Category: Other

Always Check Your Work

How many times have you had an exam returned and thought to yourself, “Well, that was stupid of me.” Many times re-reading the directions given on an exam or double-checking your work can make a substantial difference in your grade. In the “real world”, the consequences of mistakes can be more severe. This is exemplified in a comic strip titled Los Alamos from XKCD.

In the strip, the characters discuss the important implications of the test they are about to conduct, and the dire consequences of the test if their math is incorrect. One character then asks, “Is it ‘SOH CAH TOA’ or ‘COH SAH TOA’?” Always remember to check your work.


Math Games

Many of us have fallen victim to the distraction of online games. Whether playing solitaire or harvesting crops on FarmVille (which I can proudly say I have never played), there is probably something else that you should be doing. Well, tonight while I was playing online math games, I decided to turn procrastination into productivity by blogging about it.

The web site that stood out to me was The Truth Tree’s Math and Logic Board. Story problems are the main focus of this particular website (if you had told me I’d be doing story problems for fun 13 years ago I would have said you were crazy). Unlike many “math game” web sites, this one is fairly advanced, with problems ranging from logical riddles, volumes of spheres, counting, topology and more. The only downside is that the answers are not provided. There is, however, an index of similar sites on the page.

The article, Zero-Divisor Graphs of \mathbb{Z}_n and Polynomial Quotient Rings over \mathbb{Z}_n, by Daniel Endean, Kristin Henry, and Erin Manlove discusses the necessity of knowing the chromatic number of a graph as well as if it is perfect in order to understand it. It also shows several different ways that you can know when the zero-divisor graph of \mathbb{Z}_N (\Gamma(\mathbb{Z}_n)) s perfect. In its final section, the article proves that \Gamma(\mathbb{Z}_p [x]\slash<x^n>) \cong \Gamma(\mathbb{Z}_{p^n}).

The introduction of the article provides numerous definitions. For instance, a graph G is the set of vertices V(G) with a corresponding set of edges E(G) where every element in E(G) is an unordered pair of distinct vertices from V(G). The order of G is the cardinality of G, or the number of elements it contains, and is denoted |G|. The chromatic number of G is the minimum number of colors required to color G so that no adjacent vertices have the same color, and is denoted \chi(G). Furthermore, when all of the vertices of G touch, G is complete. We say that H is a subgraph of G if V(H)\subseteq V(G) and E(H)\subseteq E(G). A clique is a complete subgraph, and the clique number of a graph, denoted \omega(G), is the order of the larget clique of G. The article then moves on to discuss the ways in which \chi(G) can be derived.

Two different processes for finding \chi(G) are presented. The first is fairly straight forward. You simply find the chromatic number, and then prove that a smaller number of colors would be insufficient for the graph to be complete. The second involves showing that a graph is perfect, so that the graphs clique number can be used to find its chromatic number. According to the article, a graph is perfect if all of its subgraphs have the same chromatic number and clique number. Several theorems and corresponding proofs are provided which give specific examples of perfect graphs. Theorem 1.1 states than any graph that has no subgraphs with alternating edges and vertices is perfect. Theorem 1.2 says that for prime p, the graph \Gamma(\mathbb{Z}_{p^n}) is perfect. Theorem 1.3 states that for distinct primes p_1 and p_2, the graph \Gamma(\mathbb{Z}_{p_1p_2}) is perfect. Finally, Theorem 1.4 says that the zero-divisor graph of \mathbb{Z}_n is perfect if and only if n=p^k where p is prime or n=p_1p_2 where p_1 and p_2 are distinct primes. The last two theorems of this section show how, given a perfect graph, the chromatic number can be found using the graph’s clique number. Theorem 1.5 states that for prime p, the graph \Gamma(\mathbb{Z}_{p^n}) has chromatic number p^{\dfrac{n}{2}}-1 when n is even and p^{\dfrac{n-1}{2}} when n is odd. Finally, Theorem 1.6 says that for distinct primes p_1 and p_2, the graph \Gamma(\mathbb{Z}_{p_1p_2}) has chromatic number two.

The next section of the paper addresses zero-divisor graphs of polynomial quotient rings and defines Q such that Q=\mathbb{Z}_p[x]\slash<x^n> where p is prime and n \geq 2. The paper then uses various theorems, corollaries,  lemmas, and proofs to arrive at its final conclusion: \Gamma(\mathbb{Z}_{p^n}) \cong \Gamma(Q).

In the video, Peter Donnelly Shows How Stats Fool Juries, Peter Donnelly delivers a talk on how statistics can be deceptive. After a few jokes about the social awkwardness of statisticians, Donnelly moves on to an example. He describes a scenario where you flip a fair coin until the patter HTH emerges (H being heads and T being tails). You then flip the coin again until the pattern HTT emerges. Then he asks the audience what they believe to be true:

a) The average number of flips for HTH to emerge is less than the average number of flips for HTT to emerge.

b) The average number of flips is equal for both.

c) The opposite of a) is true.

Most people in the audience answered b). The correct answer, however, is a). This is because, as Donnelly explains, the pattern HTH can repeat itself in five flips (HTHTH), where HTT cannot (HTTHT).

This is the first example he gives of how topics in statistics can often times be deceptive. He then moves on to a more relevant example (one covered in MATH 341) of the probability of having a disease given a positive test result for a test with 99% accuracy. He illustrates that while a positive test result may make it seem that there is a 99% probability that you have the disease, the true probability depends on how many people have been tested, as well as the actual probability of having the disease. If a million people are tested and there is a .01% probability of having the disease, then there will be a much larger number of false positives (9999) than people who actually have the disease (100). Furthermore, the number of the number of people tested who have the disease, only 99% of them will have a positive test result. This makes the probability that one actually has the disease given a positive test result considerably small (less than 1%).

This example relates to how statistics can be used to deceive a jury. As a matter of fact, in the wrong hands, statistics can be incredibly dangerous. This was illustrated by a true story about a pediatrician who testified against a woman accused of killing two of her babies. The pediatrician mistakenly claimed that the chances of having two infants die from Sudden Infant Death Syndrome (SIDS) is 1 in 73,000,000. One of the many mistakes made by the pediatrician was that the probability of having two children die from SIDS is independent. The woman was convicted, and was not released until her second appeal.

See the video of Peter Donnelly’s talk below:

Who Is Kyle Burns?

Hi, my name is Kyle Burns. I am majoring in Mathematics and Economics. I chose to major in math because I have always been good at it and have enjoyed doing it for as long as I can remember.

So far at PLU, the most interesting class I have taken is Strategic Behavior (ECON 341). In Strategic Behavior I learned about game theory, which involved many different types of games as well as the strategies involved behind them. I also learned how mathematics is used to find mixed strategy equilibria for games without a pure strategy.

For my math capstone, I would like to do linear regression analysis to test the relationships between different variables in the health care industry. Initially, I am thinking of exploring the effectiveness of vaccinations, but I might have to change that based on the availability of relevant data.