Physical Sciences and Mathematics Commons^™

The Nothing That Really Matters, Szilárd Svitek

Journal of Humanistic Mathematics

Zero has (a) special role(s) in mathematics. In the current century, we take negative numbers and zero for granted, but we should also be aware that their acceptance and their emergence in mathematics, and their ubiquity today, have not come to happen as rapidly as, for example, that of natural numbers. Students can quickly become confused by the question: is zero a natural number? The answer is simple: a matter of definition. The history of zero and that of negative numbers are closely linked. It was in the calculations of debts that the negative numbers first appeared, where the state …

The Use And Development Of Mathematics Within Creative Literature, Toby S C Peres

Journal of Humanistic Mathematics

This paper presents a study on the extent to which creative literature been used as a vessel to carry forward the development of mathematical thought. The role of mathematics as a driving force for literature is highlighted, and while many examples exist that clearly show an attempt to disperse mathematical ideas, with Lewis Carroll, OuLiPo and ancient poetry considered, the argument that the sole purpose of the writings was for the sake of mathematical development is not clear-cut.

The Genesis Of A Theorem, Osvaldo Marrero

Journal of Humanistic Mathematics

We present the story of a theorem's conception and birth. The tale begins with the circumstances in which the idea sprouted; then is the question's origin; next comes the preliminary investigation, which led to the conjecture and the proof; finally, we state the theorem. Our discussion is accessible to anyone who knows mathematical induction. Therefore, this material can be used for instruction in a variety of courses. In particular, this story may be used in undergraduate courses as an example of how mathematicians do research. As a bonus, the proof by induction is not of the simplest kind, because it …

Where Do Babies Come From?, Marcio Luis Ferreira Nascimento

Journal of Humanistic Mathematics

According to European folklore, popularized by a fairy tale, storks are responsible for bringing babies to new parents. This probably came from observation in certain European countries, such as Norway, Netherlands or Germany, that storks nesting on the roofs of households were believed to bring good luck, as the possibility of new births. People love stories, but correlation simply means that there is a relationship between two factors that tells nothing about the direction of said relationship, if any. Another possibility is simple coincidence. Let us say that it’s possible that one factor causes another. It’s also possible that the …

Teaching Mathematics After Covid: A Conversation Not A Discussion, Wendy Ann Forbes, Joyce Mgombelo

Journal of Humanistic Mathematics

Inspired by Brent Davis' conceptualization of listening and conversation in his book Teaching Mathematics: Toward a Sound Alternative, we propose how we as a mathematics education community may move forward by continuing in the conversation that emerged from COVID. We encourage all involved to listen rather than assume a discussion-oriented stance. Using an enactivist lens, we look at the pandemic learning space, give an overview of the education conversation that emerged in Ontario, and offer a way to rethink Mathematics Education within the frame of a conversation. We believe that if mathematics education is to engage learners in a meaningful …

Human-Machine Collaboration In The Teaching Of Proof, Gila Hanna, Brendan P. Larvor, Xiaoheng (Kitty) Yan

Journal of Humanistic Mathematics

This paper argues that interactive theorem provers (ITPs) could play an important role in fostering students’ appreciation and understanding of proof and of mathematics in general. It shows that the ITP Lean has three features that mitigate existing difficulties in teaching and learning mathematical proof. One is that it requires students to identify a proof strategy at the start. The second is that it gives students instant feedback while allowing them to explore with maximum autonomy. The third is that elementary formal logic finds a natural place in the activity of creating proofs. The challenge in using Lean is that …

A Classification Of Musical Scales Using Binary Sequences, Thomas Hillen

Journal of Humanistic Mathematics

Every beginning music student has gone through the four main musical scales: major, natural minor, harmonic minor, and melodic minor. And some might wonder, why those four and not five, or six, or just three? Here we show that a mathematical classification can be used to identify these scales as representatives of certain scale families. Moreover, the classification reveals another scale family, which is much less known: the harmonic major scale. We find that each scale family contains exactly seven scales, which include the modes (dorian, phrygian,...) and other scales such as the Romanian, …

The Merchant And The Mathematician: Commerce And Accounting, Graziano Gentili, Luisa Simonutti, Daniele C. Struppa

Journal of Humanistic Mathematics

In this article we describe the invention of double-entry bookkeeping (or partita doppiaas it was called in Italian), as a fertile intersection between mathematics and early commerce. We focus our attention on this seemingly simple technique that requires only minimal mathematical expertise, but whose discovery is clearly the result of a mathematical way of thinking, in order to make a conceptual point about the role of mathematics as the humus from which disciplines as different as operations research, computer science, and data science have evolved.

