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http://hdl.handle.net/10012/15251
Compressible Matrix Algebras and the Distance from Projections to Nilpotents
Cramer, Zachary
In this thesis we address two problems from the fields of operator algebras and operator theory. In our first problem, we seek to obtain a description of the unital subalgebras $\mathcal{A}$ of $\mathbb{M}_n(\mathbb{C})$ with the property that $E\mathcal{A}E$ is an algebra for all idempotents $E\in\mathbb{M}_n(\mathbb{C})$. Algebras with this property are said to be \textit{idempotent compressible}. Likewise, we wish to determine which unital subalgebras of $\mathbb{M}_n(\mathbb{C})$ satisfy the analogous property for projections (i.e., self-adjoint idempotents). Such algebras are said to be \textit{projection compressible}.
We begin by constructing various examples of idempotent compressible subalgebras of $\mathbb{M}_n(\mathbb{C})$ for each integer $n\geq 3$. Using a case-by-case analysis based on reduced block upper triangular forms, we prove that our list includes all unital projection compressible subalgebras of $\mathbb{M}_3(\mathbb{C})$ up to similarity and transposition. A similar examination indicates that the same phenomenon occurs in the case of unital subalgebras of $\mathbb{M}_n(\mathbb{C})$, $n\geq 4$. We therefore demonstrate that the notions of projection compressibility and idempotent compressibility coincide for unital subalgebras of $\mathbb{M}_n(\mathbb{C})$, and obtain a complete classification of the unital algebras admitting these properties up to similarity and transposition.
In our second problem, we address the question of computing the distance from a non-zero projection to the set of nilpotent operators acting on $\mathbb{C}^n$. Building on MacDonald's results in the rank-one case, we prove that the distance from a rank $n-1$ projection to the set of nilpotents in $\mathbb{M}_n(\mathbb{C})$ is $\frac{1}{2}\sec\left(\frac{\pi}{\frac{n}{n-1}+2}\right)$. For each $n\geq 2$, we construct examples of pairs $(Q,T)$ where $Q$ is a projection of rank $n-1$ and $T\in\mathbb{M}_n(\mathbb{C})$ is a nilpotent of minimal distance to $Q$. Moreover, it is shown that any two such pairs are unitarily equivalent. We end by discussing possible extensions of these results in the case of projections of intermediate ranks.
Fri, 15 Nov 2019 00:00:00 GMThttp://hdl.handle.net/10012/152512019-11-15T00:00:00ZOn Occupancy Based Randomized Load Balancing for Large Systems with General Distributions
http://hdl.handle.net/10012/15250
On Occupancy Based Randomized Load Balancing for Large Systems with General Distributions
Vasantam, Thirupathaiah
Multi-server architectures are ubiquitous in today's information infrastructure whether for supporting cloud services, web servers, or for distributed storage. The performance of multi-server systems is highly dependent on the load distribution. This is affected by the use of load balancing strategies. Since both latency and blocking are important features, it is most reasonable to route an incoming job to a server that is lightly loaded. Hence a good load balancing policy should be dependent on the states of servers. Since obtaining information about the remaining workload of servers for every arrival is very hard, it is preferable to design load balancing policies that depend on occupancy or the number of progressing jobs of servers. Furthermore, if the system has a large number of servers, it is not practical to use the occupancy information of all the servers to dispatch or route an arrival due to high communication cost. In large-scale systems that have tens of thousands of servers, the policies which use the occupancy information of only a finite number of randomly selected servers to dispatch an arrival result in lower implementation cost than the policies which use the occupancy information of all the servers. Such policies are referred to as occupancy based randomized load balancing policies.
Motivated by cloud computing systems and web-server farms, we study two types of models. In the first model, each server is an Erlang loss server, and this model is an abstraction of Infrastructure-as-a-Service (IaaS) clouds. The second model we consider is one with processor sharing servers that is an abstraction of web-server farms which serve requests in a round-robin manner with small time granularity. The performance criterion for web-servers is the response time or the latency for the request to be processed. In most prior works, the analysis of these models was restricted to the case of exponential job length distributions and in this dissertation we study the case of general job length distributions.
To analyze the impact of a load balancing policy, we need to develop models for the system's dynamics. In this dissertation, we show that one can construct useful Markovian models. For occupancy based randomized routing policies, due to complex inter-dependencies between servers, an exact analysis is mostly intractable. However, we show that the multi-server systems that have an occupancy based randomized load balancing policy are examples of weakly interacting particle systems. In these systems, servers are interacting particles whose states lie in an uncountable state space. We develop a mean-field analysis to understand a server's behavior as the number of servers becomes large. We show that under certain assumptions, as the number of servers increases, the sequence of empirical measure-valued Markov processes which model the systems' dynamics converges to a deterministic measure-valued process referred to as the mean-field limit. We observe that the mean-field equations correspond to the dynamics of the distribution of a non-linear Markov process. A consequence of having the mean-field limit is that under minor and natural assumptions on the initial states of servers, any finite set of servers can be shown to be independent of each other as the number of servers goes to infinity. Furthermore, the mean-field limit approximates each server's distribution in the transient regime when the number of servers is large.
A salient feature of loss and processor sharing systems in the setting where their time evolution can be modeled by reversible Markov processes is that their stationary occupancy distribution is insensitive to the type of job length distribution; it depends only on the average job length but not on the type of the distribution. This property does not hold when the number of servers is finite in our context due to lack of reversibility. We show however that the fixed-point of the mean-field is insensitive to the job length distributions for all occupancy based randomized load balancing policies when the fixed-point is unique for job lengths that have exponential distributions. We also provide some deeper insights into the relationship between the mean-field and the distributions of servers and the empirical measure in the stationary regime.
