If Goldbach’s conjecture is true, then for each prime number p there is at least one pair of primes symmetric with respect to p and whose sum is 2p. In the multiplicative number theory, covering the positive integers...If Goldbach’s conjecture is true, then for each prime number p there is at least one pair of primes symmetric with respect to p and whose sum is 2p. In the multiplicative number theory, covering the positive integers with primes, during the prime factorization, may be viewed as being the outcome of a parallel system which functions properly if and only if Euler’s formula of the product of the reciprocals of the primes is true. An exact formula for the number of primes less than or equal to an arbitrary bound is given. This formula may be implemented using Wolfram’s computer package Mathematica.展开更多
The weighted Gini-Simpson quadratic index is the simplest measure of biodiversity which takes into account the relative abundance of species and some weights assigned to the species. These weights could be assigned ba...The weighted Gini-Simpson quadratic index is the simplest measure of biodiversity which takes into account the relative abundance of species and some weights assigned to the species. These weights could be assigned based on factors such as the phylogenetic distance between species, or their relative conservation values, or even the species richness or vulnerability of the habitats where these species live. In the vast majority of cases where the biodiversity is measured the species are supposed to be independent, which means that the relative proportion of a pair of species is the product of the relative proportions of the component species making up the respective pair. In the first section of the paper, the main versions of the weighted Gini-Simpson index of biodiversity for the pairs and triads of independent species are presented. In the second section of the paper, the weighted Gini-Simpson quadratic index is calculated for the general case when the species are interdependent. In this instance, the weights reflect the conservation values of the species and the distribution pattern variability of the subsets of species in the respective habitat induced by the inter-dependence between species. The third section contains a numerical example.展开更多
The distribution of biodiversity at multiple sites of a region has been traditionally investigated through the additive partitioning of the regional biodiversity, called γ-diversity, into the average within-site biod...The distribution of biodiversity at multiple sites of a region has been traditionally investigated through the additive partitioning of the regional biodiversity, called γ-diversity, into the average within-site biodiversity or α-diversity, and the biodiversity among sites, or β-diversity. The standard additive partitioning of diversity requires the use of a measure of diversity which is a concave function of the relative abundance of species, like the Shannon entropy or the Gini- Simpson index, for instance. When a phylogenetic distance between species is also taken into account, Rao’s quadratic index has been used as a measure of dissimilarity. Rao’s index, however, is not a concave function of the distribution of relative abundance of either individual species or pairs of species and, consequently, only some nonstandard additive partitionings of diversity have been given using this index. The objective of this paper is to show that the weighted quadratic index of biodiversity, a generalization of the weighted Gini-Simpson index to the pairs of species, is a concave function of the joint distribution of the relative abundance of pairs of species and, therefore, may be used in the standard additive partitioning of diversity instead of Rao’s index. The replication property of this new measure is also discussed.展开更多
The Gini-Simpson quadratic index is a classic measure of diversity, widely used by ecologists. As shown recently, however, this index is not suitable for the measurement of beta diversity when the number of species is...The Gini-Simpson quadratic index is a classic measure of diversity, widely used by ecologists. As shown recently, however, this index is not suitable for the measurement of beta diversity when the number of species is very large. The objective of this paper is to introduce the Rich- Gini-Simpson quadratic index which preserves all the qualities of the classic Gini-Simpson index but behaves very well even when the number of species is very large. The additive partitioning of species diversity using the Rich-Gini- Simpson quadratic index and an application from island biogeography are analyzed.展开更多
Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly...Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes, which implicitly proves Goldbach’s Conjecture for 2n as well.展开更多
The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact...The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes. The present paper contains explicit additional and complementary details of the proof, insisting on the existence and the number of Goldbach’s representations of even positive integers as sums of pairs of primes.展开更多
The paper contains a geometric interpretation of Andrica’s conjecture about the gap between the square roots of the consecutive primes and brings empirical evidence that the random fluctuations of the gap between the...The paper contains a geometric interpretation of Andrica’s conjecture about the gap between the square roots of the consecutive primes and brings empirical evidence that the random fluctuations of the gap between the quare roots of the consecutive primes seem to stabilize around the mean gap.展开更多
文摘If Goldbach’s conjecture is true, then for each prime number p there is at least one pair of primes symmetric with respect to p and whose sum is 2p. In the multiplicative number theory, covering the positive integers with primes, during the prime factorization, may be viewed as being the outcome of a parallel system which functions properly if and only if Euler’s formula of the product of the reciprocals of the primes is true. An exact formula for the number of primes less than or equal to an arbitrary bound is given. This formula may be implemented using Wolfram’s computer package Mathematica.
文摘The weighted Gini-Simpson quadratic index is the simplest measure of biodiversity which takes into account the relative abundance of species and some weights assigned to the species. These weights could be assigned based on factors such as the phylogenetic distance between species, or their relative conservation values, or even the species richness or vulnerability of the habitats where these species live. In the vast majority of cases where the biodiversity is measured the species are supposed to be independent, which means that the relative proportion of a pair of species is the product of the relative proportions of the component species making up the respective pair. In the first section of the paper, the main versions of the weighted Gini-Simpson index of biodiversity for the pairs and triads of independent species are presented. In the second section of the paper, the weighted Gini-Simpson quadratic index is calculated for the general case when the species are interdependent. In this instance, the weights reflect the conservation values of the species and the distribution pattern variability of the subsets of species in the respective habitat induced by the inter-dependence between species. The third section contains a numerical example.
文摘The distribution of biodiversity at multiple sites of a region has been traditionally investigated through the additive partitioning of the regional biodiversity, called γ-diversity, into the average within-site biodiversity or α-diversity, and the biodiversity among sites, or β-diversity. The standard additive partitioning of diversity requires the use of a measure of diversity which is a concave function of the relative abundance of species, like the Shannon entropy or the Gini- Simpson index, for instance. When a phylogenetic distance between species is also taken into account, Rao’s quadratic index has been used as a measure of dissimilarity. Rao’s index, however, is not a concave function of the distribution of relative abundance of either individual species or pairs of species and, consequently, only some nonstandard additive partitionings of diversity have been given using this index. The objective of this paper is to show that the weighted quadratic index of biodiversity, a generalization of the weighted Gini-Simpson index to the pairs of species, is a concave function of the joint distribution of the relative abundance of pairs of species and, therefore, may be used in the standard additive partitioning of diversity instead of Rao’s index. The replication property of this new measure is also discussed.
文摘The Gini-Simpson quadratic index is a classic measure of diversity, widely used by ecologists. As shown recently, however, this index is not suitable for the measurement of beta diversity when the number of species is very large. The objective of this paper is to introduce the Rich- Gini-Simpson quadratic index which preserves all the qualities of the classic Gini-Simpson index but behaves very well even when the number of species is very large. The additive partitioning of species diversity using the Rich-Gini- Simpson quadratic index and an application from island biogeography are analyzed.
文摘Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes, which implicitly proves Goldbach’s Conjecture for 2n as well.
文摘The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes. The present paper contains explicit additional and complementary details of the proof, insisting on the existence and the number of Goldbach’s representations of even positive integers as sums of pairs of primes.
文摘The paper contains a geometric interpretation of Andrica’s conjecture about the gap between the square roots of the consecutive primes and brings empirical evidence that the random fluctuations of the gap between the quare roots of the consecutive primes seem to stabilize around the mean gap.
