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

**5**

votes

**1**answer

700 views

### Why do mathematicians prefer one definition over the other when they both define the same concept?

Here is a basic, though very important, example:
Hilbert takes as primary the notion of “congruence” (or “equal”) between segments. His first axiom of congruence “requires the possibility of ...

**41**

votes

**17**answers

8k views

### 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 ...

**2**

votes

**3**answers

462 views

### Strong notions of general position

Hi!
I am looking for notions of general position that are stronger than linear general position.
To illustrate, 3 points in linear general position don't lie on a line. I want a notion that would ...

**4**

votes

**2**answers

1k views

### List of Charlatans in Mathematics [closed]

Recently, while looking for articles and documents to learn about the Riemann Hyopthesis, I came across a strange funny document of a chinese "mathematician" called Jiang Chun-Xuang who claimed to ...

**10**

votes

**4**answers

1k views

### How to refer to a theorem that you have shown to be wrong

I am writing a paper about a flaw that I found in a published paper. There, the statement is called “Theorem 2”. In my paper, I am reproducing the other paper’s definitions, and steps leading towards ...

**4**

votes

**1**answer

402 views

### Characterise all pairs of n/m stars that have the same inner radius

Geometry, algebra, and examples
Let n and m be integers, with 2 ≤ m < n/2. Consider the bounding polygon of an n/m star (that is, a star with n points each of which connects to the two points ±m ...

**0**

votes

**0**answers

78 views

### Asymptotic inverses of asymptotic functions

The prime number theorem states that two functions are asymptotic. Their inverses (as functions of an integral variable) are also asymptotic. In general, under what conditions are the inverses of ...

**7**

votes

**3**answers

654 views

### Formal writing: numbers under 10

I've been tasked with proofreading an Engineering/Mathematics thesis paper. I was always told that numbers under 10 should be spelled out (one, two, three, ...) but I was wondering if this rule holds ...

**4**

votes

**4**answers

465 views

### Why do we choose the standard total order on the integers?

I understand why the set of natural numbers $\mathbb N = \{ 0, 1, 2, \cdots \}$ is equipped with a total order. Indeed, every monoid has a pre-order, where $$n' \succeq n \quad \mathrm{if~and~only~if} ...

**0**

votes

**1**answer

42 views

### Multivariate expansion in terms of single variate products: what is the name for this?

In some situations we have access to a representation like this:
$ f(x,y) = \sum_i u_i(x) v_i(y) $
What is this called? (I know when you jam this into PDE get to call it 'separation of variables' ...

**11**

votes

**0**answers

337 views

### Which limit to take as a key applied math decision

The Borel-Kolmogorov paradox refers to situations where non-uniqueness in the notion of conditioning on a set of measure zero leads to apparent contradictions. As a formal matter, one requires ...

**18**

votes

**2**answers

1k views

### Center of mass from the abstract point of view, or could the ancient Greeks invent modern analysis?

This is a very open-ended question, which may or may not have a perfect answer, and for which I have a few ideas but nothing like a clear picture. However, I guess it won't hurt to ask to see if ...

**6**

votes

**4**answers

1k views

### On similar concepts in mathematics whose similarity is a non-trivial fact.

Recently, while undertaking a study of commutative algebra, I learned three concepts: (i) a local ring, (ii) a regular local ring and (iii) a regular ring.
At the end, I found myself asking this ...

**2**

votes

**1**answer

356 views

### Integer triangle

Is there a triangle whose vertices, as well as the four classical points, the centroid, the orthocenter, the incenter, and the circumcenter, all have integer coordinates?

**7**

votes

**1**answer

330 views

### Experimental mathematics: how are floating point equations discovered/converted to exact equations?

the 2005 AMS article/survey on experimental mathematics[1] by Bailey/Borwein mentions many remarkable successes in the field including new formulas for $\pi$ that were discovered via the PSLQ ...

**2**

votes

**2**answers

563 views

### Has the notion of “space” been reconsidered in 20th century?

The original title, "has the bases of geometry been reconsidered in 20th century" of this question refers to Riemann's paper "On the Hypotheses which lie at the Bases of Geometry"， an English version ...

