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Modernism happened in mathematics, too. Between about 1890 and 1930, the field turned toward dramatic, previously unimaginable abstraction, as radical innovations dispensed with inherited conventions. The recognition of non-Euclidean geometry raised the strange notion that multiple, contradictory realms of mathematics could coexist and remain equally correct, even if some seemed non-representative of the world or nonsensical in human experience. Mathematicians grew frenzied in the effort to resolve newfound paradoxes. The turn of the century saw a series of attempts to rethink math’s most basic foundations, and the 1920s witnessed explosive controversies about the role of language in mathematics. These developments do not merely constitute change in one field; they demonstrate that modernism was a broader cultural phenomenon than literary scholars have realized. This dissertation begins from the shared intellectual history of literary and mathematical modernism: a common attempt to rethink foundational axioms, a common ambivalence toward growing abstraction, and a common interest in and anxiety about form. This convergence, I argue, exposes an interpretive dilemma intrinsic to form in mathematics and writing alike. Modernist shifts in cultural assumptions gave rise to newly abstract forms that revealed form to have been always already abstract—so abstract as to estrange its own cultural function. Formalism, in literature, skates uncomfortably between profundity and superficiality, because attention to literary language in and of itself risks discounting language’s fundamental function: to express, refer, and communicate. But decades before formalism named a school of thought in literary interpretation, formalism—by that name—identified an understanding, and an interpretation, of mathematics. Mathematical formalism has roots reaching back to the 17th century, but after a long history it reached its heyday in the 1920s, when the field’s escalating abstraction had led many to question math’s grounding in the real world, i.e., to doubt its basic capacity for meaning. The influential mathematician David Hilbert argued that such a grounding could be secured via attention to form—in effect, that formalism could rescue mathematical meaning by deferring that meaning, first treating mathematics as though it were only meaningless marks on paper. This dissertation uses Hilbert’s mathematical formalism to explain the apparent contradictions of literary formalism. I demonstrate that writers and mathematicians discovered the same solutions to an urgent common problem, developing technical apparatuses (variables, equivalence relations, axiom systems) through which meaning can develop directly from patterns and rules rather than terms and symbols. Mathematics has a remarkable relationship with reality, because it is both a maddeningly ethereal realm of thought and an undeniably descriptive tool that reliably predicts physical phenomena. I argue that the bond between mathematics and science is not only a metaphor for the relationship between literature and reality, but an imitable model that multiple modernist authors knowingly seized upon and manipulated. It is in this mathematical manner that the most abstract, formal, and seemingly unworldly moments in modernism acquired relevance to history, world, and culture. They did so by describing the world without making reference to the world, expressing the shapes of reality via pattern and form. My first chapter, “Opposition, Polysemy, Pattern: Virginia Woolf’s Mathematical Generality,” begins from the more commonly understood relation between literature and mathematics: one of radical difference. In her early novel Night and Day Woolf resists the rigidity of mathematics, consistently imagining mathematical communication as a negative image of her own art, an inaccessible language that resists translation into English. But this opposition, I argue, also makes visible a likeness between the two fields, glimpsed in the “sacred pages of symbols and figures” that Woolf consistently uses to frame their common signs and signifying processes. The formalism of mathematical writing thus promises to explain many of the escalating formal complexities of Woolf’s later novels. I demonstrate that, even where she makes no direct reference to mathematics, Woolf uses the greater generality of mathematical modes of meaning to reinvent the force of ambiguity in Jacob’s Room and contradiction in To the Lighthouse, before settling into the enormous patterns that establish a formula for “the life of anybody” in The Waves. In mathematics, such generality is most meaningful in conjunction with structured systems—the patterns that make unrestricted meaning possible. I take up the problem of defining and theorizing literary structure in the following two chapters. In “The Metaphor as an Equation: The Formal Abstraction of Ezra Pound’s Precisions,” I trace Pound’s repeated descriptions of image and metaphor as instruments of equation, and I demonstrate that technical mathematical definitions of equation characterize some of Imagism’s starkest innovations. Between 1890 and 1910, mathematical understandings of equation had shifted, reaching beyond number to become newly applicable to shapes, ideas, and words. Pound, I argue, harnessed recent mathematical redefinitions of equality to place a new weight on syntax: in his early poetry, semicolons unseat verbs, and ellipsis makes possible new forms of comparison. Pound’s early metaphors invented a newly symmetric and iterable relationship between tenor and vehicle, one capable of underlying more procreant forms. And form, here, acquires a newly interdisciplinary definition: the patterns and arrangements of relations between things that can exist independent of those things.Pattern has to start from somewhere: a problem that consumed Eliot across his long career. In pure, modern mathematics, absolute knowledge and incontrovertible proof are built from patterns that always stand upon unproven assumptions—axioms. Eliot had examined those axioms in detail. In 1914, as a student at Harvard, Eliot took Bertrand Russell’s graduate course in advanced logic, intensively studying the modernist axiom systems of Gottlob Frege, Alfred North Whitehead, and Russell himself. In “From Axiom to Leap of Faith: T.S. Eliot’s Formal Systems,” I trace Eliot’s ambivalent preoccupation with logical assumption in “Prufrock” and The Waste Land and argue that, in Ash Wednesday and Four Quartets, Eliot ultimately used Russell’s formal systems to reimagine poems as infinitely intricate patterns built from finite starting points: an idea that already existed, in more troubled ways, in his earliest poetry. Eliot conceived of the most complex poetic forms as axiom systems, wherein whole aesthetic universes develop from mere handfuls of beliefs and linguistic links, just as, in mathematical axiom systems, complex meaning and sophisticated beauty develop from strikingly few assumptions, definitions, and rules. Axiom systems refer, intrinsically and everywhere, to themselves, and in this way the patterns loop back upon themselves, at once referring to themselves and anticipating their own analysis. I conclude by turning from poetry and fiction to the literary scholarship and theory engendered by modernism, generalizing the ways in which the philosophy of mathematics reframes the critical stumbling blocks of form and reference. Mathematics was a regular trope, and often a conceptual source, in the first defining works of twentieth-century formalist criticism, evincing a continuing association between the logical analysis of form and the commingling of mathematics and language. The Russian formalists, British practical critics, and American New Critics all saw mathematics as a closed linguistic system, making it uniquely useful as a descriptive analogue for literary form: mathematics offered the prime example of a system that models isolated signifying processes while lying (by general consensus) utterly outside of literature. I argue that the mathematics of form made these literary formalisms a natural and necessary response to literary modernism, determining their near-simultaneous development across different cultural contexts. When modernist form unsettled modernist content, a meta-content emerged: texts by Woolf, Pound, and Eliot all self-consciously refer to the interpretive difficulties they create. During exactly the same era, Hilbert pushed the intuitive sense of mathematics into a separate plane only to find that a new discipline derived from that distinction: metamathematics, which uses mathematics to study mathematics itself. Far from pushing the world out, formalist analysis indirectly drags the world back in, for writers and mathematicians alike. “A Few Sheets of Paper Covered with Arbitrary Symbols”: Formalism, Modernism, Mathematics thus culminates in the argument that the apparent paradoxes of formal literary interpretation—wherein formalist reading sees only the façade yet penetrates to the heart of the thing itself—are not coincidental to how we have practiced literary formalism. In fact, they are logically attendant on any application of formalist analysis, in any field. Formalism depends on the deferral of meaning but then, elaborating its own rules, bypasses that assumption to uncover a meaningful pattern, a meaning in pattern. Modernists from Woolf to Hilbert discovered and elaborated this interpretive loop. Our literary practice of formalism today remains dependent on that logical process of form.

