The gm.general-mathematics tag has no wiki summary.

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### When did people know that all real polynomials of degree greater than 2 are reducible? [migrated]

Admittedly, this may not be a research level question, but I am deeply curious about this.
Let $f(x) \in \mathbb{R}[x]$, and write $d = \deg f$. It is well known that if $\deg f > 2$, then $f$ is ...

**2**

votes

**1**answer

397 views

### What is the rate of convergence? [closed]

How quickly does the series defined by $$x_0 = 0, \ x_{n+1} = \frac{x_n^2+1}{2}$$ converge to $1$?

**9**

votes

**6**answers

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### Is this a rational function?

Is $$\sum_{n=1}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z)\ ?$$
In a slightly different vein, given a sequence of real numbers $\{a_n\}_{n=0}^\infty$, what are some necessary and sufficient ...

**124**

votes

**36**answers

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### Demonstrating that rigour is important

Any pure mathematician will from time to time discuss, or think about, the question of why we care about proofs, or to put the question in a more precise form, why we seem to be so much happier with ...

**4**

votes

**2**answers

174 views

### Find the expansion of the exact solution (beyond Taylor)

In a paper by Kitagawa & Ueda Squeezed spin states they give an argument that the minimum variance in one-axis twisting Hamiltonian scales like $V_{min} \propto S^{-2/3}$. I will shortly describe ...

**27**

votes

**2**answers

811 views

### Applications of Lawvere's fixed point theorem

Lawvere's fixed point theorem states that in a cartesian closed category, if there is a morphism $A \to X^A$ which is point-surjective (meaning that $\hom(1,A) \to \hom(1,X^A)$ is surjective), then ...

**15**

votes

**1**answer

2k views

### Why do people use “formal calculation” to describe informal calculations?

Many times, I see the word formal being used to describe a calculation that is not rigorous. I would think that such calculations should rather be termed informal than formal. What is the explanation ...

**1**

vote

**0**answers

115 views

### General procedure to find the determinant of an operator?

I want to learn to find the determinant of an operator.
I am given an operator like
$\Sigma _{\alpha\beta}=-k^2g_{\alpha\beta}+i\theta\epsilon_{\alpha\beta\gamma} k^\gamma$
$k^2=k^μk_μ$, $g^{αβ}$ ...

**17**

votes

**3**answers

982 views

### Which way for reading the proofs?

I am a master student in mathematics. For me a large part of doing mathematics is thinking about, reading and verifying the proof of theorems that I find them in my field of study. I can do this ...

**1**

vote

**6**answers

3k views

### Proofs by induction [closed]

Background
I'm interested in the issue of "explanatory" mathematical proofs and would like to try to find out what intuitions mathematicians have about induction, because there seems to be some ...

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votes

**5**answers

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### Collaboration or acknowledgment?

This post is a sequel of: When should a supervisor be a co-author?
This time the topic is about the interaction between two professional mathematicians (in particular junior-senior, but not ...

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votes

**2**answers

822 views

### What are the applications of operator algebras to other areas?

Question: What are the applications of operator algebras to other areas?
More precisely, I would like to know the results in mathematical areas outside of operator algebras which were proved by ...

**87**

votes

**20**answers

12k views

### Mathematical habits of thought and action which would be of use to non-mathematicians

Once again I come to MO for help with something I'm writing for the public.
Which habits of mathematicians -- aspects of the way we approach problems, the way we argue, the way we function as a ...

**40**

votes

**8**answers

3k views

### Published results: when to take them for granted?

Two kinds of papers. There are two kinds of papers: self-contained ones, and those relying on published results (which I believe are the vast majority).
Checking the result. Of course, one should ...

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**11**answers

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### What advanced Area of Mathematics can be delved into with only basic Calculus and Linear Algebra

Hello Mathoverflow Community,
I would really appreciate some advice on this:
All I know is Basic Calculus and Basic Linear Algebra,
I want to start learning more advanced material on my own while ...

**62**

votes

**6**answers

2k views

### Good ways to engage in mathematics outreach?

Greetings all, I have often heard that it would be good if we as a community did more in the way of mathematics outreach: more to explain what it is we do to the community at large, more to expose ...

