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7
votes
3answers
94 views

How to structure long proofs

How do you structure proofs that are longer than say half a page? I have already encountered a variety of styles (in my short math life), some of which I list below and I just hoped to hear some wise ...
5
votes
1answer
82 views

What makes “the topos $\mathbf{M}_2$” such a good counterexample?

I'd like to ask this question sooner rather than later; it might be a bit half-baked. So I'm sorry. It's just that there's a chance I'll be side-tracked from Topos Theory for a couple of months (with ...
4
votes
1answer
103 views

Lecture notes ready for $\LaTeX$

Are there on the internet lecture notes in calculus in .tex or .txt format, that is, ready to be edited/modified/re-used and compiled using $\LaTeX$? EDIT: now I am specifically asking for calculus, ...
0
votes
0answers
57 views

Complex analysis a good thing to take with algebra?

This is sort of a follow up to a question I asked awhile ago. I'm working out which courses to take my first semester of my second year of university. I've had linear algebra, calculus, and analysis ...
11
votes
4answers
882 views

How is addition different than multiplication?

Is there a fundamental difference in the things we call multiplication and those we call addition? In a field, both binary operations obey exactly the same rules (commutativity, associativity, ...
3
votes
0answers
27 views

Sufficient conditions for closed infinite pasting lemma

It's well known that the pasting lemma for infinitely many closed sets is false. It's reasonably easy to cook up examples such that for $X = \bigcup X_i$ with $X_i$ closed in $X$ such that $\left. ...
2
votes
5answers
703 views

What should I use Latex or Microsoft Word Professional? [closed]

What should I use Latex or Microsoft Word Professional for writing mathematics papers and documents and notes and courses...?
0
votes
1answer
83 views

The fourth pillar of mathematics? (analysis, algebra, geometry and …) [closed]

Many universities claim that there are three general areas in mathematics: analysis, which deals with continuous aspects of mathematics; algebra, which deals with discrete aspects; and geometry. If ...
7
votes
3answers
744 views

Masters' thesis in group theory [closed]

I would like some ideas on topics in group theory which would be suitable for a masters' thesis. What sort of problems would be suitable for this level? Because it is at masters' level, no original ...
5
votes
0answers
84 views

Reference Request: Prereqs for Lecture Notes on “Abstract Linear Algebra”

I just found this set of lecture notes on linear algebra which seems to go over several things I've been wondering about as I study linear algebra. Unfortunately there are very few exercises in the ...
0
votes
2answers
124 views

Open Problems for High School Students

I am a homeschooled rising senior in high school, and I would like to research an open problem in mathematics. I have taken a number of undergraduate-level mathematics courses, including ...
0
votes
1answer
33 views

Unbounded Operators: Notation?

For continuous a.k.a bounded operators we have $\mathcal{B}(X,Y)$ stressing on boundedness and $\mathcal{L}(X,Y)$ stressing on linearity entailing $\mathcal{C}(X,Y)$. Is there a notation for ...
1
vote
3answers
73 views

Related Methods: Lagrange Multipliers

It really pains me to ask this question, but I was working on an optimization problem and wanted to show a friend how we could also use Lagrange Multipliers to solve it. I was considering the ...
1
vote
2answers
119 views

Linear Algebra without Matrices

How far could one get in linear algebra without matrices? It seems like the more I learn, the less I actually use them, but most of the basic theorems and invariants that learned first -- and still ...
3
votes
0answers
79 views

Is there any visual animation to show the basic concept of algebraic geometry? [closed]

Is there any visual animation to show the basic concept of algebraic geometry? There are rarely pictures in textbooks, so are there any animation to show basic but important concepts?
5
votes
1answer
69 views

Defining algebras over noncommutative rings

A definition of "algebra" (that is, an associative algebra, in the sense of ring theory) generally requires a commutative base ring. But there are cases where it's reasonable to consider algebras ...
2
votes
1answer
111 views

Changing streams in PhD

I've a masters degree from a reputed Indian university in pure mathematics, with a specialization in Algebraic Number Theory. However, I'd like to apply for a PhD in computational math/theoretical ...
3
votes
2answers
145 views

Bridging the gap of understanding function terminology in math for a programmer.

