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1. Fractional diffusion limit of a kinetic equation with diffusive boundary conditions in a bounded interval

   https://doi.org/10.3233/asy-221755  

   Cesbron, L.; Mellet, A.; Puel, M. (November 2022, Asymptotic Analysis)

   We investigate the fractional diffusion approximation of a kinetic equation set in a bounded interval with diffusive reflection conditions at the boundary. In an appropriate singular limit corresponding to small Knudsen number and long time asymptotic, we show that the asymptotic density function is the unique solution of a fractional diffusion equation with Neumann boundary condition. This analysis completes a previous work by the same authors in which a limiting fractional diffusion equation was identified on the half-space, but the uniqueness of the solution (which is necessary to prove the convergence of the whole sequence) could not be established. 

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2. Determining damping terms in fractional wave equations

   Kaltenbacker, Barbara; Rundell, William (January 2022, Inverse problems)

   This paper deals with the inverse problem of recovering an arbitrary number of fractional damping terms in a wave equation. We develop several approaches on uniqueness and reconstruction, some of them relying on Tauberian theorems that provide relations between the asymptotic behaviour of solutions in time and Laplace domains. The possibility of additionally recovering space-dependent coefficients or initial data is discussed. The resulting methods for reconstructing coefficients and fractional orders in these terms are tested numerically. In addition, we provide an analysis of the forward problem consisting of a multiterm fractional wave equation. 

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3. Getting the Lay of the Land in Discrete Space: A Survey of Metric Dimension and Its Applications

   https://doi.org/10.1137/21M1409512  

   Tillquist, Richard C.; Frongillo, Rafael M.; Lladser, Manuel E. (November 2023, SIAM Review)

   The metric dimension of a graph is the smallest number of nodes required to identify allother nodes uniquely based on shortest path distances. Applications of metric dimensioninclude discovering the source of a spread in a network, canonically labeling graphs, andembedding symbolic data in low-dimensional Euclidean spaces. This survey gives a self-contained introduction to metric dimension and an overview of the quintessential resultsand applications. We discuss methods for approximating the metric dimension of generalgraphs, and specific bounds and asymptotic behavior for deterministic and random familiesof graphs. We conclude with related concepts and directions for future work. 

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4. On the cover time of dense graphs

   Frieze, Alan; Pegden, W.; Tkocz, T. (January 2019, SIAM journal on discrete mathematics)

   We consider arbitrary graphs $$G$$ with $$n$$ vertices and minimum degree at least $$\delta n$$ where $$\delta>0$$ is constant.\\ (a) If the conductance of $$G$$ is sufficiently large then we obtain an asymptotic expression for the cover time $$C_G$$ of $$G$$ as the solution to an explicit transcendental equation.\\ (b) If the conductance is not large enough to apply (a), but the mixing time of a random walk on $$G$$ is of a lesser magnitude than the cover time, then we can obtain an asymptotic deterministic estimate via a decomposition into a bounded number of dense subgraphs with high conductance. \\ (c) If $$G$$ fits neither (a) nor (b) then we give a deterministic asymptotic (2+o(1))-approximation of $$C_G$$. 

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5. Localization Game for Random Graphs

   Dudek, A; English, S.; Frieze, A.; MacRury, C.; Pralat, P. (January 2022, Discrete applied mathematics)

   We consider the localization game played on graphs in which a cop tries to determine the exact location of an invisible robber by exploiting distance probes. The corresponding graph parameter $$\zeta(G)$$ for a given graph $$G$$ is called the localization number. In this paper, we improve the bounds for dense random graphs determining an asymptotic behaviour of $$\zeta(G)$$. Moreover, we extend the argument to sparse graphs. 

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