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votes

**5**answers

2k views

### Where do surreal numbers come from and what do they mean?

I know about Conway's original discovery of the surreal numbers by way of games,
as well as Kruskal's way of viewing surreal numbers in terms of asymptotic behavior
of real-valued functions, leading ...

**3**

votes

**1**answer

531 views

### Mathematics of Computer science and AI [closed]

Computer science and Artificial Intelligence have been fertile grounds for research for decades, not only for Engineers but particularly for Mathematicians. What kinds of Mathematics have emerged ...

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votes

**2**answers

256 views

### link to a paper by Ramanujan

Hi friends,
Does anybody know of a pdf version of the following paper ?
Ramanujan, S. “Modular Equations and Approximations to Pi.” Quart. J. Pure Appl. Math. 45, 350-372, 1913-1914.
It's ...

**2**

votes

**2**answers

527 views

### Popular books written by great mathematicians [closed]

I read:
H. Poincare. Value of science
F. Klein. Development of Mathematics in the 19th Century
J.E. Littlewood. A Mathematicians Miscellany
G.H. Hardy. A Mathematician’s Apology
R. Courant, ...

**6**

votes

**1**answer

153 views

### Compiling self-referential forms

Fix $1\leq d\in\mathbb{N}$ and set $D:=\{0,1,\ldots,d-1\}$.
Consider the system of equations
\begin{equation}
x_i=c_i + \sum_{j\in D}\delta_{x_j,i}
\end{equation}
with $c_i\in D$ given and $x_i\in D$ ...

**0**

votes

**3**answers

811 views

### Simplifying finite sum over 1/(ax+b)

Can I simplify:
\begin{equation}
\sum_{x=x_0}^{x_1} \frac{1}{ax+b}
\end{equation}

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votes

**2**answers

387 views

### Are there any organized websites for seminar/conference videos?

These days, there are many conference centers and universities recording seminars and conference talks and make them available on the web. Some examples:
http://www.fields.utoronto.ca/video-archive
...

**106**

votes

**15**answers

14k views

### When should a supervisor be a co-author?

What are people's views on this? To be specific: suppose a PhD student has produced a piece of original mathematical research. Suppose that student's supervisor suggested the problem, and gave a few ...

**3**

votes

**1**answer

105 views

### Extension of an involutive automorphism

Suppose that $g$ is a complex semi-simple Lie algebra and $g'$ its reductive subalgebra.
If $\tau$ is an involutive automorphism of $g'$, can $\tau$ be extended to an involutive automorphism of $g$ ...

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votes

**6**answers

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### Does a referee have to check carefully the proof ?

I have always checked very carefully the papers I was refereeing when I wanted to suggest "accept". Actually I spend almost as much time checking the maths of a paper I referee than checking the maths ...

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votes

**5**answers

591 views

### Mathematics of privacy?

I wonder to which extent the current public debate on privacy issues (not only by state sniffing, but e.g. by microtargetting ads too an issue) offers interesting questions in mathematics?
Can we ...

**1**

vote

**1**answer

218 views

### Concept of synchronizability

This thread is about the concept of synchronizability. It's a concept I tried to formalize in its most general sense but without success. The goal of this thread is therefore to try to formalize it in ...

**2**

votes

**2**answers

165 views

### Websites for Math Shopping [closed]

I was wondering if anyone knows about good websites or stores to buy math related products. On etsy there are normal distribution plushes and famous mathematicians in coasters. However when I search ...

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vote

**2**answers

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### iterative solution better than analytic solution? [closed]

My supervisor and I were discussing a specific optimisation problem this afternoon.
To be simple: solve for $R$ in the equation $Rx=y$, where $x$, $y$ are made of samples in two difference ...

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**17**answers

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### Is rigour just a ritual that most mathematicians wish to get rid of if they could?

"No". That was my answer till this afternoon! "Mathematics without proofs isn't really mathematics at all" probably was my longer answer. Yet, I am a mathematics educator who was one of the panelists ...

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votes

**1**answer

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### Sources of Theorem drafts by the original author

When I look at first time to a theorem and I try to understand it or when I try to memorise a useful theorem I always have difficulties (I am not the only one. For example: I read a question: I always ...

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votes

**1**answer

169 views

### How to solve $e^{f(x)} + a f(x) + bx = 0$ [closed]

How should determine solutions to equations of this form?
$$e^{-f(x)} + b f(x) = ax$$
Here $f(x)>0$ is real valued. Also $a>0$, $b>0$.

