1,233 research outputs found
Tangent-point self-avoidance energies for curves
We study a two-point self-avoidance energy which is defined for all
rectifiable curves in as the double integral along the curve of .
Here stands for the radius of the (smallest) circle that is tangent to the
curve at one point and passes through another point on the curve, with obvious
natural modifications of this definition in the exceptional, non-generic cases.
It turns out that finiteness of for guarantees that
has no self-intersections or triple junctions and therefore must be
homeomorphic to the unit circle or to a closed interval. For the energy
evaluated on curves in turns out to be a knot energy separating
different knot types by infinite energy barriers and bounding the number of
knot types below a given energy value. We also establish an explicit upper
bound on the Hausdorff-distance of two curves in with finite -energy
that guarantees that these curves are ambient isotopic. This bound depends only
on and the energy values of the curves. Moreover, for all that are
larger than the critical exponent , the arclength parametrization of
is of class , with H\"{o}lder norm of the unit tangent
depending only on , the length of , and the local energy. The
exponent is optimal.Comment: 23 pages, 1 figur
Can we motivate student behavior in a first grade classroom? Reward System vs. Conventional Teaching Discipline Plan
What effect does a reward system have on first grade student behavior? Is there any way to have students behave better than they are currently? There is plenty of research conducted on different types of distracting behavior as well as different types of systems and programs that try to influence behavior. Three first grade classes were selected, observed, and data were recorded on any disruptive or unwanted behavior for three consecutive weeks. Post the first three weeks, a reward system was implanted for each class. Each class was given the opportunity to earn a “free day” in physical education class by decreasing their disruptive behavior during class. Only two of the classes were able to earn their free day
Characterizing ~submanifolds by -integrability of global curvatures
We give sufficient and necessary geometric conditions, guaranteeing that an
immersed compact closed manifold of class and of
arbitrary dimension and codimension (or, more generally, an Ahlfors-regular
compact set satisfying a mild general condition relating the size of
holes in to the flatness of measured in terms of beta
numbers) is in fact an embedded manifold of class ,
where and . The results are based on a careful analysis of
Morrey estimates for integral curvature--like energies, with integrands
expressed geometrically, in terms of functions that are designed to measure
either (a) the shape of simplices with vertices on or (b) the size of
spheres tangent to at one point and passing through another point of
.
Appropriately defined \emph{maximal functions} of such integrands turn out to
be of class for if and only if the local graph
representations of have second order derivatives in and
is embedded. There are two ingredients behind this result. One of them is an
equivalent definition of Sobolev spaces, widely used nowadays in analysis on
metric spaces. The second one is a careful analysis of local Reifenberg
flatness (and of the decay of functions measuring that flatness) for sets with
finite curvature energies. In addition, for the geometric curvature energy
involving tangent spheres we provide a nontrivial lower bound that is attained
if and only if the admissible set is a round sphere.Comment: 44 pages, 2 figures; several minor correction
Application of Developers' and Users' Dependent Factors in App Store Optimization
This paper presents an application of developers' and users' dependent
factors in the app store optimization. The application is based on two main
fields: developers' dependent factors and users' dependent factors. Developers'
dependent factors are identified as: developer name, app name, subtitle, genre,
short description, long description, content rating, system requirements, page
url, last update, what's new and price. Users' dependent factors are identified
as: download volume, average rating, rating volume and reviews. The proposed
application in its final form is modelled after mining sample data from two
leading app stores: Google Play and Apple App Store. Results from analyzing
collected data show that developer dependent elements can be better optimized.
Names and descriptions of mobile apps are not fully utilized. In Google Play
there is one significant correlation between download volume and number of
reviews, whereas in App Store there is no significant correlation between
factors
Benchmarking the cost of thread divergence in CUDA
All modern processors include a set of vector instructions. While this gives
a tremendous boost to the performance, it requires a vectorized code that can
take advantage of such instructions. As an ideal vectorization is hard to
achieve in practice, one has to decide when different instructions may be
applied to different elements of the vector operand. This is especially
important in implicit vectorization as in NVIDIA CUDA Single Instruction
Multiple Threads (SIMT) model, where the vectorization details are hidden from
the programmer. In order to assess the costs incurred by incompletely
vectorized code, we have developed a micro-benchmark that measures the
characteristics of the CUDA thread divergence model on different architectures
focusing on the loops performance
Modified log-Sobolev inequalities for convex functions on the real line. Sufficient conditions
We provide a mild sufficient condition for a probability measure on the real
line to satisfy a modified log-Sobolev inequality for convex functions,
interpolating between the classical log-Sobolev inequality and a Bobkov-Ledoux
type inequality. As a consequence we obtain dimension-free two-level
concentration results for convex function of independent random variables with
sufficiently regular tail decay. We also provide a link between modified
log-Sobolev inequalities for convex functions and weak transport-entropy
inequalities, complementing recent work by Gozlan, Roberto, Samson, and Tetali.Comment: 25 pages; changes: references and comments about recent results by
other Authors added, hypercontractive estimates in Section 3 added, a few
typos corrected; accepted for publication in Studia Mathematic
The Lavrentiev gap phenomenon for harmonic maps into spheres holds on a dense set of zero degree boundary data
We prove that for each positive integer the set of smooth, zero degree
maps which have the following three
properties:
(1) there is a unique minimizing harmonic map which agrees with on the boundary of the unit ball;
(2) this map has at least singular points in ;
(3) the Lavrentiev gap phenomenon holds for , i.e., the infimum of the
Dirichlet energies of all smooth extensions of is strictly larger than the Dirichlet
energy of the (irregular) minimizer , is
dense in the set of all smooth zero degree maps endowed with the -topology, where . This result is sharp: it fails in the topology on the set of
all smooth boundary data.Comment: 14 pages, 3 figures; minor typos etc. correcte
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
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