Mathematics Commons^™

Designing Fractal Line Pied-De-Poules: A Case Study In Algorithmic Design Mediating Between Culture And Fractal Mathematics, Loe M.G. Feijs

Journal of Humanistic Mathematics

Millions of people own and wear pied-de-poule (houndstooth) garments. The pattern has an intriguing basic figure and a typical set of symmetries. The origin of the pattern lies in a specific type of weaving. In this article I apply computational techniques to modernize this ancient decorative pattern. In particular I describe a way to enrich pied-de-poule with a fractal structure.

Although a first fractal line pied-de-poule was shown at Bridges 2015, a number of fundamental questions still remained. The following questions are addressed in this article: Does the original pied-de-poule appear as a limit case when the fractal structure is …

College Mathematics Students' Perceptions Of "Believing" Teacher Actions, Shelly Sheats Harkness, Bethany A. Noblitt

Journal of Humanistic Mathematics

Believing and doubting – two methodological processes – deserve equal attention according to Elbow (1986; 2006). When a teacher plays the doubting game in a mathematics classroom her own mathematical thinking dominates and she attempts to find flaws and errors and misconceptions in students’ mathematical thinking. When a teacher plays the believing game in a mathematics classroom she surrenders her own mathematical thinking and she attempts to find virtues and strengths and merits in students’ mathematical thinking. Paradoxically and succinctly, a teacher must believe her own mathematical thinking in order to doubt and a teacher must doubt her own mathematical …

Students Studying Students And Reasoning About Reasoning: A Qualitative Analysis, Salvatore J. Petrilli, Grant Clark, Nicholas Demarco, Jack Esposito, Brianne Giuliano, Sara Greiss, Emily Harris, Alessia Merritts, Kyle Murray, Mateusz Piekut, Brian Seidl, Scott Shannon, Nicole Silva, Christina Sullivan, Brittany Willoughby, Yile Zhou

Journal of Humanistic Mathematics

In this work, a faculty member takes a journey along with students as they enhance their understanding of how people solve mathematical problems through a mainly qualitative statistical project. Student authors of this paper registered for a problem solving seminar led by the faculty author, and then created and analyzed self-built assessment tools to explore problem solving techniques. Here we share our findings and recommendations, which we hope will inspire others to explore novel pedagogical techniques in the teaching of mathematical problem solving. We incorporate into our presentation ur voices, reflecting on how we and others solve problems.

From Carriage Wheels To Interest Rates: The Evolution Of Word Problems In Algebra Textbooks From 1901 To Today, Jemma Lorenat, Elodie Arbogast, Ethan Baer, Carla Condori Bazan, Robert Bettinger, Emily Carpenter, Hiawatha Davis Iii, Derick Grant, Olivia Howe, Neil Kelley, Maya Minier, Naima Orozco-Valdivia, Alan Peck, Carolina Saavedra, Sumesh Shiwakoty, Hunter Sidel, Carter Stripp, Josephine Terrien, Simone Wolynski, Leana Yearwood

Journal of Humanistic Mathematics

In teaching algebra, extra-mathematical word problems can bridge the gap between questions about abstract numbers and questions about everyday life. Thus, more than other aspects of elementary algebra, we would expect word problems to have changed in the recent past. This paper documents the findings of a collective research project that examined the content of such word problems over the past century. Alongside amusing and provocative examples, this paper shows how students can participate in exploratory research with primary sources from the history of mathematics.

The Mathematics Of The Astrolabe And Its History, Graziano Gentili, Luisa Simonutti, Daniele C. Struppa

Journal of Humanistic Mathematics

In this article we trace the scientific and cultural history of the astrolabe, a mechanical instrument used in the past for astronomical measurements and navigational purposes. The story of the astrolabe is interesting from several points of view, since it intertwines mathematical developments, geographical explorations, changing worldviews, and different cultures and civilizations. In our explorations we move from the early understanding of the world due to the Greeks, to the loss of their work, its rediscovery, the reception of Arab thinkers in Western natural philosophy, and, finally, to the new European culture that emerged with the end of the Middle …

College Students’ Images Of Mathematicians And Mathematical Careers, Katrina Piatek-Jimenez, Miranda Nouhan, Michaela Williams

Journal of Humanistic Mathematics

In this paper we report our findings of college students’ images of mathematicians and we reflect on different methodologies used to assess this information. The study reported in this paper was conducted in two stages. During the first stage, we asked 179 college students to “draw a mathematician” and also asked them to list five characteristics and five careers for a mathematician. In the second stage of the study, we conducted four focus group interviews with a total of twelve college students. During the focus group interviews, we showed the students 16 photos of real people and asked them to …

