For the physical vacuum free boundary problem with the sound speed being C^1/2-Holder continuous near vacuum boundaries of the one-dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the porous media equation with the same total mass when the initial datum is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in the supremum norm, and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher-order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher-order nonlinear energy estimates, and elliptic estimates.

We re-examine the constraints imposed by causality and unitarity on the low-energy effective field theory expansion of four-particle scattering amplitudes, exposing a hidden “totally positive” structure strikingly similar to the positive geometries associated with grassmannians and amplituhedra. This forces the infinite tower of higher-dimension operators to lie inside a new geometry we call the “EFT-hedron”. We initiate a systematic investigation of the boundary structure of the EFT-hedron, giving infinitely many linear and non-linear inequalities that must be satisfied by the EFT expansion in any theory. We illustrate the EFT-hedron geometry and constraints in a wide variety of examples, including new consistency conditions on the scattering amplitudes of photons and gravitons in the real world.

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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.