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5
votes
1answer
136 views

Proof of Faà di Bruno's formula using a convolution identity for Bell polynomials?

I have noticed there is an identity for Bell polynomials that can apply of Faà di Bruno's formula. This is a convolution identity that states: $$ (x \ast y)_n = \sum_{j=1}^{n-1} {n \choose j} x_j ...
2
votes
0answers
23 views

Martin Gardener's best books?

Martin Gardener wrote a lot of books, just to name a few Perplexing Puzzles and Tantalizing Teasers Mathematics, Magic and Mystery Alex's Adventures in Numberland Entertaining Mathematical ...
2
votes
0answers
69 views

Exercises in Topological K-Theory (Atiyah)

I'm currently working through Michael Atiyah's K-Theory. The main problem I'm finding with it is that it does not have any exercises. Does anyone have a good collection of exercises to go along with ...
1
vote
1answer
45 views

Rotmans Advanced Modern Algebra as first algebra book ,opinions?

Ok so I have been trying to learn abstract algebra.I have started with Dummit and Fottes Abstract algebra, but to me it seems to be burdened with informal developments,and proofs which are not ...
0
votes
1answer
46 views

Suggested Reading for Combinatorics

What are some suggestions for texts on introductory combinatorics and its applications? I would prefer if the applications would be to other branches of mathematics rather than outside of mathematics. ...
13
votes
1answer
195 views

Did Einstein introduce anything new to mathematics? [duplicate]

Newton introduced calculus, so I am wondering, did Einstein introduce anything important to mathematics?
1
vote
0answers
28 views

Degree of Map between Pseudomanifold

There are two different ways to define a degree of map. Let $M$, $N$ be smooth, and $M$ is compact, $N$ is connected. If $f\in C(M,N)$, we define $\deg f$ by smooth approximation. [Milnor/Topology ...
1
vote
4answers
68 views

Research ideas involving probability and pretty pictures?

I am interested in probability theory. I love making pretty pictures. There's an excellent probability theory researcher at my school I'd like to approach with a short term project idea, and I would ...
1
vote
0answers
32 views

How to tackle a research journal - level course in Lie Theory and Representation Theory?

I am taking a course in Lie Theory and Theory of Representations this year, where starting from the second lecture, Lie Theory is heavily bundled with Theory of Representations. It is pretty much a ...
0
votes
1answer
39 views

Something about Degree of Map

I have been told that degree of map can be defined by the combinatorial method instead of the differential one. Assume $M$, $N$ is orientiable pseduomanifold, and $M$ is compact, $N$ is connected. Let ...
1
vote
0answers
62 views

How to use a very complicated theorem for proving simpler statements without falling into a loop?

There are some too complicated theorems in mathematics which have very complicated proofs in hundreds of pages. There are few mathematicians who are aware of the entire proof of such theorems in full ...
2
votes
1answer
79 views

How can I learn to recognize the obvious questions?

I have heard and seen several references to "obvious questions", "obvious axioms" and other "obvious" things (I am not referring to obvious results!). Now, in a seminar I am taking, at the end of each ...
2
votes
1answer
115 views

How to define mathematical objects from scratch?

Are there any "Zermelo-type theories" for some other type of mathematical objects except for sets? I.e., axiomatic systems for mathematical objects formalized for first-order logic using a language ...
1
vote
3answers
81 views

Applications of derivatives outside mathematics and physics

I've been teaching calculus for several years and have some doubts about whether derivatives (and integration techniques) of common functions are useful and important outside mathematics and physics. ...
2
votes
4answers
60 views

An Impossible Ratio

I'm facing a bit of a difficulty thinking about the aspect ratio of A4 paper. The beauty of this paper size is that when it is folded in half along the longer side, it becomes A5 paper which has ...
5
votes
1answer
79 views

Research in algebra.

First of all, I don't know if this is the right place to ask about this. If not, please direct me somewhere I can get more help. I like algebra alot as a mathematics undergraduate student on his 3rd ...
2
votes
0answers
60 views

First hand written draft Or LaTex typed draft; when some one want to publish?

