152 research outputs found
The Gelfand map and symmetric products
If A is an algebra of functions on X, there are many cases when X can be
regarded as included in Hom(A,C) as the set of ring homomorphisms. In this
paper the corresponding results for the symmetric products of X are introduced.
It is shown that the symmetric product Sym^n(X) is included in Hom(A,C) as the
set of those functions that satisfy equations generalising f(xy)=f(x)f(y).
These equations are related to formulae introduced by Frobenius and, for the
relevant A, they characterise linear maps on A that are the sum of ring
homomorphisms. The main theorem is proved using an identity satisfied by
partitions of finite sets.Comment: 14 pages, Late
-function of the KdV hierarchy
In this paper we construct a family of commuting multidimensional
differential operators of order 3, which is closely related to the KdV
hierarchy. We find a common eigenfunction of this family and an algebraic
relation between these operators. Using these operators we associate a
hyperelliptic curve to any solution of the stationary KdV equation. A basic
generating function of the solutions of stationary KdV equation is introduced
as a special polarization of the equation of the hyperelliptic curve. We also
define and discuss the notion of a -function of a solution of the stationary
-KdV equation
Frobenius n-homomorphisms, transfers and branched coverings
The main purpose is to characterise continuous maps that are n-branched coverings in terms of induced maps on the rings of functions. The special properties of Frobenius nhomomorphisms between two function spaces that correspond to n-branched coverings are determined completely. Several equivalent definitions of a Frobenius n-homomorphism are compared and some of their properties are proved. An axiomatic treatment of n-transfers is given in general and properties of n-branched coverings are studied and compared with those of regular coverings
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