Digital Commons Network^™

Benford’S Law And Its Applications To Accounting, Lucy Wilson

Mathematics Student Projects

In this paper, we introduce the concept of Benford’s Law, which is the mathematical observation that in many naturally occurring numerical datasets, the leading digits are not evenly distributed. We actually find that smaller leading digits appear more often than larger ones. We will explore when and why this phenomenon occurs and how we can use statistical tests to determine how well a dataset conforms to Benford’s Law. We will also see how Benford’s Law can be used by auditors in the accounting field to catch potential fraud.

The Traveling Salesman Problem At Taylor University, Jonathan Jinoo Pawley

Mathematics Student Projects

What is the shortest route to walk to every residence hall on campus, beginning and ending with the same hall? This question can be considered by applying the Traveling Salesman Problem, an easy to understand yet hard to solve problem in the realm of discrete combinatorial optimization. The Traveling Salesman Problem is useful as an introduction to optimization problems, and it also has immensely practical applications. This paper will serve as an introduction to the computational difficulty of the Traveling Salesman Problem and will also explore various approximation algorithms. We will subsequently apply our new understanding of the theory to …

Why Micro-Funding? Why Small Businesses Are Important? Analysis Based On First Principles, Hein D. Tran, Edwin Tomy George, Vladik Kreinovich

Departmental Technical Reports (CS)

On the one hand, in economics, there is a well-known and well-studied economy of scale: when two smaller companies merge, it lowers their costs and thus, makes them more effective and therefore more competitive. At first glance, this advantage of big size would make economy dominated by big companies -- but in reality, small business remain a significant and important economic sector. Similarly, it is well known and well studied that research collaboration enhances researchers' productivity -- but still a significant portion of important results come from individual efforts. In several applications areas, there are area-specific explanations for this seemingly …

Local-Global Support For Earth Sciences: Economic Analysis, Uyen Hoang Pham, Aaron Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

Most funding for science comes from taxpayers. So, it is very important to be able to convince taxpayers that this funding is potentially beneficial for them. This task is easier in Earth sciences, e.g., in meteorology, where there are clear local benefits. The problem is that while many people support local studies focused on their region, they do not always have a good understanding of the fact that effective local benefits require also studying surrounding areas -- and what should be the optimal balance between local and (more) global studies. In this paper, on a (somewhat) simplified model of the …

Approximate Stochastic Dominance Revisited, Chon Van Le, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

According to decision theory, in general, to recommend the best of possible actions, we need to know, for each possible action, the probabilities of different outcomes, and we also need to know the decision maker's utility function -- that describes his/her preferences. For some pairs of probability distributions, however, we can make such a recommendation without knowing the exact form of the utility function -- e.g., in financial applications, we only need to know that a larger amount is preferable to a smaller one. Such situations, when we can make decisions based only on the information about probabilities, are known …

Just-In-Accuracy: Mobile Approach To Uncertainty, Martine Ceberio, Christoph Q. Lauter, Vladik Kreinovich

Departmental Technical Reports (CS)

To make a mobile device last longer, we need to limit computations to a bare minimum. One way to do that, in complex control and decision making problems, is to limit precision with which we do computations, i.e., limit the number of bits in the numbers' representation. A problem is that often, we do not know with what precision should we do computations to get the desired accuracy of the result. What we propose is to first do computations with very low precision, then, based on these computations, estimate what precision is needed to achieve the given accuracy, and then …

How To Deal With Inconsistent Intervals: Utility-Based Approach Can Overcome The Limitations Of The Purely Probability-Based Approach, Kittawit Autchariyapanitkul, Tomoe Entani, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many application areas, we rely on experts to estimate the numerical values of some quantities. Experts can provide not only the estimates themselves, they can also estimate the accuracies of their estimates -- i.e., in effect, they provide an interval of possible values of the quantity of interest. To get a more accurate estimate, it is reasonable to ask several experts -- and to take the intersection of the resulting intervals. In some cases, however, experts overestimate the accuracy of their estimates, their intervals are too narrow -- so narrow that they are inconsistent: their intersection is empty. In …

How To Make Machine Learning Financial Recommendations More Fair: Theoretical Explanation, Tho M. Nguyen, Saeid Tizpaz-Niari, Vladik Kreinovich

Departmental Technical Reports (CS)

Machine learning has been actively and successfully used to make financial decisions. In general, these systems work reasonably well. However, in some cases, these systems show unexpected bias towards minority groups -- the bias that is sometime much larger than the bias in the data on which they were trained. A recent paper analyzed whether a proper selection of hyperparameters can decrease this bias. It turned out that while the selection of hyperparameters indeed affect the system's fairness, only a few of the hyperparameters lead to consistent improvement of fairness: the number of features used for training and the number …

