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2
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1answer
21 views

Sublinear functions on a Riemannian manifold

I would like to know if there is any notion of sublinear function or subadditive function for Riemannian manifolds. Thank you!
0
votes
1answer
75 views

Is it meaningful to search for “elegant” representations of mathematical objects?

For centuries we struggled with the concept of spatial rotations. We used to represent them in many different ways: mostly, Euler Angles and matrices. Those all had drawbacks and failed in specific ...
1
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0answers
40 views

Discretizations of Differential, Geometric and Topological Notions

I have noticed a recurring theme in Graph Theory / Theoretical Computer Science (abbreviated GT and TCS throughout this post) in that notions typically belonging to differential calculus / geometry / ...
1
vote
1answer
131 views

What is the motivation behind the arbitrary union topological axiom?

1. Why is the arbitrary union axiom in the definition of topology necessary? 2. Why is it useful? Why might we expect ("intuitively") that it should be useful? 3. What is the (historical) ...
4
votes
5answers
595 views

How to Axiomize the Notion of “Continuous Space”?

EDIT (to clear up controversy and misunderstandings caused by my poor wording): Historically, Riesz's efforts to try and make rigorous a notion of a "continuous space" (as opposed to "discrete ones") ...
3
votes
1answer
47 views

What is the difference between operator theory and functional analysis?

In my undergrad mind they are the same subject because functional analysis studies functional spaces like Banach and Hilbert spaces. Operators are function, so shouldn't they be the same subject? ...
0
votes
2answers
71 views

Need Guidance for engineer taking rigorous analysis course for first time

I will be taking analysis course in a month from now. Topics are given below. I am doing engineering and had been through calculus courses but nothing like sort of analysis before. Many of my friends ...
37
votes
7answers
2k views

Is speed an important quality in a mathematician? [closed]

Is solving problems quickly an important trait for a mathematician to have? Is solving textbook/olympiad style problems quickly necessary to succeed in math? To make an analogy, is it better to be a ...
4
votes
1answer
165 views

Deceptively simple math conjectures [closed]

Why is it that some mathematical problems with seemingly simple statements end up soliciting extremely complicated and groundbreaking proofs or remain unsolved for extended periods of time? (Ex. ...
0
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2answers
59 views

Self Study of number theory

I have always wanted to learn about number theory. There is actually no one here who can teach me and it's also not in my regular school syllabus, but the greatness of number theory attracts me ...
0
votes
0answers
62 views

When is it true that $V \subseteq \overline V \subseteq U$ will hold for open sets?

Let $(X, \mathfrak{T})$ be a topological space Let $V, U \in \mathfrak{T}$, and suppose that $V \subseteq U$ Then when it is true that $V \subseteq \overline V \subseteq U$, where $\overline V$ is ...
37
votes
1answer
690 views

Mirror algorithm for computing $\pi$ and $e$ - does it hint on some connection between them?

Benoit Cloitre offered two 'mirror sequences', which allow to compute $\pi$ and $e$ in similar ways: $$u_{n+2}=u_{n+1}+\frac{u_n}{n}$$ $$v_{n+2}=\frac{v_{n+1}}{n}+v_{n}$$ $$u_1=v_1=0$$ $$u_2=v_2=...
21
votes
1answer
921 views

What Do Mathematicians Do?

The American Mathematical Society maintains a web page entitled "What Do Mathematicians Do?" which references two interesting surveys. (One of the reference links is broken, but this one works: What ...
4
votes
1answer
51 views

Predicting next headache

I am thinking of making a model or whatever name i don't know. Idea is this that I suffers from headache, say twice in a month. What i want to do is that i want to make some kind of system in which i ...
0
votes
1answer
44 views

Useful Formulas Analysis on Manifold

This is just a reference request. Does anybody has or know a book with a short handed formulary for Calculus on Manifold. I could do it, but surely someone already have done it better than I would. ...
4
votes
2answers
73 views

On the HoTT Cauchy Reals

In the Homotopy Type Theory Book there is a construction given of a kind of Cauchy reals via higher inductive type and the authors remarked, that this construction is preferred to other notions (reals ...
37
votes
3answers
1k views

This one weird thing that bugs me about summation and the like

Most of us know $$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$ Some of us know $$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$ A few of us know $$\underset{j=a}{\overset{b}{\LARGE\...
0
votes
0answers
23 views

The best known bounds for spectral radius of a graph

There are many bounds for the spectral radius of graphs in terms of no. of vertices, maximum degree, chromatic number etc. I wish to know till date what are the best lower and upper bound for the ...
10
votes
1answer
208 views

Are there parts of Integral Calculus that just *have* to be memorized?

Note : In this question I speak more from a calculation/operational point of view, as opposed to a more theoretical (Analysis) point of view. When studying Differential Calculus, I found that ...
0
votes
0answers
34 views

What are some other operators like infinite sums and products? [duplicate]

I've heard of the sigma and capital pi notations (hasn't everyone?), but I know there are some other ones, like the ones signified E, F and K. What are these? Are there any more related to infinite ...
12
votes
2answers
178 views

Relations between definite integrals not having a known closed form

Are there any known cases, when there are two (or more) definite integrals, none of them having any known closed-form expression on its own, but there is still a non-trivial$^\dagger$ elementary ...
12
votes
7answers
7k views

Why don't we use base 6 or 11?

