Polymer models and generalized Potts-Kasteleyn models

The well-established relation between Potts models withv spin values and random-cluster models (with intracluster bonding favored over intercluster bonding by a factorv) is explored, but with the random-cluster model replaced by a much generalized polymer model, implying a corresponding generalization of the Potts model. The analysis is carried out in terms a given defined functionR(), an entropy/free-energy density for the polymer model in the casev=1, expressed as a function of the density of units. The aim of the analysis is to determine the analogR _v () ofR() for general nonnegativev in terms ofR(), and thence to determine the critical value of density _vg at which gelation occurs. This critical value is independent ofv up to a valuev _P, the Potts-critical value. What is principally required ofR() is that it should show a certain given concave/convex behavior, although differentiability and another regularizing condition are required for complete conclusions. Under these conditions the unique evaluation ofR _v () in terms ofR() is given in a form known to hold for integralv but not previously extended. The analysis is carried out in terms of the Legendre transforms of these functions, in terms of which the phenomena of criticality (gelation) and Potts criticality appear very transparently andv _P is easily determined. The value ofv _P is 2 under mild conditions onR. Special interest attaches to the functionR ₀(), which is shown to be the greatest concave minorant ofR(). The naturalness of the approach is demonstrated by explicit treatment of the first-shell model.     

δ-BiHom-Jordan-李超代数的表示

本文研究δ-BiHom-Jordan-李超代数的表示.特别是详细地研究δ-BiHom-Jordan-李超代数的伴随表示、平凡表示、形变.作为应用，还讨论δ-BiHom-Jordan-李代数的导子.     

Geometric proof of a conjecture of Fulton

We give a geometric proof of a conjecture of Fulton on the multiplicities of irreducible representations in a tensor product of irreducible representations for GL(r).     

Composition formulas in the Weyl calculus

In pseudodifferential analysis, the usual composition formula, which has asymptotic value, extends that valid for differential operators. The one developed here is based instead on the decomposition of symbols (functions in Rn×Rn) as integral superpositions of homogeneous ones, of degrees lying on the complex line with real part −n. It extends the one known in the one-dimensional case in connection with automorphic pseudodifferential analysis.     

Equivalence of commutation relations

We consider C*-algebras of commutation relations over the fields ℚ _p, p = 2, 3, 5, …, ∞. We describe all the irreducible separable representations of these algebras. We prove that the algebras are not isomorphic at different p. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 406–412, December, 2008.     

On the computation of the Euler constant γ

The problem of the approximation to the Euler constant γ is considered. Some new formulas for the computation of Euler's constant are suggested. The problem of fast evaluation of γ is discussed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Torsion Endo-Trivial Modules

We prove that the group T(G) of endo-trivial modules for a noncyclic finite p-group G is detected on restriction to the family of subgroups which are either elementary Abelian of rank 2 or (almost) extraspecial. This result is closely related to the problem of finding the torsion subgroup of T(G). We give the complete structure of T(G) when G is dihedral, semi-dihedral, or quaternion.