## Research Progress

• #### Free boundary problems in Fluid and Mean Field limit (Nguyen Quoc Hung)

01 13, 2023

In 2021, we proved that the 2d-muskat equation is local/global well-posed where initial data belongs to Sobolev space H^(3/2+[log]^(2/3)).In 2022 in [2], we extended this result to the 3d-muskat equation. In [4], we proved first local wellposedness result for the 2d/3d-muskat equations with initial data f_0 \in C^1. Finally, in [1], we solved the 2d-muskat equation with H^(3/2) initial data. This result is optimal with respect to the scaling of the equation and for unbounded slopes. Our proof is the first in which a null-type structure is identified for the Muskat equation, allowing to compensate for the degeneracy of the parabolic behavior for large slopes. In [3], we used this method to prove the wellposedness of a nonlocal nonlinear equation of the roots of polynomials under differentiation.
In 2018, Sylvia Serfaty established the mean field convergence for Coulomb potential and super-Coulombic Riesz potential of points evolving along the gradient flow of their interaction energy. Serfaty's proof is based on a modulated energy method using a Coulomb or Riesz distance; and the Caffarelli-Silvestre extension theorem. In [5], we extended this result to general kernels where the Caffarelli-Silvestre extension is not available for these kernels. Moreover, our assumption on kernel contains Lenard-Jones type potentials. To get our result, we established new commutator estimates.
Publications:　　[1] Thomas Alazar, Quoc-Hung Nguyen, Endpoint Sobolev theory for the Muskat equation, Commun. Math. Phys. (2022). https://doi.org/10.1007/s00220-022-04514-7　　[2] Thomas Alazar, Quoc-Hung Nguyen, Quasilinearization of the 3D Muskat equation, and applications to the critical Cauchy problem, Advances in Math, Volume 399, 16 April 2022, 108278, https://doi.org/10.1016/j.aim.2022.108278　　[3] Thomas Alazard, Omar Lazar and Quoc-Hung Nguyen, On the dynamics of the roots of polynomials under differentiation, Journal de mathematiques pures et Appliqu'ees, V 162, June 2022, Pages 1-22,https://doi.org/10.1016/j.matpur.2022.04.001,　　[4] Ke Chen, Quoc-Hung Nguyen, and Yiran Xu, The Muskat problem with C^1 data, Trans. Amer. Math. Soc. 375 (2022), 3039-3060　　[5] Quoc-Hung Nguyen, Matthew Rosenzweig, and Sylvia Serfaty, Mean-field limits of　　[6] Riesz-type singular flows, Ars Inveniendi Analytica (2022), Paper No. 4, 45 pp, DOI: https://doi.org/10.15781/nvv7-jy87　　　　Authors:　　Nguyen Quoc Hung　　Email: qhnguyen@amss.ac.cn
• #### GraphReg: Dynamical Point Cloud Registration With Geometry-Aware Graph Signal Processing (Xiaohong Jia)

01 12, 2023

This study presents a high-accuracy, efficient, and physically induced method for 3D point cloud registration, which is the core of many important 3D vision problems. In contrast to existing physics-based methods that merely consider spatial point information and ignore surface geometry, we explore geometry aware rigid-body dynamics to regulate the particle (point) motion, which results in more precise and robust registration. Our proposed method consists of four major modules. First, we leverage the graph signal processing (GSP) framework to define a new signature, i.e., point response intensity for each point, by which we succeed in describing the local surface variation, resampling keypoints, and distinguishing different particles. Then, to address the shortcomings of current physics-based approaches that are sensitive to outliers, we accommodate the defined point response intensity to median absolute deviation (MAD) in robust statistics and adopt the X84 principle for adaptive outlier depression, ensuring a robust and stable registration. Subsequently, we propose a novel geometric invariant under rigid transformations to incorporate higher-order features of point clouds, which is further embedded for force modeling to guide the correspondence between pairwise scans credibly. Finally, we introduce an adaptive simulated annealing (ASA) method to search for the global optimum and substantially accelerate the registration process. We perform comprehensive experiments to evaluate the proposed method on various datasets captured from range scanners to LiDAR. Results demonstrate that our proposed method outperforms representative state-of-the-art approaches in terms of accuracy and is more suitable for registering large-scale point clouds. Furthermore, it is considerably faster and more robust than most competitors. Our implementation is publicly available at https://github.com/zikai1/GraphReg.
Publication:
IEEE Transactions on Image Processing, Volume 31, Pages: 7449 - 7464
https://doi.org/10.1109/TIP.2022.3223793

