Abstract
We discuss a conjecture of Ruelle concerningstable potentials on a group. For the groupsZ 2,Z 3, 4, andZ 6 any stable potential can be written as the sum of a non-negative function and a function of non-negative type. This is not true for the groupsZ k (k odd, ≧5). For the Euclidean groupR v the question is open.
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References
Ruelle, D.: Statistical mechanics — rigorous results. New York: Benjamin Inc. 1969.
Op. cit. Section 3.2.
Karlin, S.: Mathematical methods and theory in games, programming and economics, Vol. I, App. B. Reading, Mass.: Addison-Wesley 1959.
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Supported by NSF GP 7946.
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Lenard, A., Sherman, S. Stable potentials I. Commun.Math. Phys. 15, 201–207 (1969). https://doi.org/10.1007/BF01645675
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DOI: https://doi.org/10.1007/BF01645675