Homomorphism Obstructions For Satellite Maps, Allison N. Miller

Mathematics & Statistics Faculty Works

A knot in a solid torus defines a map on the set of (smooth or topological) concordance classes of knots in S³. This set admits a group structure, but a conjecture of Hedden suggests that satellite maps never induce interesting homomorphisms: we give new evidence for this conjecture in both categories. First, we use Casson-Gordon signatures to give the first obstruction to a slice pattern inducing a homomorphism on the topological concordance group, constructing examples with every winding number besides ± 1. We then provide subtle examples of satellite maps which map arbitrarily deep into the n-solvable filtration …

Are The Cans In The Store “Volume Optimized”? [Mathematics], Bukurie Gjoci

Open Educational Resources

This is one of LaGuardia’s Project Connexion STEM Team’s experiential learning activities. Project Connexion's purpose is to promote creative thinking on how to engage students in the classroom. As part of this, the STEM team developed Experiential/co-curricular activities that demonstrated to students how their work in class connects to the world around them. These activities were embedded into the syllabus to ensure the participation of all students. Each professor designed a Co-curricular activity for their courses, ensuring that the Co-curricular activity directly linked course material to the outside world.

This Calculus I Experiential Learning Project aligns with one of the …

Math 115: College Algebra For Pre-Calculus, Seth Lehman

Open Educational Resources

OER course syllabus for Math 115, College Algebra, at Queens College

Rough Numbers And Variations On The Erdős--Kac Theorem, Kai Fan

Dartmouth College Ph.D Dissertations

The study of arithmetic functions, functions with domain N and codomain C, has been a central topic in number theory. This work is dedicated to the study of the distribution of arithmetic functions of great interest in analytic and probabilistic number theory.

In the first part, we study the distribution of positive integers free of prime factors less than or equal to any given real number y>=1. Denoting by Phi(x,y) the count of these numbers up to any given x>=y, we show, by a combination of analytic methods and sieves, that Phi(x,y)<0.6x/\log y holds uniformly for all 3<=y<=sqrt{x}, improving upon an earlier result of the author in the same range. We also prove numerically explicit estimates of the de Bruijn type for Phi(x,y) which are applicable in wide ranges.

In the second part, we turn …

Rigid Body Constrained Motion Optimization And Control On Lie Groups And Their Tangent Bundles, Brennan S. Mccann

Doctoral Dissertations and Master's Theses

Rigid body motion requires formulations where rotational and translational motion are accounted for appropriately. Two Lie groups, the special orthogonal group SO(3) and the space of quaternions H, are commonly used to represent attitude. When considering rigid body pose, that is spacecraft position and attitude, the special Euclidean group SE(3) and the space of dual quaternions DH are frequently utilized. All these groups are Lie groups and Riemannian manifolds, and these identifications have profound implications for dynamics and controls. The trajectory optimization and optimal control problem on Riemannian manifolds presents significant opportunities for theoretical development. Riemannian optimization is an attractive …

Biopsychosocial Resilience Through A Complex Adaptive Systems Lens: A Narrative Review Of Nonlinear Modeling Approaches, Adam W. Kiefer, David Pincus

Psychology Faculty Articles and Research

Human resilience is often considered as static traits using a reductionist approach. More recent work has demonstrated it to be a dynamic and emergent property of complex systems. This narrative review explores human resilience through a self-organizing framework with a specific emphasis on the application of nonlinear modeling approaches. Four classes of approaches are examined: univariate dynamics, bivariate coupling, topological modeling, and network modeling. Univariate dynamics capture the temporal structure and flexibility within a single time series, while bivariate coupling approaches quantify the interaction dynamics and coordination between two time series. Topological modeling identifies bifurcations and attractor dynamics as signals …

Semidefinite Programming Bounds For Distance Distribution Of Spherical Codes, Oleg R. Musin

School of Mathematical & Statistical Sciences Faculty Publications

We present an extension of known semidefinite and linear programming upper bounds for spherical codes. We apply the main result for the distance distribution of a spherical code and show that this method can work effectively In particular, we get a shorter solution to the kissing number problem in dimension 4.

The Encyclopedia Of Neutrosophic Researchers, 5th Volume, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics, neutrosophic measure, neutrosophic precalculus, neutrosophic calculus and so on are gaining significant attention in solving many real life problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistent, and indeterminacy.

In the past years the fields of neutrosophics have been extended and applied in various fields, such as: artificial intelligence, data mining, soft computing, decision making in incomplete / indeterminate / inconsistent information systems, image processing, computational modelling, robotics, medical diagnosis, biomedical engineering, investment problems, economic forecasting, social science, humanistic and practical achievements.