Another question on this site asks why we have chosen our number system to be decimal base 10. There are others asking basically the same thing as well. I'm not really satisfied with any of the ...
24
votes
9answers
17k views

Online Math Degree Programs

Are there any real online mathematics (applied math, statistics, ...) degree programs out there? I'm full-time employed, thus not having the flexibility of attending an on campus program. I also ...
6
votes
3answers
1k views

How to stay academically active during a Mathematics Gap Year? [closed]

This year I started at the University of Cambridge to study Maths. I was very unfortunate to contract a serious infection early on in the term, and as a reult it has been mutually decided between ...
1
vote
0answers
86 views

Why are symplectic manifolds and Riemannian manifolds so different?

They seem superficially similar in some ways 1) both have a given two tensor on the tangent space 2) if we choose a hamiltonian function for the symplectic manifold then both give a canonical ...
4
votes
3answers
206 views

Why study non-T1 topological spaces?

I can understand (somewhat) why one would want to study non-Hausdorff topologies, since for example the Zariski topology is so important to algebraists, and the weak topology generated by lower ...
25
votes
2answers
983 views

Qual question archives?

Qual questions seem like a great way to study for a new topic, since they usually test slightly deeper understanding than typical questions in a textbook. Princeton has this great archive of questions ...
17
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11answers
1k views

Problems that are largely believed to be true, but are unresolved

Are there unsolved problems in math that are large believed to be true, but for reasons other then statistical justification? It seems that Goldbach should be true, but this is based on heuristic ...
70
votes
3answers
20k views

phd qualifying exams

Where can I find phd qualifying exams questions.Is there any website that keeps a collection of such problems? I need it for doing some revision of the basic topics. I know of a book but that do not ...
1
vote
2answers
100 views

What does it mean to know exact value of a number?

If we go to this page, https://en.wikipedia.org/wiki/Percolation_threshold, we find the statement "a tremendous amount of work has gone into finding exact and approximate values of the percolation ...
4
votes
1answer
120 views

What is a number theory book I can read in bed?

I am looking for a good book that is very easy going but not a "pop science" account i.e. something that goes through theory that would be on a basic undergraduate course for someone who finds the ...
18
votes
2answers
989 views

Why should a combinatorialist know category theory?

I know almost nothing about category theory (I have just skimmed the first chapters of Aluffi's algebra book), reading this question got me thinking... why should someone mostly interested in ...
718
votes
53answers
423k views

Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain, but are ...
3
votes
2answers
40 views

Hausdorff dimension via ergodic theory

This is definitely a soft question, but it was recently mentioned to me that one can study the dimension of fractals via ergodic methods. I'm familiar with ergodic theory on about the level of ...
1
vote
1answer
42 views

What is the required group theory knowledge needed to understand Verhoeff's algorithm?

The Wikipedia page tells me I need to understand permutation groups and dihedral groups. Can someone clearly outline what exactly the perquisites of understanding this is and how much time I'll take ...
10
votes
1answer
152 views

Recreational problems in set theory?

Most areas of maths that I can think of have a number of fun, recreational problems that come under their category. Nothing deep: number theoretic stuff in olympiads, integrals, limits, products, ...
0
votes
2answers
86 views

Why are there different metrics? [closed]

This is a general question and I am asking out of curiosity. There are many metrics such as Euclidean norm, sup norm etc. Can you give examples/reasons why we need all these different metric notions?
15
votes
2answers
2k views

Are infinitesimals dangerous?

Amir Alexander is a historian of mathematics. His new book is entitled "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". See here. Two questions: (1) In what sense are ...
1
vote
1answer
52 views

Correct notation for presenting solutions to equations

Let's say I have a cubic equation $(x-a)(x+b)(x-c) = 0$, and I want to represent the solutions to this equation, what is the formal/conventional way that one would arrive and state the solution to the ...
6
votes
4answers
676 views

What is… A Parsimonious History?

Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a ...
42
votes
10answers
5k views

Paradox: increasing sequence that goes to $0$?

It is $10$ o'clock, and I have a box. Inside the box is a ball marked $1$. At $10$:$30$, I will remove the ball marked $1$, and add two balls, labeled $2$ and $3$. At $10$:$45$, I will remove ...
2
votes
1answer
54 views

A nontechnical way to comprehend $\aleph_2$

This is possibly a dumb question, but I do not know where to look for an answer. Without getting technical, one can show why $card(\mathbb{N}) = card(\mathbb{Q}).$ (Typically by showing how the two ...
2
votes
2answers
47 views

Intuition for polynomial bases

In my linear algebra course I stumbled upon the following observations. We have some function $f: \Bbb{R} \to \Bbb{R}$, $f = f(x)$. $f(x)$ may be composed of elementary functions or not, but in ...
11
votes
2answers
660 views

Proving the Riemann Hypothesis and Impact on Cryptography

I was talking with a friend last night, and she raised the topic of the Clay Millennium Prize problems. I mentioned that my "favorite" problem is the Riemann Hypothesis; I explained what it posits ...
0
votes
1answer
29 views

Why do we study specifically 'normed' vector spaces?

When we study vector spaces, it is useful to define a norm on it for countless reasons. I was thinking about this recently and realised Don't all vector spaces have norms on them? If they all ...
1
vote
1answer
66 views

Books about the foundations of (calculus) functions?

I'm looking for a foundational book that builds up ideas like transcendental functions. For example, how the trigonometric functions are truly defined when plotted as continuous functions. I believe ...
1
vote
1answer
124 views

Why am I so bad at math? [closed]

Ever since I was young, I've always struggled at math. Bear in mind I could count before I entered kindgergarden, but even when I had to learn simple things like long division, I was always behind ...
27
votes
4answers
2k views

Is mathematical history written by the victors?

The question is the title of a recent piece in the Notices of the American Mathematical Society, by twelve authors (of which I am one). The contention is that traditional history of mathematics is ...