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By the theory of quasi-log schemes discussed in [7, Chapter 6] and Theorem 1.1, the fundamental theorems of the minimal model program hold for [Z, [v.sup.*][omega]].
How does the process of summation x + y = z transform into the Pythagorean theorem? There are several ways this can be done.
However, for polynomials of degree five or higher with arbitrary coefficients, the Abel-Ruffini theorem states that there is no algebraic solution.
In similar manner, in proof of Theorem 5, we obtain the result (28).
In Theorem 11 of this paper we prove that the condition "compact" is unnecessary, more precisely, if L(Z) is embedded in L(Q) or L(G), then Z must to be compact.
For example, a partial case of a corollary of Theorem 2 from [18] is the following theorem.
In this work, considering the target space Y = M a manifold and H a proper nontrivial subgroup of G, we prove the following formulation of the BorsukUlam theorem for manifolds in terms of (H, G)-coincidence.
Note that some related results which are contained and/or improved in our study are given by Anastassiou in [7, Proposition 2.9], Saker in [19, Theorem 2.9], Srivastava, Tseng, Tseng, and Lo in [21, Theorem 4], and Wong, Lian, Yeh, and Yu in [22, Theorem 2.2].
We note that if 1 [less than or equal to] p < [infinity], then by the change of variable theorem
Our result is related to Theorem 2 above, and reads as follows:
Observe that if in Theorem 10 we have m = n = 1, the statement of Theorem 10 becomes the statement of Theorem 2 in [6].