Mathematics Commons^™

A View Into Secondary Education Mathematics, Thomas Krieger Jr.

Honors Theses

Teaching methods, and the effects they can have on students, are important to consider for a classroom because when teaching you should allow for every student to have an opportunity. Every student should feel encouraged in the classroom, however not every method may allow for that. An important task for a teacher is to find out how to reach their students in their classroom; be it adapting methods or choosing when to implement one item over another. This task differs with every student that enters the classroom as no student is the same. Every students’ differences stem from their academic …

Predicting The Outcomes Of Internet-Based Cognitive Behavioral Therapy For Tinnitus: Applications Of Artificial Neural Network And Support Vector Machine, Hansapani Rodrigo, Eldré W. Beukes, Gerhard Andersson, Vinaya Manchaiah

School of Mathematical & Statistical Sciences Faculty Publications

Purpose:

Internet-based cognitive behavioral therapy (ICBT) has been found to be effective for tinnitus management, although there is limited understanding about who will benefit the most from ICBT. Traditional statistical models have largely failed to identify the nonlinear associations and hence find strong predictors of success with ICBT. This study aimed at examining the use of an artificial neural network (ANN) and support vector machine (SVM) to identify variables associated with treatment success in ICBT for tinnitus.

Method:

The study involved a secondary analysis of data from 228 individuals who had completed ICBT in previous intervention studies. A 13-point reduction …

One-Parameter Darboux-Deformed Fibonacci Numbers, Stefani C. Mancas, H. C. Rosu

Publications

One-parameter Darboux deformations are effected for the simple ODE satisfied by the continuous generalizations of the Fibonacci sequence recently discussed by Faraoni and Atieh [Symmetry 13, 200 (2021)], who promoted a formal analogy with the Friedmann equation in the FLRW homogeneous cosmology. The method allows the introduction of deformations of the continuous Fibonacci sequences, hence of Darboux-deformed Fibonacci (non integer) numbers. Considering the same ODE as a parametric oscillator equation, the Ermakov-Lewis invariants for these sequences are also discussed.

How To Choose A Law Review: An Empirical Study, Ignacio Cofone, Pierre-Jean G. Malé

Journal of Legal Education

No abstract provided.

Ideals Of Functions With Compact Support In The Integer-Valued Case, Themba Dube, Oghenetega Ighedo, Batsile Tlharesakgosi

Mathematics, Physics, and Computer Science Faculty Articles and Research

For a zero-dimensional Hausdorff space X, denote, as usual, by C(X, ℤ) the ring of continuous integer-valued functions on X. If f ∈ C(X, ℤ), denote by Z(f) the set of all points of X that are mapped to 0 by f. The set C_K(X; ℤ) = {f ∈ C(X; ℤ) | cl_X(X \ Z(f)) is compact} is the integer-valued analogue of the ideal of functions with compact support in C(X). By first working in the category of locales and then interpreting …

The History Of The Enigma Machine, Jenna Siobhan Parkinson

History Publications

The history of the Enigma machine begins with the invention of the rotor-based cipher machine in 1915. Various models for rotor-based cipher machines were developed somewhat simultaneously in different parts of the world. However, the first documented rotor machine was developed by Dutch naval officers in 1915. Nonetheless, the Enigma machine was officially invented following the end of World War I by Arthur Scherbius in 1918 (Faint, 2016).

Congruences For Consecutive Coefficients Of Gaussian Polynomials With Crank Statistics, Dennis Eichhorn, Lydia Engle, Brandt Kronholm

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we establish infinite families of congruences in consecutive arithmetic progressions modulo any odd prime ℓ for the function p ( n , m , N ) , which enumerates the partitions of n into at most m parts with no part larger than N . We also treat the function p ( n , m , ( a , b ] ) , which bounds the largest part above and below, and obtain similar infinite families of congruences.

For m ≤ 4 and ℓ = 3 , simple combinatorial statistics called "cranks" witness these congruences. We prove …

On Moments Of Certain Families Of L-Functions, Gopal Maiti

Doctoral Theses

In this thesis, I described the notion of moments of L-functions in some families. Then I discussed some problems on estimating moments of L-functions. As applications, I showed some simultaneous non-vanishing and correlation results of certain product L-functions over number fields as well as function fields.

