Publications & Presentations |

Experimental and Computational Mathematics

Geometric Proofs

“A Constructive Proof of Feuerbach’s Theorem Using a Computer Algebra System” Proceedings of 21^th Conference on Applications of Computer Algebra, Kalamata, Greece, July 2015, p.47

Feuerbach’s Theorem states that the midpoints of the three sides, the base points of the three heights, and the midpoints of the line segments between the corners of a triangle and the intersection of the heights are on a circle. This talk offers a constructive proof. It is known that algebraic expression

$x^2+y^2+ex+dy+f=0\quad\quad\quad(1)$

represents a circle centered at $(-d/2, -e/2)$ with radius

$r = \frac{d^2+e^2-4f}{4}\quad\quad\quad(2)$

provide (2) is positive. Three points among nine stated in the theorem are chosen to form a system of linear equations from (1). The values of $d, e$ and $f$ are determined by solving the equations. With the solution, (2) is shown to be positive which implies (1) indeed represents a circle. We then proceed to verify that the coordinates of the remain six points satisfy (1). Hence all nine points are on the same circle.

“A Constructive Proof of Euler’s Line Theorem Using a Computer Algebra System”
Mathematical Association of America Indiana Section Conference, Franklin College, Spring 2016

Euler’s Line Theorem states that in every triangle, the intersection of the medians, the intersection of the heights, and the center of the circumscribed circle are on a straight line. This talk offers an algebraic and algorithmic proof with the aid of a Computer Algebra System’s (CAS) symbolic computation capabilities. We will use Omega, a free online CAS Explorer in this presentation.

“An Algebraic Approach to Geometric Proof Using a Computer Algebra System” Proceedings of 19^th Conference on Applications of Computer Algebra. Edited by Jose Luis Galan Garcia. Malaga, Spain, July 2013, pp. 197-200

Geometric proof is often considered to be a challenging subject in mathematics. The traditional approach seeks a tightly knitted sequence of statements linked together by strict logic to prove that a theorem is true. Moving from one statement to the next in traditional proofs often demands clever, if not ingenious reasoning. An algebraic approach to geometric proof, however, aims to computationally produce values that imply the thesis statement of the theorem. This presentation will demonstrate the algebraic approach to geometric proof through examples.

Maxima and Minima without Calculus

“Solving Kepler’s Wine Barrel Problem Without Calculus” Mathematical Association of America Indiana Section Conference, via Zoom, Fall 2020

Kepler conducted numerical studies on his “wine barrel problem” in order to find the maximum volume held by the cylinder shaped barrels with fixed diagonal. This problem was subsequent solved analytically after the invention calculus. For volume $V=\pi d^2h/4 -\pi h^3/16$ where $d$ stands for the fixed diagonal of the cylinder, what positive value of height $h$ gives the largest volume of $V$ ? This talk presents an alternative solution to this famous problem by applying a special theorem without using calculus.

“Boosting Rocket Performance Without Calculus” 25th Conference on Applications of Computer Algebra, Montréal, Canada, July 16-20, 2019
École de technologie supérieure

Boosting a two-stage rocket’s flight performance is a non-trivial optimization problem typically solved by calculus. This presentation will show an alternative way that only requires high school mathematics, with the help of a computer algebra system (CAS). This non-calculus approach places more emphasis on problem solving through mathematical thinking, as all symboliccalculations are carried out by the CAS. It also makes a range of interesting problems readily tackled with minimum mathematical prerequisites.

“Solving Maximization/Minimization Problems by Elementary Means” Mathematical Association of America Indiana Section Conference, Indiana University East, Spring 2013

This presentation illustrates a systematic approach in which maximization/minimization problems can be solved without calculus. First, we will demonstrate how problems can be cast in certain forms so that the extreme values can be obtained immediately. Next, we will apply AM-GM inequality and its corollaries to solve problems that appear to be solvable only by calculus. The non-calculus approach makes a range of very interesting problems available to a wider audience, and at a much earlier stage of their studies in mathematics and other sciences.

