Research Progress
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Gopakumar–Vafa Type Invariants of Holomorphic Symplectic 4-Folds (Yalong Cao and collaborators)
03 18, 2024
Using reduced Gromov–Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi–Yau 3-folds, Klemm and Pandharipande for Calabi–Yau 4-folds, and Pandharipande and Zinger for Calabi–Yau 5-folds. We conjecture that our invariants are integers and give a sheaf-theoretic interpretation in terms of reduced 4-dimensional Donaldson–Thomas invariants of one-dimensional stable sheaves. We check our conjectures for the product of two K3 surfaces and for the cotangent bundle of \({\mathbb {P}}^2\). Modulo the conjectural holomorphic anomaly equation, we compute our invariants also for the Hilbert scheme of two points on a K3 surface. This yields a conjectural formula for the number of isolated genus 2 curves of minimal degree on a very general hyperk？hler 4-fold of \(K3^{[2]}\)-type. The formula may be viewed as a 4-dimensional analogue of the classical Yau–Zaslow formula concerning counts of rational curves on K3 surfaces. In the course of our computations, we also derive a new closed formula for the Fujiki constants of the Chern classes of tangent bundles of both Hilbert schemes of points on K3 surfaces and generalized Kummer varieties.
Publication:
Commun. Math. Phys. 405, 26 (2024).
https://doi.org/10.1007/s00220-023-04882-8
Author:
Yalong Cao
Morningside Center of Mathematics & Institute of Mathematics, Chinese Academy of Sciences, No. 55, Zhongguancun East Road, Beijing, 100190, China
Email: yalongcao@amss.ac.cn
Georg Oberdieck
Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden
Yukinobu Toda
Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba, 277-8583, Japan -
Gevrey Solutions of Quasi-Linear Hyperbolic Hydrostatic Navier–Stokes System (Ping Zhang and collaborators)
03 18, 2024
We study the well-posedness of a hyperbolic quasi-linear version of hydrostatic Navier–Stokes system in \(\mathbb R\times \mathbb T\) and prove the global well-posedness of the system with initial data which are small and analytic in both variables. We also prove the convergence of such analytic solutions to that of the classical hydrostatic Navier–Stokes system when the delay time converges to zero. Furthermore, we obtain a local well-posedness result in Gevrey class 2 when the initial datum is a small perturbation of some convex function.
Publication:
SIAM Journal on Mathematical Analysis, Vol. 55, Iss. 6 (2023)
https://doi.org/10.1137/22M1526290
Author:
Wei-Xi Li
School of Mathematics and Statistics, Wuhan University, 430072, Wuhan, China
Computational Science, Hubei Key Laboratory, Wuhan University, 430072, Wuhan, China.
Marius Paicu
Université Bordeaux, Institut de Mathématiques de Bordeaux, F-33405 Talence Cedex, France.
Ping Zhang
Academy of Mathematics, Systems Science, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, 100190, China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China.
Email: zp@amss.ac.cn -
In the last decade, kernel-based regularization methods (KRMs) have been widely used for stable impulse response estimation in system identification. Its favorable performance over classic maximum likelihood/prediction error methods (ML/PEM) has been verified by extensive simulations. Recently, we noticed a surprising observation: for some data sets and kernels, no matter how the hyper-parameters are tuned, the regularized least square estimate cannot have higher model fit than the least square (LS) estimate, which implies that for such cases, the regularization cannot improve the LS estimate. Therefore, this paper focuses on how to understand this observation. To this purpose, we first introduce the squared error (SE) criterion, and the corresponding oracle hyper-parameter estimator in the sense of minimizing the SE criterion. Then we find the necessary and sufficient conditions under which the regularization cannot improve the LS estimate, and we show that the probability that this happens is greater than zero. The theoretical findings are demonstrated through numerical simulations, and simultaneously the anomalous simulation outcome wherein the probability is nearly zero is elucidated, and due to the ill-conditioned nature of either the kernel matrix, the Gram matrix, or both.
Publication:
Automatica, Volume 160, February 2024, 111442
http://dx.doi.org/10.1016/j.automatica.2023.111442
Author:
Biqiang Mu
Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Email: bqmu@amss.ac.cn
Lennart Ljung
Division of Automatic Control, Department of Electrical Engineering, Link？ping University, Link？ping 58183, Sweden
Tianshi Chen
School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen 518172, China -
The generalized successive overrelaxation (GSOR) method of Bai, Parlett, and Wang [Numer. Math., 102 (2005), pp. 1–38] was designed for conventional saddle-point problems. We extend GSOR to a class of block three-by-three saddle-point problems. Based on the necessary and sufficient conditions for all roots of a real cubic polynomial to have modulus less than one, we derive convergence results under reasonable assumptions. We also analyze a class of block lower triangular preconditioners induced from GSOR and derive explicit and sharp spectral bounds for the preconditioned matrices. We report numerical experiments on test problems from the liquid crystal director model and the coupled Stokes–Darcy flow, demonstrating the usefulness of GSOR.
