title: Multiply perfect numbers of low abundancy

authors: Zhou, Qizhi

subjects: multiperfect numbers

abstract: The purpose of this thesis is to investigate the properties of multiperfect numbers with low abundancy, and to include the structure, bounds, and density of certain multiperfect numbers. As a significant result of this thesis, an exploration of the structure of an odd 4-perfect number has been made. An extension of Euler’s theorem on the structure of any odd perfect number to odd 2k-perfect numbers has also been obtained. In order to study multiperfect numbers, it is necessary to discuss the factorization of the sum of divisors, in particular for (qe), for prime q. This concept is applied to investigate multiperfect numbers with a so-called flat shape N = 2ap1 · · ·pm. If some prime divisors of N are fixed then there are finitely many flat even 3-perfect numbers. If N is a flat 4-perfect number and the exponent of 2 is not congruent to 1 (mod 12), then the exponent is even. If all odd prime divisors of N are Mersenne primes, where N is even, flat and multiperfect, then N is a perfect number. In more general cases, some necessary conditions for the divisibility by 3 of an even 4-perfect number N = 2ab are obtained, where b is an odd positive integer. Two new ideas, namely flat primes and thin primes, are introduced since these appear often in multiperfect numbers. The relative density of flat primes to all primes is given by 2 times Artin’s constant. An upper bound of the number of thin primes is T(x) less less x log2 x . The sum of the reciprocals of the thin primes is finite.

text: Multiply perfect numbers of low abundancy The purpose of this thesis is to investigate the properties of multiperfect numbers with low abundancy, and to include the structure, bounds, and density of certain multiperfect numbers. As a significant result of this thesis, an exploration of the structure of an odd 4-perfect number has been made. An extension of Euler’s theorem on the structure of any odd perfect number to odd 2k-perfect numbers has also been obtained. In order to study multiperfect numbers, it is necessary to discuss the factorization of the sum of divisors, in particular for (qe), for prime q. This concept is applied to investigate multiperfect numbers with a so-called flat shape N = 2ap1 · · ·pm. If some prime divisors of N are fixed then there are finitely many flat even 3-perfect numbers. If N is a flat 4-perfect number and the exponent of 2 is not congruent to 1 (mod 12), then the exponent is even. If all odd prime divisors of N are Mersenne primes, where N is even, flat and multiperfect, then N is a perfect number. In more general cases, some necessary conditions for the divisibility by 3 of an even 4-perfect number N = 2ab are obtained, where b is an odd positive integer. Two new ideas, namely flat primes and thin primes, are introduced since these appear often in multiperfect numbers. The relative density of flat primes to all primes is given by 2 times Artin’s constant. An upper bound of the number of thin primes is T(x) less less x log2 x . The sum of the reciprocals of the thin primes is finite.

year: 2010

title: Paddock scale nitrous oxide emissions from intensively grazed pasture: Quantification and mitigation

authors: Wecking, Anne Roswitha

subjects: Greenhouse gas

abstract: Agricultural soils are the main contributor to global emissions of the potent greenhouse gas nitrous oxide (N₂O) to the atmosphere. Nitrous oxide can absorb and transform radiative energy emitted from the sun to earth into heat. The ability of N₂O to absorb energy is part of the greenhouse effect and naturally contributes to habitable temperature conditions for life on earth. Atmospheric concentrations of N₂O have never before increased as rapidly as observed since the onset of industrialisation. High N₂O concentrations enhance the greenhouse effect and, at present, add to a warming atmosphere and a changing climate. A major pathway for N₂O production is from soils that receive large surpluses of reactive nitrogen, e.g., in the form of animal excreta or fertiliser. Once available in the soil, microbes can access and transform this nitrogen into different compounds: N₂O, for example. The current contribution of agricultural soils to global N₂O emissions is about 60% but higher in New Zealand, where soils are responsible for 94% of national N₂O emissions. However, robust quantifications are not straightforward and challenging for science to overcome. This thesis is sited at the interface of this challenge and aimed at measuring the exchange of N₂O over an intensively grazed dairy pasture in New Zealand. The core of the work was based on eddy covariance (EC), a technique that allows measuring the N₂O exchange between the soil and the atmosphere at ten times per second. Since EC measurements of N₂O over dairy grazed land are rare, main objectives of this work were: to 1) compare EC to traditional measurement approaches, i.e. static chambers, 2) quantify annual N₂O emission budgets and 3) determine the effect of farm management including pasture renewal on N₂O emissions. The findings of this thesis show that EC provided a realistic picture of the N₂O exchange at paddock scales. This means EC data identified cattle grazing, land history, farm management, soil moisture and temperature as the main factors controlling N₂O exchange throughout the year. Interestingly, using one EC system for measurements across adjacent but differently managed paddocks also allowed to compare the effect of different pasture management scenarios on N₂O emissions. This will help researchers to test different mitigation options more easily in the future, ideally, to reduce N₂O emissions and avoid further atmospheric warming. This thesis advances our understanding of the N₂O exchange process from intensively grazed land as well as it shows how limited this knowledge still is. Micrometeorological N₂O flux measurements will have to be used more frequently in the future to provide researchers with answers in the face of a rapidly changing climate on earth.