I'm a computer programmer by profession with no formal CS education. When I read in mathematics the terminology used around a function, I get confused. For example, I was reading up on some calc and ...
9
votes
3answers
213 views

Areas of contemporary Mathematical Physics

I have often heard that some developments in Physics such as Gauge Theory, String Theory, Twistor Theory, Loop Quantum Gravity etc have had a significant impact on pure Mathematics especially geometry ...
4
votes
3answers
348 views

Interesting Mathematical Fallacies [duplicate]

I recently volunteered to help with a summer math program at a local high school for which I thought would be a breeze. As it turns out, it isn't a program for those catching up (summer school) like I ...
27
votes
15answers
3k views

Ways to study mathematics while commuting

I spend approximately 3 to 4 hours on public transport everyday. I try to maximize the usage of this time by checking email etc on my phone. Are there any tips to study mathematics while commuting? ...
7
votes
3answers
148 views

Algebraically, What Does $\Bbb R$ get us?

In terms of the basic algebraic operations -- addition, negation, multiplication, division, and exponentiation -- is there any gain moving from $\Bbb Q$ to $\Bbb R$? Say we start with $\Bbb N$: ...
0
votes
0answers
41 views

Why Does $e^{ix}=\cos(x)+i\sin(x)$? [duplicate]

Something I've always wondered, but never had a good answer too (I accept there may not be one). I fully understand how to derive this, so I'm not looking for an analytic proof. But rather I cannot ...
2
votes
1answer
111 views

Best Less-Famous Texts for Forcing

There are many books, papers and lecture notes which give an introduction to forcing (e.g. Jech or Kunen's books) but here I am looking for some possibly less-famous useful comprehensive texts for ...
6
votes
6answers
251 views

Interviews of famous modern mathematicians

I was wondering, are there any good collections of interviews of famous modern mathematicians? It can be text interviews, or audio or video recordings. I am not sure what exactly I mean by "modern". ...
1
vote
3answers
57 views

Considering Vectors Geometrically

I have a few questions which a little research (searching the internet through Google) has not satiated. It seems that vectors are very important, even when considering them as the arrows which ...
4
votes
1answer
129 views

Humor in Math Textbooks

So, I was looking though a problem section in Dummit and Foote, and found this amusing "definition." I actually tried googling it, but I can't find any reference to these associated primes being ...
26
votes
8answers
2k views

Complex analysis is more “real” than real analysis

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
1
vote
2answers
126 views

Help Writing a PDF for a Math Course

I have to write up notes on a math course and would absolutely love to make my notes look very similar to this: Using libreoffice, with texmaths, does anyone know settings, i.e. text format, page ...
2
votes
1answer
57 views

Rigorous Book on Topology of Surfaces and Simplicial Complexes

In the undergraduate course I am on the topology course covered the basics of point set topogly very well and everything was done, but then moved on to simplicial complexes and surfaces which was very ...
0
votes
1answer
22 views

Definition of linear subspace

Let $k<d\in\mathbb{N}$. Given the following definition: $G=\{ S: S\text{ is }k\text{-dimensinal subspace of }\mathbb{R}^d\}$ Would you understand that $G$ contains only "homogeneous linear ...
3
votes
5answers
134 views

Axioms of Euclid

The axioms of Euclid are : Things which are equal to the same thing are also equal to one another. If equals be added to equals, the wholes are equal. If equals be subtracted from equals, the ...
5
votes
2answers
122 views

How much time per day to mathematicians usually spend working?

I was reading Poincare's wikipedia page and I noticed that Poincare only did 4 hours of hard mathematical research a day, preferring to let his subconscious have the rest of the time to attack the ...
0
votes
0answers
68 views

Multivariable calculus and real analysis in one semester. What is the best way to study for such course?

