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DNA cyclic codes over the ring F2[u, v]/ （u^2 - 1, v^3 - v, uv - vu）

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International Journal of Biomathematics 2018年第3期11卷 247-265页

作者： Hai Q. Dinh Abhay Kumar Singh Sukhamoy Pattanayak Songsak Sriboonchitta Division of Computational Mathematics and Engineering Institute for Computational Science Ton Duc Thang University Ho Chi Minh City Vietnam Faculty of Mathematics and Statistics Ton Duc Thang University Ho Chi Minh City Vietnam Department of Mathematical Sciences Kent State University 4314 Mahoning Avenue Warren Ohio 44483 USA Department of Applied Mathematics Indian Institute of Technology (ISM) Dhanbad 826004 India Faculty of Economics Chiang Mai University Chiang Mai 52000 Thailand

In this paper, our main objective is to find out the necessary and sufficient conditions for a cyclic code of arbitrary length over the ring of four elements R1 = F2 ＋ u2 （u^2 = 1） to be a reversible cyclic code. We also obtain the structure of cyclic DNA codes of odd length over the ring R = F2 [u, v]/（u^2 -1, v^3 -v, uv- vu）, which plays an important role in computational Biology. Furthermore, we establish a direct link between the elements of ring /{ and 64 codons used in the amino acids of living organisms by introducing a Gray map from R to R1. Among others, binary images of cyclic codes over R are also investigated. As applications, some cyclic DNA codes over R using the Gray map are provided.

关键词： Cyclic DNA codes reversible cyclic codes reversible-complement cyclic codes gray map.

SOLVING CALDEIRA-LEGGETT MODEL BY INCHWORM METHOD WITH FROZEN GAUSSIAN APPROXIMATION

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arXiv 2024年

作者： Wang, Geshuo Yang, Siyao Cai, Zhenning Department of Mathematics National University of Singapore Singapore119076 Singapore Committee on Computational and Applied Mathematics Department of Statistics University of Chicago ChicagoIL60637 United States

We propose an algorithm that combines the inchworm method and the frozen Gaussian approximation to simulate the Caldeira-Leggett model in which a quantum particle is coupled with thermal harmonic baths. In particular, we are interested in the real-time dynamics of the reduced density operator. In our algorithm, we use frozen Gaussian approximation to approximate the wave function as a wave packet in integral form. The desired reduced density operator is then written as a Dyson series, which is the series expression of path integrals in quantum mechanics of interacting systems. To compute the Dyson series, we further approximate each term in the series using Gaussian wave packets, and then employ the idea of the inchworm method to accelerate the convergence of the series. The inchworm method formulates the series as an integro-differential equation of "full propagators", and rewrites the infinite series on the right-hand side using these full propagators, so that the number of terms in the sum can be significantly reduced, and faster convergence can be achieved. The performance of our algorithm is verified numerically by various *** Codes 45J02, 81V99 Copyright © 2024, The Authors. All rights reserved.

关键词： Wave functions

A Continuous Finite Element Method with Homotopy VanishingViscosity for Solving the Static Eikonal Equation

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Communications in computational Physics 2022年第5期31卷 1402-1433页

作者： Yong Yang Wenrui Hao Yong-Tao Zhang College of Science Nanjing University of Aeronautics and AstronauticsNanjing 210016China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjing 211106China Department of Mathematics Pennsylvania State UniversityUniversity ParkPennsylvaniaUSA Department of Applied and Computational Mathematics and Statistics University of Notre DameNotre DameIN 46556USA

We develop a second-order continuousfinite element method for solving the static Eikonal *** is based on the vanishing viscosity approach with a homotopy method for solving the discretized nonlinear *** specifically,the homotopy method is utilized to decrease the viscosity coefficient gradually,while Newton’s method is applied to compute the solution for each viscosity ***’s method alone converges for just big enough viscosity coefficients on very coarse grids and for simple 1D examples,but the proposed method is much more robust and guarantees the convergence of the nonlinear solver for all viscosity coefficients and for all examples over all *** experiments from 1D to 3D are presented to confirm the second-order convergence and the effectiveness of the proposed method on both structured or unstructured meshes.

关键词： Eikonal equation finite element method homotopy method

ReLU Neural Networks with Linear Layers are Biased Towards Single- and Multi-Index Models

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arXiv 2023年

作者： Parkinson, Suzanna Ongie, Greg Willett, Rebecca Committee on Computational and Applied Mathematics University of Chicago ChicagoIL United States Department of Mathematical and Statistical Sciences Marquette University MilwaukeeWI United States Department of Statistics Department of Computer Science and Committee on Computational and Applied Mathematics University of Chicago ChicagoIL United States

Neural networks often operate in the overparameterized regime, in which there are far more parameters than training samples, allowing the training data to be fit perfectly. That is, training the network effectively learns an interpolating function, and properties of the interpolant affect predictions the network will make on new samples. This manuscript explores how properties of such functions learned by neural networks of depth greater than two layers. Our framework considers a family of networks of varying depths that all have the same capacity but different representation costs. The representation cost of a function induced by a neural network architecture is the minimum sum of squared weights needed for the network to represent the function;it reflects the function space bias associated with the architecture. Our results show that adding additional linear layers to the input side of a shallow ReLU network yields a representation cost favoring functions with low mixed variation – that is, it has limited variation in directions orthogonal to a low-dimensional subspace and can be well approximated by a single- or multi-index model. Such functions may be represented by the composition of a function with low two-layer representation cost and a low-rank linear operator. Our experiments confirm this behavior in standard network training regimes. They additionally show that linear layers can improve generalization and the learned network is well-aligned with the true latent low-dimensional linear subspace when data is generated using a multi-index model. Copyright © 2023, The Authors. All rights reserved.

