- What is the Bachelor of Science in Applied Mathematics (BSAM) curriculum?
- Is the BSAM degree really three years?
- How do I take a course at another Institution?
- Are there math facilitated study groups?
- Where can I find information at Applied Math advising?
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- What is the Math Club?
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For questions about the program, please contact:
Chair of Applied Mathematics
Ira Allen 319 | 617-989-4368
Student Project Abstracts
WIT Applied Math & Sciences Student Expo
April 18, 2014
Methods & Topics in Applied Math II (MATH 275)
Professor: Amanda Hattaway
Title: Modeling the Speed of Dvorak & QWERTY keyboards
Author: James Petrillo, B.S. in Applied Math ‘17
Abstract: When it comes to the computer, many people want the best and easiest product on the market. People are willing to change and embrace new technology as it comes out, except for keyboard technologies. The same QWERTY keyboard (named after the letters in the upper left side) has been used for over 100 years; other keyboard layouts have come out since then. The main goal of this work is to simulate QWERTY and Dvorak keyboard typing. In particular, I authored two C++ programs that outputs a ``time metric'', based on guidelines that are specified in the program. The results of the program simulation are that the newer Dvorak keyboard layout allows for faster typing than the QWERTY keyboard. Future suggestions for work in this area are also presented.
Title: More than Just a Game: The Statistics of Fantasy Football
Jack Reff, B.S. in Applied Math ’16
Simranjit Singh, B.S. in Applied Math ’16
A major problem that fantasy football players face is coming up with a draft strategy that yields the best team. The goal of this project is to generate a draft strategy that would give an owner a competitive team. We:
1) collected public sports data over the last five years from the NFL and ESPN grouped it by position and sub grouped it by position rank;
2) calculated points per game and point differences for each player's position and
3) used this metric use to make a list of the most valuable positions or positions an owner should draft first.
Our main conclusion is that the ideal draft strategy is: first tier half-back, tight end, second tier quarterback, defense, and third tier wide-out and kicker. The second best strategy: first tier quarterback, tight end, second tier halfback, defense, and third tier kicker and wide-out. This draft model adds value to the fantasy football industry because, unlike the popular “auction draft” type, it does not require that each team have a monetary cap and we are able to show numerically that this is a reliable strategy.
Title: Linear Random Number Congruence Generation: Tests & Comparisons
Seth Goldish, B.S. in Applied Math ’16
Ely Biggs, B.S. in Applied Math ’16
Random numbers are used in many real world applications. They can be used to solve problems numerically when an exact deterministic solution is unknown or does not exist, such as in Monte Carlo Simulations. Many methods for generating random numbers are currently in use. Each method comes with it's own set of advantages and drawbacks. An easily implemented method known as the Linear Congruence Generator (LCG) is suitable for non-critical applications, however, it is not random enough to produce accurate results in simulations. This can be seen by comparing the percent error of a simulation computed using LCG numbers with the resulting error when the same simulation is run using a more random generator, such as the Mersenne Twister (python's built-in PRNG).
Title: Breathe Strong, Live Long: A Statistical Look at Lung Cancer Deaths in Three Countries
Shylee Ezroni, B.S. in Applied Math ’16
Zach Dobbins, B.S. in Applied Math ’16
Using historical data we project lung cancer deaths over time for men and women in the U.S., China and South Africa and study trends.
Title: Apples & Hydrogen Cyanide
Author: Kristian Roca, B.S. in Applied Math ’17
Amygdalin, a toxic substance, can be found in the seeds of many common fruits, such as apples, mangoes, and cherries. When ingested orally, certain enzymes can change amygdalin's chemical structure into hydrogen cyanide (HCN). HCN has been infamously used for decades as a mean for suicide and biological warfare.
To my knowledge, no research exists addressing how often one can consume a regular "dose" of hydrogen cyanide at constant intervals.
Title: Object Value
Authors: Aaron Ciardullo, B.S. in Applied Math ’16 and B.S. in Electromech Engin ‘16,
David Lombardi, B.S. in Applied Math ’16
Kristian Roca, B.S. in Applied Math ’17
Object value is the accumulated value an item gains or loses over its lifetime. We simulated total object value over time using the normal distribution taking into account the probability that over time, an object is going to break, rendering it without value.
Operations Research (MATH 310)
Professor: Rachel Maitra
Title: Ideal Layout for a Rectangular Parking Lot
Xiaoyong Cai, B.S. in Applied Mathematics ‘17
Kaiyuen Fong, B.S. in Applied Mathematics ‘16
Abstract: The goal of our final project is to create an ideal layout for a 100’ by 200’ rectangular parking lot (The Mathematical Contest in Modeling, 1987). There are two primary concerns. One of them is to maximize the number of stalls that can be allocated inside the parking lot. The other one is to consider the convenience for drivers.