Publication:
SIAM Journal on Scientific Computing, Vol. 45, Iss. 5 (2023)
http://dx.doi.org/10.1137/22M1515884
Author:
Na Huang
Department of Applied Mathematics, College of Science, China Agricultural University, 100083 Beijing, China.
Yu-Hong Dai
LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, China.
Email: dyh@lsec.cc.ac.cn
Dominique Orban
GERAD and Department of Mathematics and Industrial Engineering, Polytechnique Montréal, QC H3C 3A7, Canada.
Michael A. Saunders
Systems Optimization Laboratory, Department of Management Science and Engineering, Stanford University, Stanford, CA 94305 USA. -
Stochastic Klein–Gordon–Schr？dinger (KGS) equations are important mathematical models and describe the interaction between scalar nucleons and neutral scalar mesons in the stochastic environment. In this paper, we propose novel structure-preserving schemes to numerically solve stochastic KGS equations with additive noise, which preserve averaged charge evolution law, averaged energy evolution law, symplecticity, and multi-symplecticity. By applying central difference, sine pseudo-spectral method, or finite element method in space and modifying finite difference in time, we present some charge and energy preserved fully-discrete schemes for the original system. In addition, combining the symplectic Runge-Kutta method in time and finite difference in space, we propose a class of multi-symplectic discretizations preserving the geometric structure of the stochastic KGS equation. Finally, numerical experiments confirm theoretical findings.
Publication:
Journal of Computational Physics, Volume 500, 1 March 2024, 112740
http://dx.doi.org/10.1016/j.jcp.2023.112740
Author:
Jialin Hong
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Email: hjl@lsec.cc.ac.cn
Baohui Hou
Department of Mathematics, Shanghai University, Shanghai 200444, PR China
Liying Sun
Academy for Multidisciplinary Studies, Capital Normal University, Beijing 100048, PR China
Xiaojing Zhang
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China -
Riesz conjugate functions theorem for harmonic quasiconformal mappings (Jinsong Liu and collaborator)
03 18, 2024
We generalize the Riesz conjugate functions theorem for planar harmonic K-quasiregular mappings (when 1<p≤2) and harmonic K-quasiconformal mappings (when 2<p<∞) in the unit disk. Moreover, if K=1, then our constant coincides with the classical analytic case. For the n dimensional case (n>2), we also obtain the Riesz conjugate functions theorem for invariant harmonic K-quasiregular mappings when 1<p≤2.
Publication:
Advances in Mathematics, Volume 434, 1 December 2023, 109321
http://dx.doi.org/10.1016/j.aim.2023.109321
Author:
Jinsong Liu
HLM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Email: liujsong@math.ac.cn
Jian-Feng Zhu
School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China -
We consider an inverse problem \boldsymbol {y}= f(\boldsymbol {Ax}) , where \boldsymbol {x}\in \mathbb {R}^{n} is the signal of interest, \boldsymbol {A} is the sensing matrix, f is a nonlinear function and \boldsymbol {y} \in \mathbb {R}^{m} is the measurement vector. In many applications, we have some level of freedom to design the sensing matrix \boldsymbol {A}, and in such circumstances we could optimize \boldsymbol {A} to achieve better reconstruction performance. As a first step towards optimal design, it is important to understand the impact of the sensing matrix on the difficulty of recovering \boldsymbol {x} from \boldsymbol {y}. In this paper, we study the performance of one of the most successful recovery methods, i.e., the expectation propagation (EP) algorithm. We define a notion of spikiness for the spectrum of \boldsymbol {A} and show the importance of this measure for the performance of EP. We show that whether a spikier spectrum can hurt or help the recovery performance depends on f. Based on our framework, we are able to show that, in phase-retrieval problems, matrices with spikier spectrums are better for EP, while in 1-bit compressed sensing problems, less spiky spectrums lead to better performance. Our results unify and substantially generalize existing results that compare Gaussian and orthogonal matrices, and provide a platform towards designing optimal sensing systems.