Exploratory Analysis and Computation

“Solving Parameter Estimation Problem Using Least Square-based Algorithms” Joint work with David Deng, Joint Mathematics Meetings (JMM), Denver, Jan 15-18, 2020

In today’s application of Data Science, a common problem concerns the estimation of unknown parameters in the governing differential equations. This presentation illustrates a machine learning approach to estimate and validate the parameters of two population models, namely, the Malthus model and Verhulst’s Logistic model.

The parameter of growth rate in the Malthus model is estimated using a least square-based algorithm applied to a training dataset with historical population data. When validating the resulting model using more contemporary dataset, we discovered inconsistency in the curve fitting of the original model.

To resolve this inconsistency, we turn to Verhulst’s Logistic model, whose governing differential equation is nonlinear, with two parameters – growth rate and saturation level. After transforming the model equation with derivatives computed through forward, centered and backward finite difference schemes, we obtain the initial values of the estimation from the least square-based algorithm. These initial values are used to extrapolate the final estimation of parameters through a second iteration of least square-based estimation. The resulting model with the estimated parameters is then ready to make predictions of population in the future.

“Maximizing the Final Speed of a Two-Stage Rocket Using a Computer Algebra System” Joint work with David Deng, Mathematical Association of America Indiana Section Conference, Wabash College, Fall 2019

A rocket consists of a payload of mass $P$ propelled by two stages of masses $m_1$ (first stage) and $m_2$ (second stage), both with structural factor $1-e$ . The exhaust speed of the first stage is $c_1$ , and of the second stage $c_2$ . The initial total mass, $m_1+m_2$ is fixed. The ratio $b = P/(m_1+ m_2)$ is assumed very small. According to the multi-stage rocket flight equation ([1]), the final speed of a two-stage rocket is

$v_f = -c_1\log(1-em_1/(m_1+m_2+P)) - c_2\log(1-em_2/(m_2+P))\quad\quad\quad(1)$

Let $a = m_2 /( m_1+m_2)$ , so that (1) becomes

$v_f = -c_1 \log(1-(e-ea)/(2+b)) - c_2\log(1-ea/(a+b))\quad\quad\quad(2)$

where $0 < a < 1, b > 0, 0 < e < 1, c_1 > 0, c_2 > 0$ .

We seek an appropriate value of $a$ to maximize $v_f$ .

The above rocket performance optimization problem is solved with the help of a Computer Algebra System (CAS) ([2]). We found that the value of $a$ for a maximum final speed is

$\sqrt{c_2b/c_1} + O(b)$

and the maximum speed is

$-(c_1+c_2)\log(1-e)-2e\sqrt{c_1c2b}/(1-e)+O(b)$ .

We compute the first order Taylor series of $a$ and $v_f$ in b to replace their complex expressions respectively. Results produced by the CAS are validated extensively through rigorous mathematical analysis.

“A Non-Iterative Method for Solving Nonlinear Equations” Proceedings of 24th Conference on Applications of Computer Algebra, Santiago de Compostela, June 18–22, 2018, p.99

Newton-Raphson method is the most commonly used iterative method for finding the root(s) of a real-valued function or nonlinear systems of equations. However, its convergence is often sensitive to the error in its initial estimation of the root(s). This talk will present a non-iterative method that mitigates non-convergence. An auxiliary initial-value problem of ordinary differential equation(s) is generated by a Computer Algebra System first, then integrated numerically over a closed interval. The solution(s) to the original systems of nonlinear equations is obtained non-iteratively at the end of the interval. A proof of the theorem serving as the base for this new method is presented at the talk. Several examples will illustrate its guaranteed convergence, a clear advantage over the Newton-Raphson method.

“Refuting a Conjecture On $x^{n}-1$ Using a Computer Algebra System” Mathematical Association of America Tri-Section Meeting of Illinois, Indiana, and Michigan Sections Conference, Valparaiso University, Spring 2018

There was a conjecture stating that the absolute value of a non-zero coefficient in the factors of $x^n-1$ is always 1. This presentation refutes this conjecture by a counter example using a computer algebra system (CAS). Furthermore, this talk will pose a similar problem to either prove or refute another conjecture regarding the uniqueness of a solution found by CAS.

