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5
votes
1answer
58 views

Why non-measurable sets exist?

What Is the reason behind the fact that $\Bbb R$ is not a complete measure space? Is it only due to the cardinality of $\Bbb R$? Or other structures on $\Bbb R$ also play a role? Can one make every ...
0
votes
0answers
16 views

Mathematics Subject GRE Apps

I was wondering if anybody could recommend any smartphone apps which would be helpful for the Mathematics Specific Subject GRE exam. Most of the resources I have found are for the general GRE. ...
2
votes
0answers
26 views

Thinking About Fractional Ideals Geometrically

So algebraic geometry gives one a way of thinking about about rings geometrically. Like prime ideals correspond to points in the spectrum of a ring, maximal ideals are closed points and so on. This ...
-2
votes
0answers
31 views

how to mupltiply two digit numbers by eleven [on hold]

I think most people might know about this, but wanted to mention... Any two digit number can be multiplied by 11, by putting the sum of those digits between them. For instance 11*11 = 121, 1+1=2 ...
6
votes
2answers
107 views

Theorems with one-line proofs [on hold]

Inspired by this very concise answer, which proves that $$\sin^2(\theta)+\cos^2(\theta) \equiv 1 $$ as follows: $f(\theta)=\cos^2\theta+\sin^2\theta \quad;$ then it's simple to see that ...
0
votes
2answers
64 views

Fields - Proof that every multiple of zero equals zero

This is one of my first proofs about fields. Please feed back and criticise in every way (including style and details). Let $(F, +, \cdot)$ be a field. Non-trivially, $\textit{associativity}$ implies ...
12
votes
16answers
1k views

Beautiful, simple proofs worthy of writing on this beautiful glass door [on hold]

What are some of the more beautiful proofs you know? I am measuring beauty in two dimensions -- first, how conceptually elegant is it and second, how aesthetically pleasing is it. Context: I work ...
3
votes
1answer
126 views

Proof that group is commutative if every element is its inverse (feedback wanted)

This is one of my first proofs about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. $e$ denotes the identity element. ...
-2
votes
2answers
130 views

Is Infinity Needed in Maths? Does Infinity Actually Exist? [on hold]

I'm asking this question as I have been having an on going online debate with a friend of mine. I claimed that Infinity does in fact exist in Maths and in Reality, as there's a whole plethora of ...
3
votes
1answer
92 views

How much set theory is necessary for serious logic?

I'm currently studying logic at my university and I have been trying to squeeze in as much set theory on the side as possible. Considering that I am spending quite a lot of time studying set theory ...
3
votes
1answer
45 views

Groups - Prove that every element equals inverse of inverse of element

This is my first proof about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. We use $^{-1}$ to denote inverse ...
3
votes
1answer
51 views

On the usage of words instead of numbers to denote numbers

[To the best of my knowledge, I did not find any previous question that deals with this issue. I hope it is not a duplicate.] I have a problem with the way in which mathematicians sometime use ...
-5
votes
0answers
55 views

What is the hardest question in mathematics? [on hold]

In general, which question in mathematics is known to be the hardest, and would probably take the most time for humanity to solve?
2
votes
1answer
62 views

What cardinals do we actually need?

I recently heard of a great variant of set theory called "Pocket Set Theory" in which the only two cardinalities are that of $\mathbb N$ and that of $\mathbb R$ (and the latter is a proper class). ...
0
votes
0answers
34 views

Book on calculus of several variables.

I'm an undergraduate student in mathematics and want to study Calculus of several variables currently this semester which involves the use of analysis,vector spaces and linear transformations. Can ...
6
votes
2answers
138 views

Self studying higher mathematics?

I'm fairly well-versed in calculus but I would like to explore beyond calculus. I have looked into the basics of some topics in higher mathematics such as group theory and abstract algebra and they ...
3
votes
6answers
574 views

How should I self-study calculus?

