27,588 research outputs found
The metric dimension for resolving several objects
A set of vertices S is a resolving set in a graph if each vertex has a unique array of distances to the vertices of S. The natural problem of finding the smallest cardinality of a resolving set in a graph has been widely studied over the years. In this paper, we wish to resolve a set of vertices (up to l vertices) instead of just one vertex with the aid of the array of distances. The smallest cardinality of a set S resolving at most l vertices is called l-set-metric dimension. We study the problem of the l-set-metric dimension in two infinite classes of graphs, namely, the two dimensional grid graphs and the n-dimensional binary hypercubes. (C) 2016 Elsevier B.V. All rights reserved
Resolving sets for Johnson and Kneser graphs
A set of vertices in a graph is a {\em resolving set} for if, for
any two vertices , there exists such that the distances . In this paper, we consider the Johnson graphs and Kneser
graphs , and obtain various constructions of resolving sets for these
graphs. As well as general constructions, we show that various interesting
combinatorial objects can be used to obtain resolving sets in these graphs,
including (for Johnson graphs) projective planes and symmetric designs, as well
as (for Kneser graphs) partial geometries, Hadamard matrices, Steiner systems
and toroidal grids.Comment: 23 pages, 2 figures, 1 tabl
p-adic path set fractals and arithmetic
This paper considers a class C(Z_p) of closed sets of the p-adic integers
obtained by graph-directed constructions analogous to those of Mauldin and
Williams over the real numbers. These sets are characterized as collections of
those p-adic integers whose p-adic expansions are describeed by paths in the
graph of a finite automaton issuing from a distinguished initial vertex. This
paper shows that this class of sets is closed under the arithmetic operations
of addition and multiplication by p-integral rational numbers. In addition the
Minkowski sum (under p-adic addition) of two set in the class is shown to also
belong to this class. These results represent purely p-adic phenomena in that
analogous closure properties do not hold over the real numbers. We also show
the existence of computable formulas for the Hausdorff dimensions of such sets.Comment: v1 24 pages; v2 added to title, 28 pages; v3, 30 pages, added
concluding section, v.4, incorporate changes requested by reviewe
Asymptotic Safety, Emergence and Minimal Length
There seems to be a common prejudice that asymptotic safety is either
incompatible with, or at best unrelated to, the other topics in the title. This
is not the case. In fact, we show that 1) the existence of a fixed point with
suitable properties is a promising way of deriving emergent properties of
gravity, and 2) there is a sense in which asymptotic safety implies a minimal
length. In so doing we also discuss possible signatures of asymptotic safety in
scattering experiments.Comment: LaTEX, 20 pages, 2 figures; v.2: minor changes, reflecting published
versio
Scale-dependent metric and causal structures in Quantum Einstein Gravity
Within the asymptotic safety scenario for gravity various conceptual issues
related to the scale dependence of the metric are analyzed. The running
effective field equations implied by the effective average action of Quantum
Einstein Gravity (QEG) and the resulting families of resolution dependent
metrics are discussed. The status of scale dependent vs. scale independent
diffeomorphisms is clarified, and the difference between isometries implemented
by scale dependent and independent Killing vectors is explained. A concept of
scale dependent causality is proposed and illustrated by various simple
examples. The possibility of assigning an "intrinsic length" to objects in a
QEG spacetime is also discussed.Comment: 52 page
Branes and Fluxes in D=5 Calabi-Yau Compactifications of M-Theory
We discuss Poincare three-brane solutions in D=5 M-Theory compactifications
on Calabi-Yau (CY) threefolds with G-fluxes. We show that the vector moduli
freeze at an attractor point. In the case with background flux only, the
spacetime geometry contains a zero volume singularity with the three-brane and
the CY space shrinking simultaneously to a point. This problem can be avoided
by including explicit three-brane sources. We consider two cases in detail: a
single brane and, when the transverse dimension is compactified on a circle, a
pair of branes with opposite tensions.Comment: 14 pages, final versio
On the noncommutative geometry of tilings
This is a chapter in an incoming book on aperiodic order. We review results
about the topology, the dynamics, and the combinatorics of aperiodically
ordered tilings obtained with the tools of noncommutative geometry
Compact manifolds with exceptional holonomy
In the classification of Riemannian holonomy groups, the exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. We outline the construction of the first known examples of compact 7- and 8-manifolds with holonomy G2 and Spin(7)
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