Author:
Mingyang Zhao
Beijing Academy of Artificial Intelligence, Beijing 100190, China
NLPR, Institute of Automation, CAS, Beijing 100190, China
Lei Ma
Beijing Academy of Artificial Intelligence, Beijing 100190, China
Institute for Artificial Intelligence, Beijing 100190, China
National Biomedical Imaging Center, Peking University, Beijing 100871, China
Xiaohong Jia
Academy of Mathematics and Systems Science, CAS, Beijing 100190, China
School of Mathematical Sciences, UCAS, Beijing 100149, China
Email: xhjia@amss.ac.cn
Dong-Ming Yan
NLPR, Institute of Automation, CAS, Beijing 100864, China
School of AI, UCAS, Beijing 101408, China
Tiejun Huang
National Engineering Research Center of Visual Technology, School of Computer Science
BAAI and Institute for Artificial Intelligence, Peking University, Beijing 100871, China
• #### The Complete SC-Invariant Affine Automorphisms of Polar Codes (Guiying Yan, Zhiming Ma)

01 12, 2023

Automorphism ensemble (AE) decoding for polar codes was proposed by decoding permuted codewords with successive cancellation (SC) decoders in parallel and hence has lower latency compared to that of successive cancellation list (SCL) decoding. However, some automorphisms are SC-invariant, thus are redundant in AE decoding. In this paper, we find a necessary and sufficient condition related to the block lowertriangular structure of transformation matrices to identify SCinvariant automorphisms. Furthermore, we provide an algorithm to determine the complete SC-invariant affine automorphisms under a specific polar code construction.
Publication:
2022 IEEE International Symposium on Information Theory (ISIT), Espoo, Finland, 2022, pp. 2368-2373, doi: 10.1109/ISIT50566.2022.9834828.
Authors:
Zicheng Ye
University of Chinese Academy of Sciences; Academy of Mathematics and Systems Science
Yuan Li
University of Chinese Academy of Sciences; Academy of Mathematics and Systems Science; Huawei Technologies Co. Ltd.
Huazi Zhang
Huawei Technologies Co. Ltd.
Rong？Li
Huawei Technologies Co. Ltd.
Jun Wang
Huawei Technologies Co. Ltd.
Guiying Yan
University of Chinese Academy of Sciences; Academy of Mathematics and Systems Science
Email: yangy@amss.ac.cn
Zhiming Ma
University of Chinese Academy of Sciences; Academy of Mathematics and Systems Science
Email: mazm@amt.ac.cn
• #### On the Message Passing Efficiency of Polar and Low-Density Parity-Check Decoders (Guiying Yan, Zhiming Ma)

01 12, 2023

This study focuses on the efficiency of message-passing-based decoding algorithms for polar and low-density parity-check (LDPC) codes. Both successive cancellation (SC) and belief propagation (BP) decoding algorithms are studied under the message-passing framework. Counter-intuitively, SC decoding demonstrates the highest decoding efficiency, although it was considered a weak decoder regarding error-correction performance. We analyze the complexity-performance tradeoff to dynamically track the decoding efficiency, where the complexity is measured by the number of messages passed (NMP), and the performance is measured by the statistical distance to the maximum a posteriori (MAP) estimate. This study offers new insight into the contribution of each message passing in decoding, and compares various decoding algorithms on a message-by-message level. The analysis corroborates recent results on terabits-per-second polar SC decoders, and might shed light on better scheduling strategies.
Publication:
2022 IEEE Global Communications Conference (GLOBECOM).
Authors:
Dawei Yin
Shandong University；Huawei Technologies Co. Ltd.
Yuan Li
University of Chinese Academy of Sciences; Academy of Mathematics and Systems Science; Huawei Technologies Co. Ltd.
Xianbin Wang
Huawei Technologies Co. Ltd.
Jiajie Tong
Huawei Technologies Co. Ltd.
Huazi Zhang
Huawei Technologies Co. Ltd.
Jun Wang
Huawei Technologies Co. Ltd.
Guanghui Wang
Shandong University
Guiying Yan
University of Chinese Academy of Sciences; Academy of Mathematics and Systems Science
Email: yangy@amss.ac.cn
Zhiming Ma
University of Chinese Academy of Sciences; Academy of Mathematics and Systems Science
Email: mazm@amt.ac.cn
• #### Deterministic Identification over Channels without CSI (Guiying Yan, Zhiming Ma)