Publication:
IEEE Transactions on Information Theory, vol. 70, no. 1, pp. 482-508, Jan. 2024
http://dx.doi.org/10.1109/TIT.2023.3307553.
Author:
Junjie Ma
Department of Statistics, Columbia University, New York City, NY, USA
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
Email: majunjie@lsec.cc.ac.cn
Ji Xu
Department of Computer Science, Columbia University, New York City, NY, USA
Arian Maleki
Department of Statistics, Columbia University, New York City, NY, USA -
On the global stability of large Fourier mode for 3-D Navier-Stokes equations (Ping Zhang and collaborator)
03 18, 2024
In this paper, we first prove the global existence of strong solutions to 3-D incompressible Navier-Stokes equations with solenoidal initial data, which writes in the cylindrical coordinates is of the form: A(r,z)\cosN\theta+B(r,z)\sinN\theta, provided that N is large enough. In particular, we prove that the corresponding solution has almost the same frequency N for any positive time. The main idea of the proof is first to write the solution in trigonometrical series in \theta variable and estimate the coefficients separately in some scale-invariant spaces, then we handle a sort of weighted sum of these norms of the coefficients in order to close the a priori estimate of the solution. Furthermore, we shall extend the above well-posedness result for initial data which is a linear combination of axisymmetric data without swirl and infinitely many large mode trigonometric series in the angular variable.
Publication:
Advances in Mathematics, Volume 438, February 2024, 109475
http://dx.doi.org/10.1016/j.aim.2023.109475
Author:
Yanlin Liu
School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University, 100875 Beijing, China
Ping Zhang
Academy of Mathematics & Systems Science and Hua Loo-Keng Center for Mathematical Sciences, Chinese Academy of Sciences, Beijing 100190, China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Email: zp@amss.ac.cn -
Turing’s model has been widely used to explain how simple, uniform structures can give rise to complex, patterned structures during the development of organisms. However, it is very hard to establish rigorous theoretical results for the dynamic evolution behavior of Turing’s model since it is described by nonlinear partial differential equations. We focus on controllability of Turing’s model by linearization and spatial discretization. This linearized model is a networked system whose agents are second order linear systems and these agents interact with each other by Laplacian dynamics on a graph. A control signal can be added to agents of choice. Under mild conditions on the parameters of the linearized Turing’s model, we prove the equivalence between controllability of the linearized Turing’s model and controllability of a Laplace dynamic system with agents of first order dynamics. When the graph is a grid graph or a cylinder grid graph, we then give precisely the minimal number of control nodes and a corresponding control node set such that the Laplace dynamic systems on these graphs with agents of first order dynamics are controllable.
Publication:
Automatica, Volume 162, April 2024, 111507
http://dx.doi.org/10.1016/j.automatica.2023.111507
Author:
Tianhao Li
Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100190, PR China
Ruichang Zhang
Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100190, PR China
Zhixin Liu
Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100190, PR China
Email: lzx@amss.ac.cn
Zhuo Zou
Fudan University, Shanghai 200433, PR China
Xiaoming Hu
Optimization and Systems Theory, KTH Royal Institute of Technology, Stockholm 10044, Sweden -
Classical continued fractions can be introduced in the field of $p$-adic numbers, where $p$-adic continued fractions offer novel perspectives on number representation and approximation. While numerous $p$-adic continued fraction expansion algorithms have been proposed by the researchers, the establishment of several excellent properties, such as the Lagrange’s Theorem for classic continued fractions, which indicates that every quadratic irrationals can be expanded periodically, remains elusive. In this paper, we introduce several new algorithms designed for expanding algebraic numbers in $\mathbb{Q}_p$ for a given prime $p$. We give an upper bound of the number of partial quotients for the expansion of rational numbers, and prove that for small primes $p$, our algorithm generates periodic continued fraction expansions for all quadratic irrationals. Experimental data demonstrates that our algorithms exhibit better performance in the periodicity of expansions for quadratic irrationals compared to the existing algorithms. Furthermore, for bigger primes $p$, we propose a potential approach to establish a $p$-adic continued fraction expansion algorithm. As before, the algorithm is designed to expand algebraic numbers in $\mathbb{Q}_p$, while generating periodic expansions for all quadratic irrationals in $\mathbb{Q}_p$.
Publication:
Math. Comp. February 5, 2024
https://doi.org/10.1090/mcom/3948
Author:
Zhaonan Wang
Key Laboratory of Mathematics Mechanization, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
Yingpu Deng
Key Laboratory of Mathematics Mechanization, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
Email: dengyp@amss.ac.cn
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