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4
votes
0answers
280 views

Isolation and self-study

A little background: I am currently a sophomore (studying mathematics) at an unknown university in the Middle East. My mother is European so it does not make sense to study mathematics in the Middle ...
1
vote
5answers
138 views

Ok, I know what does linear independence mean but why should I care?

I understand that for a set of vectors to be linearly independent, none of the vectors in the set should be a linear combination of some other vectors in that set. But why on earth should I care about ...
8
votes
0answers
175 views

Intuition behind Erdős proof of the infinitude of prime numbers

Suppose by contradiction that there are finitely many primes, namely $p_1, p_2,...,p_k$, where $k$ is a natural number. Now consider another natural number $n$, and all natural numbers $m \leq n$. ...
1
vote
0answers
23 views

Crunching some data on grades

Given a set of data that is not already paired but is known to not be independent, are there any valid methods of statistical analysis one can perform? The purpose of my analysis is to compare ...
14
votes
3answers
1k views

Improving concentration and stamina when solving difficult problems.

I am trying to improve my problem solving skills by solving olympiad problems (Putnam, IMO, etc). So far, I have discovered that problem solving is somewhat like panning for gold: you think of all the ...
6
votes
3answers
247 views

Career advice: Mathematical neuroscience

I need some advice about my career. Currently, I'm an undergraduate student of math. Since I can remember, I wanted to be a scientist, so I decided to go for applied math. The field that I'm ...
7
votes
3answers
97 views

Why do we say $n$ distinct points?

" Let's say we have $n$ distinct points... " , you see this every time you open a geometry textbook. Why not just $n$ points ? If the points are not distinct, they are not exactly $n$ points, are they ...
1
vote
1answer
147 views

Are there other models for 2 dimensional hyperbolic geometry?

I was a bit browsing the internet for models for (2-dimensional) hyperbolic geometry. and realised that besides the well known Poincare half plane model Poincare disk model Beltrami-Klein disk ...
3
votes
3answers
269 views

How to explain why free loop space, or based loop space, is infinite dimensional to non-math people.

I am giving a math talk to non-mathematicians. I was wondering how to explain how the free loop space, or based loop space, of a topological space is infinite dimensional so that a non-mathematician ...
0
votes
2answers
787 views

Practical use and applications of improper integrals

What are the most important applications of improper integrals, in particular to computer science and related fields, and to technology and engineering in general? I know that improper integrals are ...
1
vote
1answer
73 views

When is proof by contradiction necessary?

Is there a way to test a given statement can't be proved directly? Thanks.
4
votes
1answer
209 views

Do the things that you don't know in mathematics frighten you? [closed]

I have zillions of things that I don't know in mathematics. I feel I would never know any of them completely. Especially after this age (26)... and I immediately run away since I am a perfectionist. ...
0
votes
1answer
32 views

Sets of Functions

Please provide feedback to my answer to this question. Question: For all sets A,B,C if A contain in B, B contain in C and C contain in A , then A=B=C. Answer: True, since; If we let x be element of A, ...
1
vote
1answer
87 views

Prove that $\int_{-\pi}^{\pi}$ $\frac{d\theta}{1+\sin^2\theta}$ = $\pi\sqrt{2}$ using the method of Residues

Prove that $$\int_{-\pi}^{\pi}\frac{d\theta}{1+\sin^2\theta} = \pi\sqrt{2}$$ using the method of Residues How do I do this? I know I need it from $0$ to $2\pi$ but I don't know how to modify it!! ...
0
votes
0answers
28 views

Counting the roots of nonlinear systems of equations

I have a "nice" function (vector field) $$\mathbf{f}: \mathbb{R}^n \rightarrow \mathbb{R}^n$$ and I need to find how many roots (zeros) it has in a certain domain (hopefully prove that it has at most ...
0
votes
1answer
36 views

How can I prove that $_{max}|Az^{n}+b|$ =$ |A|$ + $|B|$ when |Z| $\leq$ 1?

How can I prove that $_{max}$$|Az^{n}+B|$ = $|A| + |B|$ when $|Z|$ $\leq$ 1? Remember that Z is a complex number which is why I had to include the magnitude.
0
votes
1answer
132 views

How to explain why one should use lowercase letters for variable names?

How can I explain to an Algebra I student why he should use lowercase letters when naming his variables (i.e. $q = $ number of quarters $vs.$ $Q = $ number of quarters? I am not interested in the ...
1
vote
0answers
98 views

Formal proof of famous theorem

I'm looking for a nice example of a formal proof of some well-known mathematical fact. I know about Mizar project, but I'd rather prefer something like this nice proof of $1+1=2$ which uses ...
20
votes
1answer
613 views

Why do we study Polish spaces?

In descriptive set theory, a lot of space is devoted to properties of Polish spaces. (A Polish space is a topological space, which is separable and completely metrizable.) I would like to know why ...
0
votes
1answer
53 views

What are the best ways to prepare one's self for introductory classes in proofs, analysis, and modern algebra?

I'm a second-year college student, coming from a mathematical background that includes everything up to differential equations, linear algebra, and a survey discrete mathematics. How might I go about ...
2
votes
3answers
653 views

Most important things to be proficient in before Calculus 1?

What are the main things one should be proficient in before taking Calculus 1? Please be specific.
8
votes
0answers
130 views

Galois Group of Composite Field vs. Second Isomorphism Theorem

$\DeclareMathOperator{\Gal}{Gal}$ In my abstract algebra class, we learned about how Galois groups interact with composite fields. Namely, if $K/F$ is Galois, and $L/F$ is any extension: $$\Gal(KL/L) ...
2
votes
2answers
61 views

Quick question about notation and pronunciation of indices: (i+1)st or (i+1)th?

