Some end of term work

Just about one week after my exams, I went to work on tidying up course notes and making sure that errata such as final exam study notes were polished enough to put on the blog. They’re now ready in the Course Notes section of this blog along with a bonus set of notes based on the multivariable calculus video series here.

Taught by Prof. Auroux, I decided to focus on only the vector calculus component of the lectures as the other components are well covered by my MATH 247 notes (also found on the same blog page). Specifically these notes cover the following:

In addition to these notes, I also completed one of my little side projects which is a basic proof on the irrationality of Euler’s Number, e. Check it out in the Resources section.

Over the next couple months or so, I will also be preparing to write the Putnam exam in December, so stay tuned for future interesting problems.

For undergraduates who are taking the summer off, I hope you all have a nice vacation and a good rest for the upcoming fall term.

For co-op students like me who are back to work next week, stay diligent and strong. Your efforts will appreciate someday.

Which field(s) of mathematics are you interested in?

EDIT 1: There really should be a +1 to the CO section. I just forgot to add it, when I first made the poll. Sorry!

EDIT 2: +1 to Topology as well. Please let me know if you guys have any more areas of mathematics that I’m missing!

High school life, research topics, and advice for the young aspiring mathematician. (Part 3 END)

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This will be the final post in my three part saga about mathematics and high school. You can check out parts one and two here and here respectively if you haven’t already read them. In this post I want to end off by offering the final part that was mentioned in the title, research topics (I have decided to write about advice for the upcoming mathematics undergraduate in a different series some time far in the future).

If you have been reading any of my previous posts, you might remember me talking about how a critical part of learning mathematics is knowing how to explore, be creative, and think about problems outside of the classroom. It is better to sow your tiny seeds interests early than later, because as many of you find out from learning maths, many of the areas  interconnect beautifully if you just take a while at the subtleties.

Thus, below, I have provided some topics of interest that can be researched at the high school level but delve deeply into undergraduate topics; all they need is a bit of perseverance, time, and creativity to get started. You can probably even get your teacher in on this. Ask him/her if instead of completing droll textbook problems or monotonous exercises, you could start a research topic instead! No more spending one hour every day copying down exactly what the book asks you to do or sketching and drawing precise margins (at the university level, precisely drawn graphs is one of the lesser important aspects of mathematics). Instead, mathematics turns out to be a continuous process that you can do every day, without the stress.

... unless of course, your teacher wants you know the critical skills of copying and pasting. Then, it is advisable to keep at those 'exercises'. {MS Clipart}

So without further ado, here they are (with difficulty rankings relative to high school content):

Population Dynamics

Difficulty: 7/10
Mathematics Topics: Differential Equations, Chaos Theory
About: Students can examine the different  population models such as Malthusian, logistic, and island biogeography. Investigations can be about the limitations of each model, the strengths and the ideal conditions for the model to work.
Bonus: Students can talk about how current day population models are more similar to chaotic systems than deterministic ones

Poker, Probability, and Game Theory

Difficulty: 6/10
Mathematics Topics: Game Theory, Probability, Statistics, Simulations
About: Students can investigate the best course of action to take with various poker hands, categorized by winning strategy. Approaches can be from a game theoretical, probabilistic or statistics side or all three.
Bonus: Create a computer simulation to accept or deny your theory through a null hypothesis and Monte Carlo simulations.

Rings, Fields, Groups, Vector Spaces, and Algebras

Difficulty: 9.5/10
Mathematics Topics:
Abstract Algebra, Linear Algebra
About: Students can discuss the differences and similarities between the above algebraic objects and how to construct them. Investigations can be made about various isomorphisms between algebraic objects and about interesting objects such as the power set of the real numbers being an algebra under the union and intersection operators.
Bonus: Construct your own simple algebraic object (maybe the set of all watermelons that taste like oranges; something silly) and see if you can go from a magma all the way to an algebra.

Fractals and Iteration

Difficulty: 9/10
Mathematics topics: Fractals, Chaos Theory
About: Students could take a look at the definition of a fractal and what exactly is fractal dimension. Investigations can be made about the construction of fractals under various domains, their applications (e.g. coastlines), and their self-similarity properties. Students could try to generate fractals through freeware found online.
Bonus: Students could investigate what higher dimensional fractals would look like if projected on to the 3D or 2D plane. An example of a 3D projection into the 2D plane would be a contour map.

