296,831 research outputs found
A constructive method for decomposing real representations
A constructive method for decomposing finite dimensional representations of
semisimple real Lie algebras is developed. The method is illustrated by an
example. We also discuss an implementation of the algorithm in the language of
the computer algebra system {\sf GAP}4.Comment: Final version; to appear in "Journal of Symbolic Computation
Strictly transversal slices to conjugacy classes in algebraic groups
We show that for every conjugacy class O in a connected semisimple algebraic
group G over a field of characteristic good for G one can find a special
transversal slice S to the set of conjugacy classes in G such that O intersects
S and dim O = codim S.Comment: 38 pages; minor modification
Distribution of roots of random real generalized polynomials
The average density of zeros for monic generalized polynomials,
, with real holomorphic and
real Gaussian coefficients is expressed in terms of correlation functions of
the values of the polynomial and its derivative. We obtain compact expressions
for both the regular component (generated by the complex roots) and the
singular one (real roots) of the average density of roots. The density of the
regular component goes to zero in the vicinity of the real axis like
. We present the low and high disorder asymptotic
behaviors. Then we particularize to the large limit of the average density
of complex roots of monic algebraic polynomials of the form with real independent, identically distributed
Gaussian coefficients having zero mean and dispersion . The average density tends to a simple, {\em universal}
function of and in the domain where nearly all the roots are located for
large .Comment: 17 pages, Revtex. To appear in J. Stat. Phys. Uuencoded gz-compresed
tarfile (.66MB) containing 8 Postscript figures is available by e-mail from
[email protected]
Nonnegative polynomials and their Carath\'eodory number
In 1888 Hilbert showed that every nonnegative homogeneous polynomial with
real coefficients of degree in variables is a sum of squares if and
only if (quadratic forms), (binary forms) or (ternary
quartics). In these cases, it is interesting to compute canonical expressions
for these decompositions. Starting from Carath\'eodory's Theorem, we compute
the Carath\'eodory number of Hilbert cones of nonnegative quadratic and binary
forms.Comment: 9 pages. Discrete & Computational Geometry (2014
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