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Description
In 1959, Iwasawa proved that the size of the $p$-part of the class groups of a $\mathbb{Z}_p$-extension grows as a power of $p$ with exponent ${\mu}p^m+{\lambda}\,m+\nu$ for $m$ sufficiently large. Broadly, I construct conditions to verify if a given $m$ is indeed sufficiently large. More precisely, let $CG_m^i$ (class group) be the $\epsilon_i$-eigenspace component of the $p$-Sylow subgroup of the class group of the field at the $m$-th level in a $\mathbb{Z}_p$-extension; and let $IACG^i_m$ (Iwasawa analytic class group) be ${\mathbb{Z}_p[[T]]/((1+T)^{p^m}-1,f(T,\omega^{1-i}))}$, where $f$ is the associated Iwasawa power series. It is expected that $CG_m^i$ and $IACG^i_m$ be isomorphic, providing us with a powerful connection between algebraic and analytic techniques; however, as of yet, this isomorphism is unestablished in general. I consider the existence and the properties of an exact sequence $$0\longrightarrow\ker{\longrightarrow}CG_m^i{\longrightarrow}IACG_m^i{\longrightarrow}\textrm{coker}\longrightarrow0.$$ In the case of a $\mathbb{Z}_p$-extension where the Main Conjecture is established, there exists a pseudo-isomorphism between the respective inverse limits of $CG_m^i$ and $IACG_m^i$. I consider conditions for when such a pseudo-isomorphism immediately gives the existence of the desired exact sequence, and I also consider work-around methods that preserve cardinality for otherwise. However, I primarily focus on constructing conditions to verify if a given $m$ is sufficiently large that the kernel and cokernel of the above exact sequence have become well-behaved, providing similarity of growth both in the size and in the structure of $CG_m^i$ and $IACG_m^i$; as well as conditions to determine if any such $m$ exists. The primary motivating idea is that if $IACG_m^i$ is relatively easy to work with, and if the relationship between $CG_m^i$ and $IACG_m^i$ is understood; then $CG_m^i$ becomes easier to work with. Moreover, while the motivating framework is stated concretely in terms of the cyclotomic $\mathbb{Z}_p$-extension of $p$-power roots of unity, all results are generally applicable to arbitrary $\mathbb{Z}_p$-extensions as they are developed in terms of Iwasawa-Theory-inspired, yet abstracted, algebraic results on maps between inverse limits.
ContributorsElledge, Shawn Michael (Author) / Childress, Nancy (Thesis advisor) / Bremner, Andrew (Committee member) / Fishel, Susanna (Committee member) / Jones, John (Committee member) / Paupert, Julien (Committee member) / Arizona State University (Publisher)
Created2013
Description
In 1984, Sinnott used $p$-adic measures on $\mathbb{Z}_p$ to give a new proof of the Ferrero-Washington Theorem for abelian number fields by realizing $p$-adic $L$-functions as (essentially) the $Gamma$-transform of certain $p$-adic rational function measures. Shortly afterward, Gillard and Schneps independently adapted Sinnott's techniques to the case of $p$-adic $L$-functions associated to elliptic curves with complex multiplication (CM) by realizing these $p$-adic $L$-functions as $Gamma$-transforms of certain $p$-adic rational function measures. The results in the CM case give the vanishing of the Iwasawa $mu$-invariant for certain $mathbb{Z}_p$-extensions of imaginary quadratic fields constructed from torsion points of CM elliptic curves.
In this thesis, I develop the theory of $p$-adic measures on $mathbb{Z}_p^d$, with particular interest given to the case of $d>1$. Although I introduce these measures within the context of $p$-adic integration, this study includes a strong emphasis on the interpretation of $p$-adic measures as $p$-adic power series. With this dual perspective, I describe $p$-adic analytic operations as maps on power series; the most important of these operations is the multivariate $Gamma$-transform on $p$-adic measures.
This thesis gives new significance to product measures, and in particular to the use of product measures to construct measures on $mathbb{Z}_p^2$ from measures on $mathbb{Z}_p$. I introduce a subring of pseudo-polynomial measures on $mathbb{Z}_p^2$ which is closed under the standard operations on measures, including the $Gamma$-transform. I obtain results on the Iwasawa-invariants of such pseudo-polynomial measures, and use these results to deduce certain continuity results for the $Gamma$-transform. As an application, I establish the vanishing of the Iwasawa $mu$-invariant of Yager's two-variable $p$-adic $L$-function from measure theoretic considerations.
