Physical Sciences and Mathematics Commons^™

A Quotient Of Fomin-Kirillov Algebra And Q-Lucas Polynomial, Sirous Homayouni

Mathematics Faculty Publications

We introduce a quotient of the Fomin-Kirillov algebra F K(n) denoted by F KCn (n), over the ideal generated by the edges of a complete graph on n vertices that are missing in the n-cycle graph Cn. In this quotient algebra, we establish a one-to-one correspondence between the basis elements and the set of matchings in an n-cycle graph. We prove that the Hilbert series of F KCn (n) corresponds to the q-Lucas polynomials, and the dimension of this quotient algebra is equal to the Lucas number Ln. We also find the character map of this quotient algebra over the …

Leveraging International Large-Scale Assessments: Insights From An Item-Writing Professional Development Program For High School Mathematics Teachers, Mark Lester B. Garcia, Catherine P. Vistro-Yu

Mathematics Faculty Publications

The Philippine Department of Education, despite seeing minimal improvements in the country’s score and rankings in the recent cycles of international large-scale assessments (ILSAs) such as TIMSS and PISA, maintained that it shall continue to participate in ILSAs to monitor the progress of educational effectiveness and benchmark with other countries by observing and adopting commendable practices. While legislators and other stakeholders push for increased financial resources in the education sector as part of the country’s preparations for upcoming ILSAs, one crucial aspect that is largely overlooked is teacher training. This paper presents a viable yet contentious way of upskilling teachers …

Resonant Solutions Of The Non-Linear Schrödinger Equation With Periodic Potential, Arein Duaibes, Yulia Karpeshina

Mathematics Faculty Publications

The goal is construction of stationary solutions close to non-trivial combinations of two plane waves at high energies for a periodic non-linear Schrödinger Equation in dimension two. The corresponding isoenergetic surface is described for any sufficiently large energy k2. It is shown that the isoenergetic surface corresponding to k2 is essentially different from that for the zero potential even for small potentials. We use a combination of the perturbative results obtained earlier for the linear case and a method of successive approximation.

How Generative Ai Models Such As Chatgpt Can Be (Mis)Used In Spc Practice, Education, And Research? An Exploratory Study, Fadel M. Megahed, Ying-Ju (Tessa) Chen, Joshua A. Ferris, Sven Knoth, L. Allison Jones-Farmer

Mathematics Faculty Publications

Generative Artificial Intelligence (AI) models such as OpenAI's ChatGPT have the potential to revolutionize Statistical Process Control (SPC) practice, learning, and research. However, these tools are in the early stages of development and can be easily misused or misunderstood. In this paper, we give an overview of the development of Generative AI. Specifically, we explore ChatGPT's ability to provide code, explain basic concepts, and create knowledge related to SPC practice, learning, and research. By investigating responses to structured prompts, we highlight the benefits and limitations of the results. Our study indicates that the current version of ChatGPT performs well for …

Cognitive Load Scale In Learning Formal Definition Of Limit: A Rasch Model Approach, Rina Oktaviyanthi, Ria Noviana Agus, Mark Lester B. Garcia, Kornkanok Lertdechapat

Mathematics Faculty Publications

Constructing proofs for the limit using the formal definition induces a high cognitive load. Common assessment tools, like cognitive load scales, lack specificity for the concept of limits. This research aims to validate an instrument tailored to assess cognitive load in students focused on the formal definition of limits, addressing the need for diverse strategies in education. The research employs a quantitative survey design with a Rasch model approach, utilizing a data collection instrument in the form of a questionnaire. Subsequently, the data are analyzed by focusing on three aspects: (1) item fit to the Rasch model, (2) unidimensionality, and …

Can't We Just Use Computers? Initial Efforts On Technology-Enhanced Learning Of Operations Research In A Philippine University, Lester Hao, Jeric C. Briones, Mark Anthony C. Tolentino

Mathematics Faculty Publications

Operations Research (OR) is a required course in any applied math program in the Philippines. It involves complex and computation-heavy algorithms to guide and improve decision-making in organizations and relevant contexts. Despite the advent of software that have automated the computations or the implementation of algorithms, the teaching and learning of OR may still be rife with procedures being performed manually by hand. In this paper, we report on our initial progress on a research project that aims to revise the OR syllabi in one Philippine university, by integrating more technological tools in the teaching and learning of OR for …

Id Numbers Of Lobster Graphs, Mark Anthony C. Tolentino, Luis Silvestre Jr, Richwell T. Chan Sim, Amir Jann Erikson Diga, Althea Julia R. Loyola

Mathematics Faculty Publications

No abstract provided.