Engaging The Paradoxical: Zeno's Paradoxes In Three Works Of Interactive Fiction, Michael Z. Spivey

Journal of Humanistic Mathematics

For over two millennia thinkers have wrestled with Zeno's paradoxes on space, time, motion, and the nature of infinity. In this article we compare and contrast representations of Zeno's paradoxes in three works of interactive fiction, Beyond Zork, The Chinese Room, and A Beauty Cold and Austere. Each of these works incorporates one of Zeno's paradoxes as part of a puzzle that the player must solve in order to advance and ultimately complete the story. As such, the reader must engage more deeply with the paradoxes than he or she would in a static work of fiction. …

What Gets Checked At The Door? Embracing Students' Complex Mathematical Identities, Jennifer L. Ruef

Journal of Humanistic Mathematics

Identity formation is complex, ongoing, and context specific. To be successful in mathematics classes, students must negotiate and navigate the normative identity of the class--what counts as being "good at math" (Cobb, Gresalfi & Hodge, 2009). Within the constraints of normative identity, students must also negotiate a personal doer-of-math identity: who they are within the context of this particular mathematics class. When students are compelled to suppress key aspects of their identity in order to accommodate the normative identity of the class cognitive bandwidth for learning may be impeded (Steele, 1997). Conversely, when students are guided in braiding individual identity …

John Cheever's Story "The Geometry Of Love", Robert Haas

Journal of Humanistic Mathematics

Though John Cheever was a leading writer of short fiction, his story “The Geometry of Love” has received little prior literary or mathematical comment. In this essay it is read, against the background of Cheever’s own troubled life and marriage, as a Don Quixote – like search, explicitly following the model of Euclidean geometry, and at times wildly funny, for an ideal world of truth and happiness.

Starting Our Decennial, Mark Huber, Gizem Karaali

Journal of Humanistic Mathematics

No abstract provided.

Front Matter

Journal of Humanistic Mathematics

No abstract provided.

Soccer Tournament Matrices, Lei Cao

Mathematics Colloquium Series

In this talk, I will present a combinatorial object, soccer tournament matrices, which is understandable to undergraduate students and gives a taste of combinatorial matrix theory. Consider a round-robin tournament of n teams in which each team plays every other team exactly once and where ties are allowed. A team scores 3 points for a win, 1 point for a tie, and 0 point for a loss, then each particular result leads to a soccer tournament matrix. Let T(R, 3) denote the class of all soccer tournament matrices with the row sum vector R. In this talk, I will explore …

Quantitatively Ranking Incorrect Responses To Multiple-Choice Questions Using Item Response Theory, Trevor Smith, Kyle J. Louis, Bartholemew J. Ricci Iv, Nasrine Bendjilali

College of Science & Mathematics Departmental Research

Research-based assessment instruments (RBAIs) are ubiquitous throughout both physics instruction and physics education research. The vast majority of analyses involving student responses to RBAI questions have focused on whether or not a student selects correct answers and using correctness to measure growth. This approach often undervalues the rich information that may be obtained by examining students’ particular choices of incorrect answers. In the present study, we aim to reveal some of this valuable information by quantitatively determining the relative correctness of various incorrect responses. To accomplish this, we propose an assumption that allows us to define relative correctness: students who …

Dynamical Role Of Pivotal Brain Regions In Parkinson Symptomatology Uncovered With Deep Learning, Alex A. Nguyen, Pedro D. Maia, Xiao Gao, Pablo F. Damasceno, Ashish Raj

Mathematics Faculty Publications - Archive

The release of a broad, longitudinal anatomical dataset by the Parkinson’s Progression Markers Initiative promoted a surge of machine-learning studies aimed at predicting disease onset and progression. However, the excessive number of features used in these models often conceals their relationship to the Parkinsonian symptomatology. Objectives: The aim of this study is two-fold: (i) to predict future motor and cognitive impairments up to four years from brain features acquired at baseline; and (ii) to interpret the role of pivotal brain regions responsible for different symptoms from a neurological viewpoint. Methods: We test several deep-learning neural network configurations, and report our …

Approximate And Exact Merging Of Knapsack Constraints With Cover Inequalities, Fabio Vitor, Todd Easton

Mathematics Faculty Publications

This paper presents both approximate and exact merged knapsack cover inequalities, a class of cutting planes for knapsack and multiple knapsack integer programs. These inequalities combine the information from knapsack constraints and cover inequalities. Approximate merged knapsack cover inequalities can be generated through a O(n log n) algorithm, where n is the number of variables. This class of inequalities can be strengthened to an exact version with a pseudo-polynomial time algorithm. Computational experiments demonstrate an average improvement of approximately 8% in solution time and 5% in the number of ticks from CPLEX when approximate merged knapsack cover …

A Study Of Operators On The Discrete Analogue Of Hardy Spaces On Homogeneous Trees And On Other Structures., P. Muthukumar Dr.

