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arXiv 2025年

作者： Richman, Harry Zhang, Cheng Matsen, Frederick A. Computational Biology Program Fred Hutchinson Cancer Center Seattle United States Mathematics Division National Center for Theoretical Sciences Taipei Taiwan School of Mathematical Sciences Center for Statistical Science Peking University Beijing China Department of Statistics University of Washington Seattle United States Department of Genome Sciences University of Washington Seattle United States Howard Hughes Medical Institute Fred Hutchinson Cancer Center Seattle United States

As part of work to connect phylogenetics with machine learning, there has been considerable recent interest in vector encodings of phylogenetic trees. We present a simple new "ordered leaf attachment" (OLA) method for uniquely encoding a binary, rooted phylogenetic tree topology as an integer vector. OLA encoding and decoding take linear time in the number of leaf nodes, and the set of vectors corresponding to trees is a simply-described subset of integer sequences. The OLA encoding is unique compared to other existing encodings in having these properties. The integer vector encoding induces a distance on the set of trees, and we investigate this distance in relation to the NNI and SPR *** Codes 92B10, 05C05, 05C85, 68W40 Copyright © 2025, The Authors. All rights reserved.

关键词： Vectors

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Least energy positive solutions for Schrödinger systems with critical exponent in R 3 : Mixed cooperation and competition

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Journal of Differential Equations 2025年 440卷

作者： Tianhao Liu Wenming Zou Institute of Applied Physics and Computational Mathematics Beijing 100088 China Department of Mathematical Sciences Tsinghua University Beijing 100084 China

Consider the following coupled Schrödinger system with Sobolev critical exponent − Δ u i + λ i u i = ∑ j = 1 d β i j | u j | 2 ⁎ 2 | u i | 2 ⁎ 2 − 2 u i in Ω , i = 1 , 2 , . . . , d , where d ≥ ...

Consider the following coupled Schrödinger system with Sobolev critical exponent − Δ u i + λ i u i = ∑ j = 1 d β i j | u j | 2 ⁎ 2 | u i | 2 ⁎ 2 − 2 u i in Ω , i = 1 , 2 , . . . , d , where d ≥ 2 , 2 ⁎ = 2 N N − 2 is the Sobolev critical exponent, β i i > 0 for every i and β i j = β j i for i ≠ j . The domain Ω ⊂ R N is either bounded or the whole space, and u i ∈ H 0 1 ( Ω ) or u i ∈ H 1 ( R N ) respectively. We are concerned with the mixed cooperative and competitive system, incorporating some of the “Grouping” ideas first introduced by [N. Soave: Calc. Var. Partial Differential Equations. 53 (3) (2015), 689–718.]. The existence of least energy positive solutions for N ≥ 4 has been well-studied by H. Tavares et al. [Calc. Var. Partial Differential Equations. 59, (2020); J. Funct. Anal. 283 (2022)]. In this paper, we will study the existence of solutions for N = 3 . Different from the high dimensional case N ≥ 4 , the dimension N = 3 makes the proof more challenging, requiring different analysis. When Ω is bounded, we firstly establish the existence of nonnegative solutions with m ( 1 ≤ m ≤ d ) nontrivial components. The proof is performed by mathematical induction on the number of appropriate subsystem. Then we obtain the existence of least energy positive solution for the mixed cooperative and competitive case, as well as the existence and classification result of ground state solutions for the purely cooperative case. Besides, we obtain a nonexistence result of fully nontrivial ground state under some further assumptions. When Ω is the whole space and λ 1 = ⋯ = λ d = 0 , we obtain some nonexistence results for the mixed cooperative and competitive case.

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Critical (P5, W4)-Free Graphs

arXiv

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arXiv 2025年

作者： Xia, Wen Jooken, Jorik Goedgebeur, Jan Beaton, Iain Cameron, Ben Huang, Shenwei College of Computer Science Nankai University Tianjin300071 China Tianjin Key Laboratory of Network and Data Security Technology Nankai University Tianjin300071 China Department of Computer Science KU Leuven Campus Kulak-Kortrijk Kortrijk8500 Belgium Department of Applied Mathematics Computer Science and Statistics Ghent University Ghent9000 Belgium Department of Mathematics and Statistics Acadia University WolfvilleNS Canada School of Mathematical and Computational Sciences University of Prince Edward Island CharlottetownPE Canada

A graph G is k-vertex-critical if χ(G) = k but χ(G − v) 1, H2)-free if it contains no induced subgraph isomorphic to H1 nor H2. A W4 is the graph consisting of a C4 plus an additional vertex adjacent to all the vertices of the C4. We show that there are finitely many k-vertex-critical (P5, W4)-free graphs for all k ≥ 1 and we characterize all 5-vertex-critical (P5, W4)-free graphs. Our results imply the existence of a polynomial-time certifying algorithm to decide the k-colorability of (P5, W4)-free graphs for each k ≥ 1 where the certificate is either a k-coloring or a (k + 1)-vertex-critical induced subgraph. Copyright © 2025, The Authors. All rights reserved.