Towards A More Engaged Democracy: Using Statistical Methods To Detect Election Fraud In The 2022 Philippine National Elections, Juan Miguel Cardaño, Bryan Patrick Mande, Seth William Tionko, Aldrich Ellis C. Asuncion, Jeric C. Briones

Mathematics Faculty Publications

This paper aims to show how citizens can further participate in the elections, aside from just voting. Given the accusations of fraud, this study demonstrates how existing election fraud detection methods can be used in the 2022 Philippine National Elections (PNE) context. Specifically, histograms known as 2D vote turnout distributions were first utilized to model the frequency of the winner's percentage of votes based on the voter turnout in each electoral unit, with key parameters introduced to detect fraud. Vote turnout distributions were then simulated using parametric models involving the aforementioned fraud parameters, with the goal of replicating the actual …

The Impact Of Neighborhood Socioeconomic Disadvantage On Operative Outcomes After Single-Level Lumbar Fusion, Grace Y. Ng, Ritesh Karsalia, Ryan S. Gallagher, Austin J. Borja, Jianbo Na, Scott Mcclintock, Neil R. Malhotra

Mathematics Faculty Publications

INTRODUCTION: The relationship between socioeconomic status and neurosurgical outcomes has been investigated with respect to insurance status or median household income, but few studies have considered more comprehensive measures of socioeconomic status. This study examines the relationship between Area Deprivation Index (ADI), a comprehensive measure of neighborhood socioeconomic disadvantage, and short-term postoperative outcomes after lumbar fusion surgery. METHODS: 1861 adult patients undergoing single-level, posterior-only lumbar fusion at a single, multihospital academic medical center were retrospectively enrolled. An ADI matching protocol was used to identify each patient's 9-digit zip code and the zip code-associated ADI data. Primary outcomes included 30- and …

A Mathematical App For The Conceptual Understanding Of Area And Perimeter, Jumela F. Sarmiento, Debbie Marie B. Verzosa, Maria Alva Q. Aberin, Ma. Louise Antonette De Las Penas, Agnes D. Garciano, Juan Carlo F. Mallari, Mark Anthony C. Tolentino

Mathematics Faculty Publications

This paper discusses an app that was developed to build a strong understanding of the concepts of area and perimeter in students. An important feature of the app is the three-component feature which highlights progressive learning: Explore, designed for the learning of the conceptual understanding of area and perimeter; Apply, where area and perimeter concepts are applied; and Create intended for constructing representations to develop higher order thinking skills. The pedagogical basis for the creation of the app, the game design elements employed in the app as well as the integration of the app in the classroom will be presented.

Design Of A Mobile App To Promote Understanding And Fluency In Finding The Equation Of A Line, Agnes D. Garciano, Maria Alva Q. Aberin, Ma Louise Antonette N. Delaspeñas, Juan Carlo F. Mallari, Jumela F. Sarmiento, Mark Anthony C. Tolentino, Debbie Marie B. Verzosa

Mathematics Faculty Publications

This paper focuses on the design of a mobile app called Pick or Fish that fosters comprehension and mastery of the concepts of slopes, y-intercepts and equations of lines. The app's pedagogical value lies in its potential to help students understand and become proficient in these concepts. The app is suitable for use on low-cost mobile devices. It functions within an engaging game-like setting featuring visual elements that enable students to see the effect of parameter changes on the direction of a line. The beginner and advanced levels of the app have scaffolding features that gradually introduce the students to …

