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In the study of integrable systems of ODE's arising from a Lax pair with a parameter, the constants of the motion occur as spectral curves. The specific curves depend upon the representation of the Lie algebra. In this paper a Galois theory of spectral curves is given that classifies the spectral curves from an integrable system. The spectral curves correspond to conjugacy classes of certain subgroups of the Weyl group for the Lie algebra. The theory is illustrated with the periodic Toda lattice.

This paper analyzes the applicability of the article mean citation rate measures in the Science Citation Index Journal Citation Reports (SCI JCR) to the five JCR mathematical subject categories.These measures are the traditional 2-year impact factor as well as the recently added 5-year impact factor and 5-year article influence score. Utilizing the 2008 SCI JCR, the paper compares the probability distributions of the measures in the mathematical categories to the probability distribution of a scientific model of impact factor distribution. The scientific model distribution is highly skewed, conforming to the negative binomial type, with much of the variance due to the important role of review articles in science. In contrast, the three article mean citation rate measures' distributions in the mathematical categories conformed to either the binomial or Poisson, indicating a high degree of randomness. Seeking reasons for this, the paper analyzes the bibliometric structure of Mathematics, finding it a disjointed discipline of isolated subfields with a weak central core of journals, reduced review function, and long cited half-life placing most citations beyond the measures' time limits. These combine to reduce the measures' variance to one commensurate with random error. However, the measures were found capable of identifying important journals. Using data from surveys of the Louisiana State University (LSU) faculty, the paper finds a higher level of consensus among mathematicians and others on which are the important mathematics journals than the measures indicate, positing that much of the apparent randomness may be due to the measures' inapplicability to mathematical disciplines. Moreover, tests of the stability of impact factor ranks across a 5-year time span suggested that the proper model for Mathematics is the negative binomial.

Abstract.The framed braid group on n strands is defined to be a semidirect product of the braid group B" and Z" . Framed braids represent 3-manifolds in a manner analogous to the representation of links by braids. Consider two framed braids equivalent if they represent homeomorphic 3-manifolds. The main result of this paper is a Markov type theorem giving moves that generate this equivalence relation.