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-Techniques from,coding theory are applied to study rigorously the capacity of the Hopfield associative memory. Such a memory stores n -tuple of + 1's. The components change depending on a hardlimited version of linear functions of all other components. With symmetric connections between components, a stable state is ultimately reached. By building up the connection matrix as a sum-of-outer products of m fundamental memories, one hopes to be able to recover a certain one of the no memories by using an initial n-tuple probe vector less than a Hamming distance n/2 away from the ftindamental memory. If WI fundamental memories are chosen at random, the maximum asympotic value of m in order that most of the no original memories are exactly recoverable is n/(2log n). With the added restriction that every one of the m fundamental memories be recoverable exactly, rrl can be no more than n/(4log n) asymptotically as n approaches infinity. Extensions are also considered, in particular to capacity under qnantization of the outer-product connection matrijr. This quantized memory capacity problem is closely related to the capacity of the quantized Gaussian channel.

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Abstract-The capacity region for the discrete memoryless multipleaccess channel without time synchronization at the transmitters and receivers is shown to be the same as the known capacity region for the ordinary multiple-access channel. The proof utilizes time sharing of two optimal codes for the ordinary multiple-access channel and uses maximum likeliiood decoding over shifts of the hypothesized transmitter words.

We prove two theorems that are easily conjectured, namely: (1) In a prime ring of characteristics not 2, if the iterate of two derivations is a derivation, then one of them is zero; (2) If d is a derivation of a prime ring such that, for all elements a oi the ring, ad(a)-d(a)a is central, then either the ring is commutative or d is zero. Definition.A ring R is called prime if and only if xay = Q for all aER implies x = 0 or y = 0.From this definition it follows that no nonzero element of the centroid has nonzero kernel, so that we can divide by the prime p, unless px = 0 for all x in R, in which case we call R of characteristic p.A known result that will be often used throughout this paper is given in Lemma 1. Let d be a derivation of a prime ring R and a be an element of R. If ad(x) = 0 for all xER, then either a = 0 or d is zero.Proof: In ad(x) =0 for all xER, replace x by xy. Then ad(xy) -0 = ad(x)y + axd(y) -axd(y) = 0 for all x, yER-If d is not zero, that is, if d(y) 9^0 ior some yER, then, by the definition of a prime ring, a = 0.The following lemma may have some independent interest.