title: On the ratio of the sum of divisors and Euler’s Totient Function I

authors: Broughan, Kevin A.

subjects: mathematics

abstract: We prove that the only solutions to the equation σ(n)=2φ(n) with at most three distinct prime factors are 3, 35 and 1045. Moreover, there exist at most a finite number of solutions to σ(n)=2φ(n) with  Ω(n)≤ k, and there are at most 22k+k-k squarefree solutions to  φ (n)|σ(n) if  ω(n)=k. Lastly the number of solutions to  φ(n)|φ(n)  as  x→∞ is O(x exp(-½√log x)).

text: On the ratio of the sum of divisors and Euler’s Totient Function I We prove that the only solutions to the equation σ(n)=2φ(n) with at most three distinct prime factors are 3, 35 and 1045. Moreover, there exist at most a finite number of solutions to σ(n)=2φ(n) with  Ω(n)≤ k, and there are at most 22k+k-k squarefree solutions to  φ (n)|σ(n) if  ω(n)=k. Lastly the number of solutions to  φ(n)|φ(n)  as  x→∞ is O(x exp(-½√log x)).

year: 2013

title: The effects of reinforcer magnitude in the preceding and upcoming ratios on between-ratio pausing in multiple fixed-ratio schedules

authors: Young, Rachel

subjects: Reinforcer Magnitude

abstract: Experiment 1 examined the effects of reinforcer magnitude in the preceding and upcoming ratios on between-ratio pausing during a range of FR values in a multiple FR FR schedule of reinforcement. Two components which were associated with two different discriminative stimuli were used, where one component was associated with a small reinforcer (1-s) and the other component was associated with a large reinforcer (6-s). The longest between-ratio pauses were found when the upcoming reinforcer was small (S-S and L-S transition) than when the upcoming reinforcer was large (S-L and L-L transition). In addition, consistently longer between-ratio pauses occurred when the preceding reinforcer was large and the upcoming reinforcer was small (L-S transition). Experiment 2 examined the effects of the past reinforcer magnitude on the between-ratio pause when the discriminative stimuli to differentiate the preceding and upcoming reinforcer magnitude was removed, during a range of FR values on a multiple FR FR schedule of reinforcement. When the past reinforcer magnitude had been small and the upcoming reinforcer was small or large (S-S and S-L transition) consistently shorter between-ratio pauses occurred. In comparison, when the past reinforcer had been large and the upcoming reinforcer was small or large (L-L and L-S transition) consistently longer pause durations occurred across the two transitions. Overall and running response rates were analysed in Experiments 1 and 2 and it was found that overall and running response rates consistently decreased as the FR values were increased. Overall, these experiments showed that the preceding and upcoming reinforcer magnitudes have a direct effect on the length of the between-ratio pause where increasing the FR value was observed to consistently increase the length of the between-ratio pause, when also compared across the four transitions (Small-Small, Small-Large, Large-Large, and Large-Small).

text: The effects of reinforcer magnitude in the preceding and upcoming ratios on between-ratio pausing in multiple fixed-ratio schedules Experiment 1 examined the effects of reinforcer magnitude in the preceding and upcoming ratios on between-ratio pausing during a range of FR values in a multiple FR FR schedule of reinforcement. Two components which were associated with two different discriminative stimuli were used, where one component was associated with a small reinforcer (1-s) and the other component was associated with a large reinforcer (6-s). The longest between-ratio pauses were found when the upcoming reinforcer was small (S-S and L-S transition) than when the upcoming reinforcer was large (S-L and L-L transition). In addition, consistently longer between-ratio pauses occurred when the preceding reinforcer was large and the upcoming reinforcer was small (L-S transition). Experiment 2 examined the effects of the past reinforcer magnitude on the between-ratio pause when the discriminative stimuli to differentiate the preceding and upcoming reinforcer magnitude was removed, during a range of FR values on a multiple FR FR schedule of reinforcement. When the past reinforcer magnitude had been small and the upcoming reinforcer was small or large (S-S and S-L transition) consistently shorter between-ratio pauses occurred. In comparison, when the past reinforcer had been large and the upcoming reinforcer was small or large (L-L and L-S transition) consistently longer pause durations occurred across the two transitions. Overall and running response rates were analysed in Experiments 1 and 2 and it was found that overall and running response rates consistently decreased as the FR values were increased. Overall, these experiments showed that the preceding and upcoming reinforcer magnitudes have a direct effect on the length of the between-ratio pause where increasing the FR value was observed to consistently increase the length of the between-ratio pause, when also compared across the four transitions (Small-Small, Small-Large, Large-Large, and Large-Small).

year: 2015

title: Kai hea te mana o te reo Māori?

authors: Temara, Pou

subjects: Waiata

abstract: Kai roto i tēnei waiata a Te Matua Tangata e kī ana, he mātauranga ake te ruru rāua ko te heihei i te tangata. E taea ana e te ruru rāua ko te heihei te wehewehe i ngā hāora o te pō, i te mea ko te pō te wā rērere o te ruru, ko te pō te wā e tangi ai tōna reo. He mea hanga te ruru mō te pō. Ko tā te heihei he nehu atu i te pō, he whakaō, he karanga i te awatea ki te ao. Kua eke mai te tuarā nui o awatea, kua oho te ao, kua hoki te ruru ki tōna puta, kua ketuketu ko te heihei. Ka pō, kua hoki ko te heihei ki tōna taumatua, kua oho ko te ruru.

text: Kai hea te mana o te reo Māori? Kai roto i tēnei waiata a Te Matua Tangata e kī ana, he mātauranga ake te ruru rāua ko te heihei i te tangata. E taea ana e te ruru rāua ko te heihei te wehewehe i ngā hāora o te pō, i te mea ko te pō te wā rērere o te ruru, ko te pō te wā e tangi ai tōna reo. He mea hanga te ruru mō te pō. Ko tā te heihei he nehu atu i te pō, he whakaō, he karanga i te awatea ki te ao. Kua eke mai te tuarā nui o awatea, kua oho te ao, kua hoki te ruru ki tōna puta, kua ketuketu ko te heihei. Ka pō, kua hoki ko te heihei ki tōna taumatua, kua oho ko te ruru.

year: 2011