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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
In the 1980's, Gromov and Piatetski-Shapiro introduced a technique called "hybridization'' which allowed them to produce non-arithmetic hyperbolic lattices from two non-commensurable arithmetic lattices. It has been asked whether an analogous hybridization technique exists for complex hyperbolic lattices, because certain geometric obstructions make it unclear how to adapt this technique. This thesis explores one possible construction (originally due to Hunt) in depth and uses it to produce arithmetic lattices, non-arithmetic lattices, and thin subgroups in SU(2,1).
ContributorsWells, Joseph (Author) / Paupert, Julien (Thesis advisor) / Kotschwar, Brett (Committee member) / Childress, Nancy (Committee member) / Fishel, Susanna (Committee member) / Kawski, Matthias (Committee member) / Arizona State University (Publisher)
Created2019
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 advertising agency, in its variety of forms, is one of the most powerful forces in the modern world. Its products are seen globally through various multimedia outlets and they strongly impact culture and economy. Since its conception in 1843 by Volney Palmer, the advertising agency has evolved into the recognizable—and unrecognizable—firms scattered around the world today. In the United States alone, there are roughly 13.4 thousand agencies, many of which also have branches in other countries. The evolution of the modern advertising agency coincided with, and even preceded, some of the major inflection points in history. Understanding how and why changes in advertising agencies affected these inflection points provides a glimpse of understanding into the relationship between advertising, business, and societal values.
In the pages ahead we will explore the future of the advertising industry. We will analyze our research to uncover the underlying trends pointing towards what is to come and work to apply those explanations to our understanding of advertising in the future.
In the pages ahead we will explore the future of the advertising industry. We will analyze our research to uncover the underlying trends pointing towards what is to come and work to apply those explanations to our understanding of advertising in the future.
ContributorsHarris, Chase (Co-author) / Potthoff, Zachary (Co-author) / Gray, Nancy (Thesis director) / Samper, Adriana (Committee member) / Department of Information Systems (Contributor) / Department of Marketing (Contributor) / Herberger Institute for Design and the Arts (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
This research compares shifts in a SuperSpec titanium nitride (TiN) kinetic inductance detector's (KID's) resonant frequency with accepted models for other KIDs. SuperSpec, which is being developed at the University of Colorado Boulder, is an on-chip spectrometer designed with a multiplexed readout with multiple KIDs that is set up for a broadband transmission of these measurements. It is useful for detecting radiation in the mm and sub mm wavelengths which is significant since absorption and reemission of photons by dust causes radiation from distant objects to reach us in infrared and far-infrared bands. In preparation for testing, our team installed stages designed previously by Paul Abers and his group into our cryostat and designed and installed other parts necessary for the cryostat to be able to test devices on the 250 mK stage. This work included the design and construction of additional parts, a new setup for the wiring in the cryostat, the assembly, testing, and installation of several stainless steel coaxial cables for the measurements through the devices, and other cryogenic and low pressure considerations. The SuperSpec KID was successfully tested on this 250 mK stage thus confirming that the new setup is functional. Our results are in agreement with existing models which suggest that the breaking of cooper pairs in the detector's superconductor which occurs in response to temperature, optical load, and readout power will decrease the resonant frequencies. A negative linear relationship in our results appears, as expected, since the parameters are varied only slightly so that a linear approximation is appropriate. We compared the rate at which the resonant frequency responded to temperature and found it to be close to the expected value.
