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14
votes
2answers
130 views

Bellard's exotic formula for $\pi$

In his webpage, Fabrice Bellard mentions an exotic formula for $\pi$ as follows $$\pi = \frac{1}{740025}\left(\sum_{n = 1}^{\infty}\dfrac{3P(n)}{{\displaystyle \binom{7n}{2n}2^{n - 1}}} - ...
5
votes
4answers
390 views

Is there any published research on the value of finding new proofs for old theorems?

There have been many conjectures in history of mathematics that some of them after passing long journey have resulted in lengthy and high-level-math proofs. Perelman's proof on the Poincare's ...
3
votes
1answer
65 views

Mathematical importance of the golden ratio [duplicate]

I know the golden ratio is the limit of the ratios of consecutive Fibonacci numbers and that it appears when studying many related combinatorial objects (such as the sequences of zeros and ones with ...
8
votes
1answer
163 views

How to begin self study of Mathematics?

I'm aware that this question has been asked several times, but I have specific questions hence why I'm asking again. I began to appreciate the beauty of mathematics when I glossed over the ...
1
vote
4answers
170 views

Applications of algebraic topology?

Terribly sorry if this has been asked, but I'm not about to search 382 pages of technical questions in the field. I am trying to develop a very basic understanding of what algebraic topology is ...
11
votes
2answers
502 views

How to introduce type theory to newcomer

I want to introduce (dependent) type theory to some friends having background in mathematical logic and set theory. To make this introduction easy I would like to give an informal presentation that ...
1
vote
2answers
57 views

suggest an elementary text in analysis

I have just completed my undergraduate course in mathematics but I don't feel better in analysis, there is a mugup of books in my book collection But don't know what to choose who will help me to ...
5
votes
2answers
111 views

A shirt with the imprint of a formula.

Before I began to study mathematics, a friend of mine bought me a shirt with the imprint of a formula. I did not know what these characters were and had no desire to think about it. Yesterday, I ...
6
votes
2answers
110 views

What does “rigorous proof” mean?

I have heard several times that some mathematician has given another and more rigorous for an established theorem, but I don't know what does it really mean and what differences makes it to be more ...
0
votes
1answer
419 views

Question about the mathematics in actuarial studies

I tried Google but there isn't much information on this and I would really like some insights into actuarial studies, the mathematics involved and how it compars to the mathematics in a bachelor of ...
2
votes
1answer
58 views

How far can I get with graph theory?

I am an undergraduate who had recently finished his $2$nd year. I was wondering how far can I get with Graph Theory this summer. I am studying from Bondy & Murty's book. I already finished ...
0
votes
1answer
43 views

What is the best book for coordinate geometry?

Requirements: A) not too thick , as I am reading this only to solve calculus problem . B) free on web is the best (optional) C) I don't mind if the book involves both coordinate geometry and ...
6
votes
6answers
3k views

How to interpret material conditional and explain it to freshmen?

After studying mathematics for some time, I am still confused. The material conditional “$\rightarrow$” is a logical connective in classical logic. In mathematical texts one often encounters the ...
8
votes
0answers
315 views

Overview of nonlinear analysis, differential equations (ODE and PDE), dynamical systems, and mathematical physics, and their relationships [closed]

The fields of (i) nonlinear analysis, (ii) ODE and PDE, (iii) dynamical systems, and (iv) mathematical physics (e.g., electromagnetism, general relativity, gravitation, etc,...) are very huge, ...
2
votes
2answers
128 views

Why is ${n\choose k}$ is always a product of the primes of $n$ for all $n>k$? [closed]

Let $n, k$ be two positive integers such that $n>k$. Why is ${n\choose k}$ always divisible by a prime dividing $n$ (or even a product of such primes)? Please help me understand why. I cannot seem ...
1
vote
1answer
42 views

Advice Rudin PMA

I'm currently working through Rudin's Principles of Mathematical Analysis. My background consists of mainly working through most of Apostol's Caluculus Vol. 1, Velleman's How to Prove it, Lang's ...
1
vote
0answers
22 views

Can we do better than zero padding of FFT?

