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Articles 61 - 90 of 90
Full-Text Articles in Physical Sciences and Mathematics
Computational Thinking In Mathematics And Computer Science: What Programming Does To Your Head, Al Cuoco, E. Paul Goldenberg
Computational Thinking In Mathematics And Computer Science: What Programming Does To Your Head, Al Cuoco, E. Paul Goldenberg
Journal of Humanistic Mathematics
How you think about a phenomenon certainly influences how you create a program to model it. The main point of this essay is that the influence goes both ways: creating programs influences how you think. The programs we are talking about are not just the ones we write for a computer. Programs can be implemented on a computer or with physical devices or in your mind. The implementation can bring your ideas to life. Often, though, the implementation and the ideas develop in tandem, each acting as a mirror on the other. We describe an example of how programming and …
Teaching From The Unknown, Jon Jacobsen
Teaching From The Unknown, Jon Jacobsen
Journal of Humanistic Mathematics
The goal of teaching is to transform our students' understanding, much as the goal of acting is to transform the audience's reality. In this article we use the context of mathematics to explore connections between teaching and acting and how such connections can help our students learn not only mathematics, but about the nature of mathematics.
Using Mathematical Equations To Communicate And Think About Karma, Kien H. Lim, Christopher Yakes
Using Mathematical Equations To Communicate And Think About Karma, Kien H. Lim, Christopher Yakes
Journal of Humanistic Mathematics
Two equations are presented in this article to communicate a particular understanding of karma. The first equation relates future experiences to past and present actions. Although the equation uses variables and mathematical symbols such as the integral sign and summation symbol, it reads more like a literal translation of an English sentence. Based on the key idea in the first equation, a second equation is then created to highlight the viability of using math to communicate concepts that are not readily quantifiable. Analyzing such equations can stimulate thinking, enhance understanding of spiritual concepts, raise issues, and uncover tensions between our …
Real-World Modelling To Increase Mathematical Creativity, Robert Weinhandl, Zsolt Lavicza
Real-World Modelling To Increase Mathematical Creativity, Robert Weinhandl, Zsolt Lavicza
Journal of Humanistic Mathematics
Modelling could be characterised as one of the core activities in mathematics education. However, when learning and teaching mathematics, mathematical modelling is mostly used to apply and deepen mathematical knowledge and competencies. Our educational study aims to explore how mathematical modelling, using real objects and high-quality mathematical technologies, could be utilised to acquire mathematical knowledge and competencies, and how learners could creatively use their existing knowledge. To discover the potential of mathematical modelling using real objects and high-quality mathematical technologies to acquire mathematical knowledge and competencies, and to stimulate learners' creativity, first, we combined cognitive and creative spirals and mathematical …
A Heart-Centered Stance: Receptivity To Algebra Teachers’ And Students’ Multidimensional Experiences, Nicole L. Fonger
A Heart-Centered Stance: Receptivity To Algebra Teachers’ And Students’ Multidimensional Experiences, Nicole L. Fonger
Journal of Humanistic Mathematics
The algebra classroom in urban public high schools in the United States is a complex space, ripe with many challenges and opportunities. In this paper I introduce the notion of a heart-centered stance for the teacher and the educator, and a method of engaging in creative expression for reflection and introspection toward individual change in the rich context of the high school algebra classroom. My evolving relationships with two high school algebra teachers, observations of their classrooms, as well as my own self-study and professional growth, are incorporated into this paper as I introduce and exemplify two tenets of a …
A Holistic Mathematics Curriculum Revision: An Adelphi University Case Study, Salvatore J. Petrilli
A Holistic Mathematics Curriculum Revision: An Adelphi University Case Study, Salvatore J. Petrilli
Journal of Humanistic Mathematics
Join me as I take you on a journey with the faculty of the Department of Mathematics and Computer Science at Adelphi University during our two-year re-envisioning and implementation of our mathematics curriculum. From the beginning this involved a data-driven initiative that naturally led to the revisions. Here I describe in detail the process that our department followed. In closing I end with some recommendations for interesting research directions in the field of mathematics education.
