Undergraduate Schools for Mathematics
I am currently a senior in high school, and I have been studying mathematics for about nine or ten years now in my personal time outside of school. I am not familiar with academia or in general higher...
View ArticleWhy has convexity established itself as the decisive property to assess the...
When I read somewhere about the importance of convexity for optimization most of the time it deals with the nice property of convex functions that local and global minima are the same.This is a very...
View ArticleWhy do characters span the space of class functions?
Let $G$ be a finite group and let $\chi_1, \dotsc, \chi_t$ be a complete collection of irreducible complex characters of $G$ with corresponding representations $V_1, \dotsc, V_t$ (i.e. every...
View ArticleTeaching material for a one hour refresher on probability theory (conditional...
I have been teaching a course on stochastic processes to master students in theoretical physics, and half of my students seem to have never taken a probability course and struggle with some basic...
View Articlewarping functions obey diff eq. does that imply $g_t$ obeys same diff eq?
Consider $(M,g_{t})$ equipped with a $1$-parameter family of warped metrics for real parameter $t>0$$$g_{t} = \frac{1}{\phi_t(u)^{2}}\ du^{2} + \phi_t(u)\ dv^{2}$$and suppose that the warping...
View ArticleUnderstanding the Relationship Between Measure Theory and Probability Theory?
I was at a restaurant with some of my friends who study pure math. While at the restaurant, they were having a discussion about the relationship between Measure Theory and Probability Theory. I tried...
View ArticleFormalizing the implication $dx/dt = -x \land dy/dt = y \implies dx/x = -dy/y$
As part of a solution to the PDE$$-xu_x+yu_y = u + 2x,$$through the method of characteristics, the author writes$$\frac{dx}{x} + \frac{dy}{y} = 0.$$Now, I know where such an equation comes...
View Articleaesthetics of finding solutions and asking questions [closed]
I am a first year mathematics undergraduate who would like to develop his taste for what elegant proofs and questions are. What are some textbooks or texts or blogs you would recommend? Field agnostic.
View ArticleDoes wheel theory help give meaning to some division of polynomials?
I apologise if this question is a little too vague (e.g., "meaningful"). I have included the soft-question tag for good measure.Motivation:This is motivated by one of those annoying memes on social...
View ArticleIs there is way to determine if the n-th roots of a polynomial is a polynomial?
I was this problem: $$\int\frac{dx}{\sqrt{x^4+2x^3+3x^2+2x+1}}$$I solved this question because I just knew that $(1+x+x^2)^2=x^4+2x^3+3x^2+2x+1$ but this made me wonder is there is a way to know if the...
View ArticleOn formalising logic
Recently, I became fascinated with Set Theory and I am willing to learn more. Although, there are some aspects that I would like to understand before doing it. A lot of questions concerning the...
View ArticleDo all mathematical ideas eventually find their way into the real world?
I am writing an essay for a scholarship and they asked the following question: "How will these goals enable you to help others?"Although I am majoring in pure math, this really got me to think about...
View ArticleAre real numbers defined as Banach spaces?
I was listening to this lecture by Norman Wildberger, where he explains that real numbers are constructed as equivalence classes of Cauchy sequences. He never says that these equivalence classes are...
View ArticlePast open problems with sudden and easy-to-understand solutions
What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved...
View ArticleFirst job in industry for a pure, pure mathematics folk
this is an "almost" Ph.D. in pure maths who's got a few reasons to look for a job in the industry.I know that many choices are out there for a math guy: finance, consulting, industrial research, big...
View ArticleWhy not define absolute value by...
I know that there are some discussion over the definition over absolute value in stack-exchange. However, I did not satisfied. It is generally given that $$\left| y \right| = \begin{cases}\phantom{-}y,...
View ArticleWhy do we put so much attention into curves and surfaces?
I am curious as to why in algebraic and differential geometry so much special attention is given to curves and surfaces in contras to higher dimensional manifolds or varieties. I would like to know if...
View ArticleDissertation on Integrals
I'm considering doing a dissertation on Integrals: Riemann, Henstock-Kurzweil, Lebesgue and moreI'm wondering if I can do it. This is usually a masters level dissertation (while I'm an undergraduate...
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What's new in higher dimensions?
This is a very speculative/soft question; please keep this in mind when reading it. Here "higher" means "greater than 3".What I am wondering about is what new geometrical phenomena are there in higher...
View ArticleIs it weird that my probability theory lecturer thinks that $P(E)=0$ implies...
I'm an undergraduate student in pure mathematics, and I'm taking a probability theory course based on the book A First Course in Probability by Sheldon Ross. My lecturer defined the conditional...
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