Tag Archives: math

R (and my plan) – Baiting

Image result for r coding language


At the very beginning, I want to conclude my project in Abstract Algebra. As the second round of competition (Delaware Valley Science Fair) ended on April 4th, my work with Abstract Algebra has temporarily ended as well. In the second round, I won 3rd Place in 12th Grade Mathematics. In speaking of the result, it was worse than last year when I won 2nd place. However, I clearly understood bringing a theoretically based project to a science fair was risky. So, I feel lucky that I have done what I always wanted to do, no matter what recognition I won.

Following the week after the science fair, I started to work on R, a type of computer coding language. I choose R instead of other languages because it is extensively used in statistics. Since I will most likely major in Mathematics/Statistics in college, R is one of the most helpful coding languages to learn.

First, I want to address a question asked by T. Margaret for my last blog. T. Margaret asked about my final target of the semester and what I would do as a demonstration of learning. To be honest, I am not completely confident to bring an impressive project for several reasons. First, I spent my first half of the semester on Abstract Algebra so there are no more than 2 months left for R. Second, I had no foundation on any computer programming before. Since R is my first coding language, I have to start from the most basic knowledge. And finally, there are no obvious relationships between Abstract Algebra and R. This means it will be hard or not very meaningful to combine R with Abstract Algebra. As a result, my current learning plan is to go through the most basic algorithms in R as quickly as possible. Then, I will continue my work from last year on Alzheimer’s disease.

Last year, I studied Alzheimer’s disease from a data set on Kaggle. I used models including Odds Ratio, Logistic Regression, and ROC Curves to analyze which type of people are more likely to have Alzheimer’s disease. As I handed my paper to a Statistics Doctor, she advised me to think about other models. She said Logistic Regression is a popular model in public health. However, since different subjects in my data set had a different number of testings, there was no guarantee that each data is independent of each other. For instance, Subject #1 may have 5 testings while Subject #2 has only 3, then we cannot treat each of these 8 testings as independent data point. As a result, Logistic Regression wouldn’t provide the most accurate conclusions.

My final target for the semester would be researching on a new statistic model and run it through R codes on the same data set. Even though there may not be enough time for me to write a whole report, I will bring interesting conclusions. As my demonstration of learning, I am expecting to talk about my model, my codes, and my conclusions from both years. In specific, I am interested in comparing the similarities and differences between my conclusions. As of a long term plan, I am curious about why these differences exist and if there are ways we can identify the scales of these differences.

In addition to my plans, I really want to talk about some codes I learned. However, since this is not a tutorial, talking about each function and code in R would not be helpful. In general, I learned how to set variables, perform basic algorithms, identify data type, create vectors/matrix, and draw data plots or basic functions. I have also self-learned something about “if”. Once I have a better understanding of these codes, I will certainly share my experience with you! For now, I want to introduce the software and R in general.

R (for windows): The most fundamental structure and logic of R are here. If you are using Mac or Linux, then simply google R for Mac or R for Linux. It is free.

Some advantages of R are:

1, It is free.

2, It is open source so you can install any packages easily.

3, R is easy to install and is only 50 MB.

4, R is overall easier to learn.

R Studio: This is another coding platform for R. You can’t run R Studio without having R in the first place. It goes through the same logic and process as R, however, it is neater and easier. For instance, R Studio allows you to edit your codes while R doesn’t have this function. I feel R Studio is extremely important for beginners. The most basic version is free, and is well enough for users like me!

Works Cited

Boysen, Jacob. “MRI and Alzheimers.” kaggle, www.kaggle.com/jboysen/mri-and-alzheimers. Accessed 15 Apr. 2019.

“Logistic Regression.” Carnegie Mellon University Department of Statistics, www.stat.cmu.edu/~cshalizi/uADA/12/lectures/ch12.pdf. Accessed 15 Apr. 2019.

“Plotting and Intrepretating an ROC Curve.” The Darwin Web Server, gim.unmc.edu/dxtests/roc2.htm. Accessed 15 Apr. 2019.

R Programming. Coursera, www.coursera.org/learn/r-programming. Accessed 15 Apr. 2019.

R Studio. R Studio, www.rstudio.com/. Accessed 15 Apr. 2019.

RSA Algorithm – Baiting

Image result for rsa encryption


In this blog, I would like to provide updates on the M3 Challenge and my research project in Abstract Algebra for Chester County Science Fair. Then, I will introduce the RSA Algorithm, the most used information encryption algorithm in the world and how it relates to my project.

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Last Two Weeks – Baiting

Image result for math modeling

Since my last blog, I have been working on three different projects: Modeling the Future, Science Fair (Abstract Algebra), and M3 Challenge, another mathematical modeling competition. In this blog, I will talk about all three competitions I have been working on. For Abstract Algebra, I will also use part of my paper to talk about Lagrange’s Theorem. Continue reading

Abstract Algebra: A NEW Start – Baiting

Retrieved from https://www.booktopia.com.au/abstract-algebra-gerhard-rosenberger/prod9783110250084.html

In this blog, I would like to briefly review my Fall semester and share my plan for the Spring.

In the past four months, I worked on Differential Equations and its related topics. Through following the MIT Open Course Ware, I learned different methods to solve DEs and their implications in real life.