The Prime Number Theorem As A Capstone In A Complex Analysis Course, Stephan Ramon Garcia
The Prime Number Theorem As A Capstone In A Complex Analysis Course, Stephan Ramon Garcia
Journal of Humanistic Mathematics
We present a detailed proof of the prime number theorem suitable for a typical undergraduate- or graduate-level complex analysis course. Our presentation is particularly useful for any instructor who seeks to use the prime number theorem for a series of capstone lectures, a scaffold for a series of guided exercises, or as a framework for an inquiry-based course. We require almost no knowledge of number theory, for our aim is to make a complete proof of the prime number theorem widely accessible to complex analysis instructors and their students. In particular, we highlight the potential pitfalls and subtleties that may …
Mathematical Representations In Magazine Advertisements: Have The Messages Changed In A Decade?, Jennifer Hall
Mathematical Representations In Magazine Advertisements: Have The Messages Changed In A Decade?, Jennifer Hall
Journal of Humanistic Mathematics
Although people's ideas about mathematics and mathematicians often develop from their school and home experiences, such ideas also are influenced by interactions with popular media. In this article, I report on findings from a study in which I analyzed magazine advertisements for representations of mathematics and mathematicians. Data collection took place in two phases, approximately a decade apart. In each phase, I reviewed a year’s worth of issues in each of six diverse, popular magazines for mathematical representations in advertisements. The frequency of mathematical advertisements decreased from Phase 1 to Phase 2, but the initial frequency was already extremely low, …
Peer Motivation: Getting Through Math Together, Jessica Mean, Wes Maciejewski
Peer Motivation: Getting Through Math Together, Jessica Mean, Wes Maciejewski
Journal of Humanistic Mathematics
Students have a complex relationship with mathematics. Some love it, but more often than not, the feelings are less favorable. These feelings can lead to decreased motivation which makes it difficult for students to engage with the subject as the semester progresses. Instructors also have difficulty addressing this waning motivation. In this paper, we claim peers are better able to connect with the students and this can be leveraged to better motivate students. We present an approach to having peers motivate their students. These peer interactions integrated with a mandatory mathematics course might improve students’ motivation.
In How Many Days Will He Meet His Wife?, Dipak Jadhav
In How Many Days Will He Meet His Wife?, Dipak Jadhav
Journal of Humanistic Mathematics
In how many days will he meet his wife? This is a question asked at the end of each of two problems embedded in the verses of the last chapter of the Vyavahāra-gaṇita (‘Mathematics of Transaction’) of Rājāditya of 12th century. He infuses elegance in those two problems by choosing the charming idea of a husband’s meeting with his wife after their quarrel. This paper not only presents the algorithms offered by Rājāditya to solve them on their own terms as well as on modern terms and discusses the historicity of the categories of those two problems but also …
Raise The (Proportion) Bar!, Michael Waters
Raise The (Proportion) Bar!, Michael Waters
Journal of Humanistic Mathematics
This article, drawing mainly on references to teacher preparation textbooks, proposes proportion bars as a somewhat novel graphical approach to solving simple (direct) proportion problems and to illustrate the advantages of such an approach, which include accessibility with materials at early grade levels, allowance of students to better develop number sense and estimation, facilitation of setting up proportions, allowance for conceptual understanding and motivation of the procedure for solving direct proportions, assistance with part-to-part and part-to whole comparisons, and drawing of connections among mathematical topics. The emphasis is on teaching with understanding, rather than procedural knowledge.
“It’S All For The Best”: Optimization In The History Of Science, Judith V. Grabiner
“It’S All For The Best”: Optimization In The History Of Science, Judith V. Grabiner
Journal of Humanistic Mathematics
Many problems, from optics to economics, can be solved mathematically by finding the highest, the quickest, the shortest—the best of something. This has been true from antiquity to the present. Why did we start looking for such explanations, and how and why did we conclude that we could productively do so? In this article we explore these questions and tell a story about the history of optimization. Scientific examples we use to illustrate our story include problems from ancient optics, and more modern questions in optics and classical mechanics, drawing on ideas from Newton’s and Leibniz’s calculus and from the …
Humanistic Stem: From Concept To Course, Debra T. Bourdeau, Beverly L. Wood
Humanistic Stem: From Concept To Course, Debra T. Bourdeau, Beverly L. Wood
Journal of Humanistic Mathematics
Blending perspectives from the humanities and STEM fosters the creativity of all students. The culturally implicit dichotomy between the two meta-disciplines can be overcome with carefully designed courses and programs intent on doing so. The why and how of doing so through an online course is described with qualitative evidence of the success. Future plans for a full slate of such course and a virtual community are discussed.
Ahab's Arithmetic: The Mathematics Of Moby-Dick, Sarah B. Hart
Ahab's Arithmetic: The Mathematics Of Moby-Dick, Sarah B. Hart
Journal of Humanistic Mathematics
In this article we explore mathematical allusions in Herman Melville’s novel Moby-Dick. We argue that both the quantity and sophistication of these allusions are evidence for Melville’s high level of mathematical knowledge and ability. We discuss some of the most compelling mathematical imagery, as well as giving background on the several mathematicians and mathematics books mentioned in the novel. We also include some biographical details supporting the assertion that Melville had an unusually good mathematical education.
You Can Always Count On Word Problems, Mark Huber, Gizem Karaali
You Can Always Count On Word Problems, Mark Huber, Gizem Karaali
Journal of Humanistic Mathematics
No abstract provided.
