arXiv subjects
Publications and source records attributed to Chao Qian.
Based on representation theory of Clifford algebra, Ferus, Karcher and Münzner constructed a series of isoparametric foliations. In this paper, we will survey recent studies on isoparametric hypersurfaces of OT-FKM type and investigate related geometric constructions with mean curvature flow.
Superscattering, induced by degenerate resonances, breaks the fundamental single-channel limit of scattering cross section of subwavelength structures; in principle, an arbitrarily large total cross section can be achieved via superscattering. It thus provides a unique way to strengthen the light-matter interaction at the subwavelength scale, and has many potential applications in sensing, energy harvesting, bio-imaging (such as magnetic resonance imaging), communication and optoelectronics. However, the experimental demonstration of superscattering remains an open challenge due to its vulnerability to structural imperfections and intrinsic material losses. Here we report the first experimental evidence for superscattering, by demonstrating the superscattering simultaneously in two different frequency regimes through both the far-field and near-field measurements. The underlying mechanism for the observed superscattering is the degenerate resonances of confined surface waves, by utilizing a subwavelength metasurface-based multilayer structure. Our work paves the way towards practical applications based on superscattering.
Superscattering, i.e., a phenomenon of the scattering cross section from a subwavelength object exceeding the single-channel limit, has important prospects in enhanced sensing/spectroscopy, solar cells, and biomedical imaging. Superscattering can be typically constructed only at a single frequency regime, and depends critically on the inescapable material losses. Under such realistic conditions, superscattering has not been predicted nor observed to exist simultaneously at multiple frequency regimes. Here we introduce multifrequency superscattering in a subwavelength hyperbolic structure, which can be made from artificial metamaterials or from naturally-existing materials, such as hexagonal boron nitride (BN), and show the advantage of such hyperbolic materials for reducing structural complexity. The underlying mechanism is revealed to be the multimode resonances at multiple frequency regimes as appear in BN due to the peculiar dispersion of phonon-polaritons. Importantly, the multifrequency superscattering has a high tolerance to material losses and some structural variations, bringing the concept of multifrequency superscattering closer to useful and realistic conditions.
Evolutionary algorithms (EAs) are population-based general-purpose optimization algorithms, and have been successfully applied in various real-world optimization tasks. However, previous theoretical studies often employ EAs with only a parent or offspring population and focus on specific problems. Furthermore, they often only show upper bounds on the running time, while lower bounds are also necessary to get a complete understanding of an algorithm. In this paper, we analyze the running time of the ($μ$+$λ$)-EA (a general population-based EA with mutation only) on the class of pseudo-Boolean functions with a unique global optimum. By applying the recently proposed switch analysis approach, we prove the lower bound $Ω(n \ln n+ μ+ λn\ln\ln n/ \ln n)$ for the first time. Particularly on the two widely-studied problems, OneMax and LeadingOnes, the derived lower bound discloses that the ($μ$+$λ$)-EA will be strictly slower than the (1+1)-EA when the population size $μ$ or $λ$ is above a moderate order. Our results imply that the increase of population size, while usually desired in practice, bears the risk of increasing the lower bound of the running time and thus should be carefully considered.
In this paper, two sequences of minimal isoparametric hypersurfaces are constructed via representations of Clifford algebras. Based on these, we give estimates on eigenvalues of the Laplacian of the focal submanifolds of isoparametric hypersurfaces in unit spheres. This improves results of [TY13] and [TXY14]. Eells and Lemaire [EL83] posed a problem to characterize the compact Riemannian manifold M for which there is an eigenmap from M to S^n. As another application of our constructions, the focal maps give rise to many examples of eigenmaps from minimal isoparametric hypersurfaces to unit spheres. Most importantly, by investigating the second fundamental forms of focal submanifolds of isoparametric hypersurfaces in unit spheres, we provide infinitely many counterexamples to two conjectures of Leung [Le91] (posed in 1991) on minimal submanifolds in unit spheres. Notice that these conjectures of Leung have been proved in the case that the normal connection is flat [HV01].