In this thesis, I develop the theory of $p$-adic measures on $mathbb{Z}_p^d$, with particular interest given to the case of $d>1$. Although I introduce these measures within the context of $p$-adic integration, this study includes a strong emphasis on the interpretation of $p$-adic measures as $p$-adic power series. With this dual perspective, I describe $p$-adic analytic operations as maps on power series; the most important of these operations is the multivariate $Gamma$-transform on $p$-adic measures.
This thesis gives new significance to product measures, and in particular to the use of product measures to construct measures on $mathbb{Z}_p^2$ from measures on $mathbb{Z}_p$. I introduce a subring of pseudo-polynomial measures on $mathbb{Z}_p^2$ which is closed under the standard operations on measures, including the $Gamma$-transform. I obtain results on the Iwasawa-invariants of such pseudo-polynomial measures, and use these results to deduce certain continuity results for the $Gamma$-transform. As an application, I establish the vanishing of the Iwasawa $mu$-invariant of Yager's two-variable $p$-adic $L$-function from measure theoretic considerations.
ContributorsZinzer, Scott Michael (Author) / Childress, Nancy (Thesis advisor) / Bremner, Andrew (Committee member) / Fishel, Susanna (Committee member) / Jones, John (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2015
Description
Diophantine arithmetic is one of the oldest branches of mathematics, the search
for integer or rational solutions of algebraic equations. Pythagorean triangles are
an early instance. Diophantus of Alexandria wrote the first related treatise in the
fourth century; it was an area extensively studied by the great mathematicians of the seventeenth century, including Euler and Fermat.
The modern approach is to treat the equations as defining geometric objects, curves, surfaces, etc. The theory of elliptic curves (or curves of genus 1, which are much used in modern cryptography) was developed extensively in the twentieth century, and has had great application to Diophantine equations. This theory is used in application to the problems studied in this thesis. This thesis studies some curves of high genus, and possible solutions in both rationals and in algebraic number fields, generalizes some old results and gives answers to some open problems in the literature. The methods involve known techniques together with some ingenious tricks. For example, the equations $y^2=x^6+k$, $k=-39,\,-47$, the two previously unsolved cases for $|k|<50$, are solved using algebraic number theory and the ‘elliptic Chabauty’ method. The thesis also studies the genus three quartic curves $F(x^2,y^2,z^2)=0$ where F is a homogeneous quadratic form, and extend old results of Cassels, and Bremner. It is a very delicate matter to find such curves that have no rational points, yet which do have points in odd-degree extension fields of the rationals.
The principal results of the thesis are related to surfaces where the theory is much less well known. In particular, the thesis studies some specific families of surfaces, and give a negative answer to a question in the literature regarding representation of integers n in the form $n=(x+y+z+w)(1/x+1/y+1/z+1/w).$ Further, an example, the first such known, of a quartic surface $x^4+7y^4=14z^4+18w^4$ is given with remarkable properties: it is everywhere locally solvable, yet has no non-zero rational point, despite having a point in (non-trivial) odd-degree extension fields of the rationals. The ideas here involve manipulation of the Hilbert symbol, together with the theory of elliptic curves.
for integer or rational solutions of algebraic equations. Pythagorean triangles are
an early instance. Diophantus of Alexandria wrote the first related treatise in the
fourth century; it was an area extensively studied by the great mathematicians of the seventeenth century, including Euler and Fermat.
The modern approach is to treat the equations as defining geometric objects, curves, surfaces, etc. The theory of elliptic curves (or curves of genus 1, which are much used in modern cryptography) was developed extensively in the twentieth century, and has had great application to Diophantine equations. This theory is used in application to the problems studied in this thesis. This thesis studies some curves of high genus, and possible solutions in both rationals and in algebraic number fields, generalizes some old results and gives answers to some open problems in the literature. The methods involve known techniques together with some ingenious tricks. For example, the equations $y^2=x^6+k$, $k=-39,\,-47$, the two previously unsolved cases for $|k|<50$, are solved using algebraic number theory and the ‘elliptic Chabauty’ method. The thesis also studies the genus three quartic curves $F(x^2,y^2,z^2)=0$ where F is a homogeneous quadratic form, and extend old results of Cassels, and Bremner. It is a very delicate matter to find such curves that have no rational points, yet which do have points in odd-degree extension fields of the rationals.