A Visualization App On Proving Geometric Concepts, Ma. Louise Antonette N. De Las Peñas, Debbie Marie B. Verzosa, Maria Alva Q. Aberin, Agnes D. Garciano, Jumela F. Sarmiento, Mark Anthony C. Tolentino, Juan Carlo F. Mallari

Mathematics Faculty Publications

This paper discusses the description, design and pedagogical basis of a mathematical app called Two Column Proof which provides students a framework for writing proofs of geometric statements. The app focuses on proving concepts on: properties of parallelograms, conditions that determine when a quadrilateral is a parallelogram, properties on trapezoids and kites for high school mathematics. It employs visual representations for students to understand the statement or reason of each given line in the proof, strengthening their logical and mathematical knowledge needed in the proof construction.

A Natural Pseudometric On Homotopy Groups Of Metric Spaces, Jeremy Brazas, Paul Fabel

Mathematics Faculty Publications

For a path-connected metric space (X, d), the n-th homotopy group π n ( X) inherits a natural pseudometric from the n-th iterated loop space with the uniform metric. This pseudometric gives π n ( X) the structure of a topological group and when X is compact, the induced pseudometric topology is independent of the metric d. In this paper, we study the properties of this pseudometric and how it relates to previously studied structures on π n ( X). Our main result is that the pseudometric topology agrees with the shape topology on π n ( X) if X …

The Spectrum Of Nim-Values For Achievement Games For Generating Finite Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben

Mathematics Faculty Publications

We study an impartial achievement game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The game ends when the jointly selected elements generate the group. The last player able to make a move is the winner of the game. We prove that the spectrum of nim-values of these games is {0, 1, 2, 3, 4}. This positively answers two conjectures from a previous paper by the last two authors.

How Effective Is The Efficiency Gap?, Thomas Q. Sibley

Mathematics Faculty Publications

Gerrymandering has affected U. S. politics since at least 1812. A political cartoon that year decried this tactic by then Massachusetts Governor Elbridge Gerry. (Gerrymandering is manipulating the boundaries of districts to benefit a group unfairly.)

While we may feel we know a gerrymander when we see one, finding a meaningful metric has proven challenging. This article uses elementary mathematics to investigate the efficiency gap, a recent model proposed to measure gerrymandering.

On Maps With Continuous Path Lifting, Jeremy Brazas, Atish Mitra

Mathematics Faculty Publications

We study a natural generalization of covering projections defined in terms of unique lifting properties. A map p : E -+ X has the continuous path-covering property if all paths in X lift uniquely and continuously (rel. basepoint) with respect to the compactopen topology. We show that maps with this property are closely related to fibrations with totally path-disconnected fibers and to the natural quotient topology on the homotopy groups. In particular, the class of maps with the continuous path-covering property lies properly between Hurewicz fibrations and Serre fibrations with totally path-disconnected fibers. We extend the usual classification of covering …

Free Quasitopological Groups, Jeremy Brazas, Sarah Emery

Mathematics Faculty Publications

In this paper, we study the topological structure of a universal construction related to quasitopological groups: the free quasitopological group F-q(X) on a space X. We show that free quasitopological groups may be constructed directly as quotient spaces of free semitopological monoids, which are themselves constructed by iterating product spaces equipped with the "cross topology." Using this explicit description of F-q(X), we show that for any T-1 space X, F-q(X) is the direct limit of closed subspaces F-q(X)(n) of words of length at most n. We also prove that the natural map i(n): (sic)(n)(i=0)(X boolean OR X-1)(circle times i) - …

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, …

Mathematical Mobile Apps Via Rural Casting, Ma Louise Antonette N. De Las Peñas, John Chris T. Kwong, Peter Antonio B. Banzon, Philip A. Martinez, Wilbur Isaac G. Lapada, Julian N. Eballa, John Emil C. Sebastian, Miguel D. Asido, Jan Daryl M. San Juan, Debbie Marie B. Verzosa, Jumela F. Sarmiento, Agnes D. Garciano, Mark Anthony C. Tolentino, Maria Alva Q. Aberin, Juan Carlo F. Mallari

