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: Measuring absorptive capacity in high-tech companies: Mixing qualitative and quantitative methods

authors: Sedoglavich, Vesna

subjects: mixed methods

abstract: The objective of this article is to show how mixed methods can be used to develop a deeper understanding of the construct, absorptive capacity (AC). We used qualitative data from eight case studies to identify which types of AC knowledge firms have. Then we identified variables for measuring AC. We identified two types of AC: “technological” AC and “non-core” AC. Using a Karnaugh Map and a Fuzzy Logic Design, we developed a quantitative model to measure overall AC. By focusing on interpretive integration, our results from using both qualitative and quantitative methods yielded complementary rather than contradictory findings, and the goal of “knowing more” about AC was achieved.

text: Measuring absorptive capacity in high-tech companies: Mixing qualitative and quantitative methods The objective of this article is to show how mixed methods can be used to develop a deeper understanding of the construct, absorptive capacity (AC). We used qualitative data from eight case studies to identify which types of AC knowledge firms have. Then we identified variables for measuring AC. We identified two types of AC: “technological” AC and “non-core” AC. Using a Karnaugh Map and a Fuzzy Logic Design, we developed a quantitative model to measure overall AC. By focusing on interpretive integration, our results from using both qualitative and quantitative methods yielded complementary rather than contradictory findings, and the goal of “knowing more” about AC was achieved.

year: 2014

title: Death and homelessness on the streets of Aotearoa New Zealand: Sustaining life

authors: Charvin-Fabre, Sandrine

abstract: In high-income countries, homeless people die 20 to 35 years earlier than domiciled populations. Despite this enduring reality, the issue of death inequality remains underexplored, especially in indigenous homeless contexts, facing the ongoing effects of colonisation. This study centres on the perspectives and aspirations of a Māori street homeless community, known as the “Peeps”. Its objective is twofold: to spotlight to unjust, untimely, and avoidable mortality impacting homeless communities in Aotearoa New Zealand, and to understand how life can be sustained against the adversity of street life. The thesis diverges from conventional public health approaches, which often dichotomise early deaths prevention through improved healthcare access during homelessness, and palliative care provision at the end of life of homeless people. This thesis explores “the temporal space of life prior to death”. Within that temporal realm, the omnipresent “risk of dying” that infiltrates the lives of homeless people, disrupting the linear progression toward death observed in domiciled populations, is recognised. This temporal space is understood as a zone of uncertainty and potentiality, wherein complex life-death dynamics operate, revealing broader political, societal and cultural tensions. This study encountered significant challenges, including the dearth of prior death-related research in Māori homeless contexts, and my non-Māori identity. In response, an indeterminate approach, rooted in relational ethics and complexity and systems thinking — aligned with Māori worldviews — was adopted.  

Part I of the thesis is conceptual and contextual. Part II invites the readers into the fragile “temporal space of life prior to death” and the cyclical research process. Cycle 1 focuses on understanding the local context of death inequality, through relational fieldwork and statistical analysis of coroners’ reports. This analysis uncovers a 30-year life expectancy disparity with domiciled populations, exacerbated by three-quarters of avoidable mortality, mainly due to chronic conditions and suicide. Subsequent cycles sustain relational fieldwork, using collaborative and critical performance ethnography. Cycle 2 reveals the Peeps’ resistance to access healthcare, rooted in experiences of transgenerational trauma and marginalisation of Māori beliefs and values in relation to death and dying within mainstream healthcare institutions. Collaboration with the Peeps leads to the co-creation of a drama, intended to initiate conversations with health professionals from the Peeps’ perspectives and aspirations, to improve the quality of care. This cycle also highlights the impact of disconnection processes with whānau (family, extended family) experienced by the Peeps. Progressing further, cycle 3 illuminates healing processes with whānau using Advance Care Planning, envisioned as a relational space, providing the Peeps with a sense of belonging, and spiritual continuity in the perspective of death and dying. 
 
In addition to shedding light on the profound death inequality that affect Māori homeless people, this thesis argues for homelessness to be recognised as “complex life-threatening conditions associated with serious-related health suffering”. Accordingly, the thesis advocates the shift toward comprehensive care, inclusive of a concurrent prevention and palliation approach over the course of homelessness, to mitigate its impact on the lives of Māori homeless people and enhance holistic wellbeing.

text: Death and homelessness on the streets of Aotearoa New Zealand: Sustaining life In high-income countries, homeless people die 20 to 35 years earlier than domiciled populations. Despite this enduring reality, the issue of death inequality remains underexplored, especially in indigenous homeless contexts, facing the ongoing effects of colonisation. This study centres on the perspectives and aspirations of a Māori street homeless community, known as the “Peeps”. Its objective is twofold: to spotlight to unjust, untimely, and avoidable mortality impacting homeless communities in Aotearoa New Zealand, and to understand how life can be sustained against the adversity of street life. The thesis diverges from conventional public health approaches, which often dichotomise early deaths prevention through improved healthcare access during homelessness, and palliative care provision at the end of life of homeless people. This thesis explores “the temporal space of life prior to death”. Within that temporal realm, the omnipresent “risk of dying” that infiltrates the lives of homeless people, disrupting the linear progression toward death observed in domiciled populations, is recognised. This temporal space is understood as a zone of uncertainty and potentiality, wherein complex life-death dynamics operate, revealing broader political, societal and cultural tensions. This study encountered significant challenges, including the dearth of prior death-related research in Māori homeless contexts, and my non-Māori identity. In response, an indeterminate approach, rooted in relational ethics and complexity and systems thinking — aligned with Māori worldviews — was adopted.  

Part I of the thesis is conceptual and contextual. Part II invites the readers into the fragile “temporal space of life prior to death” and the cyclical research process. Cycle 1 focuses on understanding the local context of death inequality, through relational fieldwork and statistical analysis of coroners’ reports. This analysis uncovers a 30-year life expectancy disparity with domiciled populations, exacerbated by three-quarters of avoidable mortality, mainly due to chronic conditions and suicide. Subsequent cycles sustain relational fieldwork, using collaborative and critical performance ethnography. Cycle 2 reveals the Peeps’ resistance to access healthcare, rooted in experiences of transgenerational trauma and marginalisation of Māori beliefs and values in relation to death and dying within mainstream healthcare institutions. Collaboration with the Peeps leads to the co-creation of a drama, intended to initiate conversations with health professionals from the Peeps’ perspectives and aspirations, to improve the quality of care. This cycle also highlights the impact of disconnection processes with whānau (family, extended family) experienced by the Peeps. Progressing further, cycle 3 illuminates healing processes with whānau using Advance Care Planning, envisioned as a relational space, providing the Peeps with a sense of belonging, and spiritual continuity in the perspective of death and dying. 
 
In addition to shedding light on the profound death inequality that affect Māori homeless people, this thesis argues for homelessness to be recognised as “complex life-threatening conditions associated with serious-related health suffering”. Accordingly, the thesis advocates the shift toward comprehensive care, inclusive of a concurrent prevention and palliation approach over the course of homelessness, to mitigate its impact on the lives of Māori homeless people and enhance holistic wellbeing.

year: 2024