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

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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 ...

**5**

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**5**answers

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### 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 ...

**0**

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**0**answers

282 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 ...

**7**

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**6**answers

377 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 ...

**5**

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**1**answer

212 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
...

**50**

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**8**answers

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### 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 ...

**2**

votes

**0**answers

48 views

### 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 ...

**4**

votes

**1**answer

302 views

### 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 ...

**-1**

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**1**answer

159 views

### What is an example of a non-axiomatic mathematical system? [closed]

In this wikipedia article on the foundations of mathematics, it says:
In practice, most mathematicians ... do not work from axiomatic systems
Is this correct? If so, what is an example of this?

**33**

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**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. ...

**2**

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**1**answer

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### Talking about properties of “random” elements

I asked this in MSE but did not get a satisfying answer. I apologize in advance if this is not appropriate for MO.
Suppose that we have some set X and we want to say that a "random" (or generic) ...

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628 views

### Accessible proofs of contemporary results in mathematics

Are there strong results in contemporary mathematical research (last 20 years) which have a proof which every mathematician (holding a PhD) can completely understand within a few days? -- If yes, ...

**3**

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**1**answer

364 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 ...

**8**

votes

**3**answers

540 views

### Is there any monoid in which the product of two non-invertible elements could be invertible?

I think the title speaks for itself. Thus I just explain the story behind the question. Of course, you may want to skip the story.
Story: Currently, I teach a course in linear algebra and matrices ...

**1**

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**0**answers

121 views

### Ellipses touching a circle [closed]

This is a crosspost from math.stackexchange, where the question did not find an answer.
Given a circle and two points $A$, $B$ in the plane, how do I find an ellipse with focal points $A$ and $B$ ...

**22**

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**1**answer

494 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 ...

**1**

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**2**answers

323 views

### Can one branch of mathematics be completely learned from the perspective of another branch of mathematics? [closed]

This arose from a discussion with a friend (people involved are two engineers) who argued that every result in mathematics should be transformable into another branch. For example, he argued that ...

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622 views

### Online high quality colloquium talks

In my department we're thinking about showing online lectures one day per week at lunch, as sort of a virtual colloquium appropriate to mathematics undergraduates as well as faculty. To start with ...

**7**

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**2**answers

816 views

### How should you respond to a student who asks whether a very nice physical example constitutes a proof? [closed]

"Is this really a proof?" is the exact question e-mailed to me today from an undergraduate mathematics student whom I know as a highly competent student. The one sentence question was accompanied with ...

**15**

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**3**answers

2k views

### Famous vacuously true statements

I am interested to know other examples vacuously true statements that are non-trivial. My starting example is Turan's result in regards to the Riemann hypothesis, which states
Suppose that for each ...

**13**

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**1**answer

542 views

### Relation between math and piano music

What, if any, is the relation between Cantor's function and Ligeti studio: Devil's Staircase?

**10**

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**2**answers

821 views

### What might extraterrestrial mathematics look like? [closed]

In an extensive anthropological joint research project concerning the necessities in the development of life and civilisation my group is concerned with mathematics. This forum seems to be extremely ...

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**5**answers

278 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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433 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 ...

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**4**answers

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### When did you “meet Polya”? [closed]

I guess most of us didn't meet Polya in person (this is the answer to the title)! Perhaps, it is much easier to guess that most of us have met one of his writings (or alike) on problem solving, and ...

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### What would remain of current mathematics without axiom of power set? [closed]

The power set of every infinite set is uncountable. An infinite set (as an element of the power set) cannot be defined by writing the infinite sequence of its elements but only by a finite formula. By ...

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653 views

### Where is the belly button of the Universe? [closed]

It's fine and nice and wonderful when a part of learning mathematics is chaotic, ad hoc, spontaneous, social, ...
However it would be perhaps of fundamental value to know a very central point of ...

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**2**answers

670 views

### Anything special (historical?) about surface $x\cdot y\cdot z\ +\ x+y+z=0$?

QUESTION
I wanted to introduce and develop the complex logarithm from scratch. As the result I've arrived a couple of months ago at the following identity after which the road to complex logarithm is ...

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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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### Are there proofs that you feel you did not “understand” for a long time?

Perhaps the "proofs" of ABC conjecture or newly released weak version of twin prime conjecture or alike readily come to your mind. These are not the proofs I am looking for. Indeed my question was ...

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**1**answer

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### 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 ...

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**0**answers

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### Reference needed for: Automatic generation of relevant mathematical exercises (similar to ones written by human) with the help of machine learning

I am doing research on automatic generation of relevant mathematical exercises (similar to ones written by human) with the help of machine learning.
I have found several research papers on ...

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**2**answers

259 views

### Asymptotics of a function

I hope this question is not too simple, but I would like to know the asymptotic behaviour of the following function $f: \mathbb{N}^{+} \rightarrow \mathbb{Q}$ where
$$
f(n) = \sum_{i=1}^{n} ...

**5**

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### What does a mathematician expect from mathematics education? [closed]

Consider that my question is not a personal and/or subjective question. Perhaps, you have hired a mathematics educator in your department and you are interested in finding a way to communicate with ...

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**1**answer

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### 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 ...

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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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360 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 ...

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### 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 ...

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### 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**

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**1**answer

355 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 ...

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**0**answers

72 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 ...

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610 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**

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**4**answers

414 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} ...

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**1**answer

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### 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' ...

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### 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 ...

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961 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 ...

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### 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 ...

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**1**answer

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### 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**

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### 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 ...

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### 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 ...