Mathematics Commons^™

Geogebra Applets For Fostering Conceptual Understanding In Algebra, Ma. Louise Antonette N. De Las Penas, Mark Anthony C. Tolentino, Maria Alva Q. Aberin, Agnes D. Garciano, Juan Carlo F. Mallari, Jumela F. Sarmiento, Debbie Marie B. Verzosa

Mathematics Faculty Publications

This paper discusses two GeoGebra applets, Radical and Parabola, that are designed to strengthen the conceptual understanding of specific topics in algebra. The design and pedagogical basis of the applets are presented. The integration of the applets in teaching Grade 9 mathematics in a partner high school in the Philippines is then discussed. Finally, we report feedback gathered from students and teachers during this integration. Their feedback indicates the potential of these applets for improving students’ learning of algebra.

Proving The Existence Of Equichordal Tight Fusion Frames Using The Newton–Kantorovich Theorem, Staci R. Davis

Theses and Dissertations

An equichordal tight fusion frame (ECTFF) is an example of an optimal packing of subspaces. In particular, an ECTFF is an optimal packing of points in the Grassmannian with respect to chordal distance. Equivalently, every ECTFF is an arrangement of subspaces that meet certain criteria; tightness and equichordality. The existence of an ECTFF can be rephrased as, a certain polynomial mapping as a root. Hence, one can prove the existence of an ECTFF by applying Newton–Kantorovich theorem to this polynomial mapping, given a close enough approximation of one. Newton–Kantorovich requires checking an inequality within a neighborhood of the approximate root, …

A Bayesian Approach For Lifetime Modeling And Prediction With Multi-Type Group-Shared Missing Covariates, Hao Zeng, Xuxue Sun, Kuo Wang, Yuxin Wen, Wujun Si, Mingyang Li

Engineering Faculty Articles and Research

In the field of reliability engineering, covariate information shared among product units within a specific group (e.g., a manufacturing batch, an operating region), such as operating conditions and design settings, exerts substantial influence on product lifetime prediction. The covariates shared within each group may be missing due to sensing limitations and data privacy issues. The missing covariates shared within the same group commonly encompass a variety of attribute types, such as discrete types, continuous types, or mixed types. Existing studies have mainly considered single-type missing covariates at the individual level, and they have failed to thoroughly investigate the influence of …

Optimizing Buying Strategies In Dominion, Nikolas A. Koutroulakis

Rose-Hulman Undergraduate Mathematics Journal

Dominion is a deck-building card game that simulates competing lords growing their kingdoms. Here we wish to optimize a strategy called Big Money by modeling the game as a Markov chain and utilizing the associated transition matrices to simulate the game. We provide additional analysis of a variation on this strategy known as Big Money Terminal Draw. Our results show that player's should prioritize buying provinces over improving their deck. Furthermore, we derive heuristics to guide a player's decision making for a Big Money Terminal Draw Deck. In particular, we show that buying a second Smithy is always more optimal …

Finite Element Solution Of Crack-Tip Fields For An Elastic Porous Solid With Density-Dependent Material Moduli And Preferential Stiffness, Hyun C. Yoon, S. M. Mallikarjunaiah, Dambaru Bhatta

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, the finite element solutions of crack-tip fields for an elastic porous solid with density-dependent material moduli are presented. Unlike the classical linearized case in which material parameters are globally constant under a small strain regime, the stiffness of the model presented in this paper can depend upon the density with a modeling parameter. The proposed constitutive relationship appears linear in the Cauchy stress and linearized strain independently. From a subclass of the implicit constitutive relation, the governing equation is bestowed via the balance of linear momentum, resulting in a quasi-linear partial differential equation (PDE) system. Using the …

Regular Functions On The Scaled Hypercomplex Numbers, Daniel Alpay, Ilwoo Cho

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we study the regularity of R-differentiable functions on open connected subsets of the scaled hypercomplex numbers {Ht}_t∈R by studying the kernels of suitable differential operators {∇_t}_t∈R, up to scales in the real field R.