01 12, 2023

Identification capacities of randomized and deterministic identification were proved to exceed channel capacity for Gaussian channels with channel side information (CSI). In this work, we extend deterministic identification to the block fading channels without CSI by applying identification codes for both channel estimation and user identification. We prove that identification capacity is asymptotically higher than transmission capacity even in the absence of CSI. And we also analyze the finite-length performance theoretically and numerically. The simulation results verify the feasibility of the proposed blind deterministic identification in finite blocklength regime.
Publication:
IEEE Information Theory Workshop, ITW 2022, Mumbai, India, November 1-9, 2022. pages 332-337, IEEE, 2022.
Authors:
Yuan Li
University of Chinese Academy of Sciences; Academy of Mathematics and Systems Science; Huawei Technologies Co. Ltd.
Xianbin Wang
Huawei Technologies Co. Ltd.
Huazi Zhang
Huawei Technologies Co. Ltd.
Jun Wang
Huawei Technologies Co. Ltd.
Wen Tong
Huawei Technologies Co. Ltd.
Guiying Yan
University of Chinese Academy of Sciences; Academy of Mathematics and Systems Science
Email: yangy@amss.ac.cn
Zhiming Ma
University of Chinese Academy of Sciences; Academy of Mathematics and Systems Science
Email: mazm@amt.ac.cn
• #### On Accelerating a Multilevel Correction Adaptive Finite Element Method for Kohn-Sham Equation (Hehu Xie)

01 12, 2023

Based on the numerical method proposed in Hu et al. (2018) [22] for Kohn-Sham equation, further improvement on the efficiency is obtained in this paper by i). designing a numerical method with the strategy of separately handling the nonlinear Hartree potential and exchange-correlation potential, and ii). parallelizing the algorithm in an eigenpairwise approach. The feasibility of two approaches is analyzed in detail, and the new algorithm is described completely. Compared with previous results, a significant improvement of numerical efficiency can be observed from plenty of numerical experiments, which make the new method more suitable for the practical problems.

Publication:
Journal of Computational Physics, Volume 472, 1 January 2023, 111674

Author:
Guanghui Hu
Department of Mathematics & Guangdong-Hong Kong-Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications, Faculty of Science and Technology, University of Macau, Macao S.A.R., China
Zhuhai UM Science & Technology Research Institute, Zhuhai, Guangdong, China
Hehu Xie
LSEC, ICMSEC, 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: hhxie@lsec.cc.ac.cn
Fei Xu
Faculty of Science, Beijing University of Technology, Beijing 100124, China
• #### Improving the Gilbert-Varshamov Bound by Graph Spectral Method (Zhiming Ma, Guiying Yan)

01 12, 2023

This paper improves Gilbert-Varshamov bound by graph spectral method. Gilbert graph Gq,n,d is a graph with all vectors in Fnq as vertices where two vertices are adjacent if their Hamming distance is less than d. In this paper, we calculate the eigenvalues and eigenvectors of Gq,n,d using the properties of Cayley graph. The improved bound is associated with the minimum eigenvalue of the graph. Finally, we give an algorithm to calculate the bound and linear codes which satisfy the bound.
Publication:
CSIAM Transaction on Applied Mathematics, Vol. 4, No. 1, pp. 1-12, 2023.