Local Well-Posedness Of The Cauchy Problem For A P -Adic Nagumo-Type Equation, L. F. Chacón-Cortés, C. A. Garcia-Bibiano, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

We introduce a new family of p -adic nonlinear evolution equations. We establish the local well-posedness of the Cauchy problem for these equations in Sobolev-type spaces. For a certain subfamily, we show that the blow-up phenomenon occurs and provide numerical simulations showing this phenomenon.

(R1886) Effect Of Aggregation Function In Moma-Plus Method For Obtaining Pareto Optimal Solutions, Alexandre Som, Abdoulaye Compaoré, Kounhinir Somé, Blaise Somé

Applications and Applied Mathematics: An International Journal (AAM)

In this work, we have proposed some variants of MOMA-Plus method that we have numerically tested for the resolution of nonlinear multiobjective optimization problems. This MOMA-Plus method and variants differ from each other by the choice of aggregation functions in order to reduce the number of objective functions. The theoretical results allowing us to use these aggregation functions to transform multiobjective optimization problems into single objective optimization problems are proved by two theorems. This study has highlighted the advantages of each aggregation function according to the type of Pareto front of the optimization problem. Six benchmarks test problems have been …

Which Interval-Valued Alternatives Are Possibly Optimal If We Use Hurwicz Criterion, Marina Tuyako Mizukoshi, Weldon Lodwick, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, for each alternative i, we do not know the corresponding gain xi, we only know the interval [li,ui] of possible gains. In such situations, a reasonable way to select an alternative is to choose some value α from the interval [0,1] and select the alternative i for which the Hurwicz combination α*ui + (1 − α)*li is the largest possible. In situations when we do not know the user's α, a reasonable idea is to select all alternatives that are optimal for some α. In this paper, we describe a feasible algorithm for such a selection.

(R1885) Analytical And Numerical Solutions Of A Fractional-Order Mathematical Model Of Tumor Growth For Variable Killing Rate, N. Singha, C. Nahak

Applications and Applied Mathematics: An International Journal (AAM)

This work intends to analyze the dynamics of the most aggressive form of brain tumor, glioblastomas, by following a fractional calculus approach. In describing memory preserving models, the non-local fractional derivatives not only deliver enhanced results but also acknowledge new avenues to be further explored. We suggest a mathematical model of fractional-order Burgess equation for new research perspectives of gliomas, which shall be interesting for biomedical and mathematical researchers. We replace the classical derivative with a non-integer derivative and attempt to retrieve the classical solution as a particular case. The prime motive is to acquire both analytical and numerical solutions …

(R1500) Type-I Generalized Spherical Interval Valued Fuzzy Soft Sets In Medical Diagnosis For Decision Making, M. Palanikumar, K. Arulmozhi

Applications and Applied Mathematics: An International Journal (AAM)

In the present communication, we introduce the concept of Type-I generalized spherical interval valued fuzzy soft set and define some operations. It is a generalization of the interval valued fuzzy soft set and the spherical fuzzy soft set. The spherical interval valued fuzzy soft set theory satisfies the condition that the sum of its degrees of positive, neutral, and negative membership does not exceed unity and that these parameters are assigned independently. We also propose an algorithm to solve the decision making problem based on a Type-I generalized soft set model. We introduce a similarity measure based on the Type-I …

(R1978) Heated Laminar Vertical Jet Of Psudoplastic Fluids-Against Gravity, Manisha Patel, M. G. Timol

Applications and Applied Mathematics: An International Journal (AAM)

A heated laminar jet of Pseudo-plastic fluid flowing vertically upwards from a long narrow slit into a region of the same fluid which is at a rest and at a uniform temperature is considered. The governing non-linear Partial differential equations (PDEs) for the defined flow problem are transformed into non-linear ordinary differential equations using the effective similarity technique-one parameter deductive group theory method. The obtained non-linear coupled Ordinary differential equations are solved and the results are presented by graphs. The effect of the Prandtl number and Grashof number on the velocity and temperature of the jet flow is discussed. Also, …

Data Processing Under Fuzzy Uncertainty: Towards More Accurate Algorithms, Marina Tuyako Mizukoshi, Weldon Lodwick, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Data that we process comes either from measurements or from experts -- or from the results of previous data processing that were also based on measurements and/or expert estimates. In both cases, the data is imprecise. To gauge the accuracy of the results of data processing, we need to take the corresponding data uncertainty into account. In this paper, we describe a new algorithm for taking fuzzy uncertainty into account, an algorithm that, for small number of inputs, leads to the same or even better accuracy than the previously proposed methods.

