3,168 research outputs found
Generalised Brownian Motion and Second Quantisation
A new approach to the generalised Brownian motion introduced by M. Bozejko
and R. Speicher is described, based on symmetry rather than deformation. The
symmetrisation principle is provided by Joyal's notions of tensorial and
combinatorial species. Any such species V gives rise to an endofunctor F_V of
the category of Hilbert spaces with contractions. A generalised Brownian motion
is an algebra of creation and annihilation operators acting on F_V(H) for
arbitrary Hilbert spaces H and having a prescription for the calculation of
vacuum expectations in terms of a function t on pair partitions. The positivity
is encoded by a *-semigroup of "broken pair partitions" whose representation
space with respect to t is V. The existence of the second quantisation as
functor Gamma_t from Hilbert spaces to noncommutative probability spaces is
proved to be equivalent to the multiplicative property of the function t. For a
certain one parameter interpolation between the fermionic and the free Brownian
motion it is shown that the ``field algebras'' Gamma(K) are type II_1 factors
when K is infinite dimensional.Comment: 33 pages, 5 figure
Dominating the Erdos-Moser theorem in reverse mathematics
The Erdos-Moser theorem (EM) states that every infinite tournament has an
infinite transitive subtournament. This principle plays an important role in
the understanding of the computational strength of Ramsey's theorem for pairs
(RT^2_2) by providing an alternate proof of RT^2_2 in terms of EM and the
ascending descending sequence principle (ADS). In this paper, we study the
computational weakness of EM and construct a standard model (omega-model) of
simultaneously EM, weak K\"onig's lemma and the cohesiveness principle, which
is not a model of the atomic model theorem. This separation answers a question
of Hirschfeldt, Shore and Slaman, and shows that the weakness of the
Erdos-Moser theorem goes beyond the separation of EM from ADS proven by Lerman,
Solomon and Towsner.Comment: 36 page
The Euler characteristic of the Whitehead automorphism group of a free product
A combinatorial summation identity over the lattice of labelled hypertrees is
established that allows one to gain concrete information on the Euler
characteristics of various automorphism groups of free products of groups.Comment: 19 pages, 3 figures, to appear in Trans. Amer. Math. So
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