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1
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1answer
100 views

What path to get to topology?

I am in calc 1 right now and was wondering what kind of journey is ahead of me before topology. I really want to study high level math like this but am not sure if I want to major in math. I am a pre ...
1
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0answers
27 views

Plotting the origin

I remember my high school teacher telling me that when I graph a function, I should always plot the point $(0, 1) $, $(1, 0) $ and the origin, $(0, 0) $. Always graphing the first two makes perfect ...
2
votes
2answers
141 views

How an axiomatic system is made?

An axiom is a sentence that is taken to be true without a proof. A set of (well organised) axioms is called an axiomatic system. As consequence of these axioms we get a lot of results that we call ...
6
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1answer
1k views

Introduction to Computational Topology

Question: Besides generally learning Algebraic Topology, what are some prerequisites for studying Computational Topology? Are there any accessible papers which introduce the field and the methods ...
6
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1answer
571 views

What are the active branches of number theory?

Context: I am a junior math major and am hoping to go to grad school after next year for a PhD. I have completed most of the standard undergraduate courses and have been consistently most interested ...
1
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1answer
83 views

Arthur Milgram photo

someone has photo of Arthur Milgram? (the guy from Lax-Milgram Theorem) I'm curious, why I can't find one picture of Arthur Milgram on google? Anyone has one? or where can I find one? Thanks!
21
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4answers
3k views

Is all of mathematics based on logic?

I do not understand the discipline in mathematics that is called "mathematical logic". For me, all of mathematics is based on logic and that is what makes it the exact science. Every theorem or lemma ...
1
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1answer
64 views

Do zeros present along the diagonal yield complex eigenvalues?

I was told today by a friend that having a zero along there main diagonal of a matrix will promote complex eigenvalues. I do not believe this is true because the below matrix Z has a zero present ...
0
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3answers
354 views

Not sure how to find the limit of this inequality?

I'm trying to solve the limit of this inequality. The question goes as follows: If $$4x - 9 \leq f(x) \leq x^2 - 4x + 7$$ for $x \geq 0$, find $\lim_{x\to 4} f(x)$. I'm not really sure how to go ...
0
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1answer
45 views

Does relative positive definiteness have a good name?

Often, when there are two matrices $A$ and $B$ such that $A-B$ is positive definite, this relationship is written $A \gt B$, $A \succ B$ or similar. It has the advantage of being consistent with the ...
1
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3answers
184 views

Are truth tables a valid method to prove an iff statement?

I recently had a homework assignment returned to me (for a Differential Geometry course, undergrad level) in which my instructor wrote "You cannot use truth tables to prove an if and only if statement"...
1
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0answers
101 views

Uses of the Heine-Cantor theorem?

Heine-Cantor Theorem: Let $[a,b]$ be a compact interval, $f:[a,b]\to R$ be a continuous function. Then $f$ is uniformly continuous, i.e. $\forall \epsilon>0 , \exists \delta>0$ such that $|x-y|&...
3
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1answer
89 views

Enlightening books giving a guided tour of mathematics, in a style that Gian-Carlo Rota would not mind?

I am currently reading Gian-Carlo Rota's Indiscrete Thoughts. What more can I say apart from "the man can write"? (In other words, you should really read it if you are interested in mathematics.) I ...
0
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1answer
51 views

Contour Integrals, why give them a name?

I must ask because I can't find an answer, it's a bit of a soft question and for that I apologise. Why are they called contour integrals? In all the subheadings I've checked there is no special ...
10
votes
1answer
741 views

Prerequisites for understanding G.H. Hardy's 'Divergent Series'

I picked up a copy of G.H. Hardy's 'Divergent Series' a few days ago. So far I love it, as I love the ideas associated with sequences and series, but I am finding it a bit difficult to understand. I ...
13
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7answers
1k views

Beautiful Theorems and what constitutes as beautiful [closed]

I often hear the phrase "mathematical beauty". That a proof or formula or theorem is beautiful. and I do agree I was awestruck when I first saw Euler's formula, connecting 3 seemingly unrelated ...
0
votes
1answer
50 views

Other conditions than necessary and sufficient conditions, $\Rightarrow$, $\Leftrightarrow$?

