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Articles 31 - 60 of 246
Full-Text Articles in Mathematics
Another Angle On Perspective: Solutions For Fermi Questions, May 2023, John Adam
Another Angle On Perspective: Solutions For Fermi Questions, May 2023, John Adam
Mathematics & Statistics Faculty Publications
No abstract provided.
Review Of The History Of Mathematics: A Source-Based Approach (Vol. 2), Part I, Erik R. Tou
Review Of The History Of Mathematics: A Source-Based Approach (Vol. 2), Part I, Erik R. Tou
Euleriana
Review of The History of Mathematics: A Source-Based Approach (Vol. 2), Part I, by June Barrow-Green, Jeremy Gray, and Robin Wilson. MAA Press, 2022, 330 + xiv pages.
Equidistant Sets In Spaces Of Bounded Curvature, Logan Scott Fox
Equidistant Sets In Spaces Of Bounded Curvature, Logan Scott Fox
Dissertations and Theses
Given a metric space (X,d), and two nonempty subsets A,B ⊆ X, we study the properties of the set of points of equal distance to A and B, which we call the equidistant set E(A,B). In general, the structure of the equidistant set is quite unpredictable, so we look for conditions on the ambient space, as well as the given subsets, which lead to some regularity of the properties of the equidistant set. At a minimum, we will always require that X is path connected (so that E( …
Sangaku In Multiple Geometries: Examining Japanese Temple Geometry Beyond Euclid, Nathan Hartmann
Sangaku In Multiple Geometries: Examining Japanese Temple Geometry Beyond Euclid, Nathan Hartmann
Honors College Theses
When the country of Japan was closed from the rest of the world from 1603 until
1867 during the Edo period, the field of mathematics developed in a different way
from how it developed in the rest of the world. One way we see this development
is through the sangaku, the thousands of geometric problems hung in various Shinto and Buddhist temples throughout the country. Written on wooden tablets by people from numerous walks of life, all these problems hold true within Euclidean geometry. During the 1800s, while Japan was still closed, non-Euclidean geometries began to develop across the …
How To Guard An Art Gallery: A Simple Mathematical Problem, Natalie Petruzelli
How To Guard An Art Gallery: A Simple Mathematical Problem, Natalie Petruzelli
The Review: A Journal of Undergraduate Student Research
The art gallery problem is a geometry question that seeks to find the minimum number of guards necessary to guard an art gallery based on the qualities of the museum’s shape, specifically the number of walls. Solved by Václav Chvátal in 1975, the resulting Art Gallery Theorem dictates that ⌊n/3⌋ guards are always sufficient and sometimes necessary to guard an art gallery with n walls. This theorem, along with the argument that proves it, are accessible and interesting results even to one with little to no mathematical knowledge, introducing readers to common concepts in both geometry and graph …
The Reciprocal Of The Butterfly Theorem, Ion Patrascu, Florentin Smarandache
The Reciprocal Of The Butterfly Theorem, Ion Patrascu, Florentin Smarandache
Branch Mathematics and Statistics Faculty and Staff Publications
No abstract provided.
Counting The Moduli Space Of Pentagons On Finite Projective Planes, Maxwell Hosler
Counting The Moduli Space Of Pentagons On Finite Projective Planes, Maxwell Hosler
Senior Independent Study Theses
Finite projective planes are finite incidence structures which generalize the concept of the real projective plane. In this paper, we consider structures of points embedded in these planes. In particular, we investigate pentagons in general position, meaning no three vertices are colinear. We are interested in properties of these pentagons that are preserved by collineation of the plane, and so can be conceived as properties of the equivalence class of polygons up to collineation as a whole. Amongst these are the symmetries of a pentagon and the periodicity of the pentagon under the pentagram map, and a generalization of …
From Mathematics To Medicine: A Practical Primer On Topological Data Analysis (Tda) And The Development Of Related Analytic Tools For The Functional Discovery Of Latent Structure In Fmri Data, Andrew Salch, Adam Regalski, Hassan Abdallah, Raviteja Suryadevara, Michael J. Catanzaro, Vaibhav A. Diwadkar
From Mathematics To Medicine: A Practical Primer On Topological Data Analysis (Tda) And The Development Of Related Analytic Tools For The Functional Discovery Of Latent Structure In Fmri Data, Andrew Salch, Adam Regalski, Hassan Abdallah, Raviteja Suryadevara, Michael J. Catanzaro, Vaibhav A. Diwadkar
Mathematics Faculty Research Publications
fMRI is the preeminent method for collecting signals from the human brain in vivo, for using these signals in the service of functional discovery, and relating these discoveries to anatomical structure. Numerous computational and mathematical techniques have been deployed to extract information from the fMRI signal. Yet, the application of Topological Data Analyses (TDA) remain limited to certain sub-areas such as connectomics (that is, with summarized versions of fMRI data). While connectomics is a natural and important area of application of TDA, applications of TDA in the service of extracting structure from the (non-summarized) fMRI data itself are heretofore nonexistent. …
Crocheting Mathematics Through Covid-19, Beyza C. Aslan
Crocheting Mathematics Through Covid-19, Beyza C. Aslan
Journal of Humanistic Mathematics
As it is often said, something good often comes out of most bad situations. The time I spent during COVID-19, at home and isolated with my two children, brought out one secret passion in me: crocheting. Not only did it help me pass the time in a sane and productive way, but also it gave me a new goal in life. It connected my math side with my artistic side. It gave me a new perspective to look at math, and helped me help others see math in a positive way.