Our approach is to design a layout with the highest stall density and the fewest restrictions, and then apply more constraints regarding the dimension of the parking lot, accessibility for drivers, etc. More concretely, first, assume an unbounded parking lot. This simplifies the challenge of finding the optimal angle such that the area of a stall plus the area for in and out of the stall are at minimum. Second, assume that the lot is bounded by two horizontally parallel edges at a distance of 100’ with infinite length (unbounded in one dimension). At this stage, it is straightforward to figure out the highest density combination of stall rows (the parking angle for each stall row is varied). Third, upgrade the layout from previous step by adding more constraints: two vertical lines at a distance of 200’, car turning space from one row to the others, etc. Finally, set up an ideal arrangement for entrance and exit. Enhance and adjust the layout by ensuring easy access for every driver.
In short, we have found the critical value – the optimal angle, which corresponds to the minimal area of a stall plus the space it will take to move in and out. By adapting this strategic design, the final result will accommodate the needs of both owner and consumer.
Title: Optimizing Personal Time Management
Author: Robert Moss, B.S. in Computer Science ‘14
Abstract: Personal time can be difficult to efficiently manage. Each of us manages our own schedule according to priorities specific to our lives. I separate daily tasks into a static or dynamic classification. Static tasks are responsibilities that cannot change their start or end time, and are stationary within your schedule (ex: work or class). Dynamic tasks take on a much more lenient set of restrictions because they are only constrained to their total time of completion and, in some cases, may be opted out entirely (ex: exercise or a hobby). In this research, I present a day-to-day model of specific tasks and optimize the order according to a weighted priority. The goal is to maximize the free time after static tasks are scheduled, to allow for the optimization of dynamic task placement. An implementation using the programming language Julia and AMPL will be explored.
Title: Collaborative Filtering Meetup Data – Decomposition of a Sparse Matrix
Author: Justin Hotchkiss Palermo, B.S. in Computer Information Systems ‘16
Abstract: The internet is vast, and the amount of information available to humans is overwhelming. This makes it hard to find specific information. Machine learning is used to make this data useful to human beings. Machines can take input from users, learn from it, and come to better decisions in the future. I examine how collaborative filtering can be used to make recommendations.
Meetup.com is a social networking website where users join groups, then attend events called "meetups". Users first tag themselves and supply their geographical location, and are then supplied with possible groups to join which are similar to their interests.
The data I use can be obtained at http://lsna2012.net76.net/ebsn/ Part of the challenge was dealing with 4.45 million users, 77.8 thousand tags, 42 thousand groups and 1.6 million events.
There are two overarching goals. The first is to classify meetup groups into categories, and then figure out if a person is in the category or not in the category. Typically, one would ask the user if he or she is in the category by asking for a rating of the prediction, for instance whether or not the user likes or dislikes a movie, book or song. Unfortunately, there is no explicit rating data of groups or events from meetup.com. Simply defining a baseline to determine whether or not someone belongs to a category or not is a challenge. I select specific types of groups, examine the users who have been to that meeting, then use that data to calculate a baseline which I can use to determine if someone belongs to that category, or belongs somewhere else.
I take a model-based approach to collaborative filtering, applying stochastic gradient descent to predict whether a user is likely to join a particular group to attend an event. The algorithm calculates what group a person will join based on the tags that they have labeled themselves with, then it calculates the prediction error, and iterates for each user. It then modifies its parameters and does it again, attempting to lower the prediction error.
Title: Creating an Optimal Schedule for Students at Wentworth Institute of Technology
Matthew Raison, B.S. in Applied Mathematics ‘15
Laura Boyd, B.S. in Applied Mathematics ‘15
Mohan Punjabi, B.S. in Applied Mathematics ‘15
Julie Pecukonis, B.S. in Applied Mathematics ‘15
Dylan Melnik, B.S. in Applied Mathematics ‘15
Abstract: A major problem that students face is finding a schedule that fits their needs. Our goal for this project is to devise an algorithm that generates a potential schedule such that students have as many different feasible schedules as possible. Using a previous Wentworth schedule as dummy data, we isolated several constraints for our program. After simplifying our model to a level of complexity our laptops and brains can handle, we started devising pseudo code, sensitive to class and time slot. Displaying the result as an array we consistently read out an optimized schedule for the Wentworth’s Biomedical Engineering major.