“Mira’s Wing – Mathematical Art Generated by a Computer Algebra System” Art Exhibit,Mathematical Association of America Tri-Section Meeting of Illinois, Indiana, and Michigan Sections Conference, Valparaiso University, Spring 2018

“Generating Power Summation Formulas Using a Computer Algebra System” Book of Abstracts, 23rd Conference on Application of Computer Algebra, Jerusalem, Israel, July 2017, p. 35

Mathematical induction is often used in classroom to prove various Power Summation Formulas such as

$\sum\limits_{i=1}^{n}i = \frac{n(n+1)}{2}$

$\sum\limits_{i=1}^{n}i^2 = \frac{n(n+1)(2n+1)}{6}$

$\sum\limits_{i=1}^{n}i^3 = \frac{n^2(n+1)^2}{4}$

However, how the formulas are obtained in the first place is rarely discussed.

In this presentation, we will construct the Power Summation Formulas. Specifically, a recursive algorithm is derived and its implementation in Computer Algebra generates the formulas. A closer look at this algorithm also reveals the generated formulas can also be obtained by solving an initial-value problem of difference equation symbolically.

“Computer-Algebra-Aided Chebyshev Methods for Ordinary Differential Equations”
Book of Abstracts, 23rd Conference on Application of Computer Algebra, Jerusalem, Israel, July 2017, p. 45

The solution of ordinary differential equation can be approximated by a linear combination of well-known basis functions. Using the Chebyshev Polynomials as the basis functions, the approximation can be expressed as

$y(x) = \sum\limits_{r=0}^{\infty}a_rT_r(x)\quad\quad\quad(1)$

where the $T_r(x)$ ‘s are Chebyshev Polynomials of degree $r$ , and $a_r$ ‘s are the coefficients to be determined. In practice, we seek the approximation using a truncated expression of (1), namely,

$y(x) = \sum\limits_{r=0}^{n}a_rT_r(x)\quad\quad\quad(2)$

An online Computer Algebra System (CAS) is used to generate and subsequently solve a system of equations concerning a finite number of $a_r$ ’s. The use of CAS allows the retention of more $a_r$ ’s in (2). It also obviates the need for the traditional pad and pencil computations. Examples will be given to illustrate this approach in solving initial value problems, boundary value problems as well as eigenvalue problems for ordinary differential equations whose coefficients and other terms are themselves polynomials.

“Prove Inequalities by Solving Maximum/Minimum Problems Using a Computer Algebra System“ Book of Abstracts, 20th Conference on Applications of Computer Algebra, Edited by Robert H. Lewis. New York City, US, July 2014, p. 47

This presentation offers an alternative to traditional approaches to proving non-trivial inequalities, such as applying AM-GM, Cauchy, Ho ̈lder and Minkowski inequalities. This alternative approach, as demonstrated by various examples, establishes the validity of an inequality through solving a maximization/minimization problem by commonly practiced procedures in Calculus. Since the procedures are algorithmic, a Computer Algebra System (CAS) can carry out the computation efficiently.

“Omega: A Free Computer Algebra System Explorer for Online Education” Proceedings of 19th Conference on Applications of Computer Algebra, Edited by Jose Luis Galan Garcia. Malaga, Spain, July 2013, pp. 50-54

Online courses have become the medium of choice for students who cannot otherwise attend in a traditional classroom setting. Online distance education allows students to take courses in the convenience of their home or office, at their leisure, free of the distractions of campus life, without commute, while at the same time being provided with almost instantaneous access to the instructor and course materials. However, the lack of access to Computer Algebra System, traditionally exists only in computer labs on campus can be a significant detriment to online classes. In this presentation, we will introduce Omega, a free online Computer Algebra System (CAS) Explorer that provides user with immense power in both symbolic and numeric computing. To use Omega, only a web browser is required. User composes and submits mathematical query using a calculator-like graphical user interface. Upon submission, the query is processed by Omega’s CAS engine, and the result is displayed in text or graphic formats. Omega can be accessed from desktop/laptop computers, ipad/tablets, and smartphones. It is compatible with all major web browsers. Different CAS can be plugged in as Omega’s underlying engine.

Experimental and Computational Mathematics

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