Methods for Computing Mathematical Constants

Difficulty: 8/10
Mathematics Topics: Computer Science, Simulations, Calculus
About: Students can investigate the different methods for computing mathematical constants such as $\pi$, $e$, and $\phi$ the golden ratio. Some investigation could be made about the complexity of the algorithms and how fast they converge to the constants.
Bonus: Come up with your own unique constant and an algorithm used to compute it. Talk about complexity and rate of convergence as well as if it can be represented by current day mathematical constants

Cardinality of Various Sets

Difficulty: 8.5/10
Mathematics Topics: Set Theory, Ordinals
About: Students can investigate the various cardinalities contained in sets such as the natural numbers, integers, irrational numbers, rational numbers, real numbers, and algebraic numbers. Students could also define their own set and try to determine the cardinality of that set.
Bonus: Students can devise a model for classifying the cardinalities of different sets based  on certain characteristics contained in each set.

If you have read Part 2, you might recall that I did write a research paper on a topic of my choice. From the above topics, my essay was actually about the last one, cardinalities, which you can access a copy, here. Some time in the future, I will be restoring that essay to a cleaner LaTeX typeset document. You may use it as a guide, but I highly encourage that readers take the time to learn about the above topics themselves. After all, to quote Georg Cantor:

"The essence of mathematics is in its freedom." {Image via Wikipedia}

I hope you all enjoyed my little mini-series rant about my high school experiences and if you did, I would love to hear about your experiences in the comments below. In the upcoming months, stay posted for even more content as I begin to dive straight into some serious mathematics this term and start to bring you my quality typsetted notes.

Fish and squares.

Currently I am working on two different problems, both of which I have added to the Current and Future projects page.

To summarize, these are…

1. A Fish Population Problem started by Gaurav Tiwari and his blog “My Digital Notebook”. A link to the post can be found here. A summary of the problem is as follows:

Let $F_{m}(t)$ and $F_{f}(t)$ denote the number of male and female fish, respectively, inside a tank at time t (where t is in a unit of days) and let $F(t)=F_{m}(t)+F_{f}(t)$ denote the total number of fish at time t. Initially we have $F_{m}(0)=4$ and $F_{f}(0)=3$.

Here are the conditions of the problem: A male fish can mate with any number of female fish, unless the female has already mated before. Female fish will always give birth to new offspring after 21 days of conception, with the first female always giving birth to 100 eggs, the second fish, 110 eggs, and the third, 90 eggs.

The new offspring fish will reach sexually maturity after 30 days and in each new batch, 60% of the offspring fish will die before reaching maturity. In the remaining 40%, the ratio of male to female fish is 3:2.

This second generation of fish have the following restrictions. Fish born of the same mother may note mate with each other and as with the first generation, a male fish can mate with any number of female fish, unless the female has already mated before. Every three females who give birth will always lay the eggs in the following order (from first fish to last): 100, 110, and 90. In these batches of eggs, only 40% will survive, with the new ratio of male to female being 1:1. In subsequent generations, the same restrictions found in the second generation of offspring will also apply.

Find the value of $E\left[F(365)\right]$, that is, the expected total population of fish found after one year has passed.”

2. A geometry problem that was started by a friend of a friend taking a Calculus III course at the University of  McMaster. Currently I am trying to find:

…how to calculate the volume of the solid displaced when a square plate of side length 2 is symmetrically passed through a sphere or radius 2 (the center of the square is threaded through a diameter line in the sphere).

as well as trying to attempt to solve the integral that represents said volume.

Today I have made considerable progress in both problems (for 1, see here, and for 2, see  here), but now I am at an impasse. This is why I would like to ask you, the readers, to give me some inspiration and ideas here but helping me answer the following questions:

1. For project 1, what suitable restrictions should be made so that the population can naturally be imbued with a carrying capacity?

2. For project 2, what would be a simple and intuitive way to integrate the $\arcsin$ portion of the integral.

3. For project 2, is there any way to re-write the integral to make the integration simpler?