Mathematics Faculty Publications

This paper discusses the distribution, through a digital datacasting framework, of mathematical resources for Grades 1 to 7 to two schools in a community in the Philippines. Among the mathematical resources made available, are mathematical applications (apps), which run on mobile technologies, that have been created to help in the mathematical learning of students in a remote setting. The distribution is facilitated by the RuralCasting set-top box, a developed custom set-top box capable of receiving digital TV broadcasts and providing local content access through its Wi-Fi network. This paper presents the performance of the set-top box in distributing the mobile …

The Set Chromatic Numbers Of The Middle Graph Of Tree Families, Mark Anthony C. Tolentino, Gerone Russel J. Eugenio

Mathematics Faculty Publications

The neighborhood color set of each vertex v in a vertex-colored graph G is defined as the collection of the colors of all the neighbors of v. If there are no two adjacent vertices that have equal neighborhood color sets, then the coloring is called a set coloring of G. The set coloring problem on G refers to the problem of determining its set chromatic number, which refers to the fewest colors using which a set coloring of G may be constructed. In this work, we consider the set coloring problem on graphs obtained from applying middle graph, a unary …

Practice Through Play Using Mobile Technology, Agnes D. Garciano, Debbie Marie B. Verzosa, Ma Louise Antonette N. De Las Peñas, Maria Alva Q. Aberin, Juan Carlo F. Mallari, Jumela F. Sarmiento, Mark Anthony C. Tolentino

Mathematics Faculty Publications

This paper discusses the Just Keep Solving apps that are designed based on deliberate practice model for developing mathematical skills. Features of deliberate practice include well-defined goals involving areas of weakness as determined by a knowledgeable other such as a teacher. The integration of game design features provides a positive environment wherein the learning goals critical in a deliberate practice model are emphasized. Possible strategies for integrating the apps in a classroom are also discussed using Hughes, Thomas and Scharber’s RAT (Replacement, Amplification, Transformation) framework. The games can replace traditional pen-and-paper classroom activities, amplify learning by personalizing a student’s experience …

Bridging The Mathematics Gap Through The Use Of Mathematical Apps, Ma. Louise Antonette N. De Las Penas, Debbie Marie B. Verzosa, Maria Alva Q. Aberin, Agnes D. Garciano, Juan Carlo F. Mallari, Jumela F. Sarmiento, Mark Anthony C. Tolentino

Mathematics Faculty Publications

During the COVID-19 pandemic, school campuses worldwide were forced to close, and students had to learn primarily from home. This sudden disruption is estimated to have caused significant learning loss among learners. This paper reports the use of mathematical applications (apps) to bridge the mathematical learning gaps in Grades 1 to 11 in the Philippines after the pandemic, as part of a project funded by a national government agency. The apps include those that strengthen foundational concepts in number and fraction sense in grade school mathematics, develop proving skills in geometry, promote mastery in algebraic and trigonometry through drill and …

Optimal Test Plan Of Step Stress Partially Accelerated Life Testing For Alpha Power Inverse Weibull Distribution Under Adaptive Progressive Hybrid Censored Data And Different Loss Functions, Refah Alotaibi, Ehab M. Almetwally, Qiuchen Hai, Hoda Rezk

Mathematics Faculty Publications

Accelerated life tests are used to explore the lifetime of extremely reliable items by subjecting them to elevated stress levels from stressors to cause early failures, such as temperature, voltage, pressure, and so on. The alpha power inverse Weibull (APIW) distribution is of great significance and practical applications due to its appealing characteristics, such as its flexibilities in the probability density function and the hazard rate function. We analyze the step stress partially accelerated life testing model with samples from the APIW distribution under adaptive type II progressively hybrid censoring. We first obtain the maximum likelihood estimates and two types …

Mathematical Musings On The External Anatomy Of The Novel Coronavirus. Part 4: Models Of N-Cov, Jyotirmoy Sarkar, Mamunur Rashid