Hadamard Matrices Of Orders 60 And 64 With Automorphisms Of Orders 29 And 31, Makoto Araya, Masaaki Harada, Vladimir Tonchev

Michigan Tech Publications

A classification of Hadamard matrices of order 2p + 2 with an automorphism of order p is given for p = 29 and 31. The ternary self-dual codes spanned by the newly found Hadamard matrices of order 60 with an automorphism of order 29 are computed, as well as the binary doubly even self-dual codes of length 120 with generator matrices defined by related Hadamard designs. Several new ternary near-extremal self-dual codes, as well as binary near-extremal doubly even self-dual codes with previously unknown weight enumerators are found.

Apollonian Packings And Kac-Moody Root Systems, Ian Whitehead

Mathematics & Statistics Faculty Works

We study Apollonian circle packings using the properties of a certain rank 4 indefinite Kac-Moody root system ϕ. We introduce the generating function Z(s) of a packing, an exponential series in four variables with an Apollonian symmetry group, which is a symmetric function for ϕ. By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of ϕ, with automorphic Weyl denominators, we express Z(s) in terms of Jacobi theta functions and the Siegel modular form Δ₅. We also show that the domain of convergence of Z(s) is the Tits cone of ϕ, and discover that this domain inherits the …

Using A Two-Way Engagement Community- And Family-Centered Pedagogy To Prepare Pre-Service Mathematics Teachers In A Hispanic-Serving Institution, Olga Ramirez, Mayra Ortiz Galarza, Luis M. Fernandez

School of Mathematical & Statistical Sciences Faculty Publications

Research on effective methods to prepare pre-service teachers (PSTs) in teaching mathematics to K-12 Latin* students has been gaining significant momentum. These efforts have focused, in part, on promoting pedagogical practices that recognize and incorporate the culture and language that K-12 Latin* students and their communities share. As teacher educators, we argue that if we are to further prepare PSTs to serve the needs of such increasingly diversifying K-12 student population, the same pedagogical focus on the learner’s cultural wealth should also be applied to the preparation of PSTs themselves, especially among Latin* PSTs in Hispanic Serving Institutions (HSI) like …

Functional Data Learning Using Convolutional Neural Networks, Jose Galarza, Tamer Oraby

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we show how convolutional neural networks (CNNs) can be used in regression and classification learning problems for noisy and non-noisy functional data (FD). The main idea is to transform the FD into a 28 by 28 image. We use a specific but typical architecture of a CNN to perform all the regression exercises of parameter estimation and functional form classification. First, we use some functional case studies of FD with and without random noise to showcase the strength of the new method. In particular, we use it to estimate exponential growth and decay rates, the bandwidths of …

Spacetime Geometry Of Acoustics And Electromagnetism, Lucas Burns, Tatsuya Daniel, Stephon Alexander, Justin Dressel

Mathematics, Physics, and Computer Science Faculty Articles and Research

Both acoustics and electromagnetism represent measurable fields in terms of dynamical potential fields. Electromagnetic force-fields form a spacetime bivector that is represented by a dynamical energy–momentum 4-vector potential field. Acoustic pressure and velocity fields form an energy–momentum density 4-vector field that is represented by a dynamical action scalar potential field. Surprisingly, standard field theory analyses of spin angular momentum based on these traditional potential representations contradict recent experiments, which motivates a careful reassessment of both theories. We analyze extensions of both theories that use the full geometric structure of spacetime to respect essential symmetries enforced by vacuum wave propagation. The …

Pseudo-Differential Operators On The Circle, Bernoulli Polynomials, Roger Gay, Ahmed Sebbar

Mathematics, Physics, and Computer Science Faculty Articles and Research

We show how the classical polylogarithm function Li_s (z) and its relatives, the Hurwitz zeta function and the Lerch function are all of a spectral nature, and can explain many properties of the complex powers of the Laplacian on the circle and of the distribution (x +i0)^s .We also make a relation with a result of Keiper [Fractional Calculus and its relationship to Riemann’s zeta function, Master of Science, Ohio State University, Mathematics (1975)].

