Corollary 1.2 (Vanishing theorem for normalizations).

Start with the one-dimensional theorem where a line length x inline with a line length y results in a line length z, inscribed in a circle with diameter z (Figure 3a).

Theorem 1 of Gauss was improved in 1829 by Cauchy [10] who derived more exact bounds for the moduli of the zeros of a polynomial than those given by Gauss, by proving the following.

In a similar manner, in proof of Theorem 7, we obtain the result (33).

For the case of free locally convex spaces on compact metrizable spaces Y, our Theorem 8 gives a complete description of those subspaces X of Y with the property that L(X) can be embedded as a topological vector subspace of L(Y).

Therefore the hypotheses of Theorem 3 are satisfied.

Let us observe that if H = G = [Z.sub.p], we have (|G| - r)m = (p - 1)m and therefore Theorem 1.1 generalizes Theorem A above of Volovikov.

If f [member of] [C.sup.1](R, R) and g : T [right arrow] R is delta differentiable, then the time scales chain rule, see [11, Theorem 1.90], states that

The following result is proved in [[1], Theorem 9].

Now, by Morera's theorem with aids of Holder's inequality and the dominated convergence theorem, we have (30) for [lambda] [member of] [C.sub.+].

Observe that if in Theorem 12 we have m = n = 1, the statement of Theorem 12 becomes the statement of Theorem 3 in [6].

Pirbavafa, A generalization of Geraghty's theorem in partially ordered metric spaces and applications to ordinary differential equations, Fixed Point Theory Appl., 2012, 2012:74] obtained a generalization of Geraghty's theorem.

The following theorem independently due to Kotzig [10] and Bouchet and Fouquet [3] gives a necessary and sufficient condition for a cubic graph to have a [P.sub.4]-decomposition.

Theorem 2 [9] A connected 4-regular graph G has a [P.sub.4]-decomposition if and only if [absolute value of E(G)] is divisible by 3.

Theorem Clinical Research and Biomedical Systems, a provider of centralized diagnostic services, have formed a strategic partnership to expand services available to clients worldwide.