论文标题
Lagrangian计划在$ n- $ dimensional Euclidean Space中检测加权的Fermat-Steiner-Frechet Multitree Multitree
A Lagrangian program detecting the Weighted Fermat-Steiner-Frechet multitree for a Frechet $N-$multisimplex in the $N-$dimensional Euclidean Space
论文作者
论文摘要
In this paper, we introduce the Fermat-Steiner-Frechet problem for a given $\frac{N(N+1)}{2}-$tuple of positive real numbers determining the edge lengths of an $N-$simplex in $\mathbb{R}^{N},$ in order to study its solution called the "Fermat-Steiner-Frechet multitree," which consist of a union of Fermat-Steiner Trees的所有派生的成对不一致的$ n- $单纯性在Blumenthal的意义上，Herzog，Herzog售价为$ n = 3 $，Dekster-Wilker for $ n \ ge 3. $我们获得了一种确定fermat-Steiner frechet在$ \ nathbb {r} n} $ consity of the Toryhere the Theorect of Whor the Toryher the ragr的方法。 $ n-1 $反向加权费马特问题的独立解决方案，用于$ \ mathbb {r}^{n}中的$ n- $ sissyx。 Fermat-Steiner在$ \ frac {[\ frac {n（n+1）} {2}]！} {（n+1）！} $ incruent $ n- $ nissys由$ \ frac {n（n+1）} {2} {2} {2} {2} - $ tuple of Contenter Natural数字的deker-decter-natural数字及以上的情况，并由deker-natural decter-decter-decter-dekersed and-dekerneber-theber-充满了自然的基本进化过程（最小通信网络，最小质量trasfer，最大不一致的单纯量）。此外，我们获得了反重加权费马特问题的唯一解决方案，指的是$（n+1）$权重的唯一集合，这与$ \ mathbb {r}^{n}中的$ n- $ boundare Simplex的顶点相对应。
In this paper, we introduce the Fermat-Steiner-Frechet problem for a given $\frac{N(N+1)}{2}-$tuple of positive real numbers determining the edge lengths of an $N-$simplex in $\mathbb{R}^{N},$ in order to study its solution called the "Fermat-Steiner-Frechet multitree," which consist of a union of Fermat-Steiner trees for all derived pairwise incongruent $N-$simplexes in the sense of Blumenthal, Herzog for $N=3$ and Dekster-Wilker for $N\ge 3.$ We obtain a method to determine the Fermat-Steiner Frechet multitree in $\mathbb{R}^{N}$ based on the theory of Lagrange multipliers, whose equality constraints depend on $N-1$ independent solutions of the inverse weighted Fermat problem for an $N-$simplex in $\mathbb{R}^{N}.$ A fundamental application of the Lagrangian program for the Fermat-Steiner Frechet problem in $\mathbb{R}^{N}$ is the detection of the Fermat-Steiner tree with global minimum length having $N-1$ equally weighted Fermat-Steiner points among $\frac{[\frac{N(N+1)}{2}]!}{(N+1)!}$ incongruent $N-$simplexes determined by an $\frac{N(N+1)}{2}-$tuple of consecutive natural numbers controlled by Dekster-Wilker, Blumenthal-Herzog conditions and enriched with the fundamental evolutionary processes of Nature (Minimum communication networks, minimum mass trasfer, maximum volume of incongruent simplexes). Furthermore, we obtain the unique solution of the inverse weighted Fermat problem, referring to the unique set of $(N+1)$ weights, which correspond to the vertices of an $N-$ boundary simplex in $\mathbb{R}^{N}.$
