Mathematics Commons^™

Polygon Quadrature And Dodecagonal Tessellation With Pattern Blocks, Gunhan Caglayan, Ben Kamau

Journal of Humanistic Mathematics

The age-old challenge of polygon quadrature involves converting a polygon into a square of equal area. In this educational resource, we utilize pattern blocks, commonly employed instructional aids in K-12 education across the United States, to visually demonstrate the transformation of different equilateral and regular pattern block polygons into squares. This is achieved through the application of the area conservation principle and geometric congruence/similarity reasoning.

Benefits Of Risk-Taking In Teaching Mathematics, Mehmet Kirmizi, Abigail Quansah, Zafer Buber

Journal of Humanistic Mathematics

In this paper, we, a group of graduate students in mathematics education, discuss some of the metacognitive benefits of the non-traditional teaching methods we observed employed by one of our professors. This professor’s methods challenge the common belief that well-managed class time is key for positive learning outcomes. Instead, he orients his teaching to share the exploration and sense-making phases of doing mathematics. The goal of his teaching is to share the idea that learning mathematics is a process of “refining our mathematical thinking”. We argue that this approach to teaching helps students see that mathematics is a human endeavor, …

Does Chatgpt Know Calculus?, Kris H. Green

Journal of Humanistic Mathematics

Academics and educators across the world are grappling with how OpenAI’s new software, ChatGPT, will impact teaching and learning. This essay explores ChatGPT’s response to a typical calculus problem as a way of illustrating its functionality and limitations.

The Frankensteinian Nature Of Mathematics, Ali Barahmand

Journal of Humanistic Mathematics

Frankenstein is a story about a scientist who creates a sapient creature that gets out of control and horrifies its creator by its unexpected behavior. In this note, we show that this type of undesirable behavior can reflect a part of the nature of mathematics, and that its origin is related to the ontological question of whether mathematics is invented or discovered. Based on a review of the relationship be- tween discovery and invention, we demonstrate that mathematics has similarities and differences with both discovery and invention. In the natural sciences, new instruments have to be invented to discover new …

Seating Groups And 'What A Coincidence!': Mathematics In The Making And How It Gets Presented, Peter J. Rowlett

Journal of Humanistic Mathematics

Mathematics is often presented as a neatly polished finished product, yet its development is messy and often full of mis-steps that could have been avoided with hindsight. An experience with a puzzle illustrates this conflict. The puzzle asks for the probability that a group of four and a group of two are seated adjacently within a hundred seats, and is solved using combinatorics techniques.

Logical Or Epistemological? A Study On Kepler’S 3m Cosmological Model, Po-Hung Liu

Journal of Humanistic Mathematics

Kepler published Mysterium Cosmographicum in 1597 constructing his cosmo- logical model based on the five regular polyhedra. Such a creative but weird idea was almost consistent with empirical evidence. Furthermore, following the Pythagorean belief about the connection between music and astronomy, Kepler delved into looking for what things having to do with the planetary movements have the harmonic consonances. This article claims that Kepler’s model is a 3M (mathematical, musical, and metaphysical) model and demonstrates how it had been constructed. Furthermore, I explore the reasons behind Kepler’s departure from the 3M model and his subsequent consideration of a non-circular orbit, …

On Parallels Between Words And Music, Will Turner

Journal of Humanistic Mathematics

A generalised song is a means of drawing parallels between words and music. The parallels are encoded in a mathematical structure, which is interpreted in a verbal structure and a musical structure. Here we develop a number of new techniques for drawing such parallels, in giving two examples of generalised songs, `Relation', and `Merge/Split'.

The first five partials of a note played on a piano are roughly 0,12,19,24,28 semitones above the fundamental.`Relation' is a generalised song, whose musical part is played on a piano, constructed from the mathematical relation 4.28 = 3.12 + 4.19.

