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12
votes
8answers
4k views

How to become proficient in Calculus?

It has been a while since I wanted to ask this question, but couldn't find a right forum. My question might come across as trivial, but its important to me to find an answer. Let me give you some ...
8
votes
4answers
663 views

How to start with automated theorem proving?

I'm interested in this question, but I'm not going to list my knowledge/demands but rather gear it to more general purpose; so the first thing concerns the prerequisites, i.e. How much ...
7
votes
8answers
13k views

Back to basics: What is the fastest way to multiply two digit numbers?

I been playing different math games on my android lately (for example: Math Cruncher). I've noticed that i'm unable to quickly (under 7-8 seconds) multiply two digit numbers (i.e $ 18 * 17$). So my ...
18
votes
4answers
760 views

Advice for writing good mathematics?

It's been a (far-fetched, possibly) goal of mine to some day write a math Textbook. I've been thinking about writing this question for a while, but reading an exceedingly mediocre text on Mathematical ...
5
votes
1answer
603 views

How did Target figure out a teen girl was pregnant before her father did?

First of all I do not have a mathematics degree only a B.S. in finance so please take that into account when writing an answer. Generally what type of mathematics is involved here? And specifically ...
3
votes
1answer
278 views

Proving NP-completeness intuition

When approaching a problem in NP, initially not knowing whether the problem is in P or NP-complete (or some other choice). It seems to me the only way one can go about "solving" this problem is to ...
3
votes
1answer
100 views

How can one define, in terms of equations, independence of elements in an algebraic structure defined by identities?

I have been thinking about free algebraic structures. I know the definition by a universal property. But there is a common interpretation that a free structure is one generated by a set of ...
10
votes
10answers
774 views

How to pronounce $\setminus$

A question for English speakers. When using (or reading) the symbol $\setminus$ to denote set difference — $$A\setminus B=\{x\in A|x\notin B\}$$ — how do you pronounce it? If you please, ...
7
votes
4answers
668 views

Cool/Useful Examples of Characteristic and Minimal Polynomials?

I'm teaching a Linear Algebra II undergrad course and for the section on characteristic & minimal polynomials, I really don't want to just give the students a bunch of matrices that have no ...
2
votes
3answers
178 views

“direct” ways in which a non-computable number is used?

I was wondering whether non-computable numbers are ever of "direct" use ? I understand they are immensely useful indirectly, because we need them to do analysis in the real numbers for instance. ...
6
votes
2answers
334 views

What are well-known weaknesses of CAS/math software?

I'm starting to become more functional in Mathcad, so I wanted to take the opportunity to look into known drawbacks, mathematically speaking, of CAS/computer math systems. For instance, one of the ...
19
votes
7answers
4k views

Why do introductory real analysis courses teach bottom up?

A big part of introductory real analysis courses is getting intuition for the $\epsilon-\delta$ proofs. For example, these types of proofs come up a lot when studying differentiation, continuity, and ...
6
votes
2answers
227 views

State of the progess of the automated proof checking

I recently came across a concept of automated proof checking. I am very intrigued by the idea, that in the future all the proofs could be verified by a computer. Moreover, some proofs were already ...
6
votes
1answer
151 views

Hidden structures

There is a lot of talk about "hidden structures" in the realm of mathematics: hidden structures in the ZFC system, hidden structures in the natural number system, and so on. Saunders Mac Lane ...
4
votes
1answer
207 views

Intuition of projective plane and space

What is the geometric intuition of projective plane and space? I can understand affine plane and 3 dimension affine space, for higher dimension, at least I can imagine it similarly as the 2,3 ...
30
votes
5answers
2k views

Why is it considered unlikely that there could be a contradiction in ZF/ZFC?

EDIT: No answer addresses the "bottleneck" question. It's not surprising to me because the question is vague. But I would like to know whether that is indeed the reason, or perhaps something else. ...
21
votes
7answers
2k views

Isn't the intermediate value theorem self-evident for continuous functions?

The way I understand the intermediate value theorem is this: if you have a function f that is continuous over a domain $[a,b]$ then there is a value $f(c)$, where $f(a)≤f(c)≤f(b)$, such that $a≤c≤b$. ...
2
votes
0answers
394 views

Can i research in complex analysis , PDE and differential geometry without exposure to mathematical physics?