When some one solve a math research problem and want to publish it somewhere; of course one should type it in LaTex. My Question: What should be the ideal way(time) to type it, I mean first one ...
0
votes
2answers
26 views

Discrete quantities and dimensions

This question popped into my head yesterday when my friend told me about a game which he claimed to be "$2.5$-d". Of course, I knew it was just an expression, since it wasn't really $2.5$-d; the ...
4
votes
1answer
87 views

Why are infinite sums so much harder to calculate than the associated infinite integral?

It seems that with continuous functions, we have in calculus an apparatus for "short cutting" an infinite sum. However, when we move to the discrete case, it seems that we don't have the equivalent ...
2
votes
1answer
57 views

Relationship between measure theory and real analysis

Does measure theory generalize real analysis to abstract spaces? For example, you can now talk about convergence even on unordered fields.
7
votes
1answer
118 views

Career advice in Mathematics combined with another sciences

First of all, I don't know if this is the right place to ask about careers. If not, please direct me somewhere I can get more help. Im an undergraduate student in Mathematics on my 3rd year.My ...
1
vote
2answers
72 views

Soft Question: Are sigma fields, fields?

I'm sorry if this is a foolish question but: Is a $\sigma$-field (of sets) a field (in the sense of algebra) if we only consider finite intersections and finite unions?
5
votes
0answers
64 views

Some questions about synthetic differential geometry

I've been trying to read Kock's text on synthetic differential geometry but I am getting a bit confused. For example, what does it mean to "interpret set theory in a topos"? What is a model of a ...
1
vote
0answers
40 views

Is there any relationship between localization and completion of a module?

Let $R$ be a commutative ring, $\mathfrak p$ a prime ideal of $R$ and $M$ an $R$-module. I've seen the terms 'localization' $M_\mathfrak p$ of $M$ and the completion $M_\mathfrak p$ at $\mathfrak p$ ...
3
votes
0answers
64 views

For those wanting to study theoretical computer science, is Math a degree that worth more than Computer Science?

At my University, everyone first enrol in a Science and Technology, a very general course, and then choose the specialisation (like Computer Science, Math, some Engineering, etc). As I was finishing ...
2
votes
1answer
85 views

What are the theorems in mathematics, we can prove using completely different ideas?

I know this question can have many answers. But I would like to know about your favorite theorems which can give completely different proofs. For example: When I read the book "Proof from the Book," ...
2
votes
1answer
45 views

Relationship between the diffusion equation and the heat equation

In physics we have the heat equation which describes the propagation of heat $$\dfrac{\partial u}{\partial t} = \kappa \dfrac{\partial^2 u}{\partial x^2},$$ while in biomathematics we have the ...
3
votes
2answers
60 views

Analogy to the purpose of Taylor series

I want to know an analogy to the purpose of Taylor series. I did a google search for web and videos : all talks about what Taylor series and examples of it. But no analogies. I am not a math geek and ...
3
votes
1answer
44 views

What journals/periodicals are appropriate for serious amateurs to submit findings/research?

Are there journals out there that publish work by serious amateurs (but not at the level of academic researcher). I think that MAA journal American Mathematical Monthly is a good example (I could be ...
11
votes
5answers
1k views

Humorous integration example?

I was just reading though an introductory calculus book and it has the note: NOTE When integrating quotients, do not integrate the numerator and denominator separately. This is no more valid in ...
2
votes
1answer
48 views

How to get to the heart of a subject?

(mathematical subject is intended) It's obviously not a simple question, but I thought it could lead to some interesting discussion. I was reading Lewis Campbell's biography of James Clerk Maxwell ...
2
votes
0answers
38 views

When is iterated exponentiation used and how is it defined?

I was thinking of ways to define an iterated exponentiation operation. The nice thing about addition and multiplication is that they're associative and commutative, which makes defining the sum and ...
5
votes
5answers
204 views

Book recommend for topics of Integrals in multivariable calculus.

I am an average student and have to study following topics on my own for the exam : The measure of a bounded interval in $\mathbb R^n$ , the Riemann integral of a bounded function defined on a ...
0
votes
1answer
78 views

Taking Calculus II and Calculus III at the same time?