Making Art In Math Class During The Pandemic, Larson Fairbairn, Kameelah Jackson, Ksenija Simic-Muller
Making Art In Math Class During The Pandemic, Larson Fairbairn, Kameelah Jackson, Ksenija Simic-Muller
Journal of Humanistic Mathematics
For many of us, the pandemic has changed how we teach and how we support students. This manuscript highlights creativity as a way to support for student mathematical and emotional well-being. It describes the positive impact that creative assignments in a mathematics content course for preservice K-8 teachers had on students during the early days of the pandemic. The story is told by the instructor and two former students in the course.
Quantum Symmetries In Noncommutative Geometry., Suvrajit Bhattacharjee Dr.
Quantum Symmetries In Noncommutative Geometry., Suvrajit Bhattacharjee Dr.
Doctoral Theses
No abstract provided.
Oer Curve Fitting Applied To Easter Island Stone Foundations, Cynthia Huffman Ph.D.
Oer Curve Fitting Applied To Easter Island Stone Foundations, Cynthia Huffman Ph.D.
Faculty Submissions
In this activity, curve fitting is applied to drone pictures of ruins of stone foundations of the traditional houses (hare paenga) on the island of Rapa Nui. The free mathematics application GeoGebra (geogebra.org) is used, but the activity can be adapted to other technology, such as Desmos (desmos.com). The activity can be used as a teacher demonstration or completed by students, individually or in small groups, with access to computers.
One Straight Line Addresses Another Traveling In The Same Direction On An Infinite Plane, Daniel W. Galef
One Straight Line Addresses Another Traveling In The Same Direction On An Infinite Plane, Daniel W. Galef
Journal of Humanistic Mathematics
No abstract provided.
Elementary College Geometry (2021 Ed.), Henry Africk
Elementary College Geometry (2021 Ed.), Henry Africk
Open Educational Resources
This text is intended for a brief introductory course in plane geometry. It covers the topics from elementary geometry that are most likely to be required for more advanced mathematics courses. The only prerequisite is a semester of algebra. The emphasis is on applying basic geometric principles to the numerical solution of problems. For this purpose the number of theorems and definitions is kept small. Proofs are short and intuitive, mostly in the style of those found in a typical trigonometry or precalculus text. There is little attempt to teach theorem proving or formal methods of reasoning. However the topics …
A Tropical Approach To The Brill-Noether Theory Over Hurwitz Spaces, Kaelin Cook-Powell
A Tropical Approach To The Brill-Noether Theory Over Hurwitz Spaces, Kaelin Cook-Powell
Theses and Dissertations--Mathematics
The geometry of a curve can be analyzed in many ways. One way of doing this is to study the set of all divisors on a curve of prescribed rank and degree, known as a Brill-Noether variety. A sequence of results, starting in the 1980s, answered several fundamental questions about these varieties for general curves. However, many of these questions are still unanswered if we restrict to special families of curves. This dissertation has three main goals. First, we examine Brill-Noether varieties for these special families and provide combinatorial descriptions of their irreducible components. Second, we provide a natural generalization …
Non-Singular Cubic Surfaces Over $\Mathbb{F}_{2^K}$, Fatma Karaoğlu
Non-Singular Cubic Surfaces Over $\Mathbb{F}_{2^K}$, Fatma Karaoğlu
Turkish Journal of Mathematics
We perform an opportunistic search for cubic surfaces over small fields of characteristic two. The starting point of our work is a list of surfaces complied by Dickson over the field with two elements. We consider the nonsingular ones arising in Dickson' s work for the fields of larger orders of characteristic two. We investigate the properties such as the number of lines, singularities and automorphism groups. The problem of determining the possible numbers of lines of a nonsingular cubic surface over the fields of $\mathbb{C}, \mathbb{R}, \mathbb{Q}, \mathbb{F}_q$ where q odd, $\mathbb{F}_2$ was considered by Cayley and Salmon, Schlafli, …
Oer Ellipses And Traditional Rapanui Houses On Easter Island, Cynthia Huffman Ph.D.
