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1. Combined effects of homogenization and singular perturbations: Quantitative estimates

   https://doi.org/10.3233/ASY-211709  

   Niu, Weisheng; Shen, Zhongwei (May 2021, Asymptotic Analysis)

   null (Ed.)

   We investigate quantitative estimates in periodic homogenization of second-order elliptic systems of elasticity with singular fourth-order perturbations. The convergence rates, which depend on the scale κ that represents the strength of the singular perturbation and on the length scale ε of the heterogeneities, are established. We also obtain the large-scale Lipschitz estimate, down to the scale ε and independent of κ. This large-scale estimate, when combined with small-scale estimates, yields the classical Lipschitz estimate that is uniform in both ε and κ. 

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2. Performance Analysis of Ice-Relative Upward-Looking Doppler Navigation of Underwater Vehicles Beneath Moving Sea Ice

   https://doi.org/10.3390/jmse9020174  

   Barker, Laughlin D.; Whitcomb, Louis L. (February 2021, Journal of Marine Science and Engineering)

   This paper addresses the problem of ice-relative underwater robotic vehicle navigation relative to moving or stationary contiguous sea ice. A review of previously-reported under-ice navigation methods is given, as well as motivation for the use of under-ice robotic vehicles with precision navigation capabilities. We then describe our proposed approach, which employs two or more satellite navigation beacons atop the sea ice along with other precision vehicle and ship mounted navigation sensors to estimate vehicle, ice, and ship states by means of an Extended Kalman Filter. A performances sensitivity analysis for a simulated 7.7 km under ice survey is reported. The number and the location of ice deployed satellite beacons, rotational and translational ice velocity, and separation of ship-based acoustic range sensors are varied, and their effects on estimate error and uncertainty are examined. Results suggest that increasing the number and/or separation of ice-deployed satellite beacons reduces estimate uncertainty, whereas increasing separation of ship-based acoustic range sensors has little impact on estimate uncertainty. Decreasing ice velocity is also correlated with reduced estimate uncertainty. Our analysis suggests that the proposed method is feasible and can offer scientifically useful navigation accuracy over a range of operating conditions. 

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3. Analytical Method to Approximate the Impact of Turning on the Macroscopic Fundamental Diagram

   https://doi.org/10.1177/0361198120933274  

   Xu, Guanhao; Yu, Zhengyao; Gayah, Vikash V. (September 2020, Transportation Research Record: Journal of the Transportation Research Board)

   null (Ed.)

   Network macroscopic fundamental diagrams (MFDs) have recently been shown to exist in real-world urban traffic networks. The existence of an MFD facilitates the modeling of urban traffic network dynamics at a regional level, which can be used to identify and refine large-scale network-wide control strategies. To be useful, MFD-based modeling frameworks require an estimate of the functional form of a network’s MFD. Analytical methods have been proposed to estimate a network’s MFD by abstracting the network as a single ring-road or corridor and modeling the flow–density relationship on that simplified element. However, these existing methods cannot account for the impact of turning traffic, as only a single corridor is considered. This paper proposes a method to estimate a network’s MFD when vehicles are allowed to turn into or out of a corridor. A two-ring abstraction is first used to analyze how turning will affect vehicle travel in a more general network, and then the model is further approximated using a single ring-road or corridor. This approximation is useful as it facilitates the application of existing variational theory-based methods (the stochastic method of cuts) to estimate the flow–density relationship on the corridor, while accounting for the stochastic nature of turning. Results of the approximation compared with a more realistic simulation that includes features that cannot be captured using variational theory—such as internal origins and destinations—suggest that this approximation works to estimate a network’s MFD when turning traffic is present. 

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4. Polynomial control on stability, inversion and powers of matrices on simple graphs

   https://doi.org/https://doi.org/10.1016/j.jfa.2018.09.014  

   Shin, Chang Eon; Sun, Qiyu (January 2019, Journal of functional analysis)

   Spatially distributed networks of large size arise in a variety of science and engineering problems, such as wireless sensor networks and smart power grids. Most of their features can be described by properties of their state-space matrices whose entries have indices in the vertex set of a graph. In this paper, we introduce novel algebras of Beurling type that contain matrices on a connected simple graph having polynomial off-diagonal decay, and we show that they are Banach subalgebras of the space of all bounded operators on the space of all p-summable sequences. The $$\ell^p$$-stability of state-space matrices is an essential hypothesis for the robustness of spatially distributed networks. In this paper, we establish the equivalence among -stabilities of matrices in Beurling algebras for different exponents $$p$$, with quantitative analysis for the lower stability bounds. Admission of norm-control inversion plays a crucial role in some engineering practice. In this paper, we prove that matrices in Beurling subalgebras of have norm-controlled inversion and we find a norm-controlled polynomial with close to optimal degree. Polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks. In this paper, we apply our results on norm-controlled inversion to obtain a polynomial estimate to powers of matrices in Beurling algebras. The polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary Markov chain on a spatially distributed network. 

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5. Integral Ricci curvature and the mass gap of Dirichlet Laplacians on domains

   https://doi.org/10.1002/mana.202100523  

   Ramos Olivé, Xavier; Rose, Christian; Wang, Lili; Wei, Guofang (June 2023, Mathematische Nachrichten)

   Abstract We obtain a fundamental gap estimate for classes of bounded domains with quantitative control on the boundary in a complete manifold with integral bounds on the negative part of the Ricci curvature. This extends the result of Oden, Sung, and Wang [Trans. Amer. Math. Soc. 351 (1999), no. 9, 3533–3548] to ‐Ricci curvature assumptions, . To achieve our result, it is shown that the domains under consideration are John domains, what enables us to obtain an estimate on the first nonzero Neumann eigenvalue, which is of independent interest. 

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