Mathematics Commons^™

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 …

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 …

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 …

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)].

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 …

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 …

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 …

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 …

A Finite Element Model For Hydro-Thermal Convective Flow In A Porous Medium: Effects Of Hydraulic Resistivity And Thermal Diffusivity, S. M. Mallikarjunaiah, Dambaru Bhatta

School of Mathematical & Statistical Sciences Faculty Publications

In this article, a finite element model is implemented to analyze hydro-thermal convective flow in a porous medium. The mathematical model encompasses Darcy’s law for incompressible fluid behavior, which is coupled with a convection-diffusion-type energy equation to characterize the temperature in the porous medium. The current investigation presents an efficient, stable, and accurate finite element discretization for the hydro-thermal convective flow model. The well-posedness of the proposed discrete Galerkin finite element formulation is guaranteed due to the decoupling property and the linearity of the numerical method. Computational experiments confirm the optimal convergence rates for a manufactured solution. Several numerical results …

Conditional Optimal Sets And The Quantization Coefficients For Some Uniform Distributions, Evans Nyanney, Megha Pandey, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Bucklew and Wise (1982) showed that the quantization dimension of an absolutely continuous probability measure on a given Euclidean space is constant and equals the Euclidean dimension of the space, and the quantization coefficient exists as a finite positive number. By giving different examples, in this paper, we have shown that the quantization coefficients for absolutely continuous probability measures defined on the same Euclidean space can be different. We have taken uniform distribution as a prototype of an absolutely continuous probability measure. In addition, we have also calculated the conditional optimal sets of n-points and the nth conditional quantization errors …

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 …