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4
votes
2answers
45 views

Visualizing Lie algebra of SO(3)

Let $SO(3)$ be the Lie group of 3D rotations. Rotation about z-axis by an angle $\phi$ is represented in standard basis by this matrix: $$ \begin{pmatrix} \cos \phi & -\sin\phi & 0 \\ ...
1
vote
1answer
23 views

Proof - Inverse of linear function is linear

This is my first proof related to linear functions. It refers to the linear-algebra-$\textit{linear}$ (not the calculus-$\textit{linear}$). Please comment. Theorem The inverse of a linear bijection ...
2
votes
0answers
102 views

Is mathematics invented or discovered? [on hold]

A google search yields millions of results, most of which are made by laymen who have nothing to do with math and it's "just another article" for the authors, so I assume here with so much passion in ...
4
votes
2answers
79 views

What is a geometric structure?

Every elementary book on abstract algebra usually begins with giving a definition of algebraic structures; generally speaking one or several functions on cartesian product of a point-set to the set. ...
1
vote
0answers
74 views

Any other mathematicians like Galois in recent history?

Any other mathematicians like Galois in recent history ? For "someone like Galois", I mean someone who developed a completely new theory all by himself, solved a big problem and the theory has big ...
0
votes
0answers
57 views

simple equations with deep implications [on hold]

the twitter limit of $140$ characters can stimulate the mind. in the same vein i'm wondering what are the most semantically rich mathematical statements that can be made with restricted means. i was ...
-1
votes
2answers
39 views

Proof Techniques ( Soft Question )

I've been googling around for books of methods of mathematical proofing, and I haven't had much luck finding anything reputable in book form. I do recall running by a few in a university library ( I ...
10
votes
6answers
373 views

Reference request for undergraduate complex analysis.

I am a second year student studying electrical engineering. I self-study pure mathematics and want to pursue a career as a mathematician. What are some prerequisites for studying complex analysis? ...
2
votes
3answers
53 views

how to fairly select a leader

I recently came across a rather practical problem: A large group (around 30 people) wanted to elect a new leader (someone who is not part of the group) of 4 possible candidates. Each of the ...
17
votes
6answers
2k views

What is the motivation for quaternions?

I know imaginary numbers solve $x^2 +1=0$, but what is the motivation for quaternions?
0
votes
2answers
42 views

Problem book for abstract linear algebra

Kindly suggest a good book for abstract linear algebra other than finite dimensional vector space by P R Halmos
5
votes
2answers
92 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
19 views

Mathematics Subject GRE Apps [on hold]

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. ...
4
votes
1answer
64 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
36 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 ...
7
votes
2answers
143 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
77 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 ...
13
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
134 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. ...
-1
votes
2answers
152 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 ...
4
votes
2answers
161 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
52 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
58 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
65 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
37 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
141 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
5answers
594 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
59 views

REUs for graduates? [closed]

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
130 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
75 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
181 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
74 views

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

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
40 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
103 views

General question about undergrad math classes? [closed]

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
82 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
30 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 ...
9
votes
1answer
66 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
48 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
35 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
74 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
96 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
56 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 ...
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, ...