Model Theory of Holomorphic Functions

This thesis is concerned with a conjecture of Zilber: that the complex field expanded with the exponential function should be `quasi-minimal'; that is, all its definable subsets should be countable or have countable complement. Our purpose is to study the geometry of this structure and other expansions by holomorphic functions of the complex field without having first to settle any number-theoretic problems, by treating all countable sets on an equal footing. We present axioms, modelled on those for a Zariski geometry, defining a non-first-order class of ``quasi-Zariski'' structures endowed with a dimension theory and a topology in which all countable sets are of dimension zero. We derive a quantifier elimination theorem, implying that members of the class are quasi-minimal. We look for analytic structures in this class. To an expansion of the complex field by entire holomorphic functions $\mathcal{R}$ we associate a sheaf $\mathcal{O}^{\scriptscriptstyle{\mathcal{R}}}$ of analytic germs which is closed under application of the implicit function theorem. We prove that $\mathcal{O}^{\scriptscriptstyle{\mathcal{R}}}$ is also closed under partial differentiation and that it admits Weierstrass preparation. The sheaf defines a subclass of the analytic sets which we call $\mathcal{R}$-analytic. We develop analytic geometry for this class proving a Nullstellensatz and other classical properties. We isolate a condition on the asymptotes of the varieties of certain functions in $\mathcal{R}$. If this condition is satisfied then the $\mathcal{R}$-analytic sets induce a quasi-Zariski structure under countable union. In the motivating case of the complex exponential we prove a low-dimensional case of the condition, towards the original conjecture

Defensible inferences from a nested sequence of logistic regressions: a guide for the perplexed

Employing nested sequences of models is a common practice when exploring the extent to which one set of variables mediates the impact of another set. Such an analysis in the context of logistic regression models confronts two challenges: (i) direct comparisons of coefficients across models are generally biased due to the changes in scale that accompany the changes in the set of explanatory variables, (ii) conducting a large number of tests induces a problem of multiplicity that can lead to spurious findings of significance if not heeded. This article aims to illustrate a practical strategy for conducting analyses in the face of these challenges. The challenges—and how to address them—are illustrated using a subset of the findings reported by Braun (Large-scale Assess Educ 6(4):1–52, 2018. 10.1186/s40536-018-0058-x), drawn from the Programme for the International Assessment of Adult Competencies (PIAAC), an international, large-scale assessment of adults. For each country in the dataset, a nested pair of logistic regression models was fit in order to investigate the role of Educational Attainment and Cognitive Skills in mediating the impact of family background and demographic characteristics on the location of an individual’s annual income in the national income distribution. A modified version of the Karlson–Holm–Breen (KHB) method was employed to obtain an unbiased estimate of the true differences in the coefficients between nested logistic models. In order to address the issue of multiplicity, a recent generalization of the Benjamini–Hochberg (BH) False Discovery Rate (FDR)-controlling procedure to hierarchically structured hypotheses was employed and compared to two conventional methods. The differences between the changes in coefficients calculated conventionally and with the KHB adjustment varied from negligible to very substantial. When combined with the actual magnitudes of the coefficients, we concluded that the more proximal factors indeed act as strong mediators for the background factors, but less so for Age, and hardly at all for Gender. With respect to multiplicity, applying the FDR-controlling procedure yielded results very similar to those obtained by applying a standard per-comparison procedure, but quite a few more discoveries in comparison to the Bonferroni procedure. The KHB methodology illustrated here can be applied wherever there is interest in comparing nested logistic regressions. Modifications to account for probability sampling are practicable. The categorization of variables and the order of entry should be determined by substantive considerations. On the other hand, the BH procedure is perfectly general and can be implemented to address multiplicity issues in a broad range of settings

Data in the Educational and Social Sciences: It’s Time for Some Respect

This article introduces the concept of the carrying capacity of data (CCD), defined as an integrated, evaluative judgment of the credibility of specific data-based inferences, informed by quantitative and qualitative analyses, leavened by experience. The sequential process of evaluating the CCD is represented schematically by a framework that can guide data analysis and statistical inference, as well as pedagogy. Aspects of each phase are illustrated with examples. A key initial activity in empirical work is data scrutiny, comprising consideration of data provenance and characteristics, as well as data limitations in light of the context and purpose of the study.  Relevant auxiliary information can contribute to evaluating the CCD, as can sensitivity analyses conducted at the modeling stage. It is argued that early courses in statistical methods, and the textbooks they rely on, typically give little emphasis to, or omit entirely, discussion of the importance of data scrutiny in scientific research. This inattention and lack of guided, practical experience leaves students unprepared for the real world of empirical studies. Instructors should both cultivate in their students a true respect for data and engage them in authentic empirical research involving real data, rather than the context-free data to which they are usually exposed

Students' Perspective on AI-Supported Assessment of Open-Ended Questions in Higher Education