There are about 7,000 neutrosophic researchers, within 89 countries around the …

Convergence Of The Two-Point Modulus-Based Matrix Splitting Iteration Method, Ximing Fang, Ze Gu, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we discuss the convergence of the two-point modulus-based matrix splitting iteration method for solving the linear complementarity problem. Some convergence conditions are presented from the spectral radius and the matrix norm when the system matrix is a -matrix. Besides, the quasi-optimal cases of the method are studied. Numerical experiments are provided to show the presented results.

Elementary Mathematics Curriculum: State Policy, Covid-19, And Teachers’ Control, Mona Baniahmadi, Bima Sapkota, Amy M. Olson

School of Mathematical & Statistical Sciences Faculty Publications

In the U.S., state guidance to schools in response to the COVID-19 pandemic was politicized. We used state-level political affiliation to explore whether access to curricular resources differed pre-pandemic or during pandemic remote teaching and teachers' reported control over curricular resources during pandemic teaching. We found that pre-pandemic the percentage of teachers in Republican states reported higher levels of resources overall, and use of core and teacher-created curricular resources in particular. They also reported having greater control over their curricular decision-making during the pandemic. There were no state-level differences in teachers’ level of preparation for pandemic teaching, but teachers in …

Figured Worlds Of Women Mathematics Education Scholars, Lili Zhou, Ricki L. Geller-Mckee, Brooke Max, Hyunyi Jung, Bima Sapkota, Jill Newton, Lindsay M. Keazer

School of Mathematical & Statistical Sciences Faculty Publications

Drawing on the concept of figured worlds (Holland et al., 1998), this project focuses on addressing, responding to, and understanding the self within the figured world of the mathematics education community. Specifically, we examine a group of women with diverse backgrounds in terms of race, class, and cultural contexts, who are engaged in various roles as mathematics education scholars, including teachers, teacher educators, and researchers. Using a dialogical self approach, we facilitate both internal and external discourses, exploring personal histories, narratives, and the development of evolving identities. Our findings reveal that culture and social positions, such as gender, class, and …

Rogue Waves In The Massive Thirring Model, Junchao Chen, Bo Yang, Bao-Feng Feng

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, general rogue wave solutions in the massive Thirring (MT) model are derived by using the Kadomtsev–Petviashvili (KP) hierarchy reduction method and these rational solutions are presented explicitly in terms of determinants whose matrix elements are elementary Schur polynomials. In the reduction process, three reduction conditions including one index- and two dimension-ones are proved to be consistent by only one constraint relation on parameters of tau-functions of the KP-Toda hierarchy. It is found that the rogue wave solutions in the MT model depend on two background parameters, which influence their orientation and duration. Differing from many other coupled …

Analysis And Unraveling Misconceptions: Examining Tenth-Grade High School Students’ Errors In Hots Questions On Exponential Equations And Effective Scaffolding Strategies, Nunik Indayani, Erry Hidayanto, Sisworo Sisworo

Jurnal Pendidikan Sains

The objective of this study is to elucidate the errors made by tenth-grade students at SMA Hati Bilingual Boarding School Probolinggo, Indonesia, when tackling Higher Order Thinking Skills (HOTS) questions on exponential equations, along with the scaffolding process employed to mitigate these errors. Employing qualitative research methods, four participants were chosen based on their performance in completing HOTS questions on exponential equations, followed by in-depth interviews. The identified student errors encompass basic errors, appropriate errors, missing information, and partial insights. The analysis aligns these errors predominantly with Brodie’s error taxonomy. Furthermore, the study highlights the efficacy of instructional scaffolding, utilizing …

The Identification Of Students’ Conceptual Mastery Of Fluid Statics: An Overview Of Gender, Maria Claudia Sodakain, Sentot Kusairi, Nandang Mufti, Jane Koswojo

Jurnal Pendidikan Sains

This study aimed to assess the conceptual mastery of fluid statics among students, employing a descriptive quantitative approach. The research involved 22 male and 13 female students from eleventh-grade MIPA at a private school in Surabaya, Indonesia. The primary research instrument comprised 11 multiple-choice and reasoning questions, covering concepts related to Pressure and Hydrostatic Pressure, Pascal’s Law, and Archimedes’ Principle. Results indicated a higher level of conceptual mastery among male students compared to their female counterparts. While both genders demonstrated relatively high conceptual mastery in Pressure and Hydrostatic Pressure, significant disparities were observed in their understanding of Pascal’s Law and …

Exploring The Correlation Between Metacognitive Skills And Cognitive Learning Outcomes Across Various Biology Learning Models In High Schools, Ninik Kristiani, Herawati Susilo, Aloysius Duran Corebim, Fatchur Rohman