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votes

**2**answers

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### The impact of large cardinals in mathematics [closed]

What are the main applications of large cardinals in ordinary mathematics, and what is the philosophy behind using them. In particular:
Question 1. What is the philosophy behind accepting large ...

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votes

**5**answers

1k views

### Proof or citation?

I'm writing an article. I suppose that I'll submit it to a more or less decent journal (in English). I have doubts about the following: I have a lemma (with quite a trivial proof). I don't want to ...

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votes

**1**answer

217 views

### Constructing a function from preimages

This question was inspired by Can we build a continuous function from "fibers"/preimages defined over a topological base?
Let $X,Y$ be sets and $L\subseteq \mathcal{P}(Y)$. Suppose $L$ has ...

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votes

**15**answers

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### Objects which can't be defined without making choices but which end up independent of the choice

It happens a lot of times that when one defines a new object (ring, module, space, group, algebra, morphism, whatever) out of given data one first chooses some additional structure. And sometimes ...

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votes

**1**answer

108 views

### geometry formula adjacent neighbour of a N dimensional cube [closed]

A question from a curious.
How many adjacent N dimensional cube there is when they are regularly distributed on a "grid".
By N dimensional cube I mean:
a point is a OD cube
a segment is a 1D cube
...

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votes

**6**answers

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### Negative impact of wrong or non-rigorous proofs

The recent talks of Voevodsky (for example, http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2014_IAS.pdf), which describe subtle errors in proofs by him as well as others, as well ...

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votes

**26**answers

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### Proof synopsis collection

I hate to keep going with the big lists, but the question about one-sentence summaries of topics/areas spurred this question...and I just can't help myself!
Definition (Fraleigh): A proof synopsis ...

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vote

**0**answers

318 views

### Conjectures on fractions where each digit appears once in numerator and denominator

This is a highly redacted version of a question that was asked before. Please see Criteria of considering relevance of the question to the domain of research topics for details.
Some numerical ...

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votes

**7**answers

2k views

### Books about polynomials [closed]

Hi,have you a good reference (books) for the study of polynomials with one variable or many variables ? Thanks for your help.
Don't hesitate to correct my English.

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votes

**10**answers

885 views

### Examples of “Unusual” Classifications

When one says "classification" in math, usually one of a handful of examples springs to mind:
-Classification of Finite Simple Groups with 18 infinite families and 26 sporadic examples (assuming one ...

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votes

**6**answers

429 views

### Do you have examples of such “transitive” elements?

(I've asked the same question at the MSE, so far with no answers, so I thought I'd try it here as well. If there's some clash with any site rules, please let me know and I'll abide.)
Let $A$ be a set ...

**3**

votes

**1**answer

559 views

### Equal signs with fancy marks

Some people use $\stackrel{\mathrm{def}}{=}$, $:=$ or $\stackrel{\Delta}{=}$ for definitions.
In more informal contexts, I have also seen $\stackrel{?}{=}$, for "I wish to prove this equality, which ...

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votes

**8**answers

5k views

### Have you solved problems in your sleep? [closed]

I have hit upon major (for me—relative to my trivial accomplishments)
insights in my research
in various sleep-deprived altered states of consciousness,
e.g., long solo car-drives extending ...

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votes

**5**answers

292 views

### procedure-based (as opposed to definition-based) concepts

Euler's work on divergent series was guided by computational procedures, rather than any definition of the "value" of such a series. E.g., he was happy to have half a dozen procedures that indicated ...

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votes

**33**answers

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### Structures that turn out to exhibit a symmetry even though their definition doesn't

Sometimes (often?) a structure depending on several parameters turns out to be symmetric w.r.t. interchanging two of the parameters, even though the definition gives a priori no clue of that symmetry. ...

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votes

**0**answers

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### Jacobi triple product for multidimensional lattices

The Jacobi triple product identity gives as a special case a product formula for the theta function of a 1-dimensional lattice. Is there a more general product formula for the theta function of an ...

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**9**answers

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### math circles video lectures for school children?

Hello,
I am from India. I find the mathoverflow amazing.
I have a question: Are there any good quality video lectures on school math topics?
There are a lot of high quality lectures available on ...