The rock-paper-scissor game -- which is characterized by three strategies R,P,S, satisfying the non-transitive relations S excludes P, P excludes R, and R excludes S -- serves as a simple prototype for studying more complex non-transitive systems. For well-mixed systems where interactions result in fitness reductions of the losers exceeding fitness gains of the winners, classical theory predicts that two strategies go extinct. The effects of spatial heterogeneity and dispersal rates on this outcome are analyzed using a general framework for evolutionary games in patchy landscapes. The analysis reveals that coexistence is determined by the rates at which dominant strategies invade a landscape occupied by the subordinate strategy (e.g. rock invades a landscape occupied by scissors) and the rates at which subordinate strategies get excluded in a landscape occupied by the dominant strategy (e.g. scissor gets excluded in a landscape occupied by rock). These invasion and exclusion rates correspond to eigenvalues of the linearized dynamics near single strategy equilibria. Coexistence occurs when the product of the invasion rates exceeds the product of the exclusion rates. Provided there is sufficient spatial variation in payoffs, the analysis identifies a critical dispersal rate $d^*$ required for regional persistence. For dispersal rates below $d^*$, the product of the invasion rates exceed the product of the exclusion rates and the rock-paper-scissor metacommunities persist regionally despite being extinction prone locally. For dispersal rates above $d^*$, the product of the exclusion rates exceed the product of the invasion rates and the strategies are extinction prone. These results highlight the delicate interplay between spatial heterogeneity and dispersal in mediating long-term outcomes for evolutionary games., Comment: 31pages, 5 figures

Significance Prostate cancer is the most common cancer among men. An accurate diagnosis of its severity at detection plays a major role in improving their survival. Recently, machine learning models using biomarkers identified from Raman micro-spectroscopy discriminated intraductal carcinoma of the prostate (IDC-P) from cancer tissue with a ≥85 % detection accuracy and differentiated high-grade prostatic intraepithelial neoplasia (HGPIN) from IDC-P with a ≥97.8 % accuracy. Aim To improve the classification performance of machine learning models identifying different types of prostate cancer tissue using a new dimensional reduction technique. Approach A radial basis function (RBF) kernel support vector machine (SVM) model was trained on Raman spectra of prostate tissue from a 272-patient cohort (Centre hospitalier de l'Universite de Montreal, CHUM) and tested on two independent cohorts of 76 patients [University Health Network (UHN)] and 135 patients (Centre hospitalier universitaire de Quebec-Universite Laval, CHUQc-UL). Two types of engineered features were used. Individual intensity features, i.e., Raman signal intensity measured at particular wavelengths and novel Raman spectra fitted peak features consisting of peak heights and widths. Results Combining engineered features improved classification performance for the three aforementioned classification tasks. The improvements for IDC-P/cancer classification for the UHN and CHUQc-UL testing sets in accuracy, sensitivity, specificity, and area under the curve (AUC) are (numbers in parenthesis are associated with the CHUQc-UL testing set): +4 % (+8 % ), +7 % (+9 % ), +2 % (6%), +9 (+9) with respect to the current best models. Discrimination between HGPIN and IDC-P was also improved in both testing cohorts: +2.2 % (+1.7 % ), +4.5 % (+3.6 % ), +0 % (+0 % ), +2.3 (+0). While no global improvements were obtained for the normal versus cancer classification task [+0 % (-2 % ), +0 % (-3 % ), +2 % (-2 % ), +4 (+3)], the AUC was improved in both testing sets. Conclusions Combining individual intensity features and novel Raman fitted peak features, improved the classification performance on two independent and multicenter testing sets in comparison to using only individual intensity features.

Author(s): Kapre, Rohan; Zhou, Junhan; Li, Xinzhe; Beckett, Laurel; Louie, Angelique Y | Abstract: PurposeTo demonstrate that constant coefficient of variation (CV), but nonconstant absolute variance in MRI relaxometry (T1 , T2 , R1 , R2 ) data leads to erroneous conclusions based on standard linear models such as ordinary least squares (OLS). We propose a gamma generalized linear model identity link (GGLM-ID) framework that factors the inherent CV into parameter estimates. We first examined the effects on calculations of contrast agent relaxivity before broadening to other applications such as analysis of variance (ANOVA) and liver iron content (LIC).MethodsEightnmodels including OLS and GGLM-ID were initially fit to data obtained on sulfated dextran iron oxide (SDIO) nanoparticles. Both a resampling simulation on the data as well as two separate Monte Carlo simulations (with and without concentration error) were performed to determine mean square error (MSE) and type I error rate. We then evaluated the performance of OLS/GGLM-ID on R1 repeatability and LIC data sets.ResultsOLS had an MSE of 4-5× that of GGLM-ID as well as a type I error rate of 20-30%, whereas GGLM-ID was near the nominal 5% level in the relaxivity study. Only OLS found statistically significant effects of MRI facility on relaxivity in an R1 repeatability study, but no significant differences were found in a resampling, whereas GGLM was more consistent. GGLM-ID was also superior to OLS for modeling LIC.ConclusionsOLS leads to erroneous conclusions when analyzing MRI relaxometry data. GGLM-ID factors in the inherent CV of an MRI experiment, leading to more reproducible conclusions.

The brew temperature is widely considered a key parameter affecting the final quality of coffee, with a temperature near 93 °C often described as optimal. In particular, drip brewers that do not achieve a minimum brew temperature of 92 °C within a prescribed time period fail their certification. There is little empirical evidence in terms of rigorous sensory descriptive analysis or consumer preference testing, however, to support any particular range of brew temperatures. In this study, we drip-brewed coffee to specific brew strengths, as measured by total dissolved solids (TDS), and extraction yields, as measured by percent extraction (PE), spanning the range of the classic Coffee Brewing Control Chart. Three separate brew temperatures of 87 °C, 90 °C, or 93 °C were tested, adjusting the grind size and overall brew time as necessary to achieve the target TDS and PE. Although the TDS and PE both significantly affected the sensory profile of the coffee, surprisingly the brew temperature had no appreciable impact. We conclude that brew temperature should be considered as only one of several parameters that affect the extraction dynamics, and that ultimately the sensory profile is governed by differences in TDS and PE rather than the brew temperature, at least over the range of temperatures tested.

