3,739 research outputs found
A new proof of the graph removal lemma
Let H be a fixed graph with h vertices. The graph removal lemma states that
every graph on n vertices with o(n^h) copies of H can be made H-free by
removing o(n^2) edges. We give a new proof which avoids Szemer\'edi's
regularity lemma and gives a better bound. This approach also works to give
improved bounds for the directed and multicolored analogues of the graph
removal lemma. This answers questions of Alon and Gowers.Comment: 17 page
Multimapper: Data Density Sensitive Topological Visualization
Mapper is an algorithm that summarizes the topological information contained
in a dataset and provides an insightful visualization. It takes as input a
point cloud which is possibly high-dimensional, a filter function on it and an
open cover on the range of the function. It returns the nerve simplicial
complex of the pullback of the cover. Mapper can be considered a discrete
approximation of the topological construct called Reeb space, as analysed in
the -dimensional case by [Carriere et al.,2018]. Despite its success in
obtaining insights in various fields such as in [Kamruzzaman et al., 2016],
Mapper is an ad hoc technique requiring lots of parameter tuning. There is also
no measure to quantify goodness of the resulting visualization, which often
deviates from the Reeb space in practice. In this paper, we introduce a new
cover selection scheme for data that reduces the obscuration of topological
information at both the computation and visualisation steps. To achieve this,
we replace global scale selection of cover with a scale selection scheme
sensitive to local density of data points. We also propose a method to detect
some deviations in Mapper from Reeb space via computation of persistence
features on the Mapper graph.Comment: Accepted at ICDM
Experimental study of the compression behavior of mask image projection based on stereolithography manufactured parts
The article presents the results of a series of compression tests on samples manufactured by means of the mask image projection based on stereolithography additive manufacturing technique (MIP-SL). Recent studies demonstrate the orthotropic nature of the MIP-SL materials. A research is initiated by the authors to attempt to predict the degree of anisotropy from the manufacturing parameters of the MIP-SL parts. The article focuses mainly on the development of the experimental compression tests of the first stage of the research. Special attention is paid to the four methods used to obtain the stress-strain curve of the material: strain gages, 2D Digital Image Correlation, extensometer measurements and crosshead displacement measurements. The article shows the advantages and limitations of each method. Finally, the anisotropic behaviour is verified and a testing procedure is set to obtain the constitutive parameters of the MIP-SL tested materialsPeer ReviewedPostprint (published version
Fragmentation with a Steady Source
We investigate fragmentation processes with a steady input of fragments. We
find that the size distribution approaches a stationary form which exhibits a
power law divergence in the small size limit, P(x) ~ x^{-3}. This algebraic
behavior is robust as it is independent of the details of the input as well as
the spatial dimension. The full time dependent behavior is obtained
analytically for arbitrary inputs, and is found to exhibit a universal scaling
behavior.Comment: 4 page
Growth and migration of solids in evolving protostellar disks I: Methods and Analytical tests
This series of papers investigates the early stages of planet formation by
modeling the evolution of the gas and solid content of protostellar disks from
the early T Tauri phase until complete dispersal of the gas. In this first
paper, I present a new set of simplified equations modeling the growth and
migration of various species of grains in a gaseous protostellar disk evolving
as a result of the combined effects of viscous accretion and photo-evaporation
from the central star. Using the assumption that the grain size distribution
function always maintains a power-law structure approximating the average
outcome of the exact coagulation/shattering equation, the model focuses on the
calculation of the growth rate of the largest grains only. The coupled
evolution equations for the maximum grain size, the surface density of the gas
and the surface density of solids are then presented and solved
self-consistently using a standard 1+1 dimensional formalism. I show that the
global evolution of solids is controlled by a leaky reservoir of small grains
at large radii, and propose an empirically derived evolution equation for the
total mass of solids, which can be used to estimate the total heavy element
retention efficiency in the planet formation paradigm. Consistency with
observation of the total mass of solids in the Minimum Solar Nebula augmented
with the mass of the Oort cloud sets strong upper limit on the initial grain
size distribution, as well as on the turbulent parameter \alphat. Detailed
comparisons with SED observations are presented in a following paper.Comment: Submitted to ApJ. 23 pages and 13 figure
Sketched MinDist
We consider sketch vectors of geometric objects through the \mindist
function for from a point
set . Collecting the vector of these sketch values induces a simple,
effective, and powerful distance: the Euclidean distance between these sketched
vectors. This paper shows how large this set needs to be under a variety of
shapes and scenarios. For hyperplanes we provide direct connection to the
sensitivity sample framework, so relative error can be preserved in
dimensions using . However, for other shapes, we show
we need to enforce a minimum distance parameter , and a domain size .
For the sample size then can be . For objects (e.g., trajectories) with at most pieces
this can provide stronger \emph{for all} approximations with
points. Moreover, with similar
size bounds and restrictions, such trajectories can be reconstructed exactly
using only these sketch vectors
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