Oer Ellipses And Traditional Rapanui Houses On Easter Island, Cynthia Huffman Ph.D.
Faculty Submissions
This worksheet activity is appropriate for secondary students in a class studying conic sections or students in a college algebra class. The first part of the activity gives an algebraic review of ellipses with exercises while the second part finds the equation of an ellipse corresponding to a Rapanui boat house foundation.
Supporting Our Struggling Students: Details Of A Hybrid Mathematics Summer Bridge Program, Anita White, Patrick Davis, Marti Shirley
Supporting Our Struggling Students: Details Of A Hybrid Mathematics Summer Bridge Program, Anita White, Patrick Davis, Marti Shirley
Faculty Publications & Research
It goes without saying that the schools in the consortium are used to dealing with gifted and talented students. However with such high-caliber students, we also have high expectations. What resources do we offer to the students who struggle at our institutions? This presentation will detail the setup and results of EXCEL2 - a summer bridge program offered at the Illinois Mathematics & Science Academy to help students who were unable to meet course expectations. The program operated through a hybrid online/in-person model - with instruction primarily given through video conferencing but coupled with an on-campus experience.
Oer Indoor Ellipse Multicultural (Easter Island) Activity, Cynthia Huffman Ph.D.
Oer Indoor Ellipse Multicultural (Easter Island) Activity, Cynthia Huffman Ph.D.
Faculty Submissions
This activity would fit in with a secondary or college algebra class studying conic sections, in particular ellipses, and gives students a multicultural hands-on application of the definition of an ellipse, while tracing out a scale model of the foundation of a hare paenga (boat house) from prehistoric Easter Island (Rapa Nui)..
Oer Outdoor Ellipse Multicultural (Easter Island) Activity, Cynthia Huffman Ph.D.
Oer Outdoor Ellipse Multicultural (Easter Island) Activity, Cynthia Huffman Ph.D.
Faculty Submissions
This activity would fit in with a secondary or college algebra class studying conic sections, in particular ellipses, and gives students a multicultural hands-on application of the definition of an ellipse, while tracing out a full-scale model of the foundation of a hare paenga (boat house) from prehistoric Easter Island (Rapa Nui)..
Geometry Aided Sonification, Michael Tecce
Geometry Aided Sonification, Michael Tecce
Computer Science Summer Fellows
Sonification is the process of deriving an audio representation of a time series which conveys important information about that time series. Otology and vision science have established that humans process audio information more quickly than visual information, and sonification can convey data to the visually impaired. In our work, we implement pipelines using Python/Numpy, and we handle both ordinary 1D time series and multivariate time series. For 1D time series, we find that using data to modulate the pitch or timing of preselected sounds (such as sine waves) simply and effectively captures repeating patterns and anomalies/outliers within the data. To …
Geometry Aided Sonification, Michael Tecce
Geometry Aided Sonification, Michael Tecce
Mathematics, Computer Science & Statistics Presentations
Sonification is the process of deriving an audio representation of a time series which conveys important information about that time series. Otology and vision science have established that humans process audio information more quickly than visual information, and sonification can convey data to the visually impaired. In our work, we implement pipelines using Python/Numpy, and we handle both ordinary 1D time series and multivariate time series. For 1D time series, we find that using data to modulate the pitch or timing of preselected sounds (such as sine waves) simply and effectively captures repeating patterns and anomalies/outliers within the data. To …
Studies On Polynomial Rings Through Locally Nilpotient Derivations., Nikhilesh Dasgupta Dr.
