论文标题
复杂平面中的非标准绿色能源问题
Non-standard Green energy problems in the complex plane
论文作者
论文摘要
我们考虑了复杂平面中的几个非标准离散和连续的绿色能量问题，并研究了其解决方案之间的渐近关系。在离散的环境中，我们考虑两个问题。一个具有可变粒子位置（在给定的紧凑型集合中）和可变粒子质量，另一个具有可变质量但规定的位置。允许粒子的质量在$ 0 \ 0 \ leq m \ leq r $的范围内取得任何值，其中$ r> 0 $是问题中的固定参数。相应的连续能量问题是在正值$ $ $ $ $ \ | | \ | \ leq r $的正值$μ$的空间上定义的，并在给定的紧凑型集合上受支持，并带有附加的上限，这是处方位置条件的结果。事实证明，平衡常数和平衡度量量持续变化，因为参数$ r $（弱星拓扑中的后者）的函数。在无约束的能量问题中，我们提出了一种贪婪的算法，该算法会收敛到平衡常数和平衡度量。在离散的能量问题中，表明在某些条件下，根据所考虑的问题的类型，粒子质量的最佳值由粒子的最佳位置或规定位置唯一决定。
We consider several non-standard discrete and continuous Green energy problems in the complex plane and study the asymptotic relations between their solutions. In the discrete setting, we consider two problems; one with variable particle positions (within a given compact set) and variable particle masses, the other one with variable masses but prescribed positions. The mass of a particle is allowed to take any value in the range $0\leq m\leq R$, where $R>0$ is a fixed parameter in the problem. The corresponding continuous energy problems are defined on the space of positive measures $μ$ with mass $\|μ\|\leq R$ and supported on the given compact set, with an additional upper constraint that appears as a consequence of the prescribed positions condition. It is proved that the equilibrium constant and equilibrium measure vary continuously as functions of the parameter $R$ (the latter in the weak-star topology). In the unconstrained energy problem we present a greedy algorithm that converges to the equilibrium constant and equilibrium measure. In the discrete energy problems, it is shown that under certain conditions, the optimal values of the particle masses are uniquely determined by the optimal positions or prescribed positions of the particles, depending on the type of problem considered.
