Existence, regularity and structure of confined elasticae

We consider the problem of minimizing the bending or elastic energy among Jordan curves confined in a given open set $\Omega$. We prove existence, regularity and some structural properties of minimizers. In particular, when $\Omega$ is convex we show that a minimizer is necessarily a convex curve. We also provide an example of a minimizer with self-intersections

ϵ\epsilon-Regularity and Structure of 4-dimensional Shrinking Ricci Solitons

A closed four dimensional manifold cannot possess a non-flat Ricci soliton metric with arbitrarily small $L^2$-norm of the curvature. In this paper, we localize this fact in the case of shrinking Ricci solitons by proving an $\varepsilon$-regularity theorem, thus confirming a conjecture of Cheeger-Tian. As applications, we will also derive structural results concerning the degeneration of the metrics on a family of complete non-compact four dimensional shrinking Ricci solitons without a uniform entropy lower bound. In the appendix, we provide a detailed account of the equivariant good chopping theorem when collapsing with locally bounded curvature happens

Lectures on mean curvature flow

A family of hypersurfaces evolves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow is the most natural evolution equation in extrinsic geometry, and has been extensively studied ever since the pioneering work of Brakke and Huisken. In the last 15 years, White developed a far-reaching regularity and structure theory for mean convex mean curvature flow, and Huisken-Sinestrari constructed a flow with surgery for two-convex hypersurfaces. In this course, I first give a general introduction to the mean curvature flow of hypersurfaces and then present joint work with Bruce Kleiner, where we give a streamlined and unified treatment of the theory of White and Huisken-Sinestrari. These notes are from summer schools at KIAS Seoul and SNS Pisa.Comment: Lecture notes based on arXiv:1304.0926 and arXiv:1404.233

Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with respect to the {I,H,N}^n Transform

We enumerate the inequivalent self-dual additive codes over GF(4) of blocklength n, thereby extending the sequence A090899 in The On-Line Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a well-known interpretation as quantum codes. They can also be represented by graphs, where a simple graph operation generates the orbits of equivalent codes. We highlight the regularity and structure of some graphs that correspond to codes with high distance. The codes can also be interpreted as quadratic Boolean functions, where inequivalence takes on a spectral meaning. In this context we define PAR_IHN, peak-to-average power ratio with respect to the {I,H,N}^n transform set. We prove that PAR_IHN of a Boolean function is equivalent to the the size of the maximum independent set over the associated orbit of graphs. Finally we propose a construction technique to generate Boolean functions with low PAR_IHN and algebraic degree higher than 2.Comment: Presented at Sequences and Their Applications, SETA'04, Seoul, South Korea, October 2004. 17 pages, 10 figure

Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness

In this paper, we study the pullback attractor for a general reaction-diffusion system for which the uniqueness of solutions is not assumed. We first establish some general results for a multi-valued dynamical system to have a bi-spatial pullback attractor, and then we find that the attractor can be backwards compact and composed of all the backwards bounded complete trajectories. As an application, a general reaction-diffusion system is proved to have an invariant (H, V )-pullback attractor A = {A(τ)}τ∈R. This attractor is composed of all the backwards compact complete trajectories of the system, pullback attracts bounded subsets of H in the topology of V, and moreover ∪ s6τ A(s) is precompact in V, ∀τ ∈ R. A non-autonomous Fitz-Hugh-Nagumo equation is studied as a specific example of the reaction–diffusion system.State Scholarship Fund (China)Junta de AndalucíaBrazilian-European partnership in Dynamical SystemsEuropean UnionNational Natural Science Foundation of Chin

Optimal regularity and structure of the free boundary for minimizers in cohesive zone models

We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are $C^{1, 1/2}$, and that near non-degenerate points the fracture set is $C^{1, \alpha}$, for some $\alpha \in (0, 1)$.Comment: 39 page

Exploring the movement dynamics of deception

Both the science and the everyday practice of detecting a lie rest on the same assumption: hidden cognitive states that the liar would like to remain hidden nevertheless influence observable behavior. This assumption has good evidence. The insights of professional interrogators, anecdotal evidence, and body language textbooks have all built up a sizeable catalog of non-verbal cues that have been claimed to distinguish deceptive and truthful behavior. Typically, these cues are discrete, individual behaviors—a hand touching a mouth, the rise of a brow—that distinguish lies from truths solely in terms of their frequency or duration. Research to date has failed to establish any of these non-verbal cues as a reliable marker of deception. Here we argue that perhaps this is because simple tallies of behavior can miss out on the rich but subtle organization of behavior as it unfolds over time. Research in cognitive science from a dynamical systems perspective has shown that behavior is structured across multiple timescales, with more or less regularity and structure. Using tools that are sensitive to these dynamics, we analyzed body motion data from an experiment that put participants in a realistic situation of choosing, or not, to lie to an experimenter. Our analyses indicate that when being deceptive, continuous fluctuations of movement in the upper face, and somewhat in the arms, are characterized by dynamical properties of less stability, but greater complexity. For the upper face, these distinctions are present despite no apparent differences in the overall amount of movement between deception and truth. We suggest that these unique dynamical signatures of motion are indicative of both the cognitive demands inherent to deception and the need to respond adaptively in a social context

Stochastic switching in infinite dimensions with applications to random parabolic PDE

© 2015 Society for Industrial and Applied Mathematics.We consider parabolic PDEs with randomly switching boundary conditions. In order to analyze these random PDEs, we consider more general stochastic hybrid systems and prove convergence to, and properties of, a stationary distribution. Applying these general results to the heat equation with randomly switching boundary conditions, we find explicit formulae for various statistics of the solution and obtain almost sure results about its regularity and structure. These results are of particular interest for biological applications as well as for their significant departure from behavior seen in PDEs forced by disparate Gaussian noise. Our general results also have applications to other types of stochastic hybrid systems, such as ODEs with randomly switching right-hand sides