Studies On Polynomial Rings Through Locally Nilpotient Derivations., Nikhilesh Dasgupta Dr.
Doctoral Theses
No abstract provided.
Higher Chow Cycles On The Jacobian Of Curves., Subham Sarkar Dr.
Higher Chow Cycles On The Jacobian Of Curves., Subham Sarkar Dr.
Doctoral Theses
The following formula, usually called Beilinson’s formula — though independently due to Deligne as well — describes the motivic cohomology group of a smooth projective variety X over a number field as the group of extensions in a conjectured abelian category of mixed motives, MMQ.The aim of this thesis is to describe this construction in the case of the motivic cohomology group of the Jacobian of a curve. The first work in this direction is due to Harris [Har83] and Pulte [Pul88], [Hai87]. They showed that the Abel-Jacobi image of the modified diagonal cycle on the triple product of a …
Delaunay Surfaces Expressed In Terms Of A Cartan Moving Frame, Paul Bracken
Delaunay Surfaces Expressed In Terms Of A Cartan Moving Frame, Paul Bracken
School of Mathematical & Statistical Sciences Faculty Publications
Delaunay surfaces are investigated by using a moving frame approach. These surfaces correspond to surfaces of revolution in the Euclidean three-space. A set of basic one-forms is defined. Moving frame equations can be formulated and studied. Related differential equation which depend on variables relevant to the surface are obtained. For the case of minimal and constant mean curvature surfaces, the coordinate functions can be calculated in closed form. In the case in which the mean curvature is constant, these functions can be expressed in terms of Jacobi elliptic functions.
Geometry Across The Curriculum, Corey Dunn
Geometry Across The Curriculum, Corey Dunn
Q2S Enhancing Pedagogy
This project is designed for a multicalculus class already familiar with computing the arc length of a parameterized curve in space. The activity asks the student to first recall basic facts about arc length, and then introduces the notion of measuring lengths of vectors differently, depending on where their initial point is. This is a foundational concept in metric differential geometry, and, this activity attempts to motivate this generalization of computing lengths of vectors through this arc length activity. The activity concludes with a short discussion of basic concepts of Lorentzian geometry, including the idea that lightlike vectors have length …
Cartan’S Approach To Second Order Ordinary Differential Equations, Paul Bracken
Cartan’S Approach To Second Order Ordinary Differential Equations, Paul Bracken
School of Mathematical & Statistical Sciences Faculty Publications
In his work on projective connections, Cartan discusses his theory of second order differential equations. It is the aim here to look at how a normal projective connection can be constructed and how it relates to the geometry of a single second order differential equation. The calculations are presented in some detail in order to highlight the use of gauge conditions
Shape Equations For Two-Dimensional Manifolds Through A Moving Frame Variational Approach, Paul Bracken
Shape Equations For Two-Dimensional Manifolds Through A Moving Frame Variational Approach, Paul Bracken
School of Mathematical & Statistical Sciences Faculty Publications
A variational approach is given which can be applied to functionals of a general form to determine a corresponding Euler–Lagrange or shape equation. It is the intention to formulate the theory in detail based on a moving frame approach. It is then applied to a functional of a general form which depends on both the mean and Gaussian curvatures as well as the area and volume elements of the manifold. Only the case of a two-dimensional closed manifold is considered. The first variation of the functional is calculated in terms of the variations of the basic variables of the manifold. …
Fuchsian Groups, Bob Anaya
Fuchsian Groups, Bob Anaya
Electronic Theses, Projects, and Dissertations
Fuchsian groups are discrete subgroups of isometries of the hyperbolic plane. This thesis will primarily work with the upper half-plane model, though we will provide an example in the disk model. We will define Fuchsian groups and examine their properties geometrically and algebraically. We will also discuss the relationships between fundamental regions, Dirichlet regions and Ford regions. The goal is to see how a Ford region can be constructed with isometric circles.
Unifications Of Pythagorean Triple Schema, Emily Hammes
Unifications Of Pythagorean Triple Schema, Emily Hammes
Undergraduate Honors Theses
Euclid’s Method of finding Pythagorean triples is a commonly accepted and applied technique. This study focuses on a myriad of other methods behind finding such Pythagorean triples. Specifically, we discover whether or not other ways of finding triples are special cases of Euclid’s Method.