16,716 research outputs found
An equivalence relation in finite planar spaces
AbstractThis paper is concerned with quasiparallelism relation in a finite planar space S. In particular, we prove that if no plane in S is the union of two lines quasiparallelism relation between planes is an equivalence relation. Moreover, three-dimensional affine spaces with a point at infinity and three-dimensional affine spaces with a line at infinity are characterized
Operads and Phylogenetic Trees
We construct an operad whose operations are the edge-labelled
trees used in phylogenetics. This operad is the coproduct of ,
the operad for commutative semigroups, and , the operad with unary
operations corresponding to nonnegative real numbers, where composition is
addition. We show that there is a homeomorphism between the space of -ary
operations of and , where
is the space of metric -trees introduced by Billera, Holmes
and Vogtmann. Furthermore, we show that the Markov models used to reconstruct
phylogenetic trees from genome data give coalgebras of . These
always extend to coalgebras of the larger operad ,
since Markov processes on finite sets converge to an equilibrium as time
approaches infinity. We show that for any operad , its coproduct with
contains the operad constucted by Boardman and Vogt. To
prove these results, we explicitly describe the coproduct of operads in terms
of labelled trees.Comment: 48 pages, 3 figure
Crossed simplicial groups and structured surfaces
We propose a generalization of the concept of a Ribbon graph suitable to
provide combinatorial models for marked surfaces equipped with a G-structure.
Our main insight is that the necessary combinatorics is neatly captured in the
concept of a crossed simplicial group as introduced, independently, by
Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category
leads to Ribbon graphs while other crossed simplicial groups naturally yield
different notions of structured graphs which model unoriented, N-spin, framed,
etc, surfaces. Our main result is that structured graphs provide orbicell
decompositions of the respective G-structured moduli spaces. As an application,
we show how, building on our theory of 2-Segal spaces, the resulting theory can
be used to construct categorified state sum invariants of G-structured
surfaces.Comment: 86 pages, v2: revised versio
Modal logic of planar polygons
We study the modal logic of the closure algebra , generated by the set
of all polygons in the Euclidean plane . We show that this logic
is finitely axiomatizable, is complete with respect to the class of frames we
call "crown" frames, is not first order definable, does not have the Craig
interpolation property, and its validity problem is PSPACE-complete
An Exact Relation for N=1 Orientifold Field Theories with Arbitrary Superpotential
We discuss a nonperturbative relation for orientifold parent/daughter pairs
of supersymmetric theories with an arbitrary tree-level superpotential. We show
that super-Yang-Mills (SYM) theory with matter in the adjoint representation at
N-->infinity, is equivalent to a SYM theory with matter in the antisymmetric
representation and a related superpotential. The gauge symmetry breaking
patterns match in these theories too. The moduli spaces in the limiting case of
a vanishing superpotential are also discussed. Finally we argue that there is
an exact mapping between the effective superpotentials of two finite-N theories
belonging to an orientifold pair.Comment: 14 pages, LaTex. v2: a new introductory section, a few comments and
refs. added. v3: minor changes, to appear in Nucl.Phys.
Paper folding, Riemann surfaces, and convergence of pseudo-Anosov sequences
A method is presented for constructing closed surfaces out of Euclidean
polygons with infinitely many segment identifications along the boundary. The
metric on the quotient is identified. A sufficient condition is presented which
guarantees that the Euclidean structure on the polygons induces a unique
conformal structure on the quotient surface, making it into a closed Riemann
surface. In this case, a modulus of continuity for uniformizing coordinates is
found which depends only on the geometry of the polygons and on the
identifications. An application is presented in which a uniform modulus of
continuity is obtained for a family of pseudo-Anosov homeomorphisms, making it
possible to prove that they converge to a Teichm\"uller mapping on the Riemann
sphere.Comment: 75 pages, 18 figure
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