Title: An Integer Programming Approach to Sudoku Puzzles
Author: Ries Collier, B.S. in Applied Mathematics ‘15
Abstract: Sudoku is a logic based, combinatorial puzzle which has recently become very popular internationally. The purpose is to fill in an mx n grid with the numbers 1 to n with the restrictions that two of the same number cannot be in the same row, column, or sub-grid. Because of these strict constraints, the puzzles can be easily represented by an integer programming problem searching for a feasible solution. This project is restricted, however to 9x9 Sudoku puzzles, but the method can be quickly modified to suit any mx n puzzle.
Title: The 4 C’s*, Applied: Developing a Computerized Implementation for Routing a Fleet of Specialized Snow-Removal Vehicles in a Complex Urban Layout
Eric Hart, B.S. in Applied Mathematics ‘15
Vadim Manokhin, B.S. in Computer Science ‘14
Connor Mountain, B.S. in Civil Engineering ‘16
Abstract: While developments in GPS technologies continuously improve efficiency of operations to create safe roadways for winter travel, the implementations of algorithms for effectively routing a fleet of specialized snow-removal vehicles in a complex urban environment usually do not stand up to the difficult task of dynamically analyzing and readjusting the calculated routes in the given environment with a specific set of constraints. This work develops a computerized implementation of a routing heuristic known as the “Closest Street Heuristic” using operations research techniques. We work toward resolving the capacitated arc routing problem that consists of designing a set of routes performed by vehicles of restricted capacity such that every edge or arc on a given network of arcs (e.g. such as streets in a city) is serviced while the total costs are minimized. Flexibility of the program includes the ability to specify the number of vehicles, their initial positions, number and positions of return node-points, and length and the directions of the node-connecting edges. This allows tailoring the implementation to complex existing urban layouts such as the City of Boston which is known for its one-way streets and overall intricate street layout.
Keywords: Operations Research, Winter Road Maintenance, Capacitated Arc Routing Problem, Routing Heuristic
* Critical thinking and problem solving; Communication; Collaboration; Creativity and innovation
Title: The Avalanche Algorithm
Author: Zachary E. Buzaid, B.S. in Applied Mathematics ‘15
Abstract: Backcountry skiing is a sport that enables an athlete to break away from crowded ski resorts and enter into the land of fresh snow, untouched terrain, and inherent danger. The number of backcountry-goers has grown rapidly in the past 50 years, accounting for a 700% increase in the number of avalanche deaths per year. While there is an array of gear to allows skiers to be better prepared in the event of an avalanche, there is a lack of technology aimed at preventing skiers from triggering an avalanche. This algorithm aims to fill this void, and makes backcountry-goers aware of the risk associated with the features they choose to interact with.
Tyler Carlson, B.S. in Applied Mathematics ‘15
Nora Shea, B.S. in Applied Mathematics ‘15
Abstract: Throughout the work we have done in Operations Research, scheduling problems interested us the most and we wanted to do more with this subject. We want to solve a simple scheduling problem that can apply to many people and businesses. Through our research, we learned about different algorithms and how they work. We have taken the critical path algorithm and implemented this into C++ code. With this C++ code we are able to solve a scheduling problem quickly and efficiently. The C++ code is hard coded for our specific problem but future work could lead to a more general code.
Title: Resource Allocation in the Event of a Hurricane
Sandrah Abbuah, B.S. in Applied Mathematics ‘15
Tam Nguyen, B.S. in Applied Mathematics ‘15
Ryan Ayotte, B.S. in Biomedical Engineering ‘16
Matt Flynn, B.S. in General Engineering ‘16
Abstract: This project focuses on finding the optimal order for distribution of resources to areas affected by hurricanes. This is an important topic because in the past, resource allocation has been a problem after storms such as Hurricane Katrina. The project is tested using data from Hurricane Sandy, which caused a tremendous amount of damage to the coastal cities and towns of New Jersey. Through the use of R and techniques from graph theory and network flow, we are working to achieve an optimal solution. The parameters in the model include the distance to reach each county, the damage level, and the population to find the priority of each county receiving resources. The reward function places a higher reward for distributing resources to counties with higher damage and higher population, and a lower reward with increasing distance from the supply depot.
Title: Field Goal vs. Punt: Modeling the 4
Down Decision in Football
Liam Stokinger, B.S. in Applied Mathematics ‘15
Jeremy Eiholzer, B.S. in Applied Mathematics ‘15
Abstract: Football is a game of raw athleticism, precise skill, and strategic decision-making. The coach’s role in a game is just as important, if not more so, than the performance of the players. Countless games have been won and lost based on the decision of coaches. The goal of our project is to analyze this decision: Should a team punt the ball or attempt to kick a field goal on fourth down?
