By
Theorem 1.1 below, which is the main
theorem of this paper, we see that [[W.sup.v], [v.sup.*][omega]] naturally becomes a qlc pair.
In one dimension, the Pythagorean
theorem is a simple process of summation, i.e., x + y = z.
Although the above
theorem tells about the number of zeros, it does not mention anything about the location of these zeros.
In similar manner, in proof of
Theorem 3, we obtain the result (24).
The proof of this
theorem in [7] is nontrivial and uses the deep and powerful Kolmogorov Superposition
Theorem which answered Hilbert's 13th Problem.
The later
theorem for rational a in non-explicit form was proved by S.M.
Theorem A.[17,
Theorem 1.2] Let X be a paracompact free [Z.sub.p]-space of ind X [greater than or equal to] n, and f : X [right arrow] M a continuous mapping ofX into an m-dimensional connected manifold M (orientable ifp> 2).
In 2001, Bohner and Kaymakcalan [8] (see also [11,
Theorem 6.23]) initiated the study of dynamic versions of (1.1) and proved that if T is an arbitrary time scale and f [member of] [C.sup.1.sub.rd]([[0, h].sub.T], R) with h > 0 satisfies f (0) = 0, then
In [[1],
Theorem 6], formulas (8)-(11) are analyzed and it is concluded that system (1) can be a weakly delayed system only if matrix A has one of the following three Jordan matrix forms (the case of the roots of (6) being complex conjugate is not compatible with (2) and (8)-(11)):
Using the same method as used in the proof of
Theorem 3.2 in [7]
Shiffman's proof was based on Besicovitch's result,
Theorem 1 above, on coordinate rotation, on the use of Cauchy integral formula and on the following result of Federer:
Observe that if in
Theorem 8 we have m = n = 1, the statement of
Theorem 8 becomes the statement of
Theorem 1 in [6].