Mathematics Faculty Publications

What is the shape of the novel coronavirus (n-CoV) which has turned our world upside down? Even though under a microscope, it looks dull, unattractive, and even disgusting, creative artists have attributed to it bright colors, made it look pretty, and depicted it as a thing of beauty. What can a mathematician contribute to this effort? We take a purist’s point of view by imposing on it a quasi-symmetry and then deriving some consequences. In an idealistic world, far removed from reality but still constrained by the rules of mathematics, anyone can enjoy this ethereal beauty of the mind’s creation, …

Mathematical Musings On The External Anatomy Of The Novel Coronavirus. Part 3: Spherical Triangles, Jyotirmoy Sarkar, Mamunur Rashid

Mathematics Faculty Publications

What is the shape of the novel coronavirus (n-CoV) which has turned our world upside down? Even though under a microscope, it looks dull, unattractive, and even disgusting, creative artists have attributed to it bright colors, made it look pretty, and depicted it as a thing of beauty. What can a mathematician contribute to this effort? We take a purist’s point of view by imposing on it a quasi-symmetry and then deriving some consequences. In an idealistic world, far removed from reality but still constrained by the rules of mathematics, anyone can enjoy this ethereal beauty of the mind’s creation, …

Estimating The Parameters Of A Simple Linear Regression Model Without Using Differential Calculus, Mamunur Rashid, Jyotirmoy Sarkar

Mathematics Faculty Publications

To estimate the parameters of a simple linear regression model, students who already know calculus can minimize the total squared deviations by setting its first-order partial derivatives to zero and solving simultaneously. For students who do not know calculus, most teachers/textbooks simply state the formulas without justifying them. Students accept the formulas on faith; and for given data, they evaluate the estimates using a calculator or a statistical software. In this paper, we justify the formulas without invoking calculus. We hope the users of statistics will benefit from our proposed justifications.

Visualizing Bivariate Statistics Using Ellipses Over A Scatter Plot, Mamunur Rashid, Jyotirmoy Sarkar, Siddhanta Phuyal

Mathematics Faculty Publications

A scatter plot shows the relationship between two quantitative variables 𝒙 and 𝒚. Sometimes, we can predict one variable as a linear function of the other using the least squares regression lines of 𝒚 on 𝒙 or 𝒙 on 𝒚. These two regression lines together suffice to identify the mean vector, the coefficient of determination, the correlation coefficient, and the ratio of the standard deviations (SD). So, do our proposed summary ellipses. Additionally, the inner ellipse reveals the SDs and the outer ellipse flags potential outliers.

A Generalization Of Combinatorial Identities For Stable Discrete Series Constants, Richard Ehrenborg, Sophie Moreland, Margaret Readdy

Mathematics Faculty Publications

This article is concerned with the constants that appear in Harish-Chandra’s character formula for stable discrete series of real reductive groups, although it does not require any knowledge about real reductive groups or discrete series. In Harish-Chandra’s work the only information we have about these constants is that they are uniquely determined by an inductive property. Later, Goresky–Kottwitz–MacPherson (1997) and Herb (2000) gave different formulas for these constants. In this article, we generalize these formulas to the case of arbitrary finite Coxeter groups (in this setting, discrete series no longer make sense), and give a direct proof that the two …

Mathematical Musings On The External Anatomy Of The Novel Coronavirus Part 2: Chasing After Quasi-Symmetry, Jyotirmoy Sarkar, Mamunur Rashid

Mathematics Faculty Publications

What is the shape of the novel Coronavirus which has turned our world upside down? Even though it looks dull, unattractive, and even disgusting under a microscope, creative artists have attributed to it bright colors, made it look pretty, and depicted it as a thing of beauty. What can a mathematician contribute to this effort? We take a purist’s point of view by imposing on it a quasi-symmetry and then deriving some consequences. In an idealistic world, far removed from reality but still constrained by the rules of mathematics, anyone can enjoy this ethereal beauty of the mind’s creation, beckoning …

Fantasy On A Baseball Seam, Genevieve Ahlstrom, Thomas Q. Sibley

Mathematics Faculty Publications

No abstract provided.