For this problem, we needed to sift through and pull relevant information from the 500,000+ plays run in the NFL since 2002. This data extraction resulted in a clear picture of all field goals made/missed and the distance of all punts based on field position over the last decade. Kicking a field goal is a two-result scenario: either you get three points or no points (three for a make, none for a miss). This means the expected amount of points from a kick would be three times the probability that the kick is good (for example, a 34 yard kick has an 84% chance of being good, so the expected points in this scenario would be 3x.84 = 2.53). But this doesn’t take everything into account. If one were to miss the kick, the other team would take over right where one kicked it.
A previous study has given expected points based on starting field position. To get net expected points on a kick, one must subtract the other team’s starting position expected points, should one miss, from the expected points of the kick itself (example – for the 34 yard kick, there is a 16% chance of missing the kick and the other team is expected to get .22 points if they take over at that spot. The net expected points would be the value we calculated before minus the probability of a miss times the expected points on starting field position - 2.53-(.16*.22) = 2.49). Now that we have solved for net expected points on a kick, we would need to do the same for a punt. Punt distance is normally distributed for the most part (the presence of a touchback actually makes this statement not true, but we can ignore this for our purposes).
What we can do now is find the probability that a punt will yield a more favorable result than trying to kick a field goal (this is based off of expected points from the other team’s new field position vs. expected points from going for a field goal). We can then say that if this probability is higher than 50%, a coach should choose to punt the ball from that field position. In future work, we can add the decision to just go for a first down, as well as accounting for mitigating factors like weather, time left in the game and score.
Differential Equations (MATH 625)
Professor Georgi Gospodinov
Title: The Dynamics of Raindrops
Abstract: The objective of this project was to apply one mathematical model and explore it through three arithmetical methods. In order to research how rain drops react to the air they are falling through and what effects the size of a rain drop has on its dynamics, the first step was to explore the basic movements of raindrops. We looked at a rain drop in free fall with the force of Earth’s gravity, and then, building towards real-world movement, air resistance was introduced creating the first method of solving. Next, two mathematical methods -RK4 and Eulers- are used to explore an even more real-world replication of how raindrops behave. Eulers was a preliminary method to get a basic idea of how the modeling should be done, and then RK4 took a more in depth, and accurate look at the system, modeling the raindrops with high accuracy.
Title: Forced Spring Systems
Abstract: Our forced spring system project models both a damped and undamped spring mass system from equations provided. These mathematical models are then compared with data that was gathered on both types and evaluated for differences. Variations of the normal damped oscillations were investigated and included critical damping, beats and resonance. The critical damping value was evaluated for different damping constants to find the trend and calculate the optimal constant. Beats create a bulge affect in the oscillations and this generates two distinct frequencies from the original function and the rapid occurrence of the bulge affect. Resonance explores the similarities in spring constants compared to the frequency. This explores the relationship as the frequency approaches resonance.
Title: Fourier Transform
Abstract: The Fourier transform is a useful tool in understanding problems posed by partial differential equations with unbounded spatial regions. We explored and derived properties of the Fourier Transform and its behavior with regards to convolution, and special functions such as the Heaviside Theta and Dirac Delta functions. Both of these functions are used in the one-dimensional heat equation model, which we explored as an application of the Fourier transform to the solution of a partial differential equation. We obtained numerical solution to the time evolution of temperature in an unbounded rod.
Title: Gain and Phase Shift
Abstract: This project examined gain and phase shift in electrical and mechanical systems. We were especially interested in demonstrating gain and phase shift models working in the real world. We did this by examining electrical amplifier circuits and harmonic motion in various mechanical systems. We examined how different coefficients in differential models could affect the overall behavior of the system. This enabled us to see the effects of damping and driving effects as they work in the physical world.
Title: Lead in the Body
Lead in the body is a serious health risk that enters the body through environmental inhalation, drinking liquids, and eating. Once it enters in the body it rapidly travels by the blood and is distributed into the liver and kidneys. Then it is slowly absorbed into your bones and soft tissues. After, lead is extracted through your hair, nails, sweat, and urinary system. We used a compartment model to display the flow of lead through the body showing the three compartments in which lead flows through. By using Linear Algebra and matrices we are able to compose how much lead travels through the body in each compartment. Utilizing Wolfram Mathematica and Differential Equations we are able to prove how much lead a human being can withstand in the amount of days using the equilibrium equation. Also, illustrate plots of human beings in a lead and in a lead-free environment. In the final part of our project, we can demonstrate the maximum level of lead that can travel through the body.
