8,058 research outputs found
Towards explaining the speed of -means
The -means method is a popular algorithm for clustering, known for its speed in practice. This stands in contrast to its exponential worst-case running-time. To explain the speed of the -means method, a smoothed analysis has been conducted. We sketch this smoothed analysis and a generalization to Bregman divergences
Forms of Hopf Algebras and Galois Theory
The theory of Hopf algebras is closely connected with various applications,
in particular to algebraic and formal groups. Although the
rst occurence of Hopf algebras was in algebraic topology, they are now
found in areas as remote as combinatorics and analysis. Their structure
has been studied in great detail and many of their properties are
well understood. We are interested in a systematic treatment of Hopf
algebras with the techniques of forms and descent.
The rst three paragraphs of this paper give a survey of the present
state of the theory of forms of Hopf algebras and of Hopf Galois theory
especially for separable extensions. It includes many illustrating examples
some of which cannot be found in detail in the literature. The last
two paragraphs are devoted to some new or partial results on the same
eld. There we formulate some of the open questions which should
be interesting objects for further study. We assume throughout most
of the paper that k is a base eld and do not touch upon the recent
beautiful results of Hopf Galois theory for rings of integers in algebraic
number elds as developed in [C1]
When Hopf Algebras are Frobenius Algebras
AbstractR. Larson and M. Sweedler recently proved that for free finitely generated Hopf algebras H over a principal ideal domain R the following are equivalent: (a) H has an antipode and (b) H has a nonsingular left integral. In this paper I give a generalization of this result which needs only a minor restriction, which, for example, always holds if pic(R) = 0 for the base ring R. A finitely generated projective Hopf algebra H over R has an antipode if and only if H is a Frobenius algebra with a Frobenius homomorphism ψ such that Σ h(1) ψ(h(2)) = ψ(h) · 1 for all h ϵ H. We also show that the antipode is bijective and that the ideal of left integrals is a free rank 1, R-direct summand of H
Reconstruction of Hidden Symmetries
Representations of a group in vector spaces over a field form a
category. One can reconstruct the given group from its representations to
vector spaces as the full group of monoidal automorphisms of the underlying
functor. This is a special example of Tannaka-Krein theory. This theory was
used in recent years to reconstruct quantum groups (quasitriangular Hopf
algebras) in the study of algebraic quantum field theory and other
applications.
We show that a similar study of representations in spaces with additional
structure (super vector spaces, graded vector spaces, comodules, braided
monoidal categories) produces additional symmetries, called ``hidden
symmetries''. More generally, reconstructed quantum groups tend to decompose
into a smash product of the given quantum group and a quantum group of
``hidden'' symmetries of the base category.Comment: 42 pages, amslatex, figures generated with bezier.sty, replaced to
facilitate mailin
Corrections to Scaling in the Integer Quantum Hall Effect
Finite size corrections to scaling laws in the centers of Landau levels are
studied systematically by numerical calculations. The corrections can account
for the apparent non-universality of the localization length exponent .
In the second lowest Landau level the irrelevant scaling index is
. At the center of the lowest Landau level an
additional periodic potential is found to be irrelevant with the same scaling
index. These results suggest that the localization length exponent is
universal with respect to Landau level index and an additional periodic
potential.Comment: 8 pages, RevTeX 3.0, 7 PostScript figures in uuencoded compressed tar
file include
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