Articles of soft question

Filling the gap in knowledge of algebra

Recently, I realize that my inability to solve problems, sometimes, is because I have gaps in my knowldge of algebra. For example, I recently posted a question that asked why $\sqrt{(9x^2)}$ was not $3x$ which to me was fairly embarrassing because the answer was fairly logical and something that I had missed. Furthermore, I realize […]

Is $\tan\theta\cos\theta=\sin\theta$ an identity?

A friend of mine, who is a high school teacher, called me today and asked the question above in the title. In an abstract setting, this boils down to asking whether an expression like “$f=g$” is regarded as an “identity” when one of their domains is a proper subset of the other, and the two […]

Ultimate GRE Prep

I’m planning on taking the math GRE Subject Exam in April (~11 months from today). I want to start preparing now in the hopes of scoring in the 95+ percentile. I have already taken a number of graduate courses and really need to refresh on some of the lower level stuff so here’s my plan: […]

Is “generalized” singular homology/cohomology a thing? If not, why not?

From what I understand, the singular homology groups of a topological space are defined like so: Topological Particulars. There’s a covariant functor $F : \mathbb{\Delta} \rightarrow \mathbf{Top}$ that assigns to each natural number $n$ the corresponding $n$-simplex. This yields a functor $$\mathbf{Top}(F-,-) : \Delta^{op} \times \mathbf{Top} \rightarrow \mathbf{Set}.$$ Hence to each topological space $X$, we […]

Cyrillic alphabet in math

There are a lot of variables and constants in a paper that I am writing. Is any thing wrong to use Cyrillic letters for constants and Latin and Greek letters for variables? I wonder why the letters do not be used in other papers when they need extra symbols. (Image is provided by @Ryan) $2ж+цc$ […]

Suggestions for further topics in Commutative Algebra

I am currently taking a semester long course in Commutative Algebra. We have covered a lot of dimension theory, and today finished proving Zariski’s Main Theorem, which was the professor’s original goal. However, he designed the course in such a way that a lot of basic topics have been omitted or very briefly touched upon. […]

In algebraic geometry, why do we use $\mathbb C$ instead of the algebraic closure of $\mathbb Q$?

In algebraic geometry, why do we use $\mathbb C$ instead of the algebraic closure of $\mathbb Q$? What properties of algebraic varieties use the topological completeness of our field? I’d be interested in hearing either general perspectives or specific results that might fail for $\bar{\mathbb Q}$.

What's the idea of an action of a group?

I know the formal definition of an action over a set. I’m not asking this. What I’m asking is: what’s the intuition of it? It is a way to define an algebra over a set? Since an action can exist in the most arbitrary of sets, does this mean that every set can be assumed […]

Who is a Math Historian?

In the context of classes, it is very often that discussion on the history of mathematics arises, whether it’d be on who should a lemma be attributed to or a certain event that occurred during the discovery of the proof (the elementary proof of the prime number theorem is one such example). My question is: […]

What is the deepest / most interesting known connection between Trigonometry and Statistics?

I’m teaching both at the same time to different classes in high school, so I just wondered about this. Added by OP on 16.May.2011 (Beijing time) I mean Statistics only, without Probability. In other words, Descriptive Statistics only. This rules out Buffon’s Needle Problem. The occurrence of π is counted only as a connection to […]