[ frac{partial^{2}ln vert y _{u,v} ( t ) vert }{partial upartial v} biggr] ^{2}=0, which yields the log-convexity of (vert y_{u,v} ( t ) vert ) in (( u,v ) ) on (mathbb{R}^{2}) for (tin mathbb{R}) with (tneq 0).