Jurnal Pendidikan Sains

In Malang City, a notable observation within the realm of high school biology education suggests a prevalent lack of metacognitive knowledge among teachers, hindering their efficacy in nurturing students' metacognitive capacities. Such deficiency yields implications for diminished cognitive abilities among students. Recognizing this, there emerges a pressing need for an enhanced pedagogical framework capable of fostering metacognitive skills adeptly. One promising approach involves integrating inquiry-based learning and problem- based mindful pedagogy (PBMP). This amalgamation is anticipated to synergistically harness the potential of both methodologies in bolstering metacognitive skills alongside cognitive learning outcomes. This study adopts a correlational research design, wherein …

Analysis Of Student Engagement And Perceptions Of Community Of Inquiry (Coi)-Based Blended Learning In The Chemical Separation Subject On Chromatography, Agita Dzulhajh Anggraini, Surjani Wonorahardjo, Yudhi Utomo

Jurnal Pendidikan Sains

Learning in higher education has often been conducted solely through face-to-face, teacher- centered approaches. Implementing Community of Inquiry (CoI)-based blended learning in chromatography is crucial as it offers students a novel learning experience. This pre- experimental study, utilizing a one-group pretest-posttest design, aimed to describe students’ perceptions of CoI-based blended learning in chromatography. Data were collected using a questionnaire comprising 34 statements. The results indicated that students had a neutral perception of CoI-based blended learning. While a community of inquiry has been established, it has not yet reached its full potential, as students have not fully experienced social presence in …

An Efficient Derivative Free Iterative Method And Condition Number Of Nonlinear Equations Arising From Real World Phenomena, Alatuhigha Nguni, Chacha S. Chacha, Adeline P. Mtunya

Tanzania Journal of Science

Recent researches have shown a great interest in developing novel numerical methods for solving nonlinear equations of the form  arising from real world phenomena.  However, very little attention has been given to the study of condition numbers which is an important aspect in measuring the sensitivity of the problem in response to slight perturbations in the input data. In this article, we present an efficient free derivative iterative scheme constructed by refining Newton-Raphson method standard form in which the derivative term is approximated by using finite difference scheme; hence making it derivative free. We also conducted an in-depth analysis of …

K-Distinct Lattice Paths, Eric J. Yager, Marcus Engstrom

Rose-Hulman Undergraduate Mathematics Journal

Lattice paths can be used to model scheduling and routing problems, and, therefore, identifying maximum sets of k-distinct paths is of general interest. We extend the work previously done by Gillman et. al. to determine the order of a maximum set of k-distinct lattice paths. In particular, we disprove a conjecture by Gillman that a greedy algorithm gives this maximum order and also refine an upper bound given by Brewer et. al. We illustrate that brute force is an inefficient method to determine the maximum order, as it has time complexity O(n^k).

Excursions In Vector Calculus, Diego Castano

Mathematics Colloquium Series

Vector calculus is an invaluable tool in much of physics – electromagnetism is a prime example. The use of vector calculus is highlighted in an exploration of the concept of inductance and a reconsideration of its calculation. A form of the standard equation for inductance that is more versatile is derived and applied in some examples.

Utilizing Graph Thickness Heuristics On The Earth-Moon Problem, Robert C. Weaver

Rose-Hulman Undergraduate Mathematics Journal

This paper utilizes heuristic algorithms for determining graph thickness in order to attempt to find a 10-chromatic thickness-2 graph. Doing so would eliminate 9 colors as a potential solution to the Earth-moon Problem. An empirical analysis of the algorithms made by the author are provided. Additionally, the paper lists various graphs that may or nearly have a thickness of 2, which may be solutions if one can find two planar subgraphs that partition all of the graph’s edges.

Using Assessments To Promote Growth Mindset In College Algebra, Hannah M. Lewis, Kady Schneiter, David Lane Tait

Mathematics and Statistics Faculty Publications

Scientific evidence highlights the positive impact of a growth mindset on student achievement. Students with a growth mindset view errors and obstacles as opportunities for growth and welcome challenges and the opportunity to learn from their mistakes. Much has been written about promoting growth mindset through lectures and attitudes, however, assessments can also be an important avenue for encouraging a growth mindset in students. In this paper, we describe how we used assessments to promote growth mindset in a college algebra class. In the sections that follow, we discuss the need for these assessments and the principles that underly their …

Neighboring Mapping Points Theorem, Andrei V. Malyutin, Oleg R. Musin

School of Mathematical & Statistical Sciences Faculty Publications

We introduce and study a new family of theorems extending the class of Borsuk–Ulam and topological Radon type theorems. The defining idea for this new family is to replace requirements of the form “the image of a subset that is large in some sense is a singleton” with requirements of the milder form “the image of a subset that is large in some sense is a subset that is small in some sense”. This approach covers the case of mappings Sm→Rn with m

An example of a statement from this new family is the following theorem. Let f be a …