The Roles Of Mathematical Metaphors And Gestures In The Understanding Of Abstract Mathematical Concepts, Omid Khatin-Zadeh, Zahra Eskandari, Danyal Farsani

Journal of Humanistic Mathematics

When a new mathematical idea is presented to students in terms of abstract mathematical symbols, they may have difficulty to grasp it. This difficulty arises because abstract mathematical symbols do not directly refer to concretely perceivable objects. But, when the same content is presented in the form of a graph or a gesture that depicts that graph, it is often much easier to grasp. The process of solving a complex mathematical problem can also be facilitated with the use of a graphical representation. Transforming a mathematical problem or concept into a graphical representation is a common problem solving strategy, and …

From A Doodle To A Theorem: A Case Study In Mathematical Discovery, Juan FernáNdez GonzáLez, Dirk Schlimm

Journal of Humanistic Mathematics

We present some aspects of the genesis of a geometric construction, which can be carried out with compass and straightedge, from the original idea to the published version (Fernández González 2016). The Midpoint Path Construction makes it possible to multiply the length of a line segment by a rational number between 0 and 1 by constructing only midpoints and a straight line. In the form of an interview, we explore the context and narrative behind the discovery, with first-hand insights by its author. Finally, we discuss some general aspects of this case study in the context of philosophy of mathematical …

Where Does Mathematics Come From? Really, Where?, Mark Huber, Gizem Karaali

Journal of Humanistic Mathematics

No abstract provided.

Front Matter

Journal of Humanistic Mathematics

No abstract provided.

Post-Quantum Hermite-Jensen-Mercer Inequalities, Martin Bohner, Hüseyin Budak, Hasan Kara

Mathematics and Statistics Faculty Research & Creative Works

The Jensen-Mercer inequality, which is well known in the literature, has an important place in mathematics and related disciplines. In this work, we obtain the Hermite-Jensen-Mercer inequality for post-quantum integrals by utilizing Jensen-Mercer inequalities. Then we investigate the connections between our results and those in earlier works. Moreover, we give some examples to illustrate our main results. This is the first paper about Hermite-Jensen-Mercer inequalities for post-quantum integrals.

Elements Of Higher Homotopy Groups Undetectable By Polyhedral Approximation, John K. Aceti, Jeremy Brazas

Mathematics Faculty Publications

When nontrivial local structures are present in a topological space X, a common approach to characterizing the isomorphism type of the n-th homotopy group πn(X, x0) is to consider the image of πn(X, x0) in the nth Cˇ ech homotopy group πˇ n(X, x0) under the canonical homomorphism 9n : πn(X, x0) → πˇ n(X, x0). The subgroup ker(9n) is the obstruction to this tactic as it consists of precisely those elements of πn(X, x0), which cannot be detected by polyhedral approximations to X. In this paper, we use higher dimensional analogues of Spanier groups to characterize ker(9n). In particular, …

The Asymptotic P-Poisson Equation As P →∞ In Carnot-Carathéodory Spaces, Luca Capogna, Gianmarco Giovannardi, Andrea Pinamonti, Simone Verzellesi

Mathematics Sciences: Faculty Publications

In this paper we study the asymptotic behavior of solutions to the subelliptic p-Poisson equation as p → +∞ in Carnot Carathéodory spaces. In particular, introducing a suitable notion of differentiability, we extend the celebrated result of Bhattacharya, DiBenedetto and Manfredi [BDM] and we prove that limits of such solutions solve in the sense of viscosity a hybrid first and second order PDE involving the ∞−Laplacian and the Eikonal equation.