Gauss Circle Problem Over Smooth Integers, Ankush Goswami

School of Mathematical & Statistical Sciences Faculty Publications

For a positive integer 𝑛, let 𝑟2(𝑛) be the number of representations of 𝑛 as sums of two squares (of integers), where the convention is that different signs and different orders of the summands yield distinct representations. A famous result of Gauss shows that 𝑅(𝑥) ∶= ∑ 𝑛≤𝑥 𝑟2(𝑛) ∼ 𝜋𝑥. Let 𝑃(𝑛) denote the largest prime factor of 𝑛 and let 𝑆(𝑥, 𝑦) ∶= {𝑛 ≤ 𝑥 ∶ 𝑃(𝑛) ≤ 𝑦}. In this paper, we study the asymptotic behavior of 𝑅(𝑥, 𝑦) ∶= ∑ 𝑛∈𝑆(𝑥,𝑦) 𝑟2(𝑛) for various ranges of 2 ≤ 𝑦 ≤ 𝑥. For 𝑦 in a …

An Icosahedron For Two: A Many-Sided Look At Making A Duet, Colleen T. Wahl

LASER Journal

The space around our bodies is not empty or neutral. In fact, the space around our bodies is loaded with meaning and important. When we move through it, whether it be in our daily lives or a choreographer making specific choices in order to convey a message, we activate new understandings in our lives. As a dancer and choreographer, I created a duet from improvisational climbs on an icosahedron. This article discusses choreographing from the form icosahedron and connects Laban's theories of space harmony with the activation of meaning in my life.

Two Non–*–Isomorphic *–Lie Algebra Structures On Sl(2,R) And Their Physical Origins, Luigi Accardi, Irina Ya. ArefʹEva, Yungang Lu, Igorʹ VasilʹEvich Volovich

Journal of Stochastic Analysis

No abstract provided.

Deep Neural Network-Oriented Indicator Method For Inverse Scattering Problems Using Partial Data, Yule Lin, Xiaoyi Yan, Jiguang Sun, Juan Liu

Michigan Tech Publications

We consider the inverse scattering problem to reconstruct an obstacle using partial far-field data due to one incident wave. A simple indicator function, which is negative inside the obstacle and positive outside of it, is constructed and then learned using a deep neural network (DNN). The method is easy to implement and effective as demonstrated by numerical examples. Rather than developing sophisticated network structures for the classical inverse operators, we reformulate the inverse problem as a suitable operator such that standard DNNs can learn it well. The idea of the DNN-oriented indicator method can be generalized to treat other partial …

Session 8: Machine Learning Based Behavior Of Non-Opec Global Supply In Crude Oil Price Determinism, Mofe Jeje

SDSU Data Science Symposium

Abstract

While studies on global oil price variability, occasioned by OPEC crude oil supply, is well documented in energy literature; the impact assessment of non-OPEC global oil supply on price variability, on the other hand, has not received commensurate attention. Given this gap, the primary objective of this study, therefore, is to estimate the magnitude of oil price determinism that is explained by the share of non-OPEC’s global crude oil supply. Using secondary sources of data collection method, data for target variable will be collected from the US Federal Reserve, as it relates to annual crude oil price variability, while …

Fuzzy Ideas Explain Fechner Law And Help Detect Relation Between Objects In Video, Olga Kosheleva, Vladik Kreinovich, Ahnaf Farhan

Departmental Technical Reports (CS)

How to find relation between objects in a video? If two objects are closely related -- e.g., a computer and it mouse -- then they almost always appear together, and thus, their numbers of occurrences are close. However, simply computing the differences between numbers of occurrences is not a good idea: objects with 100 and 110 occurrences are most probably related, but objects with 1 and 5 occurrences probably not, although 5 − 1 is smaller than 110 − 100. A natural idea is, instead, to compute the difference between re-scaled numbers of occurrences, for an appropriate nonlinear re-scaling. In …

Modifed Playfair For Text File Encryption And Meticulous Decryption With Arbitrary Fillers By Septenary Quadrate Pattern, N. Sugirtham, R. Sherine Jenny, B. Thiyaneswaran, S. Kumarganesh, C. Venkatesan, K. Martin Sagayam, Lam Dang, Linh Dinh, Helen Dang