My background is in signal processing and never took any course related to functional analysis or even advanced algebra. But I have a strong conviction (may be wrong) that we may be do better then ...
805
votes
29answers
52k views

Can I use my powers for good?

I hesitate to ask this question, but I read a lot of the career advice from MathOverflow and math.stackexchange, and I couldn't find anything similar. Four years after the PhD, I am pretty sure that ...
6
votes
1answer
152 views

Where is Cauchy's wrong proof?

Allegedly, Cauchy mistakingly "proved" that pointwise convergence of continuous functions is continuous. I saw this somewhere in a book, and it is also in wikipedia: Uniform convergence. In his ...
0
votes
2answers
36 views

Difference between topology and sigma-algebra axioms.

One distinct difference between axioms of topology and sigma algebra is the asymmetry between union and intersection; meaning topology is closed under finite intersections sigma-algebra closed under ...
1
vote
2answers
57 views

Integration by parts done fast

Even though I've been through countless instances where I needed to use integration by parts, to this day, I still derive it from the chain rule, identify my 'parts' and apply the formula. At some ...
6
votes
2answers
198 views

Why are square functions important in analysis?

I have been reading through chapter 1 of E.M. Stein's textbook Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. In chapter 1, Stein discusses the relationship ...
1
vote
1answer
25 views

Limit(s) of a Sequence from the decimal expansion of $\pi$

I found a statement in a book concerning the decimal expansion of $\pi$ that I do not really understand. The statement is my problem number 2, where problem number 1 really looks like a reference ...
12
votes
2answers
4k views

Why is a straight line the shortest distance between two points?

The first application I was shown of the calculus of variations was proving that the shortest distance between two points is a straight line. Define a functional measuring the length of a curve ...
9
votes
1answer
296 views

Balancing Coursework with original research

I am a first-year graduate student in a US university pursuing a PhD in mathematics. I am a bit frustrated in trying to balance coursework with original research. I saw students who spend most of ...
1
vote
0answers
15 views

Prequisites for a PDE course (Strauss)

This question is quite general. In four days I will enroll in a PDE course which will use Strauss as the textbook. However, today when I checked the course description I found that 'multivariable ...
12
votes
9answers
4k views

How to become proficient in Calculus? [closed]

It has been a while since I wanted to ask this question, but couldn't find a right forum. My question might come across as trivial, but its important to me to find an answer. Let me give you some ...
5
votes
0answers
60 views

What are some arguments/counterarguments for Zeilberger's “proof certificates”?

Here is the quote I wish to ask about: "I speculate that similar developments will occur elsewhere in mathematics, and will 'trivialize' large parts of mathematics, by reducing mathematical ...
5
votes
1answer
597 views

Why is unique ergodicity important or interesting?

I have a very simple motivational question: why do we care if a measure-preserving transformation is uniquely ergodic or not? I can appreciate that being ergodic means that a system can't really be ...
16
votes
5answers
998 views

What are some examples of induction where the base case is difficult but the inductive step is trivial?

According to Wikipedia: ...proofs by mathematical induction have two parts: the "base case" that shows that the theorem is true for a particular initial value such as n = 0 or n = 1 and ...
10
votes
3answers
947 views

Applications of the Hahn-Banach Theorems

Question: What are some interesting or useful applications of the Hahn-Banach theorem(s)? Motivation: Most of the time, I dislike most of Analysis. During a final examination, a question sparked my ...
17
votes
1answer
2k views

Can an older person become an expert in math?

Is learning math like learning a language, in that there's a certain cutoff age (pretty young) that you need to start learning by, otherwise you'll never be fluent? Is the novice older brain ...
4
votes
0answers
52 views

Experiencing a breakdown [closed]

I have been experiencing some sort of a "breakdown" recently. My brain isn't as fresh as it used to be, and I get very tired after say one hour or two of doing math. Moreover, I became very slow and ...
58
votes
17answers
8k views

Interesting “real life” applications of serious theorems

As a student one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like: "You have a ...
19
votes
8answers
2k views

How do authors make their problems/exercises for their math books? [closed]

I want to be a math professor one day, but I'm wondering how to make my own original problems to give them to my students. I think that it is a responsibility of the professor to create original and ...
3
votes
2answers
64 views

Is a simple derivation for a formula publishable?