Exact Parallel Waves In General Relativity, Cian Roche, Amir Babak Aazami, Carla Cederbaum

Mathematics

We conduct a review of the basic definitions and the principal results in the study of wavelike spacetimes, that is spacetimes whose metric models massless radiation moving at the speed of light, focusing in particular on those geometries with parallel rays. In particular, we motivate and connect their various definitions, outline their coordinate descriptions and present some classical results in their study in a language more accessible to modern readers, including the existence of “null coordinates” and the construction of Penrose limits. We also present a thorough summary of recent work on causality in pp-waves, and describe progress in addressing …

A Stronger Strong Schottky Lemma For Euclidean Buildings, Michael E. Ferguson

Dissertations, Theses, and Capstone Projects

We provide a criterion for two hyperbolic isometries of a Euclidean building to generate a free group of rank two. In particular, we extend the application of a Strong Schottky Lemma to buildings given by Alperin, Farb and Noskov. We then use this extension to obtain an infinite family of matrices that generate a free group of rank two. In doing so, we also introduce an algorithm that terminates in finite time if the lemma is applicable for pairs of certain kinds of matrices acting on the Euclidean building for the special linear group over certain discretely valued fields.

Translation Of: Familles De Surfaces Isoparamétriques Dans Les Espaces À Courbure Constante, Annali Di Mat. 17 (1938), 177–191, By Élie Cartan., Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

English title: Families of isoparametric surfaces in spaces of constant curvature

This is an English translation of the article "Familles de surfaces isoparamétriques dans les espaces à courbure constante" which was originally published in Annali di Matematica 17, 177–191 (1938), by Élie Cartan.

A note from Thomas E. Cecil, translator: This is an unofficial translation of the original paper which was written in French. All references should be made to the original paper.

Mathematics Subject Classification Numbers: 53C40, 53C42, 53B25

Generalizing Math Word Problem Solvers Via Solution Diversification, Zhenwen Liang, Jipeng Zhang, Lei Wang, Yan Wang, Jie Shao, Xiangliang Zhang

Research Collection School Of Computing and Information Systems

Current math word problem (MWP) solvers are usually Seq2Seq models trained by the (one-problem; one-solution) pairs, each of which is made of a problem description and a solution showing reasoning flow to get the correct answer. However, one MWP problem naturally has multiple solution equations. The training of an MWP solver with (one-problem; one-solution) pairs excludes other correct solutions, and thus limits the generalizability of the MWP solver. One feasible solution to this limitation is to augment multiple solutions to a given problem. However, it is difficult to collect diverse and accurate augment solutions through human efforts. In this paper, …

Combinatorial Identities Associated With A Bivariate Generating Function For Overpartition Pairs, Atul Dixit, Ankush Goswami

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We obtain a three-parameter q-series identity that generalizes two results of Chan and Mao. By specializing our identity, we derive new results of combinatorial significance in connection with N(r,s,m,n), a function counting certain overpartition pairs recently introduced by Bringmann, Lovejoy and Osburn. For example, one of our identities gives a closed-form evaluation of a double series in terms of Chebyshev polynomials of the second kind, thereby resulting in an analogue of Euler's pentagonal number theorem. Another of our results expresses a multi-sum involving N(r,s,m,n) in terms of just the partition function p(n). Using a result of Shimura we also relate …

Coloring Complexes And Combinatorial Hopf Monoids, Jacob A. White

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We generalize the notion of a coloring complex of a graph to linearized combinatorial Hopf monoids. We determine when a linearized combinatorial Hopf monoid has such a construction, and discover some inequalities that are satisfied by the quasisymmetric function invariants associated to the combinatorial Hopf monoid. We show that the collection of all such coloring complexes forms a linearized combinatorial Hopf monoid, which is the terminal object in the category of combinatorial Hopf monoids with convex characters. We also study several examples of combinatorial Hopf monoids.