关键词： Graph algorithms

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From Kernels to Features: A Multi-Scale Adaptive Theory of Feature Learning

arXiv

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arXiv 2025年

作者： Rubin, Noa Fischer, Kirsten Lindner, Javed Dahmen, David Seroussi, Inbar Ringel, Zohar Krämer, Michael Helias, Moritz The Racah Institute of Physics The Hebrew University of Jerusalem Jerusalem Israel Computational and Systems Neuroscience Jülich Research Centre Jülich Germany RWTH Aachen University Aachen Germany Department of Physics RWTH Aachen University Aachen Germany Institute for Theoretical Particle Physics and Cosmology RWTH Aachen University Aachen Germany Department of Applied Mathematics School of Mathematical Sciences Tel-Aviv University Tel-Aviv Israel

Theoretically describing feature learning in neural networks is crucial for understanding their expressive power and inductive biases, motivating various approaches. Some approaches describe network behavior after training through a simple change in kernel scale from initialization, resulting in a generalization power comparable to a Gaussian process. Conversely, in other approaches training results in the adaptation of the kernel to the data, involving complex directional changes to the kernel. While these approaches capture different facets of network behavior, their relationship and respective strengths across scaling regimes remains an open question. This work presents a theoretical framework of multi-scale adaptive feature learning bridging these approaches. Using methods from statistical mechanics, we derive analytical expressions for network output statistics which are valid across scaling regimes and in the continuum between them. A systematic expansion of the network’s probability distribution reveals that mean-field scaling requires only a saddle-point approximation, while standard scaling necessitates additional correction terms. Remarkably, we find across regimes that kernel adaptation can be reduced to an effective kernel rescaling when predicting the mean network output of a linear network. However, even in this case, the multi-scale adaptive approach captures directional feature learning effects, providing richer insights than what could be recovered from a rescaling of the kernel alone. © 2025, CC BY-NC-SA.

关键词： Contrastive Learning

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Optimization-Based String Method for Finding Minimum Energy Path

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Communications in computational Physics 2013年第7期14卷 265-275页

作者： Amit Samanta Weinan E Program in Applied and Computational Mathematics Princeton UniversityPrincetonNew JerseyUSA Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNew JerseyUSA Beijing International Center for Mathematical Research Peking UniversityBeijingChina

We present an efficient algorithm for calculating the minimum energy path(MEP)and energy barriers between local minima on a multidimensional potential energy surface(PES).Such paths play a central role in the understanding of transition pathways between metastable *** method relies on the original formulation of the string method[***.B,66,052301(2002)],*** evolve a smooth curve along a direction normal to the *** algorithm works by performing minimization steps on hyperplanes normal to the *** the problem of finding MEP on the PES is remodeled as a set of constrained minimization *** provides the flexibility of using minimization algorithms faster than the steepest descent method used in the simplified string method[***.,126(16),164103(2007)].At the same time,it provides a more direct analog of the finite temperature string *** applicability of the algorithm is demonstrated using various examples.

关键词： Rare events string method minimum energy path dislocation nucleation

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Effective Maxwell Equations from Time-dependent Density Functional Theory

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Acta Mathematica Sinica,English Series 2011年第2期27卷 339-368页

作者： Weinan E Jianfeng LU Xu YANG Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 USA Beijing International Center for Mathematical Research and School of Mathematical Sciences Peking University Beijing 100871 P. R. China Department of Mathematics Courant Institute of Mathematical Sciences New York University New York NY 10012 USA

The behavior of interacting electrons in a perfect crystal under macroscopic external electric and magnetic fields is studied. Effective Maxwell equations for the macroscopic electric and magnetic fields are derived starting from time-dependent density functional theory. Effective permittivity and permeability coefficients are obtained.