Non-Localized Physical Processes Can Help Speed Up Computations, Be It Hidden Variables In Quantum Physics Or Non-Localized Energy In General Relativity, Michael Zakharevich, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

While most physical processes are localized -- in the sense that each event can only affect events in its close vicinity -- many physicists believe that some processes are non-local. These beliefs range from more heretic -- such as hidden variables in quantum physics -- to more widely accepted, such as the non-local character of energy in General Relativity. In this paper, we attract attention to the fact that non-local processes bring in the possibility of drastically speeding up computations.

Graph Approach To Uncertainty Quantification, Hector A. Reyes, Cliff Joslyn, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional analysis of uncertainty of the result of data processing assumes that all measurement errors are independent. In reality, there may be common factor affecting these errors, so these errors may be dependent. In such cases, the independence assumption may lead to underestimation of uncertainty. In such cases, a guaranteed way to be on the safe side is to make no assumption about independence at all. In practice, however, we may have information that a few pairs of measurement errors are indeed independent -- while we still have no information about all other pairs. Alternatively, we may suspect that for …

Systems Approach Explains Why Low Heart Rate Variability Is Correlated With Depression (And Suicidal Thoughts), Francisco Zapata, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

Depression is a serious medical problem. If diagnosed early, it can usually be cured, but if left undetected, it can lead to suicidal thoughts and behavior. The early stages of depression are difficult to diagnose. Recently, researchers found a promising approach to such diagnosis -- it turns out that depression is correlated with low heart rate variability. In this paper, we show that the general systems approach can explain this empirical relation.

An Argument In Favor Of Piecewise-Constant Membership Functions, Marina Tuyako Mizukoshi, Weldon Lodwick, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Theoretically, we can have membership functions of arbitrary shape. However, in practice, at any given moment of time, we can only represent finitely many parameters in a computer. As a result, we usually restrict ourselves to finite-parametric families of membership functions. The most widely used families are piecewise linear ones, e.g., triangular and trapezoid membership functions. The problem with these families is that if we know a nonlinear relation y = f(x) between quantities, the corresponding relation between membership functions is only approximate -- since for piecewise linear membership functions for x, the resulting membership function for y is not …

Will Nanotechnology Bring In The Judgement Day?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

There are many current and prospective positive aspects of nanotechnology. However, while we look forward to its future successes, we need to keep our eyes open and be prepared for what will really be a future shock: that quantum computing – an inevitable part of nanotechnology – will enable the future folks to read all our encrypted messages and thus, learn everything that we wanted to keep secret. This will be really the Judgement Day, when all our sins will be open to everyone. How we will react to it? Will this destroy our civilization? Let us hope that the …

Standard Interval Computation Algorithm Is Not Inclusion-Monotonic: Examples, Marina Tuyako Mizukoshi, Weldon Lodwick, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When we usually process data, we, in effect, implicitly assume that we know the exact values of all the inputs. In practice, these values comes from measurements, and measurements are never absolutely accurate. In many cases, the only information about the actual (unknown) values of each input is that this value belongs to an appropriate interval. Under this interval uncertainty, we need to compute the range of all possible results of applying the data processing algorithm when the inputs are in these intervals. In general, the problem of exactly computing this range is NP-hard, which means that in feasible time, …

(R1958) On Deferred Statistical Convergence Of Fuzzy Variables, Ömer Kişi, Mehmet Gürdal, Ekrem Savaş

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, within framework credibility theory, we examine several notions of convergence and statistical convergence of fuzzy variable sequences. The convergence of fuzzy variable sequences such as the notion of convergence in credibility, convergence in distribution, convergence in mean, and convergence uniformly virtually certainly via postponed Cesàro mean and a regular matrix are researched using fuzzy variables. We investigate the connections between these concepts. Significant results on deferred statistical convergence for fuzzy variable sequences are thoroughly investigated.