On Rank-Two And Affine Cluster Algebras, Feiyang Lin
On Rank-Two And Affine Cluster Algebras, Feiyang Lin
HMC Senior Theses
Motivated by existing results about the Kronecker cluster algebra, this thesis is concerned with two families of cluster algebras, which are two different ways of generalizing the Kronecker case: rank-two cluster algebras, and cluster algebras of type An,1. Regarding rank-two cluster algebras, our main result is a conjectural bijection that would prove the equivalence of two combinatorial formulas for cluster variables of rank-two skew-symmetric cluster algebras. We identify a technical result that implies the bijection and make partial progress towards its proof. We then shift gears to study certain power series which arise as limits of ratios of …
Radial Singular Solutions To Semilinear Partial Differential Equations, Marcelo A. Almora Rios
Radial Singular Solutions To Semilinear Partial Differential Equations, Marcelo A. Almora Rios
HMC Senior Theses
We show the existence of countably many non-degenerate continua of singular radial solutions to a p-subcritical, p-Laplacian Dirichlet problem on the unit ball in R^N. This result generalizes those for the 2-Laplacian to any value p and extends recent work on the p-Laplacian by considering solutions both radial and singular.
The Slice Rank Polynomial Method, Thomas C. Martinez
The Slice Rank Polynomial Method, Thomas C. Martinez
HMC Senior Theses
Suppose you wanted to bound the maximum size of a set in which every k-tuple of elements satisfied a specific condition. How would you go about this? Introduced in 2016 by Terence Tao, the slice rank polynomial method is a recently developed approach to solving problems in extremal combinatorics using linear algebraic tools. We provide the necessary background to understand this method, as well as some applications. Finally, we investigate a generalization of the slice rank, the partition rank introduced by Eric Naslund in 2020, along with various discussions on the intuition behind the slice rank polynomial method and …
The Complexity Of Symmetry, Matthew Lemay
The Complexity Of Symmetry, Matthew Lemay
HMC Senior Theses
One of the main goals of theoretical computer science is to prove limits on how efficiently certain Boolean functions can be computed. The study of the algebraic complexity of polynomials provides an indirect approach to exploring these questions, which may prove fruitful since much is known about polynomials already from the field of algebra. This paper explores current research in establishing lower bounds on invariant rings and polynomial families. It explains the construction of an invariant ring for whom a succinct encoding would imply that NP is in P/poly. It then states a theorem about the circuit complexity partial …
On The Tropicalization Of Lines Onto Tropical Quadrics, Natasha Crepeau
On The Tropicalization Of Lines Onto Tropical Quadrics, Natasha Crepeau
HMC Senior Theses
Tropical geometry uses the minimum and addition operations to consider tropical versions of the curves, surfaces, and more generally the zero set of polynomials, called varieties, that are the objects of study in classical algebraic geometry. One known result in classical geometry is that smooth quadric surfaces in three-dimensional projective space, $\mathbb{P}^3$, are doubly ruled, and those rulings form a disjoint union of conics in $\mathbb{P}^5$. We wish to see if the same result holds for smooth tropical quadrics. We use the Fundamental Theorem of Tropical Algebraic Geometry to outline an approach to studying how lines lift onto a tropical …
Tiling Representations Of Zeckendorf Decompositions, John Lentfer
Tiling Representations Of Zeckendorf Decompositions, John Lentfer
HMC Senior Theses
Zeckendorf’s theorem states that every positive integer can be decomposed uniquely into a sum of non-consecutive Fibonacci numbers (where f1 = 1 and f2 = 2). Previous work by Grabner and Tichy (1990) and Miller and Wang (2012) has found a generalization of Zeckendorf’s theorem to a larger class of recurrent sequences, called Positive Linear Recurrence Sequences (PLRS’s). We apply well-known tiling interpretations of recurrence sequences from Benjamin and Quinn (2003) to PLRS’s. We exploit that tiling interpretation to create a new tiling interpretation specific to PLRS’s that captures the behavior of the generalized Zeckendorf’s theorem.
Towards Tropical Psi Classes, Jawahar Madan
Towards Tropical Psi Classes, Jawahar Madan
HMC Senior Theses
To help the interested reader get their initial bearings, I present a survey of prerequisite topics for understanding the budding field of tropical Gromov-Witten theory. These include the language and methods of enumerative geometry, an introduction to tropical geometry and its relation to classical geometry, an exposition of toric varieties and their correspondence to polyhedral fans, an intuitive picture of bundles and Euler classes, and finally an introduction to the moduli spaces of n-pointed stable rational curves and their tropical counterparts.