I have a somewhat silly question: What are you calling the index after the $i$th index? The $i+1$st or the $i+1$th? How do you pronounce it and how do you write it down? I like to write $i+1$ in ...
5
votes
2answers
201 views

Finite Model Theory

It seems that finite model theory is regarded (in a sense) as a computer theoretic subject. Is this the case or are there questions of interest that are of interest to mathematical logicians or more ...
13
votes
1answer
311 views

l'Hopital's questionable premise?

Historians widely report that l'Hopital's 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes contains a questionable premise expressed by an equation of type $x+dx=x$ ...
1
vote
4answers
79 views

What Type of Question is this??

Question: "Express the volume of a sphere as a function of its surface area." I know how to do that type of questions, but is there a specific name for that type of questions?
9
votes
1answer
204 views

Will learning category theory lead to a better and clearer understanding of mathematics?

I read the first chapter on a book about category theory Conceptual Mathematics:A first introduction to categories.In the preface the authors say: It has been the good fortune of the authors to live ...
0
votes
1answer
72 views

How to find the smallest prime divisors of $2^{19}-1$ and $2^{37}-1$?

How to find the smallest prime divisors of $2^{19}-1$ and $2^{37}-1$ ? I'm new to elementary number theory and I'm not sure what to do AT ALL. We're currently studying primitive roots and indices.
4
votes
2answers
75 views

Theorems which are proven by proving the existence of a formal proof without knowing the formal proof

Let $L$ be a first order language. Let $T$ be a set of sentences in $L$ and $S$ a sentence in $L$. Let's define a meta-proof to be a proof that there exists a formal proof of $S$ from $T$. Question: ...
0
votes
0answers
59 views

Invalid use of the analytic continuation of the Riemann zeta function?

Watching this video on You Tube I got the impression that some sciences (in this case physics) use the analytic continuation of the Riemann zeta function without justification. Maybe this is just my ...
1
vote
0answers
18 views

Compare Unequal Entities

This is more of like an analytic question. It could be vague but let me try to narrow it down a little. Assuming, i am working on an analysis in primary education which include factors like ...
2
votes
3answers
82 views

How 'normal' are normal spaces?

Is normality a property that is 'easily' satisfy by a given space? I mean by this: are non-normal spaces hard to construct/unnatural compared to normal spaces?
2
votes
1answer
86 views

Turing degrees of models of ZFC and naming big numbers

Before asking my question, let me give some motivation which could help getting better answers. In this MO question, Scott Aaronson was trying to use the concept of "definable number" to create ...
2
votes
3answers
523 views

How to define the magnitude of a rotation in $\mathbb{R}^n$?

Is there some well established way on how to quantify rotations in $\mathbb{R}^n$? To say which rotation is greater and which is smaller? In $\mathbb{R}^2$ the rotation is characterized by a single ...
70
votes
24answers
11k views

Mathematicians ahead of their time?

In every field there's always that person who's just years ahead of their time. For instance, Paul Morphy (born 1837) is said to have retired from chess because he found no one to match his technique ...
0
votes
1answer
23 views

Pairing of Sigma and Pi in notation

Why do $ \Sigma $ and $ \Pi $ appear together so frequently in related topics. For example: Product and Sums Borel sets, such as $ \Sigma_0^1, \Pi_0^1 $ First order formulas, such as $ \Sigma_1, ...
7
votes
2answers
658 views

Why does probability change as you change perspective?

I was trying to solve the following question: Out of 2 Boys and 2 Girls, two students are chosen to advance to the next level. What is the probability that two girls advance to the next level ...
7
votes
2answers
319 views

Good book on integral calculus (improper integrals, integrals with parameters, special functions)

Can you recommend a good book (with theoretical results with proofs, and with plenty of solved problems and examples) on the topics of improper integrals, (improper) integrals with parameters, special ...
2
votes
0answers
62 views

Instructive video content for High School kids?

I need some math Youtube channels (or any other visual media, movies maybe...) that I can recommend to High School students, not solely as a method of learning math but more to illustrate the beauty ...
1
vote
1answer
90 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 ...
4
votes
2answers
77 views

density theorems and class field theory

am I correct in thinking that the frobenius density theorem (it says that the Dirichlet density of the set of primes of K that split completely in an extension L is 1/[L:k]) is sort of one of the main ...
3
votes
1answer
241 views

Book recommendation in Foundational Mathematics

I have been navigating in this "foundational world" of mathematics for a while now ( but certainly not long enough and not deep enough ) and have read a bit about many different topics : set theory, ...
6
votes
1answer
398 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 ...
15
votes
2answers
1k views

So can anybody indicate whether it is worthwhile trying to understand what Mochizuki did?

So I am looking at some math stuff and I start looking at the abc-conjecture. Naturally I run into the name Mochizuki and so start trying to see what he did. Well, he is starting look like another ...
1
vote
2answers
100 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 ...
1
vote
2answers
82 views

Locally small category whose collection of isomorphism classes cannot be a set

For natural examples of locally small categories like the category $\mathbf{Grp}$ of groups, the isomorphism classes themselves are normally not sets. In a set theory like ZFC, even the collection of ...
1
vote
1answer
75 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!
1
vote
1answer
56 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
votes
3answers
80 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 ...
8
votes
2answers
187 views

Theory vs problems in modern math

Quick background: I'm a fourth year undergraduate entering graduate school next year. I am trying to identify areas of mathematical research in which there tends to be more emphasis on developing new ...