1,785 research outputs found
Bordered Heegaard Floer homology and graph manifolds
We perform two explicit computations of bordered Heegaard Floer invariants.
The first is the type D trimodule associated to the trivial S^1 bundle over the
pair of pants P. The second is a bimodule that is necessary for self-gluing,
when two torus boundary components of a bordered manifold are glued to each
other. Using the results of these two computations, we describe an algorithm
for computing HF-hat of any graph manifold.Comment: 59 pages, 21 figures, new version corrects typos and adds a short
discussion of grading
Solving the word problem in real time
The paper is devoted to the study of groups whose word problem can be solved by a Turing machine which operates in real time. A recent result of the first author for word hyperbolic groups is extended to prove that under certain conditions the generalised Dehn algorithms of Cannon, Goodman and Shapiro, which clearly run in linear time, can be programmed on real-time Turing machines. It follows that word-hyperbolic groups, finitely generated nilpotent groups and geometrically finite hyperbolic groups all have real-time word problems
Quantum stochastic convolution cocycles II
Schuermann's theory of quantum Levy processes, and more generally the theory
of quantum stochastic convolution cocycles, is extended to the topological
context of compact quantum groups and operator space coalgebras. Quantum
stochastic convolution cocycles on a C*-hyperbialgebra, which are
Markov-regular, completely positive and contractive, are shown to satisfy
coalgebraic quantum stochastic differential equations with completely bounded
coefficients, and the structure of their stochastic generators is obtained.
Automatic complete boundedness of a class of derivations is established,
leading to a characterisation of the stochastic generators of *-homomorphic
convolution cocycles on a C*-bialgebra. Two tentative definitions of quantum
Levy process on a compact quantum group are given and, with respect to both of
these, it is shown that an equivalent process on Fock space may be
reconstructed from the generator of the quantum Levy process. In the examples
presented, connection to the algebraic theory is emphasised by a focus on full
compact quantum groups.Comment: 32 pages, expanded introduction and updated references. The revised
version will appear in Communications in Mathematical Physic
Universal Lefschetz fibrations and Lefschetz cobordisms
We construct universal Lefschetz fibrations, defined in analogy with
classical universal bundles. We also introduce the cobordism groups of
Lefschetz fibrations, and we see how these groups are quotients of the singular
bordism groups via the universal Lefschetz fibrations.Comment: 14 pages; minor revision to match the published versio
Complete Symmetry in D2L Systems and Cellular Automata
We introduce completely symmetric D2L systems and cellular automata by means of an additional restriction on the corresponding symmetric devices. Then we show that completely symmetric D2L systems and cellular automata are still able to simulate Turing machine computations. As corollaries we obtain new characterizations of the recursively enumerable languages and of some space-bounded complexity classes
Quantum stochastic cocycles and completely bounded semigroups on operator spaces
An operator space analysis of quantum stochastic cocycles is undertaken.
These are cocycles with respect to an ampliated CCR flow, adapted to the
associated filtration of subspaces, or subalgebras. They form a noncommutative
analogue of stochastic semigroups in the sense of Skorohod. One-to-one
correspondences are established between classes of cocycle of interest and
corresponding classes of one-parameter semigroups on associated matrix spaces.
Each of these 'global' semigroups may be viewed as the expectation semigroup of
an associated quantum stochastic cocycle on the corresponding matrix space. The
classes of cocycle covered include completely positive contraction cocycles on
an operator system, or C*-algebra; completely contractive cocycles on an
operator space; and contraction operator cocycles on a Hilbert space. As
indicated by Accardi and Kozyrev, the Schur-action matrix semigroup viewpoint
circumvents technical (domain) limitations inherent in the theory of quantum
stochastic differential equations. An infinitesimal analysis of quantum
stochastic cocycles from the present wider perspective is given in a sister
paper.Comment: 32 page
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