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3
votes
2answers
358 views

Collatz conjecture - example on the opposite situation

In the "Collatz conjecture" we want to find a number that makes the process go on forever never reaching 1. I want to find an example - a problem like "Collatz conjecture" but where you have to find a ...
6
votes
3answers
324 views

Foreign undergraduate study possibilities for a student in Southeastern Europe

In the (non-EU) country I live in, the main problem with undergraduate education is that it's awfully constrained. I have only a minimal choice in choosing my courses, I cannot take graduate courses, ...
21
votes
5answers
2k views

Is it worth it to get better at contest math?

I have never done well in math competitions, and am now past the point at which I can participate in them. I am asking if it is worth it to go back and practice such types of problems until I gain ...
50
votes
7answers
2k views

How to explain to the layperson what mathematics is, why it's important, and why it's interesting [closed]

A mathematician walks into a party. No, this is not the beginning of another joke, nor of a graph theory problem, but rather the beginning of a frequent and often frustrating real-life situation. ...
8
votes
1answer
644 views

Old French papers which haven't been translated into English

So I've been studying French for a few years now and I've decided that translating an old French math paper into English would be a good exercise to further improve my French competency. I would also ...
74
votes
12answers
8k views

Can you give an example of a complex math problem that is easy to solve?

I am working on a project presentation and would like to illustrate that it is often difficult or impossible to estimate how long a task would take. I’d like to make the point by presenting three math ...
15
votes
3answers
1k views

Learning math-oriented French

I'd like to read several papers which I find interesting, but they are all in French. I have no problem with taking a traditional French class or learning it via some other method. However, I realize ...
3
votes
2answers
1k views

Mathematics needed to understand Quantum Topology+Quantum Algebra?

I've recently been given a book called Quantum Topology by Louis H. Kauffman from a friend of mine. I was wondering what branches of mathematics do I need to be able to read this? What branches of ...
0
votes
3answers
315 views

A list of all algebras?

Reading @Qiaochu Yuan's blog the other day, I came across a new term: Poisson algebra. I had never heard of it before and wondered what other algebras are out there that I'm not aware of. Neither ...
10
votes
5answers
752 views

Algebraic topology, etc. for Mac Lane's “Categories for the Working Mathematician”

[NOTE: For reasons that I hope the question below will make clear, I am interested only in answers from those who have read Mac Lane's Categories for the working mathematician [CWM], or at least have ...
2
votes
3answers
445 views

Uniform distribution on $\mathbb Z$ or $\mathbb R$

I was assisting once the course in Probability Theory where students learnt quite quickly that there are ways to assign the uniform distribution to any finite set - or even subsets of $\mathbb R$ of a ...
3
votes
2answers
247 views

Introduction to proofs with a fair amount of hand-holding?

Lately I've gotten a friend of mine interested in mathematics. He has no college-level education to speak of, but is well employed as a software engineer. So I feel he's competent to learn this stuff ...
21
votes
2answers
1k views

Grad school & success in the long run

I am an undergraduate mathematics student at some university in Turkey. I am doing well in math classes. Actually, I did not study very hard in my freshmen year and I got not so high grades from ...
8
votes
3answers
667 views

Research in plane geometry or euclidean geometry

I was doing good at school in plane geometry and trigonometry - especially in geometric proofs like proving the equality of two line segments or two angles - more than I was doing in analytic ...
18
votes
5answers
5k views

Is memorization a good skill to learn or master mathematics?

I sometimes spend inordinate amounts of time memorizing math articles or theorems/proofs or formulas. My question is "am I wasting time?" and will 'active thinking' or 'working out problems' be faster ...
13
votes
1answer
650 views

Compact = Closed + Bounded + (?)

In $\mathbb{R}^n$ we know (Heine-Borel Theorem) that a set is compact if and only if it is closed and bounded. In $C(X)$ for a compact metric space $X$, we know (corollary of Ascoli-Arzela Theorem) ...
11
votes
4answers
723 views

Visual research problems in geometry

I am considering doing research in mathematics to be my career (and my life) someday. I'm a visually oriented person in general; for example, I prefer chess over cards because when I play chess, I ...
14
votes
3answers
5k views

Studying mathematics abroad - specifically France

I have tried to avoid asking a "soft question" on here because I'm not sure if they are appreciated or not. However, this is one that has been bothering me. I was recently talking to a few friends ...
6
votes
1answer
235 views

calculus textbook avoiding “nice” numbers: all numbers are decimals with 2 or 3 sig figs

Many years ago, my father had a large number of older used textbooks. I seem to remember a calculus textbook with a somewhat unusual feature, and I am wondering if the description rings a bell with ...
1
vote
2answers
255 views

Accessible topics with a background of linear Algebra and Calculus

I have a background of a one year course in linear algebra (covering most of K&H) and two years of calculus (the first year was a one real variable course at the level of Spivak's Calculus book ...
5
votes
2answers
291 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 ...
3
votes
3answers
869 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
478 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
433 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
251 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 ...
5
votes
2answers
3k 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 ...
31
votes
2answers
2k 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
157 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 ...
34
votes
10answers
4k 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 ...
17
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
134 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 ...
41
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 ...
22
votes
2answers
656 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
903 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
337 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 ...
4
votes
3answers
1k 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
301 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 ...
17
votes
6answers
2k 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 ...
21
votes
9answers
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 ...
9
votes
4answers
6k 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
819 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
850 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 ...
9
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 a question, Are there books/papers discussing the difference/indifference/comparison ...
14
votes
1answer
838 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 ...
8
votes
2answers
431 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
188 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
98 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
583 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 ...