**0**

votes

**2**answers

683 views

### What constitutes the division between discrete and non-discrete math? Are there any math subject where it's being blurred? [closed]

I often heard about this division but always in a non-formal manner. What constitutes it? Is it a limit operation? Or a fundamental distinction between countable and uncountable sets? And what math ...

**2**

votes

**1**answer

301 views

### Computer algebra system (CAS) with good re-presenting or transformation support

Such heavy-weight transformations as expanding or factoring are provided by most of CAS-es, but what about light-weight, but a useful transformations, like "reorder some terms to make expression more ...

**6**

votes

**8**answers

2k views

### Beautiful theorems with short proof [closed]

I like to ask for beautiful mathematical theorems with short proof. A proof is short in my sense if it is at most one page assuming basic notations and very basic results a second year student will ...

**1**

vote

**1**answer

235 views

### Distance calculation in metric space

Dear All,
I want to calculate the distance between two sets in which the maximum distance between the sets are minimized. Formally problem defined as,
$\displaystyle \min_{a \in A} \max_{b \in B}$ ...

**3**

votes

**4**answers

811 views

### Another Chicken or Egg: Sequence or Series

This is a side question which is more motivated by teaching than research.
First, I am trying to convince myself that sequences appear before series (as numerical approximations to "interesting" ...

**1**

vote

**0**answers

232 views

### Allocation game optimal strategy

There are two players, Alice and Bob. There is an initial pool of 100 dollars. Alice proposes an allocation of the dollars (real numbers, not necessarily integers), and Bob can either accept or ...

**7**

votes

**0**answers

240 views

### Modelling the difficulty of mental calculation. [closed]

Are you aware of any work that tries to model the difficulty of evaluating a formula mentally (for your average, numerate, person, not a trained mental calculator)?
For instance, evaluating an ...

**1**

vote

**1**answer

461 views

### Quantifiers in function definition — is this legitimate?

I've encountered a working paper in which the author discusses two different notions about the legitimate definition of a certain functional. He expresses these notions using the universal and ...

**3**

votes

**0**answers

118 views

### Reference request: name of a transform

Define a transform on polynomials which is linear and acts on each monomial as $$\widehat{z^k} = \frac{(1+z)(2+z)\ldots(k+z)}{k!}.$$ Does anyone know whether this has a name (and therefore has been ...

**16**

votes

**3**answers

931 views

### Results that are easy to prove with a computer, but hard to prove by hand [closed]

Consider the assertion:
There is no completely multiplicative function $f:\mathbb{N}\rightarrow \{\pm 1\}$ with $\left|\sum_{n\leq x}f(n)\right|\leq 2$ for all $x\geq 0$.
One can write a very short ...

**4**

votes

**0**answers

242 views

### Seminar Notes Repository

On Seminars, people actually talk a lot more about motivations than when they write in paper. It would be a good idea if there is an online repository where people can upload notes (handwritten, ...

**11**

votes

**9**answers

1k views

### 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 ...

**11**

votes

**4**answers

831 views

### Applications of Zariski topology outside alg. geometry

Are there applications of the Zariski topology in mathematics that are not within the scope of algebraic geometry (including schemes and algebraic groups) ?
There is an older question with a similar ...

**0**

votes

**3**answers

792 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}

**12**

votes

**1**answer

535 views

### Continued fractions and projective resolutions

Hello,
This question might be vague and not thought-through enough.
If we have a real positive number $x$, we can start to write it as a continued fraction:
$x = a_0 + \frac{1}{x_1} , \ldots , ...

**6**

votes

**2**answers

517 views

### Where is the Euler/Goldbach correspondence?

I know that there is a 1965 volume containing the Euler/Goldbach correspondence, but I'm interested in looking at the original manuscripts. I'm not finding anything at University of Basel or ...

**5**

votes

**0**answers

543 views

### Open problems with practical outcome in a visible future ? [closed]

I believe that any non-trivial idea will sooner or later find application in real life.
However "sooner" is better than "later":)
If we look at famous open problems - e.g. Millennium Prize problems - ...

