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We study the equations for the quasi-momenta which characterize the wave-functions in the Bethe ansatz for the XXX-Heisenberg model. We show in a simple analytical fashion, that the usual "string hypothesis" incorrectly predicts the number of real solutions and the 1 number of complex solutions for N > 21 in the sector with two spins flipped, confirming the work of Essler et al. Two complex pair solutions drop out and form two additional real pair solutions. We also introduce a new set of variables which allows the equations to be written as a single polynomial equation in one variable. We consider in some detail the case of three spins flipped.The XXX-Heisenberg model appears to be fundamental to many of the recent interesting developments in modern quantum field theory and mathematical physics. Integrable models, conformal field theories and quantum groups are seen to arise from various limits, extensions and generalizations 1 . It is, in fact, the original model considered by Bethe 2 , which gave rise to the celebrated "Bethe ansatz" solutions. Recently, there appeared an article 3 , studying the Bethe ansatz equations in the two particle sector of the spin one-half model. It was shown that, as the number of lattice sites, N , increases past 21, two new real pair solutions appear. These solutions do not fit into the conventional scheme of classification of the solutions to the Bethe ansatz equations which causes some anxiety as to the verity of the completeness of the full set of SU (2) extended Bethe ansatz states. The full set of states, just by counting, should number 2 N . This, however, is not a problem. As the real pair solutions appear, complex pair solutions disappear, conserving the total number of states. We analyze the Bethe ansatz equations analytically and confirm their result; two complex pair solutions disappear simultaneously giving two new real pair solutions. The spin 1 2 XXX-Heisenberg model with N spins corresponds to the Hamiltonian

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A certain class of cellular automata in 1 space + 1 time dimension is shown to be closely related to quantum field theories containing Dirac fermions. In the massless case this relation can be studied analytically, while the introduction of Dirac mass requires numerical simulations. We show that in the last case the cellular automaton describes the corresponding field theory only approximately.

We apply the ~justernik-Snirelman theory, a derivative of Morse theory, to the Skyrmion-Skyrmion potential in the two-flavor Skyrme model to look for the existence of new, static, classical solutions in the sector with baryon number two. Concomitantly, we present a systematic method for obtaining an expansion of this potential, in inverse powers of the separation, when the separation is large.