I'm a high school student and recently discovered an interesting method of deriving the definition for $\arcsin z$ for the complex domain using integration. I typed it up for fun in LaTex and am ...
40
votes
14answers
3k views

What are “instantaneous” rates of change, really?

This is driving me crazy, I'm literally losing sleep over it, please help me resolve this confusion. Here's how I see it (please read the following if you can, because I address a lot of arguments ...
0
votes
1answer
93 views

Applications of $p_{n+2}+p_{n+1} \le p_1p_2…p_n , \forall n >2$?

Let $p_n$ denote the $n$-th prime number ; I know that $p_{n+2}+p_{n+1} \le p_1p_2...p_n , \forall n >2$ . I am looking for some applications of it , for example I know one application of it ...
3
votes
0answers
27 views

It is possible to talk about the degree of a transcendental equation?

When we deal with algebraic equations involving polynomial and so on we know what the degree of the equation is and this tells us how many solutions we'll find (at least in complex numbers). But this ...
5
votes
1answer
1k views

Journals that publish papers quickly

I have written two papers in Mathematics and want to get them published. Can you suggest some journals that publish quickly? Besides, how can I know if a journal is well-regarded or not? I know ...
1
vote
0answers
35 views

What is the connection between game theory and (modal) logic?

I'm interested in dynamic epistemic logic lately (reasoning about information and change in multi-agent systems). I also like game theory. I'm looking for some good resources about the connection ...
4
votes
1answer
71 views

Introductions to Ehresmann connections and Chern-Simons forms

I am looking for introductory texts on Ehresmann connections and Chern-Simons forms. I seek detailed, hands-on presentation. Please, recommend sources that employ a differential forms approach rather ...
20
votes
2answers
347 views

How to ask dumb questions [closed]

I am having trouble asking questions in seminars, conferences, and public talks. As a graduate student I often fail to keep up with the speaker and more mature members of the audience at research ...
37
votes
10answers
3k views

Mathematical literature to lose yourself in

H.M. Edwards in the preface to his book on the Riemann Zeta Function, summarises his philosophy on learning Mathematics: ...I have tried to say to students of mathematics that they should read the ...
0
votes
2answers
39 views

How to do multiplication (capital pi) in WolframAlpha?

How do i ask this in WolframAlpha: $$\prod_{i=0}^{i=10} \sin{(i)}$$ I used $\text{product}(...)$ and $\text{multiply}(...)$ or even $\text{multiplication}(...)$ but they don't seem to work. I am ...
20
votes
10answers
461 views

What are the theorems in mathematics which can be proved using completely different ideas?

I would like to know about theorems which can give different proofs using completely different techniques. For example: When I read from the book Proof from the Book, I saw there were ...
1
vote
0answers
44 views

Why is differential equations typically taught with an applied focus? [closed]

Out of all the maths I've learned, the terminology and examples given in differential equations lectures always grinds my gears. It feels like a rebellion against pure math, haha.
0
votes
2answers
39 views

Differential Equations vs Analysis I [closed]

I just got done taking Multivariable Calc and I have room for either Differential Equations or Analysis I for the next summer session. These 15 week courses are condensed into 7 weeks, which one would ...
6
votes
4answers
216 views

Examples of arguments from connectedness

Suppose $X$ is a connected topological space. A typical way that we prove a property $P(x)$ holds for all $x \in X$ is to show that $P$ is an open and a closed condition, and that $P(x)$ for some $x ...
3
votes
1answer
68 views

Soft Question about International Graduate Schools [closed]

Hello members of this site! I hope this question isn't too bothersome, but I need to ask some professionals who know what they're doing. Furthermore, I know that this is supposed to be a more ...