On Viscoelastic Fiber Jet Formation By Forcespinning At High Rotation Rate, Daniel N. Riahi, Saulo Orizaga

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We consider a nonlinear three-dimensional viscoelastic fiber jet that is generated during a forcespinning process. We provide a particular case for such a rotating jet at a high rotation rate. We use a viscoelastic constitutive model for the jet equations and then applying a new slender body approach, we continue with proper scaling and perturbation technique to develop a new model for such a jet system. We find that the profiles for jet quantities versus arc length are notably different from all those in related studies reported before for either high or low rotation rates. In particular, jet radius first …

Filling In The Gaps: A Multiple Imputation Approach To Estimating Aging Curves In Baseball, Quang Nguyen, Gregory J. Matthews

Mathematics and Statistics: Faculty Publications and Other Works

In sports, an aging curve depicts the relationship between average performance and age in athletes' careers. This paper investigates the aging curves for offensive players in the Major League Baseball. We study this problem in a missing data context and account for different types of dropouts of baseball players during their careers. In particular, the performance metric associated with the missing seasons is imputed using a multiple imputation model for multilevel data, and the aging curves are constructed based on the imputed datasets. We first perform a simulation study to evaluate the effects of different dropout mechanisms on the estimation …

C∗-Extreme Maps And Nest Algebras, Manish Kumar Dr.

Doctoral Theses

NA

Completeness Of Nominal Props, Samuel Balco, Alexander Kurz

Engineering Faculty Articles and Research

We introduce nominal string diagrams as string diagrams internal in the category of nominal sets. This leads us to define nominal PROPs and nominal monoidal theories. We show that the categories of ordinary PROPs and nominal PROPs are equivalent. This equivalence is then extended to symmetric monoidal theories and nominal monoidal theories, which allows us to transfer completeness results between ordinary and nominal calculi for string diagrams.

Music: Numbers In Motion, Graziano Gentili, Luisa Simonutti, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

Music develops and appears as we allow numbers to acquire a dynamical aspect and create, through their growth, the various keys that permit the richness of the musical texture. This idea was simply adumbrated in Plato’s work, but its importance to his philosophical worldview cannot be underestimated. In this paper we begin by discussing what is probably the first written record of an attempt to create a good temperament and then follow the Pythagoreans approach, whose problems forced musicians, over the next several centuries up to the Renaissance and early modern times, to come up with many different variations.

Introducing Systems Via Laplace Transforms, Ollie Nanyes

CODEE Journal

The purpose of this note is to show how to move from Laplace Transforms to a brief introduction to two dimensional systems of linear differential equations with only basic matrix algebra.

Determining The Proportionality Of Ischemic Stroke Risk Factors To Age, Elizabeth Hunter, John D. Kelleher

Articles

While age is an important risk factor, there are some disadvantages to including it in a stroke risk model: age can dominate the risk score and lead to over-or under-predictions in some age groups. There is evidence to suggest that some of these disadvantages are due to the non-proportionality of other risk factors with age, eg, risk factors contribute differently to stroke risk based on an individual’s age. In this paper, we present a framework to test if risk factors are proportional with age. We then apply the framework to a set of risk factors using Framingham heart study data …

The Effect Of Damping By An Environment On Emergence Of Classicality, Paul Bracken

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

The role of dissipation with respect to a microscopic superposition of quantum states is investigated by means of master equations. This has implications for the study of the emergence of classicality from the quantum level. In particular, it illustrates why it is difficult to observe a macroscopic quantum state. The role of the environment is assumed by the measuring apparatus. A pure state is reduced to a mixture in the pointer basis of the system by means of the interaction with the apparatus. It is the intention that this type of analysis will have applications to experiments which are designed …

Supplementary Files For "Adaptive Mapping Of Design Ground Snow Loads In The Conterminous United States", Jadon Wagstaff, Jesse Wheeler, Brennan Bean, Marc Maguire, Yan Sun

Browse all Datasets

Recent amendments to design ground snow load requirements in ASCE 7-22 have reduced the size of case study regions by 91% from what they were in ASCE 7-16, primarily in western states. This reduction is made possible through the development of highly accurate regional generalized additive regression models (RGAMs), stitched together with a novel smoothing scheme implemented in the R software package remap, to produce the continental- scale maps of reliability-targeted design ground snow loads available in ASCE 7-22. This approach allows for better characterizations of the changing relationship between temperature, elevation, and ground snow loads across the Conterminous United …