power law
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Sneha Dey ◽  
A. Ghorai ◽  

Exploration of dynamics of raindrops is one of the simple yet most complicated mechanical problems. Mass accretion from moist air during the motion of raindrop through resistive medium holds an arbitrary power law equation. Its integral part is the change of shape, terminal motions and terminal solutions, etc. Classical Newtonian formalism is used to formulate a mathematical model of generalized first order differential equation. We have discussed about the terminal velocity of raindrop and its variation with the extensive use of python program and library. It is found that terminal velocity 𝐯𝐓𝐜𝛂𝛃 is achieved within 20 seconds where 𝛂=, 𝛃=(𝟎,𝟏) and 𝐧=𝟎,𝟏,𝟐,𝟑,𝟒,…. Its variations due to mass accretion roughly follows the earlier predicted range 𝐠/𝟕 to 𝐠/𝟑.

2022 ◽  
Vol 16 (2) ◽  
pp. 1-23
Yiding Zhang ◽  
Xiao Wang ◽  
Nian Liu ◽  
Chuan Shi

Heterogeneous information network (HIN) embedding, aiming to project HIN into a low-dimensional space, has attracted considerable research attention. Most of the existing HIN embedding methods focus on preserving the inherent network structure and semantic correlations in Euclidean spaces. However, one fundamental problem is whether the Euclidean spaces are the intrinsic spaces of HIN? Recent researches find the complex network with hyperbolic geometry can naturally reflect some properties, e.g., hierarchical and power-law structure. In this article, we make an effort toward embedding HIN in hyperbolic spaces. We analyze the structures of three HINs and discover some properties, e.g., the power-law distribution, also exist in HINs. Therefore, we propose a novel HIN embedding model HHNE. Specifically, to capture the structure and semantic relations between nodes, HHNE employs the meta-path guided random walk to sample the sequences for each node. Then HHNE exploits the hyperbolic distance as the proximity measurement. We also derive an effective optimization strategy to update the hyperbolic embeddings iteratively. Since HHNE optimizes different relations in a single space, we further propose the extended model HHNE++. HHNE++ models different relations in different spaces, which enables it to learn complex interactions in HINs. The optimization strategy of HHNE++ is also derived to update the parameters of HHNE++ in a principle manner. The experimental results demonstrate the effectiveness of our proposed models.

2022 ◽  
Vol 40 (2) ◽  
pp. 1-24
Minghao Zhao ◽  
Qilin Deng ◽  
Kai Wang ◽  
Runze Wu ◽  
Jianrong Tao ◽  

In recent years, advances in Graph Convolutional Networks (GCNs) have given new insights into the development of social recommendation. However, many existing GCN-based social recommendation methods often directly apply GCN to capture user-item and user-user interactions, which probably have two main limitations: (a) Due to the power-law property of the degree distribution, the vanilla GCN with static normalized adjacency matrix has limitations in learning node representations, especially for the long-tail nodes; (b) multi-typed social relationships between users that are ubiquitous in the real world are rarely considered. In this article, we propose a novel Bilateral Filtering Heterogeneous Attention Network (BFHAN), which improves long-tail node representations and leverages multi-typed social relationships between user nodes. First, we propose a novel graph convolutional filter for the user-item bipartite network and extend it to the user-user homogeneous network. Further, we theoretically analyze the correlation between the convergence values of different graph convolutional filters and node degrees after stacking multiple layers. Second, we model multi-relational social interactions between users as the multiplex network and further propose a multiplex attention network to capture distinctive inter-layer influences for user representations. Last but not least, the experimental results demonstrate that our proposed method outperforms several state-of-the-art GCN-based methods for social recommendation tasks.

2022 ◽  
Vol 6 (1) ◽  
pp. 16
Bhavya Pardasani ◽  
Andrew Wetzel ◽  
Jenna Samuel

Abstract In order to investigate the role of the host halo in quenching satellite galaxies, we have characterized a single Milky Way-like host galaxy from the FIRE simulations from z = 0–1.76 by quantifying the gas density of the host halo environment with respect to distance from the host and galactocentric latitude. The gas density decreases with increasing distance from the host according to a broken power law. At earlier times (2–10 Gyr ago), the density in the inner regions of the host halo was enhanced relative to z = 0. Thus, earlier infalling satellites experienced more ram-pressure and were more efficiently quenched compared to later infalling satellites. We also find that in the inner halo (<150 kpc) the density is 2–3 times larger close to the plane of the host galaxy disk versus above or below the disk, so satellites that orbit at low galactocentric latitudes may be more efficiently quenched.

Cells ◽  
2022 ◽  
Vol 11 (2) ◽  
pp. 270
Konstantin Polev ◽  
Diana V. Kolygina ◽  
Kristiana Kandere-Grzybowska ◽  
Bartosz A. Grzybowski

Lysosomes—that is, acidic organelles known for degradation/recycling—move through the cytoplasm alternating between bursts of active transport and short, diffusive motions or even pauses. While their mobility is essential for lysosomes’ fusogenic and non-fusogenic interactions with target organelles, their movements have not been characterized in adequate detail. Here, large-scale statistical analysis of lysosomal movement trajectories reveals that lysosome trajectories in all examined cell types—both cancer and noncancerous ones—are superdiffusive and characterized by heavy-tailed distributions of run and flight lengths. Consideration of Akaike weights for various potential models (lognormal, power law, truncated power law, stretched exponential, and exponential) indicates that the experimental data are best described by the lognormal distribution, which, in turn, can be related to one of the space-search strategies particularly effective when “thorough” search needs to balance search for rare target(s) (organelles). In addition, automated, wavelet-based analysis allows for co-tracking the motions of lysosomes and the cargos they carry—particularly the nanoparticle aggregates known to cause selective lysosome disruption in cancerous cells. The methods we describe here could help study nanoparticle assemblies, viruses, and other objects transported inside various vesicle types, as well as coordinated movements of organelles/particles in the cytoplasm. Custom-written code that includes integrated workflow for our analyses is made available for academic use.

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