9,077 research outputs found
Tverberg's theorem with constraints
The topological Tverberg theorem claims that for any continuous map of the
(q-1)(d+1)-simplex to R^d there are q disjoint faces such that their images
have a non-empty intersection. This has been proved for affine maps, and if
is a prime power, but not in general.
We extend the topological Tverberg theorem in the following way: Pairs of
vertices are forced to end up in different faces. This leads to the concept of
constraint graphs. In Tverberg's theorem with constraints, we come up with a
list of constraints graphs for the topological Tverberg theorem.
The proof is based on connectivity results of chessboard-type complexes.
Moreover, Tverberg's theorem with constraints implies new lower bounds for the
number of Tverberg partitions. As a consequence, we prove Sierksma's conjecture
for , and .Comment: 16 pages, 12 figures. Accepted for publication in JCTA. Substantial
revision due to the referee
Conley Index at Infinity
The aim of this paper is to explore the possibilities of Conley index
techniques in the study of heteroclinic connections between finite and infinite
invariant sets. For this, we remind the reader of the Poincar\'e
compactification: this transformation allows to project a -dimensional
vector space on the -dimensional unit hemisphere of
and infinity on its -dimensional equator called the sphere at infinity.
Under normalizability condition, vector fields on transform into vector
fields on the Poincar\'e hemisphere whose associated flows let the equator
invariant. The dynamics on the equator reflects the dynamics at infinity, but
is now finite and may be studied by Conley index techniques. Furthermore, we
observe that some non-isolated behavior may occur around the equator, and
introduce the concept of invariant sets at infinity of isolated invariant
dynamical complement. Through the construction of an extended phase space
together which an extended flow, we are able to adapt the Conley index
techniques and prove the existence of connections to such non-isolated
invariant sets.Comment: 25 pages, 8 figure
Dense baryonic matter: constraints from recent neutron star observations
Updated constraints from neutron star masses and radii impose stronger
restrictions on the equation of state for baryonic matter at high densities and
low temperatures. The existence of two-solar-mass neutron stars rules out many
soft equations of state with prominent "exotic" compositions. The present work
reviews the conditions required for the pressure as a function of baryon
density in order to satisfy these new constraints. Several scenarios for
sufficiently stiff equations of state are evaluated. The common starting point
is a realistic description of both nuclear and neutron matter based on a chiral
effective field theory approach to the nuclear many-body problem. Possible
forms of hybrid matter featuring a quark core in the center of the star are
discussed using a three-flavor Polyakov--Nambu--Jona-Lasinio (PNJL) model. It
is found that a conventional equation of state based on nuclear chiral dynamics
meets the astrophysical constraints. Hybrid matter generally turns out to be
too soft unless additional strongly repulsive correlations, e.g. through vector
current interactions between quarks, are introduced. The extent to which
strangeness can accumulate in the equation of state is also discussed.Comment: v2; substantial revisions with respect to v1; 17 pages, 15 figure
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