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5
votes
2answers
343 views

Group actions in towers of Galois extensions

Assume we are given an extension of number fields or $\mathfrak{p}$-adic number fields $L/E/K$ where each extension is abelian and $L/K$ is only assumed Galois. Now take any element $\sigma\in ...
4
votes
3answers
1k views

Why does higher level mathematics more often than not use Greek lettering?

In high school, at least from what I've seen, mathematics courses never use Greek lettering in their description of concepts, with the notable exceptions of $\Sigma$ for summations, $\Delta$ for ...
5
votes
2answers
519 views

Programs for precocious prodigies

I am the director of my university's mathematics honors program, and we just had an inquiry from the parent of a 15 year old who has already completed most of the math courses for a standard ...
6
votes
2answers
655 views

Greens theorem: why does path orientation matter?

$$\oint_{\partial D} P\;dx + Q\;dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\;dA$$ Is it correct to interpret this equation as relating the surface area of ...
7
votes
2answers
260 views

Is this a bad way of approaching math problems?

Teacher assigns a problem. I work on it for thirty minutes, then check the textbook to see if it has similar problems. Often the textbook's problems have hints that I can use to solve the original ...
6
votes
2answers
5k views

Computational complexity of computing the determinant

The formula for the determinant of an $n$ by $ n$ matrix given by expansion of minors involves $n!$ terms. As such, computing the determinant of a given matrix of with integer entries via expansion by ...
36
votes
2answers
3k views

Development of the Idea of the Determinant

While I basically understand what a determinant is, I wonder how this idea was developed? What was the principal idea behind its origination? I would like to know this so that I can have a better ...
2
votes
1answer
162 views

The reason for different terminologies

Different authors seem to have different conventions when they define the term affine variety (similarly projective variety). For the purposes of this question let us stick with the affine case, and ...
43
votes
12answers
6k views

Chatting about mathematics (with real-time LaTeX rendering)

Do you know about some tools which can be used for online chat about mathematics? In particular, I am interested in software which would be able to render LaTeX formulas. (Since LaTeX is probably the ...
18
votes
2answers
1k views

Who are some forgotten mathematicians? [closed]

In Thomas' Calculus, he presents ''Nicole Oresme's Theorem'': $$ \sum_{n=1}^\infty {n\over 2^{n-1}}=4. $$ My first reaction was "who is this person?''. As it turns out, he was a Frenchman from the ...
3
votes
2answers
143 views

Advice for Calculus Tutoring

I am tutoring a friend in calculus. Right now, she is working on finding relative maxima and minima as well as Rolle's theorem. While she gets how to find relative maxima and minima she does not get ...
52
votes
7answers
2k views

Algebraic Intuition for Homological Algebra and Applications to More Elementary Algebra

I am taking a course next term in homological algebra (using Weibel's classic text) and am having a hard time seeing some of the big picture of the idea behind homological algebra. Now, this sort of ...
24
votes
2answers
922 views

Qual question archives?

Qual questions seem like a great way to study for a new topic, since they usually test slightly deeper understanding than typical questions in a textbook. Princeton has this great archive of questions ...
4
votes
2answers
2k views

What math is used in the theory of quantum computing?

I'd like to know what rung of the math ladder one need be on to grasp how a quantum computer computes. I realize this might not be a simple answer, so I'm just looking for an idea of the broad topics ...
5
votes
1answer
397 views

Parametrizing a conic in projective space

I am just beginning to learn algebraic geometry. An exercise in Reid, p. 24 is to prove that if $Q(x,y,z)$ is a quadratic form over a field $k$ with at least 4 elements, and $Q$ vanishes on the zero ...
2
votes
0answers
96 views

What can you do with rational solutions to linear equations?

I'm currently doing a project and for part of it I've been looking at rational solutions to linear eqautions in two vaiables. ie. ax+by=c. I'd like to add a bit about what we can use these types of ...
5
votes
3answers
3k views

Can the word “derive” be used to mean “take the derivative of”?