`Merge/Split' is a generalised song whose …

Undergraduate Mathematics Students Question And Critique Society Through Mathematical Modeling, Will Tidwell, Amy Bennett

Journal of Humanistic Mathematics

Mathematics can be used as a tool to question and critique society and, in doing so, give us more information about the world around us and how it operates. This however, is not a common perspective that is conveyed to students during their undergraduate mathematics coursework. This paper contributes to the understanding of how undergraduate mathematics students question and critique society via mathematical modeling tasks. In two courses at two universities, 27 mathematics majors and secondary preservice teachers engaged in the modeling process situated in authentic contexts to learn specific concepts and make mathematical connections across domains and disciplines. Both …

Aesthetic Approaches To Symmetric Functions, John M. Campbell

Journal of Humanistic Mathematics

Symmetry is often regarded as an integral aspect about aesthetics. This motivates the pursuit of interdisciplinary studies based on the use of subjects in mathematics concerned with symmetry in conjunction with aesthetics. What is referred to as a symmetric function in the field of algebraic combinatorics is an abstraction based on polynomials that exhibit a symmetric property, and this leads us to pursue an algebraic combinatorics-inspired exploration based on aesthetics. In particular, we use different bases and transitions between them to create aesthetically pleasing visualizations of symmetric functions. We see that these visualizations in turn raise new and interesting questions.

Mathematicians Going East, Pasha Zusmanovich

Journal of Humanistic Mathematics

I survey emigration of mathematicians from Europe, before and during WWII, to Russia. The emigration started at the end of 1920s, the time of “Great Break”, and accelerated in 1930s, after the introduction in Germany of the “non-Aryan laws”. Not everyone who wanted to emigrate managed to do so, and most of those who did spent a relatively short time in Russia, being murdered or deported, or fleeing the Russian regime. After 1937, the year of “Great Purge”, only a handful of emigrant mathematicians remained, and even fewer managed to leave a trace in the scientific milieu of their new …

Sharing Four Biscuits Between Three People: An Illustrative Example Of How Mathematics Is Intertwined With Human Values, Lovisa Sumpter, David Sumpter

Journal of Humanistic Mathematics

Despite convincing arguments by mathematicians, philosophers, sociologists and machine learning practitioners to the contrary, there remains a widespread notion amongst many members of the general public (and some practitioners) that mathematics is neutral, that it is free from human values. One reason why this notion persists is that we lack clear-cut examples that demonstrate how mathematics and values are intertwined. In this paper, we offer one such example. In particular, we show that when sharing four biscuits between three people, several possible mathematical and ethical frameworks can be used. We demonstrate that different solutions—hiding one biscuit, arbitrarily sharing the extra …

The Automathography: A Humanistic Autobiographical Writing Assignment For Mathematics Courses, Colton Sawyer, Ron Buckmire

Journal of Humanistic Mathematics

In this article, we discuss an autobiographical writing assignment that we call an “automathography” which can be used in different types of mathematics and mathematics education courses, and by extension, in other disciplines as well. This assignment can enhance student-instructor interactions, develop student communication skills, and provide outlets for student creativity by leveraging the lived experiences of students. We have deployed the assignment in different kinds of classes (general education [service] courses and major-only seminars) at different kinds of institutions (a private, open-admission, large university on the East Coast and a private, highly selective, small liberal arts college on the …

Finding Your Mathematical Roots: Inclusion And Identity Development In Mathematics, Linda Mcguire

Journal of Humanistic Mathematics

This paper details a semester-long course project that has been successfully adapted for use in mathematics courses ranging from introductory level, general-education classes to advanced courses in the mathematics major. Through creating aspirational mathematical family trees and writing mathematical autobiographies, this assignment is designed to help battle belonging uncertainty, to challenge students to self-situate in relation to the history of mathematical and scientific knowledge, and to make visible a student’s developing identity in mathematics and, more broadly, in STEM.

The construction and scaffolding of the project, assignments, examples of student work, foundational readings, assessment and outcomes, and adaptation strategies for …

Gödel's Theorem In The Continuing Education Of Mathematics Teachers, Ana J. Lemes

Journal of Humanistic Mathematics

The notion of dépaysement épistémologique (epistemological disorientation) aims to capture the sense of disorientation when a learner is led to question their prior assumptions and understandings, generating uncertainty in a context in which they thought they had certain knowledge. This article describes an activity used with a group of practicing mathematics teachers in Uruguay that integrates elements of the history of mathematics related to Gödel’s incompleteness theorem, with the aim of provoking in the participants the experience of dépaysement épistémologique. Results show that several of the teachers participating in the activity felt dépaysement épistémologique, and this feeling triggered …

Our Histories, Our Values, Our Mathematics, Mark Huber, Gizem Karaali

Journal of Humanistic Mathematics

No abstract provided.

Front Matter

Journal of Humanistic Mathematics

No abstract provided.

