Let P be an inner point of a convex N-gon ΓN ： A1A2… ANA1（N ≥ 3）, and let di,k denote the distance from the point Ai＋k to the line PAi（i = 1,2,…,N, Ai = Aj〈=〉 i ≡ j（modN））, which is called the k-Brocard...Let P be an inner point of a convex N-gon ΓN ： A1A2… ANA1（N ≥ 3）, and let di,k denote the distance from the point Ai＋k to the line PAi（i = 1,2,…,N, Ai = Aj〈=〉 i ≡ j（modN））, which is called the k-Brocard distance for P of ΓN. We have proved the following double-inequality： If P ∈ ΓN, k = N↑∩i=1∠Ai-kAiAi＋k（1 ≤ k 〈 N/2,i =1,2,…,N）, and r ≤ lnN-ln（N-1）/ln2＋2[lnN-ln（N-1）], then （1/N N↑∑↑i=1di^r, k）^1/r≤1/N coskπ/N N↑∑↑i=1｜AiAi＋k｜≤sin2kπ/2sinπ/N（1/N N↑∑↑i=1｜AiAi＋1｜^2.展开更多
In this paper, we establish a geometric inequality with a parameter and involving two n dimensional simplexes and an interior point, some applications.
This paper presents the lower bounds and upper bounds for the mean value and variance of distance distribution of non-linear codes in GF(q). By presenting several examples, it is shown that these bounds could be achie...This paper presents the lower bounds and upper bounds for the mean value and variance of distance distribution of non-linear codes in GF(q). By presenting several examples, it is shown that these bounds could be achieved.展开更多
Let C be a plane convex body. For arbitrary points , a,b ∈E ndenote by │ab│?the Euclidean length of the line-segment ab. Let a1b1 be a longest chord of C parallel to the line-segment ab. The relative distance dc(a,...Let C be a plane convex body. For arbitrary points , a,b ∈E ndenote by │ab│?the Euclidean length of the line-segment ab. Let a1b1 be a longest chord of C parallel to the line-segment ab. The relative distance dc(a,b) between the points a and b is the ratio of the Euclidean distance between a?and b?to the half of the Euclidean distance between a1 and b1. In this note we prove the triangle inequality in E2 with the relative metric dc( .,.), and apply this inequality to show that 6≤l(P)≤8, where l(P) is the perimeter of the convex polygon P measured in the metric dp( .,.). In addition, we prove that every convex hexagon has two pairs of consecutive vertices with relative distances at least 1.展开更多
文摘Let P be an inner point of a convex N-gon ΓN ： A1A2… ANA1（N ≥ 3）, and let di,k denote the distance from the point Ai＋k to the line PAi（i = 1,2,…,N, Ai = Aj〈=〉 i ≡ j（modN））, which is called the k-Brocard distance for P of ΓN. We have proved the following double-inequality： If P ∈ ΓN, k = N↑∩i=1∠Ai-kAiAi＋k（1 ≤ k 〈 N/2,i =1,2,…,N）, and r ≤ lnN-ln（N-1）/ln2＋2[lnN-ln（N-1）], then （1/N N↑∑↑i=1di^r, k）^1/r≤1/N coskπ/N N↑∑↑i=1｜AiAi＋k｜≤sin2kπ/2sinπ/N（1/N N↑∑↑i=1｜AiAi＋1｜^2.
文摘In this paper, we establish a geometric inequality with a parameter and involving two n dimensional simplexes and an interior point, some applications.
文摘This paper presents the lower bounds and upper bounds for the mean value and variance of distance distribution of non-linear codes in GF(q). By presenting several examples, it is shown that these bounds could be achieved.
文摘Let C be a plane convex body. For arbitrary points , a,b ∈E ndenote by │ab│?the Euclidean length of the line-segment ab. Let a1b1 be a longest chord of C parallel to the line-segment ab. The relative distance dc(a,b) between the points a and b is the ratio of the Euclidean distance between a?and b?to the half of the Euclidean distance between a1 and b1. In this note we prove the triangle inequality in E2 with the relative metric dc( .,.), and apply this inequality to show that 6≤l(P)≤8, where l(P) is the perimeter of the convex polygon P measured in the metric dp( .,.). In addition, we prove that every convex hexagon has two pairs of consecutive vertices with relative distances at least 1.
文摘研究加速K-medoids聚类算法,首先以PAM(partitioning around medoids)、TPAM(triangular inequality elimination criteria PAM)算法为基础给出两个加速引理,并基于中心点之间距离不等式提出两个新加速定理.同时,以O(n+K^2)额外内存空间开销辅助引理、定理的结合而提出加速SPAM(speed up PAM)聚类算法,使得K-medoids聚类算法复杂度由O(K(n-K)~2)降低至O((n-K)~2).在实际及人工模拟数据集上的实验结果表明:相对于PAM,TPAM,FKMEDOIDS(fast K-medoids)等参考算法均有改进,运行时间比PAM至少提升0.828倍.