Motivation We introduce PRINCESS, a privacy-preserving international collaboration framework for analyzing rare disease genetic data that are distributed across different continents. PRINCESS leverages Software Guard Extensions (SGX) and hardware for trustworthy computation. Unlike a traditional international collaboration model, where individual-level patient DNA are physically centralized at a single site, PRINCESS performs a secure and distributed computation over encrypted data, fulfilling institutional policies and regulations for protected health information. Results To demonstrate PRINCESS’ performance and feasibility, we conducted a family-based allelic association study for Kawasaki Disease, with data hosted in three different continents. The experimental results show that PRINCESS provides secure and accurate analyses much faster than alternative solutions, such as homomorphic encryption and garbled circuits (over 40 000× faster). Availability and Implementation https://github.com/achenfengb/PRINCESS_opensource Supplementary information Supplementary data are available at Bioinformatics online.

© 2016. Published by Oxford University Press. With expanded access to, and decreased costs of, mass spectrometry, investigators are collecting and analyzing multiple biological matrices from the same subject such as serum, plasma, tissue and urine to enhance biomarker discoveries, understanding of disease processes and identification of therapeutic targets. Commonly, each biological matrix is analyzed separately, but multivariate methods such as MANOVAs that combine information from multiple biological matrices are potentially more powerful. However, mass spectrometric data typically contain large amounts of missing values, and imputation is often used to create complete data sets for analysis. The effects of imputation on multiple biological matrix analyses have not been studied. We investigated the effects of seven imputation methods (half minimum substitution, mean substitution, k-nearest neighbors, local least squares regression, Bayesian principal components analysis, singular value decomposition and random forest), on the within-subject correlation of compounds between biological matrices and its consequences on MANOVA results. Through analysis of three real omics data sets and simulation studies, we found the amount of missing data and imputation method to substantially change the between-matrix correlation structure. The magnitude of the correlations was generally reduced in imputed data sets, and this effect increased with the amount of missing data. Significant results from MANOVA testing also were substantially affected. In particular, the number of false positives increased with the level of missing data for all imputation methods. No one imputation method was universally the best, but the simple substitution methods (Half Minimum and Mean) consistently performed poorly.

Algorithms for evaluating and optimizing the useful anomalous correlation and the anomalous signal in a SAD data set are described., A key challenge in the SAD phasing method is solving a structure when the anomalous signal-to-noise ratio is low. Here, algorithms and tools for evaluating and optimizing the useful anomalous correlation and the anomalous signal in a SAD experiment are described. A simple theoretical framework [Terwilliger et al. (2016 ▸), Acta Cryst. D72, 346–358] is used to develop methods for planning a SAD experiment, scaling SAD data sets and estimating the useful anomalous correlation and anomalous signal in a SAD data set. The phenix.plan_sad_experiment tool uses a database of solved and unsolved SAD data sets and the expected characteristics of a SAD data set to estimate the probability that the anomalous substructure will be found in the SAD experiment and the expected map quality that would be obtained if the substructure were found. The phenix.scale_and_merge tool scales unmerged SAD data from one or more crystals using local scaling and optimizes the anomalous signal by identifying the systematic differences among data sets, and the phenix.anomalous_signal tool estimates the useful anomalous correlation and anomalous signal after collecting SAD data and estimates the probability that the data set can be solved and the likely figure of merit of phasing.

This paper discusses the development of graphical user interfaces (GUIs) to illustrate sampling from a trinomial distribution by the natural extension of Galton's Quincunx to three dimensions.

Background Many studies used the older ActiGraph (7164) for physical activity measurement, but this model has been replaced with newer ones (e.g., GT3X+). The assumption that new generation models are more accurate has been questioned, especially for measuring lower intensity levels. The low-frequency extension (LFE) increases the low-intensity sensitivity of newer models, but its comparability with older models is unknown. This study compared step counts and physical activity collected with the 7164 and GT3X + using the Normal Filter and the LFE (GT3X+N and GT3X+LFE, respectively). Findings Twenty-five adults wore 2 accelerometer models simultaneously for 3Âdays and were instructed to engage in typical behaviors. Average daily step counts and minutes per day in nonwear, sedentary, light, moderate, and vigorous activity were calculated. Repeated measures ANOVAs with post-hoc pairwise comparisons were used to compare mean values. Means for the GT3X+N and 7164 were significantly different in 4 of the 6 categories (p 

We developed a generalized linear model of QTL mapping for discrete traits in line crossing experiments. Parameter estimation was achieved using two different algorithms, a mixture model-based EM (expectation–maximization) algorithm and a GEE (generalized estimating equation) algorithm under a heterogeneous residual variance model. The methods were developed using ordinal data, binary data, binomial data and Poisson data as examples. Applications of the methods to simulated as well as real data are presented. The two different algorithms were compared in the data analyses. In most situations, the two algorithms were indistinguishable, but when large QTL are located in large marker intervals, the mixture model-based EM algorithm can fail to converge to the correct solutions. Both algorithms were coded in C++ and interfaced with SAS as a user-defined SAS procedure called PROC QTL.

This research paper presents an innovative approach to anomaly detection in maritime data, focusing on the identification of anomalous vessel groups through Automatic Identification System (AIS) data. Traditional AIS anomaly detection systems are challenged by a narrow focus on individual vessel anomalies which leaves out critical collective action information. Our work seeks to overcome this limitation by introducing a graph modeling methodology that emphasizes the collective behavior of vessels, a critical aspect often overlooked in current frameworks. We work towards a system that detects unusual patterns of coordination among groups of vessels, which is particularly relevant for defense contexts where threats such as smuggling, piracy, and adversarial operations are typically carried out by networks of collaborating entities. The research contributes to the field by offering a more comprehensive understanding of maritime traffic through the lens of collective dynamics, enhancing the detection capabilities of AIS monitoring systems. The implications of our methodology are far-reaching, providing a foundational strategy for future research in anomaly detection that could be applied to various domains where group coordination plays a pivotal role in defining anomalous behavior. This paper details our graph-based approach and suggests a trajectory for subsequent investigations into the broader applications of detecting coordinated anomalies.

This paper introduces Prio+, a privacy-preserving system for the collection of aggregate statistics, with the same model and goals in mind as the original and highly influential Prio paper by Henry Corrigan-Gibbs and Dan Boneh (NSDI 2017). As in the original Prio, each client holds a private data value (e.g. number of visits to a particular website) and a small set of servers privately compute statistical functions over the set of client values (e.g. the average number of visits). To achieve security against faulty or malicious clients, unlike Prio, Prio+ clients use Boolean secret-sharing instead of zero-knowledge proofs to convince servers that their data is of the correct form and Prio+ servers execute a share conversion protocol as needed in order to properly compute over client data. This allows us to ensure that clients' data is properly formatted essentially for free, and the work shifts to novel share-conversion protocols between servers, where some care is needed to make it efficient. Our overall approach is simpler than Prio and our Prio+ strategy reduces the client's computational burden by at least two orders of magnitude (or more depending on the statistic) while keeping server costs comparable to Prio. Prio+ permits computation of exactly the same wide range of complex statistics as the original Prio protocol, including high-dimensional linear regression over private values held by clients. We report detailed benchmarks of our Prio+ implementation and compare these to both the original Go implementation of Prio and the Mozilla implementation of Prio. Our Prio+ software is open-source and released with the same license as Prio.

