The Geometry of Bifurcation Diagrams

Abstract

Applications of the mathematical formalism put forward in the previous paper are made to a number of well known lattice models in statistical mechanics. Sequences of algebraic curves are obtained from the branches of an algebraic function A , , where A: at real temperatures is the branch of A, which is the partition function per site of an m x 05 lattice section. These sequences are viewed as approximations to parts of the limiting locus C , of partition function zeros. Approximations to critical points can be obtained in a natural extension of these curves beyond the branch points of A;.

Frobenius manifolds and central invariants for the Drinfeld–Sokolov bihamiltonian structures

“…. , u n of the characteristic equation [42,43], see also [33]. The proof of Theorem 4.6 will be given in Section 5 for the A n series, in Section 6 for the B n , C n , D n series and in Section 7 for the exceptional cases.…”

The bifurcation set and logarithmic vector fields

“…Later Lyashko proved it in much more generalized form [4]. Denote the critical set and the discriminant (the set of critical values) of ~o by C and D respectively.…”

“…Especially they studied the logarithmic vector fields i) along the arrangement of hyperplanes [6,9,10] and ii) along the discriminant of semiuniversal deformation of an isolated hypersurface singularity [6,1,4,11]. Especially they studied the logarithmic vector fields i) along the arrangement of hyperplanes [6,9,10] and ii) along the discriminant of semiuniversal deformation of an isolated hypersurface singularity [6,1,4,11].…”

Configuration Spaces of the Affine Line and their Automorphism Groups

“…Using these observations and notation we can deduce from Theorem 3.17 and Corollary 4.2 the following description (cf. [40]). Proposition 4.9.…”