In this article, we discuss the Hausdorff dimensions of the distance sets and the difference sets for the graphs of continuous functions on the unit interval. We also prove that the distance set conjecture is true for...
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In this article, we discuss the Hausdorff dimensions of the distance sets and the difference sets for the graphs of continuous functions on the unit interval. We also prove that the distance set conjecture is true for a dense subset of (B([0,1]), parallel to center dot parallel to(p)). After that, we determine bounds on the dimension of the difference and distance sets for the graph of a function on an uncountable bounded domain in terms of the dimension of the graph of the function. Lastly, we determine a non-trivial lower bound for the upper box dimension of the difference set of a set in the plane and also discuss the dimension of the distance set of the product of sets.
Consider the problem of computing a function given only an oracle for its graph. For this problem, we present optimal trade-offs between serial and parallel queries. In particular, we give a function for which paralle...
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Consider the problem of computing a function given only an oracle for its graph. For this problem, we present optimal trade-offs between serial and parallel queries. In particular, we give a function for which parallel access to its own graph is exponentially more expensive than sequential access. (C) 2002 Elsevier Science (USA).
This paper offers instructional interventions designed to support undergraduate math students’ understanding of two forms of representations of Calculus concepts, mathematical language and graphs. We first discuss is...
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The purpose of this study is to investigate how students interpret expressions from calculus statements in the graphical register. To this end, I conducted 150-minute clinical interviews with 13 undergraduate mathemat...
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The purpose of this study is to investigate how students interpret expressions from calculus statements in the graphical register. To this end, I conducted 150-minute clinical interviews with 13 undergraduate mathematics students who had completed at least one calculus course. In the interviews, students evaluated six calculus statements for various real-valued functions depicted in graphs in the Cartesian plane. From my analysis of these interviews, I found four distinct interpretations of expressions in the graphical register that students used in this study while evaluating the statements using the graphs. I describe the characteristics of these four interpretations, which I refer to as (1) nominal, (2) ordinal, (3) cardinal, and (4) magnitude. For some students, the use of these interpretations supported their graphical reasoning and correct evaluations of the statements. For other students, the use of some interpretations rather than others presented obstacles to their graphical understanding of the expressions in the statement. For instance, seven of the students never used a magnitude interpretation (interpreting an expression as a distance in the graph), even when working with difference expressions. I discuss implications of these findings for teaching with graphs across levels and directions for future research.
High failure rates in Calculus I contribute to the course acting a filter, rather than a pump, for STEM disciplines. One often cited source of difficulty for students in Calculus I is their weak precalculus content kn...
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High failure rates in Calculus I contribute to the course acting a filter, rather than a pump, for STEM disciplines. One often cited source of difficulty for students in Calculus I is their weak precalculus content knowledge. In this three-paper dissertation, I investigate Calculus I students’ precalculus content knowledge and their awareness of that knowledge. In the first paper, I describe a methodology for collecting data about Calculus I students’ tendency to regulate their precalculus content knowledge and analyze the utility of quantifying self-regulated learning as a means for identifying at-risk students. In the second paper, I focus on two factors (calibration and help-seeking) to investigate the how they correlate with Calculus I students’ first exam performance. Results highlight the importance of calibration of precalculus content knowledge both directly on student success and how calibration accuracy mediates the benefits of help-seeking. Quantitative analyses of students’ precalculus content knowledge highlight Calculus I students’ difficulty with the concept of graph, despite students’ high confidence in questions related to graph. In the third paper, I conduct interviews with Calculus I students to examine their conceptions of outputs and differences of outputs of a function in the graphical context to understand nuance in how students understand and reason with graphs. Results highlight that students’ understandings of quantities and frames of references in graphs of functions can be varied and stable. Students’ understanding of quantities also impacts their understanding of other concepts such as differences of outputs and difference quotient. Results of this dissertation have implications for educators, tutor center leaders, and researchers interested in students’ understanding of graph, calibration, and help-seeking.
Highly nonlinear functions (perfect nonlinear, maximum nonlinear, etc.) on finite fields and finite (abelian or nonabelian) groups have been studied in numerous papers. Among them are absolute maximum nonlinear functi...
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Highly nonlinear functions (perfect nonlinear, maximum nonlinear, etc.) on finite fields and finite (abelian or nonabelian) groups have been studied in numerous papers. Among them are absolute maximum nonlinear functions on finite nonabelian groups introduced by Poinsot and Pott [15] in 2011. Recently, properties and constructions of absolute maximum nonlinear functions were studied in [19]. In this paper we study the characterizations of absolute maximum nonlinear functions on arbitrary finite groups. Then as an application of these characterizations, we discuss the existence of absolute maximum nonlinear functions on dihedral groups. We will prove that for a dihedral group D-2n of order 2n, where n >= 3, if there is an absolute maximum nonlinear function on D-2n, then n is an element of {3, 12, 15, 18, 30, 33, 42, 66, 138}. In particular, if there exists an absolute maximum nonlinear function from D-2n to Z(2), where Z(2) is the group of order 2, then we show that n = 12. All absolute maximum nonlinear functions from D-24 to Z(2) will be determined. (C) 2021 Elsevier Inc. All rights reserved.
The purpose of this study is to examine the characteristics of students' thinking about aspects of graphs in the context of evaluating statements about real-valued functions from Calculus. We conducted clinical in...
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The purpose of this study is to examine the characteristics of students' thinking about aspects of graphs in the context of evaluating statements about real-valued functions from Calculus. We conducted clinical interviews in which undergraduate students evaluated mathematical statements using graphs to explain their reasoning. From our data analysis, we found two ways students think about graphs, value-thinking and location-thinking. These two ways of thinking were rooted in students' attention to different attributes of points on graphs we provided: either the values represented by the points or the locations of the points in space. In this paper, we report our classification of students' thinking about aspects of graphs in terms of value-thinking and location-thinking. Our findings indicate that students' thinking about aspects of graphs accounts for key differences in their understandings of mathematical statements. We discuss some implications of our findings for instruction and curriculum development in Calculus and beyond.
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