Human Society is complex and involves interaction of many individuals in a society on different levels. Social Science and its fields study how people are organized into groups (Anthropology, Sociology), what are the core motivations of humans (Philosophy), what biological and psychological factors are involved in their behavior (Psychology), how humans compete and cooperate to achieve individual goals and society prosperity (Economics), and how a large society can organize and govern itself (Political Science). Underlying them all are laws of human social world that are the basis of human behavior, organization, interactions and decision making. Just like in Physics these laws can be scientifically studied and expressed using the tools of mathematics.The tools that have been used in the 4 papers included in the dissertation include: Social Network Analysis, Game Theory, Statistics and general tools of mathematical analysis. In the first 3 papers human organization into groups is explored by using Social Network as a base model of society. The first paper studies specific work behavior and organization in software development communities finding complex pattern of organization and providing a formula that can predict success of a project. The 2nd and 3rd papers use abstract mathematical models based on social networks and game theory to study human interactions, cooperation and success. The papers showed how social network structure plays a role in these interactions and helped to understand the very core of human interaction and how it can be modeled with game theory and what are important considerations for parameters used in creating specific game theory and network models.The 4th paper looks at human politics and elections and studies bias that can exist prior to elections with manipulation of electoral district maps to favor specific party, which is known as gerrymandering. Using tools of mathematics - functional analysis and statistics, the paper derives equations and core symmetries of biased and unbiased distribution of voters. These symmetries are analogues to physical symmetries that are the basis of physical laws.

We study the spectral theory of ergodic Schrödinger operators.The focus is on multi-dimensional Schrödinger operators that appear in quantum mechanics to model physical phenomena, e.g., electrons in a lattice or quasicrystal.We present two papers: \emph{On the spectra of separable 2D almost Mathieu operators} and \emph{Spectral estimates of dynamically-defined and amenable operator families}.The second paper is joint work with S.~Beckus.This dissertation explains what was done in those papers and how it closely relates to the research focus---to address specific problems related to the multi-dimensional case.The one-dimensional case is well developed since there are many results and many methods available, e.g., transfer matrices and Schrödinger cocycles.The multi-dimensional case on the other hand is notoriously difficult to study.In part of this dissertation, we consider a separable 2D discrete Schrödinger operator generated by 1D Almost Mathieu Operators.For fixed Diophantine frequencies we prove that for sufficiently small couplings the spectrum must be an interval.This complements a result by J.~Bourgain establishing that for fixed couplings the spectrum has gaps for some positive-measure Diophantine frequencies.The result generalizes to separable multi-dimensional discrete Schrödinger operators generated by 1D quasiperiodic operators whose potential is analytic and whose frequency is Diophantine.The proof is based on the study of the thickness of the spectrum of the Almost Mathieu Operator, and utilizes the Newhouse Gap Lemma on sums of Cantor sets.In the other part of this dissertation, we consider a class of multi-dimensional Schrödinger operators.The Hausdorff distance of the spectra is estimated by the distance of the underlying dynamical systems while the unimodular second-countable locally-compact Hausdorff group is amenable.Together with S.~Beckus it is proved that if the group has strict polynomial growth and both the group action and the coefficients are Lipschitz continuous, then the spectral estimate has a square root behavior or, equivalently, the spectrum map is $ \frac{1}{2} $-Hölder continuous.Also, the behavior can be improved resulting in the spectrum map being Lipschitz continuous if the coefficients are instead locally-constant.In 1990, the square root behavior was established by J.~Avron, P.~H.~M.~v.~Mouche, and B.~Simon for the Almost Mathieu Operator or, more generally, the quasiperiodic operators with Lipschitz continuous potentials.Our result extends the square root behavior to a bigger class of operators such as magnetic discrete Schrödinger operators with finite range and with Lipschitz continuous coefficients.

Beyond its normative relevance, the question of how institutions obtain, maintain, utilize, and lose authority is an important historical and conceptual issue. Early American politics in the colonial and revolutionary eras provides fruitful ground for exploring how the authority of institutions affects politics, and how political developments affect institutional authority. Applying game theory to questions of historical political development allows us to clearly identify strategic incentives and generalize from particular cases. In three papers, this dissertation combines historical case studies and formal models to investigate different aspects of legal authority in American political development.In the first paper, I investigate how purely formal constraints on political actors are enforced through coordination among citizens, and conditions under which such coordination fails. Specifically, this paper argues that economic power is a crucial variable for understanding the rule of law. I make this argument in two ways. First, I present a formal model to show how economic inequality undermines the rule of law. In the model, coordination among citizens increases the expense that elites face to implement an unconstitutional policy. But this constraint only binds when elites’ material resources are sufficiently low relative to those of citizens. Second, I relate the model to a historical case. In the early American republic, elites repeatedly expressed concerns about the expense of suppressing armed challenges to the constitutional order, supporting compromise instead. Amid contemporary concerns about the rule of law, this argument indicates the value of renewed attention to economic power.In the second paper, I analyze contrasting political effects of different structures of legal authority. In the process of both European and American state development, legal institutions developed from many separate venues with overlapping jurisdictions to a hierarchical structure with authority concentrated in the lawmaking institutions of the state. How did this change in the structure of legal authority affect political competition among groups in society? This paper presents a formal model to show that concentrating legal authority increases incentives for competing groups to invest in political organization. In turn, this increases political inequality: as all groups invest more effort, groups with a resource advantage are better able to make use of their advantage. The model is consistent with a case comparison of 18th century Pennsylvania and Massachusetts, two British American colonies with contrasting legal structures, patterns of political organization, and legal outcomes. The third paper provides a strategic mechanism by which the boundaries of authority between institutions incrementally change over time. Fundamental political change often occurs gradually, and it is therefore important to understand the strategic underpinnings of gradual institutional change. This paper presents a game in which players are entirely strategic and share proposal power, while showing how one player can gradually accumulate authority over time at the expense of another. The model analyzes a bargaining problem between two actors. Over an infinite number of periods, each actor has a chance each period to propose a bundle of future authority plus present policy, which creates the potential for actors to trade authority for temporary policy concessions. With some degree of noise in the division of authority, a proposal that transfers authority reduces the reservation value of the respondent, creating an incentive for the proposer to trade present policy for future authority. The model shows how ongoing, everyday political interactions gradually rearrange authority relationships among political institutions.

The following outlines my Ph.D thesis on Lagrange Inversion, Rogers-Ramanujan identities, and orthogonal polynomials. Sections $1-3$ serve as a summary of background material, much of which is based on various notes and papers either authored or co-authored by my advisor, Adriano Garsia. The subsequent sections contain related novel research. Specifically, section $4$ outlines the contents of a joint paper \cite{GG20} with Adriano Garsia, where we apply an expanded version of the combinatorial theory of orthogonal polynomials to produce new identities for the Fibonacci Polynomials. Section $5$ then details a purely combinatorial proof of $q-$Lagrange inversion, its application to Rogers-Ramanujan type identities, and more generalized Lagrange inversions involving an additional $t$ statistic.

