16,869 research outputs found
Effective Results on the Waring Problem for Finite Simple Groups
Let G be a finite quasisimple group of Lie type. We show that there are
regular semisimple elements x,y in G, x of prime order, and |y| is divisible by
at most two primes, such that the product of the conjugacy classes of x and y
contain all non-central elements of G. In fact in all but four cases, y can be
chosen to be of square-free order. Using this result, we prove an effective
version of one of the main results of Larsen, Shalev and Tiep by showing that,
given any positive integer m, if the order of a finite simple group S is at
least f(m) for a specified function f, then every element in S is a product of
two mth powers. Furthermore, the verbal width of the mth power word on any
finite simple group S is at most g(m) for a specified function g. We also show
that, given any two non-trivial words v, w, if G is a finite quasisimple group
of large enough order, then v(G)w(G) contains all non-central elements of G.Comment: Note title change from version
Mixing and non-mixing local minima of the entropy contrast for blind source separation
In this paper, both non-mixing and mixing local minima of the entropy are
analyzed from the viewpoint of blind source separation (BSS); they correspond
respectively to acceptable and spurious solutions of the BSS problem. The
contribution of this work is twofold. First, a Taylor development is used to
show that the \textit{exact} output entropy cost function has a non-mixing
minimum when this output is proportional to \textit{any} of the non-Gaussian
sources, and not only when the output is proportional to the lowest entropic
source. Second, in order to prove that mixing entropy minima exist when the
source densities are strongly multimodal, an entropy approximator is proposed.
The latter has the major advantage that an error bound can be provided. Even if
this approximator (and the associated bound) is used here in the BSS context,
it can be applied for estimating the entropy of any random variable with
multimodal density.Comment: 11 pages, 6 figures, To appear in IEEE Transactions on Information
Theor
Fractional excitations in the Luttinger liquid
We reconsider the spectrum of the Luttinger liquid (LL) usually understood in
terms of phonons (density fluctuations), and within the context of bosonization
we give an alternative representation in terms of fractional states. This
allows to make contact with Bethe Ansatz which predicts similar fractional
states. As an example we study the spinon operator in the absence of spin
rotational invariance and derive it from first principles: we find that it is
not a semion in general; a trial Jastrow wavefunction is also given for that
spinon state. Our construction of the new spectroscopy based on fractional
states leads to several new physical insights: in the low-energy limit, we find
that the continuum of gapless spin chains is due to pairs of
fractional quasiparticle-quasihole states which are the 1D counterpart of the
Laughlin FQHE quasiparticles. The holon operator for the Luttinger liquid with
spin is also derived. In the presence of a magnetic field, spin-charge
separation is not realized any longer in a LL: the holon and the spinon are
then replaced by new fractional states which we are able to describe.Comment: Revised version to appear in Physical Review B. 27 pages, 5 figures.
Expands cond-mat/9905020 (Eur.Phys.Journ.B 9, 573 (1999)
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