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1
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0answers
79 views

Role of mathematics in science

I want to know that what is the main role of mathematics in science. I mean to say is the main objective in any mathematical research should be finding some tool which may be of good use in other ...
50
votes
15answers
7k views

How do I motivate myself to do math again? [closed]

I have been thinking of asking for help for a few months now but posting in a public forum like this is intimidating. Still, I am currently in a university studying mathematics as an undergrad. I took ...
1
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0answers
128 views

On Cauchy Sequences

I would consider this a soft question because I am seeking some insight on how to work with Cauchy sequences by using the Cauchy criterion for convergence. To my understanding, the definition is ...
2
votes
2answers
401 views

Why do we call it a $\sigma$-algebra?

In simple terms, a $\sigma$-algebra is the collection of all of the things we know how to measure. Why don't we call it something that more directly suggests this, for example a 'measure space?'
3
votes
1answer
190 views

How does math education vary from school to school?

Is there a significant difference in the material taught across universities in the United States? For example, if I get a bachelor's degree in mathematics (or something related such as physics or ...
4
votes
1answer
66 views

Gaussian density function satisfies $y'=-xy$. Coincidence?

Is part of the rationale for the Gaussian distribution that the density function satisfies the differential equation $y' = -xy$? Or is this more or less incidental?
37
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9answers
2k views

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
8
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3answers
14k views

Abstract Algebra book with exercise solutions recommendations.

I am new to studying abstract algebra (and math in general). I've been reading Gilligan and Pinter's books. I am trying to improve my understanding by doing exercises. However none of the books I am ...
7
votes
1answer
2k views

Why can algebraic geometry be applied into theoretical physics?

It is to be said at the outset that I do not have much familiarity with physics beyond what is in a semi-popular book; say, the Feynman Lectures Vol 1 and 2. As I progressed in math graduate school ...
1
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1answer
34 views

Enumerations in totally ordered sets

Suppose that $A$ is a subset of the positive integers and you want to enumerate all the elements of $A$ in a sequence $a_1, a_2, \ldots$ such that $a_n < a_{n+1}$. What is the most concise and ...
1
vote
1answer
76 views

Natural Analysis: a possible new field?

I have studied real analysis for two years now, yet I often find that when applying it, (say to finding a path of shortest time) I often have to get solutions without closed forms (i.e. defined point ...
4
votes
2answers
731 views

How do I go about doing math as a hobby?

want to study math as a hobby. I find it very interesting, but I made a lot of twists and turns in college which lead me on a different path. Plus, I'm not really sure I'm gifted enough to hack it at ...
1
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0answers
18 views

A way to describe the following triplet

so first of all the tittle I think is very poor, so I apologize about that. I am working on a homework set for homology, and when I computed the image of a boundary map I got this: $(a-b,2b-c,2c)$ ...
1
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1answer
133 views

Ebooks in advanced calculus

This semester I'm taking advanced calculus and in the sources list, the lecturer recommends on "An introduction to analysis-by w. wade" and "Advanced calculus by Edwards". Both of them cost around 100 ...
2
votes
2answers
67 views

Is there a strategy to switching $\infty$ limits to $-\infty$ limits?

I asked recently in this question how to use the definition of $$e:= \lim_{x\to\infty}\left( 1+\frac1x \right)^x$$ to show that $$\lim_{x\to -\infty}\left( 1+\frac1x \right)^x = e.$$ A helpful ...
1
vote
1answer
32 views

Why does $\sum_{j,k\geq 0}\frac{(j+k)a^{j+k}}{j!k!}=\sum_{l=0}^\infty \frac{l(2a)^l}{l!}$?

I notice that $$\sum_{j,k\geq 0}\frac{(j+k)a^{j+k}}{j!k!}=2ae^{2a} = \sum_{l=0}^\infty \frac{l(2a)^l}{l!}.$$ Is there a simple intuitive explanation why these two should have the same sum, or is it ...
1
vote
1answer
125 views

Can you give me a good alternative to Rotman's Group Theory book?