In this paper we show a version of Johnson’s theorem for $\M(2,\R)$-sequences. In particular, motivated by the work of \cite{zhang2} and \cite{marx}, we show that one can identify the spectrum of Jacobi operator $J_{a,b}:\ell^2(\Z, \R)\rightarrow\ell^2(\Z,\R)$ by those energies $E \in \R$, whose cocycle map $B^{E} \in \M(2,\R)$ does not admit dominated splitting, i.e.$$\sigma(J_{a,b})=\{E\in\R:B^{E}\notin\CD\CS\}.$$ This result extends a well-known Johnson's theorem for Schr\"odinger operators, which identifies the spectrum of Schr\"odinger operator by all energy values $E \in \R$, such that the cocycle map $A^E \in \SL(2,\R)$ is not uniformly hyperbolic. Jacobi operator is a natural extension to Schr\"odinger operator, and dominated splitting generalizes the notion of uniform hyperbolicity.The biggest difference lies in the fact that for Schr\"odinger operator, the cocycle map takes value in $\SL(2,\R)$, whereas for Jacobi operator, the cocycle map takes value in $\M(2,\R)$. In particular, it is allowed to be singular, i.e. have zero determinant. This is one of the main challenges of this paper that we had to overcome. In this paper we also discuss the importance of Jacobi operators, develop definition of dominated splitting for $\M(2,\R)$-sequences, show stability theorem and finish this paper with the discussion of possibility of extending this result to dynamically defined Jacobi operators.

This work is divided into two parts which are concerned, respectively, with the combinatorics of the cardinals $\aleph_1$ and $\aleph_2$. The first part of the thesis contains the result due to the author and his advisor, Itay Neeman, that the Abraham-Rubin-Shelah Open Coloring Axiom is consistent with a large continuum; this answers a long-standing open question in forcing. Most of Part 1 appears in our submitted manuscript \cite{GN}. After surveying the relevant background in the first chapter, we proceed in the second chapter to define the notion of a Partition Product. This is a type of iteration built out of smaller ones in specific ways, roughly with memory conditions on the names and with isomorphism and coherence conditions on the various ``memories." We will prove a number of useful facts about partition products in Chapter 2. In Chapter 3, we show how to construct so-called Preassignments of Colors in the context of partition products; this forms the technical heart of Part 1. And finally, in Chapter 4, we show how to construct partition products in $L$; in particular, we construct the partition product which yields the model witnessing our theorem.Part 2 of the thesis addresses questions about a variety of combinatorial principles on $\aleph_2$. In each chapter in Part 2, we will be concerned with showing that some amount of Stationary Reflection holds at $\omega_2$, more specifically showing that various amounts of stationary reflection are compatible with other principles of wide interest. In Chapter 5, we provide an overview of these combinatorial principles and some of their history; we also spend some time collecting standard facts about Mitchell-type forcings which we will use in the subsequent chapters. Chapter 5 concludes with a proof that various Mitchell-type posets and their quotients are proper, a result which we assume is known, but which we have not encountered ourselves elsewhere.Chapters 6 and 7 address questions arising from the recent paper \emph{The Eightfold Way} by Cummings, Friedman, Magidor, Rinot, and Sinapova (see \cite{8fold}). In Chapter 6, we answer an open question asked at the end of that paper by showing that it is consistent, from a Mahlo cardinal, that the Tree Property $(\TP)$ and Approachability $(\AP)$ both fail at $\om_2$, while stationary reflection ($\RP$) holds at $\omega_2$. The authors of \cite{8fold} obtained the consistency of this same configuration from a weakly compact cardinal; our result proves the consistency of this configuration from optimal assumptions. We remark here that we present the original proof discovered by the author of this thesis. Later, the author, working with John Krueger, provided a more streamlined proof of this same result; this proof will appear in the forthcoming paper \cite{GiltonKrueger8}. Chapter 6 also includes an unrelated Easton-style lemma for preserving stationary subsets of countable cofinality; this result is due to the author and Omer Ben-Neria.In Chapter 7, we show that for any Boolean combination, $\Phi$, of $\TP$ and $\AP$, $\Phi$ is consistent with a strong form of simultaneous stationary reflection on $\omega_2$, namely that every stationary $S\seq\om_2\cap\cof(\om)$ reflects almost everywhere. This strengthens some of the results from \cite{8fold}. In Chapter 8, we return to the model from \cite{GiltonKruegerHS}, making good on a promise from the postscript therein. In \cite{GiltonKruegerHS}, the author and John Krueger originally sought to show that stationary reflection on $\om_2$ is consistent with a large continuum, and we built an involved mixed-support iteration to achieve such a model. However, we later learned from I. Neeman that such a model can be constructed by simply adding Cohen reals over the original Harrington-Shelah model (\cite{HS}). In Chapter 8 we will show that after a modification of our original preparatory iteration, we may obtain a model in which $\RP$ and $\AP$ both hold, in which $2^\om>\om_2$, and in which there are neither special Aronszajn trees on $\om_2$ nor weak Kurepa trees on $\omega_1$. This is a configuration which cannot be obtained simply by adding Cohen reals over the original Harrington-Shelah model nor by the methods of disjoint stationary sequences from \cite{GiltonKrueger8}. We hope that this demonstrates the usefulness of such a mixed support iteration.In the final chapter, we provide a list of open questions which we would like to address in future work.

This dissertation is comprised of two papers studying the topology of certain spaces of symplectic embeddings. The first paper shows how given two 4-dimensional ellipsoids whose symplectic sizes satisfy a specified inequality, a certain loop of symplectic embeddings between the two ellipsoids is noncontractible. The statement about symplectic ellipsoids is a particular case of a more general result which claims that given two convex toric domains whose first and second ECH capacities satisfy a specified inequality, one can prove that a certain loop of symplectic embeddings between the two convex toric domains is noncontractible.The second paper proves how given two 2n-dimensional symplectic ellipsoids whose symplectic sizes satisfy certain inequalities, a certain map from the n-torus to the space of symplectic embeddings from one ellipsoid to the other induces an injective map at the level of homology with mod 2 coefficients.

