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6
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1answer
1k views

Introduction to Computational Topology

Question: Besides generally learning Algebraic Topology, what are some prerequisites for studying Computational Topology? Are there any accessible papers which introduce the field and the methods ...
1
vote
0answers
161 views

Motivation for studying compass and straightedge constructions? [duplicate]

Possible Duplicate: What is the (mathematical) point of geometric constructions? Are there good motivations to study compass and straightedge constructions? More specifically I want to ...
10
votes
2answers
3k views

Undergraduate mathematics programs

It has come time for me to decide where to pursue my undergraduate education. I plan to pursue a PhD in mathematics, so accordingly my primary concern is the mathematics programs at various ...
23
votes
12answers
2k views

Alternative notation for exponents, logs and roots?

If we have $$ x^y = z $$ then we know that $$ \sqrt[y]{z} = x $$ and $$ \log_x{z} = y .$$ As a visually-oriented person I have often been dismayed that the symbols for these three operators ...
1
vote
3answers
307 views

what is/are the spectrum of operators and their applications

this is an educational question. can someone please explain with some simple examples: (1) what is/are the spectrum of operator (2) where it is useful For providing examples of spectrum of ...
4
votes
1answer
317 views

what are the uses of this identity

Consider this wonderful ( think it is) identity $$\begin{align*} &a+b(1+a) + c(1+a)(1+b) + d(1+a)(1+b)(1+c) +\cdots+l(1+a)(1+b)\cdots(1+k)\\ &\qquad= (1+a)(1+b)(1+c)\cdots(1+l)-1 ...
19
votes
2answers
2k views

motivation and use for category theory?

From reading the answers to different questions on category theory, it seems that category theory is useful as a framework for thinking about mathematics. Also, from the book Algebra by Saunders Mac ...
2
votes
3answers
505 views

How to prove the following inequalities?

thanks for your time. i am interested in various ways/techniques/tricks/methods (induction, convexity, concavity, maximum, minimum, geometry, trigonometry, ...) for proving the inequalities and their ...
5
votes
3answers
697 views

Mathematical misconceptions and how to combat them

There are a lot of common misconceptions when it comes to math. A common one that has already been addressed on this site is $1 \neq .999\cdots$, as is that imaginary numbers "do not exist". Another ...
11
votes
2answers
838 views

Supplemental number theory text to Montgomery and Vaughan

We already have a large list of the Best ever book on Number Theory, but I'm looking for a more targeted response for analytic number theory. Specifically, I'm taking a trip on which I may or may ...
49
votes
10answers
69k views

Why is Infinity multiplied by Zero not an easy Zero answer?

I did a bit of math at school and it seems like an easy one - what am I missing? $$n\times m = \underbrace{n+n+\cdots +n}_{m\text{ times}}$$ $$\quad n\times 0 = \underbrace{0 + 0 + \cdots+ ...
6
votes
1answer
229 views

Introductory texts for weak $\omega$-categories

As I'm constantly running across higher categories these days, I'm wondering what is a good starting point to get into the theory? While I am aware of nLab and the n-Category Café, I am having a real ...
46
votes
18answers
6k views

What's the goal of mathematics?

Are we just trying to prove every theorem or find theories which lead to a lot of creativity or what? I've already read G. H. Hardy Apology but I didn't get an answer from it.
1
vote
1answer
438 views

Computing explicit maps in the Mayer-Vietoris sequence

I'm trying to figure out how to actually compute maps in the Mayer-Vietoris sequence as it's quite useless unless you actually know some maps. Hatcher computes the homology for the Klein bottle $K$ on ...
13
votes
3answers
2k views

Homology and Graph Theory

What is the relationship between homology and graph theory? Can we form simplicial complexes from a graph $G$ and compute their homology groups? Are there any practical results in looking at the ...
8
votes
5answers
606 views

General Introduction to Functional and other Mathematic Notations

I've been a programmer for a good while now. Fairly experienced at a bit of math as far as coming up with algorithms and such but I am far far behind on understanding quite a deal of notation. Here ...
3
votes
2answers
195 views

Counting and Ordering of Numbers

Are there differences between 'counting' and 'ordering'? As such, the whole of rational number is countable, or they order-able too? In what cases counting and ordering are same or not?
4
votes
0answers
1k views

What is the idea of the problems in V. I. Arnold's problem book “A mathematical trivium”?"

Here is the source of most of the problems I ask here, a majority of which I am currently unable to solve. I have asked three questions related to it on this site$^{(1), (2), (3)}$ From responses ...
9
votes
2answers
599 views

Math route for someone of this background

Apologies for a soft question. I do not have a lot of time because of my job (programmer) and my wife cannot work so I cannot quit. I also did not graduate from a very prestigious school with good ...
9
votes
4answers
985 views

Uses of the 'Golden Ratio'

I have heard much about the numerous appearances of the ratio found in nature: 1.6180339887. Are there any actual mathematical uses that have been found of this number? What are its advantages? Just ...
3
votes
1answer
126 views

Motivation for product structure

I guess I understand some reasons why we should care for complex structure on manifolds, but what is the reason why product structure is studied? Does it arise naturally somewhere? The product ...
0
votes
1answer
232 views

Quantitative trading

What math subjects are used in quantitative trading firms?
6
votes
3answers
491 views

Visual representation of groups

I vaguely remember seeing something like a "picture" of various groups a while back. It was as if the elements of the group were each associated with a point and many points had segments connecting ...
2
votes
1answer
614 views

Considering math or computer science

Let me start by saying I graduated High School in 2002 and have worked as a technician, a systems administrator, and a part time web developer up till now. I have started taking courses at community ...
7
votes
8answers
1k views

Most important elementary math skills

Barring very elementary arithmetic, which skills from elementary school are essential for understanding the world better?
6
votes
1answer
298 views

Please recommend good text on complex Fourier series/analysis

I am looking for some good text/reference on complex Fourier series resp. Fourier analysis for complex (in particular holomoprhic) functions (of one variable). The more it contains on this particular ...
6
votes
3answers
799 views

Introduction to Bourbaki structures, and their relation to category theory

I just opened vol.1 of the Bourbaki treatise to take a look at how they define mathematical structure. I was amazed by its sheer complexity. Can you recommend an introductory text that wouldn't ...
22
votes
4answers
7k views

Lemma/Proposition/Theorem, which one should we pick?

