This paper mainly focuses on the optimized methods of the sprinkling irrigation for greenery patches, by maximally equalizing the amount of water sprayed on a certain area. Various models are being discussed, where the main mathematical tool is analytic geometry, employed to research the possible effects of different proposals. Firstly, the simplest models are built based on a totally ideal situation. Assuming that sprinkling spouts are spinning over plain lawn with a set of specified radii, install them in arrangements of simple geometric figures. Areas of overlapping and blank parts are being calculated and the most reasonable arrangement of all that are studied is selected. Secondly, real factors are taken into consideration separately as follows: 1. The disequilibrium of the water that drops in a line from the sprinkling center is transformed into a functional expression, whose graphs are drawn to show the water distributed over the area; 2. The plane models are changed into solid ones on the assumption that the sprinkling spouts are placed on slopes. Analytic geometry methods are employed to describe the range of sprayed water on the oblique surface. Through calculation and analysis, models can be adjusted to specific situations. Finally, the boundary problems and landscape effects are involved.

The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less than 2 plays an important role in many problems in low-dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli space of convex hyperbolic and Euclidean polyhedra can be reduced to the study of deformations of cone-manifolds by doubling a polyhedron across its faces. This deformation theory has been understood by Hodgson and Kerckhoff [7] when the singular set has no vertices, and by Weiß [32] when the cone angles are less than . We prove here an infinitesimal rigidity result valid for cone angles less than 2, stating that infinitesimal deformations which leave the dihedral angles fixed are trivial in the hyperbolic case, and reduce to some simple deformations in the Euclidean case. The method is to treat this as a problem concerning the deformation theory of singular Einstein metrics, and to apply analytic methods about elliptic operators on stratified spaces. This work is an important ingredient in the local deformation theory of cone-manifolds by the second author [22]; see also the concurrent work by Weiß [33].

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This paper is a contribution to the general problem of giving an explicit description of the basic locus in the reduction modulo $p$ of Shimura varieties. Motivated by \cite{Vollaard-Wedhorn} and \cite{Rapoport-Terstiege-Wilson}, we classify the cases where the basic locus is (in a natural way) the union of classical Deligne-Lusztig sets associated to Coxeter elements. We show that if this is satisfied, then the Newton strata and Ekedahl-Oort strata have many nice properties.

In this paper, we consider the harmonic extension problem, which is widely used in many applications of machine learning. We formulate the harmonic extension as solving a Laplace-Beltrami equation with Dirichlet boundary condition. We use the point integral method (PIM) to solve the Laplace-Beltrami equation. The basic idea of the PIM method is to approximate the Laplace equation using an integral equation, which is easy to be discretized from points. Based on the integral equation, we found that traditional graph Laplacian method (GLM) may fail to approximate the harmonic functions in the classical sense. For the Laplace-Beltrami equation with Dirichlet boundary, we can prove the convergence of the point integral method. The point integral method is also very easy to implement, which only requires a minor modification of the graph Laplacian. One important application of the harmonic extension in machine learning is semi-supervised learning. We run a popular semi-supervised learning algorithm by Zhu et al. over a couple of well-known datasets and compare the performance of the aforementioned approaches. Our experiments show the PIM performs the best. We also apply PIM to an image recovery problem and show it outperforms GLM. Finally, on a model problem of Laplace-Beltrami equation with Dirichlet boundary, we prove the convergence of the point integral method.