**1**

vote

**1**answer

217 views

### “Highly balanced” periodic functions

The function $f(x) = e^{2\pi ix}$ on the domain $\mathbb{R}/\mathbb{Z}$ has the property that, for every $n > 1$ and every $x$, $\displaystyle \sum_{i = 0}^{n-1} f(x + \frac{i}{n}) = 0$.
Other ...

**15**

votes

**7**answers

1k views

### Unexpected applications of the fact that nth degree polynomials are determined by n+1 points

I had a funny idea for proving an identity in Euclidean geometry. While it didn't end up being a very nice proof strategy in my case, I would still like to collect nice examples of where the proof ...

**2**

votes

**2**answers

963 views

### Sum of products of exponentials and polynomials

Hi,
I am looking for a closed-form expression for the finite sum of the product of an exponential function with a polynomial function --- that is, the sum
...

**4**

votes

**0**answers

642 views

### How to find a problem ??? [closed]

Hi All,
To do math, of course every one always finds some interesting problems and then try to solve them and then publish them as articles. In particular, in doing PhD, thesis advisors normally give ...

**-4**

votes

**2**answers

1k views

### what part of using vieta's formulas violates quintic non-solvability? [closed]

You can write the n roots of an n degree polynomial in terms of its n coefficients, i.e., "Vieta's" formulas.
You can solve this system of nonlinear equations using Newton's method and the Jacobian. ...

**0**

votes

**0**answers

95 views

### Repeated function resulting in quadratic time.

Let $$r(f, x) = k$$ such that
$$f^k (x) < 2 \hbox{ and } f^{k-1} (x) \geq 2$$
For example $$r(n \rightarrow n-1, 2^n) = 2^n-1; r(n \rightarrow n/2, 2^n) = n.$$
For an arbitrary $c$, we have $$r( ...

**1**

vote

**2**answers

1k views

### Normal distribution with positive SEMI-definite covariance matrix

Hi
In [1] is noted, that a covariance matrix is "positive- semi definite and symmetric". I wonder if it is possible to a multivariate normal distribution with a covariance matrix that is only ...

**2**

votes

**0**answers

1k views

### Good math speakers [closed]

Physics has had great speakers and popularizers (Carl Sagan would be the first to come to my mind). Mathematics, on the other hand, seems to be short on this. At least that is my impression... Or am I ...

**15**

votes

**3**answers

779 views

### Thom's Principle: rich structures are more numerous in low dimension

Marcel Berger states Thom's Principle as:
"rich structures are more numerous in low dimension,
and poor structures are more numerous in high dimension."
This is in
Geometry II
...

**87**

votes

**20**answers

11k 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 ...

**2**

votes

**5**answers

899 views

### Circumference of convex shapes

Here is a puzzle I found in "Mitteilungen der DMV" (roughly "Letters of the German Society of Mathematicians") issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch ...

**48**

votes

**5**answers

5k views

### Changing field of study post-PhD

I am doing my PhD in algebraic graph theory, for not much more reason than that was what was available. However, I love deep structure and theory in mathematics, and I do not particularly want to be ...

**6**

votes

**1**answer

687 views

### Stylistic question

I'm writing up a paper now where I'm the only author and have a stylistic question.
Should I write ''I'' or ''we'' as in ''I/we recall the definition...'' etc. I think this simple example will make ...

**1**

vote

**0**answers

169 views

### Finite topological dimension x local compactness

Of course, the two notions are independent one from the other, but often one of them implies the other under some additional hypotheses. For instance:
A topological vector space is finite dimensional ...

**29**

votes

**9**answers

3k views

### Dimensional Analysis in Mathematics

Is there a sensible and useful definition of units in mathematics? In other words, is there a theory of dimensional analysis for mathematics?
In physics, an extremely useful tool is the Buckingham Pi ...

**13**

votes

**14**answers

3k views

### What are some examples of “chimeras” in mathematics?

The best example I can think of at the moment is Conway's surreal number system, which
combines 2-adic behavior in-the-small with $\infty$-adic behavior in the large. The surreally
simplest element ...

**19**

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 ...