For future Internet-of-Things based Big Data applications, data collection from ubiquitous smart sensors with limited spectrum bandwidth is very challenging. On the other hand, to interpret the ...meaning behind the collected data, it is also challenging for an edge fusion center running computing tasks over large data sets with a limited computation capacity. To tackle these challenges, by exploiting the superposition property of multiple-access channel and the functional decomposition, the recently proposed technique, over-the-air computation (AirComp), enables an effective joint data collection and computation from concurrent sensor transmissions. In this paper, we focus on a single-antenna AirComp system consisting of K sensors and one receiver. We consider an optimization problem to minimize the computation mean-squared error (MSE) of the K sensors' signals at the receiver by optimizing the transmitting-receiving (Tx-Rx) policy, under the peak power constraint of each sensor. Although the problem is not convex, we derive the computation-optimal policy in closed form. Also, we comprehensively investigate the ergodic performance of the AirComp system, and the scaling laws of the average computation MSE (ACM) and the average power consumption (APC) of different Tx-Rx policies with respect to K . For the computation-optimal policy, we show that the policy has a vanishing ACM and a vanishing APC with the increasing K .
•The oscillatory behavior of drops in an ambient liquid under an impact is numerically and experimentally studied using the Basilisk C and a self-designed falling drop tower facility.•The effects of ...dimensionless parameters on the oscillatory behavior of drops were systematically studied.•A scaling law between the maximum deformation of the drops during the oscillatory behavior and the dimensionless parameters is established, which can be well interpreted by a mass-spring-damp mode.
In this work, the oscillatory behavior of drops in an ambient liquid under an impact is numerically and experimentally studied using the Basilisk C and a self-designed falling drop tower facility, respectively. The numerical results of the oscillatory drops are consistent with the experimental data. A shape parameter e is proposed to describe the oscillatory behavior of the drops. Using the energy budgets of the drops, the influence of dimensionless parameters, i.e., Reynolds number Re, Ohnesorge number Oh, Bond number Bo, and viscosity ratio η on the oscillatory amplitude and period of the shape parameter e is systematically investigated. Furthermore, a scaling law between the maximum deformation of the drops during the oscillatory behavior and the dimensionless parameters is established, which can be interpreted by a mass-spring-damp model. It is found that the established model agrees well with our numerical and experimental results without any fitting parameters when the viscosity of the ambient liquid is near or smaller than that of the drop, and the maximum deformation has a clear proportionality as Oh2Re(1+Re).
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In this article, we address a new model for the scaling law heat conduction
problem by using the scaling law vector calculus associated with the Korcak
scaling law. The scaling law heat conduction ...equations are discussed in
detail. The scaling law vector calculus formulas are proposed as an
efficiently mathematical tool to describe the Korcak scaling -law phenomena
in heat transport system.
In this article, we propose a new model for the scaling law heat conduction
equation associated with the Richardson scaling law. To find the analytical
solution for it, we present a scaling law ...series associated with the
Kohlrausch-Williams-Watts function analogous to the Fourier series. The
proposed technology is efficient to handle the Richardson scaling law
problems in mathematical physics.
In this article the non-Darcy law for the scaling law flow in porous medium
associated with the scaling law calculus with respect to the Mandelbrots
scaling law is suggested for the first time.
In this study, we propose the general calculus operators based on the
Richardson scaling law and Korcak scaling law. The Richardson-scaling-law
calculus is considered to investigate the Fourier-like ...law for the
scaling-law flow of the heat in the heat-transfer process. The
Korcak-scaling-law calculus is used to model the Darcy-like law for
describing the scaling-law flow of the fluid in porous medium. The formulas
are as the special cases of the topology calculus proposed for descriptions
of the fractal scaling-law behaviors in nature phenomena.
This paper addresses a non-traditional approach for the scaling-law
fluid-flows described by fractal scaling-law vector calculus associated with
the Mandelbrot scaling law. Their quantum equations ...were proposed to control
the fluid-flows associated with the Mandelbrot scaling law. This gives a
new insight into the descriptions for the scaling-law behaviors of the
fluid-flows in the Mandelbrot scaling-law phenomena.
The scaling law for slow earthquakes, which is a linear relationship between seismic moment and duration, was proposed 15 y ago and initiated a debate on the difference in physical processes ...governing slow vs. fast (ordinary) earthquakes. Based on new observations across a wide period range, we show that linear scaling of slow earthquakes remains valid, but as a well-defined upper bound on moment rate of ~10
Nm/s. The large gap in moment-rate between the scaling of slow and fast earthquakes remains unfilled. Slow earthquakes occur near the detectability threshold, such that we are unable to detect deformation events with lower moment rates. Observed trends within slow earthquake categories support the idea that this unobservable field is populated with events of lower moment rate. This suggests a change in perspective - that the proposed scaling should be considered as a bound, or speed limit, on slow earthquakes. We propose that slow earthquakes represent diffusional propagation, and that the bound on moment rate reflects an upper limit on the speed of those diffusional processes. Ordinary earthquakes, in contrast, occur as a coupled process between seismic wave propagation and fracture. Thus, even though both phenomena occur as shear slip, the difference of scaling reflects a difference in the physical process governing propagation.
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