Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent α>0, that characterizes the power-law scaling behavior of the structure factor S(k) as a function of wave number k≡|k| in the ...
详细信息
Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent α>0, that characterizes the power-law scaling behavior of the structure factor S(k) as a function of wave number k≡|k| in the vicinity of the origin, e.g., S(k)∼|k|α in cases where S(k) varies continuously with k as k→0. In this paper, we show that the spreadability is an effective method for determining α for quasiperiodic systems where S(k) is discontinuous and consists of a dense set of Bragg peaks. It has been shown in [Phys. Rev. E 104, 054102 (2021)] that, for media with finite α, the long-time behavior of the excess spreadability S(∞)−S(t) can be fit to a power law of the form t−(d−α)/2, where d is the space dimension, to accurately extract α for the continuous case. We first transform quasiperiodic and limit-periodic point patterns into two-phase media by mapping them onto packings of identical nonoverlapping disks, where space interior to the disks represents one phase and the space in exterior to them represents the second phase. We then compute the spectral density χ̃V(k) of the packings, and finally compute and fit the long-time behavior of their excess spreadabilities. Specifically, we show that the excess spreadability can be used to accurately extract α for the one-dimensional (1D) limit-periodic period-doubling chain (α=1) and the 1D quasicrystalline Fibonacci chain (α=3) to within 0.02% of the analytically known exact results. Moreover, we obtain a value of α=5.97±0.06 for the two-dimensional Penrose tiling and present plausible theoretical arguments strongly suggesting that α is exactly equal to six. We also show that, due to the self-similarity of the structures examined here, one can truncate the small-k region of the scattering information used to compute the spreadability and obtain an accurate value of α, with a small deviation from the untruncated case that decreases as the system size increases. This strongly suggests that one can obtain a good estimate of
We analyze the steady viscoelastic fluid flow in slowly varying contracting channels of arbitrary shape and present a theory based on the lubrication approximation for calculating the flow rate–pressure drop relation...
详细信息
We present a continuous formulation of machine learning,as a problem in the calculus of variations and differential-integral equations,in the spirit of classical numerical *** demonstrate that conventional machine lea...
详细信息
We present a continuous formulation of machine learning,as a problem in the calculus of variations and differential-integral equations,in the spirit of classical numerical *** demonstrate that conventional machine learning models and algorithms,such as the random feature model,the two-layer neural network model and the residual neural network model,can all be recovered(in a scaled form)as particular discretizations of different continuous *** also present examples of new models,such as the flow-based random feature model,and new algorithms,such as the smoothed particle method and spectral method,that arise naturally from this continuous *** discuss how the issues of generalization error and implicit regularization can be studied under this framework.
We propose a novel quadratic programming formulation for estimating the corruption levels in group synchronization, and use these estimates to solve this problem. Our objective function exploits the cycle consistency ...
详细信息
MSC Codes 90C26, 90C10, 90C17, 68Q87, 65C20Previous partial permutation synchronization (PPS) algorithms, which are commonly used for multi-object matching, often involve computation-intensive and memory-demanding mat...
详细信息
The weak-field limit of Einstein–Cartan (EC) relativity is studied. The equations of EC theory are rewritten such that they formally resemble those of Einstein General Relativity (EGR);this allows ideas from post-New...
详细信息
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...
详细信息
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.
During the last few years, neural machine translation (NMT) as a notable branch of machine translation has been increasing its popularity both in research and in practice. In particular, neural machine translation bet...
详细信息
We study how the learning rate affects the performance of a relaxed randomized Kaczmarz algorithm for solving Ax ≈ b + Ε, where Ax = b is a consistent linear system and Ε has independent mean zero random entries. W...
详细信息
The maximally random jammed (MRJ) state is the most random (i.e., disordered) configuration of strictly jammed (mechanically rigid) nonoverlapping objects. MRJ packings are hyperuniform, meaning their long-wavelength ...
详细信息
The maximally random jammed (MRJ) state is the most random (i.e., disordered) configuration of strictly jammed (mechanically rigid) nonoverlapping objects. MRJ packings are hyperuniform, meaning their long-wavelength density fluctuations are anomalously suppressed compared to typical disordered systems, i.e., their structure factors S(k) tend to zero as the wave number |k| tends to zero. Here we show that generating high-quality strictly jammed states for Euclidean space dimensions d=3,4, and 5 is of paramount importance in ensuring hyperuniformity and extracting precise values of the hyperuniformity exponent α>0 for MRJ states, defined by the power-law behavior of S(k)∼|k|α in the limit |k|→0. Moreover, we show that for fixed d it is more difficult to ensure jamming as the particle number N increases, which results in packings that are nonhyperuniform. Free-volume theory arguments suggest that the ideal MRJ state does not contain rattlers, which act as defects in numerically generated packings. As d increases, we find that the fraction of rattlers decreases substantially. Our analysis of the largest truly jammed packings suggests that the ideal MRJ packings for all dimensions d≥3 are hyperuniform with α=d−2, implying the packings become more hyperuniform as d increases. The differences in α between MRJ packings and the recently proposed Manna-class random close packed (RCP) states, which were reported to have α=0.25 in d=3 and be nonhyperuniform (α=0) for d=4 and d=5, demonstrate the vivid distinctions between the large-scale structure of RCP and MRJ states in these dimensions. Our paper clarifies the importance of the link between true jamming and hyperuniformity and motivates the development of an algorithm to produce rattler-free three-dimensional MRJ packings.
暂无评论