617 research outputs found
Elliptic Double Zeta Values
We study an elliptic analogue of multiple zeta values, the elliptic multiple
zeta values of Enriquez, which are the coefficients of the elliptic KZB
associator. Originally defined by iterated integrals on a once-punctured
complex elliptic curve, it turns out that they can also be expressed as certain
linear combinations of indefinite iterated integrals of Eisenstein series and
multiple zeta values. In this paper, we prove that the -span of
these elliptic multiple zeta values forms a -algebra, which is
naturally filtered by the length and is conjecturally graded by the weight. Our
main result is a proof of a formula for the number of -linearly
independent elliptic multiple zeta values of lengths one and two for arbitrary
weight.Comment: 22 page
Hopf algebras of endomorphisms of Hopf algebras
In the last decennia two generalizations of the Hopf algebra of symmetric
functions have appeared and shown themselves important, the Hopf algebra of
noncommutative symmetric functions NSymm and the Hopf algebra of quasisymmetric
functions QSymm. It has also become clear that it is important to understand
the noncommutative versions of such important structures as Symm the Hopf
algebra of symmetric functions. Not least because the right noncommmutative
versions are often more beautiful than the commutaive ones (not all cluttered
up with counting coefficients). NSymm and QSymm are not truly the full
noncommutative generalizations. One is maximally noncommutative but
cocommutative, the other is maximally non cocommutative but commutative. There
is a common, selfdual generalization, the Hopf algebra of permutations of
Malvenuto, Poirier, and Reutenauer (MPR). This one is, I feel, best understood
as a Hopf algebra of endomorphisms. In any case, this point of view suggests
vast generalizations leading to the Hopf algebras of endomorphisms and word
Hopf algebras with which this paper is concerned. This point of view also sheds
light on the somewhat mysterious formulas of MPR and on the question where all
the extra structure (such as autoduality) comes from. The paper concludes with
a few sections on the structure of MPR and the question of algebra retractions
of the natural inclusion of Hopf algebras of NSymm into MPR and section of the
naural projection of MPR onto QSymm.Comment: 40 pages. Revised and expanded version of a (nonarchived) preprint of
200
Relations between elliptic multiple zeta values and a special derivation algebra
We investigate relations between elliptic multiple zeta values and describe a
method to derive the number of indecomposable elements of given weight and
length. Our method is based on representing elliptic multiple zeta values as
iterated integrals over Eisenstein series and exploiting the connection with a
special derivation algebra. Its commutator relations give rise to constraints
on the iterated integrals over Eisenstein series relevant for elliptic multiple
zeta values and thereby allow to count the indecomposable representatives.
Conversely, the above connection suggests apparently new relations in the
derivation algebra. Under https://tools.aei.mpg.de/emzv we provide relations
for elliptic multiple zeta values over a wide range of weights and lengths.Comment: 43 pages, v2:replaced with published versio
Decomposition of elliptic multiple zeta values and iterated Eisenstein integrals
We describe a decomposition algorithm for elliptic multiple zeta values,
which amounts to the construction of an injective map from the algebra
of elliptic multiple zeta values to a space of iterated Eisenstein integrals.
We give many examples of this decomposition, and conclude with a short
discussion about the image of . It turns out that the failure of
surjectivity of is in some sense governed by period polynomials of
modular forms.Comment: v2, minor change
Efficient Quantum Transforms
Quantum mechanics requires the operation of quantum computers to be unitary,
and thus makes it important to have general techniques for developing fast
quantum algorithms for computing unitary transforms. A quantum routine for
computing a generalized Kronecker product is given. Applications include
re-development of the networks for computing the Walsh-Hadamard and the quantum
Fourier transform. New networks for two wavelet transforms are given. Quantum
computation of Fourier transforms for non-Abelian groups is defined. A slightly
relaxed definition is shown to simplify the analysis and the networks that
computes the transforms. Efficient networks for computing such transforms for a
class of metacyclic groups are introduced. A novel network for computing a
Fourier transform for a group used in quantum error-correction is also given.Comment: 30 pages, LaTeX2e, 7 figures include
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