Questions tagged [gm.general-mathematics]
Questions about mathematics which don't fall into the other arXiv categories. If you have a general question about mathematics but it is not research level, it's off-topic but it might be welcomed on Mathematics Stack Exchange.
343
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36
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Basic results with three or more hypotheses
Consider the following statement of the Arzela-Ascoli theorem.
Theorem. Let K be a compact topological space and let S be a subset of C(K). Then S is relatively compact if and only if S is uniformly ...
16
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7
answers
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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 ...
16
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1
answer
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Relation between math and piano music
What, if any, is the relation between Cantor's function and Ligeti studio: Devil's Staircase?
16
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0
answers
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Decidable open problems
Are there any significant open problems in mathematics which are clearly decidable (in that it is easy to write a clearly corresponding program which will eventually output either Yes or No (or ...
15
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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 ...
15
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9
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Tools from other disciplines useful to mathematics research?
Obviously, mathematics provides essential tools for physicists, biologists, economists, engineers and many others to use in their research. Equally obviously, physics, biology, economy and engineering ...
15
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4
answers
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What are some examples of understanding a space by studying the functions on this space?
In Quantum theory, groups and representations, Peter Woit writes:
A fundamental principle of modern mathematics is that the way to
understand a space $M$, given as some set of points, is to look at $...
15
votes
3
answers
992
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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
(Springer-Verlag, ...
15
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3
answers
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Card game / options pricing / Brownian bridge question
We play a game. I shuffle a deck of cards and start dealing them face up. After any card you can say "stop", at which point I pay you 1 dollar for every red card dealt and you pay me 1 for every black ...
15
votes
1
answer
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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 ...
14
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12
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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 ...
14
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4
answers
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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 ...
14
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9
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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 ...
14
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1
answer
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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 , x_n=...
14
votes
1
answer
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Research semester in math
Admittedly, this is a soft question.
My own experience in mathematical research has been of long periods of research, mostly characterized in long "blocks" and sporadic breakthrough.
How does this ...
13
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16
answers
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Oddities of evenness
Being initially a little bit perplexed by the observation that the possibility of calculating vertex potentials $\lbrace\pi_1,\dots,\pi_n\rbrace$ for weighted cycle graphs $C_n,\,2\lt n$ such that the ...
13
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16
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Individual mathematical objects whose study amounts to a (sub)discipline? [closed]
Certain mathematical objects have a theory so rich that their study
alone arguably constitutes a distinct (sub)discipline. My own list
would begin with
1) the absolute Galois group of the rationals;
...
13
votes
2
answers
923
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Is there a name for sets for which it is easier to test membership than to find members---and vice versa?
This is a question my son Bob asked me. For some sets it is relatively easy
to test for membership but a lot more difficult to find members, and for others
the reverse is true. Here is an elementary ...
13
votes
2
answers
627
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Number triangle
This question arose just out of curiosity. Note the triangle of 0-1's below, whose construction is as follows. Choose any number, say 53 as done here. The first line of the triangle is the binary ...
12
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7
answers
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Books containing new results
In Endless controversy about the correctness of significant papers, Denis Serre writes:
The research community is able to point out incorrect statements, at least among those which have some ...
12
votes
2
answers
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Is it ever unnecessary to mathematically formalize a concept?
From my understanding, mathematics sometimes gives rise to new physical/tangible laws and the converse is also true. In particular, physical phenomena give rise to new mathematics.
In all of the cases ...
11
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9
answers
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What are examples of problems we know how to solve for primes (or prime powers), but not for composites?
I am interested in seeing examples of research problems which fall into one of the two following categories:
A problem which is solved in the case of primes (or prime powers), but which remains open ...
11
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4
answers
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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 ...
11
votes
5
answers
995
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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, ...
11
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3
answers
555
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Any hints on how to prove that the function $\lvert\alpha\;\sin(A)+\sin(A+B)\rvert - \lvert\sin(B)\rvert$ is negative over the half of the total area?
I have this inequality with $0<A,B<\pi$ and a real $\lvert\alpha\rvert<1$:
$$ f(A,B):=\bigl|\alpha\;\sin(A)+\sin(A+B)\bigr| - \bigl| \sin(B)\bigr| < 0$$
Numerically, I see that ...
11
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3
answers
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How can I simplify this sum any further?