text: Paddock scale nitrous oxide emissions from intensively grazed pasture: Quantification and mitigation Agricultural soils are the main contributor to global emissions of the potent greenhouse gas nitrous oxide (N₂O) to the atmosphere. Nitrous oxide can absorb and transform radiative energy emitted from the sun to earth into heat. The ability of N₂O to absorb energy is part of the greenhouse effect and naturally contributes to habitable temperature conditions for life on earth. Atmospheric concentrations of N₂O have never before increased as rapidly as observed since the onset of industrialisation. High N₂O concentrations enhance the greenhouse effect and, at present, add to a warming atmosphere and a changing climate. A major pathway for N₂O production is from soils that receive large surpluses of reactive nitrogen, e.g., in the form of animal excreta or fertiliser. Once available in the soil, microbes can access and transform this nitrogen into different compounds: N₂O, for example. The current contribution of agricultural soils to global N₂O emissions is about 60% but higher in New Zealand, where soils are responsible for 94% of national N₂O emissions. However, robust quantifications are not straightforward and challenging for science to overcome. This thesis is sited at the interface of this challenge and aimed at measuring the exchange of N₂O over an intensively grazed dairy pasture in New Zealand. The core of the work was based on eddy covariance (EC), a technique that allows measuring the N₂O exchange between the soil and the atmosphere at ten times per second. Since EC measurements of N₂O over dairy grazed land are rare, main objectives of this work were: to 1) compare EC to traditional measurement approaches, i.e. static chambers, 2) quantify annual N₂O emission budgets and 3) determine the effect of farm management including pasture renewal on N₂O emissions. The findings of this thesis show that EC provided a realistic picture of the N₂O exchange at paddock scales. This means EC data identified cattle grazing, land history, farm management, soil moisture and temperature as the main factors controlling N₂O exchange throughout the year. Interestingly, using one EC system for measurements across adjacent but differently managed paddocks also allowed to compare the effect of different pasture management scenarios on N₂O emissions. This will help researchers to test different mitigation options more easily in the future, ideally, to reduce N₂O emissions and avoid further atmospheric warming. This thesis advances our understanding of the N₂O exchange process from intensively grazed land as well as it shows how limited this knowledge still is. Micrometeorological N₂O flux measurements will have to be used more frequently in the future to provide researchers with answers in the face of a rapidly changing climate on earth.

year: 2021

title: Waikirikiri Marae: Shared experiences of the wharemate

authors: Rua, Hare

subjects: Māori

abstract: In Te Urewera, wharemate (shelters in which the deceased receive their final farewells) have traditionally been temporary structures. In the 1980s, a new practice was introduced in the Ruātoki valley with the erection of permanent wharemate facilities. One was erected at Waikirikiri marae (tribal meeting grounds and associated buildings) in 1989. Knowledge and discussion regarding wharemate at Waikirikiri marae have changed over the years, and a whole generation has not been fortunate enough to experience tikanga (correct procedures, customary practices) that prevailed prior to the introduction of the permanent wharemate building that is there today. These changes are recorded in this paper through the shared stories of three kaumātua (elders) from Waikirikiri.

This paper is a summary of Hare Rua’s thesis study, the data for which was collected in 2009. This work forms part of the Tangi Research Programme, a collaboration between the School of Māori and Pacific Development and the Māori and Psychology Research Unit at The University of Waikato, Hamilton, New Zealand.

text: Waikirikiri Marae: Shared experiences of the wharemate In Te Urewera, wharemate (shelters in which the deceased receive their final farewells) have traditionally been temporary structures. In the 1980s, a new practice was introduced in the Ruātoki valley with the erection of permanent wharemate facilities. One was erected at Waikirikiri marae (tribal meeting grounds and associated buildings) in 1989. Knowledge and discussion regarding wharemate at Waikirikiri marae have changed over the years, and a whole generation has not been fortunate enough to experience tikanga (correct procedures, customary practices) that prevailed prior to the introduction of the permanent wharemate building that is there today. These changes are recorded in this paper through the shared stories of three kaumātua (elders) from Waikirikiri.

This paper is a summary of Hare Rua’s thesis study, the data for which was collected in 2009. This work forms part of the Tangi Research Programme, a collaboration between the School of Māori and Pacific Development and the Māori and Psychology Research Unit at The University of Waikato, Hamilton, New Zealand.

year: 2010