The principal results of the thesis are related to surfaces where the theory is much less well known. In particular, the thesis studies some specific families of surfaces, and give a negative answer to a question in the literature regarding representation of integers n in the form $n=(x+y+z+w)(1/x+1/y+1/z+1/w).$ Further, an example, the first such known, of a quartic surface $x^4+7y^4=14z^4+18w^4$ is given with remarkable properties: it is everywhere locally solvable, yet has no non-zero rational point, despite having a point in (non-trivial) odd-degree extension fields of the rationals. The ideas here involve manipulation of the Hilbert symbol, together with the theory of elliptic curves.
ContributorsNguyen, Xuan Tho (Author) / Bremner, Andrew (Thesis advisor) / Childress, Nancy (Committee member) / Jones, John (Committee member) / Quigg, John (Committee member) / Fishel, Susanna (Committee member) / Arizona State University (Publisher)
Created2019
Description
The inherent risk in testing drugs has been hotly debated since the government first started regulating the drug industry in the early 1900s. Who can assume the risks associated with trying new pharmaceuticals is unclear when looked at through society's lens. In the mid twentieth century, the US Food and Drug Administration (FDA) published several guidance documents encouraging researchers to exclude women from early clinical drug research. The motivation to publish those documents and the subsequent guidance documents in which the FDA and other regulatory offices established their standpoints on women in drug research may have been connected to current events at the time. The problem of whether women should be involved in drug research is a question of who can assume risk and who is responsible for disseminating what specific kinds of information. The problem tends to be framed as one that juxtaposes the health of women and fetuses and sets their health as in opposition. That opposition, coupled with the inherent uncertainty in testing drugs, provides for a complex set of issues surrounding consent and access to information.
ContributorsMeek, Caroline Jane (Author) / Maienschein, Jane (Thesis director) / Brian, Jennifer (Committee member) / School of Life Sciences (Contributor) / Sanford School of Social and Family Dynamics (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
Social-emotional learning (SEL) methods are beginning to receive global attention in primary school education, yet the dominant emphasis on implementing these curricula is in high-income, urbanized areas. Consequently, the unique features of developing and integrating such methods in middle- or low-income rural areas are unclear. Past studies suggest that students exposed to SEL programs show an increase in academic performance, improved ability to cope with stress, and better attitudes about themselves, others, and school, but these curricula are designed with an urban focus. The purpose of this study was to conduct a needs-based analysis to investigate components specific to a SEL curriculum contextualized to rural primary schools. A promising organization committed to rural educational development is Barefoot College, located in Tilonia, Rajasthan, India. In partnership with Barefoot, we designed an ethnographic study to identify and describe what teachers and school leaders consider the highest needs related to their students' social and emotional education. To do so, we interviewed 14 teachers and school leaders individually or in a focus group to explore their present understanding of “social-emotional learning” and the perception of their students’ social and emotional intelligence. Analysis of this data uncovered common themes among classroom behaviors and prevalent opportunities to address social and emotional well-being among students. These themes translated into the three overarching topics and eight sub-topics explored throughout the curriculum, and these opportunities guided the creation of the 21 modules within it. Through a design-based research methodology, we developed a 40-hour curriculum by implementing its various modules within seven Barefoot classrooms alongside continuous reiteration based on teacher feedback and participant observation. Through this process, we found that student engagement increased during contextualized SEL lessons as opposed to traditional methods. In addition, we found that teachers and students preferred and performed better with an activities-based approach. These findings suggest that rural educators must employ particular teaching strategies when addressing SEL, including localized content and an experiential-learning approach. Teachers reported that as their approach to SEL shifted, they began to unlock the potential to build self-aware, globally-minded students. This study concludes that social and emotional education cannot be treated in a generalized manner, as curriculum development is central to the teaching-learning process.
