Numerical study of ergodicity for the overdamped Generalized Langevin Equation with fractional noise

The Generalized Langevin Equation, in history, arises as a natural fix for the rather traditional Langevin equation when the random force is no longer memoryless. It has been proved that with fractional Gaussian noise (fGn) mostly considered by biologists, the overdamped Generalized Langevin equation satisfying fluctuation-dissipation theorem can be written as a fractional stochastic differential equation (FSDE). While the ergodicity is clear for linear forces, it remains less transparent for nonlinear forces. In this work, we present both a direct and a fast algorithm respectively to this FSDE model. The strong orders of convergence are proved for both schemes, where the role of the memory effects can be clearly observed. We verify the convergence theorems using linear forces, and then present the ergodicity study of the double well potentials in both 1D and 2D setups

A Survey on Monochromatic Connections of Graphs

The concept of monochromatic connection of graphs was introduced by Caro and Yuster in 2011. Recently, a lot of results have been published about it. In this survey, we attempt to bring together all the results that dealt with it. We begin with an introduction, and then classify the results into the following categories: monochromatic connection coloring of edge-version, monochromatic connection coloring of vertex-version, monochromatic index, monochromatic connection coloring of total-version.Comment: 26 pages, 3 figure

Massive Quiescent Cores in Orion: VI. The Internal Structures and a Candidate of Transiting Core in NGC 2024 Filament

We present a multi-wavelength observational study of the NGC 2024 filament using infrared to sub-millimeter continuum and the NH$_3$ $(1,1)$ and $(2,2)$ inversion transitions centered on FIR-3, the most massive core therein. FIR-3 is found to have no significant infrared point sources in the Spitzer/IRAC bands. But the NH$_3$ kinetic temperature map shows a peak value at the core center with $T_{\rm k}=25$ K which is significantly higher than the surrounding level ($T_{\rm k}=15-19$ K). Such internal heating signature without an infrared source suggests an ongoing core collapse possibly at a transition stage from first hydrostatic core (FHSC) to protostar. The eight dense cores in the filament have dust temperatures between 17.5 and 22 K. They are much cooler than the hot ridge ($T_{\rm d}=55$ K) around the central heating star IRS-2b. Comparison with a dust heating model suggests that the filament should have a distance of $3-5$ pc from IRS-2b. This value is much larger than the spatial extent of the hot ridge, suggesting that the filament is spatially separated from the hot region along the line of sight.Comment: 20 pages, 7 figures, 6 tables. Accepted to Ap