370 research outputs found
Abelian Functions for Cyclic Trigonal Curves of Genus Four
We discuss the theory of generalized Weierstrass and functions
defined on a trigonal curve of genus four, following earlier work on the genus
three case. The specific example of the "purely trigonal" (or "cyclic
trigonal") curve is discussed in detail, including a list of some of the associated
partial differential equations satisfied by the functions, and the
derivation of an addition formulae.Comment: 23 page
Hyperelliptic addition law
We construct an explicit form of the addition law for hyperelliptic Abelian
vector functions and . The functions and form a basis
in the field of hyperelliptic Abelian functions, i.e., any function from the
field can be expressed as a rational function of and .Comment: 18 pages, amslate
K^*(BG) rings for groups of order 32
B. Schuster \cite{SCH1} proved that the 2 Morava -theory
is evenly generated for all groups of order 32. For the four
groups with the numbers 38, 39, 40 and 41 in the Hall-Senior list \cite{H},
the ring has been shown to be generated as a -module by
transferred Euler classes. In this paper, we show this for arbitrary and
compute the ring structure of . Namely, we show that
is the quotient of a polynomial ring in 6 variables over by an
ideal for which we list explicit generators.Comment: 23 page
Polytopes, Hopf algebras and Quasi-symmetric functions
In this paper we use the technique of Hopf algebras and quasi-symmetric
functions to study the combinatorial polytopes. Consider the free abelian group
generated by all combinatorial polytopes. There are two natural
bilinear operations on this group defined by a direct product and a
join of polytopes. is a commutative
associative bigraded ring of polynomials, and is a commutative associative
threegraded ring of polynomials. The ring has the structure of a
graded Hopf algebra. It turns out that has a natural Hopf
comodule structure over . Faces operators that send a
polytope to the sum of all its -dimensional faces define on both rings
the Hopf module structures over the universal Leibnitz-Hopf algebra
. This structure gives a ring homomorphism \R\to\Qs\otimes\R,
where is or . Composing this homomorphism with
the characters of , of
, and with the counit we obtain the ring homomorphisms
f\colon\mathcal{P}\to\Qs[\alpha],
f_{\mathcal{RP}}\colon\mathcal{RP}\to\Qs[\alpha], and
\F^*:\mathcal{RP}\to\Qs, where is the Ehrenborg transformation. We
describe the images of these homomorphisms in terms of functional equations,
prove that these images are rings of polynomials over , and find the
relations between the images, the homomorphisms and the Hopf comodule
structures. For each homomorphism , and \F the images
of two polytopes coincide if and only if they have equal flag -vectors.
Therefore algebraic structures on the images give the information about flag
-vectors of polytopes.Comment: 61 page
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