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Articles 1 - 30 of 162
Full-Text Articles in Physical Sciences and Mathematics
Aspects Of Stochastic Geometric Mechanics In Molecular Biophysics, David Frost
Aspects Of Stochastic Geometric Mechanics In Molecular Biophysics, David Frost
All Dissertations
In confocal single-molecule FRET experiments, the joint distribution of FRET efficiency and donor lifetime distribution can reveal underlying molecular conformational dynamics via deviation from their theoretical Forster relationship. This shift is referred to as a dynamic shift. In this study, we investigate the influence of the free energy landscape in protein conformational dynamics on the dynamic shift by simulation of the associated continuum reaction coordinate Langevin dynamics, yielding a deeper understanding of the dynamic and structural information in the joint FRET efficiency and donor lifetime distribution. We develop novel Langevin models for the dye linker dynamics, including rotational dynamics, based …
The Mean Sum Of Squared Linking Numbers Of Random Piecewise-Linear Embeddings Of $K_N$, Yasmin Aguillon, Xingyu Cheng, Spencer Eddins, Pedro Morales
The Mean Sum Of Squared Linking Numbers Of Random Piecewise-Linear Embeddings Of $K_N$, Yasmin Aguillon, Xingyu Cheng, Spencer Eddins, Pedro Morales
Rose-Hulman Undergraduate Mathematics Journal
DNA and other polymer chains in confined spaces behave like closed loops. Arsuaga et al. \cite{AB} introduced the uniform random polygon model in order to better understand such loops in confined spaces using probabilistic and knot theoretical techniques, giving some classification on the mean squared linking number of such loops. Flapan and Kozai \cite{flapan2016linking} extended these techniques to find the mean sum of squared linking numbers for random linear embeddings of complete graphs $K_n$ and found it to have order $\Theta(n(n!))$. We further these ideas by inspecting random piecewise-linear embeddings of complete graphs and give introductory-level summaries of the ideas …
Geometry Through Architectural Design, Maureen T. Carroll, Elyn Rykken
Geometry Through Architectural Design, Maureen T. Carroll, Elyn Rykken
LASER Journal
In her 1912 geometry book, Mabel Sykes surveys complex and beautiful architectural designs from around the world to inspire exercises on geometric proof, construction and computation. In over 1800 exercises, Sykes analyzes geometric patterns from ornamental and structural features found in tile mosaics, parquet floors, Gothic windows, trusses and arches. As Sykes' writes, ``Geometry gives, as no other subject can give, an appreciation of form as it exists in the material world" . We have chosen four examples to illustrate how her appealing designs and the accompanying exercises of this hidden gem can be incorporated into any geometry course.
Could Raphael’S School Of Athens Contain Hidden Geometry?, Frode S. Larsen, Harald E. Moe
Could Raphael’S School Of Athens Contain Hidden Geometry?, Frode S. Larsen, Harald E. Moe
Journal of Humanistic Mathematics
In this article we argue that Raphael has hidden a geometric shape called a vesica piscis in his fresco The School of Athens (1510-1511). The vesica piscis, and several findings which can be interpreted as suggesting the presence of a vesica piscis in the fresco, are presented. Several of these suggestions relate to the vesica piscis drawn in the construction of an equilateral triangle in the first proposition of Euclid’s Elements. Based on findings in the fresco, we suggest that the vesica piscis should be interpreted in light of a philosophical and theological controversy which took place in Italy …
On The Superabundance Of Singular Varieties In Positive Characteristic, Jake Kettinger
On The Superabundance Of Singular Varieties In Positive Characteristic, Jake Kettinger
Department of Mathematics: Dissertations, Theses, and Student Research
The geproci property is a recent development in the world of geometry. We call a set of points Z\subseq\P_k^3 an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a and b. Examples known as grids have been known since 2011. Previously, the study of the geproci property has taken place within the characteristic 0 setting; prior to the work in this thesis, a procedure has been known for creating an (a,b)-geproci half-grid for 4\leq a\leq b, but it was not …
Exploring The Structure Of Partial Difference Sets With Denniston Parameters, Nicolas Ferree
Exploring The Structure Of Partial Difference Sets With Denniston Parameters, Nicolas Ferree
Honors Theses
In this work, we investigate the structure of particular partial difference sets (PDS) of size 70 with Denniston parameters in an elementary abelian group and in a nonelementary abelian group. We will make extensive use of character theory in our investigation and ultimately seek to understand the nature of difference sets with these parameters. To begin, we will cover some basic definitions and examples of difference sets and partial difference sets. We will then move on to some basic theorems about partial difference sets before introducing a group ring formalism and using it to explore several important constructions of partial …
The Sharp Bounds Of A Quasi-Isometry Of P-Adic Numbers In A Subset Real Plane, Kathleen Zopff
The Sharp Bounds Of A Quasi-Isometry Of P-Adic Numbers In A Subset Real Plane, Kathleen Zopff
Undergraduate Theses
P-adic numbers are numbers valued by their divisibility by high powers of some prime, p. These numbers are an important concept in number theory that are used in major ideas such as the Reimann Hypothesis and Andrew Wiles’ proof of Fermat’s last theorem, and also have applications in cryptography. In this project, we will explore various visualizations of p-adic numbers. In particular, we will look at a mapping of p-adic numbers into the real plane which constructs a fractal similar to a Sierpinski p-gon. We discuss the properties of this map and give formulas for the sharp bounds of its …
Area Activity, Admin Stem For Success
Area Activity, Admin Stem For Success
STEM for Success Showcase
Lesson plan to teach students about area including an activity plan, activity description, activity video, and additional activity materials
From A Doodle To A Theorem: A Case Study In Mathematical Discovery, Juan FernáNdez GonzáLez, Dirk Schlimm
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 …
Geometric Dissections, Daniel Robert Martin
Geometric Dissections, Daniel Robert Martin
MSU Graduate Theses
In the study of geometry, the notion of dissection and its mechanics are occasionally over-looked. We consider and trace the history and theorems surrounding geometric dissections in both recreational and academic mathematics. We explore the important advancements in this particular topic from antiquity through the nineteenth and early twentieth centuries. We conclude with an exploration of the Banach-Tarski paradox
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 …
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 …
Making Upper-Level Math Accessible To A Younger Audience, Allyson Roller
Making Upper-Level Math Accessible To A Younger Audience, Allyson Roller
WWU Honors College Senior Projects
Symmetry is all around us. It appears on fabrics and on the buildings that surround us. Believe it or not, there is actually quite a bit of math that goes into generating these patterns, which are known as the seven frieze patterns. In my work, I explain how each unique pattern is generated using different types of symmetries. I also created a PDF of a children’s book about frieze patterns to ensure that people of all ages have the opportunity to learn about seemingly complex patterns.
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.
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.
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 …
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 …
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 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)..
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)..