I've been trying to learn out of Rotman's book "An Introduction To The Theory of Groups" for the last few months, and it's rough going. I've been studying Chapters 7, 10, and 11 in particular, and ...
2
votes
2answers
2k views

Interesting Topics to Give a Seminar On?

So recently I've been attending seminars for the graduate students (and VAP's) at my local university, and after yesterday's seminar, the professor asked if I would like to give a seminar next ...
0
votes
1answer
281 views

Proving Equality of 2 Functions

I have a general question, illustrated by a specific example. The general question is how to methodically prove that two functions are equal. Much like trying to prove an "if-and-only-if" statement ...
1
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0answers
62 views

SOLVED — Equal or Equivalent Inequalities?

Suppose I have two inequalities: $2ab \leq a^{2} + b^{2}$ $0 \leq a^{2} + b^{2} - 2ab$ Now, obviously these inequalities are just rearrangements of one another. The question I have is: do I say ...
0
votes
1answer
76 views

Is it acceptable to use the shorthand “PDEs” in the title of a paper?

If I write a paper, is it acceptable for me to use a title like "analysis of PDEs related to blah blah blah" instead of "analysis of partial differential equations related to blah blah blah"? I ...
3
votes
6answers
718 views

Why do people write stochastic differential equations in differential form?

I am trying to teach myself about stochastic differential equations. In several accounts I've read, the author defines an SDE as an integral equation, in which at least one integral is a stochastic ...
3
votes
1answer
246 views

Real life applications of Maass wave forms

Explaining my work on Maass wave forms to friends and family (all non-mathematician) typically earns me blank faces. So I wonder whether there is some good example to explain their meaning to laymen. ...
7
votes
2answers
503 views

Resource for low level maths explained in high level perspectives

I would really never ask a question about resources, noting that it is a soft-question, unless I thought it was very difficult to find elsewhere, and I have looked. Furthermore, I believe that this ...
2
votes
1answer
542 views

I need tips for increasing concentration for math study

Any tips for increasing concentration? I find it difficult to concentrate for looong periods at a time, after having taken a long break from mathematics. Before I could sit for hours and do ...
4
votes
5answers
818 views

Is zero irrational?

I think of the number zero as a whole number. It can certainly be a ratio = $\frac{0}{x}, x \neq 0.$ Therefore it is rational. But any ratio equaling zero involves zero, or is irrational, ...
5
votes
2answers
836 views

Factoring Quadratics: Asterisk Method

I'm teaching my students about factoring quadratics. We've done GCF, difference of two squares, squared binomials, and grouping. One of my colleagues then found this asterisk method on line. It's ...
3
votes
1answer
157 views

Mathematics in cognitive sciences

Which areas of Maths can be formally studied in a cognitive sc major? Or: which areas of maths can support the study of brain. Some areas that seem relevant would be: mathematical logic, graph theory, ...
3
votes
0answers
119 views

Can one define informational content of a mathematical expression?

At least in physicist's thinking, information, vaguely, is something that allows one to select a subset from a set. Say, a system can be in states A and B, we have done a measurement on it ...
8
votes
1answer
248 views

How do Mathematicians send email? [closed]

I've been corresponding with people about math via email. It seems very weird to just be sending PDFs of the conversation back and forth, but there's no way to talk without typesetting. Is there an ...
4
votes
3answers
199 views

Abstract and math thinking

I'm a student in university studying math first year. I would like to know what are the best books that shape and build the right math and abstract thinking skills . I don't mind if they are general ...
4
votes
2answers
185 views

A chess problem in Kanamori's “The Higher Infinite”?

