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On some families of binary codes

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information Processing Letters 2026年 191卷

作者： Zihui Liu Key Laboratory of Mathematical Theory and Computation in Information Security Department of Mathematics Beijing Institute of Technology Beijing 100081 China

We give further results on the weight distributions of the two families of binary codes recently constructed by simplicial complexes by (Wu, Lee, 2020), and show that the converse of the above results is also correct, that is, the binary codes with such weight distributions properties must be these two families of codes. Based on the above results, we also construct another family of binary self-orthogonal codes and present their separating properties and applications to the secret sharing scheme, cryptography and other aspects of information security.

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BGG category O and -graded representation theory

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arXiv 2021年

作者： Hu, Jun Key Laboratory of Mathematical Theory and Computation in Information Security School of Mathematics and Statistics Beijing Institute of Technology Beijing100081 China

A UNIMODAL SEQUENCE WITH MODE AT A QUARTER LENGTH

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arXiv 2022年

作者： Liu, Max Y.C. Wang, David G.L. School of Mathematics and Statistics Beijing Institute of Technology Beijing102400 China School of Mathematics and Statistics MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing Institute of Technology Beijing102400 China

We show that the number A(n, m) of partitions with m even parts and largest hook length n is strongly unimodal with mode ⌊(n−1)/4⌋ for n ≥ 6. We establish this result by induction, using a 5-term recurrence due to Lin, Xiong and Yan, and two 4-term recurrences obtained by Zeilberger’s algorithm. The sequence A(n, m) is not log-concave. Using Möbius transformation and the method of interlacing zeros, we obtain that every zero of every generating function Pm A(n, m)zm lies on the left half part of the circle |z − 1| = 2. Moreover, as a direct application of Wang and Zhang’s characterization of root geometry of polynomial sequences that satisfy a recurrence of type (1, 1), we see that all these zeros are densely distributed on the half circle. Copyright © 2022, The Authors. All rights reserved.

关键词： Algebra

Optimization Models and Interpretations for Three Types of Adversarial Perturbations against Support Vector Machines

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arXiv 2022年

作者： Su, Wen Li, Qingna Cui, Chunfeng School of Mathematics and Statistics Beijing Institute of Technology Beijing100081 China School of Mathematics and Statistics Beijing Key Laboratory on MCAACI Key Laboratory of Mathematical Theory and Computation in Information Security Beijing Institute of Technology Beijing100081 China The Ministry of Education School of Mathematical Sciences Beihang University 100191 China

Adversarial perturbations have drawn great attentions in various deep neural networks. Most of them are computed by iterations and cannot be interpreted very well. In contrast, little attentions are paid to basic machine learning models such as support vector machines. In this paper, we investigate the optimization models and the interpretations for three types of adversarial perturbations against support vector machines, including sample-adversarial perturbations (sAP), class-universal adversarial perturbations (cuAP) as well as universal adversarial perturbations (uAP). For linear binary/multi classification support vector machines (SVMs), we derive the explicit solutions for sAP, cuAP and uAP (binary case), and approximate solution for uAP of multi-classification. We also obtain the upper bound of fooling rate for uAP. Such results not only increase the interpretability of the three adversarial perturbations, but also provide great convenience in computation since iterative process can be avoided. Numerical results show that our method is fast and effective in calculating three types of adversarial perturbations. Copyright © 2022, The Authors. All rights reserved.

关键词： Support vector machines

GLOBAL UNIFORM IN N ESTIMATES FOR SOLUTIONS OF A SYSTEM OF HARTREE-FOCK-BOGOLIUBOV TYPE IN THE CASE β < 1

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arXiv 2022年

作者： Chong, Jacky Dong, Xin Grillakis, Manossos Machedon, Matei Zhao, Zehua University of Texas Austin United States University of Maryland College Park United States Department of Mathematics and Statistics Beijing Institute of Technology Beijing China MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing China

We extend the results of the 2019 paper by the third and fourth author globally in time. More precisely, we prove uniform in N estimates for the solutions φ, Λ and Γ of a coupled system of Hartree–Fock–Bogoliubov type with interaction potential VN(x - y) = N3βv(Nβ(x - y)) with β Copyright © 2022, The Authors. All rights reserved.