Finally, we address the accuracy of mean-field approximations in the case of loss models. To do so we establish a functional central limit theorem under the assumption that the job lengths have exponential distributions. We show that a suitably scaled fluctuation of the stochastic empirical process around the mean-field converges to an Ornstein-Uhlenbeck process. Our analysis is also valid for the Halfin-Whitt regime in which servers are critically loaded. We then exploit the functional central limit theorem to quantify the error between the actual blocking probability of the system with a large number of servers and the blocking probability obtained from the fixed-point of the mean-field. In the Halfin-Whitt regime, the error is of the order inverse square root of the number of servers. On the other hand, for a light load regime, the error is smaller than the inverse square root of the number of servers.
Wed, 13 Nov 2019 00:00:00 GMThttp://hdl.handle.net/10012/152502019-11-13T00:00:00ZThe Influence of Social Networks on Mental and Emotional Health Resource-Seeking Behaviours Amongst Women who are Refugees from Syria
http://hdl.handle.net/10012/15249
The Influence of Social Networks on Mental and Emotional Health Resource-Seeking Behaviours Amongst Women who are Refugees from Syria
Mahajan, Shreya
Background and Objectives: While the arrival of migrants into Canada from foreign countries is not a new phenomenon, recent years have seen a larger than usual influx of individuals come into the nation. Between 2015 and 2017, Canada saw a surge of Syrian people who are refugees coming onto its soil; over 24,000 of which were women. The increasing number of individuals who are refugees directly impacts health equity for Ontario’s population and it has been reported that this group is vulnerable to declining health. Migrant women, including those classified as refugees are two to three times more likely than native Canadians to suffer from some form of depression. Other mental health challenges reported by refugees include loneliness, diminished social networks, isolation, and stress. These mental health challenges can negatively impact emotional stability as well. The focus in the present research is the role of social networks, a dimension of and opportunity afforded by social capital, in shaping the mental and emotional health of Syrian women who are refugees – specifically, resource-seeking behaviours in the treatment of mental and/or emotional health problems that they employ in improving their own mental and emotional health.
Methods: This study employs a qualitative research design to explore the stories and experiences of this relatively newly studied population. I sampled twelve participants who were adult women from Syria who were living in Canada as refugees at the time of data collection. Telephone interviews were conducted using a semi-structured approach. Eight of the twelve interviews had an Arabic interpreter present, who was able to translate English into Arabic and vice versa. Interviews were audio-recorded and transcribed using NVivo 12. Thematic coding was also done using NVivo 12 software.
Results: Findings identified that social networks can play both direct and indirect roles in influencing mental and emotional health service-seeking behaviours amongst women who are refugees from Syria. Namely, (1) families play large roles in teaching and providing information about Canada’s health system; (2) providers prevent use of resources, while family, friends, and sponsors help; and (3) women feel more welcomed into social networks in Canada than in interim countries. Additionally, (4) social networks act as alternatives to seeking formal mental and emotional health care; and (5) comparing environments from Syria and interim countries helps some women build resilience.
Conclusion: The findings of this study explored the stories and experiences of women’s transitions into Canada in relation to their mental and emotional health and social networks. This study explored the importance of considering who women seek out for support and how these groups and individuals can impact women’s decisions to seek out mental and emotional hralth care resources. Avenues for further investigation may answer questions such as: Where should resources be invested? What can be done to foster and facilitate social networks that promote mental health?
Tue, 12 Nov 2019 00:00:00 GMThttp://hdl.handle.net/10012/152492019-11-12T00:00:00ZEnvironmental Engagement and Generation Z: Evaluating and Modifying the YEEP Framework Against Research and Observations on Generation Z Youth and Youth Leaders
http://hdl.handle.net/10012/15248
Environmental Engagement and Generation Z: Evaluating and Modifying the YEEP Framework Against Research and Observations on Generation Z Youth and Youth Leaders
Watkins, Amanda M.
In October 2018, the United Nations Intergovernmental Panel on Climate Change (IPCC) announced that the global community has 12 years to stop the rapid growth of climate change and avoid increased threats of drought, flooding, and extreme heat, affecting millions of people. During this 12-year timeframe, millions of children, teenagers, and adolescents will come of age as voters and members of the workforce. Generation Z, the emerging group of youth we see today, will be the change makers during these pivotal years of climate change adaptation. Born post-Internet after 1996, the Internet Generation, iGen, Centennials, or Generation Z, consists of approximately 7.3 million people in Canada, and 65 million people in the United States. Generation Z first began to enter the labour force in 2014, when its oldest members turned 18, and they will continue to enter the labour force until 2029 and beyond, as younger members age into adulthood and complete post-secondary studies. In 2014, Riemer, Lynes and Hickman published the Youth-Based Environmental Engagement (YEEP) framework to help guide the further development of informal environmental education programs for emerging adults. This research evaluates the YEEP framework against current literature on Generation Z attitudes and behaviour, as well as an existing youth engagement program that successfully works with Generation Z participants, the Peel Environmental Youth Alliance (PEYA) in the Region of Peel, Canada. The research finds that the YEEP framework could be strengthened to successfully target Generation Z participants through a greater inclusion of questions that address the tech-savvy nature of this generation, the threats and benefits that come with this technology, as well as the generation-specific values that require an emphasis on practical and emotional skills-building opportunities.
Mon, 11 Nov 2019 00:00:00 GMThttp://hdl.handle.net/10012/152482019-11-11T00:00:00Z