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The e-positivity and schur positivity of the chromatic symmetric functions of some trees

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arXiv 2021年

作者： Wang, David G.L. Wang, Monica M.Y. School of Mathematics and Statistics Beijing Key Laboratory on MCAACI Beijing Institute of Technology Key Laboratory of Mathematical Theory and Computation in Information Security Ministry of Industry and Information Technology Beijing102400 China School of Mathematics and Statistics Beijing Institute of Technology Beijing102400 China

We investigate the e-positivity and Schur positivity of the chromatic symmetric functions of some spider graphs with three legs. We obtain the positivity classification of all broom graphs and that of most double broom graphs. The methods involve extracting particular e-coefficients of the chromatic symmetric function of these graphs with the aid of Orellana and Scott’s triple-deletion property, and using the combinatorial formula of Schur coefficients by examining certain special rim hook tabloids. We also propose some conjectures on the e-positivity and Schur positivity of trees. © 2021, CC0.

关键词： Graphic methods

Global well-posedness and scattering for fourth-order Schrödinger equations on waveguide manifolds

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arXiv 2021年

作者： Yu, Xueying Yue, Haitian Zhao, Zehua Department of Mathematics University of Washington C138 Padelford Hall Box 354350 SeattleWA98195 United States Institute of Mathematical Sciences ShanghaiTech University Pudong Shanghai China Department of Mathematics and Statistics Beijing Institute of Technology MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing China

In this paper, we study the well-posedness theory and the scattering asymptotics for fourth-order Schrödinger equations (4NLS) on waveguide manifolds (semiperiodic spaces) Rd × Tn, d ≥ 5, n = 1, 2, 3. The tori component Tn can be generalized to n-dimensional compact manifolds Mn. First, we modify Strichartz estimates for 4NLS on waveguide manifolds, with which we establish the well-posedness theory in proper function spaces via the standard contraction mapping method. Moreover, we prove the scattering asymptotics based on an interaction Morawetz-type estimate established for 4NLS on waveguides. At last, we discuss the higher dimensional analogue, the focusing scenario and give some further remarks on this research line. This result can be regarded as the waveguide analogue of Pausader [46, 47, 48] and the 4NLS analogue of Tzvetkov-Visciglia [58].MSC Codes Mathematics Subject Classification (2020) Primary: 35Q55, Secondary: 35R01, 37K06, 37L50 Copyright © 2021, The Authors. All rights reserved.

关键词： Waveguides

SINGULAR LEVY PROCESSES AND DISPERSIVE EFFECTS OF GENERALIZED SCHRÖDINGER EQUATIONS

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arXiv 2022年

作者： Sire, Yannick Yu, Xueying Yue, Haitian Zhao, Zehua Department of Mathematics Johns Hopkins University Krieger Hall 3400 N. Charles St. BaltimoreMD21218 United States Department of Mathematics University of Washington C138 Padelford Hall Box 354350 SeattleWA98195 United States Institute of Mathematical Sciences ShanghaiTech University Pudong Shanghai China 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 introduce new models for Schrödinger-type equations, which generalize standard NLS and for which different dispersion occurs depending on the directions. Our purpose is to understand dispersive properties depending on the directions of propagation, in the spirit of waveguide manifolds, but where the diffusion is of different types. We mainly consider the standard Euclidean space and the waveguide case but our arguments extend easily to other types of manifolds (like product spaces). Our approach unifies in a natural way several previous results. Those models are also generalizations of some appearing in seminal works in mathematical physics, such as relativistic strings. In particular, we prove the large data scattering on waveguide manifolds Rd × T, d ≥ 3. This result can be regarded as the analogue of [63, 64] in our setting and the waveguide analogue investigated in [28]. A key ingredient of the proof is a Morawetz-type estimate for the setting of this model. Copyright © 2022, The Authors. All rights reserved.

关键词： Waveguides

Bilevel hyperparameter optimization for support vector classification: Theoretical analysis and a solution method

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arXiv 2021年

作者： Li, Qingna Li, Zhen Zemkoho, Alain 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 School of Mathematics and Statistics Beijing Institute of Technology Beijing100081 China School of Mathematical Sciences University of Southampton SouthamptonSO17 1BJ United Kingdom

Support vector classification (SVC) is a classical and well-performed learning method for classification problems. A regularization parameter, which significantly affects the classification performance, has to be chosen and this is usually done by the cross-validation procedure. In this paper, we reformulate the hyperparameter selection problem for support vector classification as a bilevel optimization problem in which the upper-level problem minimizes the average number of misclassified data points over all the cross-validation folds, and the lower-level problems are the l1-loss SVC problems, with each one for each fold in T-fold cross-validation. The resulting bilevel optimization model is then converted to a mathematical program with equilibrium constraints (MPEC). To solve this MPEC, we propose a global relaxation cross-validation algorithm (GR-CV) based on the well-know Sholtes-type global relaxation method (GRM). It is proven to converge to a C-stationary point. Moreover, we prove that the MPEC-tailored version of the Mangasarian-Fromovitz constraint qualification (MFCQ), which is a key property to guarantee the convergence of the GRM, automatically holds at each feasible point of this MPEC. Extensive numerical results verify the efficiency of the proposed approach. In particular, compared with other methods, our algorithm enjoys superior generalization performance over almost all the data sets used in this paper. © 2021, CC BY.

关键词： Learning systems

Crystal of affine type Â−1 and Hecke algebras at a primitive 2th root of unity

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arXiv 2019年

作者： Lin, Huang Hu, Jun School of Mathematical Sciences Zhejiang University Hangzhou310027 China Key Laboratory of Mathematical Theory and Computation in Information Security School of Mathematics and Statistics Beijing Institute of Technology Beijing100081 China

Let ∈ with > 1. In this paper we give a new realization of the crystal of affine type Â−1 using the modular representation theory of the affine Hecke algebras ℌn of type A and their level two cyclotomic quotients with Hecke parameter being a primitive 2th root of unity. We construct "hat" versions of i-induction and i-restriction functors on the category RepI(ℌn) of finite dimensional integral modules over ℌn, which induce Kashiwara operators on a suitable subgroup of the Grothendieck groups of RepI(ℌn). For any simple module M ∈ RepI(ℌn), we prove that the simple submodules of res(Equation presented) which belong to (Equation presented)(∞) (Definition 5.1) occur with multiplicity two. The main results generalize the earlier work of Grojnowski and Vazirani on the relations between the crystal of (Equation presented) and the affine Hecke algebras of type A at a primitive th root of unity. Copyright © 2019, The Authors. All rights reserved.

关键词： Crystal structure