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].

The proofs of the below results are given in [[1],

Theorem 5].

We note that if 1 [less than or equal to] p < [infinity], then by the change of variable

theoremOur 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].