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The eigenvalues in (i) satisfy (0=mu_{1}

(36) Here (beta(i)) and (eta(i)) satisfy the following assumptions.

(C1) lim inf n → ∞ r n > 0. Assume that T i and S i satisfy condition (I) and F ≠ ∅.

Clearly, both f and I i satisfy the assumptions given in Theorem 4.6 and Theorem 4.7 with L 1 = L 2 = 1 6.

Clearly, both f and I i satisfy the assumptions given in Theorem 4.5 and Theorem 4.6 with L 1 = 1 5, L 2 = 1 5, respectively.

It is assumed that x i and H i satisfy the following power constraints E tr x i x i H = P i, (2).

If the ith hypothesis test is defined according to (15a) and (15b), the projection matrix A satisfies (18) for a given ε∈ 0,1), and the estimates { α ̂ i } satisfy (19) for some ζ∈[0,1], then the bound p i ≤ 1 − F d i 2 ; K, ( 1 + ε ) 2 α ̂ i 2 ζ + τ 2 (20).

In our previous section, we show that a CSMA/CA-based CRN is stable when its successful transmission probabilities {P i } satisfy a certain constraint.

Therefore, we find that I ± satisfy the ( PS ) c conditions with c ± = inf { I ± ( h ) : h ∈ E }.

Here, U is the unitary matrix obtained from the singular value decomposition (SVD) of P=U H D P U, and X is the diagonal matrix having its diagonal elements X i,i) satisfy: D_{P}^{-1}(i,i X i,i)=(mu^{-1}-D_{P}^{-1}(i,i))^, (13).

According to the corollary, the condition (i) implies the guarantee that the functions (h_{i}) satisfy the same conditions (ii), (iv) of Theorem 4 as the functions (f_{i}), (i=1,2).