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For the Rayleigh–Taylor unstable arrangement of a viscous fluid sphere embedded in a finite viscous fluid spherical shell with a rigid boundary and a radially directed acceleration, a dispersion relation is developed from a linear stability analysis using the method of normal modes. aR1 is the radially directed acceleration at the interface. ρi denotes the density, μi is the viscosity, and Ri is the radius, where i = 1 is the inner sphere and i = 2 is the outer sphere. The dispersion relation is a function of the following dimensionless variables: viscosity ratio s=μ1μ2, density ratio d=ρ1ρ2, spherical harmonic mode n, B=R1aR1ρ22μ221/3, H=R2R1, and the dimensionless growth rate α=σμ2aR12ρ21/3, where σ is the exponential growth rate. We show that the boundedness provided by the outer spherical shell has a strong influence on the instability behavior, which is reflected not only in the modulation of the growth rate but also in the selection of the most unstable modes that are physically possible. This outer boundary effect is quantified by the relative magnitude of the radius ratio H. We find that when H is close to unity, lower order harmonics are excluded from becoming the most unstable within a vast region of the parameter space. In other words, the effect of H has precedence over the other controlling parameters d, B, and a wide range of s in establishing what the lowest most unstable mode can be. When H ∼ 1, low order harmonics can become the most unstable only for s ≫ 1. However, in the limit when s → ∞, we show that the most unstable mode is n = 1 and derive the dispersion relation in this limit. The exclusion of most unstable low order harmonics caused by a finite outer boundary is not realized when the outer boundary extends beyond a certain threshold length-scale in which case all modes are equally possible depending on the value of B. 

Modeling of power distribution system components that are valid for a wide range of frequencies are crucial for highly accurate modeling of electromagnetic transient (EMT) events. This has recently become of interest due to the improvements needed for the resilient operation of distribution systems. Vector fitting (VF) is a very popular and commonly used algorithm for wide band representations of power system components in EMT simulations. In this research, we present a new multi-input rational approximation algorithm (MIAAA) and illustrate its advantages with respect to VF using examples of approximations of admittance matrices discussed in the literature. We show that MIAAA not only outperforms VF in terms of achieving better accuracy using lesser number of poles, but also has no numerical issues achieving convergence. In contrast to VF, MIAAA is not sensitive to the location of input sample points and it does not require good estimates for the location of the desired approximation poles. The novelty of this research work is the use of recent mathematical results to solve existing challenges in distribution system modeling and to develop rational approximations for power system models that intend to be optimal in terms of accuracy and performance. 

Scanned images of patent or historical documents often contain localized zigzag noise introduced by the digitizing process; yet when viewed as a whole image, global structures are apparent to humans, but not to machines. Existing denoising methods work well for natural images, but not for binary diagram images, which makes feature extraction difficult for computer vision and machine learning methods and algorithms. We propose a topological graph-based representation to tackle this denoising problem. The graph representation emphasizes the shapes and topology of diagram images, making it ideal for use in machine learning applications such as classification and matching of scientific diagram images. Our approach and algorithms provide essential structure and lay important foundation for computer vision such as scene graph-based applications, because topological relations and spatial arrangement among objects in images are captured and stored in our skeleton graph. In addition, while the parameters for almost all pixel-based methods are not adaptive, our method is robust in that it only requires one parameter and it is adaptive. Experimental comparisons with existing methods show the effectiveness of our approach.