关键词： Interpolation

Solving Caldeira-Leggett Model by Inchworm Method with Frozen Gaussian Approximation

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SSRN

SSRN 2024年

We propose an algorithm that combines the inchworm method and the frozen Gaussian approximation to simulate the Caldeira-Leggett model in which a quantum particle is coupled with thermal harmonic baths. In particular, we are interested in the real-time dynamics of the reduced density operator. In our algorithm, we use frozen Gaussian approximation to approximate the wave function as a wave packet in integral form. The desired reduced density operator is then written as a Dyson series, which is the series expression of path integrals in quantum mechanics of interacting systems. To compute the Dyson series, we further approximate each term in the series using Gaussian wave packets, and then employ the idea of the inchworm method to accelerate the convergence of the series. The inchworm method formulates the series as an integro-differential equation of ``full propagators'', and rewrites the infinite series on the right-hand side using these full propagators, so that the number of terms in the sum can be significantly reduced, and faster convergence can be achieved. The performance of our algorithm is verified numerically by various experiments. © 2024, The Authors. All rights reserved.

关键词： Wave functions

Optimal Control of PDEs under Uncertainty 1

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丛书名： SpringerBriefs in mathematics

1000年

作者： Jesús Martínez-Frutos Francisco Periago Esparza

Optimal bayesian minimax rates for unconstrained large covariance matrices

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arXiv 2017年

作者： Lee, Kyoungjae Lee, Jaeyong Department of Applied and Computational Mathematics and Statistics University of Notre Dame Department of Statistics Seoul National University

We obtain the optimal Bayesian minimax rate for the unconstrained large covariance matrix of multivariate normal sample with mean zero, when both the sample size, n, and the dimension, p, of the covariance matrix tend to infinity. Traditionally the posterior convergence rate is used to compare the frequentist asymptotic performance of priors, but defining the optimality with it is elusive. We propose a new decision theoretic framework for prior selection and define Bayesian minimax rate. Under the proposed framework, we obtain the optimal Bayesian minimax rate for the spectral norm for all rates of p. We also considered Frobenius norm, Bregman divergence and squared log-determinant loss and obtain the optimal Bayesian minimax rate under certain rate conditions on p. A simulation study is conducted to support the theoretical results. Copyright © 2017, The Authors. All rights reserved.

关键词： Covariance matrix

Machine-learning-based non-Newtonian fluid model with molecular fidelity

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Physical Review E 2020年第4期102卷 043309-043309页

作者： Huan Lei Lei Wu Weinan E Department of Computational Mathematics Science & Engineering and Department of Statistics & Probability Michigan State University East Lansing Michigan 48824 USA Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

We introduce a machine-learning-based framework for constructing continuum a non-Newtonian fluid dynamics model directly from a microscale description. Dumbbell polymer solutions are used as examples to demonstrate the essential ideas. To faithfully retain molecular fidelity, we establish a micro-macro correspondence via a set of encoders for the microscale polymer configurations and their macroscale counterparts, a set of nonlinear conformation tensors. The dynamics of these conformation tensors can be derived from the microscale model, and the relevant terms can be parametrized using machine learning. The final model, named the deep non-Newtonian model (DeePN2), takes the form of conventional non-Newtonian fluid dynamics models, with a generalized form of the objective tensor derivative that retains the microscale interpretations. Both the formulation of the dynamic equation and the neural network representation rigorously preserve the rotational invariance, which ensures the admissibility of the constructed model. Numerical results demonstrate the accuracy of DeePN2 where models based on empirical closures show limitations.

关键词： Learning Neuronal networks Complex fluids Non-Newtonian fluids Rheology Viscoelasticity Multiscale modeling

Estimating large precision matrices via modified cholesky decomposition

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arXiv 2017年

作者： Lee, Kyoungjae Lee, Jaeyong Department of Applied and Computational Mathematics and Statistics University of Notre Dame Department of Statistics Seoul National University

We introduce the k-banded Cholesky prior for estimating a high-dimensional bandable precision matrix via the modified Cholesky decomposition. The bandable assumption is imposed on the Cholesky factor of the decomposition. We obtained the P-loss convergence rate under the spectral norm and the matrix ∞norm and the minimax lower bounds. Since the P-loss convergence rate (Lee and Lee (2017)) is stronger than the posterior convergence rate, the rates obtained are also posterior convergence rates. Furthermore, when the true precision matrix is a k0-banded matrix with some finite k0, the obtained P-loss convergence rates coincide with the minimax rates. The established convergence rates are slightly slower than the minimax lower bounds, but these are the fastest rates for bandable precision matrices among the existing Bayesian approaches. A simulation study is conducted to compare the performance to the other competitive estimators in various scenarios. Copyright © 2017, The Authors. All rights reserved.

关键词： Bayesian networks