I know that $$A\Rightarrow B$$ means that $A$ is a necessary condition for $B$ and $B$ is a sufficient condition for $A$. Also, $$A\Leftrightarrow B$$ means that $A$ is necessary and sufficient ...
1
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0answers
518 views

Standard notation for the indicator function of the odd integers

Is there a commonly used notation for the indicator function of the odd integers? One candidate I think is $$1_{2\mathbb{Z}+1}(x),$$ and I could always define one, $$\chi(x)=\left\{\begin{array}{ll} 1 ...
9
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3answers
1k views

Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
5
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2answers
162 views

Geometry after Khan academy's tutorials

I always liked geometry at school, so by the side of my normal studies I've been going through the Khan academy videos. Could anyone suggest some good books that takes these geometry topics further? I'...
2
votes
1answer
54 views

Real or imaginary eigenvalues?

The question I have been lost in for a while is when will a matrix have either all real or complex eigenvalues? (Depending on dimensions of the matrix in question, complex and real eigenvalues may ...
7
votes
3answers
169 views

Is there a better way to read proofs?

I'm finishing my undergraduate degree in 6 weeks and I'm pretty happy with how my education is coming along so far. I can write proofs, solve many different problems, and I even have some idea as to ...
10
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5answers
3k views

What is Fourier Analysis on Groups and does it have “applications” to physics?

I am trying to be as specific as possible, but I am extremely unclear about this topic (Fourier Analysis on Groups). In Reed-Simon Vol II (Fourier Analysis, Self-Adjointness) there is some ...
1
vote
1answer
29 views

Would every half angle of an angle in each quadrant be in the previous quadrant?

For example, take (5pi)/4 which is in Q3, it's half angle is (5pi)/8 which is in Q2. Is this true for every angle?
4
votes
6answers
606 views

Jumping back into Calculus III

At the age of 30 I am going back to school for Electrical Engineering. Because of the way higher education works, all of my previous college coursework is being transferred, which does not allow you ...
19
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8answers
2k views

When is something “obvious”?

I try to be a good student but I often find it hard to know when something is "obvious" and when it isn't. Obviously (excuse the pun) I understand that it is specific to the level at which the writer ...
2
votes
1answer
54 views

Possible advantages and disadvantages of: Read entire section then attempt section exercises or attempt section exercises as you read small parts?

I am just curious as to the possible advantages and disadvantages to: Reading the entire section of a textbook then attempting section exercises as opposed to Read part of the section and ...
0
votes
2answers
359 views

Significance of an eigenvector being equal to a unit vector?

I was reading ahead in my math book when I came across a matrix denoted as A = $\begin{bmatrix} 1 & -1 & 0\\ 2 & -2 & 0\\ 6 & 0 & -2\\ \end{bmatrix}$. I then found the ...
3
votes
1answer
184 views

Why the $\zeta$ letter is like this? [closed]

How to write the sixth letter of the Greek alphabet $\zeta$ by hand? I cannot do it. Thanks.
3
votes
0answers
58 views

What should one do if one works on a problem and submits a paper only to find that it is already in some book?

What should one do if one works on a problem and submits a paper only to find from the editor, that the work is already in some book? The work is is not really easy to find on the internet and was ...
14
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2answers
1k views

Surprising applications of cohomology

The concept of cohomology is one of the most subtle and powerful in modern mathematics. While its application to topology and integrability is immediate (it was probably how cohomology was born in the ...
1
vote
0answers
42 views

Shortest path to frontier

I am just finishing my second semester in Abstract Algebra, we covered most of Dummit Foote, and I found part v the intro to Algebraic Geometry interesting so was looking up "roadmaps" to research ...
6
votes
1answer
360 views

difference between sequence in topological space and metric space

Reading the book about topology, I find an interesting difference between two spaces: We use net convergence $\{p_\lambda\}_{\lambda\in\Lambda}\rightarrow p$ in frontier, but use sequence ...
12
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1answer
251 views

Studies on lack of mathematical education

I am looking for studies which compare students who did not receive mathematical education beyond basic mthematics and those that learned maths upto introductory calculus, with the assumption that ...
5
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3answers
434 views

Mental Math Techniques [closed]

What are some interesting mental math techniques that you know? Here's one that I got from my Grandmother who got it from a book: To square a two-digit number (from $26$ to $49$), take the number ...
25
votes
4answers
1k views

Fractional Calculus: Motivation and Foundations.

