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 […]

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 […]

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: […]

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 […]

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) $2ж+цc$ […]

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 $\mathbb Q$? 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 $\bar{\mathbb Q}$.

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 […]

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: […]

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 […]

Intereting Posts

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If $f'$ is differentiable at $a$ then $f'$ is continuous at $(a-\delta,a+\delta)$
How to solve $4\sin \theta +3\cos \theta = 5$
Non-orthogonal projections summing to 1 in infinite-dimensional space
The cross product of two sets
Classification of operators
Infinite product equality $\prod_{n=1}^{\infty} \left(1-x^n+x^{2n}\right) = \prod_{n=1}^{\infty} \frac1{1+x^{2n-1}+x^{4n-2}}$
Kindle as a Tool for Mathematicians?
About the great result of $\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_m~dx$
Is $\frac{1}{11}+\frac{1}{111}+\frac{1}{1111}+\cdots$ an irrational number?
Does there exist a unital ring whose underlying abelian group is $\mathbb{Q}^*$?