Authors:
Zicheng Ye
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.
University of Chinese Academy of Sciences, Beijing 100049, China.
Huazi Zhang
Hangzhou Research Center, Huawei Technologies Co., Ltd., Hangzhou 310052, Zhejiang Province, China.
Rong Li
Hangzhou Research Center, Huawei Technologies Co., Ltd., Hangzhou 310052, Zhejiang Province, China.
Jun Wang
Hangzhou Research Center, Huawei Technologies Co., Ltd., Hangzhou 310052, Zhejiang Province, China.
Guiying Yan
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.
University of Chinese Academy of Sciences, Beijing 100049, China.
Email: yangy@amss.ac.cn
Zhiming Ma
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.
University of Chinese Academy of Sciences, Beijing 100049, China.
Email: mazm@amt.ac.cn
• #### Anti-Ramsey number of expansions of paths and cycles in uniform hypergraphs (Tong Li, Guiying Yan)

01 12, 2023

For an r-graph F, the anti-Ramsey number ar(n,r,F) is the minimum number c of colors such that for any edge-coloring of the complete r-graph on n vertices with at least c colors, there is a copy of F whose edges have distinct colors. Let Pk and Ck be the path and cycle with k edges in 2-graphs, respectively. In this paper, we determine ar(n,r,Pk+)and ar(n,r,Ck+) for all k≥3 and r≥3 except ar(n,r,C3+), which are extensions of several results of Gu, Li, and Shi.

Publication:
Journal of Graph Theory, Volume 101, Issue 4, December 2022, Pages: 668-685.
Authors：
Yucong Tang
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, China
Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing, China
Tong Li
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China

Guiying Yan
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China
Email: yangy@amss.ac.cn
• #### Mechanisms for Dual-role-facility Location Games: Truthfulness and Approximability (Xujin Chen, Changjun Wang)

01 11, 2023

This paper studies the dual-role-facility location game with generalized service costs, in which every agent plays a dual role of facility and customer, and is associated with a facility opening cost as his private information. The agents strategically report their opening costs to a mechanism which maps the reports to a set of selected agents and payments to them. Each selected agent opens his facility, incurs his opening cost and receives the payment the mechanism sets for him. Each unselected agent incurs a services cost that is determined by the set of selected agents in a very general way. The mechanism is truthful if under it no agent has an incentive to misreport. We provide a necessary and sufficient condition for mechanisms of the game to be truthful. This characterization particularly requires an invariant service cost for each unselected agent, which is a remarkable difference from related work in literature.
As applications of this truthfulness characterization, we focus on the classic metric-space setting, in which agents' service costs equal their distances to closest open facilities. We present truthful mechanisms that minimize or approximately minimize the maximum cost among all agents and the total cost of all agents, respectively. Moreover, when the total payment cannot exceed a given budget, we prove, for both cost-minimization objectives, lower and upper bounds on approximation ratios of truthful mechanisms that satisfy the budget constraint.
Publication:
Theoretical Computer Science, Volume 932, 6 October 2022, Pages 69-83

Author:
Xujin Chen
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China
Email: xchen@amss.ac.cn
Minming Li
Department of Computer Science, City University of Hong Kong, Hong Kong
Changjun Wang
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
Chenhao Wang
Advanced Institute of Natural Sciences, Beijing Normal University, Zhuhai, China
BNU-HKBU United International College, Zhuhai, China
Mengqi Zhang
Beijing Institute of Astronautical Systems Engineering, Beijing, China
Yingchao Zhao
Caritas Institute of Higher Education, Hong Kong
• #### Scheepers’ Conjecture and the Scheepers Diagram (Yinhe Peng)

01 11, 2023

Inductively approaching subsets by almost finite sets, we refute Scheepers’ conjecture under CH. More precisely, we prove the following.
1. Assuming CH, there is a subset of reals $X$ such that $C_p(X)$ has property ($\alpha _2$) and $X$ does not satisfy $S_1(\Gamma , \Gamma )$.
Applying the idea of approaching subsets by almost finite sets and using an analogous approaching, we complete the Scheepers Diagram.
2. $U_{fin}(\Gamma , \Gamma )$ implies $S_{fin}(\Gamma , \Omega )$.
3. $U_{fin}(\Gamma , \Omega )$ does not imply $S_{fin}(\Gamma , \Omega )$. More precisely, assuming CH, there is a subset of reals $X$ satisfying $U_{fin}(\Gamma , \Omega )$ such that $X$ does not satisfy $S_{fin}(\Gamma , \Omega )$.
These results solve three longstanding and major problems in selection principles.
Publication:
Transactions of The American Mathematical Society, Volume 376, Number 2, February 2023, Pages 1199–1229.
https://doi.org/10.1090/tran/8787

Author:
Yinhe Peng
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
Email: pengyinhe@amss.ac.cn