Broadly speaking, this thesis investigates mathematical questions motivated by computer science. The involved topics include communication avoiding algorithms, classical analysis, convex geometry, and low-rank matrix approximation. In total, the thesis consists of four self-contained sections, each adapted from papers the author has been a part of.The first two sections are both motivated by the Brascamp-Lieb inequalities, which are also often referred to as Hölder-Brascamp-Lieb inequalities. These inequalities have featured prominently in recent theoretical computer science work, due to connections to geometric complexity theory, harmonic analysis, communication-avoidance, and many other areas. Moreover, work generalizing the inequalities in various ways, such as to nonlinear versions, has been impactful to the study of differential equations.Section 1 studies the application of Hölder-Brascamp-Lieb (HBL) inequalities to the design of communication optimal algorithms. In particular, it describes optimal tiling (blocking) strategies for nested loops that lack data dependencies and exhibit affine memory access patterns. The problem roughly amounts to maximizing the volume of an object provided some of its linear images have bounded volume. The methods used are algorithmic.Another reason for the interest in these inequalities is because they are an interesting test case for non-convex optimization techniques. The optimal constant for a particular instance of the inequality is given by solving a non-convex optimization problem that is still highly structured. Of particular relevance to this thesis is that it can be formulated as a geodesically-convex problem, considered in the context of the manifold of positive definite matrices of determinant $1$. Even using the methods of Section 1, the procedure is not necessarily polynomial time, and this motivates further study of geodesic convexity.This lead to the work of Section 2, which discusses a notion of halfspace for Hadamard manifolds that is natural in the context of convex optimization. For this notion of halfspace, we generalize a classic result of Grunbaum, which itself is a corollary of Helly's theorem. Namely, given a probability distribution on the manifold, there is a point for which all halfspaces based at this point have at least 1/(n+1) of the mass, n being the dimension of the manifold. As an application, the gradient oracle complexity of geodesic convex optimization is polynomial in the parameters defining the problem. In particular it is polynomial in -log(epsilon), where epsilon is the desired error. This is a step toward the open question of whether such an algorithm exists.The remaining two sections of the paper present a different research direction, randomized numerical linear algebra. Numerical linear algebra has long been an important part of scientific computing. Due to the current trend of increasing matrix sizes and growing importance of fast, approximate solutions in industry, randomized methods are quickly increasing in popularity. Sections 3 and 4 in this thesis aim to show that randomized low-rank approximation algorithms satisfy many of the properties of classical rank-revealing factorizations.Section 3 introduces a Generalized Randomized QR-decomposition (RURV) that may be applied to arbitrary products of matrices and their inverses, without needing to explicitly compute the products or inverses. This factorization is a critical part of a communication-optimal spectral divide-and-conquer algorithm for the nonsymmetric eigenvalue problem. In this paper, we establish that this randomized QR-factorization satisfies the strong rank-revealing properties. We also formally prove its stability, making it suitable in applications. Finally, we present numerical experiments which demonstrate that our theoretical bounds capture the empirical behavior of the factorization.Section 4 concerns a Generalized LU-Factorization (GLU) for low-rank matrix approximation. We relate this to past approaches and extensively analyze its approximation properties. The established deterministic guarantees are combined with sketching ensembles satisfying Johnson-Lindenstrauss properties to present complete bounds. Particularly good performance is shown for the sub-sampled randomized Hadamard transform (SRHT) ensemble. Moreover, the factorization is shown to unify and generalize many past algorithms. It also helps to explain the effect of sketching on the growth factor during Gaussian Elimination.

Author(s): Baier, Nicholas | Abstract: In this paper, I examine novel data from long distance wh-dependencies in Seereer, an Atlantic language of Senegal. Seereer long distance wh-questions are characterized by (a) the presence of an obligatory pronoun at the edge of each embedded clause and (b) the presence of special morphology on each verb along the path of the extraction. Thus, Seereer provides striking evidence that long distance wh-movement proceeds successive cyclically through the edge of each clause. I argue that this verbal morphology spells out a valued wh-probe on C, which triggers the movement of a wh-phrase to its Spec. I show that the pronouns present at the edge of each embedded clause have the properties of copies and not of resumptive pronouns, and argue that they are in fact spelled out intermediate copies of the moved wh-phrase. I propose that such multiple copy spell out is possible precisely because they enter into a feature valuation relationship with C. Specifically, I propose that valuation of an wh-probe defines a copy as the head of an A’-chain. Thus, the application of successive cyclic movement does not result in one long chain, but instead in a series of smaller chains. This view of the structure of long A’-chains, when combined with the independently motivated principle of spelling out the heads of chains, results in the pattern of multiple copy spell out that occurs in Seereer.

This work investigates two regularization techniques designed for identifying critical points of the Yang-Mills energy.In the first half of the dissertation, we define a family of higher order functionals generalizing the Yang-Mills functional. We study the corresponding gradient flows and prove long-time existence and convergence results for subcritical dimensions as well as a bubbling criterion for critical dimensions. Consequently, we generalize the results of the convergence of Yang-Mills flow in dimensions 2 and 3 given by Rade and the bubbling criterion in dimension 4 of Struwe in the case where the initial flow data is smooth. This encompasses the contents of the author’s paper solo paper from 2014.In the second half of the dissertation we study an alternate type of regularization. In the spirit of recent work of Lamm, Malchiodi and Micallef in the setting of harmonic maps, we identify Yang-Mills connections obtained by approximations with respect to the Yang-Mills α-energy. More specifically, we show that for the SU(2) Hopf fibration over S4, for sufficiently small α values the SO(4)-invariant ADHM instanton is the unique α-critical point which has Yang-Mills α-energy lower than a specific threshold. This is an overview of the author’s solo paper from 2016.

In this paper, we introduce the notion of an Evertse–Ferretti Nevanlinna constant and compare it with the birational Nevanlinna constant introduced by the authors in a recent joint paper. We then use it to recover several previously known results. This includes a 1999 example of Faltings from his Baker’s Garden article. We also extend the theory of these Nevanlinna constants to what we call “multidivisor Nevanlinna constants,” which allow the proximity function to involve the maximum of Weil functions for finitely many divisors.

Author(s): Oh, Sung-Jin; Tataru, Daniel | Abstract: This article represents the fourth and final part of a four-paper sequence whose aim is to prove the Threshold Conjecture as well as the more general Dichotomy Theorem for the energy critical $4+1$ dimensional hyperbolic Yang--Mills equation. The Threshold Theorem asserts that topologically trivial solutions with energy below twice the ground state energy are global and scatter. The Dichotomy Theorem applies to solutions in arbitrary topological class with large energy, and provides two exclusive alternatives: Either the solution is global and scatters, or it bubbles off a soliton in either finite time or infinite time. Using the caloric gauge developed in the first paper, the continuation/scattering criteria established in the second paper, and the large data analysis in an arbitrary topological class at optimal regularity in the third paper, here we perform a blow-up analysis which shows that the failure of global well-posedness and scattering implies either the existence of a soliton with at most the same energy bubbling off, or the existence existence of a nontrivial self-similar solution. The proof is completed by showing that the latter solutions do not exist.

Given a simplicial complex $\Delta$ and its barycentric subdivision $\sdd$, we explore two homogeneous systems of parameters: one is the elementary symmetric functions inside the $\N$-graded Stanley-Reisner ring of $\Delta,$ $\k[\Delta],$ and the other is written based on a balanced coloring of $\sdd$ which lives inside the $\N^d$-graded Stanley-Reisner ring of $\sdd,$ namely, $\k[\sdd].$ The first is stable under symmetries and the other is stable under colorful automorphisms of $\sdd.$ In this paper, we develop methodology to explore comparing the resolutions of $\k[\Delta]$ and $\k[\sdd]$ over their respective parameter rings. We then prove it in the trivial case, which is simply the coinvariant algebra with alternate grading and one non-trivial case.

Let K be a number field with ring of integers O. Consider the set of n x n alternating matrices with fixed rank, rThe principal ideas behind the proof were first introduced in a paper of Katznelson where the problem of counting matrices is reduced to one of counting lattice points. Because our matrices have entries in O rather than being restricted to the set of rational integers, the lattices of Katznelson are replaced in the present work with O-modules. The generalization from the rational numbers to K renders the standard tools of lattice-theory less directly applicable, and we rely on the Minkowski map and novel arguments to, at various turns, reduce to the lattice case, or abstract results from lattices to O-modules. Ancillary results in our work include a new formula for the discriminant of a torsion-free O-module in terms of its pseudo-basis and a novel structure theorem about the set of alternating matrices whose rows lie in a specified O-module.