Doctoral Theses

In analytic function theory, the study of multiplication and composition operators has a rich structure for various analytic function spaces of the unit disk D = {z ∈ C : |z| < 1} such as the Hardy spaces Hp, the Bergman spaces Ap and the Bloch space B. This theory connects the operator theoretic properties such as boundedness, compactness, spectrum, invertibility, isometry with that of the function theoretic properties of the inducing map (symbol) such as bijectivity, boundary behaviour and vise versa In Chapter 2, we define discrete analogue of generalized Hardy spaces (Tp) and their separable subspaces (Tp,0) on a homogenous rooted tree and study some of their properties such as completeness, inclusion relations with other spaces, separability and growth estimate for functions in these spaces and their consequences. In Chapter 3, we obtain equivalent conditions for multiplication operators Mψ on Tp and Tp,0 to be bounded and compact. Furthermore, we discuss point spectrum, approximate point spectrum and spectrum of multiplication operators and discuss when a multiplication operator is an isometry. In Chapter 4, we give an equivalent conditions for the composition operator Cφ to be bounded on Tp and on Tp,0 spaces and compute their operator norms. We have considered the composition operators induced by special symbols such as univalent and multivalent maps and automorphism of a homogenous tree. We also characterize invertible composition operators and isometric composition operators on Tp and on Tp,0 spaces. Also, we discuss the compactness of Cφ on Tp spaces and finally we prove that there are no compact composition operators on Tp,0 spaces. In Chapter 5, we consider the composition operators on the Hardy-Dirichlet space H2, the space of Dirichlet series with square summable coefficients. By using the Schur test, we give some upper and lower estimates on the norm of a composition operator on H2 , for the affine-like inducing symbol ϕ(s) = c1 + cqq −s , where q ≥ 2 is a fixed integer. We also give an estimate for approximation numbers of a composition operators in our H2 setting. In Chapter 6, we study the weighted composition operators preserving the class Pα. Some of its consequences and examples of certain special cases are presented. Furthermore, we discuss about the fixed points of weighted composition operators.

A Problem Of Solute Transport In A Cylindrical Porous Media With A Fractal Structure Taking Into Account Adsorption Phenomena, B. X. Xujayorov, J. M. Maxmudov, A. I. Usmanov, B. O. Saidov

Scientific Journal of Samarkand University

The process of anomalous solute transport in a coaxial cylindrical porous media is modelled by differential equations with a fractional derivative. The problem of solute transport in a two-zone cylindrical media consisting of macro- and micropores taking into account adsorption effectshas been numericallyposed and solved. The concentration profiles of suspended particles and the adsorbed solute in the macropore and micropore, the surface of the local concentration in the micropore are determined. The influence of adsorption phenomena and the order of the derivative with respect to the coordinate, i.e. fractal dimension of the media, on the characteristics of the solute transport …

Math 220p Foundations Of Mathematics, Nicholas Vlamis

Open Educational Resources

No abstract provided.

On A Two-Speed Mathematical Model Of Two-Fluid Medium With One Pressure, X. X. Imomnazarov, S. B. Kuyliyev

Scientific Journal of Samarkand University

An overdetermined stationary system of second-order differential equations is obtained. For the two-dimensional system, a variational statement of the problem is established. It is shown that the variational problem for the system of equations of two-speed hydrodynamics is well-posed in the corresponding Sobolev space

The Cauchy Problem For The System Of The Elasticity, I. E. Niyozov, O. Karshiboyev

Scientific Journal of Samarkand University

In this paper we consider the problem of analytical continuation of solutions to the system of the elasticity in a bounded domain from their values and values of their strains on a part of the boundary of this domain, i.e., we study the Cauchy problem

A Discontinuous Galerkin Method For The Stolz–Adams Approximate Deconvolution Model For Turbulent Flows, Monika Neda, Béatrice Rivière

Mathematical Sciences Faculty Research

We consider the zeroth order model of the family of approximate deconvolution models of Stolz and Adams. We propose and analyze fully discrete schemes using discontinuous finite elements. Optimal error estimates are derived. The dependence of these estimates with respect to the Reynolds number Re is O(ReeRe), which is an improvement with respect to the classical continuous finite element method where the dependence is O(ReeRe3), Layton [1].