Total Variation Flow In R^N Dimensions With Examples Relating To Perimeters Of Level Sets, Luis Schneegans, Victoria Shumakovich

Undergraduate Research Symposium

In this project, we explore radial solutions to the Total Variation Flow (TVF) equation with the help of the Sign Fast Diffusion Equation (SFDE) and prior results in the 1-dimensional case. Specifically for radial solutions, we derive equations and explicit solutions relating to the n-dimensional case. Lastly, we look at how level sets and (time) profiles change.

Effect Of Methane Mitigation On Global Temperature Under A Permafrost Feedback, Hannah Bäck, Riley May, Divya Sree Naidu, Steffen Eikenberry

Mathematics and Statistics Student Research and Class Projects

Earth systems may fall into an undesirable system state if 1.5°C of warming is exceeded. Carbon release from substantial permafrost stocks vulnerable to near-term warming represents a positive climate feedback that may increase the risk of 1.5°C warming or greater. Methane (CH₄) is a short-lived but powerful greenhouse gas with a global warming potential 28.5 times that of carbon dioxide (CO₂) over a 100 year time span. Because permafrost thaw in the coming centuries is partly determined by the warming in the 21st century, rapid reductions in methane emissions early in the 21st century could have …

On The Qualitative Analysis Of Nonlinear Q-Fractional Delay Descriptor Systems, Abdullah Yi̇ği̇t

Turkish Journal of Mathematics

In this manuscript, we obtain some sufficient conditions for a nonlinear q fractional integro singular system with constant delays to be asymptotically admissible and a nonlinear q fractional non-singular system to be asymptotically stable. We use Lyapunov-Krasovskii functionals and some inequalities to obtain these conditions. At the same time, we present some numerical examples that confirm the sufficient conditions we obtained theoretically, with their annotated solutions and graphs.

On Lyapunov-Type Inequalities For Five Different Types Of Higher Order Boundary Value Problems, Mustafa Fahri̇ Aktaş, Bariş Berkay Erçikti

Turkish Journal of Mathematics

In this paper, we establish the uniqueness and existence of the classical solution to higher-order boundary value problems. Then, we give a new Lyapunov-type inequalities for higher order boundary value problems. Our study is based on Green’s functions corresponding to the five different types of two-point boundary value problems. In addition, some applications of the obtained inequalities are given.

Globally Generated Vector Bundles On The Del Pezzo Threefold Of Degree 6 With Picard Number 2, Takuya Nemoto

Turkish Journal of Mathematics

We classify globally generated vector bundles on a general hyperplane section of P2 × P2 embedded by the Segre embedding, considering small first Chern classes c1 = (1, 1) and c1 = (2, 1).

Extremal Functions For A Singular Super-Critical Trudinger-Moser Inequality, Juan Zhao

Turkish Journal of Mathematics

In this paper, we deal with a singular super-critical Trudinger-Moser inequality on a unit ball of Rn , n ≥ 3. For any p > 1, we set λp(B) = inf u∈W1,n 0 (B),u̸≡0 ∫ B |∇u|ndx ( ∫ B |u|pdx)n/p as an eigenvalue related to the n-Laplacian. Let S be a set of radially symmetric functions. Precisely, if β ≥ 0 and α < (1 + p nβ)n−1+n/pλp(B) , then there exists a positive constant ϵ0 such that when λ ≤ 1 + ϵ0 , sup u∈W1,n 0 (B)∩S, ∫ B |∇u|ndx−α( ∫ B |u|p|x|pβdx) np ≤1 ∫ B |x|pβ ( eαn(1+ p n β)|u| n n−1 − λ Σm k=0 |αn(1 + p nβ)u n n−1 |k k! ) dx is attained, where αn = nω1/(n−1) n−1 , ωn−1 is the surface area of the unit ball in Rn . The proof is based on the method of blow-up analysis. The case λ = 0 was studied by Yang-Zhu in [38]. de Figueiredo [11] considered the case p = 2, β ≥ 0, and α = 0 in two dimension. The case λ = 0, p = n,−1 < β < 0, and α = 0 was considered by Adimurthi-Sandeep [1]. Our results extend those of the above cases.