Exploring Winning Strategies For The Game Of Cycles, Kailee Lin
Exploring Winning Strategies For The Game Of Cycles, Kailee Lin
HMC Senior Theses
This report details my adventures exploring the Game of Cycles in search of winning strategies. I started by studying combinatorial game theory with hopes to use the Sprague-Grundy Theorem and the structure of Nimbers to gain insight for the Game of Cycles. In the second semester, I pivoted to studying specific types of boards instead. In this thesis I show that variations of the mirror-reverse strategy developed by Alvarado et al. in the original Game of Cycles paper can be used to win on additional game boards with special structure, such as lollipops, steering wheel locks, and 3-spoke trees. Additionally …
On The Inverse Hull Of A One-Sided Shift Of Finite Type, Aria Beaupre
On The Inverse Hull Of A One-Sided Shift Of Finite Type, Aria Beaupre
HMC Senior Theses
Let S be the semigroup constructed from a one-sided shift of finite type. In this thesis, we will provide the construction of H(S), the inverse hull of S, explore the properties of H(S), and begin to characterize the structure of H(S). We will also focus on a kind of one-sided shift of finite type, Markov shifts, and prove an invariant on isomorphic inverse hulls of Markov shifts.
Fractals, Fractional Derivatives, And Newton-Like Methods, Eleanor Byrnes
Fractals, Fractional Derivatives, And Newton-Like Methods, Eleanor Byrnes
HMC Senior Theses
Inspired by the fractals generated by the discretizations of the Continuous Newton Method and the notion of a fractional derivative, we ask what it would mean if such a fractional derivative were to replace the derivatives in Newton's Method. This work, largely experimental in nature, examines these new iterative methods by generating their Julia sets, computing their fractal dimension, and in certain tractable cases examining the behaviors using tools from dynamical systems.
Measuring Machine Learning Model Uncertainty With Applications To Aerial Segmentation, Kevin James Cotton
Measuring Machine Learning Model Uncertainty With Applications To Aerial Segmentation, Kevin James Cotton
CGU Theses & Dissertations
Machine learning model performance on both validation data and new data can be better measured and understood by leveraging uncertainty metrics at the time of prediction. These metrics can improve the model training process by indicating which training data need to be corrected and what part of the domain needs further annotation. The methods described have yet to reach mainstream adoption, and show great potential. Here, we survey the field of uncertainty metrics and provide a robust framework for its application to aerial segmentation. Uncertainty is divided into two types: aleatoric and epistemic. Aleatoric uncertainty arises from variations in training …
Random Matrix Theory: A Combinatorial Proof Of Wigner's Semicircle Law, Vanessa Wolf
Random Matrix Theory: A Combinatorial Proof Of Wigner's Semicircle Law, Vanessa Wolf
Scripps Senior Theses
A combinatorial proof of Wigner’s semicircle law for the Gaussian Unitary Ensemble (GUE) is presented using techniques from free probability. Motivating examples taken from the symmetric Bernoulli ensemble and the GUE show the distribution of eigenvalues of sample n x n matrices approaching Wigner’s semicircle as n get large. The concept of crossing and non-crossing pairings is developed, along with proofs of Wick’s Formula for real and complex Gaussians. It is shown that Wigner’s semicircle distribution has moments given by the Catalan numbers. Wick’s Formula and several additional lemmas (proved in sequence) lead to a "method of moments" proof that …
On Properties Of Positive Semigroups In Lattices And Totally Real Number Fields, Siki Wang
On Properties Of Positive Semigroups In Lattices And Totally Real Number Fields, Siki Wang
CMC Senior Theses
In this thesis, we give estimates on the successive minima of positive semigroups in lattices and ideals in totally real number fields. In Chapter 1 we give a brief overview of the thesis, while Chapters 2 – 4 provide expository material on some fundamental theorems about lattices, number fields and height functions, hence setting the necessary background for the original results presented in Chapter 5. The results in Chapter 5 can be summarized as follows. For a full-rank lattice L ⊂ Rd, we are concerned with the semigroup L+ ⊆ L, which denotes the set of all vectors with nonnegative …
Neither “Post-War” Nor Post-Pregnancy Paranoia: How America’S War On Drugs Continues To Perpetuate Disparate Incarceration Outcomes For Pregnant, Substance-Involved Offenders, Becca S. Zimmerman
Neither “Post-War” Nor Post-Pregnancy Paranoia: How America’S War On Drugs Continues To Perpetuate Disparate Incarceration Outcomes For Pregnant, Substance-Involved Offenders, Becca S. Zimmerman
Pitzer Senior Theses
This thesis investigates the unique interactions between pregnancy, substance involvement, and race as they relate to the War on Drugs and the hyper-incarceration of women. Using ordinary least square regression analyses and data from the Bureau of Justice Statistics’ 2016 Survey of Prison Inmates, I examine if (and how) pregnancy status, drug use, race, and their interactions influence two length of incarceration outcomes: sentence length and amount of time spent in jail between arrest and imprisonment. The results collectively indicate that pregnancy decreases length of incarceration outcomes for those offenders who are not substance-involved but not evenhandedly -- benefitting white …