Tangent-point self-avoidance energies for curves

We study a two-point self-avoidance energy $E_q$ which is defined for all rectifiable curves in $R^n$ as the double integral along the curve of $1/r^q$. Here $r$ 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 $E_q(\gamma)$ for $q\ge 2$ guarantees that $\gamma$ has no self-intersections or triple junctions and therefore must be homeomorphic to the unit circle or to a closed interval. For $q>2$ the energy $E_q$ evaluated on curves in $R^3$ 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 $R^3$ with finite $E_q$-energy that guarantees that these curves are ambient isotopic. This bound depends only on $q$ and the energy values of the curves. Moreover, for all $q$ that are larger than the critical exponent $2$, the arclength parametrization of $\gamma$ is of class $C^{1,1-2/q}$, with H\"{o}lder norm of the unit tangent depending only on $q$, the length of $\gamma$, and the local energy. The exponent $1-2/q$ 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

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 $N$ the set of smooth, zero degree maps $\psi\colon\mathbb{S}^2\to \mathbb{S}^2$ which have the following three properties: (1) there is a unique minimizing harmonic map $u\colon \mathbb{B}^3\to \mathbb{S}^2$ which agrees with $\psi$ on the boundary of the unit ball; (2) this map $u$ has at least $N$ singular points in $\mathbb{B}^3$; (3) the Lavrentiev gap phenomenon holds for $\psi$, i.e., the infimum of the Dirichlet energies $E(w)$ of all smooth extensions $w\colon \mathbb{B}^3\to\mathbb{S}^2$ of $\psi$ is strictly larger than the Dirichlet energy $\int_{\mathbb{B}^3} |\nabla u|^2$ of the (irregular) minimizer $u$, is dense in the set of all smooth zero degree maps $\phi\colon \mathbb{S}^2\to\mathbb{S}^2$ endowed with the $W^{1,p}$-topology, where $1\le p < 2$. This result is sharp: it fails in the $W^{1,2}$ 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 $p > 0$. One family is defined for 1-dimensional curves in $\R^3$ and the other one is defined for $m$-dimensional manifolds in $\R^n$. These functionals are described as integrals of appropriate integrands (strongly related to the Menger curvature) raised to power $p$. Given $p > m(m+1)$ we prove that $C^{1,\alpha}$ regularity of the set (a curve or a manifold), with $\alpha > \alpha_0 = 1 - \frac{m(m+1)}p$ implies finiteness of both curvature functionals ($m=1$ in the case of curves). We also show that $\alpha_0$ is optimal by constructing examples of $C^{1,\alpha_0}$ functions with graphs of infinite integral curvature

Integral Menger curvature for surfaces

We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a $C^{1,\lambda}$-a-priori bound for surfaces for which this functional is finite. In fact, it turns out that there is an explicit length scale $R>0$ which depends only on an upper bound $E$ for the integral Menger curvature $M_p(\Sigma)$ and the integrability exponent $p$, and \emph{not} on the surface $\Sigma$ itself; below that scale, each surface with energy smaller than $E$ looks like a nearly flat disc with the amount of bending controlled by the (local) $M_p$-energy. Moreover, integral Menger curvature can be defined a priori for surfaces with self-intersections or branch points; we prove that a posteriori all such singularities are excluded for surfaces with finite integral Menger curvature. By means of slicing and iterative arguments we bootstrap the H\"{o}lder exponent $\lambda$ up to the optimal one, $\lambda=1-(8/p)$, thus establishing a new geometric `Morrey-Sobolev' imbedding theorem. As two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral Menger curvature with a uniform bound on area, and of area minimizing surfaces subjected to a uniform bound on integral Menger curvature.Comment: 64 pages, 7 figures. Submitted. Version 2: extended comments on the relation to Lerman's and Whitehouse's work on Menger curvature