Artificial Intelligence (AI) is widely used for the assessment of multiple-choice questions. There is an increasing effort to also use it for open-ended questions. While the use of AI can benefit the learning of students, e.g. by increasing the number of feedback moments, most applications focus on saving costs by reducing the need for manual assessment. The perspective of teachers on this kind of automation has been studied extensively, the student perspective, however, is still under-researched. This paper presents the results of two surveys and a series of interviews among students to identify their perspective on AI-supported assessment and elaborate on under which conditions they would accept such technology. The results show that the majority of students (more than 80%), is, under certain conditions, open to AI-supported assessment. Most importantly, they stress that humans should still be involved in the assessment (human-in-the-loop)

Developing And Evaluating A Machine‐Scorable, Constrained Constructed‐Response Item

The use of constructed response items in large scale standardized testing has been hampered by the costs and difficulties associated with obtaining reliable scores. The advent of expert systems may signal the eventual removal of this impediment. This study investigated the accuracy with which expert systems could score a new, non‐multiple choice item type. The item type presents a faulty solution to a computer programming problem and asks the student to correct the solution. This item type was administered to a sample of high school seniors enrolled in an Advanced Placement course in Computer Science who also took the Advanced Placement Computer Science (APCS) Test. Results indicated that the expert systems were able to produce scores for between 82% and 97% of the solutions encountered and to display high agreement with a human reader on which solutions were and were not correct. Diagnoses of the specific errors produced by students were less accurate. Correlations with scores on the objective and free‐response sections of the APCS examination were moderate. Implications for additional research and for testing practice are offered.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/108589/1/ets200144.pd

Improving Transit Predictions of Known Exoplanets with TERMS

Transiting planet discoveries have largely been restricted to the short-period or low-periastron distance regimes due to the bias inherent in the geometric transit probability. Through the refinement of planetary orbital parameters, and hence reducing the size of transit windows, long-period planets become feasible targets for photometric follow-up. Here we describe the TERMS project that is monitoring these host stars at predicted transit times.Comment: 3 pages, 2 figures, to be published in ASP Conf. Proceedings: "Detection and dynamics of transiting exoplanets" 2010, OHP, France (eds.: F. Bouchy, R.F. D{\i}az, C. Moutou

Using Assessment and Feedback to Enhance Learning: Examining the Relationship Between Teachers’ Reported Use of Assessment and Feedback and Student Performance in AP Biology

This paper analyzes a national sample of teachers’ self-reported use of assessment and feedback in Advanced Placement (AP) biology classrooms. Descriptive statistics of what teachers reported doing in these two areas are discussed, followed by the identification of reported teaching practices variables that were found to be significantly related with student performance on AP biology exams. All the significant variables found are exemplars of authentic assessment and feedback practices and techniques. The study provides valuable information on the types of assessment and feedback practices currently being employed by teachers, and shows the relationship of such practices with actual student outcomes

College Students’ Critical Thinking: Assessment and Interpretation

Many colleges identify the development of critical thinking (CT) as a key learning outcome. Nonetheless, few studies examined the development of CT during college, and the instruments employed in them are often limited. This article introduces the Critical Reasoning Assessment (CRA), a new instrument based on the Reflective Judgment Model (RJM; King and Kitchener 1994) designed to engage students in analyzing ethical dilemmas while being easy to administer and score. Using the CRA, we measured the CT skills of college students in three studies, both cross-sectionally and longitudinally. The results demonstrated substantial growth in CT skills during the first year and between the first and the fourth years of college; 42% and 60% of the participants advanced to a higher level of CT by the end of their first and fourth year, respectively. This study introduces a comprehensive, theory-based, easy-to-score and interpret instrument measuring CT. Applied to longitudinal data, it adds to limited findings on CT developmental trajectories and quantifies substantively interpretable shifts in the quality of CT

The Relationship of Expert-System Scored Constrained Free-Response Items to Multiple-Choice and Open-Ended Items

This study examined the relationship of an expert- system scored constrained free-response item (requir ing the student to debug a faulty computer program) to two other item types: (1) multiple-choice and (2) free- response (requiring production of a program). Confir matory factor analysis was used to test the fit of a three-factor model to these data and to compare the fit of the model to three alternatives. These models were fit using two random-half samples, one given a faulty program containing one bug and the other a program with three bugs. A single-factor model best fit the data for the sample taking the one-bug constrained free re sponse and a two-factor model fit the data somewhat better for the second sample. In addition, the factor intercorrelations showed this item type to be highly re lated to both the free-response and multiple-choice measures.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/68260/2/10.1177_014662169001400204.pd

M dwarf stars in the light of (future) exoplanet searches

We present a brief overview of a splinter session on M dwarf stars as planet hosts that was organized as part of the Cool Stars 17 conference. The session was devoted to reviewing our current knowledge of M dwarf stars and exoplanets in order to prepare for current and future exoplanet searches focusing in low mass stars. We review the observational and theoretical challenges to characterize M dwarf stars and the importance of accurate fundamental parameters for the proper characterization of their exoplanets and our understanding on planet formation.Comment: 4 pages, 1 figure. Summary of the splinter session "M dwarf stars in the light of (future) exoplanet searches" held at the 17th Cambridge Workshop on Cool Stars, Stellar Systems, and the Sun, June 28th 2012, Barcelona, Spain. Submitted for publication in Astronomische Nachrichten - Astronomical Notes (AN) 334, Issue 1-2, Eds Klaus Strassmeier and Mercedes L\'opez-Morale