7,938 research outputs found
Propagation and interaction of chiral states in quantum gravity
We study the stability, propagation and interactions of braid states in
models of quantum gravity in which the states are four-valent spin networks
embedded in a topological three manifold and the evolution moves are given by
the dual Pachner moves. There are results for both the framed and unframed
case. We study simple braids made up of two nodes which share three edges,
which are possibly braided and twisted. We find three classes of such braids,
those which both interact and propagate, those that only propagate, and the
majority that do neither.Comment: 34 pages, 30 figures, typos corrected, 2 references added, to match
the version accepted for publication in Nucl. Phys.
Computing with space: a tangle formalism for chora and difference
What is space computing,simulation, or understanding? Converging from several sources, this seems to be something more primitive than what is meant nowadays by computation, something that was along with us since antiquity (the word "choros", "chora", denotes "space" or "place" and is seemingly the most mysterious notion from Plato, described in Timaeus 48e - 53c) which has to do with cybernetics and with the understanding of the front end visual system. It may have some unexpected applications, also. \ud
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Here, inspired by Bateson (see Supplementary Material), I explore from the mathematical side the point of view that there is no difference between the map and the territory, but instead the transformation of one into another can be understood by using a formalism of tangle diagrams
Rota-Baxter algebras, singular hypersurfaces, and renormalization on Kausz compactifications
We consider Rota-Baxter algebras of meromorphic forms with poles along a
(singular) hypersurface in a smooth projective variety and the associated
Birkhoff factorization for algebra homomorphisms from a commutative Hopf
algebra. In the case of a normal crossings divisor, the Rota-Baxter structure
simplifies considerably and the factorization becomes a simple pole
subtraction. We apply this formalism to the unrenormalized momentum space
Feynman amplitudes, viewed as (divergent) integrals in the complement of the
determinant hypersurface. We lift the integral to the Kausz compactification of
the general linear group, whose boundary divisor is normal crossings. We show
that the Kausz compactification is a Tate motive and that the boundary divisor
and the divisor that contains the boundary of the chain of integration are
mixed Tate configurations. The regularization of the integrals that we obtain
differs from the usual renormalization of physical Feynman amplitudes, and in
particular it may give mixed Tate periods in some cases that have non-mixed
Tate contributions when computed with other renormalization methods.Comment: 35 pages, LaTe
Beyond Outerplanarity
We study straight-line drawings of graphs where the vertices are placed in
convex position in the plane, i.e., convex drawings. We consider two families
of graph classes with nice convex drawings: outer -planar graphs, where each
edge is crossed by at most other edges; and, outer -quasi-planar graphs
where no edges can mutually cross. We show that the outer -planar graphs
are -degenerate, and consequently that every
outer -planar graph can be -colored, and this
bound is tight. We further show that every outer -planar graph has a
balanced separator of size . This implies that every outer -planar
graph has treewidth . For fixed , these small balanced separators
allow us to obtain a simple quasi-polynomial time algorithm to test whether a
given graph is outer -planar, i.e., none of these recognition problems are
NP-complete unless ETH fails. For the outer -quasi-planar graphs we prove
that, unlike other beyond-planar graph classes, every edge-maximal -vertex
outer -quasi planar graph has the same number of edges, namely . We also construct planar 3-trees that are not outer
-quasi-planar. Finally, we restrict outer -planar and outer
-quasi-planar drawings to \emph{closed} drawings, where the vertex sequence
on the boundary is a cycle in the graph. For each , we express closed outer
-planarity and \emph{closed outer -quasi-planarity} in extended monadic
second-order logic. Thus, closed outer -planarity is linear-time testable by
Courcelle's Theorem.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Tropical curves, graph complexes, and top weight cohomology of M_g
We study the topology of a space parametrizing stable tropical curves of
genus g with volume 1, showing that its reduced rational homology is
canonically identified with both the top weight cohomology of M_g and also with
the genus g part of the homology of Kontsevich's graph complex. Using a theorem
of Willwacher relating this graph complex to the Grothendieck-Teichmueller Lie
algebra, we deduce that H^{4g-6}(M_g;Q) is nonzero for g=3, g=5, and g at least
7. This disproves a recent conjecture of Church, Farb, and Putman as well as an
older, more general conjecture of Kontsevich. We also give an independent proof
of another theorem of Willwacher, that homology of the graph complex vanishes
in negative degrees.Comment: 31 pages. v2: streamlined exposition. Final version, to appear in J.
Amer. Math. So
Correlation filtering in financial time series
We apply a method to filter relevant information from the correlation
coefficient matrix by extracting a network of relevant interactions. This
method succeeds to generate networks with the same hierarchical structure of
the Minimum Spanning Tree but containing a larger amount of links resulting in
a richer network topology allowing loops and cliques. In Tumminello et al.
\cite{TumminielloPNAS05}, we have shown that this method, applied to a
financial portfolio of 100 stocks in the USA equity markets, is pretty
efficient in filtering relevant information about the clustering of the system
and its hierarchical structure both on the whole system and within each
cluster. In particular, we have found that triangular loops and 4 element
cliques have important and significant relations with the market structure and
properties. Here we apply this filtering procedure to the analysis of
correlation in two different kind of interest rate time series (16 Eurodollars
and 34 US interest rates).Comment: 10 pages 7 figure
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