So I already took Pre-Calc, and ended up with a B both semesters. I am an incoming senior in high school. My special-ed case manager won't let me take it because she doesn't want to see me panic ...
0
votes
1answer
29 views

Tangent bundle of the 2-sphere

I'm reading through Tu's Introduction to manifolds and today I learned about tangent bundles and vector bundles. I was surprised to learn that $TS^2$, tangent bundle of the 2-sphere, isn't trivial ...
-1
votes
0answers
57 views

REUs for graduates? [on hold]

I have done REUs for the last two summers and plan to go to graduate school in mathematics. I am graduating next year and am wondering if there are any programs for students (maybe similar to an REU) ...
4
votes
1answer
129 views

Why the whole theory about differentiable manifolds is based on open sets?

I only have studied basic topology, which means i haven't studied any about differentiable manifolds. I just skimmed pages on wikipedia. Here is a simple illustration on a basic situation. Let $E$ ...
0
votes
3answers
72 views

High dimensional vector space references

Is there any good text book or review papers that introduce high dimensional vector spaces and its peculiarities as compared to generic/low-dimensional vector spaces? For example, high dimensional ...
5
votes
1answer
179 views

Why do we study representations of groups but not fields?

Groups are great objects to work with as we all know. With surprisingly little structure, we can say fairly general things. However groups can be difficult to manage and so we look to representations ...
0
votes
0answers
21 views

Chains of Field Extension

To deal with compass-and-straightedge construction, solvability of algebraic function and integration of real/complex function, we consider the chains of field extension. More precisely, let ...
0
votes
0answers
73 views

What mathematical knowledge should a regular math. major undergraduate know? [on hold]

I knew what pure mathematics is about five years ago. And since then I am fascinated. Most of my math. knowledge are arguably understood all by myself. But now I begin to feel unsatisfactory. For ...
1
vote
0answers
39 views

Relation between $|H \lor K|$ , $|H|$ and $|K|$

Let $H$ and $K$ be subgroups of a finite group , then we know that the subgroup generated by $H \cup K$ i.e. $H \lor K$ is the smallest subgroup containing both $H$ and $K$ , then how can we relate ...
1
vote
1answer
43 views

Do you paragraph a proof?

When writing out a proof of moderate length, i.e. a proof taking less than or equal to 5 A4 papers and with normal spacing (please avoid asking the criterion for "normal"), do you tend to paragraph it ...
0
votes
2answers
101 views

General question about undergrad math classes? [on hold]

I'm currently in Real Analysis 2, and while I'm doing pretty well, it's just so much time and effort to finish all the problem sets and do well on the exams. I only have a few math classes left for ...
2
votes
0answers
34 views

Specifying types of variables in pure mathematics and applied mathematics

In pure mathematics, we can write such as "an integer $a$ ..." to specify that $a$ is a given integer or $a$ runs through the ring of integers. But in contexts where mathematics is applied ...
2
votes
0answers
41 views

Entrance exam preparation suggestions.

I will be giving my Entrance Exam for (MS in Computer Science) and this is the Syllabus. Syllabus Screenshot : http://i.imgur.com/9KUDCt3.png I am worried about the maths and reasoning part. It's ...
1
vote
1answer
81 views

Quantifier problems of equations in physics [closed]

Equations in physics are often written without quantifiers. For instance, from time to time we can see the equation $$E = mc^2$$ is casually written down. To assert that static energy equals mass ...
0
votes
1answer
28 views

Interpretation of a statistical formula involving the ratios of sample and population

Kind of an odd question, but, is this a standard equation for stats? I can't figure out for the life of me what it looks like. $\left(\frac{\text{Sample Of A}}{\text{Population Of ...
8
votes
1answer
64 views

Understanding Dynkin Diagrams - any organising ideas - are they now adequately understood?

Some 30 or so years ago JH Conway posed a question about the ubiquity of the Dynkin Diagram - not necessarily in public, but I heard him ask it. I think it was in the context of "what would be ...
1
vote
1answer
46 views

Do journals that published a proof of an important theorem $T$ publish another proof of $T$?

I want to know whether or not a journal that published a proof $P$ of an important theorem $T$ is still open to accept another proof $P'$ of $T$ such that $P'$ is greatly simpler than $P$, assuming, ...
0
votes
0answers
34 views

Is it important to study plane algebraic curves before read Fulton's book

I'm studying Fulton's algebraic curves book and I would like to know how important study plane algebraic geometry before read Fulton's book. Example of books on this subject: Algebraic Curves - ...
0
votes
2answers
73 views

What are the differences between mathematics courses taken by engineering majors and by math majors? [closed]

I am curious to know what are the differences between mathematics taken by engineering students and by math majors. Let's say in terms of the approach, depth and in the topics covered. And even within ...
3
votes
2answers
61 views

Is calling a linear-equation a linear-function, misnomer or completely wrong?

