arXiv subjects

Petrus H. Potgieter

Publications and source records attributed to Petrus H. Potgieter.

6 recordsLinked to original sources

arXiv2012

Availability of titles on peer-to-peer file sharing networks

File sharing, typically involving video or audio material in which copyright may persist and using peer-to-peer (P2P) networks like BitTorrent, has been reported to make up the bulk of Internet traffic. The free-riding problem appears in this "digital gift economy" but its users exhibit rational behaviour, subject to the characteristics of the particular network. The high demand for the Internet as a delivery channel for entertainment underlines the importance of understanding the dynamics of this market, especially when considering possible business models for future pricing or licensing regimes and for the provisioning of network capacity to support future services. The availability of specific titles on file sharing networks is the focus of this paper, with a special emphasis on the P2P protocol BitTorrent. The paper compares the incentives provided in BitTorrent to those in other file-sharing communities, including file hosting, and discusses the number of titles available in the community at any given time, with an emphasis on popular video items with ambiguous legal status.

cs.NI↗

arXiv2008

Computable counter-examples to the Brouwer fixed-point theorem

This paper is an overview of results that show the Brouwer fixed-point theorem (BFPT) to be essentially non-constructive and non-computable. The main results, the counter-examples of Orevkov and Baigger, imply that there is no procedure for finding the fixed point in general by giving an example of a computable function which does not fix any computable point. Research in reverse mathematics has shown the BFPT to be equivalent to the weak König lemma in RCA$_0$ (the system of recursive comprehension) and this result is illustrated by relating the weak König lemma directly to the Baigger example.

math.GM↗

arXiv2007

Universality and programmability of quantum computers

Manin, Feynman, and Deutsch have viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum logic circuits and quantum Turing machines has shown how these machines can simulate an arbitrary unitary transformation on a finite number of qubits. The problem of universality has been addressed most famously in a paper by Deutsch, and later by Bernstein and Vazirani as well as Kitaev and Solovay. The quantum logic circuit model, developed by Feynman and Deutsch, has been more prominent in the research literature than Deutsch's quantum Turing machines. Quantum Turing machines form a class closely related to deterministic and probabilistic Turing machines and one might hope to find a universal machine in this class. A universal machine is the basis of a notion of programmability. The extent to which universality has in fact been established by the pioneers in the field is examined and this key notion in theoretical computer science is scrutinised in quantum computing by distinguishing various connotations and concomitant results and problems.

quant-ph↗

arXiv2007

Deutsch's Universal Quantum Turing Machine (Revisited)

Deutsch, Feynman, and Manin viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum Turing machines has shown how these machines can simulate an arbitrary unitary transformation on a finite number of qubits. This interesting problem has been addressed most famously in a paper by Deutsch, and later by Bernstein and Vazirani. Quantum Turing machines form a class closely related to deterministic and probabilistic Turing machines and one might hope to find a universal machine in this class. A universal machine is the basis of a notion of programmability. The extent to which universality has in fact been established by the pioneers in the field is examined and a key notion in theoretical computer science (universality) is scrutinised. In a forthcoming paper, the authors will also consider universality in the quantum gate model.

quant-ph↗

arXiv2006

Hypercomputing the Mandelbrot Set?

The Mandelbrot set is an extremely well-known mathematical object that can be described in a quite simple way but has very interesting and non-trivial properties. This paper surveys some results that are known concerning the (non-)computability of the set. It considers two models of decidability over the reals (which have been treated much more thoroughly and technically by Hertling (2005), Blum, Shub and Smale, Brattka (2003) and Weihrauch (1999 and 2003) among others), two over the computable reals (the Russian school and hypercomputation) and a model over the rationals.

cs.CC↗

arXiv2006

Zeno machines and hypercomputation

This paper reviews the Church-Turing Thesis (or rather, theses) with reference to their origin and application and considers some models of "hypercomputation", concentrating on perhaps the most straight-forward option: Zeno machines (Turing machines with accelerating clock). The halting problem is briefly discussed in a general context and the suggestion that it is an inevitable companion of any reasonable computational model is emphasised. It is hinted that claims to have "broken the Turing barrier" could be toned down and that the important and well-founded role of Turing computability in the mathematical sciences stands unchallenged.

cs.CC↗