3,399 research outputs found
Minimal H\"older regularity implying finiteness of integral Menger curvature
We study two families of integral functionals indexed by a real number . One family is defined for 1-dimensional curves in and the other one
is defined for -dimensional manifolds in . These functionals are
described as integrals of appropriate integrands (strongly related to the
Menger curvature) raised to power . Given we prove that
regularity of the set (a curve or a manifold), with implies finiteness of both curvature functionals
( in the case of curves). We also show that is optimal by
constructing examples of functions with graphs of infinite
integral curvature
Why Do Cascade Sizes Follow a Power-Law?
We introduce random directed acyclic graph and use it to model the
information diffusion network. Subsequently, we analyze the cascade generation
model (CGM) introduced by Leskovec et al. [19]. Until now only empirical
studies of this model were done. In this paper, we present the first
theoretical proof that the sizes of cascades generated by the CGM follow the
power-law distribution, which is consistent with multiple empirical analysis of
the large social networks. We compared the assumptions of our model with the
Twitter social network and tested the goodness of approximation.Comment: 8 pages, 7 figures, accepted to WWW 201
5-dimensional contact SO(3)-manifolds and Dehn twists
In this paper the 5-dimensional contact SO(3)-manifolds are classified up to
equivariant contactomorphisms. The construction of such manifolds with singular
orbits requires the use of generalized Dehn twists.
We show as an application that all simply connected 5-manifoldswith singular
orbits are realized by a Brieskorn manifold with exponents (k,2,2,2). The
standard contact structure on such a manifold gives right-handed Dehn twists,
and a second contact structure defined in the article gives left-handed twists.Comment: 16 pages, 1 figure; simplification of arguments by restricting
classification to coorientation preserving contactomorphism
On the Equivalence Problem for Toric Contact Structures on S^3-bundles over S^2$
We study the contact equivalence problem for toric contact structures on
-bundles over . That is, given two toric contact structures, one can
ask the question: when are they equivalent as contact structures while
inequivalent as toric contact structures? In general this appears to be a
difficult problem. To find inequivalent toric contact structures that are
contact equivalent, we show that the corresponding 3-tori belong to distinct
conjugacy classes in the contactomorphism group. To show that two toric contact
structures with the same first Chern class are contact inequivalent, we use
Morse-Bott contact homology. We treat a subclass of contact structures which
include the Sasaki-Einstein contact structures studied by physicists.
In this subcase we give a complete solution to the contact equivalence problem
by showing that and are inequivalent as contact structures
if and only if .Comment: 61 page
Latent Space Model for Multi-Modal Social Data
With the emergence of social networking services, researchers enjoy the
increasing availability of large-scale heterogenous datasets capturing online
user interactions and behaviors. Traditional analysis of techno-social systems
data has focused mainly on describing either the dynamics of social
interactions, or the attributes and behaviors of the users. However,
overwhelming empirical evidence suggests that the two dimensions affect one
another, and therefore they should be jointly modeled and analyzed in a
multi-modal framework. The benefits of such an approach include the ability to
build better predictive models, leveraging social network information as well
as user behavioral signals. To this purpose, here we propose the Constrained
Latent Space Model (CLSM), a generalized framework that combines Mixed
Membership Stochastic Blockmodels (MMSB) and Latent Dirichlet Allocation (LDA)
incorporating a constraint that forces the latent space to concurrently
describe the multiple data modalities. We derive an efficient inference
algorithm based on Variational Expectation Maximization that has a
computational cost linear in the size of the network, thus making it feasible
to analyze massive social datasets. We validate the proposed framework on two
problems: prediction of social interactions from user attributes and behaviors,
and behavior prediction exploiting network information. We perform experiments
with a variety of multi-modal social systems, spanning location-based social
networks (Gowalla), social media services (Instagram, Orkut), e-commerce and
review sites (Amazon, Ciao), and finally citation networks (Cora). The results
indicate significant improvement in prediction accuracy over state of the art
methods, and demonstrate the flexibility of the proposed approach for
addressing a variety of different learning problems commonly occurring with
multi-modal social data.Comment: 12 pages, 7 figures, 2 table
On Non-Abelian Symplectic Cutting
We discuss symplectic cutting for Hamiltonian actions of non-Abelian compact
groups. By using a degeneration based on the Vinberg monoid we give, in good
cases, a global quotient description of a surgery construction introduced by
Woodward and Meinrenken, and show it can be interpreted in algebro-geometric
terms. A key ingredient is the `universal cut' of the cotangent bundle of the
group itself, which is identified with a moduli space of framed bundles on
chains of projective lines recently introduced by the authors.Comment: Various edits made, to appear in Transformation Groups. 28 pages, 8
figure
A saddle in a corner - a model of collinear triatomic chemical reactions
A geometrical model which captures the main ingredients governing atom-diatom
collinear chemical reactions is proposed. This model is neither near-integrable
nor hyperbolic, yet it is amenable to analysis using a combination of the
recently developed tools for studying systems with steep potentials and the
study of the phase space structure near a center-saddle equilibrium. The
nontrivial dependence of the reaction rates on parameters, initial conditions
and energy is thus qualitatively explained. Conditions under which the phase
space transition state theory assumptions are satisfied and conditions under
which these fail are derived
Four-vortex motion around a circular cylinder
The motion of two pairs of counter-rotating point vortices placed in a
uniform flow past a circular cylinder is studied analytically and numerically.
When the dynamics is restricted to the symmetric subspace---a case that can be
realized experimentally by placing a splitter plate in the center plane---, it
is found that there is a family of linearly stable equilibria for same-signed
vortex pairs. The nonlinear dynamics in the symmetric subspace is investigated
and several types of orbits are presented. The analysis reported here provides
new insights and reveals novel features of this four-vortex system, such as the
fact that there is no equilibrium for two pairs of vortices of opposite signs
on the opposite sides of the cylinder. (It is argued that such equilibria might
exist for vortex flows past a cylinder confined in a channel.) In addition, a
new family of opposite-signed equilibria on the normal line is reported. The
stability analysis for antisymmetric perturbations is also carried out and it
shows that all equilibria are unstable in this case.Comment: 27 pages, 13 figures, to be published in Physics of Fluid
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