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).
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  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
The proofs of the below results are given in [, Theorem
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 .