I spent most of today finishing the revisions on my proof that Dr. Deters suggested. Before I can begin proving that f is the inverse of g, I have to prove that f and g are well defined at all points in their domain (which is a vector space for each function). The most challenging part of proving that both functions were well-defined was proving that g (which is a rational polynomial function) never involves dividing by zero on its domain T. To prove this, I had to go back to the precise definition of T as established earlier in the proof and demonstrate that based on this definition, the denominator could never equal zero. Proving this took a bit of creativity, but I wrote out a good first draft of this part of the proof.
I also had to show that the range of f is a subset of T (T is the domain of g) and that the range of g is a subset of H (H is the domain of f). To show this, I again had to go back to every element of the precise axiomatic definitions of H and T. Using these definitions, I showed that any output of f and g would inevitably be a vector in T and H, respectively. Showing this required a bit more experimentation. I'm discovering that when I'm not sure how to proceed with a proof, the best course of action is often to turn to a new notebook page and begin to play around with the numbers a bit. Experimentally manipulating equations tends to give me new ideas for things to try, which eventually leads me to successful chains of reasoning. This process of experimentally manipulating equations also gives me plenty to think about (both consciously and subconsciously) when I'm away from my notebook, and I sometimes make excitingly fast progress after returning from a break. Finally, I met with Dr. Deters near the end of the day to show him the new additions I had made to the proof in LaTeX. To make the proof more concise, he suggested that I combine two of my claims into one weaker–but still sufficiently strong–claim. This way, I would neither waste energy proving unnecessary things nor claim I was proving things I wasn't. I also showed Dr. Deters the new section of the proof in which I proved that (f(g(x_1,...,x_{n+1})) = (x_1,...,x_{n+1}). He said he thought that section was quite strong, though I could make it more concise by doing as many steps as possible in the same line of equations, rather than unnecessarily creating new variables. Tomorrow, I'll focus on typing the rest of my revisions into LaTeX so that I can present the entire revised (and now much longer) proof to Dr. Deters on Monday.
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My meeting with Dr. Deters today was very productive! I have several things to adjust, but I'm off to a solid start. Dr. Deters approved the most important step of my proof, in which I prove that both sides of an essential equation are equivalent. Although we both agree that I could make this step more concise, he agrees that my mathematical reasoning is sound. Thus, I can freely use this equation anywhere else in the proof.
The main area in which I need to make my proof more rigorous is in the final step, in which I prove that one function g is the inverse of another function f. To prove this, I have to show three things: that f(g(x1, ..., xn))=(x1, ..., xn), g(f(x1, ..., xn))=(x1, ..., xn), and that g and f are well-defined at all points on their domains. To show that these functions are well-defined, I need to show that they don't break any rules of mathematics on their domains (such as dividing by zero). I also need to show that any point in the range of f is in the domain of g, and any point in the range of g is in the domain of f. Finally, I need to show that f and g are continuous at all points on their domain. I spent today beginning my revisions. I especially enjoyed revising this proof because it required me to apply concepts from each of my math classes, such as vector spaces from linear algebra and continuity from calculus. I focused today on proving that f(g(x1, ..., xn))=(x1, ..., xn), and g(f(x1, ..., xn))=(x1, ..., xn). This was a more involved process than proving that two single-variable functions are inverse of each other. To prove this, I had to move element by element. I started with the first element of the set, then moved onto all the middle elements of the set, and finally moved onto the last element of the set. To complete this section of the proof, I had to make creative use of the definition of the customized vector space onto which these functions map. I also had to make use of the equation I proved yesterday. I tremendously enjoyed seeing my work from yesterday provide the foundation for today's work! Tomorrow, I intend to finish my revisions and type them into a LaTeX document. I spent today typing into LaTeX the first draft of my proof! I've used LaTeX before for Linear Algebra, but I learned several new functions of LaTeX today, including how to number equations and how to type summation notation. I was surprised at how long it took me to convert what I had scribbled into my notebook into a LaTeX document that anyone could make sense of. It's not that the process of typing up the proof in LaTeX significantly slowed me down, but rather that the proof I had drafted in my notebook wasn't quite thorough enough to present to Dr. Deters. As I prepared the LaTeX document to present to him, I had to make sure that I had properly proven the concept for all relevant cases.
For instance, in my notebook, part of my proof only dealt with the scenario when a certain set is of length greater than 1. When the ideas in this proof are applied in real life, the length of the set in question will nearly always be greater than one, and it will often be hundreds or thousands of values long. However, the process of proving that my statements are true is slightly different if the set is of length 1, and this is a detail that I had to take into account as I was typing my proof into LaTeX. In addition, typing the proof into LaTeX also made it easier for me to evaluate my own work. As I was looking over my proof, I noticed a logical leap in my proof that made sense to me, but wasn't justified. After thinking about it more, I realized I had skipped and important step, and I spent the second half of the day adding this step into the proof. This experience illustrated to me the importance of critically looking over my own work, trying as hard as I can to find flaws that I can correct. I'm meeting with Dr. Deters to show him the draft of my proof tomorrow morning, and I look forward to hearing his feedback! Over the weekend, I met with my sponsor–Dr. Ian Deters–and we discussed the first problem I'd work on. This is a probability problem, and I'll contribute to solving it by helping develop a tool for determining whether a set of samples deviates significantly from the expected values for this set of samples.
Suppose we categorize each member of a sample with two variables. Possible variables by which to classify individuals include race, profession, or gender. To represent the entire sample, we could create a probability table with the x-axis representing each value of variable 1 (race, for instance) and the y-axis representing each value of variable 2 (profession, for instance). Each cell would contain the share of the sample that could be classified by the value in the corresponding row and column, and each cell would contain a value between 0 and 1. The total number of cells in this table equals the product of the number of values each variable can take. (If there are 5 possible values for variable 1 and 10 possible values for variable 2, then the total number of cells in this table equals 50). If information is available about the population from which the sample is taken, then each cell has an expected value. Some cells may deviate statistically significantly from the expected value, such that the probability that such deviation occurred due to random chance is less than 5%. However, with a table as large as 50 cells, the probability that at least one cell significantly deviates from the expected value is 92.3%, and this probability increases as the size of the table increases. It is important, then, to determine the number of cells that would be expected to statistically significantly deviate from their expected value for tables of various sizes. Knowing this value allows us to determine whether the table as a whole deviates in a statistically significant way from what would be expected. Today, I found the inverse of a complex function that deterministically generates probability tables. In addition, I completed a draft of a proof that the original function is a homeomorphism of the input and output vector spaces. I tremendously enjoyed the experimental process of devising a proof; this is the first time I've ever generated a proof for a real-life project! I have the proof completed in my notebook, and my task for tomorrow will be typing the proof into LaTeX and sending it to Dr. Deters for feedback. |
Jeremy MahoneyAspiring data scientist, Impromptu teacher, Lover of learning, and admirer of the universe ArchivesCategories |