The authors consider the simplest quantum mechanics model of solids, the tight binding model, and prove that in the continuum limit, the energy of tight binding model converges to that of the continuum elasticity mode...
详细信息
The authors consider the simplest quantum mechanics model of solids, the tight binding model, and prove that in the continuum limit, the energy of tight binding model converges to that of the continuum elasticity model obtained using Cauchy-Born rule. The technique in this paper is based mainly on spectral perturbation theory for large matrices.
A central problem in the mathematical analysis of fluid dynamics is the asymptotic limit of the fluid flow as viscosity goes to *** is particularly important when boundaries are present since vorticitv is typically ge...
详细信息
A central problem in the mathematical analysis of fluid dynamics is the asymptotic limit of the fluid flow as viscosity goes to *** is particularly important when boundaries are present since vorticitv is typically generated at the boundary as a result of boundary layer *** boundary laver theory,developed by Prandtl about a hundred years ago,has become a standard tool in addressing these *** at the mathematical level,there is still a lack of fundamental understanding of these questions and the validity of the boundary layer *** this article,we review recent progresses on the analysis of Prandtl’s equation and the related issue of the zero-viscosity limit for the solutions of the Navier-Stokes *** also discuss some directions where progress is expected in the near future.
We consider the nearest neighbor Ising model on the 2D square lattice and divide the lattice into 2 by 2 blocks. Each block is assigned one spin value (1 or -1) and these block spin values are kept fixed. We then impo...
详细信息
We consider the nearest neighbor Ising model on the 2D square lattice and divide the lattice into 2 by 2 blocks. Each block is assigned one spin value (1 or -1) and these block spin values are kept fixed. We then impose the majority rule and look at the effect on the phase transition that was present in the original unconstrained spin system. We find that for the checkerboard block-spin configuration, Monte Carlo simulations show that beta(c) is close to 1, which, compared to the original nearest neighbor Ising beta(c) = 0.44..., shows that the critical temperature has been reduced by more than one half For none of the other 11 block-spin configurations that ive have considered is there any indication of a phase transition in the constrained system of original spins.
Fractional-order stochastic gradient descent (FOSGD) leverages a fractional exponent to capture long-memory effects in optimization, yet its practical impact is often constrained by the difficulty of tuning and stabil...
详细信息
The rapidly growing field of single-cell transcriptomic sequencing (scRNAseq) presents challenges for data analysis due to its massive datasets. A common method in manifold learning consists in hypothesizing that data...
详细信息
We search for rotational, four-dimensional maps of standard type (x(n+1)-2x(n)+x(n-1) = epsilonf(x, epsilon)) possessing one or two polynomial integrals. There are no nontrivial maps corresponding to cubic oscillators...
We search for rotational, four-dimensional maps of standard type (x(n+1)-2x(n)+x(n-1) = epsilonf(x, epsilon)) possessing one or two polynomial integrals. There are no nontrivial maps corresponding to cubic oscillators, but we find a four-parameter family of such maps corresponding to quartic oscillators. This seems to be the only such example.
In this paper, we propose HiPoNet, an end-to-end differentiable neural network for regression, classification, and representation learning on high-dimensional point clouds. Single-cell data can have high dimensionalit...
详细信息
For an area preserving map, each chaotic orbit appears numerically to densely cover a region (an irregular component) of nonzero area. Surprisingly, the measure approximated by a long segment of such an orbit deviates...
For an area preserving map, each chaotic orbit appears numerically to densely cover a region (an irregular component) of nonzero area. Surprisingly, the measure approximated by a long segment of such an orbit deviates significantly from a constant on the irregular component. Most prominently, there are spikes in the density near the boundaries of the irregular component resulting from the stickiness of its bounding invariant circles. We show that this phenomena is transient, and therefore numerical ergodicity on the irregular component eventually obtains, though the times involved are extremely long - 10(10) iterates. A Markov model of the transport shows that the density spikes cannot be explained by the stickiness of a bounding circle of a single class - for example, a rotational circle. However, the density spikes do occur in a Markov tree model that includes the effects of islands-around-islands,
Guided by the example of gauge transformations associated with classical Yang-Mills fields, a very general class of transformations is considered. The explicit representation of these transformations involves not only...
详细信息
Guided by the example of gauge transformations associated with classical Yang-Mills fields, a very general class of transformations is considered. The explicit representation of these transformations involves not only the independent and the dependent field variables, but also a set of position-dependent parameters together with their first derivatives. The stipulation that an action integral associated with the field variables be invariant under such transformations gives rise to a set of three conditions involving the Lagrangian and its derivatives, together with derivatives of the functions that define the transformations. These invariance identities constitute an extension of the classical theorem of Noether to general transformations of this kind. An application to the case of gauge fields demonstrates the existence of two distinct types of conservation laws for such fields.
Suppose we have two chemical reactions occurring simultaneously. Then theamount y of a reactant changes due to both processes and behaves as a function of time t as y(t) =x_1e~(α_1t) + x_2e~(α_2t), where x_1, x_2, ...
详细信息
Suppose we have two chemical reactions occurring simultaneously. Then theamount y of a reactant changes due to both processes and behaves as a function of time t as y(t) =x_1e~(α_1t) + x_2e~(α_2t), where x_1, x_2, α_1, and α_2 are fixed parameters. Typically, weobserve the function y(t) for m fixed t values, perhaps t = 0, Δt, 2Δt, ..., t_(final).
暂无评论