If this is too broad, I apologise; let's keep it focused on the basics if necessary. What's the motivation and the rigorous foundations behind fractional calculus? It seems very weird & ...
3
votes
1answer
176 views

What are the differences in mathematical notation around the world? [duplicate]

I just learned that $\text{sen}\,x$ is the Portuguese notation for $\sin x$, and I was inspired to ask: What differences are there in how mathematics is written around the world? Note 1: I am likely ...
2
votes
2answers
103 views

Approaches to teaching and learning analysis

I found that studying linear algebra by getting into vector spaces and linear transformations first made things very easy. This is the approach Halmos or Axler, just to name a few, take. IMHO, the ...
2
votes
3answers
231 views

How exactly does the response “infintely many” answer the question of “how many”?

I admit that the level of this question is roughly about middle school, but this is what the question asks: The ratio of nickels to dimes to quarters is 3:8:1. If all the coins were dimes, the ...
11
votes
2answers
202 views

What's so special about $\mathbb N$?

Through recursion, inversion, extension and all kinds of simple shenanigans the successor function leads to definitions of $+, \cdot, /, x^k, \exp, \sin, \partial, \int$ and other special functions. ...
3
votes
1answer
121 views

Mathematics or physics at university

I have a strong interest in maths, and I feel that advanced physics is cool too (although I've only studied classical mechanics at high school, which is kind of boring). So I'm not sure about which ...
3
votes
1answer
119 views

What is the optimal strategy for increasing my 'mathematical maturity'? Depth or breadth? Number theory or measure theory?

My apologies for such a general question, but as a 'math enthusiast', I've often wondered what the optimal strategy is to increase my mathematical maturity. Broadly, should I aim to hunker down & ...
0
votes
1answer
129 views

Preparation for a graduate commutative algebra course based on Eisenbud

I am an undergraduate with two semesters of algebra(groups,rings, Galois theory, etc) under my belt and I am planning on going through Atiyah and MacDonald's book over the summer. Is this sufficient ...
2
votes
0answers
66 views

Revise high school material

Can you suggest me a comprehensive book to revise high school mathematics (up to besic calculus)? It should be extremely clear and complete and "scientific" (not like most high school books). Thank ...
2
votes
2answers
73 views

Material for advanced highschooler

I'm a high school student who just finished elementary school.Though since I was into math I started going through advanced math while I was in elementary school and I pretty much finished most of the ...
14
votes
2answers
444 views

The double factorial notation

The double factorial is defined as $$n!! = \begin{cases} n \cdot (n-2) \cdot (n-4) \cdots 3 \cdot 1 = \dfrac{(n+1)!}{2^{(n+1)/2}((n+1)/2)!} & \text{ If $n \in \mathbb{Z}^+$, is odd}\\ n \cdot (n-2)...
0
votes
2answers
43 views

what are some typical systems of equations generating from practical problems?

I want to know some typical forms of system of equations generating from practical problems in engineering/economics/physics,etc. Some examples or research articles would be good. Specifically, I am ...
1
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2answers
63 views

Cycloidal coincidence?

For the cycloid $$x=a(t-\sin t)\ ,\quad y=a(1-\cos t)$$ we have, as is easily seen, $$\frac{dx}{dt}=y\ .$$ Does this have any geometrical or physical significance? Or is it just a meaningless ...
3
votes
3answers
261 views

Things to do before starting math (or physics) at university

What is better to do before starting a math degree? I was thinking that maybe I should do something like: learning latex learning how to use matlab Any other suggestions?
37
votes
2answers
3k views

Development of the Idea of the Determinant

While I basically understand what a determinant is, I wonder how this idea was developed? What was the principal idea behind its origination? I would like to know this so that I can have a better ...