A little background: I'm a high school student enrolling at a local university next fall. I plan to pursue a mathematics degree, have studied this Calculus book. During the next semester, I plan to ...
1
vote
1answer
25 views

Book recommend for Tikhonov regularization

I need regularization in studying inverse partial differential problems. Therefore i want to learn regularization techniques especially Tikhonov regularization. Can you recommend any book which ...
0
votes
1answer
79 views

Reference request: non-linear analysis

I'm searching for a throughout reference that can help me to gain the necessary background (starting from undergrad real analysis courses) to understand the main methods and theorems to deal with ...
0
votes
0answers
53 views

Topics in Analysis for undergraduate level essay project

As part of my real analysis 1 course, we are being asked to write a short (3-5 page) paper on a topic connected to analysis. The professor has provided ideas such as optimal transport theory, ...
3
votes
2answers
67 views

Methods to prove that two groups are isomorphic

What are some of the more effective methods to prove that 2 groups are isomorphic. The methods that I am currently using now is to always find a function from A to B that is bijective and homomorphic ...
4
votes
0answers
68 views

Methods of constructing rapidly convergent series

It's fairly easy to see that the series $$1-\tfrac{1}{3}+\tfrac{1}{5}-\cdots=\tfrac{1}{4}\pi$$ is : 1. Convergent to the value given, and - 2. Very slowly converging, which can be seen just by ...
4
votes
1answer
93 views

Discrete math problems

I am a high school student interested in thinking about math. I don't know a lot of high-powered math (I only know up to calculus), instead I focus on discrete topics related to math Olympiads ...
2
votes
1answer
78 views

Putnam Training: “Crunch Time” Topic Selection

There is about a month left before the Putnam Exam, and it will be the last one I could take. I have looked over several problems from previous exams, and done several dozen problems from Paul Zeitz's ...
0
votes
0answers
16 views

Do you have a specific method to solve logiqual sequences or do you rely on intuiton?

I'm preparing a presentation on Logical Sequences. Here's one : $2, 6, 12, 20, 30,42, [?]$ The goal is to find the following number in the sequence. In this particular case, a possible answer is ...
108
votes
22answers
17k views

Examples of mathematical discoveries which were kept as a secret

There could be several personal, social, philosophical and even political reasons to keep a mathematical discovery as a secret. For example it is completely expected that if some mathematician find ...
0
votes
1answer
76 views

In what order should I learn linear algebra and multivariable calculus?

I took AP calculus in high school and I really enjoyed it, but when I got to my university I was upset that I couldn't take Calculus II as it didn't fit in my schedule. I feel kind of behind now that ...
4
votes
6answers
1k views

The Largest Gaps in the History of Mathematics

Edit: Based on the useful comments below. I edited the original post in order to seek for other important historical gaps in mathematics. Mathematics is full of the historical gaps. The first type ...
14
votes
6answers
223 views

Mathematical trivia (i.e. collections of anecdotes and miscellaneous (recreational) mathematics)

Can you suggest some books on mathematical trivia? I use the word "trivia" with a double meaning in this case: curious anecdotes that enlighten what the real life of mathematicians is like (like ...
8
votes
0answers
142 views

What is the overall idea of Galois theory?

I am a third year undergraduate, doing a course on Field and Galois theory. Now, while I seem to understand most of the concepts locally, I do not seem to get the 'Whole picture' of what is happening ...
8
votes
3answers
180 views

difficulty of accepting $i^2 = -1$ for first timers [duplicate]

While teaching complex numbers for those who are encounter for the first time (usually 10th grader and 11th grader), I get the question like "Can even squared number give negative results? How ...
0
votes
0answers
19 views

Limit of power function

I want to be sure if that is correct, Could you please point me to some good references for that kind of function : When $a \to +\infty$ $ x^a =\begin{cases} 1 & x = 1\\ \\ +\infty & x>1 ...
3
votes
1answer
121 views

A finite alternative to Hilbert's Hotel?

Here, I propose a finite alternative to Hilbert's hotel as the intuition for logically developing Dedekind's definition of infinite. I would like to know if this analogy entirely justifies the given ...