334 research outputs found
The Large Scale Curvature of Networks
Understanding key structural properties of large scale networks are crucial
for analyzing and optimizing their performance, and improving their reliability
and security. Here we show that these networks possess a previously unnoticed
feature, global curvature, which we argue has a major impact on core
congestion: the load at the core of a network with N nodes scales as N^2 as
compared to N^1.5 for a flat network. We substantiate this claim through
analysis of a collection of real data networks across the globe as measured and
documented by previous researchers.Comment: 4 pages, 5 figure
On residualizing homomorphisms preserving quasiconvexity
H is called a G-subgroup of a hyperbolic group G if for any finite subset M G there exists a homomorphism from G onto a non-elementary hyperbolic group G_1 that is surjective on H and injective on M. In his paper in 1993 A. Ol'shanskii gave a description of all G-subgroups in any given non-elementary hyperbolic group G. Here we show that for the same class of G-subgroups the finiteness assumption on M (under certain natural conditions) can be replaced by an assumption of quasiconvexity
Geometric Mechanics of Curved Crease Origami
Folding a sheet of paper along a curve can lead to structures seen in
decorative art and utilitarian packing boxes. Here we present a theory for the
simplest such structure: an annular circular strip that is folded along a
central circular curve to form a three-dimensional buckled structure driven by
geometrical frustration. We quantify this shape in terms of the radius of the
circle, the dihedral angle of the fold and the mechanical properties of the
sheet of paper and the fold itself. When the sheet is isometrically deformed
everywhere except along the fold itself, stiff folds result in creases with
constant curvature and oscillatory torsion. However, relatively softer folds
inherit the broken symmetry of the buckled shape with oscillatory curvature and
torsion. Our asymptotic analysis of the isometrically deformed state is
corroborated by numerical simulations which allow us to generalize our analysis
to study multiply folded structures
Rank rigidity for CAT(0) cube complexes
We prove that any group acting essentially without a fixed point at infinity
on an irreducible finite-dimensional CAT(0) cube complex contains a rank one
isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube
complexes. We derive a number of other consequences for CAT(0) cube complexes,
including a purely geometric proof of the Tits Alternative, an existence result
for regular elements in (possibly non-uniform) lattices acting on cube
complexes, and a characterization of products of trees in terms of bounded
cohomology.Comment: 39 pages, 4 figures. Revised version according to referee repor
Property A and CAT(0) cube complexes
Property A is a non-equivariant analogue of amenability defined for metric spaces. Euclidean spaces and trees are examples of spaces with Property A. Simultaneously generalising these facts, we show that finite-dimensional CAT(0) cube complexes have Property A. We do not assume that the complex is locally finite. We also prove that given a discrete group acting properly on a finite-dimensional CAT(0) cube complex the stabilisers of vertices at infinity are amenable
Degenerations of ideal hyperbolic triangulations
Let M be a cusped 3-manifold, and let T be an ideal triangulation of M. The
deformation variety D(T), a subset of which parameterises (incomplete)
hyperbolic structures obtained on M using T, is defined and compactified by
adding certain projective classes of transversely measured singular
codimension-one foliations of M. This leads to a combinatorial and geometric
variant of well-known constructions by Culler, Morgan and Shalen concerning the
character variety of a 3-manifold.Comment: 31 pages, 11 figures; minor changes; to appear in Mathematische
Zeitschrif
On the distortion of twin building lattices
We show that twin building lattices are undistorted in their ambient group;
equivalently, the orbit map of the lattice to the product of the associated
twin buildings is a quasi-isometric embedding. As a consequence, we provide an
estimate of the quasi-flat rank of these lattices, which implies that there are
infinitely many quasi-isometry classes of finitely presented simple groups. In
an appendix, we describe how non-distortion of lattices is related to the
integrability of the structural cocycle
Hyperbolic Geometry of Complex Networks
We develop a geometric framework to study the structure and function of
complex networks. We assume that hyperbolic geometry underlies these networks,
and we show that with this assumption, heterogeneous degree distributions and
strong clustering in complex networks emerge naturally as simple reflections of
the negative curvature and metric property of the underlying hyperbolic
geometry. Conversely, we show that if a network has some metric structure, and
if the network degree distribution is heterogeneous, then the network has an
effective hyperbolic geometry underneath. We then establish a mapping between
our geometric framework and statistical mechanics of complex networks. This
mapping interprets edges in a network as non-interacting fermions whose
energies are hyperbolic distances between nodes, while the auxiliary fields
coupled to edges are linear functions of these energies or distances. The
geometric network ensemble subsumes the standard configuration model and
classical random graphs as two limiting cases with degenerate geometric
structures. Finally, we show that targeted transport processes without global
topology knowledge, made possible by our geometric framework, are maximally
efficient, according to all efficiency measures, in networks with strongest
heterogeneity and clustering, and that this efficiency is remarkably robust
with respect to even catastrophic disturbances and damages to the network
structure
Minsky machines and algorithmic problems
This is a survey of using Minsky machines to study algorithmic problems in
semigroups, groups and other algebraic systems.Comment: 19 page
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