Back when I was in high school, the usage of the word "derive" to mean "take the derivative of" was really widespread. It always bothered me because I felt that the proper verb should be ...
7
votes
1answer
329 views

Finding 'verbally smallest' element of a finitely generated group

Let $G = ({\Large\ast}^n\mathbb{Z})/K$ be a group, and for each $g \in G$ define $l(g)$ as the smallest positive integer $m$ such that $g = g_1 \ldots g_m$, where each $g_i$ is a generator of $G$. Now ...
19
votes
6answers
4k views

What does Khan Academy have to offer? Depth? Rigor?

Khan Academy - http://www.khanacademy.org/ - is often cited as a great online resource for learning mathematics and other subjects. I have heard many good things about this website and was wondering ...
20
votes
4answers
1k views

A question about the definition of a neighborhood in topology

Let $X$ be a topological space, and $x \in X$ be a point. There are two prevalent conventions on how to define a neighborhood of $x$: 1) A neighborhood of $x$ is any open subset $W \subset X$ such ...
23
votes
10answers
2k views

Mathematics and Music

I have heard that, in recent years, many mathematicians as well as music theorists have applied different branches of mathematics to music. I would like to know about some books/resources relating to ...
13
votes
4answers
9k views

Difficulty level of Courant's book

I am currently studying Introduction to Calculus and Analysis by Richard Courant and Fritz John.I would like to compare Courant's book with Apostol's and Spivak's in terms of difficulty of the ...
1
vote
0answers
992 views

Working through Math 55 problem sets as self-study

I am not a professional mathematician, but have learnt Engineering Mathematics in college and worked through parts of maths textbooks myself. The latter include the first few chapters of include Real ...
5
votes
3answers
1k views

Why graph a function?

Please enlighten me as to how graphing a function helps. I can see a graph's utility with simple functions as they instantly give you value of dependent variable. But ignoring them and considering ...
17
votes
2answers
2k views

Category Theory vs. Universal Algebra - Any References?

After seeing the answer to the question Category theory, a branch of abstract algebra, I would like to ask Are there literature discussing the difference/indifference/comparison between category ...
16
votes
1answer
995 views

Proofs given in undergrad degree that need Continuum hypothesis?

Or alternative you need to assume CH is false. I know several proofs that use axiom of choice. Heine Borel theorem is the best example I can think off. Zorns lemma is heavily used in the non ...
9
votes
3answers
517 views

Different standards for writing down logical quantifiers in a formal way

What are standard ways to write mathematical expressions involving quantifiers in a (semi)formal way ? In different posts of mine concerning similar question I have encountered for a generic ...
2
votes
2answers
205 views

How formal or informal should math texts (written for different purposes) be?

When writing math articles (or just math text), do you write down mathematical expression in a formal way or describe it in words, e. g. "Let $X$ be a normed vector space. Then $X$ is called a ...
2
votes
1answer
99 views

What is the “conjugacy problem for differentiable maps”?

A couple of days ago our professor reviewed some of the exercises we had to do and one of them involved giving an example of a conjugacy class in a group. Someone gave an example that involved ...
4
votes
2answers
685 views

What is importance of the Bunyakovsky conjecture?

Bunuyakovsky conjecture states that: An irreducible polynomial $f(x)$ of degree two or higher with integer coefficients and property that $\gcd(f(1),f(2),......)=1$ generates for natural ...
10
votes
3answers
784 views

A Concrete Approach to Category Theory

Is there a way to learn Category Theory without learning so many concepts of which you have never seen examples?
4
votes
1answer
558 views

How many letters of recommendations is too many [closed]

I am planning to apply for post-doc positions soon. Most universities require three letters of recommendation and most of peers suggested that I should have at least three research letters and one ...
34
votes
6answers
1k views

Attitude towards exercises in mathematics

I'm doing self study on a couple of topics in mathematics, such as real analysis, abstract algebra, and linear algebra. From time to time, there are always a couple of exercises which I found too ...
3
votes
2answers
96 views

Pronunciation of $M(x)$ and $m(x)$

Suppose I use two functions and I denoted them by lowercase and uppercase letters $m(x)$ and $M(x)$. Of course, I have to distinguish them somehow. How do I read this? Is capital/uppercase $M$ of ...
6
votes
4answers
457 views

Intuitive results with non-intuitive proofs?

