This commentary constructs a social history of Hillbrow, an inner-city suburb in Johannesburg, South Africa, based on a review of relevant published historical, anthropological and sociological texts. We highlight the significant continuities in the social structure of the suburb, despite the radical transformations that have occurred over the last 120 years.Originally envisaged as a healthy re...

The prime graph of a finite group $G$ is denoted by $Gamma(G)$ whose vertex set is $pi(G)$ and two distinct primes $p$ and $q$ are adjacent in $Gamma(G)$, whenever $G$ contains an element with order $pq$. We say that $G$ is unrecognizable by prime graph if there is a finite group $H$ with $Gamma(H)=Gamma(G)$, in while $Hnotcong G$. In this paper, we consider finite groups with the same prime gr...

In this paper, we define the new notion of quasi-prime ideal which generalizes at once both prime ideal and primary ideal notions. Then a natural topology on the set of quasi-prime ideals of a ring is introduced which admits the Zariski topology as a subspace topology. The basic properties of the quasi-prime spectrum are studied and several interesting results are obtained. Specially, it is pro...

This paper develops a prime radix transform algorithm that can be used to calculate the discrete Fourier transform (DFT) of a data block of length P, a prime, using fixed coefficient second order recursive filter sections: this simple hardware structure allows compact, high performance DFT processors to be developed. Further “massaging” of the prime radix algorithm produces the zero factor tran...

Euclid’s proof of the infinitude of the primes is a paragon of simplicity: given a finite list of primes, multiply them together and add one. The resulting number, say N , is not divisible by any prime on the list, so any prime factor of N is a new prime. Some special cases of Dirichlet’s theorem admit a simple proof following Euclid’s model, such as the case of 1 mod 4 or 5 mod 6. (We mean by ...