关键词： Sobolev spaces

A NONCOMMUTATIVE APPROACH TO THE SCHUR POSITIVITY OF CHROMATIC SYMMETRIC FUNCTIONS

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arXiv 2023年

作者： Thibon, Jean-Yves Wang, David G.L. LIGM Université Gustave-Eiffel CNRS ENPC ESIEE-Paris 5 Boulevard Descartes Champs-sur-Marne Marne-la-Vallée77454 Cedex 2 France School of Mathematics and Statistics MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing Institute of Technology Beijing102400 China

We obtain the Schur positivity of spider graphs of the forms S(a, 2, 1) and S(a, 4, 1), which are considered to have the simpliest structures for which the Schur positivity was unknown. The proof outline has four steps. First, we find noncommutative analogs for the chromatic symmetric functions of the spider graphs S(a, b, 1). Secondly, we expand the analogs under the Λ- and R-bases, whose commutative images are the elementary and skew Schur symmetric functions, respectively. Thirdly, we recognize the Schur coefficients via the Littlewood-Richardson rule in terms of norms of multisets of Yamanouchi words. At last we establish the Schur positivity combinatorially together with the aid of computer *** Codes 05E05, 05C05, 05A15 Copyright © 2023, The Authors. All rights reserved.

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Surface Embedding of Non-Bipartite k-Extendable Graphs

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Annals of Applied Mathematics 2022年第1期38卷 1-24页

作者： Hongliang Lu David G.L.Wang School of Mathematics and Statistics Xi'an Jiaotong UniversityXi'anShaanxi 710049China School of Mathematics and Statistics&MIIT Key Laboratory ofMathematical Theory and Computation in Information Security Beijing Institute of TechnologyBeijing 102400China

For every surface,we find the minimum number k such that every non-bipartite graph that is embeddable in that surface is not *** particular,we construct a family of 3-extendable graphs which we call bow-tie *** confirms the existence of an infinite number of 3-extendable non-bipartite graphs that are embeddable in the Klein bottle.

关键词： Non-bipartite graph matching extension surface embedding.

ON SCATTERING ASYMPTOTICS FOR THE 2D CUBIC RESONANT SYSTEM

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arXiv 2022年

作者： Yang, Kailong Zhao, Zehua Chongqing National Center for Applied Mathematics Chongqing Normal University Chongqing China Department of Mathematics University of Maryland United States Department of Mathematics and Statistics Beijing Institute of Technology MIIT Key Laboratory of Mathematical Theory and Computation in Information Security China

In this paper, we prove scattering asymptotics for the 2D (discrete dimension) cubic resonant system. This scattering result was used in Zhao [39] as an assumption to obtain the scattering for cubic NLS on R2 × T2 in H1 space. Moreover, the 1D analogue is proved in Yang-Zhao [36]. Though the scheme is also tightly based on Dodson [12], the 2D case is more complicated which causes some new difficulties. One obstacle is the failure of ‘l2-estimate’ for the cubic resonances in 2D (we also discuss it in this paper, which may have its own interests). To fix this problem, we establish weaker estimates and exploit the symmetries of the resonant system to modify the proof of [36]. At last, we make a few remarks on the research line of ‘long time dynamics for NLS on waveguides’. Copyright © 2022, The Authors. All rights reserved.

关键词： Waveguides

Graded dimensions and monomial bases for the cyclotomic quiver hecke superalgebras

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arXiv 2021年

作者： Hu, Jun Shi, Lei MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing Institute of Technology Beijing102488 China School of Mathematical and Statistics Beijing Institute of Technology Beijing100081 China

MSC Codes 20C08In this paper we derive a closed formula for the (Z × Z2)-graded dimension of the cyclotomic quiver Hecke superalgebra RΛ(β) associated to an arbitrary Cartan superdatum (A, P, Π, Π∨), polynomials (Qi,j(x1, x2))i,j∈I, β ∈ Q+n and Λ ∈ P +. As applications, we obtain a necessary and sufficient condition for which e(ν) 6= 0 in RΛ(β). We construct an explicit monomial basis for the bi-weight space e(νe)RΛ(β)e(νe), where νe is a certain specific ntuple defined in (1.4). In particular, this gives rise to a monomial basis for the cyclotomic odd nilHecke algebra. Finally, we consider the case when β = α1 +α2 +· · ·+αn with α1, · · ·, αn distinct. We construct an explicit monomial basis of RΛ(β) and show that it is indecomposable in this case. Copyright © 2021, The Authors. All rights reserved.

关键词： Algebra