Are there mathematical theorems that sound trivial and are obviously true, but are really tough to rigorously show? For example I stumbled across this question. The author asks how to show that two ...
15
votes
3answers
707 views

Mathematical structures

Preamble: My previous education was focused either on classical analysis (which was given in quite old traditions, I guess) or on applied Mathematics. Since I was feeling lack of knowledge in 'modern' ...
8
votes
2answers
234 views

I feel the need to prove every result for myself

I am, at best, a novice mathematician. I started teaching myself the subject while writing my thesis in computer science. I find that I have a strong urge to prove every relationship or formula that I ...
8
votes
4answers
977 views

How To Reach The “Next Level” of Mathematics

I am a junior-high pre-algebra student. I feel that my class is holding me back, so I wanted to learn "higher-level math". So what should I learn now? What do you believe is a "next step"?
8
votes
2answers
622 views

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 ...
18
votes
4answers
1k views

“Best practice” innovative teaching in mathematics

Our department is currently revamping our first-year courses in mathematics, which are huge classes (about 500+ students) that are mostly students who will continue on to Engineering. The existing ...
2
votes
2answers
167 views

Weak Lower Bound in Apostol's “Number Theory”?

In Apostol's "Introduction to Analytic Number Theory" on page 7 he introduces Fermat numbers of the form $F_n = 2^{2^{n}} + 1$ where $n$ is a non-negative integer. He then states that The ...
4
votes
0answers
812 views

Topic appropriate for undergraduate research program

This may be not appropriate to ask this kind of question here. But please feel freely helping me. In our Maths faculty, there is not a program like Undergraduate Research Program(URP), however, the ...
2
votes
1answer
413 views

Calculating BMI (Body Mass Index)

If I'm given the following values: Weight = 15st 9lbs Height = 5.8 ft If we want to calculate the BMI, would it be: 33.30? ...
14
votes
1answer
471 views

Mathematicians who started out studying something else

I did not know much about modern mathematicians's biographies. One of my friend has just told me Edward Witten was a history student when he was undergradute. Are there any different modern ...
14
votes
10answers
7k views

What are some good iPhone/iPod Touch/iPad Apps for mathematicians? [closed]

There are lots of good apps for teaching mathematics to children but I would like to learn about apps for undergraduate/graduate/research levels. Helper questions Any algebra system (like ...
3
votes
3answers
975 views

Folland & Functional Analysis

I'm reading Folland's Real Analysis to learn some basic functional analysis. I read through his section Normed Vector Spaces and could make my way through most of the exercises I attempted. I am ...
7
votes
3answers
2k views

Experiences with Kumon

We have enrolled our 5 year old son in Kumon which is an after school math and reading enrichment program of Japanese origin. While he is learning lots of things (currently learning how to add i.e., ...
1
vote
1answer
209 views

Algebraic Topology in Simple Terms

I just wanted to clarify a few basic concepts in algebraic topology. Suppose one space is my room ($\text{Room} \ A$). Suppose the other space is another room in my house ($\text{Room} \ B$). So ...
6
votes
1answer
339 views

Concrete Categories Where Epis are Just Surjections

Before I begin let me provide some background to fix notation/make the post more readable to interested outsiders. In a category $\mathscr{C}$ we say that a morphism $X\xrightarrow{f}Y$ is an ...
19
votes
2answers
429 views

Mathematical suggestions for a long commute

I have quite a long commute on my way to my university (about 75 minutes) so I was wondering if there's a math-related podcast or something of the sort you'd recommend. Exercises are appreciated as ...