More Like this

1. Distributional Finite Element curl div Complexes and Application to Quad Curl Problems

   https://doi.org/10.1137/23M1617400  

   Chen, Long; Huang, Xuehai; Zhang, Chao (June 2025, SIAM Journal on Numerical Analysis)

   This paper addresses the challenge of constructing finite element curl div complexes in three dimensions. Tangential-normal continuity is introduced in order to develop distributional finite element curl div complexes. The spaces constructed are applied to discretize the quad curl problem, demonstrating optimal order of convergence. Furthermore, a hybridization technique is proposed, demonstrating its equivalence to nonconforming finite elements and weak Galerkin methods. 

   more » « less
2. CURL: Contrastive Unsupervised Representations for Reinforcement Learning

   Laskin, M; Srinivas, A; Abbeel, P (July 2020, In the proceedings of the International Conference on Machine Learning (ICML))
3. A div-curl inequality for orthonormal functions

   Frank, Rupert L (August 2024, Pure and Applied Functional Analysis)
4. Rotations with Constant $$\mathbf {{\text {curl }}}$$ are Constant

   https://doi.org/10.1007/s00205-022-01764-6  

   Ginster, Janusz; Acharya, Amit (June 2022, Archive for Rational Mechanics and Analysis)

   Abstract We address a problem that extends a fundamental classical result of continuum mechanics from the time of its inception, as well as answers a fundamental question in the recent, modern nonlinear elastic theory of dislocations. Interestingly, the implication of our result in the latter case is qualitatively different from its well-established analog in the linear elastic theory of dislocations. It is a classical result that if $$u\in C^2({\mathbb {R}}^n;{\mathbb {R}}^n)$$ u ∈ C 2 ( R n ; R n ) and $$\nabla u \in SO(n)$$ ∇ u ∈ S O ( n ) , it follows that u is rigid. In this article this result is generalized to matrix fields with non-vanishing $${\text {curl }}$$ curl . It is shown that every matrix field $$R\in C^2(\varOmega ;SO(3))$$ R ∈ C 2 ( Ω ; S O ( 3 ) ) such that $${\text {curl }}R = constant$$ curl R = c o n s t a n t is necessarily constant. Moreover, it is proved in arbitrary dimensions that a measurable rotation field is as regular as its distributional $${\text {curl }}$$ curl allows. In particular, a measurable matrix field $$R: \varOmega \rightarrow SO(n)$$ R : Ω → S O ( n ) , whose $${\text {curl }}$$ curl in the sense of distributions is smooth, is also smooth. 

   more » « less
5. DAGs with No Curl: Efficient DAG Structure Learning

   Yu, Yue; Gao, Tian. (January 2020, Causal Discovery & Causality-Inspired Machine learning Workshop at 34th Conference on Neural InformationProcessing Systems (NeurIPS 2020))

   null (Ed.)