We have initially hypothesized these distributions to be uniform for mathematical convenience, but all the proofs remain true under conditions similar to Property 1.
Note that Lemma 3.8 of [35] and its given proof remain fully valid for multivalued cyclic self-maps since only metric properties were used in its proof.
However, this can be relaxed: The statement of Theorem 2.3 and its proof remain valid if we only assume that the second eigenvalues (lambda _{2}(Omega )) of considered operators (mathcal {K}_{Omega }) are positive.
The proof of remained part is omitted because it is similar to the proof of Theorem 3.8 in [13].
One of the president's bullet-proof limousines remained all day in the north driveway, closer to the residence, as an extra security measure.
Armed Forces Minister Florence Parly said Wednesday's attack was proof there remained an active threat and that the 7,000-strong 7,000-strongce "waSentinelecessary than ever".
The proof of remaining part closely follows the proof of [2, Theorem 3.8], simply replacing with.
The proof of remaining part closely follows the proof of of [2, Theorem 3.8], simply replacing with.
But if there is a legal requirement to enact the terms of his will in the absence of proof that remaining pictures were illegally acquired, then making an exception to the law could in itself bring bad consequences.
The burden of proof will remain on the accused, the court ruled, saying that such a system is fair so long as it is used with "uniform application" and there is "a rational connection between the facts presumed and the facts proved".