On Strong Solvability Of One Nonlocal Boundary Value Problem For Laplace Equation In Rectangle, Telman Gasymov, Baharchin Akhmadli, Ümi̇t Ildiz

Turkish Journal of Mathematics

One nonlocal boundary value problem for the Laplace equation in a bounded domain is considered in this work. The concept of a strong solution to this problem is introduced. The correct solvability of this problem in the Sobolev spaces generated by the weighted mixed norm is proved by the Fourier method. In a classic statement, this problem has been

Invariant Symplectic Forms On Number Fields, Ahmad Rafiqi, Ayberk Zeyti̇n

Turkish Journal of Mathematics

We show that a number field of the form Q(λ) admits a symplectic form which is invariant under multiplication by λ if and only if the minimal polynomial of λ is palindromic of even degree. In particular, if λ is an algebraic integer, it is forced to be a unit. In the case when the minimal polynomial of λ is palindromic of degree 2d, we show that there is a d-dimensional space of invariant symplectic forms on Q(λ) .

On The Invariance Of Hyperstoneanness Under Lattice Isomorphisms, Banu Güntürk

Turkish Journal of Mathematics

Let X and Y be compact Hausdorff spaces with Y hyperstonean. In this paper, we prove that if C(X,R) and C(Y,R) are lattice isomorphic then these Banach spaces are linearly isometric, and, consequently, X and Y are homeomorphic, which in turn implies that X is also hyperstonean. Actually, we prove more than what is announced in the headline above. This result, in some ways, is a generalization of the well-known Banach-Stone theorem.

Some Estimates On The Spin−Submanifolds, Serhan Eker

Turkish Journal of Mathematics

In this paper, an optimal lower bound is given for the Submanifold Dirac operator in terms of the trace of an Energy−Momentum tensor, scalar curvature and mean curvature. In the equality case, it is proven that the submanifold is Einstein if the normal bundle is flat. Key words: Spin geometry, eigenvalues,

Modelling Spatial Distributions Of Sediments Fingerprinting In The Ruvu Basin Of Tanzania Without Continuous Sediment Monitoring, Charles H. Kasanzu

Tanzania Journal of Science

This study investigates various hinterlands’ contribution of sediments into the sub-basin known as Ruvu that comprises Mindu, Kibungo, Mvuha and Mindu catchments of Morogoro, Eastern Tanzania. An integrated geochemical and isotopic data were used to constrain this. Strategic sampling of basement and silty-sized sediments was conducted and the data were modelled using mass-balance computations. This approach revealed that, more than 70% of suspended sediments in the Mgeta River originate from Kigalamila, Sezima, Kingule, Mafumbo and Kibuko areas. Thus, hills north of Kisaki are major contributors of sediments with less contribution from the southern terrain. After modelling of the data from …

Machine Learning For Wireless Network Throughput Prediction, Gustavo A. Fernandez

School of Mathematical & Statistical Sciences Faculty Publications

This paper analyzes a dataset containing radio frequency (RF) measurements and Key Performance Indicators (KPIs) captured at 1876.6MHz with a bandwidth of 10MHz from an operational 4G LTE network in Nigeria. The dataset includes metrics such as RSRP (Reference Signal Received Power), which measures the power level of reference signals; RSRQ (Reference Signal Received Quality), an indicator of signal quality that provides insight into the number of users sharing the same resources; RSSI (Received Signal Strength Indicator), which gauges the total received power in a bandwidth; SINR (Signal to Interference plus Noise Ratio), a measure of signal quality considering both …

The International Crisis In Numeracy Education, Nathan D. Grawe

Numeracy

The OECD recently released results from the 2022 administration of the Programme for International Student Assessment test. As other studies suggest, pandemic mitigation policies resulted in deep learning loss including in basic mathematics which forms the foundation of numeracy. Perhaps of greater concern, however, in many countries pandemic effects amplify declining performance that dates back a decade or more. Losses of two or more years' worth of mathematics education are not uncommon among developed countries. The editorial makes an urgent call for research that identifies practical steps to reverse these trends.

Recent Studies On The Super Edge-Magic Deficiency Of Graphs, Rikio Ichishima, Susana C. Lopez, Francesc Muntaner, Yukio Takahashi

Theory & Applications of Graphs

A graph $G$ is called edge-magic if there exists a bijective function $f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left(G\right) \right\vert \right\}$ such that $f\left(u\right) + f\left(v\right) + f\left(uv\right)$ is a constant for each $uv\in E\left( G\right) $. Also, $G$ is called super edge-magic if $f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}$. Furthermore, the super edge-magic deficiency $ \mu_{s}\left(G\right)$ of a graph $G$ is defined to be either the smallest nonnegative integer $n$ with the property that $G \cup nK_{1}$ is super edge-magic or $+ \infty$ if there exists no such …