title: On integers for which the sum of divisors is the square of the squarefree core

authors: Broughan, Kevin A.

subjects: mathematics

abstract: We study integers n > 1 satisfying the relation σ(n) = γ(n) ² , where σ(n) and γ(n) are the sum of divisors and the product of distinct primes dividing n, respectively. We show that the only solution n with at most four distinct prime factors is n = 1782. We show that there is no solution which is fourth power free. We also show that the number of solutions up to x > 1 is at most x ⅟⁴⁺ᵉ for any ε > 0 and all x > xε. Further, call n primitive if no proper unitary divisor d of n satisﬁes σ(d) | γ(d) ² . We show that the number of primitive solutions to the equation up to x is less than xᵉ for x > xₑ.

text: On integers for which the sum of divisors is the square of the squarefree core We study integers n > 1 satisfying the relation σ(n) = γ(n) ² , where σ(n) and γ(n) are the sum of divisors and the product of distinct primes dividing n, respectively. We show that the only solution n with at most four distinct prime factors is n = 1782. We show that there is no solution which is fourth power free. We also show that the number of solutions up to x > 1 is at most x ⅟⁴⁺ᵉ for any ε > 0 and all x > xε. Further, call n primitive if no proper unitary divisor d of n satisﬁes σ(d) | γ(d) ² . We show that the number of primitive solutions to the equation up to x is less than xᵉ for x > xₑ.

year: 2012

title: Seeking sigma: Time-of-the-day effects on the Bitcoin network

authors: Jahanshahloo, Hossein

subjects: Social Sciences

abstract: This research investigates and tests for the presence of time-of-the-day effects on the Bitcoin network. Results indicate that NYSE trading sessions lead Bitcoin trading activity, both on the blockchain and centralised exchanges. Effects are found to have strengthened over time, however, simultaneously diminished at the weekend indicating significant exchange interactions, and that Bitcoin has developed somewhat outside its intended design parameters and is influenced by other forces such as those originating from NYSE trading. While proponents consider Bitcoin trading to be ‘24/7’, our findings suggest that both transaction and on-chain network activity are best described to be, at best, ‘12/5’, presenting significant implications for traders, with regards to centralised exchange liquidity and the speed of their transaction inclusion on the blockchain. Finally, the role and influence of both algorithm and volatility traders cannot be eliminated.

text: Seeking sigma: Time-of-the-day effects on the Bitcoin network This research investigates and tests for the presence of time-of-the-day effects on the Bitcoin network. Results indicate that NYSE trading sessions lead Bitcoin trading activity, both on the blockchain and centralised exchanges. Effects are found to have strengthened over time, however, simultaneously diminished at the weekend indicating significant exchange interactions, and that Bitcoin has developed somewhat outside its intended design parameters and is influenced by other forces such as those originating from NYSE trading. While proponents consider Bitcoin trading to be ‘24/7’, our findings suggest that both transaction and on-chain network activity are best described to be, at best, ‘12/5’, presenting significant implications for traders, with regards to centralised exchange liquidity and the speed of their transaction inclusion on the blockchain. Finally, the role and influence of both algorithm and volatility traders cannot be eliminated.

year: 2022-10-01

title: Bicultural perspectives on Māori legal research

authors: Mackinnon, Jacquelin

subjects: Māori

abstract: Increasingly, legal research projects undertaken at law school or in practice will involve Māori custom law and/or tikanga Māori.' The role of both Māori custom law and tikanga Māori is most evident in the work of the Māori Land Court in the interpretation and application of legislation relating to Māori land. Increasingly, general statutes incorporate Māori principles and values, such as those to be found in the Resource Management Act 1991, or make explicit reference to the principles of the Treaty of Waitangi. The statutory interpretation function of the Courts in relation to these, and less obvious examples, requires knowledge of tikanga Māori and/or Māori custom law. For any analysis of the work of the Courts or of the legislature, knowledge of tikanga Māori and/or custom law is required. Both Māori custom law and tikanga Māori are preserved by and accessed through the oral tradition. In addition to its role in the Courts and in relation to legislation, the most significant role played by the oral tradition is in the work of the Waitangi Tribunal. The oral tradition also plays an increasingly important part in other areas requiring research. What follows is an introduction to the oral tradition and its role in the legal system of Aotearoa/New Zealand. An understanding of the oral tradition is essential to the construction of a research path that is both ethical and effective.

text: Bicultural perspectives on Māori legal research Increasingly, legal research projects undertaken at law school or in practice will involve Māori custom law and/or tikanga Māori.' The role of both Māori custom law and tikanga Māori is most evident in the work of the Māori Land Court in the interpretation and application of legislation relating to Māori land. Increasingly, general statutes incorporate Māori principles and values, such as those to be found in the Resource Management Act 1991, or make explicit reference to the principles of the Treaty of Waitangi. The statutory interpretation function of the Courts in relation to these, and less obvious examples, requires knowledge of tikanga Māori and/or Māori custom law. For any analysis of the work of the Courts or of the legislature, knowledge of tikanga Māori and/or custom law is required. Both Māori custom law and tikanga Māori are preserved by and accessed through the oral tradition. In addition to its role in the Courts and in relation to legislation, the most significant role played by the oral tradition is in the work of the Waitangi Tribunal. The oral tradition also plays an increasingly important part in other areas requiring research. What follows is an introduction to the oral tradition and its role in the legal system of Aotearoa/New Zealand. An understanding of the oral tradition is essential to the construction of a research path that is both ethical and effective.

year: 2007