skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.


Title: Regularity Theory of Quasilinear Elliptic and Parabolic Equations in the Heisenberg Group
We review the existing literature concerning regularity for the gradient of weak solutions of the subelliptic p-Laplacian differential operator in a domain Ω in the Heisenberg group H^n, with 1 ≤ p < ∞, and of its parabolic counterpart. We present some open problems and outline some of the difficulties they present.  more » « less
Award ID(s):
2141297
PAR ID:
10519895
Author(s) / Creator(s):
; ;
Publisher / Repository:
Vietnam Journal of Mathematics
Date Published:
Journal Name:
Vietnam Journal of Mathematics
ISSN:
2305-221X
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. The proliferation of software tools and automated techniques in digital forensics has brought about some controversies regarding bias and fairness. Different biases exist and have been proven in some civil and criminal cases. In our research, we analyze and discuss these biases present in software tools and automation systems used by law enforcement organizations and in court proceedings. Furthermore, we present real-life cases and scenarios where some of these biases have determined or influenced these cases. We were also able to provide recommendations for reducing bias in software tools, which we hope will be the foundation for a framework that reduces or eliminates bias from software tools used in digital forensics. In conclusion, we anticipate that this research can help increase validation in digital forensics software tools and ensure users' trust in the tools and automation techniques. 
    more » « less
  2. Abstract The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field and an extension ω of ν to a finite extensionLofK. Then we study when the valuation ring of ω is essentially finitely generated over the valuation ring of ν. We present a necessary condition in terms of classic invariants of the extension by Hagen Knaf and show that in some particular cases, this condition is also sufficient. We also study when the corresponding extension of graded algebras is finitely generated. For this problem we present an equivalent condition (which is weaker than the one for the finite generation of the valuation rings). 
    more » « less
  3. In this article, we study polygonal symplectic billiards. We provide new results, some of which are inspired by numerical investigations. In particular, we present several polygons for which all orbits are periodic. We demonstrate their properties and derive various conjectures using two numerical implementations. 
    more » « less
  4. null (Ed.)
    We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio \begin{document}$ 1 $$\end{document}$. 
    more » « less
  5. Augmented Reality (AR) applications can enable geographically distant users to collaborate using shared video feeds or interactive 3D holograms, and may be particularly useful in the socially distant context of the Covid-19 pandemic. However, a good user experience is key for their success and could be negatively impacted by network impairments, which are an inevitable occurrence in today's best-effort Internet. In this paper, we present the findings of an empirical user study, aimed at understanding the effects of network outages, on user experience and behavior, in a collaborative AR task. We highlight how network outages affected users in different ways depending on their role in the collaborative task, and how giving users explicit information about poor network conditions helped them deal with some of these negative effects. Furthermore, we report the strategies that users themselves adopted, to deal with outages, such as batching instructions, or shifting to a different spatial referencing style when communicating with their partners. Lastly, based on our findings, we present some design implications for future remote-collaborative AR applications. 
    more » « less