Let h + ( n ) denote the class number of the maximal totally real subfield Q(cos(2π/ n )) of the field of n -th roots of unity. The goal of this paper is to show that (speculative extensions of) the Cohen-Lenstra heuristics on class groups provide support for the following conjecture: for all but finitely many pairs ( , n), where is a prime and n is a positive integer, h + ( n+1 ) = h + ( n ). In particular, this predicts that for all but finitely many primes , h + ( n ) = h + ( ) for all positive integers n. It is possible that there are no exceptional primes at all.
The control of malaria vector mosquitoes in South Africa’s affected provinces is primarily based on indoor spraying of long-lasting residual insecticides. The primary vectors in South Africa are Anopheles arabiensis and An. funestus. South Africa’s National Malaria Control Programme has adopted a malaria elimination agenda and has scaled up vector control activities accordingly. However, despite these plans, local transmission continues and is most likely because of outdoor feeding by populations of An. arabiensis. An outdoor Anopheles surveillance system has been set up in three sections of the Mamfene district in northern KwaZulu- Natal in order to assess the extent of outdoor resting An. arabiensis in Mamfene and to assess the current insecticide susceptibility status of this population. According to WHO criteria, the An. arabiensis samples tested showed evidence of resistance to deltamethrin (pyrethroid), DDT (organochlorine) and bendiocarb (carbamate), and full susceptibility to the organophosphates pirimiphos-methyl and fenitrothion. Pre-exposure to piperonyl butoxide completely nullified the deltamethrin resistance otherwise evident in these samples, supporting previous studies implicating monooxygenase-based detoxification as the primary mechanism of pyrethroid resistance. The data presented here affirm the presence of pyrethroid and DDT resistance previously detected in this population and also indicate the comparatively recent emergence of resistance to the carbamate insecticide bendiocarb. These data show that special attention and commitment needs to be given to the principles of insecticide resistance management as well as to investigations into alternative control techniques designed to target outdoor-resting An. arabiensis in northern KwaZulu-Natal.
In 1980, Carl Pomerance and J. L. Selfridge proved D. J. Newman's coprime mapping conjecture: If n is a positive integer and I is a set of n consecutive integers, then there is a bijection f W ¹1; 2; : : : ; nº ! I such that gcd.i; f .i // D 1 for 1 i n. The function f described in their theorem is called a coprime mapping. Around the same time, Roger Entringer conjectured that all trees are prime, that is, that if T is a tree with vertex set V , then there is a bijection L W V ! ¹1; 2; : : : ; jV jº such that gcd.L.x/; L.y// D 1 for all adjacent vertices x and y in V . There has been little progress so far towards a proof of this conjecture. In this paper, we extend Pomerance and Selfridge's theorem by replacing the set I with a set S of n integers in arithmetic progression and determining when there exist coprime mappings f W ¹1; 2; : : : ; nº ! S and g W ¹1; 3; : : : ; 2n 1º ! S. The rest of the paper is devoted to using coprime mappings to prove that various families of trees are prime, including palm trees, banana trees, binomial trees, and certain families of spider colonies.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.