On Estimates For The Damped Oscillatory Integrals, I. A. Ikromov, Sh. A. Muranov

Scientific Journal of Samarkand University

In this paper we consider estimates of the Fourier transform measures, concentrated on analytic hypersurfaces containing the of damping factor. The paper presents the solution of the problem S.D.Soggi and I.M. Stein about the optimal decay of the transformation Fourier measures with a damping factor for any analytic surfaces in three-dimensional Euclidean space.

Of The Cauchy Problem For Thelaplace Equations On A Plane, D. S. Shodiyev

Scientific Journal of Samarkand University

In this paper, a uniqueness theorem is proved, and an estimate of conditional stability is obtained, and an approximate solution is constructed by the method of quasi-derivation and Tikhonov regularization.

The Problem Of Integral Geometry In A Strip With Weight Function, A. X. Begmatov, A. S. Ismoilov

Scientific Journal of Samarkand University

In this work we consider the problem of reconstructing a function from a family of parabolas in the upper half-plane with a weight function having a singularity. The uniqueness of theorem for the solution of equation is proved and the inversion formula is derived. It is shown that the solution of the problem posed is weakly ill-posed, that is, stability estimates are obtained in spaces of finite smoothness.

Uniqueness And Stability Problem Of Integral Geometry With Indignation, Akram Begmatov, Zarifjon Ochilov, Abduroziq Xusanov

Scientific Journal of Samarkand University

Рассматривается новый класс задач интегральной геометрии типа Вольтерра с функцией специального веса. Доказаны теоремы единственности и существования решения, получены оценки устойчивости и формула обращения в пространствах Соболева, что свидетельствует о слабой некорректности решения задачи интегральной геометрии. Постановка задачи интегральной геометрии с возмущением на семейство парабол в полосе считается. Теорема единственности ее решения доказана в классе дважды непрерывно дифференцируемых функций с компактным носителем, а оценки устойчивости получены в пространствах конечной гладкости.

A Short Note On Extreme Points Of Certain Polytopes, Lei Cao, Ariana Hall, Selcuk Koyuncu

Mathematics Faculty Articles

We give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly substochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

Arbitrarily High-Order Unconditionally Energy Stable Schemes For Thermodynamically Consistent Gradient Flow Models, Yuezheng Gong, Jia Zhao, Qi Wang

Mathematics and Statistics Faculty Publications

We present a systematic approach to developing arbitrarily high-order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy dissipation laws. Utilizing the energy quadratization method, we formulate the gradient flow model into an equivalent form with a corresponding quadratic free energy functional. Based on the equivalent form with a quadratic energy, we propose two classes of energy stable numerical approximations. In the first approach, we use a prediction-correction strategy to improve the accuracy of linear numerical schemes. In the second approach, we adopt the Gaussian collocation method to discretize the equivalent form with a quadratic …

The Continuous, The Discrete And The Infinitesimal In Philosophy And Mathematics (Book Review), Calvin Jongsma

Faculty Work Comprehensive List

Reviewed Title: The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics by John L. Bell. Springer, 2019. 313 pp. ISBN: 9783030187064.

From Dyck Paths To Standard Young Tableaux, Juan B. Gil, Peter R. W. Mcnamara, Jordan O. Tirrell, Michael D. Weiner

Faculty Journal Articles

We present nine bijections between classes of Dyck paths and classes of standard Young tableaux (SYT). In particular, we consider SYT of flag and rectangular shapes, we give Dyck path descriptions for certain SYT of height at most 3, and we introduce a special class of labeled Dyck paths of semilength n that is shown to be in bijection with the set of all SYT with n boxes. In addition, we present bijections from certain classes of Motzkin paths to SYT. As a natural framework for some of our bijections, we introduce a class of set partitions which in some …

Towards A Novel Generalized Chinese Remainder Algorithm For Extended Rabin Cryptosystem, Justin Zhan, Peter J. Shiue, Shen C. Huang, Benjamin J. Lowe

Mathematical Sciences Faculty Research

This paper proposes a number of theorems and algorithms for the Chinese Remainder Theorem, which is used to solve a system of linear congruences, and the extended Rabin cryptosystem, which accepts a key composed of an arbitrary finite number of distinct primes. This paper further proposes methods to relax the condition on the primes with trade-offs in the time complexity. The proposed algorithms can be used to provide ciphertext indistinguishability. Finally, this paper conducts extensive experimental analysis on six large data sets. The experimental results show that the proposed algorithms are asymptotically tight to the existing decryption algorithm in the …