ContributorsBucker, Delaney Sue (Author) / Carrese, Susan (Thesis director) / Barab, Sasha (Committee member) / School of Life Sciences (Contributor, Contributor) / School of Civic & Economic Thought and Leadership (Contributor) / School of International Letters and Cultures (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
As of 2019, 30 US states have adopted abortion-specific informed consent laws that require state health departments to develop and disseminate written informational materials to patients seeking an abortion. Abortion is the only medical procedure for which states dictate the content of informed consent counseling. State abortion counseling materials have been criticized for containing inaccurate and misleading information, but overall, informed consent laws for abortion do not often receive national attention. The objective of this project was to determine the importance of informed consent laws to achieving the larger goal of dismantling the right to abortion. I found that informed consent counseling materials in most states contain a full timeline of fetal development, along with information about the risks of abortion, the risks of childbirth, and alternatives to abortion. In addition, informed consent laws for abortion are based on model legislation called the “Women’s Right to Know Act” developed by Americans United for Life (AUL). AUL calls itself the legal architect of the pro-life movement and works to pass laws at the state level that incrementally restrict abortion access so that it gradually becomes more difficult to exercise the right to abortion established by Roe v. Wade. The “Women’s Right to Know Act” is part of a larger package of model legislation called the “Women’s Protection Project,” a cluster of laws that place restrictions on abortion providers, purportedly to protect women, but actually to decrease abortion access. “Women’s Right to Know” counseling laws do not directly deny access to abortion, but they do reinforce key ideas important to the anti-abortion movement, like the concept of fetal personhood, distrust in medical professionals, the belief that pregnant people cannot be fully autonomous individuals, and the belief that abortion is not an ordinary medical procedure and requires special government oversight. “Women’s Right to Know” laws use the language of informed consent and the purported goal of protecting women to legitimize those ideas, and in doing so, they significantly undermine the right to abortion. The threat to abortion rights posed by laws like the “Women’s Right to Know” laws indicates the need to reevaluate and strengthen our ethical defense of the right to abortion.
ContributorsVenkatraman, Richa (Author) / Maienschein, Jane (Thesis director) / Brian, Jennifer (Thesis director) / Abboud, Carolina (Committee member) / Historical, Philosophical & Religious Studies (Contributor) / School of Life Sciences (Contributor, Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
Turbidity is a known problem for UV water treatment systems as suspended particles can shield contaminants from the UV radiation. UV systems that utilize a reflective radiation chamber may be able to decrease the impact of turbidity on the efficacy of the system. The purpose of this study was to determine how kaolin clay and gram flour turbidity affects inactivation of Escherichia coli (E. coli) when using a UV system with a reflective chamber. Both sources of turbidity were shown to reduce the inactivation of E. coli with increasing concentrations. Overall, it was shown that increasing kaolin clay turbidity had a consistent effect on reducing UV inactivation across UV doses. Log inactivation was reduced by 1.48 log for the low UV dose and it was reduced by at least 1.31 log for the low UV dose. Gram flour had a similar effect to the clay at the lower UV dose, reducing log inactivation by 1.58 log. At the high UV dose, there was no change in UV inactivation with an increase in turbidity. In conclusion, turbidity has a significant impact on the efficacy of UV disinfection. Therefore, removing turbidity from water is an essential process to enhance UV efficiency for the disinfection of microbial pathogens.
ContributorsMalladi, Rohith (Author) / Abbaszadegan, Morteza (Thesis director) / Alum, Absar (Committee member) / Fox, Peter (Committee member) / School of Human Evolution & Social Change (Contributor) / School of Life Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
Aquatic macroinvertebrates are important for many ecological processes within river ecosystems and, as a result, their abundance and diversity are considered indicators of water quality and ecosystem health. Macroinvertebrates can be classified into functional feeding groups (FFG) based on morphological-behavioral adaptations. FFG ratios can shift due to changes in normal disturbance patterns, such as changes in precipitation, and from human impact. Due to their increased sensitivity to environmental changes, it has become more important to protect and monitor aquatic and riparian communities in arid regions as climate change continues to intensify. Therefore, the diversity and richness of macroinvertebrate FFGs before and after monsoon and winter storm seasons were analyzed to determine the effect of flow-related disturbances. Ecosystem size was also considered, as watershed area has been shown to affect macroinvertebrate diversity. There was no strong support for flow-related disturbance or ecosystem size on macroinvertebrate diversity and richness. This may indicate a need to explore other parameters of macroinvertebrate community assembly. Establishing how disturbance affects aquatic macroinvertebrate communities will provide a key understanding as to what the stream communities will look like in the future, as anthropogenic impacts continue to affect more vulnerable ecosystems.
