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The following proposition is a consequence of [V], p.140, Theorem 2’ (the quoted theorem of Varopoulos is stronger than this statement):..
In [6] the second author discovered a general scheme, with the help of nonstandard analysis, how one can derive a theorem about the upper Banach density parallel to each
5.1. Preliminaries on twisted forms. We saw in the previous section that every quadric surface V q is an element of T.. Let X/k be a quadric surface.. The proof of Theorem 7b). First
The proof of this theorem relies on a result of Razon about fields which are PAC over subfields, on Frobenius density theorem, and on Neukirch’s recognition of p -adically closed
While Chan and Li’s proof of Horn’s Theorem and our proof of Mirsky’s Theorem have a similar structure, there is a significant difference: The matrices whose existence is guaranteed
We shall make frequent use of the following inclusion-exclusion type lemma.. The well-known proof is easy, but so short that we include it. It would then follow from Fubini’s
The main lemma in the proof of Theorem 1.1 involves estimating the resistances. A combination of the above identities will then yield lower bounds for the hitting times. We then
In this section we prove a fixed point theorem in a complete metric space by employing notion of generalized w-distance.. The following Lemma is crucial in the proof of
