20,145 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

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