1,967 research outputs found
Enumeration of non-orientable 3-manifolds using face pairing graphs and union-find
Drawing together techniques from combinatorics and computer science, we
improve the census algorithm for enumerating closed minimal P^2-irreducible
3-manifold triangulations. In particular, new constraints are proven for face
pairing graphs, and pruning techniques are improved using a modification of the
union-find algorithm. Using these results we catalogue all 136 closed
non-orientable P^2-irreducible 3-manifolds that can be formed from at most ten
tetrahedra.Comment: 37 pages, 34 figure
Cycle decompositions: from graphs to continua
We generalise a fundamental graph-theoretical fact, stating that every
element of the cycle space of a graph is a sum of edge-disjoint cycles, to
arbitrary continua. To achieve this we replace graph cycles by topological
circles, and replace the cycle space of a graph by a new homology group for
continua which is a quotient of the first singular homology group . This
homology seems to be particularly apt for studying spaces with infinitely
generated , e.g. infinite graphs or fractals.Comment: Advances in Mathematics (2011
The splitting process in free probability theory
Free cumulants were introduced by Speicher as a proper analog of classical
cumulants in Voiculescu's theory of free probability. The relation between free
moments and free cumulants is usually described in terms of Moebius calculus
over the lattice of non-crossing partitions. In this work we explore another
approach to free cumulants and to their combinatorial study using a
combinatorial Hopf algebra structure on the linear span of non-crossing
partitions. The generating series of free moments is seen as a character on
this Hopf algebra. It is characterized by solving a linear fixed point equation
that relates it to the generating series of free cumulants. These phenomena are
explained through a process similar to (though different from) the
arborification process familiar in the theory of dynamical systems, and
originating in Cayley's work
The proof-theoretic strength of Ramsey's theorem for pairs and two colors
Ramsey's theorem for -tuples and -colors () asserts
that every k-coloring of admits an infinite monochromatic
subset. We study the proof-theoretic strength of Ramsey's theorem for pairs and
two colors, namely, the set of its consequences, and show that
is conservative over . This
strengthens the proof of Chong, Slaman and Yang that does not
imply , and shows that is
finitistically reducible, in the sense of Simpson's partial realization of
Hilbert's Program. Moreover, we develop general tools to simplify the proofs of
-conservation theorems.Comment: 32 page
On Universal Prediction and Bayesian Confirmation
The Bayesian framework is a well-studied and successful framework for
inductive reasoning, which includes hypothesis testing and confirmation,
parameter estimation, sequence prediction, classification, and regression. But
standard statistical guidelines for choosing the model class and prior are not
always available or fail, in particular in complex situations. Solomonoff
completed the Bayesian framework by providing a rigorous, unique, formal, and
universal choice for the model class and the prior. We discuss in breadth how
and in which sense universal (non-i.i.d.) sequence prediction solves various
(philosophical) problems of traditional Bayesian sequence prediction. We show
that Solomonoff's model possesses many desirable properties: Strong total and
weak instantaneous bounds, and in contrast to most classical continuous prior
densities has no zero p(oste)rior problem, i.e. can confirm universal
hypotheses, is reparametrization and regrouping invariant, and avoids the
old-evidence and updating problem. It even performs well (actually better) in
non-computable environments.Comment: 24 page
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