Student Population Growth at Vanderbilt

The ultimate goal of this project is to model the increasing, decreasing, or stagnant change in the student population at Vanderbilt University over time. The general methodology by which this will be accomplished is to establish linear equations that model the population in the k+1 year of a given grade level, and look for eigenvalues. More specifically, this means setting the determinant of (A- λI) equal to 0 (where A is the matrix derived from the linear equations) in order to solve for λ. From there, the growth rate in the population of Vanderbilt Undergraduate students will be determined. Ultimately, the ratios of the populations of different grades will be concluded.

Coefficients to the population-in-year-k variables will be computed where the data is available, and derived from reasonable assumptions where it is not. For example, the data on transfers in and out of the university after the first year are readily available for the public to see (it is provided by Vanderbilt). However, the overall negative effect of transfers and dropouts on the number of incoming freshmen cannot be easily determined without extensive (and expensive) research. Therefore, those values will be assumed (the assumptions will be a source of error). A 7 by 7 matrix, A, will result from the linear equations derived for the various grades. This presents a few additional challenges in calculating the determinant, as Wolfram Alpha has a 5 by 5 limit. Therefore, Matlab, or some other computing software, will be employed to find the determinant and later put (A- λI) in RREF.  The first four variables will be the four years at Vanderbilt in year k. The next variable will be the number of graduates who have just completed their 4 years. The last two variables will be the number of dropouts per year, and the net number of transfers per year.

It is understood that it is possible for “the numbers not to work out”. At the beginning of this proposal it was stated that we were going to see if the population is growing, declining, or remaining stagnant. However, if we actually find that it is declining, we know we have done something wrong (in terms of coefficients, not mathematics, most likely). Then, we will adjust our assumptions. Conveniently, we can also do a linear regression for the past year’s grade sizes (from released VU data) to get the actual student population growth rate. Our ultimate goal is for our numbers to produce those numbers.

Finally, this problem is useful to solve, as Vanderbilt needs to predict growth trends in its student population. This is so that the school may efficiently increase capacity of student housing, faculty size, and other amenities that are necessary for a college.

Encryption/Decryption Application

For our application project we plan on approaching the subject of cryptography as it applies to linear algebra and creating a computer program that will implement different cryptographic applications. Through creating this computer application we plan on expanding on the idea of encrypting and decrypting messages we learned in class through various approaches such as shift ciphers, where you shift the letters by a specified amount, substitution ciphers (also known as Caesar ciphers), in which you have a key that is used to change from plaintext into cipher text, and Hill ciphers, where you have an alphabet of a certain size and a matrix that is applied to a certain number of characters of the plaintext at a time to change the message to cipher text.

Our program would allow the user to specify the message they wish to encrypt or decrypt and the method by which they want to do that. Depending on the method the user selects the program will encrypt or decrypt the message entered in. We will provide the user will a variety of different alphabets, including a standard 26-character alphabet and a subset of the ASCII alphabet that are printable, which would allow the use of the same ciphers to generate completely different messages.

The largest part of our project will be working with Hill ciphers using concepts in linear algebra. Our plan for the program is to allow the user of the program to specify an n by n matrix (up to size n = 10) to be applied to a message they supply. The user can either fill this matrix manually, which the program will ensure is invertible, or the program will automatically generate a random invertible matrix. We need to ensure the matrices we are using are invertible to allow for both encryption using the original matrix and decryption using the inverse of the original matrix. To do this will we be figuring out the determinants of the matrices in our program to ensure that the matrices are invertible and then also finding the decipher key matrix from an entered or generated matrix. We plan on doing this recursively in the program following the general method of reduction that we were taught in class, which lends itself to recursion. If a user decides to manually enter the matrix our program will ensure the matrix is invertible and if it isn’t will tell the user they need to enter a new matrix or use our auto-generate feature.