I love mathematics , but physics is far away from my interest . i see that recent mathematics research is strongly connected to mathematical physics which is something doesn't interest me ! I love ...
6
votes
1answer
332 views

Idea behind the factorization of the matrix $\operatorname{diag}(a,a^{-1})$ in algebraic K-Theory

If $a \in S$ is some invertible element in a ring $S$, then a computation shows $$\pmatrix{a & 0 \\ 0 & a^{-1}} = \pmatrix{1 & a \\ 0 & 1} \pmatrix{1 & 0 \\ -a^{-1} & 1} ...
6
votes
3answers
388 views

$\wedge,\cap$ and $\vee,\cup$ between Logic and Set Theory always interchangeable?

In "$\wedge,\cap,\times$ and $\vee,\cup,+$ are always interchangeable?" It has been shown that arithmetic shouldn't be included. So the new modified question is: The analogy of $\wedge,\cap$ and ...
4
votes
2answers
261 views

$\wedge,\cap,\times$ and $\vee,\cup,+$ are always interchangeable?

Update : Should have left the Arithmetic out of this question, the new modified question is posted here : $\wedge,\cap$ and $\vee,\cup$ between Logic and Set Theory always interchangeable? ...
25
votes
5answers
4k views

Is a good GRE score enough for a non-math graduate to be accepted in a decent pure mathematics graduate program?

I have a computer engineering degree , and i have studied several mathematics courses like single variable and multiples variables calculus , complex variables , probability , numerical analysis ... ...
6
votes
2answers
512 views

Organization of the Learning Process

Sorry for off topic. I'll delete this topic immediately when community decides it's useless, however if anyone finds it's interesting, share your opinion with us. I just want to know your opinion ...
14
votes
5answers
4k views

What is the best way to develop Mathematical intuition?

I want to develop my pure mathematics knowledge and would like to know what is the best way to develop mathematical intuition? I am going through exercises that ask for proofs and I don't have the ...
11
votes
1answer
369 views

What can I do with what I know? [closed]

This might be difficult for me to put into words, but bear with me because I think it's an important question. Among the many people who study math, I am one of them. I'm not particularly advanced ...
6
votes
3answers
662 views

Why is Kunen inconsistency at the top of Cantor's upper attic?

Motivation: I have reproduced part of page 396 and 397 from Handbook of Mathematical Logic below: So if we start with a concept of number and play the game of naming the largest one, does Kunen ...
2
votes
3answers
3k views

What are some good Fourier analysis books?

I have taken real analysis, but never learned Fourier analysis. What is a good book to get started? I'm not sure the Stein book would be good.
3
votes
1answer
201 views

Pure mathematics's marriage with sets

Is all of pure mathematics tightly coupled with sets ? I love mathematics but for over 2 weeks now all i have read has been somehow tied with sets. i am having such a hard time dealing with constant ...
15
votes
1answer
504 views

How do experts mentally classify indefinite integrals?

Integration is as much an art as a science. Someone who is an expert looks at an indefinite integral and classifies it in a different way than someone who is a beginner. E.g., the beginner may just ...
6
votes
3answers
457 views

Favourite proofs with a visualization

As a fan of 'visual' proofs, I love the book Visual Complex Analysis by Tristan Needham. For example, this picture http://en.wikipedia.org/wiki/File:Pythagoras_algebraic2.svg leads quickly to ...
7
votes
2answers
678 views

Why is $\zeta(2) = \frac{\pi^2}{6}$ almost equal to $\sqrt{e}$? [closed]

Why is $\zeta(2) = \frac{\pi^2}{6}$ almost equal to $\sqrt{e}$? Experimenting a bit I also found $\zeta(\frac{8}{3}) \approx e^\frac{1}{4}$, $\zeta(\frac{31}{9}) \approx e^\frac{1}{8}$ and ...
3
votes
1answer
448 views

Exponential distribution as limiting distribution

I wonder if there are well-known and studied cases involving Exponential distribution as limiting distribution. I also wonder if this would contradict Central Limit theorem.
4
votes
1answer
147 views

What is the reason to use hypergeometric functions?