This is something that confuses me. I have read a few mathematical texts and they often seem to use Lemma/Proposition/Theorem if they have a particular statement. Now, which one to use? A lemma can ...
34
votes
5answers
3k views

Why should I care about adjoint functors

I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and ...
3
votes
1answer
176 views

Hardy Spaces on the unit disk and $\mathbb R^n$

Is there a connection between the Hardy Spaces on the unit disk and on $\mathbb R^n$?. If so, can we use results from the Hardy Spaces on the unit disk to prove $(H^1)^* = \text{BMO}$? Further, what ...
10
votes
6answers
754 views

Where to go with Mathematics?

I am currently a math major in college and my main problem is that it feels directionless. My college offers little in term of variety in undergraduate math so I moved on into taking graduate courses ...
0
votes
1answer
276 views

Papers published by mathematicians

In general, are papers published by mathematicians the same thing as PhD theses? In other words, if a mathematician submitted a paper for a PhD would it get accepted? Or are papers of higher quality ...
11
votes
3answers
809 views

Differentiable at a point

My roommates and I have an argument you guys can help to settle (peace is at stake, don't let us down!) In undergrad calculus courses, one usually explains what it means for a function to be ...
2
votes
1answer
1k views

Are there any good algebraic geometry books to recommend? [duplicate]

Possible Duplicate: (undergraduate) Algebraic Geometry Textbook Recomendations I am interested in algebraic number theory and I am recently acquainted with the theory of valuations, which ...
167
votes
22answers
10k views

Why do mathematicians use single-letter variables?

I have much more experience programming than I do with advanced mathematics, so perhaps this is just a comfort thing with me, but I often get frustrated trying to follow mathematical notation. ...
4
votes
1answer
359 views

Archimedean property of $\mathbb{R}$

Theorem: If $x,y \in \mathbb{R}$ and $x > 0$, $\exists$ a positive integer $n$ such that $nx > y$ I read the proof by Rudin and understood it. I think it is very elegant and uses the LUB ...
2
votes
0answers
72 views

Comparing symbolic and analog descriptions

I've never seen the following comparison before. Let me start with a specific example: Given a finite structure with two symmetric binary relations, i.e. a graph $G$ with one vertex set $V$ and two ...
30
votes
13answers
13k views

Formerly good at math, but after 12 years I've lost most of my skills. Now I need them once again. Any advice to grow them back?

I love math, and I used to be very good at it. The correct answers came fast and intuitively. I never studied, and redid the demonstration live for the tests (sometimes inventing new ones). I was the ...
6
votes
1answer
291 views

mathematical existence

When a mathematical notion is said to exist, it seems that there is a lot of freedom in what this particularly means, a freedom which accounts I think for why some people are finitists or ...
102
votes
8answers
64k views

What is the importance of eigenvalues/eigenvectors?

What is the importance of eigenvalues/eigenvectors?
5
votes
1answer
1k views

When addition distributes over multiplication

Everyone is familiar with distributivity of multiplication over addition of real numbers. The distributivity of two binary operations sometimes goes both ways (e.g. max and min, or for lattices in ...
2
votes
2answers
5k views

Understanding $\epsilon$ transitions in a finite state automaton

I am trying to understand how $\epsilon$ transitions work. From what I've read, when you "go" to a state S that has arrows pointing outwards with $\epsilon$'s in it, you automatically go to those ...
8
votes
2answers
507 views

Pull the teeth out of Lebesgue integration

In Lebesgue integration we usually approximate the function we want to integrate with step-functions on measurable sets. How much "power" do we take away if we require that the step functions are on ...
56
votes
14answers
16k views

How to effectively and efficiently learn mathematics

How do you effectively study mathematics? How does one read a maths book instead or just staring at it for hours? (Apologies in advance if the question is ill-posed or too subjective in its current ...
78
votes
8answers
25k views

Learning mathematics as if an absolute beginner?

I dread mathematics, and I believe it's because I have come to associate mathematics with the experience of terrible teachers. All of my math teachers have been grumpy, but one in particular was the ...
4
votes
2answers
257 views

Can there be such a thing as a classification of classification theorems?

Can there be such a thing as a classification of classification theorems?
5
votes
1answer
464 views

Reading commutative diagrams?

Sorry for this whole bunch of questions. Please note, that I know what a commutative diagram is, and that I can somehow read them, at least the simpler ones. But often enough the diagrams are labelled ...
2
votes
1answer
81 views

a function of a dependent type, a section, a sheaf

I have defined this simple sheaf. Take $E:set, B:set, p:E\to B$. Let $P(B)$ be a set of subsets of $B$. Let $S$ be the set of sections of $p$. Let $F$ be a contravariant functor from $P(B)$ as a poset ...
31
votes
5answers
4k views

How to pick a thesis advisor?

This sort of question is probably in bad taste for math.stackexchange, but is probably in high demand. (I tried to start a site on Area 51 to house questions like this, but my request was closed due ...
11
votes
6answers
9k views

In what order should the following areas of mathematics be learned?

I am in a biological field (medicine) but I have genuine passion for mathematics. I want to learn it on my own , in my spare time. Mathematics , as I gather, is learned best when you have grasped ...