480 research outputs found
Additivity, subadditivity and linearity: automatic continuity and quantifier weakening
We study the interplay between additivity (as in the Cauchy functional
equation), subadditivity and linearity. We obtain automatic continuity results
in which additive or subadditive functions, under minimal regularity
conditions, are continuous and so linear. We apply our results in the context
of quantifier weakening in the theory of regular variation completing our
programme of reducing the number of hard proofs there to zero.Comment: Companion paper to: Cauchy's functional equation and extensions:
Goldie's equation and inequality, the Go{\l}\k{a}b-Schinzel equation and
Beurling's equation Updated to refer to other developments and their
publication detail
The Dirichlet Markov Ensemble
We equip the polytope of Markov matrices with the normalized
trace of the Lebesgue measure of . This probability space
provides random Markov matrices, with i.i.d. rows following the Dirichlet
distribution of mean . We show that if \bM is such a random
matrix, then the empirical distribution built from the singular values
of\sqrt{n} \bM tends as to a Wigner quarter--circle
distribution. Some computer simulations reveal striking asymptotic spectral
properties of such random matrices, still waiting for a rigorous mathematical
analysis. In particular, we believe that with probability one, the empirical
distribution of the complex spectrum of \sqrt{n} \bM tends as to
the uniform distribution on the unit disc of the complex plane, and that
moreover, the spectral gap of \bM is of order when is
large.Comment: Improved version. Accepted for publication in JMV
General regular variation, Popa groups and quantifier weakening
We introduce general regular variation, a theory of regular variation containing the existing Karamata, Bojanic-Karamata/de Haan and Beurling theories as special cases. The unifying theme is the Popa groups of our title viewed as locally compact abelian ordered topological groups, together with their Haar measure and Fourier theory. The power of this unified approach is shown by the simplification it brings to the whole area of quantifier weakening, so important in this field
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