More precisely, the following two theorems, which are the main results of the present paper, shall be proved.
A formal demonstration of this result could be obtained by studying the distribution of the residual nonlinearity (a uniform distribution is expected, but this shall be proved).
The following approximation result that we shall need can be proved as Proposition 4.1 of [9] with minor obvious modifications to accommodate the presence of (tin I:=[0,T]).
"The property, real and personal, of all persons in the State of Missouri, who shall take up arms against the United States, or who shall be directly proven to have taken an active part with their enemies in the field, is declared to be confiscated to their public use, and their slaves, if any they have, are hereby declared freemen".
The property, real and personal, of all persons in the State of Missouri who shall take up arms against the United States, and who shall be directly proven to have taken active part with their enemies in the field, is declared to be confiscated to the public use; and their slaves, if any they have, are hereby declared free.
(If I am proven wrong, tomorrow morning this post shall stay anyway).
To achieve our maim in this section, we shall use the following lemma which is proved in [13].
Proof We shall prove only (c) in Theorem 2 and other identities are proved similarly.
We shall prove only Theorems 1(1), 1(3), 2(1), and 2(4), and other identities are proved similarly and omitted.