Recently I was playing around with some numbers and I stumbled across the following formal power series:
$$\sum_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum_{l=0}^k\binom{ak}{al}\biggr)$$
I was able ...
11
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1
answer
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Smale's view of mathematical artificial intelligence
This snippet is from Smale's paper Smale, Steve (1999). "Mathematical problems for the next century". In Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B. (eds.). Mathematics: frontiers and ...
11
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2
answers
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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
...
10
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8
answers
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Most important mathematical results in last 30 years [closed]
Which results from the last 30 years, in any area of mathematics, do you think are the most important ones?
Specifically, which are the ones that will have more impact across all math and/or settle ...
10
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5
answers
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Visual representation of mathematical research interrelationships
I remember seeing a visualization in the form of a 2d (nodal) graph of all areas of academia, with math, physics and engineering over in one section, connecting in an arc to the central area of ...
10
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3
answers
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Progress in robustifying mathematics - i.e. making mathematical theorems robust to small changes in hypotheses
The idea of making a mathematical theorem robust to small changes in its hypotheses has been known for some time. In areas such as group theory reasonable progress has been made leading to the theory ...
10
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1
answer
558
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How far away can we get by multiple rounding and unit change?
This question is inspired by xkcd #2585 (Rounding):
Let $u_0,\ldots,u_n$ be positive real numbers (we can assume w.l.o.g. that $u_0=1$) or “units”.
Consider the following directed graph: its vertices ...
10
votes
1
answer
807
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Geometric meaning of unimodular matrix
Rotations are given by unitary matrices.
What is the geometric meaning of unimodular matrices that are not unitary?
10
votes
1
answer
310
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Can you get any natural number from 4 by performing given operations?
You can perform the following operations on numbers:
divide the number by 2,
add 0 or 4 at the end of the number.
Can you get any natural number from 4 by performing only these operations?
So far I ...
10
votes
1
answer
246
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Convention of Address in math journals?
The paper was written and submitted when I was in Institution A.
After (many) years, the paper is accepted when I am in Institution B.
Which address shall I put on the paper?
The current address (...
10
votes
2
answers
716
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What is the strategy for "all words valid" scrabble?
The rules for "all words valid" scrabble are exactly the same as ordinary scrabble, except that every single combination of letters is in the dictionary. To make the game deterministic, we will also ...
10
votes
1
answer
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Applications of Morley's Categoricity Theorem
I just attended a lecture by Rami Grossberg and he mentioned that he is not aware of any applications of Morley's Categoricity Theorem. This is exactly my question.
Question: Do you know of any ...
9
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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 ...
9
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4
answers
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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 ...
9
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6
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Surprising applications of the theory of games?
I am currently studying the applications of games in quantum information theory and related fields and I am aware of its uses in places like model theory and set theory. So I was curious, what are ...
9
votes
1
answer
907
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How often do you put your research into trash?
A soft question.
I am a PhD student, at early stages of my academic career; and have personally experienced the following many times. Sometimes you come up with a result, that you are not quite ...
9
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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 ...
9
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2
answers
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Naturally occurring orderings
The are many orderings that naturally occur in interesting but seemingly unrelated circumstances. Here are some examples:
The volume spectrum of orientable hyperbolic 3-manifolds has order type $\...
9
votes
1
answer
373
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Why are discreteness and smoothness in physics inversed with respect to geometry?
In a closed (say differentiable) Riemannian manifold you see only continuous features when looking at small neighbourhoods of points. From afar,
discrete features appear ((co)homology, closed ...
9
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0
answers
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List of modern points of view simplifying or clarifying classical topics
There are many modern mathematical achievements which greatly clarify or (and) simplify classical important topics. I believe a list of such achievements, among other benefits, would be a big help for ...
8
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3
answers
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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 ...
8
votes
2
answers
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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
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6
answers
685
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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 ...
8
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3
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Spaces with a quasi triangle inequality
How do you call a space with a function which is symmetric, non negative, positive definite and which satisfies a quasi-triangle inequality:
$d(x,z) \leq C( d(x,y)+d(y,z) )$
for all $x,y,z$ and some ...
8
votes
1
answer
982
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Sum of subset of geometric series: a^2^n
The formula for 1 + a + a^2 + .... where 0 < a < 1 is $\frac{1}{1-a}$, but what if you wanted to sum only those where the exponent is a power of 2? That is,
$S = a + a^2 + a^4 + a^8 + \cdots$
...