Given a closed manifold $M$ and a vector bundle $ξ$ of rank $n$ over $M$, by gluing two copies of the disc bundle of $ξ$, we can obtain a closed manifold $D(ξ, M)$, the so-called double manifold. In this paper, we firstly prove that each sphere bundle $S_r(ξ)$ of radius $r>0$ is an isoparametric hypersurface in the total space of $ξ$ equipped with a connection metric, and for $r>0$ small enough, the induced metric of $S_r(ξ)$ has positive Ricci curvature under the additional assumptions that $M$ has a metric with positive Ricci curvature and $n\geq3$. As an application, if $M$ admits a metric with positive Ricci curvature and $n\geq2$, then we construct a metric with positive Ricci curvature on $D(ξ, M)$. Moreover, under the same metric, $D(ξ, M)$ admits a natural isoparametric foliation. For a compact minimal isoparametric hypersurface $Y^{n}$ in $S^{n+1}(1)$, which separates $S^{n+1}(1)$ into $S^{n+1}_+$ and $S^{n+1}_-$, one can get double manifolds $D(S^{n+1}_+)$ and $D(S^{n+1}_-)$. Inspired by Tang, Xie and Yan's work on scalar curvature of such manifolds with isoparametric foliations(cf. \cite{TXY12}), we study Ricci curvature of them with isoparametric foliations in the last part.
In this note, we discuss the interactions between differential topology and isoparametric foliations, surveying some recent progress and open problems.
This paper extends widely the work in \cite{GT13}. Existence and non-existence results of isoparametric functions on exotic spheres and Eells-Kuiper projective planes are established. In particular, every homotopy $n$-sphere ($n>4$) carries an isoparametric function (with certain metric) with 2 points as the focal set, in strong contrast to the classification of cohomogeneity one actions on homotopy spheres \cite{St96} ( only exotic Kervaire spheres admit cohomogeneity one actions besides the standard spheres ). As an application, we improve a beautiful result of Bérard-Bergery \cite{BB77} ( see also pp.234-235 of \cite{Be78} ).
This paper gives a survey of recent progress in isoparametric functions and isoparametric hypersurfaces, mainly in two directions. (1) Isoparametric functions on Riemannian manifolds, including exotic spheres. The existences and non-existences will be considered. (2) The Yau conjecture on the first eigenvalues of the embedded minimal hypersurfaces in the unit spheres. The history and progress of the Yau conjecture on minimal isoparametric hypersurfaces will be stated.
Many optimization tasks have to be handled in noisy environments, where we cannot obtain the exact evaluation of a solution but only a noisy one. For noisy optimization tasks, evolutionary algorithms (EAs), a kind of stochastic metaheuristic search algorithm, have been widely and successfully applied. Previous work mainly focuses on empirical studying and designing EAs for noisy optimization, while, the theoretical counterpart has been little investigated. In this paper, we investigate a largely ignored question, i.e., whether an optimization problem will always become harder for EAs in a noisy environment. We prove that the answer is negative, with respect to the measurement of the expected running time. The result implies that, for optimization tasks that have already been quite hard to solve, the noise may not have a negative effect, and the easier a task the more negatively affected by the noise. On a representative problem where the noise has a strong negative effect, we examine two commonly employed mechanisms in EAs dealing with noise, the re-evaluation and the threshold selection strategies. The analysis discloses that the two strategies, however, both are not effective, i.e., they do not make the EA more noise tolerant. We then find that a small modification of the threshold selection allows it to be proven as an effective strategy for dealing with the noise in the problem.
This paper is a continuation of a paper with the same title of the last two authors. In the first part of the present paper, we give a unified geometric proof that both focal submanifolds of every isoparametric hypersurface in spheres with four distinct principal curvatures are Willmore. In the second part, we completely determine which focal submanifolds are Einstein except one case.
Evolutionary algorithms (EAs), simulating the evolution process of natural species, are used to solve optimization problems. Crossover (also called recombination), originated from simulating the chromosome exchange phenomena in zoogamy reproduction, is widely employed in EAs to generate offspring solutions, of which the effectiveness has been examined empirically in applications. However, due to the irregularity of crossover operators and the complicated interactions to mutation, crossover operators are hard to analyze and thus have few theoretical results. Therefore, analyzing crossover not only helps in understanding EAs, but also helps in developing novel techniques for analyzing sophisticated metaheuristic algorithms. In this paper, we derive the General Markov Chain Switching Theorem (GMCST) to facilitate theoretical studies of crossover-enabled EAs. The theorem allows us to analyze the running time of a sophisticated EA from an easy-to-analyze EA. Using this tool, we analyze EAs with several crossover operators on the LeadingOnes and OneMax problems, which are noticeably two well studied problems for mutation-only EAs but with few results for crossover-enabled EAs. We first derive the bounds of running time of the (2+2)-EA with crossover operators; then we study the running time gap between the mutation-only (2:2)-EA and the (2:2)-EA with crossover operators; finally, we develop strategies that apply crossover operators only when necessary, which improve from the mutation-only as well as the crossover-all-the-time (2:2)-EA. The theoretical results are verified by experiments.
In this paper, we give a finiteness result on the diffeomorphism types of curvature-adapted equifocal hypersurfaces in a simply connected compact symmetric space. Furthermore, the condition curvature-adapted can be dropped if the symmetric space is of rank one.