Simulating the Spread of the Common Cold

This modeling scenario guides students to simulate and investigate the spread of the common cold in a residence hall. An example floor plan is given, but the reader is encouraged to use a more relevant example. In groups, students run repeated simulations, collect data, derive a differential equation model, solve that equation, estimate parameter values by hand and through regression, visually evaluate the consistency of the model with their data, and present their results to the class

Lost at Sea: Introduction to Numerical Methods through Navigation

Excerpt: The ship, El Perdido, was damaged during a storm which knocked out its main and backup power generators. Before the backup generator failed, Captain Miguel Gomez sent a distress call and the crew have been able to keep El Perdido a oat, but the ship is adrift in the Pacific Ocean off the coast of California. Thankfully, a US Coast Guard rescue operation is underway after receiving the distress call. The Coast Guard has El Perdido\u27s last known position and has mapped out the surface water velocities in this area as slope fields for longitude (x) and latitude (y), which they have updated using historical data and estimated predictions. Since the search grid is small enough, this curved region on the surface of the earth is relatively at

Logistics of Mathematical Modeling-Focused Projects

This article addresses the logistics of implementing projects in an undergraduate mathematics class and is intended both for new instructors and for instructors who have had negative experiences implementing projects in the past. Project implementation is given for both lower and upper division mathematics courses with an emphasis on mathematical modeling and data collection. Projects provide tangible connections to course content which can motivate students to learn at a deeper level. Logistical pitfalls and insights are highlighted as well as descriptions of several key implementation resources. Effective assessment tools, which allowed me to smoothly adjust to student feedback, are demonstrated for a sample class. As I smoothed the transition into each project and guided students through the use of the technology, their negative feedback on projects decreased and more students noted how the projects had enhanced their understanding of the course topics. Best practices learned over the years are given along with project summaries and sample topics. These projects were implemented at a small liberal arts university, but advice is given to extend them to larger classes for broader use.Comment: 27 pages, no figures, 1 tabl

Expendable launch vehicle transportation for the space station

Logistics transportation will be a critical element in determining the Space Station Freedom's level of productivity and possible evolutionary options. The current program utilizes the Space Shuttle as the only logistics support vehicle. Augmentation of the total transportation capability by expendable launch vehicles (ELVs) may be required to meet demanding requirements and provide for enhanced manifest flexibility. The total operational concept from ground operations to final return of support hardware or its disposal is required to determine the ELV's benefits and impacts to the Space Station Freedom program. The characteristics of potential medium and large class ELVs planned to be available in the mid-1990's (both U.S. and international partners' vehicles) indicate a significant range of possible transportation systems with varying degrees of operational support capabilities. The options available for development of a support infrastructure in terms of launch vehicles, logistics carriers, transfer vehicles, and return systems is discussed

An Elementary Proof of Dodgson's Condensation Method for Calculating Determinants

In 1866, Charles Ludwidge Dodgson published a paper concerning a method for evaluating determinants called the condensation method. His paper documented a new method to calculate determinants that was based on Jacobi's Theorem. The condensation method is presented and proven here, and is demonstrated by a series of examples. The condensation method can be applied to a number of situations, including calculating eigenvalues, solving a system of linear equations, and even determining the different energy levels of a molecular system. The method is much more efficient than cofactor expansions, particularly for large matrices; for a 5 x 5 matrix, the condensation method requires about half as many calculations. Zeros appearing in the interior of a matrix can cause problems, but a way around the issue can usually be found. Overall, Dodgson's condensation method is an interesting and simple way to find determinants. This paper presents an elementary proof of Dodgson's method.Comment: 7 pages, no figure

Algae Population Self-Replenishment

This modeling scenario investigates the massive algal blooms that struck Lake Chapala, Mexico, in 1994. After reading a summary of news articles on the incident, students create an ODE system model from a verbal description of the factors, visualize this system using an executable Java applet (PPLANE) to predict overall behavior, and then analyze the nonlinear system using the Jacobian matrix, eigenvalues, phase plane, and feasibility conditions on parameters to fully describe the system behavior. Students are expected to be familiar with systems of differential equations, equilibria, jacobian matrices, and eigenvalues. Students will learn modeling from qualitative descriptions, nondimensionalization, applying feasibility conditions to parameters, and how to use technology to interactively analyze a system of differential equations

1-65-S-Algal Blooms: Algal Blooms Threatening Lake Chapala

This modeling scenario investigates the massive algal blooms that struck Lake Chapala, Mexico, starting in 1994. After reading a summary of articles written on the incidents, students are guided through the process of creating a first order differential equation from a verbal model of the factors and analyze the nonautonomous ODE using direction field, parameter evaluation, and exact solution computation to fully describe the population behavior. Students are expected to be familiar with the separable method and direction fields. Students will learn building and improving a model from qualitative descriptions, nondimensionalization, evaluating parameters, and how to use DFIELD software to interactively analyze a first order differential equation. An alternative modeling investigation of this problem leads to a nonlinear system of equations shown in modeling scenario Algae Self-Replenishmen

Interactive-graphic flowpath plotting for turbine engines

An engine cycle program capable of simulating the design and off-design performance of arbitrary turbine engines, and a computer code which, when used in conjunction with the cycle code, can predict the weight of the engines are described. A graphics subroutine was added to the code to enable the engineer to visualize the designed engine with more clarity by producing an overall view of the designed engine for output on a graphics device using IBM-370 graphics subroutines. In addition, with the engine drawn on a graphics screen, the program allows for the interactive user to make changes to the inputs to the code for the engine to be redrawn and reweighed. These improvements allow better use of the code in conjunction with the engine program

Technology requirements for an orbiting fuel depot: A necessary element of a space infrastructure

Advanced planning within NASA has identified several bold space exploration initiatives. The successful implementation of these missions will require a supporting space infrastructure which would include a fuel depot, an orbiting facility to store, transfer and process large quantities of cryogenic fluids. In order to adequately plan the technology development programs required to enable the construction and operation of a fuel depot, a multidisciplinary workshop was convened to assess critical technologies and their state of maturity. Since technology requirements depend strongly on the depot design assumptions, several depot concepts are presented with their effect on criticality ratings. Over 70 depot-related technology areas are addressed

Experimental evaluation of wall Mach number distributions of the octagonal test section proposed for NASA Lewis Research Center's altitude wind tunnel

Wall Mach number distributions were determined over a range of test-section free-stream Mach numbers from 0.2 to 0.92. The test section was slotted and had a nominal porosity of 11 percent. Reentry flaps located at the test-section exit were varied from 0 (fully closed) to 9 (fully open) degrees. Flow was bled through the test-section slots by means of a plenum evacuation system (PES) and varied from 0 to 3 percent of tunnel flow. Variations in reentry flap angle or PES flow rate had little or no effect on the Mach number distributions in the first 70 percent of the test section. However, in the aft region of the test section, flap angle and PES flow rate had a major impact on the Mach number distributions. Optimum PES flow rates were nominally 2 to 2.5 percent wtih the flaps fully closed and less than 1 percent when the flaps were fully open. The standard deviation of the test-section wall Mach numbers at the optimum PES flow rates was 0.003 or less