I am in my first year of EECS and planning on taking a lot of maths classes. I have already taken single variable calculus and linear algebra and did well in them and decided to take multivariable ...
-6
votes
1answer
139 views

Why $\epsilon$ is always small? [closed]

Whoever invented this, he must be not aware of democracy. I want $\epsilon$ to be large not small. Also, why $\theta$ is an angle?
15
votes
4answers
2k views

Is there a(n elementary) function whose derivative we cannot integrate?

Say, for example, I take a reasonably-complicated function $f(x)=\tanh[\ln(x^x)]$, and differentiate it to get $$f'(x)=\frac{4x^{2x} [1+\ln(x)]}{(x^{2x}+1)^2}.$$ Now, to integrate this, I imagine, ...
0
votes
4answers
633 views

Prove that every irrational numbers can be approximated by rational numbers. [closed]

Prove that every irrational numbers can be approximated by rational numbers. How can I do it? Ok, I admit. I heard it, I thought it is to be true. And I was a kid. Now I when I think about it, I ...
6
votes
1answer
56 views

What is the difference between a calculus and an algebra? [duplicate]

You can have a lambda calculus, the calculus of the real numbers or a logical calculus but on the other hand you could also have an algebra of sets, a Lie algebra, or a linear algebra. Is there any ...
0
votes
1answer
74 views

Algebra books for olympiad preparation

I was looking for some good books for algebra and number theory at the olympiad level. Does anybody have any suggestions? I specifically want books that work on techniques and concepts (not just ...
4
votes
3answers
282 views

When does injectivity imply surjectivity?

I'm aware of the existence of this question: Surjectivity implies injectivity However, the question is regarding a finite set $S$. I was wondering, though: What happens when $S$ is an infinite set? ...
0
votes
0answers
35 views

Why the need for a proof of the Collatz conjecture [duplicate]

I am just a mathematics student and not a professional so my knowledge is limited regarding the Collatz conjecture. But I struggle to see what could be gained from the proof of the conjecture? Is this ...
0
votes
1answer
21 views

Proof of Equality with Mixed Partials

Here is the link: Hessian I understand everything but this line: $$g(x_0 + \Delta x) − g(x_0) = \frac{dg}{dx} (ξ) \Delta x$$ i.e., $$S (X_0, \Delta x, \Delta y) = \frac{∂φ}{∂x} (ξ, y_0 + \Delta y) − ...
10
votes
1answer
111 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponenent allows for convenient ...
6
votes
3answers
223 views

Number Theory Reading List

What are the essential number theory texts that every serious student of number theory should read?
6
votes
2answers
102 views

Advice for self-studying Inequalities and Calculus

I'm interested in self-studying the following books over the next year or so: Spivak's Calculus (I'm already in Ch. 5 and it is very slow going) The Cauchy-Schwarz Master Class by J. Michael Steele ...
2
votes
4answers
344 views

What is your intuitive understanding of infinity? [duplicate]

What is your intuitive understanding of infinity? Mine is the following, I prepared it as image: Those were the main points I got to after thinking by myself about what infinity is, without ...
1
vote
1answer
68 views

“Geometric” proof of Rouche's theorem on the number of zeros?

I understand the analytic proof of Rouche's theorem as presented in Stein and Shakarchi's complex analysis - $|f(z)| > |g(z)|$ on the boundary circle C ensures that the argument principle can be ...
3
votes
1answer
83 views

Real analysis with a non-standard topology

I have recently undertaken a self study of topology and am using Munkres Topology 2nd edition as the primary text. My background(theoretical chemistry & physics) is almost entirely void of any ...
0
votes
0answers
23 views

How to find out convergence of sequence of functions in short time.

I was studying about sequence of functions and about uniform and pointwise convergence of these series. To prove or disprove the convergence of sequence of functions we have several methods like by ...
0
votes
1answer
49 views

Writing a chain of implications in English

How to write a theorem of the form $A\Rightarrow B\Rightarrow C\Rightarrow D$ where every $A$, $B$, $C$, $D$ are formulated with words (English) rather than with formulas? One idea: The next item of ...