This dissertation constructs a Morse-Bott version of embedded contact homology (ECH). The dissertation is comprised of two parts, corresponding to the two papers written by the author as a graduate student at UC Berkeley. The first part explains how to compute ECH in the Morse-Bott setting when certain transversality conditions are met and provided a certain correspondence theorem is true; and gives a large class of examples where the transversality conditions are satisfied. The second part provides the analytic foundations of the first part by giving a proof of the correspondence theorem.

The saturated de Rham-Witt complex, introduced by Bhatt-Lurie-Mathew, is a variant of the classical de Rham-Witt complex which provides a conceptual simplification of the classical construction and which is expected to produce better results for non-smooth varieties. In this paper, we introduce a generalization of the saturated de Rham-Witt complex which allows coefficients in a unit-root $F$-crystal. We define our complex by a universal property in a category of so-called de Rham-Witt modules. We prove a number of results about it, including existence, quasicoherence, and comparisons to the de Rham-Witt complex of Bhatt-Lurie-Mathew and (in the smooth case) to crystalline cohomology and the classical de Rham-Witt complex with coefficients.

This paper documents a computer-assisted procedure for rigorously analyzing small hyper-bolic 3-manifolds. Briefly, we will define a compact six-dimensional space P that parame-terizes pairs of elements of Isom(H3 ), and then construct a regular binary space partition(BSP) tree which subdivides P into subregions Pi and whose leaves are — with a few ex-ceptions — labeled with killerwords. These killerwords will encode miniature proofs that Picannot contain any points which correspond to particular choices of pairs of generators ofany torsion-free discrete group of Isom(H3 ). This tree of mini-proofs will then be used to ex-haustively isolate all possible manifolds which can have properties related to the dimensionsof P.This method has been used as a foundation for tackling multiple long-standing problemsat the interface of hyperbolic geometry and low-dimensional topology. In particular: topo-logical rigidity of hyperbolic 3-manifolds; the generalized Smale conjecture for hyperbolic3-manifolds; finding the closed 3-manifold of least volume; and the Gordon conjecture onexceptional Dehn fillings.

In this paper we introduce the notion of the osculating space of a minimal surface as a vector bundle formed from the span of the z-derivatives of a conformal parametrization of the surface. We show that a minimal surface being J-holomorphic is equivalent to the dot product being fully degenerate on the osculating space. We then show that any minimal surface lying fully in R^n with osculating space of dimension half the ambient space must either be J-holomorphic or has a complex structure that is not compatible with the Euclidean metric. Considering the dot product as a bilinear form on the osculating space, we prove some strong restrictions on the dimensionality of the null space of this bilinear form. We show that the osculating space behaves nicely with respect to decomposable minimal surfaces and that we can always decompose a minimal surface into a J-holomorphic minimal surface and a minimal surface for which the dot product is non-degenerate on the osculating space. Finally, we prove that complete finite total curvature stable surfaces of genus one in R^5 are holomorphic under the condition that a certain cover of the surface is stable and the normal bundle is not topologically trivial.

This dissertation will show the ways in which generalized geometry elucidates the study of pluriclosed flow. In their 2009 paper, Streets and Tian introduce pluriclosed flow -- a parabolic flow of pluriclosed metrics -- and classify some static solutions. In 2018, Streets expanded this into a geometrization conjecture for compact, complex surfaces. The author is able to use these tools to show an equivalence between pluriclosed flow and a non-linear, coupled Hermitian-Yang-Mills type flow. From there, the author is able to more geometrically prove a result of Streets and Warren -- an Evans-Krylov theorem for pluriclosed flow. The author is also able to use this equivalence to prove long-time existence and convergence of the flow on Bismut-flat manifolds and surfaces of non-negative Kodaira dimension.

Melanoma is an incredibly dangerous form of skin cancer. In 2023 alone, the American Cancer Society estimates that about 7,990 Americans will die from new melanoma cases. Image analysis tools that automate the diagnosis of melanoma will improve dermatologists' diagnostic accuracy and should provide a significantly better detection of melanoma. In this paper, we present an application of modern deep learning architectures using the images of skin lesions in combination with patient-level contextual information to detect the presence of melanoma. More specifically, we used the latest variant of the residual network, the ResNeSt model, to leverage its deep architecture and channel-wise attention mechanisms for melanoma detection. In parallel, we used a multi-layer perceptron model to process patient-level data, and fused its output with the extracted features from the ResNeSt model. To compare and contrast performances, we trained a similar model using a smaller, more generic, convolutional neural network in addition to the larger ResNeSt network. After properly tuning all associated hyperparameters, our primary model obtained a balanced accuracy of 0.853 and a ROC-AUC score of 0.93. The split-attention network was shown to improve the total ROC-AUC by approximately 9%, suggesting that our model was able to successfully leverage the ResNeSt design to extract the most important features and most relevant cross-feature interactions for the detection of melanoma.

In recent years, there has been growing interest in modeling volatility as a stochastic process driven by a non-Markovian process, due to empirical evidence showing that the autocorrelation function of volatility decays as a power function. This paper investigates the use of a time-scaled fractional Ornstein-Uhlenbeck process to model volatility and applies this model to derive an optimal delta hedging strategy. By incorporating non-Markovian processes into our model, we aim to provide insights into the behavior of volatility in financial markets and explore potential benefits for option pricing and risk management strategies.

In this paper, we seek to provide counter examples to two volume comparison lemmas found in \cite{GP2} if we generalize their assumptions to a lower Ricci curvature bound. Second we seek to further understand Riemannian manifolds which contain embedded submanifolds of certain focal radius. First, in papers \cite{GP1}, \cite{GP2} Grove and Petersen discussed the relationship between bounds on sectional curvature, radius, and volume and its effects on the topology of a closed Riemannian manifolds. They prove that for manifolds with a lower sectional curvature bound, an upper radius bound and almost maximal volume, one can give topological equivalence to either $S^n$ or $\mathbb{R}P^n$. It was later proved to be diffeomorphic \cite{PSW1}. Also, Grove and Petersen showed that the limit space of a convergent sequence of manifolds with maximal volume has certain geometric properties. In the proofs of these theorems they use two powerful volume comparison lemmas. We discuss why the methods used in \cite{GP2} cannot be extended in general to manifolds with a lower Ricci curvature bound by looking at some interesting counter examples. Second, in a paper by Guiljaro and Wilhelm \cite{GW1} we see a relationship between closed embedded submanifolds of maximal focal radius and topological type, and seek to understand the necessity of the submanifold being closed by providing counterexamples in which we consider Manifolds with embedded open submanifolds of certain focal radius which do not satisfy the conclusions of the results in \cite{GW1}.

