Real-time crime forecasting is important. However, accurate prediction of when and where the next crime will happen is difficult. No known physical model provides a reasonable approximation to such a complex system. Historical crime data are sparse in both space and time and the signal of interests is weak. In this work, we first present a proper representation of crime data. We then adapt the spatial temporal residual network on the well represented data to predict the distribution of crime in Los Angeles at the scale of hours in neighborhood-sized parcels. These experiments as well as comparisons with several existing approaches to prediction demonstrate the superiority of the proposed model in terms of accuracy. Finally, we present a ternarization technique to address the resource consumption issue for its deployment in real world. This work is an extension of our short conference proceeding paper [Wang et al, Arxiv 1707.03340].

We investigate the Chern–Ricci flow, an evolution equation of Hermitian metrics generalizing the Kähler–Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a Kähler–Einstein metric from the base. Some of our estimates are new even for the Kähler–Ricci flow. A consequence of our result is that, on every minimal non-Kähler surface of Kodaira dimension one, the Chern–Ricci flow converges in the sense of Gromov–Hausdorff to an orbifold Kähler–Einstein metric on a Riemann surface.

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Inspired by works of Castéras (Pac J Math 276:321–345, 2015), Li and Zhu (Calc Var Partial Differ Equ 58:1–18, 2019), Sun and Zhu (Calc Var Partial Differ Equ 60:1–26, 2021), we propose a heat flow for the mean field equation on a connected finite graph G=(V,E). Namely ∂_tϕ(u)=Δu−Q+ρ\frac{e^u}{∫_V e^u dμ} u(⋅,0)=u_0, where Δ is the standard graph Laplacian, ρ is a real number, Q:V→R is a function satisfying ∫_V Qdμ=ρ, and ϕ:R→R is one of certain smooth functions including ϕ(s)=es. We prove that for any initial data u_0 and any ρ∈R, there exists a unique solution u:V×[0,+∞)→R of the above heat flow; moreover, u(x, t) converges to some function u_∞:V→R uniformly in x∈V as t→+∞, and u_∞ is a solution of the mean field equation Δu_∞−Q+ρ\frac{e^{u_∞}}{∫_V e^{u_∞}dμ}=0. Though G is a finite graph, this result is still unexpected, even in the special case Q≡0. Our approach reads as follows: the short time existence of the heat flow follows from the ODE theory; various integral estimates give its long time existence; moreover we establish a Lojasiewicz–Simon type inequality and use it to conclude the convergence of the heat flow.