We use the contraction mapping theorem to present existence and uniqueness of solutions in a short time system non-linear Volterra integral equations certain type direct-sum H[a; b] Hilbert space V[a; b]. extend local global proposed problem. Because kernel function is transcendental on interval [a; b], results are novel very important numerical approximation.

We give a simple proof that an n×n orthogonal matrix with n ≥ 2 which cannot be written as a direct sum has at least 4n− 4 nonzero entries. 1. The result. What is the least number of nonzero entries in a real orthogonal matrix of order n? Since the identity matrix In is orthogonal the answer is clearly n. A more interesting question is: what is the least number of nonzero entries in a real orth...

In this paper we show that a direct decomposition of modules M N, with N homologically independent to the inJective hull of H, is a CS-module if and only if N is injective relative to H and both of M and N are CS-modules. As an application, we prove that a direct sum of a non-singular semisimple module and a quasi-continuous module with zero socle is quasi-continuous. This result is known for q...

We show how to efficiently simulate the sending of a message M to a receiver who has partial information about the message, so that the expected number of bits communicated in the simulation is close to the amount of additional information that the message reveals to the receiver. We use our simulation method to obtain several results in communication complexity. • We prove a new direct sum the...