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The Fourier Series: An Introduction – Baiting

In the past few weeks and Thanksgiving break, I finished Unit II and moved on to Unit III. In this unit, I will primarily focus on the Fourier Series, Laplace Transformation, their connections with Differential Equations, and their applications in reality. In this blog, I will introduce the Fourier Series of periodic functions, including the trigonometry functions, the sin and cos. Continue reading

The Spring Model Continued – Baiting


In the past two weeks, I worked on Second Order Differential Equations with constant coefficients and learned more about the Spring-Damper Model. In this blog, I will provide a brief recap of the basic knowledge, and then provide further analysis of the same model. If you are interested in my last blog, please visit here. Continue reading

Euler’s Formula – Baiting

As mentioned in my last post, I continued to develop my algebraic skills for solving differential equations during the past two weeks. In this blog, I want to introduce Euler’s Formula, the most impressive piece of math work I have ever seen. In specific, I will dive deep into the mathematical proof of this formula and explain its broad application.

Euler’s Formula consists of a simple line:


In this equation, e represents the irrational constant 2.71828…, or more specifically WeChat Screenshot_20181009230402 , and i represents WeChat Screenshot_20181009230659.png. With an irrational number and an imaginary number involved, Euler’s Formula is not easy to visualize in the regular coordinate system. In order to make this blog more comprehensive to everyone, I will offer an algebraic proof, so that we don’t have to learn the complex Cartesian and Polar coordinate system from the beginning.

To prove Euler’s Formula, we have to prove it satisfies these three fundamental properties:

  1. WeChat Screenshot_20181007153513.png(Law of exponential)
  2. WeChat Screenshot_20181007153856 (Law and definition of constant e in calculus)
  3. Taylor’s series of the left should equal to Taylor’s series on the right as the number of terms approaches infinite.

So let’s start from the first one! Remember we are now assuming Euler’s Formula is correct and testing if it [actually] follows all of the properties.

WeChat Screenshot_20181007160416.png

After proving the Euler’s Formula follows exponential identity, we can move on to proving the identity of e.

The concept of constant e was first brought up by Euler himself,  and is sometimes referred to as “Euler’s Number.” The core definition of e is the rate of change always equal to its self. If you want to know more about this mysterious number, please visit here. Due to its ingenious definition, e is used in multiple areas including charging/discharging a capacitor or calculating quantities related to half-life.

Now here is a simple and straightforward proof for Part II.

WeChat Screenshot_20181007161023

In the last part, we are proving the polynomials of e^iθ equals the polynomials of cos(x)+i*sin(x) from Taylor’s series. If you remember some relevant knowledge from Calculus II, this would also be straightforward.

Now, we have proved the Euler’s Formula. And I will briefly and broadly cover its applications. If you are interested in knowing more about any of these topics, please tell me and I will discuss them in the future!

Trigonometry, Fourier transformation, Taylor’s series, …. the span of the impact of Euler’s Formula goes on and on. With Euler’s Formula, we can easily model rotations in complex coordinate system or even explain the spiral movements of the starts; with Euler’s Formula, the trigonometry relationships become easier than ever; and most importantly, with Euler’s Formula, we can, for the first time in history, cross the wall between transcendental numbers and algebraic numbers.

Euler’s Formula is a piece of math theorem, but also a piece of art. Like Mona Lisa’s Smile by Leonardo da Vinci, Euler’s Formula contains much more than it seems. When rewritten as e^(iπ)+1=0, the simple line involves plus/minus, multiplication/division, exponential, trigonometry, complex number, the concept of zero, and transcendental numbers. Even within itself, there is a sea of knowledge waiting for me to explore.

To me, Euler’s Formula is just a beginning. On my journey of learning, more fascinating math topics will be closely studied and researched. What I will not forget, however, is the excitement after seeing this beautiful math work. Like a line of a poem that eulogizes our life, Euler’s Formula thoroughly depicts the beauty of math.




Fourier Transforms [Illustration]. (2010, Fall). Retrieved from https://slideplayer.com/slide/10994153/

Arthur Mattuck, Haynes Miller, Jeremy Orloff, and John Lewis. 18.03SC Differential Equations. Fall 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.



Leonhard Euler [Photograph]. (n.d.). Retrieved from https://www.usna.edu/Users/math/meh/euler.html

O’Connor, J. J., & Robertson, E. F. (2001, September). The number e. Retrieved October 8, 2018, from MacTutor History of Mathematics archive website: http://www-history.mcs.st-and.ac.uk/HistTopics/e.html

Bank Account Model – Baiting

In the past two weeks, I continued my learning on First Order Linear Differential Equations. In this blog, I will focus on the Bank Account Model. If you find this blog is interesting and would like to learn more from my primary source, please go here.

Bank Account Model:

When you are putting money into your bank account, what would you care about the most? While I guess the answer for me and many will be the interest. So in this model, we will look at the logarithm behind the (theoretical) interest system. Continue reading

Stopping for reflection–Summer

dice-sketch-game-theory-thoughtful-businesswoman-standing-against-concrete-wall-connected-probability-73103748As the holiday season approaches, my independent study on Game Theory is also coming to an end. In my last post, I would like to take the opportunity to reflect on my work with Game Theory this past semester, some lessons I’ve learnt, and my plan for the coming weeks. Continue reading