Mathematics and Computer Studies: Faculty Publications

Cryptography secures data and serves to ensure the confidentiality of records. Playfair is a cryptographic symmetrical algorithm that encrypts statistics based on key costs. This secret is shared with an authorized person to retrieve data. In the conventional pattern, there is an area complexity and deficiency in letters, numbers, and special characters. This hassle has been overcome in previous studies by editing pattern dimensions. The fillers used throughout the enciphering were not eliminated during the retrieval process, which resulted in the indiscrimination of the retrieved statistics. The proposed method uses a separate quadrate pattern that strengthens the Playfair cipher and …

There Is Still Plenty Of Room At The Bottom: Feynman's Vision Of Quantum Computing 65 Years Later, Alexis Lupo, Vladik Kreinovich, Victor L. Timchenko, Yuriy P. Kondratenko

Departmental Technical Reports (CS)

In 1959, Nobelist Richard Feynman gave a talk titled "There's plenty of room at the bottom", in which he emphasized that, to drastically speed up computations, we need to make computer components much smaller -- all the way to the size of molecules, atoms, and even elementary particles. At this level, physics is no longer described by deterministic Newton's mechanics, it is described by probabilistic quantum laws. Because of this, computer designers started thinking how to design a reliable computer based on non-deterministic elements -- and this thinking eventually led to the modern ideas and algorithms of quantum computing. So, …

From Quantifying And Propagating Uncertainty To Quantifying And Propagating Both Uncertainty And Reliability: Practice-Motivated Approach To Measurement Planning And Data Processing, Niklas R. Winnewisser, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

When we process data, it is important to take into account that data comes with uncertainty. There exist techniques for quantifying uncertainty and propagating this uncertainty through the data processing algorithms. However, most of these techniques do not take into account that in real world, measuring instruments are not 100% reliable -- they sometimes malfunction and produce values which are far off from the measured values of the corresponding quantities. How can we take into account both uncertainty and reliability? In this paper, we consider several possible scenarios, and we show, for each scenario, what is the natural way to …

An Overview Of The Second-Previous Memory Effect In The Strictlyalternating Donation Game, Mohamed Zayet, Essam El-Seidy, Amira Abdel-Malek, Aya Rabie

Basic Science Engineering

Game theory delves into the examination of strategic behaviour across diverse domains such as insurance, business, military, biology, andmore,with the aim of deriving optimal decisions. Recent research focusing on the alteration of memory in the donation game with simultaneous iterated rounds has spurred our interest in investigating this phenomenon within the realm of the strictly alternating donation game. This study proposes a novel decision-making approach, utilising the previous unit instead of the most recent one. Thescopenarrowsdownto16employedstrategies,each definedbyfinitetwo-stateautomata,whileaccountingforpotentialimplementationerrorsinthe computationofstrategypayoffs.Dominant strategies are determined by assessing the interaction payoffs among strategy pairs. This article centres on the calculation of equilibrium points among heteroclinic …

New Algorithmic Support For The Fundamental Theorem Of Algebra, Vitaly Zaderman

Dissertations, Theses, and Capstone Projects

Univariate polynomial root-finding is a venerated subjects of Mathematics and Computational Mathematics studied for four millenia. In 1924 Herman Weyl published a seminal root-finder and called it an algorithmic proof of the Fundamental Theorem of Algebra. Steve Smale in 1981 and Arnold Schonhage in 1982 proposed to classify such algorithmic proofs in terms of their computational complexity. This prompted extensive research in 1980s and 1990s, culminated in a divide-and-conquer polynomial root-finder by Victor Pan at ACM STOC 1995, which used a near optimal number of bit-operations. The algorithm approximates all roots of a polynomial p almost as fast as one …

Low Shear In Short-Term Impacts Endothelial Cell Traction And Alignment In Long-Term, Mohanish K. Chandurkar, Nikhil Mittal, Shaina P. Royer-Weeden, Steven D. Lehmann, Yeonwoo Rho, Sangyoon J. Han

