We study the solvability problem for the multidimensional Riccati equation −∇$u$=|∇$u$|^{q}+ω, where$q$>1 and ω is an arbitrary nonnegative function (or measure). We also discuss connections with the classical problem of the existence of positive solutions for the Schrödinger equation −Δ$u$−ω$u$=0 with nonnegative potential ω. We establish explicit criteria for the existence of global solutions on$R$^{n}in terms involving geometric (capacity) estimates or pointwise behavior of Riesz potentials, together with sharp pointwise estimates of solutions and their gradients. We also consider the corresponding nonlinear Dirichlet problem on a bounded domain, as well as more general equations of the type$−Lu=f(x, u, ∇u)$+ω where, and$L$is a uniformly elliptic operator.

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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].

In the description of isolated gravitating systems in General Relativity a spacelike timeslice has the structure of a complete Riemannian three-manifold with an asymptotically at end. Let (N; g) denote an end of such a complete Riemannian three-manifold, ie N is diffeomorphic to R3\B1 (0), and the metric on N asymptotically approaches the Euclidean metric near infinity: gij=

We introduce a new equivalence relation for complete alge- braic varieties with canonical singularities, generated by birational equivalence, by flat algebraic deformations (of varieties with canon- ical singularities), and by quasi-´etale morphisms, i.e., morphisms which are unramified in codimension 1. We denote the above equivalence by A.Q.E.D. : = Algebraic-Quasi-´ Etale- Deformation. A completely similar equivalence relation, denoted by C-Q.E.D., can be considered for compact complex spaces with canonical sin- gularities. By a recent theorem of Siu, dimension and Kodaira dimension are invariants for A.Q.E.D. of complex varieties. We address the interesting question whether conversely two al- gebraic varieties of the same dimension and with the same Ko- daira dimension are Q.E.D. - equivalent (A.Q.E.D., or at least C-Q.E.D.), the answer being positive for curves by well known results. Using Enriques’ (resp. Kodaira’s) classification we show first that the answer to the C-Q.E.D. question is positive for special algebraic surfaces (those with Kodaira dimension at most 1), resp. for compact complex surfaces with Kodaira dimension 0, 1 and even first Betti number. The appendix by S¨onke Rollenske shows that the hypothesis of even first Betti number is necessary: he proves that any sur- face which is C-Q.E.D.-equivalent to a Kodaira surface is itself a Kodaira surface. We show also that the answer to the A.Q.E.D. question is pos- itive for complex algebraic surfaces of Kodaira dimension ≤ 1. The answer to the Q.E.D. question is instead negative for sur- faces of general type: the other appendix, due to Fritz Grunewald, is devoted to showing that the (rigid) Kuga-Shavel type surfaces of general type obtained as quotients of the bidisk via discrete groups constructed from quaternion algebras belong to countably many distinct Q.E.D. equivalence classes.