In my last blog, I talked about my plan for the semester, why I want to study abstract algebra, and some basic concepts involving mapping. In this blog, I will first provide some update on Modeling the Future, a team math competition we have been working on, and how I redefine addition in group theory. I will also talk about my plan for the science fair.

Modeling the Future is a competition that requires us to pick a potential cure for a pervasive disease. Then, we need to analyze the impact of the cure on the health insurance industry. After some discussion, we picked gene therapy for Alzheimer’s Disease (AD).

As the model builder of the team, I decided to find some of the variables, they are:

C: Cost of the Therapy, C>0;

P: Price of the insurance, P>0;

K: Percentage of the cost of the therapy covered by the insurance, 0<k<1. Since gene therapies are expensive, insurance companies are not likely to covered all the costs.

Q: Quantity of insurance sold. Or, the number of people purchase the insurance. We also assume Quantity equals demand as insurance companies sell as many insurance as demanded. There is no shortage. Q>0;

R: Risk of having Alzheimer’s Disease. 0<R<1;

A: Affordability. Given that the person purchases the insurance, has the disease, and the insurance covers only part of the expensive therapy, A indicates the percentage of customers who are willing to cover the uncovered expenses. 0<A<1;

I will not talk about our model in the blog for this time as we are still in the process of finalizing it. The idea is, we need to build a supply curve of the insurance based on all these variables and some data we find, then we need to find a proper demand curve from the internet. Finally, we will use the demand-supply model from economics to decide the quantity and price of the insurance so that the company will make the most profit.

After something about MTF, let’s see what I have done to redefine addition.

Addition is one of the basic operations we use every day. In algebraic numbers, addition makes perfect sense, as we all know 1+1=2. However, the definition in Abstract Algebra or group theory is often unclear. In this more inclusive redefinition of addition, the idea applies to any set, group, or algebraic number.

If you are curious about why we need to prove all these properties, please visit here. The reason in short is we need to make sure our new definition doesn’t contradict with the conventional one. This source is particular helpful for me since it provides both the English and Chinese.

So here are all I did in the past two weeks. Since Word Press does not support some of the math notations, so I decided to write them down in a Word Document and paste them here. In the following blog, I will provide another update for Modeling the Future and talk about how I plan to prove Lagrange’s Theorem and Fermat’s Little Theorem. As you may notice in my redefinition, some concepts in math really get abstract. However, this is the beauty of mathematics, as it is only through these abstract ideas that our knowledge system becomes more and more perfect.

References

Calder, K. (n.d.). *Addition* [Image]. Retrieved from https://www.tes.com/lessons/ZeGyGhKptzKMpA/addition

*Group-like structures* [Illustration]. (n.d.). Retrieved from http://sparkandshine.net/en/algebraic-structure-primer-group-ring-field-vector-space/

Herstein, I. (1990). *Abstract
algebra* (2nd ed.). New York: Macmillan.

moneysense.ca. (2018, November 14). *New Crypto Exchange Security Scoring Model Provides Insurance Rates for Coin Owners* [Image]. Retrieved from https://www.coinwire.com/new-crypto-exchange-security-scoring-model-provides-insurance-rates-for-coin-owners

“代数结构入门：群、环、域、向量空间” [“An Introduction to the Algebra Structure: Group, Loop, Field, and Vector Space”]. *Spark & Shine*, 24 Feb. 2015, sparkandshine.net/algebraic-structure-primer-group-ring-field-vector-space/. Accessed 27 Jan. 2019.