Just after Corollary 21.17 (on p289) of Kanamori's The Higher Infinite, he outlines the direction in which he wants to take his discussion of iterated ultrapowers. However, immediately after he ...
0
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1answer
120 views

Some questions about mathematics [closed]

This question is a soft one. Well, So far I have noticed stuff that is nice in math, particularly in algebra, topology and analysis. For instance, in algebra, there is theorem that says that we can ...
3
votes
4answers
682 views

How to explain Fractional and Negative Exponents

My classmates doesn't understand Fractional and Negative exponents, since I was the top of my class, so they all came to me... Is there any way to explain it clearly to them?
7
votes
1answer
119 views

Keeping Focused: School

Mathematics is such a huge field, and I am just wondering how does a mathematician keep focused? For example to learn Stochastic Calculus you need some Analysis, Measure Theory, Probability, Set ...
9
votes
1answer
205 views

Could we reasonably classify finite groups?

So I have been reading some work on $p$-groups, and I noticed a particularly disturbing sentence: "There is no hope of finding a finite set of invariants that will define every p-group up to ...
23
votes
1answer
630 views

Is there a geometric idea behind Sylow's theorems?

I have a confession to make: none of the proofs of Sylow's theorems I saw clicked with me. My first abstract algebra courses were more on the algebraic side (without mention of group actions and ...
8
votes
1answer
113 views

Encyclopedia of categories

Is there a digest of the most widely used categories with their basic properties mentioned? So I could, say, learn from it in a couple of minutes that the category with totally ordered sets as objects ...
4
votes
2answers
371 views

How would I know if I'm good in logic?

I've always been interested in logic, but unfortunately my school contains no logicians. What are some good logic puzzles/books and how would I know if logic is right for me? Also, what can I do with ...
1
vote
3answers
195 views

how can we write abstract algorithms?

Writing pseudo-code for algorithms is common practice in the applied mathematics literature. It is also often the case that the ideal input of an algorithm is an infinite set, for example it could be ...
3
votes
1answer
321 views

Where to take Real Analysis and Linear Algebra?

I am undergraduate in economics. As you may know, most prestigious departments in economics now require their aspirants to have taken Real Analysis (and Linear Algebra, too) before entering their ...
15
votes
11answers
5k views

Undergraduate Schools for Mathematics

I am currently a senior in high school, and I have been studying mathematics for about nine or ten years now in my personal time outside of school. I am not familiar with academia or in general higher ...
7
votes
1answer
635 views

Question on questions in Spivak's Calculus?

I started reading Spivak's Calculus about a month ago and I'm at the end of chapter two, so this is not really calculus yet. However, I find the problems really difficult and the answer keys are not ...
17
votes
7answers
1k views

What is the correct spelling of Paul Erdős's name?

I am not quite sure that this question belongs to the math.stackexchange.com. What I am sure is that definitely there are people here who know the answer to my question. On the Internet and in the ...
4
votes
1answer
83 views

A persistent difference

Here's a fun math problem. I wasn't able to get it - am curious what you guys have to say. Pick a four-digit number whose digits are not all the same. From its digits form the smallest four-digit ...
3
votes
2answers
93 views

Simple question that I can't solve [duplicate]

Here is a relatively simple question that I'm unable to solve :/ There are $10000$ closed lockers in a hallway. A man begins by opening all $10000$ lockers. Next, he closes every $2^{nd}$ locker. ...
7
votes
1answer
2k views

Symmetric vs. Positive semidefinite

This may be a trivial or irrelevant question. I'm very sorry if so. Why do mathematicians use only symmetric matrices when they want positive semi/definite matrices? I mean haven't seen using ...
28
votes
18answers
1k views

What is the most surprising result that you have personally discovered?

This question is inspired by my answer to this one: Surprising identities / equations In that question, people were asked about the most surprising result that they knew. Almost all of them quoted ...
6
votes
3answers
5k views

Improbable vs Impossible?

I was wondering how mathematics in general or any of its sub fields e.g.statistics, probability, define the words Improbable and Impossible. I get their English meaning, that something is impossible ...
1
vote
0answers
59 views

If P = NP can asymmetric key exchanges still exist?

One functions are easy to compute (ie polynomial time checking) but hard to reverse. if P = NP does that mean that asymmetric key exchanges will be reduced from polynomial computation time and ...