关键词： Time-dependent density functional theory Maxwell equations effective permittivity electromagnetic fields

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Semi-Eulerian and High Order Gaussian Beam Methods for the Schrödinger Equation in the Semiclassical Regime

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Communications in computational Physics 2011年第3期9卷 668-687页

作者： Shi Jin Hao Wu Xu Yang Department of Mathematics University of WisconsinMadisonWI 53706USA Department of Mathematical Sciences Tsinghua UniversityBeijing 10084China Program in Applied and Computational Mathematics Princeton UniversityNJ 08544USA

A novel Eulerian Gaussian beam method was developed in[8]to compute the Schrödinger equation efficiently in the semiclassical *** this paper,we introduce an efficient semi-Eulerian implementation of this *** new algorithm inherits the essence of the Eulerian Gaussian beam method where the Hessian is computed through the derivatives of the complexified level set functions instead of solving the dynamic ray tracing *** difference lies in that,we solve the ray tracing equations to determine the centers of the beams and then compute quantities of interests only around these *** yields effectively a local level set implementation,and the beam summation can be carried out on the initial physical space instead of the phase *** a consequence,it reduces the computational cost and also avoids the delicate issue of beam summation around the caustics in the Eulerian Gaussian beam ***,the semi-Eulerian Gaussian beam method can be easily generalized to higher order Gaussian beam methods,which is the topic of the second part of this *** numerical examples are provided to verify the accuracy and efficiency of both the first order and higher order semi-Eulerian methods.

关键词： Schrödinger equation semi-Eulerian Gaussian beam method high order methods

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Halton-type sequences from global function fields

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Science China mathematics 2013年第7期56卷 1467-1476页

作者： NIED ERREITER Harald YEO Anderson SiangJing Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Department of Mathematics University of Salzburg School of Physical and Mathematical Sciences Nanyang Technological University

For any prime power q and any dimension s≥1, a new construction of (t, s)-sequences in base q using global function fields is presented. The construction yields an analog of Halton sequences for global function fields. It is the first general construction of (t, s)-sequences that is not directly based on the digital method. The construction can also be put into the framework of the theory of (u, e, s)-sequences that was recently introduced by Tezuka and leads in this way to better discrepancy bounds for the constructed sequences.

关键词： low-discrepancy sequence (t s)-sequence Halton sequence global function field

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Mean field limit of a dynamical model of polymer systems

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Science China mathematics 2013年第12期56卷 2591-2598页

作者： E Weinan SHEN Hao School of Mathematical Sciences and BICMR Peking University Department of Mathematics and PACM Princeton UniversityPrinceton NJ 08544 USA Program in Applied and Computational Mathematics Princeton UniversityPrinceton NJ 08544 USA

This paper provides a mathematically rigorous foundation for self-consistent mean field theory of the polymeric physics. We study a new model for dynamics of mono-polymer systems. Every polymer is regarded as a string of points which are moving randomly as Brownian motions and under elastic forces. Every two points on the same string or on two different strings also interact under a pairwise potential V. The dynamics of the system is described by a system of N coupled stochastic partial differential equations （SPDEs）. We show that the mean field limit as N -＋ c~ of the system is a self-consistent McKean-Vlasov type equation, under suitable assumptions on the initial and boundary conditions and regularity of V. We also prove that both the SPDE system of the polymers and the mean field limit equation are well-posed.

关键词： polymers mean field limit stochastic PDEs McKean-Vlasov equation

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A NUMERICAL STUDY OF THE GAUSSIAN BEAM METHODS FOR SCHR(O|¨)DINGER-POISSON EQUATIONS

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Journal of computational mathematics 2010年第2期28卷 261-272页

作者： Shi Jin Hao Wu Xu Yang Department of Mathematics University of Wisconsin Madison W1 53706 USA Department of Mathematical Sciences Tsinghua University Beijing 100084 China Department of Mathematics University of Wisconsin Madison WI 53706 USA Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544USA

As an important model in quantum semiconductor devices, the SchrSdinger-Poisson equations have generated widespread interests in both analysis and numerical simulations in recent years. In this paper, we present Gaussian beam methods for the numerical simulation of the one-dimensional Schrodinger-Poisson equations. The Gaussian beam methods for high frequency waves outperform the geometrical optics method in that the former are accurate even around caustics. The purposes of the paper are first to develop the Gaussian beam methods, based on our previous methods for the linear SchrSdinger equation, for the Schrodinger-Poisson equations, and then check their validity for this weakly-nonlinear system.

关键词： Schrodinger-Poisson equations Gaussian beam methods Vlasov-Poisson equations

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