Title: Numerical Methods for Differential Equations
Abstract: Throughout this project we explored numerical methods including Euler’s, Improved Euler’s, and Rk4. Our goal was to further understand these means of estimating and compare the three to each other to evaluate their individual accuracy. The software Wolfram Mathematica 9.0 was used for many of the calculations and proved to be an invaluable resource. To express the results of our research we took example problems and solved each using the separate methods then compared the final values in tables as well as graphically. By doing this we are able to determine the most effective and accurate method and display our findings in a clear and understandable manner.
Title: Predator and Prey Model
Author: Nghia Huynh
Abstract: Population dynamics is related to the natural growth rate of a population, given by the difference of the natural birth and natural death rate, and to the capacity of the specific habitat. This project explored population dynamics in the presence of predators, from the point of view of mathematical ecology, the study of how the populations of two species interact with one another. We employed a Predator-Prey model approach and used Wolfram Mathematica to describe and analyze the changes of the species’ population. First, we created a differential equations model that incorporated the interaction between the two species. Then we applied our model to a real example. In the final part of our project, we varied the parameters of the model to see the changes in the model’s predictions.
Title: The Pendulum
Abstract: The objective of this project was to represent the movement and behavior of a pendulum in different scenarios with varying constants. There is a massless string with a length L with a bob at the end with a mass m. When the bob is moved from its rest position and let go it swings back and forth. The time it takes the pendulum to swing from its further left position to it furthest right position is called its period. The primary force acting on the bob is the gravitational force. In addition to this force, there may be a damping force from the friction at the pivot or air resistance. We constructed an equation to describe how the angle theta of the pendulum varies as a function of time. In each part of this project different values were set for different variables and we saw how the behavior of the pendulum was affected.
Directed Research in AstroPhysics (PHYS 406)
Professor James O’Brien
Title: The Wentworth Radio Telescope
Abstract: In this project, students are engaged in a semester long design and implementation of the creation of a Wentworth Radio Telescope. Students have been working with external collaborators from amateur radio astronomy enthusiasts as well as collaborators from NASA to develop a simple yet robust radio telescope design. Initial implementation will employ multiple designs such as single disk, single antenna, and multi-disk arrays. Project descriptions, results and future work will be discussed.
Professors Derek Cascio (Industrial Design) James O’brien (Physics), Greg Sirokman (Chemistry)
Title: Sector Vector Game
Sector Vector is the result of a semester long collaboration between the departments of Physics and Industrial Design. Building upon the foundation laid by Professor’s James O’Brien and Greg Sirokman, students from Professsor Derek Cascio’s industrial design course developed the next generation of a game intended to teach students about vector mathematics. The students were responsible for every facet of the gaming experience - rule development, graphics, game pieces and production. Using visualization and iterative refinement the team has created a highly polished prototype which is already making waves in the educational community as it begins its journey to make learning fun.
Engineering Chemistry & Conceptual Physics Classes
Professor Joseph Harney
Title: Energy Transfer and Storage: An Interdisciplinary Green Technology Project
Authors: Professor Harney’s engineering chemistry and conceptual physics classes
Abstract: For this project, students design and build a windmill and two capacitors to demonstrate electricity production and storage from a green source. The system is partially built from salvaged and recycled materials. This project builds upon students’ previous class discussions on energy topics and the electromotive force. Chemistry students engineer the capacitors as an alternative to chemical batteries, contrasting capacitor vs battery disposal issues. Engineering majors help to build the windmill platform and fashion the windmill blades. Conceptual physics students finish the blades and help design the yaw and pitch controls. Since this is a simple model, safety from extreme winds have been addressed by mechanically translating the wind’s shear force to angular momentum. As the wind is increased above a set limit, the force causes the generator to pivot along the z-axis slowing the rotation and therefore the risk of damage to the system.
Special thanks to: Enric Rosiu and the Project’s lab for the DC motor and the salvaged hub, Steve Chomyszak and the Center for Manufacturing for outfitting the hub for blade attachment and Derek Cascio and the Industrial Design Department for offering leftover supplies for the overall design.
Conceptual Physics (PHYS211)
Professor Franz Rueckert
Title: Rube Goldberg Machines and Conceptual Physics
Abstract: This semester, students in Conceptual Physics constructed Rube Goldberg Machines to demonstrate various topics covered in the course. The machines featured at least 5 sections, each of which showcased a particular concept. The goal of the machines was to eventually raise a WIT flag. Students showcased their originality and creativity, using catapults, pendulums, mousetraps, and a number of other clever devices to make surprising and interesting contraptions. At the end of the course, students were challenged to join all of their machines into one combined apparatus. The machines on display are the result of that effort, representing the work of students from Industrial Design, Management, Facility Planning and Management, and Computer Information Systems.