In 1980, Murray Marshall proved that the category of Quaternionic Structures is naturally equivalent to the category of abstract Witt rings. This paper develops a combinatorial theoryfor finite Quaternionic Structures in the case where $1=-1$,by demonstrating an equivalence between finite quaternionic structures and Steiner Triple Systems (STSs) with suitable blockcolorings. Associated to these STSs are Block Intersection Graphs (BIGs)with induced vertex colorings. This equivalence allows for a classification of BIGs corresponding to the basic indecomposable Witt rings via their associated quaternionic structures. Further, this paper classifies the BIGs associatedto the Witt rings of so-called elementary type, by providingnecessary and sufficient conditions for a BIG associated toa product or group extension.

GIScience 2016 Short Paper Proceedings Space-Time Topological Graphs C. Robertson 1 Wilfrid Laurier University, 75 University Ave, Waterloo, ON Email: crobertson@wlu.ca Abstract The integration of space and time in geospatial analyses is an active research area. While methods for characterizing sequences of spatial point objects over time have evolved greatly over recent years, the treatment of space-time change in complex geographic objects such as linear features and polygons has seen much less attention. In this paper we consider the directed graph as a representation for temporal change in spatial objects, where edges are formed by spatial relationships. With this representation, the large foundation of graph- theoretic metrics and algorithms can be used for the analysis of space-time change in geographic objects. We present preliminary findings using several exemplar datasets: range expansion of polygons representing animal home ranges, the spatial path of Hurricane Katrina, and a forest insect outbreak. In particular, we investigate the dynamics of space-time change in these datasets using well-known metrics for characterizing graph structures. 1. Introduction Understanding space-time change continues to be a challenging and increasingly required task for geospatial analysis, as more spatial datasets are evolving into long-term records of spatial and temporal change (e.g, satellite archives, long-term radio collar studies). However, models of space-time change remain to be fulll developed. Stell et al. (2011) introduced a bigraph representation of space-time change, whereby objects were represented as nodes at discrete time points, and linked through relations. As well, Del Mondo et al. 2012 further developed this model with an explicit spatiotemporal graph representation, distinguishing between filial relations (based on explicit identity) and purely topological relations (derived from spatial relationships). We build on these ideas from the perspective of space-time analysis in order to analyze spatial changes in polygon objects over time. We use several space-time polygon datasets in order to explore a how space-time topological graphs can be used as a representation for spatial-temporal analysis generally. 1.1 Polygon models of space-time change Sadahiro and Umemura (2001) introduced a framework for the analysis of spatio-temporal polygon distributions based on events derived from spatial overlap relations in neighbouring time-steps. The scope of events was extended in Robertson et al. (2007) to include proximity- based spatial relations, describing various types of movement occurring over the change interval, known as spatial-temporal analysis of moving polygons (STAMP). The STAMP framework is temporally discrete, such that an appropriate temporal grain is required a priori in order to represent change. Metrics associated with space-time change include counts of event types, area change, directional change, all of which can be accumulated along paths of the graph representing continuous spatial relationships over time. In Sadahiro (2001) the graph representation of polygon change events was introduced. In this paper we build on this framework order to integrate existing graph metrics into the spatial-temporal analysis of polygon distributions.

Inspired by the paper of Klagsbrun, Mazur and Rubin [5], this thesis investigates the disparity of 2-Selmer ranks of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. In the first part, we calculate the density of quadratic twists of E with even 2-Selmer ranks under two different counting methods. First we count twists by elements inside a large convex body of the Euclidean space that contains the integer lattice of K. The second counting method is counting quadratic twists E^L by the norms of the finite part of conductors of quadratic extensions L/K. Under both counting methods we give an explicit formula for the densities, which are finite products of local factors. In the second part of the paper we give a method that uses Tate’s algorithm to calculate the size of the cokernel of the local norm maps of E at places over 2, assuming that E has good reduction. With this method we can extend Kramer’s early work on the cokernel of the local norm maps, and compute the local factors mentioned above in some additional cases.

This paper is a comprehensive study of a long observed phenomenon of increase in the stability margin and so the rate of convergence of a class of linear systems due to time delay. Our results determine (a) in what systems the delay can lead to increase in the rate of convergence, (b) the exact range of time delay for which the rate of convergence is greater than that of the delay free system, and (c) an estimate on the value of the delay that leads to the maximum rate of convergence. We also show that the ultimate bound on the maximum achievable rate of convergence via time delay is e (Euler's number) times the delay free rate. For the special case when the system matrix eigenvalues are all negative real numbers, we expand our results to show that the rate of convergence in the presence of delay depends only on the eigenvalues with minimum and maximum real parts. Moreover, we determine the exact value of the maximum rate of convergence and the corresponding maximizing time delay. The final contribution of this paper is to show the application of our results in analyzing the use of a delayed feedback to increase the rate of convergence of an agreement algorithm for networked systems.

This paper forms the first part of a series by the authors [GMV2,GMV3] concerning the structure theory of nilspaces of Antol\'in Camarena and Szegedy. A nilspace is a compact space $X$ together with closed collections of cubes $C^n(X)\subseteq X^{2^n}$, $n=1,2,\ldots$ satisfying some natural axioms. Antol\'in Camarena and Szegedy proved that from these axioms it follows that (certain) nilspaces are isomorphic (in a strong sense) to an inverse limit of nilmanifolds. The aim of our project is to provide a new self-contained treatment of this theory and give new applications to topological dynamics. This paper provides an introduction to the project from the point of view of applications to higher order Fourier analysis. We define and explain the basic definitions and constructions related to cubespaces and nilspaces and develop the weak structure theory, which is the first stage of the proof of the main structure theorem for nilspaces. Vaguely speaking, this asserts that a nilspace can be built as a finite tower of extensions where each of the successive fibers is a compact abelian group. We also make some modest innovations and extensions to this theory. In particular, we consider a class of maps that we term fibrations, which are essentially equivalent to what are termed fiber-surjective morphisms by Anatol\'in Camarena and Szegedy, and we formulate and prove a relative analogue of the weak structure theory alluded to above for these maps. These results find applications elsewhere in the project., Comment: 64 pages, minor revision based on referee's report, results and proofs have not been changed, this version is accepted for publication in J. Anal. Math

Quantile and quantile effect functions are important tools for descriptive and causal analyses due to their natural and intuitive interpretation. Existing inference methods for these functions do not apply to discrete random variables. This paper offers a simple, practical construction of simultaneous confidence bands for quantile and quantile effect functions of possibly discrete random variables. It is based on a natural transformation of simultaneous confidence bands for distribution functions, which are readily available for many problems. The construction is generic and does not depend on the nature of the underlying problem. It works in conjunction with parametric, semiparametric, and nonparametric modeling methods for observed and counterfactual distributions, and does not depend on the sampling scheme. We apply our method to characterize the distributional impact of insurance coverage on health care utilization and obtain the distributional decomposition of the racial test score gap. We find that universal insurance coverage increases the number of doctor visits across the entire distribution, and that the racial test score gap is small at early ages but grows with age due to socio economic factors affecting child development especially at the top of the distribution. These are new, interesting empirical findings that complement previous analyses that focused on mean effects only. In both applications, the outcomes of interest are discrete rendering existing inference methods invalid for obtaining uniform confidence bands for observed and counterfactual quantile functions and for their difference -- the quantile effects functions., Comment: 38 pages, 6 figures, 4 tables