ContributorsSainz, Ruby (Author) / Sabo, John (Thesis director) / Grimm, Nancy (Committee member) / Lupoli, Christina (Committee member) / School of Life Sciences (Contributor, Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
This study evaluates medical pluralism among 1.5 generation Indian American immigrants. 1.5 generation Indian Americans (N=16) were surveyed regarding their engagement in complementary and alternative medical systems (CAM), how immigration affected that, and reasons for and for not continuing the use of CAM. Results indicated most 1.5 Indian immigrants currently engage in CAM, given that their parents also engage in CAM. The top reasons respondents indicated continued engagement in CAM was that it has no side effects and is preventative. Reasons for not practicing CAM included feeling out of place, not living with parents or not believing in CAM. After immigration, most participants decreased or stopped their engagement in CAM. More women than men continued to practice CAM after immigration. From the results, it was concluded that CAM is still important to 1.5 generation Indian immigrants.
ContributorsMurugesh, Subhiksha (Author) / Stotts, Rhian (Thesis director) / Mubayi, Anuj (Committee member) / School of Human Evolution & Social Change (Contributor) / School of Life Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
Many factors are at play within the genome of an organism, contributing to much of the diversity and variation across the tree of life. While the genome is generally encoded by four nucleotides, A, C, T, and G, this code can be expanded. One particular mechanism that we examine in this thesis is modification of bases—more specifically, methylation of Adenine (m6A) within the GATC motif of Escherichia coli. These methylated adenines are especially important in a process called methyl-directed mismatch repair (MMR), a pathway responsible for repairing errors in the DNA sequence produced by replication. In this pathway, methylated adenines identify the parent strand and direct the repair proteins to correct the erroneous base in the daughter strand. While the primary role of methylated adenines at GATC sites is to direct the MMR pathway, this methylation has also been found to affect other processes, such as gene expression, the activity of transposable elements, and the timing of DNA replication. However, in the absence of MMR, the ability of these other processes to maintain adenine methylation and its targets is unknown.
To determine if the disruption of the MMR pathway results in the reduced conservation of methylated adenines as well as an increased tolerance for mutations that result in the loss or gain of new GATC sites, we surveyed individual clones isolated from experimentally evolving wild-type and MMR-deficient (mutL- ;conferring an 150x increase in mutation rate) populations of E. coli with whole-genome sequencing. Initial analysis revealed a lack of mutations affecting methylation sites (GATC tetranucleotides) in wild-type clones. However, the inherent low mutation rates conferred by the wild-type background render this result inconclusive, due to a lack of statistical power, and reveal a need for a more direct measure of changes in methylation status. Thus as a first step to comparative methylomics, we benchmarked four different methylation-calling pipelines on three biological replicates of the wildtype progenitor strain for our evolved populations.
While it is understood that these methylated sites play a role in the MMR pathway, it is not fully understood the full extent of their effect on the genome. Thus the goal of this thesis was to better understand the forces which maintain the genome, specifically concerning m6A within the GATC motif.
To determine if the disruption of the MMR pathway results in the reduced conservation of methylated adenines as well as an increased tolerance for mutations that result in the loss or gain of new GATC sites, we surveyed individual clones isolated from experimentally evolving wild-type and MMR-deficient (mutL- ;conferring an 150x increase in mutation rate) populations of E. coli with whole-genome sequencing. Initial analysis revealed a lack of mutations affecting methylation sites (GATC tetranucleotides) in wild-type clones. However, the inherent low mutation rates conferred by the wild-type background render this result inconclusive, due to a lack of statistical power, and reveal a need for a more direct measure of changes in methylation status. Thus as a first step to comparative methylomics, we benchmarked four different methylation-calling pipelines on three biological replicates of the wildtype progenitor strain for our evolved populations.
While it is understood that these methylated sites play a role in the MMR pathway, it is not fully understood the full extent of their effect on the genome. Thus the goal of this thesis was to better understand the forces which maintain the genome, specifically concerning m6A within the GATC motif.
ContributorsBoyer, Gwyneth (Author) / Lynch, Michael (Thesis director) / Behringer, Megan (Committee member) / Geiler-Samerotte, Kerry (Committee member) / School of Life Sciences (Contributor) / Department of Psychology (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05