Michigan Tech Publications

Within the vascular system, endothelial cells (ECs) are exposed to fluid shear stress (FSS), a mechanical force exerted by blood flow that is critical for regulating cellular tension and maintaining vascular homeostasis. The way ECs react to FSS varies significantly; while high, laminar FSS supports vasodilation and suppresses inflammation, low or disturbed FSS can lead to endothelial dysfunction and increase the risk of cardiovascular diseases. Yet, the adaptation of ECs to dynamically varying FSS remains poorly understood. This study focuses on the dynamic responses of ECs to brief periods of low FSS, examining its impact on endothelial traction—a measure of …

Low Shear In Short-Term Impacts Endothelial Cell Traction And Alignment In Long-Term, Mohanish Chandurkar, Nikhil Mittal, Shaina P. Royer-Weeden, Steven D. Lehmann, Yeonwoo Rho, Sangyoon J. Han

Michigan Tech Publications

Within the vascular system, endothelial cells (ECs) are exposed to fluid shear stress (FSS), a mechanical force exerted by blood flow that is critical for regulating cellular tension and maintaining vascular homeostasis. The way ECs react to FSS varies significantly; while high, laminar FSS supports vasodilation and suppresses inflammation, low or disturbed FSS can lead to endothelial dysfunction and increase the risk of cardiovascular diseases. Yet, the adaptation of ECs to dynamically varying FSS remains poorly understood. This study focuses on the dynamic responses of ECs to brief periods of low FSS, examining its impact on endothelial traction-a measure of …

New Effective Transformational Computational Methods, Jun Zhang, Ruzong Fan, Fangyang Shen, Junyi Tu

Publications and Research

Mathematics serves as a fundamental intelligent theoretic basis for computation, and mathematical analysis is very useful to develop computational methods to solve various problems in science and engineering. Integral transforms such as Laplace Transform have been playing an important role in computational methods. In this paper, we will introduce Sumudu Transform in a new computational approach, in which effective computational methods will be developed and implemented. Such computational methods are straightforward to understand, but powerful to incorporate into computational science to solve different problems automatically. We will provide computational analysis and essentiality by surveying and summarizing some related recent works, …

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 …

Tight Bounds For The N2-Chromatic Number Of Graphs, Arnold Eniego, Ian June L. Garces, Jose Rosario

Mathematics Faculty Publications

Let G be a connected graph. A vertex coloring of G is an N₂-vertex coloring if, for every vertex v, the number of different colors assigned to the vertices adjacent to v is at most two. The N₂-chromatic number of G is the maximum number of colors that can be used in an N₂vertex coloring of G. In this paper, we establish tight bounds for the N₂-chromatic number of a graph in terms of its maximum degree and its diameter, and characterize those graphs that attain these bounds.

Parameterized Algorithm For The Poset Cover Problem, Ivy D. Ordanel, Proceso L. Fernandez, Richelle Ann B. Juayong, Jhoirene B. Clemente, Henry N. Adorna

Department of Information Systems & Computer Science Faculty Publications

It is already known that the 1-Poset and 2-Poset Cover Problems are in P. In this paper, we extended the previous results and devised an algorithm for the k-Poset Cover Problem, for any k number of posets that cover the input. The algorithm runs in O(m2k n2), where m and n are the input size. With this running time, we can say that the problem belongs to XP (slicewise polynomial). The algorithm runs efficiently for small fixed k but runs exponentially for large k. While the algorithm running time has yet not to be efficient for large k, we have …

A Causal Inference Approach For Spike Train Interactions, Zach Saccomano

Dissertations, Theses, and Capstone Projects

Since the 1960s, neuroscientists have worked on the problem of estimating synaptic properties, such as connectivity and strength, from simultaneously recorded spike trains. Recent years have seen renewed interest in the problem coinciding with rapid advances in experimental technologies, including an approximate exponential increase in the number of neurons that can be recorded in parallel and perturbation techniques such as optogenetics that can be used to calibrate and validate causal hypotheses about functional connectivity. This thesis presents a mathematical examination of synaptic inference from two perspectives: (1) using in vivo data and biophysical models, we ask in what cases the …