From my college life, I remember many professors used to call a linear-equation a linear-function, however: A standard definition of linear function (or linear map) is: $$f(x+y)=f(x)+f(y),$$ ...
2
votes
1answer
91 views

Why is the powerset axiom more acceptable than the axiom of choice?

The key step in Zermelo's proof of the well ordering theorem is to use $\text{AC}$ to simultaneously choose the next elelment for all possible partial chains in prospective well orderings, but that ...
6
votes
2answers
54 views

Geometric interpretation of complex path integral

Let's say that we want to make sense of integrating a function $f: \mathbb{C}\rightarrow\mathbb{C}$ over some path $\gamma$. I can imagine two reasonable ways of doing it. First, there's the way ...
1
vote
0answers
29 views

Additional qualifications that would strengthen my profile/CV [closed]

I apologize if this isn't the right site to ask this question. I am currently doing my Masters in Mathematics, which is a 2 year course (here in India). I was thinking of strengthening my profile by ...
2
votes
0answers
64 views

Tool for converting maths writing to $\LaTeX$ [migrated]

I have a dream. I want my maths writing to magically be made into a .tex file so that I can edit it. I want to write my papers, my exams, my lecture notes, ...
3
votes
2answers
268 views

Degrees of separation between famous mathematicians

I was recently doing some reading on Wikipedia, and I noticed that if you go far enough though Isaac Newton's notable students' students' students. . . (and so on), eventually one was Augustus De ...
2
votes
0answers
44 views

Intuitive explanation for $\zeta (2)=\frac{\pi^2}{6}$ [duplicate]

Using $f(x)=x^2$ Fourier' series, the proof for $\zeta (2)=\frac{\pi^2}{6}$ is pretty straight forward. I'm wondering if there is a more intuitive explanation for the equality, one that a layman could ...
0
votes
1answer
53 views

The art of solving exercise problems [closed]

I do not know whether this question appears to be off topic or not. But I really want to know, how to be well versed in solving exercise problems prescribed at the end of each section. I understand ...
25
votes
7answers
2k views

Mathematical literature to lose yourself in

H.M. Edwards in the preface to his book on the Riemann Zeta Function, summarises his philosophy on learning Mathematics: ...I have tried to say to students of mathematics that they should read the ...
1
vote
0answers
64 views

Is there a name to this equation: $(y - a|x|^b)^2 + (cx)^2 = d$?

While doing a survey of the various equations that generate the universal love symbol, a heart curve, I find that many fit into this parametrised form: $$(y - a|x|^b)^2 + (cx)^2 = d $$ Where   ...
3
votes
1answer
57 views

How does research in math differ from research in statistics?

I'm at a crossroads where I'm considering switching my major from electrical engineering to math, because quite frankly, I'm just not getting enough math to satisfy my passion from engineering. While ...
11
votes
5answers
283 views

I need help finding a rigorous Pre-calculus textbook

I dislike modern textbooks; their cookie-cutter approach and appearance, over reliance on breaking things down into little boxes, the general spoon-feeding they engender and most of all the poor ...
10
votes
1answer
220 views

How inequalities are made

I've been solving a lot of math contest inequality problems last few days and sometimes when I solve the problem I can easily ''see'' the idea behind it's creation (for an example, one clever ...
0
votes
2answers
29 views

Splitting a matrix $A \in \mathbb{M}^{n \times n}(\mathbb{C})$by solving $Av = \lambda C v$ for some chosen $C$

If we know a matrix $A \in \mathbb{M}^{n \times n}(\mathbb{C})$ and solve $Av = \lambda v$ where we try to find $\lambda,v$, we can rewrite $A$ in a nice way. What if we choose a matrix $C$ and we ...
3
votes
0answers
75 views

What's so special about binomial coefficients that someone decided to organize them in a triangle?

I know that binomial coefficients are related to figurate numbers (which were studied by Greeks a loooong time ago, because of its connections to geometry). I also understand how the Pascal's triangle ...