ContributorsDiaz, Heriberto Chacon (Author) / Mauskopf, Philip (Thesis director) / McCartney, Martha (Committee member) / Department of Physics (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
This study estimates the capitalization effect of golf courses in Maricopa County using the hedonic pricing method. It draws upon a dataset of 574,989 residential transactions from 2000 to 2006 to examine how the aesthetic, non-golf benefits of golf courses capitalize across a gradient of proximity measures. The measures for amenity value extend beyond home adjacency and include considerations for homes within a range of discrete walkability buffers of golf courses. The models also distinguish between public and private golf courses as a proxy for the level of golf course access perceived by non-golfers. Unobserved spatial characteristics of the neighborhoods around golf courses are controlled for by increasing the extent of spatial fixed effects from city, to census tract, and finally to 2000 meter golf course ‘neighborhoods.’ The estimation results support two primary conclusions. First, golf course proximity is found to be highly valued for adjacent homes and homes up to 50 meters way from a course, still evident but minimal between 50 and 150 meters, and insignificant at all other distance ranges. Second, private golf courses do not command a higher proximity premia compared to public courses with the exception of homes within 25 to 50 meters of a course, indicating that the non-golf benefits of courses capitalize similarly, regardless of course type. The results of this study motivate further investigation into golf course features that signal access or add value to homes in the range of capitalization, particularly for near-adjacent homes between 50 and 150 meters thought previously not to capitalize.
ContributorsJoiner, Emily (Author) / Abbott, Joshua (Thesis director) / Smith, Kerry (Committee member) / Economics Program in CLAS (Contributor) / School of Sustainability (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
The current trend of interconnected devices, or the internet of things (IOT) has led to the popularization of single board computers (SBC). This is primarily due to their form-factor and low price. This has led to unique networks of devices that can have unstable network connections and minimal processing power. Many parallel program- ming libraries are intended for use in high performance computing (HPC) clusters. Unlike the IOT environment described, HPC clusters will in general look to obtain very consistent network speeds and topologies. There are a significant number of software choices that make up what is referred to as the HPC stack or parallel processing stack. My thesis focused on building an HPC stack that would run on the SCB computer name the Raspberry Pi. The intention in making this Raspberry Pi cluster is to research performance of MPI implementations in an IOT environment, which had an impact on the design choices of the cluster. This thesis is a compilation of my research efforts in creating this cluster as well as an evaluation of the software that was chosen to create the parallel processing stack.
ContributorsO'Meara, Braedon Richard (Author) / Meuth, Ryan (Thesis director) / Dasgupta, Partha (Committee member) / Computer Science and Engineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
The objective of this paper is to provide an educational diagnostic into the technology of blockchain and its application for the supply chain. Education on the topic is important to prevent misinformation on the capabilities of blockchain. Blockchain as a new technology can be confusing to grasp given the wide possibilities it can provide. This can convolute the topic by being too broad when defined. Instead, the focus will be maintained on explaining the technical details about how and why this technology works in improving the supply chain. The scope of explanation will not be limited to the solutions, but will also detail current problems. Both public and private blockchain networks will be explained and solutions they provide in supply chains. In addition, other non-blockchain systems will be described that provide important pieces in supply chain operations that blockchain cannot provide. Blockchain when applied to the supply chain provides improved consumer transparency, management of resources, logistics, trade finance, and liquidity.
ContributorsKrukar, Joel Michael (Author) / Oke, Adegoke (Thesis director) / Duarte, Brett (Committee member) / Hahn, Richard (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Department of Economics (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
The town of Guadalupe, Arizona has a long history of divided residents and high poverty rates. The high levels of poverty in the town can be attributed to numerous factors, most notably high rates of drug abuse, low high school graduation rates, and teen pregnancy. The town has named one of its most pressing issues of today to be youth disengagement. There are currently a handful of residents and community members passionate about finding a solution to this issue. After working with Guadalupe's Ending Hunger Task Force and resident youth, I set out to create a program design for a Guadalupe Youth Council. This council will contribute to combating youth disengagement. The program design will assist the task force in creating a standing youth council and deciding on the structure and role the council has in the town. I will offer learning outcomes and suggestions to the Task Force, youth council staff, and the youth of the youth council. This study contains an analysis of relevant literature, youth focus group results and data, and how the information gathered has contributed to the design of the youth council. The results of this study contain recommendations about four themes within the program design of a youth council: size, recruitment, activities and engagement, and adult support. The results also explore how the youth council will impact the power, policy, and behavior of Guadalupe youth.
ContributorsBalderas, Erica Theresa (Author) / Wang, Lili (Thesis director) / Avalos, Francisco (Committee member) / School of Community Resources and Development (Contributor) / Department of Information Systems (Contributor) / W.P. Carey School of Business (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05