I would be grateful if anyone could explain the purpose of using hypergeometric functions. If a function exists in closed form, e.g. $\sum\limits_{k \geq 0}z^k = {}_2 F_1 \bigg[{{1\; 1}\atop{1}} \vert ...
1
vote
1answer
315 views

Radical Applications of Algebraic Topology

Are there any radical applications of algebraic topology? For example, in probability theory we look at sample spaces. Suppose the sample space is a torus (for example). Would computing homology ...
5
votes
2answers
565 views

Differential Forms: High Level Approach to Real Analysis?

I am currently skimming through the differential forms book by Edwards. I was wondering whether real analysis is basically just a special case of differential forms? I am learning about flows, ...
6
votes
2answers
195 views

Good Hygiene in using Quantifiers

When using quantifiers it is probably important to pick up certain habits that Veterans agree upon as early as possible. Since it was pointed out to me by a highly esteemed member that it's ...
23
votes
4answers
901 views

Is there anything deep about Fuzzy sets/Fuzzy logic?

I keep coming across the term "fuzzy". I browsed around a bit and read a few articles.. It seems to me that there's absolutely nothing deep or foundational about fuzzy set theory or fuzzy logic. I ...
20
votes
1answer
503 views

How to find exponential objects and subobject classifiers in a given category

In a course I'm learning about Topos theory, there are a lot of exercises which require you to prove explicitly some category is an elementary topos: i.e. to construct exponentials and a subobject ...
0
votes
1answer
433 views

Where can I find a copy of Demazure and Gabriel's Introduction to algebraic geometry and algebraic groups

The question is pretty self explanatory. The book has been checked out of my university library, and I checked Amazon, and it says that the book is out of print. Also, I do not know French, so I am ...
8
votes
3answers
1k views

The most common theorems taught in Abstract Algebra

I am self learning abstract algebra. I want to know which theorems are a must to understand. Now these are limits I have to deal with (please consider when answering): I have limited ...
11
votes
2answers
722 views

Recalling Proofs

When I am able to follow a proof presented in class or in a textbook, I usually can prove the same corollary or theorem a couple days later using the same arguments. But after a week of seeing the ...
2
votes
2answers
167 views

Organising a Tournament

Imagine the following Problem. The Student Union wants to organise a tournament with 2k participants ( $k \in \mathbb{N}$ ). There are to be m rounds and in each round players should be paired ...
11
votes
2answers
291 views

Fundamental role of the Fourier Transform

I am currently learning about the Fourier Transform and the associated Fourier Analysis. So far I realize that it has a number of applications, but more than that it seems to be central to Functional ...
2
votes
1answer
2k views

What do [] mean and what does it mean if it is used in an equation?

What do the square bracket symbols mean? Are they what I hear are "sets"? And when it is in an equation, how is it interpreted? Here is an example: $$\dfrac{dy}{dx}[2x2+y(x)2]=50x+2y(dy/dx)=0$$
24
votes
7answers
2k views

Popular math books with depth

The most wonderful book I have ever read in my life was Fearless Symmetry by Avner Ash and Robert Gross, which is a good book that gives an intuition , and reasons behind the introducing fields, need ...
17
votes
6answers
732 views

Implication and Interpretation of Banach Tarski

As I understand, the Banach-Tarski paradox says a ball in 3-space may be decomposed into finitely many pieces and reassembled into two balls each of the same size as the original. Despite being called ...
2
votes
1answer
291 views

A very vague question about the cartesian product in mathematics.

Motivated by this question, I am wondering about Cartesian product analogs in various subfields of mathematics. The set theoretic Cartesian product creates an "output" set from a set of "input" sets, ...
12
votes
2answers
735 views

Is Mathematics graduation important for a Computer Scientist?

I know this might be a personal problem, but I often find some friends in the same problem as me so I think this might be helpful to them after all. I am going to graduate in Computer Science in ...
1
vote
1answer
73 views

Monty hosting a new show

I imagine the following setup. There is a contestant who has to pick one of three doors. How many prizes will be hidden is determined at random in the following way. Monty will toss a fair coin and ...
4
votes
6answers
508 views

Sleeping Mathematician (Sleeping Beauty)

I came across the following thought experiment, and I would like to understand whether the controversy around it is justified. Imagine an experiment in which a mathematician is put to sleep with some ...