How Viscosity Of An Asphalt Binder Depends On Temperature: Theoretical Explanation Of An Empirical Dependence, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

Pavement must be adequate for all the temperatures, ranging from the winter cold to the summer heat. In particular, this means that for all possible temperatures, the viscosity of the asphalt binder must stay within the desired bounds. To predict how the designed pavement will behave under different temperatures, it is desirable to have a general idea of how viscosity changes with temperature. Pavement engineers have come up with an empirical approximate formula describing this change. However, since this formula is purely empirical, with no theoretical justification, practitioners are often somewhat reluctant to depend on this formula. In this paper, …

Why In Mond -- Alternative Gravitation Theory -- A Specific Formula Works The Best: Complexity-Based Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Based on the rotation of the stars around a galaxy center, one can estimate the corresponding gravitational acceleration -- which turns out to be much larger than what Newton's theory predicts based on the masses of all visible objects. The majority of physicists believe that this discrepancy indicates the presence of "dark" matter, but this idea has some unsolved problems. An alternative idea -- known as Modified Newtonian Dynamics (MOND, for short) is that for galaxy-size distances, Newton's gravitation theory needs to be modified. One of the most effective versions of this idea uses so-called simple interpolating function. In this …

Epistemic Vs. Aleatory: Case Of Interval Uncertainty, Marina Tuyako Mizukoshi, Weldon Lodwick, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Interval computations usually deal with the case of epistemic uncertainty, when the only information that we have about a value of a quantity is that this value is contained in a given interval. However, intervals can also represent aleatory uncertainty -- when we know that each value from this interval is actually attained for some object at some moment of time. In this paper, we analyze how to take such aleatory uncertainty into account when processing data. We show that in case when different quantities are independent, we can use the same formulas for dealing with aleatory uncertainty as we …

(R2022) Mathematical Modelling Of Tuberculosis And Covid-19 Co-Infection In India: A Real Data Analysis On Concomitant Diseases, Vijai Shanker Verma, Harshita Kaushik, Archana Singh Bhadauria

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we have proposed an epidemiological model to study the dynamics of two concomitant diseases Tuberculosis (TB) and COVID-19. Here, we have formulated a deterministic compartmental model as an extended form of the classical SIS model. First, the basic reproduction number R₀ is derived and then stability analysis of the model is done. It is observed that the disease-free equilibrium is stable when R₀ is less than one and the endemic equilibrium is stable only when R₀ is greater than one. Numerical simulation is carried out to illustrate the theoretical findings and to study the …

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 (Archived)

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 …

Developing Awareness Around Language Practices In The Elementary Bilingual Mathematics Classroom, Gladys Krause, Melissa Adams Corral, Luz A. Maldonado Rodríguez

Teaching and Learning Faculty Publications

This study contributes to efforts to characterize teaching that is responsive to children’s mathematical ideas and linguistic repertoire. Building on translanguaging, defined in this article as a pedagogical practice that facilitates students’ expression of their understanding using their own language practices, and on the literature surrounding children’s mathematical thinking, we present an example of a one-on-one interview and of the circulating portion of a mathematics class from a second grade classroom. We use these examples to foreground instructional practices, for researchers and practitioners, that highlight a shift from a simplified view of conveying mathematics as instruction in symbology and formal …

New Jump Operators On Equivalence Relations, John D. Clemens, Samuel Coskey

Mathematics Faculty Publications and Presentations

We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group Γ we introduce the Γ-jump. We study the elementary properties of the Γ-jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups Γ, the Γ-jump is proper in the sense that for any Borel equivalence relation E the Γ-jump of E is strictly higher than E in the Borel reducibility hierarchy. On the other hand, there are examples of groups Γ for which the Γ-jump is not proper. To establish properness, …

Cohen-Macaulay Type Of Weighted Path Ideals, Shuai Wei

All Dissertations

In this dissertation we give a combinatorial characterization of all the weighted $r$-path suspensions for which the $f$-weighted $r$-path ideal is Cohen-Macaulay. In particular, it is shown that the $f$-weighted $r$-path ideal of a weighted $r$-path suspension is Cohen-Macaulay if and only if it is unmixed. Type is an important invariant of a Cohen-Macaulay homogeneous ideal in a polynomial ring $R$ with coefficients in a field. We compute the type of $R/I$ when $I$ is any Cohen-Macaulay $f$-weighted $r$-path ideal of any weighted $r$-path suspension, for some chosen function $f$. In particular, this computes the type for all weighted trees …