For a linear Volterra equation of scalar type in a Banach space, sufficient conditions are given for the operator norms of three associated resolvent kernels to be integrable with respect to a weight on the positive h...
In this paper, a multiscale boundary condition for the discrete unified gas kinetic scheme (DUGKS) is developed for gas flows in all flow regimes. Based on the discrete Maxwell boundary condition (DMBC), this study ad...
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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 Gauss...
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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.
A deep understanding of the mechanisms underlying many-body quantum chaos is one of the big challenges in contemporary theoretical physics. We tackle this problem in the context of a set of perturbed quadratic Sachdev...
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A deep understanding of the mechanisms underlying many-body quantum chaos is one of the big challenges in contemporary theoretical physics. We tackle this problem in the context of a set of perturbed quadratic Sachdev-Ye-Kitaev (SYK) Hamiltonians defined on graphs. This allows us to disentangle the geometrical properties of the underlying single-particle problem and the importance of the interaction terms, showing that the former is the dominant feature ensuring the single-particle to many-body chaotic transition. Our results are verified numerically with state-of-the-art numerical techniques, capable of extracting eigenvalues in a desired energy window of very large Hamiltonians. Our approach essentially provides a new way of viewing many-body chaos from a single-particle perspective.
A time-independent field theoretical framework for turbulence is suggested, based upon a variational principle for a stationary solution of the Fokker-Planck equation. We obtain a functional equation for the effective...
A time-independent field theoretical framework for turbulence is suggested, based upon a variational principle for a stationary solution of the Fokker-Planck equation. We obtain a functional equation for the effective Action of this spatial field theory and investigate its general properties and some numerical solutions. The equation is completely universal, and allows for the scale invariant solutions in the inertial range. The critical indices are not fixed at the kinematical level, but rather should be found from certain eigenvalue conditions, as in the field theory of critical phenomena. Unlike the Wyld field theory, there are no divergences in our Feynman integrals, due to some magic cancellations. The simplest possible Gaussian approximation yields crude but still reasonable results (there are deviations from Kolmogorov scaling in 3 dimensions, but at 2.7544 dimensions it would be exact). Our approach allows us to study some new problems, such as spontaneous parity breaking in 3d turbulence. It turns out that with the appropriate helicity term added to the velocity correlation function, logarithmic infrared divergences arise in our field theory which effectively eliminates these terms. In order to build a quantitative theory of turbulence, one should consider more sophisticated Ansatz for the effective Action, which would require serious numerical work.
A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are *** initialization schemes as well as...
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A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are *** initialization schemes as well as general regimes for the network width and training data size are *** the overparametrized regime,it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the *** addition,it is proved that throughout the training process the functions represented by the neural network model are uniformly close to those of a kernel *** general values of the network width and training data size,sharp estimates of the generalization error are established for target functions in the appropriate reproducing kernel Hilbert space.
In this paper we present a method to solve algebraic Riccati equations by employing a projection method based on Proper Orthogonal Decomposition. The method only requires simulations of linear systems to compute the s...
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This paper reviews Fréchet sensitivity analysis for partial differential equations with variations in distributed parameters. The Fréchet derivative provides a linear map between parametric variations and th...
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In this paper,based on some prior estimates,we show that the essential spectrum λ=0 is a bifurcation point for a superlinear elliptic equation with only local conditions,which generalizes a series of earlier results ...
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In this paper,based on some prior estimates,we show that the essential spectrum λ=0 is a bifurcation point for a superlinear elliptic equation with only local conditions,which generalizes a series of earlier results on an open problem proposed by Stuart(1983).
Rationale and Objectives: Brachial plexopathies (BPs) encompass a complex spectrum of nerve injuries affecting motor and sensory function in the upper extremities. Diagnosis is challenging due to the intricate anatomy...
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